%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP257^3 : 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 : n015.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 09:12:47 AM UTC 2026
% Result : Theorem 84.01s 19.79s
% Output : Refutation 84.01s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 10
% Syntax : Number of formulae : 46 ( 46 unt; 0 typ; 0 def)
% Number of atoms : 46 ( 45 equ; 0 cnn)
% Maximal formula atoms : 1 ( 1 avg)
% Number of connectives : 1512 ( 11 ~; 0 |; 0 &;1501 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 2 ( 1 avg)
% Number of types : 93 ( 92 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 1106 (1104 usr; 63 con; 0-8 aty)
% Number of variables : 39 ( 36 ^; 3 !; 0 ?; 39 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
real: $tType ).
thf(type_def_6,type,
int: $tType ).
thf(type_def_8,type,
rat: $tType ).
thf(type_def_9,type,
nat: $tType ).
thf(type_def_10,type,
product_prod_int_int: $tType ).
thf(type_def_11,type,
num: $tType ).
thf(type_def_12,type,
option_num: $tType ).
thf(type_def_13,type,
code_integer: $tType ).
thf(type_def_14,type,
produc6271795597528267376eger_o: $tType ).
thf(type_def_15,type,
produc8923325533196201883nteger: $tType ).
thf(type_def_16,type,
set_fi4554929511873752355omplex: $tType ).
thf(type_def_17,type,
filter6041513312241820739omplex: $tType ).
thf(type_def_18,type,
set_fi7789364187291644575l_real: $tType ).
thf(type_def_19,type,
filter2146258269922977983l_real: $tType ).
thf(type_def_20,type,
set_real: $tType ).
thf(type_def_21,type,
set_set_nat: $tType ).
thf(type_def_22,type,
set_nat: $tType ).
thf(type_def_23,type,
set_int: $tType ).
thf(type_def_24,type,
complex: $tType ).
thf(type_def_25,type,
filter_real: $tType ).
thf(type_def_26,type,
product_prod_nat_nat: $tType ).
thf(type_def_27,type,
filter_nat: $tType ).
thf(type_def_28,type,
produc4411394909380815293omplex: $tType ).
thf(type_def_29,type,
filter1242075044329608583at_nat: $tType ).
thf(type_def_30,type,
produc2422161461964618553l_real: $tType ).
thf(type_def_31,type,
set_Pr5085853215250843933omplex: $tType ).
thf(type_def_32,type,
set_Pr6218003697084177305l_real: $tType ).
thf(type_def_33,type,
set_o: $tType ).
thf(type_def_34,type,
set_complex: $tType ).
thf(type_def_35,type,
set_Product_unit: $tType ).
thf(type_def_36,type,
set_char: $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_list_int: $tType ).
thf(type_def_40,type,
set_list_nat: $tType ).
thf(type_def_41,type,
set_list_VEBT_VEBT: $tType ).
thf(type_def_42,type,
set_num: $tType ).
thf(type_def_43,type,
set_Pr958786334691620121nt_int: $tType ).
thf(type_def_44,type,
set_rat: $tType ).
thf(type_def_45,type,
set_set_complex: $tType ).
thf(type_def_46,type,
set_set_int: $tType ).
thf(type_def_47,type,
set_VEBT_VEBT: $tType ).
thf(type_def_48,type,
char: $tType ).
thf(type_def_49,type,
extended_enat: $tType ).
thf(type_def_50,type,
list_o: $tType ).
thf(type_def_51,type,
list_int: $tType ).
thf(type_def_52,type,
option_nat: $tType ).
thf(type_def_53,type,
vEBT_VEBT: $tType ).
thf(type_def_54,type,
list_VEBT_VEBT: $tType ).
thf(type_def_55,type,
list_complex: $tType ).
thf(type_def_56,type,
list_nat: $tType ).
thf(type_def_57,type,
list_real: $tType ).
thf(type_def_58,type,
list_set_nat: $tType ).
thf(type_def_59,type,
list_Code_integer: $tType ).
thf(type_def_60,type,
list_num: $tType ).
thf(type_def_61,type,
list_P4002435161011370285od_o_o: $tType ).
thf(type_def_62,type,
product_prod_o_o: $tType ).
thf(type_def_63,type,
list_P3795440434834930179_o_int: $tType ).
thf(type_def_64,type,
product_prod_o_int: $tType ).
thf(type_def_65,type,
list_P7495141550334521929T_VEBT: $tType ).
thf(type_def_66,type,
produc2504756804600209347T_VEBT: $tType ).
thf(type_def_67,type,
list_P8526636022914148096eger_o: $tType ).
thf(type_def_68,type,
list_P6011104703257516679at_nat: $tType ).
thf(type_def_69,type,
list_P1726324292696863441at_num: $tType ).
thf(type_def_70,type,
product_prod_nat_num: $tType ).
thf(type_def_71,type,
list_P3744719386663036955um_num: $tType ).
thf(type_def_72,type,
product_prod_num_num: $tType ).
thf(type_def_73,type,
list_P3126845725202233233VEBT_o: $tType ).
thf(type_def_74,type,
produc334124729049499915VEBT_o: $tType ).
thf(type_def_75,type,
list_P4547456442757143711BT_int: $tType ).
thf(type_def_76,type,
produc4894624898956917775BT_int: $tType ).
thf(type_def_77,type,
list_P7413028617227757229T_VEBT: $tType ).
thf(type_def_78,type,
produc8243902056947475879T_VEBT: $tType ).
thf(type_def_79,type,
list_P5087981734274514673_int_o: $tType ).
thf(type_def_80,type,
list_P5707943133018811711nt_int: $tType ).
thf(type_def_81,type,
list_P7524865323317820941T_VEBT: $tType ).
thf(type_def_82,type,
option4927543243414619207at_nat: $tType ).
thf(type_def_83,type,
set_Code_integer: $tType ).
thf(type_def_84,type,
produc4953844613479565601on_nat: $tType ).
thf(type_def_85,type,
produc2233624965454879586on_nat: $tType ).
thf(type_def_86,type,
produc7248412053542808358at_nat: $tType ).
thf(type_def_87,type,
produc4471711990508489141at_nat: $tType ).
thf(type_def_88,type,
produc8306885398267862888on_nat: $tType ).
thf(type_def_89,type,
produc2963631642982155120at_num: $tType ).
thf(type_def_90,type,
produc3368934014287244435at_num: $tType ).
thf(type_def_91,type,
produc3447558737645232053on_num: $tType ).
thf(type_def_92,type,
produc7036089656553540234on_num: $tType ).
thf(type_def_93,type,
produc1193250871479095198on_num: $tType ).
thf(type_def_94,type,
produc6121120109295599847at_nat: $tType ).
thf(type_def_95,type,
produc5491161045314408544at_nat: $tType ).
thf(type_def_96,type,
produc5542196010084753463at_nat: $tType ).
thf(type_def_97,type,
produc9072475918466114483BT_nat: $tType ).
thf(func_def_0,type,
produc5542196010084753463at_nat: $tType ).
thf(func_def_1,type,
produc5491161045314408544at_nat: $tType ).
thf(func_def_2,type,
produc1193250871479095198on_num: $tType ).
thf(func_def_3,type,
produc8306885398267862888on_nat: $tType ).
thf(func_def_4,type,
produc6121120109295599847at_nat: $tType ).
thf(func_def_5,type,
produc3368934014287244435at_num: $tType ).
thf(func_def_6,type,
produc4471711990508489141at_nat: $tType ).
thf(func_def_7,type,
produc7036089656553540234on_num: $tType ).
thf(func_def_8,type,
produc2233624965454879586on_nat: $tType ).
thf(func_def_9,type,
set_fi4554929511873752355omplex: $tType ).
thf(func_def_10,type,
list_P7413028617227757229T_VEBT: $tType ).
thf(func_def_11,type,
produc3447558737645232053on_num: $tType ).
thf(func_def_12,type,
produc4953844613479565601on_nat: $tType ).
thf(func_def_13,type,
produc2963631642982155120at_num: $tType ).
thf(func_def_14,type,
produc7248412053542808358at_nat: $tType ).
thf(func_def_15,type,
set_fi7789364187291644575l_real: $tType ).
thf(func_def_16,type,
filter6041513312241820739omplex: $tType ).
thf(func_def_17,type,
list_P4547456442757143711BT_int: $tType ).
thf(func_def_18,type,
list_P7524865323317820941T_VEBT: $tType ).
thf(func_def_19,type,
produc8243902056947475879T_VEBT: $tType ).
thf(func_def_20,type,
set_Pr5085853215250843933omplex: $tType ).
thf(func_def_21,type,
produc8923325533196201883nteger: $tType ).
thf(func_def_22,type,
list_P3126845725202233233VEBT_o: $tType ).
thf(func_def_23,type,
list_P7495141550334521929T_VEBT: $tType ).
thf(func_def_24,type,
filter2146258269922977983l_real: $tType ).
thf(func_def_25,type,
list_P8526636022914148096eger_o: $tType ).
thf(func_def_26,type,
option4927543243414619207at_nat: $tType ).
thf(func_def_27,type,
filter1242075044329608583at_nat: $tType ).
thf(func_def_28,type,
set_Pr6218003697084177305l_real: $tType ).
thf(func_def_29,type,
list_P3744719386663036955um_num: $tType ).
thf(func_def_30,type,
list_P1726324292696863441at_num: $tType ).
thf(func_def_31,type,
list_P6011104703257516679at_nat: $tType ).
thf(func_def_32,type,
list_P5707943133018811711nt_int: $tType ).
thf(func_def_33,type,
produc9072475918466114483BT_nat: $tType ).
thf(func_def_34,type,
produc4894624898956917775BT_int: $tType ).
thf(func_def_35,type,
set_Pr958786334691620121nt_int: $tType ).
thf(func_def_36,type,
produc4411394909380815293omplex: $tType ).
thf(func_def_37,type,
list_P5087981734274514673_int_o: $tType ).
thf(func_def_38,type,
list_P3795440434834930179_o_int: $tType ).
thf(func_def_39,type,
set_list_VEBT_VEBT: $tType ).
thf(func_def_40,type,
produc334124729049499915VEBT_o: $tType ).
thf(func_def_41,type,
produc2504756804600209347T_VEBT: $tType ).
thf(func_def_42,type,
produc6271795597528267376eger_o: $tType ).
thf(func_def_43,type,
produc2422161461964618553l_real: $tType ).
thf(func_def_44,type,
product_prod_num_num: $tType ).
thf(func_def_45,type,
product_prod_nat_num: $tType ).
thf(func_def_46,type,
product_prod_nat_nat: $tType ).
thf(func_def_47,type,
product_prod_int_int: $tType ).
thf(func_def_48,type,
list_P4002435161011370285od_o_o: $tType ).
thf(func_def_49,type,
set_list_complex: $tType ).
thf(func_def_50,type,
set_set_complex: $tType ).
thf(func_def_51,type,
list_VEBT_VEBT: $tType ).
thf(func_def_52,type,
set_list_nat: $tType ).
thf(func_def_53,type,
set_list_int: $tType ).
thf(func_def_54,type,
product_prod_o_int: $tType ).
thf(func_def_55,type,
list_set_nat: $tType ).
thf(func_def_56,type,
list_Code_integer: $tType ).
thf(func_def_57,type,
set_VEBT_VEBT: $tType ).
thf(func_def_58,type,
set_set_nat: $tType ).
thf(func_def_59,type,
set_set_int: $tType ).
thf(func_def_60,type,
set_Code_integer: $tType ).
thf(func_def_61,type,
set_Product_unit: $tType ).
thf(func_def_62,type,
list_complex: $tType ).
thf(func_def_63,type,
set_list_o: $tType ).
thf(func_def_64,type,
product_prod_o_o: $tType ).
thf(func_def_65,type,
set_complex: $tType ).
thf(func_def_66,type,
filter_real: $tType ).
thf(func_def_67,type,
option_num: $tType ).
thf(func_def_68,type,
option_nat: $tType ).
thf(func_def_69,type,
filter_nat: $tType ).
thf(func_def_70,type,
set_char: $tType ).
thf(func_def_71,type,
list_real: $tType ).
thf(func_def_72,type,
set_real: $tType ).
thf(func_def_73,type,
list_num: $tType ).
thf(func_def_74,type,
list_nat: $tType ).
thf(func_def_75,type,
list_int: $tType ).
thf(func_def_76,type,
vEBT_VEBT: $tType ).
thf(func_def_77,type,
set_rat: $tType ).
thf(func_def_78,type,
set_num: $tType ).
thf(func_def_79,type,
set_nat: $tType ).
thf(func_def_80,type,
set_int: $tType ).
thf(func_def_81,type,
code_integer: $tType ).
thf(func_def_82,type,
extended_enat: $tType ).
thf(func_def_83,type,
list_o: $tType ).
thf(func_def_84,type,
complex: $tType ).
thf(func_def_85,type,
set_o: $tType ).
thf(func_def_86,type,
char: $tType ).
thf(func_def_87,type,
real: $tType ).
thf(func_def_88,type,
rat: $tType ).
thf(func_def_89,type,
num: $tType ).
thf(func_def_90,type,
nat: $tType ).
thf(func_def_91,type,
int: $tType ).
thf(func_def_92,type,
archim7802044766580827645g_real: real > int ).
thf(func_def_93,type,
archim3151403230148437115or_rat: rat > int ).
thf(func_def_94,type,
archim6058952711729229775r_real: real > int ).
thf(func_def_95,type,
archim7778729529865785530nd_rat: rat > int ).
thf(func_def_96,type,
archim8280529875227126926d_real: real > int ).
thf(func_def_97,type,
binomial: nat > nat > nat ).
thf(func_def_98,type,
bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(func_def_99,type,
bit_and_not_num: num > num > option_num ).
thf(func_def_100,type,
bit_concat_bit: nat > int > int > int ).
thf(func_def_101,type,
bit_or_not_num_neg: num > num > num ).
thf(func_def_102,type,
bit_ri7919022796975470100ot_int: int > int ).
thf(func_def_103,type,
bit_ri6519982836138164636nteger: nat > code_integer > code_integer ).
thf(func_def_104,type,
bit_ri631733984087533419it_int: nat > int > int ).
thf(func_def_105,type,
bit_se725231765392027082nd_int: int > int > int ).
thf(func_def_106,type,
bit_se727722235901077358nd_nat: nat > nat > nat ).
thf(func_def_107,type,
bit_se8568078237143864401it_int: nat > int > int ).
thf(func_def_108,type,
bit_se8570568707652914677it_nat: nat > nat > nat ).
thf(func_def_109,type,
bit_se1345352211410354436nteger: nat > code_integer > code_integer ).
thf(func_def_110,type,
bit_se2159334234014336723it_int: nat > int > int ).
thf(func_def_111,type,
bit_se2161824704523386999it_nat: nat > nat > nat ).
thf(func_def_112,type,
bit_se2000444600071755411sk_int: nat > int ).
thf(func_def_113,type,
bit_se2002935070580805687sk_nat: nat > nat ).
thf(func_def_114,type,
bit_se1409905431419307370or_int: int > int > int ).
thf(func_def_115,type,
bit_se1412395901928357646or_nat: nat > nat > nat ).
thf(func_def_116,type,
bit_se545348938243370406it_int: nat > int > int ).
thf(func_def_117,type,
bit_se547839408752420682it_nat: nat > nat > nat ).
thf(func_def_118,type,
bit_se2793503036327961859nteger: nat > code_integer > code_integer ).
thf(func_def_119,type,
bit_se7879613467334960850it_int: nat > int > int ).
thf(func_def_120,type,
bit_se7882103937844011126it_nat: nat > nat > nat ).
thf(func_def_121,type,
bit_se2923211474154528505it_int: nat > int > int ).
thf(func_def_122,type,
bit_se2925701944663578781it_nat: nat > nat > nat ).
thf(func_def_123,type,
bit_se8260200283734997820nteger: nat > code_integer > code_integer ).
thf(func_def_124,type,
bit_se4203085406695923979it_int: nat > int > int ).
thf(func_def_125,type,
bit_se4205575877204974255it_nat: nat > nat > nat ).
thf(func_def_126,type,
bit_se6526347334894502574or_int: int > int > int ).
thf(func_def_127,type,
bit_se6528837805403552850or_nat: nat > nat > nat ).
thf(func_def_128,type,
bit_se1146084159140164899it_int: int > nat > $o ).
thf(func_def_129,type,
bit_se1148574629649215175it_nat: nat > nat > $o ).
thf(func_def_130,type,
bit_take_bit_num: nat > num > option_num ).
thf(func_def_131,type,
bit_un1837492267222099188nd_num: num > num > option_num ).
thf(func_def_132,type,
bit_un6178654185764691216or_num: num > num > option_num ).
thf(func_def_133,type,
bit_un7362597486090784418nd_num: num > num > option_num ).
thf(func_def_134,type,
bit_un2480387367778600638or_num: num > num > option_num ).
thf(func_def_135,type,
code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).
thf(func_def_136,type,
code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger ).
thf(func_def_137,type,
code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger ).
thf(func_def_138,type,
code_int_of_integer: code_integer > int ).
thf(func_def_139,type,
code_integer_of_int: int > code_integer ).
thf(func_def_140,type,
code_integer_of_num: num > code_integer ).
thf(func_def_141,type,
code_nat_of_integer: code_integer > nat ).
thf(func_def_142,type,
code_num_of_integer: code_integer > num ).
thf(func_def_143,type,
comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex ).
thf(func_def_144,type,
comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real ).
thf(func_def_145,type,
comple4887499456419720421f_real: set_real > real ).
thf(func_def_146,type,
comple7806235888213564991et_nat: set_set_nat > set_nat ).
thf(func_def_147,type,
complete_Sup_Sup_int: set_int > int ).
thf(func_def_148,type,
comple1385675409528146559p_real: set_real > real ).
thf(func_def_149,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(func_def_150,type,
arg: complex > real ).
thf(func_def_151,type,
cis: real > complex ).
thf(func_def_152,type,
cnj: complex > complex ).
thf(func_def_153,type,
complex2: real > real > complex ).
thf(func_def_154,type,
im: complex > real ).
thf(func_def_155,type,
re: complex > real ).
thf(func_def_156,type,
csqrt: complex > complex ).
thf(func_def_157,type,
imaginary_unit: complex ).
thf(func_def_158,type,
differ6690327859849518006l_real: ( real > real ) > filter_real > $o ).
thf(func_def_159,type,
has_de1759254742604945161l_real: ( real > real ) > ( real > real ) > filter_real > $o ).
thf(func_def_160,type,
has_fi5821293074295781190e_real: ( real > real ) > real > filter_real > $o ).
thf(func_def_161,type,
adjust_div: product_prod_int_int > int ).
thf(func_def_162,type,
adjust_mod: int > int > int ).
thf(func_def_163,type,
divmod_nat: nat > nat > product_prod_nat_nat ).
thf(func_def_164,type,
eucl_rel_int: int > int > product_prod_int_int > $o ).
thf(func_def_165,type,
unique5706413561485394159nteger: produc8923325533196201883nteger > $o ).
thf(func_def_166,type,
unique6319869463603278526ux_int: product_prod_int_int > $o ).
thf(func_def_167,type,
unique6322359934112328802ux_nat: product_prod_nat_nat > $o ).
thf(func_def_168,type,
unique3479559517661332726nteger: num > num > produc8923325533196201883nteger ).
thf(func_def_169,type,
unique5052692396658037445od_int: num > num > product_prod_int_int ).
thf(func_def_170,type,
unique5055182867167087721od_nat: num > num > product_prod_nat_nat ).
thf(func_def_171,type,
unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_172,type,
unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int ).
thf(func_def_173,type,
unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_174,type,
comm_s8582702949713902594nteger: code_integer > nat > code_integer ).
thf(func_def_175,type,
comm_s2602460028002588243omplex: complex > nat > complex ).
thf(func_def_176,type,
comm_s4660882817536571857er_int: int > nat > int ).
thf(func_def_177,type,
comm_s4663373288045622133er_nat: nat > nat > nat ).
thf(func_def_178,type,
comm_s4028243227959126397er_rat: rat > nat > rat ).
thf(func_def_179,type,
comm_s7457072308508201937r_real: real > nat > real ).
thf(func_def_180,type,
semiri3624122377584611663nteger: nat > code_integer ).
thf(func_def_181,type,
semiri5044797733671781792omplex: nat > complex ).
thf(func_def_182,type,
semiri1406184849735516958ct_int: nat > int ).
thf(func_def_183,type,
semiri1408675320244567234ct_nat: nat > nat ).
thf(func_def_184,type,
semiri773545260158071498ct_rat: nat > rat ).
thf(func_def_185,type,
semiri2265585572941072030t_real: nat > real ).
thf(func_def_186,type,
invers8013647133539491842omplex: complex > complex ).
thf(func_def_187,type,
inverse_inverse_rat: rat > rat ).
thf(func_def_188,type,
inverse_inverse_real: real > real ).
thf(func_def_189,type,
at_bot_real: filter_real ).
thf(func_def_190,type,
at_top_nat: filter_nat ).
thf(func_def_191,type,
at_top_real: filter_real ).
thf(func_def_192,type,
eventually_nat: ( nat > $o ) > filter_nat > $o ).
thf(func_def_193,type,
eventu5826381225784669381omplex: ( produc4411394909380815293omplex > $o ) > filter6041513312241820739omplex > $o ).
thf(func_def_194,type,
eventu1038000079068216329at_nat: ( product_prod_nat_nat > $o ) > filter1242075044329608583at_nat > $o ).
thf(func_def_195,type,
eventu3244425730907250241l_real: ( produc2422161461964618553l_real > $o ) > filter2146258269922977983l_real > $o ).
thf(func_def_196,type,
eventually_real: ( real > $o ) > filter_real > $o ).
thf(func_def_197,type,
filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).
thf(func_def_198,type,
filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).
thf(func_def_199,type,
filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).
thf(func_def_200,type,
filtermap_real_real: ( real > real ) > filter_real > filter_real ).
thf(func_def_201,type,
princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex ).
thf(func_def_202,type,
princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real ).
thf(func_def_203,type,
prod_filter_nat_nat: filter_nat > filter_nat > filter1242075044329608583at_nat ).
thf(func_def_204,type,
finite_card_o: set_o > nat ).
thf(func_def_205,type,
finite_card_complex: set_complex > nat ).
thf(func_def_206,type,
finite_card_int: set_int > nat ).
thf(func_def_207,type,
finite_card_nat: set_nat > nat ).
thf(func_def_208,type,
finite410649719033368117t_unit: set_Product_unit > nat ).
thf(func_def_209,type,
finite_card_set_nat: set_set_nat > nat ).
thf(func_def_210,type,
finite_card_char: set_char > nat ).
thf(func_def_211,type,
finite_finite_o: set_o > $o ).
thf(func_def_212,type,
finite3207457112153483333omplex: set_complex > $o ).
thf(func_def_213,type,
finite_finite_int: set_int > $o ).
thf(func_def_214,type,
finite_finite_list_o: set_list_o > $o ).
thf(func_def_215,type,
finite8712137658972009173omplex: set_list_complex > $o ).
thf(func_def_216,type,
finite3922522038869484883st_int: set_list_int > $o ).
thf(func_def_217,type,
finite8100373058378681591st_nat: set_list_nat > $o ).
thf(func_def_218,type,
finite3004134309566078307T_VEBT: set_list_VEBT_VEBT > $o ).
thf(func_def_219,type,
finite_finite_nat: set_nat > $o ).
thf(func_def_220,type,
finite_finite_num: set_num > $o ).
thf(func_def_221,type,
finite2998713641127702882nt_int: set_Pr958786334691620121nt_int > $o ).
thf(func_def_222,type,
finite_finite_rat: set_rat > $o ).
thf(func_def_223,type,
finite_finite_real: set_real > $o ).
thf(func_def_224,type,
finite6551019134538273531omplex: set_set_complex > $o ).
thf(func_def_225,type,
finite6197958912794628473et_int: set_set_int > $o ).
thf(func_def_226,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(func_def_227,type,
finite5795047828879050333T_VEBT: set_VEBT_VEBT > $o ).
thf(func_def_228,type,
bij_be1856998921033663316omplex: ( complex > complex ) > set_complex > set_complex > $o ).
thf(func_def_229,type,
bij_betw_nat_complex: ( nat > complex ) > set_nat > set_complex > $o ).
thf(func_def_230,type,
bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
thf(func_def_231,type,
comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_232,type,
comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_233,type,
comp_int_nat_int: ( int > nat ) > ( int > int ) > int > nat ).
thf(func_def_234,type,
comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(func_def_235,type,
comp_nat_real_nat: ( nat > real ) > ( nat > nat ) > nat > real ).
thf(func_def_236,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(func_def_237,type,
inj_on_nat_char: ( nat > char ) > set_nat > $o ).
thf(func_def_238,type,
inj_on_real_real: ( real > real ) > set_real > $o ).
thf(func_def_239,type,
strict1292158309912662752at_nat: ( nat > nat ) > set_nat > $o ).
thf(func_def_240,type,
the_in5290026491893676941l_real: set_real > ( real > real ) > real > real ).
thf(func_def_241,type,
gcd_Gcd_int: set_int > int ).
thf(func_def_242,type,
gcd_Gcd_nat: set_nat > nat ).
thf(func_def_243,type,
bezw: nat > nat > product_prod_int_int ).
thf(func_def_244,type,
bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_245,type,
gcd_gcd_int: int > int > int ).
thf(func_def_246,type,
gcd_gcd_nat: nat > nat > nat ).
thf(func_def_247,type,
gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_248,type,
abs_abs_Code_integer: code_integer > code_integer ).
thf(func_def_249,type,
abs_abs_complex: complex > complex ).
thf(func_def_250,type,
abs_abs_int: int > int ).
thf(func_def_251,type,
abs_abs_rat: rat > rat ).
thf(func_def_252,type,
abs_abs_real: real > real ).
thf(func_def_253,type,
minus_8727706125548526216plex_o: ( complex > $o ) > ( complex > $o ) > complex > $o ).
thf(func_def_254,type,
minus_minus_int_o: ( int > $o ) > ( int > $o ) > int > $o ).
thf(func_def_255,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(func_def_256,type,
minus_711738161318947805_int_o: ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > product_prod_int_int > $o ).
thf(func_def_257,type,
minus_minus_real_o: ( real > $o ) > ( real > $o ) > real > $o ).
thf(func_def_258,type,
minus_6910147592129066416_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > set_nat > $o ).
thf(func_def_259,type,
minus_8373710615458151222nteger: code_integer > code_integer > code_integer ).
thf(func_def_260,type,
minus_minus_complex: complex > complex > complex ).
thf(func_def_261,type,
minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_262,type,
minus_minus_int: int > int > int ).
thf(func_def_263,type,
minus_minus_nat: nat > nat > nat ).
thf(func_def_264,type,
minus_minus_rat: rat > rat > rat ).
thf(func_def_265,type,
minus_minus_real: real > real > real ).
thf(func_def_266,type,
minus_811609699411566653omplex: set_complex > set_complex > set_complex ).
thf(func_def_267,type,
minus_minus_set_int: set_int > set_int > set_int ).
thf(func_def_268,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(func_def_269,type,
minus_1052850069191792384nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
thf(func_def_270,type,
minus_minus_set_real: set_real > set_real > set_real ).
thf(func_def_271,type,
minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(func_def_272,type,
one_one_Code_integer: code_integer ).
thf(func_def_273,type,
one_one_complex: complex ).
thf(func_def_274,type,
one_on7984719198319812577d_enat: extended_enat ).
thf(func_def_275,type,
one_one_int: int ).
thf(func_def_276,type,
one_one_nat: nat ).
thf(func_def_277,type,
one_one_rat: rat ).
thf(func_def_278,type,
one_one_real: real ).
thf(func_def_279,type,
plus_p5714425477246183910nteger: code_integer > code_integer > code_integer ).
thf(func_def_280,type,
plus_plus_complex: complex > complex > complex ).
thf(func_def_281,type,
plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_282,type,
plus_plus_int: int > int > int ).
thf(func_def_283,type,
plus_plus_nat: nat > nat > nat ).
thf(func_def_284,type,
plus_plus_num: num > num > num ).
thf(func_def_285,type,
plus_plus_rat: rat > rat > rat ).
thf(func_def_286,type,
plus_plus_real: real > real > real ).
thf(func_def_287,type,
sgn_sgn_Code_integer: code_integer > code_integer ).
thf(func_def_288,type,
sgn_sgn_complex: complex > complex ).
thf(func_def_289,type,
sgn_sgn_int: int > int ).
thf(func_def_290,type,
sgn_sgn_rat: rat > rat ).
thf(func_def_291,type,
sgn_sgn_real: real > real ).
thf(func_def_292,type,
times_3573771949741848930nteger: code_integer > code_integer > code_integer ).
thf(func_def_293,type,
times_times_complex: complex > complex > complex ).
thf(func_def_294,type,
times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_295,type,
times_times_int: int > int > int ).
thf(func_def_296,type,
times_times_nat: nat > nat > nat ).
thf(func_def_297,type,
times_times_num: num > num > num ).
thf(func_def_298,type,
times_times_rat: rat > rat > rat ).
thf(func_def_299,type,
times_times_real: real > real > real ).
thf(func_def_300,type,
uminus1680532995456772888plex_o: ( complex > $o ) > complex > $o ).
thf(func_def_301,type,
uminus_uminus_int_o: ( int > $o ) > int > $o ).
thf(func_def_302,type,
uminus_uminus_nat_o: ( nat > $o ) > nat > $o ).
thf(func_def_303,type,
uminus7117520113953359693_int_o: ( product_prod_int_int > $o ) > product_prod_int_int > $o ).
thf(func_def_304,type,
uminus_uminus_real_o: ( real > $o ) > real > $o ).
thf(func_def_305,type,
uminus6401447641752708672_nat_o: ( set_nat > $o ) > set_nat > $o ).
thf(func_def_306,type,
uminus1351360451143612070nteger: code_integer > code_integer ).
thf(func_def_307,type,
uminus1482373934393186551omplex: complex > complex ).
thf(func_def_308,type,
uminus_uminus_int: int > int ).
thf(func_def_309,type,
uminus_uminus_rat: rat > rat ).
thf(func_def_310,type,
uminus_uminus_real: real > real ).
thf(func_def_311,type,
uminus8566677241136511917omplex: set_complex > set_complex ).
thf(func_def_312,type,
uminus1532241313380277803et_int: set_int > set_int ).
thf(func_def_313,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(func_def_314,type,
uminus6221592323253981072nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
thf(func_def_315,type,
uminus612125837232591019t_real: set_real > set_real ).
thf(func_def_316,type,
uminus613421341184616069et_nat: set_set_nat > set_set_nat ).
thf(func_def_317,type,
zero_z3403309356797280102nteger: code_integer ).
thf(func_def_318,type,
zero_zero_complex: complex ).
thf(func_def_319,type,
zero_z5237406670263579293d_enat: extended_enat ).
thf(func_def_320,type,
zero_zero_int: int ).
thf(func_def_321,type,
zero_zero_nat: nat ).
thf(func_def_322,type,
zero_zero_rat: rat ).
thf(func_def_323,type,
zero_zero_real: real ).
thf(func_def_324,type,
groups6621422865394947399nteger: ( complex > code_integer ) > set_complex > code_integer ).
thf(func_def_325,type,
groups7754918857620584856omplex: ( complex > complex ) > set_complex > complex ).
thf(func_def_326,type,
groups5690904116761175830ex_int: ( complex > int ) > set_complex > int ).
thf(func_def_327,type,
groups5693394587270226106ex_nat: ( complex > nat ) > set_complex > nat ).
thf(func_def_328,type,
groups5058264527183730370ex_rat: ( complex > rat ) > set_complex > rat ).
thf(func_def_329,type,
groups5808333547571424918x_real: ( complex > real ) > set_complex > real ).
thf(func_def_330,type,
groups7873554091576472773nteger: ( int > code_integer ) > set_int > code_integer ).
thf(func_def_331,type,
groups3049146728041665814omplex: ( int > complex ) > set_int > complex ).
thf(func_def_332,type,
groups4538972089207619220nt_int: ( int > int ) > set_int > int ).
thf(func_def_333,type,
groups4541462559716669496nt_nat: ( int > nat ) > set_int > nat ).
thf(func_def_334,type,
groups3906332499630173760nt_rat: ( int > rat ) > set_int > rat ).
thf(func_def_335,type,
groups8778361861064173332t_real: ( int > real ) > set_int > real ).
thf(func_def_336,type,
groups7501900531339628137nteger: ( nat > code_integer ) > set_nat > code_integer ).
thf(func_def_337,type,
groups2073611262835488442omplex: ( nat > complex ) > set_nat > complex ).
thf(func_def_338,type,
groups3539618377306564664at_int: ( nat > int ) > set_nat > int ).
thf(func_def_339,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
thf(func_def_340,type,
groups2906978787729119204at_rat: ( nat > rat ) > set_nat > rat ).
thf(func_def_341,type,
groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).
thf(func_def_342,type,
groups7713935264441627589nteger: ( real > code_integer ) > set_real > code_integer ).
thf(func_def_343,type,
groups5754745047067104278omplex: ( real > complex ) > set_real > complex ).
thf(func_def_344,type,
groups1932886352136224148al_int: ( real > int ) > set_real > int ).
thf(func_def_345,type,
groups1935376822645274424al_nat: ( real > nat ) > set_real > nat ).
thf(func_def_346,type,
groups1300246762558778688al_rat: ( real > rat ) > set_real > rat ).
thf(func_def_347,type,
groups8097168146408367636l_real: ( real > real ) > set_real > real ).
thf(func_def_348,type,
groups8255218700646806128omplex: ( set_nat > complex ) > set_set_nat > complex ).
thf(func_def_349,type,
groups8294997508430121362at_nat: ( set_nat > nat ) > set_set_nat > nat ).
thf(func_def_350,type,
groups5107569545109728110t_real: ( set_nat > real ) > set_set_nat > real ).
thf(func_def_351,type,
groups8682486955453173170nteger: ( complex > code_integer ) > set_complex > code_integer ).
thf(func_def_352,type,
groups3708469109370488835omplex: ( complex > complex ) > set_complex > complex ).
thf(func_def_353,type,
groups858564598930262913ex_int: ( complex > int ) > set_complex > int ).
thf(func_def_354,type,
groups861055069439313189ex_nat: ( complex > nat ) > set_complex > nat ).
thf(func_def_355,type,
groups225925009352817453ex_rat: ( complex > rat ) > set_complex > rat ).
thf(func_def_356,type,
groups766887009212190081x_real: ( complex > real ) > set_complex > real ).
thf(func_def_357,type,
groups3827104343326376752nteger: ( int > code_integer ) > set_int > code_integer ).
thf(func_def_358,type,
groups7440179247065528705omplex: ( int > complex ) > set_int > complex ).
thf(func_def_359,type,
groups1705073143266064639nt_int: ( int > int ) > set_int > int ).
thf(func_def_360,type,
groups1707563613775114915nt_nat: ( int > nat ) > set_int > nat ).
thf(func_def_361,type,
groups1072433553688619179nt_rat: ( int > rat ) > set_int > rat ).
thf(func_def_362,type,
groups2316167850115554303t_real: ( int > real ) > set_int > real ).
thf(func_def_363,type,
groups3455450783089532116nteger: ( nat > code_integer ) > set_nat > code_integer ).
thf(func_def_364,type,
groups6464643781859351333omplex: ( nat > complex ) > set_nat > complex ).
thf(func_def_365,type,
groups705719431365010083at_int: ( nat > int ) > set_nat > int ).
thf(func_def_366,type,
groups708209901874060359at_nat: ( nat > nat ) > set_nat > nat ).
thf(func_def_367,type,
groups73079841787564623at_rat: ( nat > rat ) > set_nat > rat ).
thf(func_def_368,type,
groups129246275422532515t_real: ( nat > real ) > set_nat > real ).
thf(func_def_369,type,
groups6225526099057966256nteger: ( real > code_integer ) > set_real > code_integer ).
thf(func_def_370,type,
groups713298508707869441omplex: ( real > complex ) > set_real > complex ).
thf(func_def_371,type,
groups4694064378042380927al_int: ( real > int ) > set_real > int ).
thf(func_def_372,type,
groups4696554848551431203al_nat: ( real > nat ) > set_real > nat ).
thf(func_def_373,type,
groups4061424788464935467al_rat: ( real > rat ) > set_real > rat ).
thf(func_def_374,type,
groups1681761925125756287l_real: ( real > real ) > set_real > real ).
thf(func_def_375,type,
groups4246057289670975065at_int: ( set_nat > int ) > set_set_nat > int ).
thf(func_def_376,type,
groups9116527308978886569_o_int: ( $o > int ) > int > list_o > int ).
thf(func_def_377,type,
the_int: ( int > $o ) > int ).
thf(func_def_378,type,
the_real: ( real > $o ) > real ).
thf(func_def_379,type,
if_Code_integer: $o > code_integer > code_integer > code_integer ).
thf(func_def_380,type,
if_complex: $o > complex > complex > complex ).
thf(func_def_381,type,
if_int: $o > int > int > int ).
thf(func_def_382,type,
if_list_int: $o > list_int > list_int > list_int ).
thf(func_def_383,type,
if_nat: $o > nat > nat > nat ).
thf(func_def_384,type,
if_num: $o > num > num > num ).
thf(func_def_385,type,
if_option_nat: $o > option_nat > option_nat > option_nat ).
thf(func_def_386,type,
if_option_num: $o > option_num > option_num > option_num ).
thf(func_def_387,type,
if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o ).
thf(func_def_388,type,
if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_389,type,
if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int ).
thf(func_def_390,type,
if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_391,type,
if_rat: $o > rat > rat > rat ).
thf(func_def_392,type,
if_real: $o > real > real > real ).
thf(func_def_393,type,
if_set_int: $o > set_int > set_int > set_int ).
thf(func_def_394,type,
if_set_nat: $o > set_nat > set_nat > set_nat ).
thf(func_def_395,type,
if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_396,type,
abs_Integ: product_prod_nat_nat > int ).
thf(func_def_397,type,
rep_Integ: int > product_prod_nat_nat ).
thf(func_def_398,type,
int_ge_less_than: int > set_Pr958786334691620121nt_int ).
thf(func_def_399,type,
int_ge_less_than2: int > set_Pr958786334691620121nt_int ).
thf(func_def_400,type,
nat2: int > nat ).
thf(func_def_401,type,
ring_1_Ints_real: set_real ).
thf(func_def_402,type,
ring_18347121197199848620nteger: int > code_integer ).
thf(func_def_403,type,
ring_17405671764205052669omplex: int > complex ).
thf(func_def_404,type,
ring_1_of_int_int: int > int ).
thf(func_def_405,type,
ring_1_of_int_rat: int > rat ).
thf(func_def_406,type,
ring_1_of_int_real: int > real ).
thf(func_def_407,type,
semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(func_def_408,type,
lattic8263393255366662781ax_int: set_int > int ).
thf(func_def_409,type,
lattic8265883725875713057ax_nat: set_nat > nat ).
thf(func_def_410,type,
bfun_nat_real: ( nat > real ) > filter_nat > $o ).
thf(func_def_411,type,
append_int: list_int > list_int > list_int ).
thf(func_def_412,type,
append_VEBT_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_413,type,
distinct_int: list_int > $o ).
thf(func_def_414,type,
drop_VEBT_VEBT: nat > list_VEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_415,type,
cons_int: int > list_int > list_int ).
thf(func_def_416,type,
cons_VEBT_VEBT: vEBT_VEBT > list_VEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_417,type,
nil_int: list_int ).
thf(func_def_418,type,
nil_VEBT_VEBT: list_VEBT_VEBT ).
thf(func_def_419,type,
set_o2: list_o > set_o ).
thf(func_def_420,type,
set_complex2: list_complex > set_complex ).
thf(func_def_421,type,
set_int2: list_int > set_int ).
thf(func_def_422,type,
set_nat2: list_nat > set_nat ).
thf(func_def_423,type,
set_real2: list_real > set_real ).
thf(func_def_424,type,
set_set_nat2: list_set_nat > set_set_nat ).
thf(func_def_425,type,
set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).
thf(func_def_426,type,
size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).
thf(func_def_427,type,
list_update_o: list_o > nat > $o > list_o ).
thf(func_def_428,type,
list_update_complex: list_complex > nat > complex > list_complex ).
thf(func_def_429,type,
list_update_int: list_int > nat > int > list_int ).
thf(func_def_430,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(func_def_431,type,
list_update_real: list_real > nat > real > list_real ).
thf(func_def_432,type,
list_update_set_nat: list_set_nat > nat > set_nat > list_set_nat ).
thf(func_def_433,type,
list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_434,type,
nth_o: list_o > nat > $o ).
thf(func_def_435,type,
nth_Code_integer: list_Code_integer > nat > code_integer ).
thf(func_def_436,type,
nth_complex: list_complex > nat > complex ).
thf(func_def_437,type,
nth_int: list_int > nat > int ).
thf(func_def_438,type,
nth_nat: list_nat > nat > nat ).
thf(func_def_439,type,
nth_num: list_num > nat > num ).
thf(func_def_440,type,
nth_Product_prod_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o ).
thf(func_def_441,type,
nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int ).
thf(func_def_442,type,
nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT ).
thf(func_def_443,type,
nth_Pr8522763379788166057eger_o: list_P8526636022914148096eger_o > nat > produc6271795597528267376eger_o ).
thf(func_def_444,type,
nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat ).
thf(func_def_445,type,
nth_Pr8326237132889035090at_num: list_P1726324292696863441at_num > nat > product_prod_nat_num ).
thf(func_def_446,type,
nth_Pr6456567536196504476um_num: list_P3744719386663036955um_num > nat > product_prod_num_num ).
thf(func_def_447,type,
nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).
thf(func_def_448,type,
nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int ).
thf(func_def_449,type,
nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT ).
thf(func_def_450,type,
nth_real: list_real > nat > real ).
thf(func_def_451,type,
nth_set_nat: list_set_nat > nat > set_nat ).
thf(func_def_452,type,
nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_453,type,
product_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).
thf(func_def_454,type,
product_o_int: list_o > list_int > list_P3795440434834930179_o_int ).
thf(func_def_455,type,
product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).
thf(func_def_456,type,
produc3607205314601156340eger_o: list_Code_integer > list_o > list_P8526636022914148096eger_o ).
thf(func_def_457,type,
product_int_o: list_int > list_o > list_P5087981734274514673_int_o ).
thf(func_def_458,type,
product_int_int: list_int > list_int > list_P5707943133018811711nt_int ).
thf(func_def_459,type,
produc662631939642741121T_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT ).
thf(func_def_460,type,
product_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).
thf(func_def_461,type,
product_nat_num: list_nat > list_num > list_P1726324292696863441at_num ).
thf(func_def_462,type,
product_num_num: list_num > list_num > list_P3744719386663036955um_num ).
thf(func_def_463,type,
product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).
thf(func_def_464,type,
produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).
thf(func_def_465,type,
produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).
thf(func_def_466,type,
replicate_o: nat > $o > list_o ).
thf(func_def_467,type,
replicate_complex: nat > complex > list_complex ).
thf(func_def_468,type,
replicate_int: nat > int > list_int ).
thf(func_def_469,type,
replicate_nat: nat > nat > list_nat ).
thf(func_def_470,type,
replicate_real: nat > real > list_real ).
thf(func_def_471,type,
replicate_set_nat: nat > set_nat > list_set_nat ).
thf(func_def_472,type,
replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_473,type,
take_VEBT_VEBT: nat > list_VEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_474,type,
upto: int > int > list_int ).
thf(func_def_475,type,
upto_aux: int > int > list_int > list_int ).
thf(func_def_476,type,
upto_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(func_def_477,type,
suc: nat > nat ).
thf(func_def_478,type,
compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(func_def_479,type,
case_nat_o: $o > ( nat > $o ) > nat > $o ).
thf(func_def_480,type,
case_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(func_def_481,type,
case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
thf(func_def_482,type,
pred: nat > nat ).
thf(func_def_483,type,
semiri4939895301339042750nteger: nat > code_integer ).
thf(func_def_484,type,
semiri8010041392384452111omplex: nat > complex ).
thf(func_def_485,type,
semiri1314217659103216013at_int: nat > int ).
thf(func_def_486,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(func_def_487,type,
semiri681578069525770553at_rat: nat > rat ).
thf(func_def_488,type,
semiri5074537144036343181t_real: nat > real ).
thf(func_def_489,type,
size_size_list_o: list_o > nat ).
thf(func_def_490,type,
size_s3445333598471063425nteger: list_Code_integer > nat ).
thf(func_def_491,type,
size_s3451745648224563538omplex: list_complex > nat ).
thf(func_def_492,type,
size_size_list_int: list_int > nat ).
thf(func_def_493,type,
size_size_list_nat: list_nat > nat ).
thf(func_def_494,type,
size_size_list_num: list_num > nat ).
thf(func_def_495,type,
size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat ).
thf(func_def_496,type,
size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat ).
thf(func_def_497,type,
size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat ).
thf(func_def_498,type,
size_s4246224855604898693_int_o: list_P5087981734274514673_int_o > nat ).
thf(func_def_499,type,
size_s5157815400016825771nt_int: list_P5707943133018811711nt_int > nat ).
thf(func_def_500,type,
size_s6639371672096860321T_VEBT: list_P7524865323317820941T_VEBT > nat ).
thf(func_def_501,type,
size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat ).
thf(func_def_502,type,
size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat ).
thf(func_def_503,type,
size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat ).
thf(func_def_504,type,
size_size_list_real: list_real > nat ).
thf(func_def_505,type,
size_s3254054031482475050et_nat: list_set_nat > nat ).
thf(func_def_506,type,
size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).
thf(func_def_507,type,
size_size_num: num > nat ).
thf(func_def_508,type,
size_size_option_nat: option_nat > nat ).
thf(func_def_509,type,
size_size_option_num: option_num > nat ).
thf(func_def_510,type,
size_s170228958280169651at_nat: option4927543243414619207at_nat > nat ).
thf(func_def_511,type,
size_size_VEBT_VEBT: vEBT_VEBT > nat ).
thf(func_def_512,type,
nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
thf(func_def_513,type,
nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_514,type,
nat_prod_encode: product_prod_nat_nat > nat ).
thf(func_def_515,type,
nat_set_decode: nat > set_nat ).
thf(func_def_516,type,
nat_set_encode: set_nat > nat ).
thf(func_def_517,type,
nat_triangle: nat > nat ).
thf(func_def_518,type,
root: nat > real > real ).
thf(func_def_519,type,
sqrt: real > real ).
thf(func_def_520,type,
bitM: num > num ).
thf(func_def_521,type,
inc: num > num ).
thf(func_def_522,type,
neg_nu8804712462038260780nteger: code_integer > code_integer ).
thf(func_def_523,type,
neg_nu7009210354673126013omplex: complex > complex ).
thf(func_def_524,type,
neg_numeral_dbl_int: int > int ).
thf(func_def_525,type,
neg_numeral_dbl_rat: rat > rat ).
thf(func_def_526,type,
neg_numeral_dbl_real: real > real ).
thf(func_def_527,type,
neg_nu7757733837767384882nteger: code_integer > code_integer ).
thf(func_def_528,type,
neg_nu6511756317524482435omplex: complex > complex ).
thf(func_def_529,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(func_def_530,type,
neg_nu3179335615603231917ec_rat: rat > rat ).
thf(func_def_531,type,
neg_nu6075765906172075777c_real: real > real ).
thf(func_def_532,type,
neg_nu5831290666863070958nteger: code_integer > code_integer ).
thf(func_def_533,type,
neg_nu8557863876264182079omplex: complex > complex ).
thf(func_def_534,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(func_def_535,type,
neg_nu5219082963157363817nc_rat: rat > rat ).
thf(func_def_536,type,
neg_nu8295874005876285629c_real: real > real ).
thf(func_def_537,type,
neg_numeral_sub_int: num > num > int ).
thf(func_def_538,type,
bit0: num > num ).
thf(func_def_539,type,
bit1: num > num ).
thf(func_def_540,type,
one: num ).
thf(func_def_541,type,
case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).
thf(func_def_542,type,
size_num: num > nat ).
thf(func_def_543,type,
num_of_nat: nat > num ).
thf(func_def_544,type,
numera6620942414471956472nteger: num > code_integer ).
thf(func_def_545,type,
numera6690914467698888265omplex: num > complex ).
thf(func_def_546,type,
numera1916890842035813515d_enat: num > extended_enat ).
thf(func_def_547,type,
numeral_numeral_int: num > int ).
thf(func_def_548,type,
numeral_numeral_nat: num > nat ).
thf(func_def_549,type,
numeral_numeral_rat: num > rat ).
thf(func_def_550,type,
numeral_numeral_real: num > real ).
thf(func_def_551,type,
pow: num > num > num ).
thf(func_def_552,type,
pred_numeral: num > nat ).
thf(func_def_553,type,
none_nat: option_nat ).
thf(func_def_554,type,
none_num: option_num ).
thf(func_def_555,type,
none_P5556105721700978146at_nat: option4927543243414619207at_nat ).
thf(func_def_556,type,
some_nat: nat > option_nat ).
thf(func_def_557,type,
some_num: num > option_num ).
thf(func_def_558,type,
some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).
thf(func_def_559,type,
case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).
thf(func_def_560,type,
case_option_int_num: int > ( num > int ) > option_num > int ).
thf(func_def_561,type,
case_option_num_num: num > ( num > num ) > option_num > num ).
thf(func_def_562,type,
case_o6005452278849405969um_num: option_num > ( num > option_num ) > option_num > option_num ).
thf(func_def_563,type,
map_option_num_num: ( num > num ) > option_num > option_num ).
thf(func_def_564,type,
size_option_nat: ( nat > nat ) > option_nat > nat ).
thf(func_def_565,type,
size_option_num: ( num > nat ) > option_num > nat ).
thf(func_def_566,type,
size_o8335143837870341156at_nat: ( product_prod_nat_nat > nat ) > option4927543243414619207at_nat > nat ).
thf(func_def_567,type,
the_nat: option_nat > nat ).
thf(func_def_568,type,
the_num: option_num > num ).
thf(func_def_569,type,
the_Pr8591224930841456533at_nat: option4927543243414619207at_nat > product_prod_nat_nat ).
thf(func_def_570,type,
bot_bot_nat: nat ).
thf(func_def_571,type,
bot_bot_set_complex: set_complex ).
thf(func_def_572,type,
bot_bot_set_int: set_int ).
thf(func_def_573,type,
bot_bot_set_nat: set_nat ).
thf(func_def_574,type,
bot_bot_set_num: set_num ).
thf(func_def_575,type,
bot_bo1796632182523588997nt_int: set_Pr958786334691620121nt_int ).
thf(func_def_576,type,
bot_bot_set_rat: set_rat ).
thf(func_def_577,type,
bot_bot_set_real: set_real ).
thf(func_def_578,type,
bot_bot_set_set_nat: set_set_nat ).
thf(func_def_579,type,
ord_less_complex_o: ( complex > $o ) > ( complex > $o ) > $o ).
thf(func_def_580,type,
ord_less_int_o: ( int > $o ) > ( int > $o ) > $o ).
thf(func_def_581,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_582,type,
ord_less_real_o: ( real > $o ) > ( real > $o ) > $o ).
thf(func_def_583,type,
ord_less_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).
thf(func_def_584,type,
ord_le6747313008572928689nteger: code_integer > code_integer > $o ).
thf(func_def_585,type,
ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).
thf(func_def_586,type,
ord_less_int: int > int > $o ).
thf(func_def_587,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_588,type,
ord_less_num: num > num > $o ).
thf(func_def_589,type,
ord_less_rat: rat > rat > $o ).
thf(func_def_590,type,
ord_less_real: real > real > $o ).
thf(func_def_591,type,
ord_le1307284697595431911nteger: set_Code_integer > set_Code_integer > $o ).
thf(func_def_592,type,
ord_less_set_complex: set_complex > set_complex > $o ).
thf(func_def_593,type,
ord_less_set_int: set_int > set_int > $o ).
thf(func_def_594,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(func_def_595,type,
ord_less_set_num: set_num > set_num > $o ).
thf(func_def_596,type,
ord_less_set_rat: set_rat > set_rat > $o ).
thf(func_def_597,type,
ord_less_set_real: set_real > set_real > $o ).
thf(func_def_598,type,
ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
thf(func_def_599,type,
ord_le4573692005234683329plex_o: ( complex > $o ) > ( complex > $o ) > $o ).
thf(func_def_600,type,
ord_less_eq_int_o: ( int > $o ) > ( int > $o ) > $o ).
thf(func_def_601,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_602,type,
ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $o ).
thf(func_def_603,type,
ord_le3964352015994296041_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).
thf(func_def_604,type,
ord_le3102999989581377725nteger: code_integer > code_integer > $o ).
thf(func_def_605,type,
ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).
thf(func_def_606,type,
ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o ).
thf(func_def_607,type,
ord_less_eq_int: int > int > $o ).
thf(func_def_608,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_609,type,
ord_less_eq_num: num > num > $o ).
thf(func_def_610,type,
ord_less_eq_rat: rat > rat > $o ).
thf(func_def_611,type,
ord_less_eq_real: real > real > $o ).
thf(func_def_612,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(func_def_613,type,
ord_le7084787975880047091nteger: set_Code_integer > set_Code_integer > $o ).
thf(func_def_614,type,
ord_le211207098394363844omplex: set_complex > set_complex > $o ).
thf(func_def_615,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(func_def_616,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(func_def_617,type,
ord_less_eq_set_num: set_num > set_num > $o ).
thf(func_def_618,type,
ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).
thf(func_def_619,type,
ord_less_eq_set_rat: set_rat > set_rat > $o ).
thf(func_def_620,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(func_def_621,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(func_def_622,type,
ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o ).
thf(func_def_623,type,
ord_max_Code_integer: code_integer > code_integer > code_integer ).
thf(func_def_624,type,
ord_max_int: int > int > int ).
thf(func_def_625,type,
ord_max_nat: nat > nat > nat ).
thf(func_def_626,type,
ord_max_num: num > num > num ).
thf(func_def_627,type,
ord_max_rat: rat > rat > rat ).
thf(func_def_628,type,
ord_max_real: real > real > real ).
thf(func_def_629,type,
ord_max_set_nat: set_nat > set_nat > set_nat ).
thf(func_def_630,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(func_def_631,type,
order_9091379641038594480t_real: ( nat > real ) > $o ).
thf(func_def_632,type,
order_mono_nat_nat: ( nat > nat ) > $o ).
thf(func_def_633,type,
order_mono_nat_real: ( nat > real ) > $o ).
thf(func_def_634,type,
order_mono_real_real: ( real > real ) > $o ).
thf(func_def_635,type,
order_5726023648592871131at_nat: ( nat > nat ) > $o ).
thf(func_def_636,type,
order_7092887310737990675l_real: ( real > real ) > $o ).
thf(func_def_637,type,
top_top_set_o: set_o ).
thf(func_def_638,type,
top_top_set_nat: set_nat ).
thf(func_def_639,type,
top_to1996260823553986621t_unit: set_Product_unit ).
thf(func_def_640,type,
top_top_set_real: set_real ).
thf(func_def_641,type,
top_top_set_char: set_char ).
thf(func_def_642,type,
power_8256067586552552935nteger: code_integer > nat > code_integer ).
thf(func_def_643,type,
power_power_complex: complex > nat > complex ).
thf(func_def_644,type,
power_power_int: int > nat > int ).
thf(func_def_645,type,
power_power_nat: nat > nat > nat ).
thf(func_def_646,type,
power_power_rat: rat > nat > rat ).
thf(func_def_647,type,
power_power_real: real > nat > real ).
thf(func_def_648,type,
produc4035269172776083154on_nat: ( nat > nat > $o ) > produc4953844613479565601on_nat > produc2233624965454879586on_nat ).
thf(func_def_649,type,
produc3209952032786966637at_nat: ( nat > nat > nat ) > produc7248412053542808358at_nat > produc4471711990508489141at_nat ).
thf(func_def_650,type,
produc8929957630744042906on_nat: ( nat > nat > nat ) > produc4953844613479565601on_nat > produc8306885398267862888on_nat ).
thf(func_def_651,type,
produc851828971589881931at_num: ( nat > num > num ) > produc2963631642982155120at_num > produc3368934014287244435at_num ).
thf(func_def_652,type,
produc3576312749637752826on_num: ( num > num > $o ) > produc3447558737645232053on_num > produc7036089656553540234on_num ).
thf(func_def_653,type,
produc5778274026573060048on_num: ( num > num > num ) > produc3447558737645232053on_num > produc1193250871479095198on_num ).
thf(func_def_654,type,
produc3994169339658061776at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > produc6121120109295599847at_nat > produc5491161045314408544at_nat ).
thf(func_def_655,type,
produc2899441246263362727at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > produc6121120109295599847at_nat > produc5542196010084753463at_nat ).
thf(func_def_656,type,
product_Pair_o_o: $o > $o > product_prod_o_o ).
thf(func_def_657,type,
product_Pair_o_int: $o > int > product_prod_o_int ).
thf(func_def_658,type,
produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT ).
thf(func_def_659,type,
produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o ).
thf(func_def_660,type,
produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger ).
thf(func_def_661,type,
product_Pair_int_int: int > int > product_prod_int_int ).
thf(func_def_662,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(func_def_663,type,
product_Pair_nat_num: nat > num > product_prod_nat_num ).
thf(func_def_664,type,
produc487386426758144856at_nat: nat > product_prod_nat_nat > produc7248412053542808358at_nat ).
thf(func_def_665,type,
produc1195630363706982562at_num: nat > product_prod_nat_num > produc2963631642982155120at_num ).
thf(func_def_666,type,
product_Pair_num_num: num > num > product_prod_num_num ).
thf(func_def_667,type,
produc5098337634421038937on_nat: option_nat > option_nat > produc4953844613479565601on_nat ).
thf(func_def_668,type,
produc8585076106096196333on_num: option_num > option_num > produc3447558737645232053on_num ).
thf(func_def_669,type,
produc488173922507101015at_nat: option4927543243414619207at_nat > option4927543243414619207at_nat > produc6121120109295599847at_nat ).
thf(func_def_670,type,
produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o ).
thf(func_def_671,type,
produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int ).
thf(func_def_672,type,
produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).
thf(func_def_673,type,
produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT ).
thf(func_def_674,type,
produc6499014454317279255nteger: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_675,type,
produc7828578312038201481er_o_o: ( code_integer > $o > $o ) > produc6271795597528267376eger_o > $o ).
thf(func_def_676,type,
produc1043322548047392435omplex: ( code_integer > $o > set_complex ) > produc6271795597528267376eger_o > set_complex ).
thf(func_def_677,type,
produc1253318751659547953et_int: ( code_integer > $o > set_int ) > produc6271795597528267376eger_o > set_int ).
thf(func_def_678,type,
produc5431169771168744661et_nat: ( code_integer > $o > set_nat ) > produc6271795597528267376eger_o > set_nat ).
thf(func_def_679,type,
produc242741666403216561t_real: ( code_integer > $o > set_real ) > produc6271795597528267376eger_o > set_real ).
thf(func_def_680,type,
produc4188289175737317920o_char: ( code_integer > $o > char ) > produc6271795597528267376eger_o > char ).
thf(func_def_681,type,
produc1553301316500091796er_int: ( code_integer > code_integer > int ) > produc8923325533196201883nteger > int ).
thf(func_def_682,type,
produc1555791787009142072er_nat: ( code_integer > code_integer > nat ) > produc8923325533196201883nteger > nat ).
thf(func_def_683,type,
produc7336495610019696514er_num: ( code_integer > code_integer > num ) > produc8923325533196201883nteger > num ).
thf(func_def_684,type,
produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).
thf(func_def_685,type,
produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_686,type,
produc6771430404735790350plex_o: ( complex > complex > $o ) > produc4411394909380815293omplex > $o ).
thf(func_def_687,type,
produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
thf(func_def_688,type,
produc8211389475949308722nt_int: ( int > int > int ) > product_prod_int_int > int ).
thf(func_def_689,type,
produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).
thf(func_def_690,type,
produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_691,type,
produc27273713700761075at_nat: ( nat > nat > product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_692,type,
produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).
thf(func_def_693,type,
produc1917071388513777916omplex: ( nat > nat > complex ) > product_prod_nat_nat > complex ).
thf(func_def_694,type,
produc6840382203811409530at_int: ( nat > nat > int ) > product_prod_nat_nat > int ).
thf(func_def_695,type,
produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).
thf(func_def_696,type,
produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_697,type,
produc6207742614233964070at_rat: ( nat > nat > rat ) > product_prod_nat_nat > rat ).
thf(func_def_698,type,
produc1703576794950452218t_real: ( nat > nat > real ) > product_prod_nat_nat > real ).
thf(func_def_699,type,
produc4927758841916487424_num_o: ( nat > num > $o ) > product_prod_nat_num > $o ).
thf(func_def_700,type,
produc478579273971653890on_num: ( nat > num > option_num ) > product_prod_nat_num > option_num ).
thf(func_def_701,type,
produc4130284055270567454et_nat: ( nat > num > set_nat ) > product_prod_nat_num > set_nat ).
thf(func_def_702,type,
produc1435849484188172666t_real: ( nat > num > set_real ) > product_prod_nat_num > set_real ).
thf(func_def_703,type,
produc5703948589228662326_num_o: ( num > num > $o ) > product_prod_num_num > $o ).
thf(func_def_704,type,
produc2866383454006189126omplex: ( num > num > set_complex ) > product_prod_num_num > set_complex ).
thf(func_def_705,type,
produc6406642877701697732et_int: ( num > num > set_int ) > product_prod_num_num > set_int ).
thf(func_def_706,type,
produc1361121860356118632et_nat: ( num > num > set_nat ) > product_prod_num_num > set_nat ).
thf(func_def_707,type,
produc8296048397933160132t_real: ( num > num > set_real ) > product_prod_num_num > set_real ).
thf(func_def_708,type,
produc5414030515140494994real_o: ( real > real > $o ) > produc2422161461964618553l_real > $o ).
thf(func_def_709,type,
produc8508995932063986495nteger: produc8923325533196201883nteger > code_integer ).
thf(func_def_710,type,
product_fst_int_int: product_prod_int_int > int ).
thf(func_def_711,type,
product_fst_nat_nat: product_prod_nat_nat > nat ).
thf(func_def_712,type,
product_snd_int_int: product_prod_int_int > int ).
thf(func_def_713,type,
product_snd_nat_nat: product_prod_nat_nat > nat ).
thf(func_def_714,type,
frct: product_prod_int_int > rat ).
thf(func_def_715,type,
normalize: product_prod_int_int > product_prod_int_int ).
thf(func_def_716,type,
of_int: int > rat ).
thf(func_def_717,type,
quotient_of: rat > product_prod_int_int ).
thf(func_def_718,type,
real_V2521375963428798218omplex: set_complex ).
thf(func_def_719,type,
real_V5970128139526366754l_real: ( real > real ) > $o ).
thf(func_def_720,type,
real_V3694042436643373181omplex: complex > complex > real ).
thf(func_def_721,type,
real_V975177566351809787t_real: real > real > real ).
thf(func_def_722,type,
real_V1022390504157884413omplex: complex > real ).
thf(func_def_723,type,
real_V7735802525324610683m_real: real > real ).
thf(func_def_724,type,
real_V4546457046886955230omplex: real > complex ).
thf(func_def_725,type,
real_V2046097035970521341omplex: real > complex > complex ).
thf(func_def_726,type,
real_V1485227260804924795R_real: real > real > real ).
thf(func_def_727,type,
divide6298287555418463151nteger: code_integer > code_integer > code_integer ).
thf(func_def_728,type,
divide1717551699836669952omplex: complex > complex > complex ).
thf(func_def_729,type,
divide_divide_int: int > int > int ).
thf(func_def_730,type,
divide_divide_nat: nat > nat > nat ).
thf(func_def_731,type,
divide_divide_rat: rat > rat > rat ).
thf(func_def_732,type,
divide_divide_real: real > real > real ).
thf(func_def_733,type,
dvd_dvd_Code_integer: code_integer > code_integer > $o ).
thf(func_def_734,type,
dvd_dvd_complex: complex > complex > $o ).
thf(func_def_735,type,
dvd_dvd_int: int > int > $o ).
thf(func_def_736,type,
dvd_dvd_nat: nat > nat > $o ).
thf(func_def_737,type,
dvd_dvd_rat: rat > rat > $o ).
thf(func_def_738,type,
dvd_dvd_real: real > real > $o ).
thf(func_def_739,type,
modulo364778990260209775nteger: code_integer > code_integer > code_integer ).
thf(func_def_740,type,
modulo_modulo_int: int > int > int ).
thf(func_def_741,type,
modulo_modulo_nat: nat > nat > nat ).
thf(func_def_742,type,
zero_n356916108424825756nteger: $o > code_integer ).
thf(func_def_743,type,
zero_n1201886186963655149omplex: $o > complex ).
thf(func_def_744,type,
zero_n2684676970156552555ol_int: $o > int ).
thf(func_def_745,type,
zero_n2687167440665602831ol_nat: $o > nat ).
thf(func_def_746,type,
zero_n2052037380579107095ol_rat: $o > rat ).
thf(func_def_747,type,
zero_n3304061248610475627l_real: $o > real ).
thf(func_def_748,type,
suminf_complex: ( nat > complex ) > complex ).
thf(func_def_749,type,
suminf_int: ( nat > int ) > int ).
thf(func_def_750,type,
suminf_nat: ( nat > nat ) > nat ).
thf(func_def_751,type,
suminf_real: ( nat > real ) > real ).
thf(func_def_752,type,
summable_complex: ( nat > complex ) > $o ).
thf(func_def_753,type,
summable_int: ( nat > int ) > $o ).
thf(func_def_754,type,
summable_nat: ( nat > nat ) > $o ).
thf(func_def_755,type,
summable_real: ( nat > real ) > $o ).
thf(func_def_756,type,
sums_complex: ( nat > complex ) > complex > $o ).
thf(func_def_757,type,
sums_int: ( nat > int ) > int > $o ).
thf(func_def_758,type,
sums_nat: ( nat > nat ) > nat > $o ).
thf(func_def_759,type,
sums_real: ( nat > real ) > real > $o ).
thf(func_def_760,type,
collect_Code_integer: ( code_integer > $o ) > set_Code_integer ).
thf(func_def_761,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(func_def_762,type,
collect_int: ( int > $o ) > set_int ).
thf(func_def_763,type,
collect_list_o: ( list_o > $o ) > set_list_o ).
thf(func_def_764,type,
collect_list_complex: ( list_complex > $o ) > set_list_complex ).
thf(func_def_765,type,
collect_list_int: ( list_int > $o ) > set_list_int ).
thf(func_def_766,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(func_def_767,type,
collec5608196760682091941T_VEBT: ( list_VEBT_VEBT > $o ) > set_list_VEBT_VEBT ).
thf(func_def_768,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(func_def_769,type,
collect_num: ( num > $o ) > set_num ).
thf(func_def_770,type,
collec8663557070575231912omplex: ( produc4411394909380815293omplex > $o ) > set_Pr5085853215250843933omplex ).
thf(func_def_771,type,
collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).
thf(func_def_772,type,
collec3799799289383736868l_real: ( produc2422161461964618553l_real > $o ) > set_Pr6218003697084177305l_real ).
thf(func_def_773,type,
collect_rat: ( rat > $o ) > set_rat ).
thf(func_def_774,type,
collect_real: ( real > $o ) > set_real ).
thf(func_def_775,type,
collect_set_complex: ( set_complex > $o ) > set_set_complex ).
thf(func_def_776,type,
collect_set_int: ( set_int > $o ) > set_set_int ).
thf(func_def_777,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(func_def_778,type,
pow_nat: set_nat > set_set_nat ).
thf(func_def_779,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(func_def_780,type,
image_int_nat: ( int > nat ) > set_int > set_nat ).
thf(func_def_781,type,
image_nat_int: ( nat > int ) > set_nat > set_int ).
thf(func_def_782,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(func_def_783,type,
image_nat_real: ( nat > real ) > set_nat > set_real ).
thf(func_def_784,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(func_def_785,type,
image_nat_char: ( nat > char ) > set_nat > set_char ).
thf(func_def_786,type,
image_5971271580939081552omplex: ( real > filter6041513312241820739omplex ) > set_real > set_fi4554929511873752355omplex ).
thf(func_def_787,type,
image_2178119161166701260l_real: ( real > filter2146258269922977983l_real ) > set_real > set_fi7789364187291644575l_real ).
thf(func_def_788,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(func_def_789,type,
image_char_nat: ( char > nat ) > set_char > set_nat ).
thf(func_def_790,type,
insert_int: int > set_int > set_int ).
thf(func_def_791,type,
insert_nat: nat > set_nat > set_nat ).
thf(func_def_792,type,
insert_real: real > set_real > set_real ).
thf(func_def_793,type,
set_fo1517530859248394432omplex: ( nat > complex > complex ) > nat > nat > complex > complex ).
thf(func_def_794,type,
set_fo2581907887559384638at_int: ( nat > int > int ) > nat > nat > int > int ).
thf(func_def_795,type,
set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).
thf(func_def_796,type,
set_fo1949268297981939178at_rat: ( nat > rat > rat ) > nat > nat > rat > rat ).
thf(func_def_797,type,
set_fo3111899725591712190t_real: ( nat > real > real ) > nat > nat > real > real ).
thf(func_def_798,type,
set_or1266510415728281911st_int: int > int > set_int ).
thf(func_def_799,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(func_def_800,type,
set_or7049704709247886629st_num: num > num > set_num ).
thf(func_def_801,type,
set_or633870826150836451st_rat: rat > rat > set_rat ).
thf(func_def_802,type,
set_or1222579329274155063t_real: real > real > set_real ).
thf(func_def_803,type,
set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).
thf(func_def_804,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(func_def_805,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(func_def_806,type,
set_ord_atLeast_nat: nat > set_nat ).
thf(func_def_807,type,
set_ord_atLeast_real: real > set_real ).
thf(func_def_808,type,
set_ord_atMost_nat: nat > set_nat ).
thf(func_def_809,type,
set_or6656581121297822940st_int: int > int > set_int ).
thf(func_def_810,type,
set_or6659071591806873216st_nat: nat > nat > set_nat ).
thf(func_def_811,type,
set_or5832277885323065728an_int: int > int > set_int ).
thf(func_def_812,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(func_def_813,type,
set_or1633881224788618240n_real: real > real > set_real ).
thf(func_def_814,type,
set_or1210151606488870762an_nat: nat > set_nat ).
thf(func_def_815,type,
set_or5849166863359141190n_real: real > set_real ).
thf(func_def_816,type,
set_ord_lessThan_int: int > set_int ).
thf(func_def_817,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(func_def_818,type,
set_ord_lessThan_num: num > set_num ).
thf(func_def_819,type,
set_ord_lessThan_rat: rat > set_rat ).
thf(func_def_820,type,
set_or5984915006950818249n_real: real > set_real ).
thf(func_def_821,type,
set_or890127255671739683et_nat: set_nat > set_set_nat ).
thf(func_def_822,type,
ascii_of: char > char ).
thf(func_def_823,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(func_def_824,type,
char_of_integer: code_integer > char ).
thf(func_def_825,type,
comm_s629917340098488124ar_nat: char > nat ).
thf(func_def_826,type,
integer_of_char: char > code_integer ).
thf(func_def_827,type,
unique3096191561947761185of_nat: nat > char ).
thf(func_def_828,type,
topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).
thf(func_def_829,type,
topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
thf(func_def_830,type,
topolo2919662092509805066nteger: ( nat > code_integer ) > $o ).
thf(func_def_831,type,
topolo4899668324122417113eq_int: ( nat > int ) > $o ).
thf(func_def_832,type,
topolo4902158794631467389eq_nat: ( nat > nat ) > $o ).
thf(func_def_833,type,
topolo1459490580787246023eq_num: ( nat > num ) > $o ).
thf(func_def_834,type,
topolo4267028734544971653eq_rat: ( nat > rat ) > $o ).
thf(func_def_835,type,
topolo6980174941875973593q_real: ( nat > real ) > $o ).
thf(func_def_836,type,
topolo7278393974255667507et_nat: ( nat > set_nat ) > $o ).
thf(func_def_837,type,
topolo4110288021797289639omplex: set_complex > $o ).
thf(func_def_838,type,
topolo4860482606490270245n_real: set_real > $o ).
thf(func_def_839,type,
topolo2177554685111907308n_real: real > set_real > filter_real ).
thf(func_def_840,type,
topolo2815343760600316023s_real: real > filter_real ).
thf(func_def_841,type,
topolo6517432010174082258omplex: ( nat > complex ) > $o ).
thf(func_def_842,type,
topolo4055970368930404560y_real: ( nat > real ) > $o ).
thf(func_def_843,type,
topolo896644834953643431omplex: filter6041513312241820739omplex ).
thf(func_def_844,type,
topolo1511823702728130853y_real: filter2146258269922977983l_real ).
thf(func_def_845,type,
arccos: real > real ).
thf(func_def_846,type,
arcosh_real: real > real ).
thf(func_def_847,type,
arcsin: real > real ).
thf(func_def_848,type,
arctan: real > real ).
thf(func_def_849,type,
arsinh_real: real > real ).
thf(func_def_850,type,
artanh_real: real > real ).
thf(func_def_851,type,
cos_complex: complex > complex ).
thf(func_def_852,type,
cos_real: real > real ).
thf(func_def_853,type,
cos_coeff: nat > real ).
thf(func_def_854,type,
cosh_real: real > real ).
thf(func_def_855,type,
cot_real: real > real ).
thf(func_def_856,type,
diffs_Code_integer: ( nat > code_integer ) > nat > code_integer ).
thf(func_def_857,type,
diffs_complex: ( nat > complex ) > nat > complex ).
thf(func_def_858,type,
diffs_int: ( nat > int ) > nat > int ).
thf(func_def_859,type,
diffs_rat: ( nat > rat ) > nat > rat ).
thf(func_def_860,type,
diffs_real: ( nat > real ) > nat > real ).
thf(func_def_861,type,
exp_complex: complex > complex ).
thf(func_def_862,type,
exp_real: real > real ).
thf(func_def_863,type,
ln_ln_real: real > real ).
thf(func_def_864,type,
log: real > real > real ).
thf(func_def_865,type,
pi: real ).
thf(func_def_866,type,
powr_real: real > real > real ).
thf(func_def_867,type,
sin_complex: complex > complex ).
thf(func_def_868,type,
sin_real: real > real ).
thf(func_def_869,type,
sin_coeff: nat > real ).
thf(func_def_870,type,
sinh_real: real > real ).
thf(func_def_871,type,
tan_complex: complex > complex ).
thf(func_def_872,type,
tan_real: real > real ).
thf(func_def_873,type,
tanh_complex: complex > complex ).
thf(func_def_874,type,
tanh_real: real > real ).
thf(func_def_875,type,
vEBT_Leaf: $o > $o > vEBT_VEBT ).
thf(func_def_876,type,
vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_877,type,
vEBT_size_VEBT: vEBT_VEBT > nat ).
thf(func_def_878,type,
vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
thf(func_def_879,type,
vEBT_VEBT_high: nat > nat > nat ).
thf(func_def_880,type,
vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
thf(func_def_881,type,
vEBT_VEBT_low: nat > nat > nat ).
thf(func_def_882,type,
vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
thf(func_def_883,type,
vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_884,type,
vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
thf(func_def_885,type,
vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_886,type,
vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
thf(func_def_887,type,
vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_888,type,
vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
thf(func_def_889,type,
vEBT_set_vebt: vEBT_VEBT > set_nat ).
thf(func_def_890,type,
vEBT_vebt_buildup: nat > vEBT_VEBT ).
thf(func_def_891,type,
vEBT_v4011308405150292612up_rel: nat > nat > $o ).
thf(func_def_892,type,
vEBT_vebt_delete: vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_893,type,
vEBT_vebt_delete_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_894,type,
vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_895,type,
vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_896,type,
vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
thf(func_def_897,type,
vEBT_VEBT_minNull: vEBT_VEBT > $o ).
thf(func_def_898,type,
vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_899,type,
vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).
thf(func_def_900,type,
vEBT_vebt_member: vEBT_VEBT > nat > $o ).
thf(func_def_901,type,
vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_902,type,
vEBT_VEBT_add: option_nat > option_nat > option_nat ).
thf(func_def_903,type,
vEBT_VEBT_greater: option_nat > option_nat > $o ).
thf(func_def_904,type,
vEBT_VEBT_less: option_nat > option_nat > $o ).
thf(func_def_905,type,
vEBT_VEBT_lesseq: option_nat > option_nat > $o ).
thf(func_def_906,type,
vEBT_VEBT_max_in_set: set_nat > nat > $o ).
thf(func_def_907,type,
vEBT_VEBT_min_in_set: set_nat > nat > $o ).
thf(func_def_908,type,
vEBT_VEBT_mul: option_nat > option_nat > option_nat ).
thf(func_def_909,type,
vEBT_V4262088993061758097ft_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).
thf(func_def_910,type,
vEBT_V819420779217536731ft_num: ( num > num > num ) > option_num > option_num > option_num ).
thf(func_def_911,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_912,type,
vEBT_VEBT_power: option_nat > option_nat > option_nat ).
thf(func_def_913,type,
vEBT_vebt_maxt: vEBT_VEBT > option_nat ).
thf(func_def_914,type,
vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_915,type,
vEBT_vebt_mint: vEBT_VEBT > option_nat ).
thf(func_def_916,type,
vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_917,type,
vEBT_is_pred_in_set: set_nat > nat > nat > $o ).
thf(func_def_918,type,
vEBT_vebt_pred: vEBT_VEBT > nat > option_nat ).
thf(func_def_919,type,
vEBT_vebt_pred_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_920,type,
vEBT_is_succ_in_set: set_nat > nat > nat > $o ).
thf(func_def_921,type,
vEBT_vebt_succ: vEBT_VEBT > nat > option_nat ).
thf(func_def_922,type,
vEBT_vebt_succ_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_923,type,
accp_nat: ( nat > nat > $o ) > nat > $o ).
thf(func_def_924,type,
accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
thf(func_def_925,type,
accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
thf(func_def_926,type,
accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).
thf(func_def_927,type,
accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).
thf(func_def_928,type,
fChoice_real: ( real > $o ) > real ).
thf(func_def_929,type,
member_o: $o > set_o > $o ).
thf(func_def_930,type,
member_complex: complex > set_complex > $o ).
thf(func_def_931,type,
member_int: int > set_int > $o ).
thf(func_def_932,type,
member_list_o: list_o > set_list_o > $o ).
thf(func_def_933,type,
member_list_int: list_int > set_list_int > $o ).
thf(func_def_934,type,
member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o ).
thf(func_def_935,type,
member_nat: nat > set_nat > $o ).
thf(func_def_936,type,
member_num: num > set_num > $o ).
thf(func_def_937,type,
member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
thf(func_def_938,type,
member_rat: rat > set_rat > $o ).
thf(func_def_939,type,
member_real: real > set_real > $o ).
thf(func_def_940,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(func_def_941,type,
member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
thf(func_def_942,type,
deg: nat ).
thf(func_def_943,type,
lx: nat ).
thf(func_def_944,type,
m: nat ).
thf(func_def_945,type,
ma: nat ).
thf(func_def_946,type,
maxi: nat ).
thf(func_def_947,type,
maxs: nat ).
thf(func_def_948,type,
mi: nat ).
thf(func_def_949,type,
na: nat ).
thf(func_def_950,type,
summary: vEBT_VEBT ).
thf(func_def_951,type,
summin: nat ).
thf(func_def_952,type,
treeList: list_VEBT_VEBT ).
thf(func_def_953,type,
xa: nat ).
thf(func_def_954,type,
ya: nat ).
thf(func_def_977,type,
sP0: vEBT_VEBT > vEBT_VEBT > nat > $o ).
thf(func_def_978,type,
sP1: vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_979,type,
sP2: vEBT_VEBT > vEBT_VEBT > nat > $o ).
thf(func_def_980,type,
sP3: nat > vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_981,type,
sP4: vEBT_VEBT > nat > vEBT_VEBT > $o ).
thf(func_def_982,type,
sP5: nat > vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_983,type,
sP6: vEBT_VEBT > option_nat > $o ).
thf(func_def_984,type,
sP7: option_nat > vEBT_VEBT > $o ).
thf(func_def_985,type,
sP8: option_nat > vEBT_VEBT > nat > $o ).
thf(func_def_986,type,
sP9: vEBT_VEBT > option_nat > nat > $o ).
thf(func_def_987,type,
sP10: vEBT_VEBT > option_nat > nat > $o ).
thf(func_def_988,type,
sP11: nat > vEBT_VEBT > option_nat > $o ).
thf(func_def_989,type,
sP12: option_nat > nat > vEBT_VEBT > $o ).
thf(func_def_990,type,
sP13: vEBT_VEBT > nat > option_nat > $o ).
thf(func_def_991,type,
sP14: nat > option_nat > vEBT_VEBT > $o ).
thf(func_def_992,type,
sP15: nat > vEBT_VEBT > option_nat > $o ).
thf(func_def_993,type,
sP16: vEBT_VEBT > vEBT_VEBT > nat > $o ).
thf(func_def_994,type,
sP17: nat > vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_995,type,
sP18: nat > vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_996,type,
sP19: vEBT_VEBT > nat > vEBT_VEBT > $o ).
thf(func_def_997,type,
sP20: nat > vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_998,type,
sP21: vEBT_VEBT > nat > vEBT_VEBT > $o ).
thf(func_def_999,type,
sP22: nat > option_nat > vEBT_VEBT > $o ).
thf(func_def_1000,type,
sP23: nat > vEBT_VEBT > option_nat > $o ).
thf(func_def_1001,type,
sP24: vEBT_VEBT > option_nat > nat > $o ).
thf(func_def_1002,type,
sP25: vEBT_VEBT > nat > option_nat > $o ).
thf(func_def_1003,type,
sP26: vEBT_VEBT > nat > option_nat > $o ).
thf(func_def_1004,type,
sP27: vEBT_VEBT > option_nat > nat > $o ).
thf(func_def_1005,type,
sP28: option_nat > vEBT_VEBT > $o ).
thf(func_def_1006,type,
sP29: option_nat > vEBT_VEBT > nat > $o ).
thf(func_def_1007,type,
sP30: nat > vEBT_VEBT > option_nat > $o ).
thf(func_def_1008,type,
sP31: vEBT_VEBT > option_nat > nat > $o ).
thf(func_def_1009,type,
sP32: option_nat > nat > vEBT_VEBT > $o ).
thf(func_def_1010,type,
sP33: vEBT_VEBT > nat > vEBT_VEBT > $o ).
thf(func_def_1011,type,
sP34: nat > vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_1012,type,
sP35: vEBT_VEBT > nat > vEBT_VEBT > $o ).
thf(func_def_1013,type,
sP36: vEBT_VEBT > vEBT_VEBT > nat > $o ).
thf(func_def_1014,type,
sK37: vEBT_VEBT > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1015,type,
sK38: vEBT_VEBT > vEBT_VEBT > option4927543243414619207at_nat ).
thf(func_def_1016,type,
sK39: vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1017,type,
sK40: vEBT_VEBT > vEBT_VEBT > nat > $o ).
thf(func_def_1018,type,
sK41: vEBT_VEBT > vEBT_VEBT > nat > $o ).
thf(func_def_1019,type,
sK42: vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1020,type,
sK43: vEBT_VEBT > vEBT_VEBT > option4927543243414619207at_nat ).
thf(func_def_1021,type,
sK44: vEBT_VEBT > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1022,type,
sK45: vEBT_VEBT > nat > vEBT_VEBT > nat ).
thf(func_def_1023,type,
sK46: vEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1024,type,
sK47: vEBT_VEBT > nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1025,type,
sK48: vEBT_VEBT > nat > vEBT_VEBT > nat ).
thf(func_def_1026,type,
sK49: vEBT_VEBT > nat > vEBT_VEBT > nat ).
thf(func_def_1027,type,
sK50: vEBT_VEBT > nat > vEBT_VEBT > nat ).
thf(func_def_1028,type,
sK51: vEBT_VEBT > nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1029,type,
sK52: vEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1030,type,
sK53: nat ).
thf(func_def_1031,type,
sK54: nat > nat > nat ).
thf(func_def_1032,type,
sK55: nat ).
thf(func_def_1033,type,
sK56: nat > vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_1034,type,
sK57: nat > vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_1035,type,
sK58: nat > nat ).
thf(func_def_1036,type,
sK59: vEBT_VEBT > nat > vEBT_VEBT > $o ).
thf(func_def_1037,type,
sK60: vEBT_VEBT > nat > vEBT_VEBT > $o ).
thf(func_def_1038,type,
sK61: nat > vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_1039,type,
sK62: nat > vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_1040,type,
sK63: vEBT_VEBT > vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1041,type,
sK64: vEBT_VEBT > vEBT_VEBT > nat > list_VEBT_VEBT ).
thf(func_def_1042,type,
sK65: vEBT_VEBT > vEBT_VEBT > nat > nat ).
thf(func_def_1043,type,
sK66: vEBT_VEBT > vEBT_VEBT > nat > nat ).
thf(func_def_1044,type,
sK67: vEBT_VEBT > vEBT_VEBT > nat > nat ).
thf(func_def_1045,type,
sK68: vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1046,type,
sK69: vEBT_VEBT > vEBT_VEBT > nat ).
thf(func_def_1047,type,
sK70: vEBT_VEBT > vEBT_VEBT > nat ).
thf(func_def_1048,type,
sK71: vEBT_VEBT > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1049,type,
sK72: vEBT_VEBT > vEBT_VEBT > nat ).
thf(func_def_1050,type,
sK73: vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1051,type,
sK74: vEBT_VEBT > vEBT_VEBT > nat ).
thf(func_def_1052,type,
sK75: vEBT_VEBT > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1053,type,
sK76: vEBT_VEBT > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1054,type,
sK77: vEBT_VEBT > vEBT_VEBT > nat ).
thf(func_def_1055,type,
sK78: vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1056,type,
sK79: option_nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1057,type,
sK80: option_nat > vEBT_VEBT > nat ).
thf(func_def_1058,type,
sK81: option_nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1059,type,
sK82: option_nat > vEBT_VEBT > nat ).
thf(func_def_1060,type,
sK83: option_nat > vEBT_VEBT > nat ).
thf(func_def_1061,type,
sK84: vEBT_VEBT > option_nat > $o ).
thf(func_def_1062,type,
sK85: vEBT_VEBT > option_nat > $o ).
thf(func_def_1063,type,
sK86: vEBT_VEBT > option_nat > nat ).
thf(func_def_1064,type,
sK87: vEBT_VEBT > option_nat > list_VEBT_VEBT ).
thf(func_def_1065,type,
sK88: vEBT_VEBT > option_nat > vEBT_VEBT ).
thf(func_def_1066,type,
sK89: nat ).
thf(func_def_1067,type,
sK90: nat > nat > nat ).
thf(func_def_1068,type,
sK91: nat ).
thf(func_def_1069,type,
sK92: ( nat > nat ) > nat ).
thf(func_def_1070,type,
sK93: ( nat > nat ) > nat ).
thf(func_def_1071,type,
sK94: ( nat > $o ) > nat ).
thf(func_def_1072,type,
sK95: ( nat > $o ) > nat ).
thf(func_def_1073,type,
sK96: nat > nat ).
thf(func_def_1074,type,
sK97: vEBT_VEBT > $o ).
thf(func_def_1075,type,
sK98: vEBT_VEBT > $o ).
thf(func_def_1076,type,
sK99: vEBT_VEBT > option_nat > nat > nat ).
thf(func_def_1077,type,
sK100: vEBT_VEBT > option_nat > nat > list_VEBT_VEBT ).
thf(func_def_1078,type,
sK101: vEBT_VEBT > option_nat > nat > vEBT_VEBT ).
thf(func_def_1079,type,
sK102: vEBT_VEBT > option_nat > nat > nat ).
thf(func_def_1080,type,
sK103: vEBT_VEBT > option_nat > nat > nat ).
thf(func_def_1081,type,
sK104: option_nat > vEBT_VEBT > nat > $o ).
thf(func_def_1082,type,
sK105: option_nat > vEBT_VEBT > nat > $o ).
thf(func_def_1083,type,
sK106: vEBT_VEBT > product_prod_nat_nat ).
thf(func_def_1084,type,
sK107: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1085,type,
sK108: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1086,type,
sK109: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1087,type,
sK110: vEBT_VEBT > nat ).
thf(func_def_1088,type,
sK111: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1089,type,
sK112: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1090,type,
sK113: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1091,type,
sK114: vEBT_VEBT > product_prod_nat_nat ).
thf(func_def_1092,type,
sK115: nat > vEBT_VEBT > option_nat > list_VEBT_VEBT ).
thf(func_def_1093,type,
sK116: nat > vEBT_VEBT > option_nat > product_prod_nat_nat ).
thf(func_def_1094,type,
sK117: nat > vEBT_VEBT > option_nat > vEBT_VEBT ).
thf(func_def_1095,type,
sK118: nat > option_nat > vEBT_VEBT > product_prod_nat_nat ).
thf(func_def_1096,type,
sK119: nat > option_nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1097,type,
sK120: nat > option_nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1098,type,
sK121: vEBT_VEBT > nat > option_nat > $o ).
thf(func_def_1099,type,
sK122: vEBT_VEBT > nat > option_nat > $o ).
thf(func_def_1100,type,
sK123: vEBT_VEBT > nat > option_nat > nat ).
thf(func_def_1101,type,
sK124: option_nat > nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1102,type,
sK125: option_nat > nat > vEBT_VEBT > nat ).
thf(func_def_1103,type,
sK126: option_nat > nat > vEBT_VEBT > nat ).
thf(func_def_1104,type,
sK127: option_nat > nat > vEBT_VEBT > nat ).
thf(func_def_1105,type,
sK128: option_nat > nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1106,type,
sK129: nat > vEBT_VEBT > option_nat > $o ).
thf(func_def_1107,type,
sK130: nat > vEBT_VEBT > option_nat > $o ).
thf(func_def_1108,type,
sK131: nat > option_nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1109,type,
sK132: nat > option_nat > vEBT_VEBT > nat ).
thf(func_def_1110,type,
sK133: nat > option_nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1111,type,
sK134: nat > nat ).
thf(func_def_1112,type,
sK135: nat > nat ).
thf(func_def_1113,type,
sK136: nat > nat > nat ).
thf(func_def_1114,type,
sK137: nat ).
thf(func_def_1115,type,
sK138: nat > nat > nat ).
thf(func_def_1116,type,
sK139: vEBT_VEBT > nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1117,type,
sK140: vEBT_VEBT > nat > vEBT_VEBT > nat ).
thf(func_def_1118,type,
sK141: vEBT_VEBT > nat > vEBT_VEBT > nat ).
thf(func_def_1119,type,
sK142: vEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1120,type,
sK143: nat > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1121,type,
sK144: nat > vEBT_VEBT > vEBT_VEBT > nat ).
thf(func_def_1122,type,
sK145: nat > vEBT_VEBT > vEBT_VEBT > nat ).
thf(func_def_1123,type,
sK146: nat > vEBT_VEBT > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1124,type,
sK147: vEBT_VEBT > nat > vEBT_VEBT > $o ).
thf(func_def_1125,type,
sK148: vEBT_VEBT > nat > vEBT_VEBT > $o ).
thf(func_def_1126,type,
sK149: nat > vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_1127,type,
sK150: nat > vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_1128,type,
sK151: nat > vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_1129,type,
sK152: nat > vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_1130,type,
sK153: nat > vEBT_VEBT > vEBT_VEBT > nat ).
thf(func_def_1131,type,
sK154: vEBT_VEBT > vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1132,type,
sK155: vEBT_VEBT > vEBT_VEBT > nat > nat ).
thf(func_def_1133,type,
sK156: vEBT_VEBT > vEBT_VEBT > nat > nat ).
thf(func_def_1134,type,
sK157: vEBT_VEBT > vEBT_VEBT > nat > nat ).
thf(func_def_1135,type,
sK158: vEBT_VEBT > vEBT_VEBT > nat > list_VEBT_VEBT ).
thf(func_def_1136,type,
sK159: nat > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1137,type,
sK160: nat > vEBT_VEBT > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1138,type,
sK161: nat > vEBT_VEBT > vEBT_VEBT > nat ).
thf(func_def_1139,type,
sK162: nat > nat > nat ).
thf(func_def_1140,type,
sK163: ( nat > nat > $o ) > nat ).
thf(func_def_1141,type,
sK164: ( nat > nat > $o ) > nat ).
thf(func_def_1142,type,
sK165: ( nat > nat > $o ) > nat ).
thf(func_def_1143,type,
sK166: ( nat > nat > $o ) > nat ).
thf(func_def_1144,type,
sK167: ( nat > nat > $o ) > nat ).
thf(func_def_1145,type,
sK168: nat > nat > nat ).
thf(func_def_1146,type,
sK169: vEBT_VEBT > option_nat > nat > product_prod_nat_nat ).
thf(func_def_1147,type,
sK170: vEBT_VEBT > option_nat > nat > vEBT_VEBT ).
thf(func_def_1148,type,
sK171: vEBT_VEBT > option_nat > nat > list_VEBT_VEBT ).
thf(func_def_1149,type,
sK172: vEBT_VEBT > nat > option_nat > list_VEBT_VEBT ).
thf(func_def_1150,type,
sK173: vEBT_VEBT > nat > option_nat > vEBT_VEBT ).
thf(func_def_1151,type,
sK174: vEBT_VEBT > nat > option_nat > nat ).
thf(func_def_1152,type,
sK175: vEBT_VEBT > nat > option_nat > $o ).
thf(func_def_1153,type,
sK176: vEBT_VEBT > nat > option_nat > $o ).
thf(func_def_1154,type,
sK177: vEBT_VEBT > option_nat > nat > vEBT_VEBT ).
thf(func_def_1155,type,
sK178: vEBT_VEBT > option_nat > nat > nat ).
thf(func_def_1156,type,
sK179: vEBT_VEBT > option_nat > nat > nat ).
thf(func_def_1157,type,
sK180: vEBT_VEBT > option_nat > nat > nat ).
thf(func_def_1158,type,
sK181: vEBT_VEBT > option_nat > nat > list_VEBT_VEBT ).
thf(func_def_1159,type,
sK182: nat > vEBT_VEBT > option_nat > $o ).
thf(func_def_1160,type,
sK183: nat > vEBT_VEBT > option_nat > $o ).
thf(func_def_1161,type,
sK184: nat > option_nat > vEBT_VEBT > $o ).
thf(func_def_1162,type,
sK185: nat > option_nat > vEBT_VEBT > $o ).
thf(func_def_1163,type,
sK186: nat > option_nat > vEBT_VEBT > nat ).
thf(func_def_1164,type,
sK187: option_nat > vEBT_VEBT > nat > product_prod_nat_nat ).
thf(func_def_1165,type,
sK188: option_nat > vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1166,type,
sK189: option_nat > vEBT_VEBT > nat > list_VEBT_VEBT ).
thf(func_def_1167,type,
sK190: nat ).
thf(func_def_1168,type,
sK191: option_nat > vEBT_VEBT > $o ).
thf(func_def_1169,type,
sK192: option_nat > vEBT_VEBT > $o ).
thf(func_def_1170,type,
sK193: option_nat > vEBT_VEBT > nat ).
thf(func_def_1171,type,
sK194: option_nat > vEBT_VEBT > nat ).
thf(func_def_1172,type,
sK195: option_nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1173,type,
sK196: option_nat > vEBT_VEBT > nat ).
thf(func_def_1174,type,
sK197: option_nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1175,type,
sK198: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1176,type,
sK199: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1177,type,
sK200: vEBT_VEBT > nat ).
thf(func_def_1178,type,
sK201: option_nat > nat > vEBT_VEBT > nat ).
thf(func_def_1179,type,
sK202: option_nat > nat > vEBT_VEBT > nat ).
thf(func_def_1180,type,
sK203: option_nat > nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1181,type,
sK204: option_nat > nat > vEBT_VEBT > nat ).
thf(func_def_1182,type,
sK205: option_nat > nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1183,type,
sK206: vEBT_VEBT > $o ).
thf(func_def_1184,type,
sK207: vEBT_VEBT > $o ).
thf(func_def_1185,type,
sK208: nat > vEBT_VEBT > option_nat > $o ).
thf(func_def_1186,type,
sK209: nat > vEBT_VEBT > option_nat > $o ).
thf(func_def_1187,type,
sK210: option_nat > vEBT_VEBT > nat > $o ).
thf(func_def_1188,type,
sK211: option_nat > vEBT_VEBT > nat > $o ).
thf(func_def_1189,type,
sK212: nat > nat ).
thf(func_def_1190,type,
sK213: vEBT_VEBT > nat ).
thf(func_def_1191,type,
sK214: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1192,type,
sK215: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1193,type,
sK216: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1194,type,
sK217: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1195,type,
sK218: vEBT_VEBT > product_prod_nat_nat ).
thf(func_def_1196,type,
sK219: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1197,type,
sK220: vEBT_VEBT > product_prod_nat_nat ).
thf(func_def_1198,type,
sK221: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1199,type,
sK222: vEBT_VEBT > vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1200,type,
sK223: vEBT_VEBT > vEBT_VEBT > nat > list_VEBT_VEBT ).
thf(func_def_1201,type,
sK224: vEBT_VEBT > vEBT_VEBT > nat > option4927543243414619207at_nat ).
thf(func_def_1202,type,
sK225: vEBT_VEBT > nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1203,type,
sK226: vEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1204,type,
sK227: vEBT_VEBT > nat > vEBT_VEBT > nat ).
thf(func_def_1205,type,
sK228: nat > vEBT_VEBT > vEBT_VEBT > option4927543243414619207at_nat ).
thf(func_def_1206,type,
sK229: nat > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1207,type,
sK230: nat > vEBT_VEBT > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1208,type,
sK231: vEBT_VEBT > nat > vEBT_VEBT > $o ).
thf(func_def_1209,type,
sK232: vEBT_VEBT > nat > vEBT_VEBT > $o ).
thf(func_def_1210,type,
sK233: vEBT_VEBT > nat > vEBT_VEBT > nat ).
thf(func_def_1211,type,
sK234: vEBT_VEBT > nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1212,type,
sK235: vEBT_VEBT > nat > vEBT_VEBT > nat ).
thf(func_def_1213,type,
sK236: vEBT_VEBT > nat > vEBT_VEBT > nat ).
thf(func_def_1214,type,
sK237: vEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1215,type,
sK238: nat > nat > nat ).
thf(func_def_1216,type,
sK239: nat > nat > nat ).
thf(func_def_1218,type,
sK241: nat > nat ).
thf(f26869,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f26868]) ).
thf(f26868,plain,
( ( plus_plus_nat @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ treeList ) @ maxs ) @ maxi )
!= ( plus_plus_nat @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ treeList ) @ maxs ) @ maxi ) ),
inference(forward_demodulation,[status(thm)],[f26867,f25368]) ).
thf(f25368,plain,
! [X0: nat] :
( ( the_nat @ ( some_nat @ X0 ) )
= X0 ),
inference(cnf_transformation,[status(thm)],[f24308]) ).
thf(f24308,plain,
! [X0: nat] :
( ( the_nat @ ( some_nat @ X0 ) )
= X0 ),
inference(rectify,[status(thm)],[f365]) ).
thf(f365,axiom,
! [X43: nat] :
( ( the_nat @ ( some_nat @ X43 ) )
= X43 ),
file('/export/starexec/sandbox/tmp/tmp.YjPWRkCLRl/DTF2THF_23017.p',fact_364_option_Osel) ).
thf(f26867,plain,
( ( plus_plus_nat @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ treeList ) @ maxs ) @ maxi )
!= ( plus_plus_nat @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ treeList ) @ maxs ) @ ( the_nat @ ( some_nat @ maxi ) ) ) ),
inference(forward_demodulation,[status(thm)],[f26865,f25695]) ).
thf(f25695,plain,
( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ maxs ) )
= ( some_nat @ maxi ) ),
inference(cnf_transformation,[status(thm)],[f17]) ).
thf(f17,axiom,
( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ maxs ) )
= ( some_nat @ maxi ) ),
file('/export/starexec/sandbox/tmp/tmp.YjPWRkCLRl/DTF2THF_23017.p',fact_16__092_060open_062Some_Amaxi_A_061_Avebt__maxt_A_ItreeList_A_B_Amaxs_J_092_060close_062) ).
thf(f26865,plain,
( ( plus_plus_nat @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ treeList ) @ maxs ) @ maxi )
!= ( plus_plus_nat @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ treeList ) @ maxs ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ maxs ) ) ) ) ),
inference(backward_demodulation,[status(thm)],[f26861,f26864]) ).
thf(f26864,plain,
( ( nth_VEBT_VEBT @ treeList @ maxs )
= ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ lx ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ lx ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) ) @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ lx ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) ) ) @ maxs ) ),
inference(forward_demodulation,[status(thm)],[f26863,f25368]) ).
thf(f26863,plain,
( ( nth_VEBT_VEBT @ treeList @ maxs )
= ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ lx ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ lx ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) ) @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ lx ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) ) ) @ ( the_nat @ ( some_nat @ maxs ) ) ) ),
inference(forward_demodulation,[status(thm)],[f26862,f26782]) ).
thf(f26782,plain,
( ( some_nat @ maxs )
= ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ lx ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) ) ) ),
inference(forward_demodulation,[status(thm)],[f26558,f25940]) ).
thf(f25940,plain,
( ( size_s6755466524823107622T_VEBT @ treeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f25363,f25775]) ).
thf(f25775,plain,
( m
= ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ),
inference(definition_unfolding,[status(thm)],[f25198,f25523]) ).
thf(f25523,plain,
( na
= ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ),
inference(cnf_transformation,[status(thm)],[f7]) ).
thf(f7,axiom,
( na
= ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.YjPWRkCLRl/DTF2THF_23017.p',fact_6__092_060open_062deg_Adiv_A2_A_061_An_092_060close_062) ).
thf(f25198,plain,
m = na,
inference(cnf_transformation,[status(thm)],[f1]) ).
thf(f1,axiom,
m = na,
file('/export/starexec/sandbox/tmp/tmp.YjPWRkCLRl/DTF2THF_23017.p',fact_0__C4_Ohyps_C_I3_J) ).
thf(f25363,plain,
( ( size_s6755466524823107622T_VEBT @ treeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ),
inference(cnf_transformation,[status(thm)],[f48]) ).
thf(f48,axiom,
( ( size_s6755466524823107622T_VEBT @ treeList )
= ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ),
file('/export/starexec/sandbox/tmp/tmp.YjPWRkCLRl/DTF2THF_23017.p',fact_47__C4_Ohyps_C_I2_J) ).
thf(f26558,plain,
( ( some_nat @ maxs )
= ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ lx ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f26139]) ).
thf(f26139,plain,
( ( some_nat @ maxs )
= ( vEBT_vebt_maxt
@ ( vEBT_vebt_delete @ summary
@ ( ^ [Y0: nat,Y1: nat] : ( divide_divide_nat @ Y0 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y1 ) )
@ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ lx )
@ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f25767,f25607,f25523,f25523]) ).
thf(f25607,plain,
( vEBT_VEBT_high
= ( ^ [Y0: nat,Y1: nat] : ( divide_divide_nat @ Y0 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f11172]) ).
thf(f11172,plain,
( vEBT_VEBT_high
= ( ^ [Y0: nat,Y1: nat] : ( divide_divide_nat @ Y0 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f11171]) ).
thf(f11171,plain,
( vEBT_VEBT_high
= ( ^ [X0: nat,X1: nat] : ( divide_divide_nat @ X0 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f6]) ).
thf(f6,axiom,
( vEBT_VEBT_high
= ( ^ [X5: nat,X6: nat] : ( divide_divide_nat @ X5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X6 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.YjPWRkCLRl/DTF2THF_23017.p',fact_5_high__def) ).
thf(f25767,plain,
( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
= ( some_nat @ maxs ) ),
inference(cnf_transformation,[status(thm)],[f19]) ).
thf(f19,axiom,
( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
= ( some_nat @ maxs ) ),
file('/export/starexec/sandbox/tmp/tmp.YjPWRkCLRl/DTF2THF_23017.p',fact_18__092_060open_062Some_Amaxs_A_061_Avebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_092_060close_062) ).
thf(f26862,plain,
( ( nth_VEBT_VEBT @ treeList @ maxs )
= ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ lx ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ lx ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) ) @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ lx ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ lx ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) ) ) ) ) ),
inference(forward_demodulation,[status(thm)],[f26448,f25940]) ).
thf(f26448,plain,
( ( nth_VEBT_VEBT @ treeList @ maxs )
= ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ lx ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ lx ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ lx ) @ ( 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 @ ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ lx ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f26017]) ).
thf(f26017,plain,
( ( nth_VEBT_VEBT @ treeList @ maxs )
= ( nth_VEBT_VEBT
@ ( list_u1324408373059187874T_VEBT @ treeList
@ ( ^ [Y0: nat,Y1: nat] : ( divide_divide_nat @ Y0 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y1 ) )
@ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ lx )
@ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
@ ( vEBT_vebt_delete
@ ( nth_VEBT_VEBT @ treeList
@ ( ^ [Y0: nat,Y1: nat] : ( divide_divide_nat @ Y0 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y1 ) )
@ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ lx )
@ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
@ ( ^ [Y0: nat,Y1: nat] : ( modulo_modulo_nat @ Y0 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y1 ) )
@ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ lx )
@ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
@ ( the_nat
@ ( vEBT_vebt_maxt
@ ( vEBT_vebt_delete @ summary
@ ( ^ [Y0: nat,Y1: nat] : ( divide_divide_nat @ Y0 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y1 ) )
@ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ lx )
@ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f25546,f25607,f25523,f25523,f25607,f25523,f25523,f25153,f25523,f25523,f25607,f25523,f25523]) ).
thf(f25153,plain,
( vEBT_VEBT_low
= ( ^ [Y0: nat,Y1: nat] : ( modulo_modulo_nat @ Y0 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f17451]) ).
thf(f17451,plain,
( vEBT_VEBT_low
= ( ^ [Y0: nat,Y1: nat] : ( modulo_modulo_nat @ Y0 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f17450]) ).
thf(f17450,plain,
( vEBT_VEBT_low
= ( ^ [X0: nat,X1: nat] : ( modulo_modulo_nat @ X0 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f1650]) ).
thf(f1650,axiom,
( vEBT_VEBT_low
= ( ^ [X5: nat,X6: nat] : ( modulo_modulo_nat @ X5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X6 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.YjPWRkCLRl/DTF2THF_23017.p',fact_1649_low__def) ).
thf(f25546,plain,
( ( nth_VEBT_VEBT @ treeList @ maxs )
= ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f13]) ).
thf(f13,axiom,
( ( nth_VEBT_VEBT @ treeList @ maxs )
= ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.YjPWRkCLRl/DTF2THF_23017.p',fact_12__092_060open_062treeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_A_B_Athe_A_Ivebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_J_A_061_AtreeList_A_B_Amaxs_092_060close_062) ).
thf(f26861,plain,
( ( plus_plus_nat @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ treeList ) @ maxs ) @ maxi )
!= ( plus_plus_nat @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ treeList ) @ maxs ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ lx ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ lx ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) ) @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ lx ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) ) ) @ maxs ) ) ) ) ),
inference(forward_demodulation,[status(thm)],[f26860,f25368]) ).
thf(f26860,plain,
( ( plus_plus_nat @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ treeList ) @ ( the_nat @ ( some_nat @ maxs ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ lx ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ lx ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) ) @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ lx ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) ) ) @ ( the_nat @ ( some_nat @ maxs ) ) ) ) ) )
!= ( plus_plus_nat @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ treeList ) @ maxs ) @ maxi ) ),
inference(forward_demodulation,[status(thm)],[f26859,f26782]) ).
thf(f26859,plain,
( ( plus_plus_nat @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ treeList ) @ maxs ) @ maxi )
!= ( plus_plus_nat @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ treeList ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ lx ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ lx ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ lx ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) ) @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ lx ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( size_s6755466524823107622T_VEBT @ treeList ) ) @ lx ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ) ) ) ) ) ) ) ) ),
inference(forward_demodulation,[status(thm)],[f26449,f25940]) ).
thf(f26449,plain,
( ( 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 @ ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ lx ) @ ( 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 @ treeList @ ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ lx ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ lx ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ lx ) @ ( 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 @ ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ lx ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
!= ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ maxs ) @ maxi ) ),
inference(beta_eta_normalization,[status(thm)],[f25961]) ).
thf(f25961,plain,
( ( 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
@ ( ^ [Y0: nat,Y1: nat] : ( divide_divide_nat @ Y0 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y1 ) )
@ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ lx )
@ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
@ ( the_nat
@ ( vEBT_vebt_maxt
@ ( nth_VEBT_VEBT
@ ( list_u1324408373059187874T_VEBT @ treeList
@ ( ^ [Y0: nat,Y1: nat] : ( divide_divide_nat @ Y0 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y1 ) )
@ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ lx )
@ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
@ ( vEBT_vebt_delete
@ ( nth_VEBT_VEBT @ treeList
@ ( ^ [Y0: nat,Y1: nat] : ( divide_divide_nat @ Y0 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y1 ) )
@ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ lx )
@ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
@ ( ^ [Y0: nat,Y1: nat] : ( modulo_modulo_nat @ Y0 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y1 ) )
@ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ lx )
@ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
@ ( the_nat
@ ( vEBT_vebt_maxt
@ ( vEBT_vebt_delete @ summary
@ ( ^ [Y0: nat,Y1: nat] : ( divide_divide_nat @ Y0 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y1 ) )
@ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ lx )
@ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
!= ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ maxs ) @ maxi ) ),
inference(definition_unfolding,[status(thm)],[f25411,f25607,f25523,f25523,f25607,f25523,f25523,f25607,f25523,f25523,f25153,f25523,f25523,f25607,f25523,f25523,f25523]) ).
thf(f25411,plain,
( ( 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 @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) ) )
!= ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) @ maxs ) @ maxi ) ),
inference(cnf_transformation,[status(thm)],[f24211]) ).
thf(f24211,plain,
( ( 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 @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) ) )
!= ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) @ maxs ) @ maxi ) ),
inference(flattening,[status(thm)],[f10240]) ).
thf(f10240,negated_conjecture,
( ( 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 @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) ) )
!= ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) @ maxs ) @ maxi ) ),
inference(negated_conjecture,[status(cth)],[f10239]) ).
thf(f10239,conjecture,
( ( 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 @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) ) )
= ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) @ maxs ) @ maxi ) ),
file('/export/starexec/sandbox/tmp/tmp.YjPWRkCLRl/DTF2THF_23017.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP257^3 : TPTP v9.2.1. Bugfixed v9.2.1.
% 0.13/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n015.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Fri Feb 27 03:51:31 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.30/0.46 ---- Original DTF file ---
% 0.30/0.46 thf(spec,logic,$$dhol).
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % File : ITP257^3 : TPTP v9.2.1. Bugfixed v9.2.1.
% 0.30/0.46 % Domain : Interactive Theorem Proving
% 0.30/0.46 % Problem : Sledgehammer problem VEBT_Delete 00798_054957
% 0.30/0.46 % Version : [Des22] axioms.
% 0.30/0.46 % English :
% 0.30/0.46
% 0.30/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.30/0.46 % : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% 0.30/0.46 % Source : [Des22]
% 0.30/0.46 % Names : 0072_VEBT_Delete_00798_054957 [Des22]
% 0.30/0.46
% 0.30/0.46 % Status : Theorem
% 0.30/0.46 % Rating : ? v9.2.1
% 0.30/0.46 % Syntax : Number of formulae : 11194 (5527 unt; 955 typ; 0 def)
% 0.30/0.46 % Number of atoms : 29042 (12268 equ; 0 cnn)
% 0.30/0.46 % Maximal formula atoms : 71 ( 2 avg)
% 0.30/0.46 % Number of connectives : 123127 (2856 ~; 522 |;1890 &;106354 @)
% 0.30/0.46 % ( 0 <=>;11505 =>; 0 <=; 0 <~>)
% 0.30/0.46 % Maximal formula depth : 39 ( 6 avg)
% 0.30/0.46 % Number of types : 93 ( 92 usr)
% 0.30/0.46 % Number of type conns : 4438 (4438 >; 0 *; 0 +; 0 <<)
% 0.30/0.46 % Number of symbols : 866 ( 863 usr; 58 con; 0-8 aty)
% 0.30/0.46 % Number of variables : 26846 (2396 ^;23715 !; 735 ?;26846 :)
% 0.30/0.46 % SPC : TH0_THM_EQU_NAR
% 0.30/0.46
% 0.30/0.46 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.30/0.46 % from the van Emde Boas Trees session in the Archive of Formal
% 0.30/0.46 % proofs -
% 0.30/0.46 % www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 0.30/0.46 % 2022-02-18 05:26:00.556
% 0.30/0.46 % Bugfixes : Added ()s around negated sides of equations.
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % Could-be-implicit typings (92)
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% 0.30/0.47 thf(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
% 0.30/0.47 size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Olist__update_001_Eo,type,
% 0.30/0.47 list_update_o: list_o > nat > $o > list_o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Olist__update_001t__Complex__Ocomplex,type,
% 0.30/0.47 list_update_complex: list_complex > nat > complex > list_complex ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Olist__update_001t__Int__Oint,type,
% 0.30/0.47 list_update_int: list_int > nat > int > list_int ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
% 0.30/0.47 list_update_nat: list_nat > nat > nat > list_nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Olist__update_001t__Real__Oreal,type,
% 0.30/0.47 list_update_real: list_real > nat > real > list_real ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Olist__update_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.30/0.47 list_update_set_nat: list_set_nat > nat > set_nat > list_set_nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Olist__update_001t__VEBT____Definitions__OVEBT,type,
% 0.30/0.47 list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Onth_001_Eo,type,
% 0.30/0.47 nth_o: list_o > nat > $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Onth_001t__Code____Numeral__Ointeger,type,
% 0.30/0.47 nth_Code_integer: list_Code_integer > nat > code_integer ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Onth_001t__Complex__Ocomplex,type,
% 0.30/0.47 nth_complex: list_complex > nat > complex ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Onth_001t__Int__Oint,type,
% 0.30/0.47 nth_int: list_int > nat > int ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Onth_001t__Nat__Onat,type,
% 0.30/0.47 nth_nat: list_nat > nat > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Onth_001t__Num__Onum,type,
% 0.30/0.47 nth_num: list_num > nat > num ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
% 0.30/0.47 nth_Product_prod_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
% 0.30/0.47 nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.30/0.47 nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
% 0.30/0.47 nth_Pr8522763379788166057eger_o: list_P8526636022914148096eger_o > nat > produc6271795597528267376eger_o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.30/0.47 nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
% 0.30/0.47 nth_Pr8326237132889035090at_num: list_P1726324292696863441at_num > nat > product_prod_nat_num ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 0.30/0.47 nth_Pr6456567536196504476um_num: list_P3744719386663036955um_num > nat > product_prod_num_num ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
% 0.30/0.47 nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
% 0.30/0.47 nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.30/0.47 nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Onth_001t__Real__Oreal,type,
% 0.30/0.47 nth_real: list_real > nat > real ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.30/0.47 nth_set_nat: list_set_nat > nat > set_nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
% 0.30/0.47 nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Oproduct_001_Eo_001_Eo,type,
% 0.30/0.47 product_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
% 0.30/0.47 product_o_int: list_o > list_int > list_P3795440434834930179_o_int ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Oproduct_001_Eo_001t__VEBT____Definitions__OVEBT,type,
% 0.30/0.47 product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Oproduct_001t__Code____Numeral__Ointeger_001_Eo,type,
% 0.30/0.47 produc3607205314601156340eger_o: list_Code_integer > list_o > list_P8526636022914148096eger_o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Oproduct_001t__Int__Oint_001_Eo,type,
% 0.30/0.47 product_int_o: list_int > list_o > list_P5087981734274514673_int_o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Oproduct_001t__Int__Oint_001t__Int__Oint,type,
% 0.30/0.47 product_int_int: list_int > list_int > list_P5707943133018811711nt_int ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Oproduct_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
% 0.30/0.47 produc662631939642741121T_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.30/0.47 product_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Num__Onum,type,
% 0.30/0.47 product_nat_num: list_nat > list_num > list_P1726324292696863441at_num ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Oproduct_001t__Num__Onum_001t__Num__Onum,type,
% 0.30/0.47 product_num_num: list_num > list_num > list_P3744719386663036955um_num ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
% 0.30/0.47 product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
% 0.30/0.47 produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 0.30/0.47 produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Oreplicate_001_Eo,type,
% 0.30/0.47 replicate_o: nat > $o > list_o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Oreplicate_001t__Complex__Ocomplex,type,
% 0.30/0.47 replicate_complex: nat > complex > list_complex ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Oreplicate_001t__Int__Oint,type,
% 0.30/0.47 replicate_int: nat > int > list_int ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
% 0.30/0.47 replicate_nat: nat > nat > list_nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Oreplicate_001t__Real__Oreal,type,
% 0.30/0.47 replicate_real: nat > real > list_real ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.30/0.47 replicate_set_nat: nat > set_nat > list_set_nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
% 0.30/0.47 replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Otake_001t__VEBT____Definitions__OVEBT,type,
% 0.30/0.47 take_VEBT_VEBT: nat > list_VEBT_VEBT > list_VEBT_VEBT ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Oupto,type,
% 0.30/0.47 upto: int > int > list_int ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Oupto__aux,type,
% 0.30/0.47 upto_aux: int > int > list_int > list_int ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_List_Oupto__rel,type,
% 0.30/0.47 upto_rel: product_prod_int_int > product_prod_int_int > $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_OSuc,type,
% 0.30/0.47 suc: nat > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.30/0.47 compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
% 0.30/0.47 case_nat_o: $o > ( nat > $o ) > nat > $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
% 0.30/0.47 case_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
% 0.30/0.47 case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Onat_Opred,type,
% 0.30/0.47 pred: nat > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
% 0.30/0.47 semiri4939895301339042750nteger: nat > code_integer ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
% 0.30/0.47 semiri8010041392384452111omplex: nat > complex ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 0.30/0.47 semiri1314217659103216013at_int: nat > int ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 0.30/0.47 semiri1316708129612266289at_nat: nat > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
% 0.30/0.47 semiri681578069525770553at_rat: nat > rat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
% 0.30/0.47 semiri5074537144036343181t_real: nat > real ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
% 0.30/0.47 size_size_list_o: list_o > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
% 0.30/0.47 size_s3445333598471063425nteger: list_Code_integer > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
% 0.30/0.47 size_s3451745648224563538omplex: list_complex > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
% 0.30/0.47 size_size_list_int: list_int > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
% 0.30/0.47 size_size_list_nat: list_nat > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
% 0.30/0.47 size_size_list_num: list_num > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
% 0.30/0.47 size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
% 0.30/0.47 size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 0.30/0.47 size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_M_Eo_J_J,type,
% 0.30/0.47 size_s4246224855604898693_int_o: list_P5087981734274514673_int_o > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 0.30/0.47 size_s5157815400016825771nt_int: list_P5707943133018811711nt_int > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 0.30/0.47 size_s6639371672096860321T_VEBT: list_P7524865323317820941T_VEBT > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
% 0.30/0.47 size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
% 0.30/0.47 size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 0.30/0.47 size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
% 0.30/0.47 size_size_list_real: list_real > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.30/0.47 size_s3254054031482475050et_nat: list_set_nat > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 0.30/0.47 size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
% 0.30/0.47 size_size_num: num > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Nat__Onat_J,type,
% 0.30/0.47 size_size_option_nat: option_nat > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
% 0.30/0.47 size_size_option_num: option_num > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.30/0.47 size_s170228958280169651at_nat: option4927543243414619207at_nat > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
% 0.30/0.47 size_size_VEBT_VEBT: vEBT_VEBT > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
% 0.30/0.47 nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
% 0.30/0.47 nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat__Bijection_Oprod__encode,type,
% 0.30/0.47 nat_prod_encode: product_prod_nat_nat > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat__Bijection_Oset__decode,type,
% 0.30/0.47 nat_set_decode: nat > set_nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat__Bijection_Oset__encode,type,
% 0.30/0.47 nat_set_encode: set_nat > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Nat__Bijection_Otriangle,type,
% 0.30/0.47 nat_triangle: nat > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_NthRoot_Oroot,type,
% 0.30/0.47 root: nat > real > real ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_NthRoot_Osqrt,type,
% 0.30/0.47 sqrt: real > real ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Num_OBitM,type,
% 0.30/0.47 bitM: num > num ).
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% 0.30/0.47
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% 0.30/0.47
% 0.30/0.47 thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
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% 0.30/0.47
% 0.30/0.47 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
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% 0.30/0.47
% 0.30/0.47 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
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% 0.30/0.47 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Rat__Orat,type,
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% 0.30/0.47 thf(sy_c_Topological__Spaces_Omonoseq_001t__Rat__Orat,type,
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% 0.30/0.47
% 0.30/0.47 thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Complex__Ocomplex,type,
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% 0.30/0.47
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% 0.30/0.47 thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
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% 0.30/0.47 thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Complex__Ocomplex,type,
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% 0.30/0.47
% 0.30/0.47 thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
% 0.30/0.47 arcosh_real: real > real ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Transcendental_Oarcsin,type,
% 0.30/0.47 arcsin: real > real ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Transcendental_Oarctan,type,
% 0.30/0.47 arctan: real > real ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
% 0.30/0.47 arsinh_real: real > real ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
% 0.30/0.47 artanh_real: real > real ).
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% 0.30/0.47 thf(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
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% 0.30/0.47 thf(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
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% 0.30/0.47
% 0.30/0.47 thf(sy_c_Transcendental_Ocos__coeff,type,
% 0.30/0.47 cos_coeff: nat > real ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
% 0.30/0.47 cosh_real: real > real ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
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% 0.30/0.47
% 0.30/0.47 thf(sy_c_Transcendental_Odiffs_001t__Code____Numeral__Ointeger,type,
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% 0.30/0.47
% 0.30/0.47 thf(sy_c_Transcendental_Odiffs_001t__Rat__Orat,type,
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% 0.30/0.47
% 0.30/0.47 thf(sy_c_Transcendental_Odiffs_001t__Real__Oreal,type,
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% 0.30/0.47
% 0.30/0.47 thf(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
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% 0.30/0.47
% 0.30/0.47 thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
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% 0.30/0.47
% 0.30/0.47 thf(sy_c_Transcendental_Olog,type,
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% 0.30/0.47
% 0.30/0.47 thf(sy_c_Transcendental_Opi,type,
% 0.30/0.47 pi: real ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
% 0.30/0.47 powr_real: real > real > real ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
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% 0.30/0.47
% 0.30/0.47 thf(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
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% 0.30/0.47
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% 0.30/0.47
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% 0.30/0.47 thf(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
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% 0.30/0.47 thf(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
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% 0.30/0.47
% 0.30/0.47 thf(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
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% 0.30/0.47
% 0.30/0.47 thf(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
% 0.30/0.47 tanh_real: real > real ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
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% 0.30/0.47
% 0.30/0.47 thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
% 0.30/0.47 vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
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% 0.30/0.47
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% 0.30/0.47
% 0.30/0.47 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
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% 0.30/0.47
% 0.30/0.47 thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
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% 0.30/0.47 thf(sy_c_VEBT__Definitions_Oset__vebt,type,
% 0.30/0.47 vEBT_set_vebt: vEBT_VEBT > set_nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
% 0.30/0.47 vEBT_vebt_buildup: nat > vEBT_VEBT ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
% 0.30/0.47 vEBT_v4011308405150292612up_rel: nat > nat > $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_VEBT__Delete_Ovebt__delete,type,
% 0.30/0.47 vEBT_vebt_delete: vEBT_VEBT > nat > vEBT_VEBT ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
% 0.30/0.47 vEBT_vebt_delete_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_VEBT__Insert_Ovebt__insert,type,
% 0.30/0.47 vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
% 0.30/0.47 vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
% 0.30/0.47 vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
% 0.30/0.47 vEBT_VEBT_minNull: vEBT_VEBT > $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
% 0.30/0.47 vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
% 0.30/0.47 vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_VEBT__Member_Ovebt__member,type,
% 0.30/0.47 vEBT_vebt_member: vEBT_VEBT > nat > $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
% 0.30/0.47 vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.30/0.47
% 0.30/0.47 thf(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
% 0.30/0.47 vEBT_VEBT_add: option_nat > option_nat > option_nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
% 0.30/0.48 vEBT_VEBT_greater: option_nat > option_nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
% 0.30/0.48 vEBT_VEBT_less: option_nat > option_nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
% 0.30/0.48 vEBT_VEBT_lesseq: option_nat > option_nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
% 0.30/0.48 vEBT_VEBT_max_in_set: set_nat > nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
% 0.30/0.48 vEBT_VEBT_min_in_set: set_nat > nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
% 0.30/0.48 vEBT_VEBT_mul: option_nat > option_nat > option_nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Nat__Onat,type,
% 0.30/0.48 vEBT_V4262088993061758097ft_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Num__Onum,type,
% 0.30/0.48 vEBT_V819420779217536731ft_num: ( num > num > num ) > option_num > option_num > option_num ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.30/0.48 vEBT_V1502963449132264192at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > option4927543243414619207at_nat > option4927543243414619207at_nat > option4927543243414619207at_nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
% 0.30/0.48 vEBT_VEBT_power: option_nat > option_nat > option_nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__MinMax_Ovebt__maxt,type,
% 0.30/0.48 vEBT_vebt_maxt: vEBT_VEBT > option_nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
% 0.30/0.48 vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__MinMax_Ovebt__mint,type,
% 0.30/0.48 vEBT_vebt_mint: vEBT_VEBT > option_nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
% 0.30/0.48 vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Pred_Ois__pred__in__set,type,
% 0.30/0.48 vEBT_is_pred_in_set: set_nat > nat > nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Pred_Ovebt__pred,type,
% 0.30/0.48 vEBT_vebt_pred: vEBT_VEBT > nat > option_nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
% 0.30/0.48 vEBT_vebt_pred_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Succ_Ois__succ__in__set,type,
% 0.30/0.48 vEBT_is_succ_in_set: set_nat > nat > nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Succ_Ovebt__succ,type,
% 0.30/0.48 vEBT_vebt_succ: vEBT_VEBT > nat > option_nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
% 0.30/0.48 vEBT_vebt_succ_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
% 0.30/0.48 accp_nat: ( nat > nat > $o ) > nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.30/0.48 accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.30/0.48 accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 0.30/0.48 accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
% 0.30/0.48 accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_fChoice_001t__Real__Oreal,type,
% 0.30/0.48 fChoice_real: ( real > $o ) > real ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_member_001_Eo,type,
% 0.30/0.48 member_o: $o > set_o > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_member_001t__Complex__Ocomplex,type,
% 0.30/0.48 member_complex: complex > set_complex > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_member_001t__Int__Oint,type,
% 0.30/0.48 member_int: int > set_int > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_member_001t__List__Olist_I_Eo_J,type,
% 0.30/0.48 member_list_o: list_o > set_list_o > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
% 0.30/0.48 member_list_int: list_int > set_list_int > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_member_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 0.30/0.48 member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_member_001t__Nat__Onat,type,
% 0.30/0.48 member_nat: nat > set_nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_member_001t__Num__Onum,type,
% 0.30/0.48 member_num: num > set_num > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.30/0.48 member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_member_001t__Rat__Orat,type,
% 0.30/0.48 member_rat: rat > set_rat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_member_001t__Real__Oreal,type,
% 0.30/0.48 member_real: real > set_real > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.30/0.48 member_set_nat: set_nat > set_set_nat > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
% 0.30/0.48 member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
% 0.30/0.48
% 0.30/0.48 thf(sy_v_deg____,type,
% 0.30/0.48 deg: nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_v_lx____,type,
% 0.30/0.48 lx: nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_v_m____,type,
% 0.30/0.48 m: nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_v_ma____,type,
% 0.30/0.48 ma: nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_v_maxi____,type,
% 0.30/0.48 maxi: nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_v_maxs____,type,
% 0.30/0.48 maxs: nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_v_mi____,type,
% 0.30/0.48 mi: nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_v_na____,type,
% 0.30/0.48 na: nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_v_summary____,type,
% 0.30/0.48 summary: vEBT_VEBT ).
% 0.30/0.48
% 0.30/0.48 thf(sy_v_summin____,type,
% 0.30/0.48 summin: nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_v_treeList____,type,
% 0.30/0.48 treeList: list_VEBT_VEBT ).
% 0.30/0.48
% 0.30/0.48 thf(sy_v_xa____,type,
% 0.30/0.48 xa: nat ).
% 0.30/0.48
% 0.30/0.48 thf(sy_v_ya____,type,
% 0.30/0.48 ya: nat ).
% 0.30/0.48
% 0.30/0.48 % Relevant facts (10202)
% 0.30/0.48 thf(fact_0__C4_Ohyps_C_I3_J,axiom,
% 0.30/0.48 m = na ).
% 0.30/0.48
% 0.30/0.48 % "4.hyps"(3)
% 0.30/0.48 thf(fact_1__092_060open_062both__member__options_A_ItreeList_A_B_Amaxs_J_Amaxi_092_060close_062,axiom,
% 0.30/0.48 vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ maxi ).
% 0.30/0.48
% 0.30/0.48 % \<open>both_member_options (treeList ! maxs) maxi\<close>
% 0.30/0.48 thf(fact_2_bit__split__inv,axiom,
% 0.30/0.48 ! [X: nat,D: nat] :
% 0.30/0.48 ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X @ D ) @ ( vEBT_VEBT_low @ X @ D ) @ D )
% 0.30/0.48 = X ) ).
% 0.30/0.48
% 0.30/0.48 % bit_split_inv
% 0.30/0.48 thf(fact_3__C4_OIH_C_I2_J,axiom,
% 0.30/0.48 ! [X: nat,Y: nat] :
% 0.30/0.48 ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ summary @ X ) @ Y )
% 0.30/0.48 = ( ( X != Y )
% 0.30/0.48 & ( vEBT_V8194947554948674370ptions @ summary @ Y ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % "4.IH"(2)
% 0.30/0.48 thf(fact_4_pow__sum,axiom,
% 0.30/0.48 ! [A: nat,B: nat] :
% 0.30/0.48 ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.48 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % pow_sum
% 0.30/0.48 thf(fact_5_high__def,axiom,
% 0.30/0.48 ( vEBT_VEBT_high
% 0.30/0.48 = ( ^ [X2: nat,N: nat] : ( divide_divide_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % high_def
% 0.30/0.48 thf(fact_6__092_060open_062deg_Adiv_A2_A_061_An_092_060close_062,axiom,
% 0.30/0.48 ( ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.48 = na ) ).
% 0.30/0.48
% 0.30/0.48 % \<open>deg div 2 = n\<close>
% 0.30/0.48 thf(fact_7__C4_Ohyps_C_I4_J,axiom,
% 0.30/0.48 ( deg
% 0.30/0.48 = ( plus_plus_nat @ na @ m ) ) ).
% 0.30/0.48
% 0.30/0.48 % "4.hyps"(4)
% 0.30/0.48 thf(fact_8__092_060open_062_092_060exists_062z_O_Aboth__member__options_A_ItreeList_A_B_Asummin_J_Az_092_060close_062,axiom,
% 0.30/0.48 ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ summin ) @ X_1 ) ).
% 0.30/0.48
% 0.30/0.48 % \<open>\<exists>z. both_member_options (treeList ! summin) z\<close>
% 0.30/0.48 thf(fact_9__092_060open_062_092_060exists_062z_O_Aboth__member__options_A_ItreeList_A_B_Amaxs_J_Az_092_060close_062,axiom,
% 0.30/0.48 ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ X_1 ) ).
% 0.30/0.48
% 0.30/0.48 % \<open>\<exists>z. both_member_options (treeList ! maxs) z\<close>
% 0.30/0.48 thf(fact_10_bit__concat__def,axiom,
% 0.30/0.48 ( vEBT_VEBT_bit_concat
% 0.30/0.48 = ( ^ [H: nat,L: nat,D2: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D2 ) ) @ L ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % bit_concat_def
% 0.30/0.48 thf(fact_11_hprolist,axiom,
% 0.30/0.48 ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) )
% 0.30/0.48 = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % hprolist
% 0.30/0.48 thf(fact_12__092_060open_062treeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_A_B_Athe_A_Ivebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_J_A_061_AtreeList_A_B_Amaxs_092_060close_062,axiom,
% 0.30/0.48 ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) )
% 0.30/0.48 = ( nth_VEBT_VEBT @ treeList @ maxs ) ) ).
% 0.30/0.48
% 0.30/0.48 % \<open>treeList [high (summin * 2 ^ n + lx) n := vebt_delete (treeList ! high (summin * 2 ^ n + lx) n) (low (summin * 2 ^ n + lx) n)] ! the (vebt_maxt (vebt_delete summary (high (summin * 2 ^ n + lx) n))) = treeList ! maxs\<close>
% 0.30/0.48 thf(fact_13__092_060open_062the_A_Ivebt__maxt_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_A_B_Athe_A_Ivebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_J_J_J_A_061_Athe_A_Ivebt__maxt_A_ItreeList_A_B_Amaxs_J_J_092_060close_062,axiom,
% 0.30/0.48 ( ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) )
% 0.30/0.48 = ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ maxs ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % \<open>the (vebt_maxt (treeList [high (summin * 2 ^ n + lx) n := vebt_delete (treeList ! high (summin * 2 ^ n + lx) n) (low (summin * 2 ^ n + lx) n)] ! the (vebt_maxt (vebt_delete summary (high (summin * 2 ^ n + lx) n))))) = the (vebt_maxt (treeList ! maxs))\<close>
% 0.30/0.48 thf(fact_14__092_060open_062both__member__options_Asummary_Amaxs_A_092_060and_062_Amaxs_A_092_060noteq_062_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_092_060close_062,axiom,
% 0.30/0.48 ( ( vEBT_V8194947554948674370ptions @ summary @ maxs )
% 0.30/0.48 & ( maxs
% 0.30/0.48 != ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % \<open>both_member_options summary maxs \<and> maxs \<noteq> high (summin * 2 ^ n + lx) n\<close>
% 0.30/0.48 thf(fact_15__092_060open_062both__member__options_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_Amaxs_092_060close_062,axiom,
% 0.30/0.48 vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ maxs ).
% 0.30/0.48
% 0.30/0.48 % \<open>both_member_options (vebt_delete summary (high (summin * 2 ^ n + lx) n)) maxs\<close>
% 0.30/0.48 thf(fact_16__092_060open_062Some_Amaxi_A_061_Avebt__maxt_A_ItreeList_A_B_Amaxs_J_092_060close_062,axiom,
% 0.30/0.48 ( ( some_nat @ maxi )
% 0.30/0.48 = ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ maxs ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % \<open>Some maxi = vebt_maxt (treeList ! maxs)\<close>
% 0.30/0.48 thf(fact_17_False,axiom,
% 0.30/0.48 ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 0.30/0.48 != none_nat ) ).
% 0.30/0.48
% 0.30/0.48 % False
% 0.30/0.48 thf(fact_18__092_060open_062Some_Amaxs_A_061_Avebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_092_060close_062,axiom,
% 0.30/0.48 ( ( some_nat @ maxs )
% 0.30/0.48 = ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % \<open>Some maxs = vebt_maxt (vebt_delete summary (high (summin * 2 ^ n + lx) n))\<close>
% 0.30/0.48 thf(fact_19_True,axiom,
% 0.30/0.48 ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 0.30/0.48 = ma ) ).
% 0.30/0.48
% 0.30/0.48 % True
% 0.30/0.48 thf(fact_20__092_060open_062length_AtreeList_A_061_Alength_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_J_092_060close_062,axiom,
% 0.30/0.48 ( ( size_s6755466524823107622T_VEBT @ treeList )
% 0.30/0.48 = ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % \<open>length treeList = length (treeList [high (summin * 2 ^ n + lx) n := vebt_delete (treeList ! high (summin * 2 ^ n + lx) n) (low (summin * 2 ^ n + lx) n)])\<close>
% 0.30/0.48 thf(fact_21_add__self__div__2,axiom,
% 0.30/0.48 ! [M: nat] :
% 0.30/0.48 ( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.48 = M ) ).
% 0.30/0.48
% 0.30/0.48 % add_self_div_2
% 0.30/0.48 thf(fact_22__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062maxs_O_ASome_Amaxs_A_061_Avebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.30/0.48 ~ ! [Maxs: nat] :
% 0.30/0.48 ( ( some_nat @ Maxs )
% 0.30/0.48 != ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % \<open>\<And>thesis. (\<And>maxs. Some maxs = vebt_maxt (vebt_delete summary (high (summin * 2 ^ n + lx) n)) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.30/0.48 thf(fact_23__092_060open_062summin_A_K_A2_A_094_An_A_L_Alx_A_060_A2_A_094_Adeg_092_060close_062,axiom,
% 0.30/0.48 ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% 0.30/0.48
% 0.30/0.48 % \<open>summin * 2 ^ n + lx < 2 ^ deg\<close>
% 0.30/0.48 thf(fact_24_minxnrel,axiom,
% 0.30/0.48 ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 0.30/0.48 != mi ) ).
% 0.30/0.48
% 0.30/0.48 % minxnrel
% 0.30/0.48 thf(fact_25_distrib__left__numeral,axiom,
% 0.30/0.48 ! [V: num,B: complex,C: complex] :
% 0.30/0.48 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B @ C ) )
% 0.30/0.48 = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % distrib_left_numeral
% 0.30/0.48 thf(fact_26_distrib__left__numeral,axiom,
% 0.30/0.48 ! [V: num,B: real,C: real] :
% 0.30/0.48 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
% 0.30/0.48 = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % distrib_left_numeral
% 0.30/0.48 thf(fact_27_distrib__left__numeral,axiom,
% 0.30/0.48 ! [V: num,B: rat,C: rat] :
% 0.30/0.48 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C ) )
% 0.30/0.48 = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % distrib_left_numeral
% 0.30/0.48 thf(fact_28_distrib__left__numeral,axiom,
% 0.30/0.48 ! [V: num,B: nat,C: nat] :
% 0.30/0.48 ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
% 0.30/0.48 = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % distrib_left_numeral
% 0.30/0.48 thf(fact_29_distrib__left__numeral,axiom,
% 0.30/0.48 ! [V: num,B: int,C: int] :
% 0.30/0.48 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
% 0.30/0.48 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % distrib_left_numeral
% 0.30/0.48 thf(fact_30_distrib__right__numeral,axiom,
% 0.30/0.48 ! [A: complex,B: complex,V: num] :
% 0.30/0.48 ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
% 0.30/0.48 = ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % distrib_right_numeral
% 0.30/0.48 thf(fact_31_distrib__right__numeral,axiom,
% 0.30/0.48 ! [A: real,B: real,V: num] :
% 0.30/0.48 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
% 0.30/0.48 = ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % distrib_right_numeral
% 0.30/0.48 thf(fact_32_distrib__right__numeral,axiom,
% 0.30/0.48 ! [A: rat,B: rat,V: num] :
% 0.30/0.48 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
% 0.30/0.48 = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % distrib_right_numeral
% 0.30/0.48 thf(fact_33_distrib__right__numeral,axiom,
% 0.30/0.48 ! [A: nat,B: nat,V: num] :
% 0.30/0.48 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
% 0.30/0.48 = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % distrib_right_numeral
% 0.30/0.48 thf(fact_34_distrib__right__numeral,axiom,
% 0.30/0.48 ! [A: int,B: int,V: num] :
% 0.30/0.48 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.48 = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % distrib_right_numeral
% 0.30/0.48 thf(fact_35_power2__sum,axiom,
% 0.30/0.48 ! [X: complex,Y: complex] :
% 0.30/0.48 ( ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.48 = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power2_sum
% 0.30/0.48 thf(fact_36_power2__sum,axiom,
% 0.30/0.48 ! [X: real,Y: real] :
% 0.30/0.48 ( ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.48 = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power2_sum
% 0.30/0.48 thf(fact_37_power2__sum,axiom,
% 0.30/0.48 ! [X: rat,Y: rat] :
% 0.30/0.48 ( ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.48 = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power2_sum
% 0.30/0.48 thf(fact_38_power2__sum,axiom,
% 0.30/0.48 ! [X: nat,Y: nat] :
% 0.30/0.48 ( ( power_power_nat @ ( plus_plus_nat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.48 = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power2_sum
% 0.30/0.48 thf(fact_39_power2__sum,axiom,
% 0.30/0.48 ! [X: int,Y: int] :
% 0.30/0.48 ( ( power_power_int @ ( plus_plus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.48 = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power2_sum
% 0.30/0.48 thf(fact_40__092_060open_062maxi_A_060_A2_A_094_An_092_060close_062,axiom,
% 0.30/0.48 ord_less_nat @ maxi @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ).
% 0.30/0.48
% 0.30/0.48 % \<open>maxi < 2 ^ n\<close>
% 0.30/0.48 thf(fact_41__092_060open_062vebt__member_A_ItreeList_A_B_Asummin_J_Alx_092_060close_062,axiom,
% 0.30/0.48 vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ summin ) @ lx ).
% 0.30/0.48
% 0.30/0.48 % \<open>vebt_member (treeList ! summin) lx\<close>
% 0.30/0.48 thf(fact_42_div__exp__eq,axiom,
% 0.30/0.48 ! [A: nat,M: nat,N2: nat] :
% 0.30/0.48 ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.48 = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % div_exp_eq
% 0.30/0.48 thf(fact_43_div__exp__eq,axiom,
% 0.30/0.48 ! [A: int,M: nat,N2: nat] :
% 0.30/0.48 ( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.48 = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % div_exp_eq
% 0.30/0.48 thf(fact_44_maxbmo,axiom,
% 0.30/0.48 ! [T: vEBT_VEBT,X: nat] :
% 0.30/0.48 ( ( ( vEBT_vebt_maxt @ T )
% 0.30/0.48 = ( some_nat @ X ) )
% 0.30/0.48 => ( vEBT_V8194947554948674370ptions @ T @ X ) ) ).
% 0.30/0.48
% 0.30/0.48 % maxbmo
% 0.30/0.48 thf(fact_45_power__shift,axiom,
% 0.30/0.48 ! [X: nat,Y: nat,Z: nat] :
% 0.30/0.48 ( ( ( power_power_nat @ X @ Y )
% 0.30/0.48 = Z )
% 0.30/0.48 = ( ( vEBT_VEBT_power @ ( some_nat @ X ) @ ( some_nat @ Y ) )
% 0.30/0.48 = ( some_nat @ Z ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_shift
% 0.30/0.48 thf(fact_46__C4_Ohyps_C_I8_J,axiom,
% 0.30/0.48 ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% 0.30/0.48
% 0.30/0.48 % "4.hyps"(8)
% 0.30/0.48 thf(fact_47__C4_Ohyps_C_I2_J,axiom,
% 0.30/0.48 ( ( size_s6755466524823107622T_VEBT @ treeList )
% 0.30/0.48 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).
% 0.30/0.48
% 0.30/0.48 % "4.hyps"(2)
% 0.30/0.48 thf(fact_48_numeral__eq__iff,axiom,
% 0.30/0.48 ! [M: num,N2: num] :
% 0.30/0.48 ( ( ( numera6690914467698888265omplex @ M )
% 0.30/0.48 = ( numera6690914467698888265omplex @ N2 ) )
% 0.30/0.48 = ( M = N2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_eq_iff
% 0.30/0.48 thf(fact_49_numeral__eq__iff,axiom,
% 0.30/0.48 ! [M: num,N2: num] :
% 0.30/0.48 ( ( ( numeral_numeral_real @ M )
% 0.30/0.48 = ( numeral_numeral_real @ N2 ) )
% 0.30/0.48 = ( M = N2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_eq_iff
% 0.30/0.48 thf(fact_50_numeral__eq__iff,axiom,
% 0.30/0.48 ! [M: num,N2: num] :
% 0.30/0.48 ( ( ( numeral_numeral_rat @ M )
% 0.30/0.48 = ( numeral_numeral_rat @ N2 ) )
% 0.30/0.48 = ( M = N2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_eq_iff
% 0.30/0.48 thf(fact_51_numeral__eq__iff,axiom,
% 0.30/0.48 ! [M: num,N2: num] :
% 0.30/0.48 ( ( ( numeral_numeral_nat @ M )
% 0.30/0.48 = ( numeral_numeral_nat @ N2 ) )
% 0.30/0.48 = ( M = N2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_eq_iff
% 0.30/0.48 thf(fact_52_numeral__eq__iff,axiom,
% 0.30/0.48 ! [M: num,N2: num] :
% 0.30/0.48 ( ( ( numeral_numeral_int @ M )
% 0.30/0.48 = ( numeral_numeral_int @ N2 ) )
% 0.30/0.48 = ( M = N2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_eq_iff
% 0.30/0.48 thf(fact_53_mimapr,axiom,
% 0.30/0.48 ord_less_nat @ mi @ ma ).
% 0.30/0.48
% 0.30/0.48 % mimapr
% 0.30/0.48 thf(fact_54__C4_Ohyps_C_I5_J,axiom,
% 0.30/0.48 ! [I: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 0.30/0.48 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I ) @ X3 ) )
% 0.30/0.48 = ( vEBT_V8194947554948674370ptions @ summary @ I ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % "4.hyps"(5)
% 0.30/0.48 thf(fact_55_high__bound__aux,axiom,
% 0.30/0.48 ! [Ma: nat,N2: nat,M: nat] :
% 0.30/0.48 ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
% 0.30/0.48 => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % high_bound_aux
% 0.30/0.48 thf(fact_56__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062maxi_O_ASome_Amaxi_A_061_Avebt__maxt_A_ItreeList_A_B_Amaxs_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.30/0.48 ~ ! [Maxi: nat] :
% 0.30/0.48 ( ( some_nat @ Maxi )
% 0.30/0.48 != ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ maxs ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % \<open>\<And>thesis. (\<And>maxi. Some maxi = vebt_maxt (treeList ! maxs) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.30/0.48 thf(fact_57__092_060open_062both__member__options_Asummary_A_Ihigh_Ama_An_J_092_060close_062,axiom,
% 0.30/0.48 vEBT_V8194947554948674370ptions @ summary @ ( vEBT_VEBT_high @ ma @ na ) ).
% 0.30/0.48
% 0.30/0.48 % \<open>both_member_options summary (high ma n)\<close>
% 0.30/0.48 thf(fact_58__092_060open_062vebt__member_Asummary_A_Ihigh_Ama_An_J_092_060close_062,axiom,
% 0.30/0.48 vEBT_vebt_member @ summary @ ( vEBT_VEBT_high @ ma @ na ) ).
% 0.30/0.48
% 0.30/0.48 % \<open>vebt_member summary (high ma n)\<close>
% 0.30/0.48 thf(fact_59_high__inv,axiom,
% 0.30/0.48 ! [X: nat,N2: nat,Y: nat] :
% 0.30/0.48 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.48 => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ X ) @ N2 )
% 0.30/0.48 = Y ) ) ).
% 0.30/0.48
% 0.30/0.48 % high_inv
% 0.30/0.48 thf(fact_60_numeral__less__iff,axiom,
% 0.30/0.48 ! [M: num,N2: num] :
% 0.30/0.48 ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
% 0.30/0.48 = ( ord_less_num @ M @ N2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_less_iff
% 0.30/0.48 thf(fact_61_numeral__less__iff,axiom,
% 0.30/0.48 ! [M: num,N2: num] :
% 0.30/0.48 ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
% 0.30/0.48 = ( ord_less_num @ M @ N2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_less_iff
% 0.30/0.48 thf(fact_62_numeral__less__iff,axiom,
% 0.30/0.48 ! [M: num,N2: num] :
% 0.30/0.48 ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
% 0.30/0.48 = ( ord_less_num @ M @ N2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_less_iff
% 0.30/0.48 thf(fact_63_numeral__less__iff,axiom,
% 0.30/0.48 ! [M: num,N2: num] :
% 0.30/0.48 ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 0.30/0.48 = ( ord_less_num @ M @ N2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_less_iff
% 0.30/0.48 thf(fact_64_mem__Collect__eq,axiom,
% 0.30/0.48 ! [A: real,P: real > $o] :
% 0.30/0.48 ( ( member_real @ A @ ( collect_real @ P ) )
% 0.30/0.48 = ( P @ A ) ) ).
% 0.30/0.48
% 0.30/0.48 % mem_Collect_eq
% 0.30/0.48 thf(fact_65_mem__Collect__eq,axiom,
% 0.30/0.48 ! [A: product_prod_int_int,P: product_prod_int_int > $o] :
% 0.30/0.48 ( ( member5262025264175285858nt_int @ A @ ( collec213857154873943460nt_int @ P ) )
% 0.30/0.48 = ( P @ A ) ) ).
% 0.30/0.48
% 0.30/0.48 % mem_Collect_eq
% 0.30/0.48 thf(fact_66_mem__Collect__eq,axiom,
% 0.30/0.48 ! [A: complex,P: complex > $o] :
% 0.30/0.48 ( ( member_complex @ A @ ( collect_complex @ P ) )
% 0.30/0.48 = ( P @ A ) ) ).
% 0.30/0.48
% 0.30/0.48 % mem_Collect_eq
% 0.30/0.48 thf(fact_67_mem__Collect__eq,axiom,
% 0.30/0.48 ! [A: set_nat,P: set_nat > $o] :
% 0.30/0.48 ( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
% 0.30/0.48 = ( P @ A ) ) ).
% 0.30/0.48
% 0.30/0.48 % mem_Collect_eq
% 0.30/0.48 thf(fact_68_mem__Collect__eq,axiom,
% 0.30/0.48 ! [A: nat,P: nat > $o] :
% 0.30/0.48 ( ( member_nat @ A @ ( collect_nat @ P ) )
% 0.30/0.48 = ( P @ A ) ) ).
% 0.30/0.48
% 0.30/0.48 % mem_Collect_eq
% 0.30/0.48 thf(fact_69_mem__Collect__eq,axiom,
% 0.30/0.48 ! [A: int,P: int > $o] :
% 0.30/0.48 ( ( member_int @ A @ ( collect_int @ P ) )
% 0.30/0.48 = ( P @ A ) ) ).
% 0.30/0.48
% 0.30/0.48 % mem_Collect_eq
% 0.30/0.48 thf(fact_70_Collect__mem__eq,axiom,
% 0.30/0.48 ! [A2: set_real] :
% 0.30/0.48 ( ( collect_real
% 0.30/0.48 @ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
% 0.30/0.48 = A2 ) ).
% 0.30/0.48
% 0.30/0.48 % Collect_mem_eq
% 0.30/0.48 thf(fact_71_Collect__mem__eq,axiom,
% 0.30/0.48 ! [A2: set_Pr958786334691620121nt_int] :
% 0.30/0.48 ( ( collec213857154873943460nt_int
% 0.30/0.48 @ ^ [X2: product_prod_int_int] : ( member5262025264175285858nt_int @ X2 @ A2 ) )
% 0.30/0.48 = A2 ) ).
% 0.30/0.48
% 0.30/0.48 % Collect_mem_eq
% 0.30/0.48 thf(fact_72_Collect__mem__eq,axiom,
% 0.30/0.48 ! [A2: set_complex] :
% 0.30/0.48 ( ( collect_complex
% 0.30/0.48 @ ^ [X2: complex] : ( member_complex @ X2 @ A2 ) )
% 0.30/0.48 = A2 ) ).
% 0.30/0.48
% 0.30/0.48 % Collect_mem_eq
% 0.30/0.48 thf(fact_73_Collect__mem__eq,axiom,
% 0.30/0.48 ! [A2: set_set_nat] :
% 0.30/0.48 ( ( collect_set_nat
% 0.30/0.48 @ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A2 ) )
% 0.30/0.48 = A2 ) ).
% 0.30/0.48
% 0.30/0.48 % Collect_mem_eq
% 0.30/0.48 thf(fact_74_Collect__mem__eq,axiom,
% 0.30/0.48 ! [A2: set_nat] :
% 0.30/0.48 ( ( collect_nat
% 0.30/0.48 @ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
% 0.30/0.48 = A2 ) ).
% 0.30/0.48
% 0.30/0.48 % Collect_mem_eq
% 0.30/0.48 thf(fact_75_Collect__mem__eq,axiom,
% 0.30/0.48 ! [A2: set_int] :
% 0.30/0.48 ( ( collect_int
% 0.30/0.48 @ ^ [X2: int] : ( member_int @ X2 @ A2 ) )
% 0.30/0.48 = A2 ) ).
% 0.30/0.48
% 0.30/0.48 % Collect_mem_eq
% 0.30/0.48 thf(fact_76_Collect__cong,axiom,
% 0.30/0.48 ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
% 0.30/0.48 ( ! [X4: product_prod_int_int] :
% 0.30/0.48 ( ( P @ X4 )
% 0.30/0.48 = ( Q @ X4 ) )
% 0.30/0.48 => ( ( collec213857154873943460nt_int @ P )
% 0.30/0.48 = ( collec213857154873943460nt_int @ Q ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Collect_cong
% 0.30/0.48 thf(fact_77_Collect__cong,axiom,
% 0.30/0.48 ! [P: complex > $o,Q: complex > $o] :
% 0.30/0.48 ( ! [X4: complex] :
% 0.30/0.48 ( ( P @ X4 )
% 0.30/0.48 = ( Q @ X4 ) )
% 0.30/0.48 => ( ( collect_complex @ P )
% 0.30/0.48 = ( collect_complex @ Q ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Collect_cong
% 0.30/0.48 thf(fact_78_Collect__cong,axiom,
% 0.30/0.48 ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.30/0.48 ( ! [X4: set_nat] :
% 0.30/0.48 ( ( P @ X4 )
% 0.30/0.48 = ( Q @ X4 ) )
% 0.30/0.48 => ( ( collect_set_nat @ P )
% 0.30/0.48 = ( collect_set_nat @ Q ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Collect_cong
% 0.30/0.48 thf(fact_79_Collect__cong,axiom,
% 0.30/0.48 ! [P: nat > $o,Q: nat > $o] :
% 0.30/0.48 ( ! [X4: nat] :
% 0.30/0.48 ( ( P @ X4 )
% 0.30/0.48 = ( Q @ X4 ) )
% 0.30/0.48 => ( ( collect_nat @ P )
% 0.30/0.48 = ( collect_nat @ Q ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Collect_cong
% 0.30/0.48 thf(fact_80_Collect__cong,axiom,
% 0.30/0.48 ! [P: int > $o,Q: int > $o] :
% 0.30/0.48 ( ! [X4: int] :
% 0.30/0.48 ( ( P @ X4 )
% 0.30/0.48 = ( Q @ X4 ) )
% 0.30/0.48 => ( ( collect_int @ P )
% 0.30/0.48 = ( collect_int @ Q ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % Collect_cong
% 0.30/0.48 thf(fact_81_low__inv,axiom,
% 0.30/0.48 ! [X: nat,N2: nat,Y: nat] :
% 0.30/0.48 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.48 => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ X ) @ N2 )
% 0.30/0.48 = X ) ) ).
% 0.30/0.48
% 0.30/0.48 % low_inv
% 0.30/0.48 thf(fact_82_mult__numeral__left__semiring__numeral,axiom,
% 0.30/0.48 ! [V: num,W: num,Z: complex] :
% 0.30/0.48 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
% 0.30/0.48 = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_numeral_left_semiring_numeral
% 0.30/0.48 thf(fact_83_mult__numeral__left__semiring__numeral,axiom,
% 0.30/0.48 ! [V: num,W: num,Z: real] :
% 0.30/0.48 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
% 0.30/0.48 = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_numeral_left_semiring_numeral
% 0.30/0.48 thf(fact_84_mult__numeral__left__semiring__numeral,axiom,
% 0.30/0.48 ! [V: num,W: num,Z: rat] :
% 0.30/0.48 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
% 0.30/0.48 = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_numeral_left_semiring_numeral
% 0.30/0.48 thf(fact_85_mult__numeral__left__semiring__numeral,axiom,
% 0.30/0.48 ! [V: num,W: num,Z: nat] :
% 0.30/0.48 ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
% 0.30/0.48 = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_numeral_left_semiring_numeral
% 0.30/0.48 thf(fact_86_mult__numeral__left__semiring__numeral,axiom,
% 0.30/0.48 ! [V: num,W: num,Z: int] :
% 0.30/0.48 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
% 0.30/0.48 = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_numeral_left_semiring_numeral
% 0.30/0.48 thf(fact_87_numeral__times__numeral,axiom,
% 0.30/0.48 ! [M: num,N2: num] :
% 0.30/0.48 ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N2 ) )
% 0.30/0.48 = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_times_numeral
% 0.30/0.48 thf(fact_88_numeral__times__numeral,axiom,
% 0.30/0.48 ! [M: num,N2: num] :
% 0.30/0.48 ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
% 0.30/0.48 = ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_times_numeral
% 0.30/0.48 thf(fact_89_numeral__times__numeral,axiom,
% 0.30/0.48 ! [M: num,N2: num] :
% 0.30/0.48 ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
% 0.30/0.48 = ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_times_numeral
% 0.30/0.48 thf(fact_90_numeral__times__numeral,axiom,
% 0.30/0.48 ! [M: num,N2: num] :
% 0.30/0.48 ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
% 0.30/0.48 = ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_times_numeral
% 0.30/0.48 thf(fact_91_numeral__times__numeral,axiom,
% 0.30/0.48 ! [M: num,N2: num] :
% 0.30/0.48 ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 0.30/0.48 = ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_times_numeral
% 0.30/0.48 thf(fact_92_add__numeral__left,axiom,
% 0.30/0.48 ! [V: num,W: num,Z: complex] :
% 0.30/0.48 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
% 0.30/0.48 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_numeral_left
% 0.30/0.48 thf(fact_93_add__numeral__left,axiom,
% 0.30/0.48 ! [V: num,W: num,Z: real] :
% 0.30/0.48 ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z ) )
% 0.30/0.48 = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_numeral_left
% 0.30/0.48 thf(fact_94_add__numeral__left,axiom,
% 0.30/0.48 ! [V: num,W: num,Z: rat] :
% 0.30/0.48 ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
% 0.30/0.48 = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_numeral_left
% 0.30/0.48 thf(fact_95_add__numeral__left,axiom,
% 0.30/0.48 ! [V: num,W: num,Z: nat] :
% 0.30/0.48 ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
% 0.30/0.48 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_numeral_left
% 0.30/0.48 thf(fact_96_add__numeral__left,axiom,
% 0.30/0.48 ! [V: num,W: num,Z: int] :
% 0.30/0.48 ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
% 0.30/0.48 = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_numeral_left
% 0.30/0.48 thf(fact_97_numeral__plus__numeral,axiom,
% 0.30/0.48 ! [M: num,N2: num] :
% 0.30/0.48 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N2 ) )
% 0.30/0.48 = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_plus_numeral
% 0.30/0.48 thf(fact_98_numeral__plus__numeral,axiom,
% 0.30/0.48 ! [M: num,N2: num] :
% 0.30/0.48 ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
% 0.30/0.48 = ( numeral_numeral_real @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_plus_numeral
% 0.30/0.48 thf(fact_99_numeral__plus__numeral,axiom,
% 0.30/0.48 ! [M: num,N2: num] :
% 0.30/0.48 ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
% 0.30/0.48 = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_plus_numeral
% 0.30/0.48 thf(fact_100_numeral__plus__numeral,axiom,
% 0.30/0.48 ! [M: num,N2: num] :
% 0.30/0.48 ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
% 0.30/0.48 = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_plus_numeral
% 0.30/0.48 thf(fact_101_numeral__plus__numeral,axiom,
% 0.30/0.48 ! [M: num,N2: num] :
% 0.30/0.48 ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 0.30/0.48 = ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_plus_numeral
% 0.30/0.48 thf(fact_102_num__double,axiom,
% 0.30/0.48 ! [N2: num] :
% 0.30/0.48 ( ( times_times_num @ ( bit0 @ one ) @ N2 )
% 0.30/0.48 = ( bit0 @ N2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % num_double
% 0.30/0.48 thf(fact_103_power__mult__numeral,axiom,
% 0.30/0.48 ! [A: nat,M: num,N2: num] :
% 0.30/0.48 ( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
% 0.30/0.48 = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_mult_numeral
% 0.30/0.48 thf(fact_104_power__mult__numeral,axiom,
% 0.30/0.48 ! [A: real,M: num,N2: num] :
% 0.30/0.48 ( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
% 0.30/0.48 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_mult_numeral
% 0.30/0.48 thf(fact_105_power__mult__numeral,axiom,
% 0.30/0.48 ! [A: int,M: num,N2: num] :
% 0.30/0.48 ( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
% 0.30/0.48 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_mult_numeral
% 0.30/0.48 thf(fact_106_power__mult__numeral,axiom,
% 0.30/0.48 ! [A: complex,M: num,N2: num] :
% 0.30/0.48 ( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
% 0.30/0.48 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_mult_numeral
% 0.30/0.48 thf(fact_107__092_060open_062summin_A_060_A2_A_094_Am_092_060close_062,axiom,
% 0.30/0.48 ord_less_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ).
% 0.30/0.48
% 0.30/0.48 % \<open>summin < 2 ^ m\<close>
% 0.30/0.48 thf(fact_108__092_060open_062maxs_A_060_A2_A_094_Am_092_060close_062,axiom,
% 0.30/0.48 ord_less_nat @ maxs @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ).
% 0.30/0.48
% 0.30/0.48 % \<open>maxs < 2 ^ m\<close>
% 0.30/0.48 thf(fact_109_power__add__numeral2,axiom,
% 0.30/0.48 ! [A: complex,M: num,N2: num,B: complex] :
% 0.30/0.48 ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
% 0.30/0.48 = ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_add_numeral2
% 0.30/0.48 thf(fact_110_power__add__numeral2,axiom,
% 0.30/0.48 ! [A: real,M: num,N2: num,B: real] :
% 0.30/0.48 ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
% 0.30/0.48 = ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_add_numeral2
% 0.30/0.48 thf(fact_111_power__add__numeral2,axiom,
% 0.30/0.48 ! [A: rat,M: num,N2: num,B: rat] :
% 0.30/0.48 ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
% 0.30/0.48 = ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_add_numeral2
% 0.30/0.48 thf(fact_112_power__add__numeral2,axiom,
% 0.30/0.48 ! [A: nat,M: num,N2: num,B: nat] :
% 0.30/0.48 ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
% 0.30/0.48 = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_add_numeral2
% 0.30/0.48 thf(fact_113_power__add__numeral2,axiom,
% 0.30/0.48 ! [A: int,M: num,N2: num,B: int] :
% 0.30/0.48 ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
% 0.30/0.48 = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_add_numeral2
% 0.30/0.48 thf(fact_114_power__add__numeral,axiom,
% 0.30/0.48 ! [A: complex,M: num,N2: num] :
% 0.30/0.48 ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N2 ) ) )
% 0.30/0.48 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_add_numeral
% 0.30/0.48 thf(fact_115_power__add__numeral,axiom,
% 0.30/0.48 ! [A: real,M: num,N2: num] :
% 0.30/0.48 ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N2 ) ) )
% 0.30/0.48 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_add_numeral
% 0.30/0.48 thf(fact_116_power__add__numeral,axiom,
% 0.30/0.48 ! [A: rat,M: num,N2: num] :
% 0.30/0.48 ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N2 ) ) )
% 0.30/0.48 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_add_numeral
% 0.30/0.48 thf(fact_117_power__add__numeral,axiom,
% 0.30/0.48 ! [A: nat,M: num,N2: num] :
% 0.30/0.48 ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N2 ) ) )
% 0.30/0.48 = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_add_numeral
% 0.30/0.48 thf(fact_118_power__add__numeral,axiom,
% 0.30/0.48 ! [A: int,M: num,N2: num] :
% 0.30/0.48 ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N2 ) ) )
% 0.30/0.48 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_add_numeral
% 0.30/0.48 thf(fact_119__C4_Ohyps_C_I7_J,axiom,
% 0.30/0.48 ord_less_eq_nat @ mi @ ma ).
% 0.30/0.48
% 0.30/0.48 % "4.hyps"(7)
% 0.30/0.48 thf(fact_120_less__divide__eq__numeral1_I1_J,axiom,
% 0.30/0.48 ! [A: real,B: real,W: num] :
% 0.30/0.48 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.48 = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_divide_eq_numeral1(1)
% 0.30/0.48 thf(fact_121_less__divide__eq__numeral1_I1_J,axiom,
% 0.30/0.48 ! [A: rat,B: rat,W: num] :
% 0.30/0.48 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 0.30/0.48 = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_divide_eq_numeral1(1)
% 0.30/0.48 thf(fact_122_divide__less__eq__numeral1_I1_J,axiom,
% 0.30/0.48 ! [B: real,W: num,A: real] :
% 0.30/0.48 ( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
% 0.30/0.48 = ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % divide_less_eq_numeral1(1)
% 0.30/0.48 thf(fact_123_divide__less__eq__numeral1_I1_J,axiom,
% 0.30/0.48 ! [B: rat,W: num,A: rat] :
% 0.30/0.48 ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
% 0.30/0.48 = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % divide_less_eq_numeral1(1)
% 0.30/0.48 thf(fact_124__092_060open_062high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_060_Alength_AtreeList_092_060close_062,axiom,
% 0.30/0.48 ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ).
% 0.30/0.48
% 0.30/0.48 % \<open>high (summin * 2 ^ n + lx) n < length treeList\<close>
% 0.30/0.48 thf(fact_125__092_060open_062Some_Asummin_A_061_Avebt__mint_Asummary_092_060close_062,axiom,
% 0.30/0.48 ( ( some_nat @ summin )
% 0.30/0.48 = ( vEBT_vebt_mint @ summary ) ) ).
% 0.30/0.48
% 0.30/0.48 % \<open>Some summin = vebt_mint summary\<close>
% 0.30/0.48 thf(fact_126_nothprolist,axiom,
% 0.30/0.48 ! [I2: nat] :
% 0.30/0.48 ( ( ( I2
% 0.30/0.48 != ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) )
% 0.30/0.48 & ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) )
% 0.30/0.48 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ I2 )
% 0.30/0.48 = ( nth_VEBT_VEBT @ treeList @ I2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nothprolist
% 0.30/0.48 thf(fact_127__C4_Ohyps_C_I6_J,axiom,
% 0.30/0.48 ( ( mi = ma )
% 0.30/0.48 => ! [X5: vEBT_VEBT] :
% 0.30/0.48 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ treeList ) )
% 0.30/0.48 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % "4.hyps"(6)
% 0.30/0.48 thf(fact_128__092_060open_062Some_Alx_A_061_Avebt__mint_A_ItreeList_A_B_Asummin_J_092_060close_062,axiom,
% 0.30/0.48 ( ( some_nat @ lx )
% 0.30/0.48 = ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ summin ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % \<open>Some lx = vebt_mint (treeList ! summin)\<close>
% 0.30/0.48 thf(fact_129__C4_Ohyps_C_I1_J,axiom,
% 0.30/0.48 vEBT_invar_vebt @ summary @ m ).
% 0.30/0.48
% 0.30/0.48 % "4.hyps"(1)
% 0.30/0.48 thf(fact_130__C4_Ohyps_C_I9_J,axiom,
% 0.30/0.48 ( ( mi != ma )
% 0.30/0.48 => ! [I: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 0.30/0.48 => ( ( ( ( vEBT_VEBT_high @ ma @ na )
% 0.30/0.48 = I )
% 0.30/0.48 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
% 0.30/0.48 & ! [X5: nat] :
% 0.30/0.48 ( ( ( ( vEBT_VEBT_high @ X5 @ na )
% 0.30/0.48 = I )
% 0.30/0.48 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I ) @ ( vEBT_VEBT_low @ X5 @ na ) ) )
% 0.30/0.48 => ( ( ord_less_nat @ mi @ X5 )
% 0.30/0.48 & ( ord_less_eq_nat @ X5 @ ma ) ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % "4.hyps"(9)
% 0.30/0.48 thf(fact_131_add__One__commute,axiom,
% 0.30/0.48 ! [N2: num] :
% 0.30/0.48 ( ( plus_plus_num @ one @ N2 )
% 0.30/0.48 = ( plus_plus_num @ N2 @ one ) ) ).
% 0.30/0.48
% 0.30/0.48 % add_One_commute
% 0.30/0.48 thf(fact_132_div__mult2__numeral__eq,axiom,
% 0.30/0.48 ! [A: nat,K: num,L2: num] :
% 0.30/0.48 ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L2 ) )
% 0.30/0.48 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % div_mult2_numeral_eq
% 0.30/0.48 thf(fact_133_div__mult2__numeral__eq,axiom,
% 0.30/0.48 ! [A: int,K: num,L2: num] :
% 0.30/0.48 ( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L2 ) )
% 0.30/0.48 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % div_mult2_numeral_eq
% 0.30/0.48 thf(fact_134_less__mult__imp__div__less,axiom,
% 0.30/0.48 ! [M: nat,I2: nat,N2: nat] :
% 0.30/0.48 ( ( ord_less_nat @ M @ ( times_times_nat @ I2 @ N2 ) )
% 0.30/0.48 => ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ I2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_mult_imp_div_less
% 0.30/0.48 thf(fact_135_less__exp,axiom,
% 0.30/0.48 ! [N2: nat] : ( ord_less_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_exp
% 0.30/0.48 thf(fact_136_is__num__normalize_I1_J,axiom,
% 0.30/0.48 ! [A: real,B: real,C: real] :
% 0.30/0.48 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.30/0.48 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % is_num_normalize(1)
% 0.30/0.48 thf(fact_137_is__num__normalize_I1_J,axiom,
% 0.30/0.48 ! [A: rat,B: rat,C: rat] :
% 0.30/0.48 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.30/0.48 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % is_num_normalize(1)
% 0.30/0.48 thf(fact_138_is__num__normalize_I1_J,axiom,
% 0.30/0.48 ! [A: int,B: int,C: int] :
% 0.30/0.48 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.30/0.48 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % is_num_normalize(1)
% 0.30/0.48 thf(fact_139_power__commuting__commutes,axiom,
% 0.30/0.48 ! [X: complex,Y: complex,N2: nat] :
% 0.30/0.48 ( ( ( times_times_complex @ X @ Y )
% 0.30/0.48 = ( times_times_complex @ Y @ X ) )
% 0.30/0.48 => ( ( times_times_complex @ ( power_power_complex @ X @ N2 ) @ Y )
% 0.30/0.48 = ( times_times_complex @ Y @ ( power_power_complex @ X @ N2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_commuting_commutes
% 0.30/0.48 thf(fact_140_power__commuting__commutes,axiom,
% 0.30/0.48 ! [X: real,Y: real,N2: nat] :
% 0.30/0.48 ( ( ( times_times_real @ X @ Y )
% 0.30/0.48 = ( times_times_real @ Y @ X ) )
% 0.30/0.48 => ( ( times_times_real @ ( power_power_real @ X @ N2 ) @ Y )
% 0.30/0.48 = ( times_times_real @ Y @ ( power_power_real @ X @ N2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_commuting_commutes
% 0.30/0.48 thf(fact_141_power__commuting__commutes,axiom,
% 0.30/0.48 ! [X: rat,Y: rat,N2: nat] :
% 0.30/0.48 ( ( ( times_times_rat @ X @ Y )
% 0.30/0.48 = ( times_times_rat @ Y @ X ) )
% 0.30/0.48 => ( ( times_times_rat @ ( power_power_rat @ X @ N2 ) @ Y )
% 0.30/0.48 = ( times_times_rat @ Y @ ( power_power_rat @ X @ N2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_commuting_commutes
% 0.30/0.48 thf(fact_142_power__commuting__commutes,axiom,
% 0.30/0.48 ! [X: nat,Y: nat,N2: nat] :
% 0.30/0.48 ( ( ( times_times_nat @ X @ Y )
% 0.30/0.48 = ( times_times_nat @ Y @ X ) )
% 0.30/0.48 => ( ( times_times_nat @ ( power_power_nat @ X @ N2 ) @ Y )
% 0.30/0.48 = ( times_times_nat @ Y @ ( power_power_nat @ X @ N2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_commuting_commutes
% 0.30/0.48 thf(fact_143_power__commuting__commutes,axiom,
% 0.30/0.48 ! [X: int,Y: int,N2: nat] :
% 0.30/0.48 ( ( ( times_times_int @ X @ Y )
% 0.30/0.48 = ( times_times_int @ Y @ X ) )
% 0.30/0.48 => ( ( times_times_int @ ( power_power_int @ X @ N2 ) @ Y )
% 0.30/0.48 = ( times_times_int @ Y @ ( power_power_int @ X @ N2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_commuting_commutes
% 0.30/0.48 thf(fact_144_power__mult__distrib,axiom,
% 0.30/0.48 ! [A: complex,B: complex,N2: nat] :
% 0.30/0.48 ( ( power_power_complex @ ( times_times_complex @ A @ B ) @ N2 )
% 0.30/0.48 = ( times_times_complex @ ( power_power_complex @ A @ N2 ) @ ( power_power_complex @ B @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_mult_distrib
% 0.30/0.48 thf(fact_145_power__mult__distrib,axiom,
% 0.30/0.48 ! [A: real,B: real,N2: nat] :
% 0.30/0.48 ( ( power_power_real @ ( times_times_real @ A @ B ) @ N2 )
% 0.30/0.48 = ( times_times_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_mult_distrib
% 0.30/0.48 thf(fact_146_power__mult__distrib,axiom,
% 0.30/0.48 ! [A: rat,B: rat,N2: nat] :
% 0.30/0.48 ( ( power_power_rat @ ( times_times_rat @ A @ B ) @ N2 )
% 0.30/0.48 = ( times_times_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_mult_distrib
% 0.30/0.48 thf(fact_147_power__mult__distrib,axiom,
% 0.30/0.48 ! [A: nat,B: nat,N2: nat] :
% 0.30/0.48 ( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N2 )
% 0.30/0.48 = ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_mult_distrib
% 0.30/0.48 thf(fact_148_power__mult__distrib,axiom,
% 0.30/0.48 ! [A: int,B: int,N2: nat] :
% 0.30/0.48 ( ( power_power_int @ ( times_times_int @ A @ B ) @ N2 )
% 0.30/0.48 = ( times_times_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_mult_distrib
% 0.30/0.48 thf(fact_149_power__commutes,axiom,
% 0.30/0.48 ! [A: complex,N2: nat] :
% 0.30/0.48 ( ( times_times_complex @ ( power_power_complex @ A @ N2 ) @ A )
% 0.30/0.48 = ( times_times_complex @ A @ ( power_power_complex @ A @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_commutes
% 0.30/0.48 thf(fact_150_power__commutes,axiom,
% 0.30/0.48 ! [A: real,N2: nat] :
% 0.30/0.48 ( ( times_times_real @ ( power_power_real @ A @ N2 ) @ A )
% 0.30/0.48 = ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_commutes
% 0.30/0.48 thf(fact_151_power__commutes,axiom,
% 0.30/0.48 ! [A: rat,N2: nat] :
% 0.30/0.48 ( ( times_times_rat @ ( power_power_rat @ A @ N2 ) @ A )
% 0.30/0.48 = ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_commutes
% 0.30/0.48 thf(fact_152_power__commutes,axiom,
% 0.30/0.48 ! [A: nat,N2: nat] :
% 0.30/0.48 ( ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ A )
% 0.30/0.48 = ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_commutes
% 0.30/0.48 thf(fact_153_power__commutes,axiom,
% 0.30/0.48 ! [A: int,N2: nat] :
% 0.30/0.48 ( ( times_times_int @ ( power_power_int @ A @ N2 ) @ A )
% 0.30/0.48 = ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_commutes
% 0.30/0.48 thf(fact_154_power__divide,axiom,
% 0.30/0.48 ! [A: complex,B: complex,N2: nat] :
% 0.30/0.48 ( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B ) @ N2 )
% 0.30/0.48 = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N2 ) @ ( power_power_complex @ B @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_divide
% 0.30/0.48 thf(fact_155_power__divide,axiom,
% 0.30/0.48 ! [A: real,B: real,N2: nat] :
% 0.30/0.48 ( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N2 )
% 0.30/0.48 = ( divide_divide_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_divide
% 0.30/0.48 thf(fact_156_power__divide,axiom,
% 0.30/0.48 ! [A: rat,B: rat,N2: nat] :
% 0.30/0.48 ( ( power_power_rat @ ( divide_divide_rat @ A @ B ) @ N2 )
% 0.30/0.48 = ( divide_divide_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_divide
% 0.30/0.48 thf(fact_157_power__mult,axiom,
% 0.30/0.48 ! [A: nat,M: nat,N2: nat] :
% 0.30/0.48 ( ( power_power_nat @ A @ ( times_times_nat @ M @ N2 ) )
% 0.30/0.48 = ( power_power_nat @ ( power_power_nat @ A @ M ) @ N2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_mult
% 0.30/0.48 thf(fact_158_power__mult,axiom,
% 0.30/0.48 ! [A: real,M: nat,N2: nat] :
% 0.30/0.48 ( ( power_power_real @ A @ ( times_times_nat @ M @ N2 ) )
% 0.30/0.48 = ( power_power_real @ ( power_power_real @ A @ M ) @ N2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_mult
% 0.30/0.48 thf(fact_159_power__mult,axiom,
% 0.30/0.48 ! [A: int,M: nat,N2: nat] :
% 0.30/0.48 ( ( power_power_int @ A @ ( times_times_nat @ M @ N2 ) )
% 0.30/0.48 = ( power_power_int @ ( power_power_int @ A @ M ) @ N2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_mult
% 0.30/0.48 thf(fact_160_power__mult,axiom,
% 0.30/0.48 ! [A: complex,M: nat,N2: nat] :
% 0.30/0.48 ( ( power_power_complex @ A @ ( times_times_nat @ M @ N2 ) )
% 0.30/0.48 = ( power_power_complex @ ( power_power_complex @ A @ M ) @ N2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_mult
% 0.30/0.48 thf(fact_161_div__mult2__eq,axiom,
% 0.30/0.48 ! [M: nat,N2: nat,Q2: nat] :
% 0.30/0.48 ( ( divide_divide_nat @ M @ ( times_times_nat @ N2 @ Q2 ) )
% 0.30/0.48 = ( divide_divide_nat @ ( divide_divide_nat @ M @ N2 ) @ Q2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % div_mult2_eq
% 0.30/0.48 thf(fact_162_mult__numeral__1__right,axiom,
% 0.30/0.48 ! [A: complex] :
% 0.30/0.48 ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
% 0.30/0.48 = A ) ).
% 0.30/0.48
% 0.30/0.48 % mult_numeral_1_right
% 0.30/0.48 thf(fact_163_mult__numeral__1__right,axiom,
% 0.30/0.48 ! [A: real] :
% 0.30/0.48 ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
% 0.30/0.48 = A ) ).
% 0.30/0.48
% 0.30/0.48 % mult_numeral_1_right
% 0.30/0.48 thf(fact_164_mult__numeral__1__right,axiom,
% 0.30/0.48 ! [A: rat] :
% 0.30/0.48 ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
% 0.30/0.48 = A ) ).
% 0.30/0.48
% 0.30/0.48 % mult_numeral_1_right
% 0.30/0.48 thf(fact_165_mult__numeral__1__right,axiom,
% 0.30/0.48 ! [A: nat] :
% 0.30/0.48 ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
% 0.30/0.48 = A ) ).
% 0.30/0.48
% 0.30/0.48 % mult_numeral_1_right
% 0.30/0.48 thf(fact_166_mult__numeral__1__right,axiom,
% 0.30/0.48 ! [A: int] :
% 0.30/0.48 ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
% 0.30/0.48 = A ) ).
% 0.30/0.48
% 0.30/0.48 % mult_numeral_1_right
% 0.30/0.48 thf(fact_167_mult__numeral__1,axiom,
% 0.30/0.48 ! [A: complex] :
% 0.30/0.48 ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
% 0.30/0.48 = A ) ).
% 0.30/0.48
% 0.30/0.48 % mult_numeral_1
% 0.30/0.48 thf(fact_168_mult__numeral__1,axiom,
% 0.30/0.48 ! [A: real] :
% 0.30/0.48 ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
% 0.30/0.48 = A ) ).
% 0.30/0.48
% 0.30/0.48 % mult_numeral_1
% 0.30/0.48 thf(fact_169_mult__numeral__1,axiom,
% 0.30/0.48 ! [A: rat] :
% 0.30/0.48 ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
% 0.30/0.48 = A ) ).
% 0.30/0.48
% 0.30/0.48 % mult_numeral_1
% 0.30/0.48 thf(fact_170_mult__numeral__1,axiom,
% 0.30/0.48 ! [A: nat] :
% 0.30/0.48 ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
% 0.30/0.48 = A ) ).
% 0.30/0.48
% 0.30/0.48 % mult_numeral_1
% 0.30/0.48 thf(fact_171_mult__numeral__1,axiom,
% 0.30/0.48 ! [A: int] :
% 0.30/0.48 ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
% 0.30/0.48 = A ) ).
% 0.30/0.48
% 0.30/0.48 % mult_numeral_1
% 0.30/0.48 thf(fact_172_numeral__Bit0,axiom,
% 0.30/0.48 ! [N2: num] :
% 0.30/0.48 ( ( numera6690914467698888265omplex @ ( bit0 @ N2 ) )
% 0.30/0.48 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_Bit0
% 0.30/0.48 thf(fact_173_numeral__Bit0,axiom,
% 0.30/0.48 ! [N2: num] :
% 0.30/0.48 ( ( numeral_numeral_real @ ( bit0 @ N2 ) )
% 0.30/0.48 = ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_Bit0
% 0.30/0.48 thf(fact_174_numeral__Bit0,axiom,
% 0.30/0.48 ! [N2: num] :
% 0.30/0.48 ( ( numeral_numeral_rat @ ( bit0 @ N2 ) )
% 0.30/0.48 = ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_Bit0
% 0.30/0.48 thf(fact_175_numeral__Bit0,axiom,
% 0.30/0.48 ! [N2: num] :
% 0.30/0.48 ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
% 0.30/0.48 = ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_Bit0
% 0.30/0.48 thf(fact_176_numeral__Bit0,axiom,
% 0.30/0.48 ! [N2: num] :
% 0.30/0.48 ( ( numeral_numeral_int @ ( bit0 @ N2 ) )
% 0.30/0.48 = ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_Bit0
% 0.30/0.48 thf(fact_177_divide__numeral__1,axiom,
% 0.30/0.48 ! [A: complex] :
% 0.30/0.48 ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
% 0.30/0.48 = A ) ).
% 0.30/0.48
% 0.30/0.48 % divide_numeral_1
% 0.30/0.48 thf(fact_178_divide__numeral__1,axiom,
% 0.30/0.48 ! [A: real] :
% 0.30/0.48 ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
% 0.30/0.48 = A ) ).
% 0.30/0.48
% 0.30/0.48 % divide_numeral_1
% 0.30/0.48 thf(fact_179_divide__numeral__1,axiom,
% 0.30/0.48 ! [A: rat] :
% 0.30/0.48 ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
% 0.30/0.48 = A ) ).
% 0.30/0.48
% 0.30/0.48 % divide_numeral_1
% 0.30/0.48 thf(fact_180_power__add,axiom,
% 0.30/0.48 ! [A: complex,M: nat,N2: nat] :
% 0.30/0.48 ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N2 ) )
% 0.30/0.48 = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_add
% 0.30/0.48 thf(fact_181_power__add,axiom,
% 0.30/0.48 ! [A: real,M: nat,N2: nat] :
% 0.30/0.48 ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N2 ) )
% 0.30/0.48 = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_add
% 0.30/0.48 thf(fact_182_power__add,axiom,
% 0.30/0.48 ! [A: rat,M: nat,N2: nat] :
% 0.30/0.48 ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N2 ) )
% 0.30/0.48 = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_add
% 0.30/0.48 thf(fact_183_power__add,axiom,
% 0.30/0.48 ! [A: nat,M: nat,N2: nat] :
% 0.30/0.48 ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N2 ) )
% 0.30/0.48 = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_add
% 0.30/0.48 thf(fact_184_power__add,axiom,
% 0.30/0.48 ! [A: int,M: nat,N2: nat] :
% 0.30/0.48 ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N2 ) )
% 0.30/0.48 = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_add
% 0.30/0.48 thf(fact_185_numeral__Bit0__div__2,axiom,
% 0.30/0.48 ! [N2: num] :
% 0.30/0.48 ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.48 = ( numeral_numeral_nat @ N2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_Bit0_div_2
% 0.30/0.48 thf(fact_186_numeral__Bit0__div__2,axiom,
% 0.30/0.48 ! [N2: num] :
% 0.30/0.48 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.48 = ( numeral_numeral_int @ N2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_Bit0_div_2
% 0.30/0.48 thf(fact_187_left__add__twice,axiom,
% 0.30/0.48 ! [A: complex,B: complex] :
% 0.30/0.48 ( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B ) )
% 0.30/0.48 = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % left_add_twice
% 0.30/0.48 thf(fact_188_left__add__twice,axiom,
% 0.30/0.48 ! [A: real,B: real] :
% 0.30/0.48 ( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.30/0.48 = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % left_add_twice
% 0.30/0.48 thf(fact_189_left__add__twice,axiom,
% 0.30/0.48 ! [A: rat,B: rat] :
% 0.30/0.48 ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.30/0.48 = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % left_add_twice
% 0.30/0.48 thf(fact_190_left__add__twice,axiom,
% 0.30/0.48 ! [A: nat,B: nat] :
% 0.30/0.48 ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.30/0.48 = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % left_add_twice
% 0.30/0.48 thf(fact_191_left__add__twice,axiom,
% 0.30/0.48 ! [A: int,B: int] :
% 0.30/0.48 ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.30/0.48 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % left_add_twice
% 0.30/0.48 thf(fact_192_mult__2__right,axiom,
% 0.30/0.48 ! [Z: complex] :
% 0.30/0.48 ( ( times_times_complex @ Z @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
% 0.30/0.48 = ( plus_plus_complex @ Z @ Z ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_2_right
% 0.30/0.48 thf(fact_193_mult__2__right,axiom,
% 0.30/0.48 ! [Z: real] :
% 0.30/0.48 ( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.30/0.48 = ( plus_plus_real @ Z @ Z ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_2_right
% 0.30/0.48 thf(fact_194_mult__2__right,axiom,
% 0.30/0.48 ! [Z: rat] :
% 0.30/0.48 ( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
% 0.30/0.48 = ( plus_plus_rat @ Z @ Z ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_2_right
% 0.30/0.48 thf(fact_195_mult__2__right,axiom,
% 0.30/0.48 ! [Z: nat] :
% 0.30/0.48 ( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.48 = ( plus_plus_nat @ Z @ Z ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_2_right
% 0.30/0.48 thf(fact_196_mult__2__right,axiom,
% 0.30/0.48 ! [Z: int] :
% 0.30/0.48 ( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.48 = ( plus_plus_int @ Z @ Z ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_2_right
% 0.30/0.48 thf(fact_197_mult__2,axiom,
% 0.30/0.48 ! [Z: complex] :
% 0.30/0.48 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z )
% 0.30/0.48 = ( plus_plus_complex @ Z @ Z ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_2
% 0.30/0.48 thf(fact_198_mult__2,axiom,
% 0.30/0.48 ! [Z: real] :
% 0.30/0.48 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
% 0.30/0.48 = ( plus_plus_real @ Z @ Z ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_2
% 0.30/0.48 thf(fact_199_mult__2,axiom,
% 0.30/0.48 ! [Z: rat] :
% 0.30/0.48 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
% 0.30/0.48 = ( plus_plus_rat @ Z @ Z ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_2
% 0.30/0.48 thf(fact_200_mult__2,axiom,
% 0.30/0.48 ! [Z: nat] :
% 0.30/0.48 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
% 0.30/0.48 = ( plus_plus_nat @ Z @ Z ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_2
% 0.30/0.48 thf(fact_201_mult__2,axiom,
% 0.30/0.48 ! [Z: int] :
% 0.30/0.48 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
% 0.30/0.48 = ( plus_plus_int @ Z @ Z ) ) ).
% 0.30/0.48
% 0.30/0.48 % mult_2
% 0.30/0.48 thf(fact_202_power2__eq__square,axiom,
% 0.30/0.48 ! [A: complex] :
% 0.30/0.48 ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.48 = ( times_times_complex @ A @ A ) ) ).
% 0.30/0.48
% 0.30/0.48 % power2_eq_square
% 0.30/0.48 thf(fact_203_power2__eq__square,axiom,
% 0.30/0.48 ! [A: real] :
% 0.30/0.48 ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.48 = ( times_times_real @ A @ A ) ) ).
% 0.30/0.48
% 0.30/0.48 % power2_eq_square
% 0.30/0.48 thf(fact_204_power2__eq__square,axiom,
% 0.30/0.48 ! [A: rat] :
% 0.30/0.48 ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.48 = ( times_times_rat @ A @ A ) ) ).
% 0.30/0.48
% 0.30/0.48 % power2_eq_square
% 0.30/0.48 thf(fact_205_power2__eq__square,axiom,
% 0.30/0.48 ! [A: nat] :
% 0.30/0.48 ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.48 = ( times_times_nat @ A @ A ) ) ).
% 0.30/0.48
% 0.30/0.48 % power2_eq_square
% 0.30/0.48 thf(fact_206_power2__eq__square,axiom,
% 0.30/0.48 ! [A: int] :
% 0.30/0.48 ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.48 = ( times_times_int @ A @ A ) ) ).
% 0.30/0.48
% 0.30/0.48 % power2_eq_square
% 0.30/0.48 thf(fact_207_power4__eq__xxxx,axiom,
% 0.30/0.48 ! [X: complex] :
% 0.30/0.48 ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.48 = ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X @ X ) @ X ) @ X ) ) ).
% 0.30/0.48
% 0.30/0.48 % power4_eq_xxxx
% 0.30/0.48 thf(fact_208_power4__eq__xxxx,axiom,
% 0.30/0.48 ! [X: real] :
% 0.30/0.48 ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.48 = ( times_times_real @ ( times_times_real @ ( times_times_real @ X @ X ) @ X ) @ X ) ) ).
% 0.30/0.48
% 0.30/0.48 % power4_eq_xxxx
% 0.30/0.48 thf(fact_209_power4__eq__xxxx,axiom,
% 0.30/0.48 ! [X: rat] :
% 0.30/0.48 ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.48 = ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X @ X ) @ X ) @ X ) ) ).
% 0.30/0.48
% 0.30/0.48 % power4_eq_xxxx
% 0.30/0.48 thf(fact_210_power4__eq__xxxx,axiom,
% 0.30/0.48 ! [X: nat] :
% 0.30/0.48 ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.48 = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).
% 0.30/0.48
% 0.30/0.48 % power4_eq_xxxx
% 0.30/0.48 thf(fact_211_power4__eq__xxxx,axiom,
% 0.30/0.48 ! [X: int] :
% 0.30/0.48 ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.48 = ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).
% 0.30/0.48
% 0.30/0.48 % power4_eq_xxxx
% 0.30/0.48 thf(fact_212_power__even__eq,axiom,
% 0.30/0.48 ! [A: nat,N2: nat] :
% 0.30/0.48 ( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.48 = ( power_power_nat @ ( power_power_nat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_even_eq
% 0.30/0.48 thf(fact_213_power__even__eq,axiom,
% 0.30/0.48 ! [A: real,N2: nat] :
% 0.30/0.48 ( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.48 = ( power_power_real @ ( power_power_real @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_even_eq
% 0.30/0.48 thf(fact_214_power__even__eq,axiom,
% 0.30/0.48 ! [A: int,N2: nat] :
% 0.30/0.48 ( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.48 = ( power_power_int @ ( power_power_int @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_even_eq
% 0.30/0.48 thf(fact_215_power__even__eq,axiom,
% 0.30/0.48 ! [A: complex,N2: nat] :
% 0.30/0.48 ( ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.48 = ( power_power_complex @ ( power_power_complex @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % power_even_eq
% 0.30/0.48 thf(fact_216_in__children__def,axiom,
% 0.30/0.48 ( vEBT_V5917875025757280293ildren
% 0.30/0.48 = ( ^ [N: nat,TreeList: list_VEBT_VEBT,X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ N ) ) @ ( vEBT_VEBT_low @ X2 @ N ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % in_children_def
% 0.30/0.48 thf(fact_217_greater__shift,axiom,
% 0.30/0.48 ( ord_less_nat
% 0.30/0.48 = ( ^ [Y2: nat,X2: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X2 ) @ ( some_nat @ Y2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % greater_shift
% 0.30/0.48 thf(fact_218_less__shift,axiom,
% 0.30/0.48 ( ord_less_nat
% 0.30/0.48 = ( ^ [X2: nat,Y2: nat] : ( vEBT_VEBT_less @ ( some_nat @ X2 ) @ ( some_nat @ Y2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % less_shift
% 0.30/0.48 thf(fact_219__092_060open_062x_A_060_A2_A_094_Adeg_092_060close_062,axiom,
% 0.30/0.48 ord_less_nat @ xa @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% 0.30/0.48
% 0.30/0.48 % \<open>x < 2 ^ deg\<close>
% 0.30/0.48 thf(fact_220_nth__list__update__eq,axiom,
% 0.30/0.48 ! [I2: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
% 0.30/0.48 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.30/0.48 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X ) @ I2 )
% 0.30/0.48 = X ) ) ).
% 0.30/0.48
% 0.30/0.48 % nth_list_update_eq
% 0.30/0.48 thf(fact_221_nth__list__update__eq,axiom,
% 0.30/0.48 ! [I2: nat,Xs: list_o,X: $o] :
% 0.30/0.48 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 0.30/0.48 => ( ( nth_o @ ( list_update_o @ Xs @ I2 @ X ) @ I2 )
% 0.30/0.48 = X ) ) ).
% 0.30/0.48
% 0.30/0.48 % nth_list_update_eq
% 0.30/0.48 thf(fact_222_nth__list__update__eq,axiom,
% 0.30/0.48 ! [I2: nat,Xs: list_int,X: int] :
% 0.30/0.48 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 0.30/0.48 => ( ( nth_int @ ( list_update_int @ Xs @ I2 @ X ) @ I2 )
% 0.30/0.48 = X ) ) ).
% 0.30/0.48
% 0.30/0.48 % nth_list_update_eq
% 0.30/0.48 thf(fact_223_option_Ocollapse,axiom,
% 0.30/0.48 ! [Option: option_nat] :
% 0.30/0.48 ( ( Option != none_nat )
% 0.30/0.48 => ( ( some_nat @ ( the_nat @ Option ) )
% 0.30/0.48 = Option ) ) ).
% 0.30/0.48
% 0.30/0.48 % option.collapse
% 0.30/0.48 thf(fact_224_option_Ocollapse,axiom,
% 0.30/0.48 ! [Option: option4927543243414619207at_nat] :
% 0.30/0.48 ( ( Option != none_P5556105721700978146at_nat )
% 0.30/0.48 => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
% 0.30/0.48 = Option ) ) ).
% 0.30/0.48
% 0.30/0.48 % option.collapse
% 0.30/0.48 thf(fact_225_option_Ocollapse,axiom,
% 0.30/0.48 ! [Option: option_num] :
% 0.30/0.48 ( ( Option != none_num )
% 0.30/0.48 => ( ( some_num @ ( the_num @ Option ) )
% 0.30/0.48 = Option ) ) ).
% 0.30/0.48
% 0.30/0.48 % option.collapse
% 0.30/0.48 thf(fact_226__092_060open_062ma_A_092_060le_062_A2_A_094_Adeg_092_060close_062,axiom,
% 0.30/0.48 ord_less_eq_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% 0.30/0.48
% 0.30/0.48 % \<open>ma \<le> 2 ^ deg\<close>
% 0.30/0.48 thf(fact_227_field__less__half__sum,axiom,
% 0.30/0.48 ! [X: real,Y: real] :
% 0.30/0.48 ( ( ord_less_real @ X @ Y )
% 0.30/0.48 => ( ord_less_real @ X @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % field_less_half_sum
% 0.30/0.48 thf(fact_228_field__less__half__sum,axiom,
% 0.30/0.48 ! [X: rat,Y: rat] :
% 0.30/0.48 ( ( ord_less_rat @ X @ Y )
% 0.30/0.48 => ( ord_less_rat @ X @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % field_less_half_sum
% 0.30/0.48 thf(fact_229_nth__list__update__neq,axiom,
% 0.30/0.48 ! [I2: nat,J: nat,Xs: list_int,X: int] :
% 0.30/0.48 ( ( I2 != J )
% 0.30/0.48 => ( ( nth_int @ ( list_update_int @ Xs @ I2 @ X ) @ J )
% 0.30/0.48 = ( nth_int @ Xs @ J ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nth_list_update_neq
% 0.30/0.48 thf(fact_230_nth__list__update__neq,axiom,
% 0.30/0.48 ! [I2: nat,J: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
% 0.30/0.48 ( ( I2 != J )
% 0.30/0.48 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X ) @ J )
% 0.30/0.48 = ( nth_VEBT_VEBT @ Xs @ J ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % nth_list_update_neq
% 0.30/0.48 thf(fact_231_list__update__id,axiom,
% 0.30/0.48 ! [Xs: list_int,I2: nat] :
% 0.30/0.48 ( ( list_update_int @ Xs @ I2 @ ( nth_int @ Xs @ I2 ) )
% 0.30/0.48 = Xs ) ).
% 0.30/0.48
% 0.30/0.48 % list_update_id
% 0.30/0.48 thf(fact_232_list__update__id,axiom,
% 0.30/0.48 ! [Xs: list_VEBT_VEBT,I2: nat] :
% 0.30/0.48 ( ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ ( nth_VEBT_VEBT @ Xs @ I2 ) )
% 0.30/0.48 = Xs ) ).
% 0.30/0.48
% 0.30/0.48 % list_update_id
% 0.30/0.48 thf(fact_233_length__list__update,axiom,
% 0.30/0.48 ! [Xs: list_VEBT_VEBT,I2: nat,X: vEBT_VEBT] :
% 0.30/0.48 ( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X ) )
% 0.30/0.48 = ( size_s6755466524823107622T_VEBT @ Xs ) ) ).
% 0.30/0.48
% 0.30/0.48 % length_list_update
% 0.30/0.48 thf(fact_234_length__list__update,axiom,
% 0.30/0.48 ! [Xs: list_o,I2: nat,X: $o] :
% 0.30/0.48 ( ( size_size_list_o @ ( list_update_o @ Xs @ I2 @ X ) )
% 0.30/0.48 = ( size_size_list_o @ Xs ) ) ).
% 0.30/0.48
% 0.30/0.48 % length_list_update
% 0.30/0.48 thf(fact_235_length__list__update,axiom,
% 0.30/0.48 ! [Xs: list_int,I2: nat,X: int] :
% 0.30/0.48 ( ( size_size_list_int @ ( list_update_int @ Xs @ I2 @ X ) )
% 0.30/0.48 = ( size_size_list_int @ Xs ) ) ).
% 0.30/0.48
% 0.30/0.48 % length_list_update
% 0.30/0.48 thf(fact_236_not__None__eq,axiom,
% 0.30/0.48 ! [X: option_nat] :
% 0.30/0.48 ( ( X != none_nat )
% 0.30/0.48 = ( ? [Y2: nat] :
% 0.30/0.48 ( X
% 0.30/0.48 = ( some_nat @ Y2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_None_eq
% 0.30/0.48 thf(fact_237_not__None__eq,axiom,
% 0.30/0.48 ! [X: option4927543243414619207at_nat] :
% 0.30/0.48 ( ( X != none_P5556105721700978146at_nat )
% 0.30/0.48 = ( ? [Y2: product_prod_nat_nat] :
% 0.30/0.48 ( X
% 0.30/0.48 = ( some_P7363390416028606310at_nat @ Y2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_None_eq
% 0.30/0.48 thf(fact_238_not__None__eq,axiom,
% 0.30/0.48 ! [X: option_num] :
% 0.30/0.48 ( ( X != none_num )
% 0.30/0.48 = ( ? [Y2: num] :
% 0.30/0.48 ( X
% 0.30/0.48 = ( some_num @ Y2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_None_eq
% 0.30/0.48 thf(fact_239_valid__member__both__member__options,axiom,
% 0.30/0.48 ! [T: vEBT_VEBT,N2: nat,X: nat] :
% 0.30/0.48 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.48 => ( ( vEBT_V8194947554948674370ptions @ T @ X )
% 0.30/0.48 => ( vEBT_vebt_member @ T @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % valid_member_both_member_options
% 0.30/0.48 thf(fact_240_both__member__options__equiv__member,axiom,
% 0.30/0.48 ! [T: vEBT_VEBT,N2: nat,X: nat] :
% 0.30/0.48 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.48 => ( ( vEBT_V8194947554948674370ptions @ T @ X )
% 0.30/0.48 = ( vEBT_vebt_member @ T @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % both_member_options_equiv_member
% 0.30/0.48 thf(fact_241_min__in__set__def,axiom,
% 0.30/0.48 ( vEBT_VEBT_min_in_set
% 0.30/0.48 = ( ^ [Xs2: set_nat,X2: nat] :
% 0.30/0.48 ( ( member_nat @ X2 @ Xs2 )
% 0.30/0.48 & ! [Y2: nat] :
% 0.30/0.48 ( ( member_nat @ Y2 @ Xs2 )
% 0.30/0.48 => ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % min_in_set_def
% 0.30/0.48 thf(fact_242_max__in__set__def,axiom,
% 0.30/0.48 ( vEBT_VEBT_max_in_set
% 0.30/0.48 = ( ^ [Xs2: set_nat,X2: nat] :
% 0.30/0.48 ( ( member_nat @ X2 @ Xs2 )
% 0.30/0.48 & ! [Y2: nat] :
% 0.30/0.48 ( ( member_nat @ Y2 @ Xs2 )
% 0.30/0.48 => ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % max_in_set_def
% 0.30/0.48 thf(fact_243__C4_OIH_C_I1_J,axiom,
% 0.30/0.48 ! [X5: vEBT_VEBT] :
% 0.30/0.48 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ treeList ) )
% 0.30/0.48 => ( ( vEBT_invar_vebt @ X5 @ na )
% 0.30/0.48 & ! [Xa: nat,Xb: nat] :
% 0.30/0.48 ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ X5 @ Xa ) @ Xb )
% 0.30/0.48 = ( ( Xa != Xb )
% 0.30/0.48 & ( vEBT_V8194947554948674370ptions @ X5 @ Xb ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % "4.IH"(1)
% 0.30/0.48 thf(fact_244__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062summin_O_ASome_Asummin_A_061_Avebt__mint_Asummary_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.30/0.48 ~ ! [Summin: nat] :
% 0.30/0.48 ( ( some_nat @ Summin )
% 0.30/0.48 != ( vEBT_vebt_mint @ summary ) ) ).
% 0.30/0.48
% 0.30/0.48 % \<open>\<And>thesis. (\<And>summin. Some summin = vebt_mint summary \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.30/0.48 thf(fact_245_inthall,axiom,
% 0.30/0.48 ! [Xs: list_nat,P: nat > $o,N2: nat] :
% 0.30/0.48 ( ! [X4: nat] :
% 0.30/0.48 ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
% 0.30/0.48 => ( P @ X4 ) )
% 0.30/0.48 => ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
% 0.30/0.48 => ( P @ ( nth_nat @ Xs @ N2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % inthall
% 0.30/0.48 thf(fact_246_inthall,axiom,
% 0.30/0.48 ! [Xs: list_real,P: real > $o,N2: nat] :
% 0.30/0.48 ( ! [X4: real] :
% 0.30/0.48 ( ( member_real @ X4 @ ( set_real2 @ Xs ) )
% 0.30/0.48 => ( P @ X4 ) )
% 0.30/0.48 => ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs ) )
% 0.30/0.48 => ( P @ ( nth_real @ Xs @ N2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % inthall
% 0.30/0.48 thf(fact_247_inthall,axiom,
% 0.30/0.48 ! [Xs: list_complex,P: complex > $o,N2: nat] :
% 0.30/0.48 ( ! [X4: complex] :
% 0.30/0.48 ( ( member_complex @ X4 @ ( set_complex2 @ Xs ) )
% 0.30/0.48 => ( P @ X4 ) )
% 0.30/0.48 => ( ( ord_less_nat @ N2 @ ( size_s3451745648224563538omplex @ Xs ) )
% 0.30/0.48 => ( P @ ( nth_complex @ Xs @ N2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % inthall
% 0.30/0.48 thf(fact_248_inthall,axiom,
% 0.30/0.48 ! [Xs: list_set_nat,P: set_nat > $o,N2: nat] :
% 0.30/0.48 ( ! [X4: set_nat] :
% 0.30/0.48 ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs ) )
% 0.30/0.48 => ( P @ X4 ) )
% 0.30/0.48 => ( ( ord_less_nat @ N2 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 0.30/0.48 => ( P @ ( nth_set_nat @ Xs @ N2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % inthall
% 0.30/0.48 thf(fact_249_inthall,axiom,
% 0.30/0.48 ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,N2: nat] :
% 0.30/0.48 ( ! [X4: vEBT_VEBT] :
% 0.30/0.48 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.30/0.48 => ( P @ X4 ) )
% 0.30/0.48 => ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.30/0.48 => ( P @ ( nth_VEBT_VEBT @ Xs @ N2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % inthall
% 0.30/0.48 thf(fact_250_inthall,axiom,
% 0.30/0.48 ! [Xs: list_o,P: $o > $o,N2: nat] :
% 0.30/0.48 ( ! [X4: $o] :
% 0.30/0.48 ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
% 0.30/0.48 => ( P @ X4 ) )
% 0.30/0.48 => ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
% 0.30/0.48 => ( P @ ( nth_o @ Xs @ N2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % inthall
% 0.30/0.48 thf(fact_251_inthall,axiom,
% 0.30/0.48 ! [Xs: list_int,P: int > $o,N2: nat] :
% 0.30/0.48 ( ! [X4: int] :
% 0.30/0.48 ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
% 0.30/0.48 => ( P @ X4 ) )
% 0.30/0.48 => ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
% 0.30/0.48 => ( P @ ( nth_int @ Xs @ N2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % inthall
% 0.30/0.48 thf(fact_252__092_060open_062x_A_061_Ami_092_060close_062,axiom,
% 0.30/0.48 xa = mi ).
% 0.30/0.48
% 0.30/0.48 % \<open>x = mi\<close>
% 0.30/0.48 thf(fact_253_maxt__member,axiom,
% 0.30/0.48 ! [T: vEBT_VEBT,N2: nat,Maxi2: nat] :
% 0.30/0.48 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.48 => ( ( ( vEBT_vebt_maxt @ T )
% 0.30/0.48 = ( some_nat @ Maxi2 ) )
% 0.30/0.48 => ( vEBT_vebt_member @ T @ Maxi2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % maxt_member
% 0.30/0.48 thf(fact_254_mint__member,axiom,
% 0.30/0.48 ! [T: vEBT_VEBT,N2: nat,Maxi2: nat] :
% 0.30/0.48 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.48 => ( ( ( vEBT_vebt_mint @ T )
% 0.30/0.48 = ( some_nat @ Maxi2 ) )
% 0.30/0.48 => ( vEBT_vebt_member @ T @ Maxi2 ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mint_member
% 0.30/0.48 thf(fact_255_option_Oinject,axiom,
% 0.30/0.48 ! [X22: nat,Y22: nat] :
% 0.30/0.48 ( ( ( some_nat @ X22 )
% 0.30/0.48 = ( some_nat @ Y22 ) )
% 0.30/0.48 = ( X22 = Y22 ) ) ).
% 0.30/0.48
% 0.30/0.48 % option.inject
% 0.30/0.48 thf(fact_256_option_Oinject,axiom,
% 0.30/0.48 ! [X22: product_prod_nat_nat,Y22: product_prod_nat_nat] :
% 0.30/0.48 ( ( ( some_P7363390416028606310at_nat @ X22 )
% 0.30/0.48 = ( some_P7363390416028606310at_nat @ Y22 ) )
% 0.30/0.48 = ( X22 = Y22 ) ) ).
% 0.30/0.48
% 0.30/0.48 % option.inject
% 0.30/0.48 thf(fact_257_option_Oinject,axiom,
% 0.30/0.48 ! [X22: num,Y22: num] :
% 0.30/0.48 ( ( ( some_num @ X22 )
% 0.30/0.48 = ( some_num @ Y22 ) )
% 0.30/0.48 = ( X22 = Y22 ) ) ).
% 0.30/0.48
% 0.30/0.48 % option.inject
% 0.30/0.48 thf(fact_258_list__update__overwrite,axiom,
% 0.30/0.48 ! [Xs: list_VEBT_VEBT,I2: nat,X: vEBT_VEBT,Y: vEBT_VEBT] :
% 0.30/0.48 ( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X ) @ I2 @ Y )
% 0.30/0.48 = ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ Y ) ) ).
% 0.30/0.48
% 0.30/0.48 % list_update_overwrite
% 0.30/0.48 thf(fact_259_maxt__corr__help,axiom,
% 0.30/0.48 ! [T: vEBT_VEBT,N2: nat,Maxi2: nat,X: nat] :
% 0.30/0.48 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.48 => ( ( ( vEBT_vebt_maxt @ T )
% 0.30/0.48 = ( some_nat @ Maxi2 ) )
% 0.30/0.48 => ( ( vEBT_vebt_member @ T @ X )
% 0.30/0.48 => ( ord_less_eq_nat @ X @ Maxi2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % maxt_corr_help
% 0.30/0.48 thf(fact_260_mint__corr__help,axiom,
% 0.30/0.48 ! [T: vEBT_VEBT,N2: nat,Mini: nat,X: nat] :
% 0.30/0.48 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.48 => ( ( ( vEBT_vebt_mint @ T )
% 0.30/0.48 = ( some_nat @ Mini ) )
% 0.30/0.48 => ( ( vEBT_vebt_member @ T @ X )
% 0.30/0.48 => ( ord_less_eq_nat @ Mini @ X ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % mint_corr_help
% 0.30/0.48 thf(fact_261_dp,axiom,
% 0.30/0.48 ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ).
% 0.30/0.48
% 0.30/0.48 % dp
% 0.30/0.48 thf(fact_262__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062lx_O_ASome_Alx_A_061_Avebt__mint_A_ItreeList_A_B_Asummin_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.30/0.48 ~ ! [Lx: nat] :
% 0.30/0.48 ( ( some_nat @ Lx )
% 0.30/0.48 != ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ summin ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % \<open>\<And>thesis. (\<And>lx. Some lx = vebt_mint (treeList ! summin) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.30/0.48 thf(fact_263__092_060open_062mi_A_092_060le_062_Ax_A_092_060and_062_Ax_A_092_060le_062_Ama_092_060close_062,axiom,
% 0.30/0.48 ( ( ord_less_eq_nat @ mi @ xa )
% 0.30/0.48 & ( ord_less_eq_nat @ xa @ ma ) ) ).
% 0.30/0.48
% 0.30/0.48 % \<open>mi \<le> x \<and> x \<le> ma\<close>
% 0.30/0.48 thf(fact_264__092_060open_062invar__vebt_A_ItreeList_A_B_Asummin_J_An_092_060close_062,axiom,
% 0.30/0.48 vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ summin ) @ na ).
% 0.30/0.48
% 0.30/0.48 % \<open>invar_vebt (treeList ! summin) n\<close>
% 0.30/0.48 thf(fact_265_member__bound,axiom,
% 0.30/0.48 ! [Tree: vEBT_VEBT,X: nat,N2: nat] :
% 0.30/0.48 ( ( vEBT_vebt_member @ Tree @ X )
% 0.30/0.48 => ( ( vEBT_invar_vebt @ Tree @ N2 )
% 0.30/0.48 => ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % member_bound
% 0.30/0.48 thf(fact_266__092_060open_062invar__vebt_A_ItreeList_A_B_Amaxs_J_An_092_060close_062,axiom,
% 0.30/0.48 vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ na ).
% 0.30/0.48
% 0.30/0.48 % \<open>invar_vebt (treeList ! maxs) n\<close>
% 0.30/0.48 thf(fact_267_numeral__le__iff,axiom,
% 0.30/0.48 ! [M: num,N2: num] :
% 0.30/0.48 ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
% 0.30/0.48 = ( ord_less_eq_num @ M @ N2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_le_iff
% 0.30/0.48 thf(fact_268_numeral__le__iff,axiom,
% 0.30/0.48 ! [M: num,N2: num] :
% 0.30/0.48 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
% 0.30/0.48 = ( ord_less_eq_num @ M @ N2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_le_iff
% 0.30/0.48 thf(fact_269_numeral__le__iff,axiom,
% 0.30/0.48 ! [M: num,N2: num] :
% 0.30/0.48 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
% 0.30/0.48 = ( ord_less_eq_num @ M @ N2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_le_iff
% 0.30/0.48 thf(fact_270_numeral__le__iff,axiom,
% 0.30/0.48 ! [M: num,N2: num] :
% 0.30/0.48 ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 0.30/0.48 = ( ord_less_eq_num @ M @ N2 ) ) ).
% 0.30/0.48
% 0.30/0.48 % numeral_le_iff
% 0.30/0.48 thf(fact_271_misiz,axiom,
% 0.30/0.48 ! [T: vEBT_VEBT,N2: nat,M: nat] :
% 0.30/0.48 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.48 => ( ( ( some_nat @ M )
% 0.30/0.48 = ( vEBT_vebt_mint @ T ) )
% 0.30/0.48 => ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % misiz
% 0.30/0.48 thf(fact_272_not__Some__eq,axiom,
% 0.30/0.48 ! [X: option_nat] :
% 0.30/0.48 ( ( ! [Y2: nat] :
% 0.30/0.48 ( X
% 0.30/0.48 != ( some_nat @ Y2 ) ) )
% 0.30/0.48 = ( X = none_nat ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_Some_eq
% 0.30/0.48 thf(fact_273_not__Some__eq,axiom,
% 0.30/0.48 ! [X: option4927543243414619207at_nat] :
% 0.30/0.48 ( ( ! [Y2: product_prod_nat_nat] :
% 0.30/0.48 ( X
% 0.30/0.48 != ( some_P7363390416028606310at_nat @ Y2 ) ) )
% 0.30/0.48 = ( X = none_P5556105721700978146at_nat ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_Some_eq
% 0.30/0.48 thf(fact_274_not__Some__eq,axiom,
% 0.30/0.48 ! [X: option_num] :
% 0.30/0.48 ( ( ! [Y2: num] :
% 0.30/0.48 ( X
% 0.30/0.48 != ( some_num @ Y2 ) ) )
% 0.30/0.48 = ( X = none_num ) ) ).
% 0.30/0.48
% 0.30/0.48 % not_Some_eq
% 0.30/0.48 thf(fact_275_member__correct,axiom,
% 0.30/0.48 ! [T: vEBT_VEBT,N2: nat,X: nat] :
% 0.30/0.48 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.48 => ( ( vEBT_vebt_member @ T @ X )
% 0.30/0.48 = ( member_nat @ X @ ( vEBT_set_vebt @ T ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % member_correct
% 0.30/0.48 thf(fact_276_list__update__beyond,axiom,
% 0.30/0.48 ! [Xs: list_VEBT_VEBT,I2: nat,X: vEBT_VEBT] :
% 0.30/0.48 ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ I2 )
% 0.30/0.48 => ( ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X )
% 0.30/0.48 = Xs ) ) ).
% 0.30/0.48
% 0.30/0.48 % list_update_beyond
% 0.30/0.48 thf(fact_277_list__update__beyond,axiom,
% 0.30/0.48 ! [Xs: list_o,I2: nat,X: $o] :
% 0.30/0.48 ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ I2 )
% 0.30/0.48 => ( ( list_update_o @ Xs @ I2 @ X )
% 0.30/0.48 = Xs ) ) ).
% 0.30/0.48
% 0.30/0.48 % list_update_beyond
% 0.30/0.48 thf(fact_278_list__update__beyond,axiom,
% 0.30/0.48 ! [Xs: list_int,I2: nat,X: int] :
% 0.30/0.48 ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ I2 )
% 0.30/0.48 => ( ( list_update_int @ Xs @ I2 @ X )
% 0.30/0.48 = Xs ) ) ).
% 0.30/0.48
% 0.30/0.48 % list_update_beyond
% 0.30/0.48 thf(fact_279_divide__le__eq__numeral1_I1_J,axiom,
% 0.30/0.48 ! [B: real,W: num,A: real] :
% 0.30/0.48 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
% 0.30/0.48 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % divide_le_eq_numeral1(1)
% 0.30/0.48 thf(fact_280_divide__le__eq__numeral1_I1_J,axiom,
% 0.30/0.48 ! [B: rat,W: num,A: rat] :
% 0.30/0.48 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
% 0.30/0.48 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % divide_le_eq_numeral1(1)
% 0.30/0.48 thf(fact_281_le__divide__eq__numeral1_I1_J,axiom,
% 0.30/0.48 ! [A: real,B: real,W: num] :
% 0.30/0.48 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.48 = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_divide_eq_numeral1(1)
% 0.30/0.48 thf(fact_282_le__divide__eq__numeral1_I1_J,axiom,
% 0.30/0.48 ! [A: rat,B: rat,W: num] :
% 0.30/0.48 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 0.30/0.48 = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% 0.30/0.48
% 0.30/0.48 % le_divide_eq_numeral1(1)
% 0.30/0.48 thf(fact_283_lesseq__shift,axiom,
% 0.30/0.48 ( ord_less_eq_nat
% 0.30/0.48 = ( ^ [X2: nat,Y2: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X2 ) @ ( some_nat @ Y2 ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % lesseq_shift
% 0.30/0.48 thf(fact_284_set__swap,axiom,
% 0.30/0.48 ! [I2: nat,Xs: list_VEBT_VEBT,J: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.30/0.48 => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.30/0.48 => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ ( nth_VEBT_VEBT @ Xs @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs @ I2 ) ) )
% 0.30/0.48 = ( set_VEBT_VEBT2 @ Xs ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % set_swap
% 0.30/0.48 thf(fact_285_set__swap,axiom,
% 0.30/0.48 ! [I2: nat,Xs: list_o,J: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 0.30/0.48 => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs ) )
% 0.30/0.48 => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs @ I2 @ ( nth_o @ Xs @ J ) ) @ J @ ( nth_o @ Xs @ I2 ) ) )
% 0.30/0.48 = ( set_o2 @ Xs ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % set_swap
% 0.30/0.48 thf(fact_286_set__swap,axiom,
% 0.30/0.48 ! [I2: nat,Xs: list_int,J: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 0.30/0.48 => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
% 0.30/0.48 => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs @ I2 @ ( nth_int @ Xs @ J ) ) @ J @ ( nth_int @ Xs @ I2 ) ) )
% 0.30/0.48 = ( set_int2 @ Xs ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % set_swap
% 0.30/0.48 thf(fact_287__092_060open_062summin_A_K_A2_A_094_An_A_L_Alx_A_061_A_Iif_Ax_A_061_Ami_Athen_Athe_A_Ivebt__mint_Asummary_J_A_K_A2_A_094_A_Ideg_Adiv_A2_J_A_L_Athe_A_Ivebt__mint_A_ItreeList_A_B_Athe_A_Ivebt__mint_Asummary_J_J_J_Aelse_Ax_J_092_060close_062,axiom,
% 0.30/0.48 ( ( ( xa = mi )
% 0.30/0.48 => ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 0.30/0.48 = ( 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 @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) ) )
% 0.30/0.48 & ( ( xa != mi )
% 0.30/0.48 => ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 0.30/0.48 = xa ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % \<open>summin * 2 ^ n + lx = (if x = mi then the (vebt_mint summary) * 2 ^ (deg div 2) + the (vebt_mint (treeList ! the (vebt_mint summary))) else x)\<close>
% 0.30/0.48 thf(fact_288_local_Opower__def,axiom,
% 0.30/0.48 ( vEBT_VEBT_power
% 0.30/0.48 = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).
% 0.30/0.48
% 0.30/0.48 % local.power_def
% 0.30/0.48 thf(fact_289_div__le__mono,axiom,
% 0.30/0.48 ! [M: nat,N2: nat,K: nat] :
% 0.30/0.48 ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.48 => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N2 @ K ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % div_le_mono
% 0.30/0.48 thf(fact_290_div__le__dividend,axiom,
% 0.30/0.48 ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ).
% 0.30/0.48
% 0.30/0.48 % div_le_dividend
% 0.30/0.48 thf(fact_291_all__set__conv__all__nth,axiom,
% 0.30/0.48 ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.30/0.48 ( ( ! [X2: vEBT_VEBT] :
% 0.30/0.48 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.30/0.48 => ( P @ X2 ) ) )
% 0.30/0.48 = ( ! [I3: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.30/0.48 => ( P @ ( nth_VEBT_VEBT @ Xs @ I3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % all_set_conv_all_nth
% 0.30/0.48 thf(fact_292_all__set__conv__all__nth,axiom,
% 0.30/0.48 ! [Xs: list_o,P: $o > $o] :
% 0.30/0.48 ( ( ! [X2: $o] :
% 0.30/0.48 ( ( member_o @ X2 @ ( set_o2 @ Xs ) )
% 0.30/0.48 => ( P @ X2 ) ) )
% 0.30/0.48 = ( ! [I3: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
% 0.30/0.48 => ( P @ ( nth_o @ Xs @ I3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % all_set_conv_all_nth
% 0.30/0.48 thf(fact_293_all__set__conv__all__nth,axiom,
% 0.30/0.48 ! [Xs: list_int,P: int > $o] :
% 0.30/0.48 ( ( ! [X2: int] :
% 0.30/0.48 ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
% 0.30/0.48 => ( P @ X2 ) ) )
% 0.30/0.48 = ( ! [I3: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
% 0.30/0.48 => ( P @ ( nth_int @ Xs @ I3 ) ) ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % all_set_conv_all_nth
% 0.30/0.48 thf(fact_294_all__nth__imp__all__set,axiom,
% 0.30/0.48 ! [Xs: list_nat,P: nat > $o,X: nat] :
% 0.30/0.48 ( ! [I4: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
% 0.30/0.48 => ( P @ ( nth_nat @ Xs @ I4 ) ) )
% 0.30/0.48 => ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 0.30/0.48 => ( P @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % all_nth_imp_all_set
% 0.30/0.48 thf(fact_295_all__nth__imp__all__set,axiom,
% 0.30/0.48 ! [Xs: list_real,P: real > $o,X: real] :
% 0.30/0.48 ( ! [I4: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs ) )
% 0.30/0.48 => ( P @ ( nth_real @ Xs @ I4 ) ) )
% 0.30/0.48 => ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 0.30/0.48 => ( P @ X ) ) ) ).
% 0.30/0.48
% 0.30/0.48 % all_nth_imp_all_set
% 0.30/0.48 thf(fact_296_all__nth__imp__all__set,axiom,
% 0.30/0.48 ! [Xs: list_complex,P: complex > $o,X: complex] :
% 0.30/0.48 ( ! [I4: nat] :
% 0.30/0.48 ( ( ord_less_nat @ I4 @ ( size_s3451745648224563538omplex @ Xs ) )
% 0.30/0.48 => ( P @ ( nth_complex @ Xs @ I4 ) ) )
% 0.30/0.49 => ( ( member_complex @ X @ ( set_complex2 @ Xs ) )
% 0.30/0.49 => ( P @ X ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % all_nth_imp_all_set
% 0.30/0.49 thf(fact_297_all__nth__imp__all__set,axiom,
% 0.30/0.49 ! [Xs: list_set_nat,P: set_nat > $o,X: set_nat] :
% 0.30/0.49 ( ! [I4: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I4 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 0.30/0.49 => ( P @ ( nth_set_nat @ Xs @ I4 ) ) )
% 0.30/0.49 => ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 0.30/0.49 => ( P @ X ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % all_nth_imp_all_set
% 0.30/0.49 thf(fact_298_all__nth__imp__all__set,axiom,
% 0.30/0.49 ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
% 0.30/0.49 ( ! [I4: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.30/0.49 => ( P @ ( nth_VEBT_VEBT @ Xs @ I4 ) ) )
% 0.30/0.49 => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.30/0.49 => ( P @ X ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % all_nth_imp_all_set
% 0.30/0.49 thf(fact_299_all__nth__imp__all__set,axiom,
% 0.30/0.49 ! [Xs: list_o,P: $o > $o,X: $o] :
% 0.30/0.49 ( ! [I4: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
% 0.30/0.49 => ( P @ ( nth_o @ Xs @ I4 ) ) )
% 0.30/0.49 => ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 0.30/0.49 => ( P @ X ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % all_nth_imp_all_set
% 0.30/0.49 thf(fact_300_all__nth__imp__all__set,axiom,
% 0.30/0.49 ! [Xs: list_int,P: int > $o,X: int] :
% 0.30/0.49 ( ! [I4: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
% 0.30/0.49 => ( P @ ( nth_int @ Xs @ I4 ) ) )
% 0.30/0.49 => ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 0.30/0.49 => ( P @ X ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % all_nth_imp_all_set
% 0.30/0.49 thf(fact_301_in__set__conv__nth,axiom,
% 0.30/0.49 ! [X: nat,Xs: list_nat] :
% 0.30/0.49 ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 0.30/0.49 = ( ? [I3: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
% 0.30/0.49 & ( ( nth_nat @ Xs @ I3 )
% 0.30/0.49 = X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % in_set_conv_nth
% 0.30/0.49 thf(fact_302_in__set__conv__nth,axiom,
% 0.30/0.49 ! [X: real,Xs: list_real] :
% 0.30/0.49 ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 0.30/0.49 = ( ? [I3: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs ) )
% 0.30/0.49 & ( ( nth_real @ Xs @ I3 )
% 0.30/0.49 = X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % in_set_conv_nth
% 0.30/0.49 thf(fact_303_in__set__conv__nth,axiom,
% 0.30/0.49 ! [X: complex,Xs: list_complex] :
% 0.30/0.49 ( ( member_complex @ X @ ( set_complex2 @ Xs ) )
% 0.30/0.49 = ( ? [I3: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs ) )
% 0.30/0.49 & ( ( nth_complex @ Xs @ I3 )
% 0.30/0.49 = X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % in_set_conv_nth
% 0.30/0.49 thf(fact_304_in__set__conv__nth,axiom,
% 0.30/0.49 ! [X: set_nat,Xs: list_set_nat] :
% 0.30/0.49 ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 0.30/0.49 = ( ? [I3: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I3 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 0.30/0.49 & ( ( nth_set_nat @ Xs @ I3 )
% 0.30/0.49 = X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % in_set_conv_nth
% 0.30/0.49 thf(fact_305_in__set__conv__nth,axiom,
% 0.30/0.49 ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 0.30/0.49 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.30/0.49 = ( ? [I3: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.30/0.49 & ( ( nth_VEBT_VEBT @ Xs @ I3 )
% 0.30/0.49 = X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % in_set_conv_nth
% 0.30/0.49 thf(fact_306_in__set__conv__nth,axiom,
% 0.30/0.49 ! [X: $o,Xs: list_o] :
% 0.30/0.49 ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 0.30/0.49 = ( ? [I3: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
% 0.30/0.49 & ( ( nth_o @ Xs @ I3 )
% 0.30/0.49 = X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % in_set_conv_nth
% 0.30/0.49 thf(fact_307_in__set__conv__nth,axiom,
% 0.30/0.49 ! [X: int,Xs: list_int] :
% 0.30/0.49 ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 0.30/0.49 = ( ? [I3: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
% 0.30/0.49 & ( ( nth_int @ Xs @ I3 )
% 0.30/0.49 = X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % in_set_conv_nth
% 0.30/0.49 thf(fact_308_list__ball__nth,axiom,
% 0.30/0.49 ! [N2: nat,Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.30/0.49 ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.30/0.49 => ( ! [X4: vEBT_VEBT] :
% 0.30/0.49 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.30/0.49 => ( P @ X4 ) )
% 0.30/0.49 => ( P @ ( nth_VEBT_VEBT @ Xs @ N2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % list_ball_nth
% 0.30/0.49 thf(fact_309_list__ball__nth,axiom,
% 0.30/0.49 ! [N2: nat,Xs: list_o,P: $o > $o] :
% 0.30/0.49 ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
% 0.30/0.49 => ( ! [X4: $o] :
% 0.30/0.49 ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
% 0.30/0.49 => ( P @ X4 ) )
% 0.30/0.49 => ( P @ ( nth_o @ Xs @ N2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % list_ball_nth
% 0.30/0.49 thf(fact_310_list__ball__nth,axiom,
% 0.30/0.49 ! [N2: nat,Xs: list_int,P: int > $o] :
% 0.30/0.49 ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
% 0.30/0.49 => ( ! [X4: int] :
% 0.30/0.49 ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
% 0.30/0.49 => ( P @ X4 ) )
% 0.30/0.49 => ( P @ ( nth_int @ Xs @ N2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % list_ball_nth
% 0.30/0.49 thf(fact_311_nth__mem,axiom,
% 0.30/0.49 ! [N2: nat,Xs: list_nat] :
% 0.30/0.49 ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
% 0.30/0.49 => ( member_nat @ ( nth_nat @ Xs @ N2 ) @ ( set_nat2 @ Xs ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nth_mem
% 0.30/0.49 thf(fact_312_nth__mem,axiom,
% 0.30/0.49 ! [N2: nat,Xs: list_real] :
% 0.30/0.49 ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs ) )
% 0.30/0.49 => ( member_real @ ( nth_real @ Xs @ N2 ) @ ( set_real2 @ Xs ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nth_mem
% 0.30/0.49 thf(fact_313_nth__mem,axiom,
% 0.30/0.49 ! [N2: nat,Xs: list_complex] :
% 0.30/0.49 ( ( ord_less_nat @ N2 @ ( size_s3451745648224563538omplex @ Xs ) )
% 0.30/0.49 => ( member_complex @ ( nth_complex @ Xs @ N2 ) @ ( set_complex2 @ Xs ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nth_mem
% 0.30/0.49 thf(fact_314_nth__mem,axiom,
% 0.30/0.49 ! [N2: nat,Xs: list_set_nat] :
% 0.30/0.49 ( ( ord_less_nat @ N2 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 0.30/0.49 => ( member_set_nat @ ( nth_set_nat @ Xs @ N2 ) @ ( set_set_nat2 @ Xs ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nth_mem
% 0.30/0.49 thf(fact_315_nth__mem,axiom,
% 0.30/0.49 ! [N2: nat,Xs: list_VEBT_VEBT] :
% 0.30/0.49 ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.30/0.49 => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs @ N2 ) @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nth_mem
% 0.30/0.49 thf(fact_316_nth__mem,axiom,
% 0.30/0.49 ! [N2: nat,Xs: list_o] :
% 0.30/0.49 ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
% 0.30/0.49 => ( member_o @ ( nth_o @ Xs @ N2 ) @ ( set_o2 @ Xs ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nth_mem
% 0.30/0.49 thf(fact_317_nth__mem,axiom,
% 0.30/0.49 ! [N2: nat,Xs: list_int] :
% 0.30/0.49 ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
% 0.30/0.49 => ( member_int @ ( nth_int @ Xs @ N2 ) @ ( set_int2 @ Xs ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nth_mem
% 0.30/0.49 thf(fact_318_set__update__memI,axiom,
% 0.30/0.49 ! [N2: nat,Xs: list_nat,X: nat] :
% 0.30/0.49 ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
% 0.30/0.49 => ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs @ N2 @ X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % set_update_memI
% 0.30/0.49 thf(fact_319_set__update__memI,axiom,
% 0.30/0.49 ! [N2: nat,Xs: list_real,X: real] :
% 0.30/0.49 ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs ) )
% 0.30/0.49 => ( member_real @ X @ ( set_real2 @ ( list_update_real @ Xs @ N2 @ X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % set_update_memI
% 0.30/0.49 thf(fact_320_set__update__memI,axiom,
% 0.30/0.49 ! [N2: nat,Xs: list_complex,X: complex] :
% 0.30/0.49 ( ( ord_less_nat @ N2 @ ( size_s3451745648224563538omplex @ Xs ) )
% 0.30/0.49 => ( member_complex @ X @ ( set_complex2 @ ( list_update_complex @ Xs @ N2 @ X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % set_update_memI
% 0.30/0.49 thf(fact_321_set__update__memI,axiom,
% 0.30/0.49 ! [N2: nat,Xs: list_set_nat,X: set_nat] :
% 0.30/0.49 ( ( ord_less_nat @ N2 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 0.30/0.49 => ( member_set_nat @ X @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ N2 @ X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % set_update_memI
% 0.30/0.49 thf(fact_322_set__update__memI,axiom,
% 0.30/0.49 ! [N2: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
% 0.30/0.49 ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.30/0.49 => ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ N2 @ X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % set_update_memI
% 0.30/0.49 thf(fact_323_set__update__memI,axiom,
% 0.30/0.49 ! [N2: nat,Xs: list_o,X: $o] :
% 0.30/0.49 ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
% 0.30/0.49 => ( member_o @ X @ ( set_o2 @ ( list_update_o @ Xs @ N2 @ X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % set_update_memI
% 0.30/0.49 thf(fact_324_set__update__memI,axiom,
% 0.30/0.49 ! [N2: nat,Xs: list_int,X: int] :
% 0.30/0.49 ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
% 0.30/0.49 => ( member_int @ X @ ( set_int2 @ ( list_update_int @ Xs @ N2 @ X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % set_update_memI
% 0.30/0.49 thf(fact_325_div__times__less__eq__dividend,axiom,
% 0.30/0.49 ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) @ M ) ).
% 0.30/0.49
% 0.30/0.49 % div_times_less_eq_dividend
% 0.30/0.49 thf(fact_326_times__div__less__eq__dividend,axiom,
% 0.30/0.49 ! [N2: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) @ M ) ).
% 0.30/0.49
% 0.30/0.49 % times_div_less_eq_dividend
% 0.30/0.49 thf(fact_327_neq__if__length__neq,axiom,
% 0.30/0.49 ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 0.30/0.49 ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.30/0.49 != ( size_s6755466524823107622T_VEBT @ Ys ) )
% 0.30/0.49 => ( Xs != Ys ) ) ).
% 0.30/0.49
% 0.30/0.49 % neq_if_length_neq
% 0.30/0.49 thf(fact_328_neq__if__length__neq,axiom,
% 0.30/0.49 ! [Xs: list_o,Ys: list_o] :
% 0.30/0.49 ( ( ( size_size_list_o @ Xs )
% 0.30/0.49 != ( size_size_list_o @ Ys ) )
% 0.30/0.49 => ( Xs != Ys ) ) ).
% 0.30/0.49
% 0.30/0.49 % neq_if_length_neq
% 0.30/0.49 thf(fact_329_neq__if__length__neq,axiom,
% 0.30/0.49 ! [Xs: list_int,Ys: list_int] :
% 0.30/0.49 ( ( ( size_size_list_int @ Xs )
% 0.30/0.49 != ( size_size_list_int @ Ys ) )
% 0.30/0.49 => ( Xs != Ys ) ) ).
% 0.30/0.49
% 0.30/0.49 % neq_if_length_neq
% 0.30/0.49 thf(fact_330_Ex__list__of__length,axiom,
% 0.30/0.49 ! [N2: nat] :
% 0.30/0.49 ? [Xs3: list_VEBT_VEBT] :
% 0.30/0.49 ( ( size_s6755466524823107622T_VEBT @ Xs3 )
% 0.30/0.49 = N2 ) ).
% 0.30/0.49
% 0.30/0.49 % Ex_list_of_length
% 0.30/0.49 thf(fact_331_Ex__list__of__length,axiom,
% 0.30/0.49 ! [N2: nat] :
% 0.30/0.49 ? [Xs3: list_o] :
% 0.30/0.49 ( ( size_size_list_o @ Xs3 )
% 0.30/0.49 = N2 ) ).
% 0.30/0.49
% 0.30/0.49 % Ex_list_of_length
% 0.30/0.49 thf(fact_332_Ex__list__of__length,axiom,
% 0.30/0.49 ! [N2: nat] :
% 0.30/0.49 ? [Xs3: list_int] :
% 0.30/0.49 ( ( size_size_list_int @ Xs3 )
% 0.30/0.49 = N2 ) ).
% 0.30/0.49
% 0.30/0.49 % Ex_list_of_length
% 0.30/0.49 thf(fact_333_list__update__swap,axiom,
% 0.30/0.49 ! [I2: nat,I5: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT,X6: vEBT_VEBT] :
% 0.30/0.49 ( ( I2 != I5 )
% 0.30/0.49 => ( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X ) @ I5 @ X6 )
% 0.30/0.49 = ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I5 @ X6 ) @ I2 @ X ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % list_update_swap
% 0.30/0.49 thf(fact_334_self__le__ge2__pow,axiom,
% 0.30/0.49 ! [K: nat,M: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.30/0.49 => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % self_le_ge2_pow
% 0.30/0.49 thf(fact_335_power2__nat__le__eq__le,axiom,
% 0.30/0.49 ! [M: nat,N2: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.49 = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % power2_nat_le_eq_le
% 0.30/0.49 thf(fact_336_power2__nat__le__imp__le,axiom,
% 0.30/0.49 ! [M: nat,N2: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N2 )
% 0.30/0.49 => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % power2_nat_le_imp_le
% 0.30/0.49 thf(fact_337_length__induct,axiom,
% 0.30/0.49 ! [P: list_VEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
% 0.30/0.49 ( ! [Xs3: list_VEBT_VEBT] :
% 0.30/0.49 ( ! [Ys2: list_VEBT_VEBT] :
% 0.30/0.49 ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys2 ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
% 0.30/0.49 => ( P @ Ys2 ) )
% 0.30/0.49 => ( P @ Xs3 ) )
% 0.30/0.49 => ( P @ Xs ) ) ).
% 0.30/0.49
% 0.30/0.49 % length_induct
% 0.30/0.49 thf(fact_338_length__induct,axiom,
% 0.30/0.49 ! [P: list_o > $o,Xs: list_o] :
% 0.30/0.49 ( ! [Xs3: list_o] :
% 0.30/0.49 ( ! [Ys2: list_o] :
% 0.30/0.49 ( ( ord_less_nat @ ( size_size_list_o @ Ys2 ) @ ( size_size_list_o @ Xs3 ) )
% 0.30/0.49 => ( P @ Ys2 ) )
% 0.30/0.49 => ( P @ Xs3 ) )
% 0.30/0.49 => ( P @ Xs ) ) ).
% 0.30/0.49
% 0.30/0.49 % length_induct
% 0.30/0.49 thf(fact_339_length__induct,axiom,
% 0.30/0.49 ! [P: list_int > $o,Xs: list_int] :
% 0.30/0.49 ( ! [Xs3: list_int] :
% 0.30/0.49 ( ! [Ys2: list_int] :
% 0.30/0.49 ( ( ord_less_nat @ ( size_size_list_int @ Ys2 ) @ ( size_size_list_int @ Xs3 ) )
% 0.30/0.49 => ( P @ Ys2 ) )
% 0.30/0.49 => ( P @ Xs3 ) )
% 0.30/0.49 => ( P @ Xs ) ) ).
% 0.30/0.49
% 0.30/0.49 % length_induct
% 0.30/0.49 thf(fact_340_combine__options__cases,axiom,
% 0.30/0.49 ! [X: option_nat,P: option_nat > option_nat > $o,Y: option_nat] :
% 0.30/0.49 ( ( ( X = none_nat )
% 0.30/0.49 => ( P @ X @ Y ) )
% 0.30/0.49 => ( ( ( Y = none_nat )
% 0.30/0.49 => ( P @ X @ Y ) )
% 0.30/0.49 => ( ! [A3: nat,B2: nat] :
% 0.30/0.49 ( ( X
% 0.30/0.49 = ( some_nat @ A3 ) )
% 0.30/0.49 => ( ( Y
% 0.30/0.49 = ( some_nat @ B2 ) )
% 0.30/0.49 => ( P @ X @ Y ) ) )
% 0.30/0.49 => ( P @ X @ Y ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % combine_options_cases
% 0.30/0.49 thf(fact_341_combine__options__cases,axiom,
% 0.30/0.49 ! [X: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
% 0.30/0.49 ( ( ( X = none_nat )
% 0.30/0.49 => ( P @ X @ Y ) )
% 0.30/0.49 => ( ( ( Y = none_P5556105721700978146at_nat )
% 0.30/0.49 => ( P @ X @ Y ) )
% 0.30/0.49 => ( ! [A3: nat,B2: product_prod_nat_nat] :
% 0.30/0.49 ( ( X
% 0.30/0.49 = ( some_nat @ A3 ) )
% 0.30/0.49 => ( ( Y
% 0.30/0.49 = ( some_P7363390416028606310at_nat @ B2 ) )
% 0.30/0.49 => ( P @ X @ Y ) ) )
% 0.30/0.49 => ( P @ X @ Y ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % combine_options_cases
% 0.30/0.49 thf(fact_342_combine__options__cases,axiom,
% 0.30/0.49 ! [X: option_nat,P: option_nat > option_num > $o,Y: option_num] :
% 0.30/0.49 ( ( ( X = none_nat )
% 0.30/0.49 => ( P @ X @ Y ) )
% 0.30/0.49 => ( ( ( Y = none_num )
% 0.30/0.49 => ( P @ X @ Y ) )
% 0.30/0.49 => ( ! [A3: nat,B2: num] :
% 0.30/0.49 ( ( X
% 0.30/0.49 = ( some_nat @ A3 ) )
% 0.30/0.49 => ( ( Y
% 0.30/0.49 = ( some_num @ B2 ) )
% 0.30/0.49 => ( P @ X @ Y ) ) )
% 0.30/0.49 => ( P @ X @ Y ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % combine_options_cases
% 0.30/0.49 thf(fact_343_combine__options__cases,axiom,
% 0.30/0.49 ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y: option_nat] :
% 0.30/0.49 ( ( ( X = none_P5556105721700978146at_nat )
% 0.30/0.49 => ( P @ X @ Y ) )
% 0.30/0.49 => ( ( ( Y = none_nat )
% 0.30/0.49 => ( P @ X @ Y ) )
% 0.30/0.49 => ( ! [A3: product_prod_nat_nat,B2: nat] :
% 0.30/0.49 ( ( X
% 0.30/0.49 = ( some_P7363390416028606310at_nat @ A3 ) )
% 0.30/0.49 => ( ( Y
% 0.30/0.49 = ( some_nat @ B2 ) )
% 0.30/0.49 => ( P @ X @ Y ) ) )
% 0.30/0.49 => ( P @ X @ Y ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % combine_options_cases
% 0.30/0.49 thf(fact_344_combine__options__cases,axiom,
% 0.30/0.49 ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
% 0.30/0.49 ( ( ( X = none_P5556105721700978146at_nat )
% 0.30/0.49 => ( P @ X @ Y ) )
% 0.30/0.49 => ( ( ( Y = none_P5556105721700978146at_nat )
% 0.30/0.49 => ( P @ X @ Y ) )
% 0.30/0.49 => ( ! [A3: product_prod_nat_nat,B2: product_prod_nat_nat] :
% 0.30/0.49 ( ( X
% 0.30/0.49 = ( some_P7363390416028606310at_nat @ A3 ) )
% 0.30/0.49 => ( ( Y
% 0.30/0.49 = ( some_P7363390416028606310at_nat @ B2 ) )
% 0.30/0.49 => ( P @ X @ Y ) ) )
% 0.30/0.49 => ( P @ X @ Y ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % combine_options_cases
% 0.30/0.49 thf(fact_345_combine__options__cases,axiom,
% 0.30/0.49 ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y: option_num] :
% 0.30/0.49 ( ( ( X = none_P5556105721700978146at_nat )
% 0.30/0.49 => ( P @ X @ Y ) )
% 0.30/0.49 => ( ( ( Y = none_num )
% 0.30/0.49 => ( P @ X @ Y ) )
% 0.30/0.49 => ( ! [A3: product_prod_nat_nat,B2: num] :
% 0.30/0.49 ( ( X
% 0.30/0.49 = ( some_P7363390416028606310at_nat @ A3 ) )
% 0.30/0.49 => ( ( Y
% 0.30/0.49 = ( some_num @ B2 ) )
% 0.30/0.49 => ( P @ X @ Y ) ) )
% 0.30/0.49 => ( P @ X @ Y ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % combine_options_cases
% 0.30/0.49 thf(fact_346_combine__options__cases,axiom,
% 0.30/0.49 ! [X: option_num,P: option_num > option_nat > $o,Y: option_nat] :
% 0.30/0.49 ( ( ( X = none_num )
% 0.30/0.49 => ( P @ X @ Y ) )
% 0.30/0.49 => ( ( ( Y = none_nat )
% 0.30/0.49 => ( P @ X @ Y ) )
% 0.30/0.49 => ( ! [A3: num,B2: nat] :
% 0.30/0.49 ( ( X
% 0.30/0.49 = ( some_num @ A3 ) )
% 0.30/0.49 => ( ( Y
% 0.30/0.49 = ( some_nat @ B2 ) )
% 0.30/0.49 => ( P @ X @ Y ) ) )
% 0.30/0.49 => ( P @ X @ Y ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % combine_options_cases
% 0.30/0.49 thf(fact_347_combine__options__cases,axiom,
% 0.30/0.49 ! [X: option_num,P: option_num > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
% 0.30/0.49 ( ( ( X = none_num )
% 0.30/0.49 => ( P @ X @ Y ) )
% 0.30/0.49 => ( ( ( Y = none_P5556105721700978146at_nat )
% 0.30/0.49 => ( P @ X @ Y ) )
% 0.30/0.49 => ( ! [A3: num,B2: product_prod_nat_nat] :
% 0.30/0.49 ( ( X
% 0.30/0.49 = ( some_num @ A3 ) )
% 0.30/0.49 => ( ( Y
% 0.30/0.49 = ( some_P7363390416028606310at_nat @ B2 ) )
% 0.30/0.49 => ( P @ X @ Y ) ) )
% 0.30/0.49 => ( P @ X @ Y ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % combine_options_cases
% 0.30/0.49 thf(fact_348_combine__options__cases,axiom,
% 0.30/0.49 ! [X: option_num,P: option_num > option_num > $o,Y: option_num] :
% 0.30/0.49 ( ( ( X = none_num )
% 0.30/0.49 => ( P @ X @ Y ) )
% 0.30/0.49 => ( ( ( Y = none_num )
% 0.30/0.49 => ( P @ X @ Y ) )
% 0.30/0.49 => ( ! [A3: num,B2: num] :
% 0.30/0.49 ( ( X
% 0.30/0.49 = ( some_num @ A3 ) )
% 0.30/0.49 => ( ( Y
% 0.30/0.49 = ( some_num @ B2 ) )
% 0.30/0.49 => ( P @ X @ Y ) ) )
% 0.30/0.49 => ( P @ X @ Y ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % combine_options_cases
% 0.30/0.49 thf(fact_349_split__option__all,axiom,
% 0.30/0.49 ( ( ^ [P2: option_nat > $o] :
% 0.30/0.49 ! [X7: option_nat] : ( P2 @ X7 ) )
% 0.30/0.49 = ( ^ [P3: option_nat > $o] :
% 0.30/0.49 ( ( P3 @ none_nat )
% 0.30/0.49 & ! [X2: nat] : ( P3 @ ( some_nat @ X2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % split_option_all
% 0.30/0.49 thf(fact_350_split__option__all,axiom,
% 0.30/0.49 ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 0.30/0.49 ! [X7: option4927543243414619207at_nat] : ( P2 @ X7 ) )
% 0.30/0.49 = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 0.30/0.49 ( ( P3 @ none_P5556105721700978146at_nat )
% 0.30/0.49 & ! [X2: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % split_option_all
% 0.30/0.49 thf(fact_351_split__option__all,axiom,
% 0.30/0.49 ( ( ^ [P2: option_num > $o] :
% 0.30/0.49 ! [X7: option_num] : ( P2 @ X7 ) )
% 0.30/0.49 = ( ^ [P3: option_num > $o] :
% 0.30/0.49 ( ( P3 @ none_num )
% 0.30/0.49 & ! [X2: num] : ( P3 @ ( some_num @ X2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % split_option_all
% 0.30/0.49 thf(fact_352_split__option__ex,axiom,
% 0.30/0.49 ( ( ^ [P2: option_nat > $o] :
% 0.30/0.49 ? [X7: option_nat] : ( P2 @ X7 ) )
% 0.30/0.49 = ( ^ [P3: option_nat > $o] :
% 0.30/0.49 ( ( P3 @ none_nat )
% 0.30/0.49 | ? [X2: nat] : ( P3 @ ( some_nat @ X2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % split_option_ex
% 0.30/0.49 thf(fact_353_split__option__ex,axiom,
% 0.30/0.49 ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 0.30/0.49 ? [X7: option4927543243414619207at_nat] : ( P2 @ X7 ) )
% 0.30/0.49 = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 0.30/0.49 ( ( P3 @ none_P5556105721700978146at_nat )
% 0.30/0.49 | ? [X2: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % split_option_ex
% 0.30/0.49 thf(fact_354_split__option__ex,axiom,
% 0.30/0.49 ( ( ^ [P2: option_num > $o] :
% 0.30/0.49 ? [X7: option_num] : ( P2 @ X7 ) )
% 0.30/0.49 = ( ^ [P3: option_num > $o] :
% 0.30/0.49 ( ( P3 @ none_num )
% 0.30/0.49 | ? [X2: num] : ( P3 @ ( some_num @ X2 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % split_option_ex
% 0.30/0.49 thf(fact_355_option_Oexhaust,axiom,
% 0.30/0.49 ! [Y: option_nat] :
% 0.30/0.49 ( ( Y != none_nat )
% 0.30/0.49 => ~ ! [X23: nat] :
% 0.30/0.49 ( Y
% 0.30/0.49 != ( some_nat @ X23 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % option.exhaust
% 0.30/0.49 thf(fact_356_option_Oexhaust,axiom,
% 0.30/0.49 ! [Y: option4927543243414619207at_nat] :
% 0.30/0.49 ( ( Y != none_P5556105721700978146at_nat )
% 0.30/0.49 => ~ ! [X23: product_prod_nat_nat] :
% 0.30/0.49 ( Y
% 0.30/0.49 != ( some_P7363390416028606310at_nat @ X23 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % option.exhaust
% 0.30/0.49 thf(fact_357_option_Oexhaust,axiom,
% 0.30/0.49 ! [Y: option_num] :
% 0.30/0.49 ( ( Y != none_num )
% 0.30/0.49 => ~ ! [X23: num] :
% 0.30/0.49 ( Y
% 0.30/0.49 != ( some_num @ X23 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % option.exhaust
% 0.30/0.49 thf(fact_358_option_OdiscI,axiom,
% 0.30/0.49 ! [Option: option_nat,X22: nat] :
% 0.30/0.49 ( ( Option
% 0.30/0.49 = ( some_nat @ X22 ) )
% 0.30/0.49 => ( Option != none_nat ) ) ).
% 0.30/0.49
% 0.30/0.49 % option.discI
% 0.30/0.49 thf(fact_359_option_OdiscI,axiom,
% 0.30/0.49 ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
% 0.30/0.49 ( ( Option
% 0.30/0.49 = ( some_P7363390416028606310at_nat @ X22 ) )
% 0.30/0.49 => ( Option != none_P5556105721700978146at_nat ) ) ).
% 0.30/0.49
% 0.30/0.49 % option.discI
% 0.30/0.49 thf(fact_360_option_OdiscI,axiom,
% 0.30/0.49 ! [Option: option_num,X22: num] :
% 0.30/0.49 ( ( Option
% 0.30/0.49 = ( some_num @ X22 ) )
% 0.30/0.49 => ( Option != none_num ) ) ).
% 0.30/0.49
% 0.30/0.49 % option.discI
% 0.30/0.49 thf(fact_361_option_Odistinct_I1_J,axiom,
% 0.30/0.49 ! [X22: nat] :
% 0.30/0.49 ( none_nat
% 0.30/0.49 != ( some_nat @ X22 ) ) ).
% 0.30/0.49
% 0.30/0.49 % option.distinct(1)
% 0.30/0.49 thf(fact_362_option_Odistinct_I1_J,axiom,
% 0.30/0.49 ! [X22: product_prod_nat_nat] :
% 0.30/0.49 ( none_P5556105721700978146at_nat
% 0.30/0.49 != ( some_P7363390416028606310at_nat @ X22 ) ) ).
% 0.30/0.49
% 0.30/0.49 % option.distinct(1)
% 0.30/0.49 thf(fact_363_option_Odistinct_I1_J,axiom,
% 0.30/0.49 ! [X22: num] :
% 0.30/0.49 ( none_num
% 0.30/0.49 != ( some_num @ X22 ) ) ).
% 0.30/0.49
% 0.30/0.49 % option.distinct(1)
% 0.30/0.49 thf(fact_364_option_Osel,axiom,
% 0.30/0.49 ! [X22: nat] :
% 0.30/0.49 ( ( the_nat @ ( some_nat @ X22 ) )
% 0.30/0.49 = X22 ) ).
% 0.30/0.49
% 0.30/0.49 % option.sel
% 0.30/0.49 thf(fact_365_option_Osel,axiom,
% 0.30/0.49 ! [X22: product_prod_nat_nat] :
% 0.30/0.49 ( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
% 0.30/0.49 = X22 ) ).
% 0.30/0.49
% 0.30/0.49 % option.sel
% 0.30/0.49 thf(fact_366_option_Osel,axiom,
% 0.30/0.49 ! [X22: num] :
% 0.30/0.49 ( ( the_num @ ( some_num @ X22 ) )
% 0.30/0.49 = X22 ) ).
% 0.30/0.49
% 0.30/0.49 % option.sel
% 0.30/0.49 thf(fact_367_option_Oexpand,axiom,
% 0.30/0.49 ! [Option: option_nat,Option2: option_nat] :
% 0.30/0.49 ( ( ( Option = none_nat )
% 0.30/0.49 = ( Option2 = none_nat ) )
% 0.30/0.49 => ( ( ( Option != none_nat )
% 0.30/0.49 => ( ( Option2 != none_nat )
% 0.30/0.49 => ( ( the_nat @ Option )
% 0.30/0.49 = ( the_nat @ Option2 ) ) ) )
% 0.30/0.49 => ( Option = Option2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % option.expand
% 0.30/0.49 thf(fact_368_option_Oexpand,axiom,
% 0.30/0.49 ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
% 0.30/0.49 ( ( ( Option = none_P5556105721700978146at_nat )
% 0.30/0.49 = ( Option2 = none_P5556105721700978146at_nat ) )
% 0.30/0.49 => ( ( ( Option != none_P5556105721700978146at_nat )
% 0.30/0.49 => ( ( Option2 != none_P5556105721700978146at_nat )
% 0.30/0.49 => ( ( the_Pr8591224930841456533at_nat @ Option )
% 0.30/0.49 = ( the_Pr8591224930841456533at_nat @ Option2 ) ) ) )
% 0.30/0.49 => ( Option = Option2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % option.expand
% 0.30/0.49 thf(fact_369_option_Oexpand,axiom,
% 0.30/0.49 ! [Option: option_num,Option2: option_num] :
% 0.30/0.49 ( ( ( Option = none_num )
% 0.30/0.49 = ( Option2 = none_num ) )
% 0.30/0.49 => ( ( ( Option != none_num )
% 0.30/0.49 => ( ( Option2 != none_num )
% 0.30/0.49 => ( ( the_num @ Option )
% 0.30/0.49 = ( the_num @ Option2 ) ) ) )
% 0.30/0.49 => ( Option = Option2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % option.expand
% 0.30/0.49 thf(fact_370_list__eq__iff__nth__eq,axiom,
% 0.30/0.49 ( ( ^ [Y3: list_VEBT_VEBT,Z2: list_VEBT_VEBT] : ( Y3 = Z2 ) )
% 0.30/0.49 = ( ^ [Xs2: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
% 0.30/0.49 ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 0.30/0.49 = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
% 0.30/0.49 & ! [I3: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.30/0.49 => ( ( nth_VEBT_VEBT @ Xs2 @ I3 )
% 0.30/0.49 = ( nth_VEBT_VEBT @ Ys3 @ I3 ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % list_eq_iff_nth_eq
% 0.30/0.49 thf(fact_371_list__eq__iff__nth__eq,axiom,
% 0.30/0.49 ( ( ^ [Y3: list_o,Z2: list_o] : ( Y3 = Z2 ) )
% 0.30/0.49 = ( ^ [Xs2: list_o,Ys3: list_o] :
% 0.30/0.49 ( ( ( size_size_list_o @ Xs2 )
% 0.30/0.49 = ( size_size_list_o @ Ys3 ) )
% 0.30/0.49 & ! [I3: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
% 0.30/0.49 => ( ( nth_o @ Xs2 @ I3 )
% 0.30/0.49 = ( nth_o @ Ys3 @ I3 ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % list_eq_iff_nth_eq
% 0.30/0.49 thf(fact_372_list__eq__iff__nth__eq,axiom,
% 0.30/0.49 ( ( ^ [Y3: list_int,Z2: list_int] : ( Y3 = Z2 ) )
% 0.30/0.49 = ( ^ [Xs2: list_int,Ys3: list_int] :
% 0.30/0.49 ( ( ( size_size_list_int @ Xs2 )
% 0.30/0.49 = ( size_size_list_int @ Ys3 ) )
% 0.30/0.49 & ! [I3: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
% 0.30/0.49 => ( ( nth_int @ Xs2 @ I3 )
% 0.30/0.49 = ( nth_int @ Ys3 @ I3 ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % list_eq_iff_nth_eq
% 0.30/0.49 thf(fact_373_Skolem__list__nth,axiom,
% 0.30/0.49 ! [K: nat,P: nat > vEBT_VEBT > $o] :
% 0.30/0.49 ( ( ! [I3: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I3 @ K )
% 0.30/0.49 => ? [X3: vEBT_VEBT] : ( P @ I3 @ X3 ) ) )
% 0.30/0.49 = ( ? [Xs2: list_VEBT_VEBT] :
% 0.30/0.49 ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 0.30/0.49 = K )
% 0.30/0.49 & ! [I3: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I3 @ K )
% 0.30/0.49 => ( P @ I3 @ ( nth_VEBT_VEBT @ Xs2 @ I3 ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Skolem_list_nth
% 0.30/0.49 thf(fact_374_Skolem__list__nth,axiom,
% 0.30/0.49 ! [K: nat,P: nat > $o > $o] :
% 0.30/0.49 ( ( ! [I3: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I3 @ K )
% 0.30/0.49 => ? [X3: $o] : ( P @ I3 @ X3 ) ) )
% 0.30/0.49 = ( ? [Xs2: list_o] :
% 0.30/0.49 ( ( ( size_size_list_o @ Xs2 )
% 0.30/0.49 = K )
% 0.30/0.49 & ! [I3: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I3 @ K )
% 0.30/0.49 => ( P @ I3 @ ( nth_o @ Xs2 @ I3 ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Skolem_list_nth
% 0.30/0.49 thf(fact_375_Skolem__list__nth,axiom,
% 0.30/0.49 ! [K: nat,P: nat > int > $o] :
% 0.30/0.49 ( ( ! [I3: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I3 @ K )
% 0.30/0.49 => ? [X3: int] : ( P @ I3 @ X3 ) ) )
% 0.30/0.49 = ( ? [Xs2: list_int] :
% 0.30/0.49 ( ( ( size_size_list_int @ Xs2 )
% 0.30/0.49 = K )
% 0.30/0.49 & ! [I3: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I3 @ K )
% 0.30/0.49 => ( P @ I3 @ ( nth_int @ Xs2 @ I3 ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % Skolem_list_nth
% 0.30/0.49 thf(fact_376_nth__equalityI,axiom,
% 0.30/0.49 ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 0.30/0.49 ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.30/0.49 = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 0.30/0.49 => ( ! [I4: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.30/0.49 => ( ( nth_VEBT_VEBT @ Xs @ I4 )
% 0.30/0.49 = ( nth_VEBT_VEBT @ Ys @ I4 ) ) )
% 0.30/0.49 => ( Xs = Ys ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nth_equalityI
% 0.30/0.49 thf(fact_377_nth__equalityI,axiom,
% 0.30/0.49 ! [Xs: list_o,Ys: list_o] :
% 0.30/0.49 ( ( ( size_size_list_o @ Xs )
% 0.30/0.49 = ( size_size_list_o @ Ys ) )
% 0.30/0.49 => ( ! [I4: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
% 0.30/0.49 => ( ( nth_o @ Xs @ I4 )
% 0.30/0.49 = ( nth_o @ Ys @ I4 ) ) )
% 0.30/0.49 => ( Xs = Ys ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nth_equalityI
% 0.30/0.49 thf(fact_378_nth__equalityI,axiom,
% 0.30/0.49 ! [Xs: list_int,Ys: list_int] :
% 0.30/0.49 ( ( ( size_size_list_int @ Xs )
% 0.30/0.49 = ( size_size_list_int @ Ys ) )
% 0.30/0.49 => ( ! [I4: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
% 0.30/0.49 => ( ( nth_int @ Xs @ I4 )
% 0.30/0.49 = ( nth_int @ Ys @ I4 ) ) )
% 0.30/0.49 => ( Xs = Ys ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nth_equalityI
% 0.30/0.49 thf(fact_379_option_Oexhaust__sel,axiom,
% 0.30/0.49 ! [Option: option_nat] :
% 0.30/0.49 ( ( Option != none_nat )
% 0.30/0.49 => ( Option
% 0.30/0.49 = ( some_nat @ ( the_nat @ Option ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % option.exhaust_sel
% 0.30/0.49 thf(fact_380_option_Oexhaust__sel,axiom,
% 0.30/0.49 ! [Option: option4927543243414619207at_nat] :
% 0.30/0.49 ( ( Option != none_P5556105721700978146at_nat )
% 0.30/0.49 => ( Option
% 0.30/0.49 = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % option.exhaust_sel
% 0.30/0.49 thf(fact_381_option_Oexhaust__sel,axiom,
% 0.30/0.49 ! [Option: option_num] :
% 0.30/0.49 ( ( Option != none_num )
% 0.30/0.49 => ( Option
% 0.30/0.49 = ( some_num @ ( the_num @ Option ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % option.exhaust_sel
% 0.30/0.49 thf(fact_382_nth__list__update,axiom,
% 0.30/0.49 ! [I2: nat,Xs: list_VEBT_VEBT,J: nat,X: vEBT_VEBT] :
% 0.30/0.49 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.30/0.49 => ( ( ( I2 = J )
% 0.30/0.49 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X ) @ J )
% 0.30/0.49 = X ) )
% 0.30/0.49 & ( ( I2 != J )
% 0.30/0.49 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X ) @ J )
% 0.30/0.49 = ( nth_VEBT_VEBT @ Xs @ J ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nth_list_update
% 0.30/0.49 thf(fact_383_nth__list__update,axiom,
% 0.30/0.49 ! [I2: nat,Xs: list_o,X: $o,J: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 0.30/0.49 => ( ( nth_o @ ( list_update_o @ Xs @ I2 @ X ) @ J )
% 0.30/0.49 = ( ( ( I2 = J )
% 0.30/0.49 => X )
% 0.30/0.49 & ( ( I2 != J )
% 0.30/0.49 => ( nth_o @ Xs @ J ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nth_list_update
% 0.30/0.49 thf(fact_384_nth__list__update,axiom,
% 0.30/0.49 ! [I2: nat,Xs: list_int,J: nat,X: int] :
% 0.30/0.49 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 0.30/0.49 => ( ( ( I2 = J )
% 0.30/0.49 => ( ( nth_int @ ( list_update_int @ Xs @ I2 @ X ) @ J )
% 0.30/0.49 = X ) )
% 0.30/0.49 & ( ( I2 != J )
% 0.30/0.49 => ( ( nth_int @ ( list_update_int @ Xs @ I2 @ X ) @ J )
% 0.30/0.49 = ( nth_int @ Xs @ J ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nth_list_update
% 0.30/0.49 thf(fact_385_list__update__same__conv,axiom,
% 0.30/0.49 ! [I2: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
% 0.30/0.49 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.30/0.49 => ( ( ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X )
% 0.30/0.49 = Xs )
% 0.30/0.49 = ( ( nth_VEBT_VEBT @ Xs @ I2 )
% 0.30/0.49 = X ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % list_update_same_conv
% 0.30/0.49 thf(fact_386_list__update__same__conv,axiom,
% 0.30/0.49 ! [I2: nat,Xs: list_o,X: $o] :
% 0.30/0.49 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 0.30/0.49 => ( ( ( list_update_o @ Xs @ I2 @ X )
% 0.30/0.49 = Xs )
% 0.30/0.49 = ( ( nth_o @ Xs @ I2 )
% 0.30/0.49 = X ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % list_update_same_conv
% 0.30/0.49 thf(fact_387_list__update__same__conv,axiom,
% 0.30/0.49 ! [I2: nat,Xs: list_int,X: int] :
% 0.30/0.49 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 0.30/0.49 => ( ( ( list_update_int @ Xs @ I2 @ X )
% 0.30/0.49 = Xs )
% 0.30/0.49 = ( ( nth_int @ Xs @ I2 )
% 0.30/0.49 = X ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % list_update_same_conv
% 0.30/0.49 thf(fact_388_field__sum__of__halves,axiom,
% 0.30/0.49 ! [X: real] :
% 0.30/0.49 ( ( plus_plus_real @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.49 = X ) ).
% 0.30/0.49
% 0.30/0.49 % field_sum_of_halves
% 0.30/0.49 thf(fact_389_field__sum__of__halves,axiom,
% 0.30/0.49 ! [X: rat] :
% 0.30/0.49 ( ( plus_plus_rat @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.30/0.49 = X ) ).
% 0.30/0.49
% 0.30/0.49 % field_sum_of_halves
% 0.30/0.49 thf(fact_390_post__member__pre__member,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,N2: nat,X: nat,Y: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.49 => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.49 => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.49 => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X ) @ Y )
% 0.30/0.49 => ( ( vEBT_vebt_member @ T @ Y )
% 0.30/0.49 | ( X = Y ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % post_member_pre_member
% 0.30/0.49 thf(fact_391_valid__insert__both__member__options__add,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,N2: nat,X: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.49 => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.49 => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X ) @ X ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % valid_insert_both_member_options_add
% 0.30/0.49 thf(fact_392_valid__insert__both__member__options__pres,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,N2: nat,X: nat,Y: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.49 => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.49 => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.49 => ( ( vEBT_V8194947554948674370ptions @ T @ X )
% 0.30/0.49 => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y ) @ X ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % valid_insert_both_member_options_pres
% 0.30/0.49 thf(fact_393_helpyd,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,N2: nat,X: nat,Y: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.49 => ( ( ( vEBT_vebt_succ @ T @ X )
% 0.30/0.49 = ( some_nat @ Y ) )
% 0.30/0.49 => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % helpyd
% 0.30/0.49 thf(fact_394_helpypredd,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,N2: nat,X: nat,Y: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.49 => ( ( ( vEBT_vebt_pred @ T @ X )
% 0.30/0.49 = ( some_nat @ Y ) )
% 0.30/0.49 => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % helpypredd
% 0.30/0.49 thf(fact_395_semiring__norm_I76_J,axiom,
% 0.30/0.49 ! [N2: num] : ( ord_less_num @ one @ ( bit0 @ N2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % semiring_norm(76)
% 0.30/0.49 thf(fact_396_both__member__options__ding,axiom,
% 0.30/0.49 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat,X: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 )
% 0.30/0.49 => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.30/0.49 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.49 => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % both_member_options_ding
% 0.30/0.49 thf(fact_397_sum__squares__bound,axiom,
% 0.30/0.49 ! [X: real,Y: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % sum_squares_bound
% 0.30/0.49 thf(fact_398_sum__squares__bound,axiom,
% 0.30/0.49 ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % sum_squares_bound
% 0.30/0.49 thf(fact_399_semiring__norm_I2_J,axiom,
% 0.30/0.49 ( ( plus_plus_num @ one @ one )
% 0.30/0.49 = ( bit0 @ one ) ) ).
% 0.30/0.49
% 0.30/0.49 % semiring_norm(2)
% 0.30/0.49 thf(fact_400_set__n__deg__not__0,axiom,
% 0.30/0.49 ! [TreeList2: list_VEBT_VEBT,N2: nat,M: nat] :
% 0.30/0.49 ( ! [X4: vEBT_VEBT] :
% 0.30/0.49 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.49 => ( vEBT_invar_vebt @ X4 @ N2 ) )
% 0.30/0.49 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.30/0.49 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.49 => ( ord_less_eq_nat @ one_one_nat @ N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % set_n_deg_not_0
% 0.30/0.49 thf(fact_401_semiring__norm_I75_J,axiom,
% 0.30/0.49 ! [M: num] :
% 0.30/0.49 ~ ( ord_less_num @ M @ one ) ).
% 0.30/0.49
% 0.30/0.49 % semiring_norm(75)
% 0.30/0.49 thf(fact_402_semiring__norm_I78_J,axiom,
% 0.30/0.49 ! [M: num,N2: num] :
% 0.30/0.49 ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
% 0.30/0.49 = ( ord_less_num @ M @ N2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % semiring_norm(78)
% 0.30/0.49 thf(fact_403_deg__deg__n,axiom,
% 0.30/0.49 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 )
% 0.30/0.49 => ( Deg = N2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % deg_deg_n
% 0.30/0.49 thf(fact_404_semiring__norm_I87_J,axiom,
% 0.30/0.49 ! [M: num,N2: num] :
% 0.30/0.49 ( ( ( bit0 @ M )
% 0.30/0.49 = ( bit0 @ N2 ) )
% 0.30/0.49 = ( M = N2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % semiring_norm(87)
% 0.30/0.49 thf(fact_405_semiring__norm_I83_J,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( one
% 0.30/0.49 != ( bit0 @ N2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % semiring_norm(83)
% 0.30/0.49 thf(fact_406_semiring__norm_I85_J,axiom,
% 0.30/0.49 ! [M: num] :
% 0.30/0.49 ( ( bit0 @ M )
% 0.30/0.49 != one ) ).
% 0.30/0.49
% 0.30/0.49 % semiring_norm(85)
% 0.30/0.49 thf(fact_407_bits__div__by__1,axiom,
% 0.30/0.49 ! [A: nat] :
% 0.30/0.49 ( ( divide_divide_nat @ A @ one_one_nat )
% 0.30/0.49 = A ) ).
% 0.30/0.49
% 0.30/0.49 % bits_div_by_1
% 0.30/0.49 thf(fact_408_bits__div__by__1,axiom,
% 0.30/0.49 ! [A: int] :
% 0.30/0.49 ( ( divide_divide_int @ A @ one_one_int )
% 0.30/0.49 = A ) ).
% 0.30/0.49
% 0.30/0.49 % bits_div_by_1
% 0.30/0.49 thf(fact_409_power__one,axiom,
% 0.30/0.49 ! [N2: nat] :
% 0.30/0.49 ( ( power_power_rat @ one_one_rat @ N2 )
% 0.30/0.49 = one_one_rat ) ).
% 0.30/0.49
% 0.30/0.49 % power_one
% 0.30/0.49 thf(fact_410_power__one,axiom,
% 0.30/0.49 ! [N2: nat] :
% 0.30/0.49 ( ( power_power_nat @ one_one_nat @ N2 )
% 0.30/0.49 = one_one_nat ) ).
% 0.30/0.49
% 0.30/0.49 % power_one
% 0.30/0.49 thf(fact_411_power__one,axiom,
% 0.30/0.49 ! [N2: nat] :
% 0.30/0.49 ( ( power_power_real @ one_one_real @ N2 )
% 0.30/0.49 = one_one_real ) ).
% 0.30/0.49
% 0.30/0.49 % power_one
% 0.30/0.49 thf(fact_412_power__one,axiom,
% 0.30/0.49 ! [N2: nat] :
% 0.30/0.49 ( ( power_power_int @ one_one_int @ N2 )
% 0.30/0.49 = one_one_int ) ).
% 0.30/0.49
% 0.30/0.49 % power_one
% 0.30/0.49 thf(fact_413_power__one,axiom,
% 0.30/0.49 ! [N2: nat] :
% 0.30/0.49 ( ( power_power_complex @ one_one_complex @ N2 )
% 0.30/0.49 = one_one_complex ) ).
% 0.30/0.49
% 0.30/0.49 % power_one
% 0.30/0.49 thf(fact_414_power__one__right,axiom,
% 0.30/0.49 ! [A: nat] :
% 0.30/0.49 ( ( power_power_nat @ A @ one_one_nat )
% 0.30/0.49 = A ) ).
% 0.30/0.49
% 0.30/0.49 % power_one_right
% 0.30/0.49 thf(fact_415_power__one__right,axiom,
% 0.30/0.49 ! [A: real] :
% 0.30/0.49 ( ( power_power_real @ A @ one_one_nat )
% 0.30/0.49 = A ) ).
% 0.30/0.49
% 0.30/0.49 % power_one_right
% 0.30/0.49 thf(fact_416_power__one__right,axiom,
% 0.30/0.49 ! [A: int] :
% 0.30/0.49 ( ( power_power_int @ A @ one_one_nat )
% 0.30/0.49 = A ) ).
% 0.30/0.49
% 0.30/0.49 % power_one_right
% 0.30/0.49 thf(fact_417_power__one__right,axiom,
% 0.30/0.49 ! [A: complex] :
% 0.30/0.49 ( ( power_power_complex @ A @ one_one_nat )
% 0.30/0.49 = A ) ).
% 0.30/0.49
% 0.30/0.49 % power_one_right
% 0.30/0.49 thf(fact_418_semiring__norm_I6_J,axiom,
% 0.30/0.49 ! [M: num,N2: num] :
% 0.30/0.49 ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
% 0.30/0.49 = ( bit0 @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % semiring_norm(6)
% 0.30/0.49 thf(fact_419_semiring__norm_I13_J,axiom,
% 0.30/0.49 ! [M: num,N2: num] :
% 0.30/0.49 ( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
% 0.30/0.49 = ( bit0 @ ( bit0 @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % semiring_norm(13)
% 0.30/0.49 thf(fact_420_semiring__norm_I11_J,axiom,
% 0.30/0.49 ! [M: num] :
% 0.30/0.49 ( ( times_times_num @ M @ one )
% 0.30/0.49 = M ) ).
% 0.30/0.49
% 0.30/0.49 % semiring_norm(11)
% 0.30/0.49 thf(fact_421_semiring__norm_I12_J,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( times_times_num @ one @ N2 )
% 0.30/0.49 = N2 ) ).
% 0.30/0.49
% 0.30/0.49 % semiring_norm(12)
% 0.30/0.49 thf(fact_422_semiring__norm_I71_J,axiom,
% 0.30/0.49 ! [M: num,N2: num] :
% 0.30/0.49 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
% 0.30/0.49 = ( ord_less_eq_num @ M @ N2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % semiring_norm(71)
% 0.30/0.49 thf(fact_423_semiring__norm_I68_J,axiom,
% 0.30/0.49 ! [N2: num] : ( ord_less_eq_num @ one @ N2 ) ).
% 0.30/0.49
% 0.30/0.49 % semiring_norm(68)
% 0.30/0.49 thf(fact_424_one__eq__numeral__iff,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( one_one_complex
% 0.30/0.49 = ( numera6690914467698888265omplex @ N2 ) )
% 0.30/0.49 = ( one = N2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_eq_numeral_iff
% 0.30/0.49 thf(fact_425_one__eq__numeral__iff,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( one_one_real
% 0.30/0.49 = ( numeral_numeral_real @ N2 ) )
% 0.30/0.49 = ( one = N2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_eq_numeral_iff
% 0.30/0.49 thf(fact_426_one__eq__numeral__iff,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( one_one_rat
% 0.30/0.49 = ( numeral_numeral_rat @ N2 ) )
% 0.30/0.49 = ( one = N2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_eq_numeral_iff
% 0.30/0.49 thf(fact_427_one__eq__numeral__iff,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( one_one_nat
% 0.30/0.49 = ( numeral_numeral_nat @ N2 ) )
% 0.30/0.49 = ( one = N2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_eq_numeral_iff
% 0.30/0.49 thf(fact_428_one__eq__numeral__iff,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( one_one_int
% 0.30/0.49 = ( numeral_numeral_int @ N2 ) )
% 0.30/0.49 = ( one = N2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_eq_numeral_iff
% 0.30/0.49 thf(fact_429_numeral__eq__one__iff,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( ( numera6690914467698888265omplex @ N2 )
% 0.30/0.49 = one_one_complex )
% 0.30/0.49 = ( N2 = one ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_eq_one_iff
% 0.30/0.49 thf(fact_430_numeral__eq__one__iff,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( ( numeral_numeral_real @ N2 )
% 0.30/0.49 = one_one_real )
% 0.30/0.49 = ( N2 = one ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_eq_one_iff
% 0.30/0.49 thf(fact_431_numeral__eq__one__iff,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( ( numeral_numeral_rat @ N2 )
% 0.30/0.49 = one_one_rat )
% 0.30/0.49 = ( N2 = one ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_eq_one_iff
% 0.30/0.49 thf(fact_432_numeral__eq__one__iff,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( ( numeral_numeral_nat @ N2 )
% 0.30/0.49 = one_one_nat )
% 0.30/0.49 = ( N2 = one ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_eq_one_iff
% 0.30/0.49 thf(fact_433_numeral__eq__one__iff,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( ( numeral_numeral_int @ N2 )
% 0.30/0.49 = one_one_int )
% 0.30/0.49 = ( N2 = one ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_eq_one_iff
% 0.30/0.49 thf(fact_434_power__inject__exp,axiom,
% 0.30/0.49 ! [A: real,M: nat,N2: nat] :
% 0.30/0.49 ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.49 => ( ( ( power_power_real @ A @ M )
% 0.30/0.49 = ( power_power_real @ A @ N2 ) )
% 0.30/0.49 = ( M = N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_inject_exp
% 0.30/0.49 thf(fact_435_power__inject__exp,axiom,
% 0.30/0.49 ! [A: rat,M: nat,N2: nat] :
% 0.30/0.49 ( ( ord_less_rat @ one_one_rat @ A )
% 0.30/0.49 => ( ( ( power_power_rat @ A @ M )
% 0.30/0.49 = ( power_power_rat @ A @ N2 ) )
% 0.30/0.49 = ( M = N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_inject_exp
% 0.30/0.49 thf(fact_436_power__inject__exp,axiom,
% 0.30/0.49 ! [A: nat,M: nat,N2: nat] :
% 0.30/0.49 ( ( ord_less_nat @ one_one_nat @ A )
% 0.30/0.49 => ( ( ( power_power_nat @ A @ M )
% 0.30/0.49 = ( power_power_nat @ A @ N2 ) )
% 0.30/0.49 = ( M = N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_inject_exp
% 0.30/0.49 thf(fact_437_power__inject__exp,axiom,
% 0.30/0.49 ! [A: int,M: nat,N2: nat] :
% 0.30/0.49 ( ( ord_less_int @ one_one_int @ A )
% 0.30/0.49 => ( ( ( power_power_int @ A @ M )
% 0.30/0.49 = ( power_power_int @ A @ N2 ) )
% 0.30/0.49 = ( M = N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_inject_exp
% 0.30/0.49 thf(fact_438_semiring__norm_I69_J,axiom,
% 0.30/0.49 ! [M: num] :
% 0.30/0.49 ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% 0.30/0.49
% 0.30/0.49 % semiring_norm(69)
% 0.30/0.49 thf(fact_439_power__strict__increasing__iff,axiom,
% 0.30/0.49 ! [B: real,X: nat,Y: nat] :
% 0.30/0.49 ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.49 => ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
% 0.30/0.49 = ( ord_less_nat @ X @ Y ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_strict_increasing_iff
% 0.30/0.49 thf(fact_440_power__strict__increasing__iff,axiom,
% 0.30/0.49 ! [B: rat,X: nat,Y: nat] :
% 0.30/0.49 ( ( ord_less_rat @ one_one_rat @ B )
% 0.30/0.49 => ( ( ord_less_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
% 0.30/0.49 = ( ord_less_nat @ X @ Y ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_strict_increasing_iff
% 0.30/0.49 thf(fact_441_power__strict__increasing__iff,axiom,
% 0.30/0.49 ! [B: nat,X: nat,Y: nat] :
% 0.30/0.49 ( ( ord_less_nat @ one_one_nat @ B )
% 0.30/0.49 => ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
% 0.30/0.49 = ( ord_less_nat @ X @ Y ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_strict_increasing_iff
% 0.30/0.49 thf(fact_442_power__strict__increasing__iff,axiom,
% 0.30/0.49 ! [B: int,X: nat,Y: nat] :
% 0.30/0.49 ( ( ord_less_int @ one_one_int @ B )
% 0.30/0.49 => ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
% 0.30/0.49 = ( ord_less_nat @ X @ Y ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_strict_increasing_iff
% 0.30/0.49 thf(fact_443_one__add__one,axiom,
% 0.30/0.49 ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
% 0.30/0.49 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_add_one
% 0.30/0.49 thf(fact_444_one__add__one,axiom,
% 0.30/0.49 ( ( plus_plus_real @ one_one_real @ one_one_real )
% 0.30/0.49 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_add_one
% 0.30/0.49 thf(fact_445_one__add__one,axiom,
% 0.30/0.49 ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
% 0.30/0.49 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_add_one
% 0.30/0.49 thf(fact_446_one__add__one,axiom,
% 0.30/0.49 ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 0.30/0.49 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_add_one
% 0.30/0.49 thf(fact_447_one__add__one,axiom,
% 0.30/0.49 ( ( plus_plus_int @ one_one_int @ one_one_int )
% 0.30/0.49 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_add_one
% 0.30/0.49 thf(fact_448_power__increasing__iff,axiom,
% 0.30/0.49 ! [B: real,X: nat,Y: nat] :
% 0.30/0.49 ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.49 => ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
% 0.30/0.49 = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_increasing_iff
% 0.30/0.49 thf(fact_449_power__increasing__iff,axiom,
% 0.30/0.49 ! [B: rat,X: nat,Y: nat] :
% 0.30/0.49 ( ( ord_less_rat @ one_one_rat @ B )
% 0.30/0.49 => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
% 0.30/0.49 = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_increasing_iff
% 0.30/0.49 thf(fact_450_power__increasing__iff,axiom,
% 0.30/0.49 ! [B: nat,X: nat,Y: nat] :
% 0.30/0.49 ( ( ord_less_nat @ one_one_nat @ B )
% 0.30/0.49 => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
% 0.30/0.49 = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_increasing_iff
% 0.30/0.49 thf(fact_451_power__increasing__iff,axiom,
% 0.30/0.49 ! [B: int,X: nat,Y: nat] :
% 0.30/0.49 ( ( ord_less_int @ one_one_int @ B )
% 0.30/0.49 => ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
% 0.30/0.49 = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_increasing_iff
% 0.30/0.49 thf(fact_452_one__plus__numeral,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N2 ) )
% 0.30/0.49 = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_plus_numeral
% 0.30/0.49 thf(fact_453_one__plus__numeral,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N2 ) )
% 0.30/0.49 = ( numeral_numeral_real @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_plus_numeral
% 0.30/0.49 thf(fact_454_one__plus__numeral,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) )
% 0.30/0.49 = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_plus_numeral
% 0.30/0.49 thf(fact_455_one__plus__numeral,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
% 0.30/0.49 = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_plus_numeral
% 0.30/0.49 thf(fact_456_one__plus__numeral,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
% 0.30/0.49 = ( numeral_numeral_int @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_plus_numeral
% 0.30/0.49 thf(fact_457_numeral__plus__one,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ one_one_complex )
% 0.30/0.49 = ( numera6690914467698888265omplex @ ( plus_plus_num @ N2 @ one ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_plus_one
% 0.30/0.49 thf(fact_458_numeral__plus__one,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ one_one_real )
% 0.30/0.49 = ( numeral_numeral_real @ ( plus_plus_num @ N2 @ one ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_plus_one
% 0.30/0.49 thf(fact_459_numeral__plus__one,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat )
% 0.30/0.49 = ( numeral_numeral_rat @ ( plus_plus_num @ N2 @ one ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_plus_one
% 0.30/0.49 thf(fact_460_numeral__plus__one,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
% 0.30/0.49 = ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_plus_one
% 0.30/0.49 thf(fact_461_numeral__plus__one,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
% 0.30/0.49 = ( numeral_numeral_int @ ( plus_plus_num @ N2 @ one ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_plus_one
% 0.30/0.49 thf(fact_462_numeral__le__one__iff,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ one_one_real )
% 0.30/0.49 = ( ord_less_eq_num @ N2 @ one ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_le_one_iff
% 0.30/0.49 thf(fact_463_numeral__le__one__iff,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat )
% 0.30/0.49 = ( ord_less_eq_num @ N2 @ one ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_le_one_iff
% 0.30/0.49 thf(fact_464_numeral__le__one__iff,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
% 0.30/0.49 = ( ord_less_eq_num @ N2 @ one ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_le_one_iff
% 0.30/0.49 thf(fact_465_numeral__le__one__iff,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
% 0.30/0.49 = ( ord_less_eq_num @ N2 @ one ) ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_le_one_iff
% 0.30/0.49 thf(fact_466_one__less__numeral__iff,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N2 ) )
% 0.30/0.49 = ( ord_less_num @ one @ N2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_less_numeral_iff
% 0.30/0.49 thf(fact_467_one__less__numeral__iff,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) )
% 0.30/0.49 = ( ord_less_num @ one @ N2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_less_numeral_iff
% 0.30/0.49 thf(fact_468_one__less__numeral__iff,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
% 0.30/0.49 = ( ord_less_num @ one @ N2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_less_numeral_iff
% 0.30/0.49 thf(fact_469_one__less__numeral__iff,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
% 0.30/0.49 = ( ord_less_num @ one @ N2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_less_numeral_iff
% 0.30/0.49 thf(fact_470_pred__correct,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,N2: nat,X: nat,Sx: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.49 => ( ( ( vEBT_vebt_pred @ T @ X )
% 0.30/0.49 = ( some_nat @ Sx ) )
% 0.30/0.49 = ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % pred_correct
% 0.30/0.49 thf(fact_471_succ__correct,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,N2: nat,X: nat,Sx: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.49 => ( ( ( vEBT_vebt_succ @ T @ X )
% 0.30/0.49 = ( some_nat @ Sx ) )
% 0.30/0.49 = ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % succ_correct
% 0.30/0.49 thf(fact_472_le__num__One__iff,axiom,
% 0.30/0.49 ! [X: num] :
% 0.30/0.49 ( ( ord_less_eq_num @ X @ one )
% 0.30/0.49 = ( X = one ) ) ).
% 0.30/0.49
% 0.30/0.49 % le_num_One_iff
% 0.30/0.49 thf(fact_473_le__numeral__extra_I4_J,axiom,
% 0.30/0.49 ord_less_eq_real @ one_one_real @ one_one_real ).
% 0.30/0.49
% 0.30/0.49 % le_numeral_extra(4)
% 0.30/0.49 thf(fact_474_le__numeral__extra_I4_J,axiom,
% 0.30/0.49 ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% 0.30/0.49
% 0.30/0.49 % le_numeral_extra(4)
% 0.30/0.49 thf(fact_475_le__numeral__extra_I4_J,axiom,
% 0.30/0.49 ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 0.30/0.49
% 0.30/0.49 % le_numeral_extra(4)
% 0.30/0.49 thf(fact_476_le__numeral__extra_I4_J,axiom,
% 0.30/0.49 ord_less_eq_int @ one_one_int @ one_one_int ).
% 0.30/0.49
% 0.30/0.49 % le_numeral_extra(4)
% 0.30/0.49 thf(fact_477_less__numeral__extra_I4_J,axiom,
% 0.30/0.49 ~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% 0.30/0.49
% 0.30/0.49 % less_numeral_extra(4)
% 0.30/0.49 thf(fact_478_less__numeral__extra_I4_J,axiom,
% 0.30/0.49 ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% 0.30/0.49
% 0.30/0.49 % less_numeral_extra(4)
% 0.30/0.49 thf(fact_479_less__numeral__extra_I4_J,axiom,
% 0.30/0.49 ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 0.30/0.49
% 0.30/0.49 % less_numeral_extra(4)
% 0.30/0.49 thf(fact_480_less__numeral__extra_I4_J,axiom,
% 0.30/0.49 ~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% 0.30/0.49
% 0.30/0.49 % less_numeral_extra(4)
% 0.30/0.49 thf(fact_481_subset__code_I1_J,axiom,
% 0.30/0.49 ! [Xs: list_real,B3: set_real] :
% 0.30/0.49 ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B3 )
% 0.30/0.49 = ( ! [X2: real] :
% 0.30/0.49 ( ( member_real @ X2 @ ( set_real2 @ Xs ) )
% 0.30/0.49 => ( member_real @ X2 @ B3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % subset_code(1)
% 0.30/0.49 thf(fact_482_subset__code_I1_J,axiom,
% 0.30/0.49 ! [Xs: list_complex,B3: set_complex] :
% 0.30/0.49 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ B3 )
% 0.30/0.49 = ( ! [X2: complex] :
% 0.30/0.49 ( ( member_complex @ X2 @ ( set_complex2 @ Xs ) )
% 0.30/0.49 => ( member_complex @ X2 @ B3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % subset_code(1)
% 0.30/0.49 thf(fact_483_subset__code_I1_J,axiom,
% 0.30/0.49 ! [Xs: list_set_nat,B3: set_set_nat] :
% 0.30/0.49 ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B3 )
% 0.30/0.49 = ( ! [X2: set_nat] :
% 0.30/0.49 ( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
% 0.30/0.49 => ( member_set_nat @ X2 @ B3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % subset_code(1)
% 0.30/0.49 thf(fact_484_subset__code_I1_J,axiom,
% 0.30/0.49 ! [Xs: list_VEBT_VEBT,B3: set_VEBT_VEBT] :
% 0.30/0.49 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ B3 )
% 0.30/0.49 = ( ! [X2: vEBT_VEBT] :
% 0.30/0.49 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.30/0.49 => ( member_VEBT_VEBT @ X2 @ B3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % subset_code(1)
% 0.30/0.49 thf(fact_485_subset__code_I1_J,axiom,
% 0.30/0.49 ! [Xs: list_int,B3: set_int] :
% 0.30/0.49 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B3 )
% 0.30/0.49 = ( ! [X2: int] :
% 0.30/0.49 ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
% 0.30/0.49 => ( member_int @ X2 @ B3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % subset_code(1)
% 0.30/0.49 thf(fact_486_subset__code_I1_J,axiom,
% 0.30/0.49 ! [Xs: list_nat,B3: set_nat] :
% 0.30/0.49 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B3 )
% 0.30/0.49 = ( ! [X2: nat] :
% 0.30/0.49 ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
% 0.30/0.49 => ( member_nat @ X2 @ B3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % subset_code(1)
% 0.30/0.49 thf(fact_487_vebt__delete_Osimps_I4_J,axiom,
% 0.30/0.49 ! [Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] :
% 0.30/0.49 ( ( vEBT_vebt_delete @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Uu )
% 0.30/0.49 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) ) ).
% 0.30/0.49
% 0.30/0.49 % vebt_delete.simps(4)
% 0.30/0.49 thf(fact_488_L2__set__mult__ineq__lemma,axiom,
% 0.30/0.49 ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C ) ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % L2_set_mult_ineq_lemma
% 0.30/0.49 thf(fact_489_one__le__numeral,axiom,
% 0.30/0.49 ! [N2: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_le_numeral
% 0.30/0.49 thf(fact_490_one__le__numeral,axiom,
% 0.30/0.49 ! [N2: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_le_numeral
% 0.30/0.49 thf(fact_491_one__le__numeral,axiom,
% 0.30/0.49 ! [N2: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_le_numeral
% 0.30/0.49 thf(fact_492_one__le__numeral,axiom,
% 0.30/0.49 ! [N2: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_le_numeral
% 0.30/0.49 thf(fact_493_not__numeral__less__one,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ one_one_real ) ).
% 0.30/0.49
% 0.30/0.49 % not_numeral_less_one
% 0.30/0.49 thf(fact_494_not__numeral__less__one,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat ) ).
% 0.30/0.49
% 0.30/0.49 % not_numeral_less_one
% 0.30/0.49 thf(fact_495_not__numeral__less__one,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat ) ).
% 0.30/0.49
% 0.30/0.49 % not_numeral_less_one
% 0.30/0.49 thf(fact_496_not__numeral__less__one,axiom,
% 0.30/0.49 ! [N2: num] :
% 0.30/0.49 ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ).
% 0.30/0.49
% 0.30/0.49 % not_numeral_less_one
% 0.30/0.49 thf(fact_497_one__plus__numeral__commute,axiom,
% 0.30/0.49 ! [X: num] :
% 0.30/0.49 ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X ) )
% 0.30/0.49 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_plus_numeral_commute
% 0.30/0.49 thf(fact_498_one__plus__numeral__commute,axiom,
% 0.30/0.49 ! [X: num] :
% 0.30/0.49 ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X ) )
% 0.30/0.49 = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_plus_numeral_commute
% 0.30/0.49 thf(fact_499_one__plus__numeral__commute,axiom,
% 0.30/0.49 ! [X: num] :
% 0.30/0.49 ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
% 0.30/0.49 = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_plus_numeral_commute
% 0.30/0.49 thf(fact_500_one__plus__numeral__commute,axiom,
% 0.30/0.49 ! [X: num] :
% 0.30/0.49 ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
% 0.30/0.49 = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_plus_numeral_commute
% 0.30/0.49 thf(fact_501_one__plus__numeral__commute,axiom,
% 0.30/0.49 ! [X: num] :
% 0.30/0.49 ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
% 0.30/0.49 = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_plus_numeral_commute
% 0.30/0.49 thf(fact_502_numeral__One,axiom,
% 0.30/0.49 ( ( numera6690914467698888265omplex @ one )
% 0.30/0.49 = one_one_complex ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_One
% 0.30/0.49 thf(fact_503_numeral__One,axiom,
% 0.30/0.49 ( ( numeral_numeral_real @ one )
% 0.30/0.49 = one_one_real ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_One
% 0.30/0.49 thf(fact_504_numeral__One,axiom,
% 0.30/0.49 ( ( numeral_numeral_rat @ one )
% 0.30/0.49 = one_one_rat ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_One
% 0.30/0.49 thf(fact_505_numeral__One,axiom,
% 0.30/0.49 ( ( numeral_numeral_nat @ one )
% 0.30/0.49 = one_one_nat ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_One
% 0.30/0.49 thf(fact_506_numeral__One,axiom,
% 0.30/0.49 ( ( numeral_numeral_int @ one )
% 0.30/0.49 = one_one_int ) ).
% 0.30/0.49
% 0.30/0.49 % numeral_One
% 0.30/0.49 thf(fact_507_one__le__power,axiom,
% 0.30/0.49 ! [A: real,N2: nat] :
% 0.30/0.49 ( ( ord_less_eq_real @ one_one_real @ A )
% 0.30/0.49 => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_le_power
% 0.30/0.49 thf(fact_508_one__le__power,axiom,
% 0.30/0.49 ! [A: rat,N2: nat] :
% 0.30/0.49 ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.30/0.49 => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_le_power
% 0.30/0.49 thf(fact_509_one__le__power,axiom,
% 0.30/0.49 ! [A: nat,N2: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.30/0.49 => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_le_power
% 0.30/0.49 thf(fact_510_one__le__power,axiom,
% 0.30/0.49 ! [A: int,N2: nat] :
% 0.30/0.49 ( ( ord_less_eq_int @ one_one_int @ A )
% 0.30/0.49 => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % one_le_power
% 0.30/0.49 thf(fact_511_left__right__inverse__power,axiom,
% 0.30/0.49 ! [X: complex,Y: complex,N2: nat] :
% 0.30/0.49 ( ( ( times_times_complex @ X @ Y )
% 0.30/0.49 = one_one_complex )
% 0.30/0.49 => ( ( times_times_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ Y @ N2 ) )
% 0.30/0.49 = one_one_complex ) ) ).
% 0.30/0.49
% 0.30/0.49 % left_right_inverse_power
% 0.30/0.49 thf(fact_512_left__right__inverse__power,axiom,
% 0.30/0.49 ! [X: real,Y: real,N2: nat] :
% 0.30/0.49 ( ( ( times_times_real @ X @ Y )
% 0.30/0.49 = one_one_real )
% 0.30/0.49 => ( ( times_times_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ N2 ) )
% 0.30/0.49 = one_one_real ) ) ).
% 0.30/0.49
% 0.30/0.49 % left_right_inverse_power
% 0.30/0.49 thf(fact_513_left__right__inverse__power,axiom,
% 0.30/0.49 ! [X: rat,Y: rat,N2: nat] :
% 0.30/0.49 ( ( ( times_times_rat @ X @ Y )
% 0.30/0.49 = one_one_rat )
% 0.30/0.49 => ( ( times_times_rat @ ( power_power_rat @ X @ N2 ) @ ( power_power_rat @ Y @ N2 ) )
% 0.30/0.49 = one_one_rat ) ) ).
% 0.30/0.49
% 0.30/0.49 % left_right_inverse_power
% 0.30/0.49 thf(fact_514_left__right__inverse__power,axiom,
% 0.30/0.49 ! [X: nat,Y: nat,N2: nat] :
% 0.30/0.49 ( ( ( times_times_nat @ X @ Y )
% 0.30/0.49 = one_one_nat )
% 0.30/0.49 => ( ( times_times_nat @ ( power_power_nat @ X @ N2 ) @ ( power_power_nat @ Y @ N2 ) )
% 0.30/0.49 = one_one_nat ) ) ).
% 0.30/0.49
% 0.30/0.49 % left_right_inverse_power
% 0.30/0.49 thf(fact_515_left__right__inverse__power,axiom,
% 0.30/0.49 ! [X: int,Y: int,N2: nat] :
% 0.30/0.49 ( ( ( times_times_int @ X @ Y )
% 0.30/0.49 = one_one_int )
% 0.30/0.49 => ( ( times_times_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ N2 ) )
% 0.30/0.49 = one_one_int ) ) ).
% 0.30/0.49
% 0.30/0.49 % left_right_inverse_power
% 0.30/0.49 thf(fact_516_power__one__over,axiom,
% 0.30/0.49 ! [A: complex,N2: nat] :
% 0.30/0.49 ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N2 )
% 0.30/0.49 = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_one_over
% 0.30/0.49 thf(fact_517_power__one__over,axiom,
% 0.30/0.49 ! [A: real,N2: nat] :
% 0.30/0.49 ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N2 )
% 0.30/0.49 = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_one_over
% 0.30/0.49 thf(fact_518_power__one__over,axiom,
% 0.30/0.49 ! [A: rat,N2: nat] :
% 0.30/0.49 ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N2 )
% 0.30/0.49 = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_one_over
% 0.30/0.49 thf(fact_519_numerals_I1_J,axiom,
% 0.30/0.49 ( ( numeral_numeral_nat @ one )
% 0.30/0.49 = one_one_nat ) ).
% 0.30/0.49
% 0.30/0.49 % numerals(1)
% 0.30/0.49 thf(fact_520_set__update__subsetI,axiom,
% 0.30/0.49 ! [Xs: list_real,A2: set_real,X: real,I2: nat] :
% 0.30/0.49 ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A2 )
% 0.30/0.49 => ( ( member_real @ X @ A2 )
% 0.30/0.49 => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I2 @ X ) ) @ A2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % set_update_subsetI
% 0.30/0.49 thf(fact_521_set__update__subsetI,axiom,
% 0.30/0.49 ! [Xs: list_complex,A2: set_complex,X: complex,I2: nat] :
% 0.30/0.49 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A2 )
% 0.30/0.49 => ( ( member_complex @ X @ A2 )
% 0.30/0.49 => ( ord_le211207098394363844omplex @ ( set_complex2 @ ( list_update_complex @ Xs @ I2 @ X ) ) @ A2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % set_update_subsetI
% 0.30/0.49 thf(fact_522_set__update__subsetI,axiom,
% 0.30/0.49 ! [Xs: list_set_nat,A2: set_set_nat,X: set_nat,I2: nat] :
% 0.30/0.49 ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ A2 )
% 0.30/0.49 => ( ( member_set_nat @ X @ A2 )
% 0.30/0.49 => ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ I2 @ X ) ) @ A2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % set_update_subsetI
% 0.30/0.49 thf(fact_523_set__update__subsetI,axiom,
% 0.30/0.49 ! [Xs: list_int,A2: set_int,X: int,I2: nat] :
% 0.30/0.49 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
% 0.30/0.49 => ( ( member_int @ X @ A2 )
% 0.30/0.49 => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I2 @ X ) ) @ A2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % set_update_subsetI
% 0.30/0.49 thf(fact_524_set__update__subsetI,axiom,
% 0.30/0.49 ! [Xs: list_VEBT_VEBT,A2: set_VEBT_VEBT,X: vEBT_VEBT,I2: nat] :
% 0.30/0.49 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
% 0.30/0.49 => ( ( member_VEBT_VEBT @ X @ A2 )
% 0.30/0.49 => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X ) ) @ A2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % set_update_subsetI
% 0.30/0.49 thf(fact_525_set__update__subsetI,axiom,
% 0.30/0.49 ! [Xs: list_nat,A2: set_nat,X: nat,I2: nat] :
% 0.30/0.49 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
% 0.30/0.49 => ( ( member_nat @ X @ A2 )
% 0.30/0.49 => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I2 @ X ) ) @ A2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % set_update_subsetI
% 0.30/0.49 thf(fact_526_power__less__power__Suc,axiom,
% 0.30/0.49 ! [A: real,N2: nat] :
% 0.30/0.49 ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.49 => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_less_power_Suc
% 0.30/0.49 thf(fact_527_power__less__power__Suc,axiom,
% 0.30/0.49 ! [A: rat,N2: nat] :
% 0.30/0.49 ( ( ord_less_rat @ one_one_rat @ A )
% 0.30/0.49 => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_less_power_Suc
% 0.30/0.49 thf(fact_528_power__less__power__Suc,axiom,
% 0.30/0.49 ! [A: nat,N2: nat] :
% 0.30/0.49 ( ( ord_less_nat @ one_one_nat @ A )
% 0.30/0.49 => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_less_power_Suc
% 0.30/0.49 thf(fact_529_power__less__power__Suc,axiom,
% 0.30/0.49 ! [A: int,N2: nat] :
% 0.30/0.49 ( ( ord_less_int @ one_one_int @ A )
% 0.30/0.49 => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_less_power_Suc
% 0.30/0.49 thf(fact_530_power__gt1__lemma,axiom,
% 0.30/0.49 ! [A: real,N2: nat] :
% 0.30/0.49 ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.49 => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_gt1_lemma
% 0.30/0.49 thf(fact_531_power__gt1__lemma,axiom,
% 0.30/0.49 ! [A: rat,N2: nat] :
% 0.30/0.49 ( ( ord_less_rat @ one_one_rat @ A )
% 0.30/0.49 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_gt1_lemma
% 0.30/0.49 thf(fact_532_power__gt1__lemma,axiom,
% 0.30/0.49 ! [A: nat,N2: nat] :
% 0.30/0.49 ( ( ord_less_nat @ one_one_nat @ A )
% 0.30/0.49 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_gt1_lemma
% 0.30/0.49 thf(fact_533_power__gt1__lemma,axiom,
% 0.30/0.49 ! [A: int,N2: nat] :
% 0.30/0.49 ( ( ord_less_int @ one_one_int @ A )
% 0.30/0.49 => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_gt1_lemma
% 0.30/0.49 thf(fact_534_power__less__imp__less__exp,axiom,
% 0.30/0.49 ! [A: real,M: nat,N2: nat] :
% 0.30/0.49 ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.49 => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) )
% 0.30/0.49 => ( ord_less_nat @ M @ N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_less_imp_less_exp
% 0.30/0.49 thf(fact_535_power__less__imp__less__exp,axiom,
% 0.30/0.49 ! [A: rat,M: nat,N2: nat] :
% 0.30/0.49 ( ( ord_less_rat @ one_one_rat @ A )
% 0.30/0.49 => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) )
% 0.30/0.49 => ( ord_less_nat @ M @ N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_less_imp_less_exp
% 0.30/0.49 thf(fact_536_power__less__imp__less__exp,axiom,
% 0.30/0.49 ! [A: nat,M: nat,N2: nat] :
% 0.30/0.49 ( ( ord_less_nat @ one_one_nat @ A )
% 0.30/0.49 => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
% 0.30/0.49 => ( ord_less_nat @ M @ N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_less_imp_less_exp
% 0.30/0.49 thf(fact_537_power__less__imp__less__exp,axiom,
% 0.30/0.49 ! [A: int,M: nat,N2: nat] :
% 0.30/0.49 ( ( ord_less_int @ one_one_int @ A )
% 0.30/0.49 => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
% 0.30/0.49 => ( ord_less_nat @ M @ N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_less_imp_less_exp
% 0.30/0.49 thf(fact_538_power__strict__increasing,axiom,
% 0.30/0.49 ! [N2: nat,N3: nat,A: real] :
% 0.30/0.49 ( ( ord_less_nat @ N2 @ N3 )
% 0.30/0.49 => ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.49 => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_strict_increasing
% 0.30/0.49 thf(fact_539_power__strict__increasing,axiom,
% 0.30/0.49 ! [N2: nat,N3: nat,A: rat] :
% 0.30/0.49 ( ( ord_less_nat @ N2 @ N3 )
% 0.30/0.49 => ( ( ord_less_rat @ one_one_rat @ A )
% 0.30/0.49 => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ A @ N3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_strict_increasing
% 0.30/0.49 thf(fact_540_power__strict__increasing,axiom,
% 0.30/0.49 ! [N2: nat,N3: nat,A: nat] :
% 0.30/0.49 ( ( ord_less_nat @ N2 @ N3 )
% 0.30/0.49 => ( ( ord_less_nat @ one_one_nat @ A )
% 0.30/0.49 => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_strict_increasing
% 0.30/0.49 thf(fact_541_power__strict__increasing,axiom,
% 0.30/0.49 ! [N2: nat,N3: nat,A: int] :
% 0.30/0.49 ( ( ord_less_nat @ N2 @ N3 )
% 0.30/0.49 => ( ( ord_less_int @ one_one_int @ A )
% 0.30/0.49 => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_strict_increasing
% 0.30/0.49 thf(fact_542_power__increasing,axiom,
% 0.30/0.49 ! [N2: nat,N3: nat,A: real] :
% 0.30/0.49 ( ( ord_less_eq_nat @ N2 @ N3 )
% 0.30/0.49 => ( ( ord_less_eq_real @ one_one_real @ A )
% 0.30/0.49 => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_increasing
% 0.30/0.49 thf(fact_543_power__increasing,axiom,
% 0.30/0.49 ! [N2: nat,N3: nat,A: rat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ N2 @ N3 )
% 0.30/0.49 => ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.30/0.49 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ A @ N3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_increasing
% 0.30/0.49 thf(fact_544_power__increasing,axiom,
% 0.30/0.49 ! [N2: nat,N3: nat,A: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ N2 @ N3 )
% 0.30/0.49 => ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.30/0.49 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_increasing
% 0.30/0.49 thf(fact_545_power__increasing,axiom,
% 0.30/0.49 ! [N2: nat,N3: nat,A: int] :
% 0.30/0.49 ( ( ord_less_eq_nat @ N2 @ N3 )
% 0.30/0.49 => ( ( ord_less_eq_int @ one_one_int @ A )
% 0.30/0.49 => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_increasing
% 0.30/0.49 thf(fact_546_power__le__imp__le__exp,axiom,
% 0.30/0.49 ! [A: real,M: nat,N2: nat] :
% 0.30/0.49 ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.49 => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) )
% 0.30/0.49 => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_le_imp_le_exp
% 0.30/0.49 thf(fact_547_power__le__imp__le__exp,axiom,
% 0.30/0.49 ! [A: rat,M: nat,N2: nat] :
% 0.30/0.49 ( ( ord_less_rat @ one_one_rat @ A )
% 0.30/0.49 => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) )
% 0.30/0.49 => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_le_imp_le_exp
% 0.30/0.49 thf(fact_548_power__le__imp__le__exp,axiom,
% 0.30/0.49 ! [A: nat,M: nat,N2: nat] :
% 0.30/0.49 ( ( ord_less_nat @ one_one_nat @ A )
% 0.30/0.49 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
% 0.30/0.49 => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_le_imp_le_exp
% 0.30/0.49 thf(fact_549_power__le__imp__le__exp,axiom,
% 0.30/0.49 ! [A: int,M: nat,N2: nat] :
% 0.30/0.49 ( ( ord_less_int @ one_one_int @ A )
% 0.30/0.49 => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
% 0.30/0.49 => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % power_le_imp_le_exp
% 0.30/0.49 thf(fact_550_one__power2,axiom,
% 0.30/0.49 ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49 = one_one_rat ) ).
% 0.30/0.49
% 0.30/0.49 % one_power2
% 0.30/0.49 thf(fact_551_one__power2,axiom,
% 0.30/0.49 ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49 = one_one_nat ) ).
% 0.30/0.49
% 0.30/0.49 % one_power2
% 0.30/0.49 thf(fact_552_one__power2,axiom,
% 0.30/0.49 ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49 = one_one_real ) ).
% 0.30/0.49
% 0.30/0.49 % one_power2
% 0.30/0.49 thf(fact_553_one__power2,axiom,
% 0.30/0.49 ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49 = one_one_int ) ).
% 0.30/0.49
% 0.30/0.49 % one_power2
% 0.30/0.49 thf(fact_554_one__power2,axiom,
% 0.30/0.49 ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.49 = one_one_complex ) ).
% 0.30/0.49
% 0.30/0.49 % one_power2
% 0.30/0.49 thf(fact_555_nat__1__add__1,axiom,
% 0.30/0.49 ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 0.30/0.49 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % nat_1_add_1
% 0.30/0.49 thf(fact_556_left__add__mult__distrib,axiom,
% 0.30/0.49 ! [I2: nat,U: nat,J: nat,K: nat] :
% 0.30/0.49 ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
% 0.30/0.49 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J ) @ U ) @ K ) ) ).
% 0.30/0.49
% 0.30/0.49 % left_add_mult_distrib
% 0.30/0.49 thf(fact_557_ex__power__ivl2,axiom,
% 0.30/0.49 ! [B: nat,K: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.30/0.49 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.30/0.49 => ? [N4: nat] :
% 0.30/0.49 ( ( ord_less_nat @ ( power_power_nat @ B @ N4 ) @ K )
% 0.30/0.49 & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ex_power_ivl2
% 0.30/0.49 thf(fact_558_ex__power__ivl1,axiom,
% 0.30/0.49 ! [B: nat,K: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.30/0.49 => ( ( ord_less_eq_nat @ one_one_nat @ K )
% 0.30/0.49 => ? [N4: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N4 ) @ K )
% 0.30/0.49 & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % ex_power_ivl1
% 0.30/0.49 thf(fact_559_four__x__squared,axiom,
% 0.30/0.49 ! [X: real] :
% 0.30/0.49 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.49 = ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % four_x_squared
% 0.30/0.49 thf(fact_560_mint__sound,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,N2: nat,X: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.49 => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
% 0.30/0.49 => ( ( vEBT_vebt_mint @ T )
% 0.30/0.49 = ( some_nat @ X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mint_sound
% 0.30/0.49 thf(fact_561_mint__corr,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,N2: nat,X: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.49 => ( ( ( vEBT_vebt_mint @ T )
% 0.30/0.49 = ( some_nat @ X ) )
% 0.30/0.49 => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mint_corr
% 0.30/0.49 thf(fact_562_maxt__sound,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,N2: nat,X: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.49 => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
% 0.30/0.49 => ( ( vEBT_vebt_maxt @ T )
% 0.30/0.49 = ( some_nat @ X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % maxt_sound
% 0.30/0.49 thf(fact_563_maxt__corr,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,N2: nat,X: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.49 => ( ( ( vEBT_vebt_maxt @ T )
% 0.30/0.49 = ( some_nat @ X ) )
% 0.30/0.49 => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % maxt_corr
% 0.30/0.49 thf(fact_564_xnin,axiom,
% 0.30/0.49 vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) ).
% 0.30/0.49
% 0.30/0.49 % xnin
% 0.30/0.49 thf(fact_565_set__vebt__set__vebt_H__valid,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,N2: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.49 => ( ( vEBT_set_vebt @ T )
% 0.30/0.49 = ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % set_vebt_set_vebt'_valid
% 0.30/0.49 thf(fact_566_invar__vebt_Ointros_I2_J,axiom,
% 0.30/0.49 ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 0.30/0.49 ( ! [X4: vEBT_VEBT] :
% 0.30/0.49 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.49 => ( vEBT_invar_vebt @ X4 @ N2 ) )
% 0.30/0.49 => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.30/0.49 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.30/0.49 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.49 => ( ( M = N2 )
% 0.30/0.49 => ( ( Deg
% 0.30/0.49 = ( plus_plus_nat @ N2 @ M ) )
% 0.30/0.49 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 0.30/0.49 => ( ! [X4: vEBT_VEBT] :
% 0.30/0.49 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.49 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
% 0.30/0.49 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % invar_vebt.intros(2)
% 0.30/0.49 thf(fact_567_both__member__options__from__chilf__to__complete__tree,axiom,
% 0.30/0.49 ! [X: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 0.30/0.49 ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.49 => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 0.30/0.49 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.49 => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % both_member_options_from_chilf_to_complete_tree
% 0.30/0.49 thf(fact_568_member__inv,axiom,
% 0.30/0.49 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 0.30/0.49 ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.30/0.49 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.49 & ( ( X = Mi )
% 0.30/0.49 | ( X = Ma )
% 0.30/0.49 | ( ( ord_less_nat @ X @ Ma )
% 0.30/0.49 & ( ord_less_nat @ Mi @ X )
% 0.30/0.49 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.49 & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % member_inv
% 0.30/0.49 thf(fact_569_tvalid,axiom,
% 0.30/0.49 vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ deg ).
% 0.30/0.49
% 0.30/0.49 % tvalid
% 0.30/0.49 thf(fact_570_both__member__options__from__complete__tree__to__child,axiom,
% 0.30/0.49 ! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 0.30/0.49 ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 0.30/0.49 => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.30/0.49 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.49 | ( X = Mi )
% 0.30/0.49 | ( X = Ma ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % both_member_options_from_complete_tree_to_child
% 0.30/0.49 thf(fact_571_VEBT_Oinject_I1_J,axiom,
% 0.30/0.49 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
% 0.30/0.49 ( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 0.30/0.49 = ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
% 0.30/0.49 = ( ( X11 = Y11 )
% 0.30/0.49 & ( X12 = Y12 )
% 0.30/0.49 & ( X13 = Y13 )
% 0.30/0.49 & ( X14 = Y14 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % VEBT.inject(1)
% 0.30/0.49 thf(fact_572_real__divide__square__eq,axiom,
% 0.30/0.49 ! [R: real,A: real] :
% 0.30/0.49 ( ( divide_divide_real @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ R ) )
% 0.30/0.49 = ( divide_divide_real @ A @ R ) ) ).
% 0.30/0.49
% 0.30/0.49 % real_divide_square_eq
% 0.30/0.49 thf(fact_573_succ__member,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,X: nat,Y: nat] :
% 0.30/0.49 ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
% 0.30/0.49 = ( ( vEBT_vebt_member @ T @ Y )
% 0.30/0.49 & ( ord_less_nat @ X @ Y )
% 0.30/0.49 & ! [Z3: nat] :
% 0.30/0.49 ( ( ( vEBT_vebt_member @ T @ Z3 )
% 0.30/0.49 & ( ord_less_nat @ X @ Z3 ) )
% 0.30/0.49 => ( ord_less_eq_nat @ Y @ Z3 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % succ_member
% 0.30/0.49 thf(fact_574_pred__member,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,X: nat,Y: nat] :
% 0.30/0.49 ( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
% 0.30/0.49 = ( ( vEBT_vebt_member @ T @ Y )
% 0.30/0.49 & ( ord_less_nat @ Y @ X )
% 0.30/0.49 & ! [Z3: nat] :
% 0.30/0.49 ( ( ( vEBT_vebt_member @ T @ Z3 )
% 0.30/0.49 & ( ord_less_nat @ Z3 @ X ) )
% 0.30/0.49 => ( ord_less_eq_nat @ Z3 @ Y ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % pred_member
% 0.30/0.49 thf(fact_575_succ__corr,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,N2: nat,X: nat,Sx: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.49 => ( ( ( vEBT_vebt_succ @ T @ X )
% 0.30/0.49 = ( some_nat @ Sx ) )
% 0.30/0.49 = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % succ_corr
% 0.30/0.49 thf(fact_576_pred__corr,axiom,
% 0.30/0.49 ! [T: vEBT_VEBT,N2: nat,X: nat,Px: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.49 => ( ( ( vEBT_vebt_pred @ T @ X )
% 0.30/0.49 = ( some_nat @ Px ) )
% 0.30/0.49 = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Px ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % pred_corr
% 0.30/0.49 thf(fact_577_mi__eq__ma__no__ch,axiom,
% 0.30/0.49 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
% 0.30/0.49 => ( ( Mi = Ma )
% 0.30/0.49 => ( ! [X5: vEBT_VEBT] :
% 0.30/0.49 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.49 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
% 0.30/0.49 & ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mi_eq_ma_no_ch
% 0.30/0.49 thf(fact_578_geqmaxNone,axiom,
% 0.30/0.49 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat,X: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N2 )
% 0.30/0.49 => ( ( ord_less_eq_nat @ Ma @ X )
% 0.30/0.49 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.30/0.49 = none_nat ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % geqmaxNone
% 0.30/0.49 thf(fact_579_insert__simp__mima,axiom,
% 0.30/0.49 ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.49 ( ( ( X = Mi )
% 0.30/0.49 | ( X = Ma ) )
% 0.30/0.49 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.49 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.30/0.49 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % insert_simp_mima
% 0.30/0.49 thf(fact_580_delt__out__of__range,axiom,
% 0.30/0.49 ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.49 ( ( ( ord_less_nat @ X @ Mi )
% 0.30/0.49 | ( ord_less_nat @ Ma @ X ) )
% 0.30/0.49 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.49 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.30/0.49 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % delt_out_of_range
% 0.30/0.49 thf(fact_581_del__single__cont,axiom,
% 0.30/0.49 ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.49 ( ( ( X = Mi )
% 0.30/0.49 & ( X = Ma ) )
% 0.30/0.49 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.49 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.30/0.49 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % del_single_cont
% 0.30/0.49 thf(fact_582_mi__ma__2__deg,axiom,
% 0.30/0.49 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N2 )
% 0.30/0.49 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 0.30/0.49 & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % mi_ma_2_deg
% 0.30/0.49 thf(fact_583_succ__min,axiom,
% 0.30/0.49 ! [Deg: nat,X: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.49 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.49 => ( ( ord_less_nat @ X @ Mi )
% 0.30/0.49 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.30/0.49 = ( some_nat @ Mi ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % succ_min
% 0.30/0.49 thf(fact_584_pred__max,axiom,
% 0.30/0.49 ! [Deg: nat,Ma: nat,X: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.49 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.49 => ( ( ord_less_nat @ Ma @ X )
% 0.30/0.49 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.30/0.49 = ( some_nat @ Ma ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % pred_max
% 0.30/0.49 thf(fact_585_summaxma,axiom,
% 0.30/0.49 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.49 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
% 0.30/0.49 => ( ( Mi != Ma )
% 0.30/0.49 => ( ( the_nat @ ( vEBT_vebt_maxt @ Summary ) )
% 0.30/0.49 = ( vEBT_VEBT_high @ Ma @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % summaxma
% 0.30/0.49 thf(fact_586_pred__list__to__short,axiom,
% 0.30/0.49 ! [Deg: nat,X: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
% 0.30/0.49 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.49 => ( ( ord_less_eq_nat @ X @ Ma )
% 0.30/0.49 => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.49 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.30/0.49 = none_nat ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % pred_list_to_short
% 0.30/0.49 thf(fact_587_succ__list__to__short,axiom,
% 0.30/0.49 ! [Deg: nat,Mi: nat,X: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
% 0.30/0.49 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.49 => ( ( ord_less_eq_nat @ Mi @ X )
% 0.30/0.49 => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.49 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.30/0.49 = none_nat ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % succ_list_to_short
% 0.30/0.49 thf(fact_588_real__arch__pow,axiom,
% 0.30/0.49 ! [X: real,Y: real] :
% 0.30/0.49 ( ( ord_less_real @ one_one_real @ X )
% 0.30/0.49 => ? [N4: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N4 ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % real_arch_pow
% 0.30/0.49 thf(fact_589_less__eq__real__def,axiom,
% 0.30/0.49 ( ord_less_eq_real
% 0.30/0.49 = ( ^ [X2: real,Y2: real] :
% 0.30/0.49 ( ( ord_less_real @ X2 @ Y2 )
% 0.30/0.49 | ( X2 = Y2 ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % less_eq_real_def
% 0.30/0.49 thf(fact_590_complete__real,axiom,
% 0.30/0.49 ! [S: set_real] :
% 0.30/0.49 ( ? [X5: real] : ( member_real @ X5 @ S )
% 0.30/0.49 => ( ? [Z4: real] :
% 0.30/0.49 ! [X4: real] :
% 0.30/0.49 ( ( member_real @ X4 @ S )
% 0.30/0.49 => ( ord_less_eq_real @ X4 @ Z4 ) )
% 0.30/0.49 => ? [Y4: real] :
% 0.30/0.49 ( ! [X5: real] :
% 0.30/0.49 ( ( member_real @ X5 @ S )
% 0.30/0.49 => ( ord_less_eq_real @ X5 @ Y4 ) )
% 0.30/0.49 & ! [Z4: real] :
% 0.30/0.49 ( ! [X4: real] :
% 0.30/0.49 ( ( member_real @ X4 @ S )
% 0.30/0.49 => ( ord_less_eq_real @ X4 @ Z4 ) )
% 0.30/0.49 => ( ord_less_eq_real @ Y4 @ Z4 ) ) ) ) ) ).
% 0.30/0.49
% 0.30/0.49 % complete_real
% 0.30/0.49 thf(fact_591_two__realpow__ge__one,axiom,
% 0.30/0.49 ! [N2: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.30/0.49
% 0.30/0.49 % two_realpow_ge_one
% 0.30/0.49 thf(fact_592_invar__vebt_Ointros_I4_J,axiom,
% 0.30/0.49 ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 0.30/0.49 ( ! [X4: vEBT_VEBT] :
% 0.30/0.49 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.49 => ( vEBT_invar_vebt @ X4 @ N2 ) )
% 0.30/0.49 => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.30/0.49 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.30/0.49 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.49 => ( ( M = N2 )
% 0.30/0.49 => ( ( Deg
% 0.30/0.49 = ( plus_plus_nat @ N2 @ M ) )
% 0.30/0.49 => ( ! [I4: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.49 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X3 ) )
% 0.30/0.49 = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
% 0.30/0.49 => ( ( ( Mi = Ma )
% 0.30/0.49 => ! [X4: vEBT_VEBT] :
% 0.30/0.49 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.49 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
% 0.30/0.49 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 0.30/0.49 => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.30/0.49 => ( ( ( Mi != Ma )
% 0.30/0.49 => ! [I4: nat] :
% 0.30/0.49 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.49 => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
% 0.30/0.49 = I4 )
% 0.30/0.49 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
% 0.30/0.49 & ! [X4: nat] :
% 0.30/0.49 ( ( ( ( vEBT_VEBT_high @ X4 @ N2 )
% 0.30/0.50 = I4 )
% 0.30/0.50 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X4 @ N2 ) ) )
% 0.30/0.50 => ( ( ord_less_nat @ Mi @ X4 )
% 0.30/0.50 & ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
% 0.30/0.50 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % invar_vebt.intros(4)
% 0.30/0.50 thf(fact_593__092_060open_062y_A_061_Asummin_A_K_A2_A_094_An_A_L_Alx_A_092_060Longrightarrow_062_Ax_A_092_060noteq_062_Ay_A_092_060and_062_Aboth__member__options_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Ay_092_060close_062,axiom,
% 0.30/0.50 ( ( ya
% 0.30/0.50 = ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) )
% 0.30/0.50 => ( ( xa != ya )
% 0.30/0.50 & ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ ya ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % \<open>y = summin * 2 ^ n + lx \<Longrightarrow> x \<noteq> y \<and> both_member_options (Node (Some (mi, ma)) deg treeList summary) y\<close>
% 0.30/0.50 thf(fact_594_del__x__mi__lets__in__not__minNull,axiom,
% 0.30/0.50 ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L2: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
% 0.30/0.50 ( ( ( X = Mi )
% 0.30/0.50 & ( ord_less_nat @ X @ Ma ) )
% 0.30/0.50 => ( ( Mi != Ma )
% 0.30/0.50 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.50 => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.50 = H2 )
% 0.30/0.50 => ( ( Xn
% 0.30/0.50 = ( 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 ) ) ) ) ) ) )
% 0.30/0.50 => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.50 = L2 )
% 0.30/0.50 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.50 => ( ( Newnode
% 0.30/0.50 = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
% 0.30/0.50 => ( ( Newlist
% 0.30/0.50 = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
% 0.30/0.50 => ( ~ ( vEBT_VEBT_minNull @ Newnode )
% 0.30/0.50 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.30/0.50 = ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % del_x_mi_lets_in_not_minNull
% 0.30/0.50 thf(fact_595_del__x__not__mi__newnode__not__nil,axiom,
% 0.30/0.50 ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L2: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.50 ( ( ( ord_less_nat @ Mi @ X )
% 0.30/0.50 & ( ord_less_eq_nat @ X @ Ma ) )
% 0.30/0.50 => ( ( Mi != Ma )
% 0.30/0.50 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.50 => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.50 = H2 )
% 0.30/0.50 => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.50 = L2 )
% 0.30/0.50 => ( ( Newnode
% 0.30/0.50 = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
% 0.30/0.50 => ( ~ ( vEBT_VEBT_minNull @ Newnode )
% 0.30/0.50 => ( ( Newlist
% 0.30/0.50 = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
% 0.30/0.50 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.50 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.30/0.50 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = 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 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % del_x_not_mi_newnode_not_nil
% 0.30/0.50 thf(fact_596_nested__mint,axiom,
% 0.30/0.50 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat,Va: nat] :
% 0.30/0.50 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N2 )
% 0.30/0.50 => ( ( N2
% 0.30/0.50 = ( suc @ ( suc @ Va ) ) )
% 0.30/0.50 => ( ~ ( ord_less_nat @ Ma @ Mi )
% 0.30/0.50 => ( ( Ma != Mi )
% 0.30/0.50 => ( 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 ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nested_mint
% 0.30/0.50 thf(fact_597_mintlistlength,axiom,
% 0.30/0.50 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
% 0.30/0.50 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N2 )
% 0.30/0.50 => ( ( Mi != Ma )
% 0.30/0.50 => ( ( ord_less_nat @ Mi @ Ma )
% 0.30/0.50 & ? [M2: nat] :
% 0.30/0.50 ( ( ( some_nat @ M2 )
% 0.30/0.50 = ( vEBT_vebt_mint @ Summary ) )
% 0.30/0.50 & ( ord_less_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mintlistlength
% 0.30/0.50 thf(fact_598_mul__def,axiom,
% 0.30/0.50 ( vEBT_VEBT_mul
% 0.30/0.50 = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).
% 0.30/0.50
% 0.30/0.50 % mul_def
% 0.30/0.50 thf(fact_599_add__def,axiom,
% 0.30/0.50 ( vEBT_VEBT_add
% 0.30/0.50 = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_def
% 0.30/0.50 thf(fact_600__092_060open_062vebt__delete_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Ax_A_061_ANode_A_ISome_A_Isummin_A_K_A2_A_094_An_A_L_Alx_M_Aif_Asummin_A_K_A2_A_094_An_A_L_Alx_A_061_Ama_Athen_Alet_Amaxs_A_061_Avebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_Ain_Aif_Amaxs_A_061_ANone_Athen_Asummin_A_K_A2_A_094_An_A_L_Alx_Aelse_A2_A_094_A_Ideg_Adiv_A2_J_A_K_Athe_Amaxs_A_L_Athe_A_Ivebt__maxt_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_A_B_Athe_Amaxs_J_J_Aelse_Ama_J_J_Adeg_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_J_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_092_060close_062,axiom,
% 0.30/0.50 ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 0.30/0.50 = ( vEBT_Node
% 0.30/0.50 @ ( some_P7363390416028606310at_nat
% 0.30/0.50 @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 0.30/0.50 @ ( if_nat
% 0.30/0.50 @ ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 0.30/0.50 = ma )
% 0.30/0.50 @ ( if_nat
% 0.30/0.50 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 0.30/0.50 = none_nat )
% 0.30/0.50 @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 0.30/0.50 @ ( 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 @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) ) ) )
% 0.30/0.50 @ ma ) ) )
% 0.30/0.50 @ deg
% 0.30/0.50 @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 0.30/0.50 @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % \<open>vebt_delete (Node (Some (mi, ma)) deg treeList summary) x = Node (Some (summin * 2 ^ n + lx, if summin * 2 ^ n + lx = ma then let maxs = vebt_maxt (vebt_delete summary (high (summin * 2 ^ n + lx) n)) in if maxs = None then summin * 2 ^ n + lx else 2 ^ (deg div 2) * the maxs + the (vebt_maxt (treeList [high (summin * 2 ^ n + lx) n := vebt_delete (treeList ! high (summin * 2 ^ n + lx) n) (low (summin * 2 ^ n + lx) n)] ! the maxs)) else ma)) deg (treeList [high (summin * 2 ^ n + lx) n := vebt_delete (treeList ! high (summin * 2 ^ n + lx) n) (low (summin * 2 ^ n + lx) n)]) (vebt_delete summary (high (summin * 2 ^ n + lx) n))\<close>
% 0.30/0.50 thf(fact_601_vebt__mint_Osimps_I3_J,axiom,
% 0.30/0.50 ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 0.30/0.50 ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 0.30/0.50 = ( some_nat @ Mi ) ) ).
% 0.30/0.50
% 0.30/0.50 % vebt_mint.simps(3)
% 0.30/0.50 thf(fact_602_vebt__maxt_Osimps_I3_J,axiom,
% 0.30/0.50 ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 0.30/0.50 ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 0.30/0.50 = ( some_nat @ Ma ) ) ).
% 0.30/0.50
% 0.30/0.50 % vebt_maxt.simps(3)
% 0.30/0.50 thf(fact_603_nat__1__eq__mult__iff,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( one_one_nat
% 0.30/0.50 = ( times_times_nat @ M @ N2 ) )
% 0.30/0.50 = ( ( M = one_one_nat )
% 0.30/0.50 & ( N2 = one_one_nat ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_1_eq_mult_iff
% 0.30/0.50 thf(fact_604_even__odd__cases,axiom,
% 0.30/0.50 ! [X: nat] :
% 0.30/0.50 ( ! [N4: nat] :
% 0.30/0.50 ( X
% 0.30/0.50 != ( plus_plus_nat @ N4 @ N4 ) )
% 0.30/0.50 => ~ ! [N4: nat] :
% 0.30/0.50 ( X
% 0.30/0.50 != ( plus_plus_nat @ N4 @ ( suc @ N4 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % even_odd_cases
% 0.30/0.50 thf(fact_605_not__min__Null__member,axiom,
% 0.30/0.50 ! [T: vEBT_VEBT] :
% 0.30/0.50 ( ~ ( vEBT_VEBT_minNull @ T )
% 0.30/0.50 => ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 ) ) ).
% 0.30/0.50
% 0.30/0.50 % not_min_Null_member
% 0.30/0.50 thf(fact_606_min__Null__member,axiom,
% 0.30/0.50 ! [T: vEBT_VEBT,X: nat] :
% 0.30/0.50 ( ( vEBT_VEBT_minNull @ T )
% 0.30/0.50 => ~ ( vEBT_vebt_member @ T @ X ) ) ).
% 0.30/0.50
% 0.30/0.50 % min_Null_member
% 0.30/0.50 thf(fact_607_set__vebt_H__def,axiom,
% 0.30/0.50 ( vEBT_VEBT_set_vebt
% 0.30/0.50 = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % set_vebt'_def
% 0.30/0.50 thf(fact_608_deg__SUcn__Node,axiom,
% 0.30/0.50 ! [Tree: vEBT_VEBT,N2: nat] :
% 0.30/0.50 ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N2 ) ) )
% 0.30/0.50 => ? [Info2: option4927543243414619207at_nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 0.30/0.50 ( Tree
% 0.30/0.50 = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N2 ) ) @ TreeList3 @ S2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % deg_SUcn_Node
% 0.30/0.50 thf(fact_609_add__shift,axiom,
% 0.30/0.50 ! [X: nat,Y: nat,Z: nat] :
% 0.30/0.50 ( ( ( plus_plus_nat @ X @ Y )
% 0.30/0.50 = Z )
% 0.30/0.50 = ( ( vEBT_VEBT_add @ ( some_nat @ X ) @ ( some_nat @ Y ) )
% 0.30/0.50 = ( some_nat @ Z ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_shift
% 0.30/0.50 thf(fact_610_mul__shift,axiom,
% 0.30/0.50 ! [X: nat,Y: nat,Z: nat] :
% 0.30/0.50 ( ( ( times_times_nat @ X @ Y )
% 0.30/0.50 = Z )
% 0.30/0.50 = ( ( vEBT_VEBT_mul @ ( some_nat @ X ) @ ( some_nat @ Y ) )
% 0.30/0.50 = ( some_nat @ Z ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mul_shift
% 0.30/0.50 thf(fact_611_minNullmin,axiom,
% 0.30/0.50 ! [T: vEBT_VEBT] :
% 0.30/0.50 ( ( vEBT_VEBT_minNull @ T )
% 0.30/0.50 => ( ( vEBT_vebt_mint @ T )
% 0.30/0.50 = none_nat ) ) ).
% 0.30/0.50
% 0.30/0.50 % minNullmin
% 0.30/0.50 thf(fact_612_minminNull,axiom,
% 0.30/0.50 ! [T: vEBT_VEBT] :
% 0.30/0.50 ( ( ( vEBT_vebt_mint @ T )
% 0.30/0.50 = none_nat )
% 0.30/0.50 => ( vEBT_VEBT_minNull @ T ) ) ).
% 0.30/0.50
% 0.30/0.50 % minminNull
% 0.30/0.50 thf(fact_613_nat_Oinject,axiom,
% 0.30/0.50 ! [X22: nat,Y22: nat] :
% 0.30/0.50 ( ( ( suc @ X22 )
% 0.30/0.50 = ( suc @ Y22 ) )
% 0.30/0.50 = ( X22 = Y22 ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat.inject
% 0.30/0.50 thf(fact_614_old_Onat_Oinject,axiom,
% 0.30/0.50 ! [Nat: nat,Nat2: nat] :
% 0.30/0.50 ( ( ( suc @ Nat )
% 0.30/0.50 = ( suc @ Nat2 ) )
% 0.30/0.50 = ( Nat = Nat2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % old.nat.inject
% 0.30/0.50 thf(fact_615_power__minus__is__div,axiom,
% 0.30/0.50 ! [B: nat,A: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ B @ A )
% 0.30/0.50 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A @ B ) )
% 0.30/0.50 = ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_minus_is_div
% 0.30/0.50 thf(fact_616_lessI,axiom,
% 0.30/0.50 ! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % lessI
% 0.30/0.50 thf(fact_617_Suc__mono,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_nat @ M @ N2 )
% 0.30/0.50 => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_mono
% 0.30/0.50 thf(fact_618_Suc__less__eq,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
% 0.30/0.50 = ( ord_less_nat @ M @ N2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_less_eq
% 0.30/0.50 thf(fact_619_Suc__le__mono,axiom,
% 0.30/0.50 ! [N2: nat,M: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M ) )
% 0.30/0.50 = ( ord_less_eq_nat @ N2 @ M ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_le_mono
% 0.30/0.50 thf(fact_620_add__Suc__right,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( plus_plus_nat @ M @ ( suc @ N2 ) )
% 0.30/0.50 = ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_Suc_right
% 0.30/0.50 thf(fact_621_nat__add__left__cancel__less,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
% 0.30/0.50 = ( ord_less_nat @ M @ N2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_add_left_cancel_less
% 0.30/0.50 thf(fact_622_diff__Suc__Suc,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N2 ) )
% 0.30/0.50 = ( minus_minus_nat @ M @ N2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_Suc_Suc
% 0.30/0.50 thf(fact_623_Suc__diff__diff,axiom,
% 0.30/0.50 ! [M: nat,N2: nat,K: nat] :
% 0.30/0.50 ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) @ ( suc @ K ) )
% 0.30/0.50 = ( minus_minus_nat @ ( minus_minus_nat @ M @ N2 ) @ K ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_diff_diff
% 0.30/0.50 thf(fact_624_nat__add__left__cancel__le,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
% 0.30/0.50 = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_add_left_cancel_le
% 0.30/0.50 thf(fact_625_diff__diff__cancel,axiom,
% 0.30/0.50 ! [I2: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ I2 @ N2 )
% 0.30/0.50 => ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I2 ) )
% 0.30/0.50 = I2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_diff_cancel
% 0.30/0.50 thf(fact_626_diff__diff__left,axiom,
% 0.30/0.50 ! [I2: nat,J: nat,K: nat] :
% 0.30/0.50 ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
% 0.30/0.50 = ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_diff_left
% 0.30/0.50 thf(fact_627_nat__mult__eq__1__iff,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( ( times_times_nat @ M @ N2 )
% 0.30/0.50 = one_one_nat )
% 0.30/0.50 = ( ( M = one_one_nat )
% 0.30/0.50 & ( N2 = one_one_nat ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_mult_eq_1_iff
% 0.30/0.50 thf(fact_628_right__diff__distrib__numeral,axiom,
% 0.30/0.50 ! [V: num,B: complex,C: complex] :
% 0.30/0.50 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( minus_minus_complex @ B @ C ) )
% 0.30/0.50 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % right_diff_distrib_numeral
% 0.30/0.50 thf(fact_629_right__diff__distrib__numeral,axiom,
% 0.30/0.50 ! [V: num,B: real,C: real] :
% 0.30/0.50 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C ) )
% 0.30/0.50 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % right_diff_distrib_numeral
% 0.30/0.50 thf(fact_630_right__diff__distrib__numeral,axiom,
% 0.30/0.50 ! [V: num,B: rat,C: rat] :
% 0.30/0.50 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B @ C ) )
% 0.30/0.50 = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % right_diff_distrib_numeral
% 0.30/0.50 thf(fact_631_right__diff__distrib__numeral,axiom,
% 0.30/0.50 ! [V: num,B: int,C: int] :
% 0.30/0.50 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
% 0.30/0.50 = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % right_diff_distrib_numeral
% 0.30/0.50 thf(fact_632_left__diff__distrib__numeral,axiom,
% 0.30/0.50 ! [A: complex,B: complex,V: num] :
% 0.30/0.50 ( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
% 0.30/0.50 = ( minus_minus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % left_diff_distrib_numeral
% 0.30/0.50 thf(fact_633_left__diff__distrib__numeral,axiom,
% 0.30/0.50 ! [A: real,B: real,V: num] :
% 0.30/0.50 ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
% 0.30/0.50 = ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % left_diff_distrib_numeral
% 0.30/0.50 thf(fact_634_left__diff__distrib__numeral,axiom,
% 0.30/0.50 ! [A: rat,B: rat,V: num] :
% 0.30/0.50 ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
% 0.30/0.50 = ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % left_diff_distrib_numeral
% 0.30/0.50 thf(fact_635_left__diff__distrib__numeral,axiom,
% 0.30/0.50 ! [A: int,B: int,V: num] :
% 0.30/0.50 ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.50 = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % left_diff_distrib_numeral
% 0.30/0.50 thf(fact_636_mult__Suc__right,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( times_times_nat @ M @ ( suc @ N2 ) )
% 0.30/0.50 = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_Suc_right
% 0.30/0.50 thf(fact_637_Nat_Odiff__diff__right,axiom,
% 0.30/0.50 ! [K: nat,J: nat,I2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ K @ J )
% 0.30/0.50 => ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
% 0.30/0.50 = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Nat.diff_diff_right
% 0.30/0.50 thf(fact_638_Nat_Oadd__diff__assoc2,axiom,
% 0.30/0.50 ! [K: nat,J: nat,I2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ K @ J )
% 0.30/0.50 => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
% 0.30/0.50 = ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Nat.add_diff_assoc2
% 0.30/0.50 thf(fact_639_Nat_Oadd__diff__assoc,axiom,
% 0.30/0.50 ! [K: nat,J: nat,I2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ K @ J )
% 0.30/0.50 => ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
% 0.30/0.50 = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Nat.add_diff_assoc
% 0.30/0.50 thf(fact_640_diff__Suc__1,axiom,
% 0.30/0.50 ! [N2: nat] :
% 0.30/0.50 ( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
% 0.30/0.50 = N2 ) ).
% 0.30/0.50
% 0.30/0.50 % diff_Suc_1
% 0.30/0.50 thf(fact_641_diff__Suc__diff__eq2,axiom,
% 0.30/0.50 ! [K: nat,J: nat,I2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ K @ J )
% 0.30/0.50 => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I2 )
% 0.30/0.50 = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_Suc_diff_eq2
% 0.30/0.50 thf(fact_642_diff__Suc__diff__eq1,axiom,
% 0.30/0.50 ! [K: nat,J: nat,I2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ K @ J )
% 0.30/0.50 => ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
% 0.30/0.50 = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_Suc_diff_eq1
% 0.30/0.50 thf(fact_643_asmt,axiom,
% 0.30/0.50 ( ( ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 0.30/0.50 = ma )
% 0.30/0.50 => ( ya
% 0.30/0.50 = ( if_nat
% 0.30/0.50 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 0.30/0.50 = none_nat )
% 0.30/0.50 @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 0.30/0.50 @ ( 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 @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) ) ) ) ) )
% 0.30/0.50 & ( ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 0.30/0.50 != ma )
% 0.30/0.50 => ( ya = ma ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % asmt
% 0.30/0.50 thf(fact_644_Suc__numeral,axiom,
% 0.30/0.50 ! [N2: num] :
% 0.30/0.50 ( ( suc @ ( numeral_numeral_nat @ N2 ) )
% 0.30/0.50 = ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_numeral
% 0.30/0.50 thf(fact_645__092_060open_062both__member__options_A_INode_A_ISome_A_Isummin_A_K_A2_A_094_An_A_L_Alx_M_Aif_Asummin_A_K_A2_A_094_An_A_L_Alx_A_061_Ama_Athen_Alet_Amaxs_A_061_Avebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_Ain_Aif_Amaxs_A_061_ANone_Athen_Asummin_A_K_A2_A_094_An_A_L_Alx_Aelse_A2_A_094_A_Ideg_Adiv_A2_J_A_K_Athe_Amaxs_A_L_Athe_A_Ivebt__maxt_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_A_B_Athe_Amaxs_J_J_Aelse_Ama_J_J_Adeg_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_J_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_J_Ay_092_060close_062,axiom,
% 0.30/0.50 ( vEBT_V8194947554948674370ptions
% 0.30/0.50 @ ( vEBT_Node
% 0.30/0.50 @ ( some_P7363390416028606310at_nat
% 0.30/0.50 @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 0.30/0.50 @ ( if_nat
% 0.30/0.50 @ ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 0.30/0.50 = ma )
% 0.30/0.50 @ ( if_nat
% 0.30/0.50 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 0.30/0.50 = none_nat )
% 0.30/0.50 @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 0.30/0.50 @ ( 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 @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) ) ) )
% 0.30/0.50 @ ma ) ) )
% 0.30/0.50 @ deg
% 0.30/0.50 @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 0.30/0.50 @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 0.30/0.50 @ ya ) ).
% 0.30/0.50
% 0.30/0.50 % \<open>both_member_options (Node (Some (summin * 2 ^ n + lx, if summin * 2 ^ n + lx = ma then let maxs = vebt_maxt (vebt_delete summary (high (summin * 2 ^ n + lx) n)) in if maxs = None then summin * 2 ^ n + lx else 2 ^ (deg div 2) * the maxs + the (vebt_maxt (treeList [high (summin * 2 ^ n + lx) n := vebt_delete (treeList ! high (summin * 2 ^ n + lx) n) (low (summin * 2 ^ n + lx) n)] ! the maxs)) else ma)) deg (treeList [high (summin * 2 ^ n + lx) n := vebt_delete (treeList ! high (summin * 2 ^ n + lx) n) (low (summin * 2 ^ n + lx) n)]) (vebt_delete summary (high (summin * 2 ^ n + lx) n))) y\<close>
% 0.30/0.50 thf(fact_646_add__2__eq__Suc_H,axiom,
% 0.30/0.50 ! [N2: nat] :
% 0.30/0.50 ( ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( suc @ ( suc @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_2_eq_Suc'
% 0.30/0.50 thf(fact_647_add__2__eq__Suc,axiom,
% 0.30/0.50 ! [N2: nat] :
% 0.30/0.50 ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.50 = ( suc @ ( suc @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_2_eq_Suc
% 0.30/0.50 thf(fact_648_div2__Suc__Suc,axiom,
% 0.30/0.50 ! [M: nat] :
% 0.30/0.50 ( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % div2_Suc_Suc
% 0.30/0.50 thf(fact_649_Suc__1,axiom,
% 0.30/0.50 ( ( suc @ one_one_nat )
% 0.30/0.50 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_1
% 0.30/0.50 thf(fact_650_del__x__not__mia,axiom,
% 0.30/0.50 ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.50 ( ( ( ord_less_nat @ Mi @ X )
% 0.30/0.50 & ( ord_less_eq_nat @ X @ Ma ) )
% 0.30/0.50 => ( ( Mi != Ma )
% 0.30/0.50 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.50 => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.50 = H2 )
% 0.30/0.50 => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.50 = L2 )
% 0.30/0.50 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.50 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.30/0.50 = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
% 0.30/0.50 @ ( vEBT_Node
% 0.30/0.50 @ ( some_P7363390416028606310at_nat
% 0.30/0.50 @ ( product_Pair_nat_nat @ Mi
% 0.30/0.50 @ ( if_nat @ ( X = Ma )
% 0.30/0.50 @ ( if_nat
% 0.30/0.50 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 0.30/0.50 = none_nat )
% 0.30/0.50 @ Mi
% 0.30/0.50 @ ( 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 ) @ L2 ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
% 0.30/0.50 @ Ma ) ) )
% 0.30/0.50 @ Deg
% 0.30/0.50 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
% 0.30/0.50 @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 0.30/0.50 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = 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 ) @ L2 ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) ) @ Summary ) ) ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % del_x_not_mia
% 0.30/0.50 thf(fact_651_del__x__not__mi__new__node__nil,axiom,
% 0.30/0.50 ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L2: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
% 0.30/0.50 ( ( ( ord_less_nat @ Mi @ X )
% 0.30/0.50 & ( ord_less_eq_nat @ X @ Ma ) )
% 0.30/0.50 => ( ( Mi != Ma )
% 0.30/0.50 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.50 => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.50 = H2 )
% 0.30/0.50 => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.50 = L2 )
% 0.30/0.50 => ( ( Newnode
% 0.30/0.50 = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
% 0.30/0.50 => ( ( vEBT_VEBT_minNull @ Newnode )
% 0.30/0.50 => ( ( Sn
% 0.30/0.50 = ( vEBT_vebt_delete @ Summary @ H2 ) )
% 0.30/0.50 => ( ( Newlist
% 0.30/0.50 = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
% 0.30/0.50 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.50 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.30/0.50 = ( vEBT_Node
% 0.30/0.50 @ ( some_P7363390416028606310at_nat
% 0.30/0.50 @ ( product_Pair_nat_nat @ Mi
% 0.30/0.50 @ ( if_nat @ ( X = Ma )
% 0.30/0.50 @ ( if_nat
% 0.30/0.50 @ ( ( vEBT_vebt_maxt @ Sn )
% 0.30/0.50 = none_nat )
% 0.30/0.50 @ Mi
% 0.30/0.50 @ ( 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 ) ) ) ) ) ) )
% 0.30/0.50 @ Ma ) ) )
% 0.30/0.50 @ Deg
% 0.30/0.50 @ Newlist
% 0.30/0.50 @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % del_x_not_mi_new_node_nil
% 0.30/0.50 thf(fact_652_del__x__not__mi,axiom,
% 0.30/0.50 ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L2: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.50 ( ( ( ord_less_nat @ Mi @ X )
% 0.30/0.50 & ( ord_less_eq_nat @ X @ Ma ) )
% 0.30/0.50 => ( ( Mi != Ma )
% 0.30/0.50 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.50 => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.50 = H2 )
% 0.30/0.50 => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.50 = L2 )
% 0.30/0.50 => ( ( Newnode
% 0.30/0.50 = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
% 0.30/0.50 => ( ( Newlist
% 0.30/0.50 = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
% 0.30/0.50 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.50 => ( ( ( vEBT_VEBT_minNull @ Newnode )
% 0.30/0.50 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.30/0.50 = ( vEBT_Node
% 0.30/0.50 @ ( some_P7363390416028606310at_nat
% 0.30/0.50 @ ( product_Pair_nat_nat @ Mi
% 0.30/0.50 @ ( if_nat @ ( X = Ma )
% 0.30/0.50 @ ( if_nat
% 0.30/0.50 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 0.30/0.50 = none_nat )
% 0.30/0.50 @ Mi
% 0.30/0.50 @ ( 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 ) ) ) ) ) ) ) )
% 0.30/0.50 @ Ma ) ) )
% 0.30/0.50 @ Deg
% 0.30/0.50 @ Newlist
% 0.30/0.50 @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
% 0.30/0.50 & ( ~ ( vEBT_VEBT_minNull @ Newnode )
% 0.30/0.50 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.30/0.50 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = 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 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % del_x_not_mi
% 0.30/0.50 thf(fact_653_del__x__mia,axiom,
% 0.30/0.50 ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.50 ( ( ( X = Mi )
% 0.30/0.50 & ( ord_less_nat @ X @ Ma ) )
% 0.30/0.50 => ( ( Mi != Ma )
% 0.30/0.50 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.50 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.30/0.50 = ( 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 ) )
% 0.30/0.50 @ ( 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 ) ) ) ) ) )
% 0.30/0.50 @ ( vEBT_Node
% 0.30/0.50 @ ( some_P7363390416028606310at_nat
% 0.30/0.50 @ ( 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 ) ) ) ) ) )
% 0.30/0.50 @ ( if_nat
% 0.30/0.50 @ ( ( 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 ) ) ) ) ) )
% 0.30/0.50 = Ma )
% 0.30/0.50 @ ( if_nat
% 0.30/0.50 @ ( ( 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 ) ) ) ) ) )
% 0.30/0.50 = none_nat )
% 0.30/0.50 @ ( 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 ) ) ) ) ) )
% 0.30/0.50 @ ( 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 ) ) ) ) ) ) ) ) ) ) ) )
% 0.30/0.50 @ Ma ) ) )
% 0.30/0.50 @ Deg
% 0.30/0.50 @ ( 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 ) ) ) ) ) )
% 0.30/0.50 @ ( 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 ) ) ) ) ) )
% 0.30/0.50 @ ( vEBT_Node
% 0.30/0.50 @ ( some_P7363390416028606310at_nat
% 0.30/0.50 @ ( 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 ) ) ) ) ) )
% 0.30/0.50 @ ( if_nat
% 0.30/0.50 @ ( ( 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 ) ) ) ) ) )
% 0.30/0.50 = Ma )
% 0.30/0.50 @ ( 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 ) ) ) ) ) ) ) )
% 0.30/0.50 @ Ma ) ) )
% 0.30/0.50 @ Deg
% 0.30/0.50 @ ( 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 ) ) ) ) ) )
% 0.30/0.50 @ Summary ) )
% 0.30/0.50 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % del_x_mia
% 0.30/0.50 thf(fact_654_del__x__mi__lets__in__minNull,axiom,
% 0.30/0.50 ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L2: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
% 0.30/0.50 ( ( ( X = Mi )
% 0.30/0.50 & ( ord_less_nat @ X @ Ma ) )
% 0.30/0.50 => ( ( Mi != Ma )
% 0.30/0.50 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.50 => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.50 = H2 )
% 0.30/0.50 => ( ( Xn
% 0.30/0.50 = ( 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 ) ) ) ) ) ) )
% 0.30/0.50 => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.50 = L2 )
% 0.30/0.50 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.50 => ( ( Newnode
% 0.30/0.50 = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
% 0.30/0.50 => ( ( Newlist
% 0.30/0.50 = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
% 0.30/0.50 => ( ( vEBT_VEBT_minNull @ Newnode )
% 0.30/0.50 => ( ( Sn
% 0.30/0.50 = ( vEBT_vebt_delete @ Summary @ H2 ) )
% 0.30/0.50 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.30/0.50 = ( vEBT_Node
% 0.30/0.50 @ ( some_P7363390416028606310at_nat
% 0.30/0.50 @ ( product_Pair_nat_nat @ Xn
% 0.30/0.50 @ ( if_nat @ ( Xn = Ma )
% 0.30/0.50 @ ( if_nat
% 0.30/0.50 @ ( ( vEBT_vebt_maxt @ Sn )
% 0.30/0.50 = none_nat )
% 0.30/0.50 @ Xn
% 0.30/0.50 @ ( 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 ) ) ) ) ) ) )
% 0.30/0.50 @ Ma ) ) )
% 0.30/0.50 @ Deg
% 0.30/0.50 @ Newlist
% 0.30/0.50 @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % del_x_mi_lets_in_minNull
% 0.30/0.50 thf(fact_655_del__x__mi__lets__in,axiom,
% 0.30/0.50 ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L2: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
% 0.30/0.50 ( ( ( X = Mi )
% 0.30/0.50 & ( ord_less_nat @ X @ Ma ) )
% 0.30/0.50 => ( ( Mi != Ma )
% 0.30/0.50 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.50 => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.50 = H2 )
% 0.30/0.50 => ( ( Xn
% 0.30/0.50 = ( 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 ) ) ) ) ) ) )
% 0.30/0.50 => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.50 = L2 )
% 0.30/0.50 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.50 => ( ( Newnode
% 0.30/0.50 = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
% 0.30/0.50 => ( ( Newlist
% 0.30/0.50 = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
% 0.30/0.50 => ( ( ( vEBT_VEBT_minNull @ Newnode )
% 0.30/0.50 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.30/0.50 = ( vEBT_Node
% 0.30/0.50 @ ( some_P7363390416028606310at_nat
% 0.30/0.50 @ ( product_Pair_nat_nat @ Xn
% 0.30/0.50 @ ( if_nat @ ( Xn = Ma )
% 0.30/0.50 @ ( if_nat
% 0.30/0.50 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 0.30/0.50 = none_nat )
% 0.30/0.50 @ Xn
% 0.30/0.50 @ ( 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 ) ) ) ) ) ) ) )
% 0.30/0.50 @ Ma ) ) )
% 0.30/0.50 @ Deg
% 0.30/0.50 @ Newlist
% 0.30/0.50 @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
% 0.30/0.50 & ( ~ ( vEBT_VEBT_minNull @ Newnode )
% 0.30/0.50 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.30/0.50 = ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % del_x_mi_lets_in
% 0.30/0.50 thf(fact_656_del__x__mi,axiom,
% 0.30/0.50 ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L2: nat] :
% 0.30/0.50 ( ( ( X = Mi )
% 0.30/0.50 & ( ord_less_nat @ X @ Ma ) )
% 0.30/0.50 => ( ( Mi != Ma )
% 0.30/0.50 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.50 => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.50 = H2 )
% 0.30/0.50 => ( ( Xn
% 0.30/0.50 = ( 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 ) ) ) ) ) ) )
% 0.30/0.50 => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.50 = L2 )
% 0.30/0.50 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.50 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.30/0.50 = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
% 0.30/0.50 @ ( vEBT_Node
% 0.30/0.50 @ ( some_P7363390416028606310at_nat
% 0.30/0.50 @ ( product_Pair_nat_nat @ Xn
% 0.30/0.50 @ ( if_nat @ ( Xn = Ma )
% 0.30/0.50 @ ( if_nat
% 0.30/0.50 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 0.30/0.50 = none_nat )
% 0.30/0.50 @ Xn
% 0.30/0.50 @ ( 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 ) @ L2 ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
% 0.30/0.50 @ Ma ) ) )
% 0.30/0.50 @ Deg
% 0.30/0.50 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
% 0.30/0.50 @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 0.30/0.50 @ ( 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 ) @ L2 ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) ) @ Summary ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % del_x_mi
% 0.30/0.50 thf(fact_657_del__in__range,axiom,
% 0.30/0.50 ! [Mi: nat,X: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.50 ( ( ( ord_less_eq_nat @ Mi @ X )
% 0.30/0.50 & ( ord_less_eq_nat @ X @ Ma ) )
% 0.30/0.50 => ( ( Mi != Ma )
% 0.30/0.50 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.50 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.30/0.50 = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.50 @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.50 @ ( vEBT_Node
% 0.30/0.50 @ ( some_P7363390416028606310at_nat
% 0.30/0.50 @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ Mi )
% 0.30/0.50 @ ( if_nat
% 0.30/0.50 @ ( ( ( X = Mi )
% 0.30/0.50 => ( ( 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 ) ) ) ) ) )
% 0.30/0.50 = Ma ) )
% 0.30/0.50 & ( ( X != Mi )
% 0.30/0.50 => ( X = Ma ) ) )
% 0.30/0.50 @ ( if_nat
% 0.30/0.50 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.50 = none_nat )
% 0.30/0.50 @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ Mi )
% 0.30/0.50 @ ( 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 @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( 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 @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
% 0.30/0.50 @ Ma ) ) )
% 0.30/0.50 @ Deg
% 0.30/0.50 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.50 @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.50 @ ( vEBT_Node
% 0.30/0.50 @ ( some_P7363390416028606310at_nat
% 0.30/0.50 @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ Mi )
% 0.30/0.50 @ ( if_nat
% 0.30/0.50 @ ( ( ( X = Mi )
% 0.30/0.50 => ( ( 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 ) ) ) ) ) )
% 0.30/0.50 = Ma ) )
% 0.30/0.50 & ( ( X != Mi )
% 0.30/0.50 => ( X = Ma ) ) )
% 0.30/0.50 @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( 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 @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.30/0.50 @ Ma ) ) )
% 0.30/0.50 @ Deg
% 0.30/0.50 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.50 @ Summary ) )
% 0.30/0.50 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % del_in_range
% 0.30/0.50 thf(fact_658_succ__less__length__list,axiom,
% 0.30/0.50 ! [Deg: nat,Mi: nat,X: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
% 0.30/0.50 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.50 => ( ( ord_less_eq_nat @ Mi @ X )
% 0.30/0.50 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.50 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.30/0.50 = ( if_option_nat
% 0.30/0.50 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.50 != none_nat )
% 0.30/0.50 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.30/0.50 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.50 @ ( if_option_nat
% 0.30/0.50 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.50 = none_nat )
% 0.30/0.50 @ none_nat
% 0.30/0.50 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % succ_less_length_list
% 0.30/0.50 thf(fact_659_succ__greatereq__min,axiom,
% 0.30/0.50 ! [Deg: nat,Mi: nat,X: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.50 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.50 => ( ( ord_less_eq_nat @ Mi @ X )
% 0.30/0.50 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.30/0.50 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.50 @ ( if_option_nat
% 0.30/0.50 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.50 != none_nat )
% 0.30/0.50 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.30/0.50 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.50 @ ( if_option_nat
% 0.30/0.50 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.50 = none_nat )
% 0.30/0.50 @ none_nat
% 0.30/0.50 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.30/0.50 @ none_nat ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % succ_greatereq_min
% 0.30/0.50 thf(fact_660_pred__lesseq__max,axiom,
% 0.30/0.50 ! [Deg: nat,X: nat,Ma: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.50 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.50 => ( ( ord_less_eq_nat @ X @ Ma )
% 0.30/0.50 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.30/0.50 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.50 @ ( if_option_nat
% 0.30/0.50 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.50 != none_nat )
% 0.30/0.50 & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.30/0.50 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.50 @ ( if_option_nat
% 0.30/0.50 @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.50 = none_nat )
% 0.30/0.50 @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
% 0.30/0.50 @ ( 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 @ X @ ( 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 @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.30/0.50 @ none_nat ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % pred_lesseq_max
% 0.30/0.50 thf(fact_661_pred__less__length__list,axiom,
% 0.30/0.50 ! [Deg: nat,X: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
% 0.30/0.50 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.50 => ( ( ord_less_eq_nat @ X @ Ma )
% 0.30/0.50 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.50 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.30/0.50 = ( if_option_nat
% 0.30/0.50 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.50 != none_nat )
% 0.30/0.50 & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.30/0.50 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.50 @ ( if_option_nat
% 0.30/0.50 @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.50 = none_nat )
% 0.30/0.50 @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
% 0.30/0.50 @ ( 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 @ X @ ( 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 @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % pred_less_length_list
% 0.30/0.50 thf(fact_662_newmaext,axiom,
% 0.30/0.50 ( ( ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 0.30/0.50 = none_nat )
% 0.30/0.50 => ( ( ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 0.30/0.50 = ma )
% 0.30/0.50 => ( ( if_nat
% 0.30/0.50 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 0.30/0.50 = none_nat )
% 0.30/0.50 @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 0.30/0.50 @ ( 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 @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) ) ) )
% 0.30/0.50 = ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) ) )
% 0.30/0.50 & ( ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 0.30/0.50 != ma )
% 0.30/0.50 => ( ma
% 0.30/0.50 = ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) ) ) ) )
% 0.30/0.50 & ( ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 0.30/0.50 != none_nat )
% 0.30/0.50 => ( ( ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 0.30/0.50 = ma )
% 0.30/0.50 => ( ( if_nat
% 0.30/0.50 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 0.30/0.50 = none_nat )
% 0.30/0.50 @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 0.30/0.50 @ ( 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 @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) ) ) )
% 0.30/0.50 = ( 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 @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) ) ) ) )
% 0.30/0.50 & ( ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 0.30/0.50 != ma )
% 0.30/0.50 => ( ma
% 0.30/0.50 = ( 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 @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % newmaext
% 0.30/0.50 thf(fact_663__092_060open_062y_A_061_Asummin_A_K_A2_A_094_An_A_L_Alx_A_092_060or_062_Ay_A_061_A_Iif_Asummin_A_K_A2_A_094_An_A_L_Alx_A_061_Ama_Athen_Alet_Amaxs_A_061_Avebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_Ain_Aif_Amaxs_A_061_ANone_Athen_Asummin_A_K_A2_A_094_An_A_L_Alx_Aelse_A2_A_094_A_Ideg_Adiv_A2_J_A_K_Athe_Amaxs_A_L_Athe_A_Ivebt__maxt_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_A_B_Athe_Amaxs_J_J_Aelse_Ama_J_A_092_060or_062_Aboth__member__options_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_A_B_Ahigh_Ay_A_Ideg_Adiv_A2_J_J_A_Ilow_Ay_A_Ideg_Adiv_A2_J_J_A_092_060and_062_Ahigh_Ay_A_Ideg_Adiv_A2_J_A_060_Alength_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_J_092_060close_062,axiom,
% 0.30/0.50 ( ( ya
% 0.30/0.50 = ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) )
% 0.30/0.50 | ( ( ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 0.30/0.50 = ma )
% 0.30/0.50 => ( ya
% 0.30/0.50 = ( if_nat
% 0.30/0.50 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 0.30/0.50 = none_nat )
% 0.30/0.50 @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 0.30/0.50 @ ( 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 @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) ) ) ) ) )
% 0.30/0.50 & ( ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 0.30/0.50 != ma )
% 0.30/0.50 => ( ya = ma ) ) )
% 0.30/0.50 | ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( vEBT_VEBT_high @ ya @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ya @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.50 & ( ord_less_nat @ ( vEBT_VEBT_high @ ya @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % \<open>y = summin * 2 ^ n + lx \<or> y = (if summin * 2 ^ n + lx = ma then let maxs = vebt_maxt (vebt_delete summary (high (summin * 2 ^ n + lx) n)) in if maxs = None then summin * 2 ^ n + lx else 2 ^ (deg div 2) * the maxs + the (vebt_maxt (treeList [high (summin * 2 ^ n + lx) n := vebt_delete (treeList ! high (summin * 2 ^ n + lx) n) (low (summin * 2 ^ n + lx) n)] ! the maxs)) else ma) \<or> both_member_options (treeList [high (summin * 2 ^ n + lx) n := vebt_delete (treeList ! high (summin * 2 ^ n + lx) n) (low (summin * 2 ^ n + lx) n)] ! high y (deg div 2)) (low y (deg div 2)) \<and> high y (deg div 2) < length (treeList [high (summin * 2 ^ n + lx) n := vebt_delete (treeList ! high (summin * 2 ^ n + lx) n) (low (summin * 2 ^ n + lx) n)])\<close>
% 0.30/0.50 thf(fact_664_Suc__diff__le,axiom,
% 0.30/0.50 ! [N2: nat,M: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.50 => ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
% 0.30/0.50 = ( suc @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_diff_le
% 0.30/0.50 thf(fact_665_diff__Suc__eq__diff__pred,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( minus_minus_nat @ M @ ( suc @ N2 ) )
% 0.30/0.50 = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_Suc_eq_diff_pred
% 0.30/0.50 thf(fact_666_Suc__diff__Suc,axiom,
% 0.30/0.50 ! [N2: nat,M: nat] :
% 0.30/0.50 ( ( ord_less_nat @ N2 @ M )
% 0.30/0.50 => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N2 ) ) )
% 0.30/0.50 = ( minus_minus_nat @ M @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_diff_Suc
% 0.30/0.50 thf(fact_667_diff__less__Suc,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ ( suc @ M ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_less_Suc
% 0.30/0.50 thf(fact_668_Suc__inject,axiom,
% 0.30/0.50 ! [X: nat,Y: nat] :
% 0.30/0.50 ( ( ( suc @ X )
% 0.30/0.50 = ( suc @ Y ) )
% 0.30/0.50 => ( X = Y ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_inject
% 0.30/0.50 thf(fact_669_n__not__Suc__n,axiom,
% 0.30/0.50 ! [N2: nat] :
% 0.30/0.50 ( N2
% 0.30/0.50 != ( suc @ N2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % n_not_Suc_n
% 0.30/0.50 thf(fact_670_diff__commute,axiom,
% 0.30/0.50 ! [I2: nat,J: nat,K: nat] :
% 0.30/0.50 ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
% 0.30/0.50 = ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_commute
% 0.30/0.50 thf(fact_671_zero__induct__lemma,axiom,
% 0.30/0.50 ! [P: nat > $o,K: nat,I2: nat] :
% 0.30/0.50 ( ( P @ K )
% 0.30/0.50 => ( ! [N4: nat] :
% 0.30/0.50 ( ( P @ ( suc @ N4 ) )
% 0.30/0.50 => ( P @ N4 ) )
% 0.30/0.50 => ( P @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % zero_induct_lemma
% 0.30/0.50 thf(fact_672_diff__less__mono2,axiom,
% 0.30/0.50 ! [M: nat,N2: nat,L2: nat] :
% 0.30/0.50 ( ( ord_less_nat @ M @ N2 )
% 0.30/0.50 => ( ( ord_less_nat @ M @ L2 )
% 0.30/0.50 => ( ord_less_nat @ ( minus_minus_nat @ L2 @ N2 ) @ ( minus_minus_nat @ L2 @ M ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_less_mono2
% 0.30/0.50 thf(fact_673_less__imp__diff__less,axiom,
% 0.30/0.50 ! [J: nat,K: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_nat @ J @ K )
% 0.30/0.50 => ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_imp_diff_less
% 0.30/0.50 thf(fact_674_Nat_OlessE,axiom,
% 0.30/0.50 ! [I2: nat,K: nat] :
% 0.30/0.50 ( ( ord_less_nat @ I2 @ K )
% 0.30/0.50 => ( ( K
% 0.30/0.50 != ( suc @ I2 ) )
% 0.30/0.50 => ~ ! [J2: nat] :
% 0.30/0.50 ( ( ord_less_nat @ I2 @ J2 )
% 0.30/0.50 => ( K
% 0.30/0.50 != ( suc @ J2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Nat.lessE
% 0.30/0.50 thf(fact_675_Suc__lessD,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_nat @ ( suc @ M ) @ N2 )
% 0.30/0.50 => ( ord_less_nat @ M @ N2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_lessD
% 0.30/0.50 thf(fact_676_Suc__lessE,axiom,
% 0.30/0.50 ! [I2: nat,K: nat] :
% 0.30/0.50 ( ( ord_less_nat @ ( suc @ I2 ) @ K )
% 0.30/0.50 => ~ ! [J2: nat] :
% 0.30/0.50 ( ( ord_less_nat @ I2 @ J2 )
% 0.30/0.50 => ( K
% 0.30/0.50 != ( suc @ J2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_lessE
% 0.30/0.50 thf(fact_677_Suc__lessI,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_nat @ M @ N2 )
% 0.30/0.50 => ( ( ( suc @ M )
% 0.30/0.50 != N2 )
% 0.30/0.50 => ( ord_less_nat @ ( suc @ M ) @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_lessI
% 0.30/0.50 thf(fact_678_less__SucE,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_nat @ M @ ( suc @ N2 ) )
% 0.30/0.50 => ( ~ ( ord_less_nat @ M @ N2 )
% 0.30/0.50 => ( M = N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_SucE
% 0.30/0.50 thf(fact_679_less__SucI,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_nat @ M @ N2 )
% 0.30/0.50 => ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_SucI
% 0.30/0.50 thf(fact_680_Ex__less__Suc,axiom,
% 0.30/0.50 ! [N2: nat,P: nat > $o] :
% 0.30/0.50 ( ( ? [I3: nat] :
% 0.30/0.50 ( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
% 0.30/0.50 & ( P @ I3 ) ) )
% 0.30/0.50 = ( ( P @ N2 )
% 0.30/0.50 | ? [I3: nat] :
% 0.30/0.50 ( ( ord_less_nat @ I3 @ N2 )
% 0.30/0.50 & ( P @ I3 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Ex_less_Suc
% 0.30/0.50 thf(fact_681_less__Suc__eq,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_nat @ M @ ( suc @ N2 ) )
% 0.30/0.50 = ( ( ord_less_nat @ M @ N2 )
% 0.30/0.50 | ( M = N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_Suc_eq
% 0.30/0.50 thf(fact_682_not__less__eq,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( ~ ( ord_less_nat @ M @ N2 ) )
% 0.30/0.50 = ( ord_less_nat @ N2 @ ( suc @ M ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % not_less_eq
% 0.30/0.50 thf(fact_683_All__less__Suc,axiom,
% 0.30/0.50 ! [N2: nat,P: nat > $o] :
% 0.30/0.50 ( ( ! [I3: nat] :
% 0.30/0.50 ( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
% 0.30/0.50 => ( P @ I3 ) ) )
% 0.30/0.50 = ( ( P @ N2 )
% 0.30/0.50 & ! [I3: nat] :
% 0.30/0.50 ( ( ord_less_nat @ I3 @ N2 )
% 0.30/0.50 => ( P @ I3 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % All_less_Suc
% 0.30/0.50 thf(fact_684_Suc__less__eq2,axiom,
% 0.30/0.50 ! [N2: nat,M: nat] :
% 0.30/0.50 ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.30/0.50 = ( ? [M3: nat] :
% 0.30/0.50 ( ( M
% 0.30/0.50 = ( suc @ M3 ) )
% 0.30/0.50 & ( ord_less_nat @ N2 @ M3 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_less_eq2
% 0.30/0.50 thf(fact_685_less__antisym,axiom,
% 0.30/0.50 ! [N2: nat,M: nat] :
% 0.30/0.50 ( ~ ( ord_less_nat @ N2 @ M )
% 0.30/0.50 => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
% 0.30/0.50 => ( M = N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_antisym
% 0.30/0.50 thf(fact_686_Suc__less__SucD,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
% 0.30/0.50 => ( ord_less_nat @ M @ N2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_less_SucD
% 0.30/0.50 thf(fact_687_less__trans__Suc,axiom,
% 0.30/0.50 ! [I2: nat,J: nat,K: nat] :
% 0.30/0.50 ( ( ord_less_nat @ I2 @ J )
% 0.30/0.50 => ( ( ord_less_nat @ J @ K )
% 0.30/0.50 => ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_trans_Suc
% 0.30/0.50 thf(fact_688_less__Suc__induct,axiom,
% 0.30/0.50 ! [I2: nat,J: nat,P: nat > nat > $o] :
% 0.30/0.50 ( ( ord_less_nat @ I2 @ J )
% 0.30/0.50 => ( ! [I4: nat] : ( P @ I4 @ ( suc @ I4 ) )
% 0.30/0.50 => ( ! [I4: nat,J2: nat,K2: nat] :
% 0.30/0.50 ( ( ord_less_nat @ I4 @ J2 )
% 0.30/0.50 => ( ( ord_less_nat @ J2 @ K2 )
% 0.30/0.50 => ( ( P @ I4 @ J2 )
% 0.30/0.50 => ( ( P @ J2 @ K2 )
% 0.30/0.50 => ( P @ I4 @ K2 ) ) ) ) )
% 0.30/0.50 => ( P @ I2 @ J ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_Suc_induct
% 0.30/0.50 thf(fact_689_strict__inc__induct,axiom,
% 0.30/0.50 ! [I2: nat,J: nat,P: nat > $o] :
% 0.30/0.50 ( ( ord_less_nat @ I2 @ J )
% 0.30/0.50 => ( ! [I4: nat] :
% 0.30/0.50 ( ( J
% 0.30/0.50 = ( suc @ I4 ) )
% 0.30/0.50 => ( P @ I4 ) )
% 0.30/0.50 => ( ! [I4: nat] :
% 0.30/0.50 ( ( ord_less_nat @ I4 @ J )
% 0.30/0.50 => ( ( P @ ( suc @ I4 ) )
% 0.30/0.50 => ( P @ I4 ) ) )
% 0.30/0.50 => ( P @ I2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % strict_inc_induct
% 0.30/0.50 thf(fact_690_not__less__less__Suc__eq,axiom,
% 0.30/0.50 ! [N2: nat,M: nat] :
% 0.30/0.50 ( ~ ( ord_less_nat @ N2 @ M )
% 0.30/0.50 => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
% 0.30/0.50 = ( N2 = M ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % not_less_less_Suc_eq
% 0.30/0.50 thf(fact_691_eq__diff__iff,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ K @ M )
% 0.30/0.50 => ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.50 => ( ( ( minus_minus_nat @ M @ K )
% 0.30/0.50 = ( minus_minus_nat @ N2 @ K ) )
% 0.30/0.50 = ( M = N2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % eq_diff_iff
% 0.30/0.50 thf(fact_692_le__diff__iff,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ K @ M )
% 0.30/0.50 => ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.50 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
% 0.30/0.50 = ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_diff_iff
% 0.30/0.50 thf(fact_693_Nat_Odiff__diff__eq,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ K @ M )
% 0.30/0.50 => ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.50 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
% 0.30/0.50 = ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Nat.diff_diff_eq
% 0.30/0.50 thf(fact_694_diff__le__mono,axiom,
% 0.30/0.50 ! [M: nat,N2: nat,L2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.50 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L2 ) @ ( minus_minus_nat @ N2 @ L2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_le_mono
% 0.30/0.50 thf(fact_695_diff__le__self,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ).
% 0.30/0.50
% 0.30/0.50 % diff_le_self
% 0.30/0.50 thf(fact_696_le__diff__iff_H,axiom,
% 0.30/0.50 ! [A: nat,C: nat,B: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ A @ C )
% 0.30/0.50 => ( ( ord_less_eq_nat @ B @ C )
% 0.30/0.50 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 0.30/0.50 = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_diff_iff'
% 0.30/0.50 thf(fact_697_diff__le__mono2,axiom,
% 0.30/0.50 ! [M: nat,N2: nat,L2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.50 => ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N2 ) @ ( minus_minus_nat @ L2 @ M ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_le_mono2
% 0.30/0.50 thf(fact_698_diff__add__inverse2,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ N2 )
% 0.30/0.50 = M ) ).
% 0.30/0.50
% 0.30/0.50 % diff_add_inverse2
% 0.30/0.50 thf(fact_699_diff__add__inverse,axiom,
% 0.30/0.50 ! [N2: nat,M: nat] :
% 0.30/0.50 ( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M ) @ N2 )
% 0.30/0.50 = M ) ).
% 0.30/0.50
% 0.30/0.50 % diff_add_inverse
% 0.30/0.50 thf(fact_700_diff__cancel2,axiom,
% 0.30/0.50 ! [M: nat,K: nat,N2: nat] :
% 0.30/0.50 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) )
% 0.30/0.50 = ( minus_minus_nat @ M @ N2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_cancel2
% 0.30/0.50 thf(fact_701_Nat_Odiff__cancel,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.50 ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
% 0.30/0.50 = ( minus_minus_nat @ M @ N2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % Nat.diff_cancel
% 0.30/0.50 thf(fact_702_Suc__leD,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
% 0.30/0.50 => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_leD
% 0.30/0.50 thf(fact_703_le__SucE,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 0.30/0.50 => ( ~ ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.50 => ( M
% 0.30/0.50 = ( suc @ N2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_SucE
% 0.30/0.50 thf(fact_704_le__SucI,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.50 => ( ord_less_eq_nat @ M @ ( suc @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_SucI
% 0.30/0.50 thf(fact_705_Suc__le__D,axiom,
% 0.30/0.50 ! [N2: nat,M4: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ ( suc @ N2 ) @ M4 )
% 0.30/0.50 => ? [M2: nat] :
% 0.30/0.50 ( M4
% 0.30/0.50 = ( suc @ M2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_le_D
% 0.30/0.50 thf(fact_706_le__Suc__eq,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 0.30/0.50 = ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.50 | ( M
% 0.30/0.50 = ( suc @ N2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_Suc_eq
% 0.30/0.50 thf(fact_707_Suc__n__not__le__n,axiom,
% 0.30/0.50 ! [N2: nat] :
% 0.30/0.50 ~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_n_not_le_n
% 0.30/0.50 thf(fact_708_not__less__eq__eq,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( ~ ( ord_less_eq_nat @ M @ N2 ) )
% 0.30/0.50 = ( ord_less_eq_nat @ ( suc @ N2 ) @ M ) ) ).
% 0.30/0.50
% 0.30/0.50 % not_less_eq_eq
% 0.30/0.50 thf(fact_709_full__nat__induct,axiom,
% 0.30/0.50 ! [P: nat > $o,N2: nat] :
% 0.30/0.50 ( ! [N4: nat] :
% 0.30/0.50 ( ! [M5: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ ( suc @ M5 ) @ N4 )
% 0.30/0.50 => ( P @ M5 ) )
% 0.30/0.50 => ( P @ N4 ) )
% 0.30/0.50 => ( P @ N2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % full_nat_induct
% 0.30/0.50 thf(fact_710_nat__induct__at__least,axiom,
% 0.30/0.50 ! [M: nat,N2: nat,P: nat > $o] :
% 0.30/0.50 ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.50 => ( ( P @ M )
% 0.30/0.50 => ( ! [N4: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ M @ N4 )
% 0.30/0.50 => ( ( P @ N4 )
% 0.30/0.50 => ( P @ ( suc @ N4 ) ) ) )
% 0.30/0.50 => ( P @ N2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_induct_at_least
% 0.30/0.50 thf(fact_711_transitive__stepwise__le,axiom,
% 0.30/0.50 ! [M: nat,N2: nat,R2: nat > nat > $o] :
% 0.30/0.50 ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.50 => ( ! [X4: nat] : ( R2 @ X4 @ X4 )
% 0.30/0.50 => ( ! [X4: nat,Y4: nat,Z5: nat] :
% 0.30/0.50 ( ( R2 @ X4 @ Y4 )
% 0.30/0.50 => ( ( R2 @ Y4 @ Z5 )
% 0.30/0.50 => ( R2 @ X4 @ Z5 ) ) )
% 0.30/0.50 => ( ! [N4: nat] : ( R2 @ N4 @ ( suc @ N4 ) )
% 0.30/0.50 => ( R2 @ M @ N2 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % transitive_stepwise_le
% 0.30/0.50 thf(fact_712_add__Suc__shift,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( plus_plus_nat @ ( suc @ M ) @ N2 )
% 0.30/0.50 = ( plus_plus_nat @ M @ ( suc @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_Suc_shift
% 0.30/0.50 thf(fact_713_add__Suc,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( plus_plus_nat @ ( suc @ M ) @ N2 )
% 0.30/0.50 = ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_Suc
% 0.30/0.50 thf(fact_714_nat__arith_Osuc1,axiom,
% 0.30/0.50 ! [A2: nat,K: nat,A: nat] :
% 0.30/0.50 ( ( A2
% 0.30/0.50 = ( plus_plus_nat @ K @ A ) )
% 0.30/0.50 => ( ( suc @ A2 )
% 0.30/0.50 = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_arith.suc1
% 0.30/0.50 thf(fact_715_diff__mult__distrib2,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.50 ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
% 0.30/0.50 = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_mult_distrib2
% 0.30/0.50 thf(fact_716_diff__mult__distrib,axiom,
% 0.30/0.50 ! [M: nat,N2: nat,K: nat] :
% 0.30/0.50 ( ( times_times_nat @ ( minus_minus_nat @ M @ N2 ) @ K )
% 0.30/0.50 = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_mult_distrib
% 0.30/0.50 thf(fact_717_Suc__mult__cancel1,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.50 ( ( ( times_times_nat @ ( suc @ K ) @ M )
% 0.30/0.50 = ( times_times_nat @ ( suc @ K ) @ N2 ) )
% 0.30/0.50 = ( M = N2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_mult_cancel1
% 0.30/0.50 thf(fact_718_add__diff__add,axiom,
% 0.30/0.50 ! [A: real,C: real,B: real,D: real] :
% 0.30/0.50 ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
% 0.30/0.50 = ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_add
% 0.30/0.50 thf(fact_719_add__diff__add,axiom,
% 0.30/0.50 ! [A: rat,C: rat,B: rat,D: rat] :
% 0.30/0.50 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) )
% 0.30/0.50 = ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ ( minus_minus_rat @ C @ D ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_add
% 0.30/0.50 thf(fact_720_add__diff__add,axiom,
% 0.30/0.50 ! [A: int,C: int,B: int,D: int] :
% 0.30/0.50 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
% 0.30/0.50 = ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_add
% 0.30/0.50 thf(fact_721_set__vebt__def,axiom,
% 0.30/0.50 ( vEBT_set_vebt
% 0.30/0.50 = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % set_vebt_def
% 0.30/0.50 thf(fact_722_numeral__code_I2_J,axiom,
% 0.30/0.50 ! [N2: num] :
% 0.30/0.50 ( ( numera6690914467698888265omplex @ ( bit0 @ N2 ) )
% 0.30/0.50 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % numeral_code(2)
% 0.30/0.50 thf(fact_723_numeral__code_I2_J,axiom,
% 0.30/0.50 ! [N2: num] :
% 0.30/0.50 ( ( numeral_numeral_real @ ( bit0 @ N2 ) )
% 0.30/0.50 = ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % numeral_code(2)
% 0.30/0.50 thf(fact_724_numeral__code_I2_J,axiom,
% 0.30/0.50 ! [N2: num] :
% 0.30/0.50 ( ( numeral_numeral_rat @ ( bit0 @ N2 ) )
% 0.30/0.50 = ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % numeral_code(2)
% 0.30/0.50 thf(fact_725_numeral__code_I2_J,axiom,
% 0.30/0.50 ! [N2: num] :
% 0.30/0.50 ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
% 0.30/0.50 = ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % numeral_code(2)
% 0.30/0.50 thf(fact_726_numeral__code_I2_J,axiom,
% 0.30/0.50 ! [N2: num] :
% 0.30/0.50 ( ( numeral_numeral_int @ ( bit0 @ N2 ) )
% 0.30/0.50 = ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % numeral_code(2)
% 0.30/0.50 thf(fact_727_lift__Suc__mono__less,axiom,
% 0.30/0.50 ! [F: nat > real,N2: nat,N5: nat] :
% 0.30/0.50 ( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.30/0.50 => ( ( ord_less_nat @ N2 @ N5 )
% 0.30/0.50 => ( ord_less_real @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % lift_Suc_mono_less
% 0.30/0.50 thf(fact_728_lift__Suc__mono__less,axiom,
% 0.30/0.50 ! [F: nat > rat,N2: nat,N5: nat] :
% 0.30/0.50 ( ! [N4: nat] : ( ord_less_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.30/0.50 => ( ( ord_less_nat @ N2 @ N5 )
% 0.30/0.50 => ( ord_less_rat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % lift_Suc_mono_less
% 0.30/0.50 thf(fact_729_lift__Suc__mono__less,axiom,
% 0.30/0.50 ! [F: nat > num,N2: nat,N5: nat] :
% 0.30/0.50 ( ! [N4: nat] : ( ord_less_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.30/0.50 => ( ( ord_less_nat @ N2 @ N5 )
% 0.30/0.50 => ( ord_less_num @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % lift_Suc_mono_less
% 0.30/0.50 thf(fact_730_lift__Suc__mono__less,axiom,
% 0.30/0.50 ! [F: nat > nat,N2: nat,N5: nat] :
% 0.30/0.50 ( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.30/0.50 => ( ( ord_less_nat @ N2 @ N5 )
% 0.30/0.50 => ( ord_less_nat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % lift_Suc_mono_less
% 0.30/0.50 thf(fact_731_lift__Suc__mono__less,axiom,
% 0.30/0.50 ! [F: nat > int,N2: nat,N5: nat] :
% 0.30/0.50 ( ! [N4: nat] : ( ord_less_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.30/0.50 => ( ( ord_less_nat @ N2 @ N5 )
% 0.30/0.50 => ( ord_less_int @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % lift_Suc_mono_less
% 0.30/0.50 thf(fact_732_lift__Suc__mono__less__iff,axiom,
% 0.30/0.50 ! [F: nat > real,N2: nat,M: nat] :
% 0.30/0.50 ( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.30/0.50 => ( ( ord_less_real @ ( F @ N2 ) @ ( F @ M ) )
% 0.30/0.50 = ( ord_less_nat @ N2 @ M ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % lift_Suc_mono_less_iff
% 0.30/0.50 thf(fact_733_lift__Suc__mono__less__iff,axiom,
% 0.30/0.50 ! [F: nat > rat,N2: nat,M: nat] :
% 0.30/0.50 ( ! [N4: nat] : ( ord_less_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.30/0.50 => ( ( ord_less_rat @ ( F @ N2 ) @ ( F @ M ) )
% 0.30/0.50 = ( ord_less_nat @ N2 @ M ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % lift_Suc_mono_less_iff
% 0.30/0.50 thf(fact_734_lift__Suc__mono__less__iff,axiom,
% 0.30/0.50 ! [F: nat > num,N2: nat,M: nat] :
% 0.30/0.50 ( ! [N4: nat] : ( ord_less_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.30/0.50 => ( ( ord_less_num @ ( F @ N2 ) @ ( F @ M ) )
% 0.30/0.50 = ( ord_less_nat @ N2 @ M ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % lift_Suc_mono_less_iff
% 0.30/0.50 thf(fact_735_lift__Suc__mono__less__iff,axiom,
% 0.30/0.50 ! [F: nat > nat,N2: nat,M: nat] :
% 0.30/0.50 ( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.30/0.50 => ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M ) )
% 0.30/0.50 = ( ord_less_nat @ N2 @ M ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % lift_Suc_mono_less_iff
% 0.30/0.50 thf(fact_736_lift__Suc__mono__less__iff,axiom,
% 0.30/0.50 ! [F: nat > int,N2: nat,M: nat] :
% 0.30/0.50 ( ! [N4: nat] : ( ord_less_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.30/0.50 => ( ( ord_less_int @ ( F @ N2 ) @ ( F @ M ) )
% 0.30/0.50 = ( ord_less_nat @ N2 @ M ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % lift_Suc_mono_less_iff
% 0.30/0.50 thf(fact_737_lift__Suc__mono__le,axiom,
% 0.30/0.50 ! [F: nat > set_nat,N2: nat,N5: nat] :
% 0.30/0.50 ( ! [N4: nat] : ( ord_less_eq_set_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.30/0.50 => ( ( ord_less_eq_nat @ N2 @ N5 )
% 0.30/0.50 => ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % lift_Suc_mono_le
% 0.30/0.50 thf(fact_738_lift__Suc__mono__le,axiom,
% 0.30/0.50 ! [F: nat > rat,N2: nat,N5: nat] :
% 0.30/0.50 ( ! [N4: nat] : ( ord_less_eq_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.30/0.50 => ( ( ord_less_eq_nat @ N2 @ N5 )
% 0.30/0.50 => ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % lift_Suc_mono_le
% 0.30/0.50 thf(fact_739_lift__Suc__mono__le,axiom,
% 0.30/0.50 ! [F: nat > num,N2: nat,N5: nat] :
% 0.30/0.50 ( ! [N4: nat] : ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.30/0.50 => ( ( ord_less_eq_nat @ N2 @ N5 )
% 0.30/0.50 => ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % lift_Suc_mono_le
% 0.30/0.50 thf(fact_740_lift__Suc__mono__le,axiom,
% 0.30/0.50 ! [F: nat > nat,N2: nat,N5: nat] :
% 0.30/0.50 ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.30/0.50 => ( ( ord_less_eq_nat @ N2 @ N5 )
% 0.30/0.50 => ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % lift_Suc_mono_le
% 0.30/0.50 thf(fact_741_lift__Suc__mono__le,axiom,
% 0.30/0.50 ! [F: nat > int,N2: nat,N5: nat] :
% 0.30/0.50 ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.30/0.50 => ( ( ord_less_eq_nat @ N2 @ N5 )
% 0.30/0.50 => ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % lift_Suc_mono_le
% 0.30/0.50 thf(fact_742_lift__Suc__antimono__le,axiom,
% 0.30/0.50 ! [F: nat > set_nat,N2: nat,N5: nat] :
% 0.30/0.50 ( ! [N4: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 0.30/0.50 => ( ( ord_less_eq_nat @ N2 @ N5 )
% 0.30/0.50 => ( ord_less_eq_set_nat @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % lift_Suc_antimono_le
% 0.30/0.50 thf(fact_743_lift__Suc__antimono__le,axiom,
% 0.30/0.50 ! [F: nat > rat,N2: nat,N5: nat] :
% 0.30/0.50 ( ! [N4: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 0.30/0.50 => ( ( ord_less_eq_nat @ N2 @ N5 )
% 0.30/0.50 => ( ord_less_eq_rat @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % lift_Suc_antimono_le
% 0.30/0.50 thf(fact_744_lift__Suc__antimono__le,axiom,
% 0.30/0.50 ! [F: nat > num,N2: nat,N5: nat] :
% 0.30/0.50 ( ! [N4: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 0.30/0.50 => ( ( ord_less_eq_nat @ N2 @ N5 )
% 0.30/0.50 => ( ord_less_eq_num @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % lift_Suc_antimono_le
% 0.30/0.50 thf(fact_745_lift__Suc__antimono__le,axiom,
% 0.30/0.50 ! [F: nat > nat,N2: nat,N5: nat] :
% 0.30/0.50 ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 0.30/0.50 => ( ( ord_less_eq_nat @ N2 @ N5 )
% 0.30/0.50 => ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % lift_Suc_antimono_le
% 0.30/0.50 thf(fact_746_lift__Suc__antimono__le,axiom,
% 0.30/0.50 ! [F: nat > int,N2: nat,N5: nat] :
% 0.30/0.50 ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 0.30/0.50 => ( ( ord_less_eq_nat @ N2 @ N5 )
% 0.30/0.50 => ( ord_less_eq_int @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % lift_Suc_antimono_le
% 0.30/0.50 thf(fact_747_diff__less__mono,axiom,
% 0.30/0.50 ! [A: nat,B: nat,C: nat] :
% 0.30/0.50 ( ( ord_less_nat @ A @ B )
% 0.30/0.50 => ( ( ord_less_eq_nat @ C @ A )
% 0.30/0.50 => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_less_mono
% 0.30/0.50 thf(fact_748_less__diff__iff,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ K @ M )
% 0.30/0.50 => ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.50 => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
% 0.30/0.50 = ( ord_less_nat @ M @ N2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_diff_iff
% 0.30/0.50 thf(fact_749_less__diff__conv,axiom,
% 0.30/0.50 ! [I2: nat,J: nat,K: nat] :
% 0.30/0.50 ( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
% 0.30/0.50 = ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_diff_conv
% 0.30/0.50 thf(fact_750_add__diff__inverse__nat,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ~ ( ord_less_nat @ M @ N2 )
% 0.30/0.50 => ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M @ N2 ) )
% 0.30/0.50 = M ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_diff_inverse_nat
% 0.30/0.50 thf(fact_751_Nat_Ole__imp__diff__is__add,axiom,
% 0.30/0.50 ! [I2: nat,J: nat,K: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.50 => ( ( ( minus_minus_nat @ J @ I2 )
% 0.30/0.50 = K )
% 0.30/0.50 = ( J
% 0.30/0.50 = ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Nat.le_imp_diff_is_add
% 0.30/0.50 thf(fact_752_Nat_Odiff__add__assoc2,axiom,
% 0.30/0.50 ! [K: nat,J: nat,I2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ K @ J )
% 0.30/0.50 => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K )
% 0.30/0.50 = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Nat.diff_add_assoc2
% 0.30/0.50 thf(fact_753_Nat_Odiff__add__assoc,axiom,
% 0.30/0.50 ! [K: nat,J: nat,I2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ K @ J )
% 0.30/0.50 => ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
% 0.30/0.50 = ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Nat.diff_add_assoc
% 0.30/0.50 thf(fact_754_Nat_Ole__diff__conv2,axiom,
% 0.30/0.50 ! [K: nat,J: nat,I2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ K @ J )
% 0.30/0.50 => ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
% 0.30/0.50 = ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Nat.le_diff_conv2
% 0.30/0.50 thf(fact_755_le__diff__conv,axiom,
% 0.30/0.50 ! [J: nat,K: nat,I2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
% 0.30/0.50 = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_diff_conv
% 0.30/0.50 thf(fact_756_le__imp__less__Suc,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.50 => ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_imp_less_Suc
% 0.30/0.50 thf(fact_757_less__eq__Suc__le,axiom,
% 0.30/0.50 ( ord_less_nat
% 0.30/0.50 = ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_eq_Suc_le
% 0.30/0.50 thf(fact_758_less__Suc__eq__le,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_nat @ M @ ( suc @ N2 ) )
% 0.30/0.50 = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_Suc_eq_le
% 0.30/0.50 thf(fact_759_le__less__Suc__eq,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.50 => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
% 0.30/0.50 = ( N2 = M ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_less_Suc_eq
% 0.30/0.50 thf(fact_760_Suc__le__lessD,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
% 0.30/0.50 => ( ord_less_nat @ M @ N2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_le_lessD
% 0.30/0.50 thf(fact_761_inc__induct,axiom,
% 0.30/0.50 ! [I2: nat,J: nat,P: nat > $o] :
% 0.30/0.50 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.50 => ( ( P @ J )
% 0.30/0.50 => ( ! [N4: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ I2 @ N4 )
% 0.30/0.50 => ( ( ord_less_nat @ N4 @ J )
% 0.30/0.50 => ( ( P @ ( suc @ N4 ) )
% 0.30/0.50 => ( P @ N4 ) ) ) )
% 0.30/0.50 => ( P @ I2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % inc_induct
% 0.30/0.50 thf(fact_762_dec__induct,axiom,
% 0.30/0.50 ! [I2: nat,J: nat,P: nat > $o] :
% 0.30/0.50 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.50 => ( ( P @ I2 )
% 0.30/0.50 => ( ! [N4: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ I2 @ N4 )
% 0.30/0.50 => ( ( ord_less_nat @ N4 @ J )
% 0.30/0.50 => ( ( P @ N4 )
% 0.30/0.50 => ( P @ ( suc @ N4 ) ) ) ) )
% 0.30/0.50 => ( P @ J ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % dec_induct
% 0.30/0.50 thf(fact_763_Suc__le__eq,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
% 0.30/0.50 = ( ord_less_nat @ M @ N2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_le_eq
% 0.30/0.50 thf(fact_764_Suc__leI,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_nat @ M @ N2 )
% 0.30/0.50 => ( ord_less_eq_nat @ ( suc @ M ) @ N2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_leI
% 0.30/0.50 thf(fact_765_less__natE,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_nat @ M @ N2 )
% 0.30/0.50 => ~ ! [Q3: nat] :
% 0.30/0.50 ( N2
% 0.30/0.50 != ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_natE
% 0.30/0.50 thf(fact_766_less__add__Suc1,axiom,
% 0.30/0.50 ! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_add_Suc1
% 0.30/0.50 thf(fact_767_less__add__Suc2,axiom,
% 0.30/0.50 ! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M @ I2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_add_Suc2
% 0.30/0.50 thf(fact_768_less__iff__Suc__add,axiom,
% 0.30/0.50 ( ord_less_nat
% 0.30/0.50 = ( ^ [M6: nat,N: nat] :
% 0.30/0.50 ? [K3: nat] :
% 0.30/0.50 ( N
% 0.30/0.50 = ( suc @ ( plus_plus_nat @ M6 @ K3 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_iff_Suc_add
% 0.30/0.50 thf(fact_769_less__imp__Suc__add,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_nat @ M @ N2 )
% 0.30/0.50 => ? [K2: nat] :
% 0.30/0.50 ( N2
% 0.30/0.50 = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_imp_Suc_add
% 0.30/0.50 thf(fact_770_Suc__mult__less__cancel1,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
% 0.30/0.50 = ( ord_less_nat @ M @ N2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_mult_less_cancel1
% 0.30/0.50 thf(fact_771_Suc__mult__le__cancel1,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
% 0.30/0.50 = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_mult_le_cancel1
% 0.30/0.50 thf(fact_772_mult__Suc,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( times_times_nat @ ( suc @ M ) @ N2 )
% 0.30/0.50 = ( plus_plus_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_Suc
% 0.30/0.50 thf(fact_773_Suc__eq__plus1,axiom,
% 0.30/0.50 ( suc
% 0.30/0.50 = ( ^ [N: nat] : ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_eq_plus1
% 0.30/0.50 thf(fact_774_plus__1__eq__Suc,axiom,
% 0.30/0.50 ( ( plus_plus_nat @ one_one_nat )
% 0.30/0.50 = suc ) ).
% 0.30/0.50
% 0.30/0.50 % plus_1_eq_Suc
% 0.30/0.50 thf(fact_775_Suc__eq__plus1__left,axiom,
% 0.30/0.50 ( suc
% 0.30/0.50 = ( plus_plus_nat @ one_one_nat ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_eq_plus1_left
% 0.30/0.50 thf(fact_776_power__numeral__even,axiom,
% 0.30/0.50 ! [Z: complex,W: num] :
% 0.30/0.50 ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 0.30/0.50 = ( times_times_complex @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_numeral_even
% 0.30/0.50 thf(fact_777_power__numeral__even,axiom,
% 0.30/0.50 ! [Z: real,W: num] :
% 0.30/0.50 ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 0.30/0.50 = ( times_times_real @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_numeral_even
% 0.30/0.50 thf(fact_778_power__numeral__even,axiom,
% 0.30/0.50 ! [Z: rat,W: num] :
% 0.30/0.50 ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 0.30/0.50 = ( times_times_rat @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_numeral_even
% 0.30/0.50 thf(fact_779_power__numeral__even,axiom,
% 0.30/0.50 ! [Z: nat,W: num] :
% 0.30/0.50 ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 0.30/0.50 = ( times_times_nat @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_numeral_even
% 0.30/0.50 thf(fact_780_power__numeral__even,axiom,
% 0.30/0.50 ! [Z: int,W: num] :
% 0.30/0.50 ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 0.30/0.50 = ( times_times_int @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_numeral_even
% 0.30/0.50 thf(fact_781_mult__diff__mult,axiom,
% 0.30/0.50 ! [X: real,Y: real,A: real,B: real] :
% 0.30/0.50 ( ( minus_minus_real @ ( times_times_real @ X @ Y ) @ ( times_times_real @ A @ B ) )
% 0.30/0.50 = ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_diff_mult
% 0.30/0.50 thf(fact_782_mult__diff__mult,axiom,
% 0.30/0.50 ! [X: rat,Y: rat,A: rat,B: rat] :
% 0.30/0.50 ( ( minus_minus_rat @ ( times_times_rat @ X @ Y ) @ ( times_times_rat @ A @ B ) )
% 0.30/0.50 = ( plus_plus_rat @ ( times_times_rat @ X @ ( minus_minus_rat @ Y @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X @ A ) @ B ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_diff_mult
% 0.30/0.50 thf(fact_783_mult__diff__mult,axiom,
% 0.30/0.50 ! [X: int,Y: int,A: int,B: int] :
% 0.30/0.50 ( ( minus_minus_int @ ( times_times_int @ X @ Y ) @ ( times_times_int @ A @ B ) )
% 0.30/0.50 = ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A ) @ B ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % mult_diff_mult
% 0.30/0.50 thf(fact_784_less__diff__conv2,axiom,
% 0.30/0.50 ! [K: nat,J: nat,I2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ K @ J )
% 0.30/0.50 => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
% 0.30/0.50 = ( ord_less_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_diff_conv2
% 0.30/0.50 thf(fact_785_power__Suc2,axiom,
% 0.30/0.50 ! [A: complex,N2: nat] :
% 0.30/0.50 ( ( power_power_complex @ A @ ( suc @ N2 ) )
% 0.30/0.50 = ( times_times_complex @ ( power_power_complex @ A @ N2 ) @ A ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_Suc2
% 0.30/0.50 thf(fact_786_power__Suc2,axiom,
% 0.30/0.50 ! [A: real,N2: nat] :
% 0.30/0.50 ( ( power_power_real @ A @ ( suc @ N2 ) )
% 0.30/0.50 = ( times_times_real @ ( power_power_real @ A @ N2 ) @ A ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_Suc2
% 0.30/0.50 thf(fact_787_power__Suc2,axiom,
% 0.30/0.50 ! [A: rat,N2: nat] :
% 0.30/0.50 ( ( power_power_rat @ A @ ( suc @ N2 ) )
% 0.30/0.50 = ( times_times_rat @ ( power_power_rat @ A @ N2 ) @ A ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_Suc2
% 0.30/0.50 thf(fact_788_power__Suc2,axiom,
% 0.30/0.50 ! [A: nat,N2: nat] :
% 0.30/0.50 ( ( power_power_nat @ A @ ( suc @ N2 ) )
% 0.30/0.50 = ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ A ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_Suc2
% 0.30/0.50 thf(fact_789_power__Suc2,axiom,
% 0.30/0.50 ! [A: int,N2: nat] :
% 0.30/0.50 ( ( power_power_int @ A @ ( suc @ N2 ) )
% 0.30/0.50 = ( times_times_int @ ( power_power_int @ A @ N2 ) @ A ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_Suc2
% 0.30/0.50 thf(fact_790_power__Suc,axiom,
% 0.30/0.50 ! [A: complex,N2: nat] :
% 0.30/0.50 ( ( power_power_complex @ A @ ( suc @ N2 ) )
% 0.30/0.50 = ( times_times_complex @ A @ ( power_power_complex @ A @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_Suc
% 0.30/0.50 thf(fact_791_power__Suc,axiom,
% 0.30/0.50 ! [A: real,N2: nat] :
% 0.30/0.50 ( ( power_power_real @ A @ ( suc @ N2 ) )
% 0.30/0.50 = ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_Suc
% 0.30/0.50 thf(fact_792_power__Suc,axiom,
% 0.30/0.50 ! [A: rat,N2: nat] :
% 0.30/0.50 ( ( power_power_rat @ A @ ( suc @ N2 ) )
% 0.30/0.50 = ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_Suc
% 0.30/0.50 thf(fact_793_power__Suc,axiom,
% 0.30/0.50 ! [A: nat,N2: nat] :
% 0.30/0.50 ( ( power_power_nat @ A @ ( suc @ N2 ) )
% 0.30/0.50 = ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_Suc
% 0.30/0.50 thf(fact_794_power__Suc,axiom,
% 0.30/0.50 ! [A: int,N2: nat] :
% 0.30/0.50 ( ( power_power_int @ A @ ( suc @ N2 ) )
% 0.30/0.50 = ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_Suc
% 0.30/0.50 thf(fact_795_Suc__div__le__mono,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ ( divide_divide_nat @ ( suc @ M ) @ N2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_div_le_mono
% 0.30/0.50 thf(fact_796_power__gt1,axiom,
% 0.30/0.50 ! [A: real,N2: nat] :
% 0.30/0.50 ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.50 => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_gt1
% 0.30/0.50 thf(fact_797_power__gt1,axiom,
% 0.30/0.50 ! [A: rat,N2: nat] :
% 0.30/0.50 ( ( ord_less_rat @ one_one_rat @ A )
% 0.30/0.50 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_gt1
% 0.30/0.50 thf(fact_798_power__gt1,axiom,
% 0.30/0.50 ! [A: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_nat @ one_one_nat @ A )
% 0.30/0.50 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_gt1
% 0.30/0.50 thf(fact_799_power__gt1,axiom,
% 0.30/0.50 ! [A: int,N2: nat] :
% 0.30/0.50 ( ( ord_less_int @ one_one_int @ A )
% 0.30/0.50 => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_gt1
% 0.30/0.50 thf(fact_800_nat__diff__add__eq2,axiom,
% 0.30/0.50 ! [I2: nat,J: nat,U: nat,M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.50 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 0.30/0.50 = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_diff_add_eq2
% 0.30/0.50 thf(fact_801_nat__diff__add__eq1,axiom,
% 0.30/0.50 ! [J: nat,I2: nat,U: nat,M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ J @ I2 )
% 0.30/0.50 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 0.30/0.50 = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_diff_add_eq1
% 0.30/0.50 thf(fact_802_nat__le__add__iff2,axiom,
% 0.30/0.50 ! [I2: nat,J: nat,U: nat,M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.50 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 0.30/0.50 = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_le_add_iff2
% 0.30/0.50 thf(fact_803_nat__le__add__iff1,axiom,
% 0.30/0.50 ! [J: nat,I2: nat,U: nat,M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ J @ I2 )
% 0.30/0.50 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 0.30/0.50 = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_le_add_iff1
% 0.30/0.50 thf(fact_804_nat__eq__add__iff2,axiom,
% 0.30/0.50 ! [I2: nat,J: nat,U: nat,M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.50 => ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M )
% 0.30/0.50 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 0.30/0.50 = ( M
% 0.30/0.50 = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_eq_add_iff2
% 0.30/0.50 thf(fact_805_nat__eq__add__iff1,axiom,
% 0.30/0.50 ! [J: nat,I2: nat,U: nat,M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ J @ I2 )
% 0.30/0.50 => ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M )
% 0.30/0.50 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 0.30/0.50 = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M )
% 0.30/0.50 = N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_eq_add_iff1
% 0.30/0.50 thf(fact_806_nat__neq__iff,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( M != N2 )
% 0.30/0.50 = ( ( ord_less_nat @ M @ N2 )
% 0.30/0.50 | ( ord_less_nat @ N2 @ M ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_neq_iff
% 0.30/0.50 thf(fact_807_less__not__refl,axiom,
% 0.30/0.50 ! [N2: nat] :
% 0.30/0.50 ~ ( ord_less_nat @ N2 @ N2 ) ).
% 0.30/0.50
% 0.30/0.50 % less_not_refl
% 0.30/0.50 thf(fact_808_less__not__refl2,axiom,
% 0.30/0.50 ! [N2: nat,M: nat] :
% 0.30/0.50 ( ( ord_less_nat @ N2 @ M )
% 0.30/0.50 => ( M != N2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_not_refl2
% 0.30/0.50 thf(fact_809_less__not__refl3,axiom,
% 0.30/0.50 ! [S3: nat,T: nat] :
% 0.30/0.50 ( ( ord_less_nat @ S3 @ T )
% 0.30/0.50 => ( S3 != T ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_not_refl3
% 0.30/0.50 thf(fact_810_less__irrefl__nat,axiom,
% 0.30/0.50 ! [N2: nat] :
% 0.30/0.50 ~ ( ord_less_nat @ N2 @ N2 ) ).
% 0.30/0.50
% 0.30/0.50 % less_irrefl_nat
% 0.30/0.50 thf(fact_811_nat__less__induct,axiom,
% 0.30/0.50 ! [P: nat > $o,N2: nat] :
% 0.30/0.50 ( ! [N4: nat] :
% 0.30/0.50 ( ! [M5: nat] :
% 0.30/0.50 ( ( ord_less_nat @ M5 @ N4 )
% 0.30/0.50 => ( P @ M5 ) )
% 0.30/0.50 => ( P @ N4 ) )
% 0.30/0.50 => ( P @ N2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_less_induct
% 0.30/0.50 thf(fact_812_infinite__descent,axiom,
% 0.30/0.50 ! [P: nat > $o,N2: nat] :
% 0.30/0.50 ( ! [N4: nat] :
% 0.30/0.50 ( ~ ( P @ N4 )
% 0.30/0.50 => ? [M5: nat] :
% 0.30/0.50 ( ( ord_less_nat @ M5 @ N4 )
% 0.30/0.50 & ~ ( P @ M5 ) ) )
% 0.30/0.50 => ( P @ N2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % infinite_descent
% 0.30/0.50 thf(fact_813_linorder__neqE__nat,axiom,
% 0.30/0.50 ! [X: nat,Y: nat] :
% 0.30/0.50 ( ( X != Y )
% 0.30/0.50 => ( ~ ( ord_less_nat @ X @ Y )
% 0.30/0.50 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % linorder_neqE_nat
% 0.30/0.50 thf(fact_814_le__refl,axiom,
% 0.30/0.50 ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% 0.30/0.50
% 0.30/0.50 % le_refl
% 0.30/0.50 thf(fact_815_le__trans,axiom,
% 0.30/0.50 ! [I2: nat,J: nat,K: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.50 => ( ( ord_less_eq_nat @ J @ K )
% 0.30/0.50 => ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_trans
% 0.30/0.50 thf(fact_816_eq__imp__le,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( M = N2 )
% 0.30/0.50 => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % eq_imp_le
% 0.30/0.50 thf(fact_817_le__antisym,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.50 => ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.50 => ( M = N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_antisym
% 0.30/0.50 thf(fact_818_nat__le__linear,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.50 | ( ord_less_eq_nat @ N2 @ M ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_le_linear
% 0.30/0.50 thf(fact_819_Nat_Oex__has__greatest__nat,axiom,
% 0.30/0.50 ! [P: nat > $o,K: nat,B: nat] :
% 0.30/0.50 ( ( P @ K )
% 0.30/0.50 => ( ! [Y4: nat] :
% 0.30/0.50 ( ( P @ Y4 )
% 0.30/0.50 => ( ord_less_eq_nat @ Y4 @ B ) )
% 0.30/0.50 => ? [X4: nat] :
% 0.30/0.50 ( ( P @ X4 )
% 0.30/0.50 & ! [Y5: nat] :
% 0.30/0.50 ( ( P @ Y5 )
% 0.30/0.50 => ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % Nat.ex_has_greatest_nat
% 0.30/0.50 thf(fact_820_size__neq__size__imp__neq,axiom,
% 0.30/0.50 ! [X: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
% 0.30/0.50 ( ( ( size_s6755466524823107622T_VEBT @ X )
% 0.30/0.50 != ( size_s6755466524823107622T_VEBT @ Y ) )
% 0.30/0.50 => ( X != Y ) ) ).
% 0.30/0.50
% 0.30/0.50 % size_neq_size_imp_neq
% 0.30/0.50 thf(fact_821_size__neq__size__imp__neq,axiom,
% 0.30/0.50 ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
% 0.30/0.50 ( ( ( size_size_VEBT_VEBT @ X )
% 0.30/0.50 != ( size_size_VEBT_VEBT @ Y ) )
% 0.30/0.50 => ( X != Y ) ) ).
% 0.30/0.50
% 0.30/0.50 % size_neq_size_imp_neq
% 0.30/0.50 thf(fact_822_size__neq__size__imp__neq,axiom,
% 0.30/0.50 ! [X: num,Y: num] :
% 0.30/0.50 ( ( ( size_size_num @ X )
% 0.30/0.50 != ( size_size_num @ Y ) )
% 0.30/0.50 => ( X != Y ) ) ).
% 0.30/0.50
% 0.30/0.50 % size_neq_size_imp_neq
% 0.30/0.50 thf(fact_823_size__neq__size__imp__neq,axiom,
% 0.30/0.50 ! [X: list_o,Y: list_o] :
% 0.30/0.50 ( ( ( size_size_list_o @ X )
% 0.30/0.50 != ( size_size_list_o @ Y ) )
% 0.30/0.50 => ( X != Y ) ) ).
% 0.30/0.50
% 0.30/0.50 % size_neq_size_imp_neq
% 0.30/0.50 thf(fact_824_size__neq__size__imp__neq,axiom,
% 0.30/0.50 ! [X: list_int,Y: list_int] :
% 0.30/0.50 ( ( ( size_size_list_int @ X )
% 0.30/0.50 != ( size_size_list_int @ Y ) )
% 0.30/0.50 => ( X != Y ) ) ).
% 0.30/0.50
% 0.30/0.50 % size_neq_size_imp_neq
% 0.30/0.50 thf(fact_825_power2__commute,axiom,
% 0.30/0.50 ! [X: complex,Y: complex] :
% 0.30/0.50 ( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( power_power_complex @ ( minus_minus_complex @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power2_commute
% 0.30/0.50 thf(fact_826_power2__commute,axiom,
% 0.30/0.50 ! [X: real,Y: real] :
% 0.30/0.50 ( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( power_power_real @ ( minus_minus_real @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power2_commute
% 0.30/0.50 thf(fact_827_power2__commute,axiom,
% 0.30/0.50 ! [X: rat,Y: rat] :
% 0.30/0.50 ( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( power_power_rat @ ( minus_minus_rat @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power2_commute
% 0.30/0.50 thf(fact_828_power2__commute,axiom,
% 0.30/0.50 ! [X: int,Y: int] :
% 0.30/0.50 ( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( power_power_int @ ( minus_minus_int @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power2_commute
% 0.30/0.50 thf(fact_829_nat__less__add__iff2,axiom,
% 0.30/0.50 ! [I2: nat,J: nat,U: nat,M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.50 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 0.30/0.50 = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_less_add_iff2
% 0.30/0.50 thf(fact_830_nat__less__add__iff1,axiom,
% 0.30/0.50 ! [J: nat,I2: nat,U: nat,M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ J @ I2 )
% 0.30/0.50 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 0.30/0.50 = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_less_add_iff1
% 0.30/0.50 thf(fact_831_div__nat__eqI,axiom,
% 0.30/0.50 ! [N2: nat,Q2: nat,M: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q2 ) @ M )
% 0.30/0.50 => ( ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q2 ) ) )
% 0.30/0.50 => ( ( divide_divide_nat @ M @ N2 )
% 0.30/0.50 = Q2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % div_nat_eqI
% 0.30/0.50 thf(fact_832_Suc__nat__number__of__add,axiom,
% 0.30/0.50 ! [V: num,N2: nat] :
% 0.30/0.50 ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N2 ) )
% 0.30/0.50 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % Suc_nat_number_of_add
% 0.30/0.50 thf(fact_833_diff__le__diff__pow,axiom,
% 0.30/0.50 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.30/0.50 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % diff_le_diff_pow
% 0.30/0.50 thf(fact_834_vebt__delete_Osimps_I7_J,axiom,
% 0.30/0.50 ! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.50 ( ( ( ( ord_less_nat @ X @ Mi )
% 0.30/0.50 | ( ord_less_nat @ Ma @ X ) )
% 0.30/0.50 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
% 0.30/0.50 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) )
% 0.30/0.50 & ( ~ ( ( ord_less_nat @ X @ Mi )
% 0.30/0.50 | ( ord_less_nat @ Ma @ X ) )
% 0.30/0.50 => ( ( ( ( X = Mi )
% 0.30/0.50 & ( X = Ma ) )
% 0.30/0.50 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
% 0.30/0.50 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) )
% 0.30/0.50 & ( ~ ( ( X = Mi )
% 0.30/0.50 & ( X = Ma ) )
% 0.30/0.50 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
% 0.30/0.50 = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.50 @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.50 @ ( vEBT_Node
% 0.30/0.50 @ ( some_P7363390416028606310at_nat
% 0.30/0.50 @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ Mi )
% 0.30/0.50 @ ( if_nat
% 0.30/0.50 @ ( ( ( X = Mi )
% 0.30/0.50 => ( ( 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 ) ) ) ) ) )
% 0.30/0.50 = Ma ) )
% 0.30/0.50 & ( ( X != Mi )
% 0.30/0.50 => ( X = Ma ) ) )
% 0.30/0.50 @ ( if_nat
% 0.30/0.50 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.50 = none_nat )
% 0.30/0.50 @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ Mi )
% 0.30/0.50 @ ( 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 @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( 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 @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( 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 @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
% 0.30/0.50 @ Ma ) ) )
% 0.30/0.50 @ ( suc @ ( suc @ Va ) )
% 0.30/0.50 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.50 @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.50 @ ( vEBT_Node
% 0.30/0.50 @ ( some_P7363390416028606310at_nat
% 0.30/0.50 @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ Mi )
% 0.30/0.50 @ ( if_nat
% 0.30/0.50 @ ( ( ( X = Mi )
% 0.30/0.50 => ( ( 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 ) ) ) ) ) )
% 0.30/0.50 = Ma ) )
% 0.30/0.50 & ( ( X != Mi )
% 0.30/0.50 => ( X = Ma ) ) )
% 0.30/0.50 @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( 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 @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.30/0.50 @ Ma ) ) )
% 0.30/0.50 @ ( suc @ ( suc @ Va ) )
% 0.30/0.50 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.50 @ Summary ) )
% 0.30/0.50 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % vebt_delete.simps(7)
% 0.30/0.50 thf(fact_835_power__odd__eq,axiom,
% 0.30/0.50 ! [A: complex,N2: nat] :
% 0.30/0.50 ( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.30/0.50 = ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_odd_eq
% 0.30/0.50 thf(fact_836_power__odd__eq,axiom,
% 0.30/0.50 ! [A: real,N2: nat] :
% 0.30/0.50 ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.30/0.50 = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_odd_eq
% 0.30/0.50 thf(fact_837_power__odd__eq,axiom,
% 0.30/0.50 ! [A: rat,N2: nat] :
% 0.30/0.50 ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.30/0.50 = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_odd_eq
% 0.30/0.50 thf(fact_838_power__odd__eq,axiom,
% 0.30/0.50 ! [A: nat,N2: nat] :
% 0.30/0.50 ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.30/0.50 = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_odd_eq
% 0.30/0.50 thf(fact_839_power__odd__eq,axiom,
% 0.30/0.50 ! [A: int,N2: nat] :
% 0.30/0.50 ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.30/0.50 = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power_odd_eq
% 0.30/0.50 thf(fact_840_power2__diff,axiom,
% 0.30/0.50 ! [X: complex,Y: complex] :
% 0.30/0.50 ( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power2_diff
% 0.30/0.50 thf(fact_841_power2__diff,axiom,
% 0.30/0.50 ! [X: real,Y: real] :
% 0.30/0.50 ( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power2_diff
% 0.30/0.50 thf(fact_842_power2__diff,axiom,
% 0.30/0.50 ! [X: rat,Y: rat] :
% 0.30/0.50 ( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power2_diff
% 0.30/0.50 thf(fact_843_power2__diff,axiom,
% 0.30/0.50 ! [X: int,Y: int] :
% 0.30/0.50 ( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.50 = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % power2_diff
% 0.30/0.50 thf(fact_844_nat__less__le,axiom,
% 0.30/0.50 ( ord_less_nat
% 0.30/0.50 = ( ^ [M6: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ M6 @ N )
% 0.30/0.50 & ( M6 != N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % nat_less_le
% 0.30/0.50 thf(fact_845_less__imp__le__nat,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_nat @ M @ N2 )
% 0.30/0.50 => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_imp_le_nat
% 0.30/0.50 thf(fact_846_le__eq__less__or__eq,axiom,
% 0.30/0.50 ( ord_less_eq_nat
% 0.30/0.50 = ( ^ [M6: nat,N: nat] :
% 0.30/0.50 ( ( ord_less_nat @ M6 @ N )
% 0.30/0.50 | ( M6 = N ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_eq_less_or_eq
% 0.30/0.50 thf(fact_847_less__or__eq__imp__le,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( ( ord_less_nat @ M @ N2 )
% 0.30/0.50 | ( M = N2 ) )
% 0.30/0.50 => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_or_eq_imp_le
% 0.30/0.50 thf(fact_848_le__neq__implies__less,axiom,
% 0.30/0.50 ! [M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.50 => ( ( M != N2 )
% 0.30/0.50 => ( ord_less_nat @ M @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % le_neq_implies_less
% 0.30/0.50 thf(fact_849_less__mono__imp__le__mono,axiom,
% 0.30/0.50 ! [F: nat > nat,I2: nat,J: nat] :
% 0.30/0.50 ( ! [I4: nat,J2: nat] :
% 0.30/0.50 ( ( ord_less_nat @ I4 @ J2 )
% 0.30/0.50 => ( ord_less_nat @ ( F @ I4 ) @ ( F @ J2 ) ) )
% 0.30/0.50 => ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.50 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_mono_imp_le_mono
% 0.30/0.50 thf(fact_850_less__add__eq__less,axiom,
% 0.30/0.50 ! [K: nat,L2: nat,M: nat,N2: nat] :
% 0.30/0.50 ( ( ord_less_nat @ K @ L2 )
% 0.30/0.50 => ( ( ( plus_plus_nat @ M @ L2 )
% 0.30/0.50 = ( plus_plus_nat @ K @ N2 ) )
% 0.30/0.50 => ( ord_less_nat @ M @ N2 ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % less_add_eq_less
% 0.30/0.50 thf(fact_851_trans__less__add2,axiom,
% 0.30/0.50 ! [I2: nat,J: nat,M: nat] :
% 0.30/0.50 ( ( ord_less_nat @ I2 @ J )
% 0.30/0.50 => ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % trans_less_add2
% 0.30/0.50 thf(fact_852_trans__less__add1,axiom,
% 0.30/0.50 ! [I2: nat,J: nat,M: nat] :
% 0.30/0.50 ( ( ord_less_nat @ I2 @ J )
% 0.30/0.50 => ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % trans_less_add1
% 0.30/0.50 thf(fact_853_add__less__mono1,axiom,
% 0.30/0.50 ! [I2: nat,J: nat,K: nat] :
% 0.30/0.50 ( ( ord_less_nat @ I2 @ J )
% 0.30/0.50 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_less_mono1
% 0.30/0.50 thf(fact_854_not__add__less2,axiom,
% 0.30/0.50 ! [J: nat,I2: nat] :
% 0.30/0.50 ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% 0.30/0.50
% 0.30/0.50 % not_add_less2
% 0.30/0.50 thf(fact_855_not__add__less1,axiom,
% 0.30/0.50 ! [I2: nat,J: nat] :
% 0.30/0.50 ~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% 0.30/0.50
% 0.30/0.50 % not_add_less1
% 0.30/0.50 thf(fact_856_add__less__mono,axiom,
% 0.30/0.50 ! [I2: nat,J: nat,K: nat,L2: nat] :
% 0.30/0.50 ( ( ord_less_nat @ I2 @ J )
% 0.30/0.50 => ( ( ord_less_nat @ K @ L2 )
% 0.30/0.50 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_less_mono
% 0.30/0.50 thf(fact_857_add__lessD1,axiom,
% 0.30/0.50 ! [I2: nat,J: nat,K: nat] :
% 0.30/0.50 ( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
% 0.30/0.50 => ( ord_less_nat @ I2 @ K ) ) ).
% 0.30/0.50
% 0.30/0.50 % add_lessD1
% 0.30/0.50 thf(fact_858_nat__le__iff__add,axiom,
% 0.30/0.50 ( ord_less_eq_nat
% 0.30/0.50 = ( ^ [M6: nat,N: nat] :
% 0.30/0.51 ? [K3: nat] :
% 0.30/0.51 ( N
% 0.30/0.51 = ( plus_plus_nat @ M6 @ K3 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_le_iff_add
% 0.30/0.51 thf(fact_859_trans__le__add2,axiom,
% 0.30/0.51 ! [I2: nat,J: nat,M: nat] :
% 0.30/0.51 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.51 => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % trans_le_add2
% 0.30/0.51 thf(fact_860_trans__le__add1,axiom,
% 0.30/0.51 ! [I2: nat,J: nat,M: nat] :
% 0.30/0.51 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.51 => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % trans_le_add1
% 0.30/0.51 thf(fact_861_add__le__mono1,axiom,
% 0.30/0.51 ! [I2: nat,J: nat,K: nat] :
% 0.30/0.51 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.51 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_le_mono1
% 0.30/0.51 thf(fact_862_add__le__mono,axiom,
% 0.30/0.51 ! [I2: nat,J: nat,K: nat,L2: nat] :
% 0.30/0.51 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.51 => ( ( ord_less_eq_nat @ K @ L2 )
% 0.30/0.51 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_le_mono
% 0.30/0.51 thf(fact_863_le__Suc__ex,axiom,
% 0.30/0.51 ! [K: nat,L2: nat] :
% 0.30/0.51 ( ( ord_less_eq_nat @ K @ L2 )
% 0.30/0.51 => ? [N4: nat] :
% 0.30/0.51 ( L2
% 0.30/0.51 = ( plus_plus_nat @ K @ N4 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % le_Suc_ex
% 0.30/0.51 thf(fact_864_add__leD2,axiom,
% 0.30/0.51 ! [M: nat,K: nat,N2: nat] :
% 0.30/0.51 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
% 0.30/0.51 => ( ord_less_eq_nat @ K @ N2 ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_leD2
% 0.30/0.51 thf(fact_865_add__leD1,axiom,
% 0.30/0.51 ! [M: nat,K: nat,N2: nat] :
% 0.30/0.51 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
% 0.30/0.51 => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_leD1
% 0.30/0.51 thf(fact_866_le__add2,axiom,
% 0.30/0.51 ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ).
% 0.30/0.51
% 0.30/0.51 % le_add2
% 0.30/0.51 thf(fact_867_le__add1,axiom,
% 0.30/0.51 ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) ) ).
% 0.30/0.51
% 0.30/0.51 % le_add1
% 0.30/0.51 thf(fact_868_add__leE,axiom,
% 0.30/0.51 ! [M: nat,K: nat,N2: nat] :
% 0.30/0.51 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
% 0.30/0.51 => ~ ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.51 => ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_leE
% 0.30/0.51 thf(fact_869_mult__le__mono2,axiom,
% 0.30/0.51 ! [I2: nat,J: nat,K: nat] :
% 0.30/0.51 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.51 => ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mult_le_mono2
% 0.30/0.51 thf(fact_870_mult__le__mono1,axiom,
% 0.30/0.51 ! [I2: nat,J: nat,K: nat] :
% 0.30/0.51 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.51 => ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mult_le_mono1
% 0.30/0.51 thf(fact_871_mult__le__mono,axiom,
% 0.30/0.51 ! [I2: nat,J: nat,K: nat,L2: nat] :
% 0.30/0.51 ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.51 => ( ( ord_less_eq_nat @ K @ L2 )
% 0.30/0.51 => ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ L2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mult_le_mono
% 0.30/0.51 thf(fact_872_le__square,axiom,
% 0.30/0.51 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 0.30/0.51
% 0.30/0.51 % le_square
% 0.30/0.51 thf(fact_873_le__cube,axiom,
% 0.30/0.51 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % le_cube
% 0.30/0.51 thf(fact_874_add__mult__distrib2,axiom,
% 0.30/0.51 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.51 ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N2 ) )
% 0.30/0.51 = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_mult_distrib2
% 0.30/0.51 thf(fact_875_add__mult__distrib,axiom,
% 0.30/0.51 ! [M: nat,N2: nat,K: nat] :
% 0.30/0.51 ( ( times_times_nat @ ( plus_plus_nat @ M @ N2 ) @ K )
% 0.30/0.51 = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_mult_distrib
% 0.30/0.51 thf(fact_876_nat__mult__1__right,axiom,
% 0.30/0.51 ! [N2: nat] :
% 0.30/0.51 ( ( times_times_nat @ N2 @ one_one_nat )
% 0.30/0.51 = N2 ) ).
% 0.30/0.51
% 0.30/0.51 % nat_mult_1_right
% 0.30/0.51 thf(fact_877_nat__mult__1,axiom,
% 0.30/0.51 ! [N2: nat] :
% 0.30/0.51 ( ( times_times_nat @ one_one_nat @ N2 )
% 0.30/0.51 = N2 ) ).
% 0.30/0.51
% 0.30/0.51 % nat_mult_1
% 0.30/0.51 thf(fact_878_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 0.30/0.51 ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
% 0.30/0.51 ( ( vEBT_V1502963449132264192at_nat @ F @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B ) )
% 0.30/0.51 = ( some_P7363390416028606310at_nat @ ( F @ A @ B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % VEBT_internal.option_shift.simps(3)
% 0.30/0.51 thf(fact_879_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 0.30/0.51 ! [F: num > num > num,A: num,B: num] :
% 0.30/0.51 ( ( vEBT_V819420779217536731ft_num @ F @ ( some_num @ A ) @ ( some_num @ B ) )
% 0.30/0.51 = ( some_num @ ( F @ A @ B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % VEBT_internal.option_shift.simps(3)
% 0.30/0.51 thf(fact_880_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 0.30/0.51 ! [F: nat > nat > nat,A: nat,B: nat] :
% 0.30/0.51 ( ( vEBT_V4262088993061758097ft_nat @ F @ ( some_nat @ A ) @ ( some_nat @ B ) )
% 0.30/0.51 = ( some_nat @ ( F @ A @ B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % VEBT_internal.option_shift.simps(3)
% 0.30/0.51 thf(fact_881_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 0.30/0.51 ! [Uu: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv: option4927543243414619207at_nat] :
% 0.30/0.51 ( ( vEBT_V1502963449132264192at_nat @ Uu @ none_P5556105721700978146at_nat @ Uv )
% 0.30/0.51 = none_P5556105721700978146at_nat ) ).
% 0.30/0.51
% 0.30/0.51 % VEBT_internal.option_shift.simps(1)
% 0.30/0.51 thf(fact_882_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 0.30/0.51 ! [Uu: num > num > num,Uv: option_num] :
% 0.30/0.51 ( ( vEBT_V819420779217536731ft_num @ Uu @ none_num @ Uv )
% 0.30/0.51 = none_num ) ).
% 0.30/0.51
% 0.30/0.51 % VEBT_internal.option_shift.simps(1)
% 0.30/0.51 thf(fact_883_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 0.30/0.51 ! [Uu: nat > nat > nat,Uv: option_nat] :
% 0.30/0.51 ( ( vEBT_V4262088993061758097ft_nat @ Uu @ none_nat @ Uv )
% 0.30/0.51 = none_nat ) ).
% 0.30/0.51
% 0.30/0.51 % VEBT_internal.option_shift.simps(1)
% 0.30/0.51 thf(fact_884_mono__nat__linear__lb,axiom,
% 0.30/0.51 ! [F: nat > nat,M: nat,K: nat] :
% 0.30/0.51 ( ! [M2: nat,N4: nat] :
% 0.30/0.51 ( ( ord_less_nat @ M2 @ N4 )
% 0.30/0.51 => ( ord_less_nat @ ( F @ M2 ) @ ( F @ N4 ) ) )
% 0.30/0.51 => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mono_nat_linear_lb
% 0.30/0.51 thf(fact_885_invar__vebt_Ointros_I3_J,axiom,
% 0.30/0.51 ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 0.30/0.51 ( ! [X4: vEBT_VEBT] :
% 0.30/0.51 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.51 => ( vEBT_invar_vebt @ X4 @ N2 ) )
% 0.30/0.51 => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.30/0.51 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.30/0.51 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.51 => ( ( M
% 0.30/0.51 = ( suc @ N2 ) )
% 0.30/0.51 => ( ( Deg
% 0.30/0.51 = ( plus_plus_nat @ N2 @ M ) )
% 0.30/0.51 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 0.30/0.51 => ( ! [X4: vEBT_VEBT] :
% 0.30/0.51 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.51 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
% 0.30/0.51 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % invar_vebt.intros(3)
% 0.30/0.51 thf(fact_886_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 0.30/0.51 ! [Uw: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V: product_prod_nat_nat] :
% 0.30/0.51 ( ( vEBT_V1502963449132264192at_nat @ Uw @ ( some_P7363390416028606310at_nat @ V ) @ none_P5556105721700978146at_nat )
% 0.30/0.51 = none_P5556105721700978146at_nat ) ).
% 0.30/0.51
% 0.30/0.51 % VEBT_internal.option_shift.simps(2)
% 0.30/0.51 thf(fact_887_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 0.30/0.51 ! [Uw: num > num > num,V: num] :
% 0.30/0.51 ( ( vEBT_V819420779217536731ft_num @ Uw @ ( some_num @ V ) @ none_num )
% 0.30/0.51 = none_num ) ).
% 0.30/0.51
% 0.30/0.51 % VEBT_internal.option_shift.simps(2)
% 0.30/0.51 thf(fact_888_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 0.30/0.51 ! [Uw: nat > nat > nat,V: nat] :
% 0.30/0.51 ( ( vEBT_V4262088993061758097ft_nat @ Uw @ ( some_nat @ V ) @ none_nat )
% 0.30/0.51 = none_nat ) ).
% 0.30/0.51
% 0.30/0.51 % VEBT_internal.option_shift.simps(2)
% 0.30/0.51 thf(fact_889_VEBT__internal_Ooption__shift_Oelims,axiom,
% 0.30/0.51 ! [X: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb2: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
% 0.30/0.51 ( ( ( vEBT_V1502963449132264192at_nat @ X @ Xa2 @ Xb2 )
% 0.30/0.51 = Y )
% 0.30/0.51 => ( ( ( Xa2 = none_P5556105721700978146at_nat )
% 0.30/0.51 => ( Y != none_P5556105721700978146at_nat ) )
% 0.30/0.51 => ( ( ? [V2: product_prod_nat_nat] :
% 0.30/0.51 ( Xa2
% 0.30/0.51 = ( some_P7363390416028606310at_nat @ V2 ) )
% 0.30/0.51 => ( ( Xb2 = none_P5556105721700978146at_nat )
% 0.30/0.51 => ( Y != none_P5556105721700978146at_nat ) ) )
% 0.30/0.51 => ~ ! [A3: product_prod_nat_nat] :
% 0.30/0.51 ( ( Xa2
% 0.30/0.51 = ( some_P7363390416028606310at_nat @ A3 ) )
% 0.30/0.51 => ! [B2: product_prod_nat_nat] :
% 0.30/0.51 ( ( Xb2
% 0.30/0.51 = ( some_P7363390416028606310at_nat @ B2 ) )
% 0.30/0.51 => ( Y
% 0.30/0.51 != ( some_P7363390416028606310at_nat @ ( X @ A3 @ B2 ) ) ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % VEBT_internal.option_shift.elims
% 0.30/0.51 thf(fact_890_VEBT__internal_Ooption__shift_Oelims,axiom,
% 0.30/0.51 ! [X: num > num > num,Xa2: option_num,Xb2: option_num,Y: option_num] :
% 0.30/0.51 ( ( ( vEBT_V819420779217536731ft_num @ X @ Xa2 @ Xb2 )
% 0.30/0.51 = Y )
% 0.30/0.51 => ( ( ( Xa2 = none_num )
% 0.30/0.51 => ( Y != none_num ) )
% 0.30/0.51 => ( ( ? [V2: num] :
% 0.30/0.51 ( Xa2
% 0.30/0.51 = ( some_num @ V2 ) )
% 0.30/0.51 => ( ( Xb2 = none_num )
% 0.30/0.51 => ( Y != none_num ) ) )
% 0.30/0.51 => ~ ! [A3: num] :
% 0.30/0.51 ( ( Xa2
% 0.30/0.51 = ( some_num @ A3 ) )
% 0.30/0.51 => ! [B2: num] :
% 0.30/0.51 ( ( Xb2
% 0.30/0.51 = ( some_num @ B2 ) )
% 0.30/0.51 => ( Y
% 0.30/0.51 != ( some_num @ ( X @ A3 @ B2 ) ) ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % VEBT_internal.option_shift.elims
% 0.30/0.51 thf(fact_891_VEBT__internal_Ooption__shift_Oelims,axiom,
% 0.30/0.51 ! [X: nat > nat > nat,Xa2: option_nat,Xb2: option_nat,Y: option_nat] :
% 0.30/0.51 ( ( ( vEBT_V4262088993061758097ft_nat @ X @ Xa2 @ Xb2 )
% 0.30/0.51 = Y )
% 0.30/0.51 => ( ( ( Xa2 = none_nat )
% 0.30/0.51 => ( Y != none_nat ) )
% 0.30/0.51 => ( ( ? [V2: nat] :
% 0.30/0.51 ( Xa2
% 0.30/0.51 = ( some_nat @ V2 ) )
% 0.30/0.51 => ( ( Xb2 = none_nat )
% 0.30/0.51 => ( Y != none_nat ) ) )
% 0.30/0.51 => ~ ! [A3: nat] :
% 0.30/0.51 ( ( Xa2
% 0.30/0.51 = ( some_nat @ A3 ) )
% 0.30/0.51 => ! [B2: nat] :
% 0.30/0.51 ( ( Xb2
% 0.30/0.51 = ( some_nat @ B2 ) )
% 0.30/0.51 => ( Y
% 0.30/0.51 != ( some_nat @ ( X @ A3 @ B2 ) ) ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % VEBT_internal.option_shift.elims
% 0.30/0.51 thf(fact_892_vebt__maxt_Osimps_I2_J,axiom,
% 0.30/0.51 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 0.30/0.51 ( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 0.30/0.51 = none_nat ) ).
% 0.30/0.51
% 0.30/0.51 % vebt_maxt.simps(2)
% 0.30/0.51 thf(fact_893_vebt__mint_Osimps_I2_J,axiom,
% 0.30/0.51 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 0.30/0.51 ( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 0.30/0.51 = none_nat ) ).
% 0.30/0.51
% 0.30/0.51 % vebt_mint.simps(2)
% 0.30/0.51 thf(fact_894_invar__vebt_Ointros_I5_J,axiom,
% 0.30/0.51 ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 0.30/0.51 ( ! [X4: vEBT_VEBT] :
% 0.30/0.51 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.51 => ( vEBT_invar_vebt @ X4 @ N2 ) )
% 0.30/0.51 => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.30/0.51 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.30/0.51 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.51 => ( ( M
% 0.30/0.51 = ( suc @ N2 ) )
% 0.30/0.51 => ( ( Deg
% 0.30/0.51 = ( plus_plus_nat @ N2 @ M ) )
% 0.30/0.51 => ( ! [I4: nat] :
% 0.30/0.51 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.51 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X3 ) )
% 0.30/0.51 = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
% 0.30/0.51 => ( ( ( Mi = Ma )
% 0.30/0.51 => ! [X4: vEBT_VEBT] :
% 0.30/0.51 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.51 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
% 0.30/0.51 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 0.30/0.51 => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.30/0.51 => ( ( ( Mi != Ma )
% 0.30/0.51 => ! [I4: nat] :
% 0.30/0.51 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.51 => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
% 0.30/0.51 = I4 )
% 0.30/0.51 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
% 0.30/0.51 & ! [X4: nat] :
% 0.30/0.51 ( ( ( ( vEBT_VEBT_high @ X4 @ N2 )
% 0.30/0.51 = I4 )
% 0.30/0.51 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X4 @ N2 ) ) )
% 0.30/0.51 => ( ( ord_less_nat @ Mi @ X4 )
% 0.30/0.51 & ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
% 0.30/0.51 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % invar_vebt.intros(5)
% 0.30/0.51 thf(fact_895_vebt__succ_Osimps_I6_J,axiom,
% 0.30/0.51 ! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.51 ( ( ( ord_less_nat @ X @ Mi )
% 0.30/0.51 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
% 0.30/0.51 = ( some_nat @ Mi ) ) )
% 0.30/0.51 & ( ~ ( ord_less_nat @ X @ Mi )
% 0.30/0.51 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
% 0.30/0.51 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.51 @ ( if_option_nat
% 0.30/0.51 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51 != none_nat )
% 0.30/0.51 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.30/0.51 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51 @ ( if_option_nat
% 0.30/0.51 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.51 = none_nat )
% 0.30/0.51 @ none_nat
% 0.30/0.51 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.30/0.51 @ none_nat ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % vebt_succ.simps(6)
% 0.30/0.51 thf(fact_896_vebt__pred_Osimps_I7_J,axiom,
% 0.30/0.51 ! [Ma: nat,X: nat,Mi: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.30/0.51 ( ( ( ord_less_nat @ Ma @ X )
% 0.30/0.51 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
% 0.30/0.51 = ( some_nat @ Ma ) ) )
% 0.30/0.51 & ( ~ ( ord_less_nat @ Ma @ X )
% 0.30/0.51 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
% 0.30/0.51 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.51 @ ( if_option_nat
% 0.30/0.51 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51 != none_nat )
% 0.30/0.51 & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.30/0.51 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51 @ ( if_option_nat
% 0.30/0.51 @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.51 = none_nat )
% 0.30/0.51 @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
% 0.30/0.51 @ ( 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 @ X @ ( 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 @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.30/0.51 @ none_nat ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % vebt_pred.simps(7)
% 0.30/0.51 thf(fact_897_vebt__member_Osimps_I5_J,axiom,
% 0.30/0.51 ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 0.30/0.51 ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
% 0.30/0.51 = ( ( X != Mi )
% 0.30/0.51 => ( ( X != Ma )
% 0.30/0.51 => ( ~ ( ord_less_nat @ X @ Mi )
% 0.30/0.51 & ( ~ ( ord_less_nat @ X @ Mi )
% 0.30/0.51 => ( ~ ( ord_less_nat @ Ma @ X )
% 0.30/0.51 & ( ~ ( ord_less_nat @ Ma @ X )
% 0.30/0.51 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.51 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.51 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % vebt_member.simps(5)
% 0.30/0.51 thf(fact_898_insert__simp__norm,axiom,
% 0.30/0.51 ! [X: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 0.30/0.51 ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.51 => ( ( ord_less_nat @ Mi @ X )
% 0.30/0.51 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.51 => ( ( X != Ma )
% 0.30/0.51 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.30/0.51 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % insert_simp_norm
% 0.30/0.51 thf(fact_899_insert__simp__excp,axiom,
% 0.30/0.51 ! [Mi: nat,Deg: nat,TreeList2: list_VEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
% 0.30/0.51 ( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.51 => ( ( ord_less_nat @ X @ Mi )
% 0.30/0.51 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.30/0.51 => ( ( X != Ma )
% 0.30/0.51 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.30/0.51 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % insert_simp_excp
% 0.30/0.51 thf(fact_900_le__add__diff__inverse2,axiom,
% 0.30/0.51 ! [B: real,A: real] :
% 0.30/0.51 ( ( ord_less_eq_real @ B @ A )
% 0.30/0.51 => ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 0.30/0.51 = A ) ) ).
% 0.30/0.51
% 0.30/0.51 % le_add_diff_inverse2
% 0.30/0.51 thf(fact_901_le__add__diff__inverse2,axiom,
% 0.30/0.51 ! [B: rat,A: rat] :
% 0.30/0.51 ( ( ord_less_eq_rat @ B @ A )
% 0.30/0.51 => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 0.30/0.51 = A ) ) ).
% 0.30/0.51
% 0.30/0.51 % le_add_diff_inverse2
% 0.30/0.51 thf(fact_902_le__add__diff__inverse2,axiom,
% 0.30/0.51 ! [B: nat,A: nat] :
% 0.30/0.51 ( ( ord_less_eq_nat @ B @ A )
% 0.30/0.51 => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
% 0.30/0.51 = A ) ) ).
% 0.30/0.51
% 0.30/0.51 % le_add_diff_inverse2
% 0.30/0.51 thf(fact_903_le__add__diff__inverse2,axiom,
% 0.30/0.51 ! [B: int,A: int] :
% 0.30/0.51 ( ( ord_less_eq_int @ B @ A )
% 0.30/0.51 => ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 0.30/0.51 = A ) ) ).
% 0.30/0.51
% 0.30/0.51 % le_add_diff_inverse2
% 0.30/0.51 thf(fact_904_le__add__diff__inverse,axiom,
% 0.30/0.51 ! [B: real,A: real] :
% 0.30/0.51 ( ( ord_less_eq_real @ B @ A )
% 0.30/0.51 => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 0.30/0.51 = A ) ) ).
% 0.30/0.51
% 0.30/0.51 % le_add_diff_inverse
% 0.30/0.51 thf(fact_905_le__add__diff__inverse,axiom,
% 0.30/0.51 ! [B: rat,A: rat] :
% 0.30/0.51 ( ( ord_less_eq_rat @ B @ A )
% 0.30/0.51 => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 0.30/0.51 = A ) ) ).
% 0.30/0.51
% 0.30/0.51 % le_add_diff_inverse
% 0.30/0.51 thf(fact_906_le__add__diff__inverse,axiom,
% 0.30/0.51 ! [B: nat,A: nat] :
% 0.30/0.51 ( ( ord_less_eq_nat @ B @ A )
% 0.30/0.51 => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 0.30/0.51 = A ) ) ).
% 0.30/0.51
% 0.30/0.51 % le_add_diff_inverse
% 0.30/0.51 thf(fact_907_le__add__diff__inverse,axiom,
% 0.30/0.51 ! [B: int,A: int] :
% 0.30/0.51 ( ( ord_less_eq_int @ B @ A )
% 0.30/0.51 => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 0.30/0.51 = A ) ) ).
% 0.30/0.51
% 0.30/0.51 % le_add_diff_inverse
% 0.30/0.51 thf(fact_908_vebt__insert_Osimps_I4_J,axiom,
% 0.30/0.51 ! [V: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 0.30/0.51 ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) @ X )
% 0.30/0.51 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) ) ).
% 0.30/0.51
% 0.30/0.51 % vebt_insert.simps(4)
% 0.30/0.51 thf(fact_909_pred__empty,axiom,
% 0.30/0.51 ! [T: vEBT_VEBT,N2: nat,X: nat] :
% 0.30/0.51 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.51 => ( ( ( vEBT_vebt_pred @ T @ X )
% 0.30/0.51 = none_nat )
% 0.30/0.51 = ( ( collect_nat
% 0.30/0.51 @ ^ [Y2: nat] :
% 0.30/0.51 ( ( vEBT_vebt_member @ T @ Y2 )
% 0.30/0.51 & ( ord_less_nat @ Y2 @ X ) ) )
% 0.30/0.51 = bot_bot_set_nat ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % pred_empty
% 0.30/0.51 thf(fact_910_succ__empty,axiom,
% 0.30/0.51 ! [T: vEBT_VEBT,N2: nat,X: nat] :
% 0.30/0.51 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.51 => ( ( ( vEBT_vebt_succ @ T @ X )
% 0.30/0.51 = none_nat )
% 0.30/0.51 = ( ( collect_nat
% 0.30/0.51 @ ^ [Y2: nat] :
% 0.30/0.51 ( ( vEBT_vebt_member @ T @ Y2 )
% 0.30/0.51 & ( ord_less_nat @ X @ Y2 ) ) )
% 0.30/0.51 = bot_bot_set_nat ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % succ_empty
% 0.30/0.51 thf(fact_911_enat__ord__number_I1_J,axiom,
% 0.30/0.51 ! [M: num,N2: num] :
% 0.30/0.51 ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
% 0.30/0.51 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % enat_ord_number(1)
% 0.30/0.51 thf(fact_912_maxt__corr__help__empty,axiom,
% 0.30/0.51 ! [T: vEBT_VEBT,N2: nat] :
% 0.30/0.51 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.51 => ( ( ( vEBT_vebt_maxt @ T )
% 0.30/0.51 = none_nat )
% 0.30/0.51 => ( ( vEBT_VEBT_set_vebt @ T )
% 0.30/0.51 = bot_bot_set_nat ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % maxt_corr_help_empty
% 0.30/0.51 thf(fact_913_mint__corr__help__empty,axiom,
% 0.30/0.51 ! [T: vEBT_VEBT,N2: nat] :
% 0.30/0.51 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.51 => ( ( ( vEBT_vebt_mint @ T )
% 0.30/0.51 = none_nat )
% 0.30/0.51 => ( ( vEBT_VEBT_set_vebt @ T )
% 0.30/0.51 = bot_bot_set_nat ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % mint_corr_help_empty
% 0.30/0.51 thf(fact_914_div__by__1,axiom,
% 0.30/0.51 ! [A: complex] :
% 0.30/0.51 ( ( divide1717551699836669952omplex @ A @ one_one_complex )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % div_by_1
% 0.30/0.51 thf(fact_915_div__by__1,axiom,
% 0.30/0.51 ! [A: real] :
% 0.30/0.51 ( ( divide_divide_real @ A @ one_one_real )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % div_by_1
% 0.30/0.51 thf(fact_916_div__by__1,axiom,
% 0.30/0.51 ! [A: rat] :
% 0.30/0.51 ( ( divide_divide_rat @ A @ one_one_rat )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % div_by_1
% 0.30/0.51 thf(fact_917_div__by__1,axiom,
% 0.30/0.51 ! [A: nat] :
% 0.30/0.51 ( ( divide_divide_nat @ A @ one_one_nat )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % div_by_1
% 0.30/0.51 thf(fact_918_div__by__1,axiom,
% 0.30/0.51 ! [A: int] :
% 0.30/0.51 ( ( divide_divide_int @ A @ one_one_int )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % div_by_1
% 0.30/0.51 thf(fact_919_max__Suc__Suc,axiom,
% 0.30/0.51 ! [M: nat,N2: nat] :
% 0.30/0.51 ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N2 ) )
% 0.30/0.51 = ( suc @ ( ord_max_nat @ M @ N2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max_Suc_Suc
% 0.30/0.51 thf(fact_920_max__number__of_I1_J,axiom,
% 0.30/0.51 ! [U: num,V: num] :
% 0.30/0.51 ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.30/0.51 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.30/0.51 = ( numeral_numeral_real @ V ) ) )
% 0.30/0.51 & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.30/0.51 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.30/0.51 = ( numeral_numeral_real @ U ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max_number_of(1)
% 0.30/0.51 thf(fact_921_max__number__of_I1_J,axiom,
% 0.30/0.51 ! [U: num,V: num] :
% 0.30/0.51 ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.30/0.51 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.30/0.51 = ( numeral_numeral_rat @ V ) ) )
% 0.30/0.51 & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.30/0.51 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.30/0.51 = ( numeral_numeral_rat @ U ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max_number_of(1)
% 0.30/0.51 thf(fact_922_max__number__of_I1_J,axiom,
% 0.30/0.51 ! [U: num,V: num] :
% 0.30/0.51 ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.30/0.51 => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.30/0.51 = ( numeral_numeral_nat @ V ) ) )
% 0.30/0.51 & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.30/0.51 => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.30/0.51 = ( numeral_numeral_nat @ U ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max_number_of(1)
% 0.30/0.51 thf(fact_923_max__number__of_I1_J,axiom,
% 0.30/0.51 ! [U: num,V: num] :
% 0.30/0.51 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.51 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.51 = ( numeral_numeral_int @ V ) ) )
% 0.30/0.51 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.51 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.51 = ( numeral_numeral_int @ U ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max_number_of(1)
% 0.30/0.51 thf(fact_924_max__0__1_I5_J,axiom,
% 0.30/0.51 ! [X: num] :
% 0.30/0.51 ( ( ord_max_real @ one_one_real @ ( numeral_numeral_real @ X ) )
% 0.30/0.51 = ( numeral_numeral_real @ X ) ) ).
% 0.30/0.51
% 0.30/0.51 % max_0_1(5)
% 0.30/0.51 thf(fact_925_max__0__1_I5_J,axiom,
% 0.30/0.51 ! [X: num] :
% 0.30/0.51 ( ( ord_max_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
% 0.30/0.51 = ( numeral_numeral_rat @ X ) ) ).
% 0.30/0.51
% 0.30/0.51 % max_0_1(5)
% 0.30/0.51 thf(fact_926_max__0__1_I5_J,axiom,
% 0.30/0.51 ! [X: num] :
% 0.30/0.51 ( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
% 0.30/0.51 = ( numeral_numeral_nat @ X ) ) ).
% 0.30/0.51
% 0.30/0.51 % max_0_1(5)
% 0.30/0.51 thf(fact_927_max__0__1_I5_J,axiom,
% 0.30/0.51 ! [X: num] :
% 0.30/0.51 ( ( ord_max_int @ one_one_int @ ( numeral_numeral_int @ X ) )
% 0.30/0.51 = ( numeral_numeral_int @ X ) ) ).
% 0.30/0.51
% 0.30/0.51 % max_0_1(5)
% 0.30/0.51 thf(fact_928_max__0__1_I6_J,axiom,
% 0.30/0.51 ! [X: num] :
% 0.30/0.51 ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ one_one_real )
% 0.30/0.51 = ( numeral_numeral_real @ X ) ) ).
% 0.30/0.51
% 0.30/0.51 % max_0_1(6)
% 0.30/0.51 thf(fact_929_max__0__1_I6_J,axiom,
% 0.30/0.51 ! [X: num] :
% 0.30/0.51 ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat )
% 0.30/0.51 = ( numeral_numeral_rat @ X ) ) ).
% 0.30/0.51
% 0.30/0.51 % max_0_1(6)
% 0.30/0.51 thf(fact_930_max__0__1_I6_J,axiom,
% 0.30/0.51 ! [X: num] :
% 0.30/0.51 ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
% 0.30/0.51 = ( numeral_numeral_nat @ X ) ) ).
% 0.30/0.51
% 0.30/0.51 % max_0_1(6)
% 0.30/0.51 thf(fact_931_max__0__1_I6_J,axiom,
% 0.30/0.51 ! [X: num] :
% 0.30/0.51 ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ one_one_int )
% 0.30/0.51 = ( numeral_numeral_int @ X ) ) ).
% 0.30/0.51
% 0.30/0.51 % max_0_1(6)
% 0.30/0.51 thf(fact_932_enat__ord__number_I2_J,axiom,
% 0.30/0.51 ! [M: num,N2: num] :
% 0.30/0.51 ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
% 0.30/0.51 = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % enat_ord_number(2)
% 0.30/0.51 thf(fact_933_add__diff__assoc__enat,axiom,
% 0.30/0.51 ! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
% 0.30/0.51 ( ( ord_le2932123472753598470d_enat @ Z @ Y )
% 0.30/0.51 => ( ( plus_p3455044024723400733d_enat @ X @ ( minus_3235023915231533773d_enat @ Y @ Z ) )
% 0.30/0.51 = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ Z ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_diff_assoc_enat
% 0.30/0.51 thf(fact_934_nat__add__max__left,axiom,
% 0.30/0.51 ! [M: nat,N2: nat,Q2: nat] :
% 0.30/0.51 ( ( plus_plus_nat @ ( ord_max_nat @ M @ N2 ) @ Q2 )
% 0.30/0.51 = ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N2 @ Q2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_add_max_left
% 0.30/0.51 thf(fact_935_nat__add__max__right,axiom,
% 0.30/0.51 ! [M: nat,N2: nat,Q2: nat] :
% 0.30/0.51 ( ( plus_plus_nat @ M @ ( ord_max_nat @ N2 @ Q2 ) )
% 0.30/0.51 = ( ord_max_nat @ ( plus_plus_nat @ M @ N2 ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_add_max_right
% 0.30/0.51 thf(fact_936_nat__mult__max__left,axiom,
% 0.30/0.51 ! [M: nat,N2: nat,Q2: nat] :
% 0.30/0.51 ( ( times_times_nat @ ( ord_max_nat @ M @ N2 ) @ Q2 )
% 0.30/0.51 = ( ord_max_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N2 @ Q2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_mult_max_left
% 0.30/0.51 thf(fact_937_nat__mult__max__right,axiom,
% 0.30/0.51 ! [M: nat,N2: nat,Q2: nat] :
% 0.30/0.51 ( ( times_times_nat @ M @ ( ord_max_nat @ N2 @ Q2 ) )
% 0.30/0.51 = ( ord_max_nat @ ( times_times_nat @ M @ N2 ) @ ( times_times_nat @ M @ Q2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_mult_max_right
% 0.30/0.51 thf(fact_938_nat__minus__add__max,axiom,
% 0.30/0.51 ! [N2: nat,M: nat] :
% 0.30/0.51 ( ( plus_plus_nat @ ( minus_minus_nat @ N2 @ M ) @ M )
% 0.30/0.51 = ( ord_max_nat @ N2 @ M ) ) ).
% 0.30/0.51
% 0.30/0.51 % nat_minus_add_max
% 0.30/0.51 thf(fact_939_linorder__neqE__linordered__idom,axiom,
% 0.30/0.51 ! [X: real,Y: real] :
% 0.30/0.51 ( ( X != Y )
% 0.30/0.51 => ( ~ ( ord_less_real @ X @ Y )
% 0.30/0.51 => ( ord_less_real @ Y @ X ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % linorder_neqE_linordered_idom
% 0.30/0.51 thf(fact_940_linorder__neqE__linordered__idom,axiom,
% 0.30/0.51 ! [X: rat,Y: rat] :
% 0.30/0.51 ( ( X != Y )
% 0.30/0.51 => ( ~ ( ord_less_rat @ X @ Y )
% 0.30/0.51 => ( ord_less_rat @ Y @ X ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % linorder_neqE_linordered_idom
% 0.30/0.51 thf(fact_941_linorder__neqE__linordered__idom,axiom,
% 0.30/0.51 ! [X: int,Y: int] :
% 0.30/0.51 ( ( X != Y )
% 0.30/0.51 => ( ~ ( ord_less_int @ X @ Y )
% 0.30/0.51 => ( ord_less_int @ Y @ X ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % linorder_neqE_linordered_idom
% 0.30/0.51 thf(fact_942_ring__class_Oring__distribs_I2_J,axiom,
% 0.30/0.51 ! [A: real,B: real,C: real] :
% 0.30/0.51 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.30/0.51 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % ring_class.ring_distribs(2)
% 0.30/0.51 thf(fact_943_ring__class_Oring__distribs_I2_J,axiom,
% 0.30/0.51 ! [A: rat,B: rat,C: rat] :
% 0.30/0.51 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.30/0.51 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % ring_class.ring_distribs(2)
% 0.30/0.51 thf(fact_944_ring__class_Oring__distribs_I2_J,axiom,
% 0.30/0.51 ! [A: int,B: int,C: int] :
% 0.30/0.51 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.30/0.51 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % ring_class.ring_distribs(2)
% 0.30/0.51 thf(fact_945_ring__class_Oring__distribs_I1_J,axiom,
% 0.30/0.51 ! [A: real,B: real,C: real] :
% 0.30/0.51 ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.30/0.51 = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % ring_class.ring_distribs(1)
% 0.30/0.51 thf(fact_946_ring__class_Oring__distribs_I1_J,axiom,
% 0.30/0.51 ! [A: rat,B: rat,C: rat] :
% 0.30/0.51 ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.30/0.51 = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % ring_class.ring_distribs(1)
% 0.30/0.51 thf(fact_947_ring__class_Oring__distribs_I1_J,axiom,
% 0.30/0.51 ! [A: int,B: int,C: int] :
% 0.30/0.51 ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.30/0.51 = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % ring_class.ring_distribs(1)
% 0.30/0.51 thf(fact_948_comm__semiring__class_Odistrib,axiom,
% 0.30/0.51 ! [A: real,B: real,C: real] :
% 0.30/0.51 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.30/0.51 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % comm_semiring_class.distrib
% 0.30/0.51 thf(fact_949_comm__semiring__class_Odistrib,axiom,
% 0.30/0.51 ! [A: rat,B: rat,C: rat] :
% 0.30/0.51 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.30/0.51 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % comm_semiring_class.distrib
% 0.30/0.51 thf(fact_950_comm__semiring__class_Odistrib,axiom,
% 0.30/0.51 ! [A: nat,B: nat,C: nat] :
% 0.30/0.51 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.30/0.51 = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % comm_semiring_class.distrib
% 0.30/0.51 thf(fact_951_comm__semiring__class_Odistrib,axiom,
% 0.30/0.51 ! [A: int,B: int,C: int] :
% 0.30/0.51 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.30/0.51 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % comm_semiring_class.distrib
% 0.30/0.51 thf(fact_952_distrib__left,axiom,
% 0.30/0.51 ! [A: real,B: real,C: real] :
% 0.30/0.51 ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.30/0.51 = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % distrib_left
% 0.30/0.51 thf(fact_953_distrib__left,axiom,
% 0.30/0.51 ! [A: rat,B: rat,C: rat] :
% 0.30/0.51 ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.30/0.51 = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % distrib_left
% 0.30/0.51 thf(fact_954_distrib__left,axiom,
% 0.30/0.51 ! [A: nat,B: nat,C: nat] :
% 0.30/0.51 ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 0.30/0.51 = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % distrib_left
% 0.30/0.51 thf(fact_955_distrib__left,axiom,
% 0.30/0.51 ! [A: int,B: int,C: int] :
% 0.30/0.51 ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.30/0.51 = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % distrib_left
% 0.30/0.51 thf(fact_956_distrib__right,axiom,
% 0.30/0.51 ! [A: real,B: real,C: real] :
% 0.30/0.51 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.30/0.51 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % distrib_right
% 0.30/0.51 thf(fact_957_distrib__right,axiom,
% 0.30/0.51 ! [A: rat,B: rat,C: rat] :
% 0.30/0.51 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.30/0.51 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % distrib_right
% 0.30/0.51 thf(fact_958_distrib__right,axiom,
% 0.30/0.51 ! [A: nat,B: nat,C: nat] :
% 0.30/0.51 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.30/0.51 = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % distrib_right
% 0.30/0.51 thf(fact_959_distrib__right,axiom,
% 0.30/0.51 ! [A: int,B: int,C: int] :
% 0.30/0.51 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.30/0.51 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % distrib_right
% 0.30/0.51 thf(fact_960_combine__common__factor,axiom,
% 0.30/0.51 ! [A: real,E: real,B: real,C: real] :
% 0.30/0.51 ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
% 0.30/0.51 = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C ) ) ).
% 0.30/0.51
% 0.30/0.51 % combine_common_factor
% 0.30/0.51 thf(fact_961_combine__common__factor,axiom,
% 0.30/0.51 ! [A: rat,E: rat,B: rat,C: rat] :
% 0.30/0.51 ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ C ) )
% 0.30/0.51 = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E ) @ C ) ) ).
% 0.30/0.51
% 0.30/0.51 % combine_common_factor
% 0.30/0.51 thf(fact_962_combine__common__factor,axiom,
% 0.30/0.51 ! [A: nat,E: nat,B: nat,C: nat] :
% 0.30/0.51 ( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
% 0.30/0.51 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% 0.30/0.51
% 0.30/0.51 % combine_common_factor
% 0.30/0.51 thf(fact_963_combine__common__factor,axiom,
% 0.30/0.51 ! [A: int,E: int,B: int,C: int] :
% 0.30/0.51 ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
% 0.30/0.51 = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).
% 0.30/0.51
% 0.30/0.51 % combine_common_factor
% 0.30/0.51 thf(fact_964_left__diff__distrib,axiom,
% 0.30/0.51 ! [A: real,B: real,C: real] :
% 0.30/0.51 ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.30/0.51 = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % left_diff_distrib
% 0.30/0.51 thf(fact_965_left__diff__distrib,axiom,
% 0.30/0.51 ! [A: rat,B: rat,C: rat] :
% 0.30/0.51 ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.30/0.51 = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % left_diff_distrib
% 0.30/0.51 thf(fact_966_left__diff__distrib,axiom,
% 0.30/0.51 ! [A: int,B: int,C: int] :
% 0.30/0.51 ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.30/0.51 = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % left_diff_distrib
% 0.30/0.51 thf(fact_967_right__diff__distrib,axiom,
% 0.30/0.51 ! [A: real,B: real,C: real] :
% 0.30/0.51 ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.30/0.51 = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % right_diff_distrib
% 0.30/0.51 thf(fact_968_right__diff__distrib,axiom,
% 0.30/0.51 ! [A: rat,B: rat,C: rat] :
% 0.30/0.51 ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 0.30/0.51 = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % right_diff_distrib
% 0.30/0.51 thf(fact_969_right__diff__distrib,axiom,
% 0.30/0.51 ! [A: int,B: int,C: int] :
% 0.30/0.51 ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.30/0.51 = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % right_diff_distrib
% 0.30/0.51 thf(fact_970_left__diff__distrib_H,axiom,
% 0.30/0.51 ! [B: real,C: real,A: real] :
% 0.30/0.51 ( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
% 0.30/0.51 = ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % left_diff_distrib'
% 0.30/0.51 thf(fact_971_left__diff__distrib_H,axiom,
% 0.30/0.51 ! [B: rat,C: rat,A: rat] :
% 0.30/0.51 ( ( times_times_rat @ ( minus_minus_rat @ B @ C ) @ A )
% 0.30/0.51 = ( minus_minus_rat @ ( times_times_rat @ B @ A ) @ ( times_times_rat @ C @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % left_diff_distrib'
% 0.30/0.51 thf(fact_972_left__diff__distrib_H,axiom,
% 0.30/0.51 ! [B: nat,C: nat,A: nat] :
% 0.30/0.51 ( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
% 0.30/0.51 = ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % left_diff_distrib'
% 0.30/0.51 thf(fact_973_left__diff__distrib_H,axiom,
% 0.30/0.51 ! [B: int,C: int,A: int] :
% 0.30/0.51 ( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
% 0.30/0.51 = ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % left_diff_distrib'
% 0.30/0.51 thf(fact_974_right__diff__distrib_H,axiom,
% 0.30/0.51 ! [A: real,B: real,C: real] :
% 0.30/0.51 ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.30/0.51 = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % right_diff_distrib'
% 0.30/0.51 thf(fact_975_right__diff__distrib_H,axiom,
% 0.30/0.51 ! [A: rat,B: rat,C: rat] :
% 0.30/0.51 ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 0.30/0.51 = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % right_diff_distrib'
% 0.30/0.51 thf(fact_976_right__diff__distrib_H,axiom,
% 0.30/0.51 ! [A: nat,B: nat,C: nat] :
% 0.30/0.51 ( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
% 0.30/0.51 = ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % right_diff_distrib'
% 0.30/0.51 thf(fact_977_right__diff__distrib_H,axiom,
% 0.30/0.51 ! [A: int,B: int,C: int] :
% 0.30/0.51 ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.30/0.51 = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % right_diff_distrib'
% 0.30/0.51 thf(fact_978_lambda__one,axiom,
% 0.30/0.51 ( ( ^ [X2: complex] : X2 )
% 0.30/0.51 = ( times_times_complex @ one_one_complex ) ) ).
% 0.30/0.51
% 0.30/0.51 % lambda_one
% 0.30/0.51 thf(fact_979_lambda__one,axiom,
% 0.30/0.51 ( ( ^ [X2: real] : X2 )
% 0.30/0.51 = ( times_times_real @ one_one_real ) ) ).
% 0.30/0.51
% 0.30/0.51 % lambda_one
% 0.30/0.51 thf(fact_980_lambda__one,axiom,
% 0.30/0.51 ( ( ^ [X2: rat] : X2 )
% 0.30/0.51 = ( times_times_rat @ one_one_rat ) ) ).
% 0.30/0.51
% 0.30/0.51 % lambda_one
% 0.30/0.51 thf(fact_981_lambda__one,axiom,
% 0.30/0.51 ( ( ^ [X2: nat] : X2 )
% 0.30/0.51 = ( times_times_nat @ one_one_nat ) ) ).
% 0.30/0.51
% 0.30/0.51 % lambda_one
% 0.30/0.51 thf(fact_982_lambda__one,axiom,
% 0.30/0.51 ( ( ^ [X2: int] : X2 )
% 0.30/0.51 = ( times_times_int @ one_one_int ) ) ).
% 0.30/0.51
% 0.30/0.51 % lambda_one
% 0.30/0.51 thf(fact_983_less__1__mult,axiom,
% 0.30/0.51 ! [M: real,N2: real] :
% 0.30/0.51 ( ( ord_less_real @ one_one_real @ M )
% 0.30/0.51 => ( ( ord_less_real @ one_one_real @ N2 )
% 0.30/0.51 => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_1_mult
% 0.30/0.51 thf(fact_984_less__1__mult,axiom,
% 0.30/0.51 ! [M: rat,N2: rat] :
% 0.30/0.51 ( ( ord_less_rat @ one_one_rat @ M )
% 0.30/0.51 => ( ( ord_less_rat @ one_one_rat @ N2 )
% 0.30/0.51 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_1_mult
% 0.30/0.51 thf(fact_985_less__1__mult,axiom,
% 0.30/0.51 ! [M: nat,N2: nat] :
% 0.30/0.51 ( ( ord_less_nat @ one_one_nat @ M )
% 0.30/0.51 => ( ( ord_less_nat @ one_one_nat @ N2 )
% 0.30/0.51 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_1_mult
% 0.30/0.51 thf(fact_986_less__1__mult,axiom,
% 0.30/0.51 ! [M: int,N2: int] :
% 0.30/0.51 ( ( ord_less_int @ one_one_int @ M )
% 0.30/0.51 => ( ( ord_less_int @ one_one_int @ N2 )
% 0.30/0.51 => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_1_mult
% 0.30/0.51 thf(fact_987_add__mono1,axiom,
% 0.30/0.51 ! [A: real,B: real] :
% 0.30/0.51 ( ( ord_less_real @ A @ B )
% 0.30/0.51 => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_mono1
% 0.30/0.51 thf(fact_988_add__mono1,axiom,
% 0.30/0.51 ! [A: rat,B: rat] :
% 0.30/0.51 ( ( ord_less_rat @ A @ B )
% 0.30/0.51 => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_mono1
% 0.30/0.51 thf(fact_989_add__mono1,axiom,
% 0.30/0.51 ! [A: nat,B: nat] :
% 0.30/0.51 ( ( ord_less_nat @ A @ B )
% 0.30/0.51 => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_mono1
% 0.30/0.51 thf(fact_990_add__mono1,axiom,
% 0.30/0.51 ! [A: int,B: int] :
% 0.30/0.51 ( ( ord_less_int @ A @ B )
% 0.30/0.51 => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_mono1
% 0.30/0.51 thf(fact_991_less__add__one,axiom,
% 0.30/0.51 ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_add_one
% 0.30/0.51 thf(fact_992_less__add__one,axiom,
% 0.30/0.51 ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_add_one
% 0.30/0.51 thf(fact_993_less__add__one,axiom,
% 0.30/0.51 ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_add_one
% 0.30/0.51 thf(fact_994_less__add__one,axiom,
% 0.30/0.51 ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_add_one
% 0.30/0.51 thf(fact_995_add__le__imp__le__diff,axiom,
% 0.30/0.51 ! [I2: real,K: real,N2: real] :
% 0.30/0.51 ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N2 )
% 0.30/0.51 => ( ord_less_eq_real @ I2 @ ( minus_minus_real @ N2 @ K ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_le_imp_le_diff
% 0.30/0.51 thf(fact_996_add__le__imp__le__diff,axiom,
% 0.30/0.51 ! [I2: rat,K: rat,N2: rat] :
% 0.30/0.51 ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ N2 )
% 0.30/0.51 => ( ord_less_eq_rat @ I2 @ ( minus_minus_rat @ N2 @ K ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_le_imp_le_diff
% 0.30/0.51 thf(fact_997_add__le__imp__le__diff,axiom,
% 0.30/0.51 ! [I2: nat,K: nat,N2: nat] :
% 0.30/0.51 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N2 )
% 0.30/0.51 => ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N2 @ K ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_le_imp_le_diff
% 0.30/0.51 thf(fact_998_add__le__imp__le__diff,axiom,
% 0.30/0.51 ! [I2: int,K: int,N2: int] :
% 0.30/0.51 ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N2 )
% 0.30/0.51 => ( ord_less_eq_int @ I2 @ ( minus_minus_int @ N2 @ K ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_le_imp_le_diff
% 0.30/0.51 thf(fact_999_add__le__add__imp__diff__le,axiom,
% 0.30/0.51 ! [I2: real,K: real,N2: real,J: real] :
% 0.30/0.51 ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N2 )
% 0.30/0.51 => ( ( ord_less_eq_real @ N2 @ ( plus_plus_real @ J @ K ) )
% 0.30/0.51 => ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N2 )
% 0.30/0.51 => ( ( ord_less_eq_real @ N2 @ ( plus_plus_real @ J @ K ) )
% 0.30/0.51 => ( ord_less_eq_real @ ( minus_minus_real @ N2 @ K ) @ J ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_le_add_imp_diff_le
% 0.30/0.51 thf(fact_1000_add__le__add__imp__diff__le,axiom,
% 0.30/0.51 ! [I2: rat,K: rat,N2: rat,J: rat] :
% 0.30/0.51 ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ N2 )
% 0.30/0.51 => ( ( ord_less_eq_rat @ N2 @ ( plus_plus_rat @ J @ K ) )
% 0.30/0.51 => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ N2 )
% 0.30/0.51 => ( ( ord_less_eq_rat @ N2 @ ( plus_plus_rat @ J @ K ) )
% 0.30/0.51 => ( ord_less_eq_rat @ ( minus_minus_rat @ N2 @ K ) @ J ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_le_add_imp_diff_le
% 0.30/0.51 thf(fact_1001_add__le__add__imp__diff__le,axiom,
% 0.30/0.51 ! [I2: nat,K: nat,N2: nat,J: nat] :
% 0.30/0.51 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N2 )
% 0.30/0.51 => ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
% 0.30/0.51 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N2 )
% 0.30/0.51 => ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
% 0.30/0.51 => ( ord_less_eq_nat @ ( minus_minus_nat @ N2 @ K ) @ J ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_le_add_imp_diff_le
% 0.30/0.51 thf(fact_1002_add__le__add__imp__diff__le,axiom,
% 0.30/0.51 ! [I2: int,K: int,N2: int,J: int] :
% 0.30/0.51 ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N2 )
% 0.30/0.51 => ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K ) )
% 0.30/0.51 => ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N2 )
% 0.30/0.51 => ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K ) )
% 0.30/0.51 => ( ord_less_eq_int @ ( minus_minus_int @ N2 @ K ) @ J ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_le_add_imp_diff_le
% 0.30/0.51 thf(fact_1003_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.30/0.51 ! [A: real,B: real] :
% 0.30/0.51 ( ~ ( ord_less_real @ A @ B )
% 0.30/0.51 => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 0.30/0.51 = A ) ) ).
% 0.30/0.51
% 0.30/0.51 % linordered_semidom_class.add_diff_inverse
% 0.30/0.51 thf(fact_1004_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.30/0.51 ! [A: rat,B: rat] :
% 0.30/0.51 ( ~ ( ord_less_rat @ A @ B )
% 0.30/0.51 => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 0.30/0.51 = A ) ) ).
% 0.30/0.51
% 0.30/0.51 % linordered_semidom_class.add_diff_inverse
% 0.30/0.51 thf(fact_1005_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.30/0.51 ! [A: nat,B: nat] :
% 0.30/0.51 ( ~ ( ord_less_nat @ A @ B )
% 0.30/0.51 => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 0.30/0.51 = A ) ) ).
% 0.30/0.51
% 0.30/0.51 % linordered_semidom_class.add_diff_inverse
% 0.30/0.51 thf(fact_1006_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.30/0.51 ! [A: int,B: int] :
% 0.30/0.51 ( ~ ( ord_less_int @ A @ B )
% 0.30/0.51 => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 0.30/0.51 = A ) ) ).
% 0.30/0.51
% 0.30/0.51 % linordered_semidom_class.add_diff_inverse
% 0.30/0.51 thf(fact_1007_eq__add__iff1,axiom,
% 0.30/0.51 ! [A: real,E: real,C: real,B: real,D: real] :
% 0.30/0.51 ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
% 0.30/0.51 = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 0.30/0.51 = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C )
% 0.30/0.51 = D ) ) ).
% 0.30/0.51
% 0.30/0.51 % eq_add_iff1
% 0.30/0.51 thf(fact_1008_eq__add__iff1,axiom,
% 0.30/0.51 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 0.30/0.51 ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
% 0.30/0.51 = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 0.30/0.51 = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C )
% 0.30/0.51 = D ) ) ).
% 0.30/0.51
% 0.30/0.51 % eq_add_iff1
% 0.30/0.51 thf(fact_1009_eq__add__iff1,axiom,
% 0.30/0.51 ! [A: int,E: int,C: int,B: int,D: int] :
% 0.30/0.51 ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
% 0.30/0.51 = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 0.30/0.51 = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
% 0.30/0.51 = D ) ) ).
% 0.30/0.51
% 0.30/0.51 % eq_add_iff1
% 0.30/0.51 thf(fact_1010_eq__add__iff2,axiom,
% 0.30/0.51 ! [A: real,E: real,C: real,B: real,D: real] :
% 0.30/0.51 ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
% 0.30/0.51 = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 0.30/0.51 = ( C
% 0.30/0.51 = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % eq_add_iff2
% 0.30/0.51 thf(fact_1011_eq__add__iff2,axiom,
% 0.30/0.51 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 0.30/0.51 ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
% 0.30/0.51 = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 0.30/0.51 = ( C
% 0.30/0.51 = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % eq_add_iff2
% 0.30/0.51 thf(fact_1012_eq__add__iff2,axiom,
% 0.30/0.51 ! [A: int,E: int,C: int,B: int,D: int] :
% 0.30/0.51 ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
% 0.30/0.51 = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 0.30/0.51 = ( C
% 0.30/0.51 = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % eq_add_iff2
% 0.30/0.51 thf(fact_1013_square__diff__square__factored,axiom,
% 0.30/0.51 ! [X: real,Y: real] :
% 0.30/0.51 ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
% 0.30/0.51 = ( times_times_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ X @ Y ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % square_diff_square_factored
% 0.30/0.51 thf(fact_1014_square__diff__square__factored,axiom,
% 0.30/0.51 ! [X: rat,Y: rat] :
% 0.30/0.51 ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
% 0.30/0.51 = ( times_times_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_rat @ X @ Y ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % square_diff_square_factored
% 0.30/0.51 thf(fact_1015_square__diff__square__factored,axiom,
% 0.30/0.51 ! [X: int,Y: int] :
% 0.30/0.51 ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
% 0.30/0.51 = ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % square_diff_square_factored
% 0.30/0.51 thf(fact_1016_VEBT__internal_OminNull_Osimps_I5_J,axiom,
% 0.30/0.51 ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
% 0.30/0.51 ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc ) ) ).
% 0.30/0.51
% 0.30/0.51 % VEBT_internal.minNull.simps(5)
% 0.30/0.51 thf(fact_1017_vebt__member_Osimps_I2_J,axiom,
% 0.30/0.51 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] :
% 0.30/0.51 ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X ) ).
% 0.30/0.51
% 0.30/0.51 % vebt_member.simps(2)
% 0.30/0.51 thf(fact_1018_VEBT__internal_OminNull_Osimps_I4_J,axiom,
% 0.30/0.51 ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ).
% 0.30/0.51
% 0.30/0.51 % VEBT_internal.minNull.simps(4)
% 0.30/0.51 thf(fact_1019_is__succ__in__set__def,axiom,
% 0.30/0.51 ( vEBT_is_succ_in_set
% 0.30/0.51 = ( ^ [Xs2: set_nat,X2: nat,Y2: nat] :
% 0.30/0.51 ( ( member_nat @ Y2 @ Xs2 )
% 0.30/0.51 & ( ord_less_nat @ X2 @ Y2 )
% 0.30/0.51 & ! [Z3: nat] :
% 0.30/0.51 ( ( member_nat @ Z3 @ Xs2 )
% 0.30/0.51 => ( ( ord_less_nat @ X2 @ Z3 )
% 0.30/0.51 => ( ord_less_eq_nat @ Y2 @ Z3 ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % is_succ_in_set_def
% 0.30/0.51 thf(fact_1020_is__pred__in__set__def,axiom,
% 0.30/0.51 ( vEBT_is_pred_in_set
% 0.30/0.51 = ( ^ [Xs2: set_nat,X2: nat,Y2: nat] :
% 0.30/0.51 ( ( member_nat @ Y2 @ Xs2 )
% 0.30/0.51 & ( ord_less_nat @ Y2 @ X2 )
% 0.30/0.51 & ! [Z3: nat] :
% 0.30/0.51 ( ( member_nat @ Z3 @ Xs2 )
% 0.30/0.51 => ( ( ord_less_nat @ Z3 @ X2 )
% 0.30/0.51 => ( ord_less_eq_nat @ Z3 @ Y2 ) ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % is_pred_in_set_def
% 0.30/0.51 thf(fact_1021_vebt__insert_Osimps_I5_J,axiom,
% 0.30/0.51 ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 0.30/0.51 ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
% 0.30/0.51 = ( if_VEBT_VEBT
% 0.30/0.51 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.51 & ~ ( ( X = Mi )
% 0.30/0.51 | ( X = Ma ) ) )
% 0.30/0.51 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ X @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
% 0.30/0.51 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % vebt_insert.simps(5)
% 0.30/0.51 thf(fact_1022_ordered__ring__class_Ole__add__iff1,axiom,
% 0.30/0.51 ! [A: real,E: real,C: real,B: real,D: real] :
% 0.30/0.51 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 0.30/0.51 = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 0.30/0.51
% 0.30/0.51 % ordered_ring_class.le_add_iff1
% 0.30/0.51 thf(fact_1023_ordered__ring__class_Ole__add__iff1,axiom,
% 0.30/0.51 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 0.30/0.51 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 0.30/0.51 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 0.30/0.51
% 0.30/0.51 % ordered_ring_class.le_add_iff1
% 0.30/0.51 thf(fact_1024_ordered__ring__class_Ole__add__iff1,axiom,
% 0.30/0.51 ! [A: int,E: int,C: int,B: int,D: int] :
% 0.30/0.51 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 0.30/0.51 = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 0.30/0.51
% 0.30/0.51 % ordered_ring_class.le_add_iff1
% 0.30/0.51 thf(fact_1025_ordered__ring__class_Ole__add__iff2,axiom,
% 0.30/0.51 ! [A: real,E: real,C: real,B: real,D: real] :
% 0.30/0.51 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 0.30/0.51 = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % ordered_ring_class.le_add_iff2
% 0.30/0.51 thf(fact_1026_ordered__ring__class_Ole__add__iff2,axiom,
% 0.30/0.51 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 0.30/0.51 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 0.30/0.51 = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % ordered_ring_class.le_add_iff2
% 0.30/0.51 thf(fact_1027_ordered__ring__class_Ole__add__iff2,axiom,
% 0.30/0.51 ! [A: int,E: int,C: int,B: int,D: int] :
% 0.30/0.51 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 0.30/0.51 = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % ordered_ring_class.le_add_iff2
% 0.30/0.51 thf(fact_1028_less__add__iff1,axiom,
% 0.30/0.51 ! [A: real,E: real,C: real,B: real,D: real] :
% 0.30/0.51 ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 0.30/0.51 = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_add_iff1
% 0.30/0.51 thf(fact_1029_less__add__iff1,axiom,
% 0.30/0.51 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 0.30/0.51 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 0.30/0.51 = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_add_iff1
% 0.30/0.51 thf(fact_1030_less__add__iff1,axiom,
% 0.30/0.51 ! [A: int,E: int,C: int,B: int,D: int] :
% 0.30/0.51 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 0.30/0.51 = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_add_iff1
% 0.30/0.51 thf(fact_1031_less__add__iff2,axiom,
% 0.30/0.51 ! [A: real,E: real,C: real,B: real,D: real] :
% 0.30/0.51 ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 0.30/0.51 = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_add_iff2
% 0.30/0.51 thf(fact_1032_less__add__iff2,axiom,
% 0.30/0.51 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 0.30/0.51 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 0.30/0.51 = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_add_iff2
% 0.30/0.51 thf(fact_1033_less__add__iff2,axiom,
% 0.30/0.51 ! [A: int,E: int,C: int,B: int,D: int] :
% 0.30/0.51 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 0.30/0.51 = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_add_iff2
% 0.30/0.51 thf(fact_1034_square__diff__one__factored,axiom,
% 0.30/0.51 ! [X: complex] :
% 0.30/0.51 ( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ one_one_complex )
% 0.30/0.51 = ( times_times_complex @ ( plus_plus_complex @ X @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % square_diff_one_factored
% 0.30/0.51 thf(fact_1035_square__diff__one__factored,axiom,
% 0.30/0.51 ! [X: real] :
% 0.30/0.51 ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
% 0.30/0.51 = ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % square_diff_one_factored
% 0.30/0.51 thf(fact_1036_square__diff__one__factored,axiom,
% 0.30/0.51 ! [X: rat] :
% 0.30/0.51 ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ one_one_rat )
% 0.30/0.51 = ( times_times_rat @ ( plus_plus_rat @ X @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % square_diff_one_factored
% 0.30/0.51 thf(fact_1037_square__diff__one__factored,axiom,
% 0.30/0.51 ! [X: int] :
% 0.30/0.51 ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
% 0.30/0.51 = ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % square_diff_one_factored
% 0.30/0.51 thf(fact_1038_vebt__succ_Osimps_I3_J,axiom,
% 0.30/0.51 ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
% 0.30/0.51 ( ( vEBT_vebt_succ @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Va )
% 0.30/0.51 = none_nat ) ).
% 0.30/0.51
% 0.30/0.51 % vebt_succ.simps(3)
% 0.30/0.51 thf(fact_1039_vebt__pred_Osimps_I4_J,axiom,
% 0.30/0.51 ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
% 0.30/0.51 ( ( vEBT_vebt_pred @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va ) @ Vb )
% 0.30/0.51 = none_nat ) ).
% 0.30/0.51
% 0.30/0.51 % vebt_pred.simps(4)
% 0.30/0.51 thf(fact_1040_real__average__minus__second,axiom,
% 0.30/0.51 ! [B: real,A: real] :
% 0.30/0.51 ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 0.30/0.51 = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % real_average_minus_second
% 0.30/0.51 thf(fact_1041_real__average__minus__first,axiom,
% 0.30/0.51 ! [A: real,B: real] :
% 0.30/0.51 ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 0.30/0.51 = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % real_average_minus_first
% 0.30/0.51 thf(fact_1042_buildup__gives__empty,axiom,
% 0.30/0.51 ! [N2: nat] :
% 0.30/0.51 ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N2 ) )
% 0.30/0.51 = bot_bot_set_nat ) ).
% 0.30/0.51
% 0.30/0.51 % buildup_gives_empty
% 0.30/0.51 thf(fact_1043_max__less__iff__conj,axiom,
% 0.30/0.51 ! [X: real,Y: real,Z: real] :
% 0.30/0.51 ( ( ord_less_real @ ( ord_max_real @ X @ Y ) @ Z )
% 0.30/0.51 = ( ( ord_less_real @ X @ Z )
% 0.30/0.51 & ( ord_less_real @ Y @ Z ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max_less_iff_conj
% 0.30/0.51 thf(fact_1044_max__less__iff__conj,axiom,
% 0.30/0.51 ! [X: rat,Y: rat,Z: rat] :
% 0.30/0.51 ( ( ord_less_rat @ ( ord_max_rat @ X @ Y ) @ Z )
% 0.30/0.51 = ( ( ord_less_rat @ X @ Z )
% 0.30/0.51 & ( ord_less_rat @ Y @ Z ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max_less_iff_conj
% 0.30/0.51 thf(fact_1045_max__less__iff__conj,axiom,
% 0.30/0.51 ! [X: num,Y: num,Z: num] :
% 0.30/0.51 ( ( ord_less_num @ ( ord_max_num @ X @ Y ) @ Z )
% 0.30/0.51 = ( ( ord_less_num @ X @ Z )
% 0.30/0.51 & ( ord_less_num @ Y @ Z ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max_less_iff_conj
% 0.30/0.51 thf(fact_1046_max__less__iff__conj,axiom,
% 0.30/0.51 ! [X: nat,Y: nat,Z: nat] :
% 0.30/0.51 ( ( ord_less_nat @ ( ord_max_nat @ X @ Y ) @ Z )
% 0.30/0.51 = ( ( ord_less_nat @ X @ Z )
% 0.30/0.51 & ( ord_less_nat @ Y @ Z ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max_less_iff_conj
% 0.30/0.51 thf(fact_1047_max__less__iff__conj,axiom,
% 0.30/0.51 ! [X: int,Y: int,Z: int] :
% 0.30/0.51 ( ( ord_less_int @ ( ord_max_int @ X @ Y ) @ Z )
% 0.30/0.51 = ( ( ord_less_int @ X @ Z )
% 0.30/0.51 & ( ord_less_int @ Y @ Z ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max_less_iff_conj
% 0.30/0.51 thf(fact_1048_max_Oabsorb4,axiom,
% 0.30/0.51 ! [A: real,B: real] :
% 0.30/0.51 ( ( ord_less_real @ A @ B )
% 0.30/0.51 => ( ( ord_max_real @ A @ B )
% 0.30/0.51 = B ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.absorb4
% 0.30/0.51 thf(fact_1049_max_Oabsorb4,axiom,
% 0.30/0.51 ! [A: rat,B: rat] :
% 0.30/0.51 ( ( ord_less_rat @ A @ B )
% 0.30/0.51 => ( ( ord_max_rat @ A @ B )
% 0.30/0.51 = B ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.absorb4
% 0.30/0.51 thf(fact_1050_max_Oabsorb4,axiom,
% 0.30/0.51 ! [A: num,B: num] :
% 0.30/0.51 ( ( ord_less_num @ A @ B )
% 0.30/0.51 => ( ( ord_max_num @ A @ B )
% 0.30/0.51 = B ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.absorb4
% 0.30/0.51 thf(fact_1051_max_Oabsorb4,axiom,
% 0.30/0.51 ! [A: nat,B: nat] :
% 0.30/0.51 ( ( ord_less_nat @ A @ B )
% 0.30/0.51 => ( ( ord_max_nat @ A @ B )
% 0.30/0.51 = B ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.absorb4
% 0.30/0.51 thf(fact_1052_max_Oabsorb4,axiom,
% 0.30/0.51 ! [A: int,B: int] :
% 0.30/0.51 ( ( ord_less_int @ A @ B )
% 0.30/0.51 => ( ( ord_max_int @ A @ B )
% 0.30/0.51 = B ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.absorb4
% 0.30/0.51 thf(fact_1053_max_Oabsorb3,axiom,
% 0.30/0.51 ! [B: real,A: real] :
% 0.30/0.51 ( ( ord_less_real @ B @ A )
% 0.30/0.51 => ( ( ord_max_real @ A @ B )
% 0.30/0.51 = A ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.absorb3
% 0.30/0.51 thf(fact_1054_max_Oabsorb3,axiom,
% 0.30/0.51 ! [B: rat,A: rat] :
% 0.30/0.51 ( ( ord_less_rat @ B @ A )
% 0.30/0.51 => ( ( ord_max_rat @ A @ B )
% 0.30/0.51 = A ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.absorb3
% 0.30/0.51 thf(fact_1055_max_Oabsorb3,axiom,
% 0.30/0.51 ! [B: num,A: num] :
% 0.30/0.51 ( ( ord_less_num @ B @ A )
% 0.30/0.51 => ( ( ord_max_num @ A @ B )
% 0.30/0.51 = A ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.absorb3
% 0.30/0.51 thf(fact_1056_max_Oabsorb3,axiom,
% 0.30/0.51 ! [B: nat,A: nat] :
% 0.30/0.51 ( ( ord_less_nat @ B @ A )
% 0.30/0.51 => ( ( ord_max_nat @ A @ B )
% 0.30/0.51 = A ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.absorb3
% 0.30/0.51 thf(fact_1057_max_Oabsorb3,axiom,
% 0.30/0.51 ! [B: int,A: int] :
% 0.30/0.51 ( ( ord_less_int @ B @ A )
% 0.30/0.51 => ( ( ord_max_int @ A @ B )
% 0.30/0.51 = A ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.absorb3
% 0.30/0.51 thf(fact_1058_max_Obounded__iff,axiom,
% 0.30/0.51 ! [B: rat,C: rat,A: rat] :
% 0.30/0.51 ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
% 0.30/0.51 = ( ( ord_less_eq_rat @ B @ A )
% 0.30/0.51 & ( ord_less_eq_rat @ C @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.bounded_iff
% 0.30/0.51 thf(fact_1059_max_Obounded__iff,axiom,
% 0.30/0.51 ! [B: num,C: num,A: num] :
% 0.30/0.51 ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
% 0.30/0.51 = ( ( ord_less_eq_num @ B @ A )
% 0.30/0.51 & ( ord_less_eq_num @ C @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.bounded_iff
% 0.30/0.51 thf(fact_1060_max_Obounded__iff,axiom,
% 0.30/0.51 ! [B: nat,C: nat,A: nat] :
% 0.30/0.51 ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
% 0.30/0.51 = ( ( ord_less_eq_nat @ B @ A )
% 0.30/0.51 & ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.bounded_iff
% 0.30/0.51 thf(fact_1061_max_Obounded__iff,axiom,
% 0.30/0.51 ! [B: int,C: int,A: int] :
% 0.30/0.51 ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
% 0.30/0.51 = ( ( ord_less_eq_int @ B @ A )
% 0.30/0.51 & ( ord_less_eq_int @ C @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.bounded_iff
% 0.30/0.51 thf(fact_1062_max_Oabsorb2,axiom,
% 0.30/0.51 ! [A: rat,B: rat] :
% 0.30/0.51 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.51 => ( ( ord_max_rat @ A @ B )
% 0.30/0.51 = B ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.absorb2
% 0.30/0.51 thf(fact_1063_max_Oabsorb2,axiom,
% 0.30/0.51 ! [A: num,B: num] :
% 0.30/0.51 ( ( ord_less_eq_num @ A @ B )
% 0.30/0.51 => ( ( ord_max_num @ A @ B )
% 0.30/0.51 = B ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.absorb2
% 0.30/0.51 thf(fact_1064_max_Oabsorb2,axiom,
% 0.30/0.51 ! [A: nat,B: nat] :
% 0.30/0.51 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.51 => ( ( ord_max_nat @ A @ B )
% 0.30/0.51 = B ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.absorb2
% 0.30/0.51 thf(fact_1065_max_Oabsorb2,axiom,
% 0.30/0.51 ! [A: int,B: int] :
% 0.30/0.51 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.51 => ( ( ord_max_int @ A @ B )
% 0.30/0.51 = B ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.absorb2
% 0.30/0.51 thf(fact_1066_max_Oabsorb1,axiom,
% 0.30/0.51 ! [B: rat,A: rat] :
% 0.30/0.51 ( ( ord_less_eq_rat @ B @ A )
% 0.30/0.51 => ( ( ord_max_rat @ A @ B )
% 0.30/0.51 = A ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.absorb1
% 0.30/0.51 thf(fact_1067_max_Oabsorb1,axiom,
% 0.30/0.51 ! [B: num,A: num] :
% 0.30/0.51 ( ( ord_less_eq_num @ B @ A )
% 0.30/0.51 => ( ( ord_max_num @ A @ B )
% 0.30/0.51 = A ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.absorb1
% 0.30/0.51 thf(fact_1068_max_Oabsorb1,axiom,
% 0.30/0.51 ! [B: nat,A: nat] :
% 0.30/0.51 ( ( ord_less_eq_nat @ B @ A )
% 0.30/0.51 => ( ( ord_max_nat @ A @ B )
% 0.30/0.51 = A ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.absorb1
% 0.30/0.51 thf(fact_1069_max_Oabsorb1,axiom,
% 0.30/0.51 ! [B: int,A: int] :
% 0.30/0.51 ( ( ord_less_eq_int @ B @ A )
% 0.30/0.51 => ( ( ord_max_int @ A @ B )
% 0.30/0.51 = A ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.absorb1
% 0.30/0.51 thf(fact_1070_Diff__eq__empty__iff,axiom,
% 0.30/0.51 ! [A2: set_int,B3: set_int] :
% 0.30/0.51 ( ( ( minus_minus_set_int @ A2 @ B3 )
% 0.30/0.51 = bot_bot_set_int )
% 0.30/0.51 = ( ord_less_eq_set_int @ A2 @ B3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % Diff_eq_empty_iff
% 0.30/0.51 thf(fact_1071_Diff__eq__empty__iff,axiom,
% 0.30/0.51 ! [A2: set_real,B3: set_real] :
% 0.30/0.51 ( ( ( minus_minus_set_real @ A2 @ B3 )
% 0.30/0.51 = bot_bot_set_real )
% 0.30/0.51 = ( ord_less_eq_set_real @ A2 @ B3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % Diff_eq_empty_iff
% 0.30/0.51 thf(fact_1072_Diff__eq__empty__iff,axiom,
% 0.30/0.51 ! [A2: set_nat,B3: set_nat] :
% 0.30/0.51 ( ( ( minus_minus_set_nat @ A2 @ B3 )
% 0.30/0.51 = bot_bot_set_nat )
% 0.30/0.51 = ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % Diff_eq_empty_iff
% 0.30/0.51 thf(fact_1073_subsetI,axiom,
% 0.30/0.51 ! [A2: set_real,B3: set_real] :
% 0.30/0.51 ( ! [X4: real] :
% 0.30/0.51 ( ( member_real @ X4 @ A2 )
% 0.30/0.51 => ( member_real @ X4 @ B3 ) )
% 0.30/0.51 => ( ord_less_eq_set_real @ A2 @ B3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % subsetI
% 0.30/0.51 thf(fact_1074_subsetI,axiom,
% 0.30/0.51 ! [A2: set_int,B3: set_int] :
% 0.30/0.51 ( ! [X4: int] :
% 0.30/0.51 ( ( member_int @ X4 @ A2 )
% 0.30/0.51 => ( member_int @ X4 @ B3 ) )
% 0.30/0.51 => ( ord_less_eq_set_int @ A2 @ B3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % subsetI
% 0.30/0.51 thf(fact_1075_subsetI,axiom,
% 0.30/0.51 ! [A2: set_complex,B3: set_complex] :
% 0.30/0.51 ( ! [X4: complex] :
% 0.30/0.51 ( ( member_complex @ X4 @ A2 )
% 0.30/0.51 => ( member_complex @ X4 @ B3 ) )
% 0.30/0.51 => ( ord_le211207098394363844omplex @ A2 @ B3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % subsetI
% 0.30/0.51 thf(fact_1076_subsetI,axiom,
% 0.30/0.51 ! [A2: set_set_nat,B3: set_set_nat] :
% 0.30/0.51 ( ! [X4: set_nat] :
% 0.30/0.51 ( ( member_set_nat @ X4 @ A2 )
% 0.30/0.51 => ( member_set_nat @ X4 @ B3 ) )
% 0.30/0.51 => ( ord_le6893508408891458716et_nat @ A2 @ B3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % subsetI
% 0.30/0.51 thf(fact_1077_subsetI,axiom,
% 0.30/0.51 ! [A2: set_nat,B3: set_nat] :
% 0.30/0.51 ( ! [X4: nat] :
% 0.30/0.51 ( ( member_nat @ X4 @ A2 )
% 0.30/0.51 => ( member_nat @ X4 @ B3 ) )
% 0.30/0.51 => ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % subsetI
% 0.30/0.51 thf(fact_1078_subset__antisym,axiom,
% 0.30/0.51 ! [A2: set_nat,B3: set_nat] :
% 0.30/0.51 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.30/0.51 => ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 0.30/0.51 => ( A2 = B3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % subset_antisym
% 0.30/0.51 thf(fact_1079_psubsetI,axiom,
% 0.30/0.51 ! [A2: set_nat,B3: set_nat] :
% 0.30/0.51 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.30/0.51 => ( ( A2 != B3 )
% 0.30/0.51 => ( ord_less_set_nat @ A2 @ B3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % psubsetI
% 0.30/0.51 thf(fact_1080_DiffI,axiom,
% 0.30/0.51 ! [C: real,A2: set_real,B3: set_real] :
% 0.30/0.51 ( ( member_real @ C @ A2 )
% 0.30/0.51 => ( ~ ( member_real @ C @ B3 )
% 0.30/0.51 => ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % DiffI
% 0.30/0.51 thf(fact_1081_DiffI,axiom,
% 0.30/0.51 ! [C: int,A2: set_int,B3: set_int] :
% 0.30/0.51 ( ( member_int @ C @ A2 )
% 0.30/0.51 => ( ~ ( member_int @ C @ B3 )
% 0.30/0.51 => ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % DiffI
% 0.30/0.51 thf(fact_1082_DiffI,axiom,
% 0.30/0.51 ! [C: complex,A2: set_complex,B3: set_complex] :
% 0.30/0.51 ( ( member_complex @ C @ A2 )
% 0.30/0.51 => ( ~ ( member_complex @ C @ B3 )
% 0.30/0.51 => ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % DiffI
% 0.30/0.51 thf(fact_1083_DiffI,axiom,
% 0.30/0.51 ! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
% 0.30/0.51 ( ( member_set_nat @ C @ A2 )
% 0.30/0.51 => ( ~ ( member_set_nat @ C @ B3 )
% 0.30/0.51 => ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B3 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % DiffI
% 0.30/0.51 thf(fact_1084_DiffI,axiom,
% 0.30/0.51 ! [C: nat,A2: set_nat,B3: set_nat] :
% 0.30/0.51 ( ( member_nat @ C @ A2 )
% 0.30/0.51 => ( ~ ( member_nat @ C @ B3 )
% 0.30/0.51 => ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % DiffI
% 0.30/0.51 thf(fact_1085_Diff__iff,axiom,
% 0.30/0.51 ! [C: real,A2: set_real,B3: set_real] :
% 0.30/0.51 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
% 0.30/0.51 = ( ( member_real @ C @ A2 )
% 0.30/0.51 & ~ ( member_real @ C @ B3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Diff_iff
% 0.30/0.51 thf(fact_1086_Diff__iff,axiom,
% 0.30/0.51 ! [C: int,A2: set_int,B3: set_int] :
% 0.30/0.51 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
% 0.30/0.51 = ( ( member_int @ C @ A2 )
% 0.30/0.51 & ~ ( member_int @ C @ B3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Diff_iff
% 0.30/0.51 thf(fact_1087_Diff__iff,axiom,
% 0.30/0.51 ! [C: complex,A2: set_complex,B3: set_complex] :
% 0.30/0.51 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 0.30/0.51 = ( ( member_complex @ C @ A2 )
% 0.30/0.51 & ~ ( member_complex @ C @ B3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Diff_iff
% 0.30/0.51 thf(fact_1088_Diff__iff,axiom,
% 0.30/0.51 ! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
% 0.30/0.51 ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B3 ) )
% 0.30/0.51 = ( ( member_set_nat @ C @ A2 )
% 0.30/0.51 & ~ ( member_set_nat @ C @ B3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Diff_iff
% 0.30/0.51 thf(fact_1089_Diff__iff,axiom,
% 0.30/0.51 ! [C: nat,A2: set_nat,B3: set_nat] :
% 0.30/0.51 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 0.30/0.51 = ( ( member_nat @ C @ A2 )
% 0.30/0.51 & ~ ( member_nat @ C @ B3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Diff_iff
% 0.30/0.51 thf(fact_1090_Diff__idemp,axiom,
% 0.30/0.51 ! [A2: set_nat,B3: set_nat] :
% 0.30/0.51 ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B3 ) @ B3 )
% 0.30/0.51 = ( minus_minus_set_nat @ A2 @ B3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % Diff_idemp
% 0.30/0.51 thf(fact_1091_subset__empty,axiom,
% 0.30/0.51 ! [A2: set_int] :
% 0.30/0.51 ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
% 0.30/0.51 = ( A2 = bot_bot_set_int ) ) ).
% 0.30/0.51
% 0.30/0.51 % subset_empty
% 0.30/0.51 thf(fact_1092_subset__empty,axiom,
% 0.30/0.51 ! [A2: set_real] :
% 0.30/0.51 ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
% 0.30/0.51 = ( A2 = bot_bot_set_real ) ) ).
% 0.30/0.51
% 0.30/0.51 % subset_empty
% 0.30/0.51 thf(fact_1093_subset__empty,axiom,
% 0.30/0.51 ! [A2: set_nat] :
% 0.30/0.51 ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
% 0.30/0.51 = ( A2 = bot_bot_set_nat ) ) ).
% 0.30/0.51
% 0.30/0.51 % subset_empty
% 0.30/0.51 thf(fact_1094_empty__subsetI,axiom,
% 0.30/0.51 ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).
% 0.30/0.51
% 0.30/0.51 % empty_subsetI
% 0.30/0.51 thf(fact_1095_empty__subsetI,axiom,
% 0.30/0.51 ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).
% 0.30/0.51
% 0.30/0.51 % empty_subsetI
% 0.30/0.51 thf(fact_1096_empty__subsetI,axiom,
% 0.30/0.51 ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% 0.30/0.51
% 0.30/0.51 % empty_subsetI
% 0.30/0.51 thf(fact_1097_Diff__cancel,axiom,
% 0.30/0.51 ! [A2: set_int] :
% 0.30/0.51 ( ( minus_minus_set_int @ A2 @ A2 )
% 0.30/0.51 = bot_bot_set_int ) ).
% 0.30/0.51
% 0.30/0.51 % Diff_cancel
% 0.30/0.51 thf(fact_1098_Diff__cancel,axiom,
% 0.30/0.51 ! [A2: set_real] :
% 0.30/0.51 ( ( minus_minus_set_real @ A2 @ A2 )
% 0.30/0.51 = bot_bot_set_real ) ).
% 0.30/0.51
% 0.30/0.51 % Diff_cancel
% 0.30/0.51 thf(fact_1099_Diff__cancel,axiom,
% 0.30/0.51 ! [A2: set_nat] :
% 0.30/0.51 ( ( minus_minus_set_nat @ A2 @ A2 )
% 0.30/0.51 = bot_bot_set_nat ) ).
% 0.30/0.51
% 0.30/0.51 % Diff_cancel
% 0.30/0.51 thf(fact_1100_empty__Diff,axiom,
% 0.30/0.51 ! [A2: set_int] :
% 0.30/0.51 ( ( minus_minus_set_int @ bot_bot_set_int @ A2 )
% 0.30/0.51 = bot_bot_set_int ) ).
% 0.30/0.51
% 0.30/0.51 % empty_Diff
% 0.30/0.51 thf(fact_1101_empty__Diff,axiom,
% 0.30/0.51 ! [A2: set_real] :
% 0.30/0.51 ( ( minus_minus_set_real @ bot_bot_set_real @ A2 )
% 0.30/0.51 = bot_bot_set_real ) ).
% 0.30/0.51
% 0.30/0.51 % empty_Diff
% 0.30/0.51 thf(fact_1102_empty__Diff,axiom,
% 0.30/0.51 ! [A2: set_nat] :
% 0.30/0.51 ( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
% 0.30/0.51 = bot_bot_set_nat ) ).
% 0.30/0.51
% 0.30/0.51 % empty_Diff
% 0.30/0.51 thf(fact_1103_Diff__empty,axiom,
% 0.30/0.51 ! [A2: set_int] :
% 0.30/0.51 ( ( minus_minus_set_int @ A2 @ bot_bot_set_int )
% 0.30/0.51 = A2 ) ).
% 0.30/0.51
% 0.30/0.51 % Diff_empty
% 0.30/0.51 thf(fact_1104_Diff__empty,axiom,
% 0.30/0.51 ! [A2: set_real] :
% 0.30/0.51 ( ( minus_minus_set_real @ A2 @ bot_bot_set_real )
% 0.30/0.51 = A2 ) ).
% 0.30/0.51
% 0.30/0.51 % Diff_empty
% 0.30/0.51 thf(fact_1105_Diff__empty,axiom,
% 0.30/0.51 ! [A2: set_nat] :
% 0.30/0.51 ( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
% 0.30/0.51 = A2 ) ).
% 0.30/0.51
% 0.30/0.51 % Diff_empty
% 0.30/0.51 thf(fact_1106_psubset__imp__ex__mem,axiom,
% 0.30/0.51 ! [A2: set_real,B3: set_real] :
% 0.30/0.51 ( ( ord_less_set_real @ A2 @ B3 )
% 0.30/0.51 => ? [B2: real] : ( member_real @ B2 @ ( minus_minus_set_real @ B3 @ A2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % psubset_imp_ex_mem
% 0.30/0.51 thf(fact_1107_psubset__imp__ex__mem,axiom,
% 0.30/0.51 ! [A2: set_int,B3: set_int] :
% 0.30/0.51 ( ( ord_less_set_int @ A2 @ B3 )
% 0.30/0.51 => ? [B2: int] : ( member_int @ B2 @ ( minus_minus_set_int @ B3 @ A2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % psubset_imp_ex_mem
% 0.30/0.51 thf(fact_1108_psubset__imp__ex__mem,axiom,
% 0.30/0.51 ! [A2: set_complex,B3: set_complex] :
% 0.30/0.51 ( ( ord_less_set_complex @ A2 @ B3 )
% 0.30/0.51 => ? [B2: complex] : ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B3 @ A2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % psubset_imp_ex_mem
% 0.30/0.51 thf(fact_1109_psubset__imp__ex__mem,axiom,
% 0.30/0.51 ! [A2: set_set_nat,B3: set_set_nat] :
% 0.30/0.51 ( ( ord_less_set_set_nat @ A2 @ B3 )
% 0.30/0.51 => ? [B2: set_nat] : ( member_set_nat @ B2 @ ( minus_2163939370556025621et_nat @ B3 @ A2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % psubset_imp_ex_mem
% 0.30/0.51 thf(fact_1110_psubset__imp__ex__mem,axiom,
% 0.30/0.51 ! [A2: set_nat,B3: set_nat] :
% 0.30/0.51 ( ( ord_less_set_nat @ A2 @ B3 )
% 0.30/0.51 => ? [B2: nat] : ( member_nat @ B2 @ ( minus_minus_set_nat @ B3 @ A2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % psubset_imp_ex_mem
% 0.30/0.51 thf(fact_1111_enat__less__induct,axiom,
% 0.30/0.51 ! [P: extended_enat > $o,N2: extended_enat] :
% 0.30/0.51 ( ! [N4: extended_enat] :
% 0.30/0.51 ( ! [M5: extended_enat] :
% 0.30/0.51 ( ( ord_le72135733267957522d_enat @ M5 @ N4 )
% 0.30/0.51 => ( P @ M5 ) )
% 0.30/0.51 => ( P @ N4 ) )
% 0.30/0.51 => ( P @ N2 ) ) ).
% 0.30/0.51
% 0.30/0.51 % enat_less_induct
% 0.30/0.51 thf(fact_1112_DiffE,axiom,
% 0.30/0.51 ! [C: real,A2: set_real,B3: set_real] :
% 0.30/0.51 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
% 0.30/0.51 => ~ ( ( member_real @ C @ A2 )
% 0.30/0.51 => ( member_real @ C @ B3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % DiffE
% 0.30/0.51 thf(fact_1113_DiffE,axiom,
% 0.30/0.51 ! [C: int,A2: set_int,B3: set_int] :
% 0.30/0.51 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
% 0.30/0.51 => ~ ( ( member_int @ C @ A2 )
% 0.30/0.51 => ( member_int @ C @ B3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % DiffE
% 0.30/0.51 thf(fact_1114_DiffE,axiom,
% 0.30/0.51 ! [C: complex,A2: set_complex,B3: set_complex] :
% 0.30/0.51 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 0.30/0.51 => ~ ( ( member_complex @ C @ A2 )
% 0.30/0.51 => ( member_complex @ C @ B3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % DiffE
% 0.30/0.51 thf(fact_1115_DiffE,axiom,
% 0.30/0.51 ! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
% 0.30/0.51 ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B3 ) )
% 0.30/0.51 => ~ ( ( member_set_nat @ C @ A2 )
% 0.30/0.51 => ( member_set_nat @ C @ B3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % DiffE
% 0.30/0.51 thf(fact_1116_DiffE,axiom,
% 0.30/0.51 ! [C: nat,A2: set_nat,B3: set_nat] :
% 0.30/0.51 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 0.30/0.51 => ~ ( ( member_nat @ C @ A2 )
% 0.30/0.51 => ( member_nat @ C @ B3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % DiffE
% 0.30/0.51 thf(fact_1117_DiffD1,axiom,
% 0.30/0.51 ! [C: real,A2: set_real,B3: set_real] :
% 0.30/0.51 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
% 0.30/0.51 => ( member_real @ C @ A2 ) ) ).
% 0.30/0.51
% 0.30/0.51 % DiffD1
% 0.30/0.51 thf(fact_1118_DiffD1,axiom,
% 0.30/0.51 ! [C: int,A2: set_int,B3: set_int] :
% 0.30/0.51 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
% 0.30/0.51 => ( member_int @ C @ A2 ) ) ).
% 0.30/0.51
% 0.30/0.51 % DiffD1
% 0.30/0.51 thf(fact_1119_DiffD1,axiom,
% 0.30/0.51 ! [C: complex,A2: set_complex,B3: set_complex] :
% 0.30/0.51 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 0.30/0.51 => ( member_complex @ C @ A2 ) ) ).
% 0.30/0.51
% 0.30/0.51 % DiffD1
% 0.30/0.51 thf(fact_1120_DiffD1,axiom,
% 0.30/0.51 ! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
% 0.30/0.51 ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B3 ) )
% 0.30/0.51 => ( member_set_nat @ C @ A2 ) ) ).
% 0.30/0.51
% 0.30/0.51 % DiffD1
% 0.30/0.51 thf(fact_1121_DiffD1,axiom,
% 0.30/0.51 ! [C: nat,A2: set_nat,B3: set_nat] :
% 0.30/0.51 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 0.30/0.51 => ( member_nat @ C @ A2 ) ) ).
% 0.30/0.51
% 0.30/0.51 % DiffD1
% 0.30/0.51 thf(fact_1122_DiffD2,axiom,
% 0.30/0.51 ! [C: real,A2: set_real,B3: set_real] :
% 0.30/0.51 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
% 0.30/0.51 => ~ ( member_real @ C @ B3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % DiffD2
% 0.30/0.51 thf(fact_1123_DiffD2,axiom,
% 0.30/0.51 ! [C: int,A2: set_int,B3: set_int] :
% 0.30/0.51 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
% 0.30/0.51 => ~ ( member_int @ C @ B3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % DiffD2
% 0.30/0.51 thf(fact_1124_DiffD2,axiom,
% 0.30/0.51 ! [C: complex,A2: set_complex,B3: set_complex] :
% 0.30/0.51 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 0.30/0.51 => ~ ( member_complex @ C @ B3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % DiffD2
% 0.30/0.51 thf(fact_1125_DiffD2,axiom,
% 0.30/0.51 ! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
% 0.30/0.51 ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B3 ) )
% 0.30/0.51 => ~ ( member_set_nat @ C @ B3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % DiffD2
% 0.30/0.51 thf(fact_1126_DiffD2,axiom,
% 0.30/0.51 ! [C: nat,A2: set_nat,B3: set_nat] :
% 0.30/0.51 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 0.30/0.51 => ~ ( member_nat @ C @ B3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % DiffD2
% 0.30/0.51 thf(fact_1127_set__diff__eq,axiom,
% 0.30/0.51 ( minus_minus_set_real
% 0.30/0.51 = ( ^ [A4: set_real,B4: set_real] :
% 0.30/0.51 ( collect_real
% 0.30/0.51 @ ^ [X2: real] :
% 0.30/0.51 ( ( member_real @ X2 @ A4 )
% 0.30/0.51 & ~ ( member_real @ X2 @ B4 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % set_diff_eq
% 0.30/0.51 thf(fact_1128_set__diff__eq,axiom,
% 0.30/0.51 ( minus_1052850069191792384nt_int
% 0.30/0.51 = ( ^ [A4: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
% 0.30/0.51 ( collec213857154873943460nt_int
% 0.30/0.51 @ ^ [X2: product_prod_int_int] :
% 0.30/0.51 ( ( member5262025264175285858nt_int @ X2 @ A4 )
% 0.30/0.51 & ~ ( member5262025264175285858nt_int @ X2 @ B4 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % set_diff_eq
% 0.30/0.51 thf(fact_1129_set__diff__eq,axiom,
% 0.30/0.51 ( minus_811609699411566653omplex
% 0.30/0.51 = ( ^ [A4: set_complex,B4: set_complex] :
% 0.30/0.51 ( collect_complex
% 0.30/0.51 @ ^ [X2: complex] :
% 0.30/0.51 ( ( member_complex @ X2 @ A4 )
% 0.30/0.51 & ~ ( member_complex @ X2 @ B4 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % set_diff_eq
% 0.30/0.51 thf(fact_1130_set__diff__eq,axiom,
% 0.30/0.51 ( minus_2163939370556025621et_nat
% 0.30/0.51 = ( ^ [A4: set_set_nat,B4: set_set_nat] :
% 0.30/0.51 ( collect_set_nat
% 0.30/0.51 @ ^ [X2: set_nat] :
% 0.30/0.51 ( ( member_set_nat @ X2 @ A4 )
% 0.30/0.51 & ~ ( member_set_nat @ X2 @ B4 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % set_diff_eq
% 0.30/0.51 thf(fact_1131_set__diff__eq,axiom,
% 0.30/0.51 ( minus_minus_set_int
% 0.30/0.51 = ( ^ [A4: set_int,B4: set_int] :
% 0.30/0.51 ( collect_int
% 0.30/0.51 @ ^ [X2: int] :
% 0.30/0.51 ( ( member_int @ X2 @ A4 )
% 0.30/0.51 & ~ ( member_int @ X2 @ B4 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % set_diff_eq
% 0.30/0.51 thf(fact_1132_set__diff__eq,axiom,
% 0.30/0.51 ( minus_minus_set_nat
% 0.30/0.51 = ( ^ [A4: set_nat,B4: set_nat] :
% 0.30/0.51 ( collect_nat
% 0.30/0.51 @ ^ [X2: nat] :
% 0.30/0.51 ( ( member_nat @ X2 @ A4 )
% 0.30/0.51 & ~ ( member_nat @ X2 @ B4 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % set_diff_eq
% 0.30/0.51 thf(fact_1133_minus__set__def,axiom,
% 0.30/0.51 ( minus_minus_set_real
% 0.30/0.51 = ( ^ [A4: set_real,B4: set_real] :
% 0.30/0.51 ( collect_real
% 0.30/0.51 @ ( minus_minus_real_o
% 0.30/0.51 @ ^ [X2: real] : ( member_real @ X2 @ A4 )
% 0.30/0.51 @ ^ [X2: real] : ( member_real @ X2 @ B4 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % minus_set_def
% 0.30/0.51 thf(fact_1134_minus__set__def,axiom,
% 0.30/0.51 ( minus_1052850069191792384nt_int
% 0.30/0.51 = ( ^ [A4: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
% 0.30/0.51 ( collec213857154873943460nt_int
% 0.30/0.51 @ ( minus_711738161318947805_int_o
% 0.30/0.51 @ ^ [X2: product_prod_int_int] : ( member5262025264175285858nt_int @ X2 @ A4 )
% 0.30/0.51 @ ^ [X2: product_prod_int_int] : ( member5262025264175285858nt_int @ X2 @ B4 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % minus_set_def
% 0.30/0.51 thf(fact_1135_minus__set__def,axiom,
% 0.30/0.51 ( minus_811609699411566653omplex
% 0.30/0.51 = ( ^ [A4: set_complex,B4: set_complex] :
% 0.30/0.51 ( collect_complex
% 0.30/0.51 @ ( minus_8727706125548526216plex_o
% 0.30/0.51 @ ^ [X2: complex] : ( member_complex @ X2 @ A4 )
% 0.30/0.51 @ ^ [X2: complex] : ( member_complex @ X2 @ B4 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % minus_set_def
% 0.30/0.51 thf(fact_1136_minus__set__def,axiom,
% 0.30/0.51 ( minus_2163939370556025621et_nat
% 0.30/0.51 = ( ^ [A4: set_set_nat,B4: set_set_nat] :
% 0.30/0.51 ( collect_set_nat
% 0.30/0.51 @ ( minus_6910147592129066416_nat_o
% 0.30/0.51 @ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A4 )
% 0.30/0.51 @ ^ [X2: set_nat] : ( member_set_nat @ X2 @ B4 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % minus_set_def
% 0.30/0.51 thf(fact_1137_minus__set__def,axiom,
% 0.30/0.51 ( minus_minus_set_int
% 0.30/0.51 = ( ^ [A4: set_int,B4: set_int] :
% 0.30/0.51 ( collect_int
% 0.30/0.51 @ ( minus_minus_int_o
% 0.30/0.51 @ ^ [X2: int] : ( member_int @ X2 @ A4 )
% 0.30/0.51 @ ^ [X2: int] : ( member_int @ X2 @ B4 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % minus_set_def
% 0.30/0.51 thf(fact_1138_minus__set__def,axiom,
% 0.30/0.51 ( minus_minus_set_nat
% 0.30/0.51 = ( ^ [A4: set_nat,B4: set_nat] :
% 0.30/0.51 ( collect_nat
% 0.30/0.51 @ ( minus_minus_nat_o
% 0.30/0.51 @ ^ [X2: nat] : ( member_nat @ X2 @ A4 )
% 0.30/0.51 @ ^ [X2: nat] : ( member_nat @ X2 @ B4 ) ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % minus_set_def
% 0.30/0.51 thf(fact_1139_not__psubset__empty,axiom,
% 0.30/0.51 ! [A2: set_nat] :
% 0.30/0.51 ~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% 0.30/0.51
% 0.30/0.51 % not_psubset_empty
% 0.30/0.51 thf(fact_1140_not__psubset__empty,axiom,
% 0.30/0.51 ! [A2: set_int] :
% 0.30/0.51 ~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).
% 0.30/0.51
% 0.30/0.51 % not_psubset_empty
% 0.30/0.51 thf(fact_1141_not__psubset__empty,axiom,
% 0.30/0.51 ! [A2: set_real] :
% 0.30/0.51 ~ ( ord_less_set_real @ A2 @ bot_bot_set_real ) ).
% 0.30/0.51
% 0.30/0.51 % not_psubset_empty
% 0.30/0.51 thf(fact_1142_in__mono,axiom,
% 0.30/0.51 ! [A2: set_real,B3: set_real,X: real] :
% 0.30/0.51 ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.30/0.51 => ( ( member_real @ X @ A2 )
% 0.30/0.51 => ( member_real @ X @ B3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % in_mono
% 0.30/0.51 thf(fact_1143_in__mono,axiom,
% 0.30/0.51 ! [A2: set_int,B3: set_int,X: int] :
% 0.30/0.51 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.30/0.51 => ( ( member_int @ X @ A2 )
% 0.30/0.51 => ( member_int @ X @ B3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % in_mono
% 0.30/0.51 thf(fact_1144_in__mono,axiom,
% 0.30/0.51 ! [A2: set_complex,B3: set_complex,X: complex] :
% 0.30/0.51 ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 0.30/0.51 => ( ( member_complex @ X @ A2 )
% 0.30/0.51 => ( member_complex @ X @ B3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % in_mono
% 0.30/0.51 thf(fact_1145_in__mono,axiom,
% 0.30/0.51 ! [A2: set_set_nat,B3: set_set_nat,X: set_nat] :
% 0.30/0.51 ( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
% 0.30/0.51 => ( ( member_set_nat @ X @ A2 )
% 0.30/0.51 => ( member_set_nat @ X @ B3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % in_mono
% 0.30/0.51 thf(fact_1146_in__mono,axiom,
% 0.30/0.51 ! [A2: set_nat,B3: set_nat,X: nat] :
% 0.30/0.51 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.30/0.51 => ( ( member_nat @ X @ A2 )
% 0.30/0.51 => ( member_nat @ X @ B3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % in_mono
% 0.30/0.51 thf(fact_1147_subsetD,axiom,
% 0.30/0.51 ! [A2: set_real,B3: set_real,C: real] :
% 0.30/0.51 ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.30/0.51 => ( ( member_real @ C @ A2 )
% 0.30/0.51 => ( member_real @ C @ B3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % subsetD
% 0.30/0.51 thf(fact_1148_subsetD,axiom,
% 0.30/0.51 ! [A2: set_int,B3: set_int,C: int] :
% 0.30/0.51 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.30/0.51 => ( ( member_int @ C @ A2 )
% 0.30/0.51 => ( member_int @ C @ B3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % subsetD
% 0.30/0.51 thf(fact_1149_subsetD,axiom,
% 0.30/0.51 ! [A2: set_complex,B3: set_complex,C: complex] :
% 0.30/0.51 ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 0.30/0.51 => ( ( member_complex @ C @ A2 )
% 0.30/0.51 => ( member_complex @ C @ B3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % subsetD
% 0.30/0.51 thf(fact_1150_subsetD,axiom,
% 0.30/0.51 ! [A2: set_set_nat,B3: set_set_nat,C: set_nat] :
% 0.30/0.51 ( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
% 0.30/0.51 => ( ( member_set_nat @ C @ A2 )
% 0.30/0.51 => ( member_set_nat @ C @ B3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % subsetD
% 0.30/0.51 thf(fact_1151_subsetD,axiom,
% 0.30/0.51 ! [A2: set_nat,B3: set_nat,C: nat] :
% 0.30/0.51 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.30/0.51 => ( ( member_nat @ C @ A2 )
% 0.30/0.51 => ( member_nat @ C @ B3 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % subsetD
% 0.30/0.51 thf(fact_1152_equalityE,axiom,
% 0.30/0.51 ! [A2: set_nat,B3: set_nat] :
% 0.30/0.51 ( ( A2 = B3 )
% 0.30/0.51 => ~ ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.30/0.51 => ~ ( ord_less_eq_set_nat @ B3 @ A2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % equalityE
% 0.30/0.51 thf(fact_1153_subset__eq,axiom,
% 0.30/0.51 ( ord_less_eq_set_real
% 0.30/0.51 = ( ^ [A4: set_real,B4: set_real] :
% 0.30/0.51 ! [X2: real] :
% 0.30/0.51 ( ( member_real @ X2 @ A4 )
% 0.30/0.51 => ( member_real @ X2 @ B4 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % subset_eq
% 0.30/0.51 thf(fact_1154_subset__eq,axiom,
% 0.30/0.51 ( ord_less_eq_set_int
% 0.30/0.51 = ( ^ [A4: set_int,B4: set_int] :
% 0.30/0.51 ! [X2: int] :
% 0.30/0.51 ( ( member_int @ X2 @ A4 )
% 0.30/0.51 => ( member_int @ X2 @ B4 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % subset_eq
% 0.30/0.51 thf(fact_1155_subset__eq,axiom,
% 0.30/0.51 ( ord_le211207098394363844omplex
% 0.30/0.51 = ( ^ [A4: set_complex,B4: set_complex] :
% 0.30/0.51 ! [X2: complex] :
% 0.30/0.51 ( ( member_complex @ X2 @ A4 )
% 0.30/0.51 => ( member_complex @ X2 @ B4 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % subset_eq
% 0.30/0.51 thf(fact_1156_subset__eq,axiom,
% 0.30/0.51 ( ord_le6893508408891458716et_nat
% 0.30/0.51 = ( ^ [A4: set_set_nat,B4: set_set_nat] :
% 0.30/0.51 ! [X2: set_nat] :
% 0.30/0.51 ( ( member_set_nat @ X2 @ A4 )
% 0.30/0.51 => ( member_set_nat @ X2 @ B4 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % subset_eq
% 0.30/0.51 thf(fact_1157_subset__eq,axiom,
% 0.30/0.51 ( ord_less_eq_set_nat
% 0.30/0.51 = ( ^ [A4: set_nat,B4: set_nat] :
% 0.30/0.51 ! [X2: nat] :
% 0.30/0.51 ( ( member_nat @ X2 @ A4 )
% 0.30/0.51 => ( member_nat @ X2 @ B4 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % subset_eq
% 0.30/0.51 thf(fact_1158_equalityD1,axiom,
% 0.30/0.51 ! [A2: set_nat,B3: set_nat] :
% 0.30/0.51 ( ( A2 = B3 )
% 0.30/0.51 => ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % equalityD1
% 0.30/0.51 thf(fact_1159_equalityD2,axiom,
% 0.30/0.51 ! [A2: set_nat,B3: set_nat] :
% 0.30/0.51 ( ( A2 = B3 )
% 0.30/0.51 => ( ord_less_eq_set_nat @ B3 @ A2 ) ) ).
% 0.30/0.51
% 0.30/0.51 % equalityD2
% 0.30/0.51 thf(fact_1160_subset__iff,axiom,
% 0.30/0.51 ( ord_less_eq_set_real
% 0.30/0.51 = ( ^ [A4: set_real,B4: set_real] :
% 0.30/0.51 ! [T2: real] :
% 0.30/0.51 ( ( member_real @ T2 @ A4 )
% 0.30/0.51 => ( member_real @ T2 @ B4 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % subset_iff
% 0.30/0.51 thf(fact_1161_subset__iff,axiom,
% 0.30/0.51 ( ord_less_eq_set_int
% 0.30/0.51 = ( ^ [A4: set_int,B4: set_int] :
% 0.30/0.51 ! [T2: int] :
% 0.30/0.51 ( ( member_int @ T2 @ A4 )
% 0.30/0.51 => ( member_int @ T2 @ B4 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % subset_iff
% 0.30/0.51 thf(fact_1162_subset__iff,axiom,
% 0.30/0.51 ( ord_le211207098394363844omplex
% 0.30/0.51 = ( ^ [A4: set_complex,B4: set_complex] :
% 0.30/0.51 ! [T2: complex] :
% 0.30/0.51 ( ( member_complex @ T2 @ A4 )
% 0.30/0.51 => ( member_complex @ T2 @ B4 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % subset_iff
% 0.30/0.51 thf(fact_1163_subset__iff,axiom,
% 0.30/0.51 ( ord_le6893508408891458716et_nat
% 0.30/0.51 = ( ^ [A4: set_set_nat,B4: set_set_nat] :
% 0.30/0.51 ! [T2: set_nat] :
% 0.30/0.51 ( ( member_set_nat @ T2 @ A4 )
% 0.30/0.51 => ( member_set_nat @ T2 @ B4 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % subset_iff
% 0.30/0.51 thf(fact_1164_subset__iff,axiom,
% 0.30/0.51 ( ord_less_eq_set_nat
% 0.30/0.51 = ( ^ [A4: set_nat,B4: set_nat] :
% 0.30/0.51 ! [T2: nat] :
% 0.30/0.51 ( ( member_nat @ T2 @ A4 )
% 0.30/0.51 => ( member_nat @ T2 @ B4 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % subset_iff
% 0.30/0.51 thf(fact_1165_subset__refl,axiom,
% 0.30/0.51 ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% 0.30/0.51
% 0.30/0.51 % subset_refl
% 0.30/0.51 thf(fact_1166_Collect__mono,axiom,
% 0.30/0.51 ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
% 0.30/0.51 ( ! [X4: product_prod_int_int] :
% 0.30/0.51 ( ( P @ X4 )
% 0.30/0.51 => ( Q @ X4 ) )
% 0.30/0.51 => ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Collect_mono
% 0.30/0.51 thf(fact_1167_Collect__mono,axiom,
% 0.30/0.51 ! [P: complex > $o,Q: complex > $o] :
% 0.30/0.51 ( ! [X4: complex] :
% 0.30/0.51 ( ( P @ X4 )
% 0.30/0.51 => ( Q @ X4 ) )
% 0.30/0.51 => ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Collect_mono
% 0.30/0.51 thf(fact_1168_Collect__mono,axiom,
% 0.30/0.51 ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.30/0.51 ( ! [X4: set_nat] :
% 0.30/0.51 ( ( P @ X4 )
% 0.30/0.51 => ( Q @ X4 ) )
% 0.30/0.51 => ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Collect_mono
% 0.30/0.51 thf(fact_1169_Collect__mono,axiom,
% 0.30/0.51 ! [P: int > $o,Q: int > $o] :
% 0.30/0.51 ( ! [X4: int] :
% 0.30/0.51 ( ( P @ X4 )
% 0.30/0.51 => ( Q @ X4 ) )
% 0.30/0.51 => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Collect_mono
% 0.30/0.51 thf(fact_1170_Collect__mono,axiom,
% 0.30/0.51 ! [P: nat > $o,Q: nat > $o] :
% 0.30/0.51 ( ! [X4: nat] :
% 0.30/0.51 ( ( P @ X4 )
% 0.30/0.51 => ( Q @ X4 ) )
% 0.30/0.51 => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Collect_mono
% 0.30/0.51 thf(fact_1171_subset__trans,axiom,
% 0.30/0.51 ! [A2: set_nat,B3: set_nat,C2: set_nat] :
% 0.30/0.51 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.30/0.51 => ( ( ord_less_eq_set_nat @ B3 @ C2 )
% 0.30/0.51 => ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % subset_trans
% 0.30/0.51 thf(fact_1172_set__eq__subset,axiom,
% 0.30/0.51 ( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
% 0.30/0.51 = ( ^ [A4: set_nat,B4: set_nat] :
% 0.30/0.51 ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 0.30/0.51 & ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % set_eq_subset
% 0.30/0.51 thf(fact_1173_Collect__mono__iff,axiom,
% 0.30/0.51 ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
% 0.30/0.51 ( ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q ) )
% 0.30/0.51 = ( ! [X2: product_prod_int_int] :
% 0.30/0.51 ( ( P @ X2 )
% 0.30/0.51 => ( Q @ X2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Collect_mono_iff
% 0.30/0.51 thf(fact_1174_Collect__mono__iff,axiom,
% 0.30/0.51 ! [P: complex > $o,Q: complex > $o] :
% 0.30/0.51 ( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
% 0.30/0.51 = ( ! [X2: complex] :
% 0.30/0.51 ( ( P @ X2 )
% 0.30/0.51 => ( Q @ X2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Collect_mono_iff
% 0.30/0.51 thf(fact_1175_Collect__mono__iff,axiom,
% 0.30/0.51 ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.30/0.51 ( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
% 0.30/0.51 = ( ! [X2: set_nat] :
% 0.30/0.51 ( ( P @ X2 )
% 0.30/0.51 => ( Q @ X2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Collect_mono_iff
% 0.30/0.51 thf(fact_1176_Collect__mono__iff,axiom,
% 0.30/0.51 ! [P: int > $o,Q: int > $o] :
% 0.30/0.51 ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
% 0.30/0.51 = ( ! [X2: int] :
% 0.30/0.51 ( ( P @ X2 )
% 0.30/0.51 => ( Q @ X2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Collect_mono_iff
% 0.30/0.51 thf(fact_1177_Collect__mono__iff,axiom,
% 0.30/0.51 ! [P: nat > $o,Q: nat > $o] :
% 0.30/0.51 ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
% 0.30/0.51 = ( ! [X2: nat] :
% 0.30/0.51 ( ( P @ X2 )
% 0.30/0.51 => ( Q @ X2 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Collect_mono_iff
% 0.30/0.51 thf(fact_1178_subset__iff__psubset__eq,axiom,
% 0.30/0.51 ( ord_less_eq_set_nat
% 0.30/0.51 = ( ^ [A4: set_nat,B4: set_nat] :
% 0.30/0.51 ( ( ord_less_set_nat @ A4 @ B4 )
% 0.30/0.51 | ( A4 = B4 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % subset_iff_psubset_eq
% 0.30/0.51 thf(fact_1179_subset__psubset__trans,axiom,
% 0.30/0.51 ! [A2: set_nat,B3: set_nat,C2: set_nat] :
% 0.30/0.51 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.30/0.51 => ( ( ord_less_set_nat @ B3 @ C2 )
% 0.30/0.51 => ( ord_less_set_nat @ A2 @ C2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % subset_psubset_trans
% 0.30/0.51 thf(fact_1180_subset__not__subset__eq,axiom,
% 0.30/0.51 ( ord_less_set_nat
% 0.30/0.51 = ( ^ [A4: set_nat,B4: set_nat] :
% 0.30/0.51 ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 0.30/0.51 & ~ ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % subset_not_subset_eq
% 0.30/0.51 thf(fact_1181_psubset__subset__trans,axiom,
% 0.30/0.51 ! [A2: set_nat,B3: set_nat,C2: set_nat] :
% 0.30/0.51 ( ( ord_less_set_nat @ A2 @ B3 )
% 0.30/0.51 => ( ( ord_less_eq_set_nat @ B3 @ C2 )
% 0.30/0.51 => ( ord_less_set_nat @ A2 @ C2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % psubset_subset_trans
% 0.30/0.51 thf(fact_1182_psubset__imp__subset,axiom,
% 0.30/0.51 ! [A2: set_nat,B3: set_nat] :
% 0.30/0.51 ( ( ord_less_set_nat @ A2 @ B3 )
% 0.30/0.51 => ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% 0.30/0.51
% 0.30/0.51 % psubset_imp_subset
% 0.30/0.51 thf(fact_1183_psubset__eq,axiom,
% 0.30/0.51 ( ord_less_set_nat
% 0.30/0.51 = ( ^ [A4: set_nat,B4: set_nat] :
% 0.30/0.51 ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 0.30/0.51 & ( A4 != B4 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % psubset_eq
% 0.30/0.51 thf(fact_1184_psubsetE,axiom,
% 0.30/0.51 ! [A2: set_nat,B3: set_nat] :
% 0.30/0.51 ( ( ord_less_set_nat @ A2 @ B3 )
% 0.30/0.51 => ~ ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.30/0.51 => ( ord_less_eq_set_nat @ B3 @ A2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % psubsetE
% 0.30/0.51 thf(fact_1185_Diff__mono,axiom,
% 0.30/0.51 ! [A2: set_nat,C2: set_nat,D3: set_nat,B3: set_nat] :
% 0.30/0.51 ( ( ord_less_eq_set_nat @ A2 @ C2 )
% 0.30/0.51 => ( ( ord_less_eq_set_nat @ D3 @ B3 )
% 0.30/0.51 => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B3 ) @ ( minus_minus_set_nat @ C2 @ D3 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % Diff_mono
% 0.30/0.51 thf(fact_1186_Diff__subset,axiom,
% 0.30/0.51 ! [A2: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B3 ) @ A2 ) ).
% 0.30/0.51
% 0.30/0.51 % Diff_subset
% 0.30/0.51 thf(fact_1187_double__diff,axiom,
% 0.30/0.51 ! [A2: set_nat,B3: set_nat,C2: set_nat] :
% 0.30/0.51 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.30/0.51 => ( ( ord_less_eq_set_nat @ B3 @ C2 )
% 0.30/0.51 => ( ( minus_minus_set_nat @ B3 @ ( minus_minus_set_nat @ C2 @ A2 ) )
% 0.30/0.51 = A2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % double_diff
% 0.30/0.51 thf(fact_1188_mult__commute__abs,axiom,
% 0.30/0.51 ! [C: real] :
% 0.30/0.51 ( ( ^ [X2: real] : ( times_times_real @ X2 @ C ) )
% 0.30/0.51 = ( times_times_real @ C ) ) ).
% 0.30/0.51
% 0.30/0.51 % mult_commute_abs
% 0.30/0.51 thf(fact_1189_mult__commute__abs,axiom,
% 0.30/0.51 ! [C: rat] :
% 0.30/0.51 ( ( ^ [X2: rat] : ( times_times_rat @ X2 @ C ) )
% 0.30/0.51 = ( times_times_rat @ C ) ) ).
% 0.30/0.51
% 0.30/0.51 % mult_commute_abs
% 0.30/0.51 thf(fact_1190_mult__commute__abs,axiom,
% 0.30/0.51 ! [C: nat] :
% 0.30/0.51 ( ( ^ [X2: nat] : ( times_times_nat @ X2 @ C ) )
% 0.30/0.51 = ( times_times_nat @ C ) ) ).
% 0.30/0.51
% 0.30/0.51 % mult_commute_abs
% 0.30/0.51 thf(fact_1191_mult__commute__abs,axiom,
% 0.30/0.51 ! [C: int] :
% 0.30/0.51 ( ( ^ [X2: int] : ( times_times_int @ X2 @ C ) )
% 0.30/0.51 = ( times_times_int @ C ) ) ).
% 0.30/0.51
% 0.30/0.51 % mult_commute_abs
% 0.30/0.51 thf(fact_1192_empty__def,axiom,
% 0.30/0.51 ( bot_bo1796632182523588997nt_int
% 0.30/0.51 = ( collec213857154873943460nt_int
% 0.30/0.51 @ ^ [X2: product_prod_int_int] : $false ) ) ).
% 0.30/0.51
% 0.30/0.51 % empty_def
% 0.30/0.51 thf(fact_1193_empty__def,axiom,
% 0.30/0.51 ( bot_bot_set_complex
% 0.30/0.51 = ( collect_complex
% 0.30/0.51 @ ^ [X2: complex] : $false ) ) ).
% 0.30/0.51
% 0.30/0.51 % empty_def
% 0.30/0.51 thf(fact_1194_empty__def,axiom,
% 0.30/0.51 ( bot_bot_set_set_nat
% 0.30/0.51 = ( collect_set_nat
% 0.30/0.51 @ ^ [X2: set_nat] : $false ) ) ).
% 0.30/0.51
% 0.30/0.51 % empty_def
% 0.30/0.51 thf(fact_1195_empty__def,axiom,
% 0.30/0.51 ( bot_bot_set_nat
% 0.30/0.51 = ( collect_nat
% 0.30/0.51 @ ^ [X2: nat] : $false ) ) ).
% 0.30/0.51
% 0.30/0.51 % empty_def
% 0.30/0.51 thf(fact_1196_empty__def,axiom,
% 0.30/0.51 ( bot_bot_set_int
% 0.30/0.51 = ( collect_int
% 0.30/0.51 @ ^ [X2: int] : $false ) ) ).
% 0.30/0.51
% 0.30/0.51 % empty_def
% 0.30/0.51 thf(fact_1197_empty__def,axiom,
% 0.30/0.51 ( bot_bot_set_real
% 0.30/0.51 = ( collect_real
% 0.30/0.51 @ ^ [X2: real] : $false ) ) ).
% 0.30/0.51
% 0.30/0.51 % empty_def
% 0.30/0.51 thf(fact_1198_less__eq__set__def,axiom,
% 0.30/0.51 ( ord_less_eq_set_real
% 0.30/0.51 = ( ^ [A4: set_real,B4: set_real] :
% 0.30/0.51 ( ord_less_eq_real_o
% 0.30/0.51 @ ^ [X2: real] : ( member_real @ X2 @ A4 )
% 0.30/0.51 @ ^ [X2: real] : ( member_real @ X2 @ B4 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_eq_set_def
% 0.30/0.51 thf(fact_1199_less__eq__set__def,axiom,
% 0.30/0.51 ( ord_less_eq_set_int
% 0.30/0.51 = ( ^ [A4: set_int,B4: set_int] :
% 0.30/0.51 ( ord_less_eq_int_o
% 0.30/0.51 @ ^ [X2: int] : ( member_int @ X2 @ A4 )
% 0.30/0.51 @ ^ [X2: int] : ( member_int @ X2 @ B4 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_eq_set_def
% 0.30/0.51 thf(fact_1200_less__eq__set__def,axiom,
% 0.30/0.51 ( ord_le211207098394363844omplex
% 0.30/0.51 = ( ^ [A4: set_complex,B4: set_complex] :
% 0.30/0.51 ( ord_le4573692005234683329plex_o
% 0.30/0.51 @ ^ [X2: complex] : ( member_complex @ X2 @ A4 )
% 0.30/0.51 @ ^ [X2: complex] : ( member_complex @ X2 @ B4 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_eq_set_def
% 0.30/0.51 thf(fact_1201_less__eq__set__def,axiom,
% 0.30/0.51 ( ord_le6893508408891458716et_nat
% 0.30/0.51 = ( ^ [A4: set_set_nat,B4: set_set_nat] :
% 0.30/0.51 ( ord_le3964352015994296041_nat_o
% 0.30/0.51 @ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A4 )
% 0.30/0.51 @ ^ [X2: set_nat] : ( member_set_nat @ X2 @ B4 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_eq_set_def
% 0.30/0.51 thf(fact_1202_less__eq__set__def,axiom,
% 0.30/0.51 ( ord_less_eq_set_nat
% 0.30/0.51 = ( ^ [A4: set_nat,B4: set_nat] :
% 0.30/0.51 ( ord_less_eq_nat_o
% 0.30/0.51 @ ^ [X2: nat] : ( member_nat @ X2 @ A4 )
% 0.30/0.51 @ ^ [X2: nat] : ( member_nat @ X2 @ B4 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_eq_set_def
% 0.30/0.51 thf(fact_1203_Collect__subset,axiom,
% 0.30/0.51 ! [A2: set_real,P: real > $o] :
% 0.30/0.51 ( ord_less_eq_set_real
% 0.30/0.51 @ ( collect_real
% 0.30/0.51 @ ^ [X2: real] :
% 0.30/0.51 ( ( member_real @ X2 @ A2 )
% 0.30/0.51 & ( P @ X2 ) ) )
% 0.30/0.51 @ A2 ) ).
% 0.30/0.51
% 0.30/0.51 % Collect_subset
% 0.30/0.51 thf(fact_1204_Collect__subset,axiom,
% 0.30/0.51 ! [A2: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
% 0.30/0.51 ( ord_le2843351958646193337nt_int
% 0.30/0.51 @ ( collec213857154873943460nt_int
% 0.30/0.51 @ ^ [X2: product_prod_int_int] :
% 0.30/0.51 ( ( member5262025264175285858nt_int @ X2 @ A2 )
% 0.30/0.51 & ( P @ X2 ) ) )
% 0.30/0.51 @ A2 ) ).
% 0.30/0.51
% 0.30/0.51 % Collect_subset
% 0.30/0.51 thf(fact_1205_Collect__subset,axiom,
% 0.30/0.51 ! [A2: set_complex,P: complex > $o] :
% 0.30/0.51 ( ord_le211207098394363844omplex
% 0.30/0.51 @ ( collect_complex
% 0.30/0.51 @ ^ [X2: complex] :
% 0.30/0.51 ( ( member_complex @ X2 @ A2 )
% 0.30/0.51 & ( P @ X2 ) ) )
% 0.30/0.51 @ A2 ) ).
% 0.30/0.51
% 0.30/0.51 % Collect_subset
% 0.30/0.51 thf(fact_1206_Collect__subset,axiom,
% 0.30/0.51 ! [A2: set_set_nat,P: set_nat > $o] :
% 0.30/0.51 ( ord_le6893508408891458716et_nat
% 0.30/0.51 @ ( collect_set_nat
% 0.30/0.51 @ ^ [X2: set_nat] :
% 0.30/0.51 ( ( member_set_nat @ X2 @ A2 )
% 0.30/0.51 & ( P @ X2 ) ) )
% 0.30/0.51 @ A2 ) ).
% 0.30/0.51
% 0.30/0.51 % Collect_subset
% 0.30/0.51 thf(fact_1207_Collect__subset,axiom,
% 0.30/0.51 ! [A2: set_int,P: int > $o] :
% 0.30/0.51 ( ord_less_eq_set_int
% 0.30/0.51 @ ( collect_int
% 0.30/0.51 @ ^ [X2: int] :
% 0.30/0.51 ( ( member_int @ X2 @ A2 )
% 0.30/0.51 & ( P @ X2 ) ) )
% 0.30/0.51 @ A2 ) ).
% 0.30/0.51
% 0.30/0.51 % Collect_subset
% 0.30/0.51 thf(fact_1208_Collect__subset,axiom,
% 0.30/0.51 ! [A2: set_nat,P: nat > $o] :
% 0.30/0.51 ( ord_less_eq_set_nat
% 0.30/0.51 @ ( collect_nat
% 0.30/0.51 @ ^ [X2: nat] :
% 0.30/0.51 ( ( member_nat @ X2 @ A2 )
% 0.30/0.51 & ( P @ X2 ) ) )
% 0.30/0.51 @ A2 ) ).
% 0.30/0.51
% 0.30/0.51 % Collect_subset
% 0.30/0.51 thf(fact_1209_max_Omono,axiom,
% 0.30/0.51 ! [C: rat,A: rat,D: rat,B: rat] :
% 0.30/0.51 ( ( ord_less_eq_rat @ C @ A )
% 0.30/0.51 => ( ( ord_less_eq_rat @ D @ B )
% 0.30/0.51 => ( ord_less_eq_rat @ ( ord_max_rat @ C @ D ) @ ( ord_max_rat @ A @ B ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.mono
% 0.30/0.51 thf(fact_1210_max_Omono,axiom,
% 0.30/0.51 ! [C: num,A: num,D: num,B: num] :
% 0.30/0.51 ( ( ord_less_eq_num @ C @ A )
% 0.30/0.51 => ( ( ord_less_eq_num @ D @ B )
% 0.30/0.51 => ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A @ B ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.mono
% 0.30/0.51 thf(fact_1211_max_Omono,axiom,
% 0.30/0.51 ! [C: nat,A: nat,D: nat,B: nat] :
% 0.30/0.51 ( ( ord_less_eq_nat @ C @ A )
% 0.30/0.51 => ( ( ord_less_eq_nat @ D @ B )
% 0.30/0.51 => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.mono
% 0.30/0.51 thf(fact_1212_max_Omono,axiom,
% 0.30/0.51 ! [C: int,A: int,D: int,B: int] :
% 0.30/0.51 ( ( ord_less_eq_int @ C @ A )
% 0.30/0.51 => ( ( ord_less_eq_int @ D @ B )
% 0.30/0.51 => ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A @ B ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.mono
% 0.30/0.51 thf(fact_1213_max_OorderE,axiom,
% 0.30/0.51 ! [B: rat,A: rat] :
% 0.30/0.51 ( ( ord_less_eq_rat @ B @ A )
% 0.30/0.51 => ( A
% 0.30/0.51 = ( ord_max_rat @ A @ B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.orderE
% 0.30/0.51 thf(fact_1214_max_OorderE,axiom,
% 0.30/0.51 ! [B: num,A: num] :
% 0.30/0.51 ( ( ord_less_eq_num @ B @ A )
% 0.30/0.51 => ( A
% 0.30/0.51 = ( ord_max_num @ A @ B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.orderE
% 0.30/0.51 thf(fact_1215_max_OorderE,axiom,
% 0.30/0.51 ! [B: nat,A: nat] :
% 0.30/0.51 ( ( ord_less_eq_nat @ B @ A )
% 0.30/0.51 => ( A
% 0.30/0.51 = ( ord_max_nat @ A @ B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.orderE
% 0.30/0.51 thf(fact_1216_max_OorderE,axiom,
% 0.30/0.51 ! [B: int,A: int] :
% 0.30/0.51 ( ( ord_less_eq_int @ B @ A )
% 0.30/0.51 => ( A
% 0.30/0.51 = ( ord_max_int @ A @ B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.orderE
% 0.30/0.51 thf(fact_1217_max_OorderI,axiom,
% 0.30/0.51 ! [A: rat,B: rat] :
% 0.30/0.51 ( ( A
% 0.30/0.51 = ( ord_max_rat @ A @ B ) )
% 0.30/0.51 => ( ord_less_eq_rat @ B @ A ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.orderI
% 0.30/0.51 thf(fact_1218_max_OorderI,axiom,
% 0.30/0.51 ! [A: num,B: num] :
% 0.30/0.51 ( ( A
% 0.30/0.51 = ( ord_max_num @ A @ B ) )
% 0.30/0.51 => ( ord_less_eq_num @ B @ A ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.orderI
% 0.30/0.51 thf(fact_1219_max_OorderI,axiom,
% 0.30/0.51 ! [A: nat,B: nat] :
% 0.30/0.51 ( ( A
% 0.30/0.51 = ( ord_max_nat @ A @ B ) )
% 0.30/0.51 => ( ord_less_eq_nat @ B @ A ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.orderI
% 0.30/0.51 thf(fact_1220_max_OorderI,axiom,
% 0.30/0.51 ! [A: int,B: int] :
% 0.30/0.51 ( ( A
% 0.30/0.51 = ( ord_max_int @ A @ B ) )
% 0.30/0.51 => ( ord_less_eq_int @ B @ A ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.orderI
% 0.30/0.51 thf(fact_1221_max_OboundedE,axiom,
% 0.30/0.51 ! [B: rat,C: rat,A: rat] :
% 0.30/0.51 ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
% 0.30/0.51 => ~ ( ( ord_less_eq_rat @ B @ A )
% 0.30/0.51 => ~ ( ord_less_eq_rat @ C @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.boundedE
% 0.30/0.51 thf(fact_1222_max_OboundedE,axiom,
% 0.30/0.51 ! [B: num,C: num,A: num] :
% 0.30/0.51 ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
% 0.30/0.51 => ~ ( ( ord_less_eq_num @ B @ A )
% 0.30/0.51 => ~ ( ord_less_eq_num @ C @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.boundedE
% 0.30/0.51 thf(fact_1223_max_OboundedE,axiom,
% 0.30/0.51 ! [B: nat,C: nat,A: nat] :
% 0.30/0.51 ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
% 0.30/0.51 => ~ ( ( ord_less_eq_nat @ B @ A )
% 0.30/0.51 => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.boundedE
% 0.30/0.51 thf(fact_1224_max_OboundedE,axiom,
% 0.30/0.51 ! [B: int,C: int,A: int] :
% 0.30/0.51 ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
% 0.30/0.51 => ~ ( ( ord_less_eq_int @ B @ A )
% 0.30/0.51 => ~ ( ord_less_eq_int @ C @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.boundedE
% 0.30/0.51 thf(fact_1225_max_OboundedI,axiom,
% 0.30/0.51 ! [B: rat,A: rat,C: rat] :
% 0.30/0.51 ( ( ord_less_eq_rat @ B @ A )
% 0.30/0.51 => ( ( ord_less_eq_rat @ C @ A )
% 0.30/0.51 => ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.boundedI
% 0.30/0.51 thf(fact_1226_max_OboundedI,axiom,
% 0.30/0.51 ! [B: num,A: num,C: num] :
% 0.30/0.51 ( ( ord_less_eq_num @ B @ A )
% 0.30/0.51 => ( ( ord_less_eq_num @ C @ A )
% 0.30/0.51 => ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.boundedI
% 0.30/0.51 thf(fact_1227_max_OboundedI,axiom,
% 0.30/0.51 ! [B: nat,A: nat,C: nat] :
% 0.30/0.51 ( ( ord_less_eq_nat @ B @ A )
% 0.30/0.51 => ( ( ord_less_eq_nat @ C @ A )
% 0.30/0.51 => ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.boundedI
% 0.30/0.51 thf(fact_1228_max_OboundedI,axiom,
% 0.30/0.51 ! [B: int,A: int,C: int] :
% 0.30/0.51 ( ( ord_less_eq_int @ B @ A )
% 0.30/0.51 => ( ( ord_less_eq_int @ C @ A )
% 0.30/0.51 => ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.boundedI
% 0.30/0.51 thf(fact_1229_max_Oorder__iff,axiom,
% 0.30/0.51 ( ord_less_eq_rat
% 0.30/0.51 = ( ^ [B5: rat,A5: rat] :
% 0.30/0.51 ( A5
% 0.30/0.51 = ( ord_max_rat @ A5 @ B5 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.order_iff
% 0.30/0.51 thf(fact_1230_max_Oorder__iff,axiom,
% 0.30/0.51 ( ord_less_eq_num
% 0.30/0.51 = ( ^ [B5: num,A5: num] :
% 0.30/0.51 ( A5
% 0.30/0.51 = ( ord_max_num @ A5 @ B5 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.order_iff
% 0.30/0.51 thf(fact_1231_max_Oorder__iff,axiom,
% 0.30/0.51 ( ord_less_eq_nat
% 0.30/0.51 = ( ^ [B5: nat,A5: nat] :
% 0.30/0.51 ( A5
% 0.30/0.51 = ( ord_max_nat @ A5 @ B5 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.order_iff
% 0.30/0.51 thf(fact_1232_max_Oorder__iff,axiom,
% 0.30/0.51 ( ord_less_eq_int
% 0.30/0.51 = ( ^ [B5: int,A5: int] :
% 0.30/0.51 ( A5
% 0.30/0.51 = ( ord_max_int @ A5 @ B5 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.order_iff
% 0.30/0.51 thf(fact_1233_max_Ocobounded1,axiom,
% 0.30/0.51 ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.cobounded1
% 0.30/0.51 thf(fact_1234_max_Ocobounded1,axiom,
% 0.30/0.51 ! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.cobounded1
% 0.30/0.51 thf(fact_1235_max_Ocobounded1,axiom,
% 0.30/0.51 ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.cobounded1
% 0.30/0.51 thf(fact_1236_max_Ocobounded1,axiom,
% 0.30/0.51 ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.cobounded1
% 0.30/0.51 thf(fact_1237_max_Ocobounded2,axiom,
% 0.30/0.51 ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.cobounded2
% 0.30/0.51 thf(fact_1238_max_Ocobounded2,axiom,
% 0.30/0.51 ! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.cobounded2
% 0.30/0.51 thf(fact_1239_max_Ocobounded2,axiom,
% 0.30/0.51 ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.cobounded2
% 0.30/0.51 thf(fact_1240_max_Ocobounded2,axiom,
% 0.30/0.51 ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.cobounded2
% 0.30/0.51 thf(fact_1241_le__max__iff__disj,axiom,
% 0.30/0.51 ! [Z: rat,X: rat,Y: rat] :
% 0.30/0.51 ( ( ord_less_eq_rat @ Z @ ( ord_max_rat @ X @ Y ) )
% 0.30/0.51 = ( ( ord_less_eq_rat @ Z @ X )
% 0.30/0.51 | ( ord_less_eq_rat @ Z @ Y ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % le_max_iff_disj
% 0.30/0.51 thf(fact_1242_le__max__iff__disj,axiom,
% 0.30/0.51 ! [Z: num,X: num,Y: num] :
% 0.30/0.51 ( ( ord_less_eq_num @ Z @ ( ord_max_num @ X @ Y ) )
% 0.30/0.51 = ( ( ord_less_eq_num @ Z @ X )
% 0.30/0.51 | ( ord_less_eq_num @ Z @ Y ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % le_max_iff_disj
% 0.30/0.51 thf(fact_1243_le__max__iff__disj,axiom,
% 0.30/0.51 ! [Z: nat,X: nat,Y: nat] :
% 0.30/0.51 ( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X @ Y ) )
% 0.30/0.51 = ( ( ord_less_eq_nat @ Z @ X )
% 0.30/0.51 | ( ord_less_eq_nat @ Z @ Y ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % le_max_iff_disj
% 0.30/0.51 thf(fact_1244_le__max__iff__disj,axiom,
% 0.30/0.51 ! [Z: int,X: int,Y: int] :
% 0.30/0.51 ( ( ord_less_eq_int @ Z @ ( ord_max_int @ X @ Y ) )
% 0.30/0.51 = ( ( ord_less_eq_int @ Z @ X )
% 0.30/0.51 | ( ord_less_eq_int @ Z @ Y ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % le_max_iff_disj
% 0.30/0.51 thf(fact_1245_max_Oabsorb__iff1,axiom,
% 0.30/0.51 ( ord_less_eq_rat
% 0.30/0.51 = ( ^ [B5: rat,A5: rat] :
% 0.30/0.51 ( ( ord_max_rat @ A5 @ B5 )
% 0.30/0.51 = A5 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.absorb_iff1
% 0.30/0.51 thf(fact_1246_max_Oabsorb__iff1,axiom,
% 0.30/0.51 ( ord_less_eq_num
% 0.30/0.51 = ( ^ [B5: num,A5: num] :
% 0.30/0.51 ( ( ord_max_num @ A5 @ B5 )
% 0.30/0.51 = A5 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.absorb_iff1
% 0.30/0.51 thf(fact_1247_max_Oabsorb__iff1,axiom,
% 0.30/0.51 ( ord_less_eq_nat
% 0.30/0.51 = ( ^ [B5: nat,A5: nat] :
% 0.30/0.51 ( ( ord_max_nat @ A5 @ B5 )
% 0.30/0.51 = A5 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.absorb_iff1
% 0.30/0.51 thf(fact_1248_max_Oabsorb__iff1,axiom,
% 0.30/0.51 ( ord_less_eq_int
% 0.30/0.51 = ( ^ [B5: int,A5: int] :
% 0.30/0.51 ( ( ord_max_int @ A5 @ B5 )
% 0.30/0.51 = A5 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.absorb_iff1
% 0.30/0.51 thf(fact_1249_max_Oabsorb__iff2,axiom,
% 0.30/0.51 ( ord_less_eq_rat
% 0.30/0.51 = ( ^ [A5: rat,B5: rat] :
% 0.30/0.51 ( ( ord_max_rat @ A5 @ B5 )
% 0.30/0.51 = B5 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.absorb_iff2
% 0.30/0.51 thf(fact_1250_max_Oabsorb__iff2,axiom,
% 0.30/0.51 ( ord_less_eq_num
% 0.30/0.51 = ( ^ [A5: num,B5: num] :
% 0.30/0.51 ( ( ord_max_num @ A5 @ B5 )
% 0.30/0.51 = B5 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.absorb_iff2
% 0.30/0.51 thf(fact_1251_max_Oabsorb__iff2,axiom,
% 0.30/0.51 ( ord_less_eq_nat
% 0.30/0.51 = ( ^ [A5: nat,B5: nat] :
% 0.30/0.51 ( ( ord_max_nat @ A5 @ B5 )
% 0.30/0.51 = B5 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.absorb_iff2
% 0.30/0.51 thf(fact_1252_max_Oabsorb__iff2,axiom,
% 0.30/0.51 ( ord_less_eq_int
% 0.30/0.51 = ( ^ [A5: int,B5: int] :
% 0.30/0.51 ( ( ord_max_int @ A5 @ B5 )
% 0.30/0.51 = B5 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.absorb_iff2
% 0.30/0.51 thf(fact_1253_max_OcoboundedI1,axiom,
% 0.30/0.51 ! [C: rat,A: rat,B: rat] :
% 0.30/0.51 ( ( ord_less_eq_rat @ C @ A )
% 0.30/0.51 => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.coboundedI1
% 0.30/0.51 thf(fact_1254_max_OcoboundedI1,axiom,
% 0.30/0.51 ! [C: num,A: num,B: num] :
% 0.30/0.51 ( ( ord_less_eq_num @ C @ A )
% 0.30/0.51 => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.coboundedI1
% 0.30/0.51 thf(fact_1255_max_OcoboundedI1,axiom,
% 0.30/0.51 ! [C: nat,A: nat,B: nat] :
% 0.30/0.51 ( ( ord_less_eq_nat @ C @ A )
% 0.30/0.51 => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.coboundedI1
% 0.30/0.51 thf(fact_1256_max_OcoboundedI1,axiom,
% 0.30/0.51 ! [C: int,A: int,B: int] :
% 0.30/0.51 ( ( ord_less_eq_int @ C @ A )
% 0.30/0.51 => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.coboundedI1
% 0.30/0.51 thf(fact_1257_max_OcoboundedI2,axiom,
% 0.30/0.51 ! [C: rat,B: rat,A: rat] :
% 0.30/0.51 ( ( ord_less_eq_rat @ C @ B )
% 0.30/0.51 => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.coboundedI2
% 0.30/0.51 thf(fact_1258_max_OcoboundedI2,axiom,
% 0.30/0.51 ! [C: num,B: num,A: num] :
% 0.30/0.51 ( ( ord_less_eq_num @ C @ B )
% 0.30/0.51 => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.coboundedI2
% 0.30/0.51 thf(fact_1259_max_OcoboundedI2,axiom,
% 0.30/0.51 ! [C: nat,B: nat,A: nat] :
% 0.30/0.51 ( ( ord_less_eq_nat @ C @ B )
% 0.30/0.51 => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.coboundedI2
% 0.30/0.51 thf(fact_1260_max_OcoboundedI2,axiom,
% 0.30/0.51 ! [C: int,B: int,A: int] :
% 0.30/0.51 ( ( ord_less_eq_int @ C @ B )
% 0.30/0.51 => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.coboundedI2
% 0.30/0.51 thf(fact_1261_less__max__iff__disj,axiom,
% 0.30/0.51 ! [Z: real,X: real,Y: real] :
% 0.30/0.51 ( ( ord_less_real @ Z @ ( ord_max_real @ X @ Y ) )
% 0.30/0.51 = ( ( ord_less_real @ Z @ X )
% 0.30/0.51 | ( ord_less_real @ Z @ Y ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_max_iff_disj
% 0.30/0.51 thf(fact_1262_less__max__iff__disj,axiom,
% 0.30/0.51 ! [Z: rat,X: rat,Y: rat] :
% 0.30/0.51 ( ( ord_less_rat @ Z @ ( ord_max_rat @ X @ Y ) )
% 0.30/0.51 = ( ( ord_less_rat @ Z @ X )
% 0.30/0.51 | ( ord_less_rat @ Z @ Y ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_max_iff_disj
% 0.30/0.51 thf(fact_1263_less__max__iff__disj,axiom,
% 0.30/0.51 ! [Z: num,X: num,Y: num] :
% 0.30/0.51 ( ( ord_less_num @ Z @ ( ord_max_num @ X @ Y ) )
% 0.30/0.51 = ( ( ord_less_num @ Z @ X )
% 0.30/0.51 | ( ord_less_num @ Z @ Y ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_max_iff_disj
% 0.30/0.51 thf(fact_1264_less__max__iff__disj,axiom,
% 0.30/0.51 ! [Z: nat,X: nat,Y: nat] :
% 0.30/0.51 ( ( ord_less_nat @ Z @ ( ord_max_nat @ X @ Y ) )
% 0.30/0.51 = ( ( ord_less_nat @ Z @ X )
% 0.30/0.51 | ( ord_less_nat @ Z @ Y ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_max_iff_disj
% 0.30/0.51 thf(fact_1265_less__max__iff__disj,axiom,
% 0.30/0.51 ! [Z: int,X: int,Y: int] :
% 0.30/0.51 ( ( ord_less_int @ Z @ ( ord_max_int @ X @ Y ) )
% 0.30/0.51 = ( ( ord_less_int @ Z @ X )
% 0.30/0.51 | ( ord_less_int @ Z @ Y ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % less_max_iff_disj
% 0.30/0.51 thf(fact_1266_max_Ostrict__boundedE,axiom,
% 0.30/0.51 ! [B: real,C: real,A: real] :
% 0.30/0.51 ( ( ord_less_real @ ( ord_max_real @ B @ C ) @ A )
% 0.30/0.51 => ~ ( ( ord_less_real @ B @ A )
% 0.30/0.51 => ~ ( ord_less_real @ C @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.strict_boundedE
% 0.30/0.51 thf(fact_1267_max_Ostrict__boundedE,axiom,
% 0.30/0.51 ! [B: rat,C: rat,A: rat] :
% 0.30/0.51 ( ( ord_less_rat @ ( ord_max_rat @ B @ C ) @ A )
% 0.30/0.51 => ~ ( ( ord_less_rat @ B @ A )
% 0.30/0.51 => ~ ( ord_less_rat @ C @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.strict_boundedE
% 0.30/0.51 thf(fact_1268_max_Ostrict__boundedE,axiom,
% 0.30/0.51 ! [B: num,C: num,A: num] :
% 0.30/0.51 ( ( ord_less_num @ ( ord_max_num @ B @ C ) @ A )
% 0.30/0.51 => ~ ( ( ord_less_num @ B @ A )
% 0.30/0.51 => ~ ( ord_less_num @ C @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.strict_boundedE
% 0.30/0.51 thf(fact_1269_max_Ostrict__boundedE,axiom,
% 0.30/0.51 ! [B: nat,C: nat,A: nat] :
% 0.30/0.51 ( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
% 0.30/0.51 => ~ ( ( ord_less_nat @ B @ A )
% 0.30/0.51 => ~ ( ord_less_nat @ C @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.strict_boundedE
% 0.30/0.51 thf(fact_1270_max_Ostrict__boundedE,axiom,
% 0.30/0.51 ! [B: int,C: int,A: int] :
% 0.30/0.51 ( ( ord_less_int @ ( ord_max_int @ B @ C ) @ A )
% 0.30/0.51 => ~ ( ( ord_less_int @ B @ A )
% 0.30/0.51 => ~ ( ord_less_int @ C @ A ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.strict_boundedE
% 0.30/0.51 thf(fact_1271_max_Ostrict__order__iff,axiom,
% 0.30/0.51 ( ord_less_real
% 0.30/0.51 = ( ^ [B5: real,A5: real] :
% 0.30/0.51 ( ( A5
% 0.30/0.51 = ( ord_max_real @ A5 @ B5 ) )
% 0.30/0.51 & ( A5 != B5 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.strict_order_iff
% 0.30/0.51 thf(fact_1272_max_Ostrict__order__iff,axiom,
% 0.30/0.51 ( ord_less_rat
% 0.30/0.51 = ( ^ [B5: rat,A5: rat] :
% 0.30/0.51 ( ( A5
% 0.30/0.51 = ( ord_max_rat @ A5 @ B5 ) )
% 0.30/0.51 & ( A5 != B5 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.strict_order_iff
% 0.30/0.51 thf(fact_1273_max_Ostrict__order__iff,axiom,
% 0.30/0.51 ( ord_less_num
% 0.30/0.51 = ( ^ [B5: num,A5: num] :
% 0.30/0.51 ( ( A5
% 0.30/0.51 = ( ord_max_num @ A5 @ B5 ) )
% 0.30/0.51 & ( A5 != B5 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.strict_order_iff
% 0.30/0.51 thf(fact_1274_max_Ostrict__order__iff,axiom,
% 0.30/0.51 ( ord_less_nat
% 0.30/0.51 = ( ^ [B5: nat,A5: nat] :
% 0.30/0.51 ( ( A5
% 0.30/0.51 = ( ord_max_nat @ A5 @ B5 ) )
% 0.30/0.51 & ( A5 != B5 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.strict_order_iff
% 0.30/0.51 thf(fact_1275_max_Ostrict__order__iff,axiom,
% 0.30/0.51 ( ord_less_int
% 0.30/0.51 = ( ^ [B5: int,A5: int] :
% 0.30/0.51 ( ( A5
% 0.30/0.51 = ( ord_max_int @ A5 @ B5 ) )
% 0.30/0.51 & ( A5 != B5 ) ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.strict_order_iff
% 0.30/0.51 thf(fact_1276_max_Ostrict__coboundedI1,axiom,
% 0.30/0.51 ! [C: real,A: real,B: real] :
% 0.30/0.51 ( ( ord_less_real @ C @ A )
% 0.30/0.51 => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.strict_coboundedI1
% 0.30/0.51 thf(fact_1277_max_Ostrict__coboundedI1,axiom,
% 0.30/0.51 ! [C: rat,A: rat,B: rat] :
% 0.30/0.51 ( ( ord_less_rat @ C @ A )
% 0.30/0.51 => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.strict_coboundedI1
% 0.30/0.51 thf(fact_1278_max_Ostrict__coboundedI1,axiom,
% 0.30/0.51 ! [C: num,A: num,B: num] :
% 0.30/0.51 ( ( ord_less_num @ C @ A )
% 0.30/0.51 => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.strict_coboundedI1
% 0.30/0.51 thf(fact_1279_max_Ostrict__coboundedI1,axiom,
% 0.30/0.51 ! [C: nat,A: nat,B: nat] :
% 0.30/0.51 ( ( ord_less_nat @ C @ A )
% 0.30/0.51 => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.strict_coboundedI1
% 0.30/0.51 thf(fact_1280_max_Ostrict__coboundedI1,axiom,
% 0.30/0.51 ! [C: int,A: int,B: int] :
% 0.30/0.51 ( ( ord_less_int @ C @ A )
% 0.30/0.51 => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.strict_coboundedI1
% 0.30/0.51 thf(fact_1281_max_Ostrict__coboundedI2,axiom,
% 0.30/0.51 ! [C: real,B: real,A: real] :
% 0.30/0.51 ( ( ord_less_real @ C @ B )
% 0.30/0.51 => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.strict_coboundedI2
% 0.30/0.51 thf(fact_1282_max_Ostrict__coboundedI2,axiom,
% 0.30/0.51 ! [C: rat,B: rat,A: rat] :
% 0.30/0.51 ( ( ord_less_rat @ C @ B )
% 0.30/0.51 => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.strict_coboundedI2
% 0.30/0.51 thf(fact_1283_max_Ostrict__coboundedI2,axiom,
% 0.30/0.51 ! [C: num,B: num,A: num] :
% 0.30/0.51 ( ( ord_less_num @ C @ B )
% 0.30/0.51 => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.strict_coboundedI2
% 0.30/0.51 thf(fact_1284_max_Ostrict__coboundedI2,axiom,
% 0.30/0.51 ! [C: nat,B: nat,A: nat] :
% 0.30/0.51 ( ( ord_less_nat @ C @ B )
% 0.30/0.51 => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.strict_coboundedI2
% 0.30/0.51 thf(fact_1285_max_Ostrict__coboundedI2,axiom,
% 0.30/0.51 ! [C: int,B: int,A: int] :
% 0.30/0.51 ( ( ord_less_int @ C @ B )
% 0.30/0.51 => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % max.strict_coboundedI2
% 0.30/0.51 thf(fact_1286_double__not__eq__Suc__double,axiom,
% 0.30/0.51 ! [M: nat,N2: nat] :
% 0.30/0.51 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.30/0.51 != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.30/0.51
% 0.30/0.51 % double_not_eq_Suc_double
% 0.30/0.51 thf(fact_1287_Suc__double__not__eq__double,axiom,
% 0.30/0.51 ! [M: nat,N2: nat] :
% 0.30/0.51 ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.51 != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.30/0.51
% 0.30/0.51 % Suc_double_not_eq_double
% 0.30/0.51 thf(fact_1288_add__diff__cancel__right_H,axiom,
% 0.30/0.51 ! [A: real,B: real] :
% 0.30/0.51 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % add_diff_cancel_right'
% 0.30/0.51 thf(fact_1289_add__diff__cancel__right_H,axiom,
% 0.30/0.51 ! [A: rat,B: rat] :
% 0.30/0.51 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % add_diff_cancel_right'
% 0.30/0.51 thf(fact_1290_add__diff__cancel__right_H,axiom,
% 0.30/0.51 ! [A: nat,B: nat] :
% 0.30/0.51 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % add_diff_cancel_right'
% 0.30/0.51 thf(fact_1291_add__diff__cancel__right_H,axiom,
% 0.30/0.51 ! [A: int,B: int] :
% 0.30/0.51 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % add_diff_cancel_right'
% 0.30/0.51 thf(fact_1292_add__diff__cancel__right,axiom,
% 0.30/0.51 ! [A: real,C: real,B: real] :
% 0.30/0.51 ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.30/0.51 = ( minus_minus_real @ A @ B ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_diff_cancel_right
% 0.30/0.51 thf(fact_1293_add__diff__cancel__right,axiom,
% 0.30/0.51 ! [A: rat,C: rat,B: rat] :
% 0.30/0.51 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.30/0.51 = ( minus_minus_rat @ A @ B ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_diff_cancel_right
% 0.30/0.51 thf(fact_1294_add__diff__cancel__right,axiom,
% 0.30/0.51 ! [A: nat,C: nat,B: nat] :
% 0.30/0.51 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.30/0.51 = ( minus_minus_nat @ A @ B ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_diff_cancel_right
% 0.30/0.51 thf(fact_1295_add__diff__cancel__right,axiom,
% 0.30/0.51 ! [A: int,C: int,B: int] :
% 0.30/0.51 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.30/0.51 = ( minus_minus_int @ A @ B ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_diff_cancel_right
% 0.30/0.51 thf(fact_1296_add__diff__cancel__left_H,axiom,
% 0.30/0.51 ! [A: real,B: real] :
% 0.30/0.51 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
% 0.30/0.51 = B ) ).
% 0.30/0.51
% 0.30/0.51 % add_diff_cancel_left'
% 0.30/0.51 thf(fact_1297_add__diff__cancel__left_H,axiom,
% 0.30/0.51 ! [A: rat,B: rat] :
% 0.30/0.51 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
% 0.30/0.51 = B ) ).
% 0.30/0.51
% 0.30/0.51 % add_diff_cancel_left'
% 0.30/0.51 thf(fact_1298_add__diff__cancel__left_H,axiom,
% 0.30/0.51 ! [A: nat,B: nat] :
% 0.30/0.51 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 0.30/0.51 = B ) ).
% 0.30/0.51
% 0.30/0.51 % add_diff_cancel_left'
% 0.30/0.51 thf(fact_1299_add__diff__cancel__left_H,axiom,
% 0.30/0.51 ! [A: int,B: int] :
% 0.30/0.51 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
% 0.30/0.51 = B ) ).
% 0.30/0.51
% 0.30/0.51 % add_diff_cancel_left'
% 0.30/0.51 thf(fact_1300_add__diff__cancel__left,axiom,
% 0.30/0.51 ! [C: real,A: real,B: real] :
% 0.30/0.51 ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.30/0.51 = ( minus_minus_real @ A @ B ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_diff_cancel_left
% 0.30/0.51 thf(fact_1301_add__diff__cancel__left,axiom,
% 0.30/0.51 ! [C: rat,A: rat,B: rat] :
% 0.30/0.51 ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.30/0.51 = ( minus_minus_rat @ A @ B ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_diff_cancel_left
% 0.30/0.51 thf(fact_1302_add__diff__cancel__left,axiom,
% 0.30/0.51 ! [C: nat,A: nat,B: nat] :
% 0.30/0.51 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.30/0.51 = ( minus_minus_nat @ A @ B ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_diff_cancel_left
% 0.30/0.51 thf(fact_1303_add__diff__cancel__left,axiom,
% 0.30/0.51 ! [C: int,A: int,B: int] :
% 0.30/0.51 ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.30/0.51 = ( minus_minus_int @ A @ B ) ) ).
% 0.30/0.51
% 0.30/0.51 % add_diff_cancel_left
% 0.30/0.51 thf(fact_1304_diff__add__cancel,axiom,
% 0.30/0.51 ! [A: real,B: real] :
% 0.30/0.51 ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % diff_add_cancel
% 0.30/0.51 thf(fact_1305_diff__add__cancel,axiom,
% 0.30/0.51 ! [A: rat,B: rat] :
% 0.30/0.51 ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % diff_add_cancel
% 0.30/0.51 thf(fact_1306_diff__add__cancel,axiom,
% 0.30/0.51 ! [A: int,B: int] :
% 0.30/0.51 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % diff_add_cancel
% 0.30/0.51 thf(fact_1307_add__diff__cancel,axiom,
% 0.30/0.51 ! [A: real,B: real] :
% 0.30/0.51 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % add_diff_cancel
% 0.30/0.51 thf(fact_1308_add__diff__cancel,axiom,
% 0.30/0.51 ! [A: rat,B: rat] :
% 0.30/0.51 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % add_diff_cancel
% 0.30/0.51 thf(fact_1309_add__diff__cancel,axiom,
% 0.30/0.51 ! [A: int,B: int] :
% 0.30/0.51 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.30/0.51 = A ) ).
% 0.30/0.51
% 0.30/0.51 % add_diff_cancel
% 0.30/0.51 thf(fact_1310_times__divide__eq__right,axiom,
% 0.30/0.51 ! [A: complex,B: complex,C: complex] :
% 0.30/0.51 ( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
% 0.30/0.51 = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ C ) ) ).
% 0.30/0.51
% 0.30/0.51 % times_divide_eq_right
% 0.30/0.51 thf(fact_1311_times__divide__eq__right,axiom,
% 0.30/0.51 ! [A: real,B: real,C: real] :
% 0.30/0.51 ( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.30/0.51 = ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% 0.30/0.51
% 0.30/0.51 % times_divide_eq_right
% 0.30/0.51 thf(fact_1312_times__divide__eq__right,axiom,
% 0.30/0.51 ! [A: rat,B: rat,C: rat] :
% 0.30/0.51 ( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.30/0.51 = ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C ) ) ).
% 0.30/0.51
% 0.30/0.51 % times_divide_eq_right
% 0.30/0.51 thf(fact_1313_divide__divide__eq__right,axiom,
% 0.30/0.51 ! [A: complex,B: complex,C: complex] :
% 0.30/0.51 ( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
% 0.30/0.52 = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_divide_eq_right
% 0.30/0.52 thf(fact_1314_divide__divide__eq__right,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real] :
% 0.30/0.52 ( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.30/0.52 = ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_divide_eq_right
% 0.30/0.52 thf(fact_1315_divide__divide__eq__right,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat] :
% 0.30/0.52 ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.30/0.52 = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_divide_eq_right
% 0.30/0.52 thf(fact_1316_add__left__cancel,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real] :
% 0.30/0.52 ( ( ( plus_plus_real @ A @ B )
% 0.30/0.52 = ( plus_plus_real @ A @ C ) )
% 0.30/0.52 = ( B = C ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_left_cancel
% 0.30/0.52 thf(fact_1317_add__left__cancel,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat] :
% 0.30/0.52 ( ( ( plus_plus_rat @ A @ B )
% 0.30/0.52 = ( plus_plus_rat @ A @ C ) )
% 0.30/0.52 = ( B = C ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_left_cancel
% 0.30/0.52 thf(fact_1318_add__left__cancel,axiom,
% 0.30/0.52 ! [A: nat,B: nat,C: nat] :
% 0.30/0.52 ( ( ( plus_plus_nat @ A @ B )
% 0.30/0.52 = ( plus_plus_nat @ A @ C ) )
% 0.30/0.52 = ( B = C ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_left_cancel
% 0.30/0.52 thf(fact_1319_add__left__cancel,axiom,
% 0.30/0.52 ! [A: int,B: int,C: int] :
% 0.30/0.52 ( ( ( plus_plus_int @ A @ B )
% 0.30/0.52 = ( plus_plus_int @ A @ C ) )
% 0.30/0.52 = ( B = C ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_left_cancel
% 0.30/0.52 thf(fact_1320_add__right__cancel,axiom,
% 0.30/0.52 ! [B: real,A: real,C: real] :
% 0.30/0.52 ( ( ( plus_plus_real @ B @ A )
% 0.30/0.52 = ( plus_plus_real @ C @ A ) )
% 0.30/0.52 = ( B = C ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_right_cancel
% 0.30/0.52 thf(fact_1321_add__right__cancel,axiom,
% 0.30/0.52 ! [B: rat,A: rat,C: rat] :
% 0.30/0.52 ( ( ( plus_plus_rat @ B @ A )
% 0.30/0.52 = ( plus_plus_rat @ C @ A ) )
% 0.30/0.52 = ( B = C ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_right_cancel
% 0.30/0.52 thf(fact_1322_add__right__cancel,axiom,
% 0.30/0.52 ! [B: nat,A: nat,C: nat] :
% 0.30/0.52 ( ( ( plus_plus_nat @ B @ A )
% 0.30/0.52 = ( plus_plus_nat @ C @ A ) )
% 0.30/0.52 = ( B = C ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_right_cancel
% 0.30/0.52 thf(fact_1323_add__right__cancel,axiom,
% 0.30/0.52 ! [B: int,A: int,C: int] :
% 0.30/0.52 ( ( ( plus_plus_int @ B @ A )
% 0.30/0.52 = ( plus_plus_int @ C @ A ) )
% 0.30/0.52 = ( B = C ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_right_cancel
% 0.30/0.52 thf(fact_1324_add__le__cancel__left,axiom,
% 0.30/0.52 ! [C: real,A: real,B: real] :
% 0.30/0.52 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.30/0.52 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_le_cancel_left
% 0.30/0.52 thf(fact_1325_add__le__cancel__left,axiom,
% 0.30/0.52 ! [C: rat,A: rat,B: rat] :
% 0.30/0.52 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.30/0.52 = ( ord_less_eq_rat @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_le_cancel_left
% 0.30/0.52 thf(fact_1326_add__le__cancel__left,axiom,
% 0.30/0.52 ! [C: nat,A: nat,B: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.30/0.52 = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_le_cancel_left
% 0.30/0.52 thf(fact_1327_add__le__cancel__left,axiom,
% 0.30/0.52 ! [C: int,A: int,B: int] :
% 0.30/0.52 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.30/0.52 = ( ord_less_eq_int @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_le_cancel_left
% 0.30/0.52 thf(fact_1328_add__le__cancel__right,axiom,
% 0.30/0.52 ! [A: real,C: real,B: real] :
% 0.30/0.52 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.30/0.52 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_le_cancel_right
% 0.30/0.52 thf(fact_1329_add__le__cancel__right,axiom,
% 0.30/0.52 ! [A: rat,C: rat,B: rat] :
% 0.30/0.52 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.30/0.52 = ( ord_less_eq_rat @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_le_cancel_right
% 0.30/0.52 thf(fact_1330_add__le__cancel__right,axiom,
% 0.30/0.52 ! [A: nat,C: nat,B: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.30/0.52 = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_le_cancel_right
% 0.30/0.52 thf(fact_1331_add__le__cancel__right,axiom,
% 0.30/0.52 ! [A: int,C: int,B: int] :
% 0.30/0.52 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.30/0.52 = ( ord_less_eq_int @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_le_cancel_right
% 0.30/0.52 thf(fact_1332_add__less__cancel__left,axiom,
% 0.30/0.52 ! [C: real,A: real,B: real] :
% 0.30/0.52 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.30/0.52 = ( ord_less_real @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_less_cancel_left
% 0.30/0.52 thf(fact_1333_add__less__cancel__left,axiom,
% 0.30/0.52 ! [C: rat,A: rat,B: rat] :
% 0.30/0.52 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.30/0.52 = ( ord_less_rat @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_less_cancel_left
% 0.30/0.52 thf(fact_1334_add__less__cancel__left,axiom,
% 0.30/0.52 ! [C: nat,A: nat,B: nat] :
% 0.30/0.52 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.30/0.52 = ( ord_less_nat @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_less_cancel_left
% 0.30/0.52 thf(fact_1335_add__less__cancel__left,axiom,
% 0.30/0.52 ! [C: int,A: int,B: int] :
% 0.30/0.52 ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.30/0.52 = ( ord_less_int @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_less_cancel_left
% 0.30/0.52 thf(fact_1336_add__less__cancel__right,axiom,
% 0.30/0.52 ! [A: real,C: real,B: real] :
% 0.30/0.52 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.30/0.52 = ( ord_less_real @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_less_cancel_right
% 0.30/0.52 thf(fact_1337_add__less__cancel__right,axiom,
% 0.30/0.52 ! [A: rat,C: rat,B: rat] :
% 0.30/0.52 ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.30/0.52 = ( ord_less_rat @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_less_cancel_right
% 0.30/0.52 thf(fact_1338_add__less__cancel__right,axiom,
% 0.30/0.52 ! [A: nat,C: nat,B: nat] :
% 0.30/0.52 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.30/0.52 = ( ord_less_nat @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_less_cancel_right
% 0.30/0.52 thf(fact_1339_add__less__cancel__right,axiom,
% 0.30/0.52 ! [A: int,C: int,B: int] :
% 0.30/0.52 ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.30/0.52 = ( ord_less_int @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_less_cancel_right
% 0.30/0.52 thf(fact_1340_mult_Oright__neutral,axiom,
% 0.30/0.52 ! [A: complex] :
% 0.30/0.52 ( ( times_times_complex @ A @ one_one_complex )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % mult.right_neutral
% 0.30/0.52 thf(fact_1341_mult_Oright__neutral,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( times_times_real @ A @ one_one_real )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % mult.right_neutral
% 0.30/0.52 thf(fact_1342_mult_Oright__neutral,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( times_times_rat @ A @ one_one_rat )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % mult.right_neutral
% 0.30/0.52 thf(fact_1343_mult_Oright__neutral,axiom,
% 0.30/0.52 ! [A: nat] :
% 0.30/0.52 ( ( times_times_nat @ A @ one_one_nat )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % mult.right_neutral
% 0.30/0.52 thf(fact_1344_mult_Oright__neutral,axiom,
% 0.30/0.52 ! [A: int] :
% 0.30/0.52 ( ( times_times_int @ A @ one_one_int )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % mult.right_neutral
% 0.30/0.52 thf(fact_1345_mult__1,axiom,
% 0.30/0.52 ! [A: complex] :
% 0.30/0.52 ( ( times_times_complex @ one_one_complex @ A )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % mult_1
% 0.30/0.52 thf(fact_1346_mult__1,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( times_times_real @ one_one_real @ A )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % mult_1
% 0.30/0.52 thf(fact_1347_mult__1,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( times_times_rat @ one_one_rat @ A )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % mult_1
% 0.30/0.52 thf(fact_1348_mult__1,axiom,
% 0.30/0.52 ! [A: nat] :
% 0.30/0.52 ( ( times_times_nat @ one_one_nat @ A )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % mult_1
% 0.30/0.52 thf(fact_1349_mult__1,axiom,
% 0.30/0.52 ! [A: int] :
% 0.30/0.52 ( ( times_times_int @ one_one_int @ A )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % mult_1
% 0.30/0.52 thf(fact_1350_times__divide__eq__left,axiom,
% 0.30/0.52 ! [B: complex,C: complex,A: complex] :
% 0.30/0.52 ( ( times_times_complex @ ( divide1717551699836669952omplex @ B @ C ) @ A )
% 0.30/0.52 = ( divide1717551699836669952omplex @ ( times_times_complex @ B @ A ) @ C ) ) ).
% 0.30/0.52
% 0.30/0.52 % times_divide_eq_left
% 0.30/0.52 thf(fact_1351_times__divide__eq__left,axiom,
% 0.30/0.52 ! [B: real,C: real,A: real] :
% 0.30/0.52 ( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.30/0.52 = ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% 0.30/0.52
% 0.30/0.52 % times_divide_eq_left
% 0.30/0.52 thf(fact_1352_times__divide__eq__left,axiom,
% 0.30/0.52 ! [B: rat,C: rat,A: rat] :
% 0.30/0.52 ( ( times_times_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.30/0.52 = ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C ) ) ).
% 0.30/0.52
% 0.30/0.52 % times_divide_eq_left
% 0.30/0.52 thf(fact_1353_divide__divide__eq__left,axiom,
% 0.30/0.52 ! [A: complex,B: complex,C: complex] :
% 0.30/0.52 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
% 0.30/0.52 = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_divide_eq_left
% 0.30/0.52 thf(fact_1354_divide__divide__eq__left,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real] :
% 0.30/0.52 ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
% 0.30/0.52 = ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_divide_eq_left
% 0.30/0.52 thf(fact_1355_divide__divide__eq__left,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat] :
% 0.30/0.52 ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
% 0.30/0.52 = ( divide_divide_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_divide_eq_left
% 0.30/0.52 thf(fact_1356_less__set__def,axiom,
% 0.30/0.52 ( ord_less_set_nat
% 0.30/0.52 = ( ^ [A4: set_nat,B4: set_nat] :
% 0.30/0.52 ( ord_less_nat_o
% 0.30/0.52 @ ^ [X2: nat] : ( member_nat @ X2 @ A4 )
% 0.30/0.52 @ ^ [X2: nat] : ( member_nat @ X2 @ B4 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_set_def
% 0.30/0.52 thf(fact_1357_less__set__def,axiom,
% 0.30/0.52 ( ord_less_set_real
% 0.30/0.52 = ( ^ [A4: set_real,B4: set_real] :
% 0.30/0.52 ( ord_less_real_o
% 0.30/0.52 @ ^ [X2: real] : ( member_real @ X2 @ A4 )
% 0.30/0.52 @ ^ [X2: real] : ( member_real @ X2 @ B4 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_set_def
% 0.30/0.52 thf(fact_1358_less__set__def,axiom,
% 0.30/0.52 ( ord_less_set_int
% 0.30/0.52 = ( ^ [A4: set_int,B4: set_int] :
% 0.30/0.52 ( ord_less_int_o
% 0.30/0.52 @ ^ [X2: int] : ( member_int @ X2 @ A4 )
% 0.30/0.52 @ ^ [X2: int] : ( member_int @ X2 @ B4 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_set_def
% 0.30/0.52 thf(fact_1359_less__set__def,axiom,
% 0.30/0.52 ( ord_less_set_complex
% 0.30/0.52 = ( ^ [A4: set_complex,B4: set_complex] :
% 0.30/0.52 ( ord_less_complex_o
% 0.30/0.52 @ ^ [X2: complex] : ( member_complex @ X2 @ A4 )
% 0.30/0.52 @ ^ [X2: complex] : ( member_complex @ X2 @ B4 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_set_def
% 0.30/0.52 thf(fact_1360_less__set__def,axiom,
% 0.30/0.52 ( ord_less_set_set_nat
% 0.30/0.52 = ( ^ [A4: set_set_nat,B4: set_set_nat] :
% 0.30/0.52 ( ord_less_set_nat_o
% 0.30/0.52 @ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A4 )
% 0.30/0.52 @ ^ [X2: set_nat] : ( member_set_nat @ X2 @ B4 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_set_def
% 0.30/0.52 thf(fact_1361_psubsetD,axiom,
% 0.30/0.52 ! [A2: set_nat,B3: set_nat,C: nat] :
% 0.30/0.52 ( ( ord_less_set_nat @ A2 @ B3 )
% 0.30/0.52 => ( ( member_nat @ C @ A2 )
% 0.30/0.52 => ( member_nat @ C @ B3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % psubsetD
% 0.30/0.52 thf(fact_1362_psubsetD,axiom,
% 0.30/0.52 ! [A2: set_real,B3: set_real,C: real] :
% 0.30/0.52 ( ( ord_less_set_real @ A2 @ B3 )
% 0.30/0.52 => ( ( member_real @ C @ A2 )
% 0.30/0.52 => ( member_real @ C @ B3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % psubsetD
% 0.30/0.52 thf(fact_1363_psubsetD,axiom,
% 0.30/0.52 ! [A2: set_int,B3: set_int,C: int] :
% 0.30/0.52 ( ( ord_less_set_int @ A2 @ B3 )
% 0.30/0.52 => ( ( member_int @ C @ A2 )
% 0.30/0.52 => ( member_int @ C @ B3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % psubsetD
% 0.30/0.52 thf(fact_1364_psubsetD,axiom,
% 0.30/0.52 ! [A2: set_complex,B3: set_complex,C: complex] :
% 0.30/0.52 ( ( ord_less_set_complex @ A2 @ B3 )
% 0.30/0.52 => ( ( member_complex @ C @ A2 )
% 0.30/0.52 => ( member_complex @ C @ B3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % psubsetD
% 0.30/0.52 thf(fact_1365_psubsetD,axiom,
% 0.30/0.52 ! [A2: set_set_nat,B3: set_set_nat,C: set_nat] :
% 0.30/0.52 ( ( ord_less_set_set_nat @ A2 @ B3 )
% 0.30/0.52 => ( ( member_set_nat @ C @ A2 )
% 0.30/0.52 => ( member_set_nat @ C @ B3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % psubsetD
% 0.30/0.52 thf(fact_1366_linordered__field__no__ub,axiom,
% 0.30/0.52 ! [X5: real] :
% 0.30/0.52 ? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).
% 0.30/0.52
% 0.30/0.52 % linordered_field_no_ub
% 0.30/0.52 thf(fact_1367_linordered__field__no__ub,axiom,
% 0.30/0.52 ! [X5: rat] :
% 0.30/0.52 ? [X_1: rat] : ( ord_less_rat @ X5 @ X_1 ) ).
% 0.30/0.52
% 0.30/0.52 % linordered_field_no_ub
% 0.30/0.52 thf(fact_1368_linordered__field__no__lb,axiom,
% 0.30/0.52 ! [X5: real] :
% 0.30/0.52 ? [Y4: real] : ( ord_less_real @ Y4 @ X5 ) ).
% 0.30/0.52
% 0.30/0.52 % linordered_field_no_lb
% 0.30/0.52 thf(fact_1369_linordered__field__no__lb,axiom,
% 0.30/0.52 ! [X5: rat] :
% 0.30/0.52 ? [Y4: rat] : ( ord_less_rat @ Y4 @ X5 ) ).
% 0.30/0.52
% 0.30/0.52 % linordered_field_no_lb
% 0.30/0.52 thf(fact_1370_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real] :
% 0.30/0.52 ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 0.30/0.52 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ab_semigroup_mult_class.mult_ac(1)
% 0.30/0.52 thf(fact_1371_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat] :
% 0.30/0.52 ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 0.30/0.52 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ab_semigroup_mult_class.mult_ac(1)
% 0.30/0.52 thf(fact_1372_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.30/0.52 ! [A: nat,B: nat,C: nat] :
% 0.30/0.52 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.30/0.52 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ab_semigroup_mult_class.mult_ac(1)
% 0.30/0.52 thf(fact_1373_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.30/0.52 ! [A: int,B: int,C: int] :
% 0.30/0.52 ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 0.30/0.52 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ab_semigroup_mult_class.mult_ac(1)
% 0.30/0.52 thf(fact_1374_mult_Oassoc,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real] :
% 0.30/0.52 ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 0.30/0.52 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult.assoc
% 0.30/0.52 thf(fact_1375_mult_Oassoc,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat] :
% 0.30/0.52 ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 0.30/0.52 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult.assoc
% 0.30/0.52 thf(fact_1376_mult_Oassoc,axiom,
% 0.30/0.52 ! [A: nat,B: nat,C: nat] :
% 0.30/0.52 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.30/0.52 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult.assoc
% 0.30/0.52 thf(fact_1377_mult_Oassoc,axiom,
% 0.30/0.52 ! [A: int,B: int,C: int] :
% 0.30/0.52 ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 0.30/0.52 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult.assoc
% 0.30/0.52 thf(fact_1378_mult_Ocommute,axiom,
% 0.30/0.52 ( times_times_real
% 0.30/0.52 = ( ^ [A5: real,B5: real] : ( times_times_real @ B5 @ A5 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult.commute
% 0.30/0.52 thf(fact_1379_mult_Ocommute,axiom,
% 0.30/0.52 ( times_times_rat
% 0.30/0.52 = ( ^ [A5: rat,B5: rat] : ( times_times_rat @ B5 @ A5 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult.commute
% 0.30/0.52 thf(fact_1380_mult_Ocommute,axiom,
% 0.30/0.52 ( times_times_nat
% 0.30/0.52 = ( ^ [A5: nat,B5: nat] : ( times_times_nat @ B5 @ A5 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult.commute
% 0.30/0.52 thf(fact_1381_mult_Ocommute,axiom,
% 0.30/0.52 ( times_times_int
% 0.30/0.52 = ( ^ [A5: int,B5: int] : ( times_times_int @ B5 @ A5 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult.commute
% 0.30/0.52 thf(fact_1382_mult_Oleft__commute,axiom,
% 0.30/0.52 ! [B: real,A: real,C: real] :
% 0.30/0.52 ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
% 0.30/0.52 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult.left_commute
% 0.30/0.52 thf(fact_1383_mult_Oleft__commute,axiom,
% 0.30/0.52 ! [B: rat,A: rat,C: rat] :
% 0.30/0.52 ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
% 0.30/0.52 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult.left_commute
% 0.30/0.52 thf(fact_1384_mult_Oleft__commute,axiom,
% 0.30/0.52 ! [B: nat,A: nat,C: nat] :
% 0.30/0.52 ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
% 0.30/0.52 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult.left_commute
% 0.30/0.52 thf(fact_1385_mult_Oleft__commute,axiom,
% 0.30/0.52 ! [B: int,A: int,C: int] :
% 0.30/0.52 ( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
% 0.30/0.52 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult.left_commute
% 0.30/0.52 thf(fact_1386_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real] :
% 0.30/0.52 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.30/0.52 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ab_semigroup_add_class.add_ac(1)
% 0.30/0.52 thf(fact_1387_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat] :
% 0.30/0.52 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.30/0.52 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ab_semigroup_add_class.add_ac(1)
% 0.30/0.52 thf(fact_1388_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.30/0.52 ! [A: nat,B: nat,C: nat] :
% 0.30/0.52 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.30/0.52 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ab_semigroup_add_class.add_ac(1)
% 0.30/0.52 thf(fact_1389_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.30/0.52 ! [A: int,B: int,C: int] :
% 0.30/0.52 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.30/0.52 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ab_semigroup_add_class.add_ac(1)
% 0.30/0.52 thf(fact_1390_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.30/0.52 ! [I2: real,J: real,K: real,L2: real] :
% 0.30/0.52 ( ( ( I2 = J )
% 0.30/0.52 & ( K = L2 ) )
% 0.30/0.52 => ( ( plus_plus_real @ I2 @ K )
% 0.30/0.52 = ( plus_plus_real @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_semiring(4)
% 0.30/0.52 thf(fact_1391_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.30/0.52 ! [I2: rat,J: rat,K: rat,L2: rat] :
% 0.30/0.52 ( ( ( I2 = J )
% 0.30/0.52 & ( K = L2 ) )
% 0.30/0.52 => ( ( plus_plus_rat @ I2 @ K )
% 0.30/0.52 = ( plus_plus_rat @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_semiring(4)
% 0.30/0.52 thf(fact_1392_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.30/0.52 ! [I2: nat,J: nat,K: nat,L2: nat] :
% 0.30/0.52 ( ( ( I2 = J )
% 0.30/0.52 & ( K = L2 ) )
% 0.30/0.52 => ( ( plus_plus_nat @ I2 @ K )
% 0.30/0.52 = ( plus_plus_nat @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_semiring(4)
% 0.30/0.52 thf(fact_1393_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.30/0.52 ! [I2: int,J: int,K: int,L2: int] :
% 0.30/0.52 ( ( ( I2 = J )
% 0.30/0.52 & ( K = L2 ) )
% 0.30/0.52 => ( ( plus_plus_int @ I2 @ K )
% 0.30/0.52 = ( plus_plus_int @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_semiring(4)
% 0.30/0.52 thf(fact_1394_group__cancel_Oadd1,axiom,
% 0.30/0.52 ! [A2: real,K: real,A: real,B: real] :
% 0.30/0.52 ( ( A2
% 0.30/0.52 = ( plus_plus_real @ K @ A ) )
% 0.30/0.52 => ( ( plus_plus_real @ A2 @ B )
% 0.30/0.52 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % group_cancel.add1
% 0.30/0.52 thf(fact_1395_group__cancel_Oadd1,axiom,
% 0.30/0.52 ! [A2: rat,K: rat,A: rat,B: rat] :
% 0.30/0.52 ( ( A2
% 0.30/0.52 = ( plus_plus_rat @ K @ A ) )
% 0.30/0.52 => ( ( plus_plus_rat @ A2 @ B )
% 0.30/0.52 = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % group_cancel.add1
% 0.30/0.52 thf(fact_1396_group__cancel_Oadd1,axiom,
% 0.30/0.52 ! [A2: nat,K: nat,A: nat,B: nat] :
% 0.30/0.52 ( ( A2
% 0.30/0.52 = ( plus_plus_nat @ K @ A ) )
% 0.30/0.52 => ( ( plus_plus_nat @ A2 @ B )
% 0.30/0.52 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % group_cancel.add1
% 0.30/0.52 thf(fact_1397_group__cancel_Oadd1,axiom,
% 0.30/0.52 ! [A2: int,K: int,A: int,B: int] :
% 0.30/0.52 ( ( A2
% 0.30/0.52 = ( plus_plus_int @ K @ A ) )
% 0.30/0.52 => ( ( plus_plus_int @ A2 @ B )
% 0.30/0.52 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % group_cancel.add1
% 0.30/0.52 thf(fact_1398_group__cancel_Oadd2,axiom,
% 0.30/0.52 ! [B3: real,K: real,B: real,A: real] :
% 0.30/0.52 ( ( B3
% 0.30/0.52 = ( plus_plus_real @ K @ B ) )
% 0.30/0.52 => ( ( plus_plus_real @ A @ B3 )
% 0.30/0.52 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % group_cancel.add2
% 0.30/0.52 thf(fact_1399_group__cancel_Oadd2,axiom,
% 0.30/0.52 ! [B3: rat,K: rat,B: rat,A: rat] :
% 0.30/0.52 ( ( B3
% 0.30/0.52 = ( plus_plus_rat @ K @ B ) )
% 0.30/0.52 => ( ( plus_plus_rat @ A @ B3 )
% 0.30/0.52 = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % group_cancel.add2
% 0.30/0.52 thf(fact_1400_group__cancel_Oadd2,axiom,
% 0.30/0.52 ! [B3: nat,K: nat,B: nat,A: nat] :
% 0.30/0.52 ( ( B3
% 0.30/0.52 = ( plus_plus_nat @ K @ B ) )
% 0.30/0.52 => ( ( plus_plus_nat @ A @ B3 )
% 0.30/0.52 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % group_cancel.add2
% 0.30/0.52 thf(fact_1401_group__cancel_Oadd2,axiom,
% 0.30/0.52 ! [B3: int,K: int,B: int,A: int] :
% 0.30/0.52 ( ( B3
% 0.30/0.52 = ( plus_plus_int @ K @ B ) )
% 0.30/0.52 => ( ( plus_plus_int @ A @ B3 )
% 0.30/0.52 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % group_cancel.add2
% 0.30/0.52 thf(fact_1402_add_Oassoc,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real] :
% 0.30/0.52 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.30/0.52 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add.assoc
% 0.30/0.52 thf(fact_1403_add_Oassoc,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat] :
% 0.30/0.52 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.30/0.52 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add.assoc
% 0.30/0.52 thf(fact_1404_add_Oassoc,axiom,
% 0.30/0.52 ! [A: nat,B: nat,C: nat] :
% 0.30/0.52 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.30/0.52 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add.assoc
% 0.30/0.52 thf(fact_1405_add_Oassoc,axiom,
% 0.30/0.52 ! [A: int,B: int,C: int] :
% 0.30/0.52 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.30/0.52 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add.assoc
% 0.30/0.52 thf(fact_1406_add_Oleft__cancel,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real] :
% 0.30/0.52 ( ( ( plus_plus_real @ A @ B )
% 0.30/0.52 = ( plus_plus_real @ A @ C ) )
% 0.30/0.52 = ( B = C ) ) ).
% 0.30/0.52
% 0.30/0.52 % add.left_cancel
% 0.30/0.52 thf(fact_1407_add_Oleft__cancel,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat] :
% 0.30/0.52 ( ( ( plus_plus_rat @ A @ B )
% 0.30/0.52 = ( plus_plus_rat @ A @ C ) )
% 0.30/0.52 = ( B = C ) ) ).
% 0.30/0.52
% 0.30/0.52 % add.left_cancel
% 0.30/0.52 thf(fact_1408_add_Oleft__cancel,axiom,
% 0.30/0.52 ! [A: int,B: int,C: int] :
% 0.30/0.52 ( ( ( plus_plus_int @ A @ B )
% 0.30/0.52 = ( plus_plus_int @ A @ C ) )
% 0.30/0.52 = ( B = C ) ) ).
% 0.30/0.52
% 0.30/0.52 % add.left_cancel
% 0.30/0.52 thf(fact_1409_add_Oright__cancel,axiom,
% 0.30/0.52 ! [B: real,A: real,C: real] :
% 0.30/0.52 ( ( ( plus_plus_real @ B @ A )
% 0.30/0.52 = ( plus_plus_real @ C @ A ) )
% 0.30/0.52 = ( B = C ) ) ).
% 0.30/0.52
% 0.30/0.52 % add.right_cancel
% 0.30/0.52 thf(fact_1410_add_Oright__cancel,axiom,
% 0.30/0.52 ! [B: rat,A: rat,C: rat] :
% 0.30/0.52 ( ( ( plus_plus_rat @ B @ A )
% 0.30/0.52 = ( plus_plus_rat @ C @ A ) )
% 0.30/0.52 = ( B = C ) ) ).
% 0.30/0.52
% 0.30/0.52 % add.right_cancel
% 0.30/0.52 thf(fact_1411_add_Oright__cancel,axiom,
% 0.30/0.52 ! [B: int,A: int,C: int] :
% 0.30/0.52 ( ( ( plus_plus_int @ B @ A )
% 0.30/0.52 = ( plus_plus_int @ C @ A ) )
% 0.30/0.52 = ( B = C ) ) ).
% 0.30/0.52
% 0.30/0.52 % add.right_cancel
% 0.30/0.52 thf(fact_1412_add_Ocommute,axiom,
% 0.30/0.52 ( plus_plus_real
% 0.30/0.52 = ( ^ [A5: real,B5: real] : ( plus_plus_real @ B5 @ A5 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add.commute
% 0.30/0.52 thf(fact_1413_add_Ocommute,axiom,
% 0.30/0.52 ( plus_plus_rat
% 0.30/0.52 = ( ^ [A5: rat,B5: rat] : ( plus_plus_rat @ B5 @ A5 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add.commute
% 0.30/0.52 thf(fact_1414_add_Ocommute,axiom,
% 0.30/0.52 ( plus_plus_nat
% 0.30/0.52 = ( ^ [A5: nat,B5: nat] : ( plus_plus_nat @ B5 @ A5 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add.commute
% 0.30/0.52 thf(fact_1415_add_Ocommute,axiom,
% 0.30/0.52 ( plus_plus_int
% 0.30/0.52 = ( ^ [A5: int,B5: int] : ( plus_plus_int @ B5 @ A5 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add.commute
% 0.30/0.52 thf(fact_1416_add_Oleft__commute,axiom,
% 0.30/0.52 ! [B: real,A: real,C: real] :
% 0.30/0.52 ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
% 0.30/0.52 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add.left_commute
% 0.30/0.52 thf(fact_1417_add_Oleft__commute,axiom,
% 0.30/0.52 ! [B: rat,A: rat,C: rat] :
% 0.30/0.52 ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
% 0.30/0.52 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add.left_commute
% 0.30/0.52 thf(fact_1418_add_Oleft__commute,axiom,
% 0.30/0.52 ! [B: nat,A: nat,C: nat] :
% 0.30/0.52 ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 0.30/0.52 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add.left_commute
% 0.30/0.52 thf(fact_1419_add_Oleft__commute,axiom,
% 0.30/0.52 ! [B: int,A: int,C: int] :
% 0.30/0.52 ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
% 0.30/0.52 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add.left_commute
% 0.30/0.52 thf(fact_1420_add__left__imp__eq,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real] :
% 0.30/0.52 ( ( ( plus_plus_real @ A @ B )
% 0.30/0.52 = ( plus_plus_real @ A @ C ) )
% 0.30/0.52 => ( B = C ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_left_imp_eq
% 0.30/0.52 thf(fact_1421_add__left__imp__eq,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat] :
% 0.30/0.52 ( ( ( plus_plus_rat @ A @ B )
% 0.30/0.52 = ( plus_plus_rat @ A @ C ) )
% 0.30/0.52 => ( B = C ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_left_imp_eq
% 0.30/0.52 thf(fact_1422_add__left__imp__eq,axiom,
% 0.30/0.52 ! [A: nat,B: nat,C: nat] :
% 0.30/0.52 ( ( ( plus_plus_nat @ A @ B )
% 0.30/0.52 = ( plus_plus_nat @ A @ C ) )
% 0.30/0.52 => ( B = C ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_left_imp_eq
% 0.30/0.52 thf(fact_1423_add__left__imp__eq,axiom,
% 0.30/0.52 ! [A: int,B: int,C: int] :
% 0.30/0.52 ( ( ( plus_plus_int @ A @ B )
% 0.30/0.52 = ( plus_plus_int @ A @ C ) )
% 0.30/0.52 => ( B = C ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_left_imp_eq
% 0.30/0.52 thf(fact_1424_add__right__imp__eq,axiom,
% 0.30/0.52 ! [B: real,A: real,C: real] :
% 0.30/0.52 ( ( ( plus_plus_real @ B @ A )
% 0.30/0.52 = ( plus_plus_real @ C @ A ) )
% 0.30/0.52 => ( B = C ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_right_imp_eq
% 0.30/0.52 thf(fact_1425_add__right__imp__eq,axiom,
% 0.30/0.52 ! [B: rat,A: rat,C: rat] :
% 0.30/0.52 ( ( ( plus_plus_rat @ B @ A )
% 0.30/0.52 = ( plus_plus_rat @ C @ A ) )
% 0.30/0.52 => ( B = C ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_right_imp_eq
% 0.30/0.52 thf(fact_1426_add__right__imp__eq,axiom,
% 0.30/0.52 ! [B: nat,A: nat,C: nat] :
% 0.30/0.52 ( ( ( plus_plus_nat @ B @ A )
% 0.30/0.52 = ( plus_plus_nat @ C @ A ) )
% 0.30/0.52 => ( B = C ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_right_imp_eq
% 0.30/0.52 thf(fact_1427_add__right__imp__eq,axiom,
% 0.30/0.52 ! [B: int,A: int,C: int] :
% 0.30/0.52 ( ( ( plus_plus_int @ B @ A )
% 0.30/0.52 = ( plus_plus_int @ C @ A ) )
% 0.30/0.52 => ( B = C ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_right_imp_eq
% 0.30/0.52 thf(fact_1428_one__reorient,axiom,
% 0.30/0.52 ! [X: complex] :
% 0.30/0.52 ( ( one_one_complex = X )
% 0.30/0.52 = ( X = one_one_complex ) ) ).
% 0.30/0.52
% 0.30/0.52 % one_reorient
% 0.30/0.52 thf(fact_1429_one__reorient,axiom,
% 0.30/0.52 ! [X: real] :
% 0.30/0.52 ( ( one_one_real = X )
% 0.30/0.52 = ( X = one_one_real ) ) ).
% 0.30/0.52
% 0.30/0.52 % one_reorient
% 0.30/0.52 thf(fact_1430_one__reorient,axiom,
% 0.30/0.52 ! [X: rat] :
% 0.30/0.52 ( ( one_one_rat = X )
% 0.30/0.52 = ( X = one_one_rat ) ) ).
% 0.30/0.52
% 0.30/0.52 % one_reorient
% 0.30/0.52 thf(fact_1431_one__reorient,axiom,
% 0.30/0.52 ! [X: nat] :
% 0.30/0.52 ( ( one_one_nat = X )
% 0.30/0.52 = ( X = one_one_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % one_reorient
% 0.30/0.52 thf(fact_1432_one__reorient,axiom,
% 0.30/0.52 ! [X: int] :
% 0.30/0.52 ( ( one_one_int = X )
% 0.30/0.52 = ( X = one_one_int ) ) ).
% 0.30/0.52
% 0.30/0.52 % one_reorient
% 0.30/0.52 thf(fact_1433_diff__right__commute,axiom,
% 0.30/0.52 ! [A: real,C: real,B: real] :
% 0.30/0.52 ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
% 0.30/0.52 = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_right_commute
% 0.30/0.52 thf(fact_1434_diff__right__commute,axiom,
% 0.30/0.52 ! [A: rat,C: rat,B: rat] :
% 0.30/0.52 ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
% 0.30/0.52 = ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_right_commute
% 0.30/0.52 thf(fact_1435_diff__right__commute,axiom,
% 0.30/0.52 ! [A: nat,C: nat,B: nat] :
% 0.30/0.52 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 0.30/0.52 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_right_commute
% 0.30/0.52 thf(fact_1436_diff__right__commute,axiom,
% 0.30/0.52 ! [A: int,C: int,B: int] :
% 0.30/0.52 ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
% 0.30/0.52 = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_right_commute
% 0.30/0.52 thf(fact_1437_diff__eq__diff__eq,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.52 ( ( ( minus_minus_real @ A @ B )
% 0.30/0.52 = ( minus_minus_real @ C @ D ) )
% 0.30/0.52 => ( ( A = B )
% 0.30/0.52 = ( C = D ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_eq_diff_eq
% 0.30/0.52 thf(fact_1438_diff__eq__diff__eq,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.52 ( ( ( minus_minus_rat @ A @ B )
% 0.30/0.52 = ( minus_minus_rat @ C @ D ) )
% 0.30/0.52 => ( ( A = B )
% 0.30/0.52 = ( C = D ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_eq_diff_eq
% 0.30/0.52 thf(fact_1439_diff__eq__diff__eq,axiom,
% 0.30/0.52 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.52 ( ( ( minus_minus_int @ A @ B )
% 0.30/0.52 = ( minus_minus_int @ C @ D ) )
% 0.30/0.52 => ( ( A = B )
% 0.30/0.52 = ( C = D ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_eq_diff_eq
% 0.30/0.52 thf(fact_1440_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.30/0.52 ! [I2: real,J: real,K: real,L2: real] :
% 0.30/0.52 ( ( ( ord_less_eq_real @ I2 @ J )
% 0.30/0.52 & ( K = L2 ) )
% 0.30/0.52 => ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_semiring(3)
% 0.30/0.52 thf(fact_1441_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.30/0.52 ! [I2: rat,J: rat,K: rat,L2: rat] :
% 0.30/0.52 ( ( ( ord_less_eq_rat @ I2 @ J )
% 0.30/0.52 & ( K = L2 ) )
% 0.30/0.52 => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_semiring(3)
% 0.30/0.52 thf(fact_1442_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.30/0.52 ! [I2: nat,J: nat,K: nat,L2: nat] :
% 0.30/0.52 ( ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.52 & ( K = L2 ) )
% 0.30/0.52 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_semiring(3)
% 0.30/0.52 thf(fact_1443_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.30/0.52 ! [I2: int,J: int,K: int,L2: int] :
% 0.30/0.52 ( ( ( ord_less_eq_int @ I2 @ J )
% 0.30/0.52 & ( K = L2 ) )
% 0.30/0.52 => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_semiring(3)
% 0.30/0.52 thf(fact_1444_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.30/0.52 ! [I2: real,J: real,K: real,L2: real] :
% 0.30/0.52 ( ( ( I2 = J )
% 0.30/0.52 & ( ord_less_eq_real @ K @ L2 ) )
% 0.30/0.52 => ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_semiring(2)
% 0.30/0.52 thf(fact_1445_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.30/0.52 ! [I2: rat,J: rat,K: rat,L2: rat] :
% 0.30/0.52 ( ( ( I2 = J )
% 0.30/0.52 & ( ord_less_eq_rat @ K @ L2 ) )
% 0.30/0.52 => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_semiring(2)
% 0.30/0.52 thf(fact_1446_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.30/0.52 ! [I2: nat,J: nat,K: nat,L2: nat] :
% 0.30/0.52 ( ( ( I2 = J )
% 0.30/0.52 & ( ord_less_eq_nat @ K @ L2 ) )
% 0.30/0.52 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_semiring(2)
% 0.30/0.52 thf(fact_1447_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.30/0.52 ! [I2: int,J: int,K: int,L2: int] :
% 0.30/0.52 ( ( ( I2 = J )
% 0.30/0.52 & ( ord_less_eq_int @ K @ L2 ) )
% 0.30/0.52 => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_semiring(2)
% 0.30/0.52 thf(fact_1448_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.30/0.52 ! [I2: real,J: real,K: real,L2: real] :
% 0.30/0.52 ( ( ( ord_less_eq_real @ I2 @ J )
% 0.30/0.52 & ( ord_less_eq_real @ K @ L2 ) )
% 0.30/0.52 => ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_semiring(1)
% 0.30/0.52 thf(fact_1449_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.30/0.52 ! [I2: rat,J: rat,K: rat,L2: rat] :
% 0.30/0.52 ( ( ( ord_less_eq_rat @ I2 @ J )
% 0.30/0.52 & ( ord_less_eq_rat @ K @ L2 ) )
% 0.30/0.52 => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_semiring(1)
% 0.30/0.52 thf(fact_1450_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.30/0.52 ! [I2: nat,J: nat,K: nat,L2: nat] :
% 0.30/0.52 ( ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.52 & ( ord_less_eq_nat @ K @ L2 ) )
% 0.30/0.52 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_semiring(1)
% 0.30/0.52 thf(fact_1451_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.30/0.52 ! [I2: int,J: int,K: int,L2: int] :
% 0.30/0.52 ( ( ( ord_less_eq_int @ I2 @ J )
% 0.30/0.52 & ( ord_less_eq_int @ K @ L2 ) )
% 0.30/0.52 => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_semiring(1)
% 0.30/0.52 thf(fact_1452_add__mono,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.52 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.52 => ( ( ord_less_eq_real @ C @ D )
% 0.30/0.52 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono
% 0.30/0.52 thf(fact_1453_add__mono,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.52 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.52 => ( ( ord_less_eq_rat @ C @ D )
% 0.30/0.52 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono
% 0.30/0.52 thf(fact_1454_add__mono,axiom,
% 0.30/0.52 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.52 => ( ( ord_less_eq_nat @ C @ D )
% 0.30/0.52 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono
% 0.30/0.52 thf(fact_1455_add__mono,axiom,
% 0.30/0.52 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.52 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.52 => ( ( ord_less_eq_int @ C @ D )
% 0.30/0.52 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono
% 0.30/0.52 thf(fact_1456_add__left__mono,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real] :
% 0.30/0.52 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.52 => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_left_mono
% 0.30/0.52 thf(fact_1457_add__left__mono,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat] :
% 0.30/0.52 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.52 => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_left_mono
% 0.30/0.52 thf(fact_1458_add__left__mono,axiom,
% 0.30/0.52 ! [A: nat,B: nat,C: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.52 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_left_mono
% 0.30/0.52 thf(fact_1459_add__left__mono,axiom,
% 0.30/0.52 ! [A: int,B: int,C: int] :
% 0.30/0.52 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.52 => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_left_mono
% 0.30/0.52 thf(fact_1460_less__eqE,axiom,
% 0.30/0.52 ! [A: nat,B: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.52 => ~ ! [C3: nat] :
% 0.30/0.52 ( B
% 0.30/0.52 != ( plus_plus_nat @ A @ C3 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_eqE
% 0.30/0.52 thf(fact_1461_add__right__mono,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real] :
% 0.30/0.52 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.52 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_right_mono
% 0.30/0.52 thf(fact_1462_add__right__mono,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat] :
% 0.30/0.52 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.52 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_right_mono
% 0.30/0.52 thf(fact_1463_add__right__mono,axiom,
% 0.30/0.52 ! [A: nat,B: nat,C: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.52 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_right_mono
% 0.30/0.52 thf(fact_1464_add__right__mono,axiom,
% 0.30/0.52 ! [A: int,B: int,C: int] :
% 0.30/0.52 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.52 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_right_mono
% 0.30/0.52 thf(fact_1465_le__iff__add,axiom,
% 0.30/0.52 ( ord_less_eq_nat
% 0.30/0.52 = ( ^ [A5: nat,B5: nat] :
% 0.30/0.52 ? [C4: nat] :
% 0.30/0.52 ( B5
% 0.30/0.52 = ( plus_plus_nat @ A5 @ C4 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_iff_add
% 0.30/0.52 thf(fact_1466_add__le__imp__le__left,axiom,
% 0.30/0.52 ! [C: real,A: real,B: real] :
% 0.30/0.52 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.30/0.52 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_le_imp_le_left
% 0.30/0.52 thf(fact_1467_add__le__imp__le__left,axiom,
% 0.30/0.52 ! [C: rat,A: rat,B: rat] :
% 0.30/0.52 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.30/0.52 => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_le_imp_le_left
% 0.30/0.52 thf(fact_1468_add__le__imp__le__left,axiom,
% 0.30/0.52 ! [C: nat,A: nat,B: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.30/0.52 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_le_imp_le_left
% 0.30/0.52 thf(fact_1469_add__le__imp__le__left,axiom,
% 0.30/0.52 ! [C: int,A: int,B: int] :
% 0.30/0.52 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.30/0.52 => ( ord_less_eq_int @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_le_imp_le_left
% 0.30/0.52 thf(fact_1470_add__le__imp__le__right,axiom,
% 0.30/0.52 ! [A: real,C: real,B: real] :
% 0.30/0.52 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.30/0.52 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_le_imp_le_right
% 0.30/0.52 thf(fact_1471_add__le__imp__le__right,axiom,
% 0.30/0.52 ! [A: rat,C: rat,B: rat] :
% 0.30/0.52 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.30/0.52 => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_le_imp_le_right
% 0.30/0.52 thf(fact_1472_add__le__imp__le__right,axiom,
% 0.30/0.52 ! [A: nat,C: nat,B: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.30/0.52 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_le_imp_le_right
% 0.30/0.52 thf(fact_1473_add__le__imp__le__right,axiom,
% 0.30/0.52 ! [A: int,C: int,B: int] :
% 0.30/0.52 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.30/0.52 => ( ord_less_eq_int @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_le_imp_le_right
% 0.30/0.52 thf(fact_1474_add__mono__thms__linordered__field_I5_J,axiom,
% 0.30/0.52 ! [I2: real,J: real,K: real,L2: real] :
% 0.30/0.52 ( ( ( ord_less_real @ I2 @ J )
% 0.30/0.52 & ( ord_less_real @ K @ L2 ) )
% 0.30/0.52 => ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_field(5)
% 0.30/0.52 thf(fact_1475_add__mono__thms__linordered__field_I5_J,axiom,
% 0.30/0.52 ! [I2: rat,J: rat,K: rat,L2: rat] :
% 0.30/0.52 ( ( ( ord_less_rat @ I2 @ J )
% 0.30/0.52 & ( ord_less_rat @ K @ L2 ) )
% 0.30/0.52 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_field(5)
% 0.30/0.52 thf(fact_1476_add__mono__thms__linordered__field_I5_J,axiom,
% 0.30/0.52 ! [I2: nat,J: nat,K: nat,L2: nat] :
% 0.30/0.52 ( ( ( ord_less_nat @ I2 @ J )
% 0.30/0.52 & ( ord_less_nat @ K @ L2 ) )
% 0.30/0.52 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_field(5)
% 0.30/0.52 thf(fact_1477_add__mono__thms__linordered__field_I5_J,axiom,
% 0.30/0.52 ! [I2: int,J: int,K: int,L2: int] :
% 0.30/0.52 ( ( ( ord_less_int @ I2 @ J )
% 0.30/0.52 & ( ord_less_int @ K @ L2 ) )
% 0.30/0.52 => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_field(5)
% 0.30/0.52 thf(fact_1478_add__mono__thms__linordered__field_I2_J,axiom,
% 0.30/0.52 ! [I2: real,J: real,K: real,L2: real] :
% 0.30/0.52 ( ( ( I2 = J )
% 0.30/0.52 & ( ord_less_real @ K @ L2 ) )
% 0.30/0.52 => ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_field(2)
% 0.30/0.52 thf(fact_1479_add__mono__thms__linordered__field_I2_J,axiom,
% 0.30/0.52 ! [I2: rat,J: rat,K: rat,L2: rat] :
% 0.30/0.52 ( ( ( I2 = J )
% 0.30/0.52 & ( ord_less_rat @ K @ L2 ) )
% 0.30/0.52 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_field(2)
% 0.30/0.52 thf(fact_1480_add__mono__thms__linordered__field_I2_J,axiom,
% 0.30/0.52 ! [I2: nat,J: nat,K: nat,L2: nat] :
% 0.30/0.52 ( ( ( I2 = J )
% 0.30/0.52 & ( ord_less_nat @ K @ L2 ) )
% 0.30/0.52 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_field(2)
% 0.30/0.52 thf(fact_1481_add__mono__thms__linordered__field_I2_J,axiom,
% 0.30/0.52 ! [I2: int,J: int,K: int,L2: int] :
% 0.30/0.52 ( ( ( I2 = J )
% 0.30/0.52 & ( ord_less_int @ K @ L2 ) )
% 0.30/0.52 => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_field(2)
% 0.30/0.52 thf(fact_1482_add__mono__thms__linordered__field_I1_J,axiom,
% 0.30/0.52 ! [I2: real,J: real,K: real,L2: real] :
% 0.30/0.52 ( ( ( ord_less_real @ I2 @ J )
% 0.30/0.52 & ( K = L2 ) )
% 0.30/0.52 => ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_field(1)
% 0.30/0.52 thf(fact_1483_add__mono__thms__linordered__field_I1_J,axiom,
% 0.30/0.52 ! [I2: rat,J: rat,K: rat,L2: rat] :
% 0.30/0.52 ( ( ( ord_less_rat @ I2 @ J )
% 0.30/0.52 & ( K = L2 ) )
% 0.30/0.52 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_field(1)
% 0.30/0.52 thf(fact_1484_add__mono__thms__linordered__field_I1_J,axiom,
% 0.30/0.52 ! [I2: nat,J: nat,K: nat,L2: nat] :
% 0.30/0.52 ( ( ( ord_less_nat @ I2 @ J )
% 0.30/0.52 & ( K = L2 ) )
% 0.30/0.52 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_field(1)
% 0.30/0.52 thf(fact_1485_add__mono__thms__linordered__field_I1_J,axiom,
% 0.30/0.52 ! [I2: int,J: int,K: int,L2: int] :
% 0.30/0.52 ( ( ( ord_less_int @ I2 @ J )
% 0.30/0.52 & ( K = L2 ) )
% 0.30/0.52 => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_field(1)
% 0.30/0.52 thf(fact_1486_add__strict__mono,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.52 ( ( ord_less_real @ A @ B )
% 0.30/0.52 => ( ( ord_less_real @ C @ D )
% 0.30/0.52 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_strict_mono
% 0.30/0.52 thf(fact_1487_add__strict__mono,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.52 ( ( ord_less_rat @ A @ B )
% 0.30/0.52 => ( ( ord_less_rat @ C @ D )
% 0.30/0.52 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_strict_mono
% 0.30/0.52 thf(fact_1488_add__strict__mono,axiom,
% 0.30/0.52 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/0.52 ( ( ord_less_nat @ A @ B )
% 0.30/0.52 => ( ( ord_less_nat @ C @ D )
% 0.30/0.52 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_strict_mono
% 0.30/0.52 thf(fact_1489_add__strict__mono,axiom,
% 0.30/0.52 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.52 ( ( ord_less_int @ A @ B )
% 0.30/0.52 => ( ( ord_less_int @ C @ D )
% 0.30/0.52 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_strict_mono
% 0.30/0.52 thf(fact_1490_add__strict__left__mono,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real] :
% 0.30/0.52 ( ( ord_less_real @ A @ B )
% 0.30/0.52 => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_strict_left_mono
% 0.30/0.52 thf(fact_1491_add__strict__left__mono,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat] :
% 0.30/0.52 ( ( ord_less_rat @ A @ B )
% 0.30/0.52 => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_strict_left_mono
% 0.30/0.52 thf(fact_1492_add__strict__left__mono,axiom,
% 0.30/0.52 ! [A: nat,B: nat,C: nat] :
% 0.30/0.52 ( ( ord_less_nat @ A @ B )
% 0.30/0.52 => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_strict_left_mono
% 0.30/0.52 thf(fact_1493_add__strict__left__mono,axiom,
% 0.30/0.52 ! [A: int,B: int,C: int] :
% 0.30/0.52 ( ( ord_less_int @ A @ B )
% 0.30/0.52 => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_strict_left_mono
% 0.30/0.52 thf(fact_1494_add__strict__right__mono,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real] :
% 0.30/0.52 ( ( ord_less_real @ A @ B )
% 0.30/0.52 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_strict_right_mono
% 0.30/0.52 thf(fact_1495_add__strict__right__mono,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat] :
% 0.30/0.52 ( ( ord_less_rat @ A @ B )
% 0.30/0.52 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_strict_right_mono
% 0.30/0.52 thf(fact_1496_add__strict__right__mono,axiom,
% 0.30/0.52 ! [A: nat,B: nat,C: nat] :
% 0.30/0.52 ( ( ord_less_nat @ A @ B )
% 0.30/0.52 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_strict_right_mono
% 0.30/0.52 thf(fact_1497_add__strict__right__mono,axiom,
% 0.30/0.52 ! [A: int,B: int,C: int] :
% 0.30/0.52 ( ( ord_less_int @ A @ B )
% 0.30/0.52 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_strict_right_mono
% 0.30/0.52 thf(fact_1498_add__less__imp__less__left,axiom,
% 0.30/0.52 ! [C: real,A: real,B: real] :
% 0.30/0.52 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.30/0.52 => ( ord_less_real @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_less_imp_less_left
% 0.30/0.52 thf(fact_1499_add__less__imp__less__left,axiom,
% 0.30/0.52 ! [C: rat,A: rat,B: rat] :
% 0.30/0.52 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.30/0.52 => ( ord_less_rat @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_less_imp_less_left
% 0.30/0.52 thf(fact_1500_add__less__imp__less__left,axiom,
% 0.30/0.52 ! [C: nat,A: nat,B: nat] :
% 0.30/0.52 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.30/0.52 => ( ord_less_nat @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_less_imp_less_left
% 0.30/0.52 thf(fact_1501_add__less__imp__less__left,axiom,
% 0.30/0.52 ! [C: int,A: int,B: int] :
% 0.30/0.52 ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.30/0.52 => ( ord_less_int @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_less_imp_less_left
% 0.30/0.52 thf(fact_1502_add__less__imp__less__right,axiom,
% 0.30/0.52 ! [A: real,C: real,B: real] :
% 0.30/0.52 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.30/0.52 => ( ord_less_real @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_less_imp_less_right
% 0.30/0.52 thf(fact_1503_add__less__imp__less__right,axiom,
% 0.30/0.52 ! [A: rat,C: rat,B: rat] :
% 0.30/0.52 ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.30/0.52 => ( ord_less_rat @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_less_imp_less_right
% 0.30/0.52 thf(fact_1504_add__less__imp__less__right,axiom,
% 0.30/0.52 ! [A: nat,C: nat,B: nat] :
% 0.30/0.52 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.30/0.52 => ( ord_less_nat @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_less_imp_less_right
% 0.30/0.52 thf(fact_1505_add__less__imp__less__right,axiom,
% 0.30/0.52 ! [A: int,C: int,B: int] :
% 0.30/0.52 ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.30/0.52 => ( ord_less_int @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_less_imp_less_right
% 0.30/0.52 thf(fact_1506_diff__eq__diff__less__eq,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.52 ( ( ( minus_minus_real @ A @ B )
% 0.30/0.52 = ( minus_minus_real @ C @ D ) )
% 0.30/0.52 => ( ( ord_less_eq_real @ A @ B )
% 0.30/0.52 = ( ord_less_eq_real @ C @ D ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_eq_diff_less_eq
% 0.30/0.52 thf(fact_1507_diff__eq__diff__less__eq,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.52 ( ( ( minus_minus_rat @ A @ B )
% 0.30/0.52 = ( minus_minus_rat @ C @ D ) )
% 0.30/0.52 => ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.52 = ( ord_less_eq_rat @ C @ D ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_eq_diff_less_eq
% 0.30/0.52 thf(fact_1508_diff__eq__diff__less__eq,axiom,
% 0.30/0.52 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.52 ( ( ( minus_minus_int @ A @ B )
% 0.30/0.52 = ( minus_minus_int @ C @ D ) )
% 0.30/0.52 => ( ( ord_less_eq_int @ A @ B )
% 0.30/0.52 = ( ord_less_eq_int @ C @ D ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_eq_diff_less_eq
% 0.30/0.52 thf(fact_1509_diff__right__mono,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real] :
% 0.30/0.52 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.52 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_right_mono
% 0.30/0.52 thf(fact_1510_diff__right__mono,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat] :
% 0.30/0.52 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.52 => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_right_mono
% 0.30/0.52 thf(fact_1511_diff__right__mono,axiom,
% 0.30/0.52 ! [A: int,B: int,C: int] :
% 0.30/0.52 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.52 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_right_mono
% 0.30/0.52 thf(fact_1512_diff__left__mono,axiom,
% 0.30/0.52 ! [B: real,A: real,C: real] :
% 0.30/0.52 ( ( ord_less_eq_real @ B @ A )
% 0.30/0.52 => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_left_mono
% 0.30/0.52 thf(fact_1513_diff__left__mono,axiom,
% 0.30/0.52 ! [B: rat,A: rat,C: rat] :
% 0.30/0.52 ( ( ord_less_eq_rat @ B @ A )
% 0.30/0.52 => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_left_mono
% 0.30/0.52 thf(fact_1514_diff__left__mono,axiom,
% 0.30/0.52 ! [B: int,A: int,C: int] :
% 0.30/0.52 ( ( ord_less_eq_int @ B @ A )
% 0.30/0.52 => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_left_mono
% 0.30/0.52 thf(fact_1515_diff__mono,axiom,
% 0.30/0.52 ! [A: real,B: real,D: real,C: real] :
% 0.30/0.52 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.52 => ( ( ord_less_eq_real @ D @ C )
% 0.30/0.52 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_mono
% 0.30/0.52 thf(fact_1516_diff__mono,axiom,
% 0.30/0.52 ! [A: rat,B: rat,D: rat,C: rat] :
% 0.30/0.52 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.52 => ( ( ord_less_eq_rat @ D @ C )
% 0.30/0.52 => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_mono
% 0.30/0.52 thf(fact_1517_diff__mono,axiom,
% 0.30/0.52 ! [A: int,B: int,D: int,C: int] :
% 0.30/0.52 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.52 => ( ( ord_less_eq_int @ D @ C )
% 0.30/0.52 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_mono
% 0.30/0.52 thf(fact_1518_diff__strict__right__mono,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real] :
% 0.30/0.52 ( ( ord_less_real @ A @ B )
% 0.30/0.52 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_strict_right_mono
% 0.30/0.52 thf(fact_1519_diff__strict__right__mono,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat] :
% 0.30/0.52 ( ( ord_less_rat @ A @ B )
% 0.30/0.52 => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_strict_right_mono
% 0.30/0.52 thf(fact_1520_diff__strict__right__mono,axiom,
% 0.30/0.52 ! [A: int,B: int,C: int] :
% 0.30/0.52 ( ( ord_less_int @ A @ B )
% 0.30/0.52 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_strict_right_mono
% 0.30/0.52 thf(fact_1521_diff__strict__left__mono,axiom,
% 0.30/0.52 ! [B: real,A: real,C: real] :
% 0.30/0.52 ( ( ord_less_real @ B @ A )
% 0.30/0.52 => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_strict_left_mono
% 0.30/0.52 thf(fact_1522_diff__strict__left__mono,axiom,
% 0.30/0.52 ! [B: rat,A: rat,C: rat] :
% 0.30/0.52 ( ( ord_less_rat @ B @ A )
% 0.30/0.52 => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_strict_left_mono
% 0.30/0.52 thf(fact_1523_diff__strict__left__mono,axiom,
% 0.30/0.52 ! [B: int,A: int,C: int] :
% 0.30/0.52 ( ( ord_less_int @ B @ A )
% 0.30/0.52 => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_strict_left_mono
% 0.30/0.52 thf(fact_1524_diff__eq__diff__less,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.52 ( ( ( minus_minus_real @ A @ B )
% 0.30/0.52 = ( minus_minus_real @ C @ D ) )
% 0.30/0.52 => ( ( ord_less_real @ A @ B )
% 0.30/0.52 = ( ord_less_real @ C @ D ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_eq_diff_less
% 0.30/0.52 thf(fact_1525_diff__eq__diff__less,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.52 ( ( ( minus_minus_rat @ A @ B )
% 0.30/0.52 = ( minus_minus_rat @ C @ D ) )
% 0.30/0.52 => ( ( ord_less_rat @ A @ B )
% 0.30/0.52 = ( ord_less_rat @ C @ D ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_eq_diff_less
% 0.30/0.52 thf(fact_1526_diff__eq__diff__less,axiom,
% 0.30/0.52 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.52 ( ( ( minus_minus_int @ A @ B )
% 0.30/0.52 = ( minus_minus_int @ C @ D ) )
% 0.30/0.52 => ( ( ord_less_int @ A @ B )
% 0.30/0.52 = ( ord_less_int @ C @ D ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_eq_diff_less
% 0.30/0.52 thf(fact_1527_diff__strict__mono,axiom,
% 0.30/0.52 ! [A: real,B: real,D: real,C: real] :
% 0.30/0.52 ( ( ord_less_real @ A @ B )
% 0.30/0.52 => ( ( ord_less_real @ D @ C )
% 0.30/0.52 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_strict_mono
% 0.30/0.52 thf(fact_1528_diff__strict__mono,axiom,
% 0.30/0.52 ! [A: rat,B: rat,D: rat,C: rat] :
% 0.30/0.52 ( ( ord_less_rat @ A @ B )
% 0.30/0.52 => ( ( ord_less_rat @ D @ C )
% 0.30/0.52 => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_strict_mono
% 0.30/0.52 thf(fact_1529_diff__strict__mono,axiom,
% 0.30/0.52 ! [A: int,B: int,D: int,C: int] :
% 0.30/0.52 ( ( ord_less_int @ A @ B )
% 0.30/0.52 => ( ( ord_less_int @ D @ C )
% 0.30/0.52 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_strict_mono
% 0.30/0.52 thf(fact_1530_comm__monoid__mult__class_Omult__1,axiom,
% 0.30/0.52 ! [A: complex] :
% 0.30/0.52 ( ( times_times_complex @ one_one_complex @ A )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % comm_monoid_mult_class.mult_1
% 0.30/0.52 thf(fact_1531_comm__monoid__mult__class_Omult__1,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( times_times_real @ one_one_real @ A )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % comm_monoid_mult_class.mult_1
% 0.30/0.52 thf(fact_1532_comm__monoid__mult__class_Omult__1,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( times_times_rat @ one_one_rat @ A )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % comm_monoid_mult_class.mult_1
% 0.30/0.52 thf(fact_1533_comm__monoid__mult__class_Omult__1,axiom,
% 0.30/0.52 ! [A: nat] :
% 0.30/0.52 ( ( times_times_nat @ one_one_nat @ A )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % comm_monoid_mult_class.mult_1
% 0.30/0.52 thf(fact_1534_comm__monoid__mult__class_Omult__1,axiom,
% 0.30/0.52 ! [A: int] :
% 0.30/0.52 ( ( times_times_int @ one_one_int @ A )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % comm_monoid_mult_class.mult_1
% 0.30/0.52 thf(fact_1535_mult_Ocomm__neutral,axiom,
% 0.30/0.52 ! [A: complex] :
% 0.30/0.52 ( ( times_times_complex @ A @ one_one_complex )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % mult.comm_neutral
% 0.30/0.52 thf(fact_1536_mult_Ocomm__neutral,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( times_times_real @ A @ one_one_real )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % mult.comm_neutral
% 0.30/0.52 thf(fact_1537_mult_Ocomm__neutral,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( times_times_rat @ A @ one_one_rat )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % mult.comm_neutral
% 0.30/0.52 thf(fact_1538_mult_Ocomm__neutral,axiom,
% 0.30/0.52 ! [A: nat] :
% 0.30/0.52 ( ( times_times_nat @ A @ one_one_nat )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % mult.comm_neutral
% 0.30/0.52 thf(fact_1539_mult_Ocomm__neutral,axiom,
% 0.30/0.52 ! [A: int] :
% 0.30/0.52 ( ( times_times_int @ A @ one_one_int )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % mult.comm_neutral
% 0.30/0.52 thf(fact_1540_times__divide__times__eq,axiom,
% 0.30/0.52 ! [X: complex,Y: complex,Z: complex,W: complex] :
% 0.30/0.52 ( ( times_times_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
% 0.30/0.52 = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ Y @ W ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % times_divide_times_eq
% 0.30/0.52 thf(fact_1541_times__divide__times__eq,axiom,
% 0.30/0.52 ! [X: real,Y: real,Z: real,W: real] :
% 0.30/0.52 ( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
% 0.30/0.52 = ( divide_divide_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ W ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % times_divide_times_eq
% 0.30/0.52 thf(fact_1542_times__divide__times__eq,axiom,
% 0.30/0.52 ! [X: rat,Y: rat,Z: rat,W: rat] :
% 0.30/0.52 ( ( times_times_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W ) )
% 0.30/0.52 = ( divide_divide_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ W ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % times_divide_times_eq
% 0.30/0.52 thf(fact_1543_divide__divide__times__eq,axiom,
% 0.30/0.52 ! [X: complex,Y: complex,Z: complex,W: complex] :
% 0.30/0.52 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
% 0.30/0.52 = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ W ) @ ( times_times_complex @ Y @ Z ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_divide_times_eq
% 0.30/0.52 thf(fact_1544_divide__divide__times__eq,axiom,
% 0.30/0.52 ! [X: real,Y: real,Z: real,W: real] :
% 0.30/0.52 ( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
% 0.30/0.52 = ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y @ Z ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_divide_times_eq
% 0.30/0.52 thf(fact_1545_divide__divide__times__eq,axiom,
% 0.30/0.52 ! [X: rat,Y: rat,Z: rat,W: rat] :
% 0.30/0.52 ( ( divide_divide_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W ) )
% 0.30/0.52 = ( divide_divide_rat @ ( times_times_rat @ X @ W ) @ ( times_times_rat @ Y @ Z ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_divide_times_eq
% 0.30/0.52 thf(fact_1546_divide__divide__eq__left_H,axiom,
% 0.30/0.52 ! [A: complex,B: complex,C: complex] :
% 0.30/0.52 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
% 0.30/0.52 = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_divide_eq_left'
% 0.30/0.52 thf(fact_1547_divide__divide__eq__left_H,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real] :
% 0.30/0.52 ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
% 0.30/0.52 = ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_divide_eq_left'
% 0.30/0.52 thf(fact_1548_divide__divide__eq__left_H,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat] :
% 0.30/0.52 ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
% 0.30/0.52 = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divide_divide_eq_left'
% 0.30/0.52 thf(fact_1549_group__cancel_Osub1,axiom,
% 0.30/0.52 ! [A2: real,K: real,A: real,B: real] :
% 0.30/0.52 ( ( A2
% 0.30/0.52 = ( plus_plus_real @ K @ A ) )
% 0.30/0.52 => ( ( minus_minus_real @ A2 @ B )
% 0.30/0.52 = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % group_cancel.sub1
% 0.30/0.52 thf(fact_1550_group__cancel_Osub1,axiom,
% 0.30/0.52 ! [A2: rat,K: rat,A: rat,B: rat] :
% 0.30/0.52 ( ( A2
% 0.30/0.52 = ( plus_plus_rat @ K @ A ) )
% 0.30/0.52 => ( ( minus_minus_rat @ A2 @ B )
% 0.30/0.52 = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % group_cancel.sub1
% 0.30/0.52 thf(fact_1551_group__cancel_Osub1,axiom,
% 0.30/0.52 ! [A2: int,K: int,A: int,B: int] :
% 0.30/0.52 ( ( A2
% 0.30/0.52 = ( plus_plus_int @ K @ A ) )
% 0.30/0.52 => ( ( minus_minus_int @ A2 @ B )
% 0.30/0.52 = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % group_cancel.sub1
% 0.30/0.52 thf(fact_1552_diff__eq__eq,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real] :
% 0.30/0.52 ( ( ( minus_minus_real @ A @ B )
% 0.30/0.52 = C )
% 0.30/0.52 = ( A
% 0.30/0.52 = ( plus_plus_real @ C @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_eq_eq
% 0.30/0.52 thf(fact_1553_diff__eq__eq,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat] :
% 0.30/0.52 ( ( ( minus_minus_rat @ A @ B )
% 0.30/0.52 = C )
% 0.30/0.52 = ( A
% 0.30/0.52 = ( plus_plus_rat @ C @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_eq_eq
% 0.30/0.52 thf(fact_1554_diff__eq__eq,axiom,
% 0.30/0.52 ! [A: int,B: int,C: int] :
% 0.30/0.52 ( ( ( minus_minus_int @ A @ B )
% 0.30/0.52 = C )
% 0.30/0.52 = ( A
% 0.30/0.52 = ( plus_plus_int @ C @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_eq_eq
% 0.30/0.52 thf(fact_1555_eq__diff__eq,axiom,
% 0.30/0.52 ! [A: real,C: real,B: real] :
% 0.30/0.52 ( ( A
% 0.30/0.52 = ( minus_minus_real @ C @ B ) )
% 0.30/0.52 = ( ( plus_plus_real @ A @ B )
% 0.30/0.52 = C ) ) ).
% 0.30/0.52
% 0.30/0.52 % eq_diff_eq
% 0.30/0.52 thf(fact_1556_eq__diff__eq,axiom,
% 0.30/0.52 ! [A: rat,C: rat,B: rat] :
% 0.30/0.52 ( ( A
% 0.30/0.52 = ( minus_minus_rat @ C @ B ) )
% 0.30/0.52 = ( ( plus_plus_rat @ A @ B )
% 0.30/0.52 = C ) ) ).
% 0.30/0.52
% 0.30/0.52 % eq_diff_eq
% 0.30/0.52 thf(fact_1557_eq__diff__eq,axiom,
% 0.30/0.52 ! [A: int,C: int,B: int] :
% 0.30/0.52 ( ( A
% 0.30/0.52 = ( minus_minus_int @ C @ B ) )
% 0.30/0.52 = ( ( plus_plus_int @ A @ B )
% 0.30/0.52 = C ) ) ).
% 0.30/0.52
% 0.30/0.52 % eq_diff_eq
% 0.30/0.52 thf(fact_1558_add__diff__eq,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real] :
% 0.30/0.52 ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.30/0.52 = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_diff_eq
% 0.30/0.52 thf(fact_1559_add__diff__eq,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat] :
% 0.30/0.52 ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 0.30/0.52 = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_diff_eq
% 0.30/0.52 thf(fact_1560_add__diff__eq,axiom,
% 0.30/0.52 ! [A: int,B: int,C: int] :
% 0.30/0.52 ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.30/0.52 = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_diff_eq
% 0.30/0.52 thf(fact_1561_diff__diff__eq2,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real] :
% 0.30/0.52 ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.30/0.52 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_diff_eq2
% 0.30/0.52 thf(fact_1562_diff__diff__eq2,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat] :
% 0.30/0.52 ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 0.30/0.52 = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_diff_eq2
% 0.30/0.52 thf(fact_1563_diff__diff__eq2,axiom,
% 0.30/0.52 ! [A: int,B: int,C: int] :
% 0.30/0.52 ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.30/0.52 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_diff_eq2
% 0.30/0.52 thf(fact_1564_diff__add__eq,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real] :
% 0.30/0.52 ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.30/0.52 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_add_eq
% 0.30/0.52 thf(fact_1565_diff__add__eq,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat] :
% 0.30/0.52 ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.30/0.52 = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_add_eq
% 0.30/0.52 thf(fact_1566_diff__add__eq,axiom,
% 0.30/0.52 ! [A: int,B: int,C: int] :
% 0.30/0.52 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.30/0.52 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_add_eq
% 0.30/0.52 thf(fact_1567_diff__add__eq__diff__diff__swap,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real] :
% 0.30/0.52 ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.30/0.52 = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_add_eq_diff_diff_swap
% 0.30/0.52 thf(fact_1568_diff__add__eq__diff__diff__swap,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat] :
% 0.30/0.52 ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.30/0.52 = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_add_eq_diff_diff_swap
% 0.30/0.52 thf(fact_1569_diff__add__eq__diff__diff__swap,axiom,
% 0.30/0.52 ! [A: int,B: int,C: int] :
% 0.30/0.52 ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.30/0.52 = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_add_eq_diff_diff_swap
% 0.30/0.52 thf(fact_1570_add__implies__diff,axiom,
% 0.30/0.52 ! [C: real,B: real,A: real] :
% 0.30/0.52 ( ( ( plus_plus_real @ C @ B )
% 0.30/0.52 = A )
% 0.30/0.52 => ( C
% 0.30/0.52 = ( minus_minus_real @ A @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_implies_diff
% 0.30/0.52 thf(fact_1571_add__implies__diff,axiom,
% 0.30/0.52 ! [C: rat,B: rat,A: rat] :
% 0.30/0.52 ( ( ( plus_plus_rat @ C @ B )
% 0.30/0.52 = A )
% 0.30/0.52 => ( C
% 0.30/0.52 = ( minus_minus_rat @ A @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_implies_diff
% 0.30/0.52 thf(fact_1572_add__implies__diff,axiom,
% 0.30/0.52 ! [C: nat,B: nat,A: nat] :
% 0.30/0.52 ( ( ( plus_plus_nat @ C @ B )
% 0.30/0.52 = A )
% 0.30/0.52 => ( C
% 0.30/0.52 = ( minus_minus_nat @ A @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_implies_diff
% 0.30/0.52 thf(fact_1573_add__implies__diff,axiom,
% 0.30/0.52 ! [C: int,B: int,A: int] :
% 0.30/0.52 ( ( ( plus_plus_int @ C @ B )
% 0.30/0.52 = A )
% 0.30/0.52 => ( C
% 0.30/0.52 = ( minus_minus_int @ A @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_implies_diff
% 0.30/0.52 thf(fact_1574_diff__diff__eq,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real] :
% 0.30/0.52 ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.30/0.52 = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_diff_eq
% 0.30/0.52 thf(fact_1575_diff__diff__eq,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat] :
% 0.30/0.52 ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.30/0.52 = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_diff_eq
% 0.30/0.52 thf(fact_1576_diff__diff__eq,axiom,
% 0.30/0.52 ! [A: nat,B: nat,C: nat] :
% 0.30/0.52 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
% 0.30/0.52 = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_diff_eq
% 0.30/0.52 thf(fact_1577_diff__diff__eq,axiom,
% 0.30/0.52 ! [A: int,B: int,C: int] :
% 0.30/0.52 ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.30/0.52 = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_diff_eq
% 0.30/0.52 thf(fact_1578_add__divide__distrib,axiom,
% 0.30/0.52 ! [A: complex,B: complex,C: complex] :
% 0.30/0.52 ( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B ) @ C )
% 0.30/0.52 = ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_divide_distrib
% 0.30/0.52 thf(fact_1579_add__divide__distrib,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real] :
% 0.30/0.52 ( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.30/0.52 = ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_divide_distrib
% 0.30/0.52 thf(fact_1580_add__divide__distrib,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat] :
% 0.30/0.52 ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.30/0.52 = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_divide_distrib
% 0.30/0.52 thf(fact_1581_diff__divide__distrib,axiom,
% 0.30/0.52 ! [A: complex,B: complex,C: complex] :
% 0.30/0.52 ( ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ B ) @ C )
% 0.30/0.52 = ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_divide_distrib
% 0.30/0.52 thf(fact_1582_diff__divide__distrib,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real] :
% 0.30/0.52 ( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.30/0.52 = ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_divide_distrib
% 0.30/0.52 thf(fact_1583_diff__divide__distrib,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat] :
% 0.30/0.52 ( ( divide_divide_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.30/0.52 = ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_divide_distrib
% 0.30/0.52 thf(fact_1584_max__add__distrib__left,axiom,
% 0.30/0.52 ! [X: real,Y: real,Z: real] :
% 0.30/0.52 ( ( plus_plus_real @ ( ord_max_real @ X @ Y ) @ Z )
% 0.30/0.52 = ( ord_max_real @ ( plus_plus_real @ X @ Z ) @ ( plus_plus_real @ Y @ Z ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % max_add_distrib_left
% 0.30/0.52 thf(fact_1585_max__add__distrib__left,axiom,
% 0.30/0.52 ! [X: rat,Y: rat,Z: rat] :
% 0.30/0.52 ( ( plus_plus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
% 0.30/0.52 = ( ord_max_rat @ ( plus_plus_rat @ X @ Z ) @ ( plus_plus_rat @ Y @ Z ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % max_add_distrib_left
% 0.30/0.52 thf(fact_1586_max__add__distrib__left,axiom,
% 0.30/0.52 ! [X: nat,Y: nat,Z: nat] :
% 0.30/0.52 ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z )
% 0.30/0.52 = ( ord_max_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y @ Z ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % max_add_distrib_left
% 0.30/0.52 thf(fact_1587_max__add__distrib__left,axiom,
% 0.30/0.52 ! [X: int,Y: int,Z: int] :
% 0.30/0.52 ( ( plus_plus_int @ ( ord_max_int @ X @ Y ) @ Z )
% 0.30/0.52 = ( ord_max_int @ ( plus_plus_int @ X @ Z ) @ ( plus_plus_int @ Y @ Z ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % max_add_distrib_left
% 0.30/0.52 thf(fact_1588_max__add__distrib__right,axiom,
% 0.30/0.52 ! [X: real,Y: real,Z: real] :
% 0.30/0.52 ( ( plus_plus_real @ X @ ( ord_max_real @ Y @ Z ) )
% 0.30/0.52 = ( ord_max_real @ ( plus_plus_real @ X @ Y ) @ ( plus_plus_real @ X @ Z ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % max_add_distrib_right
% 0.30/0.52 thf(fact_1589_max__add__distrib__right,axiom,
% 0.30/0.52 ! [X: rat,Y: rat,Z: rat] :
% 0.30/0.52 ( ( plus_plus_rat @ X @ ( ord_max_rat @ Y @ Z ) )
% 0.30/0.52 = ( ord_max_rat @ ( plus_plus_rat @ X @ Y ) @ ( plus_plus_rat @ X @ Z ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % max_add_distrib_right
% 0.30/0.52 thf(fact_1590_max__add__distrib__right,axiom,
% 0.30/0.52 ! [X: nat,Y: nat,Z: nat] :
% 0.30/0.52 ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z ) )
% 0.30/0.52 = ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % max_add_distrib_right
% 0.30/0.52 thf(fact_1591_max__add__distrib__right,axiom,
% 0.30/0.52 ! [X: int,Y: int,Z: int] :
% 0.30/0.52 ( ( plus_plus_int @ X @ ( ord_max_int @ Y @ Z ) )
% 0.30/0.52 = ( ord_max_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % max_add_distrib_right
% 0.30/0.52 thf(fact_1592_max__diff__distrib__left,axiom,
% 0.30/0.52 ! [X: real,Y: real,Z: real] :
% 0.30/0.52 ( ( minus_minus_real @ ( ord_max_real @ X @ Y ) @ Z )
% 0.30/0.52 = ( ord_max_real @ ( minus_minus_real @ X @ Z ) @ ( minus_minus_real @ Y @ Z ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % max_diff_distrib_left
% 0.30/0.52 thf(fact_1593_max__diff__distrib__left,axiom,
% 0.30/0.52 ! [X: rat,Y: rat,Z: rat] :
% 0.30/0.52 ( ( minus_minus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
% 0.30/0.52 = ( ord_max_rat @ ( minus_minus_rat @ X @ Z ) @ ( minus_minus_rat @ Y @ Z ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % max_diff_distrib_left
% 0.30/0.52 thf(fact_1594_max__diff__distrib__left,axiom,
% 0.30/0.52 ! [X: int,Y: int,Z: int] :
% 0.30/0.52 ( ( minus_minus_int @ ( ord_max_int @ X @ Y ) @ Z )
% 0.30/0.52 = ( ord_max_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y @ Z ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % max_diff_distrib_left
% 0.30/0.52 thf(fact_1595_add__less__le__mono,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.52 ( ( ord_less_real @ A @ B )
% 0.30/0.52 => ( ( ord_less_eq_real @ C @ D )
% 0.30/0.52 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_less_le_mono
% 0.30/0.52 thf(fact_1596_add__less__le__mono,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.52 ( ( ord_less_rat @ A @ B )
% 0.30/0.52 => ( ( ord_less_eq_rat @ C @ D )
% 0.30/0.52 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_less_le_mono
% 0.30/0.52 thf(fact_1597_add__less__le__mono,axiom,
% 0.30/0.52 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/0.52 ( ( ord_less_nat @ A @ B )
% 0.30/0.52 => ( ( ord_less_eq_nat @ C @ D )
% 0.30/0.52 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_less_le_mono
% 0.30/0.52 thf(fact_1598_add__less__le__mono,axiom,
% 0.30/0.52 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.52 ( ( ord_less_int @ A @ B )
% 0.30/0.52 => ( ( ord_less_eq_int @ C @ D )
% 0.30/0.52 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_less_le_mono
% 0.30/0.52 thf(fact_1599_add__le__less__mono,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.52 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.52 => ( ( ord_less_real @ C @ D )
% 0.30/0.52 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_le_less_mono
% 0.30/0.52 thf(fact_1600_add__le__less__mono,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.52 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.52 => ( ( ord_less_rat @ C @ D )
% 0.30/0.52 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_le_less_mono
% 0.30/0.52 thf(fact_1601_add__le__less__mono,axiom,
% 0.30/0.52 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.52 => ( ( ord_less_nat @ C @ D )
% 0.30/0.52 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_le_less_mono
% 0.30/0.52 thf(fact_1602_add__le__less__mono,axiom,
% 0.30/0.52 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.52 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.52 => ( ( ord_less_int @ C @ D )
% 0.30/0.52 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_le_less_mono
% 0.30/0.52 thf(fact_1603_add__mono__thms__linordered__field_I3_J,axiom,
% 0.30/0.52 ! [I2: real,J: real,K: real,L2: real] :
% 0.30/0.52 ( ( ( ord_less_real @ I2 @ J )
% 0.30/0.52 & ( ord_less_eq_real @ K @ L2 ) )
% 0.30/0.52 => ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_field(3)
% 0.30/0.52 thf(fact_1604_add__mono__thms__linordered__field_I3_J,axiom,
% 0.30/0.52 ! [I2: rat,J: rat,K: rat,L2: rat] :
% 0.30/0.52 ( ( ( ord_less_rat @ I2 @ J )
% 0.30/0.52 & ( ord_less_eq_rat @ K @ L2 ) )
% 0.30/0.52 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_field(3)
% 0.30/0.52 thf(fact_1605_add__mono__thms__linordered__field_I3_J,axiom,
% 0.30/0.52 ! [I2: nat,J: nat,K: nat,L2: nat] :
% 0.30/0.52 ( ( ( ord_less_nat @ I2 @ J )
% 0.30/0.52 & ( ord_less_eq_nat @ K @ L2 ) )
% 0.30/0.52 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_field(3)
% 0.30/0.52 thf(fact_1606_add__mono__thms__linordered__field_I3_J,axiom,
% 0.30/0.52 ! [I2: int,J: int,K: int,L2: int] :
% 0.30/0.52 ( ( ( ord_less_int @ I2 @ J )
% 0.30/0.52 & ( ord_less_eq_int @ K @ L2 ) )
% 0.30/0.52 => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_field(3)
% 0.30/0.52 thf(fact_1607_add__mono__thms__linordered__field_I4_J,axiom,
% 0.30/0.52 ! [I2: real,J: real,K: real,L2: real] :
% 0.30/0.52 ( ( ( ord_less_eq_real @ I2 @ J )
% 0.30/0.52 & ( ord_less_real @ K @ L2 ) )
% 0.30/0.52 => ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_field(4)
% 0.30/0.52 thf(fact_1608_add__mono__thms__linordered__field_I4_J,axiom,
% 0.30/0.52 ! [I2: rat,J: rat,K: rat,L2: rat] :
% 0.30/0.52 ( ( ( ord_less_eq_rat @ I2 @ J )
% 0.30/0.52 & ( ord_less_rat @ K @ L2 ) )
% 0.30/0.52 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_field(4)
% 0.30/0.52 thf(fact_1609_add__mono__thms__linordered__field_I4_J,axiom,
% 0.30/0.52 ! [I2: nat,J: nat,K: nat,L2: nat] :
% 0.30/0.52 ( ( ( ord_less_eq_nat @ I2 @ J )
% 0.30/0.52 & ( ord_less_nat @ K @ L2 ) )
% 0.30/0.52 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_field(4)
% 0.30/0.52 thf(fact_1610_add__mono__thms__linordered__field_I4_J,axiom,
% 0.30/0.52 ! [I2: int,J: int,K: int,L2: int] :
% 0.30/0.52 ( ( ( ord_less_eq_int @ I2 @ J )
% 0.30/0.52 & ( ord_less_int @ K @ L2 ) )
% 0.30/0.52 => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_mono_thms_linordered_field(4)
% 0.30/0.52 thf(fact_1611_diff__le__eq,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real] :
% 0.30/0.52 ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.30/0.52 = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_le_eq
% 0.30/0.52 thf(fact_1612_diff__le__eq,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat] :
% 0.30/0.52 ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.30/0.52 = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_le_eq
% 0.30/0.52 thf(fact_1613_diff__le__eq,axiom,
% 0.30/0.52 ! [A: int,B: int,C: int] :
% 0.30/0.52 ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.30/0.52 = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_le_eq
% 0.30/0.52 thf(fact_1614_le__diff__eq,axiom,
% 0.30/0.52 ! [A: real,C: real,B: real] :
% 0.30/0.52 ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
% 0.30/0.52 = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_diff_eq
% 0.30/0.52 thf(fact_1615_le__diff__eq,axiom,
% 0.30/0.52 ! [A: rat,C: rat,B: rat] :
% 0.30/0.52 ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 0.30/0.52 = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_diff_eq
% 0.30/0.52 thf(fact_1616_le__diff__eq,axiom,
% 0.30/0.52 ! [A: int,C: int,B: int] :
% 0.30/0.52 ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
% 0.30/0.52 = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_diff_eq
% 0.30/0.52 thf(fact_1617_diff__add,axiom,
% 0.30/0.52 ! [A: nat,B: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.52 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
% 0.30/0.52 = B ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_add
% 0.30/0.52 thf(fact_1618_le__add__diff,axiom,
% 0.30/0.52 ! [A: nat,B: nat,C: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.52 => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_add_diff
% 0.30/0.52 thf(fact_1619_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 0.30/0.52 ! [A: nat,B: nat,C: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.52 => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.30/0.52 = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ordered_cancel_comm_monoid_diff_class.le_diff_conv2
% 0.30/0.52 thf(fact_1620_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 0.30/0.52 ! [A: nat,B: nat,C: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.52 => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.30/0.52 = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc
% 0.30/0.52 thf(fact_1621_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 0.30/0.52 ! [A: nat,B: nat,C: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.52 => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
% 0.30/0.52 = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc
% 0.30/0.52 thf(fact_1622_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 0.30/0.52 ! [A: nat,B: nat,C: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.52 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
% 0.30/0.52 = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
% 0.30/0.52 thf(fact_1623_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 0.30/0.52 ! [A: nat,B: nat,C: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.52 => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
% 0.30/0.52 = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
% 0.30/0.52 thf(fact_1624_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 0.30/0.52 ! [A: nat,B: nat,C: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.52 => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.30/0.52 = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ordered_cancel_comm_monoid_diff_class.diff_diff_right
% 0.30/0.52 thf(fact_1625_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 0.30/0.52 ! [A: nat,B: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.52 => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
% 0.30/0.52 = B ) ) ).
% 0.30/0.52
% 0.30/0.52 % ordered_cancel_comm_monoid_diff_class.add_diff_inverse
% 0.30/0.52 thf(fact_1626_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 0.30/0.52 ! [A: nat,B: nat,C: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.52 => ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.52 => ( ( ( minus_minus_nat @ B @ A )
% 0.30/0.52 = C )
% 0.30/0.52 = ( B
% 0.30/0.52 = ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
% 0.30/0.52 thf(fact_1627_less__diff__eq,axiom,
% 0.30/0.52 ! [A: real,C: real,B: real] :
% 0.30/0.52 ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
% 0.30/0.52 = ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_diff_eq
% 0.30/0.52 thf(fact_1628_less__diff__eq,axiom,
% 0.30/0.52 ! [A: rat,C: rat,B: rat] :
% 0.30/0.52 ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 0.30/0.52 = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_diff_eq
% 0.30/0.52 thf(fact_1629_less__diff__eq,axiom,
% 0.30/0.52 ! [A: int,C: int,B: int] :
% 0.30/0.52 ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
% 0.30/0.52 = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_diff_eq
% 0.30/0.52 thf(fact_1630_diff__less__eq,axiom,
% 0.30/0.52 ! [A: real,B: real,C: real] :
% 0.30/0.52 ( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.30/0.52 = ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_less_eq
% 0.30/0.52 thf(fact_1631_diff__less__eq,axiom,
% 0.30/0.52 ! [A: rat,B: rat,C: rat] :
% 0.30/0.52 ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.30/0.52 = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_less_eq
% 0.30/0.52 thf(fact_1632_diff__less__eq,axiom,
% 0.30/0.52 ! [A: int,B: int,C: int] :
% 0.30/0.52 ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.30/0.52 = ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % diff_less_eq
% 0.30/0.52 thf(fact_1633_gt__half__sum,axiom,
% 0.30/0.52 ! [A: real,B: real] :
% 0.30/0.52 ( ( ord_less_real @ A @ B )
% 0.30/0.52 => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % gt_half_sum
% 0.30/0.52 thf(fact_1634_gt__half__sum,axiom,
% 0.30/0.52 ! [A: rat,B: rat] :
% 0.30/0.52 ( ( ord_less_rat @ A @ B )
% 0.30/0.52 => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % gt_half_sum
% 0.30/0.52 thf(fact_1635_less__half__sum,axiom,
% 0.30/0.52 ! [A: real,B: real] :
% 0.30/0.52 ( ( ord_less_real @ A @ B )
% 0.30/0.52 => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_half_sum
% 0.30/0.52 thf(fact_1636_less__half__sum,axiom,
% 0.30/0.52 ! [A: rat,B: rat] :
% 0.30/0.52 ( ( ord_less_rat @ A @ B )
% 0.30/0.52 => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % less_half_sum
% 0.30/0.52 thf(fact_1637_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
% 0.30/0.52 ! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
% 0.30/0.52 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList2 @ Vc ) @ X )
% 0.30/0.52 = ( ( X = Mi )
% 0.30/0.52 | ( X = Ma )
% 0.30/0.52 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.52 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.52 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % VEBT_internal.membermima.simps(4)
% 0.30/0.52 thf(fact_1638_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
% 0.30/0.52 ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT,X: nat] :
% 0.30/0.52 ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S3 ) @ X )
% 0.30/0.52 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.52 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.52 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % VEBT_internal.naive_member.simps(3)
% 0.30/0.52 thf(fact_1639_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
% 0.30/0.52 ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X: nat] :
% 0.30/0.52 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) @ X )
% 0.30/0.52 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.52 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.52 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % VEBT_internal.membermima.simps(5)
% 0.30/0.52 thf(fact_1640_divmod__step__eq,axiom,
% 0.30/0.52 ! [L2: num,R: nat,Q2: nat] :
% 0.30/0.52 ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R )
% 0.30/0.52 => ( ( unique5026877609467782581ep_nat @ L2 @ ( product_Pair_nat_nat @ Q2 @ R ) )
% 0.30/0.52 = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ one_one_nat ) @ ( minus_minus_nat @ R @ ( numeral_numeral_nat @ L2 ) ) ) ) )
% 0.30/0.52 & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R )
% 0.30/0.52 => ( ( unique5026877609467782581ep_nat @ L2 @ ( product_Pair_nat_nat @ Q2 @ R ) )
% 0.30/0.52 = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ R ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divmod_step_eq
% 0.30/0.52 thf(fact_1641_divmod__step__eq,axiom,
% 0.30/0.52 ! [L2: num,R: int,Q2: int] :
% 0.30/0.52 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R )
% 0.30/0.52 => ( ( unique5024387138958732305ep_int @ L2 @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.30/0.52 = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ one_one_int ) @ ( minus_minus_int @ R @ ( numeral_numeral_int @ L2 ) ) ) ) )
% 0.30/0.52 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R )
% 0.30/0.52 => ( ( unique5024387138958732305ep_int @ L2 @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.30/0.52 = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ R ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divmod_step_eq
% 0.30/0.52 thf(fact_1642_divmod__step__eq,axiom,
% 0.30/0.52 ! [L2: num,R: code_integer,Q2: code_integer] :
% 0.30/0.52 ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R )
% 0.30/0.52 => ( ( unique4921790084139445826nteger @ L2 @ ( produc1086072967326762835nteger @ Q2 @ R ) )
% 0.30/0.52 = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R @ ( numera6620942414471956472nteger @ L2 ) ) ) ) )
% 0.30/0.52 & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R )
% 0.30/0.52 => ( ( unique4921790084139445826nteger @ L2 @ ( produc1086072967326762835nteger @ Q2 @ R ) )
% 0.30/0.52 = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ R ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % divmod_step_eq
% 0.30/0.52 thf(fact_1643_buildup__nothing__in__leaf,axiom,
% 0.30/0.52 ! [N2: nat,X: nat] :
% 0.30/0.52 ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N2 ) @ X ) ).
% 0.30/0.52
% 0.30/0.52 % buildup_nothing_in_leaf
% 0.30/0.52 thf(fact_1644_vebt__succ_Oelims,axiom,
% 0.30/0.52 ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
% 0.30/0.52 ( ( ( vEBT_vebt_succ @ X @ Xa2 )
% 0.30/0.52 = Y )
% 0.30/0.52 => ( ! [Uu2: $o,B2: $o] :
% 0.30/0.52 ( ( X
% 0.30/0.52 = ( vEBT_Leaf @ Uu2 @ B2 ) )
% 0.30/0.52 => ( ( Xa2 = zero_zero_nat )
% 0.30/0.52 => ~ ( ( B2
% 0.30/0.52 => ( Y
% 0.30/0.52 = ( some_nat @ one_one_nat ) ) )
% 0.30/0.52 & ( ~ B2
% 0.30/0.52 => ( Y = none_nat ) ) ) ) )
% 0.30/0.52 => ( ( ? [Uv2: $o,Uw2: $o] :
% 0.30/0.52 ( X
% 0.30/0.52 = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
% 0.30/0.52 => ( ? [N4: nat] :
% 0.30/0.52 ( Xa2
% 0.30/0.52 = ( suc @ N4 ) )
% 0.30/0.52 => ( Y != none_nat ) ) )
% 0.30/0.52 => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.52 ( X
% 0.30/0.52 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
% 0.30/0.52 => ( Y != none_nat ) )
% 0.30/0.52 => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.30/0.52 ( X
% 0.30/0.52 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
% 0.30/0.52 => ( Y != none_nat ) )
% 0.30/0.52 => ( ( ? [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
% 0.30/0.52 ( X
% 0.30/0.52 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) )
% 0.30/0.52 => ( Y != none_nat ) )
% 0.30/0.52 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.52 ( ( X
% 0.30/0.52 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.30/0.52 => ~ ( ( ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.52 => ( Y
% 0.30/0.52 = ( some_nat @ Mi2 ) ) )
% 0.30/0.52 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.52 => ( Y
% 0.30/0.52 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.52 @ ( if_option_nat
% 0.30/0.52 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.52 != none_nat )
% 0.30/0.52 & ( 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 @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.30/0.52 @ ( 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 @ TreeList3 @ ( 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 ) ) ) ) ) )
% 0.30/0.52 @ ( if_option_nat
% 0.30/0.52 @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.52 = none_nat )
% 0.30/0.52 @ none_nat
% 0.30/0.52 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.30/0.52 @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % vebt_succ.elims
% 0.30/0.52 thf(fact_1645_vebt__pred_Oelims,axiom,
% 0.30/0.52 ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
% 0.30/0.52 ( ( ( vEBT_vebt_pred @ X @ Xa2 )
% 0.30/0.52 = Y )
% 0.30/0.52 => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.30/0.52 ( X
% 0.30/0.52 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.52 => ( ( Xa2 = zero_zero_nat )
% 0.30/0.52 => ( Y != none_nat ) ) )
% 0.30/0.52 => ( ! [A3: $o] :
% 0.30/0.52 ( ? [Uw2: $o] :
% 0.30/0.52 ( X
% 0.30/0.52 = ( vEBT_Leaf @ A3 @ Uw2 ) )
% 0.30/0.52 => ( ( Xa2
% 0.30/0.52 = ( suc @ zero_zero_nat ) )
% 0.30/0.52 => ~ ( ( A3
% 0.30/0.52 => ( Y
% 0.30/0.52 = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.52 & ( ~ A3
% 0.30/0.52 => ( Y = none_nat ) ) ) ) )
% 0.30/0.52 => ( ! [A3: $o,B2: $o] :
% 0.30/0.52 ( ( X
% 0.30/0.52 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.30/0.52 => ( ? [Va2: nat] :
% 0.30/0.52 ( Xa2
% 0.30/0.52 = ( suc @ ( suc @ Va2 ) ) )
% 0.30/0.52 => ~ ( ( B2
% 0.30/0.52 => ( Y
% 0.30/0.52 = ( some_nat @ one_one_nat ) ) )
% 0.30/0.52 & ( ~ B2
% 0.30/0.52 => ( ( A3
% 0.30/0.52 => ( Y
% 0.30/0.52 = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.52 & ( ~ A3
% 0.30/0.52 => ( Y = none_nat ) ) ) ) ) ) )
% 0.30/0.52 => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
% 0.30/0.52 ( X
% 0.30/0.52 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
% 0.30/0.52 => ( Y != none_nat ) )
% 0.30/0.52 => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] :
% 0.30/0.52 ( X
% 0.30/0.52 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) )
% 0.30/0.52 => ( Y != none_nat ) )
% 0.30/0.52 => ( ( ? [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
% 0.30/0.52 ( X
% 0.30/0.52 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) )
% 0.30/0.52 => ( Y != none_nat ) )
% 0.30/0.52 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.52 ( ( X
% 0.30/0.52 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.30/0.52 => ~ ( ( ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.52 => ( Y
% 0.30/0.52 = ( some_nat @ Ma2 ) ) )
% 0.30/0.52 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.52 => ( Y
% 0.30/0.52 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.52 @ ( if_option_nat
% 0.30/0.52 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.52 != none_nat )
% 0.30/0.52 & ( 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 @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.30/0.52 @ ( 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 @ TreeList3 @ ( 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 ) ) ) ) ) )
% 0.30/0.52 @ ( if_option_nat
% 0.30/0.52 @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.52 = none_nat )
% 0.30/0.52 @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
% 0.30/0.52 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.30/0.52 @ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % vebt_pred.elims
% 0.30/0.52 thf(fact_1646_discrete,axiom,
% 0.30/0.52 ( ord_less_nat
% 0.30/0.52 = ( ^ [A5: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A5 @ one_one_nat ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % discrete
% 0.30/0.52 thf(fact_1647_discrete,axiom,
% 0.30/0.52 ( ord_less_int
% 0.30/0.52 = ( ^ [A5: int] : ( ord_less_eq_int @ ( plus_plus_int @ A5 @ one_one_int ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % discrete
% 0.30/0.52 thf(fact_1648_vebt__delete_Oelims,axiom,
% 0.30/0.52 ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
% 0.30/0.52 ( ( ( vEBT_vebt_delete @ X @ Xa2 )
% 0.30/0.52 = Y )
% 0.30/0.52 => ( ! [A3: $o,B2: $o] :
% 0.30/0.52 ( ( X
% 0.30/0.52 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.30/0.52 => ( ( Xa2 = zero_zero_nat )
% 0.30/0.52 => ( Y
% 0.30/0.52 != ( vEBT_Leaf @ $false @ B2 ) ) ) )
% 0.30/0.52 => ( ! [A3: $o] :
% 0.30/0.52 ( ? [B2: $o] :
% 0.30/0.52 ( X
% 0.30/0.52 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.30/0.52 => ( ( Xa2
% 0.30/0.52 = ( suc @ zero_zero_nat ) )
% 0.30/0.52 => ( Y
% 0.30/0.52 != ( vEBT_Leaf @ A3 @ $false ) ) ) )
% 0.30/0.52 => ( ! [A3: $o,B2: $o] :
% 0.30/0.52 ( ( X
% 0.30/0.52 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.30/0.52 => ( ? [N4: nat] :
% 0.30/0.52 ( Xa2
% 0.30/0.52 = ( suc @ ( suc @ N4 ) ) )
% 0.30/0.52 => ( Y
% 0.30/0.52 != ( vEBT_Leaf @ A3 @ B2 ) ) ) )
% 0.30/0.52 => ( ! [Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.52 ( ( X
% 0.30/0.52 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.30/0.52 => ( Y
% 0.30/0.52 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) ) )
% 0.30/0.52 => ( ! [Mi2: nat,Ma2: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT] :
% 0.30/0.52 ( ( X
% 0.30/0.52 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) )
% 0.30/0.52 => ( Y
% 0.30/0.52 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) ) )
% 0.30/0.52 => ( ! [Mi2: nat,Ma2: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT] :
% 0.30/0.52 ( ( X
% 0.30/0.52 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) )
% 0.30/0.52 => ( Y
% 0.30/0.52 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) ) )
% 0.30/0.52 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.52 ( ( X
% 0.30/0.52 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.30/0.52 => ~ ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.52 | ( ord_less_nat @ Ma2 @ Xa2 ) )
% 0.30/0.52 => ( Y
% 0.30/0.52 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) )
% 0.30/0.52 & ( ~ ( ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.52 | ( ord_less_nat @ Ma2 @ Xa2 ) )
% 0.30/0.52 => ( ( ( ( Xa2 = Mi2 )
% 0.30/0.52 & ( Xa2 = Ma2 ) )
% 0.30/0.52 => ( Y
% 0.30/0.52 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) )
% 0.30/0.52 & ( ~ ( ( Xa2 = Mi2 )
% 0.30/0.52 & ( Xa2 = Ma2 ) )
% 0.30/0.52 => ( Y
% 0.30/0.52 = ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.52 @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.52 @ ( vEBT_Node
% 0.30/0.52 @ ( some_P7363390416028606310at_nat
% 0.30/0.52 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
% 0.30/0.52 @ ( if_nat
% 0.30/0.52 @ ( ( ( Xa2 = Mi2 )
% 0.30/0.52 => ( ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
% 0.30/0.52 = Ma2 ) )
% 0.30/0.52 & ( ( Xa2 != Mi2 )
% 0.30/0.52 => ( Xa2 = Ma2 ) ) )
% 0.30/0.52 @ ( if_nat
% 0.30/0.52 @ ( ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.52 = none_nat )
% 0.30/0.52 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
% 0.30/0.52 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
% 0.30/0.52 @ Ma2 ) ) )
% 0.30/0.52 @ ( suc @ ( suc @ Va2 ) )
% 0.30/0.52 @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.52 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.52 @ ( vEBT_Node
% 0.30/0.52 @ ( some_P7363390416028606310at_nat
% 0.30/0.52 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
% 0.30/0.52 @ ( if_nat
% 0.30/0.52 @ ( ( ( Xa2 = Mi2 )
% 0.30/0.52 => ( ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
% 0.30/0.52 = Ma2 ) )
% 0.30/0.52 & ( ( Xa2 != Mi2 )
% 0.30/0.52 => ( Xa2 = Ma2 ) ) )
% 0.30/0.52 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.30/0.52 @ Ma2 ) ) )
% 0.30/0.52 @ ( suc @ ( suc @ Va2 ) )
% 0.30/0.52 @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.52 @ Summary2 ) )
% 0.30/0.52 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % vebt_delete.elims
% 0.30/0.52 thf(fact_1649_low__def,axiom,
% 0.30/0.52 ( vEBT_VEBT_low
% 0.30/0.52 = ( ^ [X2: nat,N: nat] : ( modulo_modulo_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % low_def
% 0.30/0.52 thf(fact_1650_valid__0__not,axiom,
% 0.30/0.52 ! [T: vEBT_VEBT] :
% 0.30/0.52 ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % valid_0_not
% 0.30/0.52 thf(fact_1651_valid__tree__deg__neq__0,axiom,
% 0.30/0.52 ! [T: vEBT_VEBT] :
% 0.30/0.52 ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % valid_tree_deg_neq_0
% 0.30/0.52 thf(fact_1652_buildup__nothing__in__min__max,axiom,
% 0.30/0.52 ! [N2: nat,X: nat] :
% 0.30/0.52 ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N2 ) @ X ) ).
% 0.30/0.52
% 0.30/0.52 % buildup_nothing_in_min_max
% 0.30/0.52 thf(fact_1653_deg__not__0,axiom,
% 0.30/0.52 ! [T: vEBT_VEBT,N2: nat] :
% 0.30/0.52 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.52 => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 0.30/0.52
% 0.30/0.52 % deg_not_0
% 0.30/0.52 thf(fact_1654_Leaf__0__not,axiom,
% 0.30/0.52 ! [A: $o,B: $o] :
% 0.30/0.52 ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % Leaf_0_not
% 0.30/0.52 thf(fact_1655_deg1Leaf,axiom,
% 0.30/0.52 ! [T: vEBT_VEBT] :
% 0.30/0.52 ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 0.30/0.52 = ( ? [A5: $o,B5: $o] :
% 0.30/0.52 ( T
% 0.30/0.52 = ( vEBT_Leaf @ A5 @ B5 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % deg1Leaf
% 0.30/0.52 thf(fact_1656_deg__1__Leaf,axiom,
% 0.30/0.52 ! [T: vEBT_VEBT] :
% 0.30/0.52 ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 0.30/0.52 => ? [A3: $o,B2: $o] :
% 0.30/0.52 ( T
% 0.30/0.52 = ( vEBT_Leaf @ A3 @ B2 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % deg_1_Leaf
% 0.30/0.52 thf(fact_1657_deg__1__Leafy,axiom,
% 0.30/0.52 ! [T: vEBT_VEBT,N2: nat] :
% 0.30/0.52 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.52 => ( ( N2 = one_one_nat )
% 0.30/0.52 => ? [A3: $o,B2: $o] :
% 0.30/0.52 ( T
% 0.30/0.52 = ( vEBT_Leaf @ A3 @ B2 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % deg_1_Leafy
% 0.30/0.52 thf(fact_1658_both__member__options__def,axiom,
% 0.30/0.52 ( vEBT_V8194947554948674370ptions
% 0.30/0.52 = ( ^ [T2: vEBT_VEBT,X2: nat] :
% 0.30/0.52 ( ( vEBT_V5719532721284313246member @ T2 @ X2 )
% 0.30/0.52 | ( vEBT_VEBT_membermima @ T2 @ X2 ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % both_member_options_def
% 0.30/0.52 thf(fact_1659_buildup__gives__valid,axiom,
% 0.30/0.52 ! [N2: nat] :
% 0.30/0.52 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.52 => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N2 ) @ N2 ) ) ).
% 0.30/0.52
% 0.30/0.52 % buildup_gives_valid
% 0.30/0.52 thf(fact_1660_zdiv__numeral__Bit0,axiom,
% 0.30/0.52 ! [V: num,W: num] :
% 0.30/0.52 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.30/0.52 = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % zdiv_numeral_Bit0
% 0.30/0.52 thf(fact_1661_mod__mod__trivial,axiom,
% 0.30/0.52 ! [A: nat,B: nat] :
% 0.30/0.52 ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 0.30/0.52 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % mod_mod_trivial
% 0.30/0.52 thf(fact_1662_mod__mod__trivial,axiom,
% 0.30/0.52 ! [A: int,B: int] :
% 0.30/0.52 ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 0.30/0.52 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % mod_mod_trivial
% 0.30/0.52 thf(fact_1663_mod__mod__trivial,axiom,
% 0.30/0.52 ! [A: code_integer,B: code_integer] :
% 0.30/0.52 ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 0.30/0.52 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.30/0.52
% 0.30/0.52 % mod_mod_trivial
% 0.30/0.52 thf(fact_1664_member__valid__both__member__options,axiom,
% 0.30/0.52 ! [Tree: vEBT_VEBT,N2: nat,X: nat] :
% 0.30/0.52 ( ( vEBT_invar_vebt @ Tree @ N2 )
% 0.30/0.52 => ( ( vEBT_vebt_member @ Tree @ X )
% 0.30/0.52 => ( ( vEBT_V5719532721284313246member @ Tree @ X )
% 0.30/0.52 | ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % member_valid_both_member_options
% 0.30/0.52 thf(fact_1665_VEBT_Oinject_I2_J,axiom,
% 0.30/0.52 ! [X21: $o,X222: $o,Y21: $o,Y222: $o] :
% 0.30/0.52 ( ( ( vEBT_Leaf @ X21 @ X222 )
% 0.30/0.52 = ( vEBT_Leaf @ Y21 @ Y222 ) )
% 0.30/0.52 = ( ( X21 = Y21 )
% 0.30/0.52 & ( X222 = Y222 ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % VEBT.inject(2)
% 0.30/0.52 thf(fact_1666_le__zero__eq,axiom,
% 0.30/0.52 ! [N2: nat] :
% 0.30/0.52 ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
% 0.30/0.52 = ( N2 = zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % le_zero_eq
% 0.30/0.52 thf(fact_1667_not__gr__zero,axiom,
% 0.30/0.52 ! [N2: nat] :
% 0.30/0.52 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
% 0.30/0.52 = ( N2 = zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % not_gr_zero
% 0.30/0.52 thf(fact_1668_mult__cancel__right,axiom,
% 0.30/0.52 ! [A: complex,C: complex,B: complex] :
% 0.30/0.52 ( ( ( times_times_complex @ A @ C )
% 0.30/0.52 = ( times_times_complex @ B @ C ) )
% 0.30/0.52 = ( ( C = zero_zero_complex )
% 0.30/0.52 | ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_cancel_right
% 0.30/0.52 thf(fact_1669_mult__cancel__right,axiom,
% 0.30/0.52 ! [A: real,C: real,B: real] :
% 0.30/0.52 ( ( ( times_times_real @ A @ C )
% 0.30/0.52 = ( times_times_real @ B @ C ) )
% 0.30/0.52 = ( ( C = zero_zero_real )
% 0.30/0.52 | ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_cancel_right
% 0.30/0.52 thf(fact_1670_mult__cancel__right,axiom,
% 0.30/0.52 ! [A: rat,C: rat,B: rat] :
% 0.30/0.52 ( ( ( times_times_rat @ A @ C )
% 0.30/0.52 = ( times_times_rat @ B @ C ) )
% 0.30/0.52 = ( ( C = zero_zero_rat )
% 0.30/0.52 | ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_cancel_right
% 0.30/0.52 thf(fact_1671_mult__cancel__right,axiom,
% 0.30/0.52 ! [A: nat,C: nat,B: nat] :
% 0.30/0.52 ( ( ( times_times_nat @ A @ C )
% 0.30/0.52 = ( times_times_nat @ B @ C ) )
% 0.30/0.52 = ( ( C = zero_zero_nat )
% 0.30/0.52 | ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_cancel_right
% 0.30/0.52 thf(fact_1672_mult__cancel__right,axiom,
% 0.30/0.52 ! [A: int,C: int,B: int] :
% 0.30/0.52 ( ( ( times_times_int @ A @ C )
% 0.30/0.52 = ( times_times_int @ B @ C ) )
% 0.30/0.52 = ( ( C = zero_zero_int )
% 0.30/0.52 | ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_cancel_right
% 0.30/0.52 thf(fact_1673_mult__cancel__left,axiom,
% 0.30/0.52 ! [C: complex,A: complex,B: complex] :
% 0.30/0.52 ( ( ( times_times_complex @ C @ A )
% 0.30/0.52 = ( times_times_complex @ C @ B ) )
% 0.30/0.52 = ( ( C = zero_zero_complex )
% 0.30/0.52 | ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_cancel_left
% 0.30/0.52 thf(fact_1674_mult__cancel__left,axiom,
% 0.30/0.52 ! [C: real,A: real,B: real] :
% 0.30/0.52 ( ( ( times_times_real @ C @ A )
% 0.30/0.52 = ( times_times_real @ C @ B ) )
% 0.30/0.52 = ( ( C = zero_zero_real )
% 0.30/0.52 | ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_cancel_left
% 0.30/0.52 thf(fact_1675_mult__cancel__left,axiom,
% 0.30/0.52 ! [C: rat,A: rat,B: rat] :
% 0.30/0.52 ( ( ( times_times_rat @ C @ A )
% 0.30/0.52 = ( times_times_rat @ C @ B ) )
% 0.30/0.52 = ( ( C = zero_zero_rat )
% 0.30/0.52 | ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_cancel_left
% 0.30/0.52 thf(fact_1676_mult__cancel__left,axiom,
% 0.30/0.52 ! [C: nat,A: nat,B: nat] :
% 0.30/0.52 ( ( ( times_times_nat @ C @ A )
% 0.30/0.52 = ( times_times_nat @ C @ B ) )
% 0.30/0.52 = ( ( C = zero_zero_nat )
% 0.30/0.52 | ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_cancel_left
% 0.30/0.52 thf(fact_1677_mult__cancel__left,axiom,
% 0.30/0.52 ! [C: int,A: int,B: int] :
% 0.30/0.52 ( ( ( times_times_int @ C @ A )
% 0.30/0.52 = ( times_times_int @ C @ B ) )
% 0.30/0.52 = ( ( C = zero_zero_int )
% 0.30/0.52 | ( A = B ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_cancel_left
% 0.30/0.52 thf(fact_1678_mult__eq__0__iff,axiom,
% 0.30/0.52 ! [A: complex,B: complex] :
% 0.30/0.52 ( ( ( times_times_complex @ A @ B )
% 0.30/0.52 = zero_zero_complex )
% 0.30/0.52 = ( ( A = zero_zero_complex )
% 0.30/0.52 | ( B = zero_zero_complex ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_eq_0_iff
% 0.30/0.52 thf(fact_1679_mult__eq__0__iff,axiom,
% 0.30/0.52 ! [A: real,B: real] :
% 0.30/0.52 ( ( ( times_times_real @ A @ B )
% 0.30/0.52 = zero_zero_real )
% 0.30/0.52 = ( ( A = zero_zero_real )
% 0.30/0.52 | ( B = zero_zero_real ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_eq_0_iff
% 0.30/0.52 thf(fact_1680_mult__eq__0__iff,axiom,
% 0.30/0.52 ! [A: rat,B: rat] :
% 0.30/0.52 ( ( ( times_times_rat @ A @ B )
% 0.30/0.52 = zero_zero_rat )
% 0.30/0.52 = ( ( A = zero_zero_rat )
% 0.30/0.52 | ( B = zero_zero_rat ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_eq_0_iff
% 0.30/0.52 thf(fact_1681_mult__eq__0__iff,axiom,
% 0.30/0.52 ! [A: nat,B: nat] :
% 0.30/0.52 ( ( ( times_times_nat @ A @ B )
% 0.30/0.52 = zero_zero_nat )
% 0.30/0.52 = ( ( A = zero_zero_nat )
% 0.30/0.52 | ( B = zero_zero_nat ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_eq_0_iff
% 0.30/0.52 thf(fact_1682_mult__eq__0__iff,axiom,
% 0.30/0.52 ! [A: int,B: int] :
% 0.30/0.52 ( ( ( times_times_int @ A @ B )
% 0.30/0.52 = zero_zero_int )
% 0.30/0.52 = ( ( A = zero_zero_int )
% 0.30/0.52 | ( B = zero_zero_int ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % mult_eq_0_iff
% 0.30/0.52 thf(fact_1683_mult__zero__right,axiom,
% 0.30/0.52 ! [A: complex] :
% 0.30/0.52 ( ( times_times_complex @ A @ zero_zero_complex )
% 0.30/0.52 = zero_zero_complex ) ).
% 0.30/0.52
% 0.30/0.52 % mult_zero_right
% 0.30/0.52 thf(fact_1684_mult__zero__right,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( times_times_real @ A @ zero_zero_real )
% 0.30/0.52 = zero_zero_real ) ).
% 0.30/0.52
% 0.30/0.52 % mult_zero_right
% 0.30/0.52 thf(fact_1685_mult__zero__right,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( times_times_rat @ A @ zero_zero_rat )
% 0.30/0.52 = zero_zero_rat ) ).
% 0.30/0.52
% 0.30/0.52 % mult_zero_right
% 0.30/0.52 thf(fact_1686_mult__zero__right,axiom,
% 0.30/0.52 ! [A: nat] :
% 0.30/0.52 ( ( times_times_nat @ A @ zero_zero_nat )
% 0.30/0.52 = zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % mult_zero_right
% 0.30/0.52 thf(fact_1687_mult__zero__right,axiom,
% 0.30/0.52 ! [A: int] :
% 0.30/0.52 ( ( times_times_int @ A @ zero_zero_int )
% 0.30/0.52 = zero_zero_int ) ).
% 0.30/0.52
% 0.30/0.52 % mult_zero_right
% 0.30/0.52 thf(fact_1688_mult__zero__left,axiom,
% 0.30/0.52 ! [A: complex] :
% 0.30/0.52 ( ( times_times_complex @ zero_zero_complex @ A )
% 0.30/0.52 = zero_zero_complex ) ).
% 0.30/0.52
% 0.30/0.52 % mult_zero_left
% 0.30/0.52 thf(fact_1689_mult__zero__left,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( times_times_real @ zero_zero_real @ A )
% 0.30/0.52 = zero_zero_real ) ).
% 0.30/0.52
% 0.30/0.52 % mult_zero_left
% 0.30/0.52 thf(fact_1690_mult__zero__left,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( times_times_rat @ zero_zero_rat @ A )
% 0.30/0.52 = zero_zero_rat ) ).
% 0.30/0.52
% 0.30/0.52 % mult_zero_left
% 0.30/0.52 thf(fact_1691_mult__zero__left,axiom,
% 0.30/0.52 ! [A: nat] :
% 0.30/0.52 ( ( times_times_nat @ zero_zero_nat @ A )
% 0.30/0.52 = zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % mult_zero_left
% 0.30/0.52 thf(fact_1692_mult__zero__left,axiom,
% 0.30/0.52 ! [A: int] :
% 0.30/0.52 ( ( times_times_int @ zero_zero_int @ A )
% 0.30/0.52 = zero_zero_int ) ).
% 0.30/0.52
% 0.30/0.52 % mult_zero_left
% 0.30/0.52 thf(fact_1693_add__0,axiom,
% 0.30/0.52 ! [A: complex] :
% 0.30/0.52 ( ( plus_plus_complex @ zero_zero_complex @ A )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % add_0
% 0.30/0.52 thf(fact_1694_add__0,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % add_0
% 0.30/0.52 thf(fact_1695_add__0,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % add_0
% 0.30/0.52 thf(fact_1696_add__0,axiom,
% 0.30/0.52 ! [A: nat] :
% 0.30/0.52 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % add_0
% 0.30/0.52 thf(fact_1697_add__0,axiom,
% 0.30/0.52 ! [A: int] :
% 0.30/0.52 ( ( plus_plus_int @ zero_zero_int @ A )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % add_0
% 0.30/0.52 thf(fact_1698_zero__eq__add__iff__both__eq__0,axiom,
% 0.30/0.52 ! [X: nat,Y: nat] :
% 0.30/0.52 ( ( zero_zero_nat
% 0.30/0.52 = ( plus_plus_nat @ X @ Y ) )
% 0.30/0.52 = ( ( X = zero_zero_nat )
% 0.30/0.52 & ( Y = zero_zero_nat ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % zero_eq_add_iff_both_eq_0
% 0.30/0.52 thf(fact_1699_add__eq__0__iff__both__eq__0,axiom,
% 0.30/0.52 ! [X: nat,Y: nat] :
% 0.30/0.52 ( ( ( plus_plus_nat @ X @ Y )
% 0.30/0.52 = zero_zero_nat )
% 0.30/0.52 = ( ( X = zero_zero_nat )
% 0.30/0.52 & ( Y = zero_zero_nat ) ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_eq_0_iff_both_eq_0
% 0.30/0.52 thf(fact_1700_add__cancel__right__right,axiom,
% 0.30/0.52 ! [A: complex,B: complex] :
% 0.30/0.52 ( ( A
% 0.30/0.52 = ( plus_plus_complex @ A @ B ) )
% 0.30/0.52 = ( B = zero_zero_complex ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_cancel_right_right
% 0.30/0.52 thf(fact_1701_add__cancel__right__right,axiom,
% 0.30/0.52 ! [A: real,B: real] :
% 0.30/0.52 ( ( A
% 0.30/0.52 = ( plus_plus_real @ A @ B ) )
% 0.30/0.52 = ( B = zero_zero_real ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_cancel_right_right
% 0.30/0.52 thf(fact_1702_add__cancel__right__right,axiom,
% 0.30/0.52 ! [A: rat,B: rat] :
% 0.30/0.52 ( ( A
% 0.30/0.52 = ( plus_plus_rat @ A @ B ) )
% 0.30/0.52 = ( B = zero_zero_rat ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_cancel_right_right
% 0.30/0.52 thf(fact_1703_add__cancel__right__right,axiom,
% 0.30/0.52 ! [A: nat,B: nat] :
% 0.30/0.52 ( ( A
% 0.30/0.52 = ( plus_plus_nat @ A @ B ) )
% 0.30/0.52 = ( B = zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_cancel_right_right
% 0.30/0.52 thf(fact_1704_add__cancel__right__right,axiom,
% 0.30/0.52 ! [A: int,B: int] :
% 0.30/0.52 ( ( A
% 0.30/0.52 = ( plus_plus_int @ A @ B ) )
% 0.30/0.52 = ( B = zero_zero_int ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_cancel_right_right
% 0.30/0.52 thf(fact_1705_add__cancel__right__left,axiom,
% 0.30/0.52 ! [A: complex,B: complex] :
% 0.30/0.52 ( ( A
% 0.30/0.52 = ( plus_plus_complex @ B @ A ) )
% 0.30/0.52 = ( B = zero_zero_complex ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_cancel_right_left
% 0.30/0.52 thf(fact_1706_add__cancel__right__left,axiom,
% 0.30/0.52 ! [A: real,B: real] :
% 0.30/0.52 ( ( A
% 0.30/0.52 = ( plus_plus_real @ B @ A ) )
% 0.30/0.52 = ( B = zero_zero_real ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_cancel_right_left
% 0.30/0.52 thf(fact_1707_add__cancel__right__left,axiom,
% 0.30/0.52 ! [A: rat,B: rat] :
% 0.30/0.52 ( ( A
% 0.30/0.52 = ( plus_plus_rat @ B @ A ) )
% 0.30/0.52 = ( B = zero_zero_rat ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_cancel_right_left
% 0.30/0.52 thf(fact_1708_add__cancel__right__left,axiom,
% 0.30/0.52 ! [A: nat,B: nat] :
% 0.30/0.52 ( ( A
% 0.30/0.52 = ( plus_plus_nat @ B @ A ) )
% 0.30/0.52 = ( B = zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_cancel_right_left
% 0.30/0.52 thf(fact_1709_add__cancel__right__left,axiom,
% 0.30/0.52 ! [A: int,B: int] :
% 0.30/0.52 ( ( A
% 0.30/0.52 = ( plus_plus_int @ B @ A ) )
% 0.30/0.52 = ( B = zero_zero_int ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_cancel_right_left
% 0.30/0.52 thf(fact_1710_add__cancel__left__right,axiom,
% 0.30/0.52 ! [A: complex,B: complex] :
% 0.30/0.52 ( ( ( plus_plus_complex @ A @ B )
% 0.30/0.52 = A )
% 0.30/0.52 = ( B = zero_zero_complex ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_cancel_left_right
% 0.30/0.52 thf(fact_1711_add__cancel__left__right,axiom,
% 0.30/0.52 ! [A: real,B: real] :
% 0.30/0.52 ( ( ( plus_plus_real @ A @ B )
% 0.30/0.52 = A )
% 0.30/0.52 = ( B = zero_zero_real ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_cancel_left_right
% 0.30/0.52 thf(fact_1712_add__cancel__left__right,axiom,
% 0.30/0.52 ! [A: rat,B: rat] :
% 0.30/0.52 ( ( ( plus_plus_rat @ A @ B )
% 0.30/0.52 = A )
% 0.30/0.52 = ( B = zero_zero_rat ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_cancel_left_right
% 0.30/0.52 thf(fact_1713_add__cancel__left__right,axiom,
% 0.30/0.52 ! [A: nat,B: nat] :
% 0.30/0.52 ( ( ( plus_plus_nat @ A @ B )
% 0.30/0.52 = A )
% 0.30/0.52 = ( B = zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_cancel_left_right
% 0.30/0.52 thf(fact_1714_add__cancel__left__right,axiom,
% 0.30/0.52 ! [A: int,B: int] :
% 0.30/0.52 ( ( ( plus_plus_int @ A @ B )
% 0.30/0.52 = A )
% 0.30/0.52 = ( B = zero_zero_int ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_cancel_left_right
% 0.30/0.52 thf(fact_1715_add__cancel__left__left,axiom,
% 0.30/0.52 ! [B: complex,A: complex] :
% 0.30/0.52 ( ( ( plus_plus_complex @ B @ A )
% 0.30/0.52 = A )
% 0.30/0.52 = ( B = zero_zero_complex ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_cancel_left_left
% 0.30/0.52 thf(fact_1716_add__cancel__left__left,axiom,
% 0.30/0.52 ! [B: real,A: real] :
% 0.30/0.52 ( ( ( plus_plus_real @ B @ A )
% 0.30/0.52 = A )
% 0.30/0.52 = ( B = zero_zero_real ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_cancel_left_left
% 0.30/0.52 thf(fact_1717_add__cancel__left__left,axiom,
% 0.30/0.52 ! [B: rat,A: rat] :
% 0.30/0.52 ( ( ( plus_plus_rat @ B @ A )
% 0.30/0.52 = A )
% 0.30/0.52 = ( B = zero_zero_rat ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_cancel_left_left
% 0.30/0.52 thf(fact_1718_add__cancel__left__left,axiom,
% 0.30/0.52 ! [B: nat,A: nat] :
% 0.30/0.52 ( ( ( plus_plus_nat @ B @ A )
% 0.30/0.52 = A )
% 0.30/0.52 = ( B = zero_zero_nat ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_cancel_left_left
% 0.30/0.52 thf(fact_1719_add__cancel__left__left,axiom,
% 0.30/0.52 ! [B: int,A: int] :
% 0.30/0.52 ( ( ( plus_plus_int @ B @ A )
% 0.30/0.52 = A )
% 0.30/0.52 = ( B = zero_zero_int ) ) ).
% 0.30/0.52
% 0.30/0.52 % add_cancel_left_left
% 0.30/0.52 thf(fact_1720_double__zero__sym,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( zero_zero_real
% 0.30/0.52 = ( plus_plus_real @ A @ A ) )
% 0.30/0.52 = ( A = zero_zero_real ) ) ).
% 0.30/0.52
% 0.30/0.52 % double_zero_sym
% 0.30/0.52 thf(fact_1721_double__zero__sym,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( zero_zero_rat
% 0.30/0.52 = ( plus_plus_rat @ A @ A ) )
% 0.30/0.52 = ( A = zero_zero_rat ) ) ).
% 0.30/0.52
% 0.30/0.52 % double_zero_sym
% 0.30/0.52 thf(fact_1722_double__zero__sym,axiom,
% 0.30/0.52 ! [A: int] :
% 0.30/0.52 ( ( zero_zero_int
% 0.30/0.52 = ( plus_plus_int @ A @ A ) )
% 0.30/0.52 = ( A = zero_zero_int ) ) ).
% 0.30/0.52
% 0.30/0.52 % double_zero_sym
% 0.30/0.52 thf(fact_1723_add_Oright__neutral,axiom,
% 0.30/0.52 ! [A: complex] :
% 0.30/0.52 ( ( plus_plus_complex @ A @ zero_zero_complex )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % add.right_neutral
% 0.30/0.52 thf(fact_1724_add_Oright__neutral,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( plus_plus_real @ A @ zero_zero_real )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % add.right_neutral
% 0.30/0.52 thf(fact_1725_add_Oright__neutral,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % add.right_neutral
% 0.30/0.52 thf(fact_1726_add_Oright__neutral,axiom,
% 0.30/0.52 ! [A: nat] :
% 0.30/0.52 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % add.right_neutral
% 0.30/0.52 thf(fact_1727_add_Oright__neutral,axiom,
% 0.30/0.52 ! [A: int] :
% 0.30/0.52 ( ( plus_plus_int @ A @ zero_zero_int )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % add.right_neutral
% 0.30/0.52 thf(fact_1728_diff__self,axiom,
% 0.30/0.52 ! [A: complex] :
% 0.30/0.52 ( ( minus_minus_complex @ A @ A )
% 0.30/0.52 = zero_zero_complex ) ).
% 0.30/0.52
% 0.30/0.52 % diff_self
% 0.30/0.52 thf(fact_1729_diff__self,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( minus_minus_real @ A @ A )
% 0.30/0.52 = zero_zero_real ) ).
% 0.30/0.52
% 0.30/0.52 % diff_self
% 0.30/0.52 thf(fact_1730_diff__self,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( minus_minus_rat @ A @ A )
% 0.30/0.52 = zero_zero_rat ) ).
% 0.30/0.52
% 0.30/0.52 % diff_self
% 0.30/0.52 thf(fact_1731_diff__self,axiom,
% 0.30/0.52 ! [A: int] :
% 0.30/0.52 ( ( minus_minus_int @ A @ A )
% 0.30/0.52 = zero_zero_int ) ).
% 0.30/0.52
% 0.30/0.52 % diff_self
% 0.30/0.52 thf(fact_1732_diff__0__right,axiom,
% 0.30/0.52 ! [A: complex] :
% 0.30/0.52 ( ( minus_minus_complex @ A @ zero_zero_complex )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % diff_0_right
% 0.30/0.52 thf(fact_1733_diff__0__right,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( minus_minus_real @ A @ zero_zero_real )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % diff_0_right
% 0.30/0.52 thf(fact_1734_diff__0__right,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( minus_minus_rat @ A @ zero_zero_rat )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % diff_0_right
% 0.30/0.52 thf(fact_1735_diff__0__right,axiom,
% 0.30/0.52 ! [A: int] :
% 0.30/0.52 ( ( minus_minus_int @ A @ zero_zero_int )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % diff_0_right
% 0.30/0.52 thf(fact_1736_zero__diff,axiom,
% 0.30/0.52 ! [A: nat] :
% 0.30/0.52 ( ( minus_minus_nat @ zero_zero_nat @ A )
% 0.30/0.52 = zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % zero_diff
% 0.30/0.52 thf(fact_1737_diff__zero,axiom,
% 0.30/0.52 ! [A: complex] :
% 0.30/0.52 ( ( minus_minus_complex @ A @ zero_zero_complex )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % diff_zero
% 0.30/0.52 thf(fact_1738_diff__zero,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( minus_minus_real @ A @ zero_zero_real )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % diff_zero
% 0.30/0.52 thf(fact_1739_diff__zero,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( minus_minus_rat @ A @ zero_zero_rat )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % diff_zero
% 0.30/0.52 thf(fact_1740_diff__zero,axiom,
% 0.30/0.52 ! [A: nat] :
% 0.30/0.52 ( ( minus_minus_nat @ A @ zero_zero_nat )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % diff_zero
% 0.30/0.52 thf(fact_1741_diff__zero,axiom,
% 0.30/0.52 ! [A: int] :
% 0.30/0.52 ( ( minus_minus_int @ A @ zero_zero_int )
% 0.30/0.52 = A ) ).
% 0.30/0.52
% 0.30/0.52 % diff_zero
% 0.30/0.52 thf(fact_1742_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.30/0.52 ! [A: complex] :
% 0.30/0.52 ( ( minus_minus_complex @ A @ A )
% 0.30/0.52 = zero_zero_complex ) ).
% 0.30/0.52
% 0.30/0.52 % cancel_comm_monoid_add_class.diff_cancel
% 0.30/0.52 thf(fact_1743_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( minus_minus_real @ A @ A )
% 0.30/0.52 = zero_zero_real ) ).
% 0.30/0.52
% 0.30/0.52 % cancel_comm_monoid_add_class.diff_cancel
% 0.30/0.52 thf(fact_1744_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( minus_minus_rat @ A @ A )
% 0.30/0.52 = zero_zero_rat ) ).
% 0.30/0.52
% 0.30/0.52 % cancel_comm_monoid_add_class.diff_cancel
% 0.30/0.52 thf(fact_1745_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.30/0.52 ! [A: nat] :
% 0.30/0.52 ( ( minus_minus_nat @ A @ A )
% 0.30/0.52 = zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % cancel_comm_monoid_add_class.diff_cancel
% 0.30/0.52 thf(fact_1746_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.30/0.52 ! [A: int] :
% 0.30/0.52 ( ( minus_minus_int @ A @ A )
% 0.30/0.52 = zero_zero_int ) ).
% 0.30/0.52
% 0.30/0.52 % cancel_comm_monoid_add_class.diff_cancel
% 0.30/0.52 thf(fact_1747_bits__div__by__0,axiom,
% 0.30/0.52 ! [A: nat] :
% 0.30/0.52 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 0.30/0.52 = zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % bits_div_by_0
% 0.30/0.52 thf(fact_1748_bits__div__by__0,axiom,
% 0.30/0.52 ! [A: int] :
% 0.30/0.52 ( ( divide_divide_int @ A @ zero_zero_int )
% 0.30/0.52 = zero_zero_int ) ).
% 0.30/0.52
% 0.30/0.52 % bits_div_by_0
% 0.30/0.52 thf(fact_1749_bits__div__0,axiom,
% 0.30/0.52 ! [A: nat] :
% 0.30/0.52 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 0.30/0.52 = zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % bits_div_0
% 0.30/0.52 thf(fact_1750_bits__div__0,axiom,
% 0.30/0.52 ! [A: int] :
% 0.30/0.52 ( ( divide_divide_int @ zero_zero_int @ A )
% 0.30/0.52 = zero_zero_int ) ).
% 0.30/0.52
% 0.30/0.52 % bits_div_0
% 0.30/0.52 thf(fact_1751_div__by__0,axiom,
% 0.30/0.52 ! [A: complex] :
% 0.30/0.52 ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 0.30/0.52 = zero_zero_complex ) ).
% 0.30/0.52
% 0.30/0.52 % div_by_0
% 0.30/0.52 thf(fact_1752_div__by__0,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( divide_divide_real @ A @ zero_zero_real )
% 0.30/0.52 = zero_zero_real ) ).
% 0.30/0.52
% 0.30/0.52 % div_by_0
% 0.30/0.52 thf(fact_1753_div__by__0,axiom,
% 0.30/0.52 ! [A: rat] :
% 0.30/0.52 ( ( divide_divide_rat @ A @ zero_zero_rat )
% 0.30/0.52 = zero_zero_rat ) ).
% 0.30/0.52
% 0.30/0.52 % div_by_0
% 0.30/0.52 thf(fact_1754_div__by__0,axiom,
% 0.30/0.52 ! [A: nat] :
% 0.30/0.52 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 0.30/0.52 = zero_zero_nat ) ).
% 0.30/0.52
% 0.30/0.52 % div_by_0
% 0.30/0.52 thf(fact_1755_div__by__0,axiom,
% 0.30/0.52 ! [A: int] :
% 0.30/0.52 ( ( divide_divide_int @ A @ zero_zero_int )
% 0.30/0.52 = zero_zero_int ) ).
% 0.30/0.52
% 0.30/0.52 % div_by_0
% 0.30/0.52 thf(fact_1756_div__0,axiom,
% 0.30/0.52 ! [A: complex] :
% 0.30/0.52 ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
% 0.30/0.52 = zero_zero_complex ) ).
% 0.30/0.52
% 0.30/0.52 % div_0
% 0.30/0.52 thf(fact_1757_div__0,axiom,
% 0.30/0.52 ! [A: real] :
% 0.30/0.52 ( ( divide_divide_real @ zero_zero_real @ A )
% 0.30/0.53 = zero_zero_real ) ).
% 0.30/0.53
% 0.30/0.53 % div_0
% 0.30/0.53 thf(fact_1758_div__0,axiom,
% 0.30/0.53 ! [A: rat] :
% 0.30/0.53 ( ( divide_divide_rat @ zero_zero_rat @ A )
% 0.30/0.53 = zero_zero_rat ) ).
% 0.30/0.53
% 0.30/0.53 % div_0
% 0.30/0.53 thf(fact_1759_div__0,axiom,
% 0.30/0.53 ! [A: nat] :
% 0.30/0.53 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 0.30/0.53 = zero_zero_nat ) ).
% 0.30/0.53
% 0.30/0.53 % div_0
% 0.30/0.53 thf(fact_1760_div__0,axiom,
% 0.30/0.53 ! [A: int] :
% 0.30/0.53 ( ( divide_divide_int @ zero_zero_int @ A )
% 0.30/0.53 = zero_zero_int ) ).
% 0.30/0.53
% 0.30/0.53 % div_0
% 0.30/0.53 thf(fact_1761_division__ring__divide__zero,axiom,
% 0.30/0.53 ! [A: complex] :
% 0.30/0.53 ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 0.30/0.53 = zero_zero_complex ) ).
% 0.30/0.53
% 0.30/0.53 % division_ring_divide_zero
% 0.30/0.53 thf(fact_1762_division__ring__divide__zero,axiom,
% 0.30/0.53 ! [A: real] :
% 0.30/0.53 ( ( divide_divide_real @ A @ zero_zero_real )
% 0.30/0.53 = zero_zero_real ) ).
% 0.30/0.53
% 0.30/0.53 % division_ring_divide_zero
% 0.30/0.53 thf(fact_1763_division__ring__divide__zero,axiom,
% 0.30/0.53 ! [A: rat] :
% 0.30/0.53 ( ( divide_divide_rat @ A @ zero_zero_rat )
% 0.30/0.53 = zero_zero_rat ) ).
% 0.30/0.53
% 0.30/0.53 % division_ring_divide_zero
% 0.30/0.53 thf(fact_1764_divide__cancel__right,axiom,
% 0.30/0.53 ! [A: complex,C: complex,B: complex] :
% 0.30/0.53 ( ( ( divide1717551699836669952omplex @ A @ C )
% 0.30/0.53 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.30/0.53 = ( ( C = zero_zero_complex )
% 0.30/0.53 | ( A = B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_cancel_right
% 0.30/0.53 thf(fact_1765_divide__cancel__right,axiom,
% 0.30/0.53 ! [A: real,C: real,B: real] :
% 0.30/0.53 ( ( ( divide_divide_real @ A @ C )
% 0.30/0.53 = ( divide_divide_real @ B @ C ) )
% 0.30/0.53 = ( ( C = zero_zero_real )
% 0.30/0.53 | ( A = B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_cancel_right
% 0.30/0.53 thf(fact_1766_divide__cancel__right,axiom,
% 0.30/0.53 ! [A: rat,C: rat,B: rat] :
% 0.30/0.53 ( ( ( divide_divide_rat @ A @ C )
% 0.30/0.53 = ( divide_divide_rat @ B @ C ) )
% 0.30/0.53 = ( ( C = zero_zero_rat )
% 0.30/0.53 | ( A = B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_cancel_right
% 0.30/0.53 thf(fact_1767_divide__cancel__left,axiom,
% 0.30/0.53 ! [C: complex,A: complex,B: complex] :
% 0.30/0.53 ( ( ( divide1717551699836669952omplex @ C @ A )
% 0.30/0.53 = ( divide1717551699836669952omplex @ C @ B ) )
% 0.30/0.53 = ( ( C = zero_zero_complex )
% 0.30/0.53 | ( A = B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_cancel_left
% 0.30/0.53 thf(fact_1768_divide__cancel__left,axiom,
% 0.30/0.53 ! [C: real,A: real,B: real] :
% 0.30/0.53 ( ( ( divide_divide_real @ C @ A )
% 0.30/0.53 = ( divide_divide_real @ C @ B ) )
% 0.30/0.53 = ( ( C = zero_zero_real )
% 0.30/0.53 | ( A = B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_cancel_left
% 0.30/0.53 thf(fact_1769_divide__cancel__left,axiom,
% 0.30/0.53 ! [C: rat,A: rat,B: rat] :
% 0.30/0.53 ( ( ( divide_divide_rat @ C @ A )
% 0.30/0.53 = ( divide_divide_rat @ C @ B ) )
% 0.30/0.53 = ( ( C = zero_zero_rat )
% 0.30/0.53 | ( A = B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_cancel_left
% 0.30/0.53 thf(fact_1770_divide__eq__0__iff,axiom,
% 0.30/0.53 ! [A: complex,B: complex] :
% 0.30/0.53 ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.30/0.53 = zero_zero_complex )
% 0.30/0.53 = ( ( A = zero_zero_complex )
% 0.30/0.53 | ( B = zero_zero_complex ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_eq_0_iff
% 0.30/0.53 thf(fact_1771_divide__eq__0__iff,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ( divide_divide_real @ A @ B )
% 0.30/0.53 = zero_zero_real )
% 0.30/0.53 = ( ( A = zero_zero_real )
% 0.30/0.53 | ( B = zero_zero_real ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_eq_0_iff
% 0.30/0.53 thf(fact_1772_divide__eq__0__iff,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ( divide_divide_rat @ A @ B )
% 0.30/0.53 = zero_zero_rat )
% 0.30/0.53 = ( ( A = zero_zero_rat )
% 0.30/0.53 | ( B = zero_zero_rat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_eq_0_iff
% 0.30/0.53 thf(fact_1773_bits__mod__0,axiom,
% 0.30/0.53 ! [A: nat] :
% 0.30/0.53 ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 0.30/0.53 = zero_zero_nat ) ).
% 0.30/0.53
% 0.30/0.53 % bits_mod_0
% 0.30/0.53 thf(fact_1774_bits__mod__0,axiom,
% 0.30/0.53 ! [A: int] :
% 0.30/0.53 ( ( modulo_modulo_int @ zero_zero_int @ A )
% 0.30/0.53 = zero_zero_int ) ).
% 0.30/0.53
% 0.30/0.53 % bits_mod_0
% 0.30/0.53 thf(fact_1775_bits__mod__0,axiom,
% 0.30/0.53 ! [A: code_integer] :
% 0.30/0.53 ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.53 = zero_z3403309356797280102nteger ) ).
% 0.30/0.53
% 0.30/0.53 % bits_mod_0
% 0.30/0.53 thf(fact_1776_bot__nat__0_Onot__eq__extremum,axiom,
% 0.30/0.53 ! [A: nat] :
% 0.30/0.53 ( ( A != zero_zero_nat )
% 0.30/0.53 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 0.30/0.53
% 0.30/0.53 % bot_nat_0.not_eq_extremum
% 0.30/0.53 thf(fact_1777_neq0__conv,axiom,
% 0.30/0.53 ! [N2: nat] :
% 0.30/0.53 ( ( N2 != zero_zero_nat )
% 0.30/0.53 = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 0.30/0.53
% 0.30/0.53 % neq0_conv
% 0.30/0.53 thf(fact_1778_less__nat__zero__code,axiom,
% 0.30/0.53 ! [N2: nat] :
% 0.30/0.53 ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 0.30/0.53
% 0.30/0.53 % less_nat_zero_code
% 0.30/0.53 thf(fact_1779_bot__nat__0_Oextremum,axiom,
% 0.30/0.53 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.30/0.53
% 0.30/0.53 % bot_nat_0.extremum
% 0.30/0.53 thf(fact_1780_le0,axiom,
% 0.30/0.53 ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% 0.30/0.53
% 0.30/0.53 % le0
% 0.30/0.53 thf(fact_1781_mod__add__self2,axiom,
% 0.30/0.53 ! [A: nat,B: nat] :
% 0.30/0.53 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.30/0.53 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_add_self2
% 0.30/0.53 thf(fact_1782_mod__add__self2,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.30/0.53 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_add_self2
% 0.30/0.53 thf(fact_1783_mod__add__self2,axiom,
% 0.30/0.53 ! [A: code_integer,B: code_integer] :
% 0.30/0.53 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
% 0.30/0.53 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_add_self2
% 0.30/0.53 thf(fact_1784_mod__add__self1,axiom,
% 0.30/0.53 ! [B: nat,A: nat] :
% 0.30/0.53 ( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.30/0.53 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_add_self1
% 0.30/0.53 thf(fact_1785_mod__add__self1,axiom,
% 0.30/0.53 ! [B: int,A: int] :
% 0.30/0.53 ( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.30/0.53 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_add_self1
% 0.30/0.53 thf(fact_1786_mod__add__self1,axiom,
% 0.30/0.53 ! [B: code_integer,A: code_integer] :
% 0.30/0.53 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
% 0.30/0.53 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_add_self1
% 0.30/0.53 thf(fact_1787_Nat_Oadd__0__right,axiom,
% 0.30/0.53 ! [M: nat] :
% 0.30/0.53 ( ( plus_plus_nat @ M @ zero_zero_nat )
% 0.30/0.53 = M ) ).
% 0.30/0.53
% 0.30/0.53 % Nat.add_0_right
% 0.30/0.53 thf(fact_1788_add__is__0,axiom,
% 0.30/0.53 ! [M: nat,N2: nat] :
% 0.30/0.53 ( ( ( plus_plus_nat @ M @ N2 )
% 0.30/0.53 = zero_zero_nat )
% 0.30/0.53 = ( ( M = zero_zero_nat )
% 0.30/0.53 & ( N2 = zero_zero_nat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_is_0
% 0.30/0.53 thf(fact_1789_minus__mod__self2,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ B )
% 0.30/0.53 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % minus_mod_self2
% 0.30/0.53 thf(fact_1790_minus__mod__self2,axiom,
% 0.30/0.53 ! [A: code_integer,B: code_integer] :
% 0.30/0.53 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ B )
% 0.30/0.53 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % minus_mod_self2
% 0.30/0.53 thf(fact_1791_mult__cancel2,axiom,
% 0.30/0.53 ! [M: nat,K: nat,N2: nat] :
% 0.30/0.53 ( ( ( times_times_nat @ M @ K )
% 0.30/0.53 = ( times_times_nat @ N2 @ K ) )
% 0.30/0.53 = ( ( M = N2 )
% 0.30/0.53 | ( K = zero_zero_nat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_cancel2
% 0.30/0.53 thf(fact_1792_mult__cancel1,axiom,
% 0.30/0.53 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.53 ( ( ( times_times_nat @ K @ M )
% 0.30/0.53 = ( times_times_nat @ K @ N2 ) )
% 0.30/0.53 = ( ( M = N2 )
% 0.30/0.53 | ( K = zero_zero_nat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_cancel1
% 0.30/0.53 thf(fact_1793_mult__0__right,axiom,
% 0.30/0.53 ! [M: nat] :
% 0.30/0.53 ( ( times_times_nat @ M @ zero_zero_nat )
% 0.30/0.53 = zero_zero_nat ) ).
% 0.30/0.53
% 0.30/0.53 % mult_0_right
% 0.30/0.53 thf(fact_1794_mult__is__0,axiom,
% 0.30/0.53 ! [M: nat,N2: nat] :
% 0.30/0.53 ( ( ( times_times_nat @ M @ N2 )
% 0.30/0.53 = zero_zero_nat )
% 0.30/0.53 = ( ( M = zero_zero_nat )
% 0.30/0.53 | ( N2 = zero_zero_nat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_is_0
% 0.30/0.53 thf(fact_1795_diff__0__eq__0,axiom,
% 0.30/0.53 ! [N2: nat] :
% 0.30/0.53 ( ( minus_minus_nat @ zero_zero_nat @ N2 )
% 0.30/0.53 = zero_zero_nat ) ).
% 0.30/0.53
% 0.30/0.53 % diff_0_eq_0
% 0.30/0.53 thf(fact_1796_diff__self__eq__0,axiom,
% 0.30/0.53 ! [M: nat] :
% 0.30/0.53 ( ( minus_minus_nat @ M @ M )
% 0.30/0.53 = zero_zero_nat ) ).
% 0.30/0.53
% 0.30/0.53 % diff_self_eq_0
% 0.30/0.53 thf(fact_1797_mod__less,axiom,
% 0.30/0.53 ! [M: nat,N2: nat] :
% 0.30/0.53 ( ( ord_less_nat @ M @ N2 )
% 0.30/0.53 => ( ( modulo_modulo_nat @ M @ N2 )
% 0.30/0.53 = M ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_less
% 0.30/0.53 thf(fact_1798_max__0R,axiom,
% 0.30/0.53 ! [N2: nat] :
% 0.30/0.53 ( ( ord_max_nat @ N2 @ zero_zero_nat )
% 0.30/0.53 = N2 ) ).
% 0.30/0.53
% 0.30/0.53 % max_0R
% 0.30/0.53 thf(fact_1799_max__0L,axiom,
% 0.30/0.53 ! [N2: nat] :
% 0.30/0.53 ( ( ord_max_nat @ zero_zero_nat @ N2 )
% 0.30/0.53 = N2 ) ).
% 0.30/0.53
% 0.30/0.53 % max_0L
% 0.30/0.53 thf(fact_1800_max__nat_Oright__neutral,axiom,
% 0.30/0.53 ! [A: nat] :
% 0.30/0.53 ( ( ord_max_nat @ A @ zero_zero_nat )
% 0.30/0.53 = A ) ).
% 0.30/0.53
% 0.30/0.53 % max_nat.right_neutral
% 0.30/0.53 thf(fact_1801_max__nat_Oneutr__eq__iff,axiom,
% 0.30/0.53 ! [A: nat,B: nat] :
% 0.30/0.53 ( ( zero_zero_nat
% 0.30/0.53 = ( ord_max_nat @ A @ B ) )
% 0.30/0.53 = ( ( A = zero_zero_nat )
% 0.30/0.53 & ( B = zero_zero_nat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % max_nat.neutr_eq_iff
% 0.30/0.53 thf(fact_1802_max__nat_Oleft__neutral,axiom,
% 0.30/0.53 ! [A: nat] :
% 0.30/0.53 ( ( ord_max_nat @ zero_zero_nat @ A )
% 0.30/0.53 = A ) ).
% 0.30/0.53
% 0.30/0.53 % max_nat.left_neutral
% 0.30/0.53 thf(fact_1803_max__nat_Oeq__neutr__iff,axiom,
% 0.30/0.53 ! [A: nat,B: nat] :
% 0.30/0.53 ( ( ( ord_max_nat @ A @ B )
% 0.30/0.53 = zero_zero_nat )
% 0.30/0.53 = ( ( A = zero_zero_nat )
% 0.30/0.53 & ( B = zero_zero_nat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % max_nat.eq_neutr_iff
% 0.30/0.53 thf(fact_1804_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.30/0.53 ! [A: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 0.30/0.53 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_le_double_add_iff_zero_le_single_add
% 0.30/0.53 thf(fact_1805_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.30/0.53 ! [A: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 0.30/0.53 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_le_double_add_iff_zero_le_single_add
% 0.30/0.53 thf(fact_1806_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.30/0.53 ! [A: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 0.30/0.53 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_le_double_add_iff_zero_le_single_add
% 0.30/0.53 thf(fact_1807_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.30/0.53 ! [A: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 0.30/0.53 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.30/0.53
% 0.30/0.53 % double_add_le_zero_iff_single_add_le_zero
% 0.30/0.53 thf(fact_1808_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.30/0.53 ! [A: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 0.30/0.53 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.53
% 0.30/0.53 % double_add_le_zero_iff_single_add_le_zero
% 0.30/0.53 thf(fact_1809_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.30/0.53 ! [A: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 0.30/0.53 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.30/0.53
% 0.30/0.53 % double_add_le_zero_iff_single_add_le_zero
% 0.30/0.53 thf(fact_1810_le__add__same__cancel2,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.30/0.53 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % le_add_same_cancel2
% 0.30/0.53 thf(fact_1811_le__add__same__cancel2,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 0.30/0.53 = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % le_add_same_cancel2
% 0.30/0.53 thf(fact_1812_le__add__same__cancel2,axiom,
% 0.30/0.53 ! [A: nat,B: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.30/0.53 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % le_add_same_cancel2
% 0.30/0.53 thf(fact_1813_le__add__same__cancel2,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
% 0.30/0.53 = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % le_add_same_cancel2
% 0.30/0.53 thf(fact_1814_le__add__same__cancel1,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.30/0.53 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % le_add_same_cancel1
% 0.30/0.53 thf(fact_1815_le__add__same__cancel1,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.30/0.53 = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % le_add_same_cancel1
% 0.30/0.53 thf(fact_1816_le__add__same__cancel1,axiom,
% 0.30/0.53 ! [A: nat,B: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.30/0.53 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % le_add_same_cancel1
% 0.30/0.53 thf(fact_1817_le__add__same__cancel1,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.30/0.53 = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % le_add_same_cancel1
% 0.30/0.53 thf(fact_1818_add__le__same__cancel2,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.30/0.53 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_le_same_cancel2
% 0.30/0.53 thf(fact_1819_add__le__same__cancel2,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.30/0.53 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_le_same_cancel2
% 0.30/0.53 thf(fact_1820_add__le__same__cancel2,axiom,
% 0.30/0.53 ! [A: nat,B: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.30/0.53 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_le_same_cancel2
% 0.30/0.53 thf(fact_1821_add__le__same__cancel2,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.30/0.53 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_le_same_cancel2
% 0.30/0.53 thf(fact_1822_add__le__same__cancel1,axiom,
% 0.30/0.53 ! [B: real,A: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
% 0.30/0.53 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_le_same_cancel1
% 0.30/0.53 thf(fact_1823_add__le__same__cancel1,axiom,
% 0.30/0.53 ! [B: rat,A: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 0.30/0.53 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_le_same_cancel1
% 0.30/0.53 thf(fact_1824_add__le__same__cancel1,axiom,
% 0.30/0.53 ! [B: nat,A: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.30/0.53 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_le_same_cancel1
% 0.30/0.53 thf(fact_1825_add__le__same__cancel1,axiom,
% 0.30/0.53 ! [B: int,A: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.30/0.53 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_le_same_cancel1
% 0.30/0.53 thf(fact_1826_add__less__same__cancel1,axiom,
% 0.30/0.53 ! [B: real,A: real] :
% 0.30/0.53 ( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
% 0.30/0.53 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_less_same_cancel1
% 0.30/0.53 thf(fact_1827_add__less__same__cancel1,axiom,
% 0.30/0.53 ! [B: rat,A: rat] :
% 0.30/0.53 ( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 0.30/0.53 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_less_same_cancel1
% 0.30/0.53 thf(fact_1828_add__less__same__cancel1,axiom,
% 0.30/0.53 ! [B: nat,A: nat] :
% 0.30/0.53 ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.30/0.53 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_less_same_cancel1
% 0.30/0.53 thf(fact_1829_add__less__same__cancel1,axiom,
% 0.30/0.53 ! [B: int,A: int] :
% 0.30/0.53 ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.30/0.53 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_less_same_cancel1
% 0.30/0.53 thf(fact_1830_add__less__same__cancel2,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.30/0.53 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_less_same_cancel2
% 0.30/0.53 thf(fact_1831_add__less__same__cancel2,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.30/0.53 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_less_same_cancel2
% 0.30/0.53 thf(fact_1832_add__less__same__cancel2,axiom,
% 0.30/0.53 ! [A: nat,B: nat] :
% 0.30/0.53 ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.30/0.53 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_less_same_cancel2
% 0.30/0.53 thf(fact_1833_add__less__same__cancel2,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.30/0.53 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_less_same_cancel2
% 0.30/0.53 thf(fact_1834_less__add__same__cancel1,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.30/0.53 = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % less_add_same_cancel1
% 0.30/0.53 thf(fact_1835_less__add__same__cancel1,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.30/0.53 = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % less_add_same_cancel1
% 0.30/0.53 thf(fact_1836_less__add__same__cancel1,axiom,
% 0.30/0.53 ! [A: nat,B: nat] :
% 0.30/0.53 ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.30/0.53 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % less_add_same_cancel1
% 0.30/0.53 thf(fact_1837_less__add__same__cancel1,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.30/0.53 = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % less_add_same_cancel1
% 0.30/0.53 thf(fact_1838_less__add__same__cancel2,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.30/0.53 = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % less_add_same_cancel2
% 0.30/0.53 thf(fact_1839_less__add__same__cancel2,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 0.30/0.53 = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % less_add_same_cancel2
% 0.30/0.53 thf(fact_1840_less__add__same__cancel2,axiom,
% 0.30/0.53 ! [A: nat,B: nat] :
% 0.30/0.53 ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.30/0.53 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % less_add_same_cancel2
% 0.30/0.53 thf(fact_1841_less__add__same__cancel2,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
% 0.30/0.53 = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % less_add_same_cancel2
% 0.30/0.53 thf(fact_1842_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.30/0.53 ! [A: real] :
% 0.30/0.53 ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 0.30/0.53 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.30/0.53
% 0.30/0.53 % double_add_less_zero_iff_single_add_less_zero
% 0.30/0.53 thf(fact_1843_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.30/0.53 ! [A: rat] :
% 0.30/0.53 ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 0.30/0.53 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.53
% 0.30/0.53 % double_add_less_zero_iff_single_add_less_zero
% 0.30/0.53 thf(fact_1844_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.30/0.53 ! [A: int] :
% 0.30/0.53 ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 0.30/0.53 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.30/0.53
% 0.30/0.53 % double_add_less_zero_iff_single_add_less_zero
% 0.30/0.53 thf(fact_1845_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.30/0.53 ! [A: real] :
% 0.30/0.53 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 0.30/0.53 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_double_add_iff_zero_less_single_add
% 0.30/0.53 thf(fact_1846_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.30/0.53 ! [A: rat] :
% 0.30/0.53 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 0.30/0.53 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_double_add_iff_zero_less_single_add
% 0.30/0.53 thf(fact_1847_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.30/0.53 ! [A: int] :
% 0.30/0.53 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 0.30/0.53 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_double_add_iff_zero_less_single_add
% 0.30/0.53 thf(fact_1848_diff__ge__0__iff__ge,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 0.30/0.53 = ( ord_less_eq_real @ B @ A ) ) ).
% 0.30/0.53
% 0.30/0.53 % diff_ge_0_iff_ge
% 0.30/0.53 thf(fact_1849_diff__ge__0__iff__ge,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 0.30/0.53 = ( ord_less_eq_rat @ B @ A ) ) ).
% 0.30/0.53
% 0.30/0.53 % diff_ge_0_iff_ge
% 0.30/0.53 thf(fact_1850_diff__ge__0__iff__ge,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 0.30/0.53 = ( ord_less_eq_int @ B @ A ) ) ).
% 0.30/0.53
% 0.30/0.53 % diff_ge_0_iff_ge
% 0.30/0.53 thf(fact_1851_diff__gt__0__iff__gt,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 0.30/0.53 = ( ord_less_real @ B @ A ) ) ).
% 0.30/0.53
% 0.30/0.53 % diff_gt_0_iff_gt
% 0.30/0.53 thf(fact_1852_diff__gt__0__iff__gt,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 0.30/0.53 = ( ord_less_rat @ B @ A ) ) ).
% 0.30/0.53
% 0.30/0.53 % diff_gt_0_iff_gt
% 0.30/0.53 thf(fact_1853_diff__gt__0__iff__gt,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 0.30/0.53 = ( ord_less_int @ B @ A ) ) ).
% 0.30/0.53
% 0.30/0.53 % diff_gt_0_iff_gt
% 0.30/0.53 thf(fact_1854_sum__squares__eq__zero__iff,axiom,
% 0.30/0.53 ! [X: real,Y: real] :
% 0.30/0.53 ( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
% 0.30/0.53 = zero_zero_real )
% 0.30/0.53 = ( ( X = zero_zero_real )
% 0.30/0.53 & ( Y = zero_zero_real ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % sum_squares_eq_zero_iff
% 0.30/0.53 thf(fact_1855_sum__squares__eq__zero__iff,axiom,
% 0.30/0.53 ! [X: rat,Y: rat] :
% 0.30/0.53 ( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
% 0.30/0.53 = zero_zero_rat )
% 0.30/0.53 = ( ( X = zero_zero_rat )
% 0.30/0.53 & ( Y = zero_zero_rat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % sum_squares_eq_zero_iff
% 0.30/0.53 thf(fact_1856_sum__squares__eq__zero__iff,axiom,
% 0.30/0.53 ! [X: int,Y: int] :
% 0.30/0.53 ( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
% 0.30/0.53 = zero_zero_int )
% 0.30/0.53 = ( ( X = zero_zero_int )
% 0.30/0.53 & ( Y = zero_zero_int ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % sum_squares_eq_zero_iff
% 0.30/0.53 thf(fact_1857_mult__cancel__left1,axiom,
% 0.30/0.53 ! [C: complex,B: complex] :
% 0.30/0.53 ( ( C
% 0.30/0.53 = ( times_times_complex @ C @ B ) )
% 0.30/0.53 = ( ( C = zero_zero_complex )
% 0.30/0.53 | ( B = one_one_complex ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_cancel_left1
% 0.30/0.53 thf(fact_1858_mult__cancel__left1,axiom,
% 0.30/0.53 ! [C: real,B: real] :
% 0.30/0.53 ( ( C
% 0.30/0.53 = ( times_times_real @ C @ B ) )
% 0.30/0.53 = ( ( C = zero_zero_real )
% 0.30/0.53 | ( B = one_one_real ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_cancel_left1
% 0.30/0.53 thf(fact_1859_mult__cancel__left1,axiom,
% 0.30/0.53 ! [C: rat,B: rat] :
% 0.30/0.53 ( ( C
% 0.30/0.53 = ( times_times_rat @ C @ B ) )
% 0.30/0.53 = ( ( C = zero_zero_rat )
% 0.30/0.53 | ( B = one_one_rat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_cancel_left1
% 0.30/0.53 thf(fact_1860_mult__cancel__left1,axiom,
% 0.30/0.53 ! [C: int,B: int] :
% 0.30/0.53 ( ( C
% 0.30/0.53 = ( times_times_int @ C @ B ) )
% 0.30/0.53 = ( ( C = zero_zero_int )
% 0.30/0.53 | ( B = one_one_int ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_cancel_left1
% 0.30/0.53 thf(fact_1861_mult__cancel__left2,axiom,
% 0.30/0.53 ! [C: complex,A: complex] :
% 0.30/0.53 ( ( ( times_times_complex @ C @ A )
% 0.30/0.53 = C )
% 0.30/0.53 = ( ( C = zero_zero_complex )
% 0.30/0.53 | ( A = one_one_complex ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_cancel_left2
% 0.30/0.53 thf(fact_1862_mult__cancel__left2,axiom,
% 0.30/0.53 ! [C: real,A: real] :
% 0.30/0.53 ( ( ( times_times_real @ C @ A )
% 0.30/0.53 = C )
% 0.30/0.53 = ( ( C = zero_zero_real )
% 0.30/0.53 | ( A = one_one_real ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_cancel_left2
% 0.30/0.53 thf(fact_1863_mult__cancel__left2,axiom,
% 0.30/0.53 ! [C: rat,A: rat] :
% 0.30/0.53 ( ( ( times_times_rat @ C @ A )
% 0.30/0.53 = C )
% 0.30/0.53 = ( ( C = zero_zero_rat )
% 0.30/0.53 | ( A = one_one_rat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_cancel_left2
% 0.30/0.53 thf(fact_1864_mult__cancel__left2,axiom,
% 0.30/0.53 ! [C: int,A: int] :
% 0.30/0.53 ( ( ( times_times_int @ C @ A )
% 0.30/0.53 = C )
% 0.30/0.53 = ( ( C = zero_zero_int )
% 0.30/0.53 | ( A = one_one_int ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_cancel_left2
% 0.30/0.53 thf(fact_1865_mult__cancel__right1,axiom,
% 0.30/0.53 ! [C: complex,B: complex] :
% 0.30/0.53 ( ( C
% 0.30/0.53 = ( times_times_complex @ B @ C ) )
% 0.30/0.53 = ( ( C = zero_zero_complex )
% 0.30/0.53 | ( B = one_one_complex ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_cancel_right1
% 0.30/0.53 thf(fact_1866_mult__cancel__right1,axiom,
% 0.30/0.53 ! [C: real,B: real] :
% 0.30/0.53 ( ( C
% 0.30/0.53 = ( times_times_real @ B @ C ) )
% 0.30/0.53 = ( ( C = zero_zero_real )
% 0.30/0.53 | ( B = one_one_real ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_cancel_right1
% 0.30/0.53 thf(fact_1867_mult__cancel__right1,axiom,
% 0.30/0.53 ! [C: rat,B: rat] :
% 0.30/0.53 ( ( C
% 0.30/0.53 = ( times_times_rat @ B @ C ) )
% 0.30/0.53 = ( ( C = zero_zero_rat )
% 0.30/0.53 | ( B = one_one_rat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_cancel_right1
% 0.30/0.53 thf(fact_1868_mult__cancel__right1,axiom,
% 0.30/0.53 ! [C: int,B: int] :
% 0.30/0.53 ( ( C
% 0.30/0.53 = ( times_times_int @ B @ C ) )
% 0.30/0.53 = ( ( C = zero_zero_int )
% 0.30/0.53 | ( B = one_one_int ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_cancel_right1
% 0.30/0.53 thf(fact_1869_mult__cancel__right2,axiom,
% 0.30/0.53 ! [A: complex,C: complex] :
% 0.30/0.53 ( ( ( times_times_complex @ A @ C )
% 0.30/0.53 = C )
% 0.30/0.53 = ( ( C = zero_zero_complex )
% 0.30/0.53 | ( A = one_one_complex ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_cancel_right2
% 0.30/0.53 thf(fact_1870_mult__cancel__right2,axiom,
% 0.30/0.53 ! [A: real,C: real] :
% 0.30/0.53 ( ( ( times_times_real @ A @ C )
% 0.30/0.53 = C )
% 0.30/0.53 = ( ( C = zero_zero_real )
% 0.30/0.53 | ( A = one_one_real ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_cancel_right2
% 0.30/0.53 thf(fact_1871_mult__cancel__right2,axiom,
% 0.30/0.53 ! [A: rat,C: rat] :
% 0.30/0.53 ( ( ( times_times_rat @ A @ C )
% 0.30/0.53 = C )
% 0.30/0.53 = ( ( C = zero_zero_rat )
% 0.30/0.53 | ( A = one_one_rat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_cancel_right2
% 0.30/0.53 thf(fact_1872_mult__cancel__right2,axiom,
% 0.30/0.53 ! [A: int,C: int] :
% 0.30/0.53 ( ( ( times_times_int @ A @ C )
% 0.30/0.53 = C )
% 0.30/0.53 = ( ( C = zero_zero_int )
% 0.30/0.53 | ( A = one_one_int ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_cancel_right2
% 0.30/0.53 thf(fact_1873_div__mult__mult1__if,axiom,
% 0.30/0.53 ! [C: nat,A: nat,B: nat] :
% 0.30/0.53 ( ( ( C = zero_zero_nat )
% 0.30/0.53 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.30/0.53 = zero_zero_nat ) )
% 0.30/0.53 & ( ( C != zero_zero_nat )
% 0.30/0.53 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.30/0.53 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % div_mult_mult1_if
% 0.30/0.53 thf(fact_1874_div__mult__mult1__if,axiom,
% 0.30/0.53 ! [C: int,A: int,B: int] :
% 0.30/0.53 ( ( ( C = zero_zero_int )
% 0.30/0.53 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.30/0.53 = zero_zero_int ) )
% 0.30/0.53 & ( ( C != zero_zero_int )
% 0.30/0.53 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.30/0.53 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % div_mult_mult1_if
% 0.30/0.53 thf(fact_1875_div__mult__mult2,axiom,
% 0.30/0.53 ! [C: nat,A: nat,B: nat] :
% 0.30/0.53 ( ( C != zero_zero_nat )
% 0.30/0.53 => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.30/0.53 = ( divide_divide_nat @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % div_mult_mult2
% 0.30/0.53 thf(fact_1876_div__mult__mult2,axiom,
% 0.30/0.53 ! [C: int,A: int,B: int] :
% 0.30/0.53 ( ( C != zero_zero_int )
% 0.30/0.53 => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.30/0.53 = ( divide_divide_int @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % div_mult_mult2
% 0.30/0.53 thf(fact_1877_div__mult__mult1,axiom,
% 0.30/0.53 ! [C: nat,A: nat,B: nat] :
% 0.30/0.53 ( ( C != zero_zero_nat )
% 0.30/0.53 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.30/0.53 = ( divide_divide_nat @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % div_mult_mult1
% 0.30/0.53 thf(fact_1878_div__mult__mult1,axiom,
% 0.30/0.53 ! [C: int,A: int,B: int] :
% 0.30/0.53 ( ( C != zero_zero_int )
% 0.30/0.53 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.30/0.53 = ( divide_divide_int @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % div_mult_mult1
% 0.30/0.53 thf(fact_1879_nonzero__mult__div__cancel__right,axiom,
% 0.30/0.53 ! [B: complex,A: complex] :
% 0.30/0.53 ( ( B != zero_zero_complex )
% 0.30/0.53 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
% 0.30/0.53 = A ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_mult_div_cancel_right
% 0.30/0.53 thf(fact_1880_nonzero__mult__div__cancel__right,axiom,
% 0.30/0.53 ! [B: real,A: real] :
% 0.30/0.53 ( ( B != zero_zero_real )
% 0.30/0.53 => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
% 0.30/0.53 = A ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_mult_div_cancel_right
% 0.30/0.53 thf(fact_1881_nonzero__mult__div__cancel__right,axiom,
% 0.30/0.53 ! [B: rat,A: rat] :
% 0.30/0.53 ( ( B != zero_zero_rat )
% 0.30/0.53 => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
% 0.30/0.53 = A ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_mult_div_cancel_right
% 0.30/0.53 thf(fact_1882_nonzero__mult__div__cancel__right,axiom,
% 0.30/0.53 ! [B: nat,A: nat] :
% 0.30/0.53 ( ( B != zero_zero_nat )
% 0.30/0.53 => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
% 0.30/0.53 = A ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_mult_div_cancel_right
% 0.30/0.53 thf(fact_1883_nonzero__mult__div__cancel__right,axiom,
% 0.30/0.53 ! [B: int,A: int] :
% 0.30/0.53 ( ( B != zero_zero_int )
% 0.30/0.53 => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
% 0.30/0.53 = A ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_mult_div_cancel_right
% 0.30/0.53 thf(fact_1884_nonzero__mult__div__cancel__left,axiom,
% 0.30/0.53 ! [A: complex,B: complex] :
% 0.30/0.53 ( ( A != zero_zero_complex )
% 0.30/0.53 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
% 0.30/0.53 = B ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_mult_div_cancel_left
% 0.30/0.53 thf(fact_1885_nonzero__mult__div__cancel__left,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( A != zero_zero_real )
% 0.30/0.53 => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
% 0.30/0.53 = B ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_mult_div_cancel_left
% 0.30/0.53 thf(fact_1886_nonzero__mult__div__cancel__left,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( A != zero_zero_rat )
% 0.30/0.53 => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
% 0.30/0.53 = B ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_mult_div_cancel_left
% 0.30/0.53 thf(fact_1887_nonzero__mult__div__cancel__left,axiom,
% 0.30/0.53 ! [A: nat,B: nat] :
% 0.30/0.53 ( ( A != zero_zero_nat )
% 0.30/0.53 => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
% 0.30/0.53 = B ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_mult_div_cancel_left
% 0.30/0.53 thf(fact_1888_nonzero__mult__div__cancel__left,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( A != zero_zero_int )
% 0.30/0.53 => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
% 0.30/0.53 = B ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_mult_div_cancel_left
% 0.30/0.53 thf(fact_1889_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.30/0.53 ! [C: complex,A: complex,B: complex] :
% 0.30/0.53 ( ( C != zero_zero_complex )
% 0.30/0.53 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B ) )
% 0.30/0.53 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_mult_divide_mult_cancel_right2
% 0.30/0.53 thf(fact_1890_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.30/0.53 ! [C: real,A: real,B: real] :
% 0.30/0.53 ( ( C != zero_zero_real )
% 0.30/0.53 => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
% 0.30/0.53 = ( divide_divide_real @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_mult_divide_mult_cancel_right2
% 0.30/0.53 thf(fact_1891_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.30/0.53 ! [C: rat,A: rat,B: rat] :
% 0.30/0.53 ( ( C != zero_zero_rat )
% 0.30/0.53 => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
% 0.30/0.53 = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_mult_divide_mult_cancel_right2
% 0.30/0.53 thf(fact_1892_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.30/0.53 ! [C: complex,A: complex,B: complex] :
% 0.30/0.53 ( ( C != zero_zero_complex )
% 0.30/0.53 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 0.30/0.53 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_mult_divide_mult_cancel_right
% 0.30/0.53 thf(fact_1893_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.30/0.53 ! [C: real,A: real,B: real] :
% 0.30/0.53 ( ( C != zero_zero_real )
% 0.30/0.53 => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.30/0.53 = ( divide_divide_real @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_mult_divide_mult_cancel_right
% 0.30/0.53 thf(fact_1894_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.30/0.53 ! [C: rat,A: rat,B: rat] :
% 0.30/0.53 ( ( C != zero_zero_rat )
% 0.30/0.53 => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.30/0.53 = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_mult_divide_mult_cancel_right
% 0.30/0.53 thf(fact_1895_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.30/0.53 ! [C: complex,A: complex,B: complex] :
% 0.30/0.53 ( ( C != zero_zero_complex )
% 0.30/0.53 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B @ C ) )
% 0.30/0.53 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_mult_divide_mult_cancel_left2
% 0.30/0.53 thf(fact_1896_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.30/0.53 ! [C: real,A: real,B: real] :
% 0.30/0.53 ( ( C != zero_zero_real )
% 0.30/0.53 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
% 0.30/0.53 = ( divide_divide_real @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_mult_divide_mult_cancel_left2
% 0.30/0.53 thf(fact_1897_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.30/0.53 ! [C: rat,A: rat,B: rat] :
% 0.30/0.53 ( ( C != zero_zero_rat )
% 0.30/0.53 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
% 0.30/0.53 = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_mult_divide_mult_cancel_left2
% 0.30/0.53 thf(fact_1898_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.30/0.53 ! [C: complex,A: complex,B: complex] :
% 0.30/0.53 ( ( C != zero_zero_complex )
% 0.30/0.53 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.30/0.53 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_mult_divide_mult_cancel_left
% 0.30/0.53 thf(fact_1899_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.30/0.53 ! [C: real,A: real,B: real] :
% 0.30/0.53 ( ( C != zero_zero_real )
% 0.30/0.53 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.30/0.53 = ( divide_divide_real @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_mult_divide_mult_cancel_left
% 0.30/0.53 thf(fact_1900_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.30/0.53 ! [C: rat,A: rat,B: rat] :
% 0.30/0.53 ( ( C != zero_zero_rat )
% 0.30/0.53 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.30/0.53 = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_mult_divide_mult_cancel_left
% 0.30/0.53 thf(fact_1901_mult__divide__mult__cancel__left__if,axiom,
% 0.30/0.53 ! [C: complex,A: complex,B: complex] :
% 0.30/0.53 ( ( ( C = zero_zero_complex )
% 0.30/0.53 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.30/0.53 = zero_zero_complex ) )
% 0.30/0.53 & ( ( C != zero_zero_complex )
% 0.30/0.53 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.30/0.53 = ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_divide_mult_cancel_left_if
% 0.30/0.53 thf(fact_1902_mult__divide__mult__cancel__left__if,axiom,
% 0.30/0.53 ! [C: real,A: real,B: real] :
% 0.30/0.53 ( ( ( C = zero_zero_real )
% 0.30/0.53 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.30/0.53 = zero_zero_real ) )
% 0.30/0.53 & ( ( C != zero_zero_real )
% 0.30/0.53 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.30/0.53 = ( divide_divide_real @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_divide_mult_cancel_left_if
% 0.30/0.53 thf(fact_1903_mult__divide__mult__cancel__left__if,axiom,
% 0.30/0.53 ! [C: rat,A: rat,B: rat] :
% 0.30/0.53 ( ( ( C = zero_zero_rat )
% 0.30/0.53 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.30/0.53 = zero_zero_rat ) )
% 0.30/0.53 & ( ( C != zero_zero_rat )
% 0.30/0.53 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.30/0.53 = ( divide_divide_rat @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_divide_mult_cancel_left_if
% 0.30/0.53 thf(fact_1904_diff__add__zero,axiom,
% 0.30/0.53 ! [A: nat,B: nat] :
% 0.30/0.53 ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.30/0.53 = zero_zero_nat ) ).
% 0.30/0.53
% 0.30/0.53 % diff_add_zero
% 0.30/0.53 thf(fact_1905_diff__numeral__special_I9_J,axiom,
% 0.30/0.53 ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
% 0.30/0.53 = zero_zero_complex ) ).
% 0.30/0.53
% 0.30/0.53 % diff_numeral_special(9)
% 0.30/0.53 thf(fact_1906_diff__numeral__special_I9_J,axiom,
% 0.30/0.53 ( ( minus_minus_real @ one_one_real @ one_one_real )
% 0.30/0.53 = zero_zero_real ) ).
% 0.30/0.53
% 0.30/0.53 % diff_numeral_special(9)
% 0.30/0.53 thf(fact_1907_diff__numeral__special_I9_J,axiom,
% 0.30/0.53 ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
% 0.30/0.53 = zero_zero_rat ) ).
% 0.30/0.53
% 0.30/0.53 % diff_numeral_special(9)
% 0.30/0.53 thf(fact_1908_diff__numeral__special_I9_J,axiom,
% 0.30/0.53 ( ( minus_minus_int @ one_one_int @ one_one_int )
% 0.30/0.53 = zero_zero_int ) ).
% 0.30/0.53
% 0.30/0.53 % diff_numeral_special(9)
% 0.30/0.53 thf(fact_1909_div__self,axiom,
% 0.30/0.53 ! [A: complex] :
% 0.30/0.53 ( ( A != zero_zero_complex )
% 0.30/0.53 => ( ( divide1717551699836669952omplex @ A @ A )
% 0.30/0.53 = one_one_complex ) ) ).
% 0.30/0.53
% 0.30/0.53 % div_self
% 0.30/0.53 thf(fact_1910_div__self,axiom,
% 0.30/0.53 ! [A: real] :
% 0.30/0.53 ( ( A != zero_zero_real )
% 0.30/0.53 => ( ( divide_divide_real @ A @ A )
% 0.30/0.53 = one_one_real ) ) ).
% 0.30/0.53
% 0.30/0.53 % div_self
% 0.30/0.53 thf(fact_1911_div__self,axiom,
% 0.30/0.53 ! [A: rat] :
% 0.30/0.53 ( ( A != zero_zero_rat )
% 0.30/0.53 => ( ( divide_divide_rat @ A @ A )
% 0.30/0.53 = one_one_rat ) ) ).
% 0.30/0.53
% 0.30/0.53 % div_self
% 0.30/0.53 thf(fact_1912_div__self,axiom,
% 0.30/0.53 ! [A: nat] :
% 0.30/0.53 ( ( A != zero_zero_nat )
% 0.30/0.53 => ( ( divide_divide_nat @ A @ A )
% 0.30/0.53 = one_one_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % div_self
% 0.30/0.53 thf(fact_1913_div__self,axiom,
% 0.30/0.53 ! [A: int] :
% 0.30/0.53 ( ( A != zero_zero_int )
% 0.30/0.53 => ( ( divide_divide_int @ A @ A )
% 0.30/0.53 = one_one_int ) ) ).
% 0.30/0.53
% 0.30/0.53 % div_self
% 0.30/0.53 thf(fact_1914_zero__eq__1__divide__iff,axiom,
% 0.30/0.53 ! [A: real] :
% 0.30/0.53 ( ( zero_zero_real
% 0.30/0.53 = ( divide_divide_real @ one_one_real @ A ) )
% 0.30/0.53 = ( A = zero_zero_real ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_eq_1_divide_iff
% 0.30/0.53 thf(fact_1915_zero__eq__1__divide__iff,axiom,
% 0.30/0.53 ! [A: rat] :
% 0.30/0.53 ( ( zero_zero_rat
% 0.30/0.53 = ( divide_divide_rat @ one_one_rat @ A ) )
% 0.30/0.53 = ( A = zero_zero_rat ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_eq_1_divide_iff
% 0.30/0.53 thf(fact_1916_one__divide__eq__0__iff,axiom,
% 0.30/0.53 ! [A: real] :
% 0.30/0.53 ( ( ( divide_divide_real @ one_one_real @ A )
% 0.30/0.53 = zero_zero_real )
% 0.30/0.53 = ( A = zero_zero_real ) ) ).
% 0.30/0.53
% 0.30/0.53 % one_divide_eq_0_iff
% 0.30/0.53 thf(fact_1917_one__divide__eq__0__iff,axiom,
% 0.30/0.53 ! [A: rat] :
% 0.30/0.53 ( ( ( divide_divide_rat @ one_one_rat @ A )
% 0.30/0.53 = zero_zero_rat )
% 0.30/0.53 = ( A = zero_zero_rat ) ) ).
% 0.30/0.53
% 0.30/0.53 % one_divide_eq_0_iff
% 0.30/0.53 thf(fact_1918_eq__divide__eq__1,axiom,
% 0.30/0.53 ! [B: real,A: real] :
% 0.30/0.53 ( ( one_one_real
% 0.30/0.53 = ( divide_divide_real @ B @ A ) )
% 0.30/0.53 = ( ( A != zero_zero_real )
% 0.30/0.53 & ( A = B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % eq_divide_eq_1
% 0.30/0.53 thf(fact_1919_eq__divide__eq__1,axiom,
% 0.30/0.53 ! [B: rat,A: rat] :
% 0.30/0.53 ( ( one_one_rat
% 0.30/0.53 = ( divide_divide_rat @ B @ A ) )
% 0.30/0.53 = ( ( A != zero_zero_rat )
% 0.30/0.53 & ( A = B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % eq_divide_eq_1
% 0.30/0.53 thf(fact_1920_divide__eq__eq__1,axiom,
% 0.30/0.53 ! [B: real,A: real] :
% 0.30/0.53 ( ( ( divide_divide_real @ B @ A )
% 0.30/0.53 = one_one_real )
% 0.30/0.53 = ( ( A != zero_zero_real )
% 0.30/0.53 & ( A = B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_eq_eq_1
% 0.30/0.53 thf(fact_1921_divide__eq__eq__1,axiom,
% 0.30/0.53 ! [B: rat,A: rat] :
% 0.30/0.53 ( ( ( divide_divide_rat @ B @ A )
% 0.30/0.53 = one_one_rat )
% 0.30/0.53 = ( ( A != zero_zero_rat )
% 0.30/0.53 & ( A = B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_eq_eq_1
% 0.30/0.53 thf(fact_1922_divide__self__if,axiom,
% 0.30/0.53 ! [A: complex] :
% 0.30/0.53 ( ( ( A = zero_zero_complex )
% 0.30/0.53 => ( ( divide1717551699836669952omplex @ A @ A )
% 0.30/0.53 = zero_zero_complex ) )
% 0.30/0.53 & ( ( A != zero_zero_complex )
% 0.30/0.53 => ( ( divide1717551699836669952omplex @ A @ A )
% 0.30/0.53 = one_one_complex ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_self_if
% 0.30/0.53 thf(fact_1923_divide__self__if,axiom,
% 0.30/0.53 ! [A: real] :
% 0.30/0.53 ( ( ( A = zero_zero_real )
% 0.30/0.53 => ( ( divide_divide_real @ A @ A )
% 0.30/0.53 = zero_zero_real ) )
% 0.30/0.53 & ( ( A != zero_zero_real )
% 0.30/0.53 => ( ( divide_divide_real @ A @ A )
% 0.30/0.53 = one_one_real ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_self_if
% 0.30/0.53 thf(fact_1924_divide__self__if,axiom,
% 0.30/0.53 ! [A: rat] :
% 0.30/0.53 ( ( ( A = zero_zero_rat )
% 0.30/0.53 => ( ( divide_divide_rat @ A @ A )
% 0.30/0.53 = zero_zero_rat ) )
% 0.30/0.53 & ( ( A != zero_zero_rat )
% 0.30/0.53 => ( ( divide_divide_rat @ A @ A )
% 0.30/0.53 = one_one_rat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_self_if
% 0.30/0.53 thf(fact_1925_divide__self,axiom,
% 0.30/0.53 ! [A: complex] :
% 0.30/0.53 ( ( A != zero_zero_complex )
% 0.30/0.53 => ( ( divide1717551699836669952omplex @ A @ A )
% 0.30/0.53 = one_one_complex ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_self
% 0.30/0.53 thf(fact_1926_divide__self,axiom,
% 0.30/0.53 ! [A: real] :
% 0.30/0.53 ( ( A != zero_zero_real )
% 0.30/0.53 => ( ( divide_divide_real @ A @ A )
% 0.30/0.53 = one_one_real ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_self
% 0.30/0.53 thf(fact_1927_divide__self,axiom,
% 0.30/0.53 ! [A: rat] :
% 0.30/0.53 ( ( A != zero_zero_rat )
% 0.30/0.53 => ( ( divide_divide_rat @ A @ A )
% 0.30/0.53 = one_one_rat ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_self
% 0.30/0.53 thf(fact_1928_one__eq__divide__iff,axiom,
% 0.30/0.53 ! [A: complex,B: complex] :
% 0.30/0.53 ( ( one_one_complex
% 0.30/0.53 = ( divide1717551699836669952omplex @ A @ B ) )
% 0.30/0.53 = ( ( B != zero_zero_complex )
% 0.30/0.53 & ( A = B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % one_eq_divide_iff
% 0.30/0.53 thf(fact_1929_one__eq__divide__iff,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( one_one_real
% 0.30/0.53 = ( divide_divide_real @ A @ B ) )
% 0.30/0.53 = ( ( B != zero_zero_real )
% 0.30/0.53 & ( A = B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % one_eq_divide_iff
% 0.30/0.53 thf(fact_1930_one__eq__divide__iff,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( one_one_rat
% 0.30/0.53 = ( divide_divide_rat @ A @ B ) )
% 0.30/0.53 = ( ( B != zero_zero_rat )
% 0.30/0.53 & ( A = B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % one_eq_divide_iff
% 0.30/0.53 thf(fact_1931_divide__eq__1__iff,axiom,
% 0.30/0.53 ! [A: complex,B: complex] :
% 0.30/0.53 ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.30/0.53 = one_one_complex )
% 0.30/0.53 = ( ( B != zero_zero_complex )
% 0.30/0.53 & ( A = B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_eq_1_iff
% 0.30/0.53 thf(fact_1932_divide__eq__1__iff,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ( divide_divide_real @ A @ B )
% 0.30/0.53 = one_one_real )
% 0.30/0.53 = ( ( B != zero_zero_real )
% 0.30/0.53 & ( A = B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_eq_1_iff
% 0.30/0.53 thf(fact_1933_divide__eq__1__iff,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ( divide_divide_rat @ A @ B )
% 0.30/0.53 = one_one_rat )
% 0.30/0.53 = ( ( B != zero_zero_rat )
% 0.30/0.53 & ( A = B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_eq_1_iff
% 0.30/0.53 thf(fact_1934_power__0__Suc,axiom,
% 0.30/0.53 ! [N2: nat] :
% 0.30/0.53 ( ( power_power_rat @ zero_zero_rat @ ( suc @ N2 ) )
% 0.30/0.53 = zero_zero_rat ) ).
% 0.30/0.53
% 0.30/0.53 % power_0_Suc
% 0.30/0.53 thf(fact_1935_power__0__Suc,axiom,
% 0.30/0.53 ! [N2: nat] :
% 0.30/0.53 ( ( power_power_nat @ zero_zero_nat @ ( suc @ N2 ) )
% 0.30/0.53 = zero_zero_nat ) ).
% 0.30/0.53
% 0.30/0.53 % power_0_Suc
% 0.30/0.53 thf(fact_1936_power__0__Suc,axiom,
% 0.30/0.53 ! [N2: nat] :
% 0.30/0.53 ( ( power_power_real @ zero_zero_real @ ( suc @ N2 ) )
% 0.30/0.53 = zero_zero_real ) ).
% 0.30/0.53
% 0.30/0.53 % power_0_Suc
% 0.30/0.53 thf(fact_1937_power__0__Suc,axiom,
% 0.30/0.53 ! [N2: nat] :
% 0.30/0.53 ( ( power_power_int @ zero_zero_int @ ( suc @ N2 ) )
% 0.30/0.53 = zero_zero_int ) ).
% 0.30/0.53
% 0.30/0.53 % power_0_Suc
% 0.30/0.53 thf(fact_1938_power__0__Suc,axiom,
% 0.30/0.53 ! [N2: nat] :
% 0.30/0.53 ( ( power_power_complex @ zero_zero_complex @ ( suc @ N2 ) )
% 0.30/0.53 = zero_zero_complex ) ).
% 0.30/0.53
% 0.30/0.53 % power_0_Suc
% 0.30/0.53 thf(fact_1939_power__zero__numeral,axiom,
% 0.30/0.53 ! [K: num] :
% 0.30/0.53 ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
% 0.30/0.53 = zero_zero_rat ) ).
% 0.30/0.53
% 0.30/0.53 % power_zero_numeral
% 0.30/0.53 thf(fact_1940_power__zero__numeral,axiom,
% 0.30/0.53 ! [K: num] :
% 0.30/0.53 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
% 0.30/0.53 = zero_zero_nat ) ).
% 0.30/0.53
% 0.30/0.53 % power_zero_numeral
% 0.30/0.53 thf(fact_1941_power__zero__numeral,axiom,
% 0.30/0.53 ! [K: num] :
% 0.30/0.53 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
% 0.30/0.53 = zero_zero_real ) ).
% 0.30/0.53
% 0.30/0.53 % power_zero_numeral
% 0.30/0.53 thf(fact_1942_power__zero__numeral,axiom,
% 0.30/0.53 ! [K: num] :
% 0.30/0.53 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
% 0.30/0.53 = zero_zero_int ) ).
% 0.30/0.53
% 0.30/0.53 % power_zero_numeral
% 0.30/0.53 thf(fact_1943_power__zero__numeral,axiom,
% 0.30/0.53 ! [K: num] :
% 0.30/0.53 ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
% 0.30/0.53 = zero_zero_complex ) ).
% 0.30/0.53
% 0.30/0.53 % power_zero_numeral
% 0.30/0.53 thf(fact_1944_mod__mult__self1__is__0,axiom,
% 0.30/0.53 ! [B: nat,A: nat] :
% 0.30/0.53 ( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
% 0.30/0.53 = zero_zero_nat ) ).
% 0.30/0.53
% 0.30/0.53 % mod_mult_self1_is_0
% 0.30/0.53 thf(fact_1945_mod__mult__self1__is__0,axiom,
% 0.30/0.53 ! [B: int,A: int] :
% 0.30/0.53 ( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
% 0.30/0.53 = zero_zero_int ) ).
% 0.30/0.53
% 0.30/0.53 % mod_mult_self1_is_0
% 0.30/0.53 thf(fact_1946_mod__mult__self1__is__0,axiom,
% 0.30/0.53 ! [B: code_integer,A: code_integer] :
% 0.30/0.53 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
% 0.30/0.53 = zero_z3403309356797280102nteger ) ).
% 0.30/0.53
% 0.30/0.53 % mod_mult_self1_is_0
% 0.30/0.53 thf(fact_1947_mod__mult__self2__is__0,axiom,
% 0.30/0.53 ! [A: nat,B: nat] :
% 0.30/0.53 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
% 0.30/0.53 = zero_zero_nat ) ).
% 0.30/0.53
% 0.30/0.53 % mod_mult_self2_is_0
% 0.30/0.53 thf(fact_1948_mod__mult__self2__is__0,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
% 0.30/0.53 = zero_zero_int ) ).
% 0.30/0.53
% 0.30/0.53 % mod_mult_self2_is_0
% 0.30/0.53 thf(fact_1949_mod__mult__self2__is__0,axiom,
% 0.30/0.53 ! [A: code_integer,B: code_integer] :
% 0.30/0.53 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
% 0.30/0.53 = zero_z3403309356797280102nteger ) ).
% 0.30/0.53
% 0.30/0.53 % mod_mult_self2_is_0
% 0.30/0.53 thf(fact_1950_bits__mod__by__1,axiom,
% 0.30/0.53 ! [A: nat] :
% 0.30/0.53 ( ( modulo_modulo_nat @ A @ one_one_nat )
% 0.30/0.53 = zero_zero_nat ) ).
% 0.30/0.53
% 0.30/0.53 % bits_mod_by_1
% 0.30/0.53 thf(fact_1951_bits__mod__by__1,axiom,
% 0.30/0.53 ! [A: int] :
% 0.30/0.53 ( ( modulo_modulo_int @ A @ one_one_int )
% 0.30/0.53 = zero_zero_int ) ).
% 0.30/0.53
% 0.30/0.53 % bits_mod_by_1
% 0.30/0.53 thf(fact_1952_bits__mod__by__1,axiom,
% 0.30/0.53 ! [A: code_integer] :
% 0.30/0.53 ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 0.30/0.53 = zero_z3403309356797280102nteger ) ).
% 0.30/0.53
% 0.30/0.53 % bits_mod_by_1
% 0.30/0.53 thf(fact_1953_mod__by__1,axiom,
% 0.30/0.53 ! [A: nat] :
% 0.30/0.53 ( ( modulo_modulo_nat @ A @ one_one_nat )
% 0.30/0.53 = zero_zero_nat ) ).
% 0.30/0.53
% 0.30/0.53 % mod_by_1
% 0.30/0.53 thf(fact_1954_mod__by__1,axiom,
% 0.30/0.53 ! [A: int] :
% 0.30/0.53 ( ( modulo_modulo_int @ A @ one_one_int )
% 0.30/0.53 = zero_zero_int ) ).
% 0.30/0.53
% 0.30/0.53 % mod_by_1
% 0.30/0.53 thf(fact_1955_mod__by__1,axiom,
% 0.30/0.53 ! [A: code_integer] :
% 0.30/0.53 ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 0.30/0.53 = zero_z3403309356797280102nteger ) ).
% 0.30/0.53
% 0.30/0.53 % mod_by_1
% 0.30/0.53 thf(fact_1956_power__Suc0__right,axiom,
% 0.30/0.53 ! [A: nat] :
% 0.30/0.53 ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
% 0.30/0.53 = A ) ).
% 0.30/0.53
% 0.30/0.53 % power_Suc0_right
% 0.30/0.53 thf(fact_1957_power__Suc0__right,axiom,
% 0.30/0.53 ! [A: real] :
% 0.30/0.53 ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
% 0.30/0.53 = A ) ).
% 0.30/0.53
% 0.30/0.53 % power_Suc0_right
% 0.30/0.53 thf(fact_1958_power__Suc0__right,axiom,
% 0.30/0.53 ! [A: int] :
% 0.30/0.53 ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
% 0.30/0.53 = A ) ).
% 0.30/0.53
% 0.30/0.53 % power_Suc0_right
% 0.30/0.53 thf(fact_1959_power__Suc0__right,axiom,
% 0.30/0.53 ! [A: complex] :
% 0.30/0.53 ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
% 0.30/0.53 = A ) ).
% 0.30/0.53
% 0.30/0.53 % power_Suc0_right
% 0.30/0.53 thf(fact_1960_bits__mod__div__trivial,axiom,
% 0.30/0.53 ! [A: nat,B: nat] :
% 0.30/0.53 ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 0.30/0.53 = zero_zero_nat ) ).
% 0.30/0.53
% 0.30/0.53 % bits_mod_div_trivial
% 0.30/0.53 thf(fact_1961_bits__mod__div__trivial,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 0.30/0.53 = zero_zero_int ) ).
% 0.30/0.53
% 0.30/0.53 % bits_mod_div_trivial
% 0.30/0.53 thf(fact_1962_bits__mod__div__trivial,axiom,
% 0.30/0.53 ! [A: code_integer,B: code_integer] :
% 0.30/0.53 ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 0.30/0.53 = zero_z3403309356797280102nteger ) ).
% 0.30/0.53
% 0.30/0.53 % bits_mod_div_trivial
% 0.30/0.53 thf(fact_1963_mod__div__trivial,axiom,
% 0.30/0.53 ! [A: nat,B: nat] :
% 0.30/0.53 ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 0.30/0.53 = zero_zero_nat ) ).
% 0.30/0.53
% 0.30/0.53 % mod_div_trivial
% 0.30/0.53 thf(fact_1964_mod__div__trivial,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 0.30/0.53 = zero_zero_int ) ).
% 0.30/0.53
% 0.30/0.53 % mod_div_trivial
% 0.30/0.53 thf(fact_1965_mod__div__trivial,axiom,
% 0.30/0.53 ! [A: code_integer,B: code_integer] :
% 0.30/0.53 ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 0.30/0.53 = zero_z3403309356797280102nteger ) ).
% 0.30/0.53
% 0.30/0.53 % mod_div_trivial
% 0.30/0.53 thf(fact_1966_mod__mult__self4,axiom,
% 0.30/0.53 ! [B: nat,C: nat,A: nat] :
% 0.30/0.53 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 0.30/0.53 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_mult_self4
% 0.30/0.53 thf(fact_1967_mod__mult__self4,axiom,
% 0.30/0.53 ! [B: int,C: int,A: int] :
% 0.30/0.53 ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 0.30/0.53 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_mult_self4
% 0.30/0.53 thf(fact_1968_mod__mult__self4,axiom,
% 0.30/0.53 ! [B: code_integer,C: code_integer,A: code_integer] :
% 0.30/0.53 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
% 0.30/0.53 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_mult_self4
% 0.30/0.53 thf(fact_1969_mod__mult__self3,axiom,
% 0.30/0.53 ! [C: nat,B: nat,A: nat] :
% 0.30/0.53 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 0.30/0.53 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_mult_self3
% 0.30/0.53 thf(fact_1970_mod__mult__self3,axiom,
% 0.30/0.53 ! [C: int,B: int,A: int] :
% 0.30/0.53 ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 0.30/0.53 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_mult_self3
% 0.30/0.53 thf(fact_1971_mod__mult__self3,axiom,
% 0.30/0.53 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.30/0.53 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
% 0.30/0.53 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_mult_self3
% 0.30/0.53 thf(fact_1972_mod__mult__self2,axiom,
% 0.30/0.53 ! [A: nat,B: nat,C: nat] :
% 0.30/0.53 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 0.30/0.53 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_mult_self2
% 0.30/0.53 thf(fact_1973_mod__mult__self2,axiom,
% 0.30/0.53 ! [A: int,B: int,C: int] :
% 0.30/0.53 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 0.30/0.53 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_mult_self2
% 0.30/0.53 thf(fact_1974_mod__mult__self2,axiom,
% 0.30/0.53 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.53 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
% 0.30/0.53 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_mult_self2
% 0.30/0.53 thf(fact_1975_mod__mult__self1,axiom,
% 0.30/0.53 ! [A: nat,C: nat,B: nat] :
% 0.30/0.53 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 0.30/0.53 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_mult_self1
% 0.30/0.53 thf(fact_1976_mod__mult__self1,axiom,
% 0.30/0.53 ! [A: int,C: int,B: int] :
% 0.30/0.53 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 0.30/0.53 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_mult_self1
% 0.30/0.53 thf(fact_1977_mod__mult__self1,axiom,
% 0.30/0.53 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.53 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
% 0.30/0.53 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_mult_self1
% 0.30/0.53 thf(fact_1978_less__Suc0,axiom,
% 0.30/0.53 ! [N2: nat] :
% 0.30/0.53 ( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 0.30/0.53 = ( N2 = zero_zero_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % less_Suc0
% 0.30/0.53 thf(fact_1979_zero__less__Suc,axiom,
% 0.30/0.53 ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_Suc
% 0.30/0.53 thf(fact_1980_max__0__1_I4_J,axiom,
% 0.30/0.53 ! [X: num] :
% 0.30/0.53 ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ zero_zero_real )
% 0.30/0.53 = ( numeral_numeral_real @ X ) ) ).
% 0.30/0.53
% 0.30/0.53 % max_0_1(4)
% 0.30/0.53 thf(fact_1981_max__0__1_I4_J,axiom,
% 0.30/0.53 ! [X: num] :
% 0.30/0.53 ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ zero_zero_rat )
% 0.30/0.53 = ( numeral_numeral_rat @ X ) ) ).
% 0.30/0.53
% 0.30/0.53 % max_0_1(4)
% 0.30/0.53 thf(fact_1982_max__0__1_I4_J,axiom,
% 0.30/0.53 ! [X: num] :
% 0.30/0.53 ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ zero_zero_nat )
% 0.30/0.53 = ( numeral_numeral_nat @ X ) ) ).
% 0.30/0.53
% 0.30/0.53 % max_0_1(4)
% 0.30/0.53 thf(fact_1983_max__0__1_I4_J,axiom,
% 0.30/0.53 ! [X: num] :
% 0.30/0.53 ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ zero_zero_int )
% 0.30/0.53 = ( numeral_numeral_int @ X ) ) ).
% 0.30/0.53
% 0.30/0.53 % max_0_1(4)
% 0.30/0.53 thf(fact_1984_max__0__1_I3_J,axiom,
% 0.30/0.53 ! [X: num] :
% 0.30/0.53 ( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X ) )
% 0.30/0.53 = ( numeral_numeral_real @ X ) ) ).
% 0.30/0.53
% 0.30/0.53 % max_0_1(3)
% 0.30/0.53 thf(fact_1985_max__0__1_I3_J,axiom,
% 0.30/0.53 ! [X: num] :
% 0.30/0.53 ( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X ) )
% 0.30/0.53 = ( numeral_numeral_rat @ X ) ) ).
% 0.30/0.53
% 0.30/0.53 % max_0_1(3)
% 0.30/0.53 thf(fact_1986_max__0__1_I3_J,axiom,
% 0.30/0.53 ! [X: num] :
% 0.30/0.53 ( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X ) )
% 0.30/0.53 = ( numeral_numeral_nat @ X ) ) ).
% 0.30/0.53
% 0.30/0.53 % max_0_1(3)
% 0.30/0.53 thf(fact_1987_max__0__1_I3_J,axiom,
% 0.30/0.53 ! [X: num] :
% 0.30/0.53 ( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X ) )
% 0.30/0.53 = ( numeral_numeral_int @ X ) ) ).
% 0.30/0.53
% 0.30/0.53 % max_0_1(3)
% 0.30/0.53 thf(fact_1988_one__eq__mult__iff,axiom,
% 0.30/0.53 ! [M: nat,N2: nat] :
% 0.30/0.53 ( ( ( suc @ zero_zero_nat )
% 0.30/0.53 = ( times_times_nat @ M @ N2 ) )
% 0.30/0.53 = ( ( M
% 0.30/0.53 = ( suc @ zero_zero_nat ) )
% 0.30/0.53 & ( N2
% 0.30/0.53 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % one_eq_mult_iff
% 0.30/0.53 thf(fact_1989_mult__eq__1__iff,axiom,
% 0.30/0.53 ! [M: nat,N2: nat] :
% 0.30/0.53 ( ( ( times_times_nat @ M @ N2 )
% 0.30/0.53 = ( suc @ zero_zero_nat ) )
% 0.30/0.53 = ( ( M
% 0.30/0.53 = ( suc @ zero_zero_nat ) )
% 0.30/0.53 & ( N2
% 0.30/0.53 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_eq_1_iff
% 0.30/0.53 thf(fact_1990_add__gr__0,axiom,
% 0.30/0.53 ! [M: nat,N2: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N2 ) )
% 0.30/0.53 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.53 | ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % add_gr_0
% 0.30/0.53 thf(fact_1991_max__0__1_I1_J,axiom,
% 0.30/0.53 ( ( ord_max_real @ zero_zero_real @ one_one_real )
% 0.30/0.53 = one_one_real ) ).
% 0.30/0.53
% 0.30/0.53 % max_0_1(1)
% 0.30/0.53 thf(fact_1992_max__0__1_I1_J,axiom,
% 0.30/0.53 ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
% 0.30/0.53 = one_one_rat ) ).
% 0.30/0.53
% 0.30/0.53 % max_0_1(1)
% 0.30/0.53 thf(fact_1993_max__0__1_I1_J,axiom,
% 0.30/0.53 ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
% 0.30/0.53 = one_one_nat ) ).
% 0.30/0.53
% 0.30/0.53 % max_0_1(1)
% 0.30/0.53 thf(fact_1994_max__0__1_I1_J,axiom,
% 0.30/0.53 ( ( ord_max_int @ zero_zero_int @ one_one_int )
% 0.30/0.53 = one_one_int ) ).
% 0.30/0.53
% 0.30/0.53 % max_0_1(1)
% 0.30/0.53 thf(fact_1995_max__0__1_I2_J,axiom,
% 0.30/0.53 ( ( ord_max_real @ one_one_real @ zero_zero_real )
% 0.30/0.53 = one_one_real ) ).
% 0.30/0.53
% 0.30/0.53 % max_0_1(2)
% 0.30/0.53 thf(fact_1996_max__0__1_I2_J,axiom,
% 0.30/0.53 ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
% 0.30/0.53 = one_one_rat ) ).
% 0.30/0.53
% 0.30/0.53 % max_0_1(2)
% 0.30/0.53 thf(fact_1997_max__0__1_I2_J,axiom,
% 0.30/0.53 ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
% 0.30/0.53 = one_one_nat ) ).
% 0.30/0.53
% 0.30/0.53 % max_0_1(2)
% 0.30/0.53 thf(fact_1998_max__0__1_I2_J,axiom,
% 0.30/0.53 ( ( ord_max_int @ one_one_int @ zero_zero_int )
% 0.30/0.53 = one_one_int ) ).
% 0.30/0.53
% 0.30/0.53 % max_0_1(2)
% 0.30/0.53 thf(fact_1999_div__by__Suc__0,axiom,
% 0.30/0.53 ! [M: nat] :
% 0.30/0.53 ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.30/0.53 = M ) ).
% 0.30/0.53
% 0.30/0.53 % div_by_Suc_0
% 0.30/0.53 thf(fact_2000_mult__less__cancel2,axiom,
% 0.30/0.53 ! [M: nat,K: nat,N2: nat] :
% 0.30/0.53 ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
% 0.30/0.53 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.53 & ( ord_less_nat @ M @ N2 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_less_cancel2
% 0.30/0.53 thf(fact_2001_nat__0__less__mult__iff,axiom,
% 0.30/0.53 ! [M: nat,N2: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N2 ) )
% 0.30/0.53 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.53 & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nat_0_less_mult_iff
% 0.30/0.53 thf(fact_2002_nat__mult__less__cancel__disj,axiom,
% 0.30/0.53 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.53 ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 0.30/0.53 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.53 & ( ord_less_nat @ M @ N2 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nat_mult_less_cancel_disj
% 0.30/0.53 thf(fact_2003_power__Suc__0,axiom,
% 0.30/0.53 ! [N2: nat] :
% 0.30/0.53 ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 0.30/0.53 = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_Suc_0
% 0.30/0.53 thf(fact_2004_nat__power__eq__Suc__0__iff,axiom,
% 0.30/0.53 ! [X: nat,M: nat] :
% 0.30/0.53 ( ( ( power_power_nat @ X @ M )
% 0.30/0.53 = ( suc @ zero_zero_nat ) )
% 0.30/0.53 = ( ( M = zero_zero_nat )
% 0.30/0.53 | ( X
% 0.30/0.53 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nat_power_eq_Suc_0_iff
% 0.30/0.53 thf(fact_2005_zero__less__diff,axiom,
% 0.30/0.53 ! [N2: nat,M: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M ) )
% 0.30/0.53 = ( ord_less_nat @ M @ N2 ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_diff
% 0.30/0.53 thf(fact_2006_div__less,axiom,
% 0.30/0.53 ! [M: nat,N2: nat] :
% 0.30/0.53 ( ( ord_less_nat @ M @ N2 )
% 0.30/0.53 => ( ( divide_divide_nat @ M @ N2 )
% 0.30/0.53 = zero_zero_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % div_less
% 0.30/0.53 thf(fact_2007_nat__zero__less__power__iff,axiom,
% 0.30/0.53 ! [X: nat,N2: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N2 ) )
% 0.30/0.53 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.30/0.53 | ( N2 = zero_zero_nat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nat_zero_less_power_iff
% 0.30/0.53 thf(fact_2008_diff__is__0__eq_H,axiom,
% 0.30/0.53 ! [M: nat,N2: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.53 => ( ( minus_minus_nat @ M @ N2 )
% 0.30/0.53 = zero_zero_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % diff_is_0_eq'
% 0.30/0.53 thf(fact_2009_diff__is__0__eq,axiom,
% 0.30/0.53 ! [M: nat,N2: nat] :
% 0.30/0.53 ( ( ( minus_minus_nat @ M @ N2 )
% 0.30/0.53 = zero_zero_nat )
% 0.30/0.53 = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.30/0.53
% 0.30/0.53 % diff_is_0_eq
% 0.30/0.53 thf(fact_2010_less__one,axiom,
% 0.30/0.53 ! [N2: nat] :
% 0.30/0.53 ( ( ord_less_nat @ N2 @ one_one_nat )
% 0.30/0.53 = ( N2 = zero_zero_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % less_one
% 0.30/0.53 thf(fact_2011_nat__mult__div__cancel__disj,axiom,
% 0.30/0.53 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.53 ( ( ( K = zero_zero_nat )
% 0.30/0.53 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 0.30/0.53 = zero_zero_nat ) )
% 0.30/0.53 & ( ( K != zero_zero_nat )
% 0.30/0.53 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 0.30/0.53 = ( divide_divide_nat @ M @ N2 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nat_mult_div_cancel_disj
% 0.30/0.53 thf(fact_2012_mod__by__Suc__0,axiom,
% 0.30/0.53 ! [M: nat] :
% 0.30/0.53 ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.30/0.53 = zero_zero_nat ) ).
% 0.30/0.53
% 0.30/0.53 % mod_by_Suc_0
% 0.30/0.53 thf(fact_2013_zero__le__divide__1__iff,axiom,
% 0.30/0.53 ! [A: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 0.30/0.53 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_le_divide_1_iff
% 0.30/0.53 thf(fact_2014_zero__le__divide__1__iff,axiom,
% 0.30/0.53 ! [A: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 0.30/0.53 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_le_divide_1_iff
% 0.30/0.53 thf(fact_2015_divide__le__0__1__iff,axiom,
% 0.30/0.53 ! [A: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 0.30/0.53 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_le_0_1_iff
% 0.30/0.53 thf(fact_2016_divide__le__0__1__iff,axiom,
% 0.30/0.53 ! [A: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 0.30/0.53 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_le_0_1_iff
% 0.30/0.53 thf(fact_2017_eq__divide__eq__numeral1_I1_J,axiom,
% 0.30/0.53 ! [A: complex,B: complex,W: num] :
% 0.30/0.53 ( ( A
% 0.30/0.53 = ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) ) )
% 0.30/0.53 = ( ( ( ( numera6690914467698888265omplex @ W )
% 0.30/0.53 != zero_zero_complex )
% 0.30/0.53 => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) )
% 0.30/0.53 = B ) )
% 0.30/0.53 & ( ( ( numera6690914467698888265omplex @ W )
% 0.30/0.53 = zero_zero_complex )
% 0.30/0.53 => ( A = zero_zero_complex ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % eq_divide_eq_numeral1(1)
% 0.30/0.53 thf(fact_2018_eq__divide__eq__numeral1_I1_J,axiom,
% 0.30/0.53 ! [A: real,B: real,W: num] :
% 0.30/0.53 ( ( A
% 0.30/0.53 = ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 0.30/0.53 = ( ( ( ( numeral_numeral_real @ W )
% 0.30/0.53 != zero_zero_real )
% 0.30/0.53 => ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
% 0.30/0.53 = B ) )
% 0.30/0.53 & ( ( ( numeral_numeral_real @ W )
% 0.30/0.53 = zero_zero_real )
% 0.30/0.53 => ( A = zero_zero_real ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % eq_divide_eq_numeral1(1)
% 0.30/0.53 thf(fact_2019_eq__divide__eq__numeral1_I1_J,axiom,
% 0.30/0.53 ! [A: rat,B: rat,W: num] :
% 0.30/0.53 ( ( A
% 0.30/0.53 = ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 0.30/0.53 = ( ( ( ( numeral_numeral_rat @ W )
% 0.30/0.53 != zero_zero_rat )
% 0.30/0.53 => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
% 0.30/0.53 = B ) )
% 0.30/0.53 & ( ( ( numeral_numeral_rat @ W )
% 0.30/0.53 = zero_zero_rat )
% 0.30/0.53 => ( A = zero_zero_rat ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % eq_divide_eq_numeral1(1)
% 0.30/0.53 thf(fact_2020_divide__eq__eq__numeral1_I1_J,axiom,
% 0.30/0.53 ! [B: complex,W: num,A: complex] :
% 0.30/0.53 ( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) )
% 0.30/0.53 = A )
% 0.30/0.53 = ( ( ( ( numera6690914467698888265omplex @ W )
% 0.30/0.53 != zero_zero_complex )
% 0.30/0.53 => ( B
% 0.30/0.53 = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) ) )
% 0.30/0.53 & ( ( ( numera6690914467698888265omplex @ W )
% 0.30/0.53 = zero_zero_complex )
% 0.30/0.53 => ( A = zero_zero_complex ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_eq_eq_numeral1(1)
% 0.30/0.53 thf(fact_2021_divide__eq__eq__numeral1_I1_J,axiom,
% 0.30/0.53 ! [B: real,W: num,A: real] :
% 0.30/0.53 ( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) )
% 0.30/0.53 = A )
% 0.30/0.53 = ( ( ( ( numeral_numeral_real @ W )
% 0.30/0.53 != zero_zero_real )
% 0.30/0.53 => ( B
% 0.30/0.53 = ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
% 0.30/0.53 & ( ( ( numeral_numeral_real @ W )
% 0.30/0.53 = zero_zero_real )
% 0.30/0.53 => ( A = zero_zero_real ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_eq_eq_numeral1(1)
% 0.30/0.53 thf(fact_2022_divide__eq__eq__numeral1_I1_J,axiom,
% 0.30/0.53 ! [B: rat,W: num,A: rat] :
% 0.30/0.53 ( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) )
% 0.30/0.53 = A )
% 0.30/0.53 = ( ( ( ( numeral_numeral_rat @ W )
% 0.30/0.53 != zero_zero_rat )
% 0.30/0.53 => ( B
% 0.30/0.53 = ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
% 0.30/0.53 & ( ( ( numeral_numeral_rat @ W )
% 0.30/0.53 = zero_zero_rat )
% 0.30/0.53 => ( A = zero_zero_rat ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_eq_eq_numeral1(1)
% 0.30/0.53 thf(fact_2023_divide__less__0__1__iff,axiom,
% 0.30/0.53 ! [A: real] :
% 0.30/0.53 ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 0.30/0.53 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_less_0_1_iff
% 0.30/0.53 thf(fact_2024_divide__less__0__1__iff,axiom,
% 0.30/0.53 ! [A: rat] :
% 0.30/0.53 ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 0.30/0.53 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_less_0_1_iff
% 0.30/0.53 thf(fact_2025_divide__less__eq__1__neg,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.53 => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.30/0.53 = ( ord_less_real @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_less_eq_1_neg
% 0.30/0.53 thf(fact_2026_divide__less__eq__1__neg,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.53 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.30/0.53 = ( ord_less_rat @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_less_eq_1_neg
% 0.30/0.53 thf(fact_2027_divide__less__eq__1__pos,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.53 => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.30/0.53 = ( ord_less_real @ B @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_less_eq_1_pos
% 0.30/0.53 thf(fact_2028_divide__less__eq__1__pos,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.53 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.30/0.53 = ( ord_less_rat @ B @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_less_eq_1_pos
% 0.30/0.53 thf(fact_2029_less__divide__eq__1__neg,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.53 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.30/0.53 = ( ord_less_real @ B @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % less_divide_eq_1_neg
% 0.30/0.53 thf(fact_2030_less__divide__eq__1__neg,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.53 => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.30/0.53 = ( ord_less_rat @ B @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % less_divide_eq_1_neg
% 0.30/0.53 thf(fact_2031_less__divide__eq__1__pos,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.53 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.30/0.53 = ( ord_less_real @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % less_divide_eq_1_pos
% 0.30/0.53 thf(fact_2032_less__divide__eq__1__pos,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.53 => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.30/0.53 = ( ord_less_rat @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % less_divide_eq_1_pos
% 0.30/0.53 thf(fact_2033_zero__less__divide__1__iff,axiom,
% 0.30/0.53 ! [A: real] :
% 0.30/0.53 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 0.30/0.53 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_divide_1_iff
% 0.30/0.53 thf(fact_2034_zero__less__divide__1__iff,axiom,
% 0.30/0.53 ! [A: rat] :
% 0.30/0.53 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 0.30/0.53 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_divide_1_iff
% 0.30/0.53 thf(fact_2035_div__mult__self4,axiom,
% 0.30/0.53 ! [B: nat,C: nat,A: nat] :
% 0.30/0.53 ( ( B != zero_zero_nat )
% 0.30/0.53 => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 0.30/0.53 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % div_mult_self4
% 0.30/0.53 thf(fact_2036_div__mult__self4,axiom,
% 0.30/0.53 ! [B: int,C: int,A: int] :
% 0.30/0.53 ( ( B != zero_zero_int )
% 0.30/0.53 => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 0.30/0.53 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % div_mult_self4
% 0.30/0.53 thf(fact_2037_div__mult__self3,axiom,
% 0.30/0.53 ! [B: nat,C: nat,A: nat] :
% 0.30/0.53 ( ( B != zero_zero_nat )
% 0.30/0.53 => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 0.30/0.53 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % div_mult_self3
% 0.30/0.53 thf(fact_2038_div__mult__self3,axiom,
% 0.30/0.53 ! [B: int,C: int,A: int] :
% 0.30/0.53 ( ( B != zero_zero_int )
% 0.30/0.53 => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 0.30/0.53 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % div_mult_self3
% 0.30/0.53 thf(fact_2039_div__mult__self2,axiom,
% 0.30/0.53 ! [B: nat,A: nat,C: nat] :
% 0.30/0.53 ( ( B != zero_zero_nat )
% 0.30/0.53 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 0.30/0.53 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % div_mult_self2
% 0.30/0.53 thf(fact_2040_div__mult__self2,axiom,
% 0.30/0.53 ! [B: int,A: int,C: int] :
% 0.30/0.53 ( ( B != zero_zero_int )
% 0.30/0.53 => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 0.30/0.53 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % div_mult_self2
% 0.30/0.53 thf(fact_2041_div__mult__self1,axiom,
% 0.30/0.53 ! [B: nat,A: nat,C: nat] :
% 0.30/0.53 ( ( B != zero_zero_nat )
% 0.30/0.53 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 0.30/0.53 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % div_mult_self1
% 0.30/0.53 thf(fact_2042_div__mult__self1,axiom,
% 0.30/0.53 ! [B: int,A: int,C: int] :
% 0.30/0.53 ( ( B != zero_zero_int )
% 0.30/0.53 => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 0.30/0.53 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % div_mult_self1
% 0.30/0.53 thf(fact_2043_nonzero__divide__mult__cancel__right,axiom,
% 0.30/0.53 ! [B: complex,A: complex] :
% 0.30/0.53 ( ( B != zero_zero_complex )
% 0.30/0.53 => ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
% 0.30/0.53 = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_divide_mult_cancel_right
% 0.30/0.53 thf(fact_2044_nonzero__divide__mult__cancel__right,axiom,
% 0.30/0.53 ! [B: real,A: real] :
% 0.30/0.53 ( ( B != zero_zero_real )
% 0.30/0.53 => ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
% 0.30/0.53 = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_divide_mult_cancel_right
% 0.30/0.53 thf(fact_2045_nonzero__divide__mult__cancel__right,axiom,
% 0.30/0.53 ! [B: rat,A: rat] :
% 0.30/0.53 ( ( B != zero_zero_rat )
% 0.30/0.53 => ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
% 0.30/0.53 = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_divide_mult_cancel_right
% 0.30/0.53 thf(fact_2046_nonzero__divide__mult__cancel__left,axiom,
% 0.30/0.53 ! [A: complex,B: complex] :
% 0.30/0.53 ( ( A != zero_zero_complex )
% 0.30/0.53 => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
% 0.30/0.53 = ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_divide_mult_cancel_left
% 0.30/0.53 thf(fact_2047_nonzero__divide__mult__cancel__left,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( A != zero_zero_real )
% 0.30/0.53 => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
% 0.30/0.53 = ( divide_divide_real @ one_one_real @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_divide_mult_cancel_left
% 0.30/0.53 thf(fact_2048_nonzero__divide__mult__cancel__left,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( A != zero_zero_rat )
% 0.30/0.53 => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
% 0.30/0.53 = ( divide_divide_rat @ one_one_rat @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nonzero_divide_mult_cancel_left
% 0.30/0.53 thf(fact_2049_power__eq__0__iff,axiom,
% 0.30/0.53 ! [A: rat,N2: nat] :
% 0.30/0.53 ( ( ( power_power_rat @ A @ N2 )
% 0.30/0.53 = zero_zero_rat )
% 0.30/0.53 = ( ( A = zero_zero_rat )
% 0.30/0.53 & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_eq_0_iff
% 0.30/0.53 thf(fact_2050_power__eq__0__iff,axiom,
% 0.30/0.53 ! [A: nat,N2: nat] :
% 0.30/0.53 ( ( ( power_power_nat @ A @ N2 )
% 0.30/0.53 = zero_zero_nat )
% 0.30/0.53 = ( ( A = zero_zero_nat )
% 0.30/0.53 & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_eq_0_iff
% 0.30/0.53 thf(fact_2051_power__eq__0__iff,axiom,
% 0.30/0.53 ! [A: real,N2: nat] :
% 0.30/0.53 ( ( ( power_power_real @ A @ N2 )
% 0.30/0.53 = zero_zero_real )
% 0.30/0.53 = ( ( A = zero_zero_real )
% 0.30/0.53 & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_eq_0_iff
% 0.30/0.53 thf(fact_2052_power__eq__0__iff,axiom,
% 0.30/0.53 ! [A: int,N2: nat] :
% 0.30/0.53 ( ( ( power_power_int @ A @ N2 )
% 0.30/0.53 = zero_zero_int )
% 0.30/0.53 = ( ( A = zero_zero_int )
% 0.30/0.53 & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_eq_0_iff
% 0.30/0.53 thf(fact_2053_power__eq__0__iff,axiom,
% 0.30/0.53 ! [A: complex,N2: nat] :
% 0.30/0.53 ( ( ( power_power_complex @ A @ N2 )
% 0.30/0.53 = zero_zero_complex )
% 0.30/0.53 = ( ( A = zero_zero_complex )
% 0.30/0.53 & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_eq_0_iff
% 0.30/0.53 thf(fact_2054_Suc__pred,axiom,
% 0.30/0.53 ! [N2: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.53 => ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
% 0.30/0.53 = N2 ) ) ).
% 0.30/0.53
% 0.30/0.53 % Suc_pred
% 0.30/0.53 thf(fact_2055_one__le__mult__iff,axiom,
% 0.30/0.53 ! [M: nat,N2: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) )
% 0.30/0.53 = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.30/0.53 & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % one_le_mult_iff
% 0.30/0.53 thf(fact_2056_mult__le__cancel2,axiom,
% 0.30/0.53 ! [M: nat,K: nat,N2: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
% 0.30/0.53 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.53 => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_le_cancel2
% 0.30/0.53 thf(fact_2057_nat__mult__le__cancel__disj,axiom,
% 0.30/0.53 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 0.30/0.53 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.53 => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % nat_mult_le_cancel_disj
% 0.30/0.53 thf(fact_2058_div__mult__self1__is__m,axiom,
% 0.30/0.53 ! [N2: nat,M: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.53 => ( ( divide_divide_nat @ ( times_times_nat @ N2 @ M ) @ N2 )
% 0.30/0.53 = M ) ) ).
% 0.30/0.53
% 0.30/0.53 % div_mult_self1_is_m
% 0.30/0.53 thf(fact_2059_div__mult__self__is__m,axiom,
% 0.30/0.53 ! [N2: nat,M: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.53 => ( ( divide_divide_nat @ ( times_times_nat @ M @ N2 ) @ N2 )
% 0.30/0.53 = M ) ) ).
% 0.30/0.53
% 0.30/0.53 % div_mult_self_is_m
% 0.30/0.53 thf(fact_2060_Suc__mod__mult__self4,axiom,
% 0.30/0.53 ! [N2: nat,K: nat,M: nat] :
% 0.30/0.53 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N2 @ K ) @ M ) ) @ N2 )
% 0.30/0.53 = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).
% 0.30/0.53
% 0.30/0.53 % Suc_mod_mult_self4
% 0.30/0.53 thf(fact_2061_Suc__mod__mult__self3,axiom,
% 0.30/0.53 ! [K: nat,N2: nat,M: nat] :
% 0.30/0.53 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N2 ) @ M ) ) @ N2 )
% 0.30/0.53 = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).
% 0.30/0.53
% 0.30/0.53 % Suc_mod_mult_self3
% 0.30/0.53 thf(fact_2062_Suc__mod__mult__self2,axiom,
% 0.30/0.53 ! [M: nat,N2: nat,K: nat] :
% 0.30/0.53 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N2 @ K ) ) ) @ N2 )
% 0.30/0.53 = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).
% 0.30/0.53
% 0.30/0.53 % Suc_mod_mult_self2
% 0.30/0.53 thf(fact_2063_Suc__mod__mult__self1,axiom,
% 0.30/0.53 ! [M: nat,K: nat,N2: nat] :
% 0.30/0.53 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N2 ) ) ) @ N2 )
% 0.30/0.53 = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).
% 0.30/0.53
% 0.30/0.53 % Suc_mod_mult_self1
% 0.30/0.53 thf(fact_2064_divide__le__eq__1__neg,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.53 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.30/0.53 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_le_eq_1_neg
% 0.30/0.53 thf(fact_2065_divide__le__eq__1__neg,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.53 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.30/0.53 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_le_eq_1_neg
% 0.30/0.53 thf(fact_2066_divide__le__eq__1__pos,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.53 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.30/0.53 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_le_eq_1_pos
% 0.30/0.53 thf(fact_2067_divide__le__eq__1__pos,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.53 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.30/0.53 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % divide_le_eq_1_pos
% 0.30/0.53 thf(fact_2068_le__divide__eq__1__neg,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.53 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.30/0.53 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % le_divide_eq_1_neg
% 0.30/0.53 thf(fact_2069_le__divide__eq__1__neg,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.53 => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.30/0.53 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % le_divide_eq_1_neg
% 0.30/0.53 thf(fact_2070_le__divide__eq__1__pos,axiom,
% 0.30/0.53 ! [A: real,B: real] :
% 0.30/0.53 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.53 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.30/0.53 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % le_divide_eq_1_pos
% 0.30/0.53 thf(fact_2071_le__divide__eq__1__pos,axiom,
% 0.30/0.53 ! [A: rat,B: rat] :
% 0.30/0.53 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.53 => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.30/0.53 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % le_divide_eq_1_pos
% 0.30/0.53 thf(fact_2072_power__strict__decreasing__iff,axiom,
% 0.30/0.53 ! [B: real,M: nat,N2: nat] :
% 0.30/0.53 ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/0.53 => ( ( ord_less_real @ B @ one_one_real )
% 0.30/0.53 => ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N2 ) )
% 0.30/0.53 = ( ord_less_nat @ N2 @ M ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_strict_decreasing_iff
% 0.30/0.53 thf(fact_2073_power__strict__decreasing__iff,axiom,
% 0.30/0.53 ! [B: rat,M: nat,N2: nat] :
% 0.30/0.53 ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.30/0.53 => ( ( ord_less_rat @ B @ one_one_rat )
% 0.30/0.53 => ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N2 ) )
% 0.30/0.53 = ( ord_less_nat @ N2 @ M ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_strict_decreasing_iff
% 0.30/0.53 thf(fact_2074_power__strict__decreasing__iff,axiom,
% 0.30/0.53 ! [B: nat,M: nat,N2: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.53 => ( ( ord_less_nat @ B @ one_one_nat )
% 0.30/0.53 => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N2 ) )
% 0.30/0.53 = ( ord_less_nat @ N2 @ M ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_strict_decreasing_iff
% 0.30/0.53 thf(fact_2075_power__strict__decreasing__iff,axiom,
% 0.30/0.53 ! [B: int,M: nat,N2: nat] :
% 0.30/0.53 ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.53 => ( ( ord_less_int @ B @ one_one_int )
% 0.30/0.53 => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N2 ) )
% 0.30/0.53 = ( ord_less_nat @ N2 @ M ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_strict_decreasing_iff
% 0.30/0.53 thf(fact_2076_zero__eq__power2,axiom,
% 0.30/0.53 ! [A: rat] :
% 0.30/0.53 ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.53 = zero_zero_rat )
% 0.30/0.53 = ( A = zero_zero_rat ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_eq_power2
% 0.30/0.53 thf(fact_2077_zero__eq__power2,axiom,
% 0.30/0.53 ! [A: nat] :
% 0.30/0.53 ( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.53 = zero_zero_nat )
% 0.30/0.53 = ( A = zero_zero_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_eq_power2
% 0.30/0.53 thf(fact_2078_zero__eq__power2,axiom,
% 0.30/0.53 ! [A: real] :
% 0.30/0.53 ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.53 = zero_zero_real )
% 0.30/0.53 = ( A = zero_zero_real ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_eq_power2
% 0.30/0.53 thf(fact_2079_zero__eq__power2,axiom,
% 0.30/0.53 ! [A: int] :
% 0.30/0.53 ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.53 = zero_zero_int )
% 0.30/0.53 = ( A = zero_zero_int ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_eq_power2
% 0.30/0.53 thf(fact_2080_zero__eq__power2,axiom,
% 0.30/0.53 ! [A: complex] :
% 0.30/0.53 ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.53 = zero_zero_complex )
% 0.30/0.53 = ( A = zero_zero_complex ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_eq_power2
% 0.30/0.53 thf(fact_2081_power__mono__iff,axiom,
% 0.30/0.53 ! [A: real,B: real,N2: nat] :
% 0.30/0.53 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.53 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.53 => ( ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) )
% 0.30/0.53 = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_mono_iff
% 0.30/0.53 thf(fact_2082_power__mono__iff,axiom,
% 0.30/0.53 ! [A: rat,B: rat,N2: nat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.30/0.53 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.53 => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) )
% 0.30/0.53 = ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_mono_iff
% 0.30/0.53 thf(fact_2083_power__mono__iff,axiom,
% 0.30/0.53 ! [A: nat,B: nat,N2: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.30/0.53 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.53 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
% 0.30/0.53 = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_mono_iff
% 0.30/0.53 thf(fact_2084_power__mono__iff,axiom,
% 0.30/0.53 ! [A: int,B: int,N2: nat] :
% 0.30/0.53 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.30/0.53 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.53 => ( ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
% 0.30/0.53 = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_mono_iff
% 0.30/0.53 thf(fact_2085_one__mod__two__eq__one,axiom,
% 0.30/0.53 ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.53 = one_one_nat ) ).
% 0.30/0.53
% 0.30/0.53 % one_mod_two_eq_one
% 0.30/0.53 thf(fact_2086_one__mod__two__eq__one,axiom,
% 0.30/0.53 ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.53 = one_one_int ) ).
% 0.30/0.53
% 0.30/0.53 % one_mod_two_eq_one
% 0.30/0.53 thf(fact_2087_one__mod__two__eq__one,axiom,
% 0.30/0.53 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.53 = one_one_Code_integer ) ).
% 0.30/0.53
% 0.30/0.53 % one_mod_two_eq_one
% 0.30/0.53 thf(fact_2088_bits__one__mod__two__eq__one,axiom,
% 0.30/0.53 ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.53 = one_one_nat ) ).
% 0.30/0.53
% 0.30/0.53 % bits_one_mod_two_eq_one
% 0.30/0.53 thf(fact_2089_bits__one__mod__two__eq__one,axiom,
% 0.30/0.53 ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.53 = one_one_int ) ).
% 0.30/0.53
% 0.30/0.53 % bits_one_mod_two_eq_one
% 0.30/0.53 thf(fact_2090_bits__one__mod__two__eq__one,axiom,
% 0.30/0.53 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.53 = one_one_Code_integer ) ).
% 0.30/0.53
% 0.30/0.53 % bits_one_mod_two_eq_one
% 0.30/0.53 thf(fact_2091_mod2__Suc__Suc,axiom,
% 0.30/0.53 ! [M: nat] :
% 0.30/0.53 ( ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.53 = ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod2_Suc_Suc
% 0.30/0.53 thf(fact_2092_Suc__diff__1,axiom,
% 0.30/0.53 ! [N2: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.53 => ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
% 0.30/0.53 = N2 ) ) ).
% 0.30/0.53
% 0.30/0.53 % Suc_diff_1
% 0.30/0.53 thf(fact_2093_Suc__times__numeral__mod__eq,axiom,
% 0.30/0.53 ! [K: num,N2: nat] :
% 0.30/0.53 ( ( ( numeral_numeral_nat @ K )
% 0.30/0.53 != one_one_nat )
% 0.30/0.53 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N2 ) ) @ ( numeral_numeral_nat @ K ) )
% 0.30/0.53 = one_one_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % Suc_times_numeral_mod_eq
% 0.30/0.53 thf(fact_2094_one__div__two__eq__zero,axiom,
% 0.30/0.53 ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.53 = zero_zero_nat ) ).
% 0.30/0.53
% 0.30/0.53 % one_div_two_eq_zero
% 0.30/0.53 thf(fact_2095_one__div__two__eq__zero,axiom,
% 0.30/0.53 ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.53 = zero_zero_int ) ).
% 0.30/0.53
% 0.30/0.53 % one_div_two_eq_zero
% 0.30/0.53 thf(fact_2096_bits__1__div__2,axiom,
% 0.30/0.53 ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.53 = zero_zero_nat ) ).
% 0.30/0.53
% 0.30/0.53 % bits_1_div_2
% 0.30/0.53 thf(fact_2097_bits__1__div__2,axiom,
% 0.30/0.53 ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.53 = zero_zero_int ) ).
% 0.30/0.53
% 0.30/0.53 % bits_1_div_2
% 0.30/0.53 thf(fact_2098_power2__eq__iff__nonneg,axiom,
% 0.30/0.53 ! [X: real,Y: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.30/0.53 => ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.53 = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.53 = ( X = Y ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power2_eq_iff_nonneg
% 0.30/0.53 thf(fact_2099_power2__eq__iff__nonneg,axiom,
% 0.30/0.53 ! [X: rat,Y: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.30/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.30/0.53 => ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.53 = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.53 = ( X = Y ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power2_eq_iff_nonneg
% 0.30/0.53 thf(fact_2100_power2__eq__iff__nonneg,axiom,
% 0.30/0.53 ! [X: nat,Y: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.30/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 0.30/0.53 => ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.53 = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.53 = ( X = Y ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power2_eq_iff_nonneg
% 0.30/0.53 thf(fact_2101_power2__eq__iff__nonneg,axiom,
% 0.30/0.53 ! [X: int,Y: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.30/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.30/0.53 => ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.53 = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.53 = ( X = Y ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power2_eq_iff_nonneg
% 0.30/0.53 thf(fact_2102_power2__less__eq__zero__iff,axiom,
% 0.30/0.53 ! [A: real] :
% 0.30/0.53 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
% 0.30/0.53 = ( A = zero_zero_real ) ) ).
% 0.30/0.53
% 0.30/0.53 % power2_less_eq_zero_iff
% 0.30/0.53 thf(fact_2103_power2__less__eq__zero__iff,axiom,
% 0.30/0.53 ! [A: rat] :
% 0.30/0.53 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
% 0.30/0.53 = ( A = zero_zero_rat ) ) ).
% 0.30/0.53
% 0.30/0.53 % power2_less_eq_zero_iff
% 0.30/0.53 thf(fact_2104_power2__less__eq__zero__iff,axiom,
% 0.30/0.53 ! [A: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 0.30/0.53 = ( A = zero_zero_int ) ) ).
% 0.30/0.53
% 0.30/0.53 % power2_less_eq_zero_iff
% 0.30/0.53 thf(fact_2105_zero__less__power2,axiom,
% 0.30/0.53 ! [A: real] :
% 0.30/0.53 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.53 = ( A != zero_zero_real ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_power2
% 0.30/0.53 thf(fact_2106_zero__less__power2,axiom,
% 0.30/0.53 ! [A: rat] :
% 0.30/0.53 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.53 = ( A != zero_zero_rat ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_power2
% 0.30/0.53 thf(fact_2107_zero__less__power2,axiom,
% 0.30/0.53 ! [A: int] :
% 0.30/0.53 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.53 = ( A != zero_zero_int ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_less_power2
% 0.30/0.53 thf(fact_2108_power__decreasing__iff,axiom,
% 0.30/0.53 ! [B: real,M: nat,N2: nat] :
% 0.30/0.53 ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/0.53 => ( ( ord_less_real @ B @ one_one_real )
% 0.30/0.53 => ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N2 ) )
% 0.30/0.53 = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_decreasing_iff
% 0.30/0.53 thf(fact_2109_power__decreasing__iff,axiom,
% 0.30/0.53 ! [B: rat,M: nat,N2: nat] :
% 0.30/0.53 ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.30/0.53 => ( ( ord_less_rat @ B @ one_one_rat )
% 0.30/0.53 => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N2 ) )
% 0.30/0.53 = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_decreasing_iff
% 0.30/0.53 thf(fact_2110_power__decreasing__iff,axiom,
% 0.30/0.53 ! [B: nat,M: nat,N2: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.53 => ( ( ord_less_nat @ B @ one_one_nat )
% 0.30/0.53 => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N2 ) )
% 0.30/0.53 = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_decreasing_iff
% 0.30/0.53 thf(fact_2111_power__decreasing__iff,axiom,
% 0.30/0.53 ! [B: int,M: nat,N2: nat] :
% 0.30/0.53 ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.53 => ( ( ord_less_int @ B @ one_one_int )
% 0.30/0.53 => ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N2 ) )
% 0.30/0.53 = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_decreasing_iff
% 0.30/0.53 thf(fact_2112_sum__power2__eq__zero__iff,axiom,
% 0.30/0.53 ! [X: rat,Y: rat] :
% 0.30/0.53 ( ( ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.53 = zero_zero_rat )
% 0.30/0.53 = ( ( X = zero_zero_rat )
% 0.30/0.53 & ( Y = zero_zero_rat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % sum_power2_eq_zero_iff
% 0.30/0.53 thf(fact_2113_sum__power2__eq__zero__iff,axiom,
% 0.30/0.53 ! [X: real,Y: real] :
% 0.30/0.53 ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.53 = zero_zero_real )
% 0.30/0.53 = ( ( X = zero_zero_real )
% 0.30/0.53 & ( Y = zero_zero_real ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % sum_power2_eq_zero_iff
% 0.30/0.53 thf(fact_2114_sum__power2__eq__zero__iff,axiom,
% 0.30/0.53 ! [X: int,Y: int] :
% 0.30/0.53 ( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.53 = zero_zero_int )
% 0.30/0.53 = ( ( X = zero_zero_int )
% 0.30/0.53 & ( Y = zero_zero_int ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % sum_power2_eq_zero_iff
% 0.30/0.53 thf(fact_2115_not__mod__2__eq__1__eq__0,axiom,
% 0.30/0.53 ! [A: nat] :
% 0.30/0.53 ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.53 != one_one_nat )
% 0.30/0.53 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.53 = zero_zero_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % not_mod_2_eq_1_eq_0
% 0.30/0.53 thf(fact_2116_not__mod__2__eq__1__eq__0,axiom,
% 0.30/0.53 ! [A: int] :
% 0.30/0.53 ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.53 != one_one_int )
% 0.30/0.53 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.53 = zero_zero_int ) ) ).
% 0.30/0.53
% 0.30/0.53 % not_mod_2_eq_1_eq_0
% 0.30/0.53 thf(fact_2117_not__mod__2__eq__1__eq__0,axiom,
% 0.30/0.53 ! [A: code_integer] :
% 0.30/0.53 ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.53 != one_one_Code_integer )
% 0.30/0.53 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.53 = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.53
% 0.30/0.53 % not_mod_2_eq_1_eq_0
% 0.30/0.53 thf(fact_2118_not__mod__2__eq__0__eq__1,axiom,
% 0.30/0.53 ! [A: nat] :
% 0.30/0.53 ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.53 != zero_zero_nat )
% 0.30/0.53 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.53 = one_one_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % not_mod_2_eq_0_eq_1
% 0.30/0.53 thf(fact_2119_not__mod__2__eq__0__eq__1,axiom,
% 0.30/0.53 ! [A: int] :
% 0.30/0.53 ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.53 != zero_zero_int )
% 0.30/0.53 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.53 = one_one_int ) ) ).
% 0.30/0.53
% 0.30/0.53 % not_mod_2_eq_0_eq_1
% 0.30/0.53 thf(fact_2120_not__mod__2__eq__0__eq__1,axiom,
% 0.30/0.53 ! [A: code_integer] :
% 0.30/0.53 ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.53 != zero_z3403309356797280102nteger )
% 0.30/0.53 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.53 = one_one_Code_integer ) ) ).
% 0.30/0.53
% 0.30/0.53 % not_mod_2_eq_0_eq_1
% 0.30/0.53 thf(fact_2121_not__mod2__eq__Suc__0__eq__0,axiom,
% 0.30/0.53 ! [N2: nat] :
% 0.30/0.53 ( ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.53 != ( suc @ zero_zero_nat ) )
% 0.30/0.53 = ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.53 = zero_zero_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % not_mod2_eq_Suc_0_eq_0
% 0.30/0.53 thf(fact_2122_add__self__mod__2,axiom,
% 0.30/0.53 ! [M: nat] :
% 0.30/0.53 ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.53 = zero_zero_nat ) ).
% 0.30/0.53
% 0.30/0.53 % add_self_mod_2
% 0.30/0.53 thf(fact_2123_mod2__gr__0,axiom,
% 0.30/0.53 ! [M: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.53 = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.53 = one_one_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod2_gr_0
% 0.30/0.53 thf(fact_2124_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.30/0.53 ! [B: nat,A: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.53 => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B ) @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.30/0.53 thf(fact_2125_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.30/0.53 ! [B: int,A: int] :
% 0.30/0.53 ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.53 => ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.30/0.53 thf(fact_2126_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.30/0.53 ! [B: code_integer,A: code_integer] :
% 0.30/0.53 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.30/0.53 => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.30/0.53 thf(fact_2127_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.30/0.53 ! [A: code_integer,B: code_integer] :
% 0.30/0.53 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.53 => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).
% 0.30/0.53
% 0.30/0.53 % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.30/0.53 thf(fact_2128_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.30/0.53 ! [A: nat,B: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.53 => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B ) @ A ) ) ).
% 0.30/0.53
% 0.30/0.53 % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.30/0.53 thf(fact_2129_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.53 => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ A ) ) ).
% 0.30/0.53
% 0.30/0.53 % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.30/0.53 thf(fact_2130_mod__Suc,axiom,
% 0.30/0.53 ! [M: nat,N2: nat] :
% 0.30/0.53 ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N2 ) )
% 0.30/0.53 = N2 )
% 0.30/0.53 => ( ( modulo_modulo_nat @ ( suc @ M ) @ N2 )
% 0.30/0.53 = zero_zero_nat ) )
% 0.30/0.53 & ( ( ( suc @ ( modulo_modulo_nat @ M @ N2 ) )
% 0.30/0.53 != N2 )
% 0.30/0.53 => ( ( modulo_modulo_nat @ ( suc @ M ) @ N2 )
% 0.30/0.53 = ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_Suc
% 0.30/0.53 thf(fact_2131_VEBT_Osize_I4_J,axiom,
% 0.30/0.53 ! [X21: $o,X222: $o] :
% 0.30/0.53 ( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 0.30/0.53 = zero_zero_nat ) ).
% 0.30/0.53
% 0.30/0.53 % VEBT.size(4)
% 0.30/0.53 thf(fact_2132_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
% 0.30/0.53 ! [Uu: $o,Uv: $o,Uw: nat] :
% 0.30/0.53 ~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ).
% 0.30/0.53
% 0.30/0.53 % VEBT_internal.membermima.simps(1)
% 0.30/0.53 thf(fact_2133_mod__eq__0D,axiom,
% 0.30/0.53 ! [M: nat,D: nat] :
% 0.30/0.53 ( ( ( modulo_modulo_nat @ M @ D )
% 0.30/0.53 = zero_zero_nat )
% 0.30/0.53 => ? [Q3: nat] :
% 0.30/0.53 ( M
% 0.30/0.53 = ( times_times_nat @ D @ Q3 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_eq_0D
% 0.30/0.53 thf(fact_2134_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
% 0.30/0.53 ! [A: $o,B: $o,X: nat] :
% 0.30/0.53 ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X )
% 0.30/0.53 = ( ( ( X = zero_zero_nat )
% 0.30/0.53 => A )
% 0.30/0.53 & ( ( X != zero_zero_nat )
% 0.30/0.53 => ( ( ( X = one_one_nat )
% 0.30/0.53 => B )
% 0.30/0.53 & ( X = one_one_nat ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % VEBT_internal.naive_member.simps(1)
% 0.30/0.53 thf(fact_2135_mod__eq__self__iff__div__eq__0,axiom,
% 0.30/0.53 ! [A: nat,B: nat] :
% 0.30/0.53 ( ( ( modulo_modulo_nat @ A @ B )
% 0.30/0.53 = A )
% 0.30/0.53 = ( ( divide_divide_nat @ A @ B )
% 0.30/0.53 = zero_zero_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_eq_self_iff_div_eq_0
% 0.30/0.53 thf(fact_2136_mod__eq__self__iff__div__eq__0,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( ( modulo_modulo_int @ A @ B )
% 0.30/0.53 = A )
% 0.30/0.53 = ( ( divide_divide_int @ A @ B )
% 0.30/0.53 = zero_zero_int ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_eq_self_iff_div_eq_0
% 0.30/0.53 thf(fact_2137_mod__eq__self__iff__div__eq__0,axiom,
% 0.30/0.53 ! [A: code_integer,B: code_integer] :
% 0.30/0.53 ( ( ( modulo364778990260209775nteger @ A @ B )
% 0.30/0.53 = A )
% 0.30/0.53 = ( ( divide6298287555418463151nteger @ A @ B )
% 0.30/0.53 = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_eq_self_iff_div_eq_0
% 0.30/0.53 thf(fact_2138_VEBT__internal_Ovalid_H_Ocases,axiom,
% 0.30/0.53 ! [X: produc9072475918466114483BT_nat] :
% 0.30/0.53 ( ! [Uu2: $o,Uv2: $o,D4: nat] :
% 0.30/0.53 ( X
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D4 ) )
% 0.30/0.53 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
% 0.30/0.53 ( X
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Deg3 ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % VEBT_internal.valid'.cases
% 0.30/0.53 thf(fact_2139_mod__less__divisor,axiom,
% 0.30/0.53 ! [N2: nat,M: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.53 => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N2 ) @ N2 ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_less_divisor
% 0.30/0.53 thf(fact_2140_vebt__delete_Osimps_I1_J,axiom,
% 0.30/0.53 ! [A: $o,B: $o] :
% 0.30/0.53 ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat )
% 0.30/0.53 = ( vEBT_Leaf @ $false @ B ) ) ).
% 0.30/0.53
% 0.30/0.53 % vebt_delete.simps(1)
% 0.30/0.53 thf(fact_2141_VEBT__internal_Onaive__member_Ocases,axiom,
% 0.30/0.53 ! [X: produc9072475918466114483BT_nat] :
% 0.30/0.53 ( ! [A3: $o,B2: $o,X4: nat] :
% 0.30/0.53 ( X
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ X4 ) )
% 0.30/0.53 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
% 0.30/0.53 ( X
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
% 0.30/0.53 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT,X4: nat] :
% 0.30/0.53 ( X
% 0.30/0.53 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) @ X4 ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % VEBT_internal.naive_member.cases
% 0.30/0.53 thf(fact_2142_vebt__buildup_Osimps_I1_J,axiom,
% 0.30/0.53 ( ( vEBT_vebt_buildup @ zero_zero_nat )
% 0.30/0.53 = ( vEBT_Leaf @ $false @ $false ) ) ).
% 0.30/0.53
% 0.30/0.53 % vebt_buildup.simps(1)
% 0.30/0.53 thf(fact_2143_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.30/0.53 ! [B: code_integer,A: code_integer] :
% 0.30/0.53 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.30/0.53 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.30/0.53 thf(fact_2144_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.30/0.53 ! [B: nat,A: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.53 => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.30/0.53 thf(fact_2145_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.30/0.53 ! [B: int,A: int] :
% 0.30/0.53 ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.53 => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.30/0.53 thf(fact_2146_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.30/0.53 ! [A: code_integer,B: code_integer] :
% 0.30/0.53 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.53 => ( ( ord_le6747313008572928689nteger @ A @ B )
% 0.30/0.53 => ( ( modulo364778990260209775nteger @ A @ B )
% 0.30/0.53 = A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % unique_euclidean_semiring_numeral_class.mod_less
% 0.30/0.53 thf(fact_2147_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.30/0.53 ! [A: nat,B: nat] :
% 0.30/0.53 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.53 => ( ( ord_less_nat @ A @ B )
% 0.30/0.53 => ( ( modulo_modulo_nat @ A @ B )
% 0.30/0.53 = A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % unique_euclidean_semiring_numeral_class.mod_less
% 0.30/0.53 thf(fact_2148_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.30/0.53 ! [A: int,B: int] :
% 0.30/0.53 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.53 => ( ( ord_less_int @ A @ B )
% 0.30/0.53 => ( ( modulo_modulo_int @ A @ B )
% 0.30/0.53 = A ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % unique_euclidean_semiring_numeral_class.mod_less
% 0.30/0.53 thf(fact_2149_cong__exp__iff__simps_I2_J,axiom,
% 0.30/0.53 ! [N2: num,Q2: num] :
% 0.30/0.53 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.30/0.53 = zero_zero_nat )
% 0.30/0.53 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.30/0.53 = zero_zero_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % cong_exp_iff_simps(2)
% 0.30/0.53 thf(fact_2150_cong__exp__iff__simps_I2_J,axiom,
% 0.30/0.53 ! [N2: num,Q2: num] :
% 0.30/0.53 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.30/0.53 = zero_zero_int )
% 0.30/0.53 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q2 ) )
% 0.30/0.53 = zero_zero_int ) ) ).
% 0.30/0.53
% 0.30/0.53 % cong_exp_iff_simps(2)
% 0.30/0.53 thf(fact_2151_cong__exp__iff__simps_I2_J,axiom,
% 0.30/0.53 ! [N2: num,Q2: num] :
% 0.30/0.53 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.30/0.53 = zero_z3403309356797280102nteger )
% 0.30/0.53 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.30/0.53 = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.53
% 0.30/0.53 % cong_exp_iff_simps(2)
% 0.30/0.53 thf(fact_2152_cong__exp__iff__simps_I1_J,axiom,
% 0.30/0.53 ! [N2: num] :
% 0.30/0.53 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ one ) )
% 0.30/0.53 = zero_zero_nat ) ).
% 0.30/0.53
% 0.30/0.53 % cong_exp_iff_simps(1)
% 0.30/0.53 thf(fact_2153_cong__exp__iff__simps_I1_J,axiom,
% 0.30/0.53 ! [N2: num] :
% 0.30/0.53 ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ one ) )
% 0.30/0.53 = zero_zero_int ) ).
% 0.30/0.53
% 0.30/0.53 % cong_exp_iff_simps(1)
% 0.30/0.53 thf(fact_2154_cong__exp__iff__simps_I1_J,axiom,
% 0.30/0.53 ! [N2: num] :
% 0.30/0.53 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ one ) )
% 0.30/0.53 = zero_z3403309356797280102nteger ) ).
% 0.30/0.53
% 0.30/0.53 % cong_exp_iff_simps(1)
% 0.30/0.53 thf(fact_2155_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
% 0.30/0.53 ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
% 0.30/0.53 ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) ).
% 0.30/0.53
% 0.30/0.53 % VEBT_internal.naive_member.simps(2)
% 0.30/0.53 thf(fact_2156_VEBT__internal_Ooption__shift_Ocases,axiom,
% 0.30/0.53 ! [X: produc8306885398267862888on_nat] :
% 0.30/0.53 ( ! [Uu2: nat > nat > nat,Uv2: option_nat] :
% 0.30/0.53 ( X
% 0.30/0.53 != ( produc8929957630744042906on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
% 0.30/0.53 => ( ! [Uw2: nat > nat > nat,V2: nat] :
% 0.30/0.53 ( X
% 0.30/0.53 != ( produc8929957630744042906on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
% 0.30/0.53 => ~ ! [F2: nat > nat > nat,A3: nat,B2: nat] :
% 0.30/0.53 ( X
% 0.30/0.53 != ( produc8929957630744042906on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A3 ) @ ( some_nat @ B2 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % VEBT_internal.option_shift.cases
% 0.30/0.53 thf(fact_2157_VEBT__internal_Ooption__shift_Ocases,axiom,
% 0.30/0.53 ! [X: produc5542196010084753463at_nat] :
% 0.30/0.53 ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
% 0.30/0.53 ( X
% 0.30/0.53 != ( produc2899441246263362727at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
% 0.30/0.53 => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V2: product_prod_nat_nat] :
% 0.30/0.53 ( X
% 0.30/0.53 != ( produc2899441246263362727at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
% 0.30/0.53 => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A3: product_prod_nat_nat,B2: product_prod_nat_nat] :
% 0.30/0.53 ( X
% 0.30/0.53 != ( produc2899441246263362727at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A3 ) @ ( some_P7363390416028606310at_nat @ B2 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % VEBT_internal.option_shift.cases
% 0.30/0.53 thf(fact_2158_VEBT__internal_Ooption__shift_Ocases,axiom,
% 0.30/0.53 ! [X: produc1193250871479095198on_num] :
% 0.30/0.53 ( ! [Uu2: num > num > num,Uv2: option_num] :
% 0.30/0.53 ( X
% 0.30/0.53 != ( produc5778274026573060048on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
% 0.30/0.53 => ( ! [Uw2: num > num > num,V2: num] :
% 0.30/0.53 ( X
% 0.30/0.53 != ( produc5778274026573060048on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
% 0.30/0.53 => ~ ! [F2: num > num > num,A3: num,B2: num] :
% 0.30/0.53 ( X
% 0.30/0.53 != ( produc5778274026573060048on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ A3 ) @ ( some_num @ B2 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % VEBT_internal.option_shift.cases
% 0.30/0.53 thf(fact_2159_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 0.30/0.53 ! [X: produc2233624965454879586on_nat] :
% 0.30/0.53 ( ! [Uu2: nat > nat > $o,Uv2: option_nat] :
% 0.30/0.53 ( X
% 0.30/0.53 != ( produc4035269172776083154on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
% 0.30/0.53 => ( ! [Uw2: nat > nat > $o,V2: nat] :
% 0.30/0.53 ( X
% 0.30/0.53 != ( produc4035269172776083154on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
% 0.30/0.53 => ~ ! [F2: nat > nat > $o,X4: nat,Y4: nat] :
% 0.30/0.53 ( X
% 0.30/0.53 != ( produc4035269172776083154on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ X4 ) @ ( some_nat @ Y4 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % VEBT_internal.option_comp_shift.cases
% 0.30/0.53 thf(fact_2160_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 0.30/0.53 ! [X: produc5491161045314408544at_nat] :
% 0.30/0.53 ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > $o,Uv2: option4927543243414619207at_nat] :
% 0.30/0.53 ( X
% 0.30/0.53 != ( produc3994169339658061776at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
% 0.30/0.53 => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > $o,V2: product_prod_nat_nat] :
% 0.30/0.53 ( X
% 0.30/0.53 != ( produc3994169339658061776at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
% 0.30/0.53 => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,X4: product_prod_nat_nat,Y4: product_prod_nat_nat] :
% 0.30/0.53 ( X
% 0.30/0.53 != ( produc3994169339658061776at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X4 ) @ ( some_P7363390416028606310at_nat @ Y4 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % VEBT_internal.option_comp_shift.cases
% 0.30/0.53 thf(fact_2161_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 0.30/0.53 ! [X: produc7036089656553540234on_num] :
% 0.30/0.53 ( ! [Uu2: num > num > $o,Uv2: option_num] :
% 0.30/0.53 ( X
% 0.30/0.53 != ( produc3576312749637752826on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
% 0.30/0.53 => ( ! [Uw2: num > num > $o,V2: num] :
% 0.30/0.53 ( X
% 0.30/0.53 != ( produc3576312749637752826on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
% 0.30/0.53 => ~ ! [F2: num > num > $o,X4: num,Y4: num] :
% 0.30/0.53 ( X
% 0.30/0.53 != ( produc3576312749637752826on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ X4 ) @ ( some_num @ Y4 ) ) ) ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % VEBT_internal.option_comp_shift.cases
% 0.30/0.53 thf(fact_2162_mod__le__divisor,axiom,
% 0.30/0.53 ! [N2: nat,M: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.53 => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N2 ) @ N2 ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_le_divisor
% 0.30/0.53 thf(fact_2163_invar__vebt_Ointros_I1_J,axiom,
% 0.30/0.53 ! [A: $o,B: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % invar_vebt.intros(1)
% 0.30/0.53 thf(fact_2164_vebt__delete_Osimps_I2_J,axiom,
% 0.30/0.53 ! [A: $o,B: $o] :
% 0.30/0.53 ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) )
% 0.30/0.53 = ( vEBT_Leaf @ A @ $false ) ) ).
% 0.30/0.53
% 0.30/0.53 % vebt_delete.simps(2)
% 0.30/0.53 thf(fact_2165_power__0__left,axiom,
% 0.30/0.53 ! [N2: nat] :
% 0.30/0.53 ( ( ( N2 = zero_zero_nat )
% 0.30/0.53 => ( ( power_power_rat @ zero_zero_rat @ N2 )
% 0.30/0.53 = one_one_rat ) )
% 0.30/0.53 & ( ( N2 != zero_zero_nat )
% 0.30/0.53 => ( ( power_power_rat @ zero_zero_rat @ N2 )
% 0.30/0.53 = zero_zero_rat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_0_left
% 0.30/0.53 thf(fact_2166_power__0__left,axiom,
% 0.30/0.53 ! [N2: nat] :
% 0.30/0.53 ( ( ( N2 = zero_zero_nat )
% 0.30/0.53 => ( ( power_power_nat @ zero_zero_nat @ N2 )
% 0.30/0.53 = one_one_nat ) )
% 0.30/0.53 & ( ( N2 != zero_zero_nat )
% 0.30/0.53 => ( ( power_power_nat @ zero_zero_nat @ N2 )
% 0.30/0.53 = zero_zero_nat ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_0_left
% 0.30/0.53 thf(fact_2167_power__0__left,axiom,
% 0.30/0.53 ! [N2: nat] :
% 0.30/0.53 ( ( ( N2 = zero_zero_nat )
% 0.30/0.53 => ( ( power_power_real @ zero_zero_real @ N2 )
% 0.30/0.53 = one_one_real ) )
% 0.30/0.53 & ( ( N2 != zero_zero_nat )
% 0.30/0.53 => ( ( power_power_real @ zero_zero_real @ N2 )
% 0.30/0.53 = zero_zero_real ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_0_left
% 0.30/0.53 thf(fact_2168_power__0__left,axiom,
% 0.30/0.53 ! [N2: nat] :
% 0.30/0.53 ( ( ( N2 = zero_zero_nat )
% 0.30/0.53 => ( ( power_power_int @ zero_zero_int @ N2 )
% 0.30/0.53 = one_one_int ) )
% 0.30/0.53 & ( ( N2 != zero_zero_nat )
% 0.30/0.53 => ( ( power_power_int @ zero_zero_int @ N2 )
% 0.30/0.53 = zero_zero_int ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_0_left
% 0.30/0.53 thf(fact_2169_power__0__left,axiom,
% 0.30/0.53 ! [N2: nat] :
% 0.30/0.53 ( ( ( N2 = zero_zero_nat )
% 0.30/0.53 => ( ( power_power_complex @ zero_zero_complex @ N2 )
% 0.30/0.53 = one_one_complex ) )
% 0.30/0.53 & ( ( N2 != zero_zero_nat )
% 0.30/0.53 => ( ( power_power_complex @ zero_zero_complex @ N2 )
% 0.30/0.53 = zero_zero_complex ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % power_0_left
% 0.30/0.53 thf(fact_2170_zero__power,axiom,
% 0.30/0.53 ! [N2: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.53 => ( ( power_power_rat @ zero_zero_rat @ N2 )
% 0.30/0.53 = zero_zero_rat ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_power
% 0.30/0.53 thf(fact_2171_zero__power,axiom,
% 0.30/0.53 ! [N2: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.53 => ( ( power_power_nat @ zero_zero_nat @ N2 )
% 0.30/0.53 = zero_zero_nat ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_power
% 0.30/0.53 thf(fact_2172_zero__power,axiom,
% 0.30/0.53 ! [N2: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.53 => ( ( power_power_real @ zero_zero_real @ N2 )
% 0.30/0.53 = zero_zero_real ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_power
% 0.30/0.53 thf(fact_2173_zero__power,axiom,
% 0.30/0.53 ! [N2: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.53 => ( ( power_power_int @ zero_zero_int @ N2 )
% 0.30/0.53 = zero_zero_int ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_power
% 0.30/0.53 thf(fact_2174_zero__power,axiom,
% 0.30/0.53 ! [N2: nat] :
% 0.30/0.53 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.53 => ( ( power_power_complex @ zero_zero_complex @ N2 )
% 0.30/0.53 = zero_zero_complex ) ) ).
% 0.30/0.53
% 0.30/0.53 % zero_power
% 0.30/0.53 thf(fact_2175_mod__mult__right__eq,axiom,
% 0.30/0.53 ! [A: nat,B: nat,C: nat] :
% 0.30/0.53 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 0.30/0.53 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_mult_right_eq
% 0.30/0.53 thf(fact_2176_mod__mult__right__eq,axiom,
% 0.30/0.53 ! [A: int,B: int,C: int] :
% 0.30/0.53 ( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.30/0.53 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_mult_right_eq
% 0.30/0.53 thf(fact_2177_mod__mult__right__eq,axiom,
% 0.30/0.53 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.53 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.30/0.53 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_mult_right_eq
% 0.30/0.53 thf(fact_2178_mod__mult__left__eq,axiom,
% 0.30/0.53 ! [A: nat,C: nat,B: nat] :
% 0.30/0.53 ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
% 0.30/0.53 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_mult_left_eq
% 0.30/0.53 thf(fact_2179_mod__mult__left__eq,axiom,
% 0.30/0.53 ! [A: int,C: int,B: int] :
% 0.30/0.53 ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 0.30/0.53 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_mult_left_eq
% 0.30/0.53 thf(fact_2180_mod__mult__left__eq,axiom,
% 0.30/0.53 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.53 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 0.30/0.53 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 0.30/0.53
% 0.30/0.53 % mod_mult_left_eq
% 0.30/0.53 thf(fact_2181_mult__mod__right,axiom,
% 0.30/0.53 ! [C: nat,A: nat,B: nat] :
% 0.30/0.53 ( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 0.30/0.53 = ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_mod_right
% 0.30/0.53 thf(fact_2182_mult__mod__right,axiom,
% 0.30/0.53 ! [C: int,A: int,B: int] :
% 0.30/0.53 ( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 0.30/0.53 = ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ).
% 0.30/0.53
% 0.30/0.53 % mult_mod_right
% 0.30/0.53 thf(fact_2183_mult__mod__right,axiom,
% 0.30/0.53 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.30/0.53 ( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.30/0.54 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_mod_right
% 0.30/0.54 thf(fact_2184_mod__mult__mult2,axiom,
% 0.30/0.54 ! [A: nat,C: nat,B: nat] :
% 0.30/0.54 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.30/0.54 = ( times_times_nat @ ( modulo_modulo_nat @ A @ B ) @ C ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_mult_mult2
% 0.30/0.54 thf(fact_2185_mod__mult__mult2,axiom,
% 0.30/0.54 ! [A: int,C: int,B: int] :
% 0.30/0.54 ( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.30/0.54 = ( times_times_int @ ( modulo_modulo_int @ A @ B ) @ C ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_mult_mult2
% 0.30/0.54 thf(fact_2186_mod__mult__mult2,axiom,
% 0.30/0.54 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.54 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.30/0.54 = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_mult_mult2
% 0.30/0.54 thf(fact_2187_mod__mult__cong,axiom,
% 0.30/0.54 ! [A: nat,C: nat,A6: nat,B: nat,B6: nat] :
% 0.30/0.54 ( ( ( modulo_modulo_nat @ A @ C )
% 0.30/0.54 = ( modulo_modulo_nat @ A6 @ C ) )
% 0.30/0.54 => ( ( ( modulo_modulo_nat @ B @ C )
% 0.30/0.54 = ( modulo_modulo_nat @ B6 @ C ) )
% 0.30/0.54 => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.30/0.54 = ( modulo_modulo_nat @ ( times_times_nat @ A6 @ B6 ) @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_mult_cong
% 0.30/0.54 thf(fact_2188_mod__mult__cong,axiom,
% 0.30/0.54 ! [A: int,C: int,A6: int,B: int,B6: int] :
% 0.30/0.54 ( ( ( modulo_modulo_int @ A @ C )
% 0.30/0.54 = ( modulo_modulo_int @ A6 @ C ) )
% 0.30/0.54 => ( ( ( modulo_modulo_int @ B @ C )
% 0.30/0.54 = ( modulo_modulo_int @ B6 @ C ) )
% 0.30/0.54 => ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C )
% 0.30/0.54 = ( modulo_modulo_int @ ( times_times_int @ A6 @ B6 ) @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_mult_cong
% 0.30/0.54 thf(fact_2189_mod__mult__cong,axiom,
% 0.30/0.54 ! [A: code_integer,C: code_integer,A6: code_integer,B: code_integer,B6: code_integer] :
% 0.30/0.54 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.30/0.54 = ( modulo364778990260209775nteger @ A6 @ C ) )
% 0.30/0.54 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 0.30/0.54 = ( modulo364778990260209775nteger @ B6 @ C ) )
% 0.30/0.54 => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.30/0.54 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A6 @ B6 ) @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_mult_cong
% 0.30/0.54 thf(fact_2190_mod__mult__eq,axiom,
% 0.30/0.54 ! [A: nat,C: nat,B: nat] :
% 0.30/0.54 ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 0.30/0.54 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_mult_eq
% 0.30/0.54 thf(fact_2191_mod__mult__eq,axiom,
% 0.30/0.54 ! [A: int,C: int,B: int] :
% 0.30/0.54 ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.30/0.54 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_mult_eq
% 0.30/0.54 thf(fact_2192_mod__mult__eq,axiom,
% 0.30/0.54 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.54 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.30/0.54 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_mult_eq
% 0.30/0.54 thf(fact_2193_mod__add__right__eq,axiom,
% 0.30/0.54 ! [A: nat,B: nat,C: nat] :
% 0.30/0.54 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 0.30/0.54 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_add_right_eq
% 0.30/0.54 thf(fact_2194_mod__add__right__eq,axiom,
% 0.30/0.54 ! [A: int,B: int,C: int] :
% 0.30/0.54 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.30/0.54 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_add_right_eq
% 0.30/0.54 thf(fact_2195_mod__add__right__eq,axiom,
% 0.30/0.54 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.54 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.30/0.54 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_add_right_eq
% 0.30/0.54 thf(fact_2196_mod__add__left__eq,axiom,
% 0.30/0.54 ! [A: nat,C: nat,B: nat] :
% 0.30/0.54 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
% 0.30/0.54 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_add_left_eq
% 0.30/0.54 thf(fact_2197_mod__add__left__eq,axiom,
% 0.30/0.54 ! [A: int,C: int,B: int] :
% 0.30/0.54 ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 0.30/0.54 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_add_left_eq
% 0.30/0.54 thf(fact_2198_mod__add__left__eq,axiom,
% 0.30/0.54 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.54 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 0.30/0.54 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_add_left_eq
% 0.30/0.54 thf(fact_2199_mod__add__cong,axiom,
% 0.30/0.54 ! [A: nat,C: nat,A6: nat,B: nat,B6: nat] :
% 0.30/0.54 ( ( ( modulo_modulo_nat @ A @ C )
% 0.30/0.54 = ( modulo_modulo_nat @ A6 @ C ) )
% 0.30/0.54 => ( ( ( modulo_modulo_nat @ B @ C )
% 0.30/0.54 = ( modulo_modulo_nat @ B6 @ C ) )
% 0.30/0.54 => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.30/0.54 = ( modulo_modulo_nat @ ( plus_plus_nat @ A6 @ B6 ) @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_add_cong
% 0.30/0.54 thf(fact_2200_mod__add__cong,axiom,
% 0.30/0.54 ! [A: int,C: int,A6: int,B: int,B6: int] :
% 0.30/0.54 ( ( ( modulo_modulo_int @ A @ C )
% 0.30/0.54 = ( modulo_modulo_int @ A6 @ C ) )
% 0.30/0.54 => ( ( ( modulo_modulo_int @ B @ C )
% 0.30/0.54 = ( modulo_modulo_int @ B6 @ C ) )
% 0.30/0.54 => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.30/0.54 = ( modulo_modulo_int @ ( plus_plus_int @ A6 @ B6 ) @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_add_cong
% 0.30/0.54 thf(fact_2201_mod__add__cong,axiom,
% 0.30/0.54 ! [A: code_integer,C: code_integer,A6: code_integer,B: code_integer,B6: code_integer] :
% 0.30/0.54 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.30/0.54 = ( modulo364778990260209775nteger @ A6 @ C ) )
% 0.30/0.54 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 0.30/0.54 = ( modulo364778990260209775nteger @ B6 @ C ) )
% 0.30/0.54 => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.30/0.54 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A6 @ B6 ) @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_add_cong
% 0.30/0.54 thf(fact_2202_mod__add__eq,axiom,
% 0.30/0.54 ! [A: nat,C: nat,B: nat] :
% 0.30/0.54 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 0.30/0.54 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_add_eq
% 0.30/0.54 thf(fact_2203_mod__add__eq,axiom,
% 0.30/0.54 ! [A: int,C: int,B: int] :
% 0.30/0.54 ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.30/0.54 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_add_eq
% 0.30/0.54 thf(fact_2204_mod__add__eq,axiom,
% 0.30/0.54 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.54 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.30/0.54 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_add_eq
% 0.30/0.54 thf(fact_2205_mod__diff__right__eq,axiom,
% 0.30/0.54 ! [A: int,B: int,C: int] :
% 0.30/0.54 ( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.30/0.54 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_diff_right_eq
% 0.30/0.54 thf(fact_2206_mod__diff__right__eq,axiom,
% 0.30/0.54 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.54 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.30/0.54 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_diff_right_eq
% 0.30/0.54 thf(fact_2207_mod__diff__left__eq,axiom,
% 0.30/0.54 ! [A: int,C: int,B: int] :
% 0.30/0.54 ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 0.30/0.54 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_diff_left_eq
% 0.30/0.54 thf(fact_2208_mod__diff__left__eq,axiom,
% 0.30/0.54 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.54 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 0.30/0.54 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_diff_left_eq
% 0.30/0.54 thf(fact_2209_mod__diff__cong,axiom,
% 0.30/0.54 ! [A: int,C: int,A6: int,B: int,B6: int] :
% 0.30/0.54 ( ( ( modulo_modulo_int @ A @ C )
% 0.30/0.54 = ( modulo_modulo_int @ A6 @ C ) )
% 0.30/0.54 => ( ( ( modulo_modulo_int @ B @ C )
% 0.30/0.54 = ( modulo_modulo_int @ B6 @ C ) )
% 0.30/0.54 => ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.30/0.54 = ( modulo_modulo_int @ ( minus_minus_int @ A6 @ B6 ) @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_diff_cong
% 0.30/0.54 thf(fact_2210_mod__diff__cong,axiom,
% 0.30/0.54 ! [A: code_integer,C: code_integer,A6: code_integer,B: code_integer,B6: code_integer] :
% 0.30/0.54 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.30/0.54 = ( modulo364778990260209775nteger @ A6 @ C ) )
% 0.30/0.54 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 0.30/0.54 = ( modulo364778990260209775nteger @ B6 @ C ) )
% 0.30/0.54 => ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
% 0.30/0.54 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A6 @ B6 ) @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_diff_cong
% 0.30/0.54 thf(fact_2211_mod__diff__eq,axiom,
% 0.30/0.54 ! [A: int,C: int,B: int] :
% 0.30/0.54 ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.30/0.54 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_diff_eq
% 0.30/0.54 thf(fact_2212_mod__diff__eq,axiom,
% 0.30/0.54 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.54 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.30/0.54 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_diff_eq
% 0.30/0.54 thf(fact_2213_power__mod,axiom,
% 0.30/0.54 ! [A: nat,B: nat,N2: nat] :
% 0.30/0.54 ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B ) @ N2 ) @ B )
% 0.30/0.54 = ( modulo_modulo_nat @ ( power_power_nat @ A @ N2 ) @ B ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_mod
% 0.30/0.54 thf(fact_2214_power__mod,axiom,
% 0.30/0.54 ! [A: int,B: int,N2: nat] :
% 0.30/0.54 ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B ) @ N2 ) @ B )
% 0.30/0.54 = ( modulo_modulo_int @ ( power_power_int @ A @ N2 ) @ B ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_mod
% 0.30/0.54 thf(fact_2215_power__mod,axiom,
% 0.30/0.54 ! [A: code_integer,B: code_integer,N2: nat] :
% 0.30/0.54 ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B ) @ N2 ) @ B )
% 0.30/0.54 = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ B ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_mod
% 0.30/0.54 thf(fact_2216_vebt__member_Osimps_I1_J,axiom,
% 0.30/0.54 ! [A: $o,B: $o,X: nat] :
% 0.30/0.54 ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B ) @ X )
% 0.30/0.54 = ( ( ( X = zero_zero_nat )
% 0.30/0.54 => A )
% 0.30/0.54 & ( ( X != zero_zero_nat )
% 0.30/0.54 => ( ( ( X = one_one_nat )
% 0.30/0.54 => B )
% 0.30/0.54 & ( X = one_one_nat ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % vebt_member.simps(1)
% 0.30/0.54 thf(fact_2217_mod__Suc__Suc__eq,axiom,
% 0.30/0.54 ! [M: nat,N2: nat] :
% 0.30/0.54 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) ) @ N2 )
% 0.30/0.54 = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N2 ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_Suc_Suc_eq
% 0.30/0.54 thf(fact_2218_mod__Suc__eq,axiom,
% 0.30/0.54 ! [M: nat,N2: nat] :
% 0.30/0.54 ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) @ N2 )
% 0.30/0.54 = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_Suc_eq
% 0.30/0.54 thf(fact_2219_VEBT_Odistinct_I1_J,axiom,
% 0.30/0.54 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] :
% 0.30/0.54 ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 0.30/0.54 != ( vEBT_Leaf @ X21 @ X222 ) ) ).
% 0.30/0.54
% 0.30/0.54 % VEBT.distinct(1)
% 0.30/0.54 thf(fact_2220_VEBT_Oexhaust,axiom,
% 0.30/0.54 ! [Y: vEBT_VEBT] :
% 0.30/0.54 ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
% 0.30/0.54 ( Y
% 0.30/0.54 != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
% 0.30/0.54 => ~ ! [X212: $o,X223: $o] :
% 0.30/0.54 ( Y
% 0.30/0.54 != ( vEBT_Leaf @ X212 @ X223 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % VEBT.exhaust
% 0.30/0.54 thf(fact_2221_vebt__buildup_Osimps_I2_J,axiom,
% 0.30/0.54 ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
% 0.30/0.54 = ( vEBT_Leaf @ $false @ $false ) ) ).
% 0.30/0.54
% 0.30/0.54 % vebt_buildup.simps(2)
% 0.30/0.54 thf(fact_2222_mod__less__eq__dividend,axiom,
% 0.30/0.54 ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N2 ) @ M ) ).
% 0.30/0.54
% 0.30/0.54 % mod_less_eq_dividend
% 0.30/0.54 thf(fact_2223_vebt__insert_Osimps_I1_J,axiom,
% 0.30/0.54 ! [X: nat,A: $o,B: $o] :
% 0.30/0.54 ( ( ( X = zero_zero_nat )
% 0.30/0.54 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
% 0.30/0.54 = ( vEBT_Leaf @ $true @ B ) ) )
% 0.30/0.54 & ( ( X != zero_zero_nat )
% 0.30/0.54 => ( ( ( X = one_one_nat )
% 0.30/0.54 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
% 0.30/0.54 = ( vEBT_Leaf @ A @ $true ) ) )
% 0.30/0.54 & ( ( X != one_one_nat )
% 0.30/0.54 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
% 0.30/0.54 = ( vEBT_Leaf @ A @ B ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % vebt_insert.simps(1)
% 0.30/0.54 thf(fact_2224_vebt__pred_Osimps_I1_J,axiom,
% 0.30/0.54 ! [Uu: $o,Uv: $o] :
% 0.30/0.54 ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat )
% 0.30/0.54 = none_nat ) ).
% 0.30/0.54
% 0.30/0.54 % vebt_pred.simps(1)
% 0.30/0.54 thf(fact_2225_le__numeral__extra_I3_J,axiom,
% 0.30/0.54 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 0.30/0.54
% 0.30/0.54 % le_numeral_extra(3)
% 0.30/0.54 thf(fact_2226_le__numeral__extra_I3_J,axiom,
% 0.30/0.54 ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% 0.30/0.54
% 0.30/0.54 % le_numeral_extra(3)
% 0.30/0.54 thf(fact_2227_le__numeral__extra_I3_J,axiom,
% 0.30/0.54 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 0.30/0.54
% 0.30/0.54 % le_numeral_extra(3)
% 0.30/0.54 thf(fact_2228_le__numeral__extra_I3_J,axiom,
% 0.30/0.54 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 0.30/0.54
% 0.30/0.54 % le_numeral_extra(3)
% 0.30/0.54 thf(fact_2229_zero__le,axiom,
% 0.30/0.54 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 0.30/0.54
% 0.30/0.54 % zero_le
% 0.30/0.54 thf(fact_2230_less__numeral__extra_I3_J,axiom,
% 0.30/0.54 ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 0.30/0.54
% 0.30/0.54 % less_numeral_extra(3)
% 0.30/0.54 thf(fact_2231_less__numeral__extra_I3_J,axiom,
% 0.30/0.54 ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% 0.30/0.54
% 0.30/0.54 % less_numeral_extra(3)
% 0.30/0.54 thf(fact_2232_less__numeral__extra_I3_J,axiom,
% 0.30/0.54 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 0.30/0.54
% 0.30/0.54 % less_numeral_extra(3)
% 0.30/0.54 thf(fact_2233_less__numeral__extra_I3_J,axiom,
% 0.30/0.54 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.30/0.54
% 0.30/0.54 % less_numeral_extra(3)
% 0.30/0.54 thf(fact_2234_field__lbound__gt__zero,axiom,
% 0.30/0.54 ! [D1: real,D22: real] :
% 0.30/0.54 ( ( ord_less_real @ zero_zero_real @ D1 )
% 0.30/0.54 => ( ( ord_less_real @ zero_zero_real @ D22 )
% 0.30/0.54 => ? [E2: real] :
% 0.30/0.54 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.30/0.54 & ( ord_less_real @ E2 @ D1 )
% 0.30/0.54 & ( ord_less_real @ E2 @ D22 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % field_lbound_gt_zero
% 0.30/0.54 thf(fact_2235_field__lbound__gt__zero,axiom,
% 0.30/0.54 ! [D1: rat,D22: rat] :
% 0.30/0.54 ( ( ord_less_rat @ zero_zero_rat @ D1 )
% 0.30/0.54 => ( ( ord_less_rat @ zero_zero_rat @ D22 )
% 0.30/0.54 => ? [E2: rat] :
% 0.30/0.54 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 0.30/0.54 & ( ord_less_rat @ E2 @ D1 )
% 0.30/0.54 & ( ord_less_rat @ E2 @ D22 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % field_lbound_gt_zero
% 0.30/0.54 thf(fact_2236_gr__zeroI,axiom,
% 0.30/0.54 ! [N2: nat] :
% 0.30/0.54 ( ( N2 != zero_zero_nat )
% 0.30/0.54 => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 0.30/0.54
% 0.30/0.54 % gr_zeroI
% 0.30/0.54 thf(fact_2237_not__less__zero,axiom,
% 0.30/0.54 ! [N2: nat] :
% 0.30/0.54 ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 0.30/0.54
% 0.30/0.54 % not_less_zero
% 0.30/0.54 thf(fact_2238_gr__implies__not__zero,axiom,
% 0.30/0.54 ! [M: nat,N2: nat] :
% 0.30/0.54 ( ( ord_less_nat @ M @ N2 )
% 0.30/0.54 => ( N2 != zero_zero_nat ) ) ).
% 0.30/0.54
% 0.30/0.54 % gr_implies_not_zero
% 0.30/0.54 thf(fact_2239_zero__less__iff__neq__zero,axiom,
% 0.30/0.54 ! [N2: nat] :
% 0.30/0.54 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.54 = ( N2 != zero_zero_nat ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_less_iff_neq_zero
% 0.30/0.54 thf(fact_2240_zero__neq__numeral,axiom,
% 0.30/0.54 ! [N2: num] :
% 0.30/0.54 ( zero_zero_complex
% 0.30/0.54 != ( numera6690914467698888265omplex @ N2 ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_neq_numeral
% 0.30/0.54 thf(fact_2241_zero__neq__numeral,axiom,
% 0.30/0.54 ! [N2: num] :
% 0.30/0.54 ( zero_zero_real
% 0.30/0.54 != ( numeral_numeral_real @ N2 ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_neq_numeral
% 0.30/0.54 thf(fact_2242_zero__neq__numeral,axiom,
% 0.30/0.54 ! [N2: num] :
% 0.30/0.54 ( zero_zero_rat
% 0.30/0.54 != ( numeral_numeral_rat @ N2 ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_neq_numeral
% 0.30/0.54 thf(fact_2243_zero__neq__numeral,axiom,
% 0.30/0.54 ! [N2: num] :
% 0.30/0.54 ( zero_zero_nat
% 0.30/0.54 != ( numeral_numeral_nat @ N2 ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_neq_numeral
% 0.30/0.54 thf(fact_2244_zero__neq__numeral,axiom,
% 0.30/0.54 ! [N2: num] :
% 0.30/0.54 ( zero_zero_int
% 0.30/0.54 != ( numeral_numeral_int @ N2 ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_neq_numeral
% 0.30/0.54 thf(fact_2245_cong__exp__iff__simps_I9_J,axiom,
% 0.30/0.54 ! [M: num,Q2: num,N2: num] :
% 0.30/0.54 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.30/0.54 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 0.30/0.54 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.30/0.54 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % cong_exp_iff_simps(9)
% 0.30/0.54 thf(fact_2246_cong__exp__iff__simps_I9_J,axiom,
% 0.30/0.54 ! [M: num,Q2: num,N2: num] :
% 0.30/0.54 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.30/0.54 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 0.30/0.54 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 0.30/0.54 = ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % cong_exp_iff_simps(9)
% 0.30/0.54 thf(fact_2247_cong__exp__iff__simps_I9_J,axiom,
% 0.30/0.54 ! [M: num,Q2: num,N2: num] :
% 0.30/0.54 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.30/0.54 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 0.30/0.54 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.30/0.54 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % cong_exp_iff_simps(9)
% 0.30/0.54 thf(fact_2248_cong__exp__iff__simps_I4_J,axiom,
% 0.30/0.54 ! [M: num,N2: num] :
% 0.30/0.54 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
% 0.30/0.54 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ one ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % cong_exp_iff_simps(4)
% 0.30/0.54 thf(fact_2249_cong__exp__iff__simps_I4_J,axiom,
% 0.30/0.54 ! [M: num,N2: num] :
% 0.30/0.54 ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
% 0.30/0.54 = ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ one ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % cong_exp_iff_simps(4)
% 0.30/0.54 thf(fact_2250_cong__exp__iff__simps_I4_J,axiom,
% 0.30/0.54 ! [M: num,N2: num] :
% 0.30/0.54 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
% 0.30/0.54 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % cong_exp_iff_simps(4)
% 0.30/0.54 thf(fact_2251_mult__right__cancel,axiom,
% 0.30/0.54 ! [C: complex,A: complex,B: complex] :
% 0.30/0.54 ( ( C != zero_zero_complex )
% 0.30/0.54 => ( ( ( times_times_complex @ A @ C )
% 0.30/0.54 = ( times_times_complex @ B @ C ) )
% 0.30/0.54 = ( A = B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_right_cancel
% 0.30/0.54 thf(fact_2252_mult__right__cancel,axiom,
% 0.30/0.54 ! [C: real,A: real,B: real] :
% 0.30/0.54 ( ( C != zero_zero_real )
% 0.30/0.54 => ( ( ( times_times_real @ A @ C )
% 0.30/0.54 = ( times_times_real @ B @ C ) )
% 0.30/0.54 = ( A = B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_right_cancel
% 0.30/0.54 thf(fact_2253_mult__right__cancel,axiom,
% 0.30/0.54 ! [C: rat,A: rat,B: rat] :
% 0.30/0.54 ( ( C != zero_zero_rat )
% 0.30/0.54 => ( ( ( times_times_rat @ A @ C )
% 0.30/0.54 = ( times_times_rat @ B @ C ) )
% 0.30/0.54 = ( A = B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_right_cancel
% 0.30/0.54 thf(fact_2254_mult__right__cancel,axiom,
% 0.30/0.54 ! [C: nat,A: nat,B: nat] :
% 0.30/0.54 ( ( C != zero_zero_nat )
% 0.30/0.54 => ( ( ( times_times_nat @ A @ C )
% 0.30/0.54 = ( times_times_nat @ B @ C ) )
% 0.30/0.54 = ( A = B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_right_cancel
% 0.30/0.54 thf(fact_2255_mult__right__cancel,axiom,
% 0.30/0.54 ! [C: int,A: int,B: int] :
% 0.30/0.54 ( ( C != zero_zero_int )
% 0.30/0.54 => ( ( ( times_times_int @ A @ C )
% 0.30/0.54 = ( times_times_int @ B @ C ) )
% 0.30/0.54 = ( A = B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_right_cancel
% 0.30/0.54 thf(fact_2256_mult__left__cancel,axiom,
% 0.30/0.54 ! [C: complex,A: complex,B: complex] :
% 0.30/0.54 ( ( C != zero_zero_complex )
% 0.30/0.54 => ( ( ( times_times_complex @ C @ A )
% 0.30/0.54 = ( times_times_complex @ C @ B ) )
% 0.30/0.54 = ( A = B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_left_cancel
% 0.30/0.54 thf(fact_2257_mult__left__cancel,axiom,
% 0.30/0.54 ! [C: real,A: real,B: real] :
% 0.30/0.54 ( ( C != zero_zero_real )
% 0.30/0.54 => ( ( ( times_times_real @ C @ A )
% 0.30/0.54 = ( times_times_real @ C @ B ) )
% 0.30/0.54 = ( A = B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_left_cancel
% 0.30/0.54 thf(fact_2258_mult__left__cancel,axiom,
% 0.30/0.54 ! [C: rat,A: rat,B: rat] :
% 0.30/0.54 ( ( C != zero_zero_rat )
% 0.30/0.54 => ( ( ( times_times_rat @ C @ A )
% 0.30/0.54 = ( times_times_rat @ C @ B ) )
% 0.30/0.54 = ( A = B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_left_cancel
% 0.30/0.54 thf(fact_2259_mult__left__cancel,axiom,
% 0.30/0.54 ! [C: nat,A: nat,B: nat] :
% 0.30/0.54 ( ( C != zero_zero_nat )
% 0.30/0.54 => ( ( ( times_times_nat @ C @ A )
% 0.30/0.54 = ( times_times_nat @ C @ B ) )
% 0.30/0.54 = ( A = B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_left_cancel
% 0.30/0.54 thf(fact_2260_mult__left__cancel,axiom,
% 0.30/0.54 ! [C: int,A: int,B: int] :
% 0.30/0.54 ( ( C != zero_zero_int )
% 0.30/0.54 => ( ( ( times_times_int @ C @ A )
% 0.30/0.54 = ( times_times_int @ C @ B ) )
% 0.30/0.54 = ( A = B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_left_cancel
% 0.30/0.54 thf(fact_2261_no__zero__divisors,axiom,
% 0.30/0.54 ! [A: complex,B: complex] :
% 0.30/0.54 ( ( A != zero_zero_complex )
% 0.30/0.54 => ( ( B != zero_zero_complex )
% 0.30/0.54 => ( ( times_times_complex @ A @ B )
% 0.30/0.54 != zero_zero_complex ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % no_zero_divisors
% 0.30/0.54 thf(fact_2262_no__zero__divisors,axiom,
% 0.30/0.54 ! [A: real,B: real] :
% 0.30/0.54 ( ( A != zero_zero_real )
% 0.30/0.54 => ( ( B != zero_zero_real )
% 0.30/0.54 => ( ( times_times_real @ A @ B )
% 0.30/0.54 != zero_zero_real ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % no_zero_divisors
% 0.30/0.54 thf(fact_2263_no__zero__divisors,axiom,
% 0.30/0.54 ! [A: rat,B: rat] :
% 0.30/0.54 ( ( A != zero_zero_rat )
% 0.30/0.54 => ( ( B != zero_zero_rat )
% 0.30/0.54 => ( ( times_times_rat @ A @ B )
% 0.30/0.54 != zero_zero_rat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % no_zero_divisors
% 0.30/0.54 thf(fact_2264_no__zero__divisors,axiom,
% 0.30/0.54 ! [A: nat,B: nat] :
% 0.30/0.54 ( ( A != zero_zero_nat )
% 0.30/0.54 => ( ( B != zero_zero_nat )
% 0.30/0.54 => ( ( times_times_nat @ A @ B )
% 0.30/0.54 != zero_zero_nat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % no_zero_divisors
% 0.30/0.54 thf(fact_2265_no__zero__divisors,axiom,
% 0.30/0.54 ! [A: int,B: int] :
% 0.30/0.54 ( ( A != zero_zero_int )
% 0.30/0.54 => ( ( B != zero_zero_int )
% 0.30/0.54 => ( ( times_times_int @ A @ B )
% 0.30/0.54 != zero_zero_int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % no_zero_divisors
% 0.30/0.54 thf(fact_2266_divisors__zero,axiom,
% 0.30/0.54 ! [A: complex,B: complex] :
% 0.30/0.54 ( ( ( times_times_complex @ A @ B )
% 0.30/0.54 = zero_zero_complex )
% 0.30/0.54 => ( ( A = zero_zero_complex )
% 0.30/0.54 | ( B = zero_zero_complex ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divisors_zero
% 0.30/0.54 thf(fact_2267_divisors__zero,axiom,
% 0.30/0.54 ! [A: real,B: real] :
% 0.30/0.54 ( ( ( times_times_real @ A @ B )
% 0.30/0.54 = zero_zero_real )
% 0.30/0.54 => ( ( A = zero_zero_real )
% 0.30/0.54 | ( B = zero_zero_real ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divisors_zero
% 0.30/0.54 thf(fact_2268_divisors__zero,axiom,
% 0.30/0.54 ! [A: rat,B: rat] :
% 0.30/0.54 ( ( ( times_times_rat @ A @ B )
% 0.30/0.54 = zero_zero_rat )
% 0.30/0.54 => ( ( A = zero_zero_rat )
% 0.30/0.54 | ( B = zero_zero_rat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divisors_zero
% 0.30/0.54 thf(fact_2269_divisors__zero,axiom,
% 0.30/0.54 ! [A: nat,B: nat] :
% 0.30/0.54 ( ( ( times_times_nat @ A @ B )
% 0.30/0.54 = zero_zero_nat )
% 0.30/0.54 => ( ( A = zero_zero_nat )
% 0.30/0.54 | ( B = zero_zero_nat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divisors_zero
% 0.30/0.54 thf(fact_2270_divisors__zero,axiom,
% 0.30/0.54 ! [A: int,B: int] :
% 0.30/0.54 ( ( ( times_times_int @ A @ B )
% 0.30/0.54 = zero_zero_int )
% 0.30/0.54 => ( ( A = zero_zero_int )
% 0.30/0.54 | ( B = zero_zero_int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divisors_zero
% 0.30/0.54 thf(fact_2271_mult__not__zero,axiom,
% 0.30/0.54 ! [A: complex,B: complex] :
% 0.30/0.54 ( ( ( times_times_complex @ A @ B )
% 0.30/0.54 != zero_zero_complex )
% 0.30/0.54 => ( ( A != zero_zero_complex )
% 0.30/0.54 & ( B != zero_zero_complex ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_not_zero
% 0.30/0.54 thf(fact_2272_mult__not__zero,axiom,
% 0.30/0.54 ! [A: real,B: real] :
% 0.30/0.54 ( ( ( times_times_real @ A @ B )
% 0.30/0.54 != zero_zero_real )
% 0.30/0.54 => ( ( A != zero_zero_real )
% 0.30/0.54 & ( B != zero_zero_real ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_not_zero
% 0.30/0.54 thf(fact_2273_mult__not__zero,axiom,
% 0.30/0.54 ! [A: rat,B: rat] :
% 0.30/0.54 ( ( ( times_times_rat @ A @ B )
% 0.30/0.54 != zero_zero_rat )
% 0.30/0.54 => ( ( A != zero_zero_rat )
% 0.30/0.54 & ( B != zero_zero_rat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_not_zero
% 0.30/0.54 thf(fact_2274_mult__not__zero,axiom,
% 0.30/0.54 ! [A: nat,B: nat] :
% 0.30/0.54 ( ( ( times_times_nat @ A @ B )
% 0.30/0.54 != zero_zero_nat )
% 0.30/0.54 => ( ( A != zero_zero_nat )
% 0.30/0.54 & ( B != zero_zero_nat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_not_zero
% 0.30/0.54 thf(fact_2275_mult__not__zero,axiom,
% 0.30/0.54 ! [A: int,B: int] :
% 0.30/0.54 ( ( ( times_times_int @ A @ B )
% 0.30/0.54 != zero_zero_int )
% 0.30/0.54 => ( ( A != zero_zero_int )
% 0.30/0.54 & ( B != zero_zero_int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_not_zero
% 0.30/0.54 thf(fact_2276_add_Ogroup__left__neutral,axiom,
% 0.30/0.54 ! [A: complex] :
% 0.30/0.54 ( ( plus_plus_complex @ zero_zero_complex @ A )
% 0.30/0.54 = A ) ).
% 0.30/0.54
% 0.30/0.54 % add.group_left_neutral
% 0.30/0.54 thf(fact_2277_add_Ogroup__left__neutral,axiom,
% 0.30/0.54 ! [A: real] :
% 0.30/0.54 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.30/0.54 = A ) ).
% 0.30/0.54
% 0.30/0.54 % add.group_left_neutral
% 0.30/0.54 thf(fact_2278_add_Ogroup__left__neutral,axiom,
% 0.30/0.54 ! [A: rat] :
% 0.30/0.54 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 0.30/0.54 = A ) ).
% 0.30/0.54
% 0.30/0.54 % add.group_left_neutral
% 0.30/0.54 thf(fact_2279_add_Ogroup__left__neutral,axiom,
% 0.30/0.54 ! [A: int] :
% 0.30/0.54 ( ( plus_plus_int @ zero_zero_int @ A )
% 0.30/0.54 = A ) ).
% 0.30/0.54
% 0.30/0.54 % add.group_left_neutral
% 0.30/0.54 thf(fact_2280_add_Ocomm__neutral,axiom,
% 0.30/0.54 ! [A: complex] :
% 0.30/0.54 ( ( plus_plus_complex @ A @ zero_zero_complex )
% 0.30/0.54 = A ) ).
% 0.30/0.54
% 0.30/0.54 % add.comm_neutral
% 0.30/0.54 thf(fact_2281_add_Ocomm__neutral,axiom,
% 0.30/0.54 ! [A: real] :
% 0.30/0.54 ( ( plus_plus_real @ A @ zero_zero_real )
% 0.30/0.54 = A ) ).
% 0.30/0.54
% 0.30/0.54 % add.comm_neutral
% 0.30/0.54 thf(fact_2282_add_Ocomm__neutral,axiom,
% 0.30/0.54 ! [A: rat] :
% 0.30/0.54 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.30/0.54 = A ) ).
% 0.30/0.54
% 0.30/0.54 % add.comm_neutral
% 0.30/0.54 thf(fact_2283_add_Ocomm__neutral,axiom,
% 0.30/0.54 ! [A: nat] :
% 0.30/0.54 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.30/0.54 = A ) ).
% 0.30/0.54
% 0.30/0.54 % add.comm_neutral
% 0.30/0.54 thf(fact_2284_add_Ocomm__neutral,axiom,
% 0.30/0.54 ! [A: int] :
% 0.30/0.54 ( ( plus_plus_int @ A @ zero_zero_int )
% 0.30/0.54 = A ) ).
% 0.30/0.54
% 0.30/0.54 % add.comm_neutral
% 0.30/0.54 thf(fact_2285_comm__monoid__add__class_Oadd__0,axiom,
% 0.30/0.54 ! [A: complex] :
% 0.30/0.54 ( ( plus_plus_complex @ zero_zero_complex @ A )
% 0.30/0.54 = A ) ).
% 0.30/0.54
% 0.30/0.54 % comm_monoid_add_class.add_0
% 0.30/0.54 thf(fact_2286_comm__monoid__add__class_Oadd__0,axiom,
% 0.30/0.54 ! [A: real] :
% 0.30/0.54 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.30/0.54 = A ) ).
% 0.30/0.54
% 0.30/0.54 % comm_monoid_add_class.add_0
% 0.30/0.54 thf(fact_2287_comm__monoid__add__class_Oadd__0,axiom,
% 0.30/0.54 ! [A: rat] :
% 0.30/0.54 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 0.30/0.54 = A ) ).
% 0.30/0.54
% 0.30/0.54 % comm_monoid_add_class.add_0
% 0.30/0.54 thf(fact_2288_comm__monoid__add__class_Oadd__0,axiom,
% 0.30/0.54 ! [A: nat] :
% 0.30/0.54 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 0.30/0.54 = A ) ).
% 0.30/0.54
% 0.30/0.54 % comm_monoid_add_class.add_0
% 0.30/0.54 thf(fact_2289_comm__monoid__add__class_Oadd__0,axiom,
% 0.30/0.54 ! [A: int] :
% 0.30/0.54 ( ( plus_plus_int @ zero_zero_int @ A )
% 0.30/0.54 = A ) ).
% 0.30/0.54
% 0.30/0.54 % comm_monoid_add_class.add_0
% 0.30/0.54 thf(fact_2290_zero__neq__one,axiom,
% 0.30/0.54 zero_zero_complex != one_one_complex ).
% 0.30/0.54
% 0.30/0.54 % zero_neq_one
% 0.30/0.54 thf(fact_2291_zero__neq__one,axiom,
% 0.30/0.54 zero_zero_real != one_one_real ).
% 0.30/0.54
% 0.30/0.54 % zero_neq_one
% 0.30/0.54 thf(fact_2292_zero__neq__one,axiom,
% 0.30/0.54 zero_zero_rat != one_one_rat ).
% 0.30/0.54
% 0.30/0.54 % zero_neq_one
% 0.30/0.54 thf(fact_2293_zero__neq__one,axiom,
% 0.30/0.54 zero_zero_nat != one_one_nat ).
% 0.30/0.54
% 0.30/0.54 % zero_neq_one
% 0.30/0.54 thf(fact_2294_zero__neq__one,axiom,
% 0.30/0.54 zero_zero_int != one_one_int ).
% 0.30/0.54
% 0.30/0.54 % zero_neq_one
% 0.30/0.54 thf(fact_2295_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
% 0.30/0.54 ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
% 0.30/0.54 ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz ) ).
% 0.30/0.54
% 0.30/0.54 % VEBT_internal.membermima.simps(2)
% 0.30/0.54 thf(fact_2296_eq__iff__diff__eq__0,axiom,
% 0.30/0.54 ( ( ^ [Y3: complex,Z2: complex] : ( Y3 = Z2 ) )
% 0.30/0.54 = ( ^ [A5: complex,B5: complex] :
% 0.30/0.54 ( ( minus_minus_complex @ A5 @ B5 )
% 0.30/0.54 = zero_zero_complex ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % eq_iff_diff_eq_0
% 0.30/0.54 thf(fact_2297_eq__iff__diff__eq__0,axiom,
% 0.30/0.54 ( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
% 0.30/0.54 = ( ^ [A5: real,B5: real] :
% 0.30/0.54 ( ( minus_minus_real @ A5 @ B5 )
% 0.30/0.54 = zero_zero_real ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % eq_iff_diff_eq_0
% 0.30/0.54 thf(fact_2298_eq__iff__diff__eq__0,axiom,
% 0.30/0.54 ( ( ^ [Y3: rat,Z2: rat] : ( Y3 = Z2 ) )
% 0.30/0.54 = ( ^ [A5: rat,B5: rat] :
% 0.30/0.54 ( ( minus_minus_rat @ A5 @ B5 )
% 0.30/0.54 = zero_zero_rat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % eq_iff_diff_eq_0
% 0.30/0.54 thf(fact_2299_eq__iff__diff__eq__0,axiom,
% 0.30/0.54 ( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
% 0.30/0.54 = ( ^ [A5: int,B5: int] :
% 0.30/0.54 ( ( minus_minus_int @ A5 @ B5 )
% 0.30/0.54 = zero_zero_int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % eq_iff_diff_eq_0
% 0.30/0.54 thf(fact_2300_power__not__zero,axiom,
% 0.30/0.54 ! [A: rat,N2: nat] :
% 0.30/0.54 ( ( A != zero_zero_rat )
% 0.30/0.54 => ( ( power_power_rat @ A @ N2 )
% 0.30/0.54 != zero_zero_rat ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_not_zero
% 0.30/0.54 thf(fact_2301_power__not__zero,axiom,
% 0.30/0.54 ! [A: nat,N2: nat] :
% 0.30/0.54 ( ( A != zero_zero_nat )
% 0.30/0.54 => ( ( power_power_nat @ A @ N2 )
% 0.30/0.54 != zero_zero_nat ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_not_zero
% 0.30/0.54 thf(fact_2302_power__not__zero,axiom,
% 0.30/0.54 ! [A: real,N2: nat] :
% 0.30/0.54 ( ( A != zero_zero_real )
% 0.30/0.54 => ( ( power_power_real @ A @ N2 )
% 0.30/0.54 != zero_zero_real ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_not_zero
% 0.30/0.54 thf(fact_2303_power__not__zero,axiom,
% 0.30/0.54 ! [A: int,N2: nat] :
% 0.30/0.54 ( ( A != zero_zero_int )
% 0.30/0.54 => ( ( power_power_int @ A @ N2 )
% 0.30/0.54 != zero_zero_int ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_not_zero
% 0.30/0.54 thf(fact_2304_power__not__zero,axiom,
% 0.30/0.54 ! [A: complex,N2: nat] :
% 0.30/0.54 ( ( A != zero_zero_complex )
% 0.30/0.54 => ( ( power_power_complex @ A @ N2 )
% 0.30/0.54 != zero_zero_complex ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_not_zero
% 0.30/0.54 thf(fact_2305_num_Osize_I4_J,axiom,
% 0.30/0.54 ( ( size_size_num @ one )
% 0.30/0.54 = zero_zero_nat ) ).
% 0.30/0.54
% 0.30/0.54 % num.size(4)
% 0.30/0.54 thf(fact_2306_VEBT__internal_OminNull_Osimps_I3_J,axiom,
% 0.30/0.54 ! [Uu: $o] :
% 0.30/0.54 ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).
% 0.30/0.54
% 0.30/0.54 % VEBT_internal.minNull.simps(3)
% 0.30/0.54 thf(fact_2307_VEBT__internal_OminNull_Osimps_I2_J,axiom,
% 0.30/0.54 ! [Uv: $o] :
% 0.30/0.54 ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).
% 0.30/0.54
% 0.30/0.54 % VEBT_internal.minNull.simps(2)
% 0.30/0.54 thf(fact_2308_VEBT__internal_OminNull_Osimps_I1_J,axiom,
% 0.30/0.54 vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% 0.30/0.54
% 0.30/0.54 % VEBT_internal.minNull.simps(1)
% 0.30/0.54 thf(fact_2309_vebt__buildup_Ocases,axiom,
% 0.30/0.54 ! [X: nat] :
% 0.30/0.54 ( ( X != zero_zero_nat )
% 0.30/0.54 => ( ( X
% 0.30/0.54 != ( suc @ zero_zero_nat ) )
% 0.30/0.54 => ~ ! [Va2: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( suc @ ( suc @ Va2 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % vebt_buildup.cases
% 0.30/0.54 thf(fact_2310_nat_Odistinct_I1_J,axiom,
% 0.30/0.54 ! [X22: nat] :
% 0.30/0.54 ( zero_zero_nat
% 0.30/0.54 != ( suc @ X22 ) ) ).
% 0.30/0.54
% 0.30/0.54 % nat.distinct(1)
% 0.30/0.54 thf(fact_2311_old_Onat_Odistinct_I2_J,axiom,
% 0.30/0.54 ! [Nat2: nat] :
% 0.30/0.54 ( ( suc @ Nat2 )
% 0.30/0.54 != zero_zero_nat ) ).
% 0.30/0.54
% 0.30/0.54 % old.nat.distinct(2)
% 0.30/0.54 thf(fact_2312_old_Onat_Odistinct_I1_J,axiom,
% 0.30/0.54 ! [Nat2: nat] :
% 0.30/0.54 ( zero_zero_nat
% 0.30/0.54 != ( suc @ Nat2 ) ) ).
% 0.30/0.54
% 0.30/0.54 % old.nat.distinct(1)
% 0.30/0.54 thf(fact_2313_nat_OdiscI,axiom,
% 0.30/0.54 ! [Nat: nat,X22: nat] :
% 0.30/0.54 ( ( Nat
% 0.30/0.54 = ( suc @ X22 ) )
% 0.30/0.54 => ( Nat != zero_zero_nat ) ) ).
% 0.30/0.54
% 0.30/0.54 % nat.discI
% 0.30/0.54 thf(fact_2314_old_Onat_Oexhaust,axiom,
% 0.30/0.54 ! [Y: nat] :
% 0.30/0.54 ( ( Y != zero_zero_nat )
% 0.30/0.54 => ~ ! [Nat3: nat] :
% 0.30/0.54 ( Y
% 0.30/0.54 != ( suc @ Nat3 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % old.nat.exhaust
% 0.30/0.54 thf(fact_2315_nat__induct,axiom,
% 0.30/0.54 ! [P: nat > $o,N2: nat] :
% 0.30/0.54 ( ( P @ zero_zero_nat )
% 0.30/0.54 => ( ! [N4: nat] :
% 0.30/0.54 ( ( P @ N4 )
% 0.30/0.54 => ( P @ ( suc @ N4 ) ) )
% 0.30/0.54 => ( P @ N2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % nat_induct
% 0.30/0.54 thf(fact_2316_diff__induct,axiom,
% 0.30/0.54 ! [P: nat > nat > $o,M: nat,N2: nat] :
% 0.30/0.54 ( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
% 0.30/0.54 => ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
% 0.30/0.54 => ( ! [X4: nat,Y4: nat] :
% 0.30/0.54 ( ( P @ X4 @ Y4 )
% 0.30/0.54 => ( P @ ( suc @ X4 ) @ ( suc @ Y4 ) ) )
% 0.30/0.54 => ( P @ M @ N2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % diff_induct
% 0.30/0.54 thf(fact_2317_zero__induct,axiom,
% 0.30/0.54 ! [P: nat > $o,K: nat] :
% 0.30/0.54 ( ( P @ K )
% 0.30/0.54 => ( ! [N4: nat] :
% 0.30/0.54 ( ( P @ ( suc @ N4 ) )
% 0.30/0.54 => ( P @ N4 ) )
% 0.30/0.54 => ( P @ zero_zero_nat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_induct
% 0.30/0.54 thf(fact_2318_Suc__neq__Zero,axiom,
% 0.30/0.54 ! [M: nat] :
% 0.30/0.54 ( ( suc @ M )
% 0.30/0.54 != zero_zero_nat ) ).
% 0.30/0.54
% 0.30/0.54 % Suc_neq_Zero
% 0.30/0.54 thf(fact_2319_Zero__neq__Suc,axiom,
% 0.30/0.54 ! [M: nat] :
% 0.30/0.54 ( zero_zero_nat
% 0.30/0.54 != ( suc @ M ) ) ).
% 0.30/0.54
% 0.30/0.54 % Zero_neq_Suc
% 0.30/0.54 thf(fact_2320_Zero__not__Suc,axiom,
% 0.30/0.54 ! [M: nat] :
% 0.30/0.54 ( zero_zero_nat
% 0.30/0.54 != ( suc @ M ) ) ).
% 0.30/0.54
% 0.30/0.54 % Zero_not_Suc
% 0.30/0.54 thf(fact_2321_not0__implies__Suc,axiom,
% 0.30/0.54 ! [N2: nat] :
% 0.30/0.54 ( ( N2 != zero_zero_nat )
% 0.30/0.54 => ? [M2: nat] :
% 0.30/0.54 ( N2
% 0.30/0.54 = ( suc @ M2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % not0_implies_Suc
% 0.30/0.54 thf(fact_2322_bot__nat__0_Oextremum__strict,axiom,
% 0.30/0.54 ! [A: nat] :
% 0.30/0.54 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 0.30/0.54
% 0.30/0.54 % bot_nat_0.extremum_strict
% 0.30/0.54 thf(fact_2323_gr0I,axiom,
% 0.30/0.54 ! [N2: nat] :
% 0.30/0.54 ( ( N2 != zero_zero_nat )
% 0.30/0.54 => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 0.30/0.54
% 0.30/0.54 % gr0I
% 0.30/0.54 thf(fact_2324_not__gr0,axiom,
% 0.30/0.54 ! [N2: nat] :
% 0.30/0.54 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
% 0.30/0.54 = ( N2 = zero_zero_nat ) ) ).
% 0.30/0.54
% 0.30/0.54 % not_gr0
% 0.30/0.54 thf(fact_2325_not__less0,axiom,
% 0.30/0.54 ! [N2: nat] :
% 0.30/0.54 ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 0.30/0.54
% 0.30/0.54 % not_less0
% 0.30/0.54 thf(fact_2326_less__zeroE,axiom,
% 0.30/0.54 ! [N2: nat] :
% 0.30/0.54 ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 0.30/0.54
% 0.30/0.54 % less_zeroE
% 0.30/0.54 thf(fact_2327_gr__implies__not0,axiom,
% 0.30/0.54 ! [M: nat,N2: nat] :
% 0.30/0.54 ( ( ord_less_nat @ M @ N2 )
% 0.30/0.54 => ( N2 != zero_zero_nat ) ) ).
% 0.30/0.54
% 0.30/0.54 % gr_implies_not0
% 0.30/0.54 thf(fact_2328_infinite__descent0,axiom,
% 0.30/0.54 ! [P: nat > $o,N2: nat] :
% 0.30/0.54 ( ( P @ zero_zero_nat )
% 0.30/0.54 => ( ! [N4: nat] :
% 0.30/0.54 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 0.30/0.54 => ( ~ ( P @ N4 )
% 0.30/0.54 => ? [M5: nat] :
% 0.30/0.54 ( ( ord_less_nat @ M5 @ N4 )
% 0.30/0.54 & ~ ( P @ M5 ) ) ) )
% 0.30/0.54 => ( P @ N2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % infinite_descent0
% 0.30/0.54 thf(fact_2329_less__eq__nat_Osimps_I1_J,axiom,
% 0.30/0.54 ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% 0.30/0.54
% 0.30/0.54 % less_eq_nat.simps(1)
% 0.30/0.54 thf(fact_2330_bot__nat__0_Oextremum__unique,axiom,
% 0.30/0.54 ! [A: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.30/0.54 = ( A = zero_zero_nat ) ) ).
% 0.30/0.54
% 0.30/0.54 % bot_nat_0.extremum_unique
% 0.30/0.54 thf(fact_2331_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.30/0.54 ! [A: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.30/0.54 => ( A = zero_zero_nat ) ) ).
% 0.30/0.54
% 0.30/0.54 % bot_nat_0.extremum_uniqueI
% 0.30/0.54 thf(fact_2332_le__0__eq,axiom,
% 0.30/0.54 ! [N2: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
% 0.30/0.54 = ( N2 = zero_zero_nat ) ) ).
% 0.30/0.54
% 0.30/0.54 % le_0_eq
% 0.30/0.54 thf(fact_2333_add__eq__self__zero,axiom,
% 0.30/0.54 ! [M: nat,N2: nat] :
% 0.30/0.54 ( ( ( plus_plus_nat @ M @ N2 )
% 0.30/0.54 = M )
% 0.30/0.54 => ( N2 = zero_zero_nat ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_eq_self_zero
% 0.30/0.54 thf(fact_2334_plus__nat_Oadd__0,axiom,
% 0.30/0.54 ! [N2: nat] :
% 0.30/0.54 ( ( plus_plus_nat @ zero_zero_nat @ N2 )
% 0.30/0.54 = N2 ) ).
% 0.30/0.54
% 0.30/0.54 % plus_nat.add_0
% 0.30/0.54 thf(fact_2335_mult__0,axiom,
% 0.30/0.54 ! [N2: nat] :
% 0.30/0.54 ( ( times_times_nat @ zero_zero_nat @ N2 )
% 0.30/0.54 = zero_zero_nat ) ).
% 0.30/0.54
% 0.30/0.54 % mult_0
% 0.30/0.54 thf(fact_2336_nat__mult__eq__cancel__disj,axiom,
% 0.30/0.54 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.54 ( ( ( times_times_nat @ K @ M )
% 0.30/0.54 = ( times_times_nat @ K @ N2 ) )
% 0.30/0.54 = ( ( K = zero_zero_nat )
% 0.30/0.54 | ( M = N2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % nat_mult_eq_cancel_disj
% 0.30/0.54 thf(fact_2337_minus__nat_Odiff__0,axiom,
% 0.30/0.54 ! [M: nat] :
% 0.30/0.54 ( ( minus_minus_nat @ M @ zero_zero_nat )
% 0.30/0.54 = M ) ).
% 0.30/0.54
% 0.30/0.54 % minus_nat.diff_0
% 0.30/0.54 thf(fact_2338_diffs0__imp__equal,axiom,
% 0.30/0.54 ! [M: nat,N2: nat] :
% 0.30/0.54 ( ( ( minus_minus_nat @ M @ N2 )
% 0.30/0.54 = zero_zero_nat )
% 0.30/0.54 => ( ( ( minus_minus_nat @ N2 @ M )
% 0.30/0.54 = zero_zero_nat )
% 0.30/0.54 => ( M = N2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % diffs0_imp_equal
% 0.30/0.54 thf(fact_2339_mod__geq,axiom,
% 0.30/0.54 ! [M: nat,N2: nat] :
% 0.30/0.54 ( ~ ( ord_less_nat @ M @ N2 )
% 0.30/0.54 => ( ( modulo_modulo_nat @ M @ N2 )
% 0.30/0.54 = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_geq
% 0.30/0.54 thf(fact_2340_nat__mod__eq__iff,axiom,
% 0.30/0.54 ! [X: nat,N2: nat,Y: nat] :
% 0.30/0.54 ( ( ( modulo_modulo_nat @ X @ N2 )
% 0.30/0.54 = ( modulo_modulo_nat @ Y @ N2 ) )
% 0.30/0.54 = ( ? [Q1: nat,Q22: nat] :
% 0.30/0.54 ( ( plus_plus_nat @ X @ ( times_times_nat @ N2 @ Q1 ) )
% 0.30/0.54 = ( plus_plus_nat @ Y @ ( times_times_nat @ N2 @ Q22 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % nat_mod_eq_iff
% 0.30/0.54 thf(fact_2341_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.30/0.54 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.30/0.54 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.30/0.54 => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.30/0.54 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.30/0.54 thf(fact_2342_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.30/0.54 ! [C: nat,A: nat,B: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.54 => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.30/0.54 = ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.30/0.54 thf(fact_2343_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.30/0.54 ! [C: int,A: int,B: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.54 => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
% 0.30/0.54 = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.30/0.54 thf(fact_2344_split__mod,axiom,
% 0.30/0.54 ! [P: nat > $o,M: nat,N2: nat] :
% 0.30/0.54 ( ( P @ ( modulo_modulo_nat @ M @ N2 ) )
% 0.30/0.54 = ( ( ( N2 = zero_zero_nat )
% 0.30/0.54 => ( P @ M ) )
% 0.30/0.54 & ( ( N2 != zero_zero_nat )
% 0.30/0.54 => ! [I3: nat,J3: nat] :
% 0.30/0.54 ( ( ord_less_nat @ J3 @ N2 )
% 0.30/0.54 => ( ( M
% 0.30/0.54 = ( plus_plus_nat @ ( times_times_nat @ N2 @ I3 ) @ J3 ) )
% 0.30/0.54 => ( P @ J3 ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % split_mod
% 0.30/0.54 thf(fact_2345_power__eq__iff__eq__base,axiom,
% 0.30/0.54 ! [N2: nat,A: real,B: real] :
% 0.30/0.54 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.54 => ( ( ( power_power_real @ A @ N2 )
% 0.30/0.54 = ( power_power_real @ B @ N2 ) )
% 0.30/0.54 = ( A = B ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_eq_iff_eq_base
% 0.30/0.54 thf(fact_2346_power__eq__iff__eq__base,axiom,
% 0.30/0.54 ! [N2: nat,A: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.30/0.54 => ( ( ( power_power_rat @ A @ N2 )
% 0.30/0.54 = ( power_power_rat @ B @ N2 ) )
% 0.30/0.54 = ( A = B ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_eq_iff_eq_base
% 0.30/0.54 thf(fact_2347_power__eq__iff__eq__base,axiom,
% 0.30/0.54 ! [N2: nat,A: nat,B: nat] :
% 0.30/0.54 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.30/0.54 => ( ( ( power_power_nat @ A @ N2 )
% 0.30/0.54 = ( power_power_nat @ B @ N2 ) )
% 0.30/0.54 = ( A = B ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_eq_iff_eq_base
% 0.30/0.54 thf(fact_2348_power__eq__iff__eq__base,axiom,
% 0.30/0.54 ! [N2: nat,A: int,B: int] :
% 0.30/0.54 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.30/0.54 => ( ( ( power_power_int @ A @ N2 )
% 0.30/0.54 = ( power_power_int @ B @ N2 ) )
% 0.30/0.54 = ( A = B ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_eq_iff_eq_base
% 0.30/0.54 thf(fact_2349_power__eq__imp__eq__base,axiom,
% 0.30/0.54 ! [A: real,N2: nat,B: real] :
% 0.30/0.54 ( ( ( power_power_real @ A @ N2 )
% 0.30/0.54 = ( power_power_real @ B @ N2 ) )
% 0.30/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.54 => ( A = B ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_eq_imp_eq_base
% 0.30/0.54 thf(fact_2350_power__eq__imp__eq__base,axiom,
% 0.30/0.54 ! [A: rat,N2: nat,B: rat] :
% 0.30/0.54 ( ( ( power_power_rat @ A @ N2 )
% 0.30/0.54 = ( power_power_rat @ B @ N2 ) )
% 0.30/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.54 => ( A = B ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_eq_imp_eq_base
% 0.30/0.54 thf(fact_2351_power__eq__imp__eq__base,axiom,
% 0.30/0.54 ! [A: nat,N2: nat,B: nat] :
% 0.30/0.54 ( ( ( power_power_nat @ A @ N2 )
% 0.30/0.54 = ( power_power_nat @ B @ N2 ) )
% 0.30/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.54 => ( A = B ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_eq_imp_eq_base
% 0.30/0.54 thf(fact_2352_power__eq__imp__eq__base,axiom,
% 0.30/0.54 ! [A: int,N2: nat,B: int] :
% 0.30/0.54 ( ( ( power_power_int @ A @ N2 )
% 0.30/0.54 = ( power_power_int @ B @ N2 ) )
% 0.30/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.54 => ( A = B ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_eq_imp_eq_base
% 0.30/0.54 thf(fact_2353_lambda__zero,axiom,
% 0.30/0.54 ( ( ^ [H: complex] : zero_zero_complex )
% 0.30/0.54 = ( times_times_complex @ zero_zero_complex ) ) ).
% 0.30/0.54
% 0.30/0.54 % lambda_zero
% 0.30/0.54 thf(fact_2354_lambda__zero,axiom,
% 0.30/0.54 ( ( ^ [H: real] : zero_zero_real )
% 0.30/0.54 = ( times_times_real @ zero_zero_real ) ) ).
% 0.30/0.54
% 0.30/0.54 % lambda_zero
% 0.30/0.54 thf(fact_2355_lambda__zero,axiom,
% 0.30/0.54 ( ( ^ [H: rat] : zero_zero_rat )
% 0.30/0.54 = ( times_times_rat @ zero_zero_rat ) ) ).
% 0.30/0.54
% 0.30/0.54 % lambda_zero
% 0.30/0.54 thf(fact_2356_lambda__zero,axiom,
% 0.30/0.54 ( ( ^ [H: nat] : zero_zero_nat )
% 0.30/0.54 = ( times_times_nat @ zero_zero_nat ) ) ).
% 0.30/0.54
% 0.30/0.54 % lambda_zero
% 0.30/0.54 thf(fact_2357_lambda__zero,axiom,
% 0.30/0.54 ( ( ^ [H: int] : zero_zero_int )
% 0.30/0.54 = ( times_times_int @ zero_zero_int ) ) ).
% 0.30/0.54
% 0.30/0.54 % lambda_zero
% 0.30/0.54 thf(fact_2358_VEBT__internal_Omembermima_Ocases,axiom,
% 0.30/0.54 ! [X: produc9072475918466114483BT_nat] :
% 0.30/0.54 ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
% 0.30/0.54 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
% 0.30/0.54 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X4: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X4 ) )
% 0.30/0.54 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ X4 ) )
% 0.30/0.54 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT,X4: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ X4 ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % VEBT_internal.membermima.cases
% 0.30/0.54 thf(fact_2359_vebt__delete_Ocases,axiom,
% 0.30/0.54 ! [X: produc9072475918466114483BT_nat] :
% 0.30/0.54 ( ! [A3: $o,B2: $o] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ zero_zero_nat ) )
% 0.30/0.54 => ( ! [A3: $o,B2: $o] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ zero_zero_nat ) ) )
% 0.30/0.54 => ( ! [A3: $o,B2: $o,N4: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ N4 ) ) ) )
% 0.30/0.54 => ( ! [Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu2: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) @ Uu2 ) )
% 0.30/0.54 => ( ! [Mi2: nat,Ma2: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X4: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) @ X4 ) )
% 0.30/0.54 => ( ! [Mi2: nat,Ma2: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X4: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) @ X4 ) )
% 0.30/0.54 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X4 ) ) ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % vebt_delete.cases
% 0.30/0.54 thf(fact_2360_vebt__member_Ocases,axiom,
% 0.30/0.54 ! [X: produc9072475918466114483BT_nat] :
% 0.30/0.54 ( ! [A3: $o,B2: $o,X4: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ X4 ) )
% 0.30/0.54 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X4: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X4 ) )
% 0.30/0.54 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X4: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X4 ) )
% 0.30/0.54 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X4 ) )
% 0.30/0.54 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X4 ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % vebt_member.cases
% 0.30/0.54 thf(fact_2361_vebt__insert_Ocases,axiom,
% 0.30/0.54 ! [X: produc9072475918466114483BT_nat] :
% 0.30/0.54 ( ! [A3: $o,B2: $o,X4: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ X4 ) )
% 0.30/0.54 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,X4: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) @ X4 ) )
% 0.30/0.54 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,X4: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) @ X4 ) )
% 0.30/0.54 => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ X4 ) )
% 0.30/0.54 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X4 ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % vebt_insert.cases
% 0.30/0.54 thf(fact_2362_vebt__succ_Ocases,axiom,
% 0.30/0.54 ! [X: produc9072475918466114483BT_nat] :
% 0.30/0.54 ( ! [Uu2: $o,B2: $o] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B2 ) @ zero_zero_nat ) )
% 0.30/0.54 => ( ! [Uv2: $o,Uw2: $o,N4: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N4 ) ) )
% 0.30/0.54 => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Va3 ) )
% 0.30/0.54 => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Ve ) )
% 0.30/0.54 => ( ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi ) )
% 0.30/0.54 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X4 ) ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % vebt_succ.cases
% 0.30/0.54 thf(fact_2363_vebt__pred_Ocases,axiom,
% 0.30/0.54 ! [X: produc9072475918466114483BT_nat] :
% 0.30/0.54 ( ! [Uu2: $o,Uv2: $o] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) )
% 0.30/0.54 => ( ! [A3: $o,Uw2: $o] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) )
% 0.30/0.54 => ( ! [A3: $o,B2: $o,Va2: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ Va2 ) ) ) )
% 0.30/0.54 => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Vb2 ) )
% 0.30/0.54 => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) @ Vf ) )
% 0.30/0.54 => ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj ) )
% 0.30/0.54 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X4 ) ) ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % vebt_pred.cases
% 0.30/0.54 thf(fact_2364_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
% 0.30/0.54 ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
% 0.30/0.54 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va @ Vb ) @ X )
% 0.30/0.54 = ( ( X = Mi )
% 0.30/0.54 | ( X = Ma ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % VEBT_internal.membermima.simps(3)
% 0.30/0.54 thf(fact_2365_divmod__digit__0_I2_J,axiom,
% 0.30/0.54 ! [B: nat,A: nat] :
% 0.30/0.54 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.54 => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.30/0.54 => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
% 0.30/0.54 = ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divmod_digit_0(2)
% 0.30/0.54 thf(fact_2366_divmod__digit__0_I2_J,axiom,
% 0.30/0.54 ! [B: int,A: int] :
% 0.30/0.54 ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.54 => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.30/0.54 => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
% 0.30/0.54 = ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divmod_digit_0(2)
% 0.30/0.54 thf(fact_2367_divmod__digit__0_I2_J,axiom,
% 0.30/0.54 ! [B: code_integer,A: code_integer] :
% 0.30/0.54 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.30/0.54 => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.30/0.54 => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
% 0.30/0.54 = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % divmod_digit_0(2)
% 0.30/0.54 thf(fact_2368_cong__exp__iff__simps_I6_J,axiom,
% 0.30/0.54 ! [Q2: num,N2: num] :
% 0.30/0.54 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.30/0.54 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % cong_exp_iff_simps(6)
% 0.30/0.54 thf(fact_2369_cong__exp__iff__simps_I6_J,axiom,
% 0.30/0.54 ! [Q2: num,N2: num] :
% 0.30/0.54 ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.30/0.54 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % cong_exp_iff_simps(6)
% 0.30/0.54 thf(fact_2370_cong__exp__iff__simps_I6_J,axiom,
% 0.30/0.54 ! [Q2: num,N2: num] :
% 0.30/0.54 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.30/0.54 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % cong_exp_iff_simps(6)
% 0.30/0.54 thf(fact_2371_cong__exp__iff__simps_I8_J,axiom,
% 0.30/0.54 ! [M: num,Q2: num] :
% 0.30/0.54 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.30/0.54 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % cong_exp_iff_simps(8)
% 0.30/0.54 thf(fact_2372_cong__exp__iff__simps_I8_J,axiom,
% 0.30/0.54 ! [M: num,Q2: num] :
% 0.30/0.54 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.30/0.54 != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % cong_exp_iff_simps(8)
% 0.30/0.54 thf(fact_2373_cong__exp__iff__simps_I8_J,axiom,
% 0.30/0.54 ! [M: num,Q2: num] :
% 0.30/0.54 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.30/0.54 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % cong_exp_iff_simps(8)
% 0.30/0.54 thf(fact_2374_mod__eqE,axiom,
% 0.30/0.54 ! [A: int,C: int,B: int] :
% 0.30/0.54 ( ( ( modulo_modulo_int @ A @ C )
% 0.30/0.54 = ( modulo_modulo_int @ B @ C ) )
% 0.30/0.54 => ~ ! [D4: int] :
% 0.30/0.54 ( B
% 0.30/0.54 != ( plus_plus_int @ A @ ( times_times_int @ C @ D4 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_eqE
% 0.30/0.54 thf(fact_2375_mod__eqE,axiom,
% 0.30/0.54 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.54 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.30/0.54 = ( modulo364778990260209775nteger @ B @ C ) )
% 0.30/0.54 => ~ ! [D4: code_integer] :
% 0.30/0.54 ( B
% 0.30/0.54 != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D4 ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_eqE
% 0.30/0.54 thf(fact_2376_div__add1__eq,axiom,
% 0.30/0.54 ! [A: nat,B: nat,C: nat] :
% 0.30/0.54 ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.30/0.54 = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % div_add1_eq
% 0.30/0.54 thf(fact_2377_div__add1__eq,axiom,
% 0.30/0.54 ! [A: int,B: int,C: int] :
% 0.30/0.54 ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.30/0.54 = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % div_add1_eq
% 0.30/0.54 thf(fact_2378_div__add1__eq,axiom,
% 0.30/0.54 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.54 ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.30/0.54 = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % div_add1_eq
% 0.30/0.54 thf(fact_2379_Suc__times__mod__eq,axiom,
% 0.30/0.54 ! [M: nat,N2: nat] :
% 0.30/0.54 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.30/0.54 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N2 ) ) @ M )
% 0.30/0.54 = one_one_nat ) ) ).
% 0.30/0.54
% 0.30/0.54 % Suc_times_mod_eq
% 0.30/0.54 thf(fact_2380_mod__induct,axiom,
% 0.30/0.54 ! [P: nat > $o,N2: nat,P4: nat,M: nat] :
% 0.30/0.54 ( ( P @ N2 )
% 0.30/0.54 => ( ( ord_less_nat @ N2 @ P4 )
% 0.30/0.54 => ( ( ord_less_nat @ M @ P4 )
% 0.30/0.54 => ( ! [N4: nat] :
% 0.30/0.54 ( ( ord_less_nat @ N4 @ P4 )
% 0.30/0.54 => ( ( P @ N4 )
% 0.30/0.54 => ( P @ ( modulo_modulo_nat @ ( suc @ N4 ) @ P4 ) ) ) )
% 0.30/0.54 => ( P @ M ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_induct
% 0.30/0.54 thf(fact_2381_mod__Suc__le__divisor,axiom,
% 0.30/0.54 ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N2 ) ) @ N2 ) ).
% 0.30/0.54
% 0.30/0.54 % mod_Suc_le_divisor
% 0.30/0.54 thf(fact_2382_nat__mod__eq__lemma,axiom,
% 0.30/0.54 ! [X: nat,N2: nat,Y: nat] :
% 0.30/0.54 ( ( ( modulo_modulo_nat @ X @ N2 )
% 0.30/0.54 = ( modulo_modulo_nat @ Y @ N2 ) )
% 0.30/0.54 => ( ( ord_less_eq_nat @ Y @ X )
% 0.30/0.54 => ? [Q3: nat] :
% 0.30/0.54 ( X
% 0.30/0.54 = ( plus_plus_nat @ Y @ ( times_times_nat @ N2 @ Q3 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % nat_mod_eq_lemma
% 0.30/0.54 thf(fact_2383_mod__eq__nat2E,axiom,
% 0.30/0.54 ! [M: nat,Q2: nat,N2: nat] :
% 0.30/0.54 ( ( ( modulo_modulo_nat @ M @ Q2 )
% 0.30/0.54 = ( modulo_modulo_nat @ N2 @ Q2 ) )
% 0.30/0.54 => ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.54 => ~ ! [S2: nat] :
% 0.30/0.54 ( N2
% 0.30/0.54 != ( plus_plus_nat @ M @ ( times_times_nat @ Q2 @ S2 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_eq_nat2E
% 0.30/0.54 thf(fact_2384_mod__eq__nat1E,axiom,
% 0.30/0.54 ! [M: nat,Q2: nat,N2: nat] :
% 0.30/0.54 ( ( ( modulo_modulo_nat @ M @ Q2 )
% 0.30/0.54 = ( modulo_modulo_nat @ N2 @ Q2 ) )
% 0.30/0.54 => ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.54 => ~ ! [S2: nat] :
% 0.30/0.54 ( M
% 0.30/0.54 != ( plus_plus_nat @ N2 @ ( times_times_nat @ Q2 @ S2 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_eq_nat1E
% 0.30/0.54 thf(fact_2385_power__strict__mono,axiom,
% 0.30/0.54 ! [A: real,B: real,N2: nat] :
% 0.30/0.54 ( ( ord_less_real @ A @ B )
% 0.30/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.54 => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_strict_mono
% 0.30/0.54 thf(fact_2386_power__strict__mono,axiom,
% 0.30/0.54 ! [A: rat,B: rat,N2: nat] :
% 0.30/0.54 ( ( ord_less_rat @ A @ B )
% 0.30/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.54 => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_strict_mono
% 0.30/0.54 thf(fact_2387_power__strict__mono,axiom,
% 0.30/0.54 ! [A: nat,B: nat,N2: nat] :
% 0.30/0.54 ( ( ord_less_nat @ A @ B )
% 0.30/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.54 => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_strict_mono
% 0.30/0.54 thf(fact_2388_power__strict__mono,axiom,
% 0.30/0.54 ! [A: int,B: int,N2: nat] :
% 0.30/0.54 ( ( ord_less_int @ A @ B )
% 0.30/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.54 => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % power_strict_mono
% 0.30/0.54 thf(fact_2389_mod__if,axiom,
% 0.30/0.54 ( modulo_modulo_nat
% 0.30/0.54 = ( ^ [M6: nat,N: nat] : ( if_nat @ ( ord_less_nat @ M6 @ N ) @ M6 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M6 @ N ) @ N ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mod_if
% 0.30/0.54 thf(fact_2390_vebt__delete_Osimps_I3_J,axiom,
% 0.30/0.54 ! [A: $o,B: $o,N2: nat] :
% 0.30/0.54 ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N2 ) ) )
% 0.30/0.54 = ( vEBT_Leaf @ A @ B ) ) ).
% 0.30/0.54
% 0.30/0.54 % vebt_delete.simps(3)
% 0.30/0.54 thf(fact_2391_le__mod__geq,axiom,
% 0.30/0.54 ! [N2: nat,M: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.54 => ( ( modulo_modulo_nat @ M @ N2 )
% 0.30/0.54 = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % le_mod_geq
% 0.30/0.54 thf(fact_2392_vebt__maxt_Osimps_I1_J,axiom,
% 0.30/0.54 ! [B: $o,A: $o] :
% 0.30/0.54 ( ( B
% 0.30/0.54 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 0.30/0.54 = ( some_nat @ one_one_nat ) ) )
% 0.30/0.54 & ( ~ B
% 0.30/0.54 => ( ( A
% 0.30/0.54 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 0.30/0.54 = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.54 & ( ~ A
% 0.30/0.54 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 0.30/0.54 = none_nat ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % vebt_maxt.simps(1)
% 0.30/0.54 thf(fact_2393_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 0.30/0.54 ! [A: nat,B: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.54 => ( ( ord_less_nat @ A @ B )
% 0.30/0.54 => ( ( divide_divide_nat @ A @ B )
% 0.30/0.54 = zero_zero_nat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % unique_euclidean_semiring_numeral_class.div_less
% 0.30/0.54 thf(fact_2394_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 0.30/0.54 ! [A: int,B: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.54 => ( ( ord_less_int @ A @ B )
% 0.30/0.54 => ( ( divide_divide_int @ A @ B )
% 0.30/0.54 = zero_zero_int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % unique_euclidean_semiring_numeral_class.div_less
% 0.30/0.54 thf(fact_2395_div__positive,axiom,
% 0.30/0.54 ! [B: nat,A: nat] :
% 0.30/0.54 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.54 => ( ( ord_less_eq_nat @ B @ A )
% 0.30/0.54 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % div_positive
% 0.30/0.54 thf(fact_2396_div__positive,axiom,
% 0.30/0.54 ! [B: int,A: int] :
% 0.30/0.54 ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.54 => ( ( ord_less_eq_int @ B @ A )
% 0.30/0.54 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % div_positive
% 0.30/0.54 thf(fact_2397_vebt__mint_Osimps_I1_J,axiom,
% 0.30/0.54 ! [A: $o,B: $o] :
% 0.30/0.54 ( ( A
% 0.30/0.54 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 0.30/0.54 = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.54 & ( ~ A
% 0.30/0.54 => ( ( B
% 0.30/0.54 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 0.30/0.54 = ( some_nat @ one_one_nat ) ) )
% 0.30/0.54 & ( ~ B
% 0.30/0.54 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 0.30/0.54 = none_nat ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % vebt_mint.simps(1)
% 0.30/0.54 thf(fact_2398_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 0.30/0.54 ! [C: nat,A: nat,B: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.54 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.30/0.54 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % unique_euclidean_semiring_numeral_class.div_mult2_eq
% 0.30/0.54 thf(fact_2399_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 0.30/0.54 ! [C: int,A: int,B: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.54 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.30/0.54 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % unique_euclidean_semiring_numeral_class.div_mult2_eq
% 0.30/0.54 thf(fact_2400_vebt__pred_Osimps_I2_J,axiom,
% 0.30/0.54 ! [A: $o,Uw: $o] :
% 0.30/0.54 ( ( A
% 0.30/0.54 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
% 0.30/0.54 = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.54 & ( ~ A
% 0.30/0.54 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
% 0.30/0.54 = none_nat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % vebt_pred.simps(2)
% 0.30/0.54 thf(fact_2401_vebt__succ_Osimps_I1_J,axiom,
% 0.30/0.54 ! [B: $o,Uu: $o] :
% 0.30/0.54 ( ( B
% 0.30/0.54 => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
% 0.30/0.54 = ( some_nat @ one_one_nat ) ) )
% 0.30/0.54 & ( ~ B
% 0.30/0.54 => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
% 0.30/0.54 = none_nat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % vebt_succ.simps(1)
% 0.30/0.54 thf(fact_2402_zero__le__numeral,axiom,
% 0.30/0.54 ! [N2: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_le_numeral
% 0.30/0.54 thf(fact_2403_zero__le__numeral,axiom,
% 0.30/0.54 ! [N2: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N2 ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_le_numeral
% 0.30/0.54 thf(fact_2404_zero__le__numeral,axiom,
% 0.30/0.54 ! [N2: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_le_numeral
% 0.30/0.54 thf(fact_2405_zero__le__numeral,axiom,
% 0.30/0.54 ! [N2: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_le_numeral
% 0.30/0.54 thf(fact_2406_not__numeral__le__zero,axiom,
% 0.30/0.54 ! [N2: num] :
% 0.30/0.54 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).
% 0.30/0.54
% 0.30/0.54 % not_numeral_le_zero
% 0.30/0.54 thf(fact_2407_not__numeral__le__zero,axiom,
% 0.30/0.54 ! [N2: num] :
% 0.30/0.54 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ zero_zero_rat ) ).
% 0.30/0.54
% 0.30/0.54 % not_numeral_le_zero
% 0.30/0.54 thf(fact_2408_not__numeral__le__zero,axiom,
% 0.30/0.54 ! [N2: num] :
% 0.30/0.54 ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).
% 0.30/0.54
% 0.30/0.54 % not_numeral_le_zero
% 0.30/0.54 thf(fact_2409_not__numeral__le__zero,axiom,
% 0.30/0.54 ! [N2: num] :
% 0.30/0.54 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).
% 0.30/0.54
% 0.30/0.54 % not_numeral_le_zero
% 0.30/0.54 thf(fact_2410_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.30/0.54 ! [A: real,B: real,C: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.30/0.54 thf(fact_2411_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.30/0.54 ! [A: rat,B: rat,C: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.30/0.54 thf(fact_2412_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.30/0.54 ! [A: nat,B: nat,C: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.54 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.30/0.54 thf(fact_2413_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.30/0.54 ! [A: int,B: int,C: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.54 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.30/0.54 thf(fact_2414_zero__le__mult__iff,axiom,
% 0.30/0.54 ! [A: real,B: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.30/0.54 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.54 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.30/0.54 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.54 & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_le_mult_iff
% 0.30/0.54 thf(fact_2415_zero__le__mult__iff,axiom,
% 0.30/0.54 ! [A: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.30/0.54 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.54 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 0.30/0.54 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.54 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_le_mult_iff
% 0.30/0.54 thf(fact_2416_zero__le__mult__iff,axiom,
% 0.30/0.54 ! [A: int,B: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 0.30/0.54 = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.54 & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 0.30/0.54 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.54 & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_le_mult_iff
% 0.30/0.54 thf(fact_2417_mult__nonneg__nonpos2,axiom,
% 0.30/0.54 ! [A: real,B: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.54 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.30/0.54 => ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_nonneg_nonpos2
% 0.30/0.54 thf(fact_2418_mult__nonneg__nonpos2,axiom,
% 0.30/0.54 ! [A: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.54 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_nonneg_nonpos2
% 0.30/0.54 thf(fact_2419_mult__nonneg__nonpos2,axiom,
% 0.30/0.54 ! [A: nat,B: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.54 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.30/0.54 => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_nonneg_nonpos2
% 0.30/0.54 thf(fact_2420_mult__nonneg__nonpos2,axiom,
% 0.30/0.54 ! [A: int,B: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.54 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.30/0.54 => ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_nonneg_nonpos2
% 0.30/0.54 thf(fact_2421_mult__nonpos__nonneg,axiom,
% 0.30/0.54 ! [A: real,B: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.54 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_nonpos_nonneg
% 0.30/0.54 thf(fact_2422_mult__nonpos__nonneg,axiom,
% 0.30/0.54 ! [A: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.30/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_nonpos_nonneg
% 0.30/0.54 thf(fact_2423_mult__nonpos__nonneg,axiom,
% 0.30/0.54 ! [A: nat,B: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.30/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.30/0.54 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_nonpos_nonneg
% 0.30/0.54 thf(fact_2424_mult__nonpos__nonneg,axiom,
% 0.30/0.54 ! [A: int,B: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.30/0.54 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_nonpos_nonneg
% 0.30/0.54 thf(fact_2425_mult__nonneg__nonpos,axiom,
% 0.30/0.54 ! [A: real,B: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.54 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.30/0.54 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_nonneg_nonpos
% 0.30/0.54 thf(fact_2426_mult__nonneg__nonpos,axiom,
% 0.30/0.54 ! [A: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.54 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_nonneg_nonpos
% 0.30/0.54 thf(fact_2427_mult__nonneg__nonpos,axiom,
% 0.30/0.54 ! [A: nat,B: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.54 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.30/0.54 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_nonneg_nonpos
% 0.30/0.54 thf(fact_2428_mult__nonneg__nonpos,axiom,
% 0.30/0.54 ! [A: int,B: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.54 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.30/0.54 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_nonneg_nonpos
% 0.30/0.54 thf(fact_2429_mult__nonneg__nonneg,axiom,
% 0.30/0.54 ! [A: real,B: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.54 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_nonneg_nonneg
% 0.30/0.54 thf(fact_2430_mult__nonneg__nonneg,axiom,
% 0.30/0.54 ! [A: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.30/0.54 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_nonneg_nonneg
% 0.30/0.54 thf(fact_2431_mult__nonneg__nonneg,axiom,
% 0.30/0.54 ! [A: nat,B: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.30/0.54 => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_nonneg_nonneg
% 0.30/0.54 thf(fact_2432_mult__nonneg__nonneg,axiom,
% 0.30/0.54 ! [A: int,B: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.30/0.54 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_nonneg_nonneg
% 0.30/0.54 thf(fact_2433_split__mult__neg__le,axiom,
% 0.30/0.54 ! [A: real,B: real] :
% 0.30/0.54 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.54 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.30/0.54 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.54 & ( ord_less_eq_real @ zero_zero_real @ B ) ) )
% 0.30/0.54 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% 0.30/0.54
% 0.30/0.54 % split_mult_neg_le
% 0.30/0.54 thf(fact_2434_split__mult__neg__le,axiom,
% 0.30/0.54 ! [A: rat,B: rat] :
% 0.30/0.54 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.54 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 0.30/0.54 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.54 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
% 0.30/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).
% 0.30/0.54
% 0.30/0.54 % split_mult_neg_le
% 0.30/0.54 thf(fact_2435_split__mult__neg__le,axiom,
% 0.30/0.54 ! [A: nat,B: nat] :
% 0.30/0.54 ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.54 & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
% 0.30/0.54 | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.30/0.54 & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
% 0.30/0.54 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% 0.30/0.54
% 0.30/0.54 % split_mult_neg_le
% 0.30/0.54 thf(fact_2436_split__mult__neg__le,axiom,
% 0.30/0.54 ! [A: int,B: int] :
% 0.30/0.54 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.54 & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 0.30/0.54 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.54 & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
% 0.30/0.54 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% 0.30/0.54
% 0.30/0.54 % split_mult_neg_le
% 0.30/0.54 thf(fact_2437_mult__le__0__iff,axiom,
% 0.30/0.54 ! [A: real,B: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 0.30/0.54 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.54 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.30/0.54 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.54 & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_0_iff
% 0.30/0.54 thf(fact_2438_mult__le__0__iff,axiom,
% 0.30/0.54 ! [A: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 0.30/0.54 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.54 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 0.30/0.54 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.54 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_0_iff
% 0.30/0.54 thf(fact_2439_mult__le__0__iff,axiom,
% 0.30/0.54 ! [A: int,B: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 0.30/0.54 = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.54 & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 0.30/0.54 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.54 & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_le_0_iff
% 0.30/0.54 thf(fact_2440_mult__right__mono,axiom,
% 0.30/0.54 ! [A: real,B: real,C: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_right_mono
% 0.30/0.54 thf(fact_2441_mult__right__mono,axiom,
% 0.30/0.54 ! [A: rat,B: rat,C: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_right_mono
% 0.30/0.54 thf(fact_2442_mult__right__mono,axiom,
% 0.30/0.54 ! [A: nat,B: nat,C: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.54 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_right_mono
% 0.30/0.54 thf(fact_2443_mult__right__mono,axiom,
% 0.30/0.54 ! [A: int,B: int,C: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.54 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_right_mono
% 0.30/0.54 thf(fact_2444_mult__right__mono__neg,axiom,
% 0.30/0.54 ! [B: real,A: real,C: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ B @ A )
% 0.30/0.54 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_right_mono_neg
% 0.30/0.54 thf(fact_2445_mult__right__mono__neg,axiom,
% 0.30/0.54 ! [B: rat,A: rat,C: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ B @ A )
% 0.30/0.54 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_right_mono_neg
% 0.30/0.54 thf(fact_2446_mult__right__mono__neg,axiom,
% 0.30/0.54 ! [B: int,A: int,C: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ B @ A )
% 0.30/0.54 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.30/0.54 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_right_mono_neg
% 0.30/0.54 thf(fact_2447_mult__left__mono,axiom,
% 0.30/0.54 ! [A: real,B: real,C: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_left_mono
% 0.30/0.54 thf(fact_2448_mult__left__mono,axiom,
% 0.30/0.54 ! [A: rat,B: rat,C: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_left_mono
% 0.30/0.54 thf(fact_2449_mult__left__mono,axiom,
% 0.30/0.54 ! [A: nat,B: nat,C: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.54 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_left_mono
% 0.30/0.54 thf(fact_2450_mult__left__mono,axiom,
% 0.30/0.54 ! [A: int,B: int,C: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.54 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_left_mono
% 0.30/0.54 thf(fact_2451_mult__nonpos__nonpos,axiom,
% 0.30/0.54 ! [A: real,B: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.54 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.30/0.54 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_nonpos_nonpos
% 0.30/0.54 thf(fact_2452_mult__nonpos__nonpos,axiom,
% 0.30/0.54 ! [A: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.54 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_nonpos_nonpos
% 0.30/0.54 thf(fact_2453_mult__nonpos__nonpos,axiom,
% 0.30/0.54 ! [A: int,B: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.54 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.30/0.54 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_nonpos_nonpos
% 0.30/0.54 thf(fact_2454_mult__left__mono__neg,axiom,
% 0.30/0.54 ! [B: real,A: real,C: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ B @ A )
% 0.30/0.54 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_left_mono_neg
% 0.30/0.54 thf(fact_2455_mult__left__mono__neg,axiom,
% 0.30/0.54 ! [B: rat,A: rat,C: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ B @ A )
% 0.30/0.54 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_left_mono_neg
% 0.30/0.54 thf(fact_2456_mult__left__mono__neg,axiom,
% 0.30/0.54 ! [B: int,A: int,C: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ B @ A )
% 0.30/0.54 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.30/0.54 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_left_mono_neg
% 0.30/0.54 thf(fact_2457_split__mult__pos__le,axiom,
% 0.30/0.54 ! [A: real,B: real] :
% 0.30/0.54 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.54 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.30/0.54 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.54 & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 0.30/0.54 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % split_mult_pos_le
% 0.30/0.54 thf(fact_2458_split__mult__pos__le,axiom,
% 0.30/0.54 ! [A: rat,B: rat] :
% 0.30/0.54 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.54 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 0.30/0.54 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.54 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 0.30/0.54 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % split_mult_pos_le
% 0.30/0.54 thf(fact_2459_split__mult__pos__le,axiom,
% 0.30/0.54 ! [A: int,B: int] :
% 0.30/0.54 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.54 & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 0.30/0.54 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.54 & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 0.30/0.54 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % split_mult_pos_le
% 0.30/0.54 thf(fact_2460_zero__le__square,axiom,
% 0.30/0.54 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_le_square
% 0.30/0.54 thf(fact_2461_zero__le__square,axiom,
% 0.30/0.54 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_le_square
% 0.30/0.54 thf(fact_2462_zero__le__square,axiom,
% 0.30/0.54 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_le_square
% 0.30/0.54 thf(fact_2463_mult__mono_H,axiom,
% 0.30/0.54 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.54 => ( ( ord_less_eq_real @ C @ D )
% 0.30/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_mono'
% 0.30/0.54 thf(fact_2464_mult__mono_H,axiom,
% 0.30/0.54 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.54 => ( ( ord_less_eq_rat @ C @ D )
% 0.30/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_mono'
% 0.30/0.54 thf(fact_2465_mult__mono_H,axiom,
% 0.30/0.54 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.54 => ( ( ord_less_eq_nat @ C @ D )
% 0.30/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.54 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_mono'
% 0.30/0.54 thf(fact_2466_mult__mono_H,axiom,
% 0.30/0.54 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.54 => ( ( ord_less_eq_int @ C @ D )
% 0.30/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.54 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_mono'
% 0.30/0.54 thf(fact_2467_mult__mono,axiom,
% 0.30/0.54 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.54 => ( ( ord_less_eq_real @ C @ D )
% 0.30/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_mono
% 0.30/0.54 thf(fact_2468_mult__mono,axiom,
% 0.30/0.54 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.54 => ( ( ord_less_eq_rat @ C @ D )
% 0.30/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.30/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_mono
% 0.30/0.54 thf(fact_2469_mult__mono,axiom,
% 0.30/0.54 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.54 => ( ( ord_less_eq_nat @ C @ D )
% 0.30/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.30/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.54 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_mono
% 0.30/0.54 thf(fact_2470_mult__mono,axiom,
% 0.30/0.54 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.54 => ( ( ord_less_eq_int @ C @ D )
% 0.30/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.30/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.54 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_mono
% 0.30/0.54 thf(fact_2471_zero__less__numeral,axiom,
% 0.30/0.54 ! [N2: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_less_numeral
% 0.30/0.54 thf(fact_2472_zero__less__numeral,axiom,
% 0.30/0.54 ! [N2: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N2 ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_less_numeral
% 0.30/0.54 thf(fact_2473_zero__less__numeral,axiom,
% 0.30/0.54 ! [N2: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_less_numeral
% 0.30/0.54 thf(fact_2474_zero__less__numeral,axiom,
% 0.30/0.54 ! [N2: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_less_numeral
% 0.30/0.54 thf(fact_2475_not__numeral__less__zero,axiom,
% 0.30/0.54 ! [N2: num] :
% 0.30/0.54 ~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).
% 0.30/0.54
% 0.30/0.54 % not_numeral_less_zero
% 0.30/0.54 thf(fact_2476_not__numeral__less__zero,axiom,
% 0.30/0.54 ! [N2: num] :
% 0.30/0.54 ~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ zero_zero_rat ) ).
% 0.30/0.54
% 0.30/0.54 % not_numeral_less_zero
% 0.30/0.54 thf(fact_2477_not__numeral__less__zero,axiom,
% 0.30/0.54 ! [N2: num] :
% 0.30/0.54 ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).
% 0.30/0.54
% 0.30/0.54 % not_numeral_less_zero
% 0.30/0.54 thf(fact_2478_not__numeral__less__zero,axiom,
% 0.30/0.54 ! [N2: num] :
% 0.30/0.54 ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).
% 0.30/0.54
% 0.30/0.54 % not_numeral_less_zero
% 0.30/0.54 thf(fact_2479_add__nonpos__eq__0__iff,axiom,
% 0.30/0.54 ! [X: real,Y: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.30/0.54 => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 0.30/0.54 => ( ( ( plus_plus_real @ X @ Y )
% 0.30/0.54 = zero_zero_real )
% 0.30/0.54 = ( ( X = zero_zero_real )
% 0.30/0.54 & ( Y = zero_zero_real ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_nonpos_eq_0_iff
% 0.30/0.54 thf(fact_2480_add__nonpos__eq__0__iff,axiom,
% 0.30/0.54 ! [X: rat,Y: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.30/0.54 => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
% 0.30/0.54 => ( ( ( plus_plus_rat @ X @ Y )
% 0.30/0.54 = zero_zero_rat )
% 0.30/0.54 = ( ( X = zero_zero_rat )
% 0.30/0.54 & ( Y = zero_zero_rat ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_nonpos_eq_0_iff
% 0.30/0.54 thf(fact_2481_add__nonpos__eq__0__iff,axiom,
% 0.30/0.54 ! [X: nat,Y: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ X @ zero_zero_nat )
% 0.30/0.54 => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
% 0.30/0.54 => ( ( ( plus_plus_nat @ X @ Y )
% 0.30/0.54 = zero_zero_nat )
% 0.30/0.54 = ( ( X = zero_zero_nat )
% 0.30/0.54 & ( Y = zero_zero_nat ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_nonpos_eq_0_iff
% 0.30/0.54 thf(fact_2482_add__nonpos__eq__0__iff,axiom,
% 0.30/0.54 ! [X: int,Y: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ X @ zero_zero_int )
% 0.30/0.54 => ( ( ord_less_eq_int @ Y @ zero_zero_int )
% 0.30/0.54 => ( ( ( plus_plus_int @ X @ Y )
% 0.30/0.54 = zero_zero_int )
% 0.30/0.54 = ( ( X = zero_zero_int )
% 0.30/0.54 & ( Y = zero_zero_int ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_nonpos_eq_0_iff
% 0.30/0.54 thf(fact_2483_add__nonneg__eq__0__iff,axiom,
% 0.30/0.54 ! [X: real,Y: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.30/0.54 => ( ( ( plus_plus_real @ X @ Y )
% 0.30/0.54 = zero_zero_real )
% 0.30/0.54 = ( ( X = zero_zero_real )
% 0.30/0.54 & ( Y = zero_zero_real ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_nonneg_eq_0_iff
% 0.30/0.54 thf(fact_2484_add__nonneg__eq__0__iff,axiom,
% 0.30/0.54 ! [X: rat,Y: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.30/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.30/0.54 => ( ( ( plus_plus_rat @ X @ Y )
% 0.30/0.54 = zero_zero_rat )
% 0.30/0.54 = ( ( X = zero_zero_rat )
% 0.30/0.54 & ( Y = zero_zero_rat ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_nonneg_eq_0_iff
% 0.30/0.54 thf(fact_2485_add__nonneg__eq__0__iff,axiom,
% 0.30/0.54 ! [X: nat,Y: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.30/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 0.30/0.54 => ( ( ( plus_plus_nat @ X @ Y )
% 0.30/0.54 = zero_zero_nat )
% 0.30/0.54 = ( ( X = zero_zero_nat )
% 0.30/0.54 & ( Y = zero_zero_nat ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_nonneg_eq_0_iff
% 0.30/0.54 thf(fact_2486_add__nonneg__eq__0__iff,axiom,
% 0.30/0.54 ! [X: int,Y: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.30/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.30/0.54 => ( ( ( plus_plus_int @ X @ Y )
% 0.30/0.54 = zero_zero_int )
% 0.30/0.54 = ( ( X = zero_zero_int )
% 0.30/0.54 & ( Y = zero_zero_int ) ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_nonneg_eq_0_iff
% 0.30/0.54 thf(fact_2487_add__nonpos__nonpos,axiom,
% 0.30/0.54 ! [A: real,B: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.54 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.30/0.54 => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_nonpos_nonpos
% 0.30/0.54 thf(fact_2488_add__nonpos__nonpos,axiom,
% 0.30/0.54 ! [A: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.54 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_nonpos_nonpos
% 0.30/0.54 thf(fact_2489_add__nonpos__nonpos,axiom,
% 0.30/0.54 ! [A: nat,B: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.30/0.54 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.30/0.54 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_nonpos_nonpos
% 0.30/0.54 thf(fact_2490_add__nonpos__nonpos,axiom,
% 0.30/0.54 ! [A: int,B: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.54 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.30/0.54 => ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_nonpos_nonpos
% 0.30/0.54 thf(fact_2491_add__nonneg__nonneg,axiom,
% 0.30/0.54 ! [A: real,B: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.54 => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_nonneg_nonneg
% 0.30/0.54 thf(fact_2492_add__nonneg__nonneg,axiom,
% 0.30/0.54 ! [A: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.30/0.54 => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_nonneg_nonneg
% 0.30/0.54 thf(fact_2493_add__nonneg__nonneg,axiom,
% 0.30/0.54 ! [A: nat,B: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.30/0.54 => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_nonneg_nonneg
% 0.30/0.54 thf(fact_2494_add__nonneg__nonneg,axiom,
% 0.30/0.54 ! [A: int,B: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.30/0.54 => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_nonneg_nonneg
% 0.30/0.54 thf(fact_2495_add__increasing2,axiom,
% 0.30/0.54 ! [C: real,B: real,A: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ( ord_less_eq_real @ B @ A )
% 0.30/0.54 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_increasing2
% 0.30/0.54 thf(fact_2496_add__increasing2,axiom,
% 0.30/0.54 ! [C: rat,B: rat,A: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ( ord_less_eq_rat @ B @ A )
% 0.30/0.54 => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_increasing2
% 0.30/0.54 thf(fact_2497_add__increasing2,axiom,
% 0.30/0.54 ! [C: nat,B: nat,A: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.54 => ( ( ord_less_eq_nat @ B @ A )
% 0.30/0.54 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_increasing2
% 0.30/0.54 thf(fact_2498_add__increasing2,axiom,
% 0.30/0.54 ! [C: int,B: int,A: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.54 => ( ( ord_less_eq_int @ B @ A )
% 0.30/0.54 => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_increasing2
% 0.30/0.54 thf(fact_2499_add__decreasing2,axiom,
% 0.30/0.54 ! [C: real,A: real,B: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ( ord_less_eq_real @ A @ B )
% 0.30/0.54 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_decreasing2
% 0.30/0.54 thf(fact_2500_add__decreasing2,axiom,
% 0.30/0.54 ! [C: rat,A: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.54 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_decreasing2
% 0.30/0.54 thf(fact_2501_add__decreasing2,axiom,
% 0.30/0.54 ! [C: nat,A: nat,B: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 0.30/0.54 => ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.54 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_decreasing2
% 0.30/0.54 thf(fact_2502_add__decreasing2,axiom,
% 0.30/0.54 ! [C: int,A: int,B: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.30/0.54 => ( ( ord_less_eq_int @ A @ B )
% 0.30/0.54 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_decreasing2
% 0.30/0.54 thf(fact_2503_add__increasing,axiom,
% 0.30/0.54 ! [A: real,B: real,C: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.54 => ( ( ord_less_eq_real @ B @ C )
% 0.30/0.54 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_increasing
% 0.30/0.54 thf(fact_2504_add__increasing,axiom,
% 0.30/0.54 ! [A: rat,B: rat,C: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.54 => ( ( ord_less_eq_rat @ B @ C )
% 0.30/0.54 => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_increasing
% 0.30/0.54 thf(fact_2505_add__increasing,axiom,
% 0.30/0.54 ! [A: nat,B: nat,C: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.54 => ( ( ord_less_eq_nat @ B @ C )
% 0.30/0.54 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_increasing
% 0.30/0.54 thf(fact_2506_add__increasing,axiom,
% 0.30/0.54 ! [A: int,B: int,C: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.54 => ( ( ord_less_eq_int @ B @ C )
% 0.30/0.54 => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_increasing
% 0.30/0.54 thf(fact_2507_add__decreasing,axiom,
% 0.30/0.54 ! [A: real,C: real,B: real] :
% 0.30/0.54 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.54 => ( ( ord_less_eq_real @ C @ B )
% 0.30/0.54 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_decreasing
% 0.30/0.54 thf(fact_2508_add__decreasing,axiom,
% 0.30/0.54 ! [A: rat,C: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.54 => ( ( ord_less_eq_rat @ C @ B )
% 0.30/0.54 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_decreasing
% 0.30/0.54 thf(fact_2509_add__decreasing,axiom,
% 0.30/0.54 ! [A: nat,C: nat,B: nat] :
% 0.30/0.54 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.30/0.54 => ( ( ord_less_eq_nat @ C @ B )
% 0.30/0.54 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_decreasing
% 0.30/0.54 thf(fact_2510_add__decreasing,axiom,
% 0.30/0.54 ! [A: int,C: int,B: int] :
% 0.30/0.54 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.54 => ( ( ord_less_eq_int @ C @ B )
% 0.30/0.54 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_decreasing
% 0.30/0.54 thf(fact_2511_not__one__le__zero,axiom,
% 0.30/0.54 ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% 0.30/0.54
% 0.30/0.54 % not_one_le_zero
% 0.30/0.54 thf(fact_2512_not__one__le__zero,axiom,
% 0.30/0.54 ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% 0.30/0.54
% 0.30/0.54 % not_one_le_zero
% 0.30/0.54 thf(fact_2513_not__one__le__zero,axiom,
% 0.30/0.54 ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.30/0.54
% 0.30/0.54 % not_one_le_zero
% 0.30/0.54 thf(fact_2514_not__one__le__zero,axiom,
% 0.30/0.54 ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% 0.30/0.54
% 0.30/0.54 % not_one_le_zero
% 0.30/0.54 thf(fact_2515_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.30/0.54 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 0.30/0.54
% 0.30/0.54 % linordered_nonzero_semiring_class.zero_le_one
% 0.30/0.54 thf(fact_2516_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.30/0.54 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 0.30/0.54
% 0.30/0.54 % linordered_nonzero_semiring_class.zero_le_one
% 0.30/0.54 thf(fact_2517_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.30/0.54 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.30/0.54
% 0.30/0.54 % linordered_nonzero_semiring_class.zero_le_one
% 0.30/0.54 thf(fact_2518_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.30/0.54 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 0.30/0.54
% 0.30/0.54 % linordered_nonzero_semiring_class.zero_le_one
% 0.30/0.54 thf(fact_2519_zero__less__one__class_Ozero__le__one,axiom,
% 0.30/0.54 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 0.30/0.54
% 0.30/0.54 % zero_less_one_class.zero_le_one
% 0.30/0.54 thf(fact_2520_zero__less__one__class_Ozero__le__one,axiom,
% 0.30/0.54 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 0.30/0.54
% 0.30/0.54 % zero_less_one_class.zero_le_one
% 0.30/0.54 thf(fact_2521_zero__less__one__class_Ozero__le__one,axiom,
% 0.30/0.54 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.30/0.54
% 0.30/0.54 % zero_less_one_class.zero_le_one
% 0.30/0.54 thf(fact_2522_zero__less__one__class_Ozero__le__one,axiom,
% 0.30/0.54 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 0.30/0.54
% 0.30/0.54 % zero_less_one_class.zero_le_one
% 0.30/0.54 thf(fact_2523_mult__neg__neg,axiom,
% 0.30/0.54 ! [A: real,B: real] :
% 0.30/0.54 ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.54 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.30/0.54 => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_neg_neg
% 0.30/0.54 thf(fact_2524_mult__neg__neg,axiom,
% 0.30/0.54 ! [A: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.54 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_neg_neg
% 0.30/0.54 thf(fact_2525_mult__neg__neg,axiom,
% 0.30/0.54 ! [A: int,B: int] :
% 0.30/0.54 ( ( ord_less_int @ A @ zero_zero_int )
% 0.30/0.54 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.30/0.54 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_neg_neg
% 0.30/0.54 thf(fact_2526_not__square__less__zero,axiom,
% 0.30/0.54 ! [A: real] :
% 0.30/0.54 ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% 0.30/0.54
% 0.30/0.54 % not_square_less_zero
% 0.30/0.54 thf(fact_2527_not__square__less__zero,axiom,
% 0.30/0.54 ! [A: rat] :
% 0.30/0.54 ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).
% 0.30/0.54
% 0.30/0.54 % not_square_less_zero
% 0.30/0.54 thf(fact_2528_not__square__less__zero,axiom,
% 0.30/0.54 ! [A: int] :
% 0.30/0.54 ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% 0.30/0.54
% 0.30/0.54 % not_square_less_zero
% 0.30/0.54 thf(fact_2529_mult__less__0__iff,axiom,
% 0.30/0.54 ! [A: real,B: real] :
% 0.30/0.54 ( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 0.30/0.54 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.54 & ( ord_less_real @ B @ zero_zero_real ) )
% 0.30/0.54 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.54 & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_less_0_iff
% 0.30/0.54 thf(fact_2530_mult__less__0__iff,axiom,
% 0.30/0.54 ! [A: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 0.30/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.54 & ( ord_less_rat @ B @ zero_zero_rat ) )
% 0.30/0.54 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.54 & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_less_0_iff
% 0.30/0.54 thf(fact_2531_mult__less__0__iff,axiom,
% 0.30/0.54 ! [A: int,B: int] :
% 0.30/0.54 ( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 0.30/0.54 = ( ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.54 & ( ord_less_int @ B @ zero_zero_int ) )
% 0.30/0.54 | ( ( ord_less_int @ A @ zero_zero_int )
% 0.30/0.54 & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_less_0_iff
% 0.30/0.54 thf(fact_2532_mult__neg__pos,axiom,
% 0.30/0.54 ! [A: real,B: real] :
% 0.30/0.54 ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.54 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/0.54 => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_neg_pos
% 0.30/0.54 thf(fact_2533_mult__neg__pos,axiom,
% 0.30/0.54 ! [A: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.54 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.30/0.54 => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_neg_pos
% 0.30/0.54 thf(fact_2534_mult__neg__pos,axiom,
% 0.30/0.54 ! [A: nat,B: nat] :
% 0.30/0.54 ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.54 => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_neg_pos
% 0.30/0.54 thf(fact_2535_mult__neg__pos,axiom,
% 0.30/0.54 ! [A: int,B: int] :
% 0.30/0.54 ( ( ord_less_int @ A @ zero_zero_int )
% 0.30/0.54 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.54 => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_neg_pos
% 0.30/0.54 thf(fact_2536_mult__pos__neg,axiom,
% 0.30/0.54 ! [A: real,B: real] :
% 0.30/0.54 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.54 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.30/0.54 => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_pos_neg
% 0.30/0.54 thf(fact_2537_mult__pos__neg,axiom,
% 0.30/0.54 ! [A: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.54 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_pos_neg
% 0.30/0.54 thf(fact_2538_mult__pos__neg,axiom,
% 0.30/0.54 ! [A: nat,B: nat] :
% 0.30/0.54 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.54 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.30/0.54 => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_pos_neg
% 0.30/0.54 thf(fact_2539_mult__pos__neg,axiom,
% 0.30/0.54 ! [A: int,B: int] :
% 0.30/0.54 ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.54 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.30/0.54 => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_pos_neg
% 0.30/0.54 thf(fact_2540_mult__pos__pos,axiom,
% 0.30/0.54 ! [A: real,B: real] :
% 0.30/0.54 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.54 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/0.54 => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_pos_pos
% 0.30/0.54 thf(fact_2541_mult__pos__pos,axiom,
% 0.30/0.54 ! [A: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.54 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.30/0.54 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_pos_pos
% 0.30/0.54 thf(fact_2542_mult__pos__pos,axiom,
% 0.30/0.54 ! [A: nat,B: nat] :
% 0.30/0.54 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.54 => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_pos_pos
% 0.30/0.54 thf(fact_2543_mult__pos__pos,axiom,
% 0.30/0.54 ! [A: int,B: int] :
% 0.30/0.54 ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.54 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.54 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_pos_pos
% 0.30/0.54 thf(fact_2544_mult__pos__neg2,axiom,
% 0.30/0.54 ! [A: real,B: real] :
% 0.30/0.54 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.54 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.30/0.54 => ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_pos_neg2
% 0.30/0.54 thf(fact_2545_mult__pos__neg2,axiom,
% 0.30/0.54 ! [A: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.54 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_pos_neg2
% 0.30/0.54 thf(fact_2546_mult__pos__neg2,axiom,
% 0.30/0.54 ! [A: nat,B: nat] :
% 0.30/0.54 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.54 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.30/0.54 => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_pos_neg2
% 0.30/0.54 thf(fact_2547_mult__pos__neg2,axiom,
% 0.30/0.54 ! [A: int,B: int] :
% 0.30/0.54 ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.54 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.30/0.54 => ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_pos_neg2
% 0.30/0.54 thf(fact_2548_zero__less__mult__iff,axiom,
% 0.30/0.54 ! [A: real,B: real] :
% 0.30/0.54 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.30/0.54 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.54 & ( ord_less_real @ zero_zero_real @ B ) )
% 0.30/0.54 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.54 & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_less_mult_iff
% 0.30/0.54 thf(fact_2549_zero__less__mult__iff,axiom,
% 0.30/0.54 ! [A: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.30/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.54 & ( ord_less_rat @ zero_zero_rat @ B ) )
% 0.30/0.54 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.54 & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_less_mult_iff
% 0.30/0.54 thf(fact_2550_zero__less__mult__iff,axiom,
% 0.30/0.54 ! [A: int,B: int] :
% 0.30/0.54 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 0.30/0.54 = ( ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.54 & ( ord_less_int @ zero_zero_int @ B ) )
% 0.30/0.54 | ( ( ord_less_int @ A @ zero_zero_int )
% 0.30/0.54 & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_less_mult_iff
% 0.30/0.54 thf(fact_2551_zero__less__mult__pos,axiom,
% 0.30/0.54 ! [A: real,B: real] :
% 0.30/0.54 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.30/0.54 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.54 => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_less_mult_pos
% 0.30/0.54 thf(fact_2552_zero__less__mult__pos,axiom,
% 0.30/0.54 ! [A: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.30/0.54 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.54 => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_less_mult_pos
% 0.30/0.54 thf(fact_2553_zero__less__mult__pos,axiom,
% 0.30/0.54 ! [A: nat,B: nat] :
% 0.30/0.54 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.54 => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_less_mult_pos
% 0.30/0.54 thf(fact_2554_zero__less__mult__pos,axiom,
% 0.30/0.54 ! [A: int,B: int] :
% 0.30/0.54 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 0.30/0.54 => ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.54 => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_less_mult_pos
% 0.30/0.54 thf(fact_2555_zero__less__mult__pos2,axiom,
% 0.30/0.54 ! [B: real,A: real] :
% 0.30/0.54 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
% 0.30/0.54 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.54 => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_less_mult_pos2
% 0.30/0.54 thf(fact_2556_zero__less__mult__pos2,axiom,
% 0.30/0.54 ! [B: rat,A: rat] :
% 0.30/0.54 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
% 0.30/0.54 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.54 => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_less_mult_pos2
% 0.30/0.54 thf(fact_2557_zero__less__mult__pos2,axiom,
% 0.30/0.54 ! [B: nat,A: nat] :
% 0.30/0.54 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.54 => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_less_mult_pos2
% 0.30/0.54 thf(fact_2558_zero__less__mult__pos2,axiom,
% 0.30/0.54 ! [B: int,A: int] :
% 0.30/0.54 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
% 0.30/0.54 => ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.54 => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % zero_less_mult_pos2
% 0.30/0.54 thf(fact_2559_mult__less__cancel__left__neg,axiom,
% 0.30/0.54 ! [C: real,A: real,B: real] :
% 0.30/0.54 ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.30/0.54 = ( ord_less_real @ B @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_less_cancel_left_neg
% 0.30/0.54 thf(fact_2560_mult__less__cancel__left__neg,axiom,
% 0.30/0.54 ! [C: rat,A: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.30/0.54 = ( ord_less_rat @ B @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_less_cancel_left_neg
% 0.30/0.54 thf(fact_2561_mult__less__cancel__left__neg,axiom,
% 0.30/0.54 ! [C: int,A: int,B: int] :
% 0.30/0.54 ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.54 => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.30/0.54 = ( ord_less_int @ B @ A ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_less_cancel_left_neg
% 0.30/0.54 thf(fact_2562_mult__less__cancel__left__pos,axiom,
% 0.30/0.54 ! [C: real,A: real,B: real] :
% 0.30/0.54 ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.30/0.54 = ( ord_less_real @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_less_cancel_left_pos
% 0.30/0.54 thf(fact_2563_mult__less__cancel__left__pos,axiom,
% 0.30/0.54 ! [C: rat,A: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.30/0.54 = ( ord_less_rat @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_less_cancel_left_pos
% 0.30/0.54 thf(fact_2564_mult__less__cancel__left__pos,axiom,
% 0.30/0.54 ! [C: int,A: int,B: int] :
% 0.30/0.54 ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.54 => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.30/0.54 = ( ord_less_int @ A @ B ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_less_cancel_left_pos
% 0.30/0.54 thf(fact_2565_mult__strict__left__mono__neg,axiom,
% 0.30/0.54 ! [B: real,A: real,C: real] :
% 0.30/0.54 ( ( ord_less_real @ B @ A )
% 0.30/0.54 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_strict_left_mono_neg
% 0.30/0.54 thf(fact_2566_mult__strict__left__mono__neg,axiom,
% 0.30/0.54 ! [B: rat,A: rat,C: rat] :
% 0.30/0.54 ( ( ord_less_rat @ B @ A )
% 0.30/0.54 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_strict_left_mono_neg
% 0.30/0.54 thf(fact_2567_mult__strict__left__mono__neg,axiom,
% 0.30/0.54 ! [B: int,A: int,C: int] :
% 0.30/0.54 ( ( ord_less_int @ B @ A )
% 0.30/0.54 => ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.54 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_strict_left_mono_neg
% 0.30/0.54 thf(fact_2568_mult__strict__left__mono,axiom,
% 0.30/0.54 ! [A: real,B: real,C: real] :
% 0.30/0.54 ( ( ord_less_real @ A @ B )
% 0.30/0.54 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_strict_left_mono
% 0.30/0.54 thf(fact_2569_mult__strict__left__mono,axiom,
% 0.30/0.54 ! [A: rat,B: rat,C: rat] :
% 0.30/0.54 ( ( ord_less_rat @ A @ B )
% 0.30/0.54 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_strict_left_mono
% 0.30/0.54 thf(fact_2570_mult__strict__left__mono,axiom,
% 0.30/0.54 ! [A: nat,B: nat,C: nat] :
% 0.30/0.54 ( ( ord_less_nat @ A @ B )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.30/0.54 => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_strict_left_mono
% 0.30/0.54 thf(fact_2571_mult__strict__left__mono,axiom,
% 0.30/0.54 ! [A: int,B: int,C: int] :
% 0.30/0.54 ( ( ord_less_int @ A @ B )
% 0.30/0.54 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.54 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_strict_left_mono
% 0.30/0.54 thf(fact_2572_mult__less__cancel__left__disj,axiom,
% 0.30/0.54 ! [C: real,A: real,B: real] :
% 0.30/0.54 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.30/0.54 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54 & ( ord_less_real @ A @ B ) )
% 0.30/0.54 | ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54 & ( ord_less_real @ B @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_less_cancel_left_disj
% 0.30/0.54 thf(fact_2573_mult__less__cancel__left__disj,axiom,
% 0.30/0.54 ! [C: rat,A: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.30/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54 & ( ord_less_rat @ A @ B ) )
% 0.30/0.54 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_less_cancel_left_disj
% 0.30/0.54 thf(fact_2574_mult__less__cancel__left__disj,axiom,
% 0.30/0.54 ! [C: int,A: int,B: int] :
% 0.30/0.54 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.30/0.54 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.54 & ( ord_less_int @ A @ B ) )
% 0.30/0.54 | ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.54 & ( ord_less_int @ B @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_less_cancel_left_disj
% 0.30/0.54 thf(fact_2575_mult__strict__right__mono__neg,axiom,
% 0.30/0.54 ! [B: real,A: real,C: real] :
% 0.30/0.54 ( ( ord_less_real @ B @ A )
% 0.30/0.54 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_strict_right_mono_neg
% 0.30/0.54 thf(fact_2576_mult__strict__right__mono__neg,axiom,
% 0.30/0.54 ! [B: rat,A: rat,C: rat] :
% 0.30/0.54 ( ( ord_less_rat @ B @ A )
% 0.30/0.54 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_strict_right_mono_neg
% 0.30/0.54 thf(fact_2577_mult__strict__right__mono__neg,axiom,
% 0.30/0.54 ! [B: int,A: int,C: int] :
% 0.30/0.54 ( ( ord_less_int @ B @ A )
% 0.30/0.54 => ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.54 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_strict_right_mono_neg
% 0.30/0.54 thf(fact_2578_mult__strict__right__mono,axiom,
% 0.30/0.54 ! [A: real,B: real,C: real] :
% 0.30/0.54 ( ( ord_less_real @ A @ B )
% 0.30/0.54 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_strict_right_mono
% 0.30/0.54 thf(fact_2579_mult__strict__right__mono,axiom,
% 0.30/0.54 ! [A: rat,B: rat,C: rat] :
% 0.30/0.54 ( ( ord_less_rat @ A @ B )
% 0.30/0.54 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_strict_right_mono
% 0.30/0.54 thf(fact_2580_mult__strict__right__mono,axiom,
% 0.30/0.54 ! [A: nat,B: nat,C: nat] :
% 0.30/0.54 ( ( ord_less_nat @ A @ B )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.30/0.54 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_strict_right_mono
% 0.30/0.54 thf(fact_2581_mult__strict__right__mono,axiom,
% 0.30/0.54 ! [A: int,B: int,C: int] :
% 0.30/0.54 ( ( ord_less_int @ A @ B )
% 0.30/0.54 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.54 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_strict_right_mono
% 0.30/0.54 thf(fact_2582_mult__less__cancel__right__disj,axiom,
% 0.30/0.54 ! [A: real,C: real,B: real] :
% 0.30/0.54 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.30/0.54 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54 & ( ord_less_real @ A @ B ) )
% 0.30/0.54 | ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.54 & ( ord_less_real @ B @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_less_cancel_right_disj
% 0.30/0.54 thf(fact_2583_mult__less__cancel__right__disj,axiom,
% 0.30/0.54 ! [A: rat,C: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.30/0.54 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54 & ( ord_less_rat @ A @ B ) )
% 0.30/0.54 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.54 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_less_cancel_right_disj
% 0.30/0.54 thf(fact_2584_mult__less__cancel__right__disj,axiom,
% 0.30/0.54 ! [A: int,C: int,B: int] :
% 0.30/0.54 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.30/0.54 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.54 & ( ord_less_int @ A @ B ) )
% 0.30/0.54 | ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.54 & ( ord_less_int @ B @ A ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % mult_less_cancel_right_disj
% 0.30/0.54 thf(fact_2585_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.30/0.54 ! [A: real,B: real,C: real] :
% 0.30/0.54 ( ( ord_less_real @ A @ B )
% 0.30/0.54 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.54 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.30/0.54 thf(fact_2586_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.30/0.54 ! [A: rat,B: rat,C: rat] :
% 0.30/0.54 ( ( ord_less_rat @ A @ B )
% 0.30/0.54 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.54 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.30/0.54 thf(fact_2587_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.30/0.54 ! [A: nat,B: nat,C: nat] :
% 0.30/0.54 ( ( ord_less_nat @ A @ B )
% 0.30/0.54 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.30/0.54 => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.30/0.54 thf(fact_2588_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.30/0.54 ! [A: int,B: int,C: int] :
% 0.30/0.54 ( ( ord_less_int @ A @ B )
% 0.30/0.54 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.54 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.30/0.54 thf(fact_2589_add__less__zeroD,axiom,
% 0.30/0.54 ! [X: real,Y: real] :
% 0.30/0.54 ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
% 0.30/0.54 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.30/0.54 | ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_less_zeroD
% 0.30/0.54 thf(fact_2590_add__less__zeroD,axiom,
% 0.30/0.54 ! [X: rat,Y: rat] :
% 0.30/0.54 ( ( ord_less_rat @ ( plus_plus_rat @ X @ Y ) @ zero_zero_rat )
% 0.30/0.54 => ( ( ord_less_rat @ X @ zero_zero_rat )
% 0.30/0.54 | ( ord_less_rat @ Y @ zero_zero_rat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_less_zeroD
% 0.30/0.54 thf(fact_2591_add__less__zeroD,axiom,
% 0.30/0.54 ! [X: int,Y: int] :
% 0.30/0.54 ( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
% 0.30/0.54 => ( ( ord_less_int @ X @ zero_zero_int )
% 0.30/0.54 | ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_less_zeroD
% 0.30/0.54 thf(fact_2592_add__neg__neg,axiom,
% 0.30/0.54 ! [A: real,B: real] :
% 0.30/0.54 ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.54 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.30/0.54 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_neg_neg
% 0.30/0.54 thf(fact_2593_add__neg__neg,axiom,
% 0.30/0.54 ! [A: rat,B: rat] :
% 0.30/0.54 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.54 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.30/0.54 => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.30/0.54
% 0.30/0.54 % add_neg_neg
% 0.30/0.54 thf(fact_2594_add__neg__neg,axiom,
% 0.30/0.54 ! [A: nat,B: nat] :
% 0.30/0.54 ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.30/0.55 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.30/0.55 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_neg_neg
% 0.30/0.55 thf(fact_2595_add__neg__neg,axiom,
% 0.30/0.55 ! [A: int,B: int] :
% 0.30/0.55 ( ( ord_less_int @ A @ zero_zero_int )
% 0.30/0.55 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.30/0.55 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_neg_neg
% 0.30/0.55 thf(fact_2596_add__pos__pos,axiom,
% 0.30/0.55 ! [A: real,B: real] :
% 0.30/0.55 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.55 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/0.55 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_pos_pos
% 0.30/0.55 thf(fact_2597_add__pos__pos,axiom,
% 0.30/0.55 ! [A: rat,B: rat] :
% 0.30/0.55 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.55 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.30/0.55 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_pos_pos
% 0.30/0.55 thf(fact_2598_add__pos__pos,axiom,
% 0.30/0.55 ! [A: nat,B: nat] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.55 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.55 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_pos_pos
% 0.30/0.55 thf(fact_2599_add__pos__pos,axiom,
% 0.30/0.55 ! [A: int,B: int] :
% 0.30/0.55 ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.55 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.55 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_pos_pos
% 0.30/0.55 thf(fact_2600_canonically__ordered__monoid__add__class_OlessE,axiom,
% 0.30/0.55 ! [A: nat,B: nat] :
% 0.30/0.55 ( ( ord_less_nat @ A @ B )
% 0.30/0.55 => ~ ! [C3: nat] :
% 0.30/0.55 ( ( B
% 0.30/0.55 = ( plus_plus_nat @ A @ C3 ) )
% 0.30/0.55 => ( C3 = zero_zero_nat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % canonically_ordered_monoid_add_class.lessE
% 0.30/0.55 thf(fact_2601_pos__add__strict,axiom,
% 0.30/0.55 ! [A: real,B: real,C: real] :
% 0.30/0.55 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.55 => ( ( ord_less_real @ B @ C )
% 0.30/0.55 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % pos_add_strict
% 0.30/0.55 thf(fact_2602_pos__add__strict,axiom,
% 0.30/0.55 ! [A: rat,B: rat,C: rat] :
% 0.30/0.55 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.55 => ( ( ord_less_rat @ B @ C )
% 0.30/0.55 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % pos_add_strict
% 0.30/0.55 thf(fact_2603_pos__add__strict,axiom,
% 0.30/0.55 ! [A: nat,B: nat,C: nat] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.55 => ( ( ord_less_nat @ B @ C )
% 0.30/0.55 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % pos_add_strict
% 0.30/0.55 thf(fact_2604_pos__add__strict,axiom,
% 0.30/0.55 ! [A: int,B: int,C: int] :
% 0.30/0.55 ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.55 => ( ( ord_less_int @ B @ C )
% 0.30/0.55 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % pos_add_strict
% 0.30/0.55 thf(fact_2605_less__numeral__extra_I1_J,axiom,
% 0.30/0.55 ord_less_real @ zero_zero_real @ one_one_real ).
% 0.30/0.55
% 0.30/0.55 % less_numeral_extra(1)
% 0.30/0.55 thf(fact_2606_less__numeral__extra_I1_J,axiom,
% 0.30/0.55 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 0.30/0.55
% 0.30/0.55 % less_numeral_extra(1)
% 0.30/0.55 thf(fact_2607_less__numeral__extra_I1_J,axiom,
% 0.30/0.55 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 0.30/0.55
% 0.30/0.55 % less_numeral_extra(1)
% 0.30/0.55 thf(fact_2608_less__numeral__extra_I1_J,axiom,
% 0.30/0.55 ord_less_int @ zero_zero_int @ one_one_int ).
% 0.30/0.55
% 0.30/0.55 % less_numeral_extra(1)
% 0.30/0.55 thf(fact_2609_zero__less__one,axiom,
% 0.30/0.55 ord_less_real @ zero_zero_real @ one_one_real ).
% 0.30/0.55
% 0.30/0.55 % zero_less_one
% 0.30/0.55 thf(fact_2610_zero__less__one,axiom,
% 0.30/0.55 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 0.30/0.55
% 0.30/0.55 % zero_less_one
% 0.30/0.55 thf(fact_2611_zero__less__one,axiom,
% 0.30/0.55 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 0.30/0.55
% 0.30/0.55 % zero_less_one
% 0.30/0.55 thf(fact_2612_zero__less__one,axiom,
% 0.30/0.55 ord_less_int @ zero_zero_int @ one_one_int ).
% 0.30/0.55
% 0.30/0.55 % zero_less_one
% 0.30/0.55 thf(fact_2613_not__one__less__zero,axiom,
% 0.30/0.55 ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% 0.30/0.55
% 0.30/0.55 % not_one_less_zero
% 0.30/0.55 thf(fact_2614_not__one__less__zero,axiom,
% 0.30/0.55 ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% 0.30/0.55
% 0.30/0.55 % not_one_less_zero
% 0.30/0.55 thf(fact_2615_not__one__less__zero,axiom,
% 0.30/0.55 ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.30/0.55
% 0.30/0.55 % not_one_less_zero
% 0.30/0.55 thf(fact_2616_not__one__less__zero,axiom,
% 0.30/0.55 ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% 0.30/0.55
% 0.30/0.55 % not_one_less_zero
% 0.30/0.55 thf(fact_2617_le__iff__diff__le__0,axiom,
% 0.30/0.55 ( ord_less_eq_real
% 0.30/0.55 = ( ^ [A5: real,B5: real] : ( ord_less_eq_real @ ( minus_minus_real @ A5 @ B5 ) @ zero_zero_real ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % le_iff_diff_le_0
% 0.30/0.55 thf(fact_2618_le__iff__diff__le__0,axiom,
% 0.30/0.55 ( ord_less_eq_rat
% 0.30/0.55 = ( ^ [A5: rat,B5: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A5 @ B5 ) @ zero_zero_rat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % le_iff_diff_le_0
% 0.30/0.55 thf(fact_2619_le__iff__diff__le__0,axiom,
% 0.30/0.55 ( ord_less_eq_int
% 0.30/0.55 = ( ^ [A5: int,B5: int] : ( ord_less_eq_int @ ( minus_minus_int @ A5 @ B5 ) @ zero_zero_int ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % le_iff_diff_le_0
% 0.30/0.55 thf(fact_2620_divide__right__mono__neg,axiom,
% 0.30/0.55 ! [A: real,B: real,C: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.55 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.55 => ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_right_mono_neg
% 0.30/0.55 thf(fact_2621_divide__right__mono__neg,axiom,
% 0.30/0.55 ! [A: rat,B: rat,C: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.55 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.30/0.55 => ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_right_mono_neg
% 0.30/0.55 thf(fact_2622_divide__nonpos__nonpos,axiom,
% 0.30/0.55 ! [X: real,Y: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.30/0.55 => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 0.30/0.55 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_nonpos_nonpos
% 0.30/0.55 thf(fact_2623_divide__nonpos__nonpos,axiom,
% 0.30/0.55 ! [X: rat,Y: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.30/0.55 => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
% 0.30/0.55 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_nonpos_nonpos
% 0.30/0.55 thf(fact_2624_divide__nonpos__nonneg,axiom,
% 0.30/0.55 ! [X: real,Y: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.30/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.30/0.55 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_nonpos_nonneg
% 0.30/0.55 thf(fact_2625_divide__nonpos__nonneg,axiom,
% 0.30/0.55 ! [X: rat,Y: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.30/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.30/0.55 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_nonpos_nonneg
% 0.30/0.55 thf(fact_2626_divide__nonneg__nonpos,axiom,
% 0.30/0.55 ! [X: real,Y: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.55 => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 0.30/0.55 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_nonneg_nonpos
% 0.30/0.55 thf(fact_2627_divide__nonneg__nonpos,axiom,
% 0.30/0.55 ! [X: rat,Y: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.30/0.55 => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
% 0.30/0.55 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_nonneg_nonpos
% 0.30/0.55 thf(fact_2628_divide__nonneg__nonneg,axiom,
% 0.30/0.55 ! [X: real,Y: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.30/0.55 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_nonneg_nonneg
% 0.30/0.55 thf(fact_2629_divide__nonneg__nonneg,axiom,
% 0.30/0.55 ! [X: rat,Y: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.30/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.30/0.55 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_nonneg_nonneg
% 0.30/0.55 thf(fact_2630_zero__le__divide__iff,axiom,
% 0.30/0.55 ! [A: real,B: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 0.30/0.55 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.55 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.30/0.55 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.55 & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zero_le_divide_iff
% 0.30/0.55 thf(fact_2631_zero__le__divide__iff,axiom,
% 0.30/0.55 ! [A: rat,B: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 0.30/0.55 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.55 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 0.30/0.55 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.55 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zero_le_divide_iff
% 0.30/0.55 thf(fact_2632_divide__right__mono,axiom,
% 0.30/0.55 ! [A: real,B: real,C: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.55 => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_right_mono
% 0.30/0.55 thf(fact_2633_divide__right__mono,axiom,
% 0.30/0.55 ! [A: rat,B: rat,C: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.55 => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_right_mono
% 0.30/0.55 thf(fact_2634_divide__le__0__iff,axiom,
% 0.30/0.55 ! [A: real,B: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 0.30/0.55 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.55 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.30/0.55 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.55 & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_le_0_iff
% 0.30/0.55 thf(fact_2635_divide__le__0__iff,axiom,
% 0.30/0.55 ! [A: rat,B: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 0.30/0.55 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.55 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 0.30/0.55 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.55 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_le_0_iff
% 0.30/0.55 thf(fact_2636_less__iff__diff__less__0,axiom,
% 0.30/0.55 ( ord_less_real
% 0.30/0.55 = ( ^ [A5: real,B5: real] : ( ord_less_real @ ( minus_minus_real @ A5 @ B5 ) @ zero_zero_real ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % less_iff_diff_less_0
% 0.30/0.55 thf(fact_2637_less__iff__diff__less__0,axiom,
% 0.30/0.55 ( ord_less_rat
% 0.30/0.55 = ( ^ [A5: rat,B5: rat] : ( ord_less_rat @ ( minus_minus_rat @ A5 @ B5 ) @ zero_zero_rat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % less_iff_diff_less_0
% 0.30/0.55 thf(fact_2638_less__iff__diff__less__0,axiom,
% 0.30/0.55 ( ord_less_int
% 0.30/0.55 = ( ^ [A5: int,B5: int] : ( ord_less_int @ ( minus_minus_int @ A5 @ B5 ) @ zero_zero_int ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % less_iff_diff_less_0
% 0.30/0.55 thf(fact_2639_power__mono,axiom,
% 0.30/0.55 ! [A: real,B: real,N2: nat] :
% 0.30/0.55 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.55 => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power_mono
% 0.30/0.55 thf(fact_2640_power__mono,axiom,
% 0.30/0.55 ! [A: rat,B: rat,N2: nat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.55 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power_mono
% 0.30/0.55 thf(fact_2641_power__mono,axiom,
% 0.30/0.55 ! [A: nat,B: nat,N2: nat] :
% 0.30/0.55 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.55 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power_mono
% 0.30/0.55 thf(fact_2642_power__mono,axiom,
% 0.30/0.55 ! [A: int,B: int,N2: nat] :
% 0.30/0.55 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.55 => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power_mono
% 0.30/0.55 thf(fact_2643_zero__le__power,axiom,
% 0.30/0.55 ! [A: real,N2: nat] :
% 0.30/0.55 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.55 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zero_le_power
% 0.30/0.55 thf(fact_2644_zero__le__power,axiom,
% 0.30/0.55 ! [A: rat,N2: nat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.55 => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zero_le_power
% 0.30/0.55 thf(fact_2645_zero__le__power,axiom,
% 0.30/0.55 ! [A: nat,N2: nat] :
% 0.30/0.55 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.55 => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zero_le_power
% 0.30/0.55 thf(fact_2646_zero__le__power,axiom,
% 0.30/0.55 ! [A: int,N2: nat] :
% 0.30/0.55 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.55 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zero_le_power
% 0.30/0.55 thf(fact_2647_divide__neg__neg,axiom,
% 0.30/0.55 ! [X: real,Y: real] :
% 0.30/0.55 ( ( ord_less_real @ X @ zero_zero_real )
% 0.30/0.55 => ( ( ord_less_real @ Y @ zero_zero_real )
% 0.30/0.55 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_neg_neg
% 0.30/0.55 thf(fact_2648_divide__neg__neg,axiom,
% 0.30/0.55 ! [X: rat,Y: rat] :
% 0.30/0.55 ( ( ord_less_rat @ X @ zero_zero_rat )
% 0.30/0.55 => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 0.30/0.55 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_neg_neg
% 0.30/0.55 thf(fact_2649_divide__neg__pos,axiom,
% 0.30/0.55 ! [X: real,Y: real] :
% 0.30/0.55 ( ( ord_less_real @ X @ zero_zero_real )
% 0.30/0.55 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.30/0.55 => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_neg_pos
% 0.30/0.55 thf(fact_2650_divide__neg__pos,axiom,
% 0.30/0.55 ! [X: rat,Y: rat] :
% 0.30/0.55 ( ( ord_less_rat @ X @ zero_zero_rat )
% 0.30/0.55 => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.30/0.55 => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_neg_pos
% 0.30/0.55 thf(fact_2651_divide__pos__neg,axiom,
% 0.30/0.55 ! [X: real,Y: real] :
% 0.30/0.55 ( ( ord_less_real @ zero_zero_real @ X )
% 0.30/0.55 => ( ( ord_less_real @ Y @ zero_zero_real )
% 0.30/0.55 => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_pos_neg
% 0.30/0.55 thf(fact_2652_divide__pos__neg,axiom,
% 0.30/0.55 ! [X: rat,Y: rat] :
% 0.30/0.55 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.30/0.55 => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 0.30/0.55 => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_pos_neg
% 0.30/0.55 thf(fact_2653_divide__pos__pos,axiom,
% 0.30/0.55 ! [X: real,Y: real] :
% 0.30/0.55 ( ( ord_less_real @ zero_zero_real @ X )
% 0.30/0.55 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.30/0.55 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_pos_pos
% 0.30/0.55 thf(fact_2654_divide__pos__pos,axiom,
% 0.30/0.55 ! [X: rat,Y: rat] :
% 0.30/0.55 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.30/0.55 => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.30/0.55 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_pos_pos
% 0.30/0.55 thf(fact_2655_divide__less__0__iff,axiom,
% 0.30/0.55 ! [A: real,B: real] :
% 0.30/0.55 ( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 0.30/0.55 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.55 & ( ord_less_real @ B @ zero_zero_real ) )
% 0.30/0.55 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.55 & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_less_0_iff
% 0.30/0.55 thf(fact_2656_divide__less__0__iff,axiom,
% 0.30/0.55 ! [A: rat,B: rat] :
% 0.30/0.55 ( ( ord_less_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 0.30/0.55 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.55 & ( ord_less_rat @ B @ zero_zero_rat ) )
% 0.30/0.55 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.55 & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_less_0_iff
% 0.30/0.55 thf(fact_2657_divide__less__cancel,axiom,
% 0.30/0.55 ! [A: real,C: real,B: real] :
% 0.30/0.55 ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 0.30/0.55 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55 => ( ord_less_real @ A @ B ) )
% 0.30/0.55 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.55 => ( ord_less_real @ B @ A ) )
% 0.30/0.55 & ( C != zero_zero_real ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_less_cancel
% 0.30/0.55 thf(fact_2658_divide__less__cancel,axiom,
% 0.30/0.55 ! [A: rat,C: rat,B: rat] :
% 0.30/0.55 ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 0.30/0.55 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55 => ( ord_less_rat @ A @ B ) )
% 0.30/0.55 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.55 => ( ord_less_rat @ B @ A ) )
% 0.30/0.55 & ( C != zero_zero_rat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_less_cancel
% 0.30/0.55 thf(fact_2659_zero__less__divide__iff,axiom,
% 0.30/0.55 ! [A: real,B: real] :
% 0.30/0.55 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 0.30/0.55 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.55 & ( ord_less_real @ zero_zero_real @ B ) )
% 0.30/0.55 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.55 & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zero_less_divide_iff
% 0.30/0.55 thf(fact_2660_zero__less__divide__iff,axiom,
% 0.30/0.55 ! [A: rat,B: rat] :
% 0.30/0.55 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 0.30/0.55 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.55 & ( ord_less_rat @ zero_zero_rat @ B ) )
% 0.30/0.55 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.55 & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zero_less_divide_iff
% 0.30/0.55 thf(fact_2661_divide__strict__right__mono,axiom,
% 0.30/0.55 ! [A: real,B: real,C: real] :
% 0.30/0.55 ( ( ord_less_real @ A @ B )
% 0.30/0.55 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_strict_right_mono
% 0.30/0.55 thf(fact_2662_divide__strict__right__mono,axiom,
% 0.30/0.55 ! [A: rat,B: rat,C: rat] :
% 0.30/0.55 ( ( ord_less_rat @ A @ B )
% 0.30/0.55 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55 => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_strict_right_mono
% 0.30/0.55 thf(fact_2663_divide__strict__right__mono__neg,axiom,
% 0.30/0.55 ! [B: real,A: real,C: real] :
% 0.30/0.55 ( ( ord_less_real @ B @ A )
% 0.30/0.55 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.55 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_strict_right_mono_neg
% 0.30/0.55 thf(fact_2664_divide__strict__right__mono__neg,axiom,
% 0.30/0.55 ! [B: rat,A: rat,C: rat] :
% 0.30/0.55 ( ( ord_less_rat @ B @ A )
% 0.30/0.55 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.55 => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_strict_right_mono_neg
% 0.30/0.55 thf(fact_2665_zero__less__power,axiom,
% 0.30/0.55 ! [A: real,N2: nat] :
% 0.30/0.55 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.55 => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zero_less_power
% 0.30/0.55 thf(fact_2666_zero__less__power,axiom,
% 0.30/0.55 ! [A: rat,N2: nat] :
% 0.30/0.55 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.55 => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zero_less_power
% 0.30/0.55 thf(fact_2667_zero__less__power,axiom,
% 0.30/0.55 ! [A: nat,N2: nat] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.55 => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zero_less_power
% 0.30/0.55 thf(fact_2668_zero__less__power,axiom,
% 0.30/0.55 ! [A: int,N2: nat] :
% 0.30/0.55 ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.55 => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % zero_less_power
% 0.30/0.55 thf(fact_2669_nonzero__eq__divide__eq,axiom,
% 0.30/0.55 ! [C: complex,A: complex,B: complex] :
% 0.30/0.55 ( ( C != zero_zero_complex )
% 0.30/0.55 => ( ( A
% 0.30/0.55 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.30/0.55 = ( ( times_times_complex @ A @ C )
% 0.30/0.55 = B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nonzero_eq_divide_eq
% 0.30/0.55 thf(fact_2670_nonzero__eq__divide__eq,axiom,
% 0.30/0.55 ! [C: real,A: real,B: real] :
% 0.30/0.55 ( ( C != zero_zero_real )
% 0.30/0.55 => ( ( A
% 0.30/0.55 = ( divide_divide_real @ B @ C ) )
% 0.30/0.55 = ( ( times_times_real @ A @ C )
% 0.30/0.55 = B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nonzero_eq_divide_eq
% 0.30/0.55 thf(fact_2671_nonzero__eq__divide__eq,axiom,
% 0.30/0.55 ! [C: rat,A: rat,B: rat] :
% 0.30/0.55 ( ( C != zero_zero_rat )
% 0.30/0.55 => ( ( A
% 0.30/0.55 = ( divide_divide_rat @ B @ C ) )
% 0.30/0.55 = ( ( times_times_rat @ A @ C )
% 0.30/0.55 = B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nonzero_eq_divide_eq
% 0.30/0.55 thf(fact_2672_nonzero__divide__eq__eq,axiom,
% 0.30/0.55 ! [C: complex,B: complex,A: complex] :
% 0.30/0.55 ( ( C != zero_zero_complex )
% 0.30/0.55 => ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.30/0.55 = A )
% 0.30/0.55 = ( B
% 0.30/0.55 = ( times_times_complex @ A @ C ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nonzero_divide_eq_eq
% 0.30/0.55 thf(fact_2673_nonzero__divide__eq__eq,axiom,
% 0.30/0.55 ! [C: real,B: real,A: real] :
% 0.30/0.55 ( ( C != zero_zero_real )
% 0.30/0.55 => ( ( ( divide_divide_real @ B @ C )
% 0.30/0.55 = A )
% 0.30/0.55 = ( B
% 0.30/0.55 = ( times_times_real @ A @ C ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nonzero_divide_eq_eq
% 0.30/0.55 thf(fact_2674_nonzero__divide__eq__eq,axiom,
% 0.30/0.55 ! [C: rat,B: rat,A: rat] :
% 0.30/0.55 ( ( C != zero_zero_rat )
% 0.30/0.55 => ( ( ( divide_divide_rat @ B @ C )
% 0.30/0.55 = A )
% 0.30/0.55 = ( B
% 0.30/0.55 = ( times_times_rat @ A @ C ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nonzero_divide_eq_eq
% 0.30/0.55 thf(fact_2675_eq__divide__imp,axiom,
% 0.30/0.55 ! [C: complex,A: complex,B: complex] :
% 0.30/0.55 ( ( C != zero_zero_complex )
% 0.30/0.55 => ( ( ( times_times_complex @ A @ C )
% 0.30/0.55 = B )
% 0.30/0.55 => ( A
% 0.30/0.55 = ( divide1717551699836669952omplex @ B @ C ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % eq_divide_imp
% 0.30/0.55 thf(fact_2676_eq__divide__imp,axiom,
% 0.30/0.55 ! [C: real,A: real,B: real] :
% 0.30/0.55 ( ( C != zero_zero_real )
% 0.30/0.55 => ( ( ( times_times_real @ A @ C )
% 0.30/0.55 = B )
% 0.30/0.55 => ( A
% 0.30/0.55 = ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % eq_divide_imp
% 0.30/0.55 thf(fact_2677_eq__divide__imp,axiom,
% 0.30/0.55 ! [C: rat,A: rat,B: rat] :
% 0.30/0.55 ( ( C != zero_zero_rat )
% 0.30/0.55 => ( ( ( times_times_rat @ A @ C )
% 0.30/0.55 = B )
% 0.30/0.55 => ( A
% 0.30/0.55 = ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % eq_divide_imp
% 0.30/0.55 thf(fact_2678_divide__eq__imp,axiom,
% 0.30/0.55 ! [C: complex,B: complex,A: complex] :
% 0.30/0.55 ( ( C != zero_zero_complex )
% 0.30/0.55 => ( ( B
% 0.30/0.55 = ( times_times_complex @ A @ C ) )
% 0.30/0.55 => ( ( divide1717551699836669952omplex @ B @ C )
% 0.30/0.55 = A ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_eq_imp
% 0.30/0.55 thf(fact_2679_divide__eq__imp,axiom,
% 0.30/0.55 ! [C: real,B: real,A: real] :
% 0.30/0.55 ( ( C != zero_zero_real )
% 0.30/0.55 => ( ( B
% 0.30/0.55 = ( times_times_real @ A @ C ) )
% 0.30/0.55 => ( ( divide_divide_real @ B @ C )
% 0.30/0.55 = A ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_eq_imp
% 0.30/0.55 thf(fact_2680_divide__eq__imp,axiom,
% 0.30/0.55 ! [C: rat,B: rat,A: rat] :
% 0.30/0.55 ( ( C != zero_zero_rat )
% 0.30/0.55 => ( ( B
% 0.30/0.55 = ( times_times_rat @ A @ C ) )
% 0.30/0.55 => ( ( divide_divide_rat @ B @ C )
% 0.30/0.55 = A ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_eq_imp
% 0.30/0.55 thf(fact_2681_eq__divide__eq,axiom,
% 0.30/0.55 ! [A: complex,B: complex,C: complex] :
% 0.30/0.55 ( ( A
% 0.30/0.55 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.30/0.55 = ( ( ( C != zero_zero_complex )
% 0.30/0.55 => ( ( times_times_complex @ A @ C )
% 0.30/0.55 = B ) )
% 0.30/0.55 & ( ( C = zero_zero_complex )
% 0.30/0.55 => ( A = zero_zero_complex ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % eq_divide_eq
% 0.30/0.55 thf(fact_2682_eq__divide__eq,axiom,
% 0.30/0.55 ! [A: real,B: real,C: real] :
% 0.30/0.55 ( ( A
% 0.30/0.55 = ( divide_divide_real @ B @ C ) )
% 0.30/0.55 = ( ( ( C != zero_zero_real )
% 0.30/0.55 => ( ( times_times_real @ A @ C )
% 0.30/0.55 = B ) )
% 0.30/0.55 & ( ( C = zero_zero_real )
% 0.30/0.55 => ( A = zero_zero_real ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % eq_divide_eq
% 0.30/0.55 thf(fact_2683_eq__divide__eq,axiom,
% 0.30/0.55 ! [A: rat,B: rat,C: rat] :
% 0.30/0.55 ( ( A
% 0.30/0.55 = ( divide_divide_rat @ B @ C ) )
% 0.30/0.55 = ( ( ( C != zero_zero_rat )
% 0.30/0.55 => ( ( times_times_rat @ A @ C )
% 0.30/0.55 = B ) )
% 0.30/0.55 & ( ( C = zero_zero_rat )
% 0.30/0.55 => ( A = zero_zero_rat ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % eq_divide_eq
% 0.30/0.55 thf(fact_2684_divide__eq__eq,axiom,
% 0.30/0.55 ! [B: complex,C: complex,A: complex] :
% 0.30/0.55 ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.30/0.55 = A )
% 0.30/0.55 = ( ( ( C != zero_zero_complex )
% 0.30/0.55 => ( B
% 0.30/0.55 = ( times_times_complex @ A @ C ) ) )
% 0.30/0.55 & ( ( C = zero_zero_complex )
% 0.30/0.55 => ( A = zero_zero_complex ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_eq_eq
% 0.30/0.55 thf(fact_2685_divide__eq__eq,axiom,
% 0.30/0.55 ! [B: real,C: real,A: real] :
% 0.30/0.55 ( ( ( divide_divide_real @ B @ C )
% 0.30/0.55 = A )
% 0.30/0.55 = ( ( ( C != zero_zero_real )
% 0.30/0.55 => ( B
% 0.30/0.55 = ( times_times_real @ A @ C ) ) )
% 0.30/0.55 & ( ( C = zero_zero_real )
% 0.30/0.55 => ( A = zero_zero_real ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_eq_eq
% 0.30/0.55 thf(fact_2686_divide__eq__eq,axiom,
% 0.30/0.55 ! [B: rat,C: rat,A: rat] :
% 0.30/0.55 ( ( ( divide_divide_rat @ B @ C )
% 0.30/0.55 = A )
% 0.30/0.55 = ( ( ( C != zero_zero_rat )
% 0.30/0.55 => ( B
% 0.30/0.55 = ( times_times_rat @ A @ C ) ) )
% 0.30/0.55 & ( ( C = zero_zero_rat )
% 0.30/0.55 => ( A = zero_zero_rat ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_eq_eq
% 0.30/0.55 thf(fact_2687_frac__eq__eq,axiom,
% 0.30/0.55 ! [Y: complex,Z: complex,X: complex,W: complex] :
% 0.30/0.55 ( ( Y != zero_zero_complex )
% 0.30/0.55 => ( ( Z != zero_zero_complex )
% 0.30/0.55 => ( ( ( divide1717551699836669952omplex @ X @ Y )
% 0.30/0.55 = ( divide1717551699836669952omplex @ W @ Z ) )
% 0.30/0.55 = ( ( times_times_complex @ X @ Z )
% 0.30/0.55 = ( times_times_complex @ W @ Y ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % frac_eq_eq
% 0.30/0.55 thf(fact_2688_frac__eq__eq,axiom,
% 0.30/0.55 ! [Y: real,Z: real,X: real,W: real] :
% 0.30/0.55 ( ( Y != zero_zero_real )
% 0.30/0.55 => ( ( Z != zero_zero_real )
% 0.30/0.55 => ( ( ( divide_divide_real @ X @ Y )
% 0.30/0.55 = ( divide_divide_real @ W @ Z ) )
% 0.30/0.55 = ( ( times_times_real @ X @ Z )
% 0.30/0.55 = ( times_times_real @ W @ Y ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % frac_eq_eq
% 0.30/0.55 thf(fact_2689_frac__eq__eq,axiom,
% 0.30/0.55 ! [Y: rat,Z: rat,X: rat,W: rat] :
% 0.30/0.55 ( ( Y != zero_zero_rat )
% 0.30/0.55 => ( ( Z != zero_zero_rat )
% 0.30/0.55 => ( ( ( divide_divide_rat @ X @ Y )
% 0.30/0.55 = ( divide_divide_rat @ W @ Z ) )
% 0.30/0.55 = ( ( times_times_rat @ X @ Z )
% 0.30/0.55 = ( times_times_rat @ W @ Y ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % frac_eq_eq
% 0.30/0.55 thf(fact_2690_right__inverse__eq,axiom,
% 0.30/0.55 ! [B: complex,A: complex] :
% 0.30/0.55 ( ( B != zero_zero_complex )
% 0.30/0.55 => ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.30/0.55 = one_one_complex )
% 0.30/0.55 = ( A = B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % right_inverse_eq
% 0.30/0.55 thf(fact_2691_right__inverse__eq,axiom,
% 0.30/0.55 ! [B: real,A: real] :
% 0.30/0.55 ( ( B != zero_zero_real )
% 0.30/0.55 => ( ( ( divide_divide_real @ A @ B )
% 0.30/0.55 = one_one_real )
% 0.30/0.55 = ( A = B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % right_inverse_eq
% 0.30/0.55 thf(fact_2692_right__inverse__eq,axiom,
% 0.30/0.55 ! [B: rat,A: rat] :
% 0.30/0.55 ( ( B != zero_zero_rat )
% 0.30/0.55 => ( ( ( divide_divide_rat @ A @ B )
% 0.30/0.55 = one_one_rat )
% 0.30/0.55 = ( A = B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % right_inverse_eq
% 0.30/0.55 thf(fact_2693_power__0,axiom,
% 0.30/0.55 ! [A: rat] :
% 0.30/0.55 ( ( power_power_rat @ A @ zero_zero_nat )
% 0.30/0.55 = one_one_rat ) ).
% 0.30/0.55
% 0.30/0.55 % power_0
% 0.30/0.55 thf(fact_2694_power__0,axiom,
% 0.30/0.55 ! [A: nat] :
% 0.30/0.55 ( ( power_power_nat @ A @ zero_zero_nat )
% 0.30/0.55 = one_one_nat ) ).
% 0.30/0.55
% 0.30/0.55 % power_0
% 0.30/0.55 thf(fact_2695_power__0,axiom,
% 0.30/0.55 ! [A: real] :
% 0.30/0.55 ( ( power_power_real @ A @ zero_zero_nat )
% 0.30/0.55 = one_one_real ) ).
% 0.30/0.55
% 0.30/0.55 % power_0
% 0.30/0.55 thf(fact_2696_power__0,axiom,
% 0.30/0.55 ! [A: int] :
% 0.30/0.55 ( ( power_power_int @ A @ zero_zero_nat )
% 0.30/0.55 = one_one_int ) ).
% 0.30/0.55
% 0.30/0.55 % power_0
% 0.30/0.55 thf(fact_2697_power__0,axiom,
% 0.30/0.55 ! [A: complex] :
% 0.30/0.55 ( ( power_power_complex @ A @ zero_zero_nat )
% 0.30/0.55 = one_one_complex ) ).
% 0.30/0.55
% 0.30/0.55 % power_0
% 0.30/0.55 thf(fact_2698_divmod__digit__0_I1_J,axiom,
% 0.30/0.55 ! [B: nat,A: nat] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.55 => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.30/0.55 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 0.30/0.55 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divmod_digit_0(1)
% 0.30/0.55 thf(fact_2699_divmod__digit__0_I1_J,axiom,
% 0.30/0.55 ! [B: int,A: int] :
% 0.30/0.55 ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.55 => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.30/0.55 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 0.30/0.55 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divmod_digit_0(1)
% 0.30/0.55 thf(fact_2700_divmod__digit__0_I1_J,axiom,
% 0.30/0.55 ! [B: code_integer,A: code_integer] :
% 0.30/0.55 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.30/0.55 => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.30/0.55 => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 0.30/0.55 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divmod_digit_0(1)
% 0.30/0.55 thf(fact_2701_Ex__less__Suc2,axiom,
% 0.30/0.55 ! [N2: nat,P: nat > $o] :
% 0.30/0.55 ( ( ? [I3: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
% 0.30/0.55 & ( P @ I3 ) ) )
% 0.30/0.55 = ( ( P @ zero_zero_nat )
% 0.30/0.55 | ? [I3: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I3 @ N2 )
% 0.30/0.55 & ( P @ ( suc @ I3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Ex_less_Suc2
% 0.30/0.55 thf(fact_2702_gr0__conv__Suc,axiom,
% 0.30/0.55 ! [N2: nat] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.55 = ( ? [M6: nat] :
% 0.30/0.55 ( N2
% 0.30/0.55 = ( suc @ M6 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gr0_conv_Suc
% 0.30/0.55 thf(fact_2703_All__less__Suc2,axiom,
% 0.30/0.55 ! [N2: nat,P: nat > $o] :
% 0.30/0.55 ( ( ! [I3: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
% 0.30/0.55 => ( P @ I3 ) ) )
% 0.30/0.55 = ( ( P @ zero_zero_nat )
% 0.30/0.55 & ! [I3: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I3 @ N2 )
% 0.30/0.55 => ( P @ ( suc @ I3 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % All_less_Suc2
% 0.30/0.55 thf(fact_2704_gr0__implies__Suc,axiom,
% 0.30/0.55 ! [N2: nat] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.55 => ? [M2: nat] :
% 0.30/0.55 ( N2
% 0.30/0.55 = ( suc @ M2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % gr0_implies_Suc
% 0.30/0.55 thf(fact_2705_less__Suc__eq__0__disj,axiom,
% 0.30/0.55 ! [M: nat,N2: nat] :
% 0.30/0.55 ( ( ord_less_nat @ M @ ( suc @ N2 ) )
% 0.30/0.55 = ( ( M = zero_zero_nat )
% 0.30/0.55 | ? [J3: nat] :
% 0.30/0.55 ( ( M
% 0.30/0.55 = ( suc @ J3 ) )
% 0.30/0.55 & ( ord_less_nat @ J3 @ N2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % less_Suc_eq_0_disj
% 0.30/0.55 thf(fact_2706_one__is__add,axiom,
% 0.30/0.55 ! [M: nat,N2: nat] :
% 0.30/0.55 ( ( ( suc @ zero_zero_nat )
% 0.30/0.55 = ( plus_plus_nat @ M @ N2 ) )
% 0.30/0.55 = ( ( ( M
% 0.30/0.55 = ( suc @ zero_zero_nat ) )
% 0.30/0.55 & ( N2 = zero_zero_nat ) )
% 0.30/0.55 | ( ( M = zero_zero_nat )
% 0.30/0.55 & ( N2
% 0.30/0.55 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % one_is_add
% 0.30/0.55 thf(fact_2707_add__is__1,axiom,
% 0.30/0.55 ! [M: nat,N2: nat] :
% 0.30/0.55 ( ( ( plus_plus_nat @ M @ N2 )
% 0.30/0.55 = ( suc @ zero_zero_nat ) )
% 0.30/0.55 = ( ( ( M
% 0.30/0.55 = ( suc @ zero_zero_nat ) )
% 0.30/0.55 & ( N2 = zero_zero_nat ) )
% 0.30/0.55 | ( ( M = zero_zero_nat )
% 0.30/0.55 & ( N2
% 0.30/0.55 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_is_1
% 0.30/0.55 thf(fact_2708_option_Osize_I4_J,axiom,
% 0.30/0.55 ! [X22: nat] :
% 0.30/0.55 ( ( size_size_option_nat @ ( some_nat @ X22 ) )
% 0.30/0.55 = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.55
% 0.30/0.55 % option.size(4)
% 0.30/0.55 thf(fact_2709_option_Osize_I4_J,axiom,
% 0.30/0.55 ! [X22: product_prod_nat_nat] :
% 0.30/0.55 ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
% 0.30/0.55 = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.55
% 0.30/0.55 % option.size(4)
% 0.30/0.55 thf(fact_2710_option_Osize_I4_J,axiom,
% 0.30/0.55 ! [X22: num] :
% 0.30/0.55 ( ( size_size_option_num @ ( some_num @ X22 ) )
% 0.30/0.55 = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.55
% 0.30/0.55 % option.size(4)
% 0.30/0.55 thf(fact_2711_ex__least__nat__le,axiom,
% 0.30/0.55 ! [P: nat > $o,N2: nat] :
% 0.30/0.55 ( ( P @ N2 )
% 0.30/0.55 => ( ~ ( P @ zero_zero_nat )
% 0.30/0.55 => ? [K2: nat] :
% 0.30/0.55 ( ( ord_less_eq_nat @ K2 @ N2 )
% 0.30/0.55 & ! [I: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I @ K2 )
% 0.30/0.55 => ~ ( P @ I ) )
% 0.30/0.55 & ( P @ K2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % ex_least_nat_le
% 0.30/0.55 thf(fact_2712_less__imp__add__positive,axiom,
% 0.30/0.55 ! [I2: nat,J: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I2 @ J )
% 0.30/0.55 => ? [K2: nat] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ K2 )
% 0.30/0.55 & ( ( plus_plus_nat @ I2 @ K2 )
% 0.30/0.55 = J ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % less_imp_add_positive
% 0.30/0.55 thf(fact_2713_option_Osize_I3_J,axiom,
% 0.30/0.55 ( ( size_size_option_nat @ none_nat )
% 0.30/0.55 = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.55
% 0.30/0.55 % option.size(3)
% 0.30/0.55 thf(fact_2714_option_Osize_I3_J,axiom,
% 0.30/0.55 ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
% 0.30/0.55 = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.55
% 0.30/0.55 % option.size(3)
% 0.30/0.55 thf(fact_2715_option_Osize_I3_J,axiom,
% 0.30/0.55 ( ( size_size_option_num @ none_num )
% 0.30/0.55 = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.55
% 0.30/0.55 % option.size(3)
% 0.30/0.55 thf(fact_2716_mult__less__mono1,axiom,
% 0.30/0.55 ! [I2: nat,J: nat,K: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I2 @ J )
% 0.30/0.55 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.55 => ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_less_mono1
% 0.30/0.55 thf(fact_2717_mult__less__mono2,axiom,
% 0.30/0.55 ! [I2: nat,J: nat,K: nat] :
% 0.30/0.55 ( ( ord_less_nat @ I2 @ J )
% 0.30/0.55 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.55 => ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_less_mono2
% 0.30/0.55 thf(fact_2718_nat__mult__eq__cancel1,axiom,
% 0.30/0.55 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.55 => ( ( ( times_times_nat @ K @ M )
% 0.30/0.55 = ( times_times_nat @ K @ N2 ) )
% 0.30/0.55 = ( M = N2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nat_mult_eq_cancel1
% 0.30/0.55 thf(fact_2719_nat__mult__less__cancel1,axiom,
% 0.30/0.55 ! [K: nat,M: nat,N2: nat] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.55 => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 0.30/0.55 = ( ord_less_nat @ M @ N2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nat_mult_less_cancel1
% 0.30/0.55 thf(fact_2720_diff__less,axiom,
% 0.30/0.55 ! [N2: nat,M: nat] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.55 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.55 => ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % diff_less
% 0.30/0.55 thf(fact_2721_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 0.30/0.55 ! [M: nat,N2: nat] :
% 0.30/0.55 ( ( ( divide_divide_nat @ M @ N2 )
% 0.30/0.55 = zero_zero_nat )
% 0.30/0.55 = ( ( ord_less_nat @ M @ N2 )
% 0.30/0.55 | ( N2 = zero_zero_nat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % Euclidean_Division.div_eq_0_iff
% 0.30/0.55 thf(fact_2722_One__nat__def,axiom,
% 0.30/0.55 ( one_one_nat
% 0.30/0.55 = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.55
% 0.30/0.55 % One_nat_def
% 0.30/0.55 thf(fact_2723_nat__power__less__imp__less,axiom,
% 0.30/0.55 ! [I2: nat,M: nat,N2: nat] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ I2 )
% 0.30/0.55 => ( ( ord_less_nat @ ( power_power_nat @ I2 @ M ) @ ( power_power_nat @ I2 @ N2 ) )
% 0.30/0.55 => ( ord_less_nat @ M @ N2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % nat_power_less_imp_less
% 0.30/0.55 thf(fact_2724_diff__add__0,axiom,
% 0.30/0.55 ! [N2: nat,M: nat] :
% 0.30/0.55 ( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) )
% 0.30/0.55 = zero_zero_nat ) ).
% 0.30/0.55
% 0.30/0.55 % diff_add_0
% 0.30/0.55 thf(fact_2725_bits__stable__imp__add__self,axiom,
% 0.30/0.55 ! [A: nat] :
% 0.30/0.55 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.55 = A )
% 0.30/0.55 => ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.55 = zero_zero_nat ) ) ).
% 0.30/0.55
% 0.30/0.55 % bits_stable_imp_add_self
% 0.30/0.55 thf(fact_2726_bits__stable__imp__add__self,axiom,
% 0.30/0.55 ! [A: int] :
% 0.30/0.55 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.55 = A )
% 0.30/0.55 => ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.30/0.55 = zero_zero_int ) ) ).
% 0.30/0.55
% 0.30/0.55 % bits_stable_imp_add_self
% 0.30/0.55 thf(fact_2727_bits__stable__imp__add__self,axiom,
% 0.30/0.55 ! [A: code_integer] :
% 0.30/0.55 ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.55 = A )
% 0.30/0.55 => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 0.30/0.55 = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.55
% 0.30/0.55 % bits_stable_imp_add_self
% 0.30/0.55 thf(fact_2728_mult__eq__self__implies__10,axiom,
% 0.30/0.55 ! [M: nat,N2: nat] :
% 0.30/0.55 ( ( M
% 0.30/0.55 = ( times_times_nat @ M @ N2 ) )
% 0.30/0.55 => ( ( N2 = one_one_nat )
% 0.30/0.55 | ( M = zero_zero_nat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_eq_self_implies_10
% 0.30/0.55 thf(fact_2729_vebt__insert_Osimps_I2_J,axiom,
% 0.30/0.55 ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X: nat] :
% 0.30/0.55 ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S3 ) @ X )
% 0.30/0.55 = ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S3 ) ) ).
% 0.30/0.55
% 0.30/0.55 % vebt_insert.simps(2)
% 0.30/0.55 thf(fact_2730_vebt__pred_Osimps_I3_J,axiom,
% 0.30/0.55 ! [B: $o,A: $o,Va: nat] :
% 0.30/0.55 ( ( B
% 0.30/0.55 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
% 0.30/0.55 = ( some_nat @ one_one_nat ) ) )
% 0.30/0.55 & ( ~ B
% 0.30/0.55 => ( ( A
% 0.30/0.55 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
% 0.30/0.55 = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.55 & ( ~ A
% 0.30/0.55 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
% 0.30/0.55 = none_nat ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % vebt_pred.simps(3)
% 0.30/0.55 thf(fact_2731_mod__double__modulus,axiom,
% 0.30/0.55 ! [M: code_integer,X: code_integer] :
% 0.30/0.55 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
% 0.30/0.55 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 0.30/0.55 => ( ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.55 = ( modulo364778990260209775nteger @ X @ M ) )
% 0.30/0.55 | ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.55 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X @ M ) @ M ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mod_double_modulus
% 0.30/0.55 thf(fact_2732_mod__double__modulus,axiom,
% 0.30/0.55 ! [M: nat,X: nat] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.30/0.55 => ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.55 = ( modulo_modulo_nat @ X @ M ) )
% 0.30/0.55 | ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.55 = ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M ) @ M ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mod_double_modulus
% 0.30/0.55 thf(fact_2733_mod__double__modulus,axiom,
% 0.30/0.55 ! [M: int,X: int] :
% 0.30/0.55 ( ( ord_less_int @ zero_zero_int @ M )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.30/0.55 => ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.55 = ( modulo_modulo_int @ X @ M ) )
% 0.30/0.55 | ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.55 = ( plus_plus_int @ ( modulo_modulo_int @ X @ M ) @ M ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mod_double_modulus
% 0.30/0.55 thf(fact_2734_divmod__digit__1_I2_J,axiom,
% 0.30/0.55 ! [A: code_integer,B: code_integer] :
% 0.30/0.55 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.55 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.30/0.55 => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 0.30/0.55 => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.30/0.55 = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divmod_digit_1(2)
% 0.30/0.55 thf(fact_2735_divmod__digit__1_I2_J,axiom,
% 0.30/0.55 ! [A: nat,B: nat] :
% 0.30/0.55 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.55 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.55 => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 0.30/0.55 => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.30/0.55 = ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divmod_digit_1(2)
% 0.30/0.55 thf(fact_2736_divmod__digit__1_I2_J,axiom,
% 0.30/0.55 ! [A: int,B: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.55 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.55 => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 0.30/0.55 => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.30/0.55 = ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divmod_digit_1(2)
% 0.30/0.55 thf(fact_2737_div__mult1__eq,axiom,
% 0.30/0.55 ! [A: nat,B: nat,C: nat] :
% 0.30/0.55 ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.30/0.55 = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % div_mult1_eq
% 0.30/0.55 thf(fact_2738_div__mult1__eq,axiom,
% 0.30/0.55 ! [A: int,B: int,C: int] :
% 0.30/0.55 ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C )
% 0.30/0.55 = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % div_mult1_eq
% 0.30/0.55 thf(fact_2739_div__mult1__eq,axiom,
% 0.30/0.55 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.55 ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.30/0.55 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % div_mult1_eq
% 0.30/0.55 thf(fact_2740_mult__div__mod__eq,axiom,
% 0.30/0.55 ! [B: nat,A: nat] :
% 0.30/0.55 ( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
% 0.30/0.55 = A ) ).
% 0.30/0.55
% 0.30/0.55 % mult_div_mod_eq
% 0.30/0.55 thf(fact_2741_mult__div__mod__eq,axiom,
% 0.30/0.55 ! [B: int,A: int] :
% 0.30/0.55 ( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
% 0.30/0.55 = A ) ).
% 0.30/0.55
% 0.30/0.55 % mult_div_mod_eq
% 0.30/0.55 thf(fact_2742_mult__div__mod__eq,axiom,
% 0.30/0.55 ! [B: code_integer,A: code_integer] :
% 0.30/0.55 ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.30/0.55 = A ) ).
% 0.30/0.55
% 0.30/0.55 % mult_div_mod_eq
% 0.30/0.55 thf(fact_2743_mod__mult__div__eq,axiom,
% 0.30/0.55 ! [A: nat,B: nat] :
% 0.30/0.55 ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
% 0.30/0.55 = A ) ).
% 0.30/0.55
% 0.30/0.55 % mod_mult_div_eq
% 0.30/0.55 thf(fact_2744_mod__mult__div__eq,axiom,
% 0.30/0.55 ! [A: int,B: int] :
% 0.30/0.55 ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
% 0.30/0.55 = A ) ).
% 0.30/0.55
% 0.30/0.55 % mod_mult_div_eq
% 0.30/0.55 thf(fact_2745_mod__mult__div__eq,axiom,
% 0.30/0.55 ! [A: code_integer,B: code_integer] :
% 0.30/0.55 ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
% 0.30/0.55 = A ) ).
% 0.30/0.55
% 0.30/0.55 % mod_mult_div_eq
% 0.30/0.55 thf(fact_2746_mod__div__mult__eq,axiom,
% 0.30/0.55 ! [A: nat,B: nat] :
% 0.30/0.55 ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
% 0.30/0.55 = A ) ).
% 0.30/0.55
% 0.30/0.55 % mod_div_mult_eq
% 0.30/0.55 thf(fact_2747_mod__div__mult__eq,axiom,
% 0.30/0.55 ! [A: int,B: int] :
% 0.30/0.55 ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
% 0.30/0.55 = A ) ).
% 0.30/0.55
% 0.30/0.55 % mod_div_mult_eq
% 0.30/0.55 thf(fact_2748_mod__div__mult__eq,axiom,
% 0.30/0.55 ! [A: code_integer,B: code_integer] :
% 0.30/0.55 ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
% 0.30/0.55 = A ) ).
% 0.30/0.55
% 0.30/0.55 % mod_div_mult_eq
% 0.30/0.55 thf(fact_2749_div__mult__mod__eq,axiom,
% 0.30/0.55 ! [A: nat,B: nat] :
% 0.30/0.55 ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
% 0.30/0.55 = A ) ).
% 0.30/0.55
% 0.30/0.55 % div_mult_mod_eq
% 0.30/0.55 thf(fact_2750_div__mult__mod__eq,axiom,
% 0.30/0.55 ! [A: int,B: int] :
% 0.30/0.55 ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
% 0.30/0.55 = A ) ).
% 0.30/0.55
% 0.30/0.55 % div_mult_mod_eq
% 0.30/0.55 thf(fact_2751_div__mult__mod__eq,axiom,
% 0.30/0.55 ! [A: code_integer,B: code_integer] :
% 0.30/0.55 ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.30/0.55 = A ) ).
% 0.30/0.55
% 0.30/0.55 % div_mult_mod_eq
% 0.30/0.55 thf(fact_2752_mod__div__decomp,axiom,
% 0.30/0.55 ! [A: nat,B: nat] :
% 0.30/0.55 ( A
% 0.30/0.55 = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mod_div_decomp
% 0.30/0.55 thf(fact_2753_mod__div__decomp,axiom,
% 0.30/0.55 ! [A: int,B: int] :
% 0.30/0.55 ( A
% 0.30/0.55 = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mod_div_decomp
% 0.30/0.55 thf(fact_2754_mod__div__decomp,axiom,
% 0.30/0.55 ! [A: code_integer,B: code_integer] :
% 0.30/0.55 ( A
% 0.30/0.55 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mod_div_decomp
% 0.30/0.55 thf(fact_2755_cancel__div__mod__rules_I1_J,axiom,
% 0.30/0.55 ! [A: nat,B: nat,C: nat] :
% 0.30/0.55 ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
% 0.30/0.55 = ( plus_plus_nat @ A @ C ) ) ).
% 0.30/0.55
% 0.30/0.55 % cancel_div_mod_rules(1)
% 0.30/0.55 thf(fact_2756_cancel__div__mod__rules_I1_J,axiom,
% 0.30/0.55 ! [A: int,B: int,C: int] :
% 0.30/0.55 ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
% 0.30/0.55 = ( plus_plus_int @ A @ C ) ) ).
% 0.30/0.55
% 0.30/0.55 % cancel_div_mod_rules(1)
% 0.30/0.55 thf(fact_2757_cancel__div__mod__rules_I1_J,axiom,
% 0.30/0.55 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.55 ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
% 0.30/0.55 = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 0.30/0.55
% 0.30/0.55 % cancel_div_mod_rules(1)
% 0.30/0.55 thf(fact_2758_cancel__div__mod__rules_I2_J,axiom,
% 0.30/0.55 ! [B: nat,A: nat,C: nat] :
% 0.30/0.55 ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
% 0.30/0.55 = ( plus_plus_nat @ A @ C ) ) ).
% 0.30/0.55
% 0.30/0.55 % cancel_div_mod_rules(2)
% 0.30/0.55 thf(fact_2759_cancel__div__mod__rules_I2_J,axiom,
% 0.30/0.55 ! [B: int,A: int,C: int] :
% 0.30/0.55 ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
% 0.30/0.55 = ( plus_plus_int @ A @ C ) ) ).
% 0.30/0.55
% 0.30/0.55 % cancel_div_mod_rules(2)
% 0.30/0.55 thf(fact_2760_cancel__div__mod__rules_I2_J,axiom,
% 0.30/0.55 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.55 ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
% 0.30/0.55 = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 0.30/0.55
% 0.30/0.55 % cancel_div_mod_rules(2)
% 0.30/0.55 thf(fact_2761_minus__mult__div__eq__mod,axiom,
% 0.30/0.55 ! [A: nat,B: nat] :
% 0.30/0.55 ( ( minus_minus_nat @ A @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
% 0.30/0.55 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.30/0.55
% 0.30/0.55 % minus_mult_div_eq_mod
% 0.30/0.55 thf(fact_2762_minus__mult__div__eq__mod,axiom,
% 0.30/0.55 ! [A: int,B: int] :
% 0.30/0.55 ( ( minus_minus_int @ A @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
% 0.30/0.55 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.30/0.55
% 0.30/0.55 % minus_mult_div_eq_mod
% 0.30/0.55 thf(fact_2763_minus__mult__div__eq__mod,axiom,
% 0.30/0.55 ! [A: code_integer,B: code_integer] :
% 0.30/0.55 ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
% 0.30/0.55 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.30/0.55
% 0.30/0.55 % minus_mult_div_eq_mod
% 0.30/0.55 thf(fact_2764_minus__mod__eq__mult__div,axiom,
% 0.30/0.55 ! [A: nat,B: nat] :
% 0.30/0.55 ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
% 0.30/0.55 = ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % minus_mod_eq_mult_div
% 0.30/0.55 thf(fact_2765_minus__mod__eq__mult__div,axiom,
% 0.30/0.55 ! [A: int,B: int] :
% 0.30/0.55 ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
% 0.30/0.55 = ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % minus_mod_eq_mult_div
% 0.30/0.55 thf(fact_2766_minus__mod__eq__mult__div,axiom,
% 0.30/0.55 ! [A: code_integer,B: code_integer] :
% 0.30/0.55 ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.30/0.55 = ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % minus_mod_eq_mult_div
% 0.30/0.55 thf(fact_2767_minus__mod__eq__div__mult,axiom,
% 0.30/0.55 ! [A: nat,B: nat] :
% 0.30/0.55 ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
% 0.30/0.55 = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ).
% 0.30/0.55
% 0.30/0.55 % minus_mod_eq_div_mult
% 0.30/0.55 thf(fact_2768_minus__mod__eq__div__mult,axiom,
% 0.30/0.55 ! [A: int,B: int] :
% 0.30/0.55 ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
% 0.30/0.55 = ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ).
% 0.30/0.55
% 0.30/0.55 % minus_mod_eq_div_mult
% 0.30/0.55 thf(fact_2769_minus__mod__eq__div__mult,axiom,
% 0.30/0.55 ! [A: code_integer,B: code_integer] :
% 0.30/0.55 ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.30/0.55 = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) ) ).
% 0.30/0.55
% 0.30/0.55 % minus_mod_eq_div_mult
% 0.30/0.55 thf(fact_2770_minus__div__mult__eq__mod,axiom,
% 0.30/0.55 ! [A: nat,B: nat] :
% 0.30/0.55 ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
% 0.30/0.55 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.30/0.55
% 0.30/0.55 % minus_div_mult_eq_mod
% 0.30/0.55 thf(fact_2771_minus__div__mult__eq__mod,axiom,
% 0.30/0.55 ! [A: int,B: int] :
% 0.30/0.55 ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
% 0.30/0.55 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.30/0.55
% 0.30/0.55 % minus_div_mult_eq_mod
% 0.30/0.55 thf(fact_2772_minus__div__mult__eq__mod,axiom,
% 0.30/0.55 ! [A: code_integer,B: code_integer] :
% 0.30/0.55 ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
% 0.30/0.55 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.30/0.55
% 0.30/0.55 % minus_div_mult_eq_mod
% 0.30/0.55 thf(fact_2773_mod__mult2__eq,axiom,
% 0.30/0.55 ! [M: nat,N2: nat,Q2: nat] :
% 0.30/0.55 ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N2 @ Q2 ) )
% 0.30/0.55 = ( plus_plus_nat @ ( times_times_nat @ N2 @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N2 ) @ Q2 ) ) @ ( modulo_modulo_nat @ M @ N2 ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mod_mult2_eq
% 0.30/0.55 thf(fact_2774_modulo__nat__def,axiom,
% 0.30/0.55 ( modulo_modulo_nat
% 0.30/0.55 = ( ^ [M6: nat,N: nat] : ( minus_minus_nat @ M6 @ ( times_times_nat @ ( divide_divide_nat @ M6 @ N ) @ N ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % modulo_nat_def
% 0.30/0.55 thf(fact_2775_VEBT__internal_OminNull_Ocases,axiom,
% 0.30/0.55 ! [X: vEBT_VEBT] :
% 0.30/0.55 ( ( X
% 0.30/0.55 != ( vEBT_Leaf @ $false @ $false ) )
% 0.30/0.55 => ( ! [Uv2: $o] :
% 0.30/0.55 ( X
% 0.30/0.55 != ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.30/0.55 => ( ! [Uu2: $o] :
% 0.30/0.55 ( X
% 0.30/0.55 != ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.30/0.55 => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.30/0.55 ( X
% 0.30/0.55 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 0.30/0.55 => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.55 ( X
% 0.30/0.55 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % VEBT_internal.minNull.cases
% 0.30/0.55 thf(fact_2776_div__geq,axiom,
% 0.30/0.55 ! [N2: nat,M: nat] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.55 => ( ~ ( ord_less_nat @ M @ N2 )
% 0.30/0.55 => ( ( divide_divide_nat @ M @ N2 )
% 0.30/0.55 = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % div_geq
% 0.30/0.55 thf(fact_2777_VEBT__internal_OminNull_Oelims_I3_J,axiom,
% 0.30/0.55 ! [X: vEBT_VEBT] :
% 0.30/0.55 ( ~ ( vEBT_VEBT_minNull @ X )
% 0.30/0.55 => ( ! [Uv2: $o] :
% 0.30/0.55 ( X
% 0.30/0.55 != ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.30/0.55 => ( ! [Uu2: $o] :
% 0.30/0.55 ( X
% 0.30/0.55 != ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.30/0.55 => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.55 ( X
% 0.30/0.55 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % VEBT_internal.minNull.elims(3)
% 0.30/0.55 thf(fact_2778_vebt__succ_Osimps_I2_J,axiom,
% 0.30/0.55 ! [Uv: $o,Uw: $o,N2: nat] :
% 0.30/0.55 ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) )
% 0.30/0.55 = none_nat ) ).
% 0.30/0.55
% 0.30/0.55 % vebt_succ.simps(2)
% 0.30/0.55 thf(fact_2779_VEBT__internal_OminNull_Oelims_I2_J,axiom,
% 0.30/0.55 ! [X: vEBT_VEBT] :
% 0.30/0.55 ( ( vEBT_VEBT_minNull @ X )
% 0.30/0.55 => ( ( X
% 0.30/0.55 != ( vEBT_Leaf @ $false @ $false ) )
% 0.30/0.55 => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.30/0.55 ( X
% 0.30/0.55 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % VEBT_internal.minNull.elims(2)
% 0.30/0.55 thf(fact_2780_mult__le__cancel__left,axiom,
% 0.30/0.55 ! [C: real,A: real,B: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.30/0.55 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55 => ( ord_less_eq_real @ A @ B ) )
% 0.30/0.55 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.55 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_le_cancel_left
% 0.30/0.55 thf(fact_2781_mult__le__cancel__left,axiom,
% 0.30/0.55 ! [C: rat,A: rat,B: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.30/0.55 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55 => ( ord_less_eq_rat @ A @ B ) )
% 0.30/0.55 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.55 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_le_cancel_left
% 0.30/0.55 thf(fact_2782_mult__le__cancel__left,axiom,
% 0.30/0.55 ! [C: int,A: int,B: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.30/0.55 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.55 => ( ord_less_eq_int @ A @ B ) )
% 0.30/0.55 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.55 => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_le_cancel_left
% 0.30/0.55 thf(fact_2783_mult__le__cancel__right,axiom,
% 0.30/0.55 ! [A: real,C: real,B: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.30/0.55 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55 => ( ord_less_eq_real @ A @ B ) )
% 0.30/0.55 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.55 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_le_cancel_right
% 0.30/0.55 thf(fact_2784_mult__le__cancel__right,axiom,
% 0.30/0.55 ! [A: rat,C: rat,B: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.30/0.55 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55 => ( ord_less_eq_rat @ A @ B ) )
% 0.30/0.55 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.55 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_le_cancel_right
% 0.30/0.55 thf(fact_2785_mult__le__cancel__right,axiom,
% 0.30/0.55 ! [A: int,C: int,B: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.30/0.55 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.55 => ( ord_less_eq_int @ A @ B ) )
% 0.30/0.55 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.55 => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_le_cancel_right
% 0.30/0.55 thf(fact_2786_mult__left__less__imp__less,axiom,
% 0.30/0.55 ! [C: real,A: real,B: real] :
% 0.30/0.55 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.30/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.55 => ( ord_less_real @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_left_less_imp_less
% 0.30/0.55 thf(fact_2787_mult__left__less__imp__less,axiom,
% 0.30/0.55 ! [C: rat,A: rat,B: rat] :
% 0.30/0.55 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.30/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.55 => ( ord_less_rat @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_left_less_imp_less
% 0.30/0.55 thf(fact_2788_mult__left__less__imp__less,axiom,
% 0.30/0.55 ! [C: nat,A: nat,B: nat] :
% 0.30/0.55 ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.30/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.55 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_left_less_imp_less
% 0.30/0.55 thf(fact_2789_mult__left__less__imp__less,axiom,
% 0.30/0.55 ! [C: int,A: int,B: int] :
% 0.30/0.55 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.55 => ( ord_less_int @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_left_less_imp_less
% 0.30/0.55 thf(fact_2790_mult__strict__mono,axiom,
% 0.30/0.55 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.55 ( ( ord_less_real @ A @ B )
% 0.30/0.55 => ( ( ord_less_real @ C @ D )
% 0.30/0.55 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.55 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_strict_mono
% 0.30/0.55 thf(fact_2791_mult__strict__mono,axiom,
% 0.30/0.55 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.55 ( ( ord_less_rat @ A @ B )
% 0.30/0.55 => ( ( ord_less_rat @ C @ D )
% 0.30/0.55 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.30/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.55 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_strict_mono
% 0.30/0.55 thf(fact_2792_mult__strict__mono,axiom,
% 0.30/0.55 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/0.55 ( ( ord_less_nat @ A @ B )
% 0.30/0.55 => ( ( ord_less_nat @ C @ D )
% 0.30/0.55 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.55 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_strict_mono
% 0.30/0.55 thf(fact_2793_mult__strict__mono,axiom,
% 0.30/0.55 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.55 ( ( ord_less_int @ A @ B )
% 0.30/0.55 => ( ( ord_less_int @ C @ D )
% 0.30/0.55 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.55 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_strict_mono
% 0.30/0.55 thf(fact_2794_mult__less__cancel__left,axiom,
% 0.30/0.55 ! [C: real,A: real,B: real] :
% 0.30/0.55 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.30/0.55 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.55 => ( ord_less_real @ A @ B ) )
% 0.30/0.55 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.55 => ( ord_less_real @ B @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_less_cancel_left
% 0.30/0.55 thf(fact_2795_mult__less__cancel__left,axiom,
% 0.30/0.55 ! [C: rat,A: rat,B: rat] :
% 0.30/0.55 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.30/0.55 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.55 => ( ord_less_rat @ A @ B ) )
% 0.30/0.55 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.30/0.55 => ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_less_cancel_left
% 0.30/0.55 thf(fact_2796_mult__less__cancel__left,axiom,
% 0.30/0.55 ! [C: int,A: int,B: int] :
% 0.30/0.55 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.30/0.55 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.55 => ( ord_less_int @ A @ B ) )
% 0.30/0.55 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.30/0.55 => ( ord_less_int @ B @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_less_cancel_left
% 0.30/0.55 thf(fact_2797_mult__right__less__imp__less,axiom,
% 0.30/0.55 ! [A: real,C: real,B: real] :
% 0.30/0.55 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.30/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.55 => ( ord_less_real @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_right_less_imp_less
% 0.30/0.55 thf(fact_2798_mult__right__less__imp__less,axiom,
% 0.30/0.55 ! [A: rat,C: rat,B: rat] :
% 0.30/0.55 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.30/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.55 => ( ord_less_rat @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_right_less_imp_less
% 0.30/0.55 thf(fact_2799_mult__right__less__imp__less,axiom,
% 0.30/0.55 ! [A: nat,C: nat,B: nat] :
% 0.30/0.55 ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.30/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.55 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_right_less_imp_less
% 0.30/0.55 thf(fact_2800_mult__right__less__imp__less,axiom,
% 0.30/0.55 ! [A: int,C: int,B: int] :
% 0.30/0.55 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.55 => ( ord_less_int @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_right_less_imp_less
% 0.30/0.55 thf(fact_2801_mult__strict__mono_H,axiom,
% 0.30/0.55 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.55 ( ( ord_less_real @ A @ B )
% 0.30/0.55 => ( ( ord_less_real @ C @ D )
% 0.30/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.55 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_strict_mono'
% 0.30/0.55 thf(fact_2802_mult__strict__mono_H,axiom,
% 0.30/0.55 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.55 ( ( ord_less_rat @ A @ B )
% 0.30/0.55 => ( ( ord_less_rat @ C @ D )
% 0.30/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.55 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_strict_mono'
% 0.30/0.55 thf(fact_2803_mult__strict__mono_H,axiom,
% 0.30/0.55 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/0.55 ( ( ord_less_nat @ A @ B )
% 0.30/0.55 => ( ( ord_less_nat @ C @ D )
% 0.30/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.55 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_strict_mono'
% 0.30/0.55 thf(fact_2804_mult__strict__mono_H,axiom,
% 0.30/0.55 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.55 ( ( ord_less_int @ A @ B )
% 0.30/0.55 => ( ( ord_less_int @ C @ D )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.55 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_strict_mono'
% 0.30/0.55 thf(fact_2805_mult__less__cancel__right,axiom,
% 0.30/0.55 ! [A: real,C: real,B: real] :
% 0.30/0.55 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.30/0.55 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.55 => ( ord_less_real @ A @ B ) )
% 0.30/0.55 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.30/0.55 => ( ord_less_real @ B @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_less_cancel_right
% 0.30/0.55 thf(fact_2806_mult__less__cancel__right,axiom,
% 0.30/0.55 ! [A: rat,C: rat,B: rat] :
% 0.30/0.55 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.30/0.55 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.55 => ( ord_less_rat @ A @ B ) )
% 0.30/0.55 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.30/0.55 => ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_less_cancel_right
% 0.30/0.55 thf(fact_2807_mult__less__cancel__right,axiom,
% 0.30/0.55 ! [A: int,C: int,B: int] :
% 0.30/0.55 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.30/0.55 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.55 => ( ord_less_int @ A @ B ) )
% 0.30/0.55 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.30/0.55 => ( ord_less_int @ B @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_less_cancel_right
% 0.30/0.55 thf(fact_2808_mult__le__cancel__left__neg,axiom,
% 0.30/0.55 ! [C: real,A: real,B: real] :
% 0.30/0.55 ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.55 => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.30/0.55 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_le_cancel_left_neg
% 0.30/0.55 thf(fact_2809_mult__le__cancel__left__neg,axiom,
% 0.30/0.55 ! [C: rat,A: rat,B: rat] :
% 0.30/0.55 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.55 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.30/0.55 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_le_cancel_left_neg
% 0.30/0.55 thf(fact_2810_mult__le__cancel__left__neg,axiom,
% 0.30/0.55 ! [C: int,A: int,B: int] :
% 0.30/0.55 ( ( ord_less_int @ C @ zero_zero_int )
% 0.30/0.55 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.30/0.55 = ( ord_less_eq_int @ B @ A ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_le_cancel_left_neg
% 0.30/0.55 thf(fact_2811_mult__le__cancel__left__pos,axiom,
% 0.30/0.55 ! [C: real,A: real,B: real] :
% 0.30/0.55 ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55 => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.30/0.55 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_le_cancel_left_pos
% 0.30/0.55 thf(fact_2812_mult__le__cancel__left__pos,axiom,
% 0.30/0.55 ! [C: rat,A: rat,B: rat] :
% 0.30/0.55 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.30/0.55 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_le_cancel_left_pos
% 0.30/0.55 thf(fact_2813_mult__le__cancel__left__pos,axiom,
% 0.30/0.55 ! [C: int,A: int,B: int] :
% 0.30/0.55 ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.55 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.30/0.55 = ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_le_cancel_left_pos
% 0.30/0.55 thf(fact_2814_mult__left__le__imp__le,axiom,
% 0.30/0.55 ! [C: real,A: real,B: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.30/0.55 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_left_le_imp_le
% 0.30/0.55 thf(fact_2815_mult__left__le__imp__le,axiom,
% 0.30/0.55 ! [C: rat,A: rat,B: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.30/0.55 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55 => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_left_le_imp_le
% 0.30/0.55 thf(fact_2816_mult__left__le__imp__le,axiom,
% 0.30/0.55 ! [C: nat,A: nat,B: nat] :
% 0.30/0.55 ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.30/0.55 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.30/0.55 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_left_le_imp_le
% 0.30/0.55 thf(fact_2817_mult__left__le__imp__le,axiom,
% 0.30/0.55 ! [C: int,A: int,B: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.30/0.55 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.55 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_left_le_imp_le
% 0.30/0.55 thf(fact_2818_mult__right__le__imp__le,axiom,
% 0.30/0.55 ! [A: real,C: real,B: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.30/0.55 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_right_le_imp_le
% 0.30/0.55 thf(fact_2819_mult__right__le__imp__le,axiom,
% 0.30/0.55 ! [A: rat,C: rat,B: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.30/0.55 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55 => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_right_le_imp_le
% 0.30/0.55 thf(fact_2820_mult__right__le__imp__le,axiom,
% 0.30/0.55 ! [A: nat,C: nat,B: nat] :
% 0.30/0.55 ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.30/0.55 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.30/0.55 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_right_le_imp_le
% 0.30/0.55 thf(fact_2821_mult__right__le__imp__le,axiom,
% 0.30/0.55 ! [A: int,C: int,B: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.30/0.55 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.55 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_right_le_imp_le
% 0.30/0.55 thf(fact_2822_mult__le__less__imp__less,axiom,
% 0.30/0.55 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ A @ B )
% 0.30/0.55 => ( ( ord_less_real @ C @ D )
% 0.30/0.55 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.55 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_le_less_imp_less
% 0.30/0.55 thf(fact_2823_mult__le__less__imp__less,axiom,
% 0.30/0.55 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.55 => ( ( ord_less_rat @ C @ D )
% 0.30/0.55 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.30/0.55 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_le_less_imp_less
% 0.30/0.55 thf(fact_2824_mult__le__less__imp__less,axiom,
% 0.30/0.55 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/0.55 ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.55 => ( ( ord_less_nat @ C @ D )
% 0.30/0.55 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.55 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_le_less_imp_less
% 0.30/0.55 thf(fact_2825_mult__le__less__imp__less,axiom,
% 0.30/0.55 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ A @ B )
% 0.30/0.55 => ( ( ord_less_int @ C @ D )
% 0.30/0.55 => ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.55 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_le_less_imp_less
% 0.30/0.55 thf(fact_2826_mult__less__le__imp__less,axiom,
% 0.30/0.55 ! [A: real,B: real,C: real,D: real] :
% 0.30/0.55 ( ( ord_less_real @ A @ B )
% 0.30/0.55 => ( ( ord_less_eq_real @ C @ D )
% 0.30/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.55 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_less_le_imp_less
% 0.30/0.55 thf(fact_2827_mult__less__le__imp__less,axiom,
% 0.30/0.55 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.55 ( ( ord_less_rat @ A @ B )
% 0.30/0.55 => ( ( ord_less_eq_rat @ C @ D )
% 0.30/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.55 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_less_le_imp_less
% 0.30/0.55 thf(fact_2828_mult__less__le__imp__less,axiom,
% 0.30/0.55 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/0.55 ( ( ord_less_nat @ A @ B )
% 0.30/0.55 => ( ( ord_less_eq_nat @ C @ D )
% 0.30/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.55 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.30/0.55 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_less_le_imp_less
% 0.30/0.55 thf(fact_2829_mult__less__le__imp__less,axiom,
% 0.30/0.55 ! [A: int,B: int,C: int,D: int] :
% 0.30/0.55 ( ( ord_less_int @ A @ B )
% 0.30/0.55 => ( ( ord_less_eq_int @ C @ D )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.55 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.30/0.55 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_less_le_imp_less
% 0.30/0.55 thf(fact_2830_field__le__epsilon,axiom,
% 0.30/0.55 ! [X: real,Y: real] :
% 0.30/0.55 ( ! [E2: real] :
% 0.30/0.55 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.30/0.55 => ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E2 ) ) )
% 0.30/0.55 => ( ord_less_eq_real @ X @ Y ) ) ).
% 0.30/0.55
% 0.30/0.55 % field_le_epsilon
% 0.30/0.55 thf(fact_2831_field__le__epsilon,axiom,
% 0.30/0.55 ! [X: rat,Y: rat] :
% 0.30/0.55 ( ! [E2: rat] :
% 0.30/0.55 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 0.30/0.55 => ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y @ E2 ) ) )
% 0.30/0.55 => ( ord_less_eq_rat @ X @ Y ) ) ).
% 0.30/0.55
% 0.30/0.55 % field_le_epsilon
% 0.30/0.55 thf(fact_2832_add__neg__nonpos,axiom,
% 0.30/0.55 ! [A: real,B: real] :
% 0.30/0.55 ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.55 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.30/0.55 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_neg_nonpos
% 0.30/0.55 thf(fact_2833_add__neg__nonpos,axiom,
% 0.30/0.55 ! [A: rat,B: rat] :
% 0.30/0.55 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.55 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.30/0.55 => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_neg_nonpos
% 0.30/0.55 thf(fact_2834_add__neg__nonpos,axiom,
% 0.30/0.55 ! [A: nat,B: nat] :
% 0.30/0.55 ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.30/0.55 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.30/0.55 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_neg_nonpos
% 0.30/0.55 thf(fact_2835_add__neg__nonpos,axiom,
% 0.30/0.55 ! [A: int,B: int] :
% 0.30/0.55 ( ( ord_less_int @ A @ zero_zero_int )
% 0.30/0.55 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.30/0.55 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_neg_nonpos
% 0.30/0.55 thf(fact_2836_add__nonneg__pos,axiom,
% 0.30/0.55 ! [A: real,B: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.55 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/0.55 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_nonneg_pos
% 0.30/0.55 thf(fact_2837_add__nonneg__pos,axiom,
% 0.30/0.55 ! [A: rat,B: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.55 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.30/0.55 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_nonneg_pos
% 0.30/0.55 thf(fact_2838_add__nonneg__pos,axiom,
% 0.30/0.55 ! [A: nat,B: nat] :
% 0.30/0.55 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.55 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.55 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_nonneg_pos
% 0.30/0.55 thf(fact_2839_add__nonneg__pos,axiom,
% 0.30/0.55 ! [A: int,B: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.55 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.55 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_nonneg_pos
% 0.30/0.55 thf(fact_2840_add__nonpos__neg,axiom,
% 0.30/0.55 ! [A: real,B: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.55 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.30/0.55 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_nonpos_neg
% 0.30/0.55 thf(fact_2841_add__nonpos__neg,axiom,
% 0.30/0.55 ! [A: rat,B: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.55 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.30/0.55 => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_nonpos_neg
% 0.30/0.55 thf(fact_2842_add__nonpos__neg,axiom,
% 0.30/0.55 ! [A: nat,B: nat] :
% 0.30/0.55 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.30/0.55 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.30/0.55 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_nonpos_neg
% 0.30/0.55 thf(fact_2843_add__nonpos__neg,axiom,
% 0.30/0.55 ! [A: int,B: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.55 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.30/0.55 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_nonpos_neg
% 0.30/0.55 thf(fact_2844_add__pos__nonneg,axiom,
% 0.30/0.55 ! [A: real,B: real] :
% 0.30/0.55 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.55 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_pos_nonneg
% 0.30/0.55 thf(fact_2845_add__pos__nonneg,axiom,
% 0.30/0.55 ! [A: rat,B: rat] :
% 0.30/0.55 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.30/0.55 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_pos_nonneg
% 0.30/0.55 thf(fact_2846_add__pos__nonneg,axiom,
% 0.30/0.55 ! [A: nat,B: nat] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.30/0.55 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_pos_nonneg
% 0.30/0.55 thf(fact_2847_add__pos__nonneg,axiom,
% 0.30/0.55 ! [A: int,B: int] :
% 0.30/0.55 ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.30/0.55 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_pos_nonneg
% 0.30/0.55 thf(fact_2848_add__strict__increasing,axiom,
% 0.30/0.55 ! [A: real,B: real,C: real] :
% 0.30/0.55 ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.55 => ( ( ord_less_eq_real @ B @ C )
% 0.30/0.55 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_strict_increasing
% 0.30/0.55 thf(fact_2849_add__strict__increasing,axiom,
% 0.30/0.55 ! [A: rat,B: rat,C: rat] :
% 0.30/0.55 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.55 => ( ( ord_less_eq_rat @ B @ C )
% 0.30/0.55 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_strict_increasing
% 0.30/0.55 thf(fact_2850_add__strict__increasing,axiom,
% 0.30/0.55 ! [A: nat,B: nat,C: nat] :
% 0.30/0.55 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.30/0.55 => ( ( ord_less_eq_nat @ B @ C )
% 0.30/0.55 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_strict_increasing
% 0.30/0.55 thf(fact_2851_add__strict__increasing,axiom,
% 0.30/0.55 ! [A: int,B: int,C: int] :
% 0.30/0.55 ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.55 => ( ( ord_less_eq_int @ B @ C )
% 0.30/0.55 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_strict_increasing
% 0.30/0.55 thf(fact_2852_add__strict__increasing2,axiom,
% 0.30/0.55 ! [A: real,B: real,C: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.55 => ( ( ord_less_real @ B @ C )
% 0.30/0.55 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_strict_increasing2
% 0.30/0.55 thf(fact_2853_add__strict__increasing2,axiom,
% 0.30/0.55 ! [A: rat,B: rat,C: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.55 => ( ( ord_less_rat @ B @ C )
% 0.30/0.55 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_strict_increasing2
% 0.30/0.55 thf(fact_2854_add__strict__increasing2,axiom,
% 0.30/0.55 ! [A: nat,B: nat,C: nat] :
% 0.30/0.55 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.55 => ( ( ord_less_nat @ B @ C )
% 0.30/0.55 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_strict_increasing2
% 0.30/0.55 thf(fact_2855_add__strict__increasing2,axiom,
% 0.30/0.55 ! [A: int,B: int,C: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.55 => ( ( ord_less_int @ B @ C )
% 0.30/0.55 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % add_strict_increasing2
% 0.30/0.55 thf(fact_2856_frac__le,axiom,
% 0.30/0.55 ! [Y: real,X: real,W: real,Z: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.30/0.55 => ( ( ord_less_eq_real @ X @ Y )
% 0.30/0.55 => ( ( ord_less_real @ zero_zero_real @ W )
% 0.30/0.55 => ( ( ord_less_eq_real @ W @ Z )
% 0.30/0.55 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % frac_le
% 0.30/0.55 thf(fact_2857_frac__le,axiom,
% 0.30/0.55 ! [Y: rat,X: rat,W: rat,Z: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.30/0.55 => ( ( ord_less_eq_rat @ X @ Y )
% 0.30/0.55 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 0.30/0.55 => ( ( ord_less_eq_rat @ W @ Z )
% 0.30/0.55 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % frac_le
% 0.30/0.55 thf(fact_2858_frac__less,axiom,
% 0.30/0.55 ! [X: real,Y: real,W: real,Z: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.55 => ( ( ord_less_real @ X @ Y )
% 0.30/0.55 => ( ( ord_less_real @ zero_zero_real @ W )
% 0.30/0.55 => ( ( ord_less_eq_real @ W @ Z )
% 0.30/0.55 => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % frac_less
% 0.30/0.55 thf(fact_2859_frac__less,axiom,
% 0.30/0.55 ! [X: rat,Y: rat,W: rat,Z: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.30/0.55 => ( ( ord_less_rat @ X @ Y )
% 0.30/0.55 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 0.30/0.55 => ( ( ord_less_eq_rat @ W @ Z )
% 0.30/0.55 => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % frac_less
% 0.30/0.55 thf(fact_2860_frac__less2,axiom,
% 0.30/0.55 ! [X: real,Y: real,W: real,Z: real] :
% 0.30/0.55 ( ( ord_less_real @ zero_zero_real @ X )
% 0.30/0.55 => ( ( ord_less_eq_real @ X @ Y )
% 0.30/0.55 => ( ( ord_less_real @ zero_zero_real @ W )
% 0.30/0.55 => ( ( ord_less_real @ W @ Z )
% 0.30/0.55 => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % frac_less2
% 0.30/0.55 thf(fact_2861_frac__less2,axiom,
% 0.30/0.55 ! [X: rat,Y: rat,W: rat,Z: rat] :
% 0.30/0.55 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.30/0.55 => ( ( ord_less_eq_rat @ X @ Y )
% 0.30/0.55 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 0.30/0.55 => ( ( ord_less_rat @ W @ Z )
% 0.30/0.55 => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % frac_less2
% 0.30/0.55 thf(fact_2862_divide__le__cancel,axiom,
% 0.30/0.55 ! [A: real,C: real,B: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 0.30/0.55 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.55 => ( ord_less_eq_real @ A @ B ) )
% 0.30/0.55 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.55 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_le_cancel
% 0.30/0.55 thf(fact_2863_divide__le__cancel,axiom,
% 0.30/0.55 ! [A: rat,C: rat,B: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 0.30/0.55 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.55 => ( ord_less_eq_rat @ A @ B ) )
% 0.30/0.55 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.55 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_le_cancel
% 0.30/0.55 thf(fact_2864_divide__nonneg__neg,axiom,
% 0.30/0.55 ! [X: real,Y: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.55 => ( ( ord_less_real @ Y @ zero_zero_real )
% 0.30/0.55 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_nonneg_neg
% 0.30/0.55 thf(fact_2865_divide__nonneg__neg,axiom,
% 0.30/0.55 ! [X: rat,Y: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.30/0.55 => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 0.30/0.55 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_nonneg_neg
% 0.30/0.55 thf(fact_2866_divide__nonneg__pos,axiom,
% 0.30/0.55 ! [X: real,Y: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.55 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.30/0.55 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_nonneg_pos
% 0.30/0.55 thf(fact_2867_divide__nonneg__pos,axiom,
% 0.30/0.55 ! [X: rat,Y: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.30/0.55 => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.30/0.55 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_nonneg_pos
% 0.30/0.55 thf(fact_2868_divide__nonpos__neg,axiom,
% 0.30/0.55 ! [X: real,Y: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.30/0.55 => ( ( ord_less_real @ Y @ zero_zero_real )
% 0.30/0.55 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_nonpos_neg
% 0.30/0.55 thf(fact_2869_divide__nonpos__neg,axiom,
% 0.30/0.55 ! [X: rat,Y: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.30/0.55 => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 0.30/0.55 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_nonpos_neg
% 0.30/0.55 thf(fact_2870_divide__nonpos__pos,axiom,
% 0.30/0.55 ! [X: real,Y: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.30/0.55 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.30/0.55 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_nonpos_pos
% 0.30/0.55 thf(fact_2871_divide__nonpos__pos,axiom,
% 0.30/0.55 ! [X: rat,Y: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.30/0.55 => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.30/0.55 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % divide_nonpos_pos
% 0.30/0.55 thf(fact_2872_sum__squares__le__zero__iff,axiom,
% 0.30/0.55 ! [X: real,Y: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
% 0.30/0.55 = ( ( X = zero_zero_real )
% 0.30/0.55 & ( Y = zero_zero_real ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % sum_squares_le_zero_iff
% 0.30/0.55 thf(fact_2873_sum__squares__le__zero__iff,axiom,
% 0.30/0.55 ! [X: rat,Y: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
% 0.30/0.55 = ( ( X = zero_zero_rat )
% 0.30/0.55 & ( Y = zero_zero_rat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % sum_squares_le_zero_iff
% 0.30/0.55 thf(fact_2874_sum__squares__le__zero__iff,axiom,
% 0.30/0.55 ! [X: int,Y: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
% 0.30/0.55 = ( ( X = zero_zero_int )
% 0.30/0.55 & ( Y = zero_zero_int ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % sum_squares_le_zero_iff
% 0.30/0.55 thf(fact_2875_sum__squares__ge__zero,axiom,
% 0.30/0.55 ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % sum_squares_ge_zero
% 0.30/0.55 thf(fact_2876_sum__squares__ge__zero,axiom,
% 0.30/0.55 ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % sum_squares_ge_zero
% 0.30/0.55 thf(fact_2877_sum__squares__ge__zero,axiom,
% 0.30/0.55 ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % sum_squares_ge_zero
% 0.30/0.55 thf(fact_2878_mult__left__le,axiom,
% 0.30/0.55 ! [C: real,A: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ C @ one_one_real )
% 0.30/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.55 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_left_le
% 0.30/0.55 thf(fact_2879_mult__left__le,axiom,
% 0.30/0.55 ! [C: rat,A: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ C @ one_one_rat )
% 0.30/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.55 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_left_le
% 0.30/0.55 thf(fact_2880_mult__left__le,axiom,
% 0.30/0.55 ! [C: nat,A: nat] :
% 0.30/0.55 ( ( ord_less_eq_nat @ C @ one_one_nat )
% 0.30/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.55 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_left_le
% 0.30/0.55 thf(fact_2881_mult__left__le,axiom,
% 0.30/0.55 ! [C: int,A: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ C @ one_one_int )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.55 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_left_le
% 0.30/0.55 thf(fact_2882_mult__le__one,axiom,
% 0.30/0.55 ! [A: real,B: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ A @ one_one_real )
% 0.30/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.55 => ( ( ord_less_eq_real @ B @ one_one_real )
% 0.30/0.55 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_le_one
% 0.30/0.55 thf(fact_2883_mult__le__one,axiom,
% 0.30/0.55 ! [A: rat,B: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.30/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.30/0.55 => ( ( ord_less_eq_rat @ B @ one_one_rat )
% 0.30/0.55 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_le_one
% 0.30/0.55 thf(fact_2884_mult__le__one,axiom,
% 0.30/0.55 ! [A: nat,B: nat] :
% 0.30/0.55 ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.30/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.30/0.55 => ( ( ord_less_eq_nat @ B @ one_one_nat )
% 0.30/0.55 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_le_one
% 0.30/0.55 thf(fact_2885_mult__le__one,axiom,
% 0.30/0.55 ! [A: int,B: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ A @ one_one_int )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.30/0.55 => ( ( ord_less_eq_int @ B @ one_one_int )
% 0.30/0.55 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_le_one
% 0.30/0.55 thf(fact_2886_mult__right__le__one__le,axiom,
% 0.30/0.55 ! [X: real,Y: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.30/0.55 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.30/0.55 => ( ord_less_eq_real @ ( times_times_real @ X @ Y ) @ X ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_right_le_one_le
% 0.30/0.55 thf(fact_2887_mult__right__le__one__le,axiom,
% 0.30/0.55 ! [X: rat,Y: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.30/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.30/0.55 => ( ( ord_less_eq_rat @ Y @ one_one_rat )
% 0.30/0.55 => ( ord_less_eq_rat @ ( times_times_rat @ X @ Y ) @ X ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_right_le_one_le
% 0.30/0.55 thf(fact_2888_mult__right__le__one__le,axiom,
% 0.30/0.55 ! [X: int,Y: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.30/0.55 => ( ( ord_less_eq_int @ Y @ one_one_int )
% 0.30/0.55 => ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_right_le_one_le
% 0.30/0.55 thf(fact_2889_mult__left__le__one__le,axiom,
% 0.30/0.55 ! [X: real,Y: real] :
% 0.30/0.55 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.30/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.30/0.55 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.30/0.55 => ( ord_less_eq_real @ ( times_times_real @ Y @ X ) @ X ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_left_le_one_le
% 0.30/0.55 thf(fact_2890_mult__left__le__one__le,axiom,
% 0.30/0.55 ! [X: rat,Y: rat] :
% 0.30/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.30/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.30/0.55 => ( ( ord_less_eq_rat @ Y @ one_one_rat )
% 0.30/0.55 => ( ord_less_eq_rat @ ( times_times_rat @ Y @ X ) @ X ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_left_le_one_le
% 0.30/0.55 thf(fact_2891_mult__left__le__one__le,axiom,
% 0.30/0.55 ! [X: int,Y: int] :
% 0.30/0.55 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.30/0.55 => ( ( ord_less_eq_int @ Y @ one_one_int )
% 0.30/0.55 => ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % mult_left_le_one_le
% 0.30/0.55 thf(fact_2892_power__less__imp__less__base,axiom,
% 0.30/0.55 ! [A: real,N2: nat,B: real] :
% 0.30/0.55 ( ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) )
% 0.30/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.55 => ( ord_less_real @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power_less_imp_less_base
% 0.30/0.55 thf(fact_2893_power__less__imp__less__base,axiom,
% 0.30/0.55 ! [A: rat,N2: nat,B: rat] :
% 0.30/0.55 ( ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) )
% 0.30/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.30/0.55 => ( ord_less_rat @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power_less_imp_less_base
% 0.30/0.55 thf(fact_2894_power__less__imp__less__base,axiom,
% 0.30/0.55 ! [A: nat,N2: nat,B: nat] :
% 0.30/0.55 ( ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
% 0.30/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.30/0.55 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power_less_imp_less_base
% 0.30/0.55 thf(fact_2895_power__less__imp__less__base,axiom,
% 0.30/0.55 ! [A: int,N2: nat,B: int] :
% 0.30/0.55 ( ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
% 0.30/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.30/0.55 => ( ord_less_int @ A @ B ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % power_less_imp_less_base
% 0.30/0.55 thf(fact_2896_sum__squares__gt__zero__iff,axiom,
% 0.30/0.55 ! [X: real,Y: real] :
% 0.30/0.55 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) )
% 0.30/0.55 = ( ( X != zero_zero_real )
% 0.30/0.55 | ( Y != zero_zero_real ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % sum_squares_gt_zero_iff
% 0.30/0.55 thf(fact_2897_sum__squares__gt__zero__iff,axiom,
% 0.30/0.55 ! [X: rat,Y: rat] :
% 0.30/0.55 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) )
% 0.30/0.55 = ( ( X != zero_zero_rat )
% 0.30/0.55 | ( Y != zero_zero_rat ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % sum_squares_gt_zero_iff
% 0.30/0.55 thf(fact_2898_sum__squares__gt__zero__iff,axiom,
% 0.30/0.55 ! [X: int,Y: int] :
% 0.30/0.55 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
% 0.30/0.55 = ( ( X != zero_zero_int )
% 0.30/0.55 | ( Y != zero_zero_int ) ) ) ).
% 0.30/0.55
% 0.30/0.55 % sum_squares_gt_zero_iff
% 0.30/0.55 thf(fact_2899_not__sum__squares__lt__zero,axiom,
% 0.30/0.55 ! [X: real,Y: real] :
% 0.30/0.55 ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).
% 0.30/0.55
% 0.30/0.55 % not_sum_squares_lt_zero
% 0.30/0.55 thf(fact_2900_not__sum__squares__lt__zero,axiom,
% 0.30/0.55 ! [X: rat,Y: rat] :
% 0.30/0.55 ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).
% 0.30/0.55
% 0.30/0.55 % not_sum_squares_lt_zero
% 0.30/0.55 thf(fact_2901_not__sum__squares__lt__zero,axiom,
% 0.30/0.55 ! [X: int,Y: int] :
% 0.30/0.55 ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% 0.30/0.55
% 0.30/0.55 % not_sum_squares_lt_zero
% 0.30/0.55 thf(fact_2902_zero__less__two,axiom,
% 0.30/0.55 ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% 0.30/0.55
% 0.30/0.55 % zero_less_two
% 0.30/0.55 thf(fact_2903_zero__less__two,axiom,
% 0.30/0.55 ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% 0.30/0.55
% 0.30/0.55 % zero_less_two
% 0.30/0.55 thf(fact_2904_zero__less__two,axiom,
% 0.30/0.55 ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% 0.30/0.55
% 0.30/0.55 % zero_less_two
% 0.30/0.55 thf(fact_2905_zero__less__two,axiom,
% 0.30/0.55 ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% 0.30/0.55
% 0.30/0.55 % zero_less_two
% 0.30/0.55 thf(fact_2906_divide__strict__left__mono__neg,axiom,
% 0.30/0.55 ! [A: real,B: real,C: real] :
% 0.30/0.55 ( ( ord_less_real @ A @ B )
% 0.30/0.55 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.55 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.32/0.55 => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_strict_left_mono_neg
% 0.32/0.55 thf(fact_2907_divide__strict__left__mono__neg,axiom,
% 0.32/0.55 ! [A: rat,B: rat,C: rat] :
% 0.32/0.55 ( ( ord_less_rat @ A @ B )
% 0.32/0.55 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.55 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.32/0.55 => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_strict_left_mono_neg
% 0.32/0.55 thf(fact_2908_divide__strict__left__mono,axiom,
% 0.32/0.55 ! [B: real,A: real,C: real] :
% 0.32/0.55 ( ( ord_less_real @ B @ A )
% 0.32/0.55 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.55 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.32/0.55 => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_strict_left_mono
% 0.32/0.55 thf(fact_2909_divide__strict__left__mono,axiom,
% 0.32/0.55 ! [B: rat,A: rat,C: rat] :
% 0.32/0.55 ( ( ord_less_rat @ B @ A )
% 0.32/0.55 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.55 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.32/0.55 => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_strict_left_mono
% 0.32/0.55 thf(fact_2910_mult__imp__less__div__pos,axiom,
% 0.32/0.55 ! [Y: real,Z: real,X: real] :
% 0.32/0.55 ( ( ord_less_real @ zero_zero_real @ Y )
% 0.32/0.55 => ( ( ord_less_real @ ( times_times_real @ Z @ Y ) @ X )
% 0.32/0.55 => ( ord_less_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % mult_imp_less_div_pos
% 0.32/0.55 thf(fact_2911_mult__imp__less__div__pos,axiom,
% 0.32/0.55 ! [Y: rat,Z: rat,X: rat] :
% 0.32/0.55 ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.32/0.55 => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y ) @ X )
% 0.32/0.55 => ( ord_less_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % mult_imp_less_div_pos
% 0.32/0.55 thf(fact_2912_mult__imp__div__pos__less,axiom,
% 0.32/0.55 ! [Y: real,X: real,Z: real] :
% 0.32/0.55 ( ( ord_less_real @ zero_zero_real @ Y )
% 0.32/0.55 => ( ( ord_less_real @ X @ ( times_times_real @ Z @ Y ) )
% 0.32/0.55 => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % mult_imp_div_pos_less
% 0.32/0.55 thf(fact_2913_mult__imp__div__pos__less,axiom,
% 0.32/0.55 ! [Y: rat,X: rat,Z: rat] :
% 0.32/0.55 ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.32/0.55 => ( ( ord_less_rat @ X @ ( times_times_rat @ Z @ Y ) )
% 0.32/0.55 => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % mult_imp_div_pos_less
% 0.32/0.55 thf(fact_2914_pos__less__divide__eq,axiom,
% 0.32/0.55 ! [C: real,A: real,B: real] :
% 0.32/0.55 ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.55 => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.32/0.55 = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % pos_less_divide_eq
% 0.32/0.55 thf(fact_2915_pos__less__divide__eq,axiom,
% 0.32/0.55 ! [C: rat,A: rat,B: rat] :
% 0.32/0.55 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.55 => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.55 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % pos_less_divide_eq
% 0.32/0.55 thf(fact_2916_pos__divide__less__eq,axiom,
% 0.32/0.55 ! [C: real,B: real,A: real] :
% 0.32/0.55 ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.55 => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.32/0.55 = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % pos_divide_less_eq
% 0.32/0.55 thf(fact_2917_pos__divide__less__eq,axiom,
% 0.32/0.55 ! [C: rat,B: rat,A: rat] :
% 0.32/0.55 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.55 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.32/0.55 = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % pos_divide_less_eq
% 0.32/0.55 thf(fact_2918_neg__less__divide__eq,axiom,
% 0.32/0.55 ! [C: real,A: real,B: real] :
% 0.32/0.55 ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.55 => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.32/0.55 = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % neg_less_divide_eq
% 0.32/0.55 thf(fact_2919_neg__less__divide__eq,axiom,
% 0.32/0.55 ! [C: rat,A: rat,B: rat] :
% 0.32/0.55 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.55 => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.55 = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % neg_less_divide_eq
% 0.32/0.55 thf(fact_2920_neg__divide__less__eq,axiom,
% 0.32/0.55 ! [C: real,B: real,A: real] :
% 0.32/0.55 ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.55 => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.32/0.55 = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % neg_divide_less_eq
% 0.32/0.55 thf(fact_2921_neg__divide__less__eq,axiom,
% 0.32/0.55 ! [C: rat,B: rat,A: rat] :
% 0.32/0.55 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.55 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.32/0.55 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % neg_divide_less_eq
% 0.32/0.55 thf(fact_2922_less__divide__eq,axiom,
% 0.32/0.55 ! [A: real,B: real,C: real] :
% 0.32/0.55 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.32/0.55 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.55 => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.32/0.55 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.55 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.55 => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.32/0.55 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.55 => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_divide_eq
% 0.32/0.55 thf(fact_2923_less__divide__eq,axiom,
% 0.32/0.55 ! [A: rat,B: rat,C: rat] :
% 0.32/0.55 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.55 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.55 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.32/0.55 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.55 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.55 => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.32/0.55 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.55 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_divide_eq
% 0.32/0.55 thf(fact_2924_divide__less__eq,axiom,
% 0.32/0.55 ! [B: real,C: real,A: real] :
% 0.32/0.55 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.32/0.55 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.55 => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.32/0.55 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.55 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.55 => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.32/0.55 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.55 => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_less_eq
% 0.32/0.55 thf(fact_2925_divide__less__eq,axiom,
% 0.32/0.55 ! [B: rat,C: rat,A: rat] :
% 0.32/0.55 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.32/0.55 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.55 => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.32/0.55 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.55 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.55 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.32/0.55 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.55 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_less_eq
% 0.32/0.55 thf(fact_2926_power__le__one,axiom,
% 0.32/0.55 ! [A: real,N2: nat] :
% 0.32/0.55 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.55 => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.32/0.55 => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ one_one_real ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_le_one
% 0.32/0.55 thf(fact_2927_power__le__one,axiom,
% 0.32/0.55 ! [A: rat,N2: nat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.55 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.32/0.55 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ one_one_rat ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_le_one
% 0.32/0.55 thf(fact_2928_power__le__one,axiom,
% 0.32/0.55 ! [A: nat,N2: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.55 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.32/0.55 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ one_one_nat ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_le_one
% 0.32/0.55 thf(fact_2929_power__le__one,axiom,
% 0.32/0.55 ! [A: int,N2: nat] :
% 0.32/0.55 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.55 => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.32/0.55 => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ one_one_int ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_le_one
% 0.32/0.55 thf(fact_2930_eq__divide__eq__numeral_I1_J,axiom,
% 0.32/0.55 ! [W: num,B: complex,C: complex] :
% 0.32/0.55 ( ( ( numera6690914467698888265omplex @ W )
% 0.32/0.55 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.32/0.55 = ( ( ( C != zero_zero_complex )
% 0.32/0.55 => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C )
% 0.32/0.55 = B ) )
% 0.32/0.55 & ( ( C = zero_zero_complex )
% 0.32/0.55 => ( ( numera6690914467698888265omplex @ W )
% 0.32/0.55 = zero_zero_complex ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % eq_divide_eq_numeral(1)
% 0.32/0.55 thf(fact_2931_eq__divide__eq__numeral_I1_J,axiom,
% 0.32/0.55 ! [W: num,B: real,C: real] :
% 0.32/0.55 ( ( ( numeral_numeral_real @ W )
% 0.32/0.55 = ( divide_divide_real @ B @ C ) )
% 0.32/0.55 = ( ( ( C != zero_zero_real )
% 0.32/0.55 => ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
% 0.32/0.55 = B ) )
% 0.32/0.55 & ( ( C = zero_zero_real )
% 0.32/0.55 => ( ( numeral_numeral_real @ W )
% 0.32/0.55 = zero_zero_real ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % eq_divide_eq_numeral(1)
% 0.32/0.55 thf(fact_2932_eq__divide__eq__numeral_I1_J,axiom,
% 0.32/0.55 ! [W: num,B: rat,C: rat] :
% 0.32/0.55 ( ( ( numeral_numeral_rat @ W )
% 0.32/0.55 = ( divide_divide_rat @ B @ C ) )
% 0.32/0.55 = ( ( ( C != zero_zero_rat )
% 0.32/0.55 => ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
% 0.32/0.55 = B ) )
% 0.32/0.55 & ( ( C = zero_zero_rat )
% 0.32/0.55 => ( ( numeral_numeral_rat @ W )
% 0.32/0.55 = zero_zero_rat ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % eq_divide_eq_numeral(1)
% 0.32/0.55 thf(fact_2933_divide__eq__eq__numeral_I1_J,axiom,
% 0.32/0.55 ! [B: complex,C: complex,W: num] :
% 0.32/0.55 ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.32/0.55 = ( numera6690914467698888265omplex @ W ) )
% 0.32/0.55 = ( ( ( C != zero_zero_complex )
% 0.32/0.55 => ( B
% 0.32/0.55 = ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C ) ) )
% 0.32/0.55 & ( ( C = zero_zero_complex )
% 0.32/0.55 => ( ( numera6690914467698888265omplex @ W )
% 0.32/0.55 = zero_zero_complex ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_eq_eq_numeral(1)
% 0.32/0.55 thf(fact_2934_divide__eq__eq__numeral_I1_J,axiom,
% 0.32/0.55 ! [B: real,C: real,W: num] :
% 0.32/0.55 ( ( ( divide_divide_real @ B @ C )
% 0.32/0.55 = ( numeral_numeral_real @ W ) )
% 0.32/0.55 = ( ( ( C != zero_zero_real )
% 0.32/0.55 => ( B
% 0.32/0.55 = ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.32/0.55 & ( ( C = zero_zero_real )
% 0.32/0.55 => ( ( numeral_numeral_real @ W )
% 0.32/0.55 = zero_zero_real ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_eq_eq_numeral(1)
% 0.32/0.55 thf(fact_2935_divide__eq__eq__numeral_I1_J,axiom,
% 0.32/0.55 ! [B: rat,C: rat,W: num] :
% 0.32/0.55 ( ( ( divide_divide_rat @ B @ C )
% 0.32/0.55 = ( numeral_numeral_rat @ W ) )
% 0.32/0.55 = ( ( ( C != zero_zero_rat )
% 0.32/0.55 => ( B
% 0.32/0.55 = ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.32/0.55 & ( ( C = zero_zero_rat )
% 0.32/0.55 => ( ( numeral_numeral_rat @ W )
% 0.32/0.55 = zero_zero_rat ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_eq_eq_numeral(1)
% 0.32/0.55 thf(fact_2936_divide__less__eq__1,axiom,
% 0.32/0.55 ! [B: real,A: real] :
% 0.32/0.55 ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.32/0.55 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.55 & ( ord_less_real @ B @ A ) )
% 0.32/0.55 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.55 & ( ord_less_real @ A @ B ) )
% 0.32/0.55 | ( A = zero_zero_real ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_less_eq_1
% 0.32/0.55 thf(fact_2937_divide__less__eq__1,axiom,
% 0.32/0.55 ! [B: rat,A: rat] :
% 0.32/0.55 ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.32/0.55 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.55 & ( ord_less_rat @ B @ A ) )
% 0.32/0.55 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.55 & ( ord_less_rat @ A @ B ) )
% 0.32/0.55 | ( A = zero_zero_rat ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_less_eq_1
% 0.32/0.55 thf(fact_2938_less__divide__eq__1,axiom,
% 0.32/0.55 ! [B: real,A: real] :
% 0.32/0.55 ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.32/0.55 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.55 & ( ord_less_real @ A @ B ) )
% 0.32/0.55 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.55 & ( ord_less_real @ B @ A ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_divide_eq_1
% 0.32/0.55 thf(fact_2939_less__divide__eq__1,axiom,
% 0.32/0.55 ! [B: rat,A: rat] :
% 0.32/0.55 ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.32/0.55 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.55 & ( ord_less_rat @ A @ B ) )
% 0.32/0.55 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.55 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_divide_eq_1
% 0.32/0.55 thf(fact_2940_divide__add__eq__iff,axiom,
% 0.32/0.55 ! [Z: complex,X: complex,Y: complex] :
% 0.32/0.55 ( ( Z != zero_zero_complex )
% 0.32/0.55 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y )
% 0.32/0.55 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_add_eq_iff
% 0.32/0.55 thf(fact_2941_divide__add__eq__iff,axiom,
% 0.32/0.55 ! [Z: real,X: real,Y: real] :
% 0.32/0.55 ( ( Z != zero_zero_real )
% 0.32/0.55 => ( ( plus_plus_real @ ( divide_divide_real @ X @ Z ) @ Y )
% 0.32/0.55 = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_add_eq_iff
% 0.32/0.55 thf(fact_2942_divide__add__eq__iff,axiom,
% 0.32/0.55 ! [Z: rat,X: rat,Y: rat] :
% 0.32/0.55 ( ( Z != zero_zero_rat )
% 0.32/0.55 => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
% 0.32/0.55 = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_add_eq_iff
% 0.32/0.55 thf(fact_2943_add__divide__eq__iff,axiom,
% 0.32/0.55 ! [Z: complex,X: complex,Y: complex] :
% 0.32/0.55 ( ( Z != zero_zero_complex )
% 0.32/0.55 => ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z ) )
% 0.32/0.55 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ Y ) @ Z ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_divide_eq_iff
% 0.32/0.55 thf(fact_2944_add__divide__eq__iff,axiom,
% 0.32/0.55 ! [Z: real,X: real,Y: real] :
% 0.32/0.55 ( ( Z != zero_zero_real )
% 0.32/0.55 => ( ( plus_plus_real @ X @ ( divide_divide_real @ Y @ Z ) )
% 0.32/0.55 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_divide_eq_iff
% 0.32/0.55 thf(fact_2945_add__divide__eq__iff,axiom,
% 0.32/0.55 ! [Z: rat,X: rat,Y: rat] :
% 0.32/0.55 ( ( Z != zero_zero_rat )
% 0.32/0.55 => ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
% 0.32/0.55 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_divide_eq_iff
% 0.32/0.55 thf(fact_2946_add__num__frac,axiom,
% 0.32/0.55 ! [Y: complex,Z: complex,X: complex] :
% 0.32/0.55 ( ( Y != zero_zero_complex )
% 0.32/0.55 => ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X @ Y ) )
% 0.32/0.55 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_num_frac
% 0.32/0.55 thf(fact_2947_add__num__frac,axiom,
% 0.32/0.55 ! [Y: real,Z: real,X: real] :
% 0.32/0.55 ( ( Y != zero_zero_real )
% 0.32/0.55 => ( ( plus_plus_real @ Z @ ( divide_divide_real @ X @ Y ) )
% 0.32/0.55 = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_num_frac
% 0.32/0.55 thf(fact_2948_add__num__frac,axiom,
% 0.32/0.55 ! [Y: rat,Z: rat,X: rat] :
% 0.32/0.55 ( ( Y != zero_zero_rat )
% 0.32/0.55 => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X @ Y ) )
% 0.32/0.55 = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_num_frac
% 0.32/0.55 thf(fact_2949_add__frac__num,axiom,
% 0.32/0.55 ! [Y: complex,X: complex,Z: complex] :
% 0.32/0.55 ( ( Y != zero_zero_complex )
% 0.32/0.55 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ Z )
% 0.32/0.55 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_frac_num
% 0.32/0.55 thf(fact_2950_add__frac__num,axiom,
% 0.32/0.55 ! [Y: real,X: real,Z: real] :
% 0.32/0.55 ( ( Y != zero_zero_real )
% 0.32/0.55 => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ Z )
% 0.32/0.55 = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_frac_num
% 0.32/0.55 thf(fact_2951_add__frac__num,axiom,
% 0.32/0.55 ! [Y: rat,X: rat,Z: rat] :
% 0.32/0.55 ( ( Y != zero_zero_rat )
% 0.32/0.55 => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ Z )
% 0.32/0.55 = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_frac_num
% 0.32/0.55 thf(fact_2952_add__frac__eq,axiom,
% 0.32/0.55 ! [Y: complex,Z: complex,X: complex,W: complex] :
% 0.32/0.55 ( ( Y != zero_zero_complex )
% 0.32/0.55 => ( ( Z != zero_zero_complex )
% 0.32/0.55 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
% 0.32/0.55 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_frac_eq
% 0.32/0.55 thf(fact_2953_add__frac__eq,axiom,
% 0.32/0.55 ! [Y: real,Z: real,X: real,W: real] :
% 0.32/0.55 ( ( Y != zero_zero_real )
% 0.32/0.55 => ( ( Z != zero_zero_real )
% 0.32/0.55 => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
% 0.32/0.55 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_frac_eq
% 0.32/0.55 thf(fact_2954_add__frac__eq,axiom,
% 0.32/0.55 ! [Y: rat,Z: rat,X: rat,W: rat] :
% 0.32/0.55 ( ( Y != zero_zero_rat )
% 0.32/0.55 => ( ( Z != zero_zero_rat )
% 0.32/0.55 => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
% 0.32/0.55 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_frac_eq
% 0.32/0.55 thf(fact_2955_add__divide__eq__if__simps_I1_J,axiom,
% 0.32/0.55 ! [Z: complex,A: complex,B: complex] :
% 0.32/0.55 ( ( ( Z = zero_zero_complex )
% 0.32/0.55 => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 0.32/0.55 = A ) )
% 0.32/0.55 & ( ( Z != zero_zero_complex )
% 0.32/0.55 => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 0.32/0.55 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_divide_eq_if_simps(1)
% 0.32/0.55 thf(fact_2956_add__divide__eq__if__simps_I1_J,axiom,
% 0.32/0.55 ! [Z: real,A: real,B: real] :
% 0.32/0.55 ( ( ( Z = zero_zero_real )
% 0.32/0.55 => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 0.32/0.55 = A ) )
% 0.32/0.55 & ( ( Z != zero_zero_real )
% 0.32/0.55 => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 0.32/0.55 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_divide_eq_if_simps(1)
% 0.32/0.55 thf(fact_2957_add__divide__eq__if__simps_I1_J,axiom,
% 0.32/0.55 ! [Z: rat,A: rat,B: rat] :
% 0.32/0.55 ( ( ( Z = zero_zero_rat )
% 0.32/0.55 => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 0.32/0.55 = A ) )
% 0.32/0.55 & ( ( Z != zero_zero_rat )
% 0.32/0.55 => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 0.32/0.55 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_divide_eq_if_simps(1)
% 0.32/0.55 thf(fact_2958_add__divide__eq__if__simps_I2_J,axiom,
% 0.32/0.55 ! [Z: complex,A: complex,B: complex] :
% 0.32/0.55 ( ( ( Z = zero_zero_complex )
% 0.32/0.55 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 0.32/0.55 = B ) )
% 0.32/0.55 & ( ( Z != zero_zero_complex )
% 0.32/0.55 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 0.32/0.55 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_divide_eq_if_simps(2)
% 0.32/0.55 thf(fact_2959_add__divide__eq__if__simps_I2_J,axiom,
% 0.32/0.55 ! [Z: real,A: real,B: real] :
% 0.32/0.55 ( ( ( Z = zero_zero_real )
% 0.32/0.55 => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 0.32/0.55 = B ) )
% 0.32/0.55 & ( ( Z != zero_zero_real )
% 0.32/0.55 => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 0.32/0.55 = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_divide_eq_if_simps(2)
% 0.32/0.55 thf(fact_2960_add__divide__eq__if__simps_I2_J,axiom,
% 0.32/0.55 ! [Z: rat,A: rat,B: rat] :
% 0.32/0.55 ( ( ( Z = zero_zero_rat )
% 0.32/0.55 => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 0.32/0.55 = B ) )
% 0.32/0.55 & ( ( Z != zero_zero_rat )
% 0.32/0.55 => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 0.32/0.55 = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_divide_eq_if_simps(2)
% 0.32/0.55 thf(fact_2961_power__le__imp__le__base,axiom,
% 0.32/0.55 ! [A: real,N2: nat,B: real] :
% 0.32/0.55 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ ( power_power_real @ B @ ( suc @ N2 ) ) )
% 0.32/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.32/0.55 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_le_imp_le_base
% 0.32/0.55 thf(fact_2962_power__le__imp__le__base,axiom,
% 0.32/0.55 ! [A: rat,N2: nat,B: rat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ ( power_power_rat @ B @ ( suc @ N2 ) ) )
% 0.32/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.32/0.55 => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_le_imp_le_base
% 0.32/0.55 thf(fact_2963_power__le__imp__le__base,axiom,
% 0.32/0.55 ! [A: nat,N2: nat,B: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ ( power_power_nat @ B @ ( suc @ N2 ) ) )
% 0.32/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.32/0.55 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_le_imp_le_base
% 0.32/0.55 thf(fact_2964_power__le__imp__le__base,axiom,
% 0.32/0.55 ! [A: int,N2: nat,B: int] :
% 0.32/0.55 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ ( power_power_int @ B @ ( suc @ N2 ) ) )
% 0.32/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.55 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_le_imp_le_base
% 0.32/0.55 thf(fact_2965_power__inject__base,axiom,
% 0.32/0.55 ! [A: real,N2: nat,B: real] :
% 0.32/0.55 ( ( ( power_power_real @ A @ ( suc @ N2 ) )
% 0.32/0.55 = ( power_power_real @ B @ ( suc @ N2 ) ) )
% 0.32/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.32/0.55 => ( A = B ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_inject_base
% 0.32/0.55 thf(fact_2966_power__inject__base,axiom,
% 0.32/0.55 ! [A: rat,N2: nat,B: rat] :
% 0.32/0.55 ( ( ( power_power_rat @ A @ ( suc @ N2 ) )
% 0.32/0.55 = ( power_power_rat @ B @ ( suc @ N2 ) ) )
% 0.32/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.32/0.55 => ( A = B ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_inject_base
% 0.32/0.55 thf(fact_2967_power__inject__base,axiom,
% 0.32/0.55 ! [A: nat,N2: nat,B: nat] :
% 0.32/0.55 ( ( ( power_power_nat @ A @ ( suc @ N2 ) )
% 0.32/0.55 = ( power_power_nat @ B @ ( suc @ N2 ) ) )
% 0.32/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.55 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.32/0.55 => ( A = B ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_inject_base
% 0.32/0.55 thf(fact_2968_power__inject__base,axiom,
% 0.32/0.55 ! [A: int,N2: nat,B: int] :
% 0.32/0.55 ( ( ( power_power_int @ A @ ( suc @ N2 ) )
% 0.32/0.55 = ( power_power_int @ B @ ( suc @ N2 ) ) )
% 0.32/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.55 => ( A = B ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_inject_base
% 0.32/0.55 thf(fact_2969_div__add__self2,axiom,
% 0.32/0.55 ! [B: nat,A: nat] :
% 0.32/0.55 ( ( B != zero_zero_nat )
% 0.32/0.55 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.32/0.55 = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % div_add_self2
% 0.32/0.55 thf(fact_2970_div__add__self2,axiom,
% 0.32/0.55 ! [B: int,A: int] :
% 0.32/0.55 ( ( B != zero_zero_int )
% 0.32/0.55 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.32/0.55 = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % div_add_self2
% 0.32/0.55 thf(fact_2971_div__add__self1,axiom,
% 0.32/0.55 ! [B: nat,A: nat] :
% 0.32/0.55 ( ( B != zero_zero_nat )
% 0.32/0.55 => ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.32/0.55 = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % div_add_self1
% 0.32/0.55 thf(fact_2972_div__add__self1,axiom,
% 0.32/0.55 ! [B: int,A: int] :
% 0.32/0.55 ( ( B != zero_zero_int )
% 0.32/0.55 => ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.32/0.55 = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % div_add_self1
% 0.32/0.55 thf(fact_2973_divide__diff__eq__iff,axiom,
% 0.32/0.55 ! [Z: complex,X: complex,Y: complex] :
% 0.32/0.55 ( ( Z != zero_zero_complex )
% 0.32/0.55 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y )
% 0.32/0.55 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_diff_eq_iff
% 0.32/0.55 thf(fact_2974_divide__diff__eq__iff,axiom,
% 0.32/0.55 ! [Z: real,X: real,Y: real] :
% 0.32/0.55 ( ( Z != zero_zero_real )
% 0.32/0.55 => ( ( minus_minus_real @ ( divide_divide_real @ X @ Z ) @ Y )
% 0.32/0.55 = ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_diff_eq_iff
% 0.32/0.55 thf(fact_2975_divide__diff__eq__iff,axiom,
% 0.32/0.55 ! [Z: rat,X: rat,Y: rat] :
% 0.32/0.55 ( ( Z != zero_zero_rat )
% 0.32/0.55 => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
% 0.32/0.55 = ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % divide_diff_eq_iff
% 0.32/0.55 thf(fact_2976_diff__divide__eq__iff,axiom,
% 0.32/0.55 ! [Z: complex,X: complex,Y: complex] :
% 0.32/0.55 ( ( Z != zero_zero_complex )
% 0.32/0.55 => ( ( minus_minus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z ) )
% 0.32/0.55 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ Y ) @ Z ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % diff_divide_eq_iff
% 0.32/0.55 thf(fact_2977_diff__divide__eq__iff,axiom,
% 0.32/0.55 ! [Z: real,X: real,Y: real] :
% 0.32/0.55 ( ( Z != zero_zero_real )
% 0.32/0.55 => ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z ) )
% 0.32/0.55 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % diff_divide_eq_iff
% 0.32/0.55 thf(fact_2978_diff__divide__eq__iff,axiom,
% 0.32/0.55 ! [Z: rat,X: rat,Y: rat] :
% 0.32/0.55 ( ( Z != zero_zero_rat )
% 0.32/0.55 => ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
% 0.32/0.55 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % diff_divide_eq_iff
% 0.32/0.55 thf(fact_2979_diff__frac__eq,axiom,
% 0.32/0.55 ! [Y: complex,Z: complex,X: complex,W: complex] :
% 0.32/0.55 ( ( Y != zero_zero_complex )
% 0.32/0.55 => ( ( Z != zero_zero_complex )
% 0.32/0.55 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
% 0.32/0.55 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % diff_frac_eq
% 0.32/0.55 thf(fact_2980_diff__frac__eq,axiom,
% 0.32/0.55 ! [Y: real,Z: real,X: real,W: real] :
% 0.32/0.55 ( ( Y != zero_zero_real )
% 0.32/0.55 => ( ( Z != zero_zero_real )
% 0.32/0.55 => ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
% 0.32/0.55 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % diff_frac_eq
% 0.32/0.55 thf(fact_2981_diff__frac__eq,axiom,
% 0.32/0.55 ! [Y: rat,Z: rat,X: rat,W: rat] :
% 0.32/0.55 ( ( Y != zero_zero_rat )
% 0.32/0.55 => ( ( Z != zero_zero_rat )
% 0.32/0.55 => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
% 0.32/0.55 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % diff_frac_eq
% 0.32/0.55 thf(fact_2982_add__divide__eq__if__simps_I4_J,axiom,
% 0.32/0.55 ! [Z: complex,A: complex,B: complex] :
% 0.32/0.55 ( ( ( Z = zero_zero_complex )
% 0.32/0.55 => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 0.32/0.55 = A ) )
% 0.32/0.55 & ( ( Z != zero_zero_complex )
% 0.32/0.55 => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 0.32/0.55 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_divide_eq_if_simps(4)
% 0.32/0.55 thf(fact_2983_add__divide__eq__if__simps_I4_J,axiom,
% 0.32/0.55 ! [Z: real,A: real,B: real] :
% 0.32/0.55 ( ( ( Z = zero_zero_real )
% 0.32/0.55 => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 0.32/0.55 = A ) )
% 0.32/0.55 & ( ( Z != zero_zero_real )
% 0.32/0.55 => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 0.32/0.55 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_divide_eq_if_simps(4)
% 0.32/0.55 thf(fact_2984_add__divide__eq__if__simps_I4_J,axiom,
% 0.32/0.55 ! [Z: rat,A: rat,B: rat] :
% 0.32/0.55 ( ( ( Z = zero_zero_rat )
% 0.32/0.55 => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 0.32/0.55 = A ) )
% 0.32/0.55 & ( ( Z != zero_zero_rat )
% 0.32/0.55 => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 0.32/0.55 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_divide_eq_if_simps(4)
% 0.32/0.55 thf(fact_2985_numeral__1__eq__Suc__0,axiom,
% 0.32/0.55 ( ( numeral_numeral_nat @ one )
% 0.32/0.55 = ( suc @ zero_zero_nat ) ) ).
% 0.32/0.55
% 0.32/0.55 % numeral_1_eq_Suc_0
% 0.32/0.55 thf(fact_2986_num_Osize_I5_J,axiom,
% 0.32/0.55 ! [X22: num] :
% 0.32/0.55 ( ( size_size_num @ ( bit0 @ X22 ) )
% 0.32/0.55 = ( plus_plus_nat @ ( size_size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % num.size(5)
% 0.32/0.55 thf(fact_2987_ex__least__nat__less,axiom,
% 0.32/0.55 ! [P: nat > $o,N2: nat] :
% 0.32/0.55 ( ( P @ N2 )
% 0.32/0.55 => ( ~ ( P @ zero_zero_nat )
% 0.32/0.55 => ? [K2: nat] :
% 0.32/0.55 ( ( ord_less_nat @ K2 @ N2 )
% 0.32/0.55 & ! [I: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ I @ K2 )
% 0.32/0.55 => ~ ( P @ I ) )
% 0.32/0.55 & ( P @ ( suc @ K2 ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % ex_least_nat_less
% 0.32/0.55 thf(fact_2988_one__less__mult,axiom,
% 0.32/0.55 ! [N2: nat,M: nat] :
% 0.32/0.55 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 0.32/0.55 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.32/0.55 => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % one_less_mult
% 0.32/0.55 thf(fact_2989_n__less__m__mult__n,axiom,
% 0.32/0.55 ! [N2: nat,M: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.55 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.32/0.55 => ( ord_less_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % n_less_m_mult_n
% 0.32/0.55 thf(fact_2990_n__less__n__mult__m,axiom,
% 0.32/0.55 ! [N2: nat,M: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.55 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.32/0.55 => ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % n_less_n_mult_m
% 0.32/0.55 thf(fact_2991_diff__Suc__less,axiom,
% 0.32/0.55 ! [N2: nat,I2: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.55 => ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) @ N2 ) ) ).
% 0.32/0.55
% 0.32/0.55 % diff_Suc_less
% 0.32/0.56 thf(fact_2992_power__gt__expt,axiom,
% 0.32/0.56 ! [N2: nat,K: nat] :
% 0.32/0.56 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 0.32/0.56 => ( ord_less_nat @ K @ ( power_power_nat @ N2 @ K ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_gt_expt
% 0.32/0.56 thf(fact_2993_nat__mult__le__cancel1,axiom,
% 0.32/0.56 ! [K: nat,M: nat,N2: nat] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.56 => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 0.32/0.56 = ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % nat_mult_le_cancel1
% 0.32/0.56 thf(fact_2994_nat__induct__non__zero,axiom,
% 0.32/0.56 ! [N2: nat,P: nat > $o] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.56 => ( ( P @ one_one_nat )
% 0.32/0.56 => ( ! [N4: nat] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 0.32/0.56 => ( ( P @ N4 )
% 0.32/0.56 => ( P @ ( suc @ N4 ) ) ) )
% 0.32/0.56 => ( P @ N2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % nat_induct_non_zero
% 0.32/0.56 thf(fact_2995_length__pos__if__in__set,axiom,
% 0.32/0.56 ! [X: nat,Xs: list_nat] :
% 0.32/0.56 ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 0.32/0.56 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % length_pos_if_in_set
% 0.32/0.56 thf(fact_2996_length__pos__if__in__set,axiom,
% 0.32/0.56 ! [X: real,Xs: list_real] :
% 0.32/0.56 ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 0.32/0.56 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % length_pos_if_in_set
% 0.32/0.56 thf(fact_2997_length__pos__if__in__set,axiom,
% 0.32/0.56 ! [X: complex,Xs: list_complex] :
% 0.32/0.56 ( ( member_complex @ X @ ( set_complex2 @ Xs ) )
% 0.32/0.56 => ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % length_pos_if_in_set
% 0.32/0.56 thf(fact_2998_length__pos__if__in__set,axiom,
% 0.32/0.56 ! [X: set_nat,Xs: list_set_nat] :
% 0.32/0.56 ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 0.32/0.56 => ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % length_pos_if_in_set
% 0.32/0.56 thf(fact_2999_length__pos__if__in__set,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 0.32/0.56 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.32/0.56 => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % length_pos_if_in_set
% 0.32/0.56 thf(fact_3000_length__pos__if__in__set,axiom,
% 0.32/0.56 ! [X: $o,Xs: list_o] :
% 0.32/0.56 ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 0.32/0.56 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % length_pos_if_in_set
% 0.32/0.56 thf(fact_3001_length__pos__if__in__set,axiom,
% 0.32/0.56 ! [X: int,Xs: list_int] :
% 0.32/0.56 ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 0.32/0.56 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % length_pos_if_in_set
% 0.32/0.56 thf(fact_3002_nat__one__le__power,axiom,
% 0.32/0.56 ! [I2: nat,N2: nat] :
% 0.32/0.56 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I2 )
% 0.32/0.56 => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I2 @ N2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % nat_one_le_power
% 0.32/0.56 thf(fact_3003_nat__diff__split,axiom,
% 0.32/0.56 ! [P: nat > $o,A: nat,B: nat] :
% 0.32/0.56 ( ( P @ ( minus_minus_nat @ A @ B ) )
% 0.32/0.56 = ( ( ( ord_less_nat @ A @ B )
% 0.32/0.56 => ( P @ zero_zero_nat ) )
% 0.32/0.56 & ! [D2: nat] :
% 0.32/0.56 ( ( A
% 0.32/0.56 = ( plus_plus_nat @ B @ D2 ) )
% 0.32/0.56 => ( P @ D2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % nat_diff_split
% 0.32/0.56 thf(fact_3004_nat__diff__split__asm,axiom,
% 0.32/0.56 ! [P: nat > $o,A: nat,B: nat] :
% 0.32/0.56 ( ( P @ ( minus_minus_nat @ A @ B ) )
% 0.32/0.56 = ( ~ ( ( ( ord_less_nat @ A @ B )
% 0.32/0.56 & ~ ( P @ zero_zero_nat ) )
% 0.32/0.56 | ? [D2: nat] :
% 0.32/0.56 ( ( A
% 0.32/0.56 = ( plus_plus_nat @ B @ D2 ) )
% 0.32/0.56 & ~ ( P @ D2 ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % nat_diff_split_asm
% 0.32/0.56 thf(fact_3005_div__le__mono2,axiom,
% 0.32/0.56 ! [M: nat,N2: nat,K: nat] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.56 => ( ( ord_less_eq_nat @ M @ N2 )
% 0.32/0.56 => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N2 ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_le_mono2
% 0.32/0.56 thf(fact_3006_div__greater__zero__iff,axiom,
% 0.32/0.56 ! [M: nat,N2: nat] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N2 ) )
% 0.32/0.56 = ( ( ord_less_eq_nat @ N2 @ M )
% 0.32/0.56 & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_greater_zero_iff
% 0.32/0.56 thf(fact_3007_nat__mult__div__cancel1,axiom,
% 0.32/0.56 ! [K: nat,M: nat,N2: nat] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.56 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 0.32/0.56 = ( divide_divide_nat @ M @ N2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % nat_mult_div_cancel1
% 0.32/0.56 thf(fact_3008_div__less__iff__less__mult,axiom,
% 0.32/0.56 ! [Q2: nat,M: nat,N2: nat] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ Q2 )
% 0.32/0.56 => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N2 )
% 0.32/0.56 = ( ord_less_nat @ M @ ( times_times_nat @ N2 @ Q2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_less_iff_less_mult
% 0.32/0.56 thf(fact_3009_div__less__dividend,axiom,
% 0.32/0.56 ! [N2: nat,M: nat] :
% 0.32/0.56 ( ( ord_less_nat @ one_one_nat @ N2 )
% 0.32/0.56 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.56 => ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_less_dividend
% 0.32/0.56 thf(fact_3010_div__eq__dividend__iff,axiom,
% 0.32/0.56 ! [M: nat,N2: nat] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.56 => ( ( ( divide_divide_nat @ M @ N2 )
% 0.32/0.56 = M )
% 0.32/0.56 = ( N2 = one_one_nat ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_eq_dividend_iff
% 0.32/0.56 thf(fact_3011_divmod__digit__1_I1_J,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.32/0.56 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.32/0.56 => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 0.32/0.56 => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
% 0.32/0.56 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % divmod_digit_1(1)
% 0.32/0.56 thf(fact_3012_divmod__digit__1_I1_J,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.56 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.32/0.56 => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 0.32/0.56 => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_nat )
% 0.32/0.56 = ( divide_divide_nat @ A @ B ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % divmod_digit_1(1)
% 0.32/0.56 thf(fact_3013_divmod__digit__1_I1_J,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.56 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.56 => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 0.32/0.56 => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_int )
% 0.32/0.56 = ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % divmod_digit_1(1)
% 0.32/0.56 thf(fact_3014_vebt__insert_Osimps_I3_J,axiom,
% 0.32/0.56 ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X: nat] :
% 0.32/0.56 ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) @ X )
% 0.32/0.56 = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) ) ).
% 0.32/0.56
% 0.32/0.56 % vebt_insert.simps(3)
% 0.32/0.56 thf(fact_3015_vebt__member_Osimps_I3_J,axiom,
% 0.32/0.56 ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
% 0.32/0.56 ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X ) ).
% 0.32/0.56
% 0.32/0.56 % vebt_member.simps(3)
% 0.32/0.56 thf(fact_3016_vebt__mint_Ocases,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT] :
% 0.32/0.56 ( ! [A3: $o,B2: $o] :
% 0.32/0.56 ( X
% 0.32/0.56 != ( vEBT_Leaf @ A3 @ B2 ) )
% 0.32/0.56 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.32/0.56 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % vebt_mint.cases
% 0.32/0.56 thf(fact_3017_VEBT__internal_OminNull_Oelims_I1_J,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Y: $o] :
% 0.32/0.56 ( ( ( vEBT_VEBT_minNull @ X )
% 0.32/0.56 = Y )
% 0.32/0.56 => ( ( ( X
% 0.32/0.56 = ( vEBT_Leaf @ $false @ $false ) )
% 0.32/0.56 => ~ Y )
% 0.32/0.56 => ( ( ? [Uv2: $o] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.32/0.56 => Y )
% 0.32/0.56 => ( ( ? [Uu2: $o] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.32/0.56 => Y )
% 0.32/0.56 => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 0.32/0.56 => ~ Y )
% 0.32/0.56 => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 0.32/0.56 => Y ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % VEBT_internal.minNull.elims(1)
% 0.32/0.56 thf(fact_3018_vebt__maxt_Oelims,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Y: option_nat] :
% 0.32/0.56 ( ( ( vEBT_vebt_maxt @ X )
% 0.32/0.56 = Y )
% 0.32/0.56 => ( ! [A3: $o,B2: $o] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.32/0.56 => ~ ( ( B2
% 0.32/0.56 => ( Y
% 0.32/0.56 = ( some_nat @ one_one_nat ) ) )
% 0.32/0.56 & ( ~ B2
% 0.32/0.56 => ( ( A3
% 0.32/0.56 => ( Y
% 0.32/0.56 = ( some_nat @ zero_zero_nat ) ) )
% 0.32/0.56 & ( ~ A3
% 0.32/0.56 => ( Y = none_nat ) ) ) ) ) )
% 0.32/0.56 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.32/0.56 => ( Y != none_nat ) )
% 0.32/0.56 => ~ ! [Mi2: nat,Ma2: nat] :
% 0.32/0.56 ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.32/0.56 => ( Y
% 0.32/0.56 != ( some_nat @ Ma2 ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % vebt_maxt.elims
% 0.32/0.56 thf(fact_3019_vebt__mint_Oelims,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Y: option_nat] :
% 0.32/0.56 ( ( ( vEBT_vebt_mint @ X )
% 0.32/0.56 = Y )
% 0.32/0.56 => ( ! [A3: $o,B2: $o] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.32/0.56 => ~ ( ( A3
% 0.32/0.56 => ( Y
% 0.32/0.56 = ( some_nat @ zero_zero_nat ) ) )
% 0.32/0.56 & ( ~ A3
% 0.32/0.56 => ( ( B2
% 0.32/0.56 => ( Y
% 0.32/0.56 = ( some_nat @ one_one_nat ) ) )
% 0.32/0.56 & ( ~ B2
% 0.32/0.56 => ( Y = none_nat ) ) ) ) ) )
% 0.32/0.56 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.32/0.56 => ( Y != none_nat ) )
% 0.32/0.56 => ~ ! [Mi2: nat] :
% 0.32/0.56 ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.32/0.56 => ( Y
% 0.32/0.56 != ( some_nat @ Mi2 ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % vebt_mint.elims
% 0.32/0.56 thf(fact_3020_mult__le__cancel__left1,axiom,
% 0.32/0.56 ! [C: real,B: real] :
% 0.32/0.56 ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
% 0.32/0.56 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.56 => ( ord_less_eq_real @ one_one_real @ B ) )
% 0.32/0.56 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.56 => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_le_cancel_left1
% 0.32/0.56 thf(fact_3021_mult__le__cancel__left1,axiom,
% 0.32/0.56 ! [C: rat,B: rat] :
% 0.32/0.56 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
% 0.32/0.56 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.56 => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 0.32/0.56 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.56 => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_le_cancel_left1
% 0.32/0.56 thf(fact_3022_mult__le__cancel__left1,axiom,
% 0.32/0.56 ! [C: int,B: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
% 0.32/0.56 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.56 => ( ord_less_eq_int @ one_one_int @ B ) )
% 0.32/0.56 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.32/0.56 => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_le_cancel_left1
% 0.32/0.56 thf(fact_3023_mult__le__cancel__left2,axiom,
% 0.32/0.56 ! [C: real,A: real] :
% 0.32/0.56 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
% 0.32/0.56 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.56 => ( ord_less_eq_real @ A @ one_one_real ) )
% 0.32/0.56 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.56 => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_le_cancel_left2
% 0.32/0.56 thf(fact_3024_mult__le__cancel__left2,axiom,
% 0.32/0.56 ! [C: rat,A: rat] :
% 0.32/0.56 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
% 0.32/0.56 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.56 => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 0.32/0.56 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.56 => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_le_cancel_left2
% 0.32/0.56 thf(fact_3025_mult__le__cancel__left2,axiom,
% 0.32/0.56 ! [C: int,A: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
% 0.32/0.56 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.56 => ( ord_less_eq_int @ A @ one_one_int ) )
% 0.32/0.56 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.32/0.56 => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_le_cancel_left2
% 0.32/0.56 thf(fact_3026_mult__le__cancel__right1,axiom,
% 0.32/0.56 ! [C: real,B: real] :
% 0.32/0.56 ( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
% 0.32/0.56 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.56 => ( ord_less_eq_real @ one_one_real @ B ) )
% 0.32/0.56 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.56 => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_le_cancel_right1
% 0.32/0.56 thf(fact_3027_mult__le__cancel__right1,axiom,
% 0.32/0.56 ! [C: rat,B: rat] :
% 0.32/0.56 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
% 0.32/0.56 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.56 => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 0.32/0.56 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.56 => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_le_cancel_right1
% 0.32/0.56 thf(fact_3028_mult__le__cancel__right1,axiom,
% 0.32/0.56 ! [C: int,B: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
% 0.32/0.56 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.56 => ( ord_less_eq_int @ one_one_int @ B ) )
% 0.32/0.56 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.32/0.56 => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_le_cancel_right1
% 0.32/0.56 thf(fact_3029_mult__le__cancel__right2,axiom,
% 0.32/0.56 ! [A: real,C: real] :
% 0.32/0.56 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
% 0.32/0.56 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.56 => ( ord_less_eq_real @ A @ one_one_real ) )
% 0.32/0.56 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.56 => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_le_cancel_right2
% 0.32/0.56 thf(fact_3030_mult__le__cancel__right2,axiom,
% 0.32/0.56 ! [A: rat,C: rat] :
% 0.32/0.56 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
% 0.32/0.56 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.56 => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 0.32/0.56 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.56 => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_le_cancel_right2
% 0.32/0.56 thf(fact_3031_mult__le__cancel__right2,axiom,
% 0.32/0.56 ! [A: int,C: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
% 0.32/0.56 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.56 => ( ord_less_eq_int @ A @ one_one_int ) )
% 0.32/0.56 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.32/0.56 => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_le_cancel_right2
% 0.32/0.56 thf(fact_3032_mult__less__cancel__left1,axiom,
% 0.32/0.56 ! [C: real,B: real] :
% 0.32/0.56 ( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
% 0.32/0.56 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.56 => ( ord_less_real @ one_one_real @ B ) )
% 0.32/0.56 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.32/0.56 => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_less_cancel_left1
% 0.32/0.56 thf(fact_3033_mult__less__cancel__left1,axiom,
% 0.32/0.56 ! [C: rat,B: rat] :
% 0.32/0.56 ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
% 0.32/0.56 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.56 => ( ord_less_rat @ one_one_rat @ B ) )
% 0.32/0.56 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.32/0.56 => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_less_cancel_left1
% 0.32/0.56 thf(fact_3034_mult__less__cancel__left1,axiom,
% 0.32/0.56 ! [C: int,B: int] :
% 0.32/0.56 ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
% 0.32/0.56 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.56 => ( ord_less_int @ one_one_int @ B ) )
% 0.32/0.56 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.32/0.56 => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_less_cancel_left1
% 0.32/0.56 thf(fact_3035_mult__less__cancel__left2,axiom,
% 0.32/0.56 ! [C: real,A: real] :
% 0.32/0.56 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
% 0.32/0.56 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.56 => ( ord_less_real @ A @ one_one_real ) )
% 0.32/0.56 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.32/0.56 => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_less_cancel_left2
% 0.32/0.56 thf(fact_3036_mult__less__cancel__left2,axiom,
% 0.32/0.56 ! [C: rat,A: rat] :
% 0.32/0.56 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
% 0.32/0.56 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.56 => ( ord_less_rat @ A @ one_one_rat ) )
% 0.32/0.56 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.32/0.56 => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_less_cancel_left2
% 0.32/0.56 thf(fact_3037_mult__less__cancel__left2,axiom,
% 0.32/0.56 ! [C: int,A: int] :
% 0.32/0.56 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
% 0.32/0.56 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.56 => ( ord_less_int @ A @ one_one_int ) )
% 0.32/0.56 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.32/0.56 => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_less_cancel_left2
% 0.32/0.56 thf(fact_3038_mult__less__cancel__right1,axiom,
% 0.32/0.56 ! [C: real,B: real] :
% 0.32/0.56 ( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
% 0.32/0.56 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.56 => ( ord_less_real @ one_one_real @ B ) )
% 0.32/0.56 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.32/0.56 => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_less_cancel_right1
% 0.32/0.56 thf(fact_3039_mult__less__cancel__right1,axiom,
% 0.32/0.56 ! [C: rat,B: rat] :
% 0.32/0.56 ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
% 0.32/0.56 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.56 => ( ord_less_rat @ one_one_rat @ B ) )
% 0.32/0.56 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.32/0.56 => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_less_cancel_right1
% 0.32/0.56 thf(fact_3040_mult__less__cancel__right1,axiom,
% 0.32/0.56 ! [C: int,B: int] :
% 0.32/0.56 ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
% 0.32/0.56 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.56 => ( ord_less_int @ one_one_int @ B ) )
% 0.32/0.56 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.32/0.56 => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_less_cancel_right1
% 0.32/0.56 thf(fact_3041_mult__less__cancel__right2,axiom,
% 0.32/0.56 ! [A: real,C: real] :
% 0.32/0.56 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
% 0.32/0.56 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.56 => ( ord_less_real @ A @ one_one_real ) )
% 0.32/0.56 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.32/0.56 => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_less_cancel_right2
% 0.32/0.56 thf(fact_3042_mult__less__cancel__right2,axiom,
% 0.32/0.56 ! [A: rat,C: rat] :
% 0.32/0.56 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
% 0.32/0.56 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.56 => ( ord_less_rat @ A @ one_one_rat ) )
% 0.32/0.56 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.32/0.56 => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_less_cancel_right2
% 0.32/0.56 thf(fact_3043_mult__less__cancel__right2,axiom,
% 0.32/0.56 ! [A: int,C: int] :
% 0.32/0.56 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
% 0.32/0.56 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.56 => ( ord_less_int @ A @ one_one_int ) )
% 0.32/0.56 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.32/0.56 => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_less_cancel_right2
% 0.32/0.56 thf(fact_3044_field__le__mult__one__interval,axiom,
% 0.32/0.56 ! [X: real,Y: real] :
% 0.32/0.56 ( ! [Z5: real] :
% 0.32/0.56 ( ( ord_less_real @ zero_zero_real @ Z5 )
% 0.32/0.56 => ( ( ord_less_real @ Z5 @ one_one_real )
% 0.32/0.56 => ( ord_less_eq_real @ ( times_times_real @ Z5 @ X ) @ Y ) ) )
% 0.32/0.56 => ( ord_less_eq_real @ X @ Y ) ) ).
% 0.32/0.56
% 0.32/0.56 % field_le_mult_one_interval
% 0.32/0.56 thf(fact_3045_field__le__mult__one__interval,axiom,
% 0.32/0.56 ! [X: rat,Y: rat] :
% 0.32/0.56 ( ! [Z5: rat] :
% 0.32/0.56 ( ( ord_less_rat @ zero_zero_rat @ Z5 )
% 0.32/0.56 => ( ( ord_less_rat @ Z5 @ one_one_rat )
% 0.32/0.56 => ( ord_less_eq_rat @ ( times_times_rat @ Z5 @ X ) @ Y ) ) )
% 0.32/0.56 => ( ord_less_eq_rat @ X @ Y ) ) ).
% 0.32/0.56
% 0.32/0.56 % field_le_mult_one_interval
% 0.32/0.56 thf(fact_3046_divide__left__mono__neg,axiom,
% 0.32/0.56 ! [A: real,B: real,C: real] :
% 0.32/0.56 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.56 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.32/0.56 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.32/0.56 => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % divide_left_mono_neg
% 0.32/0.56 thf(fact_3047_divide__left__mono__neg,axiom,
% 0.32/0.56 ! [A: rat,B: rat,C: rat] :
% 0.32/0.56 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.56 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.32/0.56 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.32/0.56 => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % divide_left_mono_neg
% 0.32/0.56 thf(fact_3048_mult__imp__le__div__pos,axiom,
% 0.32/0.56 ! [Y: real,Z: real,X: real] :
% 0.32/0.56 ( ( ord_less_real @ zero_zero_real @ Y )
% 0.32/0.56 => ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y ) @ X )
% 0.32/0.56 => ( ord_less_eq_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_imp_le_div_pos
% 0.32/0.56 thf(fact_3049_mult__imp__le__div__pos,axiom,
% 0.32/0.56 ! [Y: rat,Z: rat,X: rat] :
% 0.32/0.56 ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.32/0.56 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y ) @ X )
% 0.32/0.56 => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_imp_le_div_pos
% 0.32/0.56 thf(fact_3050_mult__imp__div__pos__le,axiom,
% 0.32/0.56 ! [Y: real,X: real,Z: real] :
% 0.32/0.56 ( ( ord_less_real @ zero_zero_real @ Y )
% 0.32/0.56 => ( ( ord_less_eq_real @ X @ ( times_times_real @ Z @ Y ) )
% 0.32/0.56 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_imp_div_pos_le
% 0.32/0.56 thf(fact_3051_mult__imp__div__pos__le,axiom,
% 0.32/0.56 ! [Y: rat,X: rat,Z: rat] :
% 0.32/0.56 ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.32/0.56 => ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z @ Y ) )
% 0.32/0.56 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_imp_div_pos_le
% 0.32/0.56 thf(fact_3052_pos__le__divide__eq,axiom,
% 0.32/0.56 ! [C: real,A: real,B: real] :
% 0.32/0.56 ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.56 => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.32/0.56 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % pos_le_divide_eq
% 0.32/0.56 thf(fact_3053_pos__le__divide__eq,axiom,
% 0.32/0.56 ! [C: rat,A: rat,B: rat] :
% 0.32/0.56 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.56 => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.56 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % pos_le_divide_eq
% 0.32/0.56 thf(fact_3054_pos__divide__le__eq,axiom,
% 0.32/0.56 ! [C: real,B: real,A: real] :
% 0.32/0.56 ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.56 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.32/0.56 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % pos_divide_le_eq
% 0.32/0.56 thf(fact_3055_pos__divide__le__eq,axiom,
% 0.32/0.56 ! [C: rat,B: rat,A: rat] :
% 0.32/0.56 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.56 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.32/0.56 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % pos_divide_le_eq
% 0.32/0.56 thf(fact_3056_neg__le__divide__eq,axiom,
% 0.32/0.56 ! [C: real,A: real,B: real] :
% 0.32/0.56 ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.56 => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.32/0.56 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % neg_le_divide_eq
% 0.32/0.56 thf(fact_3057_neg__le__divide__eq,axiom,
% 0.32/0.56 ! [C: rat,A: rat,B: rat] :
% 0.32/0.56 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.56 => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.56 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % neg_le_divide_eq
% 0.32/0.56 thf(fact_3058_neg__divide__le__eq,axiom,
% 0.32/0.56 ! [C: real,B: real,A: real] :
% 0.32/0.56 ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.56 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.32/0.56 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % neg_divide_le_eq
% 0.32/0.56 thf(fact_3059_neg__divide__le__eq,axiom,
% 0.32/0.56 ! [C: rat,B: rat,A: rat] :
% 0.32/0.56 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.56 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.32/0.56 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % neg_divide_le_eq
% 0.32/0.56 thf(fact_3060_divide__left__mono,axiom,
% 0.32/0.56 ! [B: real,A: real,C: real] :
% 0.32/0.56 ( ( ord_less_eq_real @ B @ A )
% 0.32/0.56 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.56 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.32/0.56 => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % divide_left_mono
% 0.32/0.56 thf(fact_3061_divide__left__mono,axiom,
% 0.32/0.56 ! [B: rat,A: rat,C: rat] :
% 0.32/0.56 ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.56 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.56 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.32/0.56 => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % divide_left_mono
% 0.32/0.56 thf(fact_3062_le__divide__eq,axiom,
% 0.32/0.56 ! [A: real,B: real,C: real] :
% 0.32/0.56 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.32/0.56 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.56 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.32/0.56 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.56 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.56 => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.32/0.56 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.56 => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % le_divide_eq
% 0.32/0.56 thf(fact_3063_le__divide__eq,axiom,
% 0.32/0.56 ! [A: rat,B: rat,C: rat] :
% 0.32/0.56 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.56 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.56 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.32/0.56 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.56 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.56 => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.32/0.56 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.56 => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % le_divide_eq
% 0.32/0.56 thf(fact_3064_divide__le__eq,axiom,
% 0.32/0.56 ! [B: real,C: real,A: real] :
% 0.32/0.56 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.32/0.56 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.56 => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.32/0.56 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.56 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.56 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.32/0.56 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.56 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % divide_le_eq
% 0.32/0.56 thf(fact_3065_divide__le__eq,axiom,
% 0.32/0.56 ! [B: rat,C: rat,A: rat] :
% 0.32/0.56 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.32/0.56 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.56 => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.32/0.56 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.56 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.56 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.32/0.56 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.56 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % divide_le_eq
% 0.32/0.56 thf(fact_3066_divide__le__eq__1,axiom,
% 0.32/0.56 ! [B: real,A: real] :
% 0.32/0.56 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.32/0.56 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.56 & ( ord_less_eq_real @ B @ A ) )
% 0.32/0.56 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.56 & ( ord_less_eq_real @ A @ B ) )
% 0.32/0.56 | ( A = zero_zero_real ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % divide_le_eq_1
% 0.32/0.56 thf(fact_3067_divide__le__eq__1,axiom,
% 0.32/0.56 ! [B: rat,A: rat] :
% 0.32/0.56 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.32/0.56 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.56 & ( ord_less_eq_rat @ B @ A ) )
% 0.32/0.56 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.56 & ( ord_less_eq_rat @ A @ B ) )
% 0.32/0.56 | ( A = zero_zero_rat ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % divide_le_eq_1
% 0.32/0.56 thf(fact_3068_le__divide__eq__1,axiom,
% 0.32/0.56 ! [B: real,A: real] :
% 0.32/0.56 ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.32/0.56 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.56 & ( ord_less_eq_real @ A @ B ) )
% 0.32/0.56 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.56 & ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % le_divide_eq_1
% 0.32/0.56 thf(fact_3069_le__divide__eq__1,axiom,
% 0.32/0.56 ! [B: rat,A: rat] :
% 0.32/0.56 ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.32/0.56 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.56 & ( ord_less_eq_rat @ A @ B ) )
% 0.32/0.56 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.56 & ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % le_divide_eq_1
% 0.32/0.56 thf(fact_3070_convex__bound__le,axiom,
% 0.32/0.56 ! [X: real,A: real,Y: real,U: real,V: real] :
% 0.32/0.56 ( ( ord_less_eq_real @ X @ A )
% 0.32/0.56 => ( ( ord_less_eq_real @ Y @ A )
% 0.32/0.56 => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 0.32/0.56 => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 0.32/0.56 => ( ( ( plus_plus_real @ U @ V )
% 0.32/0.56 = one_one_real )
% 0.32/0.56 => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % convex_bound_le
% 0.32/0.56 thf(fact_3071_convex__bound__le,axiom,
% 0.32/0.56 ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
% 0.32/0.56 ( ( ord_less_eq_rat @ X @ A )
% 0.32/0.56 => ( ( ord_less_eq_rat @ Y @ A )
% 0.32/0.56 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 0.32/0.56 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 0.32/0.56 => ( ( ( plus_plus_rat @ U @ V )
% 0.32/0.56 = one_one_rat )
% 0.32/0.56 => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % convex_bound_le
% 0.32/0.56 thf(fact_3072_convex__bound__le,axiom,
% 0.32/0.56 ! [X: int,A: int,Y: int,U: int,V: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ X @ A )
% 0.32/0.56 => ( ( ord_less_eq_int @ Y @ A )
% 0.32/0.56 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.32/0.56 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 0.32/0.56 => ( ( ( plus_plus_int @ U @ V )
% 0.32/0.56 = one_one_int )
% 0.32/0.56 => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % convex_bound_le
% 0.32/0.56 thf(fact_3073_less__divide__eq__numeral_I1_J,axiom,
% 0.32/0.56 ! [W: num,B: real,C: real] :
% 0.32/0.56 ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
% 0.32/0.56 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.56 => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 0.32/0.56 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.56 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.56 => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.32/0.56 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.56 => ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % less_divide_eq_numeral(1)
% 0.32/0.56 thf(fact_3074_less__divide__eq__numeral_I1_J,axiom,
% 0.32/0.56 ! [W: num,B: rat,C: rat] :
% 0.32/0.56 ( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.56 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.56 => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 0.32/0.56 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.56 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.56 => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.32/0.56 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.56 => ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % less_divide_eq_numeral(1)
% 0.32/0.56 thf(fact_3075_divide__less__eq__numeral_I1_J,axiom,
% 0.32/0.56 ! [B: real,C: real,W: num] :
% 0.32/0.56 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
% 0.32/0.56 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.56 => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.32/0.56 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.56 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.56 => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 0.32/0.56 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.56 => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % divide_less_eq_numeral(1)
% 0.32/0.56 thf(fact_3076_divide__less__eq__numeral_I1_J,axiom,
% 0.32/0.56 ! [B: rat,C: rat,W: num] :
% 0.32/0.56 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
% 0.32/0.56 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.56 => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.32/0.56 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.56 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.56 => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 0.32/0.56 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.56 => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % divide_less_eq_numeral(1)
% 0.32/0.56 thf(fact_3077_frac__le__eq,axiom,
% 0.32/0.56 ! [Y: real,Z: real,X: real,W: real] :
% 0.32/0.56 ( ( Y != zero_zero_real )
% 0.32/0.56 => ( ( Z != zero_zero_real )
% 0.32/0.56 => ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
% 0.32/0.56 = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % frac_le_eq
% 0.32/0.56 thf(fact_3078_frac__le__eq,axiom,
% 0.32/0.56 ! [Y: rat,Z: rat,X: rat,W: rat] :
% 0.32/0.56 ( ( Y != zero_zero_rat )
% 0.32/0.56 => ( ( Z != zero_zero_rat )
% 0.32/0.56 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
% 0.32/0.56 = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % frac_le_eq
% 0.32/0.56 thf(fact_3079_power__Suc__less,axiom,
% 0.32/0.56 ! [A: real,N2: nat] :
% 0.32/0.56 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.56 => ( ( ord_less_real @ A @ one_one_real )
% 0.32/0.56 => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) @ ( power_power_real @ A @ N2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_Suc_less
% 0.32/0.56 thf(fact_3080_power__Suc__less,axiom,
% 0.32/0.56 ! [A: rat,N2: nat] :
% 0.32/0.56 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.56 => ( ( ord_less_rat @ A @ one_one_rat )
% 0.32/0.56 => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) @ ( power_power_rat @ A @ N2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_Suc_less
% 0.32/0.56 thf(fact_3081_power__Suc__less,axiom,
% 0.32/0.56 ! [A: nat,N2: nat] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.56 => ( ( ord_less_nat @ A @ one_one_nat )
% 0.32/0.56 => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_Suc_less
% 0.32/0.56 thf(fact_3082_power__Suc__less,axiom,
% 0.32/0.56 ! [A: int,N2: nat] :
% 0.32/0.56 ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.56 => ( ( ord_less_int @ A @ one_one_int )
% 0.32/0.56 => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) @ ( power_power_int @ A @ N2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_Suc_less
% 0.32/0.56 thf(fact_3083_frac__less__eq,axiom,
% 0.32/0.56 ! [Y: real,Z: real,X: real,W: real] :
% 0.32/0.56 ( ( Y != zero_zero_real )
% 0.32/0.56 => ( ( Z != zero_zero_real )
% 0.32/0.56 => ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
% 0.32/0.56 = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % frac_less_eq
% 0.32/0.56 thf(fact_3084_frac__less__eq,axiom,
% 0.32/0.56 ! [Y: rat,Z: rat,X: rat,W: rat] :
% 0.32/0.56 ( ( Y != zero_zero_rat )
% 0.32/0.56 => ( ( Z != zero_zero_rat )
% 0.32/0.56 => ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
% 0.32/0.56 = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % frac_less_eq
% 0.32/0.56 thf(fact_3085_power__Suc__le__self,axiom,
% 0.32/0.56 ! [A: real,N2: nat] :
% 0.32/0.56 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.56 => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.32/0.56 => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_Suc_le_self
% 0.32/0.56 thf(fact_3086_power__Suc__le__self,axiom,
% 0.32/0.56 ! [A: rat,N2: nat] :
% 0.32/0.56 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.56 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.32/0.56 => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_Suc_le_self
% 0.32/0.56 thf(fact_3087_power__Suc__le__self,axiom,
% 0.32/0.56 ! [A: nat,N2: nat] :
% 0.32/0.56 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.56 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.32/0.56 => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_Suc_le_self
% 0.32/0.56 thf(fact_3088_power__Suc__le__self,axiom,
% 0.32/0.56 ! [A: int,N2: nat] :
% 0.32/0.56 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.56 => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.32/0.56 => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_Suc_le_self
% 0.32/0.56 thf(fact_3089_power__Suc__less__one,axiom,
% 0.32/0.56 ! [A: real,N2: nat] :
% 0.32/0.56 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.56 => ( ( ord_less_real @ A @ one_one_real )
% 0.32/0.56 => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ one_one_real ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_Suc_less_one
% 0.32/0.56 thf(fact_3090_power__Suc__less__one,axiom,
% 0.32/0.56 ! [A: rat,N2: nat] :
% 0.32/0.56 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.56 => ( ( ord_less_rat @ A @ one_one_rat )
% 0.32/0.56 => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ one_one_rat ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_Suc_less_one
% 0.32/0.56 thf(fact_3091_power__Suc__less__one,axiom,
% 0.32/0.56 ! [A: nat,N2: nat] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.56 => ( ( ord_less_nat @ A @ one_one_nat )
% 0.32/0.56 => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ one_one_nat ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_Suc_less_one
% 0.32/0.56 thf(fact_3092_power__Suc__less__one,axiom,
% 0.32/0.56 ! [A: int,N2: nat] :
% 0.32/0.56 ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.56 => ( ( ord_less_int @ A @ one_one_int )
% 0.32/0.56 => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ one_one_int ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_Suc_less_one
% 0.32/0.56 thf(fact_3093_power__strict__decreasing,axiom,
% 0.32/0.56 ! [N2: nat,N3: nat,A: real] :
% 0.32/0.56 ( ( ord_less_nat @ N2 @ N3 )
% 0.32/0.56 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.56 => ( ( ord_less_real @ A @ one_one_real )
% 0.32/0.56 => ( ord_less_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_strict_decreasing
% 0.32/0.56 thf(fact_3094_power__strict__decreasing,axiom,
% 0.32/0.56 ! [N2: nat,N3: nat,A: rat] :
% 0.32/0.56 ( ( ord_less_nat @ N2 @ N3 )
% 0.32/0.56 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.56 => ( ( ord_less_rat @ A @ one_one_rat )
% 0.32/0.56 => ( ord_less_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_strict_decreasing
% 0.32/0.56 thf(fact_3095_power__strict__decreasing,axiom,
% 0.32/0.56 ! [N2: nat,N3: nat,A: nat] :
% 0.32/0.56 ( ( ord_less_nat @ N2 @ N3 )
% 0.32/0.56 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.56 => ( ( ord_less_nat @ A @ one_one_nat )
% 0.32/0.56 => ( ord_less_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_strict_decreasing
% 0.32/0.56 thf(fact_3096_power__strict__decreasing,axiom,
% 0.32/0.56 ! [N2: nat,N3: nat,A: int] :
% 0.32/0.56 ( ( ord_less_nat @ N2 @ N3 )
% 0.32/0.56 => ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.56 => ( ( ord_less_int @ A @ one_one_int )
% 0.32/0.56 => ( ord_less_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_strict_decreasing
% 0.32/0.56 thf(fact_3097_zero__power2,axiom,
% 0.32/0.56 ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.56 = zero_zero_rat ) ).
% 0.32/0.56
% 0.32/0.56 % zero_power2
% 0.32/0.56 thf(fact_3098_zero__power2,axiom,
% 0.32/0.56 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.56 = zero_zero_nat ) ).
% 0.32/0.56
% 0.32/0.56 % zero_power2
% 0.32/0.56 thf(fact_3099_zero__power2,axiom,
% 0.32/0.56 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.56 = zero_zero_real ) ).
% 0.32/0.56
% 0.32/0.56 % zero_power2
% 0.32/0.56 thf(fact_3100_zero__power2,axiom,
% 0.32/0.56 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.56 = zero_zero_int ) ).
% 0.32/0.56
% 0.32/0.56 % zero_power2
% 0.32/0.56 thf(fact_3101_zero__power2,axiom,
% 0.32/0.56 ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.56 = zero_zero_complex ) ).
% 0.32/0.56
% 0.32/0.56 % zero_power2
% 0.32/0.56 thf(fact_3102_power__decreasing,axiom,
% 0.32/0.56 ! [N2: nat,N3: nat,A: real] :
% 0.32/0.56 ( ( ord_less_eq_nat @ N2 @ N3 )
% 0.32/0.56 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.56 => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.32/0.56 => ( ord_less_eq_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_decreasing
% 0.32/0.56 thf(fact_3103_power__decreasing,axiom,
% 0.32/0.56 ! [N2: nat,N3: nat,A: rat] :
% 0.32/0.56 ( ( ord_less_eq_nat @ N2 @ N3 )
% 0.32/0.56 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.56 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.32/0.56 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_decreasing
% 0.32/0.56 thf(fact_3104_power__decreasing,axiom,
% 0.32/0.56 ! [N2: nat,N3: nat,A: nat] :
% 0.32/0.56 ( ( ord_less_eq_nat @ N2 @ N3 )
% 0.32/0.56 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.56 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.32/0.56 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_decreasing
% 0.32/0.56 thf(fact_3105_power__decreasing,axiom,
% 0.32/0.56 ! [N2: nat,N3: nat,A: int] :
% 0.32/0.56 ( ( ord_less_eq_nat @ N2 @ N3 )
% 0.32/0.56 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.56 => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.32/0.56 => ( ord_less_eq_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_decreasing
% 0.32/0.56 thf(fact_3106_self__le__power,axiom,
% 0.32/0.56 ! [A: real,N2: nat] :
% 0.32/0.56 ( ( ord_less_eq_real @ one_one_real @ A )
% 0.32/0.56 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.56 => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % self_le_power
% 0.32/0.56 thf(fact_3107_self__le__power,axiom,
% 0.32/0.56 ! [A: rat,N2: nat] :
% 0.32/0.56 ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.32/0.56 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.56 => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % self_le_power
% 0.32/0.56 thf(fact_3108_self__le__power,axiom,
% 0.32/0.56 ! [A: nat,N2: nat] :
% 0.32/0.56 ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.32/0.56 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.56 => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % self_le_power
% 0.32/0.56 thf(fact_3109_self__le__power,axiom,
% 0.32/0.56 ! [A: int,N2: nat] :
% 0.32/0.56 ( ( ord_less_eq_int @ one_one_int @ A )
% 0.32/0.56 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.56 => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % self_le_power
% 0.32/0.56 thf(fact_3110_numeral__2__eq__2,axiom,
% 0.32/0.56 ( ( numeral_numeral_nat @ ( bit0 @ one ) )
% 0.32/0.56 = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % numeral_2_eq_2
% 0.32/0.56 thf(fact_3111_one__less__power,axiom,
% 0.32/0.56 ! [A: real,N2: nat] :
% 0.32/0.56 ( ( ord_less_real @ one_one_real @ A )
% 0.32/0.56 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.56 => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % one_less_power
% 0.32/0.56 thf(fact_3112_one__less__power,axiom,
% 0.32/0.56 ! [A: rat,N2: nat] :
% 0.32/0.56 ( ( ord_less_rat @ one_one_rat @ A )
% 0.32/0.56 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.56 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % one_less_power
% 0.32/0.56 thf(fact_3113_one__less__power,axiom,
% 0.32/0.56 ! [A: nat,N2: nat] :
% 0.32/0.56 ( ( ord_less_nat @ one_one_nat @ A )
% 0.32/0.56 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.56 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % one_less_power
% 0.32/0.56 thf(fact_3114_one__less__power,axiom,
% 0.32/0.56 ! [A: int,N2: nat] :
% 0.32/0.56 ( ( ord_less_int @ one_one_int @ A )
% 0.32/0.56 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.56 => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % one_less_power
% 0.32/0.56 thf(fact_3115_pos2,axiom,
% 0.32/0.56 ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% 0.32/0.56
% 0.32/0.56 % pos2
% 0.32/0.56 thf(fact_3116_power__diff,axiom,
% 0.32/0.56 ! [A: complex,N2: nat,M: nat] :
% 0.32/0.56 ( ( A != zero_zero_complex )
% 0.32/0.56 => ( ( ord_less_eq_nat @ N2 @ M )
% 0.32/0.56 => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N2 ) )
% 0.32/0.56 = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N2 ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_diff
% 0.32/0.56 thf(fact_3117_power__diff,axiom,
% 0.32/0.56 ! [A: real,N2: nat,M: nat] :
% 0.32/0.56 ( ( A != zero_zero_real )
% 0.32/0.56 => ( ( ord_less_eq_nat @ N2 @ M )
% 0.32/0.56 => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N2 ) )
% 0.32/0.56 = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_diff
% 0.32/0.56 thf(fact_3118_power__diff,axiom,
% 0.32/0.56 ! [A: rat,N2: nat,M: nat] :
% 0.32/0.56 ( ( A != zero_zero_rat )
% 0.32/0.56 => ( ( ord_less_eq_nat @ N2 @ M )
% 0.32/0.56 => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N2 ) )
% 0.32/0.56 = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_diff
% 0.32/0.56 thf(fact_3119_power__diff,axiom,
% 0.32/0.56 ! [A: nat,N2: nat,M: nat] :
% 0.32/0.56 ( ( A != zero_zero_nat )
% 0.32/0.56 => ( ( ord_less_eq_nat @ N2 @ M )
% 0.32/0.56 => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N2 ) )
% 0.32/0.56 = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_diff
% 0.32/0.56 thf(fact_3120_power__diff,axiom,
% 0.32/0.56 ! [A: int,N2: nat,M: nat] :
% 0.32/0.56 ( ( A != zero_zero_int )
% 0.32/0.56 => ( ( ord_less_eq_nat @ N2 @ M )
% 0.32/0.56 => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N2 ) )
% 0.32/0.56 = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_diff
% 0.32/0.56 thf(fact_3121_div__if,axiom,
% 0.32/0.56 ( divide_divide_nat
% 0.32/0.56 = ( ^ [M6: nat,N: nat] :
% 0.32/0.56 ( if_nat
% 0.32/0.56 @ ( ( ord_less_nat @ M6 @ N )
% 0.32/0.56 | ( N = zero_zero_nat ) )
% 0.32/0.56 @ zero_zero_nat
% 0.32/0.56 @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M6 @ N ) @ N ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_if
% 0.32/0.56 thf(fact_3122_Suc__pred_H,axiom,
% 0.32/0.56 ! [N2: nat] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.56 => ( N2
% 0.32/0.56 = ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % Suc_pred'
% 0.32/0.56 thf(fact_3123_Suc__diff__eq__diff__pred,axiom,
% 0.32/0.56 ! [N2: nat,M: nat] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.56 => ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
% 0.32/0.56 = ( minus_minus_nat @ M @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % Suc_diff_eq_diff_pred
% 0.32/0.56 thf(fact_3124_less__eq__div__iff__mult__less__eq,axiom,
% 0.32/0.56 ! [Q2: nat,M: nat,N2: nat] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ Q2 )
% 0.32/0.56 => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N2 @ Q2 ) )
% 0.32/0.56 = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q2 ) @ N2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % less_eq_div_iff_mult_less_eq
% 0.32/0.56 thf(fact_3125_dividend__less__times__div,axiom,
% 0.32/0.56 ! [N2: nat,M: nat] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.56 => ( ord_less_nat @ M @ ( plus_plus_nat @ N2 @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dividend_less_times_div
% 0.32/0.56 thf(fact_3126_dividend__less__div__times,axiom,
% 0.32/0.56 ! [N2: nat,M: nat] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.56 => ( ord_less_nat @ M @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dividend_less_div_times
% 0.32/0.56 thf(fact_3127_split__div,axiom,
% 0.32/0.56 ! [P: nat > $o,M: nat,N2: nat] :
% 0.32/0.56 ( ( P @ ( divide_divide_nat @ M @ N2 ) )
% 0.32/0.56 = ( ( ( N2 = zero_zero_nat )
% 0.32/0.56 => ( P @ zero_zero_nat ) )
% 0.32/0.56 & ( ( N2 != zero_zero_nat )
% 0.32/0.56 => ! [I3: nat,J3: nat] :
% 0.32/0.56 ( ( ord_less_nat @ J3 @ N2 )
% 0.32/0.56 => ( ( M
% 0.32/0.56 = ( plus_plus_nat @ ( times_times_nat @ N2 @ I3 ) @ J3 ) )
% 0.32/0.56 => ( P @ I3 ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % split_div
% 0.32/0.56 thf(fact_3128_add__eq__if,axiom,
% 0.32/0.56 ( plus_plus_nat
% 0.32/0.56 = ( ^ [M6: nat,N: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % add_eq_if
% 0.32/0.56 thf(fact_3129_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 0.32/0.56 ( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
% 0.32/0.56 = Y )
% 0.32/0.56 => ( ! [A3: $o,B2: $o] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.32/0.56 => ( Y
% 0.32/0.56 = ( ~ ( ( ( Xa2 = zero_zero_nat )
% 0.32/0.56 => A3 )
% 0.32/0.56 & ( ( Xa2 != zero_zero_nat )
% 0.32/0.56 => ( ( ( Xa2 = one_one_nat )
% 0.32/0.56 => B2 )
% 0.32/0.56 & ( Xa2 = one_one_nat ) ) ) ) ) ) )
% 0.32/0.56 => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.32/0.56 => Y )
% 0.32/0.56 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.32/0.56 ( ? [S2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
% 0.32/0.56 => ( Y
% 0.32/0.56 = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ 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 ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % VEBT_internal.naive_member.elims(1)
% 0.32/0.56 thf(fact_3130_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xa2: nat] :
% 0.32/0.56 ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
% 0.32/0.56 => ( ! [A3: $o,B2: $o] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.32/0.56 => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.32/0.56 => A3 )
% 0.32/0.56 & ( ( Xa2 != zero_zero_nat )
% 0.32/0.56 => ( ( ( Xa2 = one_one_nat )
% 0.32/0.56 => B2 )
% 0.32/0.56 & ( Xa2 = one_one_nat ) ) ) ) )
% 0.32/0.56 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.32/0.56 ( ? [S2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
% 0.32/0.56 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ 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 ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % VEBT_internal.naive_member.elims(2)
% 0.32/0.56 thf(fact_3131_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xa2: nat] :
% 0.32/0.56 ( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
% 0.32/0.56 => ( ! [A3: $o,B2: $o] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.32/0.56 => ( ( ( Xa2 = zero_zero_nat )
% 0.32/0.56 => A3 )
% 0.32/0.56 & ( ( Xa2 != zero_zero_nat )
% 0.32/0.56 => ( ( ( Xa2 = one_one_nat )
% 0.32/0.56 => B2 )
% 0.32/0.56 & ( Xa2 = one_one_nat ) ) ) ) )
% 0.32/0.56 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.32/0.56 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.32/0.56 ( ? [S2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
% 0.32/0.56 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ 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 ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % VEBT_internal.naive_member.elims(3)
% 0.32/0.56 thf(fact_3132_mult__eq__if,axiom,
% 0.32/0.56 ( times_times_nat
% 0.32/0.56 = ( ^ [M6: nat,N: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N @ ( times_times_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_eq_if
% 0.32/0.56 thf(fact_3133_vebt__member_Osimps_I4_J,axiom,
% 0.32/0.56 ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
% 0.32/0.56 ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X ) ).
% 0.32/0.56
% 0.32/0.56 % vebt_member.simps(4)
% 0.32/0.56 thf(fact_3134_vebt__delete_Osimps_I5_J,axiom,
% 0.32/0.56 ! [Mi: nat,Ma: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT,X: nat] :
% 0.32/0.56 ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) @ X )
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) ) ).
% 0.32/0.56
% 0.32/0.56 % vebt_delete.simps(5)
% 0.32/0.56 thf(fact_3135_vebt__pred_Osimps_I5_J,axiom,
% 0.32/0.56 ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
% 0.32/0.56 ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve2 ) @ Vf2 )
% 0.32/0.56 = none_nat ) ).
% 0.32/0.56
% 0.32/0.56 % vebt_pred.simps(5)
% 0.32/0.56 thf(fact_3136_vebt__succ_Osimps_I4_J,axiom,
% 0.32/0.56 ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve2: nat] :
% 0.32/0.56 ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve2 )
% 0.32/0.56 = none_nat ) ).
% 0.32/0.56
% 0.32/0.56 % vebt_succ.simps(4)
% 0.32/0.56 thf(fact_3137_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 0.32/0.56 ( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
% 0.32/0.56 = Y )
% 0.32/0.56 => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.32/0.56 => Y )
% 0.32/0.56 => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.32/0.56 => Y )
% 0.32/0.56 => ( ! [Mi2: nat,Ma2: nat] :
% 0.32/0.56 ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.32/0.56 => ( Y
% 0.32/0.56 = ( ~ ( ( Xa2 = Mi2 )
% 0.32/0.56 | ( Xa2 = Ma2 ) ) ) ) )
% 0.32/0.56 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.32/0.56 ( ? [Vc2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.32/0.56 => ( Y
% 0.32/0.56 = ( ~ ( ( Xa2 = Mi2 )
% 0.32/0.56 | ( Xa2 = Ma2 )
% 0.32/0.56 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ 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 ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) )
% 0.32/0.56 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.32/0.56 ( ? [Vd2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 0.32/0.56 => ( Y
% 0.32/0.56 = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ 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 ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % VEBT_internal.membermima.elims(1)
% 0.32/0.56 thf(fact_3138_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xa2: nat] :
% 0.32/0.56 ( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
% 0.32/0.56 => ( ! [Uu2: $o,Uv2: $o] :
% 0.32/0.56 ( X
% 0.32/0.56 != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.32/0.56 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.32/0.56 => ( ! [Mi2: nat,Ma2: nat] :
% 0.32/0.56 ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.32/0.56 => ( ( Xa2 = Mi2 )
% 0.32/0.56 | ( Xa2 = Ma2 ) ) )
% 0.32/0.56 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.32/0.56 ( ? [Vc2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.32/0.56 => ( ( Xa2 = Mi2 )
% 0.32/0.56 | ( Xa2 = Ma2 )
% 0.32/0.56 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ 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 ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 0.32/0.56 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.32/0.56 ( ? [Vd2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 0.32/0.56 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ 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 ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % VEBT_internal.membermima.elims(3)
% 0.32/0.56 thf(fact_3139_le__divide__eq__numeral_I1_J,axiom,
% 0.32/0.56 ! [W: num,B: real,C: real] :
% 0.32/0.56 ( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
% 0.32/0.56 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.56 => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 0.32/0.56 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.56 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.56 => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.32/0.56 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.56 => ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % le_divide_eq_numeral(1)
% 0.32/0.56 thf(fact_3140_le__divide__eq__numeral_I1_J,axiom,
% 0.32/0.56 ! [W: num,B: rat,C: rat] :
% 0.32/0.56 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.56 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.56 => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 0.32/0.56 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.56 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.56 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.32/0.56 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.56 => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % le_divide_eq_numeral(1)
% 0.32/0.56 thf(fact_3141_divide__le__eq__numeral_I1_J,axiom,
% 0.32/0.56 ! [B: real,C: real,W: num] :
% 0.32/0.56 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
% 0.32/0.56 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.56 => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.32/0.56 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.56 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.56 => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 0.32/0.56 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.56 => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % divide_le_eq_numeral(1)
% 0.32/0.56 thf(fact_3142_divide__le__eq__numeral_I1_J,axiom,
% 0.32/0.56 ! [B: rat,C: rat,W: num] :
% 0.32/0.56 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
% 0.32/0.56 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.56 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.32/0.56 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.56 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.56 => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 0.32/0.56 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.56 => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % divide_le_eq_numeral(1)
% 0.32/0.56 thf(fact_3143_convex__bound__lt,axiom,
% 0.32/0.56 ! [X: real,A: real,Y: real,U: real,V: real] :
% 0.32/0.56 ( ( ord_less_real @ X @ A )
% 0.32/0.56 => ( ( ord_less_real @ Y @ A )
% 0.32/0.56 => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 0.32/0.56 => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 0.32/0.56 => ( ( ( plus_plus_real @ U @ V )
% 0.32/0.56 = one_one_real )
% 0.32/0.56 => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % convex_bound_lt
% 0.32/0.56 thf(fact_3144_convex__bound__lt,axiom,
% 0.32/0.56 ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
% 0.32/0.56 ( ( ord_less_rat @ X @ A )
% 0.32/0.56 => ( ( ord_less_rat @ Y @ A )
% 0.32/0.56 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 0.32/0.56 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 0.32/0.56 => ( ( ( plus_plus_rat @ U @ V )
% 0.32/0.56 = one_one_rat )
% 0.32/0.56 => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % convex_bound_lt
% 0.32/0.56 thf(fact_3145_convex__bound__lt,axiom,
% 0.32/0.56 ! [X: int,A: int,Y: int,U: int,V: int] :
% 0.32/0.56 ( ( ord_less_int @ X @ A )
% 0.32/0.56 => ( ( ord_less_int @ Y @ A )
% 0.32/0.56 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.32/0.56 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 0.32/0.56 => ( ( ( plus_plus_int @ U @ V )
% 0.32/0.56 = one_one_int )
% 0.32/0.56 => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % convex_bound_lt
% 0.32/0.56 thf(fact_3146_half__gt__zero__iff,axiom,
% 0.32/0.56 ! [A: real] :
% 0.32/0.56 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.56 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.32/0.56
% 0.32/0.56 % half_gt_zero_iff
% 0.32/0.56 thf(fact_3147_half__gt__zero__iff,axiom,
% 0.32/0.56 ! [A: rat] :
% 0.32/0.56 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.32/0.56 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.32/0.56
% 0.32/0.56 % half_gt_zero_iff
% 0.32/0.56 thf(fact_3148_half__gt__zero,axiom,
% 0.32/0.56 ! [A: real] :
% 0.32/0.56 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.56 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % half_gt_zero
% 0.32/0.56 thf(fact_3149_half__gt__zero,axiom,
% 0.32/0.56 ! [A: rat] :
% 0.32/0.56 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.56 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % half_gt_zero
% 0.32/0.56 thf(fact_3150_scaling__mono,axiom,
% 0.32/0.56 ! [U: real,V: real,R: real,S3: real] :
% 0.32/0.56 ( ( ord_less_eq_real @ U @ V )
% 0.32/0.56 => ( ( ord_less_eq_real @ zero_zero_real @ R )
% 0.32/0.56 => ( ( ord_less_eq_real @ R @ S3 )
% 0.32/0.56 => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R @ ( minus_minus_real @ V @ U ) ) @ S3 ) ) @ V ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % scaling_mono
% 0.32/0.56 thf(fact_3151_scaling__mono,axiom,
% 0.32/0.56 ! [U: rat,V: rat,R: rat,S3: rat] :
% 0.32/0.56 ( ( ord_less_eq_rat @ U @ V )
% 0.32/0.56 => ( ( ord_less_eq_rat @ zero_zero_rat @ R )
% 0.32/0.56 => ( ( ord_less_eq_rat @ R @ S3 )
% 0.32/0.56 => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R @ ( minus_minus_rat @ V @ U ) ) @ S3 ) ) @ V ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % scaling_mono
% 0.32/0.56 thf(fact_3152_power2__le__imp__le,axiom,
% 0.32/0.56 ! [X: real,Y: real] :
% 0.32/0.56 ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.56 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.32/0.56 => ( ord_less_eq_real @ X @ Y ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power2_le_imp_le
% 0.32/0.56 thf(fact_3153_power2__le__imp__le,axiom,
% 0.32/0.56 ! [X: rat,Y: rat] :
% 0.32/0.56 ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.56 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.32/0.56 => ( ord_less_eq_rat @ X @ Y ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power2_le_imp_le
% 0.32/0.56 thf(fact_3154_power2__le__imp__le,axiom,
% 0.32/0.56 ! [X: nat,Y: nat] :
% 0.32/0.56 ( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.56 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 0.32/0.56 => ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power2_le_imp_le
% 0.32/0.56 thf(fact_3155_power2__le__imp__le,axiom,
% 0.32/0.56 ! [X: int,Y: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.56 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.32/0.56 => ( ord_less_eq_int @ X @ Y ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power2_le_imp_le
% 0.32/0.56 thf(fact_3156_power2__eq__imp__eq,axiom,
% 0.32/0.56 ! [X: real,Y: real] :
% 0.32/0.56 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.56 = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.56 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.56 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.32/0.56 => ( X = Y ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power2_eq_imp_eq
% 0.32/0.56 thf(fact_3157_power2__eq__imp__eq,axiom,
% 0.32/0.56 ! [X: rat,Y: rat] :
% 0.32/0.56 ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.56 = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.56 => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.32/0.56 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.32/0.56 => ( X = Y ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power2_eq_imp_eq
% 0.32/0.56 thf(fact_3158_power2__eq__imp__eq,axiom,
% 0.32/0.56 ! [X: nat,Y: nat] :
% 0.32/0.56 ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.56 = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.56 => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.32/0.56 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 0.32/0.56 => ( X = Y ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power2_eq_imp_eq
% 0.32/0.56 thf(fact_3159_power2__eq__imp__eq,axiom,
% 0.32/0.56 ! [X: int,Y: int] :
% 0.32/0.56 ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.56 = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.56 => ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.56 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.32/0.56 => ( X = Y ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power2_eq_imp_eq
% 0.32/0.56 thf(fact_3160_zero__le__power2,axiom,
% 0.32/0.56 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zero_le_power2
% 0.32/0.56 thf(fact_3161_zero__le__power2,axiom,
% 0.32/0.56 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zero_le_power2
% 0.32/0.56 thf(fact_3162_zero__le__power2,axiom,
% 0.32/0.56 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zero_le_power2
% 0.32/0.56 thf(fact_3163_power2__less__0,axiom,
% 0.32/0.56 ! [A: real] :
% 0.32/0.56 ~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% 0.32/0.56
% 0.32/0.56 % power2_less_0
% 0.32/0.56 thf(fact_3164_power2__less__0,axiom,
% 0.32/0.56 ! [A: rat] :
% 0.32/0.56 ~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).
% 0.32/0.56
% 0.32/0.56 % power2_less_0
% 0.32/0.56 thf(fact_3165_power2__less__0,axiom,
% 0.32/0.56 ! [A: int] :
% 0.32/0.56 ~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% 0.32/0.56
% 0.32/0.56 % power2_less_0
% 0.32/0.56 thf(fact_3166_exp__add__not__zero__imp__right,axiom,
% 0.32/0.56 ! [M: nat,N2: nat] :
% 0.32/0.56 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
% 0.32/0.56 != zero_zero_nat )
% 0.32/0.56 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.56 != zero_zero_nat ) ) ).
% 0.32/0.56
% 0.32/0.56 % exp_add_not_zero_imp_right
% 0.32/0.56 thf(fact_3167_exp__add__not__zero__imp__right,axiom,
% 0.32/0.56 ! [M: nat,N2: nat] :
% 0.32/0.56 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
% 0.32/0.56 != zero_zero_int )
% 0.32/0.56 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.56 != zero_zero_int ) ) ).
% 0.32/0.56
% 0.32/0.56 % exp_add_not_zero_imp_right
% 0.32/0.56 thf(fact_3168_exp__add__not__zero__imp__left,axiom,
% 0.32/0.56 ! [M: nat,N2: nat] :
% 0.32/0.56 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
% 0.32/0.56 != zero_zero_nat )
% 0.32/0.56 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.32/0.56 != zero_zero_nat ) ) ).
% 0.32/0.56
% 0.32/0.56 % exp_add_not_zero_imp_left
% 0.32/0.56 thf(fact_3169_exp__add__not__zero__imp__left,axiom,
% 0.32/0.56 ! [M: nat,N2: nat] :
% 0.32/0.56 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
% 0.32/0.56 != zero_zero_int )
% 0.32/0.56 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 0.32/0.56 != zero_zero_int ) ) ).
% 0.32/0.56
% 0.32/0.56 % exp_add_not_zero_imp_left
% 0.32/0.56 thf(fact_3170_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 0.32/0.56 ! [N2: nat,M: nat] :
% 0.32/0.56 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.56 != zero_zero_nat )
% 0.32/0.56 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) )
% 0.32/0.56 != zero_zero_nat ) ) ).
% 0.32/0.56
% 0.32/0.56 % exp_not_zero_imp_exp_diff_not_zero
% 0.32/0.56 thf(fact_3171_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 0.32/0.56 ! [N2: nat,M: nat] :
% 0.32/0.56 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.56 != zero_zero_int )
% 0.32/0.56 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) )
% 0.32/0.56 != zero_zero_int ) ) ).
% 0.32/0.56
% 0.32/0.56 % exp_not_zero_imp_exp_diff_not_zero
% 0.32/0.56 thf(fact_3172_power__diff__power__eq,axiom,
% 0.32/0.56 ! [A: nat,N2: nat,M: nat] :
% 0.32/0.56 ( ( A != zero_zero_nat )
% 0.32/0.56 => ( ( ( ord_less_eq_nat @ N2 @ M )
% 0.32/0.56 => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
% 0.32/0.56 = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N2 ) ) ) )
% 0.32/0.56 & ( ~ ( ord_less_eq_nat @ N2 @ M )
% 0.32/0.56 => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
% 0.32/0.56 = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_diff_power_eq
% 0.32/0.56 thf(fact_3173_power__diff__power__eq,axiom,
% 0.32/0.56 ! [A: int,N2: nat,M: nat] :
% 0.32/0.56 ( ( A != zero_zero_int )
% 0.32/0.56 => ( ( ( ord_less_eq_nat @ N2 @ M )
% 0.32/0.56 => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
% 0.32/0.56 = ( power_power_int @ A @ ( minus_minus_nat @ M @ N2 ) ) ) )
% 0.32/0.56 & ( ~ ( ord_less_eq_nat @ N2 @ M )
% 0.32/0.56 => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
% 0.32/0.56 = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_diff_power_eq
% 0.32/0.56 thf(fact_3174_less__2__cases__iff,axiom,
% 0.32/0.56 ! [N2: nat] :
% 0.32/0.56 ( ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.56 = ( ( N2 = zero_zero_nat )
% 0.32/0.56 | ( N2
% 0.32/0.56 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % less_2_cases_iff
% 0.32/0.56 thf(fact_3175_less__2__cases,axiom,
% 0.32/0.56 ! [N2: nat] :
% 0.32/0.56 ( ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.56 => ( ( N2 = zero_zero_nat )
% 0.32/0.56 | ( N2
% 0.32/0.56 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % less_2_cases
% 0.32/0.56 thf(fact_3176_nat__induct2,axiom,
% 0.32/0.56 ! [P: nat > $o,N2: nat] :
% 0.32/0.56 ( ( P @ zero_zero_nat )
% 0.32/0.56 => ( ( P @ one_one_nat )
% 0.32/0.56 => ( ! [N4: nat] :
% 0.32/0.56 ( ( P @ N4 )
% 0.32/0.56 => ( P @ ( plus_plus_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.56 => ( P @ N2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % nat_induct2
% 0.32/0.56 thf(fact_3177_power__eq__if,axiom,
% 0.32/0.56 ( power_power_complex
% 0.32/0.56 = ( ^ [P5: complex,M6: nat] : ( if_complex @ ( M6 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P5 @ ( power_power_complex @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_eq_if
% 0.32/0.56 thf(fact_3178_power__eq__if,axiom,
% 0.32/0.56 ( power_power_real
% 0.32/0.56 = ( ^ [P5: real,M6: nat] : ( if_real @ ( M6 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P5 @ ( power_power_real @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_eq_if
% 0.32/0.56 thf(fact_3179_power__eq__if,axiom,
% 0.32/0.56 ( power_power_rat
% 0.32/0.56 = ( ^ [P5: rat,M6: nat] : ( if_rat @ ( M6 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P5 @ ( power_power_rat @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_eq_if
% 0.32/0.56 thf(fact_3180_power__eq__if,axiom,
% 0.32/0.56 ( power_power_nat
% 0.32/0.56 = ( ^ [P5: nat,M6: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P5 @ ( power_power_nat @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_eq_if
% 0.32/0.56 thf(fact_3181_power__eq__if,axiom,
% 0.32/0.56 ( power_power_int
% 0.32/0.56 = ( ^ [P5: int,M6: nat] : ( if_int @ ( M6 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P5 @ ( power_power_int @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_eq_if
% 0.32/0.56 thf(fact_3182_power__minus__mult,axiom,
% 0.32/0.56 ! [N2: nat,A: complex] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.56 => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
% 0.32/0.56 = ( power_power_complex @ A @ N2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_minus_mult
% 0.32/0.56 thf(fact_3183_power__minus__mult,axiom,
% 0.32/0.56 ! [N2: nat,A: real] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.56 => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
% 0.32/0.56 = ( power_power_real @ A @ N2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_minus_mult
% 0.32/0.56 thf(fact_3184_power__minus__mult,axiom,
% 0.32/0.56 ! [N2: nat,A: rat] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.56 => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
% 0.32/0.56 = ( power_power_rat @ A @ N2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_minus_mult
% 0.32/0.56 thf(fact_3185_power__minus__mult,axiom,
% 0.32/0.56 ! [N2: nat,A: nat] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.56 => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
% 0.32/0.56 = ( power_power_nat @ A @ N2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_minus_mult
% 0.32/0.56 thf(fact_3186_power__minus__mult,axiom,
% 0.32/0.56 ! [N2: nat,A: int] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.56 => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
% 0.32/0.56 = ( power_power_int @ A @ N2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power_minus_mult
% 0.32/0.56 thf(fact_3187_split__div_H,axiom,
% 0.32/0.56 ! [P: nat > $o,M: nat,N2: nat] :
% 0.32/0.56 ( ( P @ ( divide_divide_nat @ M @ N2 ) )
% 0.32/0.56 = ( ( ( N2 = zero_zero_nat )
% 0.32/0.56 & ( P @ zero_zero_nat ) )
% 0.32/0.56 | ? [Q4: nat] :
% 0.32/0.56 ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q4 ) @ M )
% 0.32/0.56 & ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q4 ) ) )
% 0.32/0.56 & ( P @ Q4 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % split_div'
% 0.32/0.56 thf(fact_3188_le__div__geq,axiom,
% 0.32/0.56 ! [N2: nat,M: nat] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.56 => ( ( ord_less_eq_nat @ N2 @ M )
% 0.32/0.56 => ( ( divide_divide_nat @ M @ N2 )
% 0.32/0.56 = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % le_div_geq
% 0.32/0.56 thf(fact_3189_div__exp__mod__exp__eq,axiom,
% 0.32/0.56 ! [A: nat,N2: nat,M: nat] :
% 0.32/0.56 ( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.32/0.56 = ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_exp_mod_exp_eq
% 0.32/0.56 thf(fact_3190_div__exp__mod__exp__eq,axiom,
% 0.32/0.56 ! [A: int,N2: nat,M: nat] :
% 0.32/0.56 ( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 0.32/0.56 = ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_exp_mod_exp_eq
% 0.32/0.56 thf(fact_3191_div__exp__mod__exp__eq,axiom,
% 0.32/0.56 ! [A: code_integer,N2: nat,M: nat] :
% 0.32/0.56 ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.32/0.56 = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_exp_mod_exp_eq
% 0.32/0.56 thf(fact_3192_vebt__delete_Osimps_I6_J,axiom,
% 0.32/0.56 ! [Mi: nat,Ma: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT,X: nat] :
% 0.32/0.56 ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) @ X )
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) ) ).
% 0.32/0.56
% 0.32/0.56 % vebt_delete.simps(6)
% 0.32/0.56 thf(fact_3193_vebt__pred_Osimps_I6_J,axiom,
% 0.32/0.56 ! [V: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
% 0.32/0.56 ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 )
% 0.32/0.56 = none_nat ) ).
% 0.32/0.56
% 0.32/0.56 % vebt_pred.simps(6)
% 0.32/0.56 thf(fact_3194_vebt__succ_Osimps_I5_J,axiom,
% 0.32/0.56 ! [V: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
% 0.32/0.56 ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 )
% 0.32/0.56 = none_nat ) ).
% 0.32/0.56
% 0.32/0.56 % vebt_succ.simps(5)
% 0.32/0.56 thf(fact_3195_power2__less__imp__less,axiom,
% 0.32/0.56 ! [X: real,Y: real] :
% 0.32/0.56 ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.56 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.32/0.56 => ( ord_less_real @ X @ Y ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power2_less_imp_less
% 0.32/0.56 thf(fact_3196_power2__less__imp__less,axiom,
% 0.32/0.56 ! [X: rat,Y: rat] :
% 0.32/0.56 ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.56 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.32/0.56 => ( ord_less_rat @ X @ Y ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power2_less_imp_less
% 0.32/0.56 thf(fact_3197_power2__less__imp__less,axiom,
% 0.32/0.56 ! [X: nat,Y: nat] :
% 0.32/0.56 ( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.56 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 0.32/0.56 => ( ord_less_nat @ X @ Y ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power2_less_imp_less
% 0.32/0.56 thf(fact_3198_power2__less__imp__less,axiom,
% 0.32/0.56 ! [X: int,Y: int] :
% 0.32/0.56 ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.56 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.32/0.56 => ( ord_less_int @ X @ Y ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % power2_less_imp_less
% 0.32/0.56 thf(fact_3199_sum__power2__le__zero__iff,axiom,
% 0.32/0.56 ! [X: real,Y: real] :
% 0.32/0.56 ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
% 0.32/0.56 = ( ( X = zero_zero_real )
% 0.32/0.56 & ( Y = zero_zero_real ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum_power2_le_zero_iff
% 0.32/0.56 thf(fact_3200_sum__power2__le__zero__iff,axiom,
% 0.32/0.56 ! [X: rat,Y: rat] :
% 0.32/0.56 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
% 0.32/0.56 = ( ( X = zero_zero_rat )
% 0.32/0.56 & ( Y = zero_zero_rat ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum_power2_le_zero_iff
% 0.32/0.56 thf(fact_3201_sum__power2__le__zero__iff,axiom,
% 0.32/0.56 ! [X: int,Y: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
% 0.32/0.56 = ( ( X = zero_zero_int )
% 0.32/0.56 & ( Y = zero_zero_int ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum_power2_le_zero_iff
% 0.32/0.56 thf(fact_3202_sum__power2__ge__zero,axiom,
% 0.32/0.56 ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum_power2_ge_zero
% 0.32/0.56 thf(fact_3203_sum__power2__ge__zero,axiom,
% 0.32/0.56 ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum_power2_ge_zero
% 0.32/0.56 thf(fact_3204_sum__power2__ge__zero,axiom,
% 0.32/0.56 ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum_power2_ge_zero
% 0.32/0.56 thf(fact_3205_sum__power2__gt__zero__iff,axiom,
% 0.32/0.56 ! [X: real,Y: real] :
% 0.32/0.56 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.56 = ( ( X != zero_zero_real )
% 0.32/0.56 | ( Y != zero_zero_real ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum_power2_gt_zero_iff
% 0.32/0.56 thf(fact_3206_sum__power2__gt__zero__iff,axiom,
% 0.32/0.56 ! [X: rat,Y: rat] :
% 0.32/0.56 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.56 = ( ( X != zero_zero_rat )
% 0.32/0.56 | ( Y != zero_zero_rat ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum_power2_gt_zero_iff
% 0.32/0.56 thf(fact_3207_sum__power2__gt__zero__iff,axiom,
% 0.32/0.56 ! [X: int,Y: int] :
% 0.32/0.56 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.56 = ( ( X != zero_zero_int )
% 0.32/0.56 | ( Y != zero_zero_int ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum_power2_gt_zero_iff
% 0.32/0.56 thf(fact_3208_not__sum__power2__lt__zero,axiom,
% 0.32/0.56 ! [X: real,Y: real] :
% 0.32/0.56 ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% 0.32/0.56
% 0.32/0.56 % not_sum_power2_lt_zero
% 0.32/0.56 thf(fact_3209_not__sum__power2__lt__zero,axiom,
% 0.32/0.56 ! [X: rat,Y: rat] :
% 0.32/0.56 ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).
% 0.32/0.56
% 0.32/0.56 % not_sum_power2_lt_zero
% 0.32/0.56 thf(fact_3210_not__sum__power2__lt__zero,axiom,
% 0.32/0.56 ! [X: int,Y: int] :
% 0.32/0.56 ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% 0.32/0.56
% 0.32/0.56 % not_sum_power2_lt_zero
% 0.32/0.56 thf(fact_3211_zero__le__even__power_H,axiom,
% 0.32/0.56 ! [A: real,N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zero_le_even_power'
% 0.32/0.56 thf(fact_3212_zero__le__even__power_H,axiom,
% 0.32/0.56 ! [A: rat,N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zero_le_even_power'
% 0.32/0.56 thf(fact_3213_zero__le__even__power_H,axiom,
% 0.32/0.56 ! [A: int,N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zero_le_even_power'
% 0.32/0.56 thf(fact_3214_nat__bit__induct,axiom,
% 0.32/0.56 ! [P: nat > $o,N2: nat] :
% 0.32/0.56 ( ( P @ zero_zero_nat )
% 0.32/0.56 => ( ! [N4: nat] :
% 0.32/0.56 ( ( P @ N4 )
% 0.32/0.56 => ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 0.32/0.56 => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
% 0.32/0.56 => ( ! [N4: nat] :
% 0.32/0.56 ( ( P @ N4 )
% 0.32/0.56 => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
% 0.32/0.56 => ( P @ N2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % nat_bit_induct
% 0.32/0.56 thf(fact_3215_Suc__n__div__2__gt__zero,axiom,
% 0.32/0.56 ! [N2: nat] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.56 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % Suc_n_div_2_gt_zero
% 0.32/0.56 thf(fact_3216_div__2__gt__zero,axiom,
% 0.32/0.56 ! [N2: nat] :
% 0.32/0.56 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 0.32/0.56 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_2_gt_zero
% 0.32/0.56 thf(fact_3217_mult__exp__mod__exp__eq,axiom,
% 0.32/0.56 ! [M: nat,N2: nat,A: nat] :
% 0.32/0.56 ( ( ord_less_eq_nat @ M @ N2 )
% 0.32/0.56 => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.56 = ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_exp_mod_exp_eq
% 0.32/0.56 thf(fact_3218_mult__exp__mod__exp__eq,axiom,
% 0.32/0.56 ! [M: nat,N2: nat,A: int] :
% 0.32/0.56 ( ( ord_less_eq_nat @ M @ N2 )
% 0.32/0.56 => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.56 = ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_exp_mod_exp_eq
% 0.32/0.56 thf(fact_3219_mult__exp__mod__exp__eq,axiom,
% 0.32/0.56 ! [M: nat,N2: nat,A: code_integer] :
% 0.32/0.56 ( ( ord_less_eq_nat @ M @ N2 )
% 0.32/0.56 => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.56 = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_exp_mod_exp_eq
% 0.32/0.56 thf(fact_3220_odd__0__le__power__imp__0__le,axiom,
% 0.32/0.56 ! [A: real,N2: nat] :
% 0.32/0.56 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 0.32/0.56 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.32/0.56
% 0.32/0.56 % odd_0_le_power_imp_0_le
% 0.32/0.56 thf(fact_3221_odd__0__le__power__imp__0__le,axiom,
% 0.32/0.56 ! [A: rat,N2: nat] :
% 0.32/0.56 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 0.32/0.56 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.32/0.56
% 0.32/0.56 % odd_0_le_power_imp_0_le
% 0.32/0.56 thf(fact_3222_odd__0__le__power__imp__0__le,axiom,
% 0.32/0.56 ! [A: int,N2: nat] :
% 0.32/0.56 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 0.32/0.56 => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.32/0.56
% 0.32/0.56 % odd_0_le_power_imp_0_le
% 0.32/0.56 thf(fact_3223_odd__power__less__zero,axiom,
% 0.32/0.56 ! [A: real,N2: nat] :
% 0.32/0.56 ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.56 => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_real ) ) ).
% 0.32/0.56
% 0.32/0.56 % odd_power_less_zero
% 0.32/0.56 thf(fact_3224_odd__power__less__zero,axiom,
% 0.32/0.56 ! [A: rat,N2: nat] :
% 0.32/0.56 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.56 => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_rat ) ) ).
% 0.32/0.56
% 0.32/0.56 % odd_power_less_zero
% 0.32/0.56 thf(fact_3225_odd__power__less__zero,axiom,
% 0.32/0.56 ! [A: int,N2: nat] :
% 0.32/0.56 ( ( ord_less_int @ A @ zero_zero_int )
% 0.32/0.56 => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_int ) ) ).
% 0.32/0.56
% 0.32/0.56 % odd_power_less_zero
% 0.32/0.56 thf(fact_3226_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
% 0.32/0.56 ! [X: nat,N2: nat,M: nat] :
% 0.32/0.56 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
% 0.32/0.56 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.56 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.56 => ( ord_less_nat @ ( vEBT_VEBT_high @ X @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % VEBT_internal.exp_split_high_low(1)
% 0.32/0.56 thf(fact_3227_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
% 0.32/0.56 ! [X: nat,N2: nat,M: nat] :
% 0.32/0.56 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
% 0.32/0.56 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.56 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.56 => ( ord_less_nat @ ( vEBT_VEBT_low @ X @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % VEBT_internal.exp_split_high_low(2)
% 0.32/0.56 thf(fact_3228_vebt__member_Oelims_I2_J,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xa2: nat] :
% 0.32/0.56 ( ( vEBT_vebt_member @ X @ Xa2 )
% 0.32/0.56 => ( ! [A3: $o,B2: $o] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.32/0.56 => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.32/0.56 => A3 )
% 0.32/0.56 & ( ( Xa2 != zero_zero_nat )
% 0.32/0.56 => ( ( ( Xa2 = one_one_nat )
% 0.32/0.56 => B2 )
% 0.32/0.56 & ( Xa2 = one_one_nat ) ) ) ) )
% 0.32/0.56 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
% 0.32/0.56 ( ? [Summary2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.32/0.56 => ~ ( ( Xa2 != Mi2 )
% 0.32/0.56 => ( ( Xa2 != Ma2 )
% 0.32/0.56 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.32/0.56 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.32/0.56 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.32/0.56 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.32/0.56 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % vebt_member.elims(2)
% 0.32/0.56 thf(fact_3229_vebt__member_Oelims_I3_J,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xa2: nat] :
% 0.32/0.56 ( ~ ( vEBT_vebt_member @ X @ Xa2 )
% 0.32/0.56 => ( ! [A3: $o,B2: $o] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.32/0.56 => ( ( ( Xa2 = zero_zero_nat )
% 0.32/0.56 => A3 )
% 0.32/0.56 & ( ( Xa2 != zero_zero_nat )
% 0.32/0.56 => ( ( ( Xa2 = one_one_nat )
% 0.32/0.56 => B2 )
% 0.32/0.56 & ( Xa2 = one_one_nat ) ) ) ) )
% 0.32/0.56 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.32/0.56 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.32/0.56 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.32/0.56 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
% 0.32/0.56 ( ? [Summary2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.32/0.56 => ( ( Xa2 != Mi2 )
% 0.32/0.56 => ( ( Xa2 != Ma2 )
% 0.32/0.56 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.32/0.56 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.32/0.56 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.32/0.56 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.32/0.56 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % vebt_member.elims(3)
% 0.32/0.56 thf(fact_3230_vebt__member_Oelims_I1_J,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 0.32/0.56 ( ( ( vEBT_vebt_member @ X @ Xa2 )
% 0.32/0.56 = Y )
% 0.32/0.56 => ( ! [A3: $o,B2: $o] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.32/0.56 => ( Y
% 0.32/0.56 = ( ~ ( ( ( Xa2 = zero_zero_nat )
% 0.32/0.56 => A3 )
% 0.32/0.56 & ( ( Xa2 != zero_zero_nat )
% 0.32/0.56 => ( ( ( Xa2 = one_one_nat )
% 0.32/0.56 => B2 )
% 0.32/0.56 & ( Xa2 = one_one_nat ) ) ) ) ) ) )
% 0.32/0.56 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.32/0.56 => Y )
% 0.32/0.56 => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.32/0.56 => Y )
% 0.32/0.56 => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.32/0.56 => Y )
% 0.32/0.56 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
% 0.32/0.56 ( ? [Summary2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.32/0.56 => ( Y
% 0.32/0.56 = ( ~ ( ( Xa2 != Mi2 )
% 0.32/0.56 => ( ( Xa2 != Ma2 )
% 0.32/0.56 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.32/0.56 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.32/0.56 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.32/0.56 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.32/0.56 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % vebt_member.elims(1)
% 0.32/0.56 thf(fact_3231_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xa2: nat] :
% 0.32/0.56 ( ( vEBT_VEBT_membermima @ X @ Xa2 )
% 0.32/0.56 => ( ! [Mi2: nat,Ma2: nat] :
% 0.32/0.56 ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.32/0.56 => ~ ( ( Xa2 = Mi2 )
% 0.32/0.56 | ( Xa2 = Ma2 ) ) )
% 0.32/0.56 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.32/0.56 ( ? [Vc2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.32/0.56 => ~ ( ( Xa2 = Mi2 )
% 0.32/0.56 | ( Xa2 = Ma2 )
% 0.32/0.56 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ 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 ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 0.32/0.56 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.32/0.56 ( ? [Vd2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 0.32/0.56 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ 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 ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % VEBT_internal.membermima.elims(2)
% 0.32/0.56 thf(fact_3232_arith__geo__mean,axiom,
% 0.32/0.56 ! [U: real,X: real,Y: real] :
% 0.32/0.56 ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.56 = ( times_times_real @ X @ Y ) )
% 0.32/0.56 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.56 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.32/0.56 => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % arith_geo_mean
% 0.32/0.56 thf(fact_3233_arith__geo__mean,axiom,
% 0.32/0.56 ! [U: rat,X: rat,Y: rat] :
% 0.32/0.56 ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.56 = ( times_times_rat @ X @ Y ) )
% 0.32/0.56 => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.32/0.56 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.32/0.56 => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % arith_geo_mean
% 0.32/0.56 thf(fact_3234_invar__vebt_Osimps,axiom,
% 0.32/0.56 ( vEBT_invar_vebt
% 0.32/0.56 = ( ^ [A1: vEBT_VEBT,A22: nat] :
% 0.32/0.56 ( ( ? [A5: $o,B5: $o] :
% 0.32/0.56 ( A1
% 0.32/0.56 = ( vEBT_Leaf @ A5 @ B5 ) )
% 0.32/0.56 & ( A22
% 0.32/0.56 = ( suc @ zero_zero_nat ) ) )
% 0.32/0.56 | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT] :
% 0.32/0.56 ( ( A1
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
% 0.32/0.56 & ! [X2: vEBT_VEBT] :
% 0.32/0.56 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.32/0.56 => ( vEBT_invar_vebt @ X2 @ N ) )
% 0.32/0.56 & ( vEBT_invar_vebt @ Summary3 @ N )
% 0.32/0.56 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.32/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.56 & ( A22
% 0.32/0.56 = ( plus_plus_nat @ N @ N ) )
% 0.32/0.56 & ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X3 )
% 0.32/0.56 & ! [X2: vEBT_VEBT] :
% 0.32/0.56 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.32/0.56 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 0.32/0.56 | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT] :
% 0.32/0.56 ( ( A1
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
% 0.32/0.56 & ! [X2: vEBT_VEBT] :
% 0.32/0.56 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.32/0.56 => ( vEBT_invar_vebt @ X2 @ N ) )
% 0.32/0.56 & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N ) )
% 0.32/0.56 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.32/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
% 0.32/0.56 & ( A22
% 0.32/0.56 = ( plus_plus_nat @ N @ ( suc @ N ) ) )
% 0.32/0.56 & ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X3 )
% 0.32/0.56 & ! [X2: vEBT_VEBT] :
% 0.32/0.56 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.32/0.56 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 0.32/0.56 | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 0.32/0.56 ( ( A1
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
% 0.32/0.56 & ! [X2: vEBT_VEBT] :
% 0.32/0.56 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.32/0.56 => ( vEBT_invar_vebt @ X2 @ N ) )
% 0.32/0.56 & ( vEBT_invar_vebt @ Summary3 @ N )
% 0.32/0.56 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.32/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.56 & ( A22
% 0.32/0.56 = ( plus_plus_nat @ N @ N ) )
% 0.32/0.56 & ! [I3: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.56 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X3 ) )
% 0.32/0.56 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
% 0.32/0.56 & ( ( Mi3 = Ma3 )
% 0.32/0.56 => ! [X2: vEBT_VEBT] :
% 0.32/0.56 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.32/0.56 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 0.32/0.56 & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.32/0.56 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
% 0.32/0.56 & ( ( Mi3 != Ma3 )
% 0.32/0.56 => ! [I3: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.56 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N )
% 0.32/0.56 = I3 )
% 0.32/0.56 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) )
% 0.32/0.56 & ! [X2: nat] :
% 0.32/0.56 ( ( ( ( vEBT_VEBT_high @ X2 @ N )
% 0.32/0.56 = I3 )
% 0.32/0.56 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X2 @ N ) ) )
% 0.32/0.56 => ( ( ord_less_nat @ Mi3 @ X2 )
% 0.32/0.56 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) )
% 0.32/0.56 | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 0.32/0.56 ( ( A1
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
% 0.32/0.56 & ! [X2: vEBT_VEBT] :
% 0.32/0.56 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.32/0.56 => ( vEBT_invar_vebt @ X2 @ N ) )
% 0.32/0.56 & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N ) )
% 0.32/0.56 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.32/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
% 0.32/0.56 & ( A22
% 0.32/0.56 = ( plus_plus_nat @ N @ ( suc @ N ) ) )
% 0.32/0.56 & ! [I3: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
% 0.32/0.56 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X3 ) )
% 0.32/0.56 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
% 0.32/0.56 & ( ( Mi3 = Ma3 )
% 0.32/0.56 => ! [X2: vEBT_VEBT] :
% 0.32/0.56 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.32/0.56 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 0.32/0.56 & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.32/0.56 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
% 0.32/0.56 & ( ( Mi3 != Ma3 )
% 0.32/0.56 => ! [I3: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
% 0.32/0.56 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N )
% 0.32/0.56 = I3 )
% 0.32/0.56 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) )
% 0.32/0.56 & ! [X2: nat] :
% 0.32/0.56 ( ( ( ( vEBT_VEBT_high @ X2 @ N )
% 0.32/0.56 = I3 )
% 0.32/0.56 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X2 @ N ) ) )
% 0.32/0.56 => ( ( ord_less_nat @ Mi3 @ X2 )
% 0.32/0.56 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % invar_vebt.simps
% 0.32/0.56 thf(fact_3235_invar__vebt_Ocases,axiom,
% 0.32/0.56 ! [A12: vEBT_VEBT,A23: nat] :
% 0.32/0.56 ( ( vEBT_invar_vebt @ A12 @ A23 )
% 0.32/0.56 => ( ( ? [A3: $o,B2: $o] :
% 0.32/0.56 ( A12
% 0.32/0.56 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.32/0.56 => ( A23
% 0.32/0.56 != ( suc @ zero_zero_nat ) ) )
% 0.32/0.56 => ( ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat] :
% 0.32/0.56 ( ( A12
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.32/0.56 => ( ( A23 = Deg2 )
% 0.32/0.56 => ( ! [X5: vEBT_VEBT] :
% 0.32/0.56 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_invar_vebt @ X5 @ N4 ) )
% 0.32/0.56 => ( ( vEBT_invar_vebt @ Summary2 @ M2 )
% 0.32/0.56 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.32/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 0.32/0.56 => ( ( M2 = N4 )
% 0.32/0.56 => ( ( Deg2
% 0.32/0.56 = ( plus_plus_nat @ N4 @ M2 ) )
% 0.32/0.56 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
% 0.32/0.56 => ~ ! [X5: vEBT_VEBT] :
% 0.32/0.56 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.56 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
% 0.32/0.56 => ( ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat] :
% 0.32/0.56 ( ( A12
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.32/0.56 => ( ( A23 = Deg2 )
% 0.32/0.56 => ( ! [X5: vEBT_VEBT] :
% 0.32/0.56 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_invar_vebt @ X5 @ N4 ) )
% 0.32/0.56 => ( ( vEBT_invar_vebt @ Summary2 @ M2 )
% 0.32/0.56 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.32/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 0.32/0.56 => ( ( M2
% 0.32/0.56 = ( suc @ N4 ) )
% 0.32/0.56 => ( ( Deg2
% 0.32/0.56 = ( plus_plus_nat @ N4 @ M2 ) )
% 0.32/0.56 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
% 0.32/0.56 => ~ ! [X5: vEBT_VEBT] :
% 0.32/0.56 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.56 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
% 0.32/0.56 => ( ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 0.32/0.56 ( ( A12
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.32/0.56 => ( ( A23 = Deg2 )
% 0.32/0.56 => ( ! [X5: vEBT_VEBT] :
% 0.32/0.56 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_invar_vebt @ X5 @ N4 ) )
% 0.32/0.56 => ( ( vEBT_invar_vebt @ Summary2 @ M2 )
% 0.32/0.56 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.32/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 0.32/0.56 => ( ( M2 = N4 )
% 0.32/0.56 => ( ( Deg2
% 0.32/0.56 = ( plus_plus_nat @ N4 @ M2 ) )
% 0.32/0.56 => ( ! [I: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 0.32/0.56 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ X3 ) )
% 0.32/0.56 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
% 0.32/0.56 => ( ( ( Mi2 = Ma2 )
% 0.32/0.56 => ! [X5: vEBT_VEBT] :
% 0.32/0.56 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.56 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
% 0.32/0.56 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 0.32/0.56 => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.32/0.56 => ~ ( ( Mi2 != Ma2 )
% 0.32/0.56 => ! [I: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 0.32/0.56 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N4 )
% 0.32/0.56 = I )
% 0.32/0.56 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ ( vEBT_VEBT_low @ Ma2 @ N4 ) ) )
% 0.32/0.56 & ! [X5: nat] :
% 0.32/0.56 ( ( ( ( vEBT_VEBT_high @ X5 @ N4 )
% 0.32/0.56 = I )
% 0.32/0.56 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ ( vEBT_VEBT_low @ X5 @ N4 ) ) )
% 0.32/0.56 => ( ( ord_less_nat @ Mi2 @ X5 )
% 0.32/0.56 & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 0.32/0.56 => ~ ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 0.32/0.56 ( ( A12
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.32/0.56 => ( ( A23 = Deg2 )
% 0.32/0.56 => ( ! [X5: vEBT_VEBT] :
% 0.32/0.56 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_invar_vebt @ X5 @ N4 ) )
% 0.32/0.56 => ( ( vEBT_invar_vebt @ Summary2 @ M2 )
% 0.32/0.56 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.32/0.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 0.32/0.56 => ( ( M2
% 0.32/0.56 = ( suc @ N4 ) )
% 0.32/0.56 => ( ( Deg2
% 0.32/0.56 = ( plus_plus_nat @ N4 @ M2 ) )
% 0.32/0.56 => ( ! [I: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 0.32/0.56 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ X3 ) )
% 0.32/0.56 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
% 0.32/0.56 => ( ( ( Mi2 = Ma2 )
% 0.32/0.56 => ! [X5: vEBT_VEBT] :
% 0.32/0.56 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.56 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
% 0.32/0.56 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 0.32/0.56 => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.32/0.56 => ~ ( ( Mi2 != Ma2 )
% 0.32/0.56 => ! [I: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 0.32/0.56 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N4 )
% 0.32/0.56 = I )
% 0.32/0.56 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ ( vEBT_VEBT_low @ Ma2 @ N4 ) ) )
% 0.32/0.56 & ! [X5: nat] :
% 0.32/0.56 ( ( ( ( vEBT_VEBT_high @ X5 @ N4 )
% 0.32/0.56 = I )
% 0.32/0.56 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ ( vEBT_VEBT_low @ X5 @ N4 ) ) )
% 0.32/0.56 => ( ( ord_less_nat @ Mi2 @ X5 )
% 0.32/0.56 & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % invar_vebt.cases
% 0.32/0.56 thf(fact_3236_vebt__insert_Oelims,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
% 0.32/0.56 ( ( ( vEBT_vebt_insert @ X @ Xa2 )
% 0.32/0.56 = Y )
% 0.32/0.56 => ( ! [A3: $o,B2: $o] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.32/0.56 => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.32/0.56 => ( Y
% 0.32/0.56 = ( vEBT_Leaf @ $true @ B2 ) ) )
% 0.32/0.56 & ( ( Xa2 != zero_zero_nat )
% 0.32/0.56 => ( ( ( Xa2 = one_one_nat )
% 0.32/0.56 => ( Y
% 0.32/0.56 = ( vEBT_Leaf @ A3 @ $true ) ) )
% 0.32/0.56 & ( ( Xa2 != one_one_nat )
% 0.32/0.56 => ( Y
% 0.32/0.56 = ( vEBT_Leaf @ A3 @ B2 ) ) ) ) ) ) )
% 0.32/0.56 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) )
% 0.32/0.56 => ( Y
% 0.32/0.56 != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) ) )
% 0.32/0.56 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) )
% 0.32/0.56 => ( Y
% 0.32/0.56 != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) ) )
% 0.32/0.56 => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
% 0.32/0.56 => ( Y
% 0.32/0.56 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) ) )
% 0.32/0.56 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.32/0.56 => ( Y
% 0.32/0.56 != ( if_VEBT_VEBT
% 0.32/0.56 @ ( ( 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 @ TreeList3 ) )
% 0.32/0.56 & ~ ( ( Xa2 = Mi2 )
% 0.32/0.56 | ( Xa2 = Ma2 ) ) )
% 0.32/0.56 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 ) )
% 0.32/0.56 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % vebt_insert.elims
% 0.32/0.56 thf(fact_3237_verit__le__mono__div,axiom,
% 0.32/0.56 ! [A2: nat,B3: nat,N2: nat] :
% 0.32/0.56 ( ( ord_less_nat @ A2 @ B3 )
% 0.32/0.56 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.56 => ( ord_less_eq_nat
% 0.32/0.56 @ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N2 )
% 0.32/0.56 @ ( if_nat
% 0.32/0.56 @ ( ( modulo_modulo_nat @ B3 @ N2 )
% 0.32/0.56 = zero_zero_nat )
% 0.32/0.56 @ one_one_nat
% 0.32/0.56 @ zero_zero_nat ) )
% 0.32/0.56 @ ( divide_divide_nat @ B3 @ N2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % verit_le_mono_div
% 0.32/0.56 thf(fact_3238_inrange,axiom,
% 0.32/0.56 ! [T: vEBT_VEBT,N2: nat] :
% 0.32/0.56 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.32/0.56 => ( 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 ) ) @ N2 ) @ one_one_nat ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % inrange
% 0.32/0.56 thf(fact_3239_set__bit__0,axiom,
% 0.32/0.56 ! [A: int] :
% 0.32/0.56 ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
% 0.32/0.56 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % set_bit_0
% 0.32/0.56 thf(fact_3240_set__bit__0,axiom,
% 0.32/0.56 ! [A: nat] :
% 0.32/0.56 ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
% 0.32/0.56 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % set_bit_0
% 0.32/0.56 thf(fact_3241_vebt__succ_Opelims,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
% 0.32/0.56 ( ( ( vEBT_vebt_succ @ X @ Xa2 )
% 0.32/0.56 = Y )
% 0.32/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.32/0.56 => ( ! [Uu2: $o,B2: $o] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Leaf @ Uu2 @ B2 ) )
% 0.32/0.56 => ( ( Xa2 = zero_zero_nat )
% 0.32/0.56 => ( ( ( B2
% 0.32/0.56 => ( Y
% 0.32/0.56 = ( some_nat @ one_one_nat ) ) )
% 0.32/0.56 & ( ~ B2
% 0.32/0.56 => ( Y = none_nat ) ) )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B2 ) @ zero_zero_nat ) ) ) ) )
% 0.32/0.56 => ( ! [Uv2: $o,Uw2: $o] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
% 0.32/0.56 => ! [N4: nat] :
% 0.32/0.56 ( ( Xa2
% 0.32/0.56 = ( suc @ N4 ) )
% 0.32/0.56 => ( ( Y = none_nat )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N4 ) ) ) ) ) )
% 0.32/0.56 => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
% 0.32/0.56 => ( ( Y = none_nat )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
% 0.32/0.56 => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
% 0.32/0.56 => ( ( Y = none_nat )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa2 ) ) ) )
% 0.32/0.56 => ( ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) )
% 0.32/0.56 => ( ( Y = none_nat )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Xa2 ) ) ) )
% 0.32/0.56 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.32/0.56 => ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
% 0.32/0.56 => ( Y
% 0.32/0.56 = ( some_nat @ Mi2 ) ) )
% 0.32/0.56 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.32/0.56 => ( Y
% 0.32/0.56 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 @ ( if_option_nat
% 0.32/0.56 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 != none_nat )
% 0.32/0.56 & ( 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 @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.32/0.56 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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 @ TreeList3 @ ( 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 ) ) ) ) ) )
% 0.32/0.56 @ ( if_option_nat
% 0.32/0.56 @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.56 = none_nat )
% 0.32/0.56 @ none_nat
% 0.32/0.56 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.32/0.56 @ none_nat ) ) ) )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % vebt_succ.pelims
% 0.32/0.56 thf(fact_3242_vebt__pred_Opelims,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
% 0.32/0.56 ( ( ( vEBT_vebt_pred @ X @ Xa2 )
% 0.32/0.56 = Y )
% 0.32/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.32/0.56 => ( ! [Uu2: $o,Uv2: $o] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.32/0.56 => ( ( Xa2 = zero_zero_nat )
% 0.32/0.56 => ( ( Y = none_nat )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
% 0.32/0.56 => ( ! [A3: $o,Uw2: $o] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Leaf @ A3 @ Uw2 ) )
% 0.32/0.56 => ( ( Xa2
% 0.32/0.56 = ( suc @ zero_zero_nat ) )
% 0.32/0.56 => ( ( ( A3
% 0.32/0.56 => ( Y
% 0.32/0.56 = ( some_nat @ zero_zero_nat ) ) )
% 0.32/0.56 & ( ~ A3
% 0.32/0.56 => ( Y = none_nat ) ) )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
% 0.32/0.56 => ( ! [A3: $o,B2: $o] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.32/0.56 => ! [Va2: nat] :
% 0.32/0.56 ( ( Xa2
% 0.32/0.56 = ( suc @ ( suc @ Va2 ) ) )
% 0.32/0.56 => ( ( ( B2
% 0.32/0.56 => ( Y
% 0.32/0.56 = ( some_nat @ one_one_nat ) ) )
% 0.32/0.56 & ( ~ B2
% 0.32/0.56 => ( ( A3
% 0.32/0.56 => ( Y
% 0.32/0.56 = ( some_nat @ zero_zero_nat ) ) )
% 0.32/0.56 & ( ~ A3
% 0.32/0.56 => ( Y = none_nat ) ) ) ) )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) )
% 0.32/0.56 => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
% 0.32/0.56 => ( ( Y = none_nat )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Xa2 ) ) ) )
% 0.32/0.56 => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) )
% 0.32/0.56 => ( ( Y = none_nat )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) @ Xa2 ) ) ) )
% 0.32/0.56 => ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) )
% 0.32/0.56 => ( ( Y = none_nat )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Xa2 ) ) ) )
% 0.32/0.56 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.32/0.56 => ( ( ( ( ord_less_nat @ Ma2 @ Xa2 )
% 0.32/0.56 => ( Y
% 0.32/0.56 = ( some_nat @ Ma2 ) ) )
% 0.32/0.56 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.32/0.56 => ( Y
% 0.32/0.56 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 @ ( if_option_nat
% 0.32/0.56 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 != none_nat )
% 0.32/0.56 & ( 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 @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.32/0.56 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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 @ TreeList3 @ ( 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 ) ) ) ) ) )
% 0.32/0.56 @ ( if_option_nat
% 0.32/0.56 @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.56 = none_nat )
% 0.32/0.56 @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
% 0.32/0.56 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.32/0.56 @ none_nat ) ) ) )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % vebt_pred.pelims
% 0.32/0.56 thf(fact_3243_double__eq__0__iff,axiom,
% 0.32/0.56 ! [A: real] :
% 0.32/0.56 ( ( ( plus_plus_real @ A @ A )
% 0.32/0.56 = zero_zero_real )
% 0.32/0.56 = ( A = zero_zero_real ) ) ).
% 0.32/0.56
% 0.32/0.56 % double_eq_0_iff
% 0.32/0.56 thf(fact_3244_double__eq__0__iff,axiom,
% 0.32/0.56 ! [A: rat] :
% 0.32/0.56 ( ( ( plus_plus_rat @ A @ A )
% 0.32/0.56 = zero_zero_rat )
% 0.32/0.56 = ( A = zero_zero_rat ) ) ).
% 0.32/0.56
% 0.32/0.56 % double_eq_0_iff
% 0.32/0.56 thf(fact_3245_double__eq__0__iff,axiom,
% 0.32/0.56 ! [A: int] :
% 0.32/0.56 ( ( ( plus_plus_int @ A @ A )
% 0.32/0.56 = zero_zero_int )
% 0.32/0.56 = ( A = zero_zero_int ) ) ).
% 0.32/0.56
% 0.32/0.56 % double_eq_0_iff
% 0.32/0.56 thf(fact_3246_vebt__delete_Opelims,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
% 0.32/0.56 ( ( ( vEBT_vebt_delete @ X @ Xa2 )
% 0.32/0.56 = Y )
% 0.32/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.32/0.56 => ( ! [A3: $o,B2: $o] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.32/0.56 => ( ( Xa2 = zero_zero_nat )
% 0.32/0.56 => ( ( Y
% 0.32/0.56 = ( vEBT_Leaf @ $false @ B2 ) )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ zero_zero_nat ) ) ) ) )
% 0.32/0.56 => ( ! [A3: $o,B2: $o] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.32/0.56 => ( ( Xa2
% 0.32/0.56 = ( suc @ zero_zero_nat ) )
% 0.32/0.56 => ( ( Y
% 0.32/0.56 = ( vEBT_Leaf @ A3 @ $false ) )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
% 0.32/0.56 => ( ! [A3: $o,B2: $o] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.32/0.56 => ! [N4: nat] :
% 0.32/0.56 ( ( Xa2
% 0.32/0.56 = ( suc @ ( suc @ N4 ) ) )
% 0.32/0.56 => ( ( Y
% 0.32/0.56 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ N4 ) ) ) ) ) ) )
% 0.32/0.56 => ( ! [Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.32/0.56 => ( ( Y
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) )
% 0.32/0.56 => ( ! [Mi2: nat,Ma2: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) )
% 0.32/0.56 => ( ( Y
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) @ Xa2 ) ) ) )
% 0.32/0.56 => ( ! [Mi2: nat,Ma2: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) )
% 0.32/0.56 => ( ( Y
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) @ Xa2 ) ) ) )
% 0.32/0.56 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.32/0.56 => ( ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
% 0.32/0.56 | ( ord_less_nat @ Ma2 @ Xa2 ) )
% 0.32/0.56 => ( Y
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) )
% 0.32/0.56 & ( ~ ( ( ord_less_nat @ Xa2 @ Mi2 )
% 0.32/0.56 | ( ord_less_nat @ Ma2 @ Xa2 ) )
% 0.32/0.56 => ( ( ( ( Xa2 = Mi2 )
% 0.32/0.56 & ( Xa2 = Ma2 ) )
% 0.32/0.56 => ( Y
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) )
% 0.32/0.56 & ( ~ ( ( Xa2 = Mi2 )
% 0.32/0.56 & ( Xa2 = Ma2 ) )
% 0.32/0.56 => ( Y
% 0.32/0.56 = ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 @ ( vEBT_Node
% 0.32/0.56 @ ( some_P7363390416028606310at_nat
% 0.32/0.56 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
% 0.32/0.56 @ ( if_nat
% 0.32/0.56 @ ( ( ( Xa2 = Mi2 )
% 0.32/0.56 => ( ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
% 0.32/0.56 = Ma2 ) )
% 0.32/0.56 & ( ( Xa2 != Mi2 )
% 0.32/0.56 => ( Xa2 = Ma2 ) ) )
% 0.32/0.56 @ ( if_nat
% 0.32/0.56 @ ( ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 = none_nat )
% 0.32/0.56 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
% 0.32/0.56 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
% 0.32/0.56 @ Ma2 ) ) )
% 0.32/0.56 @ ( suc @ ( suc @ Va2 ) )
% 0.32/0.56 @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 @ ( vEBT_Node
% 0.32/0.56 @ ( some_P7363390416028606310at_nat
% 0.32/0.56 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
% 0.32/0.56 @ ( if_nat
% 0.32/0.56 @ ( ( ( Xa2 = Mi2 )
% 0.32/0.56 => ( ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
% 0.32/0.56 = Ma2 ) )
% 0.32/0.56 & ( ( Xa2 != Mi2 )
% 0.32/0.56 => ( Xa2 = Ma2 ) ) )
% 0.32/0.56 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.32/0.56 @ Ma2 ) ) )
% 0.32/0.56 @ ( suc @ ( suc @ Va2 ) )
% 0.32/0.56 @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 @ Summary2 ) )
% 0.32/0.56 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) ) ) ) ) )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % vebt_delete.pelims
% 0.32/0.56 thf(fact_3247_unset__bit__0,axiom,
% 0.32/0.56 ! [A: int] :
% 0.32/0.56 ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
% 0.32/0.56 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % unset_bit_0
% 0.32/0.56 thf(fact_3248_unset__bit__0,axiom,
% 0.32/0.56 ! [A: nat] :
% 0.32/0.56 ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
% 0.32/0.56 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % unset_bit_0
% 0.32/0.56 thf(fact_3249_div__mod__decomp,axiom,
% 0.32/0.56 ! [A2: nat,N2: nat] :
% 0.32/0.56 ( A2
% 0.32/0.56 = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N2 ) @ N2 ) @ ( modulo_modulo_nat @ A2 @ N2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_mod_decomp
% 0.32/0.56 thf(fact_3250_unset__bit__Suc,axiom,
% 0.32/0.56 ! [N2: nat,A: code_integer] :
% 0.32/0.56 ( ( bit_se8260200283734997820nteger @ ( suc @ N2 ) @ A )
% 0.32/0.56 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % unset_bit_Suc
% 0.32/0.56 thf(fact_3251_unset__bit__Suc,axiom,
% 0.32/0.56 ! [N2: nat,A: int] :
% 0.32/0.56 ( ( bit_se4203085406695923979it_int @ ( suc @ N2 ) @ A )
% 0.32/0.56 = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % unset_bit_Suc
% 0.32/0.56 thf(fact_3252_unset__bit__Suc,axiom,
% 0.32/0.56 ! [N2: nat,A: nat] :
% 0.32/0.56 ( ( bit_se4205575877204974255it_nat @ ( suc @ N2 ) @ A )
% 0.32/0.56 = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % unset_bit_Suc
% 0.32/0.56 thf(fact_3253_verit__eq__simplify_I8_J,axiom,
% 0.32/0.56 ! [X22: num,Y22: num] :
% 0.32/0.56 ( ( ( bit0 @ X22 )
% 0.32/0.56 = ( bit0 @ Y22 ) )
% 0.32/0.56 = ( X22 = Y22 ) ) ).
% 0.32/0.56
% 0.32/0.56 % verit_eq_simplify(8)
% 0.32/0.56 thf(fact_3254_div__neg__neg__trivial,axiom,
% 0.32/0.56 ! [K: int,L2: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.32/0.56 => ( ( ord_less_int @ L2 @ K )
% 0.32/0.56 => ( ( divide_divide_int @ K @ L2 )
% 0.32/0.56 = zero_zero_int ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_neg_neg_trivial
% 0.32/0.56 thf(fact_3255_div__pos__pos__trivial,axiom,
% 0.32/0.56 ! [K: int,L2: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.56 => ( ( ord_less_int @ K @ L2 )
% 0.32/0.56 => ( ( divide_divide_int @ K @ L2 )
% 0.32/0.56 = zero_zero_int ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_pos_pos_trivial
% 0.32/0.56 thf(fact_3256_i0__less,axiom,
% 0.32/0.56 ! [N2: extended_enat] :
% 0.32/0.56 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 )
% 0.32/0.56 = ( N2 != zero_z5237406670263579293d_enat ) ) ).
% 0.32/0.56
% 0.32/0.56 % i0_less
% 0.32/0.56 thf(fact_3257_idiff__0__right,axiom,
% 0.32/0.56 ! [N2: extended_enat] :
% 0.32/0.56 ( ( minus_3235023915231533773d_enat @ N2 @ zero_z5237406670263579293d_enat )
% 0.32/0.56 = N2 ) ).
% 0.32/0.56
% 0.32/0.56 % idiff_0_right
% 0.32/0.56 thf(fact_3258_idiff__0,axiom,
% 0.32/0.56 ! [N2: extended_enat] :
% 0.32/0.56 ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N2 )
% 0.32/0.56 = zero_z5237406670263579293d_enat ) ).
% 0.32/0.56
% 0.32/0.56 % idiff_0
% 0.32/0.56 thf(fact_3259_not__real__square__gt__zero,axiom,
% 0.32/0.56 ! [X: real] :
% 0.32/0.56 ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
% 0.32/0.56 = ( X = zero_zero_real ) ) ).
% 0.32/0.56
% 0.32/0.56 % not_real_square_gt_zero
% 0.32/0.56 thf(fact_3260_zmod__numeral__Bit0,axiom,
% 0.32/0.56 ! [V: num,W: num] :
% 0.32/0.56 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.32/0.56 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zmod_numeral_Bit0
% 0.32/0.56 thf(fact_3261_half__negative__int__iff,axiom,
% 0.32/0.56 ! [K: int] :
% 0.32/0.56 ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 0.32/0.56 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.32/0.56
% 0.32/0.56 % half_negative_int_iff
% 0.32/0.56 thf(fact_3262_half__nonnegative__int__iff,axiom,
% 0.32/0.56 ! [K: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.32/0.56 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.32/0.56
% 0.32/0.56 % half_nonnegative_int_iff
% 0.32/0.56 thf(fact_3263_split__zdiv,axiom,
% 0.32/0.56 ! [P: int > $o,N2: int,K: int] :
% 0.32/0.56 ( ( P @ ( divide_divide_int @ N2 @ K ) )
% 0.32/0.56 = ( ( ( K = zero_zero_int )
% 0.32/0.56 => ( P @ zero_zero_int ) )
% 0.32/0.56 & ( ( ord_less_int @ zero_zero_int @ K )
% 0.32/0.56 => ! [I3: int,J3: int] :
% 0.32/0.56 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.32/0.56 & ( ord_less_int @ J3 @ K )
% 0.32/0.56 & ( N2
% 0.32/0.56 = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
% 0.32/0.56 => ( P @ I3 ) ) )
% 0.32/0.56 & ( ( ord_less_int @ K @ zero_zero_int )
% 0.32/0.56 => ! [I3: int,J3: int] :
% 0.32/0.56 ( ( ( ord_less_int @ K @ J3 )
% 0.32/0.56 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.32/0.56 & ( N2
% 0.32/0.56 = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
% 0.32/0.56 => ( P @ I3 ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % split_zdiv
% 0.32/0.56 thf(fact_3264_zdiv__mono1,axiom,
% 0.32/0.56 ! [A: int,A6: int,B: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ A @ A6 )
% 0.32/0.56 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.56 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A6 @ B ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zdiv_mono1
% 0.32/0.56 thf(fact_3265_zdiv__mono2,axiom,
% 0.32/0.56 ! [A: int,B6: int,B: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.56 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 0.32/0.56 => ( ( ord_less_eq_int @ B6 @ B )
% 0.32/0.56 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B6 ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zdiv_mono2
% 0.32/0.56 thf(fact_3266_div__pos__geq,axiom,
% 0.32/0.56 ! [L2: int,K: int] :
% 0.32/0.56 ( ( ord_less_int @ zero_zero_int @ L2 )
% 0.32/0.56 => ( ( ord_less_eq_int @ L2 @ K )
% 0.32/0.56 => ( ( divide_divide_int @ K @ L2 )
% 0.32/0.56 = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L2 ) @ L2 ) @ one_one_int ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_pos_geq
% 0.32/0.56 thf(fact_3267_zdiv__eq__0__iff,axiom,
% 0.32/0.56 ! [I2: int,K: int] :
% 0.32/0.56 ( ( ( divide_divide_int @ I2 @ K )
% 0.32/0.56 = zero_zero_int )
% 0.32/0.56 = ( ( K = zero_zero_int )
% 0.32/0.56 | ( ( ord_less_eq_int @ zero_zero_int @ I2 )
% 0.32/0.56 & ( ord_less_int @ I2 @ K ) )
% 0.32/0.56 | ( ( ord_less_eq_int @ I2 @ zero_zero_int )
% 0.32/0.56 & ( ord_less_int @ K @ I2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zdiv_eq_0_iff
% 0.32/0.56 thf(fact_3268_int__div__neg__eq,axiom,
% 0.32/0.56 ! [A: int,B: int,Q2: int,R: int] :
% 0.32/0.56 ( ( A
% 0.32/0.56 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 0.32/0.56 => ( ( ord_less_eq_int @ R @ zero_zero_int )
% 0.32/0.56 => ( ( ord_less_int @ B @ R )
% 0.32/0.56 => ( ( divide_divide_int @ A @ B )
% 0.32/0.56 = Q2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % int_div_neg_eq
% 0.32/0.56 thf(fact_3269_int__div__pos__eq,axiom,
% 0.32/0.56 ! [A: int,B: int,Q2: int,R: int] :
% 0.32/0.56 ( ( A
% 0.32/0.56 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 0.32/0.56 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 0.32/0.56 => ( ( ord_less_int @ R @ B )
% 0.32/0.56 => ( ( divide_divide_int @ A @ B )
% 0.32/0.56 = Q2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % int_div_pos_eq
% 0.32/0.56 thf(fact_3270_zdiv__mono1__neg,axiom,
% 0.32/0.56 ! [A: int,A6: int,B: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ A @ A6 )
% 0.32/0.56 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.32/0.56 => ( ord_less_eq_int @ ( divide_divide_int @ A6 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zdiv_mono1_neg
% 0.32/0.56 thf(fact_3271_zdiv__mono2__neg,axiom,
% 0.32/0.56 ! [A: int,B6: int,B: int] :
% 0.32/0.56 ( ( ord_less_int @ A @ zero_zero_int )
% 0.32/0.56 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 0.32/0.56 => ( ( ord_less_eq_int @ B6 @ B )
% 0.32/0.56 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B6 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zdiv_mono2_neg
% 0.32/0.56 thf(fact_3272_div__int__pos__iff,axiom,
% 0.32/0.56 ! [K: int,L2: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L2 ) )
% 0.32/0.56 = ( ( K = zero_zero_int )
% 0.32/0.56 | ( L2 = zero_zero_int )
% 0.32/0.56 | ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.56 & ( ord_less_eq_int @ zero_zero_int @ L2 ) )
% 0.32/0.56 | ( ( ord_less_int @ K @ zero_zero_int )
% 0.32/0.56 & ( ord_less_int @ L2 @ zero_zero_int ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_int_pos_iff
% 0.32/0.56 thf(fact_3273_split__neg__lemma,axiom,
% 0.32/0.56 ! [K: int,P: int > int > $o,N2: int] :
% 0.32/0.56 ( ( ord_less_int @ K @ zero_zero_int )
% 0.32/0.56 => ( ( P @ ( divide_divide_int @ N2 @ K ) @ ( modulo_modulo_int @ N2 @ K ) )
% 0.32/0.56 = ( ! [I3: int,J3: int] :
% 0.32/0.56 ( ( ( ord_less_int @ K @ J3 )
% 0.32/0.56 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.32/0.56 & ( N2
% 0.32/0.56 = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
% 0.32/0.56 => ( P @ I3 @ J3 ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % split_neg_lemma
% 0.32/0.56 thf(fact_3274_split__pos__lemma,axiom,
% 0.32/0.56 ! [K: int,P: int > int > $o,N2: int] :
% 0.32/0.56 ( ( ord_less_int @ zero_zero_int @ K )
% 0.32/0.56 => ( ( P @ ( divide_divide_int @ N2 @ K ) @ ( modulo_modulo_int @ N2 @ K ) )
% 0.32/0.56 = ( ! [I3: int,J3: int] :
% 0.32/0.56 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.32/0.56 & ( ord_less_int @ J3 @ K )
% 0.32/0.56 & ( N2
% 0.32/0.56 = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
% 0.32/0.56 => ( P @ I3 @ J3 ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % split_pos_lemma
% 0.32/0.56 thf(fact_3275_div__positive__int,axiom,
% 0.32/0.56 ! [L2: int,K: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ L2 @ K )
% 0.32/0.56 => ( ( ord_less_int @ zero_zero_int @ L2 )
% 0.32/0.56 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_positive_int
% 0.32/0.56 thf(fact_3276_verit__le__mono__div__int,axiom,
% 0.32/0.56 ! [A2: int,B3: int,N2: int] :
% 0.32/0.56 ( ( ord_less_int @ A2 @ B3 )
% 0.32/0.56 => ( ( ord_less_int @ zero_zero_int @ N2 )
% 0.32/0.56 => ( ord_less_eq_int
% 0.32/0.56 @ ( plus_plus_int @ ( divide_divide_int @ A2 @ N2 )
% 0.32/0.56 @ ( if_int
% 0.32/0.56 @ ( ( modulo_modulo_int @ B3 @ N2 )
% 0.32/0.56 = zero_zero_int )
% 0.32/0.56 @ one_one_int
% 0.32/0.56 @ zero_zero_int ) )
% 0.32/0.56 @ ( divide_divide_int @ B3 @ N2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % verit_le_mono_div_int
% 0.32/0.56 thf(fact_3277_div__nonneg__neg__le0,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.56 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.32/0.56 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_nonneg_neg_le0
% 0.32/0.56 thf(fact_3278_div__nonpos__pos__le0,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.32/0.56 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.56 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_nonpos_pos_le0
% 0.32/0.56 thf(fact_3279_pos__imp__zdiv__pos__iff,axiom,
% 0.32/0.56 ! [K: int,I2: int] :
% 0.32/0.56 ( ( ord_less_int @ zero_zero_int @ K )
% 0.32/0.56 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I2 @ K ) )
% 0.32/0.56 = ( ord_less_eq_int @ K @ I2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % pos_imp_zdiv_pos_iff
% 0.32/0.56 thf(fact_3280_neg__imp__zdiv__nonneg__iff,axiom,
% 0.32/0.56 ! [B: int,A: int] :
% 0.32/0.56 ( ( ord_less_int @ B @ zero_zero_int )
% 0.32/0.56 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 0.32/0.56 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % neg_imp_zdiv_nonneg_iff
% 0.32/0.56 thf(fact_3281_pos__imp__zdiv__nonneg__iff,axiom,
% 0.32/0.56 ! [B: int,A: int] :
% 0.32/0.56 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.56 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 0.32/0.56 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % pos_imp_zdiv_nonneg_iff
% 0.32/0.56 thf(fact_3282_nonneg1__imp__zdiv__pos__iff,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.56 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 0.32/0.56 = ( ( ord_less_eq_int @ B @ A )
% 0.32/0.56 & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % nonneg1_imp_zdiv_pos_iff
% 0.32/0.56 thf(fact_3283_zmod__zmult2__eq,axiom,
% 0.32/0.56 ! [C: int,A: int,B: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.56 => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
% 0.32/0.56 = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % zmod_zmult2_eq
% 0.32/0.56 thf(fact_3284_pos__imp__zdiv__neg__iff,axiom,
% 0.32/0.56 ! [B: int,A: int] :
% 0.32/0.56 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.56 => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 0.32/0.56 = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % pos_imp_zdiv_neg_iff
% 0.32/0.56 thf(fact_3285_neg__imp__zdiv__neg__iff,axiom,
% 0.32/0.56 ! [B: int,A: int] :
% 0.32/0.56 ( ( ord_less_int @ B @ zero_zero_int )
% 0.32/0.56 => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 0.32/0.56 = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % neg_imp_zdiv_neg_iff
% 0.32/0.56 thf(fact_3286_div__neg__pos__less0,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( ord_less_int @ A @ zero_zero_int )
% 0.32/0.56 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.56 => ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_neg_pos_less0
% 0.32/0.56 thf(fact_3287_zdiv__mono__strict,axiom,
% 0.32/0.56 ! [A2: int,B3: int,N2: int] :
% 0.32/0.56 ( ( ord_less_int @ A2 @ B3 )
% 0.32/0.56 => ( ( ord_less_int @ zero_zero_int @ N2 )
% 0.32/0.56 => ( ( ( modulo_modulo_int @ A2 @ N2 )
% 0.32/0.56 = zero_zero_int )
% 0.32/0.56 => ( ( ( modulo_modulo_int @ B3 @ N2 )
% 0.32/0.57 = zero_zero_int )
% 0.32/0.57 => ( ord_less_int @ ( divide_divide_int @ A2 @ N2 ) @ ( divide_divide_int @ B3 @ N2 ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % zdiv_mono_strict
% 0.32/0.57 thf(fact_3288_int__div__less__self,axiom,
% 0.32/0.57 ! [X: int,K: int] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ X )
% 0.32/0.57 => ( ( ord_less_int @ one_one_int @ K )
% 0.32/0.57 => ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % int_div_less_self
% 0.32/0.57 thf(fact_3289_zdiv__zmult2__eq,axiom,
% 0.32/0.57 ! [C: int,A: int,B: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.57 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.32/0.57 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % zdiv_zmult2_eq
% 0.32/0.57 thf(fact_3290_div__mod__decomp__int,axiom,
% 0.32/0.57 ! [A2: int,N2: int] :
% 0.32/0.57 ( A2
% 0.32/0.57 = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ N2 ) @ N2 ) @ ( modulo_modulo_int @ A2 @ N2 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_mod_decomp_int
% 0.32/0.57 thf(fact_3291_neg__zmod__mult__2,axiom,
% 0.32/0.57 ! [A: int,B: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.32/0.57 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.57 = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % neg_zmod_mult_2
% 0.32/0.57 thf(fact_3292_pos__zmod__mult__2,axiom,
% 0.32/0.57 ! [A: int,B: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.57 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.57 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ A ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % pos_zmod_mult_2
% 0.32/0.57 thf(fact_3293_not__iless0,axiom,
% 0.32/0.57 ! [N2: extended_enat] :
% 0.32/0.57 ~ ( ord_le72135733267957522d_enat @ N2 @ zero_z5237406670263579293d_enat ) ).
% 0.32/0.57
% 0.32/0.57 % not_iless0
% 0.32/0.57 thf(fact_3294_enat__0__less__mult__iff,axiom,
% 0.32/0.57 ! [M: extended_enat,N2: extended_enat] :
% 0.32/0.57 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N2 ) )
% 0.32/0.57 = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
% 0.32/0.57 & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % enat_0_less_mult_iff
% 0.32/0.57 thf(fact_3295_all__nat__less,axiom,
% 0.32/0.57 ! [N2: nat,P: nat > $o] :
% 0.32/0.57 ( ( ! [M6: nat] :
% 0.32/0.57 ( ( ord_less_eq_nat @ M6 @ N2 )
% 0.32/0.57 => ( P @ M6 ) ) )
% 0.32/0.57 = ( ! [X2: nat] :
% 0.32/0.57 ( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 0.32/0.57 => ( P @ X2 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % all_nat_less
% 0.32/0.57 thf(fact_3296_ex__nat__less,axiom,
% 0.32/0.57 ! [N2: nat,P: nat > $o] :
% 0.32/0.57 ( ( ? [M6: nat] :
% 0.32/0.57 ( ( ord_less_eq_nat @ M6 @ N2 )
% 0.32/0.57 & ( P @ M6 ) ) )
% 0.32/0.57 = ( ? [X2: nat] :
% 0.32/0.57 ( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 0.32/0.57 & ( P @ X2 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % ex_nat_less
% 0.32/0.57 thf(fact_3297_not__exp__less__eq__0__int,axiom,
% 0.32/0.57 ! [N2: nat] :
% 0.32/0.57 ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_int ) ).
% 0.32/0.57
% 0.32/0.57 % not_exp_less_eq_0_int
% 0.32/0.57 thf(fact_3298_pos__zdiv__mult__2,axiom,
% 0.32/0.57 ! [A: int,B: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.57 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.57 = ( divide_divide_int @ B @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % pos_zdiv_mult_2
% 0.32/0.57 thf(fact_3299_neg__zdiv__mult__2,axiom,
% 0.32/0.57 ! [A: int,B: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.32/0.57 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.57 = ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % neg_zdiv_mult_2
% 0.32/0.57 thf(fact_3300_verit__la__disequality,axiom,
% 0.32/0.57 ! [A: rat,B: rat] :
% 0.32/0.57 ( ( A = B )
% 0.32/0.57 | ~ ( ord_less_eq_rat @ A @ B )
% 0.32/0.57 | ~ ( ord_less_eq_rat @ B @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % verit_la_disequality
% 0.32/0.57 thf(fact_3301_verit__la__disequality,axiom,
% 0.32/0.57 ! [A: num,B: num] :
% 0.32/0.57 ( ( A = B )
% 0.32/0.57 | ~ ( ord_less_eq_num @ A @ B )
% 0.32/0.57 | ~ ( ord_less_eq_num @ B @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % verit_la_disequality
% 0.32/0.57 thf(fact_3302_verit__la__disequality,axiom,
% 0.32/0.57 ! [A: nat,B: nat] :
% 0.32/0.57 ( ( A = B )
% 0.32/0.57 | ~ ( ord_less_eq_nat @ A @ B )
% 0.32/0.57 | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % verit_la_disequality
% 0.32/0.57 thf(fact_3303_verit__la__disequality,axiom,
% 0.32/0.57 ! [A: int,B: int] :
% 0.32/0.57 ( ( A = B )
% 0.32/0.57 | ~ ( ord_less_eq_int @ A @ B )
% 0.32/0.57 | ~ ( ord_less_eq_int @ B @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % verit_la_disequality
% 0.32/0.57 thf(fact_3304_verit__comp__simplify1_I2_J,axiom,
% 0.32/0.57 ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% 0.32/0.57
% 0.32/0.57 % verit_comp_simplify1(2)
% 0.32/0.57 thf(fact_3305_verit__comp__simplify1_I2_J,axiom,
% 0.32/0.57 ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 0.32/0.57
% 0.32/0.57 % verit_comp_simplify1(2)
% 0.32/0.57 thf(fact_3306_verit__comp__simplify1_I2_J,axiom,
% 0.32/0.57 ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 0.32/0.57
% 0.32/0.57 % verit_comp_simplify1(2)
% 0.32/0.57 thf(fact_3307_verit__comp__simplify1_I2_J,axiom,
% 0.32/0.57 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.32/0.57
% 0.32/0.57 % verit_comp_simplify1(2)
% 0.32/0.57 thf(fact_3308_verit__comp__simplify1_I2_J,axiom,
% 0.32/0.57 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 0.32/0.57
% 0.32/0.57 % verit_comp_simplify1(2)
% 0.32/0.57 thf(fact_3309_verit__comp__simplify1_I1_J,axiom,
% 0.32/0.57 ! [A: real] :
% 0.32/0.57 ~ ( ord_less_real @ A @ A ) ).
% 0.32/0.57
% 0.32/0.57 % verit_comp_simplify1(1)
% 0.32/0.57 thf(fact_3310_verit__comp__simplify1_I1_J,axiom,
% 0.32/0.57 ! [A: rat] :
% 0.32/0.57 ~ ( ord_less_rat @ A @ A ) ).
% 0.32/0.57
% 0.32/0.57 % verit_comp_simplify1(1)
% 0.32/0.57 thf(fact_3311_verit__comp__simplify1_I1_J,axiom,
% 0.32/0.57 ! [A: num] :
% 0.32/0.57 ~ ( ord_less_num @ A @ A ) ).
% 0.32/0.57
% 0.32/0.57 % verit_comp_simplify1(1)
% 0.32/0.57 thf(fact_3312_verit__comp__simplify1_I1_J,axiom,
% 0.32/0.57 ! [A: nat] :
% 0.32/0.57 ~ ( ord_less_nat @ A @ A ) ).
% 0.32/0.57
% 0.32/0.57 % verit_comp_simplify1(1)
% 0.32/0.57 thf(fact_3313_verit__comp__simplify1_I1_J,axiom,
% 0.32/0.57 ! [A: int] :
% 0.32/0.57 ~ ( ord_less_int @ A @ A ) ).
% 0.32/0.57
% 0.32/0.57 % verit_comp_simplify1(1)
% 0.32/0.57 thf(fact_3314_realpow__pos__nth2,axiom,
% 0.32/0.57 ! [A: real,N2: nat] :
% 0.32/0.57 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.57 => ? [R3: real] :
% 0.32/0.57 ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.32/0.57 & ( ( power_power_real @ R3 @ ( suc @ N2 ) )
% 0.32/0.57 = A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % realpow_pos_nth2
% 0.32/0.57 thf(fact_3315_real__arch__pow__inv,axiom,
% 0.32/0.57 ! [Y: real,X: real] :
% 0.32/0.57 ( ( ord_less_real @ zero_zero_real @ Y )
% 0.32/0.57 => ( ( ord_less_real @ X @ one_one_real )
% 0.32/0.57 => ? [N4: nat] : ( ord_less_real @ ( power_power_real @ X @ N4 ) @ Y ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % real_arch_pow_inv
% 0.32/0.57 thf(fact_3316_int__power__div__base,axiom,
% 0.32/0.57 ! [M: nat,K: int] :
% 0.32/0.57 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.57 => ( ( ord_less_int @ zero_zero_int @ K )
% 0.32/0.57 => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
% 0.32/0.57 = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % int_power_div_base
% 0.32/0.57 thf(fact_3317_realpow__pos__nth__unique,axiom,
% 0.32/0.57 ! [N2: nat,A: real] :
% 0.32/0.57 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.57 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.57 => ? [X4: real] :
% 0.32/0.57 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.32/0.57 & ( ( power_power_real @ X4 @ N2 )
% 0.32/0.57 = A )
% 0.32/0.57 & ! [Y5: real] :
% 0.32/0.57 ( ( ( ord_less_real @ zero_zero_real @ Y5 )
% 0.32/0.57 & ( ( power_power_real @ Y5 @ N2 )
% 0.32/0.57 = A ) )
% 0.32/0.57 => ( Y5 = X4 ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % realpow_pos_nth_unique
% 0.32/0.57 thf(fact_3318_realpow__pos__nth,axiom,
% 0.32/0.57 ! [N2: nat,A: real] :
% 0.32/0.57 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.57 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.57 => ? [R3: real] :
% 0.32/0.57 ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.32/0.57 & ( ( power_power_real @ R3 @ N2 )
% 0.32/0.57 = A ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % realpow_pos_nth
% 0.32/0.57 thf(fact_3319_verit__comp__simplify1_I3_J,axiom,
% 0.32/0.57 ! [B6: real,A6: real] :
% 0.32/0.57 ( ( ~ ( ord_less_eq_real @ B6 @ A6 ) )
% 0.32/0.57 = ( ord_less_real @ A6 @ B6 ) ) ).
% 0.32/0.57
% 0.32/0.57 % verit_comp_simplify1(3)
% 0.32/0.57 thf(fact_3320_verit__comp__simplify1_I3_J,axiom,
% 0.32/0.57 ! [B6: rat,A6: rat] :
% 0.32/0.57 ( ( ~ ( ord_less_eq_rat @ B6 @ A6 ) )
% 0.32/0.57 = ( ord_less_rat @ A6 @ B6 ) ) ).
% 0.32/0.57
% 0.32/0.57 % verit_comp_simplify1(3)
% 0.32/0.57 thf(fact_3321_verit__comp__simplify1_I3_J,axiom,
% 0.32/0.57 ! [B6: num,A6: num] :
% 0.32/0.57 ( ( ~ ( ord_less_eq_num @ B6 @ A6 ) )
% 0.32/0.57 = ( ord_less_num @ A6 @ B6 ) ) ).
% 0.32/0.57
% 0.32/0.57 % verit_comp_simplify1(3)
% 0.32/0.57 thf(fact_3322_verit__comp__simplify1_I3_J,axiom,
% 0.32/0.57 ! [B6: nat,A6: nat] :
% 0.32/0.57 ( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
% 0.32/0.57 = ( ord_less_nat @ A6 @ B6 ) ) ).
% 0.32/0.57
% 0.32/0.57 % verit_comp_simplify1(3)
% 0.32/0.57 thf(fact_3323_verit__comp__simplify1_I3_J,axiom,
% 0.32/0.57 ! [B6: int,A6: int] :
% 0.32/0.57 ( ( ~ ( ord_less_eq_int @ B6 @ A6 ) )
% 0.32/0.57 = ( ord_less_int @ A6 @ B6 ) ) ).
% 0.32/0.57
% 0.32/0.57 % verit_comp_simplify1(3)
% 0.32/0.57 thf(fact_3324_verit__sum__simplify,axiom,
% 0.32/0.57 ! [A: complex] :
% 0.32/0.57 ( ( plus_plus_complex @ A @ zero_zero_complex )
% 0.32/0.57 = A ) ).
% 0.32/0.57
% 0.32/0.57 % verit_sum_simplify
% 0.32/0.57 thf(fact_3325_verit__sum__simplify,axiom,
% 0.32/0.57 ! [A: real] :
% 0.32/0.57 ( ( plus_plus_real @ A @ zero_zero_real )
% 0.32/0.57 = A ) ).
% 0.32/0.57
% 0.32/0.57 % verit_sum_simplify
% 0.32/0.57 thf(fact_3326_verit__sum__simplify,axiom,
% 0.32/0.57 ! [A: rat] :
% 0.32/0.57 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.32/0.57 = A ) ).
% 0.32/0.57
% 0.32/0.57 % verit_sum_simplify
% 0.32/0.57 thf(fact_3327_verit__sum__simplify,axiom,
% 0.32/0.57 ! [A: nat] :
% 0.32/0.57 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.32/0.57 = A ) ).
% 0.32/0.57
% 0.32/0.57 % verit_sum_simplify
% 0.32/0.57 thf(fact_3328_verit__sum__simplify,axiom,
% 0.32/0.57 ! [A: int] :
% 0.32/0.57 ( ( plus_plus_int @ A @ zero_zero_int )
% 0.32/0.57 = A ) ).
% 0.32/0.57
% 0.32/0.57 % verit_sum_simplify
% 0.32/0.57 thf(fact_3329_verit__eq__simplify_I10_J,axiom,
% 0.32/0.57 ! [X22: num] :
% 0.32/0.57 ( one
% 0.32/0.57 != ( bit0 @ X22 ) ) ).
% 0.32/0.57
% 0.32/0.57 % verit_eq_simplify(10)
% 0.32/0.57 thf(fact_3330_max__def__raw,axiom,
% 0.32/0.57 ( ord_max_set_nat
% 0.32/0.57 = ( ^ [A5: set_nat,B5: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A5 @ B5 ) @ B5 @ A5 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_def_raw
% 0.32/0.57 thf(fact_3331_max__def__raw,axiom,
% 0.32/0.57 ( ord_max_rat
% 0.32/0.57 = ( ^ [A5: rat,B5: rat] : ( if_rat @ ( ord_less_eq_rat @ A5 @ B5 ) @ B5 @ A5 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_def_raw
% 0.32/0.57 thf(fact_3332_max__def__raw,axiom,
% 0.32/0.57 ( ord_max_num
% 0.32/0.57 = ( ^ [A5: num,B5: num] : ( if_num @ ( ord_less_eq_num @ A5 @ B5 ) @ B5 @ A5 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_def_raw
% 0.32/0.57 thf(fact_3333_max__def__raw,axiom,
% 0.32/0.57 ( ord_max_nat
% 0.32/0.57 = ( ^ [A5: nat,B5: nat] : ( if_nat @ ( ord_less_eq_nat @ A5 @ B5 ) @ B5 @ A5 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_def_raw
% 0.32/0.57 thf(fact_3334_max__def__raw,axiom,
% 0.32/0.57 ( ord_max_int
% 0.32/0.57 = ( ^ [A5: int,B5: int] : ( if_int @ ( ord_less_eq_int @ A5 @ B5 ) @ B5 @ A5 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_def_raw
% 0.32/0.57 thf(fact_3335_div__less__mono,axiom,
% 0.32/0.57 ! [A2: nat,B3: nat,N2: nat] :
% 0.32/0.57 ( ( ord_less_nat @ A2 @ B3 )
% 0.32/0.57 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.57 => ( ( ( modulo_modulo_nat @ A2 @ N2 )
% 0.32/0.57 = zero_zero_nat )
% 0.32/0.57 => ( ( ( modulo_modulo_nat @ B3 @ N2 )
% 0.32/0.57 = zero_zero_nat )
% 0.32/0.57 => ( ord_less_nat @ ( divide_divide_nat @ A2 @ N2 ) @ ( divide_divide_nat @ B3 @ N2 ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_less_mono
% 0.32/0.57 thf(fact_3336_set__bit__Suc,axiom,
% 0.32/0.57 ! [N2: nat,A: code_integer] :
% 0.32/0.57 ( ( bit_se2793503036327961859nteger @ ( suc @ N2 ) @ A )
% 0.32/0.57 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % set_bit_Suc
% 0.32/0.57 thf(fact_3337_set__bit__Suc,axiom,
% 0.32/0.57 ! [N2: nat,A: int] :
% 0.32/0.57 ( ( bit_se7879613467334960850it_int @ ( suc @ N2 ) @ A )
% 0.32/0.57 = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % set_bit_Suc
% 0.32/0.57 thf(fact_3338_set__bit__Suc,axiom,
% 0.32/0.57 ! [N2: nat,A: nat] :
% 0.32/0.57 ( ( bit_se7882103937844011126it_nat @ ( suc @ N2 ) @ A )
% 0.32/0.57 = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % set_bit_Suc
% 0.32/0.57 thf(fact_3339_vebt__insert_Opelims,axiom,
% 0.32/0.57 ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
% 0.32/0.57 ( ( ( vEBT_vebt_insert @ X @ Xa2 )
% 0.32/0.57 = Y )
% 0.32/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.32/0.57 => ( ! [A3: $o,B2: $o] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.32/0.57 => ( ( ( ( Xa2 = zero_zero_nat )
% 0.32/0.57 => ( Y
% 0.32/0.57 = ( vEBT_Leaf @ $true @ B2 ) ) )
% 0.32/0.57 & ( ( Xa2 != zero_zero_nat )
% 0.32/0.57 => ( ( ( Xa2 = one_one_nat )
% 0.32/0.57 => ( Y
% 0.32/0.57 = ( vEBT_Leaf @ A3 @ $true ) ) )
% 0.32/0.57 & ( ( Xa2 != one_one_nat )
% 0.32/0.57 => ( Y
% 0.32/0.57 = ( vEBT_Leaf @ A3 @ B2 ) ) ) ) ) )
% 0.32/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa2 ) ) ) )
% 0.32/0.57 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) )
% 0.32/0.57 => ( ( Y
% 0.32/0.57 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) )
% 0.32/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) @ Xa2 ) ) ) )
% 0.32/0.57 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) )
% 0.32/0.57 => ( ( Y
% 0.32/0.57 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) )
% 0.32/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) @ Xa2 ) ) ) )
% 0.32/0.57 => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
% 0.32/0.57 => ( ( Y
% 0.32/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
% 0.32/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) )
% 0.32/0.57 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.32/0.57 => ( ( Y
% 0.32/0.57 = ( if_VEBT_VEBT
% 0.32/0.57 @ ( ( 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 @ TreeList3 ) )
% 0.32/0.57 & ~ ( ( Xa2 = Mi2 )
% 0.32/0.57 | ( Xa2 = Ma2 ) ) )
% 0.32/0.57 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 ) )
% 0.32/0.57 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) )
% 0.32/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % vebt_insert.pelims
% 0.32/0.57 thf(fact_3340_vebt__member_Opelims_I1_J,axiom,
% 0.32/0.57 ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 0.32/0.57 ( ( ( vEBT_vebt_member @ X @ Xa2 )
% 0.32/0.57 = Y )
% 0.32/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.32/0.57 => ( ! [A3: $o,B2: $o] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.32/0.57 => ( ( Y
% 0.32/0.57 = ( ( ( Xa2 = zero_zero_nat )
% 0.32/0.57 => A3 )
% 0.32/0.57 & ( ( Xa2 != zero_zero_nat )
% 0.32/0.57 => ( ( ( Xa2 = one_one_nat )
% 0.32/0.57 => B2 )
% 0.32/0.57 & ( Xa2 = one_one_nat ) ) ) ) )
% 0.32/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa2 ) ) ) )
% 0.32/0.57 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.32/0.57 => ( ~ Y
% 0.32/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
% 0.32/0.57 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.32/0.57 => ( ~ Y
% 0.32/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
% 0.32/0.57 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.32/0.57 => ( ~ Y
% 0.32/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
% 0.32/0.57 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.32/0.57 => ( ( Y
% 0.32/0.57 = ( ( Xa2 != Mi2 )
% 0.32/0.57 => ( ( Xa2 != Ma2 )
% 0.32/0.57 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.32/0.57 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.32/0.57 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.32/0.57 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.32/0.57 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.57 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 0.32/0.57 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) )
% 0.32/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % vebt_member.pelims(1)
% 0.32/0.57 thf(fact_3341_vebt__member_Opelims_I3_J,axiom,
% 0.32/0.57 ! [X: vEBT_VEBT,Xa2: nat] :
% 0.32/0.57 ( ~ ( vEBT_vebt_member @ X @ Xa2 )
% 0.32/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.32/0.57 => ( ! [A3: $o,B2: $o] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.32/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa2 ) )
% 0.32/0.57 => ( ( ( Xa2 = zero_zero_nat )
% 0.32/0.57 => A3 )
% 0.32/0.57 & ( ( Xa2 != zero_zero_nat )
% 0.32/0.57 => ( ( ( Xa2 = one_one_nat )
% 0.32/0.57 => B2 )
% 0.32/0.57 & ( Xa2 = one_one_nat ) ) ) ) ) )
% 0.32/0.57 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.32/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) )
% 0.32/0.57 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.32/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) )
% 0.32/0.57 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.32/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
% 0.32/0.57 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.32/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) )
% 0.32/0.57 => ( ( Xa2 != Mi2 )
% 0.32/0.57 => ( ( Xa2 != Ma2 )
% 0.32/0.57 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.32/0.57 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.32/0.57 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.32/0.57 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.32/0.57 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.57 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 0.32/0.57 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % vebt_member.pelims(3)
% 0.32/0.57 thf(fact_3342_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
% 0.32/0.57 ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 0.32/0.57 ( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
% 0.32/0.57 = Y )
% 0.32/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.32/0.57 => ( ! [A3: $o,B2: $o] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.32/0.57 => ( ( Y
% 0.32/0.57 = ( ( ( Xa2 = zero_zero_nat )
% 0.32/0.57 => A3 )
% 0.32/0.57 & ( ( Xa2 != zero_zero_nat )
% 0.32/0.57 => ( ( ( Xa2 = one_one_nat )
% 0.32/0.57 => B2 )
% 0.32/0.57 & ( Xa2 = one_one_nat ) ) ) ) )
% 0.32/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa2 ) ) ) )
% 0.32/0.57 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.32/0.57 => ( ~ Y
% 0.32/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
% 0.32/0.57 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
% 0.32/0.57 => ( ( Y
% 0.32/0.57 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.57 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 0.32/0.57 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 0.32/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) @ Xa2 ) ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % VEBT_internal.naive_member.pelims(1)
% 0.32/0.57 thf(fact_3343_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
% 0.32/0.57 ! [X: vEBT_VEBT,Xa2: nat] :
% 0.32/0.57 ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
% 0.32/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.32/0.57 => ( ! [A3: $o,B2: $o] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.32/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa2 ) )
% 0.32/0.57 => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.32/0.57 => A3 )
% 0.32/0.57 & ( ( Xa2 != zero_zero_nat )
% 0.32/0.57 => ( ( ( Xa2 = one_one_nat )
% 0.32/0.57 => B2 )
% 0.32/0.57 & ( Xa2 = one_one_nat ) ) ) ) ) )
% 0.32/0.57 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
% 0.32/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) @ Xa2 ) )
% 0.32/0.57 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.57 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 0.32/0.57 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % VEBT_internal.naive_member.pelims(2)
% 0.32/0.57 thf(fact_3344_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
% 0.32/0.57 ! [X: vEBT_VEBT,Xa2: nat] :
% 0.32/0.57 ( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
% 0.32/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.32/0.57 => ( ! [A3: $o,B2: $o] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.32/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa2 ) )
% 0.32/0.57 => ( ( ( Xa2 = zero_zero_nat )
% 0.32/0.57 => A3 )
% 0.32/0.57 & ( ( Xa2 != zero_zero_nat )
% 0.32/0.57 => ( ( ( Xa2 = one_one_nat )
% 0.32/0.57 => B2 )
% 0.32/0.57 & ( Xa2 = one_one_nat ) ) ) ) ) )
% 0.32/0.57 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.32/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) )
% 0.32/0.57 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
% 0.32/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) @ Xa2 ) )
% 0.32/0.57 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.57 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 0.32/0.57 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % VEBT_internal.naive_member.pelims(3)
% 0.32/0.57 thf(fact_3345_vebt__member_Opelims_I2_J,axiom,
% 0.32/0.57 ! [X: vEBT_VEBT,Xa2: nat] :
% 0.32/0.57 ( ( vEBT_vebt_member @ X @ Xa2 )
% 0.32/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.32/0.57 => ( ! [A3: $o,B2: $o] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.32/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa2 ) )
% 0.32/0.57 => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.32/0.57 => A3 )
% 0.32/0.57 & ( ( Xa2 != zero_zero_nat )
% 0.32/0.57 => ( ( ( Xa2 = one_one_nat )
% 0.32/0.57 => B2 )
% 0.32/0.57 & ( Xa2 = one_one_nat ) ) ) ) ) )
% 0.32/0.57 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.32/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) )
% 0.32/0.57 => ~ ( ( Xa2 != Mi2 )
% 0.32/0.57 => ( ( Xa2 != Ma2 )
% 0.32/0.57 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.32/0.57 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.32/0.57 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.32/0.57 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.32/0.57 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.57 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 0.32/0.57 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % vebt_member.pelims(2)
% 0.32/0.57 thf(fact_3346_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
% 0.32/0.57 ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 0.32/0.57 ( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
% 0.32/0.57 = Y )
% 0.32/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.32/0.57 => ( ! [Uu2: $o,Uv2: $o] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.32/0.57 => ( ~ Y
% 0.32/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
% 0.32/0.57 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.32/0.57 => ( ~ Y
% 0.32/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) ) )
% 0.32/0.57 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.32/0.57 => ( ( Y
% 0.32/0.57 = ( ( Xa2 = Mi2 )
% 0.32/0.57 | ( Xa2 = Ma2 ) ) )
% 0.32/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) ) ) )
% 0.32/0.57 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.32/0.57 => ( ( Y
% 0.32/0.57 = ( ( Xa2 = Mi2 )
% 0.32/0.57 | ( Xa2 = Ma2 )
% 0.32/0.57 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.57 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 0.32/0.57 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 0.32/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) ) ) )
% 0.32/0.57 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 0.32/0.57 => ( ( Y
% 0.32/0.57 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.57 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 0.32/0.57 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 0.32/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % VEBT_internal.membermima.pelims(1)
% 0.32/0.57 thf(fact_3347_set__bit__nonnegative__int__iff,axiom,
% 0.32/0.57 ! [N2: nat,K: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N2 @ K ) )
% 0.32/0.57 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.32/0.57
% 0.32/0.57 % set_bit_nonnegative_int_iff
% 0.32/0.57 thf(fact_3348_unset__bit__nonnegative__int__iff,axiom,
% 0.32/0.57 ! [N2: nat,K: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N2 @ K ) )
% 0.32/0.57 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.32/0.57
% 0.32/0.57 % unset_bit_nonnegative_int_iff
% 0.32/0.57 thf(fact_3349_unset__bit__negative__int__iff,axiom,
% 0.32/0.57 ! [N2: nat,K: int] :
% 0.32/0.57 ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N2 @ K ) @ zero_zero_int )
% 0.32/0.57 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.32/0.57
% 0.32/0.57 % unset_bit_negative_int_iff
% 0.32/0.57 thf(fact_3350_set__bit__negative__int__iff,axiom,
% 0.32/0.57 ! [N2: nat,K: int] :
% 0.32/0.57 ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N2 @ K ) @ zero_zero_int )
% 0.32/0.57 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.32/0.57
% 0.32/0.57 % set_bit_negative_int_iff
% 0.32/0.57 thf(fact_3351_zle__add1__eq__le,axiom,
% 0.32/0.57 ! [W: int,Z: int] :
% 0.32/0.57 ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 0.32/0.57 = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.32/0.57
% 0.32/0.57 % zle_add1_eq_le
% 0.32/0.57 thf(fact_3352_zle__diff1__eq,axiom,
% 0.32/0.57 ! [W: int,Z: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
% 0.32/0.57 = ( ord_less_int @ W @ Z ) ) ).
% 0.32/0.57
% 0.32/0.57 % zle_diff1_eq
% 0.32/0.57 thf(fact_3353_mod__neg__neg__trivial,axiom,
% 0.32/0.57 ! [K: int,L2: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.32/0.57 => ( ( ord_less_int @ L2 @ K )
% 0.32/0.57 => ( ( modulo_modulo_int @ K @ L2 )
% 0.32/0.57 = K ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mod_neg_neg_trivial
% 0.32/0.57 thf(fact_3354_mod__pos__pos__trivial,axiom,
% 0.32/0.57 ! [K: int,L2: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.57 => ( ( ord_less_int @ K @ L2 )
% 0.32/0.57 => ( ( modulo_modulo_int @ K @ L2 )
% 0.32/0.57 = K ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mod_pos_pos_trivial
% 0.32/0.57 thf(fact_3355_add1__zle__eq,axiom,
% 0.32/0.57 ! [W: int,Z: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
% 0.32/0.57 = ( ord_less_int @ W @ Z ) ) ).
% 0.32/0.57
% 0.32/0.57 % add1_zle_eq
% 0.32/0.57 thf(fact_3356_le__imp__0__less,axiom,
% 0.32/0.57 ! [Z: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.32/0.57 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % le_imp_0_less
% 0.32/0.57 thf(fact_3357_split__zmod,axiom,
% 0.32/0.57 ! [P: int > $o,N2: int,K: int] :
% 0.32/0.57 ( ( P @ ( modulo_modulo_int @ N2 @ K ) )
% 0.32/0.57 = ( ( ( K = zero_zero_int )
% 0.32/0.57 => ( P @ N2 ) )
% 0.32/0.57 & ( ( ord_less_int @ zero_zero_int @ K )
% 0.32/0.57 => ! [I3: int,J3: int] :
% 0.32/0.57 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.32/0.57 & ( ord_less_int @ J3 @ K )
% 0.32/0.57 & ( N2
% 0.32/0.57 = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
% 0.32/0.57 => ( P @ J3 ) ) )
% 0.32/0.57 & ( ( ord_less_int @ K @ zero_zero_int )
% 0.32/0.57 => ! [I3: int,J3: int] :
% 0.32/0.57 ( ( ( ord_less_int @ K @ J3 )
% 0.32/0.57 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.32/0.57 & ( N2
% 0.32/0.57 = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
% 0.32/0.57 => ( P @ J3 ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % split_zmod
% 0.32/0.57 thf(fact_3358_mod__pos__geq,axiom,
% 0.32/0.57 ! [L2: int,K: int] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ L2 )
% 0.32/0.57 => ( ( ord_less_eq_int @ L2 @ K )
% 0.32/0.57 => ( ( modulo_modulo_int @ K @ L2 )
% 0.32/0.57 = ( modulo_modulo_int @ ( minus_minus_int @ K @ L2 ) @ L2 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mod_pos_geq
% 0.32/0.57 thf(fact_3359_q__pos__lemma,axiom,
% 0.32/0.57 ! [B6: int,Q5: int,R4: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q5 ) @ R4 ) )
% 0.32/0.57 => ( ( ord_less_int @ R4 @ B6 )
% 0.32/0.57 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 0.32/0.57 => ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % q_pos_lemma
% 0.32/0.57 thf(fact_3360_int__less__induct,axiom,
% 0.32/0.57 ! [I2: int,K: int,P: int > $o] :
% 0.32/0.57 ( ( ord_less_int @ I2 @ K )
% 0.32/0.57 => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
% 0.32/0.57 => ( ! [I4: int] :
% 0.32/0.57 ( ( ord_less_int @ I4 @ K )
% 0.32/0.57 => ( ( P @ I4 )
% 0.32/0.57 => ( P @ ( minus_minus_int @ I4 @ one_one_int ) ) ) )
% 0.32/0.57 => ( P @ I2 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % int_less_induct
% 0.32/0.57 thf(fact_3361_neg__mod__conj,axiom,
% 0.32/0.57 ! [B: int,A: int] :
% 0.32/0.57 ( ( ord_less_int @ B @ zero_zero_int )
% 0.32/0.57 => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
% 0.32/0.57 & ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % neg_mod_conj
% 0.32/0.57 thf(fact_3362_pos__mod__conj,axiom,
% 0.32/0.57 ! [B: int,A: int] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.57 => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
% 0.32/0.57 & ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % pos_mod_conj
% 0.32/0.57 thf(fact_3363_int__mod__neg__eq,axiom,
% 0.32/0.57 ! [A: int,B: int,Q2: int,R: int] :
% 0.32/0.57 ( ( A
% 0.32/0.57 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 0.32/0.57 => ( ( ord_less_eq_int @ R @ zero_zero_int )
% 0.32/0.57 => ( ( ord_less_int @ B @ R )
% 0.32/0.57 => ( ( modulo_modulo_int @ A @ B )
% 0.32/0.57 = R ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % int_mod_neg_eq
% 0.32/0.57 thf(fact_3364_int__mod__pos__eq,axiom,
% 0.32/0.57 ! [A: int,B: int,Q2: int,R: int] :
% 0.32/0.57 ( ( A
% 0.32/0.57 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 0.32/0.57 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 0.32/0.57 => ( ( ord_less_int @ R @ B )
% 0.32/0.57 => ( ( modulo_modulo_int @ A @ B )
% 0.32/0.57 = R ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % int_mod_pos_eq
% 0.32/0.57 thf(fact_3365_zless__imp__add1__zle,axiom,
% 0.32/0.57 ! [W: int,Z: int] :
% 0.32/0.57 ( ( ord_less_int @ W @ Z )
% 0.32/0.57 => ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% 0.32/0.57
% 0.32/0.57 % zless_imp_add1_zle
% 0.32/0.57 thf(fact_3366_zdiv__mono2__lemma,axiom,
% 0.32/0.57 ! [B: int,Q2: int,R: int,B6: int,Q5: int,R4: int] :
% 0.32/0.57 ( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R )
% 0.32/0.57 = ( plus_plus_int @ ( times_times_int @ B6 @ Q5 ) @ R4 ) )
% 0.32/0.57 => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q5 ) @ R4 ) )
% 0.32/0.57 => ( ( ord_less_int @ R4 @ B6 )
% 0.32/0.57 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 0.32/0.57 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 0.32/0.57 => ( ( ord_less_eq_int @ B6 @ B )
% 0.32/0.57 => ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % zdiv_mono2_lemma
% 0.32/0.57 thf(fact_3367_zmod__trivial__iff,axiom,
% 0.32/0.57 ! [I2: int,K: int] :
% 0.32/0.57 ( ( ( modulo_modulo_int @ I2 @ K )
% 0.32/0.57 = I2 )
% 0.32/0.57 = ( ( K = zero_zero_int )
% 0.32/0.57 | ( ( ord_less_eq_int @ zero_zero_int @ I2 )
% 0.32/0.57 & ( ord_less_int @ I2 @ K ) )
% 0.32/0.57 | ( ( ord_less_eq_int @ I2 @ zero_zero_int )
% 0.32/0.57 & ( ord_less_int @ K @ I2 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % zmod_trivial_iff
% 0.32/0.57 thf(fact_3368_zdiv__mono2__neg__lemma,axiom,
% 0.32/0.57 ! [B: int,Q2: int,R: int,B6: int,Q5: int,R4: int] :
% 0.32/0.57 ( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R )
% 0.32/0.57 = ( plus_plus_int @ ( times_times_int @ B6 @ Q5 ) @ R4 ) )
% 0.32/0.57 => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q5 ) @ R4 ) @ zero_zero_int )
% 0.32/0.57 => ( ( ord_less_int @ R @ B )
% 0.32/0.57 => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 0.32/0.57 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 0.32/0.57 => ( ( ord_less_eq_int @ B6 @ B )
% 0.32/0.57 => ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % zdiv_mono2_neg_lemma
% 0.32/0.57 thf(fact_3369_int__one__le__iff__zero__less,axiom,
% 0.32/0.57 ! [Z: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ one_one_int @ Z )
% 0.32/0.57 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.32/0.57
% 0.32/0.57 % int_one_le_iff_zero_less
% 0.32/0.57 thf(fact_3370_unique__quotient__lemma,axiom,
% 0.32/0.57 ! [B: int,Q5: int,R4: int,Q2: int,R: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 0.32/0.57 => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 0.32/0.57 => ( ( ord_less_int @ R4 @ B )
% 0.32/0.57 => ( ( ord_less_int @ R @ B )
% 0.32/0.57 => ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % unique_quotient_lemma
% 0.32/0.57 thf(fact_3371_neg__mod__sign,axiom,
% 0.32/0.57 ! [L2: int,K: int] :
% 0.32/0.57 ( ( ord_less_int @ L2 @ zero_zero_int )
% 0.32/0.57 => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L2 ) @ zero_zero_int ) ) ).
% 0.32/0.57
% 0.32/0.57 % neg_mod_sign
% 0.32/0.57 thf(fact_3372_Euclidean__Division_Opos__mod__sign,axiom,
% 0.32/0.57 ! [L2: int,K: int] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ L2 )
% 0.32/0.57 => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L2 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % Euclidean_Division.pos_mod_sign
% 0.32/0.57 thf(fact_3373_unique__quotient__lemma__neg,axiom,
% 0.32/0.57 ! [B: int,Q5: int,R4: int,Q2: int,R: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 0.32/0.57 => ( ( ord_less_eq_int @ R @ zero_zero_int )
% 0.32/0.57 => ( ( ord_less_int @ B @ R )
% 0.32/0.57 => ( ( ord_less_int @ B @ R4 )
% 0.32/0.57 => ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % unique_quotient_lemma_neg
% 0.32/0.57 thf(fact_3374_mod__pos__neg__trivial,axiom,
% 0.32/0.57 ! [K: int,L2: int] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ K )
% 0.32/0.57 => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L2 ) @ zero_zero_int )
% 0.32/0.57 => ( ( modulo_modulo_int @ K @ L2 )
% 0.32/0.57 = ( plus_plus_int @ K @ L2 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mod_pos_neg_trivial
% 0.32/0.57 thf(fact_3375_minus__int__code_I1_J,axiom,
% 0.32/0.57 ! [K: int] :
% 0.32/0.57 ( ( minus_minus_int @ K @ zero_zero_int )
% 0.32/0.57 = K ) ).
% 0.32/0.57
% 0.32/0.57 % minus_int_code(1)
% 0.32/0.57 thf(fact_3376_Euclidean__Division_Opos__mod__bound,axiom,
% 0.32/0.57 ! [L2: int,K: int] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ L2 )
% 0.32/0.57 => ( ord_less_int @ ( modulo_modulo_int @ K @ L2 ) @ L2 ) ) ).
% 0.32/0.57
% 0.32/0.57 % Euclidean_Division.pos_mod_bound
% 0.32/0.57 thf(fact_3377_neg__mod__bound,axiom,
% 0.32/0.57 ! [L2: int,K: int] :
% 0.32/0.57 ( ( ord_less_int @ L2 @ zero_zero_int )
% 0.32/0.57 => ( ord_less_int @ L2 @ ( modulo_modulo_int @ K @ L2 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % neg_mod_bound
% 0.32/0.57 thf(fact_3378_less__int__code_I1_J,axiom,
% 0.32/0.57 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.32/0.57
% 0.32/0.57 % less_int_code(1)
% 0.32/0.57 thf(fact_3379_zmult__zless__mono2,axiom,
% 0.32/0.57 ! [I2: int,J: int,K: int] :
% 0.32/0.57 ( ( ord_less_int @ I2 @ J )
% 0.32/0.57 => ( ( ord_less_int @ zero_zero_int @ K )
% 0.32/0.57 => ( ord_less_int @ ( times_times_int @ K @ I2 ) @ ( times_times_int @ K @ J ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % zmult_zless_mono2
% 0.32/0.57 thf(fact_3380_pos__zmult__eq__1__iff,axiom,
% 0.32/0.57 ! [M: int,N2: int] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ M )
% 0.32/0.57 => ( ( ( times_times_int @ M @ N2 )
% 0.32/0.57 = one_one_int )
% 0.32/0.57 = ( ( M = one_one_int )
% 0.32/0.57 & ( N2 = one_one_int ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % pos_zmult_eq_1_iff
% 0.32/0.57 thf(fact_3381_odd__less__0__iff,axiom,
% 0.32/0.57 ! [Z: int] :
% 0.32/0.57 ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
% 0.32/0.57 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.32/0.57
% 0.32/0.57 % odd_less_0_iff
% 0.32/0.57 thf(fact_3382_zless__add1__eq,axiom,
% 0.32/0.57 ! [W: int,Z: int] :
% 0.32/0.57 ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 0.32/0.57 = ( ( ord_less_int @ W @ Z )
% 0.32/0.57 | ( W = Z ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % zless_add1_eq
% 0.32/0.57 thf(fact_3383_int__gr__induct,axiom,
% 0.32/0.57 ! [K: int,I2: int,P: int > $o] :
% 0.32/0.57 ( ( ord_less_int @ K @ I2 )
% 0.32/0.57 => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
% 0.32/0.57 => ( ! [I4: int] :
% 0.32/0.57 ( ( ord_less_int @ K @ I4 )
% 0.32/0.57 => ( ( P @ I4 )
% 0.32/0.57 => ( P @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
% 0.32/0.57 => ( P @ I2 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % int_gr_induct
% 0.32/0.57 thf(fact_3384_zero__one__enat__neq_I1_J,axiom,
% 0.32/0.57 zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% 0.32/0.57
% 0.32/0.57 % zero_one_enat_neq(1)
% 0.32/0.57 thf(fact_3385_odd__nonzero,axiom,
% 0.32/0.57 ! [Z: int] :
% 0.32/0.57 ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
% 0.32/0.57 != zero_zero_int ) ).
% 0.32/0.57
% 0.32/0.57 % odd_nonzero
% 0.32/0.57 thf(fact_3386_set__bit__greater__eq,axiom,
% 0.32/0.57 ! [K: int,N2: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N2 @ K ) ) ).
% 0.32/0.57
% 0.32/0.57 % set_bit_greater_eq
% 0.32/0.57 thf(fact_3387_unset__bit__less__eq,axiom,
% 0.32/0.57 ! [N2: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N2 @ K ) @ K ) ).
% 0.32/0.57
% 0.32/0.57 % unset_bit_less_eq
% 0.32/0.57 thf(fact_3388_int__distrib_I4_J,axiom,
% 0.32/0.57 ! [W: int,Z1: int,Z22: int] :
% 0.32/0.57 ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
% 0.32/0.57 = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % int_distrib(4)
% 0.32/0.57 thf(fact_3389_int__distrib_I3_J,axiom,
% 0.32/0.57 ! [Z1: int,Z22: int,W: int] :
% 0.32/0.57 ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
% 0.32/0.57 = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % int_distrib(3)
% 0.32/0.57 thf(fact_3390_int__le__induct,axiom,
% 0.32/0.57 ! [I2: int,K: int,P: int > $o] :
% 0.32/0.57 ( ( ord_less_eq_int @ I2 @ K )
% 0.32/0.57 => ( ( P @ K )
% 0.32/0.57 => ( ! [I4: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ I4 @ K )
% 0.32/0.57 => ( ( P @ I4 )
% 0.32/0.57 => ( P @ ( minus_minus_int @ I4 @ one_one_int ) ) ) )
% 0.32/0.57 => ( P @ I2 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % int_le_induct
% 0.32/0.57 thf(fact_3391_int__ge__induct,axiom,
% 0.32/0.57 ! [K: int,I2: int,P: int > $o] :
% 0.32/0.57 ( ( ord_less_eq_int @ K @ I2 )
% 0.32/0.57 => ( ( P @ K )
% 0.32/0.57 => ( ! [I4: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ K @ I4 )
% 0.32/0.57 => ( ( P @ I4 )
% 0.32/0.57 => ( P @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
% 0.32/0.57 => ( P @ I2 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % int_ge_induct
% 0.32/0.57 thf(fact_3392_int__induct,axiom,
% 0.32/0.57 ! [P: int > $o,K: int,I2: int] :
% 0.32/0.57 ( ( P @ K )
% 0.32/0.57 => ( ! [I4: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ K @ I4 )
% 0.32/0.57 => ( ( P @ I4 )
% 0.32/0.57 => ( P @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
% 0.32/0.57 => ( ! [I4: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ I4 @ K )
% 0.32/0.57 => ( ( P @ I4 )
% 0.32/0.57 => ( P @ ( minus_minus_int @ I4 @ one_one_int ) ) ) )
% 0.32/0.57 => ( P @ I2 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % int_induct
% 0.32/0.57 thf(fact_3393_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
% 0.32/0.57 ! [X: vEBT_VEBT,Xa2: nat] :
% 0.32/0.57 ( ( vEBT_VEBT_membermima @ X @ Xa2 )
% 0.32/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.32/0.57 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.32/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
% 0.32/0.57 => ~ ( ( Xa2 = Mi2 )
% 0.32/0.57 | ( Xa2 = Ma2 ) ) ) )
% 0.32/0.57 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.32/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
% 0.32/0.57 => ~ ( ( Xa2 = Mi2 )
% 0.32/0.57 | ( Xa2 = Ma2 )
% 0.32/0.57 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.57 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 0.32/0.57 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 0.32/0.57 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 0.32/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) )
% 0.32/0.57 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.57 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 0.32/0.57 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % VEBT_internal.membermima.pelims(2)
% 0.32/0.57 thf(fact_3394_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
% 0.32/0.57 ! [X: vEBT_VEBT,Xa2: nat] :
% 0.32/0.57 ( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
% 0.32/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.32/0.57 => ( ! [Uu2: $o,Uv2: $o] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.32/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) )
% 0.32/0.57 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.32/0.57 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) )
% 0.32/0.57 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.32/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
% 0.32/0.57 => ( ( Xa2 = Mi2 )
% 0.32/0.57 | ( Xa2 = Ma2 ) ) ) )
% 0.32/0.57 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.32/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
% 0.32/0.57 => ( ( Xa2 = Mi2 )
% 0.32/0.57 | ( Xa2 = Ma2 )
% 0.32/0.57 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.57 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 0.32/0.57 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 0.32/0.57 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.32/0.57 ( ( X
% 0.32/0.57 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 0.32/0.57 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) )
% 0.32/0.57 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.57 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 0.32/0.57 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % VEBT_internal.membermima.pelims(3)
% 0.32/0.57 thf(fact_3395_atLeastatMost__empty,axiom,
% 0.32/0.57 ! [B: rat,A: rat] :
% 0.32/0.57 ( ( ord_less_rat @ B @ A )
% 0.32/0.57 => ( ( set_or633870826150836451st_rat @ A @ B )
% 0.32/0.57 = bot_bot_set_rat ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_empty
% 0.32/0.57 thf(fact_3396_atLeastatMost__empty,axiom,
% 0.32/0.57 ! [B: num,A: num] :
% 0.32/0.57 ( ( ord_less_num @ B @ A )
% 0.32/0.57 => ( ( set_or7049704709247886629st_num @ A @ B )
% 0.32/0.57 = bot_bot_set_num ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_empty
% 0.32/0.57 thf(fact_3397_atLeastatMost__empty,axiom,
% 0.32/0.57 ! [B: nat,A: nat] :
% 0.32/0.57 ( ( ord_less_nat @ B @ A )
% 0.32/0.57 => ( ( set_or1269000886237332187st_nat @ A @ B )
% 0.32/0.57 = bot_bot_set_nat ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_empty
% 0.32/0.57 thf(fact_3398_atLeastatMost__empty,axiom,
% 0.32/0.57 ! [B: int,A: int] :
% 0.32/0.57 ( ( ord_less_int @ B @ A )
% 0.32/0.57 => ( ( set_or1266510415728281911st_int @ A @ B )
% 0.32/0.57 = bot_bot_set_int ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_empty
% 0.32/0.57 thf(fact_3399_atLeastatMost__empty,axiom,
% 0.32/0.57 ! [B: real,A: real] :
% 0.32/0.57 ( ( ord_less_real @ B @ A )
% 0.32/0.57 => ( ( set_or1222579329274155063t_real @ A @ B )
% 0.32/0.57 = bot_bot_set_real ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_empty
% 0.32/0.57 thf(fact_3400_atLeastatMost__subset__iff,axiom,
% 0.32/0.57 ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
% 0.32/0.57 ( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
% 0.32/0.57 = ( ~ ( ord_less_eq_set_nat @ A @ B )
% 0.32/0.57 | ( ( ord_less_eq_set_nat @ C @ A )
% 0.32/0.57 & ( ord_less_eq_set_nat @ B @ D ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_subset_iff
% 0.32/0.57 thf(fact_3401_atLeastatMost__subset__iff,axiom,
% 0.32/0.57 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.57 ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 0.32/0.57 = ( ~ ( ord_less_eq_rat @ A @ B )
% 0.32/0.57 | ( ( ord_less_eq_rat @ C @ A )
% 0.32/0.57 & ( ord_less_eq_rat @ B @ D ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_subset_iff
% 0.32/0.57 thf(fact_3402_atLeastatMost__subset__iff,axiom,
% 0.32/0.57 ! [A: num,B: num,C: num,D: num] :
% 0.32/0.57 ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 0.32/0.57 = ( ~ ( ord_less_eq_num @ A @ B )
% 0.32/0.57 | ( ( ord_less_eq_num @ C @ A )
% 0.32/0.57 & ( ord_less_eq_num @ B @ D ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_subset_iff
% 0.32/0.57 thf(fact_3403_atLeastatMost__subset__iff,axiom,
% 0.32/0.57 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.57 ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 0.32/0.57 = ( ~ ( ord_less_eq_nat @ A @ B )
% 0.32/0.57 | ( ( ord_less_eq_nat @ C @ A )
% 0.32/0.57 & ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_subset_iff
% 0.32/0.57 thf(fact_3404_atLeastatMost__subset__iff,axiom,
% 0.32/0.57 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.57 ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 0.32/0.57 = ( ~ ( ord_less_eq_int @ A @ B )
% 0.32/0.57 | ( ( ord_less_eq_int @ C @ A )
% 0.32/0.57 & ( ord_less_eq_int @ B @ D ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_subset_iff
% 0.32/0.57 thf(fact_3405_atLeastatMost__subset__iff,axiom,
% 0.32/0.57 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.57 ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 0.32/0.57 = ( ~ ( ord_less_eq_real @ A @ B )
% 0.32/0.57 | ( ( ord_less_eq_real @ C @ A )
% 0.32/0.57 & ( ord_less_eq_real @ B @ D ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_subset_iff
% 0.32/0.57 thf(fact_3406_atLeastatMost__empty__iff2,axiom,
% 0.32/0.57 ! [A: set_nat,B: set_nat] :
% 0.32/0.57 ( ( bot_bot_set_set_nat
% 0.32/0.57 = ( set_or4548717258645045905et_nat @ A @ B ) )
% 0.32/0.57 = ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_empty_iff2
% 0.32/0.57 thf(fact_3407_atLeastatMost__empty__iff2,axiom,
% 0.32/0.57 ! [A: rat,B: rat] :
% 0.32/0.57 ( ( bot_bot_set_rat
% 0.32/0.57 = ( set_or633870826150836451st_rat @ A @ B ) )
% 0.32/0.57 = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_empty_iff2
% 0.32/0.57 thf(fact_3408_atLeastatMost__empty__iff2,axiom,
% 0.32/0.57 ! [A: num,B: num] :
% 0.32/0.57 ( ( bot_bot_set_num
% 0.32/0.57 = ( set_or7049704709247886629st_num @ A @ B ) )
% 0.32/0.57 = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_empty_iff2
% 0.32/0.57 thf(fact_3409_atLeastatMost__empty__iff2,axiom,
% 0.32/0.57 ! [A: nat,B: nat] :
% 0.32/0.57 ( ( bot_bot_set_nat
% 0.32/0.57 = ( set_or1269000886237332187st_nat @ A @ B ) )
% 0.32/0.57 = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_empty_iff2
% 0.32/0.57 thf(fact_3410_atLeastatMost__empty__iff2,axiom,
% 0.32/0.57 ! [A: int,B: int] :
% 0.32/0.57 ( ( bot_bot_set_int
% 0.32/0.57 = ( set_or1266510415728281911st_int @ A @ B ) )
% 0.32/0.57 = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_empty_iff2
% 0.32/0.57 thf(fact_3411_atLeastatMost__empty__iff2,axiom,
% 0.32/0.57 ! [A: real,B: real] :
% 0.32/0.57 ( ( bot_bot_set_real
% 0.32/0.57 = ( set_or1222579329274155063t_real @ A @ B ) )
% 0.32/0.57 = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_empty_iff2
% 0.32/0.57 thf(fact_3412_atLeastatMost__empty__iff,axiom,
% 0.32/0.57 ! [A: set_nat,B: set_nat] :
% 0.32/0.57 ( ( ( set_or4548717258645045905et_nat @ A @ B )
% 0.32/0.57 = bot_bot_set_set_nat )
% 0.32/0.57 = ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_empty_iff
% 0.32/0.57 thf(fact_3413_atLeastatMost__empty__iff,axiom,
% 0.32/0.57 ! [A: rat,B: rat] :
% 0.32/0.57 ( ( ( set_or633870826150836451st_rat @ A @ B )
% 0.32/0.57 = bot_bot_set_rat )
% 0.32/0.57 = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_empty_iff
% 0.32/0.57 thf(fact_3414_atLeastatMost__empty__iff,axiom,
% 0.32/0.57 ! [A: num,B: num] :
% 0.32/0.57 ( ( ( set_or7049704709247886629st_num @ A @ B )
% 0.32/0.57 = bot_bot_set_num )
% 0.32/0.57 = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_empty_iff
% 0.32/0.57 thf(fact_3415_atLeastatMost__empty__iff,axiom,
% 0.32/0.57 ! [A: nat,B: nat] :
% 0.32/0.57 ( ( ( set_or1269000886237332187st_nat @ A @ B )
% 0.32/0.57 = bot_bot_set_nat )
% 0.32/0.57 = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_empty_iff
% 0.32/0.57 thf(fact_3416_atLeastatMost__empty__iff,axiom,
% 0.32/0.57 ! [A: int,B: int] :
% 0.32/0.57 ( ( ( set_or1266510415728281911st_int @ A @ B )
% 0.32/0.57 = bot_bot_set_int )
% 0.32/0.57 = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_empty_iff
% 0.32/0.57 thf(fact_3417_atLeastatMost__empty__iff,axiom,
% 0.32/0.57 ! [A: real,B: real] :
% 0.32/0.57 ( ( ( set_or1222579329274155063t_real @ A @ B )
% 0.32/0.57 = bot_bot_set_real )
% 0.32/0.57 = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_empty_iff
% 0.32/0.57 thf(fact_3418_decr__mult__lemma,axiom,
% 0.32/0.57 ! [D: int,P: int > $o,K: int] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ D )
% 0.32/0.57 => ( ! [X4: int] :
% 0.32/0.57 ( ( P @ X4 )
% 0.32/0.57 => ( P @ ( minus_minus_int @ X4 @ D ) ) )
% 0.32/0.57 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.57 => ! [X5: int] :
% 0.32/0.57 ( ( P @ X5 )
% 0.32/0.57 => ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % decr_mult_lemma
% 0.32/0.57 thf(fact_3419_incr__mult__lemma,axiom,
% 0.32/0.57 ! [D: int,P: int > $o,K: int] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ D )
% 0.32/0.57 => ( ! [X4: int] :
% 0.32/0.57 ( ( P @ X4 )
% 0.32/0.57 => ( P @ ( plus_plus_int @ X4 @ D ) ) )
% 0.32/0.57 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.57 => ! [X5: int] :
% 0.32/0.57 ( ( P @ X5 )
% 0.32/0.57 => ( P @ ( plus_plus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % incr_mult_lemma
% 0.32/0.57 thf(fact_3420_Icc__eq__Icc,axiom,
% 0.32/0.57 ! [L2: set_nat,H2: set_nat,L3: set_nat,H3: set_nat] :
% 0.32/0.57 ( ( ( set_or4548717258645045905et_nat @ L2 @ H2 )
% 0.32/0.57 = ( set_or4548717258645045905et_nat @ L3 @ H3 ) )
% 0.32/0.57 = ( ( ( L2 = L3 )
% 0.32/0.57 & ( H2 = H3 ) )
% 0.32/0.57 | ( ~ ( ord_less_eq_set_nat @ L2 @ H2 )
% 0.32/0.57 & ~ ( ord_less_eq_set_nat @ L3 @ H3 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % Icc_eq_Icc
% 0.32/0.57 thf(fact_3421_Icc__eq__Icc,axiom,
% 0.32/0.57 ! [L2: rat,H2: rat,L3: rat,H3: rat] :
% 0.32/0.57 ( ( ( set_or633870826150836451st_rat @ L2 @ H2 )
% 0.32/0.57 = ( set_or633870826150836451st_rat @ L3 @ H3 ) )
% 0.32/0.57 = ( ( ( L2 = L3 )
% 0.32/0.57 & ( H2 = H3 ) )
% 0.32/0.57 | ( ~ ( ord_less_eq_rat @ L2 @ H2 )
% 0.32/0.57 & ~ ( ord_less_eq_rat @ L3 @ H3 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % Icc_eq_Icc
% 0.32/0.57 thf(fact_3422_Icc__eq__Icc,axiom,
% 0.32/0.57 ! [L2: num,H2: num,L3: num,H3: num] :
% 0.32/0.57 ( ( ( set_or7049704709247886629st_num @ L2 @ H2 )
% 0.32/0.57 = ( set_or7049704709247886629st_num @ L3 @ H3 ) )
% 0.32/0.57 = ( ( ( L2 = L3 )
% 0.32/0.57 & ( H2 = H3 ) )
% 0.32/0.57 | ( ~ ( ord_less_eq_num @ L2 @ H2 )
% 0.32/0.57 & ~ ( ord_less_eq_num @ L3 @ H3 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % Icc_eq_Icc
% 0.32/0.57 thf(fact_3423_Icc__eq__Icc,axiom,
% 0.32/0.57 ! [L2: nat,H2: nat,L3: nat,H3: nat] :
% 0.32/0.57 ( ( ( set_or1269000886237332187st_nat @ L2 @ H2 )
% 0.32/0.57 = ( set_or1269000886237332187st_nat @ L3 @ H3 ) )
% 0.32/0.57 = ( ( ( L2 = L3 )
% 0.32/0.57 & ( H2 = H3 ) )
% 0.32/0.57 | ( ~ ( ord_less_eq_nat @ L2 @ H2 )
% 0.32/0.57 & ~ ( ord_less_eq_nat @ L3 @ H3 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % Icc_eq_Icc
% 0.32/0.57 thf(fact_3424_Icc__eq__Icc,axiom,
% 0.32/0.57 ! [L2: int,H2: int,L3: int,H3: int] :
% 0.32/0.57 ( ( ( set_or1266510415728281911st_int @ L2 @ H2 )
% 0.32/0.57 = ( set_or1266510415728281911st_int @ L3 @ H3 ) )
% 0.32/0.57 = ( ( ( L2 = L3 )
% 0.32/0.57 & ( H2 = H3 ) )
% 0.32/0.57 | ( ~ ( ord_less_eq_int @ L2 @ H2 )
% 0.32/0.57 & ~ ( ord_less_eq_int @ L3 @ H3 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % Icc_eq_Icc
% 0.32/0.57 thf(fact_3425_Icc__eq__Icc,axiom,
% 0.32/0.57 ! [L2: real,H2: real,L3: real,H3: real] :
% 0.32/0.57 ( ( ( set_or1222579329274155063t_real @ L2 @ H2 )
% 0.32/0.57 = ( set_or1222579329274155063t_real @ L3 @ H3 ) )
% 0.32/0.57 = ( ( ( L2 = L3 )
% 0.32/0.57 & ( H2 = H3 ) )
% 0.32/0.57 | ( ~ ( ord_less_eq_real @ L2 @ H2 )
% 0.32/0.57 & ~ ( ord_less_eq_real @ L3 @ H3 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % Icc_eq_Icc
% 0.32/0.57 thf(fact_3426_atLeastAtMost__iff,axiom,
% 0.32/0.57 ! [I2: set_nat,L2: set_nat,U: set_nat] :
% 0.32/0.57 ( ( member_set_nat @ I2 @ ( set_or4548717258645045905et_nat @ L2 @ U ) )
% 0.32/0.57 = ( ( ord_less_eq_set_nat @ L2 @ I2 )
% 0.32/0.57 & ( ord_less_eq_set_nat @ I2 @ U ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastAtMost_iff
% 0.32/0.57 thf(fact_3427_atLeastAtMost__iff,axiom,
% 0.32/0.57 ! [I2: rat,L2: rat,U: rat] :
% 0.32/0.57 ( ( member_rat @ I2 @ ( set_or633870826150836451st_rat @ L2 @ U ) )
% 0.32/0.57 = ( ( ord_less_eq_rat @ L2 @ I2 )
% 0.32/0.57 & ( ord_less_eq_rat @ I2 @ U ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastAtMost_iff
% 0.32/0.57 thf(fact_3428_atLeastAtMost__iff,axiom,
% 0.32/0.57 ! [I2: num,L2: num,U: num] :
% 0.32/0.57 ( ( member_num @ I2 @ ( set_or7049704709247886629st_num @ L2 @ U ) )
% 0.32/0.57 = ( ( ord_less_eq_num @ L2 @ I2 )
% 0.32/0.57 & ( ord_less_eq_num @ I2 @ U ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastAtMost_iff
% 0.32/0.57 thf(fact_3429_atLeastAtMost__iff,axiom,
% 0.32/0.57 ! [I2: nat,L2: nat,U: nat] :
% 0.32/0.57 ( ( member_nat @ I2 @ ( set_or1269000886237332187st_nat @ L2 @ U ) )
% 0.32/0.57 = ( ( ord_less_eq_nat @ L2 @ I2 )
% 0.32/0.57 & ( ord_less_eq_nat @ I2 @ U ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastAtMost_iff
% 0.32/0.57 thf(fact_3430_atLeastAtMost__iff,axiom,
% 0.32/0.57 ! [I2: int,L2: int,U: int] :
% 0.32/0.57 ( ( member_int @ I2 @ ( set_or1266510415728281911st_int @ L2 @ U ) )
% 0.32/0.57 = ( ( ord_less_eq_int @ L2 @ I2 )
% 0.32/0.57 & ( ord_less_eq_int @ I2 @ U ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastAtMost_iff
% 0.32/0.57 thf(fact_3431_atLeastAtMost__iff,axiom,
% 0.32/0.57 ! [I2: real,L2: real,U: real] :
% 0.32/0.57 ( ( member_real @ I2 @ ( set_or1222579329274155063t_real @ L2 @ U ) )
% 0.32/0.57 = ( ( ord_less_eq_real @ L2 @ I2 )
% 0.32/0.57 & ( ord_less_eq_real @ I2 @ U ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastAtMost_iff
% 0.32/0.57 thf(fact_3432_aset_I2_J,axiom,
% 0.32/0.57 ! [D3: int,A2: set_int,P: int > $o,Q: int > $o] :
% 0.32/0.57 ( ! [X4: int] :
% 0.32/0.57 ( ! [Xa: int] :
% 0.32/0.57 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 => ! [Xb: int] :
% 0.32/0.57 ( ( member_int @ Xb @ A2 )
% 0.32/0.57 => ( X4
% 0.32/0.57 != ( minus_minus_int @ Xb @ Xa ) ) ) )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 => ( P @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 0.32/0.57 => ( ! [X4: int] :
% 0.32/0.57 ( ! [Xa: int] :
% 0.32/0.57 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 => ! [Xb: int] :
% 0.32/0.57 ( ( member_int @ Xb @ A2 )
% 0.32/0.57 => ( X4
% 0.32/0.57 != ( minus_minus_int @ Xb @ Xa ) ) ) )
% 0.32/0.57 => ( ( Q @ X4 )
% 0.32/0.57 => ( Q @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 0.32/0.57 => ! [X5: int] :
% 0.32/0.57 ( ! [Xa3: int] :
% 0.32/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 => ! [Xb3: int] :
% 0.32/0.57 ( ( member_int @ Xb3 @ A2 )
% 0.32/0.57 => ( X5
% 0.32/0.57 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.57 => ( ( ( P @ X5 )
% 0.32/0.57 | ( Q @ X5 ) )
% 0.32/0.57 => ( ( P @ ( plus_plus_int @ X5 @ D3 ) )
% 0.32/0.57 | ( Q @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % aset(2)
% 0.32/0.57 thf(fact_3433_aset_I1_J,axiom,
% 0.32/0.57 ! [D3: int,A2: set_int,P: int > $o,Q: int > $o] :
% 0.32/0.57 ( ! [X4: int] :
% 0.32/0.57 ( ! [Xa: int] :
% 0.32/0.57 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 => ! [Xb: int] :
% 0.32/0.57 ( ( member_int @ Xb @ A2 )
% 0.32/0.57 => ( X4
% 0.32/0.57 != ( minus_minus_int @ Xb @ Xa ) ) ) )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 => ( P @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 0.32/0.57 => ( ! [X4: int] :
% 0.32/0.57 ( ! [Xa: int] :
% 0.32/0.57 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 => ! [Xb: int] :
% 0.32/0.57 ( ( member_int @ Xb @ A2 )
% 0.32/0.57 => ( X4
% 0.32/0.57 != ( minus_minus_int @ Xb @ Xa ) ) ) )
% 0.32/0.57 => ( ( Q @ X4 )
% 0.32/0.57 => ( Q @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 0.32/0.57 => ! [X5: int] :
% 0.32/0.57 ( ! [Xa3: int] :
% 0.32/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 => ! [Xb3: int] :
% 0.32/0.57 ( ( member_int @ Xb3 @ A2 )
% 0.32/0.57 => ( X5
% 0.32/0.57 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.57 => ( ( ( P @ X5 )
% 0.32/0.57 & ( Q @ X5 ) )
% 0.32/0.57 => ( ( P @ ( plus_plus_int @ X5 @ D3 ) )
% 0.32/0.57 & ( Q @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % aset(1)
% 0.32/0.57 thf(fact_3434_bset_I2_J,axiom,
% 0.32/0.57 ! [D3: int,B3: set_int,P: int > $o,Q: int > $o] :
% 0.32/0.57 ( ! [X4: int] :
% 0.32/0.57 ( ! [Xa: int] :
% 0.32/0.57 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 => ! [Xb: int] :
% 0.32/0.57 ( ( member_int @ Xb @ B3 )
% 0.32/0.57 => ( X4
% 0.32/0.57 != ( plus_plus_int @ Xb @ Xa ) ) ) )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 => ( P @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 0.32/0.57 => ( ! [X4: int] :
% 0.32/0.57 ( ! [Xa: int] :
% 0.32/0.57 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 => ! [Xb: int] :
% 0.32/0.57 ( ( member_int @ Xb @ B3 )
% 0.32/0.57 => ( X4
% 0.32/0.57 != ( plus_plus_int @ Xb @ Xa ) ) ) )
% 0.32/0.57 => ( ( Q @ X4 )
% 0.32/0.57 => ( Q @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 0.32/0.57 => ! [X5: int] :
% 0.32/0.57 ( ! [Xa3: int] :
% 0.32/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 => ! [Xb3: int] :
% 0.32/0.57 ( ( member_int @ Xb3 @ B3 )
% 0.32/0.57 => ( X5
% 0.32/0.57 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.57 => ( ( ( P @ X5 )
% 0.32/0.57 | ( Q @ X5 ) )
% 0.32/0.57 => ( ( P @ ( minus_minus_int @ X5 @ D3 ) )
% 0.32/0.57 | ( Q @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % bset(2)
% 0.32/0.57 thf(fact_3435_bset_I1_J,axiom,
% 0.32/0.57 ! [D3: int,B3: set_int,P: int > $o,Q: int > $o] :
% 0.32/0.57 ( ! [X4: int] :
% 0.32/0.57 ( ! [Xa: int] :
% 0.32/0.57 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 => ! [Xb: int] :
% 0.32/0.57 ( ( member_int @ Xb @ B3 )
% 0.32/0.57 => ( X4
% 0.32/0.57 != ( plus_plus_int @ Xb @ Xa ) ) ) )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 => ( P @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 0.32/0.57 => ( ! [X4: int] :
% 0.32/0.57 ( ! [Xa: int] :
% 0.32/0.57 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 => ! [Xb: int] :
% 0.32/0.57 ( ( member_int @ Xb @ B3 )
% 0.32/0.57 => ( X4
% 0.32/0.57 != ( plus_plus_int @ Xb @ Xa ) ) ) )
% 0.32/0.57 => ( ( Q @ X4 )
% 0.32/0.57 => ( Q @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 0.32/0.57 => ! [X5: int] :
% 0.32/0.57 ( ! [Xa3: int] :
% 0.32/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 => ! [Xb3: int] :
% 0.32/0.57 ( ( member_int @ Xb3 @ B3 )
% 0.32/0.57 => ( X5
% 0.32/0.57 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.57 => ( ( ( P @ X5 )
% 0.32/0.57 & ( Q @ X5 ) )
% 0.32/0.57 => ( ( P @ ( minus_minus_int @ X5 @ D3 ) )
% 0.32/0.57 & ( Q @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % bset(1)
% 0.32/0.57 thf(fact_3436_pinf_I1_J,axiom,
% 0.32/0.57 ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
% 0.32/0.57 ( ? [Z4: real] :
% 0.32/0.57 ! [X4: real] :
% 0.32/0.57 ( ( ord_less_real @ Z4 @ X4 )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 = ( P6 @ X4 ) ) )
% 0.32/0.57 => ( ? [Z4: real] :
% 0.32/0.57 ! [X4: real] :
% 0.32/0.57 ( ( ord_less_real @ Z4 @ X4 )
% 0.32/0.57 => ( ( Q @ X4 )
% 0.32/0.57 = ( Q6 @ X4 ) ) )
% 0.32/0.57 => ? [Z5: real] :
% 0.32/0.57 ! [X5: real] :
% 0.32/0.57 ( ( ord_less_real @ Z5 @ X5 )
% 0.32/0.57 => ( ( ( P @ X5 )
% 0.32/0.57 & ( Q @ X5 ) )
% 0.32/0.57 = ( ( P6 @ X5 )
% 0.32/0.57 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(1)
% 0.32/0.57 thf(fact_3437_pinf_I1_J,axiom,
% 0.32/0.57 ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 0.32/0.57 ( ? [Z4: rat] :
% 0.32/0.57 ! [X4: rat] :
% 0.32/0.57 ( ( ord_less_rat @ Z4 @ X4 )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 = ( P6 @ X4 ) ) )
% 0.32/0.57 => ( ? [Z4: rat] :
% 0.32/0.57 ! [X4: rat] :
% 0.32/0.57 ( ( ord_less_rat @ Z4 @ X4 )
% 0.32/0.57 => ( ( Q @ X4 )
% 0.32/0.57 = ( Q6 @ X4 ) ) )
% 0.32/0.57 => ? [Z5: rat] :
% 0.32/0.57 ! [X5: rat] :
% 0.32/0.57 ( ( ord_less_rat @ Z5 @ X5 )
% 0.32/0.57 => ( ( ( P @ X5 )
% 0.32/0.57 & ( Q @ X5 ) )
% 0.32/0.57 = ( ( P6 @ X5 )
% 0.32/0.57 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(1)
% 0.32/0.57 thf(fact_3438_pinf_I1_J,axiom,
% 0.32/0.57 ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
% 0.32/0.57 ( ? [Z4: num] :
% 0.32/0.57 ! [X4: num] :
% 0.32/0.57 ( ( ord_less_num @ Z4 @ X4 )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 = ( P6 @ X4 ) ) )
% 0.32/0.57 => ( ? [Z4: num] :
% 0.32/0.57 ! [X4: num] :
% 0.32/0.57 ( ( ord_less_num @ Z4 @ X4 )
% 0.32/0.57 => ( ( Q @ X4 )
% 0.32/0.57 = ( Q6 @ X4 ) ) )
% 0.32/0.57 => ? [Z5: num] :
% 0.32/0.57 ! [X5: num] :
% 0.32/0.57 ( ( ord_less_num @ Z5 @ X5 )
% 0.32/0.57 => ( ( ( P @ X5 )
% 0.32/0.57 & ( Q @ X5 ) )
% 0.32/0.57 = ( ( P6 @ X5 )
% 0.32/0.57 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(1)
% 0.32/0.57 thf(fact_3439_pinf_I1_J,axiom,
% 0.32/0.57 ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 0.32/0.57 ( ? [Z4: nat] :
% 0.32/0.57 ! [X4: nat] :
% 0.32/0.57 ( ( ord_less_nat @ Z4 @ X4 )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 = ( P6 @ X4 ) ) )
% 0.32/0.57 => ( ? [Z4: nat] :
% 0.32/0.57 ! [X4: nat] :
% 0.32/0.57 ( ( ord_less_nat @ Z4 @ X4 )
% 0.32/0.57 => ( ( Q @ X4 )
% 0.32/0.57 = ( Q6 @ X4 ) ) )
% 0.32/0.57 => ? [Z5: nat] :
% 0.32/0.57 ! [X5: nat] :
% 0.32/0.57 ( ( ord_less_nat @ Z5 @ X5 )
% 0.32/0.57 => ( ( ( P @ X5 )
% 0.32/0.57 & ( Q @ X5 ) )
% 0.32/0.57 = ( ( P6 @ X5 )
% 0.32/0.57 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(1)
% 0.32/0.57 thf(fact_3440_pinf_I1_J,axiom,
% 0.32/0.57 ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
% 0.32/0.57 ( ? [Z4: int] :
% 0.32/0.57 ! [X4: int] :
% 0.32/0.57 ( ( ord_less_int @ Z4 @ X4 )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 = ( P6 @ X4 ) ) )
% 0.32/0.57 => ( ? [Z4: int] :
% 0.32/0.57 ! [X4: int] :
% 0.32/0.57 ( ( ord_less_int @ Z4 @ X4 )
% 0.32/0.57 => ( ( Q @ X4 )
% 0.32/0.57 = ( Q6 @ X4 ) ) )
% 0.32/0.57 => ? [Z5: int] :
% 0.32/0.57 ! [X5: int] :
% 0.32/0.57 ( ( ord_less_int @ Z5 @ X5 )
% 0.32/0.57 => ( ( ( P @ X5 )
% 0.32/0.57 & ( Q @ X5 ) )
% 0.32/0.57 = ( ( P6 @ X5 )
% 0.32/0.57 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(1)
% 0.32/0.57 thf(fact_3441_pinf_I2_J,axiom,
% 0.32/0.57 ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
% 0.32/0.57 ( ? [Z4: real] :
% 0.32/0.57 ! [X4: real] :
% 0.32/0.57 ( ( ord_less_real @ Z4 @ X4 )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 = ( P6 @ X4 ) ) )
% 0.32/0.57 => ( ? [Z4: real] :
% 0.32/0.57 ! [X4: real] :
% 0.32/0.57 ( ( ord_less_real @ Z4 @ X4 )
% 0.32/0.57 => ( ( Q @ X4 )
% 0.32/0.57 = ( Q6 @ X4 ) ) )
% 0.32/0.57 => ? [Z5: real] :
% 0.32/0.57 ! [X5: real] :
% 0.32/0.57 ( ( ord_less_real @ Z5 @ X5 )
% 0.32/0.57 => ( ( ( P @ X5 )
% 0.32/0.57 | ( Q @ X5 ) )
% 0.32/0.57 = ( ( P6 @ X5 )
% 0.32/0.57 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(2)
% 0.32/0.57 thf(fact_3442_pinf_I2_J,axiom,
% 0.32/0.57 ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 0.32/0.57 ( ? [Z4: rat] :
% 0.32/0.57 ! [X4: rat] :
% 0.32/0.57 ( ( ord_less_rat @ Z4 @ X4 )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 = ( P6 @ X4 ) ) )
% 0.32/0.57 => ( ? [Z4: rat] :
% 0.32/0.57 ! [X4: rat] :
% 0.32/0.57 ( ( ord_less_rat @ Z4 @ X4 )
% 0.32/0.57 => ( ( Q @ X4 )
% 0.32/0.57 = ( Q6 @ X4 ) ) )
% 0.32/0.57 => ? [Z5: rat] :
% 0.32/0.57 ! [X5: rat] :
% 0.32/0.57 ( ( ord_less_rat @ Z5 @ X5 )
% 0.32/0.57 => ( ( ( P @ X5 )
% 0.32/0.57 | ( Q @ X5 ) )
% 0.32/0.57 = ( ( P6 @ X5 )
% 0.32/0.57 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(2)
% 0.32/0.57 thf(fact_3443_pinf_I2_J,axiom,
% 0.32/0.57 ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
% 0.32/0.57 ( ? [Z4: num] :
% 0.32/0.57 ! [X4: num] :
% 0.32/0.57 ( ( ord_less_num @ Z4 @ X4 )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 = ( P6 @ X4 ) ) )
% 0.32/0.57 => ( ? [Z4: num] :
% 0.32/0.57 ! [X4: num] :
% 0.32/0.57 ( ( ord_less_num @ Z4 @ X4 )
% 0.32/0.57 => ( ( Q @ X4 )
% 0.32/0.57 = ( Q6 @ X4 ) ) )
% 0.32/0.57 => ? [Z5: num] :
% 0.32/0.57 ! [X5: num] :
% 0.32/0.57 ( ( ord_less_num @ Z5 @ X5 )
% 0.32/0.57 => ( ( ( P @ X5 )
% 0.32/0.57 | ( Q @ X5 ) )
% 0.32/0.57 = ( ( P6 @ X5 )
% 0.32/0.57 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(2)
% 0.32/0.57 thf(fact_3444_pinf_I2_J,axiom,
% 0.32/0.57 ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 0.32/0.57 ( ? [Z4: nat] :
% 0.32/0.57 ! [X4: nat] :
% 0.32/0.57 ( ( ord_less_nat @ Z4 @ X4 )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 = ( P6 @ X4 ) ) )
% 0.32/0.57 => ( ? [Z4: nat] :
% 0.32/0.57 ! [X4: nat] :
% 0.32/0.57 ( ( ord_less_nat @ Z4 @ X4 )
% 0.32/0.57 => ( ( Q @ X4 )
% 0.32/0.57 = ( Q6 @ X4 ) ) )
% 0.32/0.57 => ? [Z5: nat] :
% 0.32/0.57 ! [X5: nat] :
% 0.32/0.57 ( ( ord_less_nat @ Z5 @ X5 )
% 0.32/0.57 => ( ( ( P @ X5 )
% 0.32/0.57 | ( Q @ X5 ) )
% 0.32/0.57 = ( ( P6 @ X5 )
% 0.32/0.57 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(2)
% 0.32/0.57 thf(fact_3445_pinf_I2_J,axiom,
% 0.32/0.57 ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
% 0.32/0.57 ( ? [Z4: int] :
% 0.32/0.57 ! [X4: int] :
% 0.32/0.57 ( ( ord_less_int @ Z4 @ X4 )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 = ( P6 @ X4 ) ) )
% 0.32/0.57 => ( ? [Z4: int] :
% 0.32/0.57 ! [X4: int] :
% 0.32/0.57 ( ( ord_less_int @ Z4 @ X4 )
% 0.32/0.57 => ( ( Q @ X4 )
% 0.32/0.57 = ( Q6 @ X4 ) ) )
% 0.32/0.57 => ? [Z5: int] :
% 0.32/0.57 ! [X5: int] :
% 0.32/0.57 ( ( ord_less_int @ Z5 @ X5 )
% 0.32/0.57 => ( ( ( P @ X5 )
% 0.32/0.57 | ( Q @ X5 ) )
% 0.32/0.57 = ( ( P6 @ X5 )
% 0.32/0.57 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(2)
% 0.32/0.57 thf(fact_3446_pinf_I3_J,axiom,
% 0.32/0.57 ! [T: real] :
% 0.32/0.57 ? [Z5: real] :
% 0.32/0.57 ! [X5: real] :
% 0.32/0.57 ( ( ord_less_real @ Z5 @ X5 )
% 0.32/0.57 => ( X5 != T ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(3)
% 0.32/0.57 thf(fact_3447_pinf_I3_J,axiom,
% 0.32/0.57 ! [T: rat] :
% 0.32/0.57 ? [Z5: rat] :
% 0.32/0.57 ! [X5: rat] :
% 0.32/0.57 ( ( ord_less_rat @ Z5 @ X5 )
% 0.32/0.57 => ( X5 != T ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(3)
% 0.32/0.57 thf(fact_3448_pinf_I3_J,axiom,
% 0.32/0.57 ! [T: num] :
% 0.32/0.57 ? [Z5: num] :
% 0.32/0.57 ! [X5: num] :
% 0.32/0.57 ( ( ord_less_num @ Z5 @ X5 )
% 0.32/0.57 => ( X5 != T ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(3)
% 0.32/0.57 thf(fact_3449_pinf_I3_J,axiom,
% 0.32/0.57 ! [T: nat] :
% 0.32/0.57 ? [Z5: nat] :
% 0.32/0.57 ! [X5: nat] :
% 0.32/0.57 ( ( ord_less_nat @ Z5 @ X5 )
% 0.32/0.57 => ( X5 != T ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(3)
% 0.32/0.57 thf(fact_3450_pinf_I3_J,axiom,
% 0.32/0.57 ! [T: int] :
% 0.32/0.57 ? [Z5: int] :
% 0.32/0.57 ! [X5: int] :
% 0.32/0.57 ( ( ord_less_int @ Z5 @ X5 )
% 0.32/0.57 => ( X5 != T ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(3)
% 0.32/0.57 thf(fact_3451_pinf_I4_J,axiom,
% 0.32/0.57 ! [T: real] :
% 0.32/0.57 ? [Z5: real] :
% 0.32/0.57 ! [X5: real] :
% 0.32/0.57 ( ( ord_less_real @ Z5 @ X5 )
% 0.32/0.57 => ( X5 != T ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(4)
% 0.32/0.57 thf(fact_3452_pinf_I4_J,axiom,
% 0.32/0.57 ! [T: rat] :
% 0.32/0.57 ? [Z5: rat] :
% 0.32/0.57 ! [X5: rat] :
% 0.32/0.57 ( ( ord_less_rat @ Z5 @ X5 )
% 0.32/0.57 => ( X5 != T ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(4)
% 0.32/0.57 thf(fact_3453_pinf_I4_J,axiom,
% 0.32/0.57 ! [T: num] :
% 0.32/0.57 ? [Z5: num] :
% 0.32/0.57 ! [X5: num] :
% 0.32/0.57 ( ( ord_less_num @ Z5 @ X5 )
% 0.32/0.57 => ( X5 != T ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(4)
% 0.32/0.57 thf(fact_3454_pinf_I4_J,axiom,
% 0.32/0.57 ! [T: nat] :
% 0.32/0.57 ? [Z5: nat] :
% 0.32/0.57 ! [X5: nat] :
% 0.32/0.57 ( ( ord_less_nat @ Z5 @ X5 )
% 0.32/0.57 => ( X5 != T ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(4)
% 0.32/0.57 thf(fact_3455_pinf_I4_J,axiom,
% 0.32/0.57 ! [T: int] :
% 0.32/0.57 ? [Z5: int] :
% 0.32/0.57 ! [X5: int] :
% 0.32/0.57 ( ( ord_less_int @ Z5 @ X5 )
% 0.32/0.57 => ( X5 != T ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(4)
% 0.32/0.57 thf(fact_3456_pinf_I5_J,axiom,
% 0.32/0.57 ! [T: real] :
% 0.32/0.57 ? [Z5: real] :
% 0.32/0.57 ! [X5: real] :
% 0.32/0.57 ( ( ord_less_real @ Z5 @ X5 )
% 0.32/0.57 => ~ ( ord_less_real @ X5 @ T ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(5)
% 0.32/0.57 thf(fact_3457_pinf_I5_J,axiom,
% 0.32/0.57 ! [T: rat] :
% 0.32/0.57 ? [Z5: rat] :
% 0.32/0.57 ! [X5: rat] :
% 0.32/0.57 ( ( ord_less_rat @ Z5 @ X5 )
% 0.32/0.57 => ~ ( ord_less_rat @ X5 @ T ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(5)
% 0.32/0.57 thf(fact_3458_pinf_I5_J,axiom,
% 0.32/0.57 ! [T: num] :
% 0.32/0.57 ? [Z5: num] :
% 0.32/0.57 ! [X5: num] :
% 0.32/0.57 ( ( ord_less_num @ Z5 @ X5 )
% 0.32/0.57 => ~ ( ord_less_num @ X5 @ T ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(5)
% 0.32/0.57 thf(fact_3459_pinf_I5_J,axiom,
% 0.32/0.57 ! [T: nat] :
% 0.32/0.57 ? [Z5: nat] :
% 0.32/0.57 ! [X5: nat] :
% 0.32/0.57 ( ( ord_less_nat @ Z5 @ X5 )
% 0.32/0.57 => ~ ( ord_less_nat @ X5 @ T ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(5)
% 0.32/0.57 thf(fact_3460_pinf_I5_J,axiom,
% 0.32/0.57 ! [T: int] :
% 0.32/0.57 ? [Z5: int] :
% 0.32/0.57 ! [X5: int] :
% 0.32/0.57 ( ( ord_less_int @ Z5 @ X5 )
% 0.32/0.57 => ~ ( ord_less_int @ X5 @ T ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(5)
% 0.32/0.57 thf(fact_3461_pinf_I7_J,axiom,
% 0.32/0.57 ! [T: real] :
% 0.32/0.57 ? [Z5: real] :
% 0.32/0.57 ! [X5: real] :
% 0.32/0.57 ( ( ord_less_real @ Z5 @ X5 )
% 0.32/0.57 => ( ord_less_real @ T @ X5 ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(7)
% 0.32/0.57 thf(fact_3462_pinf_I7_J,axiom,
% 0.32/0.57 ! [T: rat] :
% 0.32/0.57 ? [Z5: rat] :
% 0.32/0.57 ! [X5: rat] :
% 0.32/0.57 ( ( ord_less_rat @ Z5 @ X5 )
% 0.32/0.57 => ( ord_less_rat @ T @ X5 ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(7)
% 0.32/0.57 thf(fact_3463_pinf_I7_J,axiom,
% 0.32/0.57 ! [T: num] :
% 0.32/0.57 ? [Z5: num] :
% 0.32/0.57 ! [X5: num] :
% 0.32/0.57 ( ( ord_less_num @ Z5 @ X5 )
% 0.32/0.57 => ( ord_less_num @ T @ X5 ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(7)
% 0.32/0.57 thf(fact_3464_pinf_I7_J,axiom,
% 0.32/0.57 ! [T: nat] :
% 0.32/0.57 ? [Z5: nat] :
% 0.32/0.57 ! [X5: nat] :
% 0.32/0.57 ( ( ord_less_nat @ Z5 @ X5 )
% 0.32/0.57 => ( ord_less_nat @ T @ X5 ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(7)
% 0.32/0.57 thf(fact_3465_pinf_I7_J,axiom,
% 0.32/0.57 ! [T: int] :
% 0.32/0.57 ? [Z5: int] :
% 0.32/0.57 ! [X5: int] :
% 0.32/0.57 ( ( ord_less_int @ Z5 @ X5 )
% 0.32/0.57 => ( ord_less_int @ T @ X5 ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(7)
% 0.32/0.57 thf(fact_3466_minf_I1_J,axiom,
% 0.32/0.57 ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
% 0.32/0.57 ( ? [Z4: real] :
% 0.32/0.57 ! [X4: real] :
% 0.32/0.57 ( ( ord_less_real @ X4 @ Z4 )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 = ( P6 @ X4 ) ) )
% 0.32/0.57 => ( ? [Z4: real] :
% 0.32/0.57 ! [X4: real] :
% 0.32/0.57 ( ( ord_less_real @ X4 @ Z4 )
% 0.32/0.57 => ( ( Q @ X4 )
% 0.32/0.57 = ( Q6 @ X4 ) ) )
% 0.32/0.57 => ? [Z5: real] :
% 0.32/0.57 ! [X5: real] :
% 0.32/0.57 ( ( ord_less_real @ X5 @ Z5 )
% 0.32/0.57 => ( ( ( P @ X5 )
% 0.32/0.57 & ( Q @ X5 ) )
% 0.32/0.57 = ( ( P6 @ X5 )
% 0.32/0.57 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(1)
% 0.32/0.57 thf(fact_3467_minf_I1_J,axiom,
% 0.32/0.57 ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 0.32/0.57 ( ? [Z4: rat] :
% 0.32/0.57 ! [X4: rat] :
% 0.32/0.57 ( ( ord_less_rat @ X4 @ Z4 )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 = ( P6 @ X4 ) ) )
% 0.32/0.57 => ( ? [Z4: rat] :
% 0.32/0.57 ! [X4: rat] :
% 0.32/0.57 ( ( ord_less_rat @ X4 @ Z4 )
% 0.32/0.57 => ( ( Q @ X4 )
% 0.32/0.57 = ( Q6 @ X4 ) ) )
% 0.32/0.57 => ? [Z5: rat] :
% 0.32/0.57 ! [X5: rat] :
% 0.32/0.57 ( ( ord_less_rat @ X5 @ Z5 )
% 0.32/0.57 => ( ( ( P @ X5 )
% 0.32/0.57 & ( Q @ X5 ) )
% 0.32/0.57 = ( ( P6 @ X5 )
% 0.32/0.57 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(1)
% 0.32/0.57 thf(fact_3468_minf_I1_J,axiom,
% 0.32/0.57 ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
% 0.32/0.57 ( ? [Z4: num] :
% 0.32/0.57 ! [X4: num] :
% 0.32/0.57 ( ( ord_less_num @ X4 @ Z4 )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 = ( P6 @ X4 ) ) )
% 0.32/0.57 => ( ? [Z4: num] :
% 0.32/0.57 ! [X4: num] :
% 0.32/0.57 ( ( ord_less_num @ X4 @ Z4 )
% 0.32/0.57 => ( ( Q @ X4 )
% 0.32/0.57 = ( Q6 @ X4 ) ) )
% 0.32/0.57 => ? [Z5: num] :
% 0.32/0.57 ! [X5: num] :
% 0.32/0.57 ( ( ord_less_num @ X5 @ Z5 )
% 0.32/0.57 => ( ( ( P @ X5 )
% 0.32/0.57 & ( Q @ X5 ) )
% 0.32/0.57 = ( ( P6 @ X5 )
% 0.32/0.57 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(1)
% 0.32/0.57 thf(fact_3469_minf_I1_J,axiom,
% 0.32/0.57 ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 0.32/0.57 ( ? [Z4: nat] :
% 0.32/0.57 ! [X4: nat] :
% 0.32/0.57 ( ( ord_less_nat @ X4 @ Z4 )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 = ( P6 @ X4 ) ) )
% 0.32/0.57 => ( ? [Z4: nat] :
% 0.32/0.57 ! [X4: nat] :
% 0.32/0.57 ( ( ord_less_nat @ X4 @ Z4 )
% 0.32/0.57 => ( ( Q @ X4 )
% 0.32/0.57 = ( Q6 @ X4 ) ) )
% 0.32/0.57 => ? [Z5: nat] :
% 0.32/0.57 ! [X5: nat] :
% 0.32/0.57 ( ( ord_less_nat @ X5 @ Z5 )
% 0.32/0.57 => ( ( ( P @ X5 )
% 0.32/0.57 & ( Q @ X5 ) )
% 0.32/0.57 = ( ( P6 @ X5 )
% 0.32/0.57 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(1)
% 0.32/0.57 thf(fact_3470_minf_I1_J,axiom,
% 0.32/0.57 ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
% 0.32/0.57 ( ? [Z4: int] :
% 0.32/0.57 ! [X4: int] :
% 0.32/0.57 ( ( ord_less_int @ X4 @ Z4 )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 = ( P6 @ X4 ) ) )
% 0.32/0.57 => ( ? [Z4: int] :
% 0.32/0.57 ! [X4: int] :
% 0.32/0.57 ( ( ord_less_int @ X4 @ Z4 )
% 0.32/0.57 => ( ( Q @ X4 )
% 0.32/0.57 = ( Q6 @ X4 ) ) )
% 0.32/0.57 => ? [Z5: int] :
% 0.32/0.57 ! [X5: int] :
% 0.32/0.57 ( ( ord_less_int @ X5 @ Z5 )
% 0.32/0.57 => ( ( ( P @ X5 )
% 0.32/0.57 & ( Q @ X5 ) )
% 0.32/0.57 = ( ( P6 @ X5 )
% 0.32/0.57 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(1)
% 0.32/0.57 thf(fact_3471_minf_I2_J,axiom,
% 0.32/0.57 ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
% 0.32/0.57 ( ? [Z4: real] :
% 0.32/0.57 ! [X4: real] :
% 0.32/0.57 ( ( ord_less_real @ X4 @ Z4 )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 = ( P6 @ X4 ) ) )
% 0.32/0.57 => ( ? [Z4: real] :
% 0.32/0.57 ! [X4: real] :
% 0.32/0.57 ( ( ord_less_real @ X4 @ Z4 )
% 0.32/0.57 => ( ( Q @ X4 )
% 0.32/0.57 = ( Q6 @ X4 ) ) )
% 0.32/0.57 => ? [Z5: real] :
% 0.32/0.57 ! [X5: real] :
% 0.32/0.57 ( ( ord_less_real @ X5 @ Z5 )
% 0.32/0.57 => ( ( ( P @ X5 )
% 0.32/0.57 | ( Q @ X5 ) )
% 0.32/0.57 = ( ( P6 @ X5 )
% 0.32/0.57 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(2)
% 0.32/0.57 thf(fact_3472_minf_I2_J,axiom,
% 0.32/0.57 ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 0.32/0.57 ( ? [Z4: rat] :
% 0.32/0.57 ! [X4: rat] :
% 0.32/0.57 ( ( ord_less_rat @ X4 @ Z4 )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 = ( P6 @ X4 ) ) )
% 0.32/0.57 => ( ? [Z4: rat] :
% 0.32/0.57 ! [X4: rat] :
% 0.32/0.57 ( ( ord_less_rat @ X4 @ Z4 )
% 0.32/0.57 => ( ( Q @ X4 )
% 0.32/0.57 = ( Q6 @ X4 ) ) )
% 0.32/0.57 => ? [Z5: rat] :
% 0.32/0.57 ! [X5: rat] :
% 0.32/0.57 ( ( ord_less_rat @ X5 @ Z5 )
% 0.32/0.57 => ( ( ( P @ X5 )
% 0.32/0.57 | ( Q @ X5 ) )
% 0.32/0.57 = ( ( P6 @ X5 )
% 0.32/0.57 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(2)
% 0.32/0.57 thf(fact_3473_minf_I2_J,axiom,
% 0.32/0.57 ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
% 0.32/0.57 ( ? [Z4: num] :
% 0.32/0.57 ! [X4: num] :
% 0.32/0.57 ( ( ord_less_num @ X4 @ Z4 )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 = ( P6 @ X4 ) ) )
% 0.32/0.57 => ( ? [Z4: num] :
% 0.32/0.57 ! [X4: num] :
% 0.32/0.57 ( ( ord_less_num @ X4 @ Z4 )
% 0.32/0.57 => ( ( Q @ X4 )
% 0.32/0.57 = ( Q6 @ X4 ) ) )
% 0.32/0.57 => ? [Z5: num] :
% 0.32/0.57 ! [X5: num] :
% 0.32/0.57 ( ( ord_less_num @ X5 @ Z5 )
% 0.32/0.57 => ( ( ( P @ X5 )
% 0.32/0.57 | ( Q @ X5 ) )
% 0.32/0.57 = ( ( P6 @ X5 )
% 0.32/0.57 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(2)
% 0.32/0.57 thf(fact_3474_minf_I2_J,axiom,
% 0.32/0.57 ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 0.32/0.57 ( ? [Z4: nat] :
% 0.32/0.57 ! [X4: nat] :
% 0.32/0.57 ( ( ord_less_nat @ X4 @ Z4 )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 = ( P6 @ X4 ) ) )
% 0.32/0.57 => ( ? [Z4: nat] :
% 0.32/0.57 ! [X4: nat] :
% 0.32/0.57 ( ( ord_less_nat @ X4 @ Z4 )
% 0.32/0.57 => ( ( Q @ X4 )
% 0.32/0.57 = ( Q6 @ X4 ) ) )
% 0.32/0.57 => ? [Z5: nat] :
% 0.32/0.57 ! [X5: nat] :
% 0.32/0.57 ( ( ord_less_nat @ X5 @ Z5 )
% 0.32/0.57 => ( ( ( P @ X5 )
% 0.32/0.57 | ( Q @ X5 ) )
% 0.32/0.57 = ( ( P6 @ X5 )
% 0.32/0.57 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(2)
% 0.32/0.57 thf(fact_3475_minf_I2_J,axiom,
% 0.32/0.57 ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
% 0.32/0.57 ( ? [Z4: int] :
% 0.32/0.57 ! [X4: int] :
% 0.32/0.57 ( ( ord_less_int @ X4 @ Z4 )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 = ( P6 @ X4 ) ) )
% 0.32/0.57 => ( ? [Z4: int] :
% 0.32/0.57 ! [X4: int] :
% 0.32/0.57 ( ( ord_less_int @ X4 @ Z4 )
% 0.32/0.57 => ( ( Q @ X4 )
% 0.32/0.57 = ( Q6 @ X4 ) ) )
% 0.32/0.57 => ? [Z5: int] :
% 0.32/0.57 ! [X5: int] :
% 0.32/0.57 ( ( ord_less_int @ X5 @ Z5 )
% 0.32/0.57 => ( ( ( P @ X5 )
% 0.32/0.57 | ( Q @ X5 ) )
% 0.32/0.57 = ( ( P6 @ X5 )
% 0.32/0.57 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(2)
% 0.32/0.57 thf(fact_3476_minf_I3_J,axiom,
% 0.32/0.57 ! [T: real] :
% 0.32/0.57 ? [Z5: real] :
% 0.32/0.57 ! [X5: real] :
% 0.32/0.57 ( ( ord_less_real @ X5 @ Z5 )
% 0.32/0.57 => ( X5 != T ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(3)
% 0.32/0.57 thf(fact_3477_minf_I3_J,axiom,
% 0.32/0.57 ! [T: rat] :
% 0.32/0.57 ? [Z5: rat] :
% 0.32/0.57 ! [X5: rat] :
% 0.32/0.57 ( ( ord_less_rat @ X5 @ Z5 )
% 0.32/0.57 => ( X5 != T ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(3)
% 0.32/0.57 thf(fact_3478_minf_I3_J,axiom,
% 0.32/0.57 ! [T: num] :
% 0.32/0.57 ? [Z5: num] :
% 0.32/0.57 ! [X5: num] :
% 0.32/0.57 ( ( ord_less_num @ X5 @ Z5 )
% 0.32/0.57 => ( X5 != T ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(3)
% 0.32/0.57 thf(fact_3479_minf_I3_J,axiom,
% 0.32/0.57 ! [T: nat] :
% 0.32/0.57 ? [Z5: nat] :
% 0.32/0.57 ! [X5: nat] :
% 0.32/0.57 ( ( ord_less_nat @ X5 @ Z5 )
% 0.32/0.57 => ( X5 != T ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(3)
% 0.32/0.57 thf(fact_3480_minf_I3_J,axiom,
% 0.32/0.57 ! [T: int] :
% 0.32/0.57 ? [Z5: int] :
% 0.32/0.57 ! [X5: int] :
% 0.32/0.57 ( ( ord_less_int @ X5 @ Z5 )
% 0.32/0.57 => ( X5 != T ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(3)
% 0.32/0.57 thf(fact_3481_minf_I4_J,axiom,
% 0.32/0.57 ! [T: real] :
% 0.32/0.57 ? [Z5: real] :
% 0.32/0.57 ! [X5: real] :
% 0.32/0.57 ( ( ord_less_real @ X5 @ Z5 )
% 0.32/0.57 => ( X5 != T ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(4)
% 0.32/0.57 thf(fact_3482_minf_I4_J,axiom,
% 0.32/0.57 ! [T: rat] :
% 0.32/0.57 ? [Z5: rat] :
% 0.32/0.57 ! [X5: rat] :
% 0.32/0.57 ( ( ord_less_rat @ X5 @ Z5 )
% 0.32/0.57 => ( X5 != T ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(4)
% 0.32/0.57 thf(fact_3483_minf_I4_J,axiom,
% 0.32/0.57 ! [T: num] :
% 0.32/0.57 ? [Z5: num] :
% 0.32/0.57 ! [X5: num] :
% 0.32/0.57 ( ( ord_less_num @ X5 @ Z5 )
% 0.32/0.57 => ( X5 != T ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(4)
% 0.32/0.57 thf(fact_3484_minf_I4_J,axiom,
% 0.32/0.57 ! [T: nat] :
% 0.32/0.57 ? [Z5: nat] :
% 0.32/0.57 ! [X5: nat] :
% 0.32/0.57 ( ( ord_less_nat @ X5 @ Z5 )
% 0.32/0.57 => ( X5 != T ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(4)
% 0.32/0.57 thf(fact_3485_minf_I4_J,axiom,
% 0.32/0.57 ! [T: int] :
% 0.32/0.57 ? [Z5: int] :
% 0.32/0.57 ! [X5: int] :
% 0.32/0.57 ( ( ord_less_int @ X5 @ Z5 )
% 0.32/0.57 => ( X5 != T ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(4)
% 0.32/0.57 thf(fact_3486_minf_I5_J,axiom,
% 0.32/0.57 ! [T: real] :
% 0.32/0.57 ? [Z5: real] :
% 0.32/0.57 ! [X5: real] :
% 0.32/0.57 ( ( ord_less_real @ X5 @ Z5 )
% 0.32/0.57 => ( ord_less_real @ X5 @ T ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(5)
% 0.32/0.57 thf(fact_3487_minf_I5_J,axiom,
% 0.32/0.57 ! [T: rat] :
% 0.32/0.57 ? [Z5: rat] :
% 0.32/0.57 ! [X5: rat] :
% 0.32/0.57 ( ( ord_less_rat @ X5 @ Z5 )
% 0.32/0.57 => ( ord_less_rat @ X5 @ T ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(5)
% 0.32/0.57 thf(fact_3488_minf_I5_J,axiom,
% 0.32/0.57 ! [T: num] :
% 0.32/0.57 ? [Z5: num] :
% 0.32/0.57 ! [X5: num] :
% 0.32/0.57 ( ( ord_less_num @ X5 @ Z5 )
% 0.32/0.57 => ( ord_less_num @ X5 @ T ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(5)
% 0.32/0.57 thf(fact_3489_minf_I5_J,axiom,
% 0.32/0.57 ! [T: nat] :
% 0.32/0.57 ? [Z5: nat] :
% 0.32/0.57 ! [X5: nat] :
% 0.32/0.57 ( ( ord_less_nat @ X5 @ Z5 )
% 0.32/0.57 => ( ord_less_nat @ X5 @ T ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(5)
% 0.32/0.57 thf(fact_3490_minf_I5_J,axiom,
% 0.32/0.57 ! [T: int] :
% 0.32/0.57 ? [Z5: int] :
% 0.32/0.57 ! [X5: int] :
% 0.32/0.57 ( ( ord_less_int @ X5 @ Z5 )
% 0.32/0.57 => ( ord_less_int @ X5 @ T ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(5)
% 0.32/0.57 thf(fact_3491_minf_I7_J,axiom,
% 0.32/0.57 ! [T: real] :
% 0.32/0.57 ? [Z5: real] :
% 0.32/0.57 ! [X5: real] :
% 0.32/0.57 ( ( ord_less_real @ X5 @ Z5 )
% 0.32/0.57 => ~ ( ord_less_real @ T @ X5 ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(7)
% 0.32/0.57 thf(fact_3492_minf_I7_J,axiom,
% 0.32/0.57 ! [T: rat] :
% 0.32/0.57 ? [Z5: rat] :
% 0.32/0.57 ! [X5: rat] :
% 0.32/0.57 ( ( ord_less_rat @ X5 @ Z5 )
% 0.32/0.57 => ~ ( ord_less_rat @ T @ X5 ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(7)
% 0.32/0.57 thf(fact_3493_minf_I7_J,axiom,
% 0.32/0.57 ! [T: num] :
% 0.32/0.57 ? [Z5: num] :
% 0.32/0.57 ! [X5: num] :
% 0.32/0.57 ( ( ord_less_num @ X5 @ Z5 )
% 0.32/0.57 => ~ ( ord_less_num @ T @ X5 ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(7)
% 0.32/0.57 thf(fact_3494_minf_I7_J,axiom,
% 0.32/0.57 ! [T: nat] :
% 0.32/0.57 ? [Z5: nat] :
% 0.32/0.57 ! [X5: nat] :
% 0.32/0.57 ( ( ord_less_nat @ X5 @ Z5 )
% 0.32/0.57 => ~ ( ord_less_nat @ T @ X5 ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(7)
% 0.32/0.57 thf(fact_3495_minf_I7_J,axiom,
% 0.32/0.57 ! [T: int] :
% 0.32/0.57 ? [Z5: int] :
% 0.32/0.57 ! [X5: int] :
% 0.32/0.57 ( ( ord_less_int @ X5 @ Z5 )
% 0.32/0.57 => ~ ( ord_less_int @ T @ X5 ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(7)
% 0.32/0.57 thf(fact_3496_bounded__Max__nat,axiom,
% 0.32/0.57 ! [P: nat > $o,X: nat,M7: nat] :
% 0.32/0.57 ( ( P @ X )
% 0.32/0.57 => ( ! [X4: nat] :
% 0.32/0.57 ( ( P @ X4 )
% 0.32/0.57 => ( ord_less_eq_nat @ X4 @ M7 ) )
% 0.32/0.57 => ~ ! [M2: nat] :
% 0.32/0.57 ( ( P @ M2 )
% 0.32/0.57 => ~ ! [X5: nat] :
% 0.32/0.57 ( ( P @ X5 )
% 0.32/0.57 => ( ord_less_eq_nat @ X5 @ M2 ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % bounded_Max_nat
% 0.32/0.57 thf(fact_3497_fold__atLeastAtMost__nat_Ocases,axiom,
% 0.32/0.57 ! [X: produc3368934014287244435at_num] :
% 0.32/0.57 ~ ! [F2: nat > num > num,A3: nat,B2: nat,Acc: num] :
% 0.32/0.57 ( X
% 0.32/0.57 != ( produc851828971589881931at_num @ F2 @ ( produc1195630363706982562at_num @ A3 @ ( product_Pair_nat_num @ B2 @ Acc ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % fold_atLeastAtMost_nat.cases
% 0.32/0.57 thf(fact_3498_fold__atLeastAtMost__nat_Ocases,axiom,
% 0.32/0.57 ! [X: produc4471711990508489141at_nat] :
% 0.32/0.57 ~ ! [F2: nat > nat > nat,A3: nat,B2: nat,Acc: nat] :
% 0.32/0.57 ( X
% 0.32/0.57 != ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A3 @ ( product_Pair_nat_nat @ B2 @ Acc ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % fold_atLeastAtMost_nat.cases
% 0.32/0.57 thf(fact_3499_periodic__finite__ex,axiom,
% 0.32/0.57 ! [D: int,P: int > $o] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ D )
% 0.32/0.57 => ( ! [X4: int,K2: int] :
% 0.32/0.57 ( ( P @ X4 )
% 0.32/0.57 = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
% 0.32/0.57 => ( ( ? [X3: int] : ( P @ X3 ) )
% 0.32/0.57 = ( ? [X2: int] :
% 0.32/0.57 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
% 0.32/0.57 & ( P @ X2 ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % periodic_finite_ex
% 0.32/0.57 thf(fact_3500_bset_I3_J,axiom,
% 0.32/0.57 ! [D3: int,T: int,B3: set_int] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.32/0.57 => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B3 )
% 0.32/0.57 => ! [X5: int] :
% 0.32/0.57 ( ! [Xa3: int] :
% 0.32/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 => ! [Xb3: int] :
% 0.32/0.57 ( ( member_int @ Xb3 @ B3 )
% 0.32/0.57 => ( X5
% 0.32/0.57 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.57 => ( ( X5 = T )
% 0.32/0.57 => ( ( minus_minus_int @ X5 @ D3 )
% 0.32/0.57 = T ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % bset(3)
% 0.32/0.57 thf(fact_3501_bset_I4_J,axiom,
% 0.32/0.57 ! [D3: int,T: int,B3: set_int] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.32/0.57 => ( ( member_int @ T @ B3 )
% 0.32/0.57 => ! [X5: int] :
% 0.32/0.57 ( ! [Xa3: int] :
% 0.32/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 => ! [Xb3: int] :
% 0.32/0.57 ( ( member_int @ Xb3 @ B3 )
% 0.32/0.57 => ( X5
% 0.32/0.57 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.57 => ( ( X5 != T )
% 0.32/0.57 => ( ( minus_minus_int @ X5 @ D3 )
% 0.32/0.57 != T ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % bset(4)
% 0.32/0.57 thf(fact_3502_bset_I5_J,axiom,
% 0.32/0.57 ! [D3: int,B3: set_int,T: int] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.32/0.57 => ! [X5: int] :
% 0.32/0.57 ( ! [Xa3: int] :
% 0.32/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 => ! [Xb3: int] :
% 0.32/0.57 ( ( member_int @ Xb3 @ B3 )
% 0.32/0.57 => ( X5
% 0.32/0.57 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.57 => ( ( ord_less_int @ X5 @ T )
% 0.32/0.57 => ( ord_less_int @ ( minus_minus_int @ X5 @ D3 ) @ T ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % bset(5)
% 0.32/0.57 thf(fact_3503_bset_I7_J,axiom,
% 0.32/0.57 ! [D3: int,T: int,B3: set_int] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.32/0.57 => ( ( member_int @ T @ B3 )
% 0.32/0.57 => ! [X5: int] :
% 0.32/0.57 ( ! [Xa3: int] :
% 0.32/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 => ! [Xb3: int] :
% 0.32/0.57 ( ( member_int @ Xb3 @ B3 )
% 0.32/0.57 => ( X5
% 0.32/0.57 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.57 => ( ( ord_less_int @ T @ X5 )
% 0.32/0.57 => ( ord_less_int @ T @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % bset(7)
% 0.32/0.57 thf(fact_3504_aset_I3_J,axiom,
% 0.32/0.57 ! [D3: int,T: int,A2: set_int] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.32/0.57 => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
% 0.32/0.57 => ! [X5: int] :
% 0.32/0.57 ( ! [Xa3: int] :
% 0.32/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 => ! [Xb3: int] :
% 0.32/0.57 ( ( member_int @ Xb3 @ A2 )
% 0.32/0.57 => ( X5
% 0.32/0.57 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.57 => ( ( X5 = T )
% 0.32/0.57 => ( ( plus_plus_int @ X5 @ D3 )
% 0.32/0.57 = T ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % aset(3)
% 0.32/0.57 thf(fact_3505_aset_I4_J,axiom,
% 0.32/0.57 ! [D3: int,T: int,A2: set_int] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.32/0.57 => ( ( member_int @ T @ A2 )
% 0.32/0.57 => ! [X5: int] :
% 0.32/0.57 ( ! [Xa3: int] :
% 0.32/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 => ! [Xb3: int] :
% 0.32/0.57 ( ( member_int @ Xb3 @ A2 )
% 0.32/0.57 => ( X5
% 0.32/0.57 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.57 => ( ( X5 != T )
% 0.32/0.57 => ( ( plus_plus_int @ X5 @ D3 )
% 0.32/0.57 != T ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % aset(4)
% 0.32/0.57 thf(fact_3506_aset_I5_J,axiom,
% 0.32/0.57 ! [D3: int,T: int,A2: set_int] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.32/0.57 => ( ( member_int @ T @ A2 )
% 0.32/0.57 => ! [X5: int] :
% 0.32/0.57 ( ! [Xa3: int] :
% 0.32/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 => ! [Xb3: int] :
% 0.32/0.57 ( ( member_int @ Xb3 @ A2 )
% 0.32/0.57 => ( X5
% 0.32/0.57 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.57 => ( ( ord_less_int @ X5 @ T )
% 0.32/0.57 => ( ord_less_int @ ( plus_plus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % aset(5)
% 0.32/0.57 thf(fact_3507_aset_I7_J,axiom,
% 0.32/0.57 ! [D3: int,A2: set_int,T: int] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.32/0.57 => ! [X5: int] :
% 0.32/0.57 ( ! [Xa3: int] :
% 0.32/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 => ! [Xb3: int] :
% 0.32/0.57 ( ( member_int @ Xb3 @ A2 )
% 0.32/0.57 => ( X5
% 0.32/0.57 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.57 => ( ( ord_less_int @ T @ X5 )
% 0.32/0.57 => ( ord_less_int @ T @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % aset(7)
% 0.32/0.57 thf(fact_3508_bset_I6_J,axiom,
% 0.32/0.57 ! [D3: int,B3: set_int,T: int] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.32/0.57 => ! [X5: int] :
% 0.32/0.57 ( ! [Xa3: int] :
% 0.32/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 => ! [Xb3: int] :
% 0.32/0.57 ( ( member_int @ Xb3 @ B3 )
% 0.32/0.57 => ( X5
% 0.32/0.57 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.57 => ( ( ord_less_eq_int @ X5 @ T )
% 0.32/0.57 => ( ord_less_eq_int @ ( minus_minus_int @ X5 @ D3 ) @ T ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % bset(6)
% 0.32/0.57 thf(fact_3509_bset_I8_J,axiom,
% 0.32/0.57 ! [D3: int,T: int,B3: set_int] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.32/0.57 => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B3 )
% 0.32/0.57 => ! [X5: int] :
% 0.32/0.57 ( ! [Xa3: int] :
% 0.32/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 => ! [Xb3: int] :
% 0.32/0.57 ( ( member_int @ Xb3 @ B3 )
% 0.32/0.57 => ( X5
% 0.32/0.57 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.57 => ( ( ord_less_eq_int @ T @ X5 )
% 0.32/0.57 => ( ord_less_eq_int @ T @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % bset(8)
% 0.32/0.57 thf(fact_3510_aset_I6_J,axiom,
% 0.32/0.57 ! [D3: int,T: int,A2: set_int] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.32/0.57 => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
% 0.32/0.57 => ! [X5: int] :
% 0.32/0.57 ( ! [Xa3: int] :
% 0.32/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 => ! [Xb3: int] :
% 0.32/0.57 ( ( member_int @ Xb3 @ A2 )
% 0.32/0.57 => ( X5
% 0.32/0.57 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.57 => ( ( ord_less_eq_int @ X5 @ T )
% 0.32/0.57 => ( ord_less_eq_int @ ( plus_plus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % aset(6)
% 0.32/0.57 thf(fact_3511_aset_I8_J,axiom,
% 0.32/0.57 ! [D3: int,A2: set_int,T: int] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.32/0.57 => ! [X5: int] :
% 0.32/0.57 ( ! [Xa3: int] :
% 0.32/0.57 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 => ! [Xb3: int] :
% 0.32/0.57 ( ( member_int @ Xb3 @ A2 )
% 0.32/0.57 => ( X5
% 0.32/0.57 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.57 => ( ( ord_less_eq_int @ T @ X5 )
% 0.32/0.57 => ( ord_less_eq_int @ T @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % aset(8)
% 0.32/0.57 thf(fact_3512_cppi,axiom,
% 0.32/0.57 ! [D3: int,P: int > $o,P6: int > $o,A2: set_int] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.32/0.57 => ( ? [Z4: int] :
% 0.32/0.57 ! [X4: int] :
% 0.32/0.57 ( ( ord_less_int @ Z4 @ X4 )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 = ( P6 @ X4 ) ) )
% 0.32/0.57 => ( ! [X4: int] :
% 0.32/0.57 ( ! [Xa: int] :
% 0.32/0.57 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 => ! [Xb: int] :
% 0.32/0.57 ( ( member_int @ Xb @ A2 )
% 0.32/0.57 => ( X4
% 0.32/0.57 != ( minus_minus_int @ Xb @ Xa ) ) ) )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 => ( P @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 0.32/0.57 => ( ! [X4: int,K2: int] :
% 0.32/0.57 ( ( P6 @ X4 )
% 0.32/0.57 = ( P6 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D3 ) ) ) )
% 0.32/0.57 => ( ( ? [X3: int] : ( P @ X3 ) )
% 0.32/0.57 = ( ? [X2: int] :
% 0.32/0.57 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 & ( P6 @ X2 ) )
% 0.32/0.57 | ? [X2: int] :
% 0.32/0.57 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 & ? [Y2: int] :
% 0.32/0.57 ( ( member_int @ Y2 @ A2 )
% 0.32/0.57 & ( P @ ( minus_minus_int @ Y2 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % cppi
% 0.32/0.57 thf(fact_3513_cpmi,axiom,
% 0.32/0.57 ! [D3: int,P: int > $o,P6: int > $o,B3: set_int] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ D3 )
% 0.32/0.57 => ( ? [Z4: int] :
% 0.32/0.57 ! [X4: int] :
% 0.32/0.57 ( ( ord_less_int @ X4 @ Z4 )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 = ( P6 @ X4 ) ) )
% 0.32/0.57 => ( ! [X4: int] :
% 0.32/0.57 ( ! [Xa: int] :
% 0.32/0.57 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 => ! [Xb: int] :
% 0.32/0.57 ( ( member_int @ Xb @ B3 )
% 0.32/0.57 => ( X4
% 0.32/0.57 != ( plus_plus_int @ Xb @ Xa ) ) ) )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 => ( P @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 0.32/0.57 => ( ! [X4: int,K2: int] :
% 0.32/0.57 ( ( P6 @ X4 )
% 0.32/0.57 = ( P6 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D3 ) ) ) )
% 0.32/0.57 => ( ( ? [X3: int] : ( P @ X3 ) )
% 0.32/0.57 = ( ? [X2: int] :
% 0.32/0.57 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 & ( P6 @ X2 ) )
% 0.32/0.57 | ? [X2: int] :
% 0.32/0.57 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.57 & ? [Y2: int] :
% 0.32/0.57 ( ( member_int @ Y2 @ B3 )
% 0.32/0.57 & ( P @ ( plus_plus_int @ Y2 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % cpmi
% 0.32/0.57 thf(fact_3514_pinf_I6_J,axiom,
% 0.32/0.57 ! [T: real] :
% 0.32/0.57 ? [Z5: real] :
% 0.32/0.57 ! [X5: real] :
% 0.32/0.57 ( ( ord_less_real @ Z5 @ X5 )
% 0.32/0.57 => ~ ( ord_less_eq_real @ X5 @ T ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(6)
% 0.32/0.57 thf(fact_3515_pinf_I6_J,axiom,
% 0.32/0.57 ! [T: rat] :
% 0.32/0.57 ? [Z5: rat] :
% 0.32/0.57 ! [X5: rat] :
% 0.32/0.57 ( ( ord_less_rat @ Z5 @ X5 )
% 0.32/0.57 => ~ ( ord_less_eq_rat @ X5 @ T ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(6)
% 0.32/0.57 thf(fact_3516_pinf_I6_J,axiom,
% 0.32/0.57 ! [T: num] :
% 0.32/0.57 ? [Z5: num] :
% 0.32/0.57 ! [X5: num] :
% 0.32/0.57 ( ( ord_less_num @ Z5 @ X5 )
% 0.32/0.57 => ~ ( ord_less_eq_num @ X5 @ T ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(6)
% 0.32/0.57 thf(fact_3517_pinf_I6_J,axiom,
% 0.32/0.57 ! [T: nat] :
% 0.32/0.57 ? [Z5: nat] :
% 0.32/0.57 ! [X5: nat] :
% 0.32/0.57 ( ( ord_less_nat @ Z5 @ X5 )
% 0.32/0.57 => ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(6)
% 0.32/0.57 thf(fact_3518_pinf_I6_J,axiom,
% 0.32/0.57 ! [T: int] :
% 0.32/0.57 ? [Z5: int] :
% 0.32/0.57 ! [X5: int] :
% 0.32/0.57 ( ( ord_less_int @ Z5 @ X5 )
% 0.32/0.57 => ~ ( ord_less_eq_int @ X5 @ T ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(6)
% 0.32/0.57 thf(fact_3519_pinf_I8_J,axiom,
% 0.32/0.57 ! [T: real] :
% 0.32/0.57 ? [Z5: real] :
% 0.32/0.57 ! [X5: real] :
% 0.32/0.57 ( ( ord_less_real @ Z5 @ X5 )
% 0.32/0.57 => ( ord_less_eq_real @ T @ X5 ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(8)
% 0.32/0.57 thf(fact_3520_pinf_I8_J,axiom,
% 0.32/0.57 ! [T: rat] :
% 0.32/0.57 ? [Z5: rat] :
% 0.32/0.57 ! [X5: rat] :
% 0.32/0.57 ( ( ord_less_rat @ Z5 @ X5 )
% 0.32/0.57 => ( ord_less_eq_rat @ T @ X5 ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(8)
% 0.32/0.57 thf(fact_3521_pinf_I8_J,axiom,
% 0.32/0.57 ! [T: num] :
% 0.32/0.57 ? [Z5: num] :
% 0.32/0.57 ! [X5: num] :
% 0.32/0.57 ( ( ord_less_num @ Z5 @ X5 )
% 0.32/0.57 => ( ord_less_eq_num @ T @ X5 ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(8)
% 0.32/0.57 thf(fact_3522_pinf_I8_J,axiom,
% 0.32/0.57 ! [T: nat] :
% 0.32/0.57 ? [Z5: nat] :
% 0.32/0.57 ! [X5: nat] :
% 0.32/0.57 ( ( ord_less_nat @ Z5 @ X5 )
% 0.32/0.57 => ( ord_less_eq_nat @ T @ X5 ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(8)
% 0.32/0.57 thf(fact_3523_pinf_I8_J,axiom,
% 0.32/0.57 ! [T: int] :
% 0.32/0.57 ? [Z5: int] :
% 0.32/0.57 ! [X5: int] :
% 0.32/0.57 ( ( ord_less_int @ Z5 @ X5 )
% 0.32/0.57 => ( ord_less_eq_int @ T @ X5 ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(8)
% 0.32/0.57 thf(fact_3524_minf_I6_J,axiom,
% 0.32/0.57 ! [T: real] :
% 0.32/0.57 ? [Z5: real] :
% 0.32/0.57 ! [X5: real] :
% 0.32/0.57 ( ( ord_less_real @ X5 @ Z5 )
% 0.32/0.57 => ( ord_less_eq_real @ X5 @ T ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(6)
% 0.32/0.57 thf(fact_3525_minf_I6_J,axiom,
% 0.32/0.57 ! [T: rat] :
% 0.32/0.57 ? [Z5: rat] :
% 0.32/0.57 ! [X5: rat] :
% 0.32/0.57 ( ( ord_less_rat @ X5 @ Z5 )
% 0.32/0.57 => ( ord_less_eq_rat @ X5 @ T ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(6)
% 0.32/0.57 thf(fact_3526_minf_I6_J,axiom,
% 0.32/0.57 ! [T: num] :
% 0.32/0.57 ? [Z5: num] :
% 0.32/0.57 ! [X5: num] :
% 0.32/0.57 ( ( ord_less_num @ X5 @ Z5 )
% 0.32/0.57 => ( ord_less_eq_num @ X5 @ T ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(6)
% 0.32/0.57 thf(fact_3527_minf_I6_J,axiom,
% 0.32/0.57 ! [T: nat] :
% 0.32/0.57 ? [Z5: nat] :
% 0.32/0.57 ! [X5: nat] :
% 0.32/0.57 ( ( ord_less_nat @ X5 @ Z5 )
% 0.32/0.57 => ( ord_less_eq_nat @ X5 @ T ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(6)
% 0.32/0.57 thf(fact_3528_minf_I6_J,axiom,
% 0.32/0.57 ! [T: int] :
% 0.32/0.57 ? [Z5: int] :
% 0.32/0.57 ! [X5: int] :
% 0.32/0.57 ( ( ord_less_int @ X5 @ Z5 )
% 0.32/0.57 => ( ord_less_eq_int @ X5 @ T ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(6)
% 0.32/0.57 thf(fact_3529_minf_I8_J,axiom,
% 0.32/0.57 ! [T: real] :
% 0.32/0.57 ? [Z5: real] :
% 0.32/0.57 ! [X5: real] :
% 0.32/0.57 ( ( ord_less_real @ X5 @ Z5 )
% 0.32/0.57 => ~ ( ord_less_eq_real @ T @ X5 ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(8)
% 0.32/0.57 thf(fact_3530_minf_I8_J,axiom,
% 0.32/0.57 ! [T: rat] :
% 0.32/0.57 ? [Z5: rat] :
% 0.32/0.57 ! [X5: rat] :
% 0.32/0.57 ( ( ord_less_rat @ X5 @ Z5 )
% 0.32/0.57 => ~ ( ord_less_eq_rat @ T @ X5 ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(8)
% 0.32/0.57 thf(fact_3531_minf_I8_J,axiom,
% 0.32/0.57 ! [T: num] :
% 0.32/0.57 ? [Z5: num] :
% 0.32/0.57 ! [X5: num] :
% 0.32/0.57 ( ( ord_less_num @ X5 @ Z5 )
% 0.32/0.57 => ~ ( ord_less_eq_num @ T @ X5 ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(8)
% 0.32/0.57 thf(fact_3532_minf_I8_J,axiom,
% 0.32/0.57 ! [T: nat] :
% 0.32/0.57 ? [Z5: nat] :
% 0.32/0.57 ! [X5: nat] :
% 0.32/0.57 ( ( ord_less_nat @ X5 @ Z5 )
% 0.32/0.57 => ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(8)
% 0.32/0.57 thf(fact_3533_minf_I8_J,axiom,
% 0.32/0.57 ! [T: int] :
% 0.32/0.57 ? [Z5: int] :
% 0.32/0.57 ! [X5: int] :
% 0.32/0.57 ( ( ord_less_int @ X5 @ Z5 )
% 0.32/0.57 => ~ ( ord_less_eq_int @ T @ X5 ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(8)
% 0.32/0.57 thf(fact_3534_inf__period_I2_J,axiom,
% 0.32/0.57 ! [P: real > $o,D3: real,Q: real > $o] :
% 0.32/0.57 ( ! [X4: real,K2: real] :
% 0.32/0.57 ( ( P @ X4 )
% 0.32/0.57 = ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K2 @ D3 ) ) ) )
% 0.32/0.57 => ( ! [X4: real,K2: real] :
% 0.32/0.57 ( ( Q @ X4 )
% 0.32/0.57 = ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K2 @ D3 ) ) ) )
% 0.32/0.57 => ! [X5: real,K4: real] :
% 0.32/0.57 ( ( ( P @ X5 )
% 0.32/0.57 | ( Q @ X5 ) )
% 0.32/0.57 = ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) )
% 0.32/0.57 | ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % inf_period(2)
% 0.32/0.57 thf(fact_3535_inf__period_I2_J,axiom,
% 0.32/0.57 ! [P: rat > $o,D3: rat,Q: rat > $o] :
% 0.32/0.57 ( ! [X4: rat,K2: rat] :
% 0.32/0.57 ( ( P @ X4 )
% 0.32/0.57 = ( P @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K2 @ D3 ) ) ) )
% 0.32/0.57 => ( ! [X4: rat,K2: rat] :
% 0.32/0.57 ( ( Q @ X4 )
% 0.32/0.57 = ( Q @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K2 @ D3 ) ) ) )
% 0.32/0.57 => ! [X5: rat,K4: rat] :
% 0.32/0.57 ( ( ( P @ X5 )
% 0.32/0.57 | ( Q @ X5 ) )
% 0.32/0.57 = ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) )
% 0.32/0.57 | ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % inf_period(2)
% 0.32/0.57 thf(fact_3536_inf__period_I2_J,axiom,
% 0.32/0.57 ! [P: int > $o,D3: int,Q: int > $o] :
% 0.32/0.57 ( ! [X4: int,K2: int] :
% 0.32/0.57 ( ( P @ X4 )
% 0.32/0.57 = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D3 ) ) ) )
% 0.32/0.57 => ( ! [X4: int,K2: int] :
% 0.32/0.57 ( ( Q @ X4 )
% 0.32/0.57 = ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D3 ) ) ) )
% 0.32/0.57 => ! [X5: int,K4: int] :
% 0.32/0.57 ( ( ( P @ X5 )
% 0.32/0.57 | ( Q @ X5 ) )
% 0.32/0.57 = ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) )
% 0.32/0.57 | ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % inf_period(2)
% 0.32/0.57 thf(fact_3537_inf__period_I1_J,axiom,
% 0.32/0.57 ! [P: real > $o,D3: real,Q: real > $o] :
% 0.32/0.57 ( ! [X4: real,K2: real] :
% 0.32/0.57 ( ( P @ X4 )
% 0.32/0.57 = ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K2 @ D3 ) ) ) )
% 0.32/0.57 => ( ! [X4: real,K2: real] :
% 0.32/0.57 ( ( Q @ X4 )
% 0.32/0.57 = ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K2 @ D3 ) ) ) )
% 0.32/0.57 => ! [X5: real,K4: real] :
% 0.32/0.57 ( ( ( P @ X5 )
% 0.32/0.57 & ( Q @ X5 ) )
% 0.32/0.57 = ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) )
% 0.32/0.57 & ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % inf_period(1)
% 0.32/0.57 thf(fact_3538_inf__period_I1_J,axiom,
% 0.32/0.57 ! [P: rat > $o,D3: rat,Q: rat > $o] :
% 0.32/0.57 ( ! [X4: rat,K2: rat] :
% 0.32/0.57 ( ( P @ X4 )
% 0.32/0.57 = ( P @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K2 @ D3 ) ) ) )
% 0.32/0.57 => ( ! [X4: rat,K2: rat] :
% 0.32/0.57 ( ( Q @ X4 )
% 0.32/0.57 = ( Q @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K2 @ D3 ) ) ) )
% 0.32/0.57 => ! [X5: rat,K4: rat] :
% 0.32/0.57 ( ( ( P @ X5 )
% 0.32/0.57 & ( Q @ X5 ) )
% 0.32/0.57 = ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) )
% 0.32/0.57 & ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % inf_period(1)
% 0.32/0.57 thf(fact_3539_inf__period_I1_J,axiom,
% 0.32/0.57 ! [P: int > $o,D3: int,Q: int > $o] :
% 0.32/0.57 ( ! [X4: int,K2: int] :
% 0.32/0.57 ( ( P @ X4 )
% 0.32/0.57 = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D3 ) ) ) )
% 0.32/0.57 => ( ! [X4: int,K2: int] :
% 0.32/0.57 ( ( Q @ X4 )
% 0.32/0.57 = ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D3 ) ) ) )
% 0.32/0.57 => ! [X5: int,K4: int] :
% 0.32/0.57 ( ( ( P @ X5 )
% 0.32/0.57 & ( Q @ X5 ) )
% 0.32/0.57 = ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) )
% 0.32/0.57 & ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % inf_period(1)
% 0.32/0.57 thf(fact_3540_atLeastatMost__psubset__iff,axiom,
% 0.32/0.57 ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
% 0.32/0.57 ( ( ord_less_set_set_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
% 0.32/0.57 = ( ( ~ ( ord_less_eq_set_nat @ A @ B )
% 0.32/0.57 | ( ( ord_less_eq_set_nat @ C @ A )
% 0.32/0.57 & ( ord_less_eq_set_nat @ B @ D )
% 0.32/0.57 & ( ( ord_less_set_nat @ C @ A )
% 0.32/0.57 | ( ord_less_set_nat @ B @ D ) ) ) )
% 0.32/0.57 & ( ord_less_eq_set_nat @ C @ D ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_psubset_iff
% 0.32/0.57 thf(fact_3541_atLeastatMost__psubset__iff,axiom,
% 0.32/0.57 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.57 ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 0.32/0.57 = ( ( ~ ( ord_less_eq_rat @ A @ B )
% 0.32/0.57 | ( ( ord_less_eq_rat @ C @ A )
% 0.32/0.57 & ( ord_less_eq_rat @ B @ D )
% 0.32/0.57 & ( ( ord_less_rat @ C @ A )
% 0.32/0.57 | ( ord_less_rat @ B @ D ) ) ) )
% 0.32/0.57 & ( ord_less_eq_rat @ C @ D ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_psubset_iff
% 0.32/0.57 thf(fact_3542_atLeastatMost__psubset__iff,axiom,
% 0.32/0.57 ! [A: num,B: num,C: num,D: num] :
% 0.32/0.57 ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 0.32/0.57 = ( ( ~ ( ord_less_eq_num @ A @ B )
% 0.32/0.57 | ( ( ord_less_eq_num @ C @ A )
% 0.32/0.57 & ( ord_less_eq_num @ B @ D )
% 0.32/0.57 & ( ( ord_less_num @ C @ A )
% 0.32/0.57 | ( ord_less_num @ B @ D ) ) ) )
% 0.32/0.57 & ( ord_less_eq_num @ C @ D ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_psubset_iff
% 0.32/0.57 thf(fact_3543_atLeastatMost__psubset__iff,axiom,
% 0.32/0.57 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.57 ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 0.32/0.57 = ( ( ~ ( ord_less_eq_nat @ A @ B )
% 0.32/0.57 | ( ( ord_less_eq_nat @ C @ A )
% 0.32/0.57 & ( ord_less_eq_nat @ B @ D )
% 0.32/0.57 & ( ( ord_less_nat @ C @ A )
% 0.32/0.57 | ( ord_less_nat @ B @ D ) ) ) )
% 0.32/0.57 & ( ord_less_eq_nat @ C @ D ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_psubset_iff
% 0.32/0.57 thf(fact_3544_atLeastatMost__psubset__iff,axiom,
% 0.32/0.57 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.57 ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 0.32/0.57 = ( ( ~ ( ord_less_eq_int @ A @ B )
% 0.32/0.57 | ( ( ord_less_eq_int @ C @ A )
% 0.32/0.57 & ( ord_less_eq_int @ B @ D )
% 0.32/0.57 & ( ( ord_less_int @ C @ A )
% 0.32/0.57 | ( ord_less_int @ B @ D ) ) ) )
% 0.32/0.57 & ( ord_less_eq_int @ C @ D ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_psubset_iff
% 0.32/0.57 thf(fact_3545_atLeastatMost__psubset__iff,axiom,
% 0.32/0.57 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.57 ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 0.32/0.57 = ( ( ~ ( ord_less_eq_real @ A @ B )
% 0.32/0.57 | ( ( ord_less_eq_real @ C @ A )
% 0.32/0.57 & ( ord_less_eq_real @ B @ D )
% 0.32/0.57 & ( ( ord_less_real @ C @ A )
% 0.32/0.57 | ( ord_less_real @ B @ D ) ) ) )
% 0.32/0.57 & ( ord_less_eq_real @ C @ D ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % atLeastatMost_psubset_iff
% 0.32/0.57 thf(fact_3546_plusinfinity,axiom,
% 0.32/0.57 ! [D: int,P6: int > $o,P: int > $o] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ D )
% 0.32/0.57 => ( ! [X4: int,K2: int] :
% 0.32/0.57 ( ( P6 @ X4 )
% 0.32/0.57 = ( P6 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
% 0.32/0.57 => ( ? [Z4: int] :
% 0.32/0.57 ! [X4: int] :
% 0.32/0.57 ( ( ord_less_int @ Z4 @ X4 )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 = ( P6 @ X4 ) ) )
% 0.32/0.57 => ( ? [X_12: int] : ( P6 @ X_12 )
% 0.32/0.57 => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % plusinfinity
% 0.32/0.57 thf(fact_3547_minusinfinity,axiom,
% 0.32/0.57 ! [D: int,P1: int > $o,P: int > $o] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ D )
% 0.32/0.57 => ( ! [X4: int,K2: int] :
% 0.32/0.57 ( ( P1 @ X4 )
% 0.32/0.57 = ( P1 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
% 0.32/0.57 => ( ? [Z4: int] :
% 0.32/0.57 ! [X4: int] :
% 0.32/0.57 ( ( ord_less_int @ X4 @ Z4 )
% 0.32/0.57 => ( ( P @ X4 )
% 0.32/0.57 = ( P1 @ X4 ) ) )
% 0.32/0.57 => ( ? [X_12: int] : ( P1 @ X_12 )
% 0.32/0.57 => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % minusinfinity
% 0.32/0.57 thf(fact_3548_flip__bit__Suc,axiom,
% 0.32/0.57 ! [N2: nat,A: code_integer] :
% 0.32/0.57 ( ( bit_se1345352211410354436nteger @ ( suc @ N2 ) @ A )
% 0.32/0.57 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % flip_bit_Suc
% 0.32/0.57 thf(fact_3549_flip__bit__Suc,axiom,
% 0.32/0.57 ! [N2: nat,A: int] :
% 0.32/0.57 ( ( bit_se2159334234014336723it_int @ ( suc @ N2 ) @ A )
% 0.32/0.57 = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % flip_bit_Suc
% 0.32/0.57 thf(fact_3550_flip__bit__Suc,axiom,
% 0.32/0.57 ! [N2: nat,A: nat] :
% 0.32/0.57 ( ( bit_se2161824704523386999it_nat @ ( suc @ N2 ) @ A )
% 0.32/0.57 = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % flip_bit_Suc
% 0.32/0.57 thf(fact_3551_Bolzano,axiom,
% 0.32/0.57 ! [A: real,B: real,P: real > real > $o] :
% 0.32/0.57 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.57 => ( ! [A3: real,B2: real,C3: real] :
% 0.32/0.57 ( ( P @ A3 @ B2 )
% 0.32/0.57 => ( ( P @ B2 @ C3 )
% 0.32/0.57 => ( ( ord_less_eq_real @ A3 @ B2 )
% 0.32/0.57 => ( ( ord_less_eq_real @ B2 @ C3 )
% 0.32/0.57 => ( P @ A3 @ C3 ) ) ) ) )
% 0.32/0.57 => ( ! [X4: real] :
% 0.32/0.57 ( ( ord_less_eq_real @ A @ X4 )
% 0.32/0.57 => ( ( ord_less_eq_real @ X4 @ B )
% 0.32/0.57 => ? [D5: real] :
% 0.32/0.57 ( ( ord_less_real @ zero_zero_real @ D5 )
% 0.32/0.57 & ! [A3: real,B2: real] :
% 0.32/0.57 ( ( ( ord_less_eq_real @ A3 @ X4 )
% 0.32/0.57 & ( ord_less_eq_real @ X4 @ B2 )
% 0.32/0.57 & ( ord_less_real @ ( minus_minus_real @ B2 @ A3 ) @ D5 ) )
% 0.32/0.57 => ( P @ A3 @ B2 ) ) ) ) )
% 0.32/0.57 => ( P @ A @ B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % Bolzano
% 0.32/0.57 thf(fact_3552_mult__le__cancel__iff2,axiom,
% 0.32/0.57 ! [Z: real,X: real,Y: real] :
% 0.32/0.57 ( ( ord_less_real @ zero_zero_real @ Z )
% 0.32/0.57 => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ ( times_times_real @ Z @ Y ) )
% 0.32/0.57 = ( ord_less_eq_real @ X @ Y ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mult_le_cancel_iff2
% 0.32/0.57 thf(fact_3553_mult__le__cancel__iff2,axiom,
% 0.32/0.57 ! [Z: rat,X: rat,Y: rat] :
% 0.32/0.57 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 0.32/0.57 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ ( times_times_rat @ Z @ Y ) )
% 0.32/0.57 = ( ord_less_eq_rat @ X @ Y ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mult_le_cancel_iff2
% 0.32/0.57 thf(fact_3554_mult__le__cancel__iff2,axiom,
% 0.32/0.57 ! [Z: int,X: int,Y: int] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.32/0.57 => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y ) )
% 0.32/0.57 = ( ord_less_eq_int @ X @ Y ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mult_le_cancel_iff2
% 0.32/0.57 thf(fact_3555_mult__le__cancel__iff1,axiom,
% 0.32/0.57 ! [Z: real,X: real,Y: real] :
% 0.32/0.57 ( ( ord_less_real @ zero_zero_real @ Z )
% 0.32/0.57 => ( ( ord_less_eq_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
% 0.32/0.57 = ( ord_less_eq_real @ X @ Y ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mult_le_cancel_iff1
% 0.32/0.57 thf(fact_3556_mult__le__cancel__iff1,axiom,
% 0.32/0.57 ! [Z: rat,X: rat,Y: rat] :
% 0.32/0.57 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 0.32/0.57 => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
% 0.32/0.57 = ( ord_less_eq_rat @ X @ Y ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mult_le_cancel_iff1
% 0.32/0.57 thf(fact_3557_mult__le__cancel__iff1,axiom,
% 0.32/0.57 ! [Z: int,X: int,Y: int] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.32/0.57 => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
% 0.32/0.57 = ( ord_less_eq_int @ X @ Y ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mult_le_cancel_iff1
% 0.32/0.57 thf(fact_3558_divides__aux__eq,axiom,
% 0.32/0.57 ! [Q2: nat,R: nat] :
% 0.32/0.57 ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q2 @ R ) )
% 0.32/0.57 = ( R = zero_zero_nat ) ) ).
% 0.32/0.57
% 0.32/0.57 % divides_aux_eq
% 0.32/0.57 thf(fact_3559_divides__aux__eq,axiom,
% 0.32/0.57 ! [Q2: int,R: int] :
% 0.32/0.57 ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.32/0.57 = ( R = zero_zero_int ) ) ).
% 0.32/0.57
% 0.32/0.57 % divides_aux_eq
% 0.32/0.57 thf(fact_3560_neg__eucl__rel__int__mult__2,axiom,
% 0.32/0.57 ! [B: int,A: int,Q2: int,R: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.32/0.57 => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.32/0.57 => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q2 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R ) @ one_one_int ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % neg_eucl_rel_int_mult_2
% 0.32/0.57 thf(fact_3561_flip__bit__nonnegative__int__iff,axiom,
% 0.32/0.57 ! [N2: nat,K: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N2 @ K ) )
% 0.32/0.57 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.32/0.57
% 0.32/0.57 % flip_bit_nonnegative_int_iff
% 0.32/0.57 thf(fact_3562_flip__bit__negative__int__iff,axiom,
% 0.32/0.57 ! [N2: nat,K: int] :
% 0.32/0.57 ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N2 @ K ) @ zero_zero_int )
% 0.32/0.57 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.32/0.57
% 0.32/0.57 % flip_bit_negative_int_iff
% 0.32/0.57 thf(fact_3563_div__int__unique,axiom,
% 0.32/0.57 ! [K: int,L2: int,Q2: int,R: int] :
% 0.32/0.57 ( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.32/0.57 => ( ( divide_divide_int @ K @ L2 )
% 0.32/0.57 = Q2 ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_int_unique
% 0.32/0.57 thf(fact_3564_eucl__rel__int,axiom,
% 0.32/0.57 ! [K: int,L2: int] : ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L2 ) @ ( modulo_modulo_int @ K @ L2 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % eucl_rel_int
% 0.32/0.57 thf(fact_3565_eucl__rel__int__iff,axiom,
% 0.32/0.57 ! [K: int,L2: int,Q2: int,R: int] :
% 0.32/0.57 ( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.32/0.57 = ( ( K
% 0.32/0.57 = ( plus_plus_int @ ( times_times_int @ L2 @ Q2 ) @ R ) )
% 0.32/0.57 & ( ( ord_less_int @ zero_zero_int @ L2 )
% 0.32/0.57 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 0.32/0.57 & ( ord_less_int @ R @ L2 ) ) )
% 0.32/0.57 & ( ~ ( ord_less_int @ zero_zero_int @ L2 )
% 0.32/0.57 => ( ( ( ord_less_int @ L2 @ zero_zero_int )
% 0.32/0.57 => ( ( ord_less_int @ L2 @ R )
% 0.32/0.57 & ( ord_less_eq_int @ R @ zero_zero_int ) ) )
% 0.32/0.57 & ( ~ ( ord_less_int @ L2 @ zero_zero_int )
% 0.32/0.57 => ( Q2 = zero_zero_int ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % eucl_rel_int_iff
% 0.32/0.57 thf(fact_3566_mult__less__iff1,axiom,
% 0.32/0.57 ! [Z: real,X: real,Y: real] :
% 0.32/0.57 ( ( ord_less_real @ zero_zero_real @ Z )
% 0.32/0.57 => ( ( ord_less_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
% 0.32/0.57 = ( ord_less_real @ X @ Y ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mult_less_iff1
% 0.32/0.57 thf(fact_3567_mult__less__iff1,axiom,
% 0.32/0.57 ! [Z: rat,X: rat,Y: rat] :
% 0.32/0.57 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 0.32/0.57 => ( ( ord_less_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
% 0.32/0.57 = ( ord_less_rat @ X @ Y ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mult_less_iff1
% 0.32/0.57 thf(fact_3568_mult__less__iff1,axiom,
% 0.32/0.57 ! [Z: int,X: int,Y: int] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.32/0.57 => ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
% 0.32/0.57 = ( ord_less_int @ X @ Y ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mult_less_iff1
% 0.32/0.57 thf(fact_3569_pos__eucl__rel__int__mult__2,axiom,
% 0.32/0.57 ! [B: int,A: int,Q2: int,R: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.57 => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.32/0.57 => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q2 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % pos_eucl_rel_int_mult_2
% 0.32/0.57 thf(fact_3570_product__nth,axiom,
% 0.32/0.57 ! [N2: nat,Xs: list_num,Ys: list_num] :
% 0.32/0.57 ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_num @ Xs ) @ ( size_size_list_num @ Ys ) ) )
% 0.32/0.57 => ( ( nth_Pr6456567536196504476um_num @ ( product_num_num @ Xs @ Ys ) @ N2 )
% 0.32/0.57 = ( product_Pair_num_num @ ( nth_num @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_num @ Ys ) ) ) @ ( nth_num @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_num @ Ys ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % product_nth
% 0.32/0.57 thf(fact_3571_product__nth,axiom,
% 0.32/0.57 ! [N2: nat,Xs: list_nat,Ys: list_num] :
% 0.32/0.57 ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_num @ Ys ) ) )
% 0.32/0.57 => ( ( nth_Pr8326237132889035090at_num @ ( product_nat_num @ Xs @ Ys ) @ N2 )
% 0.32/0.57 = ( product_Pair_nat_num @ ( nth_nat @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_num @ Ys ) ) ) @ ( nth_num @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_num @ Ys ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % product_nth
% 0.32/0.57 thf(fact_3572_product__nth,axiom,
% 0.32/0.57 ! [N2: nat,Xs: list_nat,Ys: list_nat] :
% 0.32/0.57 ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
% 0.32/0.57 => ( ( nth_Pr7617993195940197384at_nat @ ( product_nat_nat @ Xs @ Ys ) @ N2 )
% 0.32/0.57 = ( product_Pair_nat_nat @ ( nth_nat @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % product_nth
% 0.32/0.57 thf(fact_3573_product__nth,axiom,
% 0.32/0.57 ! [N2: nat,Xs: list_Code_integer,Ys: list_o] :
% 0.32/0.57 ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s3445333598471063425nteger @ Xs ) @ ( size_size_list_o @ Ys ) ) )
% 0.32/0.57 => ( ( nth_Pr8522763379788166057eger_o @ ( produc3607205314601156340eger_o @ Xs @ Ys ) @ N2 )
% 0.32/0.57 = ( produc6677183202524767010eger_o @ ( nth_Code_integer @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % product_nth
% 0.32/0.57 thf(fact_3574_product__nth,axiom,
% 0.32/0.57 ! [N2: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 0.32/0.57 ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
% 0.32/0.57 => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys ) @ N2 )
% 0.32/0.57 = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % product_nth
% 0.32/0.57 thf(fact_3575_product__nth,axiom,
% 0.32/0.57 ! [N2: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
% 0.32/0.57 ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys ) ) )
% 0.32/0.57 => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys ) @ N2 )
% 0.32/0.57 = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % product_nth
% 0.32/0.57 thf(fact_3576_product__nth,axiom,
% 0.32/0.57 ! [N2: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
% 0.32/0.57 ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys ) ) )
% 0.32/0.57 => ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys ) @ N2 )
% 0.32/0.57 = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % product_nth
% 0.32/0.57 thf(fact_3577_product__nth,axiom,
% 0.32/0.57 ! [N2: nat,Xs: list_o,Ys: list_VEBT_VEBT] :
% 0.32/0.57 ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
% 0.32/0.57 => ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs @ Ys ) @ N2 )
% 0.32/0.57 = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs @ ( divide_divide_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % product_nth
% 0.32/0.57 thf(fact_3578_product__nth,axiom,
% 0.32/0.57 ! [N2: nat,Xs: list_o,Ys: list_o] :
% 0.32/0.57 ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) ) )
% 0.32/0.57 => ( ( nth_Product_prod_o_o @ ( product_o_o @ Xs @ Ys ) @ N2 )
% 0.32/0.57 = ( product_Pair_o_o @ ( nth_o @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % product_nth
% 0.32/0.57 thf(fact_3579_product__nth,axiom,
% 0.32/0.57 ! [N2: nat,Xs: list_o,Ys: list_int] :
% 0.32/0.57 ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys ) ) )
% 0.32/0.57 => ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs @ Ys ) @ N2 )
% 0.32/0.57 = ( product_Pair_o_int @ ( nth_o @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % product_nth
% 0.32/0.57 thf(fact_3580_obtain__set__pred,axiom,
% 0.32/0.57 ! [Z: nat,X: nat,A2: set_nat] :
% 0.32/0.57 ( ( ord_less_nat @ Z @ X )
% 0.32/0.57 => ( ( vEBT_VEBT_min_in_set @ A2 @ Z )
% 0.32/0.57 => ( ( finite_finite_nat @ A2 )
% 0.32/0.57 => ? [X_1: nat] : ( vEBT_is_pred_in_set @ A2 @ X @ X_1 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % obtain_set_pred
% 0.32/0.57 thf(fact_3581_obtain__set__succ,axiom,
% 0.32/0.57 ! [X: nat,Z: nat,A2: set_nat,B3: set_nat] :
% 0.32/0.57 ( ( ord_less_nat @ X @ Z )
% 0.32/0.57 => ( ( vEBT_VEBT_max_in_set @ A2 @ Z )
% 0.32/0.57 => ( ( finite_finite_nat @ B3 )
% 0.32/0.57 => ( ( A2 = B3 )
% 0.32/0.57 => ? [X_1: nat] : ( vEBT_is_succ_in_set @ A2 @ X @ X_1 ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % obtain_set_succ
% 0.32/0.57 thf(fact_3582_set__vebt__finite,axiom,
% 0.32/0.57 ! [T: vEBT_VEBT,N2: nat] :
% 0.32/0.57 ( ( vEBT_invar_vebt @ T @ N2 )
% 0.32/0.57 => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % set_vebt_finite
% 0.32/0.57 thf(fact_3583_succ__none__empty,axiom,
% 0.32/0.57 ! [Xs: set_nat,A: nat] :
% 0.32/0.57 ( ~ ? [X_1: nat] : ( vEBT_is_succ_in_set @ Xs @ A @ X_1 )
% 0.32/0.57 => ( ( finite_finite_nat @ Xs )
% 0.32/0.57 => ~ ? [X5: nat] :
% 0.32/0.57 ( ( member_nat @ X5 @ Xs )
% 0.32/0.57 & ( ord_less_nat @ A @ X5 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % succ_none_empty
% 0.32/0.57 thf(fact_3584_pred__none__empty,axiom,
% 0.32/0.58 ! [Xs: set_nat,A: nat] :
% 0.32/0.58 ( ~ ? [X_1: nat] : ( vEBT_is_pred_in_set @ Xs @ A @ X_1 )
% 0.32/0.58 => ( ( finite_finite_nat @ Xs )
% 0.32/0.58 => ~ ? [X5: nat] :
% 0.32/0.58 ( ( member_nat @ X5 @ Xs )
% 0.32/0.58 & ( ord_less_nat @ X5 @ A ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % pred_none_empty
% 0.32/0.58 thf(fact_3585_old_Oprod_Oinject,axiom,
% 0.32/0.58 ! [A: code_integer,B: $o,A6: code_integer,B6: $o] :
% 0.32/0.58 ( ( ( produc6677183202524767010eger_o @ A @ B )
% 0.32/0.58 = ( produc6677183202524767010eger_o @ A6 @ B6 ) )
% 0.32/0.58 = ( ( A = A6 )
% 0.32/0.58 & ( B = B6 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % old.prod.inject
% 0.32/0.58 thf(fact_3586_old_Oprod_Oinject,axiom,
% 0.32/0.58 ! [A: num,B: num,A6: num,B6: num] :
% 0.32/0.58 ( ( ( product_Pair_num_num @ A @ B )
% 0.32/0.58 = ( product_Pair_num_num @ A6 @ B6 ) )
% 0.32/0.58 = ( ( A = A6 )
% 0.32/0.58 & ( B = B6 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % old.prod.inject
% 0.32/0.58 thf(fact_3587_old_Oprod_Oinject,axiom,
% 0.32/0.58 ! [A: nat,B: num,A6: nat,B6: num] :
% 0.32/0.58 ( ( ( product_Pair_nat_num @ A @ B )
% 0.32/0.58 = ( product_Pair_nat_num @ A6 @ B6 ) )
% 0.32/0.58 = ( ( A = A6 )
% 0.32/0.58 & ( B = B6 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % old.prod.inject
% 0.32/0.58 thf(fact_3588_old_Oprod_Oinject,axiom,
% 0.32/0.58 ! [A: nat,B: nat,A6: nat,B6: nat] :
% 0.32/0.58 ( ( ( product_Pair_nat_nat @ A @ B )
% 0.32/0.58 = ( product_Pair_nat_nat @ A6 @ B6 ) )
% 0.32/0.58 = ( ( A = A6 )
% 0.32/0.58 & ( B = B6 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % old.prod.inject
% 0.32/0.58 thf(fact_3589_old_Oprod_Oinject,axiom,
% 0.32/0.58 ! [A: int,B: int,A6: int,B6: int] :
% 0.32/0.58 ( ( ( product_Pair_int_int @ A @ B )
% 0.32/0.58 = ( product_Pair_int_int @ A6 @ B6 ) )
% 0.32/0.58 = ( ( A = A6 )
% 0.32/0.58 & ( B = B6 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % old.prod.inject
% 0.32/0.58 thf(fact_3590_prod_Oinject,axiom,
% 0.32/0.58 ! [X1: code_integer,X22: $o,Y1: code_integer,Y22: $o] :
% 0.32/0.58 ( ( ( produc6677183202524767010eger_o @ X1 @ X22 )
% 0.32/0.58 = ( produc6677183202524767010eger_o @ Y1 @ Y22 ) )
% 0.32/0.58 = ( ( X1 = Y1 )
% 0.32/0.58 & ( X22 = Y22 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % prod.inject
% 0.32/0.58 thf(fact_3591_prod_Oinject,axiom,
% 0.32/0.58 ! [X1: num,X22: num,Y1: num,Y22: num] :
% 0.32/0.58 ( ( ( product_Pair_num_num @ X1 @ X22 )
% 0.32/0.58 = ( product_Pair_num_num @ Y1 @ Y22 ) )
% 0.32/0.58 = ( ( X1 = Y1 )
% 0.32/0.58 & ( X22 = Y22 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % prod.inject
% 0.32/0.58 thf(fact_3592_prod_Oinject,axiom,
% 0.32/0.58 ! [X1: nat,X22: num,Y1: nat,Y22: num] :
% 0.32/0.58 ( ( ( product_Pair_nat_num @ X1 @ X22 )
% 0.32/0.58 = ( product_Pair_nat_num @ Y1 @ Y22 ) )
% 0.32/0.58 = ( ( X1 = Y1 )
% 0.32/0.58 & ( X22 = Y22 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % prod.inject
% 0.32/0.58 thf(fact_3593_prod_Oinject,axiom,
% 0.32/0.58 ! [X1: nat,X22: nat,Y1: nat,Y22: nat] :
% 0.32/0.58 ( ( ( product_Pair_nat_nat @ X1 @ X22 )
% 0.32/0.58 = ( product_Pair_nat_nat @ Y1 @ Y22 ) )
% 0.32/0.58 = ( ( X1 = Y1 )
% 0.32/0.58 & ( X22 = Y22 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % prod.inject
% 0.32/0.58 thf(fact_3594_prod_Oinject,axiom,
% 0.32/0.58 ! [X1: int,X22: int,Y1: int,Y22: int] :
% 0.32/0.58 ( ( ( product_Pair_int_int @ X1 @ X22 )
% 0.32/0.58 = ( product_Pair_int_int @ Y1 @ Y22 ) )
% 0.32/0.58 = ( ( X1 = Y1 )
% 0.32/0.58 & ( X22 = Y22 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % prod.inject
% 0.32/0.58 thf(fact_3595_List_Ofinite__set,axiom,
% 0.32/0.58 ! [Xs: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) ) ).
% 0.32/0.58
% 0.32/0.58 % List.finite_set
% 0.32/0.58 thf(fact_3596_List_Ofinite__set,axiom,
% 0.32/0.58 ! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% 0.32/0.58
% 0.32/0.58 % List.finite_set
% 0.32/0.58 thf(fact_3597_List_Ofinite__set,axiom,
% 0.32/0.58 ! [Xs: list_int] : ( finite_finite_int @ ( set_int2 @ Xs ) ) ).
% 0.32/0.58
% 0.32/0.58 % List.finite_set
% 0.32/0.58 thf(fact_3598_List_Ofinite__set,axiom,
% 0.32/0.58 ! [Xs: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs ) ) ).
% 0.32/0.58
% 0.32/0.58 % List.finite_set
% 0.32/0.58 thf(fact_3599_infinite__Icc__iff,axiom,
% 0.32/0.58 ! [A: rat,B: rat] :
% 0.32/0.58 ( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) )
% 0.32/0.58 = ( ord_less_rat @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % infinite_Icc_iff
% 0.32/0.58 thf(fact_3600_infinite__Icc__iff,axiom,
% 0.32/0.58 ! [A: real,B: real] :
% 0.32/0.58 ( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) )
% 0.32/0.58 = ( ord_less_real @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % infinite_Icc_iff
% 0.32/0.58 thf(fact_3601_length__product,axiom,
% 0.32/0.58 ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 0.32/0.58 ( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys ) )
% 0.32/0.58 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % length_product
% 0.32/0.58 thf(fact_3602_length__product,axiom,
% 0.32/0.58 ! [Xs: list_VEBT_VEBT,Ys: list_o] :
% 0.32/0.58 ( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys ) )
% 0.32/0.58 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % length_product
% 0.32/0.58 thf(fact_3603_length__product,axiom,
% 0.32/0.58 ! [Xs: list_VEBT_VEBT,Ys: list_int] :
% 0.32/0.58 ( ( size_s3661962791536183091BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys ) )
% 0.32/0.58 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % length_product
% 0.32/0.58 thf(fact_3604_length__product,axiom,
% 0.32/0.58 ! [Xs: list_o,Ys: list_VEBT_VEBT] :
% 0.32/0.58 ( ( size_s4313452262239582901T_VEBT @ ( product_o_VEBT_VEBT @ Xs @ Ys ) )
% 0.32/0.58 = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % length_product
% 0.32/0.58 thf(fact_3605_length__product,axiom,
% 0.32/0.58 ! [Xs: list_o,Ys: list_o] :
% 0.32/0.58 ( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs @ Ys ) )
% 0.32/0.58 = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % length_product
% 0.32/0.58 thf(fact_3606_length__product,axiom,
% 0.32/0.58 ! [Xs: list_o,Ys: list_int] :
% 0.32/0.58 ( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs @ Ys ) )
% 0.32/0.58 = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % length_product
% 0.32/0.58 thf(fact_3607_length__product,axiom,
% 0.32/0.58 ! [Xs: list_int,Ys: list_VEBT_VEBT] :
% 0.32/0.58 ( ( size_s6639371672096860321T_VEBT @ ( produc662631939642741121T_VEBT @ Xs @ Ys ) )
% 0.32/0.58 = ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % length_product
% 0.32/0.58 thf(fact_3608_length__product,axiom,
% 0.32/0.58 ! [Xs: list_int,Ys: list_o] :
% 0.32/0.58 ( ( size_s4246224855604898693_int_o @ ( product_int_o @ Xs @ Ys ) )
% 0.32/0.58 = ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % length_product
% 0.32/0.58 thf(fact_3609_length__product,axiom,
% 0.32/0.58 ! [Xs: list_int,Ys: list_int] :
% 0.32/0.58 ( ( size_s5157815400016825771nt_int @ ( product_int_int @ Xs @ Ys ) )
% 0.32/0.58 = ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % length_product
% 0.32/0.58 thf(fact_3610_finite__nat__set__iff__bounded,axiom,
% 0.32/0.58 ( finite_finite_nat
% 0.32/0.58 = ( ^ [N6: set_nat] :
% 0.32/0.58 ? [M6: nat] :
% 0.32/0.58 ! [X2: nat] :
% 0.32/0.58 ( ( member_nat @ X2 @ N6 )
% 0.32/0.58 => ( ord_less_nat @ X2 @ M6 ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_nat_set_iff_bounded
% 0.32/0.58 thf(fact_3611_bounded__nat__set__is__finite,axiom,
% 0.32/0.58 ! [N3: set_nat,N2: nat] :
% 0.32/0.58 ( ! [X4: nat] :
% 0.32/0.58 ( ( member_nat @ X4 @ N3 )
% 0.32/0.58 => ( ord_less_nat @ X4 @ N2 ) )
% 0.32/0.58 => ( finite_finite_nat @ N3 ) ) ).
% 0.32/0.58
% 0.32/0.58 % bounded_nat_set_is_finite
% 0.32/0.58 thf(fact_3612_finite__nat__set__iff__bounded__le,axiom,
% 0.32/0.58 ( finite_finite_nat
% 0.32/0.58 = ( ^ [N6: set_nat] :
% 0.32/0.58 ? [M6: nat] :
% 0.32/0.58 ! [X2: nat] :
% 0.32/0.58 ( ( member_nat @ X2 @ N6 )
% 0.32/0.58 => ( ord_less_eq_nat @ X2 @ M6 ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_nat_set_iff_bounded_le
% 0.32/0.58 thf(fact_3613_finite__list,axiom,
% 0.32/0.58 ! [A2: set_VEBT_VEBT] :
% 0.32/0.58 ( ( finite5795047828879050333T_VEBT @ A2 )
% 0.32/0.58 => ? [Xs3: list_VEBT_VEBT] :
% 0.32/0.58 ( ( set_VEBT_VEBT2 @ Xs3 )
% 0.32/0.58 = A2 ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_list
% 0.32/0.58 thf(fact_3614_finite__list,axiom,
% 0.32/0.58 ! [A2: set_nat] :
% 0.32/0.58 ( ( finite_finite_nat @ A2 )
% 0.32/0.58 => ? [Xs3: list_nat] :
% 0.32/0.58 ( ( set_nat2 @ Xs3 )
% 0.32/0.58 = A2 ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_list
% 0.32/0.58 thf(fact_3615_finite__list,axiom,
% 0.32/0.58 ! [A2: set_int] :
% 0.32/0.58 ( ( finite_finite_int @ A2 )
% 0.32/0.58 => ? [Xs3: list_int] :
% 0.32/0.58 ( ( set_int2 @ Xs3 )
% 0.32/0.58 = A2 ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_list
% 0.32/0.58 thf(fact_3616_finite__list,axiom,
% 0.32/0.58 ! [A2: set_complex] :
% 0.32/0.58 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.58 => ? [Xs3: list_complex] :
% 0.32/0.58 ( ( set_complex2 @ Xs3 )
% 0.32/0.58 = A2 ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_list
% 0.32/0.58 thf(fact_3617_finite__M__bounded__by__nat,axiom,
% 0.32/0.58 ! [P: nat > $o,I2: nat] :
% 0.32/0.58 ( finite_finite_nat
% 0.32/0.58 @ ( collect_nat
% 0.32/0.58 @ ^ [K3: nat] :
% 0.32/0.58 ( ( P @ K3 )
% 0.32/0.58 & ( ord_less_nat @ K3 @ I2 ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_M_bounded_by_nat
% 0.32/0.58 thf(fact_3618_finite__less__ub,axiom,
% 0.32/0.58 ! [F: nat > nat,U: nat] :
% 0.32/0.58 ( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F @ N4 ) )
% 0.32/0.58 => ( finite_finite_nat
% 0.32/0.58 @ ( collect_nat
% 0.32/0.58 @ ^ [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ U ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_less_ub
% 0.32/0.58 thf(fact_3619_finite__lists__length__eq,axiom,
% 0.32/0.58 ! [A2: set_complex,N2: nat] :
% 0.32/0.58 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.58 => ( finite8712137658972009173omplex
% 0.32/0.58 @ ( collect_list_complex
% 0.32/0.58 @ ^ [Xs2: list_complex] :
% 0.32/0.58 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
% 0.32/0.58 & ( ( size_s3451745648224563538omplex @ Xs2 )
% 0.32/0.58 = N2 ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_lists_length_eq
% 0.32/0.58 thf(fact_3620_finite__lists__length__eq,axiom,
% 0.32/0.58 ! [A2: set_VEBT_VEBT,N2: nat] :
% 0.32/0.58 ( ( finite5795047828879050333T_VEBT @ A2 )
% 0.32/0.58 => ( finite3004134309566078307T_VEBT
% 0.32/0.58 @ ( collec5608196760682091941T_VEBT
% 0.32/0.58 @ ^ [Xs2: list_VEBT_VEBT] :
% 0.32/0.58 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
% 0.32/0.58 & ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 0.32/0.58 = N2 ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_lists_length_eq
% 0.32/0.58 thf(fact_3621_finite__lists__length__eq,axiom,
% 0.32/0.58 ! [A2: set_o,N2: nat] :
% 0.32/0.58 ( ( finite_finite_o @ A2 )
% 0.32/0.58 => ( finite_finite_list_o
% 0.32/0.58 @ ( collect_list_o
% 0.32/0.58 @ ^ [Xs2: list_o] :
% 0.32/0.58 ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A2 )
% 0.32/0.58 & ( ( size_size_list_o @ Xs2 )
% 0.32/0.58 = N2 ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_lists_length_eq
% 0.32/0.58 thf(fact_3622_finite__lists__length__eq,axiom,
% 0.32/0.58 ! [A2: set_int,N2: nat] :
% 0.32/0.58 ( ( finite_finite_int @ A2 )
% 0.32/0.58 => ( finite3922522038869484883st_int
% 0.32/0.58 @ ( collect_list_int
% 0.32/0.58 @ ^ [Xs2: list_int] :
% 0.32/0.58 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
% 0.32/0.58 & ( ( size_size_list_int @ Xs2 )
% 0.32/0.58 = N2 ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_lists_length_eq
% 0.32/0.58 thf(fact_3623_finite__lists__length__eq,axiom,
% 0.32/0.58 ! [A2: set_nat,N2: nat] :
% 0.32/0.58 ( ( finite_finite_nat @ A2 )
% 0.32/0.58 => ( finite8100373058378681591st_nat
% 0.32/0.58 @ ( collect_list_nat
% 0.32/0.58 @ ^ [Xs2: list_nat] :
% 0.32/0.58 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
% 0.32/0.58 & ( ( size_size_list_nat @ Xs2 )
% 0.32/0.58 = N2 ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_lists_length_eq
% 0.32/0.58 thf(fact_3624_infinite__Icc,axiom,
% 0.32/0.58 ! [A: rat,B: rat] :
% 0.32/0.58 ( ( ord_less_rat @ A @ B )
% 0.32/0.58 => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % infinite_Icc
% 0.32/0.58 thf(fact_3625_infinite__Icc,axiom,
% 0.32/0.58 ! [A: real,B: real] :
% 0.32/0.58 ( ( ord_less_real @ A @ B )
% 0.32/0.58 => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % infinite_Icc
% 0.32/0.58 thf(fact_3626_finite__lists__length__le,axiom,
% 0.32/0.58 ! [A2: set_complex,N2: nat] :
% 0.32/0.58 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.58 => ( finite8712137658972009173omplex
% 0.32/0.58 @ ( collect_list_complex
% 0.32/0.58 @ ^ [Xs2: list_complex] :
% 0.32/0.58 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
% 0.32/0.58 & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs2 ) @ N2 ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_lists_length_le
% 0.32/0.58 thf(fact_3627_finite__lists__length__le,axiom,
% 0.32/0.58 ! [A2: set_VEBT_VEBT,N2: nat] :
% 0.32/0.58 ( ( finite5795047828879050333T_VEBT @ A2 )
% 0.32/0.58 => ( finite3004134309566078307T_VEBT
% 0.32/0.58 @ ( collec5608196760682091941T_VEBT
% 0.32/0.58 @ ^ [Xs2: list_VEBT_VEBT] :
% 0.32/0.58 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
% 0.32/0.58 & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ N2 ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_lists_length_le
% 0.32/0.58 thf(fact_3628_finite__lists__length__le,axiom,
% 0.32/0.58 ! [A2: set_o,N2: nat] :
% 0.32/0.58 ( ( finite_finite_o @ A2 )
% 0.32/0.58 => ( finite_finite_list_o
% 0.32/0.58 @ ( collect_list_o
% 0.32/0.58 @ ^ [Xs2: list_o] :
% 0.32/0.58 ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A2 )
% 0.32/0.58 & ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ N2 ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_lists_length_le
% 0.32/0.58 thf(fact_3629_finite__lists__length__le,axiom,
% 0.32/0.58 ! [A2: set_int,N2: nat] :
% 0.32/0.58 ( ( finite_finite_int @ A2 )
% 0.32/0.58 => ( finite3922522038869484883st_int
% 0.32/0.58 @ ( collect_list_int
% 0.32/0.58 @ ^ [Xs2: list_int] :
% 0.32/0.58 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
% 0.32/0.58 & ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ N2 ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_lists_length_le
% 0.32/0.58 thf(fact_3630_finite__lists__length__le,axiom,
% 0.32/0.58 ! [A2: set_nat,N2: nat] :
% 0.32/0.58 ( ( finite_finite_nat @ A2 )
% 0.32/0.58 => ( finite8100373058378681591st_nat
% 0.32/0.58 @ ( collect_list_nat
% 0.32/0.58 @ ^ [Xs2: list_nat] :
% 0.32/0.58 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
% 0.32/0.58 & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N2 ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_lists_length_le
% 0.32/0.58 thf(fact_3631_Pair__inject,axiom,
% 0.32/0.58 ! [A: code_integer,B: $o,A6: code_integer,B6: $o] :
% 0.32/0.58 ( ( ( produc6677183202524767010eger_o @ A @ B )
% 0.32/0.58 = ( produc6677183202524767010eger_o @ A6 @ B6 ) )
% 0.32/0.58 => ~ ( ( A = A6 )
% 0.32/0.58 => ( B = ( ~ B6 ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % Pair_inject
% 0.32/0.58 thf(fact_3632_Pair__inject,axiom,
% 0.32/0.58 ! [A: num,B: num,A6: num,B6: num] :
% 0.32/0.58 ( ( ( product_Pair_num_num @ A @ B )
% 0.32/0.58 = ( product_Pair_num_num @ A6 @ B6 ) )
% 0.32/0.58 => ~ ( ( A = A6 )
% 0.32/0.58 => ( B != B6 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % Pair_inject
% 0.32/0.58 thf(fact_3633_Pair__inject,axiom,
% 0.32/0.58 ! [A: nat,B: num,A6: nat,B6: num] :
% 0.32/0.58 ( ( ( product_Pair_nat_num @ A @ B )
% 0.32/0.58 = ( product_Pair_nat_num @ A6 @ B6 ) )
% 0.32/0.58 => ~ ( ( A = A6 )
% 0.32/0.58 => ( B != B6 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % Pair_inject
% 0.32/0.58 thf(fact_3634_Pair__inject,axiom,
% 0.32/0.58 ! [A: nat,B: nat,A6: nat,B6: nat] :
% 0.32/0.58 ( ( ( product_Pair_nat_nat @ A @ B )
% 0.32/0.58 = ( product_Pair_nat_nat @ A6 @ B6 ) )
% 0.32/0.58 => ~ ( ( A = A6 )
% 0.32/0.58 => ( B != B6 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % Pair_inject
% 0.32/0.58 thf(fact_3635_Pair__inject,axiom,
% 0.32/0.58 ! [A: int,B: int,A6: int,B6: int] :
% 0.32/0.58 ( ( ( product_Pair_int_int @ A @ B )
% 0.32/0.58 = ( product_Pair_int_int @ A6 @ B6 ) )
% 0.32/0.58 => ~ ( ( A = A6 )
% 0.32/0.58 => ( B != B6 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % Pair_inject
% 0.32/0.58 thf(fact_3636_prod__cases,axiom,
% 0.32/0.58 ! [P: produc6271795597528267376eger_o > $o,P4: produc6271795597528267376eger_o] :
% 0.32/0.58 ( ! [A3: code_integer,B2: $o] : ( P @ ( produc6677183202524767010eger_o @ A3 @ B2 ) )
% 0.32/0.58 => ( P @ P4 ) ) ).
% 0.32/0.58
% 0.32/0.58 % prod_cases
% 0.32/0.58 thf(fact_3637_prod__cases,axiom,
% 0.32/0.58 ! [P: product_prod_num_num > $o,P4: product_prod_num_num] :
% 0.32/0.58 ( ! [A3: num,B2: num] : ( P @ ( product_Pair_num_num @ A3 @ B2 ) )
% 0.32/0.58 => ( P @ P4 ) ) ).
% 0.32/0.58
% 0.32/0.58 % prod_cases
% 0.32/0.58 thf(fact_3638_prod__cases,axiom,
% 0.32/0.58 ! [P: product_prod_nat_num > $o,P4: product_prod_nat_num] :
% 0.32/0.58 ( ! [A3: nat,B2: num] : ( P @ ( product_Pair_nat_num @ A3 @ B2 ) )
% 0.32/0.58 => ( P @ P4 ) ) ).
% 0.32/0.58
% 0.32/0.58 % prod_cases
% 0.32/0.58 thf(fact_3639_prod__cases,axiom,
% 0.32/0.58 ! [P: product_prod_nat_nat > $o,P4: product_prod_nat_nat] :
% 0.32/0.58 ( ! [A3: nat,B2: nat] : ( P @ ( product_Pair_nat_nat @ A3 @ B2 ) )
% 0.32/0.58 => ( P @ P4 ) ) ).
% 0.32/0.58
% 0.32/0.58 % prod_cases
% 0.32/0.58 thf(fact_3640_prod__cases,axiom,
% 0.32/0.58 ! [P: product_prod_int_int > $o,P4: product_prod_int_int] :
% 0.32/0.58 ( ! [A3: int,B2: int] : ( P @ ( product_Pair_int_int @ A3 @ B2 ) )
% 0.32/0.58 => ( P @ P4 ) ) ).
% 0.32/0.58
% 0.32/0.58 % prod_cases
% 0.32/0.58 thf(fact_3641_surj__pair,axiom,
% 0.32/0.58 ! [P4: produc6271795597528267376eger_o] :
% 0.32/0.58 ? [X4: code_integer,Y4: $o] :
% 0.32/0.58 ( P4
% 0.32/0.58 = ( produc6677183202524767010eger_o @ X4 @ Y4 ) ) ).
% 0.32/0.58
% 0.32/0.58 % surj_pair
% 0.32/0.58 thf(fact_3642_surj__pair,axiom,
% 0.32/0.58 ! [P4: product_prod_num_num] :
% 0.32/0.58 ? [X4: num,Y4: num] :
% 0.32/0.58 ( P4
% 0.32/0.58 = ( product_Pair_num_num @ X4 @ Y4 ) ) ).
% 0.32/0.58
% 0.32/0.58 % surj_pair
% 0.32/0.58 thf(fact_3643_surj__pair,axiom,
% 0.32/0.58 ! [P4: product_prod_nat_num] :
% 0.32/0.58 ? [X4: nat,Y4: num] :
% 0.32/0.58 ( P4
% 0.32/0.58 = ( product_Pair_nat_num @ X4 @ Y4 ) ) ).
% 0.32/0.58
% 0.32/0.58 % surj_pair
% 0.32/0.58 thf(fact_3644_surj__pair,axiom,
% 0.32/0.58 ! [P4: product_prod_nat_nat] :
% 0.32/0.58 ? [X4: nat,Y4: nat] :
% 0.32/0.58 ( P4
% 0.32/0.58 = ( product_Pair_nat_nat @ X4 @ Y4 ) ) ).
% 0.32/0.58
% 0.32/0.58 % surj_pair
% 0.32/0.58 thf(fact_3645_surj__pair,axiom,
% 0.32/0.58 ! [P4: product_prod_int_int] :
% 0.32/0.58 ? [X4: int,Y4: int] :
% 0.32/0.58 ( P4
% 0.32/0.58 = ( product_Pair_int_int @ X4 @ Y4 ) ) ).
% 0.32/0.58
% 0.32/0.58 % surj_pair
% 0.32/0.58 thf(fact_3646_old_Oprod_Oexhaust,axiom,
% 0.32/0.58 ! [Y: produc6271795597528267376eger_o] :
% 0.32/0.58 ~ ! [A3: code_integer,B2: $o] :
% 0.32/0.58 ( Y
% 0.32/0.58 != ( produc6677183202524767010eger_o @ A3 @ B2 ) ) ).
% 0.32/0.58
% 0.32/0.58 % old.prod.exhaust
% 0.32/0.58 thf(fact_3647_old_Oprod_Oexhaust,axiom,
% 0.32/0.58 ! [Y: product_prod_num_num] :
% 0.32/0.58 ~ ! [A3: num,B2: num] :
% 0.32/0.58 ( Y
% 0.32/0.58 != ( product_Pair_num_num @ A3 @ B2 ) ) ).
% 0.32/0.58
% 0.32/0.58 % old.prod.exhaust
% 0.32/0.58 thf(fact_3648_old_Oprod_Oexhaust,axiom,
% 0.32/0.58 ! [Y: product_prod_nat_num] :
% 0.32/0.58 ~ ! [A3: nat,B2: num] :
% 0.32/0.58 ( Y
% 0.32/0.58 != ( product_Pair_nat_num @ A3 @ B2 ) ) ).
% 0.32/0.58
% 0.32/0.58 % old.prod.exhaust
% 0.32/0.58 thf(fact_3649_old_Oprod_Oexhaust,axiom,
% 0.32/0.58 ! [Y: product_prod_nat_nat] :
% 0.32/0.58 ~ ! [A3: nat,B2: nat] :
% 0.32/0.58 ( Y
% 0.32/0.58 != ( product_Pair_nat_nat @ A3 @ B2 ) ) ).
% 0.32/0.58
% 0.32/0.58 % old.prod.exhaust
% 0.32/0.58 thf(fact_3650_old_Oprod_Oexhaust,axiom,
% 0.32/0.58 ! [Y: product_prod_int_int] :
% 0.32/0.58 ~ ! [A3: int,B2: int] :
% 0.32/0.58 ( Y
% 0.32/0.58 != ( product_Pair_int_int @ A3 @ B2 ) ) ).
% 0.32/0.58
% 0.32/0.58 % old.prod.exhaust
% 0.32/0.58 thf(fact_3651_finite__Collect__le__nat,axiom,
% 0.32/0.58 ! [K: nat] :
% 0.32/0.58 ( finite_finite_nat
% 0.32/0.58 @ ( collect_nat
% 0.32/0.58 @ ^ [N: nat] : ( ord_less_eq_nat @ N @ K ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_Collect_le_nat
% 0.32/0.58 thf(fact_3652_finite__Collect__less__nat,axiom,
% 0.32/0.58 ! [K: nat] :
% 0.32/0.58 ( finite_finite_nat
% 0.32/0.58 @ ( collect_nat
% 0.32/0.58 @ ^ [N: nat] : ( ord_less_nat @ N @ K ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_Collect_less_nat
% 0.32/0.58 thf(fact_3653_finite__Collect__subsets,axiom,
% 0.32/0.58 ! [A2: set_int] :
% 0.32/0.58 ( ( finite_finite_int @ A2 )
% 0.32/0.58 => ( finite6197958912794628473et_int
% 0.32/0.58 @ ( collect_set_int
% 0.32/0.58 @ ^ [B4: set_int] : ( ord_less_eq_set_int @ B4 @ A2 ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_Collect_subsets
% 0.32/0.58 thf(fact_3654_finite__Collect__subsets,axiom,
% 0.32/0.58 ! [A2: set_complex] :
% 0.32/0.58 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.58 => ( finite6551019134538273531omplex
% 0.32/0.58 @ ( collect_set_complex
% 0.32/0.58 @ ^ [B4: set_complex] : ( ord_le211207098394363844omplex @ B4 @ A2 ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_Collect_subsets
% 0.32/0.58 thf(fact_3655_finite__Collect__subsets,axiom,
% 0.32/0.58 ! [A2: set_nat] :
% 0.32/0.58 ( ( finite_finite_nat @ A2 )
% 0.32/0.58 => ( finite1152437895449049373et_nat
% 0.32/0.58 @ ( collect_set_nat
% 0.32/0.58 @ ^ [B4: set_nat] : ( ord_less_eq_set_nat @ B4 @ A2 ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_Collect_subsets
% 0.32/0.58 thf(fact_3656_finite__roots__unity,axiom,
% 0.32/0.58 ! [N2: nat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ one_one_nat @ N2 )
% 0.32/0.58 => ( finite_finite_real
% 0.32/0.58 @ ( collect_real
% 0.32/0.58 @ ^ [Z3: real] :
% 0.32/0.58 ( ( power_power_real @ Z3 @ N2 )
% 0.32/0.58 = one_one_real ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_roots_unity
% 0.32/0.58 thf(fact_3657_finite__roots__unity,axiom,
% 0.32/0.58 ! [N2: nat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ one_one_nat @ N2 )
% 0.32/0.58 => ( finite3207457112153483333omplex
% 0.32/0.58 @ ( collect_complex
% 0.32/0.58 @ ^ [Z3: complex] :
% 0.32/0.58 ( ( power_power_complex @ Z3 @ N2 )
% 0.32/0.58 = one_one_complex ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_roots_unity
% 0.32/0.58 thf(fact_3658_finite__Diff,axiom,
% 0.32/0.58 ! [A2: set_int,B3: set_int] :
% 0.32/0.58 ( ( finite_finite_int @ A2 )
% 0.32/0.58 => ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_Diff
% 0.32/0.58 thf(fact_3659_finite__Diff,axiom,
% 0.32/0.58 ! [A2: set_complex,B3: set_complex] :
% 0.32/0.58 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.58 => ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_Diff
% 0.32/0.58 thf(fact_3660_finite__Diff,axiom,
% 0.32/0.58 ! [A2: set_nat,B3: set_nat] :
% 0.32/0.58 ( ( finite_finite_nat @ A2 )
% 0.32/0.58 => ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_Diff
% 0.32/0.58 thf(fact_3661_finite__Diff2,axiom,
% 0.32/0.58 ! [B3: set_int,A2: set_int] :
% 0.32/0.58 ( ( finite_finite_int @ B3 )
% 0.32/0.58 => ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B3 ) )
% 0.32/0.58 = ( finite_finite_int @ A2 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_Diff2
% 0.32/0.58 thf(fact_3662_finite__Diff2,axiom,
% 0.32/0.58 ! [B3: set_complex,A2: set_complex] :
% 0.32/0.58 ( ( finite3207457112153483333omplex @ B3 )
% 0.32/0.58 => ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 0.32/0.58 = ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_Diff2
% 0.32/0.58 thf(fact_3663_finite__Diff2,axiom,
% 0.32/0.58 ! [B3: set_nat,A2: set_nat] :
% 0.32/0.58 ( ( finite_finite_nat @ B3 )
% 0.32/0.58 => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 0.32/0.58 = ( finite_finite_nat @ A2 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_Diff2
% 0.32/0.58 thf(fact_3664_finite__Collect__conjI,axiom,
% 0.32/0.58 ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
% 0.32/0.58 ( ( ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
% 0.32/0.58 | ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ Q ) ) )
% 0.32/0.58 => ( finite2998713641127702882nt_int
% 0.32/0.58 @ ( collec213857154873943460nt_int
% 0.32/0.58 @ ^ [X2: product_prod_int_int] :
% 0.32/0.58 ( ( P @ X2 )
% 0.32/0.58 & ( Q @ X2 ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_Collect_conjI
% 0.32/0.58 thf(fact_3665_finite__Collect__conjI,axiom,
% 0.32/0.58 ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.32/0.58 ( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
% 0.32/0.58 | ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) )
% 0.32/0.58 => ( finite1152437895449049373et_nat
% 0.32/0.58 @ ( collect_set_nat
% 0.32/0.58 @ ^ [X2: set_nat] :
% 0.32/0.58 ( ( P @ X2 )
% 0.32/0.58 & ( Q @ X2 ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_Collect_conjI
% 0.32/0.58 thf(fact_3666_finite__Collect__conjI,axiom,
% 0.32/0.58 ! [P: nat > $o,Q: nat > $o] :
% 0.32/0.58 ( ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.32/0.58 | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
% 0.32/0.58 => ( finite_finite_nat
% 0.32/0.58 @ ( collect_nat
% 0.32/0.58 @ ^ [X2: nat] :
% 0.32/0.58 ( ( P @ X2 )
% 0.32/0.58 & ( Q @ X2 ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_Collect_conjI
% 0.32/0.58 thf(fact_3667_finite__Collect__conjI,axiom,
% 0.32/0.58 ! [P: int > $o,Q: int > $o] :
% 0.32/0.58 ( ( ( finite_finite_int @ ( collect_int @ P ) )
% 0.32/0.58 | ( finite_finite_int @ ( collect_int @ Q ) ) )
% 0.32/0.58 => ( finite_finite_int
% 0.32/0.58 @ ( collect_int
% 0.32/0.58 @ ^ [X2: int] :
% 0.32/0.58 ( ( P @ X2 )
% 0.32/0.58 & ( Q @ X2 ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_Collect_conjI
% 0.32/0.58 thf(fact_3668_finite__Collect__conjI,axiom,
% 0.32/0.58 ! [P: complex > $o,Q: complex > $o] :
% 0.32/0.58 ( ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
% 0.32/0.58 | ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) )
% 0.32/0.58 => ( finite3207457112153483333omplex
% 0.32/0.58 @ ( collect_complex
% 0.32/0.58 @ ^ [X2: complex] :
% 0.32/0.58 ( ( P @ X2 )
% 0.32/0.58 & ( Q @ X2 ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_Collect_conjI
% 0.32/0.58 thf(fact_3669_finite__Collect__disjI,axiom,
% 0.32/0.58 ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
% 0.32/0.58 ( ( finite2998713641127702882nt_int
% 0.32/0.58 @ ( collec213857154873943460nt_int
% 0.32/0.58 @ ^ [X2: product_prod_int_int] :
% 0.32/0.58 ( ( P @ X2 )
% 0.32/0.58 | ( Q @ X2 ) ) ) )
% 0.32/0.58 = ( ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
% 0.32/0.58 & ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ Q ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_Collect_disjI
% 0.32/0.58 thf(fact_3670_finite__Collect__disjI,axiom,
% 0.32/0.58 ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.32/0.58 ( ( finite1152437895449049373et_nat
% 0.32/0.58 @ ( collect_set_nat
% 0.32/0.58 @ ^ [X2: set_nat] :
% 0.32/0.58 ( ( P @ X2 )
% 0.32/0.58 | ( Q @ X2 ) ) ) )
% 0.32/0.58 = ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
% 0.32/0.58 & ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_Collect_disjI
% 0.32/0.58 thf(fact_3671_finite__Collect__disjI,axiom,
% 0.32/0.58 ! [P: nat > $o,Q: nat > $o] :
% 0.32/0.58 ( ( finite_finite_nat
% 0.32/0.58 @ ( collect_nat
% 0.32/0.58 @ ^ [X2: nat] :
% 0.32/0.58 ( ( P @ X2 )
% 0.32/0.58 | ( Q @ X2 ) ) ) )
% 0.32/0.58 = ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.32/0.58 & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_Collect_disjI
% 0.32/0.58 thf(fact_3672_finite__Collect__disjI,axiom,
% 0.32/0.58 ! [P: int > $o,Q: int > $o] :
% 0.32/0.58 ( ( finite_finite_int
% 0.32/0.58 @ ( collect_int
% 0.32/0.58 @ ^ [X2: int] :
% 0.32/0.58 ( ( P @ X2 )
% 0.32/0.58 | ( Q @ X2 ) ) ) )
% 0.32/0.58 = ( ( finite_finite_int @ ( collect_int @ P ) )
% 0.32/0.58 & ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_Collect_disjI
% 0.32/0.58 thf(fact_3673_finite__Collect__disjI,axiom,
% 0.32/0.58 ! [P: complex > $o,Q: complex > $o] :
% 0.32/0.58 ( ( finite3207457112153483333omplex
% 0.32/0.58 @ ( collect_complex
% 0.32/0.58 @ ^ [X2: complex] :
% 0.32/0.58 ( ( P @ X2 )
% 0.32/0.58 | ( Q @ X2 ) ) ) )
% 0.32/0.58 = ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
% 0.32/0.58 & ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_Collect_disjI
% 0.32/0.58 thf(fact_3674_finite__interval__int1,axiom,
% 0.32/0.58 ! [A: int,B: int] :
% 0.32/0.58 ( finite_finite_int
% 0.32/0.58 @ ( collect_int
% 0.32/0.58 @ ^ [I3: int] :
% 0.32/0.58 ( ( ord_less_eq_int @ A @ I3 )
% 0.32/0.58 & ( ord_less_eq_int @ I3 @ B ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_interval_int1
% 0.32/0.58 thf(fact_3675_finite__interval__int4,axiom,
% 0.32/0.58 ! [A: int,B: int] :
% 0.32/0.58 ( finite_finite_int
% 0.32/0.58 @ ( collect_int
% 0.32/0.58 @ ^ [I3: int] :
% 0.32/0.58 ( ( ord_less_int @ A @ I3 )
% 0.32/0.58 & ( ord_less_int @ I3 @ B ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_interval_int4
% 0.32/0.58 thf(fact_3676_finite__interval__int3,axiom,
% 0.32/0.58 ! [A: int,B: int] :
% 0.32/0.58 ( finite_finite_int
% 0.32/0.58 @ ( collect_int
% 0.32/0.58 @ ^ [I3: int] :
% 0.32/0.58 ( ( ord_less_int @ A @ I3 )
% 0.32/0.58 & ( ord_less_eq_int @ I3 @ B ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_interval_int3
% 0.32/0.58 thf(fact_3677_finite__interval__int2,axiom,
% 0.32/0.58 ! [A: int,B: int] :
% 0.32/0.58 ( finite_finite_int
% 0.32/0.58 @ ( collect_int
% 0.32/0.58 @ ^ [I3: int] :
% 0.32/0.58 ( ( ord_less_eq_int @ A @ I3 )
% 0.32/0.58 & ( ord_less_int @ I3 @ B ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_interval_int2
% 0.32/0.58 thf(fact_3678_finite__maxlen,axiom,
% 0.32/0.58 ! [M7: set_list_VEBT_VEBT] :
% 0.32/0.58 ( ( finite3004134309566078307T_VEBT @ M7 )
% 0.32/0.58 => ? [N4: nat] :
% 0.32/0.58 ! [X5: list_VEBT_VEBT] :
% 0.32/0.58 ( ( member2936631157270082147T_VEBT @ X5 @ M7 )
% 0.32/0.58 => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X5 ) @ N4 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_maxlen
% 0.32/0.58 thf(fact_3679_finite__maxlen,axiom,
% 0.32/0.58 ! [M7: set_list_o] :
% 0.32/0.58 ( ( finite_finite_list_o @ M7 )
% 0.32/0.58 => ? [N4: nat] :
% 0.32/0.58 ! [X5: list_o] :
% 0.32/0.58 ( ( member_list_o @ X5 @ M7 )
% 0.32/0.58 => ( ord_less_nat @ ( size_size_list_o @ X5 ) @ N4 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_maxlen
% 0.32/0.58 thf(fact_3680_finite__maxlen,axiom,
% 0.32/0.58 ! [M7: set_list_int] :
% 0.32/0.58 ( ( finite3922522038869484883st_int @ M7 )
% 0.32/0.58 => ? [N4: nat] :
% 0.32/0.58 ! [X5: list_int] :
% 0.32/0.58 ( ( member_list_int @ X5 @ M7 )
% 0.32/0.58 => ( ord_less_nat @ ( size_size_list_int @ X5 ) @ N4 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_maxlen
% 0.32/0.58 thf(fact_3681_pigeonhole__infinite__rel,axiom,
% 0.32/0.58 ! [A2: set_real,B3: set_nat,R2: real > nat > $o] :
% 0.32/0.58 ( ~ ( finite_finite_real @ A2 )
% 0.32/0.58 => ( ( finite_finite_nat @ B3 )
% 0.32/0.58 => ( ! [X4: real] :
% 0.32/0.58 ( ( member_real @ X4 @ A2 )
% 0.32/0.58 => ? [Xa: nat] :
% 0.32/0.58 ( ( member_nat @ Xa @ B3 )
% 0.32/0.58 & ( R2 @ X4 @ Xa ) ) )
% 0.32/0.58 => ? [X4: nat] :
% 0.32/0.58 ( ( member_nat @ X4 @ B3 )
% 0.32/0.58 & ~ ( finite_finite_real
% 0.32/0.58 @ ( collect_real
% 0.32/0.58 @ ^ [A5: real] :
% 0.32/0.58 ( ( member_real @ A5 @ A2 )
% 0.32/0.58 & ( R2 @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % pigeonhole_infinite_rel
% 0.32/0.58 thf(fact_3682_pigeonhole__infinite__rel,axiom,
% 0.32/0.58 ! [A2: set_real,B3: set_int,R2: real > int > $o] :
% 0.32/0.58 ( ~ ( finite_finite_real @ A2 )
% 0.32/0.58 => ( ( finite_finite_int @ B3 )
% 0.32/0.58 => ( ! [X4: real] :
% 0.32/0.58 ( ( member_real @ X4 @ A2 )
% 0.32/0.58 => ? [Xa: int] :
% 0.32/0.58 ( ( member_int @ Xa @ B3 )
% 0.32/0.58 & ( R2 @ X4 @ Xa ) ) )
% 0.32/0.58 => ? [X4: int] :
% 0.32/0.58 ( ( member_int @ X4 @ B3 )
% 0.32/0.58 & ~ ( finite_finite_real
% 0.32/0.58 @ ( collect_real
% 0.32/0.58 @ ^ [A5: real] :
% 0.32/0.58 ( ( member_real @ A5 @ A2 )
% 0.32/0.58 & ( R2 @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % pigeonhole_infinite_rel
% 0.32/0.58 thf(fact_3683_pigeonhole__infinite__rel,axiom,
% 0.32/0.58 ! [A2: set_real,B3: set_complex,R2: real > complex > $o] :
% 0.32/0.58 ( ~ ( finite_finite_real @ A2 )
% 0.32/0.58 => ( ( finite3207457112153483333omplex @ B3 )
% 0.32/0.58 => ( ! [X4: real] :
% 0.32/0.58 ( ( member_real @ X4 @ A2 )
% 0.32/0.58 => ? [Xa: complex] :
% 0.32/0.58 ( ( member_complex @ Xa @ B3 )
% 0.32/0.58 & ( R2 @ X4 @ Xa ) ) )
% 0.32/0.58 => ? [X4: complex] :
% 0.32/0.58 ( ( member_complex @ X4 @ B3 )
% 0.32/0.58 & ~ ( finite_finite_real
% 0.32/0.58 @ ( collect_real
% 0.32/0.58 @ ^ [A5: real] :
% 0.32/0.58 ( ( member_real @ A5 @ A2 )
% 0.32/0.58 & ( R2 @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % pigeonhole_infinite_rel
% 0.32/0.58 thf(fact_3684_pigeonhole__infinite__rel,axiom,
% 0.32/0.58 ! [A2: set_nat,B3: set_nat,R2: nat > nat > $o] :
% 0.32/0.58 ( ~ ( finite_finite_nat @ A2 )
% 0.32/0.58 => ( ( finite_finite_nat @ B3 )
% 0.32/0.58 => ( ! [X4: nat] :
% 0.32/0.58 ( ( member_nat @ X4 @ A2 )
% 0.32/0.58 => ? [Xa: nat] :
% 0.32/0.58 ( ( member_nat @ Xa @ B3 )
% 0.32/0.58 & ( R2 @ X4 @ Xa ) ) )
% 0.32/0.58 => ? [X4: nat] :
% 0.32/0.58 ( ( member_nat @ X4 @ B3 )
% 0.32/0.58 & ~ ( finite_finite_nat
% 0.32/0.58 @ ( collect_nat
% 0.32/0.58 @ ^ [A5: nat] :
% 0.32/0.58 ( ( member_nat @ A5 @ A2 )
% 0.32/0.58 & ( R2 @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % pigeonhole_infinite_rel
% 0.32/0.58 thf(fact_3685_pigeonhole__infinite__rel,axiom,
% 0.32/0.58 ! [A2: set_nat,B3: set_int,R2: nat > int > $o] :
% 0.32/0.58 ( ~ ( finite_finite_nat @ A2 )
% 0.32/0.58 => ( ( finite_finite_int @ B3 )
% 0.32/0.58 => ( ! [X4: nat] :
% 0.32/0.58 ( ( member_nat @ X4 @ A2 )
% 0.32/0.58 => ? [Xa: int] :
% 0.32/0.58 ( ( member_int @ Xa @ B3 )
% 0.32/0.58 & ( R2 @ X4 @ Xa ) ) )
% 0.32/0.58 => ? [X4: int] :
% 0.32/0.58 ( ( member_int @ X4 @ B3 )
% 0.32/0.58 & ~ ( finite_finite_nat
% 0.32/0.58 @ ( collect_nat
% 0.32/0.58 @ ^ [A5: nat] :
% 0.32/0.58 ( ( member_nat @ A5 @ A2 )
% 0.32/0.58 & ( R2 @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % pigeonhole_infinite_rel
% 0.32/0.58 thf(fact_3686_pigeonhole__infinite__rel,axiom,
% 0.32/0.58 ! [A2: set_nat,B3: set_complex,R2: nat > complex > $o] :
% 0.32/0.58 ( ~ ( finite_finite_nat @ A2 )
% 0.32/0.58 => ( ( finite3207457112153483333omplex @ B3 )
% 0.32/0.58 => ( ! [X4: nat] :
% 0.32/0.58 ( ( member_nat @ X4 @ A2 )
% 0.32/0.58 => ? [Xa: complex] :
% 0.32/0.58 ( ( member_complex @ Xa @ B3 )
% 0.32/0.58 & ( R2 @ X4 @ Xa ) ) )
% 0.32/0.58 => ? [X4: complex] :
% 0.32/0.58 ( ( member_complex @ X4 @ B3 )
% 0.32/0.58 & ~ ( finite_finite_nat
% 0.32/0.58 @ ( collect_nat
% 0.32/0.58 @ ^ [A5: nat] :
% 0.32/0.58 ( ( member_nat @ A5 @ A2 )
% 0.32/0.58 & ( R2 @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % pigeonhole_infinite_rel
% 0.32/0.58 thf(fact_3687_pigeonhole__infinite__rel,axiom,
% 0.32/0.58 ! [A2: set_int,B3: set_nat,R2: int > nat > $o] :
% 0.32/0.58 ( ~ ( finite_finite_int @ A2 )
% 0.32/0.58 => ( ( finite_finite_nat @ B3 )
% 0.32/0.58 => ( ! [X4: int] :
% 0.32/0.58 ( ( member_int @ X4 @ A2 )
% 0.32/0.58 => ? [Xa: nat] :
% 0.32/0.58 ( ( member_nat @ Xa @ B3 )
% 0.32/0.58 & ( R2 @ X4 @ Xa ) ) )
% 0.32/0.58 => ? [X4: nat] :
% 0.32/0.58 ( ( member_nat @ X4 @ B3 )
% 0.32/0.58 & ~ ( finite_finite_int
% 0.32/0.58 @ ( collect_int
% 0.32/0.58 @ ^ [A5: int] :
% 0.32/0.58 ( ( member_int @ A5 @ A2 )
% 0.32/0.58 & ( R2 @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % pigeonhole_infinite_rel
% 0.32/0.58 thf(fact_3688_pigeonhole__infinite__rel,axiom,
% 0.32/0.58 ! [A2: set_int,B3: set_int,R2: int > int > $o] :
% 0.32/0.58 ( ~ ( finite_finite_int @ A2 )
% 0.32/0.58 => ( ( finite_finite_int @ B3 )
% 0.32/0.58 => ( ! [X4: int] :
% 0.32/0.58 ( ( member_int @ X4 @ A2 )
% 0.32/0.58 => ? [Xa: int] :
% 0.32/0.58 ( ( member_int @ Xa @ B3 )
% 0.32/0.58 & ( R2 @ X4 @ Xa ) ) )
% 0.32/0.58 => ? [X4: int] :
% 0.32/0.58 ( ( member_int @ X4 @ B3 )
% 0.32/0.58 & ~ ( finite_finite_int
% 0.32/0.58 @ ( collect_int
% 0.32/0.58 @ ^ [A5: int] :
% 0.32/0.58 ( ( member_int @ A5 @ A2 )
% 0.32/0.58 & ( R2 @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % pigeonhole_infinite_rel
% 0.32/0.58 thf(fact_3689_pigeonhole__infinite__rel,axiom,
% 0.32/0.58 ! [A2: set_int,B3: set_complex,R2: int > complex > $o] :
% 0.32/0.58 ( ~ ( finite_finite_int @ A2 )
% 0.32/0.58 => ( ( finite3207457112153483333omplex @ B3 )
% 0.32/0.58 => ( ! [X4: int] :
% 0.32/0.58 ( ( member_int @ X4 @ A2 )
% 0.32/0.58 => ? [Xa: complex] :
% 0.32/0.58 ( ( member_complex @ Xa @ B3 )
% 0.32/0.58 & ( R2 @ X4 @ Xa ) ) )
% 0.32/0.58 => ? [X4: complex] :
% 0.32/0.58 ( ( member_complex @ X4 @ B3 )
% 0.32/0.58 & ~ ( finite_finite_int
% 0.32/0.58 @ ( collect_int
% 0.32/0.58 @ ^ [A5: int] :
% 0.32/0.58 ( ( member_int @ A5 @ A2 )
% 0.32/0.58 & ( R2 @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % pigeonhole_infinite_rel
% 0.32/0.58 thf(fact_3690_pigeonhole__infinite__rel,axiom,
% 0.32/0.58 ! [A2: set_complex,B3: set_nat,R2: complex > nat > $o] :
% 0.32/0.58 ( ~ ( finite3207457112153483333omplex @ A2 )
% 0.32/0.58 => ( ( finite_finite_nat @ B3 )
% 0.32/0.58 => ( ! [X4: complex] :
% 0.32/0.58 ( ( member_complex @ X4 @ A2 )
% 0.32/0.58 => ? [Xa: nat] :
% 0.32/0.58 ( ( member_nat @ Xa @ B3 )
% 0.32/0.58 & ( R2 @ X4 @ Xa ) ) )
% 0.32/0.58 => ? [X4: nat] :
% 0.32/0.58 ( ( member_nat @ X4 @ B3 )
% 0.32/0.58 & ~ ( finite3207457112153483333omplex
% 0.32/0.58 @ ( collect_complex
% 0.32/0.58 @ ^ [A5: complex] :
% 0.32/0.58 ( ( member_complex @ A5 @ A2 )
% 0.32/0.58 & ( R2 @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % pigeonhole_infinite_rel
% 0.32/0.58 thf(fact_3691_not__finite__existsD,axiom,
% 0.32/0.58 ! [P: product_prod_int_int > $o] :
% 0.32/0.58 ( ~ ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
% 0.32/0.58 => ? [X_1: product_prod_int_int] : ( P @ X_1 ) ) ).
% 0.32/0.58
% 0.32/0.58 % not_finite_existsD
% 0.32/0.58 thf(fact_3692_not__finite__existsD,axiom,
% 0.32/0.58 ! [P: set_nat > $o] :
% 0.32/0.58 ( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
% 0.32/0.58 => ? [X_1: set_nat] : ( P @ X_1 ) ) ).
% 0.32/0.58
% 0.32/0.58 % not_finite_existsD
% 0.32/0.58 thf(fact_3693_not__finite__existsD,axiom,
% 0.32/0.58 ! [P: nat > $o] :
% 0.32/0.58 ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.32/0.58 => ? [X_1: nat] : ( P @ X_1 ) ) ).
% 0.32/0.58
% 0.32/0.58 % not_finite_existsD
% 0.32/0.58 thf(fact_3694_not__finite__existsD,axiom,
% 0.32/0.58 ! [P: int > $o] :
% 0.32/0.58 ( ~ ( finite_finite_int @ ( collect_int @ P ) )
% 0.32/0.58 => ? [X_1: int] : ( P @ X_1 ) ) ).
% 0.32/0.58
% 0.32/0.58 % not_finite_existsD
% 0.32/0.58 thf(fact_3695_not__finite__existsD,axiom,
% 0.32/0.58 ! [P: complex > $o] :
% 0.32/0.58 ( ~ ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
% 0.32/0.58 => ? [X_1: complex] : ( P @ X_1 ) ) ).
% 0.32/0.58
% 0.32/0.58 % not_finite_existsD
% 0.32/0.58 thf(fact_3696_finite__has__maximal2,axiom,
% 0.32/0.58 ! [A2: set_real,A: real] :
% 0.32/0.58 ( ( finite_finite_real @ A2 )
% 0.32/0.58 => ( ( member_real @ A @ A2 )
% 0.32/0.58 => ? [X4: real] :
% 0.32/0.58 ( ( member_real @ X4 @ A2 )
% 0.32/0.58 & ( ord_less_eq_real @ A @ X4 )
% 0.32/0.58 & ! [Xa: real] :
% 0.32/0.58 ( ( member_real @ Xa @ A2 )
% 0.32/0.58 => ( ( ord_less_eq_real @ X4 @ Xa )
% 0.32/0.58 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_has_maximal2
% 0.32/0.58 thf(fact_3697_finite__has__maximal2,axiom,
% 0.32/0.58 ! [A2: set_set_nat,A: set_nat] :
% 0.32/0.58 ( ( finite1152437895449049373et_nat @ A2 )
% 0.32/0.58 => ( ( member_set_nat @ A @ A2 )
% 0.32/0.58 => ? [X4: set_nat] :
% 0.32/0.58 ( ( member_set_nat @ X4 @ A2 )
% 0.32/0.58 & ( ord_less_eq_set_nat @ A @ X4 )
% 0.32/0.58 & ! [Xa: set_nat] :
% 0.32/0.58 ( ( member_set_nat @ Xa @ A2 )
% 0.32/0.58 => ( ( ord_less_eq_set_nat @ X4 @ Xa )
% 0.32/0.58 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_has_maximal2
% 0.32/0.58 thf(fact_3698_finite__has__maximal2,axiom,
% 0.32/0.58 ! [A2: set_rat,A: rat] :
% 0.32/0.58 ( ( finite_finite_rat @ A2 )
% 0.32/0.58 => ( ( member_rat @ A @ A2 )
% 0.32/0.58 => ? [X4: rat] :
% 0.32/0.58 ( ( member_rat @ X4 @ A2 )
% 0.32/0.58 & ( ord_less_eq_rat @ A @ X4 )
% 0.32/0.58 & ! [Xa: rat] :
% 0.32/0.58 ( ( member_rat @ Xa @ A2 )
% 0.32/0.58 => ( ( ord_less_eq_rat @ X4 @ Xa )
% 0.32/0.58 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_has_maximal2
% 0.32/0.58 thf(fact_3699_finite__has__maximal2,axiom,
% 0.32/0.58 ! [A2: set_num,A: num] :
% 0.32/0.58 ( ( finite_finite_num @ A2 )
% 0.32/0.58 => ( ( member_num @ A @ A2 )
% 0.32/0.58 => ? [X4: num] :
% 0.32/0.58 ( ( member_num @ X4 @ A2 )
% 0.32/0.58 & ( ord_less_eq_num @ A @ X4 )
% 0.32/0.58 & ! [Xa: num] :
% 0.32/0.58 ( ( member_num @ Xa @ A2 )
% 0.32/0.58 => ( ( ord_less_eq_num @ X4 @ Xa )
% 0.32/0.58 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_has_maximal2
% 0.32/0.58 thf(fact_3700_finite__has__maximal2,axiom,
% 0.32/0.58 ! [A2: set_nat,A: nat] :
% 0.32/0.58 ( ( finite_finite_nat @ A2 )
% 0.32/0.58 => ( ( member_nat @ A @ A2 )
% 0.32/0.58 => ? [X4: nat] :
% 0.32/0.58 ( ( member_nat @ X4 @ A2 )
% 0.32/0.58 & ( ord_less_eq_nat @ A @ X4 )
% 0.32/0.58 & ! [Xa: nat] :
% 0.32/0.58 ( ( member_nat @ Xa @ A2 )
% 0.32/0.58 => ( ( ord_less_eq_nat @ X4 @ Xa )
% 0.32/0.58 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_has_maximal2
% 0.32/0.58 thf(fact_3701_finite__has__maximal2,axiom,
% 0.32/0.58 ! [A2: set_int,A: int] :
% 0.32/0.58 ( ( finite_finite_int @ A2 )
% 0.32/0.58 => ( ( member_int @ A @ A2 )
% 0.32/0.58 => ? [X4: int] :
% 0.32/0.58 ( ( member_int @ X4 @ A2 )
% 0.32/0.58 & ( ord_less_eq_int @ A @ X4 )
% 0.32/0.58 & ! [Xa: int] :
% 0.32/0.58 ( ( member_int @ Xa @ A2 )
% 0.32/0.58 => ( ( ord_less_eq_int @ X4 @ Xa )
% 0.32/0.58 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_has_maximal2
% 0.32/0.58 thf(fact_3702_finite__has__minimal2,axiom,
% 0.32/0.58 ! [A2: set_real,A: real] :
% 0.32/0.58 ( ( finite_finite_real @ A2 )
% 0.32/0.58 => ( ( member_real @ A @ A2 )
% 0.32/0.58 => ? [X4: real] :
% 0.32/0.58 ( ( member_real @ X4 @ A2 )
% 0.32/0.58 & ( ord_less_eq_real @ X4 @ A )
% 0.32/0.58 & ! [Xa: real] :
% 0.32/0.58 ( ( member_real @ Xa @ A2 )
% 0.32/0.58 => ( ( ord_less_eq_real @ Xa @ X4 )
% 0.32/0.58 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_has_minimal2
% 0.32/0.58 thf(fact_3703_finite__has__minimal2,axiom,
% 0.32/0.58 ! [A2: set_set_nat,A: set_nat] :
% 0.32/0.58 ( ( finite1152437895449049373et_nat @ A2 )
% 0.32/0.58 => ( ( member_set_nat @ A @ A2 )
% 0.32/0.58 => ? [X4: set_nat] :
% 0.32/0.58 ( ( member_set_nat @ X4 @ A2 )
% 0.32/0.58 & ( ord_less_eq_set_nat @ X4 @ A )
% 0.32/0.58 & ! [Xa: set_nat] :
% 0.32/0.58 ( ( member_set_nat @ Xa @ A2 )
% 0.32/0.58 => ( ( ord_less_eq_set_nat @ Xa @ X4 )
% 0.32/0.58 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_has_minimal2
% 0.32/0.58 thf(fact_3704_finite__has__minimal2,axiom,
% 0.32/0.58 ! [A2: set_rat,A: rat] :
% 0.32/0.58 ( ( finite_finite_rat @ A2 )
% 0.32/0.58 => ( ( member_rat @ A @ A2 )
% 0.32/0.58 => ? [X4: rat] :
% 0.32/0.58 ( ( member_rat @ X4 @ A2 )
% 0.32/0.58 & ( ord_less_eq_rat @ X4 @ A )
% 0.32/0.58 & ! [Xa: rat] :
% 0.32/0.58 ( ( member_rat @ Xa @ A2 )
% 0.32/0.58 => ( ( ord_less_eq_rat @ Xa @ X4 )
% 0.32/0.58 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_has_minimal2
% 0.32/0.58 thf(fact_3705_finite__has__minimal2,axiom,
% 0.32/0.58 ! [A2: set_num,A: num] :
% 0.32/0.58 ( ( finite_finite_num @ A2 )
% 0.32/0.58 => ( ( member_num @ A @ A2 )
% 0.32/0.58 => ? [X4: num] :
% 0.32/0.58 ( ( member_num @ X4 @ A2 )
% 0.32/0.58 & ( ord_less_eq_num @ X4 @ A )
% 0.32/0.58 & ! [Xa: num] :
% 0.32/0.58 ( ( member_num @ Xa @ A2 )
% 0.32/0.58 => ( ( ord_less_eq_num @ Xa @ X4 )
% 0.32/0.58 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_has_minimal2
% 0.32/0.58 thf(fact_3706_finite__has__minimal2,axiom,
% 0.32/0.58 ! [A2: set_nat,A: nat] :
% 0.32/0.58 ( ( finite_finite_nat @ A2 )
% 0.32/0.58 => ( ( member_nat @ A @ A2 )
% 0.32/0.58 => ? [X4: nat] :
% 0.32/0.58 ( ( member_nat @ X4 @ A2 )
% 0.32/0.58 & ( ord_less_eq_nat @ X4 @ A )
% 0.32/0.58 & ! [Xa: nat] :
% 0.32/0.58 ( ( member_nat @ Xa @ A2 )
% 0.32/0.58 => ( ( ord_less_eq_nat @ Xa @ X4 )
% 0.32/0.58 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_has_minimal2
% 0.32/0.58 thf(fact_3707_finite__has__minimal2,axiom,
% 0.32/0.58 ! [A2: set_int,A: int] :
% 0.32/0.58 ( ( finite_finite_int @ A2 )
% 0.32/0.58 => ( ( member_int @ A @ A2 )
% 0.32/0.58 => ? [X4: int] :
% 0.32/0.58 ( ( member_int @ X4 @ A2 )
% 0.32/0.58 & ( ord_less_eq_int @ X4 @ A )
% 0.32/0.58 & ! [Xa: int] :
% 0.32/0.58 ( ( member_int @ Xa @ A2 )
% 0.32/0.58 => ( ( ord_less_eq_int @ Xa @ X4 )
% 0.32/0.58 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_has_minimal2
% 0.32/0.58 thf(fact_3708_finite__subset,axiom,
% 0.32/0.58 ! [A2: set_int,B3: set_int] :
% 0.32/0.58 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.32/0.58 => ( ( finite_finite_int @ B3 )
% 0.32/0.58 => ( finite_finite_int @ A2 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_subset
% 0.32/0.58 thf(fact_3709_finite__subset,axiom,
% 0.32/0.58 ! [A2: set_complex,B3: set_complex] :
% 0.32/0.58 ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 0.32/0.58 => ( ( finite3207457112153483333omplex @ B3 )
% 0.32/0.58 => ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_subset
% 0.32/0.58 thf(fact_3710_finite__subset,axiom,
% 0.32/0.58 ! [A2: set_nat,B3: set_nat] :
% 0.32/0.58 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.32/0.58 => ( ( finite_finite_nat @ B3 )
% 0.32/0.58 => ( finite_finite_nat @ A2 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_subset
% 0.32/0.58 thf(fact_3711_infinite__super,axiom,
% 0.32/0.58 ! [S: set_int,T3: set_int] :
% 0.32/0.58 ( ( ord_less_eq_set_int @ S @ T3 )
% 0.32/0.58 => ( ~ ( finite_finite_int @ S )
% 0.32/0.58 => ~ ( finite_finite_int @ T3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % infinite_super
% 0.32/0.58 thf(fact_3712_infinite__super,axiom,
% 0.32/0.58 ! [S: set_complex,T3: set_complex] :
% 0.32/0.58 ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.32/0.58 => ( ~ ( finite3207457112153483333omplex @ S )
% 0.32/0.58 => ~ ( finite3207457112153483333omplex @ T3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % infinite_super
% 0.32/0.58 thf(fact_3713_infinite__super,axiom,
% 0.32/0.58 ! [S: set_nat,T3: set_nat] :
% 0.32/0.58 ( ( ord_less_eq_set_nat @ S @ T3 )
% 0.32/0.58 => ( ~ ( finite_finite_nat @ S )
% 0.32/0.58 => ~ ( finite_finite_nat @ T3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % infinite_super
% 0.32/0.58 thf(fact_3714_rev__finite__subset,axiom,
% 0.32/0.58 ! [B3: set_int,A2: set_int] :
% 0.32/0.58 ( ( finite_finite_int @ B3 )
% 0.32/0.58 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.32/0.58 => ( finite_finite_int @ A2 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % rev_finite_subset
% 0.32/0.58 thf(fact_3715_rev__finite__subset,axiom,
% 0.32/0.58 ! [B3: set_complex,A2: set_complex] :
% 0.32/0.58 ( ( finite3207457112153483333omplex @ B3 )
% 0.32/0.58 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 0.32/0.58 => ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % rev_finite_subset
% 0.32/0.58 thf(fact_3716_rev__finite__subset,axiom,
% 0.32/0.58 ! [B3: set_nat,A2: set_nat] :
% 0.32/0.58 ( ( finite_finite_nat @ B3 )
% 0.32/0.58 => ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.32/0.58 => ( finite_finite_nat @ A2 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % rev_finite_subset
% 0.32/0.58 thf(fact_3717_Diff__infinite__finite,axiom,
% 0.32/0.58 ! [T3: set_int,S: set_int] :
% 0.32/0.58 ( ( finite_finite_int @ T3 )
% 0.32/0.58 => ( ~ ( finite_finite_int @ S )
% 0.32/0.58 => ~ ( finite_finite_int @ ( minus_minus_set_int @ S @ T3 ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % Diff_infinite_finite
% 0.32/0.58 thf(fact_3718_Diff__infinite__finite,axiom,
% 0.32/0.58 ! [T3: set_complex,S: set_complex] :
% 0.32/0.58 ( ( finite3207457112153483333omplex @ T3 )
% 0.32/0.58 => ( ~ ( finite3207457112153483333omplex @ S )
% 0.32/0.58 => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S @ T3 ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % Diff_infinite_finite
% 0.32/0.58 thf(fact_3719_Diff__infinite__finite,axiom,
% 0.32/0.58 ! [T3: set_nat,S: set_nat] :
% 0.32/0.58 ( ( finite_finite_nat @ T3 )
% 0.32/0.58 => ( ~ ( finite_finite_nat @ S )
% 0.32/0.58 => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T3 ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % Diff_infinite_finite
% 0.32/0.58 thf(fact_3720_finite__psubset__induct,axiom,
% 0.32/0.58 ! [A2: set_nat,P: set_nat > $o] :
% 0.32/0.58 ( ( finite_finite_nat @ A2 )
% 0.32/0.58 => ( ! [A7: set_nat] :
% 0.32/0.58 ( ( finite_finite_nat @ A7 )
% 0.32/0.58 => ( ! [B7: set_nat] :
% 0.32/0.58 ( ( ord_less_set_nat @ B7 @ A7 )
% 0.32/0.58 => ( P @ B7 ) )
% 0.32/0.58 => ( P @ A7 ) ) )
% 0.32/0.58 => ( P @ A2 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_psubset_induct
% 0.32/0.58 thf(fact_3721_finite__psubset__induct,axiom,
% 0.32/0.58 ! [A2: set_int,P: set_int > $o] :
% 0.32/0.58 ( ( finite_finite_int @ A2 )
% 0.32/0.58 => ( ! [A7: set_int] :
% 0.32/0.58 ( ( finite_finite_int @ A7 )
% 0.32/0.58 => ( ! [B7: set_int] :
% 0.32/0.58 ( ( ord_less_set_int @ B7 @ A7 )
% 0.32/0.58 => ( P @ B7 ) )
% 0.32/0.58 => ( P @ A7 ) ) )
% 0.32/0.58 => ( P @ A2 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_psubset_induct
% 0.32/0.58 thf(fact_3722_finite__psubset__induct,axiom,
% 0.32/0.58 ! [A2: set_complex,P: set_complex > $o] :
% 0.32/0.58 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.58 => ( ! [A7: set_complex] :
% 0.32/0.58 ( ( finite3207457112153483333omplex @ A7 )
% 0.32/0.58 => ( ! [B7: set_complex] :
% 0.32/0.58 ( ( ord_less_set_complex @ B7 @ A7 )
% 0.32/0.58 => ( P @ B7 ) )
% 0.32/0.58 => ( P @ A7 ) ) )
% 0.32/0.58 => ( P @ A2 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_psubset_induct
% 0.32/0.58 thf(fact_3723_finite__has__minimal,axiom,
% 0.32/0.58 ! [A2: set_real] :
% 0.32/0.58 ( ( finite_finite_real @ A2 )
% 0.32/0.58 => ( ( A2 != bot_bot_set_real )
% 0.32/0.58 => ? [X4: real] :
% 0.32/0.58 ( ( member_real @ X4 @ A2 )
% 0.32/0.58 & ! [Xa: real] :
% 0.32/0.58 ( ( member_real @ Xa @ A2 )
% 0.32/0.58 => ( ( ord_less_eq_real @ Xa @ X4 )
% 0.32/0.58 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_has_minimal
% 0.32/0.58 thf(fact_3724_finite__has__minimal,axiom,
% 0.32/0.58 ! [A2: set_set_nat] :
% 0.32/0.58 ( ( finite1152437895449049373et_nat @ A2 )
% 0.32/0.58 => ( ( A2 != bot_bot_set_set_nat )
% 0.32/0.58 => ? [X4: set_nat] :
% 0.32/0.58 ( ( member_set_nat @ X4 @ A2 )
% 0.32/0.58 & ! [Xa: set_nat] :
% 0.32/0.58 ( ( member_set_nat @ Xa @ A2 )
% 0.32/0.58 => ( ( ord_less_eq_set_nat @ Xa @ X4 )
% 0.32/0.58 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_has_minimal
% 0.32/0.58 thf(fact_3725_finite__has__minimal,axiom,
% 0.32/0.58 ! [A2: set_rat] :
% 0.32/0.58 ( ( finite_finite_rat @ A2 )
% 0.32/0.58 => ( ( A2 != bot_bot_set_rat )
% 0.32/0.58 => ? [X4: rat] :
% 0.32/0.58 ( ( member_rat @ X4 @ A2 )
% 0.32/0.58 & ! [Xa: rat] :
% 0.32/0.58 ( ( member_rat @ Xa @ A2 )
% 0.32/0.58 => ( ( ord_less_eq_rat @ Xa @ X4 )
% 0.32/0.58 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_has_minimal
% 0.32/0.58 thf(fact_3726_finite__has__minimal,axiom,
% 0.32/0.58 ! [A2: set_num] :
% 0.32/0.58 ( ( finite_finite_num @ A2 )
% 0.32/0.58 => ( ( A2 != bot_bot_set_num )
% 0.32/0.58 => ? [X4: num] :
% 0.32/0.58 ( ( member_num @ X4 @ A2 )
% 0.32/0.58 & ! [Xa: num] :
% 0.32/0.58 ( ( member_num @ Xa @ A2 )
% 0.32/0.58 => ( ( ord_less_eq_num @ Xa @ X4 )
% 0.32/0.58 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_has_minimal
% 0.32/0.58 thf(fact_3727_finite__has__minimal,axiom,
% 0.32/0.58 ! [A2: set_nat] :
% 0.32/0.58 ( ( finite_finite_nat @ A2 )
% 0.32/0.58 => ( ( A2 != bot_bot_set_nat )
% 0.32/0.58 => ? [X4: nat] :
% 0.32/0.58 ( ( member_nat @ X4 @ A2 )
% 0.32/0.58 & ! [Xa: nat] :
% 0.32/0.58 ( ( member_nat @ Xa @ A2 )
% 0.32/0.58 => ( ( ord_less_eq_nat @ Xa @ X4 )
% 0.32/0.58 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_has_minimal
% 0.32/0.58 thf(fact_3728_finite__has__minimal,axiom,
% 0.32/0.58 ! [A2: set_int] :
% 0.32/0.58 ( ( finite_finite_int @ A2 )
% 0.32/0.58 => ( ( A2 != bot_bot_set_int )
% 0.32/0.58 => ? [X4: int] :
% 0.32/0.58 ( ( member_int @ X4 @ A2 )
% 0.32/0.58 & ! [Xa: int] :
% 0.32/0.58 ( ( member_int @ Xa @ A2 )
% 0.32/0.58 => ( ( ord_less_eq_int @ Xa @ X4 )
% 0.32/0.58 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_has_minimal
% 0.32/0.58 thf(fact_3729_finite__has__maximal,axiom,
% 0.32/0.58 ! [A2: set_real] :
% 0.32/0.58 ( ( finite_finite_real @ A2 )
% 0.32/0.58 => ( ( A2 != bot_bot_set_real )
% 0.32/0.58 => ? [X4: real] :
% 0.32/0.58 ( ( member_real @ X4 @ A2 )
% 0.32/0.58 & ! [Xa: real] :
% 0.32/0.58 ( ( member_real @ Xa @ A2 )
% 0.32/0.58 => ( ( ord_less_eq_real @ X4 @ Xa )
% 0.32/0.58 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_has_maximal
% 0.32/0.58 thf(fact_3730_finite__has__maximal,axiom,
% 0.32/0.58 ! [A2: set_set_nat] :
% 0.32/0.58 ( ( finite1152437895449049373et_nat @ A2 )
% 0.32/0.58 => ( ( A2 != bot_bot_set_set_nat )
% 0.32/0.58 => ? [X4: set_nat] :
% 0.32/0.58 ( ( member_set_nat @ X4 @ A2 )
% 0.32/0.58 & ! [Xa: set_nat] :
% 0.32/0.58 ( ( member_set_nat @ Xa @ A2 )
% 0.32/0.58 => ( ( ord_less_eq_set_nat @ X4 @ Xa )
% 0.32/0.58 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_has_maximal
% 0.32/0.58 thf(fact_3731_finite__has__maximal,axiom,
% 0.32/0.58 ! [A2: set_rat] :
% 0.32/0.58 ( ( finite_finite_rat @ A2 )
% 0.32/0.58 => ( ( A2 != bot_bot_set_rat )
% 0.32/0.58 => ? [X4: rat] :
% 0.32/0.58 ( ( member_rat @ X4 @ A2 )
% 0.32/0.58 & ! [Xa: rat] :
% 0.32/0.58 ( ( member_rat @ Xa @ A2 )
% 0.32/0.58 => ( ( ord_less_eq_rat @ X4 @ Xa )
% 0.32/0.58 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_has_maximal
% 0.32/0.58 thf(fact_3732_finite__has__maximal,axiom,
% 0.32/0.58 ! [A2: set_num] :
% 0.32/0.58 ( ( finite_finite_num @ A2 )
% 0.32/0.58 => ( ( A2 != bot_bot_set_num )
% 0.32/0.58 => ? [X4: num] :
% 0.32/0.58 ( ( member_num @ X4 @ A2 )
% 0.32/0.58 & ! [Xa: num] :
% 0.32/0.58 ( ( member_num @ Xa @ A2 )
% 0.32/0.58 => ( ( ord_less_eq_num @ X4 @ Xa )
% 0.32/0.58 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_has_maximal
% 0.32/0.58 thf(fact_3733_finite__has__maximal,axiom,
% 0.32/0.58 ! [A2: set_nat] :
% 0.32/0.58 ( ( finite_finite_nat @ A2 )
% 0.32/0.58 => ( ( A2 != bot_bot_set_nat )
% 0.32/0.58 => ? [X4: nat] :
% 0.32/0.58 ( ( member_nat @ X4 @ A2 )
% 0.32/0.58 & ! [Xa: nat] :
% 0.32/0.58 ( ( member_nat @ Xa @ A2 )
% 0.32/0.58 => ( ( ord_less_eq_nat @ X4 @ Xa )
% 0.32/0.58 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_has_maximal
% 0.32/0.58 thf(fact_3734_finite__has__maximal,axiom,
% 0.32/0.58 ! [A2: set_int] :
% 0.32/0.58 ( ( finite_finite_int @ A2 )
% 0.32/0.58 => ( ( A2 != bot_bot_set_int )
% 0.32/0.58 => ? [X4: int] :
% 0.32/0.58 ( ( member_int @ X4 @ A2 )
% 0.32/0.58 & ! [Xa: int] :
% 0.32/0.58 ( ( member_int @ Xa @ A2 )
% 0.32/0.58 => ( ( ord_less_eq_int @ X4 @ Xa )
% 0.32/0.58 => ( X4 = Xa ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_has_maximal
% 0.32/0.58 thf(fact_3735_arcosh__1,axiom,
% 0.32/0.58 ( ( arcosh_real @ one_one_real )
% 0.32/0.58 = zero_zero_real ) ).
% 0.32/0.58
% 0.32/0.58 % arcosh_1
% 0.32/0.58 thf(fact_3736_finite__nth__roots,axiom,
% 0.32/0.58 ! [N2: nat,C: complex] :
% 0.32/0.58 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.58 => ( finite3207457112153483333omplex
% 0.32/0.58 @ ( collect_complex
% 0.32/0.58 @ ^ [Z3: complex] :
% 0.32/0.58 ( ( power_power_complex @ Z3 @ N2 )
% 0.32/0.58 = C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % finite_nth_roots
% 0.32/0.58 thf(fact_3737_prod_Ofinite__Collect__op,axiom,
% 0.32/0.58 ! [I6: set_real,X: real > complex,Y: real > complex] :
% 0.32/0.58 ( ( finite_finite_real
% 0.32/0.58 @ ( collect_real
% 0.32/0.58 @ ^ [I3: real] :
% 0.32/0.58 ( ( member_real @ I3 @ I6 )
% 0.32/0.58 & ( ( X @ I3 )
% 0.32/0.58 != one_one_complex ) ) ) )
% 0.32/0.58 => ( ( finite_finite_real
% 0.32/0.58 @ ( collect_real
% 0.32/0.58 @ ^ [I3: real] :
% 0.32/0.58 ( ( member_real @ I3 @ I6 )
% 0.32/0.58 & ( ( Y @ I3 )
% 0.32/0.58 != one_one_complex ) ) ) )
% 0.32/0.58 => ( finite_finite_real
% 0.32/0.58 @ ( collect_real
% 0.32/0.58 @ ^ [I3: real] :
% 0.32/0.58 ( ( member_real @ I3 @ I6 )
% 0.32/0.58 & ( ( times_times_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
% 0.32/0.58 != one_one_complex ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % prod.finite_Collect_op
% 0.32/0.58 thf(fact_3738_prod_Ofinite__Collect__op,axiom,
% 0.32/0.58 ! [I6: set_nat,X: nat > complex,Y: nat > complex] :
% 0.32/0.58 ( ( finite_finite_nat
% 0.32/0.58 @ ( collect_nat
% 0.32/0.58 @ ^ [I3: nat] :
% 0.32/0.58 ( ( member_nat @ I3 @ I6 )
% 0.32/0.58 & ( ( X @ I3 )
% 0.32/0.58 != one_one_complex ) ) ) )
% 0.32/0.58 => ( ( finite_finite_nat
% 0.32/0.58 @ ( collect_nat
% 0.32/0.58 @ ^ [I3: nat] :
% 0.32/0.58 ( ( member_nat @ I3 @ I6 )
% 0.32/0.58 & ( ( Y @ I3 )
% 0.32/0.58 != one_one_complex ) ) ) )
% 0.32/0.58 => ( finite_finite_nat
% 0.32/0.58 @ ( collect_nat
% 0.32/0.58 @ ^ [I3: nat] :
% 0.32/0.58 ( ( member_nat @ I3 @ I6 )
% 0.32/0.58 & ( ( times_times_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
% 0.32/0.58 != one_one_complex ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % prod.finite_Collect_op
% 0.32/0.58 thf(fact_3739_prod_Ofinite__Collect__op,axiom,
% 0.32/0.58 ! [I6: set_int,X: int > complex,Y: int > complex] :
% 0.32/0.58 ( ( finite_finite_int
% 0.32/0.58 @ ( collect_int
% 0.32/0.58 @ ^ [I3: int] :
% 0.32/0.58 ( ( member_int @ I3 @ I6 )
% 0.32/0.58 & ( ( X @ I3 )
% 0.32/0.58 != one_one_complex ) ) ) )
% 0.32/0.58 => ( ( finite_finite_int
% 0.32/0.58 @ ( collect_int
% 0.32/0.58 @ ^ [I3: int] :
% 0.32/0.58 ( ( member_int @ I3 @ I6 )
% 0.32/0.58 & ( ( Y @ I3 )
% 0.32/0.58 != one_one_complex ) ) ) )
% 0.32/0.58 => ( finite_finite_int
% 0.32/0.58 @ ( collect_int
% 0.32/0.58 @ ^ [I3: int] :
% 0.32/0.58 ( ( member_int @ I3 @ I6 )
% 0.32/0.58 & ( ( times_times_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
% 0.32/0.58 != one_one_complex ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % prod.finite_Collect_op
% 0.32/0.58 thf(fact_3740_prod_Ofinite__Collect__op,axiom,
% 0.32/0.58 ! [I6: set_complex,X: complex > complex,Y: complex > complex] :
% 0.32/0.58 ( ( finite3207457112153483333omplex
% 0.32/0.58 @ ( collect_complex
% 0.32/0.58 @ ^ [I3: complex] :
% 0.32/0.58 ( ( member_complex @ I3 @ I6 )
% 0.32/0.58 & ( ( X @ I3 )
% 0.32/0.58 != one_one_complex ) ) ) )
% 0.32/0.58 => ( ( finite3207457112153483333omplex
% 0.32/0.58 @ ( collect_complex
% 0.32/0.58 @ ^ [I3: complex] :
% 0.32/0.58 ( ( member_complex @ I3 @ I6 )
% 0.32/0.58 & ( ( Y @ I3 )
% 0.32/0.58 != one_one_complex ) ) ) )
% 0.32/0.58 => ( finite3207457112153483333omplex
% 0.32/0.58 @ ( collect_complex
% 0.32/0.58 @ ^ [I3: complex] :
% 0.32/0.58 ( ( member_complex @ I3 @ I6 )
% 0.32/0.58 & ( ( times_times_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
% 0.32/0.58 != one_one_complex ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % prod.finite_Collect_op
% 0.32/0.58 thf(fact_3741_prod_Ofinite__Collect__op,axiom,
% 0.32/0.58 ! [I6: set_real,X: real > real,Y: real > real] :
% 0.32/0.58 ( ( finite_finite_real
% 0.32/0.58 @ ( collect_real
% 0.32/0.58 @ ^ [I3: real] :
% 0.32/0.58 ( ( member_real @ I3 @ I6 )
% 0.32/0.58 & ( ( X @ I3 )
% 0.32/0.58 != one_one_real ) ) ) )
% 0.32/0.58 => ( ( finite_finite_real
% 0.32/0.58 @ ( collect_real
% 0.32/0.58 @ ^ [I3: real] :
% 0.32/0.58 ( ( member_real @ I3 @ I6 )
% 0.32/0.58 & ( ( Y @ I3 )
% 0.32/0.58 != one_one_real ) ) ) )
% 0.32/0.58 => ( finite_finite_real
% 0.32/0.58 @ ( collect_real
% 0.32/0.58 @ ^ [I3: real] :
% 0.32/0.58 ( ( member_real @ I3 @ I6 )
% 0.32/0.58 & ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
% 0.32/0.58 != one_one_real ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % prod.finite_Collect_op
% 0.32/0.58 thf(fact_3742_prod_Ofinite__Collect__op,axiom,
% 0.32/0.58 ! [I6: set_nat,X: nat > real,Y: nat > real] :
% 0.32/0.58 ( ( finite_finite_nat
% 0.32/0.58 @ ( collect_nat
% 0.32/0.58 @ ^ [I3: nat] :
% 0.32/0.58 ( ( member_nat @ I3 @ I6 )
% 0.32/0.58 & ( ( X @ I3 )
% 0.32/0.58 != one_one_real ) ) ) )
% 0.32/0.58 => ( ( finite_finite_nat
% 0.32/0.58 @ ( collect_nat
% 0.32/0.58 @ ^ [I3: nat] :
% 0.32/0.58 ( ( member_nat @ I3 @ I6 )
% 0.32/0.58 & ( ( Y @ I3 )
% 0.32/0.58 != one_one_real ) ) ) )
% 0.32/0.58 => ( finite_finite_nat
% 0.32/0.58 @ ( collect_nat
% 0.32/0.58 @ ^ [I3: nat] :
% 0.32/0.58 ( ( member_nat @ I3 @ I6 )
% 0.32/0.58 & ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
% 0.32/0.58 != one_one_real ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % prod.finite_Collect_op
% 0.32/0.58 thf(fact_3743_prod_Ofinite__Collect__op,axiom,
% 0.32/0.58 ! [I6: set_int,X: int > real,Y: int > real] :
% 0.32/0.58 ( ( finite_finite_int
% 0.32/0.58 @ ( collect_int
% 0.32/0.58 @ ^ [I3: int] :
% 0.32/0.58 ( ( member_int @ I3 @ I6 )
% 0.32/0.58 & ( ( X @ I3 )
% 0.32/0.58 != one_one_real ) ) ) )
% 0.32/0.58 => ( ( finite_finite_int
% 0.32/0.58 @ ( collect_int
% 0.32/0.58 @ ^ [I3: int] :
% 0.32/0.58 ( ( member_int @ I3 @ I6 )
% 0.32/0.58 & ( ( Y @ I3 )
% 0.32/0.58 != one_one_real ) ) ) )
% 0.32/0.58 => ( finite_finite_int
% 0.32/0.58 @ ( collect_int
% 0.32/0.58 @ ^ [I3: int] :
% 0.32/0.58 ( ( member_int @ I3 @ I6 )
% 0.32/0.58 & ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
% 0.32/0.58 != one_one_real ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % prod.finite_Collect_op
% 0.32/0.58 thf(fact_3744_prod_Ofinite__Collect__op,axiom,
% 0.32/0.58 ! [I6: set_complex,X: complex > real,Y: complex > real] :
% 0.32/0.58 ( ( finite3207457112153483333omplex
% 0.32/0.58 @ ( collect_complex
% 0.32/0.58 @ ^ [I3: complex] :
% 0.32/0.58 ( ( member_complex @ I3 @ I6 )
% 0.32/0.58 & ( ( X @ I3 )
% 0.32/0.58 != one_one_real ) ) ) )
% 0.32/0.58 => ( ( finite3207457112153483333omplex
% 0.32/0.58 @ ( collect_complex
% 0.32/0.58 @ ^ [I3: complex] :
% 0.32/0.58 ( ( member_complex @ I3 @ I6 )
% 0.32/0.58 & ( ( Y @ I3 )
% 0.32/0.58 != one_one_real ) ) ) )
% 0.32/0.58 => ( finite3207457112153483333omplex
% 0.32/0.58 @ ( collect_complex
% 0.32/0.58 @ ^ [I3: complex] :
% 0.32/0.58 ( ( member_complex @ I3 @ I6 )
% 0.32/0.58 & ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
% 0.32/0.58 != one_one_real ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % prod.finite_Collect_op
% 0.32/0.58 thf(fact_3745_prod_Ofinite__Collect__op,axiom,
% 0.32/0.58 ! [I6: set_real,X: real > rat,Y: real > rat] :
% 0.32/0.58 ( ( finite_finite_real
% 0.32/0.58 @ ( collect_real
% 0.32/0.58 @ ^ [I3: real] :
% 0.32/0.58 ( ( member_real @ I3 @ I6 )
% 0.32/0.58 & ( ( X @ I3 )
% 0.32/0.58 != one_one_rat ) ) ) )
% 0.32/0.58 => ( ( finite_finite_real
% 0.32/0.58 @ ( collect_real
% 0.32/0.58 @ ^ [I3: real] :
% 0.32/0.58 ( ( member_real @ I3 @ I6 )
% 0.32/0.58 & ( ( Y @ I3 )
% 0.32/0.58 != one_one_rat ) ) ) )
% 0.32/0.58 => ( finite_finite_real
% 0.32/0.58 @ ( collect_real
% 0.32/0.58 @ ^ [I3: real] :
% 0.32/0.58 ( ( member_real @ I3 @ I6 )
% 0.32/0.58 & ( ( times_times_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
% 0.32/0.58 != one_one_rat ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % prod.finite_Collect_op
% 0.32/0.58 thf(fact_3746_prod_Ofinite__Collect__op,axiom,
% 0.32/0.58 ! [I6: set_nat,X: nat > rat,Y: nat > rat] :
% 0.32/0.58 ( ( finite_finite_nat
% 0.32/0.58 @ ( collect_nat
% 0.32/0.58 @ ^ [I3: nat] :
% 0.32/0.58 ( ( member_nat @ I3 @ I6 )
% 0.32/0.58 & ( ( X @ I3 )
% 0.32/0.58 != one_one_rat ) ) ) )
% 0.32/0.58 => ( ( finite_finite_nat
% 0.32/0.58 @ ( collect_nat
% 0.32/0.58 @ ^ [I3: nat] :
% 0.32/0.58 ( ( member_nat @ I3 @ I6 )
% 0.32/0.58 & ( ( Y @ I3 )
% 0.32/0.58 != one_one_rat ) ) ) )
% 0.32/0.58 => ( finite_finite_nat
% 0.32/0.58 @ ( collect_nat
% 0.32/0.58 @ ^ [I3: nat] :
% 0.32/0.58 ( ( member_nat @ I3 @ I6 )
% 0.32/0.58 & ( ( times_times_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
% 0.32/0.58 != one_one_rat ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % prod.finite_Collect_op
% 0.32/0.58 thf(fact_3747_sum_Ofinite__Collect__op,axiom,
% 0.32/0.58 ! [I6: set_real,X: real > complex,Y: real > complex] :
% 0.32/0.58 ( ( finite_finite_real
% 0.32/0.58 @ ( collect_real
% 0.32/0.58 @ ^ [I3: real] :
% 0.32/0.58 ( ( member_real @ I3 @ I6 )
% 0.32/0.58 & ( ( X @ I3 )
% 0.32/0.58 != zero_zero_complex ) ) ) )
% 0.32/0.58 => ( ( finite_finite_real
% 0.32/0.58 @ ( collect_real
% 0.32/0.58 @ ^ [I3: real] :
% 0.32/0.58 ( ( member_real @ I3 @ I6 )
% 0.32/0.58 & ( ( Y @ I3 )
% 0.32/0.58 != zero_zero_complex ) ) ) )
% 0.32/0.58 => ( finite_finite_real
% 0.32/0.58 @ ( collect_real
% 0.32/0.58 @ ^ [I3: real] :
% 0.32/0.58 ( ( member_real @ I3 @ I6 )
% 0.32/0.58 & ( ( plus_plus_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
% 0.32/0.58 != zero_zero_complex ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % sum.finite_Collect_op
% 0.32/0.58 thf(fact_3748_sum_Ofinite__Collect__op,axiom,
% 0.32/0.58 ! [I6: set_nat,X: nat > complex,Y: nat > complex] :
% 0.32/0.58 ( ( finite_finite_nat
% 0.32/0.58 @ ( collect_nat
% 0.32/0.58 @ ^ [I3: nat] :
% 0.32/0.58 ( ( member_nat @ I3 @ I6 )
% 0.32/0.58 & ( ( X @ I3 )
% 0.32/0.58 != zero_zero_complex ) ) ) )
% 0.32/0.58 => ( ( finite_finite_nat
% 0.32/0.58 @ ( collect_nat
% 0.32/0.58 @ ^ [I3: nat] :
% 0.32/0.58 ( ( member_nat @ I3 @ I6 )
% 0.32/0.58 & ( ( Y @ I3 )
% 0.32/0.58 != zero_zero_complex ) ) ) )
% 0.32/0.58 => ( finite_finite_nat
% 0.32/0.58 @ ( collect_nat
% 0.32/0.58 @ ^ [I3: nat] :
% 0.32/0.58 ( ( member_nat @ I3 @ I6 )
% 0.32/0.58 & ( ( plus_plus_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
% 0.32/0.58 != zero_zero_complex ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % sum.finite_Collect_op
% 0.32/0.58 thf(fact_3749_sum_Ofinite__Collect__op,axiom,
% 0.32/0.58 ! [I6: set_int,X: int > complex,Y: int > complex] :
% 0.32/0.58 ( ( finite_finite_int
% 0.32/0.58 @ ( collect_int
% 0.32/0.58 @ ^ [I3: int] :
% 0.32/0.58 ( ( member_int @ I3 @ I6 )
% 0.32/0.58 & ( ( X @ I3 )
% 0.32/0.58 != zero_zero_complex ) ) ) )
% 0.32/0.58 => ( ( finite_finite_int
% 0.32/0.58 @ ( collect_int
% 0.32/0.58 @ ^ [I3: int] :
% 0.32/0.58 ( ( member_int @ I3 @ I6 )
% 0.32/0.58 & ( ( Y @ I3 )
% 0.32/0.58 != zero_zero_complex ) ) ) )
% 0.32/0.58 => ( finite_finite_int
% 0.32/0.58 @ ( collect_int
% 0.32/0.58 @ ^ [I3: int] :
% 0.32/0.58 ( ( member_int @ I3 @ I6 )
% 0.32/0.58 & ( ( plus_plus_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
% 0.32/0.58 != zero_zero_complex ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % sum.finite_Collect_op
% 0.32/0.58 thf(fact_3750_sum_Ofinite__Collect__op,axiom,
% 0.32/0.58 ! [I6: set_complex,X: complex > complex,Y: complex > complex] :
% 0.32/0.58 ( ( finite3207457112153483333omplex
% 0.32/0.58 @ ( collect_complex
% 0.32/0.58 @ ^ [I3: complex] :
% 0.32/0.58 ( ( member_complex @ I3 @ I6 )
% 0.32/0.58 & ( ( X @ I3 )
% 0.32/0.58 != zero_zero_complex ) ) ) )
% 0.32/0.58 => ( ( finite3207457112153483333omplex
% 0.32/0.58 @ ( collect_complex
% 0.32/0.58 @ ^ [I3: complex] :
% 0.32/0.58 ( ( member_complex @ I3 @ I6 )
% 0.32/0.58 & ( ( Y @ I3 )
% 0.32/0.58 != zero_zero_complex ) ) ) )
% 0.32/0.58 => ( finite3207457112153483333omplex
% 0.32/0.58 @ ( collect_complex
% 0.32/0.58 @ ^ [I3: complex] :
% 0.32/0.58 ( ( member_complex @ I3 @ I6 )
% 0.32/0.58 & ( ( plus_plus_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
% 0.32/0.58 != zero_zero_complex ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % sum.finite_Collect_op
% 0.32/0.58 thf(fact_3751_sum_Ofinite__Collect__op,axiom,
% 0.32/0.58 ! [I6: set_real,X: real > real,Y: real > real] :
% 0.32/0.58 ( ( finite_finite_real
% 0.32/0.58 @ ( collect_real
% 0.32/0.58 @ ^ [I3: real] :
% 0.32/0.58 ( ( member_real @ I3 @ I6 )
% 0.32/0.58 & ( ( X @ I3 )
% 0.32/0.58 != zero_zero_real ) ) ) )
% 0.32/0.58 => ( ( finite_finite_real
% 0.32/0.58 @ ( collect_real
% 0.32/0.58 @ ^ [I3: real] :
% 0.32/0.58 ( ( member_real @ I3 @ I6 )
% 0.32/0.58 & ( ( Y @ I3 )
% 0.32/0.58 != zero_zero_real ) ) ) )
% 0.32/0.58 => ( finite_finite_real
% 0.32/0.58 @ ( collect_real
% 0.32/0.58 @ ^ [I3: real] :
% 0.32/0.58 ( ( member_real @ I3 @ I6 )
% 0.32/0.58 & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
% 0.32/0.58 != zero_zero_real ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % sum.finite_Collect_op
% 0.32/0.58 thf(fact_3752_sum_Ofinite__Collect__op,axiom,
% 0.32/0.58 ! [I6: set_nat,X: nat > real,Y: nat > real] :
% 0.32/0.58 ( ( finite_finite_nat
% 0.32/0.58 @ ( collect_nat
% 0.32/0.58 @ ^ [I3: nat] :
% 0.32/0.58 ( ( member_nat @ I3 @ I6 )
% 0.32/0.58 & ( ( X @ I3 )
% 0.32/0.58 != zero_zero_real ) ) ) )
% 0.32/0.58 => ( ( finite_finite_nat
% 0.32/0.58 @ ( collect_nat
% 0.32/0.58 @ ^ [I3: nat] :
% 0.32/0.58 ( ( member_nat @ I3 @ I6 )
% 0.32/0.58 & ( ( Y @ I3 )
% 0.32/0.58 != zero_zero_real ) ) ) )
% 0.32/0.58 => ( finite_finite_nat
% 0.32/0.58 @ ( collect_nat
% 0.32/0.58 @ ^ [I3: nat] :
% 0.32/0.58 ( ( member_nat @ I3 @ I6 )
% 0.32/0.58 & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
% 0.32/0.58 != zero_zero_real ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % sum.finite_Collect_op
% 0.32/0.58 thf(fact_3753_sum_Ofinite__Collect__op,axiom,
% 0.32/0.58 ! [I6: set_int,X: int > real,Y: int > real] :
% 0.32/0.58 ( ( finite_finite_int
% 0.32/0.58 @ ( collect_int
% 0.32/0.58 @ ^ [I3: int] :
% 0.32/0.58 ( ( member_int @ I3 @ I6 )
% 0.32/0.58 & ( ( X @ I3 )
% 0.32/0.58 != zero_zero_real ) ) ) )
% 0.32/0.58 => ( ( finite_finite_int
% 0.32/0.58 @ ( collect_int
% 0.32/0.58 @ ^ [I3: int] :
% 0.32/0.58 ( ( member_int @ I3 @ I6 )
% 0.32/0.58 & ( ( Y @ I3 )
% 0.32/0.58 != zero_zero_real ) ) ) )
% 0.32/0.58 => ( finite_finite_int
% 0.32/0.58 @ ( collect_int
% 0.32/0.58 @ ^ [I3: int] :
% 0.32/0.58 ( ( member_int @ I3 @ I6 )
% 0.32/0.58 & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
% 0.32/0.58 != zero_zero_real ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % sum.finite_Collect_op
% 0.32/0.58 thf(fact_3754_sum_Ofinite__Collect__op,axiom,
% 0.32/0.58 ! [I6: set_complex,X: complex > real,Y: complex > real] :
% 0.32/0.58 ( ( finite3207457112153483333omplex
% 0.32/0.58 @ ( collect_complex
% 0.32/0.58 @ ^ [I3: complex] :
% 0.32/0.58 ( ( member_complex @ I3 @ I6 )
% 0.32/0.58 & ( ( X @ I3 )
% 0.32/0.58 != zero_zero_real ) ) ) )
% 0.32/0.58 => ( ( finite3207457112153483333omplex
% 0.32/0.58 @ ( collect_complex
% 0.32/0.58 @ ^ [I3: complex] :
% 0.32/0.58 ( ( member_complex @ I3 @ I6 )
% 0.32/0.58 & ( ( Y @ I3 )
% 0.32/0.58 != zero_zero_real ) ) ) )
% 0.32/0.58 => ( finite3207457112153483333omplex
% 0.32/0.58 @ ( collect_complex
% 0.32/0.58 @ ^ [I3: complex] :
% 0.32/0.58 ( ( member_complex @ I3 @ I6 )
% 0.32/0.58 & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
% 0.32/0.58 != zero_zero_real ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % sum.finite_Collect_op
% 0.32/0.58 thf(fact_3755_sum_Ofinite__Collect__op,axiom,
% 0.32/0.58 ! [I6: set_real,X: real > rat,Y: real > rat] :
% 0.32/0.58 ( ( finite_finite_real
% 0.32/0.58 @ ( collect_real
% 0.32/0.58 @ ^ [I3: real] :
% 0.32/0.58 ( ( member_real @ I3 @ I6 )
% 0.32/0.58 & ( ( X @ I3 )
% 0.32/0.58 != zero_zero_rat ) ) ) )
% 0.32/0.58 => ( ( finite_finite_real
% 0.32/0.58 @ ( collect_real
% 0.32/0.58 @ ^ [I3: real] :
% 0.32/0.58 ( ( member_real @ I3 @ I6 )
% 0.32/0.58 & ( ( Y @ I3 )
% 0.32/0.58 != zero_zero_rat ) ) ) )
% 0.32/0.58 => ( finite_finite_real
% 0.32/0.58 @ ( collect_real
% 0.32/0.58 @ ^ [I3: real] :
% 0.32/0.58 ( ( member_real @ I3 @ I6 )
% 0.32/0.58 & ( ( plus_plus_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
% 0.32/0.58 != zero_zero_rat ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % sum.finite_Collect_op
% 0.32/0.58 thf(fact_3756_sum_Ofinite__Collect__op,axiom,
% 0.32/0.58 ! [I6: set_nat,X: nat > rat,Y: nat > rat] :
% 0.32/0.58 ( ( finite_finite_nat
% 0.32/0.58 @ ( collect_nat
% 0.32/0.58 @ ^ [I3: nat] :
% 0.32/0.58 ( ( member_nat @ I3 @ I6 )
% 0.32/0.58 & ( ( X @ I3 )
% 0.32/0.58 != zero_zero_rat ) ) ) )
% 0.32/0.58 => ( ( finite_finite_nat
% 0.32/0.58 @ ( collect_nat
% 0.32/0.58 @ ^ [I3: nat] :
% 0.32/0.58 ( ( member_nat @ I3 @ I6 )
% 0.32/0.58 & ( ( Y @ I3 )
% 0.32/0.58 != zero_zero_rat ) ) ) )
% 0.32/0.58 => ( finite_finite_nat
% 0.32/0.58 @ ( collect_nat
% 0.32/0.58 @ ^ [I3: nat] :
% 0.32/0.58 ( ( member_nat @ I3 @ I6 )
% 0.32/0.58 & ( ( plus_plus_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
% 0.32/0.58 != zero_zero_rat ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % sum.finite_Collect_op
% 0.32/0.58 thf(fact_3757_gcd__nat__induct,axiom,
% 0.32/0.58 ! [P: nat > nat > $o,M: nat,N2: nat] :
% 0.32/0.58 ( ! [M2: nat] : ( P @ M2 @ zero_zero_nat )
% 0.32/0.58 => ( ! [M2: nat,N4: nat] :
% 0.32/0.58 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 0.32/0.58 => ( ( P @ N4 @ ( modulo_modulo_nat @ M2 @ N4 ) )
% 0.32/0.58 => ( P @ M2 @ N4 ) ) )
% 0.32/0.58 => ( P @ M @ N2 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % gcd_nat_induct
% 0.32/0.58 thf(fact_3758_Euclid__induct,axiom,
% 0.32/0.58 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.32/0.58 ( ! [A3: nat,B2: nat] :
% 0.32/0.58 ( ( P @ A3 @ B2 )
% 0.32/0.58 = ( P @ B2 @ A3 ) )
% 0.32/0.58 => ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
% 0.32/0.58 => ( ! [A3: nat,B2: nat] :
% 0.32/0.58 ( ( P @ A3 @ B2 )
% 0.32/0.58 => ( P @ A3 @ ( plus_plus_nat @ A3 @ B2 ) ) )
% 0.32/0.58 => ( P @ A @ B ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % Euclid_induct
% 0.32/0.58 thf(fact_3759_concat__bit__Suc,axiom,
% 0.32/0.58 ! [N2: nat,K: int,L2: int] :
% 0.32/0.58 ( ( bit_concat_bit @ ( suc @ N2 ) @ K @ L2 )
% 0.32/0.58 = ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N2 @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L2 ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % concat_bit_Suc
% 0.32/0.58 thf(fact_3760_dbl__simps_I3_J,axiom,
% 0.32/0.58 ( ( neg_nu7009210354673126013omplex @ one_one_complex )
% 0.32/0.58 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dbl_simps(3)
% 0.32/0.58 thf(fact_3761_dbl__simps_I3_J,axiom,
% 0.32/0.58 ( ( neg_numeral_dbl_real @ one_one_real )
% 0.32/0.58 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dbl_simps(3)
% 0.32/0.58 thf(fact_3762_dbl__simps_I3_J,axiom,
% 0.32/0.58 ( ( neg_numeral_dbl_rat @ one_one_rat )
% 0.32/0.58 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dbl_simps(3)
% 0.32/0.58 thf(fact_3763_dbl__simps_I3_J,axiom,
% 0.32/0.58 ( ( neg_numeral_dbl_int @ one_one_int )
% 0.32/0.58 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dbl_simps(3)
% 0.32/0.58 thf(fact_3764_even__succ__mod__exp,axiom,
% 0.32/0.58 ! [A: nat,N2: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.58 => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.58 = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_succ_mod_exp
% 0.32/0.58 thf(fact_3765_even__succ__mod__exp,axiom,
% 0.32/0.58 ! [A: int,N2: nat] :
% 0.32/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.58 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.58 = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_succ_mod_exp
% 0.32/0.58 thf(fact_3766_even__succ__mod__exp,axiom,
% 0.32/0.58 ! [A: code_integer,N2: nat] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.58 => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.58 = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_succ_mod_exp
% 0.32/0.58 thf(fact_3767_even__succ__div__exp,axiom,
% 0.32/0.58 ! [A: code_integer,N2: nat] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.58 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.58 = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_succ_div_exp
% 0.32/0.58 thf(fact_3768_even__succ__div__exp,axiom,
% 0.32/0.58 ! [A: nat,N2: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.58 => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.58 = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_succ_div_exp
% 0.32/0.58 thf(fact_3769_even__succ__div__exp,axiom,
% 0.32/0.58 ! [A: int,N2: nat] :
% 0.32/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.58 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.58 = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_succ_div_exp
% 0.32/0.58 thf(fact_3770_order__refl,axiom,
% 0.32/0.58 ! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% 0.32/0.58
% 0.32/0.58 % order_refl
% 0.32/0.58 thf(fact_3771_order__refl,axiom,
% 0.32/0.58 ! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).
% 0.32/0.58
% 0.32/0.58 % order_refl
% 0.32/0.58 thf(fact_3772_order__refl,axiom,
% 0.32/0.58 ! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% 0.32/0.58
% 0.32/0.58 % order_refl
% 0.32/0.58 thf(fact_3773_order__refl,axiom,
% 0.32/0.58 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 0.32/0.58
% 0.32/0.58 % order_refl
% 0.32/0.58 thf(fact_3774_order__refl,axiom,
% 0.32/0.58 ! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% 0.32/0.58
% 0.32/0.58 % order_refl
% 0.32/0.58 thf(fact_3775_dual__order_Orefl,axiom,
% 0.32/0.58 ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.refl
% 0.32/0.58 thf(fact_3776_dual__order_Orefl,axiom,
% 0.32/0.58 ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.refl
% 0.32/0.58 thf(fact_3777_dual__order_Orefl,axiom,
% 0.32/0.58 ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.refl
% 0.32/0.58 thf(fact_3778_dual__order_Orefl,axiom,
% 0.32/0.58 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.refl
% 0.32/0.58 thf(fact_3779_dual__order_Orefl,axiom,
% 0.32/0.58 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.refl
% 0.32/0.58 thf(fact_3780_nat__dvd__1__iff__1,axiom,
% 0.32/0.58 ! [M: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ M @ one_one_nat )
% 0.32/0.58 = ( M = one_one_nat ) ) ).
% 0.32/0.58
% 0.32/0.58 % nat_dvd_1_iff_1
% 0.32/0.58 thf(fact_3781_dvd__add__triv__right__iff,axiom,
% 0.32/0.58 ! [A: code_integer,B: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
% 0.32/0.58 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_add_triv_right_iff
% 0.32/0.58 thf(fact_3782_dvd__add__triv__right__iff,axiom,
% 0.32/0.58 ! [A: real,B: real] :
% 0.32/0.58 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.32/0.58 = ( dvd_dvd_real @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_add_triv_right_iff
% 0.32/0.58 thf(fact_3783_dvd__add__triv__right__iff,axiom,
% 0.32/0.58 ! [A: rat,B: rat] :
% 0.32/0.58 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 0.32/0.58 = ( dvd_dvd_rat @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_add_triv_right_iff
% 0.32/0.58 thf(fact_3784_dvd__add__triv__right__iff,axiom,
% 0.32/0.58 ! [A: nat,B: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.32/0.58 = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_add_triv_right_iff
% 0.32/0.58 thf(fact_3785_dvd__add__triv__right__iff,axiom,
% 0.32/0.58 ! [A: int,B: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
% 0.32/0.58 = ( dvd_dvd_int @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_add_triv_right_iff
% 0.32/0.58 thf(fact_3786_dvd__add__triv__left__iff,axiom,
% 0.32/0.58 ! [A: code_integer,B: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.32/0.58 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_add_triv_left_iff
% 0.32/0.58 thf(fact_3787_dvd__add__triv__left__iff,axiom,
% 0.32/0.58 ! [A: real,B: real] :
% 0.32/0.58 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.32/0.58 = ( dvd_dvd_real @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_add_triv_left_iff
% 0.32/0.58 thf(fact_3788_dvd__add__triv__left__iff,axiom,
% 0.32/0.58 ! [A: rat,B: rat] :
% 0.32/0.58 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.32/0.58 = ( dvd_dvd_rat @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_add_triv_left_iff
% 0.32/0.58 thf(fact_3789_dvd__add__triv__left__iff,axiom,
% 0.32/0.58 ! [A: nat,B: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.32/0.58 = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_add_triv_left_iff
% 0.32/0.58 thf(fact_3790_dvd__add__triv__left__iff,axiom,
% 0.32/0.58 ! [A: int,B: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.32/0.58 = ( dvd_dvd_int @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_add_triv_left_iff
% 0.32/0.58 thf(fact_3791_dvd__1__left,axiom,
% 0.32/0.58 ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_1_left
% 0.32/0.58 thf(fact_3792_dvd__1__iff__1,axiom,
% 0.32/0.58 ! [M: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.32/0.58 = ( M
% 0.32/0.58 = ( suc @ zero_zero_nat ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_1_iff_1
% 0.32/0.58 thf(fact_3793_div__dvd__div,axiom,
% 0.32/0.58 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.58 => ( ( dvd_dvd_Code_integer @ A @ C )
% 0.32/0.58 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B @ A ) @ ( divide6298287555418463151nteger @ C @ A ) )
% 0.32/0.58 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % div_dvd_div
% 0.32/0.58 thf(fact_3794_div__dvd__div,axiom,
% 0.32/0.58 ! [A: nat,B: nat,C: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.58 => ( ( dvd_dvd_nat @ A @ C )
% 0.32/0.58 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C @ A ) )
% 0.32/0.58 = ( dvd_dvd_nat @ B @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % div_dvd_div
% 0.32/0.58 thf(fact_3795_div__dvd__div,axiom,
% 0.32/0.58 ! [A: int,B: int,C: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ A @ B )
% 0.32/0.58 => ( ( dvd_dvd_int @ A @ C )
% 0.32/0.58 => ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C @ A ) )
% 0.32/0.58 = ( dvd_dvd_int @ B @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % div_dvd_div
% 0.32/0.58 thf(fact_3796_nat__mult__dvd__cancel__disj,axiom,
% 0.32/0.58 ! [K: nat,M: nat,N2: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 0.32/0.58 = ( ( K = zero_zero_nat )
% 0.32/0.58 | ( dvd_dvd_nat @ M @ N2 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % nat_mult_dvd_cancel_disj
% 0.32/0.58 thf(fact_3797_concat__bit__0,axiom,
% 0.32/0.58 ! [K: int,L2: int] :
% 0.32/0.58 ( ( bit_concat_bit @ zero_zero_nat @ K @ L2 )
% 0.32/0.58 = L2 ) ).
% 0.32/0.58
% 0.32/0.58 % concat_bit_0
% 0.32/0.58 thf(fact_3798_dbl__simps_I2_J,axiom,
% 0.32/0.58 ( ( neg_nu7009210354673126013omplex @ zero_zero_complex )
% 0.32/0.58 = zero_zero_complex ) ).
% 0.32/0.58
% 0.32/0.58 % dbl_simps(2)
% 0.32/0.58 thf(fact_3799_dbl__simps_I2_J,axiom,
% 0.32/0.58 ( ( neg_numeral_dbl_real @ zero_zero_real )
% 0.32/0.58 = zero_zero_real ) ).
% 0.32/0.58
% 0.32/0.58 % dbl_simps(2)
% 0.32/0.58 thf(fact_3800_dbl__simps_I2_J,axiom,
% 0.32/0.58 ( ( neg_numeral_dbl_rat @ zero_zero_rat )
% 0.32/0.58 = zero_zero_rat ) ).
% 0.32/0.58
% 0.32/0.58 % dbl_simps(2)
% 0.32/0.58 thf(fact_3801_dbl__simps_I2_J,axiom,
% 0.32/0.58 ( ( neg_numeral_dbl_int @ zero_zero_int )
% 0.32/0.58 = zero_zero_int ) ).
% 0.32/0.58
% 0.32/0.58 % dbl_simps(2)
% 0.32/0.58 thf(fact_3802_dvd__times__right__cancel__iff,axiom,
% 0.32/0.58 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.58 ( ( A != zero_z3403309356797280102nteger )
% 0.32/0.58 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
% 0.32/0.58 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_times_right_cancel_iff
% 0.32/0.58 thf(fact_3803_dvd__times__right__cancel__iff,axiom,
% 0.32/0.58 ! [A: nat,B: nat,C: nat] :
% 0.32/0.58 ( ( A != zero_zero_nat )
% 0.32/0.58 => ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
% 0.32/0.58 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_times_right_cancel_iff
% 0.32/0.58 thf(fact_3804_dvd__times__right__cancel__iff,axiom,
% 0.32/0.58 ! [A: int,B: int,C: int] :
% 0.32/0.58 ( ( A != zero_zero_int )
% 0.32/0.58 => ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
% 0.32/0.58 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_times_right_cancel_iff
% 0.32/0.58 thf(fact_3805_dvd__times__left__cancel__iff,axiom,
% 0.32/0.58 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.58 ( ( A != zero_z3403309356797280102nteger )
% 0.32/0.58 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) )
% 0.32/0.58 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_times_left_cancel_iff
% 0.32/0.58 thf(fact_3806_dvd__times__left__cancel__iff,axiom,
% 0.32/0.58 ! [A: nat,B: nat,C: nat] :
% 0.32/0.58 ( ( A != zero_zero_nat )
% 0.32/0.58 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
% 0.32/0.58 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_times_left_cancel_iff
% 0.32/0.58 thf(fact_3807_dvd__times__left__cancel__iff,axiom,
% 0.32/0.58 ! [A: int,B: int,C: int] :
% 0.32/0.58 ( ( A != zero_zero_int )
% 0.32/0.58 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
% 0.32/0.58 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_times_left_cancel_iff
% 0.32/0.58 thf(fact_3808_dvd__mult__cancel__right,axiom,
% 0.32/0.58 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.32/0.58 = ( ( C = zero_z3403309356797280102nteger )
% 0.32/0.58 | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_mult_cancel_right
% 0.32/0.58 thf(fact_3809_dvd__mult__cancel__right,axiom,
% 0.32/0.58 ! [A: complex,C: complex,B: complex] :
% 0.32/0.58 ( ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 0.32/0.58 = ( ( C = zero_zero_complex )
% 0.32/0.58 | ( dvd_dvd_complex @ A @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_mult_cancel_right
% 0.32/0.58 thf(fact_3810_dvd__mult__cancel__right,axiom,
% 0.32/0.58 ! [A: real,C: real,B: real] :
% 0.32/0.58 ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.32/0.58 = ( ( C = zero_zero_real )
% 0.32/0.58 | ( dvd_dvd_real @ A @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_mult_cancel_right
% 0.32/0.58 thf(fact_3811_dvd__mult__cancel__right,axiom,
% 0.32/0.58 ! [A: rat,C: rat,B: rat] :
% 0.32/0.58 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.32/0.58 = ( ( C = zero_zero_rat )
% 0.32/0.58 | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_mult_cancel_right
% 0.32/0.58 thf(fact_3812_dvd__mult__cancel__right,axiom,
% 0.32/0.58 ! [A: int,C: int,B: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.32/0.58 = ( ( C = zero_zero_int )
% 0.32/0.58 | ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_mult_cancel_right
% 0.32/0.58 thf(fact_3813_dvd__mult__cancel__left,axiom,
% 0.32/0.58 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.32/0.58 = ( ( C = zero_z3403309356797280102nteger )
% 0.32/0.58 | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_mult_cancel_left
% 0.32/0.58 thf(fact_3814_dvd__mult__cancel__left,axiom,
% 0.32/0.58 ! [C: complex,A: complex,B: complex] :
% 0.32/0.58 ( ( dvd_dvd_complex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.32/0.58 = ( ( C = zero_zero_complex )
% 0.32/0.58 | ( dvd_dvd_complex @ A @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_mult_cancel_left
% 0.32/0.58 thf(fact_3815_dvd__mult__cancel__left,axiom,
% 0.32/0.58 ! [C: real,A: real,B: real] :
% 0.32/0.58 ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.32/0.58 = ( ( C = zero_zero_real )
% 0.32/0.58 | ( dvd_dvd_real @ A @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_mult_cancel_left
% 0.32/0.58 thf(fact_3816_dvd__mult__cancel__left,axiom,
% 0.32/0.58 ! [C: rat,A: rat,B: rat] :
% 0.32/0.58 ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.58 = ( ( C = zero_zero_rat )
% 0.32/0.58 | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_mult_cancel_left
% 0.32/0.58 thf(fact_3817_dvd__mult__cancel__left,axiom,
% 0.32/0.58 ! [C: int,A: int,B: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.32/0.58 = ( ( C = zero_zero_int )
% 0.32/0.58 | ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_mult_cancel_left
% 0.32/0.58 thf(fact_3818_dvd__add__times__triv__right__iff,axiom,
% 0.32/0.58 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C @ A ) ) )
% 0.32/0.58 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_add_times_triv_right_iff
% 0.32/0.58 thf(fact_3819_dvd__add__times__triv__right__iff,axiom,
% 0.32/0.58 ! [A: real,B: real,C: real] :
% 0.32/0.58 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ ( times_times_real @ C @ A ) ) )
% 0.32/0.58 = ( dvd_dvd_real @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_add_times_triv_right_iff
% 0.32/0.58 thf(fact_3820_dvd__add__times__triv__right__iff,axiom,
% 0.32/0.58 ! [A: rat,B: rat,C: rat] :
% 0.32/0.58 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A ) ) )
% 0.32/0.58 = ( dvd_dvd_rat @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_add_times_triv_right_iff
% 0.32/0.58 thf(fact_3821_dvd__add__times__triv__right__iff,axiom,
% 0.32/0.58 ! [A: nat,B: nat,C: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
% 0.32/0.58 = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_add_times_triv_right_iff
% 0.32/0.58 thf(fact_3822_dvd__add__times__triv__right__iff,axiom,
% 0.32/0.58 ! [A: int,B: int,C: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
% 0.32/0.58 = ( dvd_dvd_int @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_add_times_triv_right_iff
% 0.32/0.58 thf(fact_3823_dvd__add__times__triv__left__iff,axiom,
% 0.32/0.58 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B ) )
% 0.32/0.58 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_add_times_triv_left_iff
% 0.32/0.58 thf(fact_3824_dvd__add__times__triv__left__iff,axiom,
% 0.32/0.58 ! [A: real,C: real,B: real] :
% 0.32/0.58 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B ) )
% 0.32/0.58 = ( dvd_dvd_real @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_add_times_triv_left_iff
% 0.32/0.58 thf(fact_3825_dvd__add__times__triv__left__iff,axiom,
% 0.32/0.58 ! [A: rat,C: rat,B: rat] :
% 0.32/0.58 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B ) )
% 0.32/0.58 = ( dvd_dvd_rat @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_add_times_triv_left_iff
% 0.32/0.58 thf(fact_3826_dvd__add__times__triv__left__iff,axiom,
% 0.32/0.58 ! [A: nat,C: nat,B: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
% 0.32/0.58 = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_add_times_triv_left_iff
% 0.32/0.58 thf(fact_3827_dvd__add__times__triv__left__iff,axiom,
% 0.32/0.58 ! [A: int,C: int,B: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
% 0.32/0.58 = ( dvd_dvd_int @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_add_times_triv_left_iff
% 0.32/0.58 thf(fact_3828_unit__prod,axiom,
% 0.32/0.58 ! [A: code_integer,B: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.58 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.58 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_prod
% 0.32/0.58 thf(fact_3829_unit__prod,axiom,
% 0.32/0.58 ! [A: nat,B: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.58 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.58 => ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_prod
% 0.32/0.58 thf(fact_3830_unit__prod,axiom,
% 0.32/0.58 ! [A: int,B: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.58 => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.58 => ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_prod
% 0.32/0.58 thf(fact_3831_dvd__mult__div__cancel,axiom,
% 0.32/0.58 ! [A: code_integer,B: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.58 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ A ) )
% 0.32/0.58 = B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_mult_div_cancel
% 0.32/0.58 thf(fact_3832_dvd__mult__div__cancel,axiom,
% 0.32/0.58 ! [A: nat,B: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.58 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
% 0.32/0.58 = B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_mult_div_cancel
% 0.32/0.58 thf(fact_3833_dvd__mult__div__cancel,axiom,
% 0.32/0.58 ! [A: int,B: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ A @ B )
% 0.32/0.58 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
% 0.32/0.58 = B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_mult_div_cancel
% 0.32/0.58 thf(fact_3834_dvd__div__mult__self,axiom,
% 0.32/0.58 ! [A: code_integer,B: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.58 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
% 0.32/0.58 = B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_div_mult_self
% 0.32/0.58 thf(fact_3835_dvd__div__mult__self,axiom,
% 0.32/0.58 ! [A: nat,B: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.58 => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
% 0.32/0.58 = B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_div_mult_self
% 0.32/0.58 thf(fact_3836_dvd__div__mult__self,axiom,
% 0.32/0.58 ! [A: int,B: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ A @ B )
% 0.32/0.58 => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
% 0.32/0.58 = B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_div_mult_self
% 0.32/0.58 thf(fact_3837_div__add,axiom,
% 0.32/0.58 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ C @ A )
% 0.32/0.58 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.58 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.32/0.58 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % div_add
% 0.32/0.58 thf(fact_3838_div__add,axiom,
% 0.32/0.58 ! [C: nat,A: nat,B: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ C @ A )
% 0.32/0.58 => ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.58 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.32/0.58 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % div_add
% 0.32/0.58 thf(fact_3839_div__add,axiom,
% 0.32/0.58 ! [C: int,A: int,B: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ C @ A )
% 0.32/0.58 => ( ( dvd_dvd_int @ C @ B )
% 0.32/0.58 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.32/0.58 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % div_add
% 0.32/0.58 thf(fact_3840_unit__div__1__div__1,axiom,
% 0.32/0.58 ! [A: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.58 => ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 0.32/0.58 = A ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_div_1_div_1
% 0.32/0.58 thf(fact_3841_unit__div__1__div__1,axiom,
% 0.32/0.58 ! [A: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.58 => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
% 0.32/0.58 = A ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_div_1_div_1
% 0.32/0.58 thf(fact_3842_unit__div__1__div__1,axiom,
% 0.32/0.58 ! [A: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.58 => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
% 0.32/0.58 = A ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_div_1_div_1
% 0.32/0.58 thf(fact_3843_unit__div__1__unit,axiom,
% 0.32/0.58 ! [A: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.58 => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_div_1_unit
% 0.32/0.58 thf(fact_3844_unit__div__1__unit,axiom,
% 0.32/0.58 ! [A: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.58 => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_div_1_unit
% 0.32/0.58 thf(fact_3845_unit__div__1__unit,axiom,
% 0.32/0.58 ! [A: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.58 => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_div_1_unit
% 0.32/0.58 thf(fact_3846_unit__div,axiom,
% 0.32/0.58 ! [A: code_integer,B: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.58 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.58 => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_div
% 0.32/0.58 thf(fact_3847_unit__div,axiom,
% 0.32/0.58 ! [A: nat,B: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.58 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.58 => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_div
% 0.32/0.58 thf(fact_3848_unit__div,axiom,
% 0.32/0.58 ! [A: int,B: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.58 => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.58 => ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_div
% 0.32/0.58 thf(fact_3849_div__diff,axiom,
% 0.32/0.58 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ C @ A )
% 0.32/0.58 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.58 => ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
% 0.32/0.58 = ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % div_diff
% 0.32/0.58 thf(fact_3850_div__diff,axiom,
% 0.32/0.58 ! [C: int,A: int,B: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ C @ A )
% 0.32/0.58 => ( ( dvd_dvd_int @ C @ B )
% 0.32/0.58 => ( ( divide_divide_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.32/0.58 = ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % div_diff
% 0.32/0.58 thf(fact_3851_concat__bit__nonnegative__iff,axiom,
% 0.32/0.58 ! [N2: nat,K: int,L2: int] :
% 0.32/0.58 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N2 @ K @ L2 ) )
% 0.32/0.58 = ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ).
% 0.32/0.58
% 0.32/0.58 % concat_bit_nonnegative_iff
% 0.32/0.58 thf(fact_3852_concat__bit__negative__iff,axiom,
% 0.32/0.58 ! [N2: nat,K: int,L2: int] :
% 0.32/0.58 ( ( ord_less_int @ ( bit_concat_bit @ N2 @ K @ L2 ) @ zero_zero_int )
% 0.32/0.58 = ( ord_less_int @ L2 @ zero_zero_int ) ) ).
% 0.32/0.58
% 0.32/0.58 % concat_bit_negative_iff
% 0.32/0.58 thf(fact_3853_dbl__simps_I5_J,axiom,
% 0.32/0.58 ! [K: num] :
% 0.32/0.58 ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
% 0.32/0.58 = ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dbl_simps(5)
% 0.32/0.58 thf(fact_3854_dbl__simps_I5_J,axiom,
% 0.32/0.58 ! [K: num] :
% 0.32/0.58 ( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
% 0.32/0.58 = ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dbl_simps(5)
% 0.32/0.58 thf(fact_3855_dbl__simps_I5_J,axiom,
% 0.32/0.58 ! [K: num] :
% 0.32/0.58 ( ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) )
% 0.32/0.58 = ( numeral_numeral_rat @ ( bit0 @ K ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dbl_simps(5)
% 0.32/0.58 thf(fact_3856_dbl__simps_I5_J,axiom,
% 0.32/0.58 ! [K: num] :
% 0.32/0.58 ( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
% 0.32/0.58 = ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dbl_simps(5)
% 0.32/0.58 thf(fact_3857_even__Suc__Suc__iff,axiom,
% 0.32/0.58 ! [N2: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N2 ) ) )
% 0.32/0.58 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_Suc_Suc_iff
% 0.32/0.58 thf(fact_3858_even__Suc,axiom,
% 0.32/0.58 ! [N2: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) )
% 0.32/0.58 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_Suc
% 0.32/0.58 thf(fact_3859_unit__div__mult__self,axiom,
% 0.32/0.58 ! [A: code_integer,B: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.58 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
% 0.32/0.58 = B ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_div_mult_self
% 0.32/0.58 thf(fact_3860_unit__div__mult__self,axiom,
% 0.32/0.58 ! [A: nat,B: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.58 => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
% 0.32/0.58 = B ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_div_mult_self
% 0.32/0.58 thf(fact_3861_unit__div__mult__self,axiom,
% 0.32/0.58 ! [A: int,B: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.58 => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
% 0.32/0.58 = B ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_div_mult_self
% 0.32/0.58 thf(fact_3862_unit__mult__div__div,axiom,
% 0.32/0.58 ! [A: code_integer,B: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.58 => ( ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 0.32/0.58 = ( divide6298287555418463151nteger @ B @ A ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_mult_div_div
% 0.32/0.58 thf(fact_3863_unit__mult__div__div,axiom,
% 0.32/0.58 ! [A: nat,B: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.58 => ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
% 0.32/0.58 = ( divide_divide_nat @ B @ A ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_mult_div_div
% 0.32/0.58 thf(fact_3864_unit__mult__div__div,axiom,
% 0.32/0.58 ! [A: int,B: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.58 => ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
% 0.32/0.58 = ( divide_divide_int @ B @ A ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_mult_div_div
% 0.32/0.58 thf(fact_3865_pow__divides__pow__iff,axiom,
% 0.32/0.58 ! [N2: nat,A: nat,B: nat] :
% 0.32/0.58 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.58 => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
% 0.32/0.58 = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % pow_divides_pow_iff
% 0.32/0.58 thf(fact_3866_pow__divides__pow__iff,axiom,
% 0.32/0.58 ! [N2: nat,A: int,B: int] :
% 0.32/0.58 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.58 => ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
% 0.32/0.58 = ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % pow_divides_pow_iff
% 0.32/0.58 thf(fact_3867_even__mult__iff,axiom,
% 0.32/0.58 ! [A: code_integer,B: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.32/0.58 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_mult_iff
% 0.32/0.58 thf(fact_3868_even__mult__iff,axiom,
% 0.32/0.58 ! [A: nat,B: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) )
% 0.32/0.58 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_mult_iff
% 0.32/0.58 thf(fact_3869_even__mult__iff,axiom,
% 0.32/0.58 ! [A: int,B: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) )
% 0.32/0.58 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_mult_iff
% 0.32/0.58 thf(fact_3870_even__add,axiom,
% 0.32/0.58 ! [A: code_integer,B: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.32/0.58 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_add
% 0.32/0.58 thf(fact_3871_even__add,axiom,
% 0.32/0.58 ! [A: nat,B: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
% 0.32/0.58 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_add
% 0.32/0.58 thf(fact_3872_even__add,axiom,
% 0.32/0.58 ! [A: int,B: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
% 0.32/0.58 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_add
% 0.32/0.58 thf(fact_3873_odd__add,axiom,
% 0.32/0.58 ! [A: code_integer,B: code_integer] :
% 0.32/0.58 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) )
% 0.32/0.58 = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.58 != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % odd_add
% 0.32/0.58 thf(fact_3874_odd__add,axiom,
% 0.32/0.58 ! [A: nat,B: nat] :
% 0.32/0.58 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
% 0.32/0.58 = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.58 != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % odd_add
% 0.32/0.58 thf(fact_3875_odd__add,axiom,
% 0.32/0.58 ! [A: int,B: int] :
% 0.32/0.58 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
% 0.32/0.58 = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.58 != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % odd_add
% 0.32/0.58 thf(fact_3876_even__mod__2__iff,axiom,
% 0.32/0.58 ! [A: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.58 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_mod_2_iff
% 0.32/0.58 thf(fact_3877_even__mod__2__iff,axiom,
% 0.32/0.58 ! [A: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.32/0.58 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_mod_2_iff
% 0.32/0.58 thf(fact_3878_even__mod__2__iff,axiom,
% 0.32/0.58 ! [A: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 0.32/0.58 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_mod_2_iff
% 0.32/0.58 thf(fact_3879_even__Suc__div__two,axiom,
% 0.32/0.58 ! [N2: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.58 => ( ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.58 = ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_Suc_div_two
% 0.32/0.58 thf(fact_3880_odd__Suc__div__two,axiom,
% 0.32/0.58 ! [N2: nat] :
% 0.32/0.58 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.58 => ( ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.58 = ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % odd_Suc_div_two
% 0.32/0.58 thf(fact_3881_zero__le__power__eq__numeral,axiom,
% 0.32/0.58 ! [A: real,W: num] :
% 0.32/0.58 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.32/0.58 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.58 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.58 & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % zero_le_power_eq_numeral
% 0.32/0.58 thf(fact_3882_zero__le__power__eq__numeral,axiom,
% 0.32/0.58 ! [A: rat,W: num] :
% 0.32/0.58 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.32/0.58 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.58 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.58 & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % zero_le_power_eq_numeral
% 0.32/0.58 thf(fact_3883_zero__le__power__eq__numeral,axiom,
% 0.32/0.58 ! [A: int,W: num] :
% 0.32/0.58 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.32/0.58 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.58 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.58 & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % zero_le_power_eq_numeral
% 0.32/0.58 thf(fact_3884_power__less__zero__eq__numeral,axiom,
% 0.32/0.58 ! [A: real,W: num] :
% 0.32/0.58 ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 0.32/0.58 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.58 & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % power_less_zero_eq_numeral
% 0.32/0.58 thf(fact_3885_power__less__zero__eq__numeral,axiom,
% 0.32/0.58 ! [A: rat,W: num] :
% 0.32/0.58 ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 0.32/0.58 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.58 & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % power_less_zero_eq_numeral
% 0.32/0.58 thf(fact_3886_power__less__zero__eq__numeral,axiom,
% 0.32/0.58 ! [A: int,W: num] :
% 0.32/0.58 ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 0.32/0.58 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.58 & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % power_less_zero_eq_numeral
% 0.32/0.58 thf(fact_3887_power__less__zero__eq,axiom,
% 0.32/0.58 ! [A: real,N2: nat] :
% 0.32/0.58 ( ( ord_less_real @ ( power_power_real @ A @ N2 ) @ zero_zero_real )
% 0.32/0.58 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.58 & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % power_less_zero_eq
% 0.32/0.58 thf(fact_3888_power__less__zero__eq,axiom,
% 0.32/0.58 ! [A: rat,N2: nat] :
% 0.32/0.58 ( ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ zero_zero_rat )
% 0.32/0.58 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.58 & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % power_less_zero_eq
% 0.32/0.58 thf(fact_3889_power__less__zero__eq,axiom,
% 0.32/0.58 ! [A: int,N2: nat] :
% 0.32/0.58 ( ( ord_less_int @ ( power_power_int @ A @ N2 ) @ zero_zero_int )
% 0.32/0.58 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.58 & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % power_less_zero_eq
% 0.32/0.58 thf(fact_3890_even__plus__one__iff,axiom,
% 0.32/0.58 ! [A: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
% 0.32/0.58 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_plus_one_iff
% 0.32/0.58 thf(fact_3891_even__plus__one__iff,axiom,
% 0.32/0.58 ! [A: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
% 0.32/0.58 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_plus_one_iff
% 0.32/0.58 thf(fact_3892_even__plus__one__iff,axiom,
% 0.32/0.58 ! [A: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
% 0.32/0.58 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_plus_one_iff
% 0.32/0.58 thf(fact_3893_even__diff,axiom,
% 0.32/0.58 ! [A: code_integer,B: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B ) )
% 0.32/0.58 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_diff
% 0.32/0.58 thf(fact_3894_even__diff,axiom,
% 0.32/0.58 ! [A: int,B: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
% 0.32/0.58 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_diff
% 0.32/0.58 thf(fact_3895_odd__Suc__minus__one,axiom,
% 0.32/0.58 ! [N2: nat] :
% 0.32/0.58 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.58 => ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
% 0.32/0.58 = N2 ) ) ).
% 0.32/0.58
% 0.32/0.58 % odd_Suc_minus_one
% 0.32/0.58 thf(fact_3896_even__diff__nat,axiom,
% 0.32/0.58 ! [M: nat,N2: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) )
% 0.32/0.58 = ( ( ord_less_nat @ M @ N2 )
% 0.32/0.58 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_diff_nat
% 0.32/0.58 thf(fact_3897_zero__less__power__eq__numeral,axiom,
% 0.32/0.58 ! [A: real,W: num] :
% 0.32/0.58 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.32/0.58 = ( ( ( numeral_numeral_nat @ W )
% 0.32/0.58 = zero_zero_nat )
% 0.32/0.58 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.58 & ( A != zero_zero_real ) )
% 0.32/0.58 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.58 & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % zero_less_power_eq_numeral
% 0.32/0.58 thf(fact_3898_zero__less__power__eq__numeral,axiom,
% 0.32/0.58 ! [A: rat,W: num] :
% 0.32/0.58 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.32/0.58 = ( ( ( numeral_numeral_nat @ W )
% 0.32/0.58 = zero_zero_nat )
% 0.32/0.58 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.58 & ( A != zero_zero_rat ) )
% 0.32/0.58 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.58 & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % zero_less_power_eq_numeral
% 0.32/0.58 thf(fact_3899_zero__less__power__eq__numeral,axiom,
% 0.32/0.58 ! [A: int,W: num] :
% 0.32/0.58 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.32/0.58 = ( ( ( numeral_numeral_nat @ W )
% 0.32/0.58 = zero_zero_nat )
% 0.32/0.58 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.58 & ( A != zero_zero_int ) )
% 0.32/0.58 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.58 & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % zero_less_power_eq_numeral
% 0.32/0.58 thf(fact_3900_even__succ__div__2,axiom,
% 0.32/0.58 ! [A: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.58 = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_succ_div_2
% 0.32/0.58 thf(fact_3901_even__succ__div__2,axiom,
% 0.32/0.58 ! [A: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.58 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_succ_div_2
% 0.32/0.58 thf(fact_3902_even__succ__div__2,axiom,
% 0.32/0.58 ! [A: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.58 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_succ_div_2
% 0.32/0.58 thf(fact_3903_even__succ__div__two,axiom,
% 0.32/0.58 ! [A: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.58 = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_succ_div_two
% 0.32/0.58 thf(fact_3904_even__succ__div__two,axiom,
% 0.32/0.58 ! [A: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.58 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_succ_div_two
% 0.32/0.58 thf(fact_3905_even__succ__div__two,axiom,
% 0.32/0.58 ! [A: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.58 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_succ_div_two
% 0.32/0.58 thf(fact_3906_odd__succ__div__two,axiom,
% 0.32/0.58 ! [A: code_integer] :
% 0.32/0.58 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.58 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % odd_succ_div_two
% 0.32/0.58 thf(fact_3907_odd__succ__div__two,axiom,
% 0.32/0.58 ! [A: nat] :
% 0.32/0.58 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.58 = ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % odd_succ_div_two
% 0.32/0.58 thf(fact_3908_odd__succ__div__two,axiom,
% 0.32/0.58 ! [A: int] :
% 0.32/0.58 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.58 = ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % odd_succ_div_two
% 0.32/0.58 thf(fact_3909_even__power,axiom,
% 0.32/0.58 ! [A: code_integer,N2: nat] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N2 ) )
% 0.32/0.58 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_power
% 0.32/0.58 thf(fact_3910_even__power,axiom,
% 0.32/0.58 ! [A: nat,N2: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N2 ) )
% 0.32/0.58 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_power
% 0.32/0.58 thf(fact_3911_even__power,axiom,
% 0.32/0.58 ! [A: int,N2: nat] :
% 0.32/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N2 ) )
% 0.32/0.58 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_power
% 0.32/0.58 thf(fact_3912_odd__two__times__div__two__nat,axiom,
% 0.32/0.58 ! [N2: nat] :
% 0.32/0.58 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.58 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.58 = ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % odd_two_times_div_two_nat
% 0.32/0.58 thf(fact_3913_odd__two__times__div__two__succ,axiom,
% 0.32/0.58 ! [A: code_integer] :
% 0.32/0.58 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
% 0.32/0.58 = A ) ) ).
% 0.32/0.58
% 0.32/0.58 % odd_two_times_div_two_succ
% 0.32/0.58 thf(fact_3914_odd__two__times__div__two__succ,axiom,
% 0.32/0.58 ! [A: nat] :
% 0.32/0.58 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
% 0.32/0.58 = A ) ) ).
% 0.32/0.58
% 0.32/0.58 % odd_two_times_div_two_succ
% 0.32/0.58 thf(fact_3915_odd__two__times__div__two__succ,axiom,
% 0.32/0.58 ! [A: int] :
% 0.32/0.58 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
% 0.32/0.58 = A ) ) ).
% 0.32/0.58
% 0.32/0.58 % odd_two_times_div_two_succ
% 0.32/0.58 thf(fact_3916_power__le__zero__eq__numeral,axiom,
% 0.32/0.58 ! [A: real,W: num] :
% 0.32/0.58 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 0.32/0.58 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 0.32/0.58 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.58 & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.32/0.58 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.58 & ( A = zero_zero_real ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % power_le_zero_eq_numeral
% 0.32/0.58 thf(fact_3917_power__le__zero__eq__numeral,axiom,
% 0.32/0.58 ! [A: rat,W: num] :
% 0.32/0.58 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 0.32/0.58 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 0.32/0.58 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.58 & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.32/0.58 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.58 & ( A = zero_zero_rat ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % power_le_zero_eq_numeral
% 0.32/0.58 thf(fact_3918_power__le__zero__eq__numeral,axiom,
% 0.32/0.58 ! [A: int,W: num] :
% 0.32/0.58 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 0.32/0.58 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 0.32/0.58 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.58 & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.32/0.58 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.58 & ( A = zero_zero_int ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % power_le_zero_eq_numeral
% 0.32/0.58 thf(fact_3919_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.32/0.58 ! [N2: nat] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) @ one_one_Code_integer ) )
% 0.32/0.58 = ( N2 = zero_zero_nat ) ) ).
% 0.32/0.58
% 0.32/0.58 % semiring_parity_class.even_mask_iff
% 0.32/0.58 thf(fact_3920_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.32/0.58 ! [N2: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) )
% 0.32/0.58 = ( N2 = zero_zero_nat ) ) ).
% 0.32/0.58
% 0.32/0.58 % semiring_parity_class.even_mask_iff
% 0.32/0.58 thf(fact_3921_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.32/0.58 ! [N2: nat] :
% 0.32/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) )
% 0.32/0.58 = ( N2 = zero_zero_nat ) ) ).
% 0.32/0.58
% 0.32/0.58 % semiring_parity_class.even_mask_iff
% 0.32/0.58 thf(fact_3922_dvd__triv__right,axiom,
% 0.32/0.58 ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ A ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_triv_right
% 0.32/0.58 thf(fact_3923_dvd__triv__right,axiom,
% 0.32/0.58 ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_triv_right
% 0.32/0.58 thf(fact_3924_dvd__triv__right,axiom,
% 0.32/0.58 ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_triv_right
% 0.32/0.58 thf(fact_3925_dvd__triv__right,axiom,
% 0.32/0.58 ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_triv_right
% 0.32/0.58 thf(fact_3926_dvd__triv__right,axiom,
% 0.32/0.58 ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_triv_right
% 0.32/0.58 thf(fact_3927_dvd__mult__right,axiom,
% 0.32/0.58 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.32/0.58 => ( dvd_dvd_Code_integer @ B @ C ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_mult_right
% 0.32/0.58 thf(fact_3928_dvd__mult__right,axiom,
% 0.32/0.58 ! [A: real,B: real,C: real] :
% 0.32/0.58 ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
% 0.32/0.58 => ( dvd_dvd_real @ B @ C ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_mult_right
% 0.32/0.58 thf(fact_3929_dvd__mult__right,axiom,
% 0.32/0.58 ! [A: rat,B: rat,C: rat] :
% 0.32/0.58 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
% 0.32/0.58 => ( dvd_dvd_rat @ B @ C ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_mult_right
% 0.32/0.58 thf(fact_3930_dvd__mult__right,axiom,
% 0.32/0.58 ! [A: nat,B: nat,C: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.32/0.58 => ( dvd_dvd_nat @ B @ C ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_mult_right
% 0.32/0.58 thf(fact_3931_dvd__mult__right,axiom,
% 0.32/0.58 ! [A: int,B: int,C: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 0.32/0.58 => ( dvd_dvd_int @ B @ C ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_mult_right
% 0.32/0.58 thf(fact_3932_mult__dvd__mono,axiom,
% 0.32/0.58 ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.58 => ( ( dvd_dvd_Code_integer @ C @ D )
% 0.32/0.58 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % mult_dvd_mono
% 0.32/0.58 thf(fact_3933_mult__dvd__mono,axiom,
% 0.32/0.58 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.58 ( ( dvd_dvd_real @ A @ B )
% 0.32/0.58 => ( ( dvd_dvd_real @ C @ D )
% 0.32/0.58 => ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % mult_dvd_mono
% 0.32/0.58 thf(fact_3934_mult__dvd__mono,axiom,
% 0.32/0.58 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.58 ( ( dvd_dvd_rat @ A @ B )
% 0.32/0.58 => ( ( dvd_dvd_rat @ C @ D )
% 0.32/0.58 => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % mult_dvd_mono
% 0.32/0.58 thf(fact_3935_mult__dvd__mono,axiom,
% 0.32/0.58 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.58 => ( ( dvd_dvd_nat @ C @ D )
% 0.32/0.58 => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % mult_dvd_mono
% 0.32/0.58 thf(fact_3936_mult__dvd__mono,axiom,
% 0.32/0.58 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ A @ B )
% 0.32/0.58 => ( ( dvd_dvd_int @ C @ D )
% 0.32/0.58 => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % mult_dvd_mono
% 0.32/0.58 thf(fact_3937_dvd__triv__left,axiom,
% 0.32/0.58 ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_triv_left
% 0.32/0.58 thf(fact_3938_dvd__triv__left,axiom,
% 0.32/0.58 ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_triv_left
% 0.32/0.58 thf(fact_3939_dvd__triv__left,axiom,
% 0.32/0.58 ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_triv_left
% 0.32/0.58 thf(fact_3940_dvd__triv__left,axiom,
% 0.32/0.58 ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_triv_left
% 0.32/0.58 thf(fact_3941_dvd__triv__left,axiom,
% 0.32/0.58 ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_triv_left
% 0.32/0.58 thf(fact_3942_dvd__mult__left,axiom,
% 0.32/0.58 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.32/0.58 => ( dvd_dvd_Code_integer @ A @ C ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_mult_left
% 0.32/0.58 thf(fact_3943_dvd__mult__left,axiom,
% 0.32/0.58 ! [A: real,B: real,C: real] :
% 0.32/0.58 ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
% 0.32/0.58 => ( dvd_dvd_real @ A @ C ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_mult_left
% 0.32/0.58 thf(fact_3944_dvd__mult__left,axiom,
% 0.32/0.58 ! [A: rat,B: rat,C: rat] :
% 0.32/0.58 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
% 0.32/0.58 => ( dvd_dvd_rat @ A @ C ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_mult_left
% 0.32/0.58 thf(fact_3945_dvd__mult__left,axiom,
% 0.32/0.58 ! [A: nat,B: nat,C: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.32/0.58 => ( dvd_dvd_nat @ A @ C ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_mult_left
% 0.32/0.58 thf(fact_3946_dvd__mult__left,axiom,
% 0.32/0.58 ! [A: int,B: int,C: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 0.32/0.58 => ( dvd_dvd_int @ A @ C ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_mult_left
% 0.32/0.58 thf(fact_3947_dvd__mult2,axiom,
% 0.32/0.58 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.58 => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_mult2
% 0.32/0.58 thf(fact_3948_dvd__mult2,axiom,
% 0.32/0.58 ! [A: real,B: real,C: real] :
% 0.32/0.59 ( ( dvd_dvd_real @ A @ B )
% 0.32/0.59 => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_mult2
% 0.32/0.59 thf(fact_3949_dvd__mult2,axiom,
% 0.32/0.59 ! [A: rat,B: rat,C: rat] :
% 0.32/0.59 ( ( dvd_dvd_rat @ A @ B )
% 0.32/0.59 => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_mult2
% 0.32/0.59 thf(fact_3950_dvd__mult2,axiom,
% 0.32/0.59 ! [A: nat,B: nat,C: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.59 => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_mult2
% 0.32/0.59 thf(fact_3951_dvd__mult2,axiom,
% 0.32/0.59 ! [A: int,B: int,C: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ A @ B )
% 0.32/0.59 => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_mult2
% 0.32/0.59 thf(fact_3952_dvd__mult,axiom,
% 0.32/0.59 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ A @ C )
% 0.32/0.59 => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_mult
% 0.32/0.59 thf(fact_3953_dvd__mult,axiom,
% 0.32/0.59 ! [A: real,C: real,B: real] :
% 0.32/0.59 ( ( dvd_dvd_real @ A @ C )
% 0.32/0.59 => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_mult
% 0.32/0.59 thf(fact_3954_dvd__mult,axiom,
% 0.32/0.59 ! [A: rat,C: rat,B: rat] :
% 0.32/0.59 ( ( dvd_dvd_rat @ A @ C )
% 0.32/0.59 => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_mult
% 0.32/0.59 thf(fact_3955_dvd__mult,axiom,
% 0.32/0.59 ! [A: nat,C: nat,B: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ A @ C )
% 0.32/0.59 => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_mult
% 0.32/0.59 thf(fact_3956_dvd__mult,axiom,
% 0.32/0.59 ! [A: int,C: int,B: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ A @ C )
% 0.32/0.59 => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_mult
% 0.32/0.59 thf(fact_3957_dvd__def,axiom,
% 0.32/0.59 ( dvd_dvd_Code_integer
% 0.32/0.59 = ( ^ [B5: code_integer,A5: code_integer] :
% 0.32/0.59 ? [K3: code_integer] :
% 0.32/0.59 ( A5
% 0.32/0.59 = ( times_3573771949741848930nteger @ B5 @ K3 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_def
% 0.32/0.59 thf(fact_3958_dvd__def,axiom,
% 0.32/0.59 ( dvd_dvd_real
% 0.32/0.59 = ( ^ [B5: real,A5: real] :
% 0.32/0.59 ? [K3: real] :
% 0.32/0.59 ( A5
% 0.32/0.59 = ( times_times_real @ B5 @ K3 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_def
% 0.32/0.59 thf(fact_3959_dvd__def,axiom,
% 0.32/0.59 ( dvd_dvd_rat
% 0.32/0.59 = ( ^ [B5: rat,A5: rat] :
% 0.32/0.59 ? [K3: rat] :
% 0.32/0.59 ( A5
% 0.32/0.59 = ( times_times_rat @ B5 @ K3 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_def
% 0.32/0.59 thf(fact_3960_dvd__def,axiom,
% 0.32/0.59 ( dvd_dvd_nat
% 0.32/0.59 = ( ^ [B5: nat,A5: nat] :
% 0.32/0.59 ? [K3: nat] :
% 0.32/0.59 ( A5
% 0.32/0.59 = ( times_times_nat @ B5 @ K3 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_def
% 0.32/0.59 thf(fact_3961_dvd__def,axiom,
% 0.32/0.59 ( dvd_dvd_int
% 0.32/0.59 = ( ^ [B5: int,A5: int] :
% 0.32/0.59 ? [K3: int] :
% 0.32/0.59 ( A5
% 0.32/0.59 = ( times_times_int @ B5 @ K3 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_def
% 0.32/0.59 thf(fact_3962_dvdI,axiom,
% 0.32/0.59 ! [A: code_integer,B: code_integer,K: code_integer] :
% 0.32/0.59 ( ( A
% 0.32/0.59 = ( times_3573771949741848930nteger @ B @ K ) )
% 0.32/0.59 => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvdI
% 0.32/0.59 thf(fact_3963_dvdI,axiom,
% 0.32/0.59 ! [A: real,B: real,K: real] :
% 0.32/0.59 ( ( A
% 0.32/0.59 = ( times_times_real @ B @ K ) )
% 0.32/0.59 => ( dvd_dvd_real @ B @ A ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvdI
% 0.32/0.59 thf(fact_3964_dvdI,axiom,
% 0.32/0.59 ! [A: rat,B: rat,K: rat] :
% 0.32/0.59 ( ( A
% 0.32/0.59 = ( times_times_rat @ B @ K ) )
% 0.32/0.59 => ( dvd_dvd_rat @ B @ A ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvdI
% 0.32/0.59 thf(fact_3965_dvdI,axiom,
% 0.32/0.59 ! [A: nat,B: nat,K: nat] :
% 0.32/0.59 ( ( A
% 0.32/0.59 = ( times_times_nat @ B @ K ) )
% 0.32/0.59 => ( dvd_dvd_nat @ B @ A ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvdI
% 0.32/0.59 thf(fact_3966_dvdI,axiom,
% 0.32/0.59 ! [A: int,B: int,K: int] :
% 0.32/0.59 ( ( A
% 0.32/0.59 = ( times_times_int @ B @ K ) )
% 0.32/0.59 => ( dvd_dvd_int @ B @ A ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvdI
% 0.32/0.59 thf(fact_3967_dvdE,axiom,
% 0.32/0.59 ! [B: code_integer,A: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.32/0.59 => ~ ! [K2: code_integer] :
% 0.32/0.59 ( A
% 0.32/0.59 != ( times_3573771949741848930nteger @ B @ K2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvdE
% 0.32/0.59 thf(fact_3968_dvdE,axiom,
% 0.32/0.59 ! [B: real,A: real] :
% 0.32/0.59 ( ( dvd_dvd_real @ B @ A )
% 0.32/0.59 => ~ ! [K2: real] :
% 0.32/0.59 ( A
% 0.32/0.59 != ( times_times_real @ B @ K2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvdE
% 0.32/0.59 thf(fact_3969_dvdE,axiom,
% 0.32/0.59 ! [B: rat,A: rat] :
% 0.32/0.59 ( ( dvd_dvd_rat @ B @ A )
% 0.32/0.59 => ~ ! [K2: rat] :
% 0.32/0.59 ( A
% 0.32/0.59 != ( times_times_rat @ B @ K2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvdE
% 0.32/0.59 thf(fact_3970_dvdE,axiom,
% 0.32/0.59 ! [B: nat,A: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ B @ A )
% 0.32/0.59 => ~ ! [K2: nat] :
% 0.32/0.59 ( A
% 0.32/0.59 != ( times_times_nat @ B @ K2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvdE
% 0.32/0.59 thf(fact_3971_dvdE,axiom,
% 0.32/0.59 ! [B: int,A: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ B @ A )
% 0.32/0.59 => ~ ! [K2: int] :
% 0.32/0.59 ( A
% 0.32/0.59 != ( times_times_int @ B @ K2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvdE
% 0.32/0.59 thf(fact_3972_division__decomp,axiom,
% 0.32/0.59 ! [A: nat,B: nat,C: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.32/0.59 => ? [B8: nat,C5: nat] :
% 0.32/0.59 ( ( A
% 0.32/0.59 = ( times_times_nat @ B8 @ C5 ) )
% 0.32/0.59 & ( dvd_dvd_nat @ B8 @ B )
% 0.32/0.59 & ( dvd_dvd_nat @ C5 @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % division_decomp
% 0.32/0.59 thf(fact_3973_division__decomp,axiom,
% 0.32/0.59 ! [A: int,B: int,C: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
% 0.32/0.59 => ? [B8: int,C5: int] :
% 0.32/0.59 ( ( A
% 0.32/0.59 = ( times_times_int @ B8 @ C5 ) )
% 0.32/0.59 & ( dvd_dvd_int @ B8 @ B )
% 0.32/0.59 & ( dvd_dvd_int @ C5 @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % division_decomp
% 0.32/0.59 thf(fact_3974_dvd__productE,axiom,
% 0.32/0.59 ! [P4: nat,A: nat,B: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ P4 @ ( times_times_nat @ A @ B ) )
% 0.32/0.59 => ~ ! [X4: nat,Y4: nat] :
% 0.32/0.59 ( ( P4
% 0.32/0.59 = ( times_times_nat @ X4 @ Y4 ) )
% 0.32/0.59 => ( ( dvd_dvd_nat @ X4 @ A )
% 0.32/0.59 => ~ ( dvd_dvd_nat @ Y4 @ B ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_productE
% 0.32/0.59 thf(fact_3975_dvd__productE,axiom,
% 0.32/0.59 ! [P4: int,A: int,B: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ P4 @ ( times_times_int @ A @ B ) )
% 0.32/0.59 => ~ ! [X4: int,Y4: int] :
% 0.32/0.59 ( ( P4
% 0.32/0.59 = ( times_times_int @ X4 @ Y4 ) )
% 0.32/0.59 => ( ( dvd_dvd_int @ X4 @ A )
% 0.32/0.59 => ~ ( dvd_dvd_int @ Y4 @ B ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_productE
% 0.32/0.59 thf(fact_3976_dvd__add__right__iff,axiom,
% 0.32/0.59 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 0.32/0.59 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_add_right_iff
% 0.32/0.59 thf(fact_3977_dvd__add__right__iff,axiom,
% 0.32/0.59 ! [A: real,B: real,C: real] :
% 0.32/0.59 ( ( dvd_dvd_real @ A @ B )
% 0.32/0.59 => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.32/0.59 = ( dvd_dvd_real @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_add_right_iff
% 0.32/0.59 thf(fact_3978_dvd__add__right__iff,axiom,
% 0.32/0.59 ! [A: rat,B: rat,C: rat] :
% 0.32/0.59 ( ( dvd_dvd_rat @ A @ B )
% 0.32/0.59 => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.32/0.59 = ( dvd_dvd_rat @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_add_right_iff
% 0.32/0.59 thf(fact_3979_dvd__add__right__iff,axiom,
% 0.32/0.59 ! [A: nat,B: nat,C: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.59 => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 0.32/0.59 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_add_right_iff
% 0.32/0.59 thf(fact_3980_dvd__add__right__iff,axiom,
% 0.32/0.59 ! [A: int,B: int,C: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ A @ B )
% 0.32/0.59 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.32/0.59 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_add_right_iff
% 0.32/0.59 thf(fact_3981_dvd__add__left__iff,axiom,
% 0.32/0.59 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ A @ C )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 0.32/0.59 = ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_add_left_iff
% 0.32/0.59 thf(fact_3982_dvd__add__left__iff,axiom,
% 0.32/0.59 ! [A: real,C: real,B: real] :
% 0.32/0.59 ( ( dvd_dvd_real @ A @ C )
% 0.32/0.59 => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.32/0.59 = ( dvd_dvd_real @ A @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_add_left_iff
% 0.32/0.59 thf(fact_3983_dvd__add__left__iff,axiom,
% 0.32/0.59 ! [A: rat,C: rat,B: rat] :
% 0.32/0.59 ( ( dvd_dvd_rat @ A @ C )
% 0.32/0.59 => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.32/0.59 = ( dvd_dvd_rat @ A @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_add_left_iff
% 0.32/0.59 thf(fact_3984_dvd__add__left__iff,axiom,
% 0.32/0.59 ! [A: nat,C: nat,B: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ A @ C )
% 0.32/0.59 => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 0.32/0.59 = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_add_left_iff
% 0.32/0.59 thf(fact_3985_dvd__add__left__iff,axiom,
% 0.32/0.59 ! [A: int,C: int,B: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ A @ C )
% 0.32/0.59 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.32/0.59 = ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_add_left_iff
% 0.32/0.59 thf(fact_3986_dvd__add,axiom,
% 0.32/0.59 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ A @ C )
% 0.32/0.59 => ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_add
% 0.32/0.59 thf(fact_3987_dvd__add,axiom,
% 0.32/0.59 ! [A: real,B: real,C: real] :
% 0.32/0.59 ( ( dvd_dvd_real @ A @ B )
% 0.32/0.59 => ( ( dvd_dvd_real @ A @ C )
% 0.32/0.59 => ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_add
% 0.32/0.59 thf(fact_3988_dvd__add,axiom,
% 0.32/0.59 ! [A: rat,B: rat,C: rat] :
% 0.32/0.59 ( ( dvd_dvd_rat @ A @ B )
% 0.32/0.59 => ( ( dvd_dvd_rat @ A @ C )
% 0.32/0.59 => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_add
% 0.32/0.59 thf(fact_3989_dvd__add,axiom,
% 0.32/0.59 ! [A: nat,B: nat,C: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.59 => ( ( dvd_dvd_nat @ A @ C )
% 0.32/0.59 => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_add
% 0.32/0.59 thf(fact_3990_dvd__add,axiom,
% 0.32/0.59 ! [A: int,B: int,C: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ A @ B )
% 0.32/0.59 => ( ( dvd_dvd_int @ A @ C )
% 0.32/0.59 => ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_add
% 0.32/0.59 thf(fact_3991_one__dvd,axiom,
% 0.32/0.59 ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).
% 0.32/0.59
% 0.32/0.59 % one_dvd
% 0.32/0.59 thf(fact_3992_one__dvd,axiom,
% 0.32/0.59 ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).
% 0.32/0.59
% 0.32/0.59 % one_dvd
% 0.32/0.59 thf(fact_3993_one__dvd,axiom,
% 0.32/0.59 ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).
% 0.32/0.59
% 0.32/0.59 % one_dvd
% 0.32/0.59 thf(fact_3994_one__dvd,axiom,
% 0.32/0.59 ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).
% 0.32/0.59
% 0.32/0.59 % one_dvd
% 0.32/0.59 thf(fact_3995_one__dvd,axiom,
% 0.32/0.59 ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% 0.32/0.59
% 0.32/0.59 % one_dvd
% 0.32/0.59 thf(fact_3996_one__dvd,axiom,
% 0.32/0.59 ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% 0.32/0.59
% 0.32/0.59 % one_dvd
% 0.32/0.59 thf(fact_3997_unit__imp__dvd,axiom,
% 0.32/0.59 ! [B: code_integer,A: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.59 => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_imp_dvd
% 0.32/0.59 thf(fact_3998_unit__imp__dvd,axiom,
% 0.32/0.59 ! [B: nat,A: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.59 => ( dvd_dvd_nat @ B @ A ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_imp_dvd
% 0.32/0.59 thf(fact_3999_unit__imp__dvd,axiom,
% 0.32/0.59 ! [B: int,A: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.59 => ( dvd_dvd_int @ B @ A ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_imp_dvd
% 0.32/0.59 thf(fact_4000_dvd__unit__imp__unit,axiom,
% 0.32/0.59 ! [A: code_integer,B: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.59 => ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_unit_imp_unit
% 0.32/0.59 thf(fact_4001_dvd__unit__imp__unit,axiom,
% 0.32/0.59 ! [A: nat,B: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.59 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.59 => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_unit_imp_unit
% 0.32/0.59 thf(fact_4002_dvd__unit__imp__unit,axiom,
% 0.32/0.59 ! [A: int,B: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ A @ B )
% 0.32/0.59 => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.59 => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_unit_imp_unit
% 0.32/0.59 thf(fact_4003_dvd__diff__commute,axiom,
% 0.32/0.59 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ C @ B ) )
% 0.32/0.59 = ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ B @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_diff_commute
% 0.32/0.59 thf(fact_4004_dvd__diff__commute,axiom,
% 0.32/0.59 ! [A: int,C: int,B: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ A @ ( minus_minus_int @ C @ B ) )
% 0.32/0.59 = ( dvd_dvd_int @ A @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_diff_commute
% 0.32/0.59 thf(fact_4005_dvd__diff,axiom,
% 0.32/0.59 ! [X: code_integer,Y: code_integer,Z: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ X @ Y )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ X @ Z )
% 0.32/0.59 => ( dvd_dvd_Code_integer @ X @ ( minus_8373710615458151222nteger @ Y @ Z ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_diff
% 0.32/0.59 thf(fact_4006_dvd__diff,axiom,
% 0.32/0.59 ! [X: real,Y: real,Z: real] :
% 0.32/0.59 ( ( dvd_dvd_real @ X @ Y )
% 0.32/0.59 => ( ( dvd_dvd_real @ X @ Z )
% 0.32/0.59 => ( dvd_dvd_real @ X @ ( minus_minus_real @ Y @ Z ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_diff
% 0.32/0.59 thf(fact_4007_dvd__diff,axiom,
% 0.32/0.59 ! [X: rat,Y: rat,Z: rat] :
% 0.32/0.59 ( ( dvd_dvd_rat @ X @ Y )
% 0.32/0.59 => ( ( dvd_dvd_rat @ X @ Z )
% 0.32/0.59 => ( dvd_dvd_rat @ X @ ( minus_minus_rat @ Y @ Z ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_diff
% 0.32/0.59 thf(fact_4008_dvd__diff,axiom,
% 0.32/0.59 ! [X: int,Y: int,Z: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ X @ Y )
% 0.32/0.59 => ( ( dvd_dvd_int @ X @ Z )
% 0.32/0.59 => ( dvd_dvd_int @ X @ ( minus_minus_int @ Y @ Z ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_diff
% 0.32/0.59 thf(fact_4009_div__div__div__same,axiom,
% 0.32/0.59 ! [D: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ D @ B )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.32/0.59 => ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ D ) @ ( divide6298287555418463151nteger @ B @ D ) )
% 0.32/0.59 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_div_div_same
% 0.32/0.59 thf(fact_4010_div__div__div__same,axiom,
% 0.32/0.59 ! [D: nat,B: nat,A: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ D @ B )
% 0.32/0.59 => ( ( dvd_dvd_nat @ B @ A )
% 0.32/0.59 => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B @ D ) )
% 0.32/0.59 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_div_div_same
% 0.32/0.59 thf(fact_4011_div__div__div__same,axiom,
% 0.32/0.59 ! [D: int,B: int,A: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ D @ B )
% 0.32/0.59 => ( ( dvd_dvd_int @ B @ A )
% 0.32/0.59 => ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B @ D ) )
% 0.32/0.59 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_div_div_same
% 0.32/0.59 thf(fact_4012_dvd__div__eq__cancel,axiom,
% 0.32/0.59 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.59 ( ( ( divide6298287555418463151nteger @ A @ C )
% 0.32/0.59 = ( divide6298287555418463151nteger @ B @ C ) )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ C @ A )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.59 => ( A = B ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_eq_cancel
% 0.32/0.59 thf(fact_4013_dvd__div__eq__cancel,axiom,
% 0.32/0.59 ! [A: complex,C: complex,B: complex] :
% 0.32/0.59 ( ( ( divide1717551699836669952omplex @ A @ C )
% 0.32/0.59 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.32/0.59 => ( ( dvd_dvd_complex @ C @ A )
% 0.32/0.59 => ( ( dvd_dvd_complex @ C @ B )
% 0.32/0.59 => ( A = B ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_eq_cancel
% 0.32/0.59 thf(fact_4014_dvd__div__eq__cancel,axiom,
% 0.32/0.59 ! [A: real,C: real,B: real] :
% 0.32/0.59 ( ( ( divide_divide_real @ A @ C )
% 0.32/0.59 = ( divide_divide_real @ B @ C ) )
% 0.32/0.59 => ( ( dvd_dvd_real @ C @ A )
% 0.32/0.59 => ( ( dvd_dvd_real @ C @ B )
% 0.32/0.59 => ( A = B ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_eq_cancel
% 0.32/0.59 thf(fact_4015_dvd__div__eq__cancel,axiom,
% 0.32/0.59 ! [A: rat,C: rat,B: rat] :
% 0.32/0.59 ( ( ( divide_divide_rat @ A @ C )
% 0.32/0.59 = ( divide_divide_rat @ B @ C ) )
% 0.32/0.59 => ( ( dvd_dvd_rat @ C @ A )
% 0.32/0.59 => ( ( dvd_dvd_rat @ C @ B )
% 0.32/0.59 => ( A = B ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_eq_cancel
% 0.32/0.59 thf(fact_4016_dvd__div__eq__cancel,axiom,
% 0.32/0.59 ! [A: nat,C: nat,B: nat] :
% 0.32/0.59 ( ( ( divide_divide_nat @ A @ C )
% 0.32/0.59 = ( divide_divide_nat @ B @ C ) )
% 0.32/0.59 => ( ( dvd_dvd_nat @ C @ A )
% 0.32/0.59 => ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.59 => ( A = B ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_eq_cancel
% 0.32/0.59 thf(fact_4017_dvd__div__eq__cancel,axiom,
% 0.32/0.59 ! [A: int,C: int,B: int] :
% 0.32/0.59 ( ( ( divide_divide_int @ A @ C )
% 0.32/0.59 = ( divide_divide_int @ B @ C ) )
% 0.32/0.59 => ( ( dvd_dvd_int @ C @ A )
% 0.32/0.59 => ( ( dvd_dvd_int @ C @ B )
% 0.32/0.59 => ( A = B ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_eq_cancel
% 0.32/0.59 thf(fact_4018_dvd__div__eq__iff,axiom,
% 0.32/0.59 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ C @ A )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.59 => ( ( ( divide6298287555418463151nteger @ A @ C )
% 0.32/0.59 = ( divide6298287555418463151nteger @ B @ C ) )
% 0.32/0.59 = ( A = B ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_eq_iff
% 0.32/0.59 thf(fact_4019_dvd__div__eq__iff,axiom,
% 0.32/0.59 ! [C: complex,A: complex,B: complex] :
% 0.32/0.59 ( ( dvd_dvd_complex @ C @ A )
% 0.32/0.59 => ( ( dvd_dvd_complex @ C @ B )
% 0.32/0.59 => ( ( ( divide1717551699836669952omplex @ A @ C )
% 0.32/0.59 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.32/0.59 = ( A = B ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_eq_iff
% 0.32/0.59 thf(fact_4020_dvd__div__eq__iff,axiom,
% 0.32/0.59 ! [C: real,A: real,B: real] :
% 0.32/0.59 ( ( dvd_dvd_real @ C @ A )
% 0.32/0.59 => ( ( dvd_dvd_real @ C @ B )
% 0.32/0.59 => ( ( ( divide_divide_real @ A @ C )
% 0.32/0.59 = ( divide_divide_real @ B @ C ) )
% 0.32/0.59 = ( A = B ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_eq_iff
% 0.32/0.59 thf(fact_4021_dvd__div__eq__iff,axiom,
% 0.32/0.59 ! [C: rat,A: rat,B: rat] :
% 0.32/0.59 ( ( dvd_dvd_rat @ C @ A )
% 0.32/0.59 => ( ( dvd_dvd_rat @ C @ B )
% 0.32/0.59 => ( ( ( divide_divide_rat @ A @ C )
% 0.32/0.59 = ( divide_divide_rat @ B @ C ) )
% 0.32/0.59 = ( A = B ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_eq_iff
% 0.32/0.59 thf(fact_4022_dvd__div__eq__iff,axiom,
% 0.32/0.59 ! [C: nat,A: nat,B: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ C @ A )
% 0.32/0.59 => ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.59 => ( ( ( divide_divide_nat @ A @ C )
% 0.32/0.59 = ( divide_divide_nat @ B @ C ) )
% 0.32/0.59 = ( A = B ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_eq_iff
% 0.32/0.59 thf(fact_4023_dvd__div__eq__iff,axiom,
% 0.32/0.59 ! [C: int,A: int,B: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ C @ A )
% 0.32/0.59 => ( ( dvd_dvd_int @ C @ B )
% 0.32/0.59 => ( ( ( divide_divide_int @ A @ C )
% 0.32/0.59 = ( divide_divide_int @ B @ C ) )
% 0.32/0.59 = ( A = B ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_eq_iff
% 0.32/0.59 thf(fact_4024_dvd__power__same,axiom,
% 0.32/0.59 ! [X: code_integer,Y: code_integer,N2: nat] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ X @ Y )
% 0.32/0.59 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N2 ) @ ( power_8256067586552552935nteger @ Y @ N2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_power_same
% 0.32/0.59 thf(fact_4025_dvd__power__same,axiom,
% 0.32/0.59 ! [X: nat,Y: nat,N2: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ X @ Y )
% 0.32/0.59 => ( dvd_dvd_nat @ ( power_power_nat @ X @ N2 ) @ ( power_power_nat @ Y @ N2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_power_same
% 0.32/0.59 thf(fact_4026_dvd__power__same,axiom,
% 0.32/0.59 ! [X: real,Y: real,N2: nat] :
% 0.32/0.59 ( ( dvd_dvd_real @ X @ Y )
% 0.32/0.59 => ( dvd_dvd_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ N2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_power_same
% 0.32/0.59 thf(fact_4027_dvd__power__same,axiom,
% 0.32/0.59 ! [X: int,Y: int,N2: nat] :
% 0.32/0.59 ( ( dvd_dvd_int @ X @ Y )
% 0.32/0.59 => ( dvd_dvd_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ N2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_power_same
% 0.32/0.59 thf(fact_4028_dvd__power__same,axiom,
% 0.32/0.59 ! [X: complex,Y: complex,N2: nat] :
% 0.32/0.59 ( ( dvd_dvd_complex @ X @ Y )
% 0.32/0.59 => ( dvd_dvd_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ Y @ N2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_power_same
% 0.32/0.59 thf(fact_4029_mod__mod__cancel,axiom,
% 0.32/0.59 ! [C: nat,B: nat,A: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.59 => ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ C )
% 0.32/0.59 = ( modulo_modulo_nat @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % mod_mod_cancel
% 0.32/0.59 thf(fact_4030_mod__mod__cancel,axiom,
% 0.32/0.59 ! [C: int,B: int,A: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ C @ B )
% 0.32/0.59 => ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ C )
% 0.32/0.59 = ( modulo_modulo_int @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % mod_mod_cancel
% 0.32/0.59 thf(fact_4031_mod__mod__cancel,axiom,
% 0.32/0.59 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.59 => ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C )
% 0.32/0.59 = ( modulo364778990260209775nteger @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % mod_mod_cancel
% 0.32/0.59 thf(fact_4032_dvd__mod,axiom,
% 0.32/0.59 ! [K: nat,M: nat,N2: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ K @ M )
% 0.32/0.59 => ( ( dvd_dvd_nat @ K @ N2 )
% 0.32/0.59 => ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_mod
% 0.32/0.59 thf(fact_4033_dvd__mod,axiom,
% 0.32/0.59 ! [K: int,M: int,N2: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ K @ M )
% 0.32/0.59 => ( ( dvd_dvd_int @ K @ N2 )
% 0.32/0.59 => ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N2 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_mod
% 0.32/0.59 thf(fact_4034_dvd__mod,axiom,
% 0.32/0.59 ! [K: code_integer,M: code_integer,N2: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ K @ M )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ K @ N2 )
% 0.32/0.59 => ( dvd_dvd_Code_integer @ K @ ( modulo364778990260209775nteger @ M @ N2 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_mod
% 0.32/0.59 thf(fact_4035_dvd__diff__nat,axiom,
% 0.32/0.59 ! [K: nat,M: nat,N2: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ K @ M )
% 0.32/0.59 => ( ( dvd_dvd_nat @ K @ N2 )
% 0.32/0.59 => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_diff_nat
% 0.32/0.59 thf(fact_4036_zdvd__zdiffD,axiom,
% 0.32/0.59 ! [K: int,M: int,N2: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N2 ) )
% 0.32/0.59 => ( ( dvd_dvd_int @ K @ N2 )
% 0.32/0.59 => ( dvd_dvd_int @ K @ M ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % zdvd_zdiffD
% 0.32/0.59 thf(fact_4037_subset__divisors__dvd,axiom,
% 0.32/0.59 ! [A: complex,B: complex] :
% 0.32/0.59 ( ( ord_le211207098394363844omplex
% 0.32/0.59 @ ( collect_complex
% 0.32/0.59 @ ^ [C4: complex] : ( dvd_dvd_complex @ C4 @ A ) )
% 0.32/0.59 @ ( collect_complex
% 0.32/0.59 @ ^ [C4: complex] : ( dvd_dvd_complex @ C4 @ B ) ) )
% 0.32/0.59 = ( dvd_dvd_complex @ A @ B ) ) ).
% 0.32/0.59
% 0.32/0.59 % subset_divisors_dvd
% 0.32/0.59 thf(fact_4038_subset__divisors__dvd,axiom,
% 0.32/0.59 ! [A: int,B: int] :
% 0.32/0.59 ( ( ord_less_eq_set_int
% 0.32/0.59 @ ( collect_int
% 0.32/0.59 @ ^ [C4: int] : ( dvd_dvd_int @ C4 @ A ) )
% 0.32/0.59 @ ( collect_int
% 0.32/0.59 @ ^ [C4: int] : ( dvd_dvd_int @ C4 @ B ) ) )
% 0.32/0.59 = ( dvd_dvd_int @ A @ B ) ) ).
% 0.32/0.59
% 0.32/0.59 % subset_divisors_dvd
% 0.32/0.59 thf(fact_4039_subset__divisors__dvd,axiom,
% 0.32/0.59 ! [A: code_integer,B: code_integer] :
% 0.32/0.59 ( ( ord_le7084787975880047091nteger
% 0.32/0.59 @ ( collect_Code_integer
% 0.32/0.59 @ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ A ) )
% 0.32/0.59 @ ( collect_Code_integer
% 0.32/0.59 @ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ B ) ) )
% 0.32/0.59 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.32/0.59
% 0.32/0.59 % subset_divisors_dvd
% 0.32/0.59 thf(fact_4040_subset__divisors__dvd,axiom,
% 0.32/0.59 ! [A: nat,B: nat] :
% 0.32/0.59 ( ( ord_less_eq_set_nat
% 0.32/0.59 @ ( collect_nat
% 0.32/0.59 @ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ A ) )
% 0.32/0.59 @ ( collect_nat
% 0.32/0.59 @ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ B ) ) )
% 0.32/0.59 = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.32/0.59
% 0.32/0.59 % subset_divisors_dvd
% 0.32/0.59 thf(fact_4041_concat__bit__assoc,axiom,
% 0.32/0.59 ! [N2: nat,K: int,M: nat,L2: int,R: int] :
% 0.32/0.59 ( ( bit_concat_bit @ N2 @ K @ ( bit_concat_bit @ M @ L2 @ R ) )
% 0.32/0.59 = ( bit_concat_bit @ ( plus_plus_nat @ M @ N2 ) @ ( bit_concat_bit @ N2 @ K @ L2 ) @ R ) ) ).
% 0.32/0.59
% 0.32/0.59 % concat_bit_assoc
% 0.32/0.59 thf(fact_4042_strict__subset__divisors__dvd,axiom,
% 0.32/0.59 ! [A: complex,B: complex] :
% 0.32/0.59 ( ( ord_less_set_complex
% 0.32/0.59 @ ( collect_complex
% 0.32/0.59 @ ^ [C4: complex] : ( dvd_dvd_complex @ C4 @ A ) )
% 0.32/0.59 @ ( collect_complex
% 0.32/0.59 @ ^ [C4: complex] : ( dvd_dvd_complex @ C4 @ B ) ) )
% 0.32/0.59 = ( ( dvd_dvd_complex @ A @ B )
% 0.32/0.59 & ~ ( dvd_dvd_complex @ B @ A ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % strict_subset_divisors_dvd
% 0.32/0.59 thf(fact_4043_strict__subset__divisors__dvd,axiom,
% 0.32/0.59 ! [A: nat,B: nat] :
% 0.32/0.59 ( ( ord_less_set_nat
% 0.32/0.59 @ ( collect_nat
% 0.32/0.59 @ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ A ) )
% 0.32/0.59 @ ( collect_nat
% 0.32/0.59 @ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ B ) ) )
% 0.32/0.59 = ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.59 & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % strict_subset_divisors_dvd
% 0.32/0.59 thf(fact_4044_strict__subset__divisors__dvd,axiom,
% 0.32/0.59 ! [A: int,B: int] :
% 0.32/0.59 ( ( ord_less_set_int
% 0.32/0.59 @ ( collect_int
% 0.32/0.59 @ ^ [C4: int] : ( dvd_dvd_int @ C4 @ A ) )
% 0.32/0.59 @ ( collect_int
% 0.32/0.59 @ ^ [C4: int] : ( dvd_dvd_int @ C4 @ B ) ) )
% 0.32/0.59 = ( ( dvd_dvd_int @ A @ B )
% 0.32/0.59 & ~ ( dvd_dvd_int @ B @ A ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % strict_subset_divisors_dvd
% 0.32/0.59 thf(fact_4045_strict__subset__divisors__dvd,axiom,
% 0.32/0.59 ! [A: code_integer,B: code_integer] :
% 0.32/0.59 ( ( ord_le1307284697595431911nteger
% 0.32/0.59 @ ( collect_Code_integer
% 0.32/0.59 @ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ A ) )
% 0.32/0.59 @ ( collect_Code_integer
% 0.32/0.59 @ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ B ) ) )
% 0.32/0.59 = ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.59 & ~ ( dvd_dvd_Code_integer @ B @ A ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % strict_subset_divisors_dvd
% 0.32/0.59 thf(fact_4046_not__is__unit__0,axiom,
% 0.32/0.59 ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).
% 0.32/0.59
% 0.32/0.59 % not_is_unit_0
% 0.32/0.59 thf(fact_4047_not__is__unit__0,axiom,
% 0.32/0.59 ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% 0.32/0.59
% 0.32/0.59 % not_is_unit_0
% 0.32/0.59 thf(fact_4048_not__is__unit__0,axiom,
% 0.32/0.59 ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% 0.32/0.59
% 0.32/0.59 % not_is_unit_0
% 0.32/0.59 thf(fact_4049_minf_I10_J,axiom,
% 0.32/0.59 ! [D: code_integer,S3: code_integer] :
% 0.32/0.59 ? [Z5: code_integer] :
% 0.32/0.59 ! [X5: code_integer] :
% 0.32/0.59 ( ( ord_le6747313008572928689nteger @ X5 @ Z5 )
% 0.32/0.59 => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S3 ) ) )
% 0.32/0.59 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S3 ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % minf(10)
% 0.32/0.59 thf(fact_4050_minf_I10_J,axiom,
% 0.32/0.59 ! [D: real,S3: real] :
% 0.32/0.59 ? [Z5: real] :
% 0.32/0.59 ! [X5: real] :
% 0.32/0.59 ( ( ord_less_real @ X5 @ Z5 )
% 0.32/0.59 => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S3 ) ) )
% 0.32/0.59 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S3 ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % minf(10)
% 0.32/0.59 thf(fact_4051_minf_I10_J,axiom,
% 0.32/0.59 ! [D: rat,S3: rat] :
% 0.32/0.59 ? [Z5: rat] :
% 0.32/0.59 ! [X5: rat] :
% 0.32/0.59 ( ( ord_less_rat @ X5 @ Z5 )
% 0.32/0.59 => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S3 ) ) )
% 0.32/0.59 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S3 ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % minf(10)
% 0.32/0.59 thf(fact_4052_minf_I10_J,axiom,
% 0.32/0.59 ! [D: nat,S3: nat] :
% 0.32/0.59 ? [Z5: nat] :
% 0.32/0.59 ! [X5: nat] :
% 0.32/0.59 ( ( ord_less_nat @ X5 @ Z5 )
% 0.32/0.59 => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S3 ) ) )
% 0.32/0.59 = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S3 ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % minf(10)
% 0.32/0.59 thf(fact_4053_minf_I10_J,axiom,
% 0.32/0.59 ! [D: int,S3: int] :
% 0.32/0.59 ? [Z5: int] :
% 0.32/0.59 ! [X5: int] :
% 0.32/0.59 ( ( ord_less_int @ X5 @ Z5 )
% 0.32/0.59 => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S3 ) ) )
% 0.32/0.59 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S3 ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % minf(10)
% 0.32/0.59 thf(fact_4054_minf_I9_J,axiom,
% 0.32/0.59 ! [D: code_integer,S3: code_integer] :
% 0.32/0.59 ? [Z5: code_integer] :
% 0.32/0.59 ! [X5: code_integer] :
% 0.32/0.59 ( ( ord_le6747313008572928689nteger @ X5 @ Z5 )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S3 ) )
% 0.32/0.59 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S3 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % minf(9)
% 0.32/0.59 thf(fact_4055_minf_I9_J,axiom,
% 0.32/0.59 ! [D: real,S3: real] :
% 0.32/0.59 ? [Z5: real] :
% 0.32/0.59 ! [X5: real] :
% 0.32/0.59 ( ( ord_less_real @ X5 @ Z5 )
% 0.32/0.59 => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S3 ) )
% 0.32/0.59 = ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S3 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % minf(9)
% 0.32/0.59 thf(fact_4056_minf_I9_J,axiom,
% 0.32/0.59 ! [D: rat,S3: rat] :
% 0.32/0.59 ? [Z5: rat] :
% 0.32/0.59 ! [X5: rat] :
% 0.32/0.59 ( ( ord_less_rat @ X5 @ Z5 )
% 0.32/0.59 => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S3 ) )
% 0.32/0.59 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S3 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % minf(9)
% 0.32/0.59 thf(fact_4057_minf_I9_J,axiom,
% 0.32/0.59 ! [D: nat,S3: nat] :
% 0.32/0.59 ? [Z5: nat] :
% 0.32/0.59 ! [X5: nat] :
% 0.32/0.59 ( ( ord_less_nat @ X5 @ Z5 )
% 0.32/0.59 => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S3 ) )
% 0.32/0.59 = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S3 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % minf(9)
% 0.32/0.59 thf(fact_4058_minf_I9_J,axiom,
% 0.32/0.59 ! [D: int,S3: int] :
% 0.32/0.59 ? [Z5: int] :
% 0.32/0.59 ! [X5: int] :
% 0.32/0.59 ( ( ord_less_int @ X5 @ Z5 )
% 0.32/0.59 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S3 ) )
% 0.32/0.59 = ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S3 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % minf(9)
% 0.32/0.59 thf(fact_4059_pinf_I10_J,axiom,
% 0.32/0.59 ! [D: code_integer,S3: code_integer] :
% 0.32/0.59 ? [Z5: code_integer] :
% 0.32/0.59 ! [X5: code_integer] :
% 0.32/0.59 ( ( ord_le6747313008572928689nteger @ Z5 @ X5 )
% 0.32/0.59 => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S3 ) ) )
% 0.32/0.59 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S3 ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % pinf(10)
% 0.32/0.59 thf(fact_4060_pinf_I10_J,axiom,
% 0.32/0.59 ! [D: real,S3: real] :
% 0.32/0.59 ? [Z5: real] :
% 0.32/0.59 ! [X5: real] :
% 0.32/0.59 ( ( ord_less_real @ Z5 @ X5 )
% 0.32/0.59 => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S3 ) ) )
% 0.32/0.59 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S3 ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % pinf(10)
% 0.32/0.59 thf(fact_4061_pinf_I10_J,axiom,
% 0.32/0.59 ! [D: rat,S3: rat] :
% 0.32/0.59 ? [Z5: rat] :
% 0.32/0.59 ! [X5: rat] :
% 0.32/0.59 ( ( ord_less_rat @ Z5 @ X5 )
% 0.32/0.59 => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S3 ) ) )
% 0.32/0.59 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S3 ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % pinf(10)
% 0.32/0.59 thf(fact_4062_pinf_I10_J,axiom,
% 0.32/0.59 ! [D: nat,S3: nat] :
% 0.32/0.59 ? [Z5: nat] :
% 0.32/0.59 ! [X5: nat] :
% 0.32/0.59 ( ( ord_less_nat @ Z5 @ X5 )
% 0.32/0.59 => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S3 ) ) )
% 0.32/0.59 = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S3 ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % pinf(10)
% 0.32/0.59 thf(fact_4063_pinf_I10_J,axiom,
% 0.32/0.59 ! [D: int,S3: int] :
% 0.32/0.59 ? [Z5: int] :
% 0.32/0.59 ! [X5: int] :
% 0.32/0.59 ( ( ord_less_int @ Z5 @ X5 )
% 0.32/0.59 => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S3 ) ) )
% 0.32/0.59 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S3 ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % pinf(10)
% 0.32/0.59 thf(fact_4064_pinf_I9_J,axiom,
% 0.32/0.59 ! [D: code_integer,S3: code_integer] :
% 0.32/0.59 ? [Z5: code_integer] :
% 0.32/0.59 ! [X5: code_integer] :
% 0.32/0.59 ( ( ord_le6747313008572928689nteger @ Z5 @ X5 )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S3 ) )
% 0.32/0.59 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S3 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % pinf(9)
% 0.32/0.59 thf(fact_4065_pinf_I9_J,axiom,
% 0.32/0.59 ! [D: real,S3: real] :
% 0.32/0.59 ? [Z5: real] :
% 0.32/0.59 ! [X5: real] :
% 0.32/0.59 ( ( ord_less_real @ Z5 @ X5 )
% 0.32/0.59 => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S3 ) )
% 0.32/0.59 = ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S3 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % pinf(9)
% 0.32/0.59 thf(fact_4066_pinf_I9_J,axiom,
% 0.32/0.59 ! [D: rat,S3: rat] :
% 0.32/0.59 ? [Z5: rat] :
% 0.32/0.59 ! [X5: rat] :
% 0.32/0.59 ( ( ord_less_rat @ Z5 @ X5 )
% 0.32/0.59 => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S3 ) )
% 0.32/0.59 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S3 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % pinf(9)
% 0.32/0.59 thf(fact_4067_pinf_I9_J,axiom,
% 0.32/0.59 ! [D: nat,S3: nat] :
% 0.32/0.59 ? [Z5: nat] :
% 0.32/0.59 ! [X5: nat] :
% 0.32/0.59 ( ( ord_less_nat @ Z5 @ X5 )
% 0.32/0.59 => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S3 ) )
% 0.32/0.59 = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S3 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % pinf(9)
% 0.32/0.59 thf(fact_4068_pinf_I9_J,axiom,
% 0.32/0.59 ! [D: int,S3: int] :
% 0.32/0.59 ? [Z5: int] :
% 0.32/0.59 ! [X5: int] :
% 0.32/0.59 ( ( ord_less_int @ Z5 @ X5 )
% 0.32/0.59 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S3 ) )
% 0.32/0.59 = ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S3 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % pinf(9)
% 0.32/0.59 thf(fact_4069_dvd__div__eq__0__iff,axiom,
% 0.32/0.59 ! [B: code_integer,A: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.32/0.59 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 0.32/0.59 = zero_z3403309356797280102nteger )
% 0.32/0.59 = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_eq_0_iff
% 0.32/0.59 thf(fact_4070_dvd__div__eq__0__iff,axiom,
% 0.32/0.59 ! [B: complex,A: complex] :
% 0.32/0.59 ( ( dvd_dvd_complex @ B @ A )
% 0.32/0.59 => ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.32/0.59 = zero_zero_complex )
% 0.32/0.59 = ( A = zero_zero_complex ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_eq_0_iff
% 0.32/0.59 thf(fact_4071_dvd__div__eq__0__iff,axiom,
% 0.32/0.59 ! [B: real,A: real] :
% 0.32/0.59 ( ( dvd_dvd_real @ B @ A )
% 0.32/0.59 => ( ( ( divide_divide_real @ A @ B )
% 0.32/0.59 = zero_zero_real )
% 0.32/0.59 = ( A = zero_zero_real ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_eq_0_iff
% 0.32/0.59 thf(fact_4072_dvd__div__eq__0__iff,axiom,
% 0.32/0.59 ! [B: rat,A: rat] :
% 0.32/0.59 ( ( dvd_dvd_rat @ B @ A )
% 0.32/0.59 => ( ( ( divide_divide_rat @ A @ B )
% 0.32/0.59 = zero_zero_rat )
% 0.32/0.59 = ( A = zero_zero_rat ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_eq_0_iff
% 0.32/0.59 thf(fact_4073_dvd__div__eq__0__iff,axiom,
% 0.32/0.59 ! [B: nat,A: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ B @ A )
% 0.32/0.59 => ( ( ( divide_divide_nat @ A @ B )
% 0.32/0.59 = zero_zero_nat )
% 0.32/0.59 = ( A = zero_zero_nat ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_eq_0_iff
% 0.32/0.59 thf(fact_4074_dvd__div__eq__0__iff,axiom,
% 0.32/0.59 ! [B: int,A: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ B @ A )
% 0.32/0.59 => ( ( ( divide_divide_int @ A @ B )
% 0.32/0.59 = zero_zero_int )
% 0.32/0.59 = ( A = zero_zero_int ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_eq_0_iff
% 0.32/0.59 thf(fact_4075_is__unit__mult__iff,axiom,
% 0.32/0.59 ! [A: code_integer,B: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer )
% 0.32/0.59 = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.59 & ( dvd_dvd_Code_integer @ B @ one_one_Code_integer ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % is_unit_mult_iff
% 0.32/0.59 thf(fact_4076_is__unit__mult__iff,axiom,
% 0.32/0.59 ! [A: nat,B: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
% 0.32/0.59 = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.59 & ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % is_unit_mult_iff
% 0.32/0.59 thf(fact_4077_is__unit__mult__iff,axiom,
% 0.32/0.59 ! [A: int,B: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
% 0.32/0.59 = ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.59 & ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % is_unit_mult_iff
% 0.32/0.59 thf(fact_4078_dvd__mult__unit__iff,axiom,
% 0.32/0.59 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.32/0.59 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_mult_unit_iff
% 0.32/0.59 thf(fact_4079_dvd__mult__unit__iff,axiom,
% 0.32/0.59 ! [B: nat,A: nat,C: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.59 => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
% 0.32/0.59 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_mult_unit_iff
% 0.32/0.59 thf(fact_4080_dvd__mult__unit__iff,axiom,
% 0.32/0.59 ! [B: int,A: int,C: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.59 => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
% 0.32/0.59 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_mult_unit_iff
% 0.32/0.59 thf(fact_4081_mult__unit__dvd__iff,axiom,
% 0.32/0.59 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.32/0.59 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % mult_unit_dvd_iff
% 0.32/0.59 thf(fact_4082_mult__unit__dvd__iff,axiom,
% 0.32/0.59 ! [B: nat,A: nat,C: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.59 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.32/0.59 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % mult_unit_dvd_iff
% 0.32/0.59 thf(fact_4083_mult__unit__dvd__iff,axiom,
% 0.32/0.59 ! [B: int,A: int,C: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.59 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 0.32/0.59 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % mult_unit_dvd_iff
% 0.32/0.59 thf(fact_4084_dvd__mult__unit__iff_H,axiom,
% 0.32/0.59 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.32/0.59 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_mult_unit_iff'
% 0.32/0.59 thf(fact_4085_dvd__mult__unit__iff_H,axiom,
% 0.32/0.59 ! [B: nat,A: nat,C: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.59 => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.32/0.59 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_mult_unit_iff'
% 0.32/0.59 thf(fact_4086_dvd__mult__unit__iff_H,axiom,
% 0.32/0.59 ! [B: int,A: int,C: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.59 => ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
% 0.32/0.59 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_mult_unit_iff'
% 0.32/0.59 thf(fact_4087_mult__unit__dvd__iff_H,axiom,
% 0.32/0.59 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.32/0.59 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % mult_unit_dvd_iff'
% 0.32/0.59 thf(fact_4088_mult__unit__dvd__iff_H,axiom,
% 0.32/0.59 ! [A: nat,B: nat,C: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.59 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.32/0.59 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % mult_unit_dvd_iff'
% 0.32/0.59 thf(fact_4089_mult__unit__dvd__iff_H,axiom,
% 0.32/0.59 ! [A: int,B: int,C: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.59 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 0.32/0.59 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % mult_unit_dvd_iff'
% 0.32/0.59 thf(fact_4090_unit__mult__left__cancel,axiom,
% 0.32/0.59 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.59 => ( ( ( times_3573771949741848930nteger @ A @ B )
% 0.32/0.59 = ( times_3573771949741848930nteger @ A @ C ) )
% 0.32/0.59 = ( B = C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_mult_left_cancel
% 0.32/0.59 thf(fact_4091_unit__mult__left__cancel,axiom,
% 0.32/0.59 ! [A: nat,B: nat,C: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.59 => ( ( ( times_times_nat @ A @ B )
% 0.32/0.59 = ( times_times_nat @ A @ C ) )
% 0.32/0.59 = ( B = C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_mult_left_cancel
% 0.32/0.59 thf(fact_4092_unit__mult__left__cancel,axiom,
% 0.32/0.59 ! [A: int,B: int,C: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.59 => ( ( ( times_times_int @ A @ B )
% 0.32/0.59 = ( times_times_int @ A @ C ) )
% 0.32/0.59 = ( B = C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_mult_left_cancel
% 0.32/0.59 thf(fact_4093_unit__mult__right__cancel,axiom,
% 0.32/0.59 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.59 => ( ( ( times_3573771949741848930nteger @ B @ A )
% 0.32/0.59 = ( times_3573771949741848930nteger @ C @ A ) )
% 0.32/0.59 = ( B = C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_mult_right_cancel
% 0.32/0.59 thf(fact_4094_unit__mult__right__cancel,axiom,
% 0.32/0.59 ! [A: nat,B: nat,C: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.59 => ( ( ( times_times_nat @ B @ A )
% 0.32/0.59 = ( times_times_nat @ C @ A ) )
% 0.32/0.59 = ( B = C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_mult_right_cancel
% 0.32/0.59 thf(fact_4095_unit__mult__right__cancel,axiom,
% 0.32/0.59 ! [A: int,B: int,C: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.59 => ( ( ( times_times_int @ B @ A )
% 0.32/0.59 = ( times_times_int @ C @ A ) )
% 0.32/0.59 = ( B = C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_mult_right_cancel
% 0.32/0.59 thf(fact_4096_div__mult__div__if__dvd,axiom,
% 0.32/0.59 ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ D @ C )
% 0.32/0.59 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ ( divide6298287555418463151nteger @ C @ D ) )
% 0.32/0.59 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_mult_div_if_dvd
% 0.32/0.59 thf(fact_4097_div__mult__div__if__dvd,axiom,
% 0.32/0.59 ! [B: nat,A: nat,D: nat,C: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ B @ A )
% 0.32/0.59 => ( ( dvd_dvd_nat @ D @ C )
% 0.32/0.59 => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C @ D ) )
% 0.32/0.59 = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_mult_div_if_dvd
% 0.32/0.59 thf(fact_4098_div__mult__div__if__dvd,axiom,
% 0.32/0.59 ! [B: int,A: int,D: int,C: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ B @ A )
% 0.32/0.59 => ( ( dvd_dvd_int @ D @ C )
% 0.32/0.59 => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C @ D ) )
% 0.32/0.59 = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_mult_div_if_dvd
% 0.32/0.59 thf(fact_4099_dvd__mult__imp__div,axiom,
% 0.32/0.59 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B )
% 0.32/0.59 => ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_mult_imp_div
% 0.32/0.59 thf(fact_4100_dvd__mult__imp__div,axiom,
% 0.32/0.59 ! [A: nat,C: nat,B: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B )
% 0.32/0.59 => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_mult_imp_div
% 0.32/0.59 thf(fact_4101_dvd__mult__imp__div,axiom,
% 0.32/0.59 ! [A: int,C: int,B: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B )
% 0.32/0.59 => ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_mult_imp_div
% 0.32/0.59 thf(fact_4102_dvd__div__mult2__eq,axiom,
% 0.32/0.59 ! [B: code_integer,C: code_integer,A: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ C ) @ A )
% 0.32/0.59 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.32/0.59 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_mult2_eq
% 0.32/0.59 thf(fact_4103_dvd__div__mult2__eq,axiom,
% 0.32/0.59 ! [B: nat,C: nat,A: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A )
% 0.32/0.59 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.32/0.59 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_mult2_eq
% 0.32/0.59 thf(fact_4104_dvd__div__mult2__eq,axiom,
% 0.32/0.59 ! [B: int,C: int,A: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ ( times_times_int @ B @ C ) @ A )
% 0.32/0.59 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.32/0.59 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_mult2_eq
% 0.32/0.59 thf(fact_4105_div__div__eq__right,axiom,
% 0.32/0.59 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.32/0.59 => ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.32/0.59 = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_div_eq_right
% 0.32/0.59 thf(fact_4106_div__div__eq__right,axiom,
% 0.32/0.59 ! [C: nat,B: nat,A: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.59 => ( ( dvd_dvd_nat @ B @ A )
% 0.32/0.59 => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.32/0.59 = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_div_eq_right
% 0.32/0.59 thf(fact_4107_div__div__eq__right,axiom,
% 0.32/0.59 ! [C: int,B: int,A: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ C @ B )
% 0.32/0.59 => ( ( dvd_dvd_int @ B @ A )
% 0.32/0.59 => ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.32/0.59 = ( times_times_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_div_eq_right
% 0.32/0.59 thf(fact_4108_div__mult__swap,axiom,
% 0.32/0.59 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.59 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.32/0.59 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_mult_swap
% 0.32/0.59 thf(fact_4109_div__mult__swap,axiom,
% 0.32/0.59 ! [C: nat,B: nat,A: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.59 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.32/0.59 = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_mult_swap
% 0.32/0.59 thf(fact_4110_div__mult__swap,axiom,
% 0.32/0.59 ! [C: int,B: int,A: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ C @ B )
% 0.32/0.59 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.32/0.59 = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_mult_swap
% 0.32/0.59 thf(fact_4111_dvd__div__mult,axiom,
% 0.32/0.59 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.59 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ C ) @ A )
% 0.32/0.59 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B @ A ) @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_mult
% 0.32/0.59 thf(fact_4112_dvd__div__mult,axiom,
% 0.32/0.59 ! [C: nat,B: nat,A: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.59 => ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A )
% 0.32/0.59 = ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_mult
% 0.32/0.59 thf(fact_4113_dvd__div__mult,axiom,
% 0.32/0.59 ! [C: int,B: int,A: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ C @ B )
% 0.32/0.59 => ( ( times_times_int @ ( divide_divide_int @ B @ C ) @ A )
% 0.32/0.59 = ( divide_divide_int @ ( times_times_int @ B @ A ) @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_mult
% 0.32/0.59 thf(fact_4114_div__plus__div__distrib__dvd__right,axiom,
% 0.32/0.59 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.59 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.32/0.59 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_plus_div_distrib_dvd_right
% 0.32/0.59 thf(fact_4115_div__plus__div__distrib__dvd__right,axiom,
% 0.32/0.59 ! [C: nat,B: nat,A: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.59 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.32/0.59 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_plus_div_distrib_dvd_right
% 0.32/0.59 thf(fact_4116_div__plus__div__distrib__dvd__right,axiom,
% 0.32/0.59 ! [C: int,B: int,A: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ C @ B )
% 0.32/0.59 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.32/0.59 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_plus_div_distrib_dvd_right
% 0.32/0.59 thf(fact_4117_div__plus__div__distrib__dvd__left,axiom,
% 0.32/0.59 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ C @ A )
% 0.32/0.59 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.32/0.59 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_plus_div_distrib_dvd_left
% 0.32/0.59 thf(fact_4118_div__plus__div__distrib__dvd__left,axiom,
% 0.32/0.59 ! [C: nat,A: nat,B: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ C @ A )
% 0.32/0.59 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.32/0.59 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_plus_div_distrib_dvd_left
% 0.32/0.59 thf(fact_4119_div__plus__div__distrib__dvd__left,axiom,
% 0.32/0.59 ! [C: int,A: int,B: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ C @ A )
% 0.32/0.59 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.32/0.59 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_plus_div_distrib_dvd_left
% 0.32/0.59 thf(fact_4120_dvd__div__unit__iff,axiom,
% 0.32/0.59 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B ) )
% 0.32/0.59 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_unit_iff
% 0.32/0.59 thf(fact_4121_dvd__div__unit__iff,axiom,
% 0.32/0.59 ! [B: nat,A: nat,C: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.59 => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B ) )
% 0.32/0.59 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_unit_iff
% 0.32/0.59 thf(fact_4122_dvd__div__unit__iff,axiom,
% 0.32/0.59 ! [B: int,A: int,C: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.59 => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B ) )
% 0.32/0.59 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_unit_iff
% 0.32/0.59 thf(fact_4123_div__unit__dvd__iff,axiom,
% 0.32/0.59 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 0.32/0.59 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_unit_dvd_iff
% 0.32/0.59 thf(fact_4124_div__unit__dvd__iff,axiom,
% 0.32/0.59 ! [B: nat,A: nat,C: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.59 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 0.32/0.59 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_unit_dvd_iff
% 0.32/0.59 thf(fact_4125_div__unit__dvd__iff,axiom,
% 0.32/0.59 ! [B: int,A: int,C: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.59 => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 0.32/0.59 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_unit_dvd_iff
% 0.32/0.59 thf(fact_4126_unit__div__cancel,axiom,
% 0.32/0.59 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.59 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 0.32/0.59 = ( divide6298287555418463151nteger @ C @ A ) )
% 0.32/0.59 = ( B = C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_div_cancel
% 0.32/0.59 thf(fact_4127_unit__div__cancel,axiom,
% 0.32/0.59 ! [A: nat,B: nat,C: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.59 => ( ( ( divide_divide_nat @ B @ A )
% 0.32/0.59 = ( divide_divide_nat @ C @ A ) )
% 0.32/0.59 = ( B = C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_div_cancel
% 0.32/0.59 thf(fact_4128_unit__div__cancel,axiom,
% 0.32/0.59 ! [A: int,B: int,C: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.59 => ( ( ( divide_divide_int @ B @ A )
% 0.32/0.59 = ( divide_divide_int @ C @ A ) )
% 0.32/0.59 = ( B = C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_div_cancel
% 0.32/0.59 thf(fact_4129_div__power,axiom,
% 0.32/0.59 ! [B: code_integer,A: code_integer,N2: nat] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.32/0.59 => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B ) @ N2 )
% 0.32/0.59 = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( power_8256067586552552935nteger @ B @ N2 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_power
% 0.32/0.59 thf(fact_4130_div__power,axiom,
% 0.32/0.59 ! [B: nat,A: nat,N2: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ B @ A )
% 0.32/0.59 => ( ( power_power_nat @ ( divide_divide_nat @ A @ B ) @ N2 )
% 0.32/0.59 = ( divide_divide_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_power
% 0.32/0.59 thf(fact_4131_div__power,axiom,
% 0.32/0.59 ! [B: int,A: int,N2: nat] :
% 0.32/0.59 ( ( dvd_dvd_int @ B @ A )
% 0.32/0.59 => ( ( power_power_int @ ( divide_divide_int @ A @ B ) @ N2 )
% 0.32/0.59 = ( divide_divide_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_power
% 0.32/0.59 thf(fact_4132_dvd__power__le,axiom,
% 0.32/0.59 ! [X: code_integer,Y: code_integer,N2: nat,M: nat] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ X @ Y )
% 0.32/0.59 => ( ( ord_less_eq_nat @ N2 @ M )
% 0.32/0.59 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N2 ) @ ( power_8256067586552552935nteger @ Y @ M ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_power_le
% 0.32/0.59 thf(fact_4133_dvd__power__le,axiom,
% 0.32/0.59 ! [X: nat,Y: nat,N2: nat,M: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ X @ Y )
% 0.32/0.59 => ( ( ord_less_eq_nat @ N2 @ M )
% 0.32/0.59 => ( dvd_dvd_nat @ ( power_power_nat @ X @ N2 ) @ ( power_power_nat @ Y @ M ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_power_le
% 0.32/0.59 thf(fact_4134_dvd__power__le,axiom,
% 0.32/0.59 ! [X: real,Y: real,N2: nat,M: nat] :
% 0.32/0.59 ( ( dvd_dvd_real @ X @ Y )
% 0.32/0.59 => ( ( ord_less_eq_nat @ N2 @ M )
% 0.32/0.59 => ( dvd_dvd_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ M ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_power_le
% 0.32/0.59 thf(fact_4135_dvd__power__le,axiom,
% 0.32/0.59 ! [X: int,Y: int,N2: nat,M: nat] :
% 0.32/0.59 ( ( dvd_dvd_int @ X @ Y )
% 0.32/0.59 => ( ( ord_less_eq_nat @ N2 @ M )
% 0.32/0.59 => ( dvd_dvd_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ M ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_power_le
% 0.32/0.59 thf(fact_4136_dvd__power__le,axiom,
% 0.32/0.59 ! [X: complex,Y: complex,N2: nat,M: nat] :
% 0.32/0.59 ( ( dvd_dvd_complex @ X @ Y )
% 0.32/0.59 => ( ( ord_less_eq_nat @ N2 @ M )
% 0.32/0.59 => ( dvd_dvd_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ Y @ M ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_power_le
% 0.32/0.59 thf(fact_4137_power__le__dvd,axiom,
% 0.32/0.59 ! [A: code_integer,N2: nat,B: code_integer,M: nat] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) @ B )
% 0.32/0.59 => ( ( ord_less_eq_nat @ M @ N2 )
% 0.32/0.59 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % power_le_dvd
% 0.32/0.59 thf(fact_4138_power__le__dvd,axiom,
% 0.32/0.59 ! [A: nat,N2: nat,B: nat,M: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ B )
% 0.32/0.59 => ( ( ord_less_eq_nat @ M @ N2 )
% 0.32/0.59 => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % power_le_dvd
% 0.32/0.59 thf(fact_4139_power__le__dvd,axiom,
% 0.32/0.59 ! [A: real,N2: nat,B: real,M: nat] :
% 0.32/0.59 ( ( dvd_dvd_real @ ( power_power_real @ A @ N2 ) @ B )
% 0.32/0.59 => ( ( ord_less_eq_nat @ M @ N2 )
% 0.32/0.59 => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % power_le_dvd
% 0.32/0.59 thf(fact_4140_power__le__dvd,axiom,
% 0.32/0.59 ! [A: int,N2: nat,B: int,M: nat] :
% 0.32/0.59 ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ B )
% 0.32/0.59 => ( ( ord_less_eq_nat @ M @ N2 )
% 0.32/0.59 => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % power_le_dvd
% 0.32/0.59 thf(fact_4141_power__le__dvd,axiom,
% 0.32/0.59 ! [A: complex,N2: nat,B: complex,M: nat] :
% 0.32/0.59 ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N2 ) @ B )
% 0.32/0.59 => ( ( ord_less_eq_nat @ M @ N2 )
% 0.32/0.59 => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % power_le_dvd
% 0.32/0.59 thf(fact_4142_le__imp__power__dvd,axiom,
% 0.32/0.59 ! [M: nat,N2: nat,A: code_integer] :
% 0.32/0.59 ( ( ord_less_eq_nat @ M @ N2 )
% 0.32/0.59 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % le_imp_power_dvd
% 0.32/0.59 thf(fact_4143_le__imp__power__dvd,axiom,
% 0.32/0.59 ! [M: nat,N2: nat,A: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ M @ N2 )
% 0.32/0.59 => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % le_imp_power_dvd
% 0.32/0.59 thf(fact_4144_le__imp__power__dvd,axiom,
% 0.32/0.59 ! [M: nat,N2: nat,A: real] :
% 0.32/0.59 ( ( ord_less_eq_nat @ M @ N2 )
% 0.32/0.59 => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % le_imp_power_dvd
% 0.32/0.59 thf(fact_4145_le__imp__power__dvd,axiom,
% 0.32/0.59 ! [M: nat,N2: nat,A: int] :
% 0.32/0.59 ( ( ord_less_eq_nat @ M @ N2 )
% 0.32/0.59 => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % le_imp_power_dvd
% 0.32/0.59 thf(fact_4146_le__imp__power__dvd,axiom,
% 0.32/0.59 ! [M: nat,N2: nat,A: complex] :
% 0.32/0.59 ( ( ord_less_eq_nat @ M @ N2 )
% 0.32/0.59 => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % le_imp_power_dvd
% 0.32/0.59 thf(fact_4147_dvd__minus__mod,axiom,
% 0.32/0.59 ! [B: nat,A: nat] : ( dvd_dvd_nat @ B @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_minus_mod
% 0.32/0.59 thf(fact_4148_dvd__minus__mod,axiom,
% 0.32/0.59 ! [B: int,A: int] : ( dvd_dvd_int @ B @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_minus_mod
% 0.32/0.59 thf(fact_4149_dvd__minus__mod,axiom,
% 0.32/0.59 ! [B: code_integer,A: code_integer] : ( dvd_dvd_Code_integer @ B @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_minus_mod
% 0.32/0.59 thf(fact_4150_mod__eq__dvd__iff,axiom,
% 0.32/0.59 ! [A: int,C: int,B: int] :
% 0.32/0.59 ( ( ( modulo_modulo_int @ A @ C )
% 0.32/0.59 = ( modulo_modulo_int @ B @ C ) )
% 0.32/0.59 = ( dvd_dvd_int @ C @ ( minus_minus_int @ A @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % mod_eq_dvd_iff
% 0.32/0.59 thf(fact_4151_mod__eq__dvd__iff,axiom,
% 0.32/0.59 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.59 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.32/0.59 = ( modulo364778990260209775nteger @ B @ C ) )
% 0.32/0.59 = ( dvd_dvd_Code_integer @ C @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % mod_eq_dvd_iff
% 0.32/0.59 thf(fact_4152_nat__dvd__not__less,axiom,
% 0.32/0.59 ! [M: nat,N2: nat] :
% 0.32/0.59 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.59 => ( ( ord_less_nat @ M @ N2 )
% 0.32/0.59 => ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % nat_dvd_not_less
% 0.32/0.59 thf(fact_4153_dvd__pos__nat,axiom,
% 0.32/0.59 ! [N2: nat,M: nat] :
% 0.32/0.59 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.59 => ( ( dvd_dvd_nat @ M @ N2 )
% 0.32/0.59 => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_pos_nat
% 0.32/0.59 thf(fact_4154_dvd__minus__self,axiom,
% 0.32/0.59 ! [M: nat,N2: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ M ) )
% 0.32/0.59 = ( ( ord_less_nat @ N2 @ M )
% 0.32/0.59 | ( dvd_dvd_nat @ M @ N2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_minus_self
% 0.32/0.59 thf(fact_4155_bezout__lemma__nat,axiom,
% 0.32/0.59 ! [D: nat,A: nat,B: nat,X: nat,Y: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ D @ A )
% 0.32/0.59 => ( ( dvd_dvd_nat @ D @ B )
% 0.32/0.59 => ( ( ( ( times_times_nat @ A @ X )
% 0.32/0.59 = ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D ) )
% 0.32/0.59 | ( ( times_times_nat @ B @ X )
% 0.32/0.59 = ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D ) ) )
% 0.32/0.59 => ? [X4: nat,Y4: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ D @ A )
% 0.32/0.59 & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
% 0.32/0.59 & ( ( ( times_times_nat @ A @ X4 )
% 0.32/0.59 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y4 ) @ D ) )
% 0.32/0.59 | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X4 )
% 0.32/0.59 = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D ) ) ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % bezout_lemma_nat
% 0.32/0.59 thf(fact_4156_bezout__add__nat,axiom,
% 0.32/0.59 ! [A: nat,B: nat] :
% 0.32/0.59 ? [D4: nat,X4: nat,Y4: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ D4 @ A )
% 0.32/0.59 & ( dvd_dvd_nat @ D4 @ B )
% 0.32/0.59 & ( ( ( times_times_nat @ A @ X4 )
% 0.32/0.59 = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D4 ) )
% 0.32/0.59 | ( ( times_times_nat @ B @ X4 )
% 0.32/0.59 = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % bezout_add_nat
% 0.32/0.59 thf(fact_4157_less__eq__dvd__minus,axiom,
% 0.32/0.59 ! [M: nat,N2: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ M @ N2 )
% 0.32/0.59 => ( ( dvd_dvd_nat @ M @ N2 )
% 0.32/0.59 = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % less_eq_dvd_minus
% 0.32/0.59 thf(fact_4158_dvd__diffD1,axiom,
% 0.32/0.59 ! [K: nat,M: nat,N2: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
% 0.32/0.59 => ( ( dvd_dvd_nat @ K @ M )
% 0.32/0.59 => ( ( ord_less_eq_nat @ N2 @ M )
% 0.32/0.59 => ( dvd_dvd_nat @ K @ N2 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_diffD1
% 0.32/0.59 thf(fact_4159_dvd__diffD,axiom,
% 0.32/0.59 ! [K: nat,M: nat,N2: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
% 0.32/0.59 => ( ( dvd_dvd_nat @ K @ N2 )
% 0.32/0.59 => ( ( ord_less_eq_nat @ N2 @ M )
% 0.32/0.59 => ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_diffD
% 0.32/0.59 thf(fact_4160_zdvd__not__zless,axiom,
% 0.32/0.59 ! [M: int,N2: int] :
% 0.32/0.59 ( ( ord_less_int @ zero_zero_int @ M )
% 0.32/0.59 => ( ( ord_less_int @ M @ N2 )
% 0.32/0.59 => ~ ( dvd_dvd_int @ N2 @ M ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % zdvd_not_zless
% 0.32/0.59 thf(fact_4161_bezout1__nat,axiom,
% 0.32/0.59 ! [A: nat,B: nat] :
% 0.32/0.59 ? [D4: nat,X4: nat,Y4: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ D4 @ A )
% 0.32/0.59 & ( dvd_dvd_nat @ D4 @ B )
% 0.32/0.59 & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X4 ) @ ( times_times_nat @ B @ Y4 ) )
% 0.32/0.59 = D4 )
% 0.32/0.59 | ( ( minus_minus_nat @ ( times_times_nat @ B @ X4 ) @ ( times_times_nat @ A @ Y4 ) )
% 0.32/0.59 = D4 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % bezout1_nat
% 0.32/0.59 thf(fact_4162_dbl__def,axiom,
% 0.32/0.59 ( neg_numeral_dbl_real
% 0.32/0.59 = ( ^ [X2: real] : ( plus_plus_real @ X2 @ X2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dbl_def
% 0.32/0.59 thf(fact_4163_dbl__def,axiom,
% 0.32/0.59 ( neg_numeral_dbl_rat
% 0.32/0.59 = ( ^ [X2: rat] : ( plus_plus_rat @ X2 @ X2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dbl_def
% 0.32/0.59 thf(fact_4164_dbl__def,axiom,
% 0.32/0.59 ( neg_numeral_dbl_int
% 0.32/0.59 = ( ^ [X2: int] : ( plus_plus_int @ X2 @ X2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dbl_def
% 0.32/0.59 thf(fact_4165_finite__divisors__int,axiom,
% 0.32/0.59 ! [I2: int] :
% 0.32/0.59 ( ( I2 != zero_zero_int )
% 0.32/0.59 => ( finite_finite_int
% 0.32/0.59 @ ( collect_int
% 0.32/0.59 @ ^ [D2: int] : ( dvd_dvd_int @ D2 @ I2 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % finite_divisors_int
% 0.32/0.59 thf(fact_4166_div2__even__ext__nat,axiom,
% 0.32/0.59 ! [X: nat,Y: nat] :
% 0.32/0.59 ( ( ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.59 = ( divide_divide_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.59 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X )
% 0.32/0.59 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y ) )
% 0.32/0.59 => ( X = Y ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div2_even_ext_nat
% 0.32/0.59 thf(fact_4167_unity__coeff__ex,axiom,
% 0.32/0.59 ! [P: code_integer > $o,L2: code_integer] :
% 0.32/0.59 ( ( ? [X2: code_integer] : ( P @ ( times_3573771949741848930nteger @ L2 @ X2 ) ) )
% 0.32/0.59 = ( ? [X2: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ L2 @ ( plus_p5714425477246183910nteger @ X2 @ zero_z3403309356797280102nteger ) )
% 0.32/0.59 & ( P @ X2 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unity_coeff_ex
% 0.32/0.59 thf(fact_4168_unity__coeff__ex,axiom,
% 0.32/0.59 ! [P: complex > $o,L2: complex] :
% 0.32/0.59 ( ( ? [X2: complex] : ( P @ ( times_times_complex @ L2 @ X2 ) ) )
% 0.32/0.59 = ( ? [X2: complex] :
% 0.32/0.59 ( ( dvd_dvd_complex @ L2 @ ( plus_plus_complex @ X2 @ zero_zero_complex ) )
% 0.32/0.59 & ( P @ X2 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unity_coeff_ex
% 0.32/0.59 thf(fact_4169_unity__coeff__ex,axiom,
% 0.32/0.59 ! [P: real > $o,L2: real] :
% 0.32/0.59 ( ( ? [X2: real] : ( P @ ( times_times_real @ L2 @ X2 ) ) )
% 0.32/0.59 = ( ? [X2: real] :
% 0.32/0.59 ( ( dvd_dvd_real @ L2 @ ( plus_plus_real @ X2 @ zero_zero_real ) )
% 0.32/0.59 & ( P @ X2 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unity_coeff_ex
% 0.32/0.59 thf(fact_4170_unity__coeff__ex,axiom,
% 0.32/0.59 ! [P: rat > $o,L2: rat] :
% 0.32/0.59 ( ( ? [X2: rat] : ( P @ ( times_times_rat @ L2 @ X2 ) ) )
% 0.32/0.59 = ( ? [X2: rat] :
% 0.32/0.59 ( ( dvd_dvd_rat @ L2 @ ( plus_plus_rat @ X2 @ zero_zero_rat ) )
% 0.32/0.59 & ( P @ X2 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unity_coeff_ex
% 0.32/0.59 thf(fact_4171_unity__coeff__ex,axiom,
% 0.32/0.59 ! [P: nat > $o,L2: nat] :
% 0.32/0.59 ( ( ? [X2: nat] : ( P @ ( times_times_nat @ L2 @ X2 ) ) )
% 0.32/0.59 = ( ? [X2: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ L2 @ ( plus_plus_nat @ X2 @ zero_zero_nat ) )
% 0.32/0.59 & ( P @ X2 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unity_coeff_ex
% 0.32/0.59 thf(fact_4172_unity__coeff__ex,axiom,
% 0.32/0.59 ! [P: int > $o,L2: int] :
% 0.32/0.59 ( ( ? [X2: int] : ( P @ ( times_times_int @ L2 @ X2 ) ) )
% 0.32/0.59 = ( ? [X2: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ L2 @ ( plus_plus_int @ X2 @ zero_zero_int ) )
% 0.32/0.59 & ( P @ X2 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unity_coeff_ex
% 0.32/0.59 thf(fact_4173_unit__dvdE,axiom,
% 0.32/0.59 ! [A: code_integer,B: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.59 => ~ ( ( A != zero_z3403309356797280102nteger )
% 0.32/0.59 => ! [C3: code_integer] :
% 0.32/0.59 ( B
% 0.32/0.59 != ( times_3573771949741848930nteger @ A @ C3 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_dvdE
% 0.32/0.59 thf(fact_4174_unit__dvdE,axiom,
% 0.32/0.59 ! [A: nat,B: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.59 => ~ ( ( A != zero_zero_nat )
% 0.32/0.59 => ! [C3: nat] :
% 0.32/0.59 ( B
% 0.32/0.59 != ( times_times_nat @ A @ C3 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_dvdE
% 0.32/0.59 thf(fact_4175_unit__dvdE,axiom,
% 0.32/0.59 ! [A: int,B: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.59 => ~ ( ( A != zero_zero_int )
% 0.32/0.59 => ! [C3: int] :
% 0.32/0.59 ( B
% 0.32/0.59 != ( times_times_int @ A @ C3 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_dvdE
% 0.32/0.59 thf(fact_4176_dvd__div__div__eq__mult,axiom,
% 0.32/0.59 ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 0.32/0.59 ( ( A != zero_z3403309356797280102nteger )
% 0.32/0.59 => ( ( C != zero_z3403309356797280102nteger )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ C @ D )
% 0.32/0.59 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 0.32/0.59 = ( divide6298287555418463151nteger @ D @ C ) )
% 0.32/0.59 = ( ( times_3573771949741848930nteger @ B @ C )
% 0.32/0.59 = ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_div_eq_mult
% 0.32/0.59 thf(fact_4177_dvd__div__div__eq__mult,axiom,
% 0.32/0.59 ! [A: nat,C: nat,B: nat,D: nat] :
% 0.32/0.59 ( ( A != zero_zero_nat )
% 0.32/0.59 => ( ( C != zero_zero_nat )
% 0.32/0.59 => ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.59 => ( ( dvd_dvd_nat @ C @ D )
% 0.32/0.59 => ( ( ( divide_divide_nat @ B @ A )
% 0.32/0.59 = ( divide_divide_nat @ D @ C ) )
% 0.32/0.59 = ( ( times_times_nat @ B @ C )
% 0.32/0.59 = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_div_eq_mult
% 0.32/0.59 thf(fact_4178_dvd__div__div__eq__mult,axiom,
% 0.32/0.59 ! [A: int,C: int,B: int,D: int] :
% 0.32/0.59 ( ( A != zero_zero_int )
% 0.32/0.59 => ( ( C != zero_zero_int )
% 0.32/0.59 => ( ( dvd_dvd_int @ A @ B )
% 0.32/0.59 => ( ( dvd_dvd_int @ C @ D )
% 0.32/0.59 => ( ( ( divide_divide_int @ B @ A )
% 0.32/0.59 = ( divide_divide_int @ D @ C ) )
% 0.32/0.59 = ( ( times_times_int @ B @ C )
% 0.32/0.59 = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_div_eq_mult
% 0.32/0.59 thf(fact_4179_dvd__div__iff__mult,axiom,
% 0.32/0.59 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.59 ( ( C != zero_z3403309356797280102nteger )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.32/0.59 = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_iff_mult
% 0.32/0.59 thf(fact_4180_dvd__div__iff__mult,axiom,
% 0.32/0.59 ! [C: nat,B: nat,A: nat] :
% 0.32/0.59 ( ( C != zero_zero_nat )
% 0.32/0.59 => ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.59 => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.32/0.59 = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_iff_mult
% 0.32/0.59 thf(fact_4181_dvd__div__iff__mult,axiom,
% 0.32/0.59 ! [C: int,B: int,A: int] :
% 0.32/0.59 ( ( C != zero_zero_int )
% 0.32/0.59 => ( ( dvd_dvd_int @ C @ B )
% 0.32/0.59 => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.32/0.59 = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_iff_mult
% 0.32/0.59 thf(fact_4182_div__dvd__iff__mult,axiom,
% 0.32/0.59 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.59 ( ( B != zero_z3403309356797280102nteger )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 0.32/0.59 = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_dvd_iff_mult
% 0.32/0.59 thf(fact_4183_div__dvd__iff__mult,axiom,
% 0.32/0.59 ! [B: nat,A: nat,C: nat] :
% 0.32/0.59 ( ( B != zero_zero_nat )
% 0.32/0.59 => ( ( dvd_dvd_nat @ B @ A )
% 0.32/0.59 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 0.32/0.59 = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_dvd_iff_mult
% 0.32/0.59 thf(fact_4184_div__dvd__iff__mult,axiom,
% 0.32/0.59 ! [B: int,A: int,C: int] :
% 0.32/0.59 ( ( B != zero_zero_int )
% 0.32/0.59 => ( ( dvd_dvd_int @ B @ A )
% 0.32/0.59 => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 0.32/0.59 = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_dvd_iff_mult
% 0.32/0.59 thf(fact_4185_dvd__div__eq__mult,axiom,
% 0.32/0.59 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.59 ( ( A != zero_z3403309356797280102nteger )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.59 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 0.32/0.59 = C )
% 0.32/0.59 = ( B
% 0.32/0.59 = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_eq_mult
% 0.32/0.59 thf(fact_4186_dvd__div__eq__mult,axiom,
% 0.32/0.59 ! [A: nat,B: nat,C: nat] :
% 0.32/0.59 ( ( A != zero_zero_nat )
% 0.32/0.59 => ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.59 => ( ( ( divide_divide_nat @ B @ A )
% 0.32/0.59 = C )
% 0.32/0.59 = ( B
% 0.32/0.59 = ( times_times_nat @ C @ A ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_eq_mult
% 0.32/0.59 thf(fact_4187_dvd__div__eq__mult,axiom,
% 0.32/0.59 ! [A: int,B: int,C: int] :
% 0.32/0.59 ( ( A != zero_zero_int )
% 0.32/0.59 => ( ( dvd_dvd_int @ A @ B )
% 0.32/0.59 => ( ( ( divide_divide_int @ B @ A )
% 0.32/0.59 = C )
% 0.32/0.59 = ( B
% 0.32/0.59 = ( times_times_int @ C @ A ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_div_eq_mult
% 0.32/0.59 thf(fact_4188_even__numeral,axiom,
% 0.32/0.59 ! [N2: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % even_numeral
% 0.32/0.59 thf(fact_4189_even__numeral,axiom,
% 0.32/0.59 ! [N2: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % even_numeral
% 0.32/0.59 thf(fact_4190_even__numeral,axiom,
% 0.32/0.59 ! [N2: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % even_numeral
% 0.32/0.59 thf(fact_4191_unit__div__eq__0__iff,axiom,
% 0.32/0.59 ! [B: code_integer,A: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.59 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 0.32/0.59 = zero_z3403309356797280102nteger )
% 0.32/0.59 = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_div_eq_0_iff
% 0.32/0.59 thf(fact_4192_unit__div__eq__0__iff,axiom,
% 0.32/0.59 ! [B: nat,A: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.59 => ( ( ( divide_divide_nat @ A @ B )
% 0.32/0.59 = zero_zero_nat )
% 0.32/0.59 = ( A = zero_zero_nat ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_div_eq_0_iff
% 0.32/0.59 thf(fact_4193_unit__div__eq__0__iff,axiom,
% 0.32/0.59 ! [B: int,A: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.59 => ( ( ( divide_divide_int @ A @ B )
% 0.32/0.59 = zero_zero_int )
% 0.32/0.59 = ( A = zero_zero_int ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_div_eq_0_iff
% 0.32/0.59 thf(fact_4194_inf__period_I3_J,axiom,
% 0.32/0.59 ! [D: code_integer,D3: code_integer,T: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ D @ D3 )
% 0.32/0.59 => ! [X5: code_integer,K4: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ T ) )
% 0.32/0.59 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K4 @ D3 ) ) @ T ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % inf_period(3)
% 0.32/0.59 thf(fact_4195_inf__period_I3_J,axiom,
% 0.32/0.59 ! [D: real,D3: real,T: real] :
% 0.32/0.59 ( ( dvd_dvd_real @ D @ D3 )
% 0.32/0.59 => ! [X5: real,K4: real] :
% 0.32/0.59 ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) )
% 0.32/0.59 = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) @ T ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % inf_period(3)
% 0.32/0.59 thf(fact_4196_inf__period_I3_J,axiom,
% 0.32/0.59 ! [D: rat,D3: rat,T: rat] :
% 0.32/0.59 ( ( dvd_dvd_rat @ D @ D3 )
% 0.32/0.59 => ! [X5: rat,K4: rat] :
% 0.32/0.59 ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) )
% 0.32/0.59 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) @ T ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % inf_period(3)
% 0.32/0.59 thf(fact_4197_inf__period_I3_J,axiom,
% 0.32/0.59 ! [D: int,D3: int,T: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ D @ D3 )
% 0.32/0.59 => ! [X5: int,K4: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 0.32/0.59 = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) @ T ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % inf_period(3)
% 0.32/0.59 thf(fact_4198_inf__period_I4_J,axiom,
% 0.32/0.59 ! [D: code_integer,D3: code_integer,T: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ D @ D3 )
% 0.32/0.59 => ! [X5: code_integer,K4: code_integer] :
% 0.32/0.59 ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ T ) ) )
% 0.32/0.59 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K4 @ D3 ) ) @ T ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % inf_period(4)
% 0.32/0.59 thf(fact_4199_inf__period_I4_J,axiom,
% 0.32/0.59 ! [D: real,D3: real,T: real] :
% 0.32/0.59 ( ( dvd_dvd_real @ D @ D3 )
% 0.32/0.59 => ! [X5: real,K4: real] :
% 0.32/0.59 ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) ) )
% 0.32/0.59 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) @ T ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % inf_period(4)
% 0.32/0.59 thf(fact_4200_inf__period_I4_J,axiom,
% 0.32/0.59 ! [D: rat,D3: rat,T: rat] :
% 0.32/0.59 ( ( dvd_dvd_rat @ D @ D3 )
% 0.32/0.59 => ! [X5: rat,K4: rat] :
% 0.32/0.59 ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) ) )
% 0.32/0.59 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) @ T ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % inf_period(4)
% 0.32/0.59 thf(fact_4201_inf__period_I4_J,axiom,
% 0.32/0.59 ! [D: int,D3: int,T: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ D @ D3 )
% 0.32/0.59 => ! [X5: int,K4: int] :
% 0.32/0.59 ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) ) )
% 0.32/0.59 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) @ T ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % inf_period(4)
% 0.32/0.59 thf(fact_4202_is__unit__div__mult2__eq,axiom,
% 0.32/0.59 ! [B: code_integer,C: code_integer,A: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 0.32/0.59 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.32/0.59 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % is_unit_div_mult2_eq
% 0.32/0.59 thf(fact_4203_is__unit__div__mult2__eq,axiom,
% 0.32/0.59 ! [B: nat,C: nat,A: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.59 => ( ( dvd_dvd_nat @ C @ one_one_nat )
% 0.32/0.59 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.32/0.59 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % is_unit_div_mult2_eq
% 0.32/0.59 thf(fact_4204_is__unit__div__mult2__eq,axiom,
% 0.32/0.59 ! [B: int,C: int,A: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.59 => ( ( dvd_dvd_int @ C @ one_one_int )
% 0.32/0.59 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.32/0.59 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % is_unit_div_mult2_eq
% 0.32/0.59 thf(fact_4205_unit__div__mult__swap,axiom,
% 0.32/0.59 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 0.32/0.59 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.32/0.59 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_div_mult_swap
% 0.32/0.59 thf(fact_4206_unit__div__mult__swap,axiom,
% 0.32/0.59 ! [C: nat,A: nat,B: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ C @ one_one_nat )
% 0.32/0.59 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.32/0.59 = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_div_mult_swap
% 0.32/0.59 thf(fact_4207_unit__div__mult__swap,axiom,
% 0.32/0.59 ! [C: int,A: int,B: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ C @ one_one_int )
% 0.32/0.59 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.32/0.59 = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_div_mult_swap
% 0.32/0.59 thf(fact_4208_unit__div__commute,axiom,
% 0.32/0.59 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.59 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 0.32/0.59 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_div_commute
% 0.32/0.59 thf(fact_4209_unit__div__commute,axiom,
% 0.32/0.59 ! [B: nat,A: nat,C: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.59 => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 0.32/0.59 = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_div_commute
% 0.32/0.59 thf(fact_4210_unit__div__commute,axiom,
% 0.32/0.59 ! [B: int,A: int,C: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.59 => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ C )
% 0.32/0.59 = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_div_commute
% 0.32/0.59 thf(fact_4211_div__mult__unit2,axiom,
% 0.32/0.59 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 0.32/0.59 => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.32/0.59 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.32/0.59 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_mult_unit2
% 0.32/0.59 thf(fact_4212_div__mult__unit2,axiom,
% 0.32/0.59 ! [C: nat,B: nat,A: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ C @ one_one_nat )
% 0.32/0.59 => ( ( dvd_dvd_nat @ B @ A )
% 0.32/0.59 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.32/0.59 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_mult_unit2
% 0.32/0.59 thf(fact_4213_div__mult__unit2,axiom,
% 0.32/0.59 ! [C: int,B: int,A: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ C @ one_one_int )
% 0.32/0.59 => ( ( dvd_dvd_int @ B @ A )
% 0.32/0.59 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.32/0.59 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % div_mult_unit2
% 0.32/0.59 thf(fact_4214_unit__eq__div2,axiom,
% 0.32/0.59 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.59 => ( ( A
% 0.32/0.59 = ( divide6298287555418463151nteger @ C @ B ) )
% 0.32/0.59 = ( ( times_3573771949741848930nteger @ A @ B )
% 0.32/0.59 = C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_eq_div2
% 0.32/0.59 thf(fact_4215_unit__eq__div2,axiom,
% 0.32/0.59 ! [B: nat,A: nat,C: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.59 => ( ( A
% 0.32/0.59 = ( divide_divide_nat @ C @ B ) )
% 0.32/0.59 = ( ( times_times_nat @ A @ B )
% 0.32/0.59 = C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_eq_div2
% 0.32/0.59 thf(fact_4216_unit__eq__div2,axiom,
% 0.32/0.59 ! [B: int,A: int,C: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.59 => ( ( A
% 0.32/0.59 = ( divide_divide_int @ C @ B ) )
% 0.32/0.59 = ( ( times_times_int @ A @ B )
% 0.32/0.59 = C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_eq_div2
% 0.32/0.59 thf(fact_4217_unit__eq__div1,axiom,
% 0.32/0.59 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.59 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 0.32/0.59 = C )
% 0.32/0.59 = ( A
% 0.32/0.59 = ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_eq_div1
% 0.32/0.59 thf(fact_4218_unit__eq__div1,axiom,
% 0.32/0.59 ! [B: nat,A: nat,C: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.59 => ( ( ( divide_divide_nat @ A @ B )
% 0.32/0.59 = C )
% 0.32/0.59 = ( A
% 0.32/0.59 = ( times_times_nat @ C @ B ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_eq_div1
% 0.32/0.59 thf(fact_4219_unit__eq__div1,axiom,
% 0.32/0.59 ! [B: int,A: int,C: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.59 => ( ( ( divide_divide_int @ A @ B )
% 0.32/0.59 = C )
% 0.32/0.59 = ( A
% 0.32/0.59 = ( times_times_int @ C @ B ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_eq_div1
% 0.32/0.59 thf(fact_4220_is__unit__power__iff,axiom,
% 0.32/0.59 ! [A: code_integer,N2: nat] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) @ one_one_Code_integer )
% 0.32/0.59 = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.59 | ( N2 = zero_zero_nat ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % is_unit_power_iff
% 0.32/0.59 thf(fact_4221_is__unit__power__iff,axiom,
% 0.32/0.59 ! [A: nat,N2: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ one_one_nat )
% 0.32/0.59 = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.59 | ( N2 = zero_zero_nat ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % is_unit_power_iff
% 0.32/0.59 thf(fact_4222_is__unit__power__iff,axiom,
% 0.32/0.59 ! [A: int,N2: nat] :
% 0.32/0.59 ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ one_one_int )
% 0.32/0.59 = ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.59 | ( N2 = zero_zero_nat ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % is_unit_power_iff
% 0.32/0.59 thf(fact_4223_unit__imp__mod__eq__0,axiom,
% 0.32/0.59 ! [B: nat,A: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.59 => ( ( modulo_modulo_nat @ A @ B )
% 0.32/0.59 = zero_zero_nat ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_imp_mod_eq_0
% 0.32/0.59 thf(fact_4224_unit__imp__mod__eq__0,axiom,
% 0.32/0.59 ! [B: int,A: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.59 => ( ( modulo_modulo_int @ A @ B )
% 0.32/0.59 = zero_zero_int ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_imp_mod_eq_0
% 0.32/0.59 thf(fact_4225_unit__imp__mod__eq__0,axiom,
% 0.32/0.59 ! [B: code_integer,A: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.59 => ( ( modulo364778990260209775nteger @ A @ B )
% 0.32/0.59 = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.59
% 0.32/0.59 % unit_imp_mod_eq_0
% 0.32/0.59 thf(fact_4226_dvd__imp__le,axiom,
% 0.32/0.59 ! [K: nat,N2: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ K @ N2 )
% 0.32/0.59 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.59 => ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_imp_le
% 0.32/0.59 thf(fact_4227_dvd__mult__cancel,axiom,
% 0.32/0.59 ! [K: nat,M: nat,N2: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 0.32/0.59 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.59 => ( dvd_dvd_nat @ M @ N2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_mult_cancel
% 0.32/0.59 thf(fact_4228_nat__mult__dvd__cancel1,axiom,
% 0.32/0.59 ! [K: nat,M: nat,N2: nat] :
% 0.32/0.59 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.59 => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 0.32/0.59 = ( dvd_dvd_nat @ M @ N2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % nat_mult_dvd_cancel1
% 0.32/0.59 thf(fact_4229_bezout__add__strong__nat,axiom,
% 0.32/0.59 ! [A: nat,B: nat] :
% 0.32/0.59 ( ( A != zero_zero_nat )
% 0.32/0.59 => ? [D4: nat,X4: nat,Y4: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ D4 @ A )
% 0.32/0.59 & ( dvd_dvd_nat @ D4 @ B )
% 0.32/0.59 & ( ( times_times_nat @ A @ X4 )
% 0.32/0.59 = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % bezout_add_strong_nat
% 0.32/0.59 thf(fact_4230_zdvd__imp__le,axiom,
% 0.32/0.59 ! [Z: int,N2: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ Z @ N2 )
% 0.32/0.59 => ( ( ord_less_int @ zero_zero_int @ N2 )
% 0.32/0.59 => ( ord_less_eq_int @ Z @ N2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % zdvd_imp_le
% 0.32/0.59 thf(fact_4231_nle__le,axiom,
% 0.32/0.59 ! [A: rat,B: rat] :
% 0.32/0.59 ( ( ~ ( ord_less_eq_rat @ A @ B ) )
% 0.32/0.59 = ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.59 & ( B != A ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % nle_le
% 0.32/0.59 thf(fact_4232_nle__le,axiom,
% 0.32/0.59 ! [A: num,B: num] :
% 0.32/0.59 ( ( ~ ( ord_less_eq_num @ A @ B ) )
% 0.32/0.59 = ( ( ord_less_eq_num @ B @ A )
% 0.32/0.59 & ( B != A ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % nle_le
% 0.32/0.59 thf(fact_4233_nle__le,axiom,
% 0.32/0.59 ! [A: nat,B: nat] :
% 0.32/0.59 ( ( ~ ( ord_less_eq_nat @ A @ B ) )
% 0.32/0.59 = ( ( ord_less_eq_nat @ B @ A )
% 0.32/0.59 & ( B != A ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % nle_le
% 0.32/0.59 thf(fact_4234_nle__le,axiom,
% 0.32/0.59 ! [A: int,B: int] :
% 0.32/0.59 ( ( ~ ( ord_less_eq_int @ A @ B ) )
% 0.32/0.59 = ( ( ord_less_eq_int @ B @ A )
% 0.32/0.59 & ( B != A ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % nle_le
% 0.32/0.59 thf(fact_4235_le__cases3,axiom,
% 0.32/0.59 ! [X: rat,Y: rat,Z: rat] :
% 0.32/0.59 ( ( ( ord_less_eq_rat @ X @ Y )
% 0.32/0.59 => ~ ( ord_less_eq_rat @ Y @ Z ) )
% 0.32/0.59 => ( ( ( ord_less_eq_rat @ Y @ X )
% 0.32/0.59 => ~ ( ord_less_eq_rat @ X @ Z ) )
% 0.32/0.59 => ( ( ( ord_less_eq_rat @ X @ Z )
% 0.32/0.59 => ~ ( ord_less_eq_rat @ Z @ Y ) )
% 0.32/0.59 => ( ( ( ord_less_eq_rat @ Z @ Y )
% 0.32/0.59 => ~ ( ord_less_eq_rat @ Y @ X ) )
% 0.32/0.59 => ( ( ( ord_less_eq_rat @ Y @ Z )
% 0.32/0.59 => ~ ( ord_less_eq_rat @ Z @ X ) )
% 0.32/0.59 => ~ ( ( ord_less_eq_rat @ Z @ X )
% 0.32/0.59 => ~ ( ord_less_eq_rat @ X @ Y ) ) ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % le_cases3
% 0.32/0.59 thf(fact_4236_le__cases3,axiom,
% 0.32/0.59 ! [X: num,Y: num,Z: num] :
% 0.32/0.59 ( ( ( ord_less_eq_num @ X @ Y )
% 0.32/0.59 => ~ ( ord_less_eq_num @ Y @ Z ) )
% 0.32/0.59 => ( ( ( ord_less_eq_num @ Y @ X )
% 0.32/0.59 => ~ ( ord_less_eq_num @ X @ Z ) )
% 0.32/0.59 => ( ( ( ord_less_eq_num @ X @ Z )
% 0.32/0.59 => ~ ( ord_less_eq_num @ Z @ Y ) )
% 0.32/0.59 => ( ( ( ord_less_eq_num @ Z @ Y )
% 0.32/0.59 => ~ ( ord_less_eq_num @ Y @ X ) )
% 0.32/0.59 => ( ( ( ord_less_eq_num @ Y @ Z )
% 0.32/0.59 => ~ ( ord_less_eq_num @ Z @ X ) )
% 0.32/0.59 => ~ ( ( ord_less_eq_num @ Z @ X )
% 0.32/0.59 => ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % le_cases3
% 0.32/0.59 thf(fact_4237_le__cases3,axiom,
% 0.32/0.59 ! [X: nat,Y: nat,Z: nat] :
% 0.32/0.59 ( ( ( ord_less_eq_nat @ X @ Y )
% 0.32/0.59 => ~ ( ord_less_eq_nat @ Y @ Z ) )
% 0.32/0.59 => ( ( ( ord_less_eq_nat @ Y @ X )
% 0.32/0.59 => ~ ( ord_less_eq_nat @ X @ Z ) )
% 0.32/0.59 => ( ( ( ord_less_eq_nat @ X @ Z )
% 0.32/0.59 => ~ ( ord_less_eq_nat @ Z @ Y ) )
% 0.32/0.59 => ( ( ( ord_less_eq_nat @ Z @ Y )
% 0.32/0.59 => ~ ( ord_less_eq_nat @ Y @ X ) )
% 0.32/0.59 => ( ( ( ord_less_eq_nat @ Y @ Z )
% 0.32/0.59 => ~ ( ord_less_eq_nat @ Z @ X ) )
% 0.32/0.59 => ~ ( ( ord_less_eq_nat @ Z @ X )
% 0.32/0.59 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % le_cases3
% 0.32/0.59 thf(fact_4238_le__cases3,axiom,
% 0.32/0.59 ! [X: int,Y: int,Z: int] :
% 0.32/0.59 ( ( ( ord_less_eq_int @ X @ Y )
% 0.32/0.59 => ~ ( ord_less_eq_int @ Y @ Z ) )
% 0.32/0.59 => ( ( ( ord_less_eq_int @ Y @ X )
% 0.32/0.59 => ~ ( ord_less_eq_int @ X @ Z ) )
% 0.32/0.59 => ( ( ( ord_less_eq_int @ X @ Z )
% 0.32/0.59 => ~ ( ord_less_eq_int @ Z @ Y ) )
% 0.32/0.59 => ( ( ( ord_less_eq_int @ Z @ Y )
% 0.32/0.59 => ~ ( ord_less_eq_int @ Y @ X ) )
% 0.32/0.59 => ( ( ( ord_less_eq_int @ Y @ Z )
% 0.32/0.59 => ~ ( ord_less_eq_int @ Z @ X ) )
% 0.32/0.59 => ~ ( ( ord_less_eq_int @ Z @ X )
% 0.32/0.59 => ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % le_cases3
% 0.32/0.59 thf(fact_4239_order__class_Oorder__eq__iff,axiom,
% 0.32/0.59 ( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
% 0.32/0.59 = ( ^ [X2: set_nat,Y2: set_nat] :
% 0.32/0.59 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.32/0.59 & ( ord_less_eq_set_nat @ Y2 @ X2 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_class.order_eq_iff
% 0.32/0.59 thf(fact_4240_order__class_Oorder__eq__iff,axiom,
% 0.32/0.59 ( ( ^ [Y3: rat,Z2: rat] : ( Y3 = Z2 ) )
% 0.32/0.59 = ( ^ [X2: rat,Y2: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X2 @ Y2 )
% 0.32/0.59 & ( ord_less_eq_rat @ Y2 @ X2 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_class.order_eq_iff
% 0.32/0.59 thf(fact_4241_order__class_Oorder__eq__iff,axiom,
% 0.32/0.59 ( ( ^ [Y3: num,Z2: num] : ( Y3 = Z2 ) )
% 0.32/0.59 = ( ^ [X2: num,Y2: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X2 @ Y2 )
% 0.32/0.59 & ( ord_less_eq_num @ Y2 @ X2 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_class.order_eq_iff
% 0.32/0.59 thf(fact_4242_order__class_Oorder__eq__iff,axiom,
% 0.32/0.59 ( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
% 0.32/0.59 = ( ^ [X2: nat,Y2: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.32/0.59 & ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_class.order_eq_iff
% 0.32/0.59 thf(fact_4243_order__class_Oorder__eq__iff,axiom,
% 0.32/0.59 ( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
% 0.32/0.59 = ( ^ [X2: int,Y2: int] :
% 0.32/0.59 ( ( ord_less_eq_int @ X2 @ Y2 )
% 0.32/0.59 & ( ord_less_eq_int @ Y2 @ X2 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_class.order_eq_iff
% 0.32/0.59 thf(fact_4244_ord__eq__le__trans,axiom,
% 0.32/0.59 ! [A: set_nat,B: set_nat,C: set_nat] :
% 0.32/0.59 ( ( A = B )
% 0.32/0.59 => ( ( ord_less_eq_set_nat @ B @ C )
% 0.32/0.59 => ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_eq_le_trans
% 0.32/0.59 thf(fact_4245_ord__eq__le__trans,axiom,
% 0.32/0.59 ! [A: rat,B: rat,C: rat] :
% 0.32/0.59 ( ( A = B )
% 0.32/0.59 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.59 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_eq_le_trans
% 0.32/0.59 thf(fact_4246_ord__eq__le__trans,axiom,
% 0.32/0.59 ! [A: num,B: num,C: num] :
% 0.32/0.59 ( ( A = B )
% 0.32/0.59 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.59 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_eq_le_trans
% 0.32/0.59 thf(fact_4247_ord__eq__le__trans,axiom,
% 0.32/0.59 ! [A: nat,B: nat,C: nat] :
% 0.32/0.59 ( ( A = B )
% 0.32/0.59 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/0.59 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_eq_le_trans
% 0.32/0.59 thf(fact_4248_ord__eq__le__trans,axiom,
% 0.32/0.59 ! [A: int,B: int,C: int] :
% 0.32/0.59 ( ( A = B )
% 0.32/0.59 => ( ( ord_less_eq_int @ B @ C )
% 0.32/0.59 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_eq_le_trans
% 0.32/0.59 thf(fact_4249_ord__le__eq__trans,axiom,
% 0.32/0.59 ! [A: set_nat,B: set_nat,C: set_nat] :
% 0.32/0.59 ( ( ord_less_eq_set_nat @ A @ B )
% 0.32/0.59 => ( ( B = C )
% 0.32/0.59 => ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_le_eq_trans
% 0.32/0.59 thf(fact_4250_ord__le__eq__trans,axiom,
% 0.32/0.59 ! [A: rat,B: rat,C: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.59 => ( ( B = C )
% 0.32/0.59 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_le_eq_trans
% 0.32/0.59 thf(fact_4251_ord__le__eq__trans,axiom,
% 0.32/0.59 ! [A: num,B: num,C: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.59 => ( ( B = C )
% 0.32/0.59 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_le_eq_trans
% 0.32/0.59 thf(fact_4252_ord__le__eq__trans,axiom,
% 0.32/0.59 ! [A: nat,B: nat,C: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.59 => ( ( B = C )
% 0.32/0.59 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_le_eq_trans
% 0.32/0.59 thf(fact_4253_ord__le__eq__trans,axiom,
% 0.32/0.59 ! [A: int,B: int,C: int] :
% 0.32/0.59 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.59 => ( ( B = C )
% 0.32/0.59 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_le_eq_trans
% 0.32/0.59 thf(fact_4254_order__antisym,axiom,
% 0.32/0.59 ! [X: set_nat,Y: set_nat] :
% 0.32/0.59 ( ( ord_less_eq_set_nat @ X @ Y )
% 0.32/0.59 => ( ( ord_less_eq_set_nat @ Y @ X )
% 0.32/0.59 => ( X = Y ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_antisym
% 0.32/0.59 thf(fact_4255_order__antisym,axiom,
% 0.32/0.59 ! [X: rat,Y: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X @ Y )
% 0.32/0.59 => ( ( ord_less_eq_rat @ Y @ X )
% 0.32/0.59 => ( X = Y ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_antisym
% 0.32/0.59 thf(fact_4256_order__antisym,axiom,
% 0.32/0.59 ! [X: num,Y: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X @ Y )
% 0.32/0.59 => ( ( ord_less_eq_num @ Y @ X )
% 0.32/0.59 => ( X = Y ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_antisym
% 0.32/0.59 thf(fact_4257_order__antisym,axiom,
% 0.32/0.59 ! [X: nat,Y: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ X @ Y )
% 0.32/0.59 => ( ( ord_less_eq_nat @ Y @ X )
% 0.32/0.59 => ( X = Y ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_antisym
% 0.32/0.59 thf(fact_4258_order__antisym,axiom,
% 0.32/0.59 ! [X: int,Y: int] :
% 0.32/0.59 ( ( ord_less_eq_int @ X @ Y )
% 0.32/0.59 => ( ( ord_less_eq_int @ Y @ X )
% 0.32/0.59 => ( X = Y ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_antisym
% 0.32/0.59 thf(fact_4259_order_Otrans,axiom,
% 0.32/0.59 ! [A: set_nat,B: set_nat,C: set_nat] :
% 0.32/0.59 ( ( ord_less_eq_set_nat @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_set_nat @ B @ C )
% 0.32/0.59 => ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.trans
% 0.32/0.59 thf(fact_4260_order_Otrans,axiom,
% 0.32/0.59 ! [A: rat,B: rat,C: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.59 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.trans
% 0.32/0.59 thf(fact_4261_order_Otrans,axiom,
% 0.32/0.59 ! [A: num,B: num,C: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.59 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.trans
% 0.32/0.59 thf(fact_4262_order_Otrans,axiom,
% 0.32/0.59 ! [A: nat,B: nat,C: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/0.59 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.trans
% 0.32/0.59 thf(fact_4263_order_Otrans,axiom,
% 0.32/0.59 ! [A: int,B: int,C: int] :
% 0.32/0.59 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_int @ B @ C )
% 0.32/0.59 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.trans
% 0.32/0.59 thf(fact_4264_order__trans,axiom,
% 0.32/0.59 ! [X: set_nat,Y: set_nat,Z: set_nat] :
% 0.32/0.59 ( ( ord_less_eq_set_nat @ X @ Y )
% 0.32/0.59 => ( ( ord_less_eq_set_nat @ Y @ Z )
% 0.32/0.59 => ( ord_less_eq_set_nat @ X @ Z ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_trans
% 0.32/0.59 thf(fact_4265_order__trans,axiom,
% 0.32/0.59 ! [X: rat,Y: rat,Z: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X @ Y )
% 0.32/0.59 => ( ( ord_less_eq_rat @ Y @ Z )
% 0.32/0.59 => ( ord_less_eq_rat @ X @ Z ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_trans
% 0.32/0.59 thf(fact_4266_order__trans,axiom,
% 0.32/0.59 ! [X: num,Y: num,Z: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X @ Y )
% 0.32/0.59 => ( ( ord_less_eq_num @ Y @ Z )
% 0.32/0.59 => ( ord_less_eq_num @ X @ Z ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_trans
% 0.32/0.59 thf(fact_4267_order__trans,axiom,
% 0.32/0.59 ! [X: nat,Y: nat,Z: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ X @ Y )
% 0.32/0.59 => ( ( ord_less_eq_nat @ Y @ Z )
% 0.32/0.59 => ( ord_less_eq_nat @ X @ Z ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_trans
% 0.32/0.59 thf(fact_4268_order__trans,axiom,
% 0.32/0.59 ! [X: int,Y: int,Z: int] :
% 0.32/0.59 ( ( ord_less_eq_int @ X @ Y )
% 0.32/0.59 => ( ( ord_less_eq_int @ Y @ Z )
% 0.32/0.59 => ( ord_less_eq_int @ X @ Z ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_trans
% 0.32/0.59 thf(fact_4269_linorder__wlog,axiom,
% 0.32/0.59 ! [P: rat > rat > $o,A: rat,B: rat] :
% 0.32/0.59 ( ! [A3: rat,B2: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A3 @ B2 )
% 0.32/0.59 => ( P @ A3 @ B2 ) )
% 0.32/0.59 => ( ! [A3: rat,B2: rat] :
% 0.32/0.59 ( ( P @ B2 @ A3 )
% 0.32/0.59 => ( P @ A3 @ B2 ) )
% 0.32/0.59 => ( P @ A @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % linorder_wlog
% 0.32/0.59 thf(fact_4270_linorder__wlog,axiom,
% 0.32/0.59 ! [P: num > num > $o,A: num,B: num] :
% 0.32/0.59 ( ! [A3: num,B2: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ A3 @ B2 )
% 0.32/0.59 => ( P @ A3 @ B2 ) )
% 0.32/0.59 => ( ! [A3: num,B2: num] :
% 0.32/0.59 ( ( P @ B2 @ A3 )
% 0.32/0.59 => ( P @ A3 @ B2 ) )
% 0.32/0.59 => ( P @ A @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % linorder_wlog
% 0.32/0.59 thf(fact_4271_linorder__wlog,axiom,
% 0.32/0.59 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.32/0.59 ( ! [A3: nat,B2: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ A3 @ B2 )
% 0.32/0.59 => ( P @ A3 @ B2 ) )
% 0.32/0.59 => ( ! [A3: nat,B2: nat] :
% 0.32/0.59 ( ( P @ B2 @ A3 )
% 0.32/0.59 => ( P @ A3 @ B2 ) )
% 0.32/0.59 => ( P @ A @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % linorder_wlog
% 0.32/0.59 thf(fact_4272_linorder__wlog,axiom,
% 0.32/0.59 ! [P: int > int > $o,A: int,B: int] :
% 0.32/0.59 ( ! [A3: int,B2: int] :
% 0.32/0.59 ( ( ord_less_eq_int @ A3 @ B2 )
% 0.32/0.59 => ( P @ A3 @ B2 ) )
% 0.32/0.59 => ( ! [A3: int,B2: int] :
% 0.32/0.59 ( ( P @ B2 @ A3 )
% 0.32/0.59 => ( P @ A3 @ B2 ) )
% 0.32/0.59 => ( P @ A @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % linorder_wlog
% 0.32/0.59 thf(fact_4273_dual__order_Oeq__iff,axiom,
% 0.32/0.59 ( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
% 0.32/0.59 = ( ^ [A5: set_nat,B5: set_nat] :
% 0.32/0.59 ( ( ord_less_eq_set_nat @ B5 @ A5 )
% 0.32/0.59 & ( ord_less_eq_set_nat @ A5 @ B5 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.eq_iff
% 0.32/0.59 thf(fact_4274_dual__order_Oeq__iff,axiom,
% 0.32/0.59 ( ( ^ [Y3: rat,Z2: rat] : ( Y3 = Z2 ) )
% 0.32/0.59 = ( ^ [A5: rat,B5: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ B5 @ A5 )
% 0.32/0.59 & ( ord_less_eq_rat @ A5 @ B5 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.eq_iff
% 0.32/0.59 thf(fact_4275_dual__order_Oeq__iff,axiom,
% 0.32/0.59 ( ( ^ [Y3: num,Z2: num] : ( Y3 = Z2 ) )
% 0.32/0.59 = ( ^ [A5: num,B5: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ B5 @ A5 )
% 0.32/0.59 & ( ord_less_eq_num @ A5 @ B5 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.eq_iff
% 0.32/0.59 thf(fact_4276_dual__order_Oeq__iff,axiom,
% 0.32/0.59 ( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
% 0.32/0.59 = ( ^ [A5: nat,B5: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ B5 @ A5 )
% 0.32/0.59 & ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.eq_iff
% 0.32/0.59 thf(fact_4277_dual__order_Oeq__iff,axiom,
% 0.32/0.59 ( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
% 0.32/0.59 = ( ^ [A5: int,B5: int] :
% 0.32/0.59 ( ( ord_less_eq_int @ B5 @ A5 )
% 0.32/0.59 & ( ord_less_eq_int @ A5 @ B5 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.eq_iff
% 0.32/0.59 thf(fact_4278_dual__order_Oantisym,axiom,
% 0.32/0.59 ! [B: set_nat,A: set_nat] :
% 0.32/0.59 ( ( ord_less_eq_set_nat @ B @ A )
% 0.32/0.59 => ( ( ord_less_eq_set_nat @ A @ B )
% 0.32/0.59 => ( A = B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.antisym
% 0.32/0.59 thf(fact_4279_dual__order_Oantisym,axiom,
% 0.32/0.59 ! [B: rat,A: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.59 => ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.59 => ( A = B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.antisym
% 0.32/0.59 thf(fact_4280_dual__order_Oantisym,axiom,
% 0.32/0.59 ! [B: num,A: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ B @ A )
% 0.32/0.59 => ( ( ord_less_eq_num @ A @ B )
% 0.32/0.59 => ( A = B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.antisym
% 0.32/0.59 thf(fact_4281_dual__order_Oantisym,axiom,
% 0.32/0.59 ! [B: nat,A: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ B @ A )
% 0.32/0.59 => ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.59 => ( A = B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.antisym
% 0.32/0.59 thf(fact_4282_dual__order_Oantisym,axiom,
% 0.32/0.59 ! [B: int,A: int] :
% 0.32/0.59 ( ( ord_less_eq_int @ B @ A )
% 0.32/0.59 => ( ( ord_less_eq_int @ A @ B )
% 0.32/0.59 => ( A = B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.antisym
% 0.32/0.59 thf(fact_4283_dual__order_Otrans,axiom,
% 0.32/0.59 ! [B: set_nat,A: set_nat,C: set_nat] :
% 0.32/0.59 ( ( ord_less_eq_set_nat @ B @ A )
% 0.32/0.59 => ( ( ord_less_eq_set_nat @ C @ B )
% 0.32/0.59 => ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.trans
% 0.32/0.59 thf(fact_4284_dual__order_Otrans,axiom,
% 0.32/0.59 ! [B: rat,A: rat,C: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.59 => ( ( ord_less_eq_rat @ C @ B )
% 0.32/0.59 => ( ord_less_eq_rat @ C @ A ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.trans
% 0.32/0.59 thf(fact_4285_dual__order_Otrans,axiom,
% 0.32/0.59 ! [B: num,A: num,C: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ B @ A )
% 0.32/0.59 => ( ( ord_less_eq_num @ C @ B )
% 0.32/0.59 => ( ord_less_eq_num @ C @ A ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.trans
% 0.32/0.59 thf(fact_4286_dual__order_Otrans,axiom,
% 0.32/0.59 ! [B: nat,A: nat,C: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ B @ A )
% 0.32/0.59 => ( ( ord_less_eq_nat @ C @ B )
% 0.32/0.59 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.trans
% 0.32/0.59 thf(fact_4287_dual__order_Otrans,axiom,
% 0.32/0.59 ! [B: int,A: int,C: int] :
% 0.32/0.59 ( ( ord_less_eq_int @ B @ A )
% 0.32/0.59 => ( ( ord_less_eq_int @ C @ B )
% 0.32/0.59 => ( ord_less_eq_int @ C @ A ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.trans
% 0.32/0.59 thf(fact_4288_antisym,axiom,
% 0.32/0.59 ! [A: set_nat,B: set_nat] :
% 0.32/0.59 ( ( ord_less_eq_set_nat @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_set_nat @ B @ A )
% 0.32/0.59 => ( A = B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % antisym
% 0.32/0.59 thf(fact_4289_antisym,axiom,
% 0.32/0.59 ! [A: rat,B: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.59 => ( A = B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % antisym
% 0.32/0.59 thf(fact_4290_antisym,axiom,
% 0.32/0.59 ! [A: num,B: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_num @ B @ A )
% 0.32/0.59 => ( A = B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % antisym
% 0.32/0.59 thf(fact_4291_antisym,axiom,
% 0.32/0.59 ! [A: nat,B: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_nat @ B @ A )
% 0.32/0.59 => ( A = B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % antisym
% 0.32/0.59 thf(fact_4292_antisym,axiom,
% 0.32/0.59 ! [A: int,B: int] :
% 0.32/0.59 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_int @ B @ A )
% 0.32/0.59 => ( A = B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % antisym
% 0.32/0.59 thf(fact_4293_Orderings_Oorder__eq__iff,axiom,
% 0.32/0.59 ( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
% 0.32/0.59 = ( ^ [A5: set_nat,B5: set_nat] :
% 0.32/0.59 ( ( ord_less_eq_set_nat @ A5 @ B5 )
% 0.32/0.59 & ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % Orderings.order_eq_iff
% 0.32/0.59 thf(fact_4294_Orderings_Oorder__eq__iff,axiom,
% 0.32/0.59 ( ( ^ [Y3: rat,Z2: rat] : ( Y3 = Z2 ) )
% 0.32/0.59 = ( ^ [A5: rat,B5: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A5 @ B5 )
% 0.32/0.59 & ( ord_less_eq_rat @ B5 @ A5 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % Orderings.order_eq_iff
% 0.32/0.59 thf(fact_4295_Orderings_Oorder__eq__iff,axiom,
% 0.32/0.59 ( ( ^ [Y3: num,Z2: num] : ( Y3 = Z2 ) )
% 0.32/0.59 = ( ^ [A5: num,B5: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ A5 @ B5 )
% 0.32/0.59 & ( ord_less_eq_num @ B5 @ A5 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % Orderings.order_eq_iff
% 0.32/0.59 thf(fact_4296_Orderings_Oorder__eq__iff,axiom,
% 0.32/0.59 ( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
% 0.32/0.59 = ( ^ [A5: nat,B5: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ A5 @ B5 )
% 0.32/0.59 & ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % Orderings.order_eq_iff
% 0.32/0.59 thf(fact_4297_Orderings_Oorder__eq__iff,axiom,
% 0.32/0.59 ( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
% 0.32/0.59 = ( ^ [A5: int,B5: int] :
% 0.32/0.59 ( ( ord_less_eq_int @ A5 @ B5 )
% 0.32/0.59 & ( ord_less_eq_int @ B5 @ A5 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % Orderings.order_eq_iff
% 0.32/0.59 thf(fact_4298_order__subst1,axiom,
% 0.32/0.59 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.59 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_subst1
% 0.32/0.59 thf(fact_4299_order__subst1,axiom,
% 0.32/0.59 ! [A: rat,F: num > rat,B: num,C: num] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.59 => ( ! [X4: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_subst1
% 0.32/0.59 thf(fact_4300_order__subst1,axiom,
% 0.32/0.59 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/0.59 => ( ! [X4: nat,Y4: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_subst1
% 0.32/0.59 thf(fact_4301_order__subst1,axiom,
% 0.32/0.59 ! [A: rat,F: int > rat,B: int,C: int] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_int @ B @ C )
% 0.32/0.59 => ( ! [X4: int,Y4: int] :
% 0.32/0.59 ( ( ord_less_eq_int @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_subst1
% 0.32/0.59 thf(fact_4302_order__subst1,axiom,
% 0.32/0.59 ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.32/0.59 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.59 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_subst1
% 0.32/0.59 thf(fact_4303_order__subst1,axiom,
% 0.32/0.59 ! [A: num,F: num > num,B: num,C: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.59 => ( ! [X4: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_subst1
% 0.32/0.59 thf(fact_4304_order__subst1,axiom,
% 0.32/0.59 ! [A: num,F: nat > num,B: nat,C: nat] :
% 0.32/0.59 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/0.59 => ( ! [X4: nat,Y4: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_subst1
% 0.32/0.59 thf(fact_4305_order__subst1,axiom,
% 0.32/0.59 ! [A: num,F: int > num,B: int,C: int] :
% 0.32/0.59 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_int @ B @ C )
% 0.32/0.59 => ( ! [X4: int,Y4: int] :
% 0.32/0.59 ( ( ord_less_eq_int @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_subst1
% 0.32/0.59 thf(fact_4306_order__subst1,axiom,
% 0.32/0.59 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.59 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_subst1
% 0.32/0.59 thf(fact_4307_order__subst1,axiom,
% 0.32/0.59 ! [A: nat,F: num > nat,B: num,C: num] :
% 0.32/0.59 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.59 => ( ! [X4: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_subst1
% 0.32/0.59 thf(fact_4308_order__subst2,axiom,
% 0.32/0.59 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_subst2
% 0.32/0.59 thf(fact_4309_order__subst2,axiom,
% 0.32/0.59 ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_subst2
% 0.32/0.59 thf(fact_4310_order__subst2,axiom,
% 0.32/0.59 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_subst2
% 0.32/0.59 thf(fact_4311_order__subst2,axiom,
% 0.32/0.59 ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_subst2
% 0.32/0.59 thf(fact_4312_order__subst2,axiom,
% 0.32/0.59 ! [A: num,B: num,F: num > rat,C: rat] :
% 0.32/0.59 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X4: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_subst2
% 0.32/0.59 thf(fact_4313_order__subst2,axiom,
% 0.32/0.59 ! [A: num,B: num,F: num > num,C: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X4: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_subst2
% 0.32/0.59 thf(fact_4314_order__subst2,axiom,
% 0.32/0.59 ! [A: num,B: num,F: num > nat,C: nat] :
% 0.32/0.59 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X4: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_subst2
% 0.32/0.59 thf(fact_4315_order__subst2,axiom,
% 0.32/0.59 ! [A: num,B: num,F: num > int,C: int] :
% 0.32/0.59 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X4: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_subst2
% 0.32/0.59 thf(fact_4316_order__subst2,axiom,
% 0.32/0.59 ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X4: nat,Y4: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_subst2
% 0.32/0.59 thf(fact_4317_order__subst2,axiom,
% 0.32/0.59 ! [A: nat,B: nat,F: nat > num,C: num] :
% 0.32/0.59 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X4: nat,Y4: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_subst2
% 0.32/0.59 thf(fact_4318_order__eq__refl,axiom,
% 0.32/0.59 ! [X: set_nat,Y: set_nat] :
% 0.32/0.59 ( ( X = Y )
% 0.32/0.59 => ( ord_less_eq_set_nat @ X @ Y ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_eq_refl
% 0.32/0.59 thf(fact_4319_order__eq__refl,axiom,
% 0.32/0.59 ! [X: rat,Y: rat] :
% 0.32/0.59 ( ( X = Y )
% 0.32/0.59 => ( ord_less_eq_rat @ X @ Y ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_eq_refl
% 0.32/0.59 thf(fact_4320_order__eq__refl,axiom,
% 0.32/0.59 ! [X: num,Y: num] :
% 0.32/0.59 ( ( X = Y )
% 0.32/0.59 => ( ord_less_eq_num @ X @ Y ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_eq_refl
% 0.32/0.59 thf(fact_4321_order__eq__refl,axiom,
% 0.32/0.59 ! [X: nat,Y: nat] :
% 0.32/0.59 ( ( X = Y )
% 0.32/0.59 => ( ord_less_eq_nat @ X @ Y ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_eq_refl
% 0.32/0.59 thf(fact_4322_order__eq__refl,axiom,
% 0.32/0.59 ! [X: int,Y: int] :
% 0.32/0.59 ( ( X = Y )
% 0.32/0.59 => ( ord_less_eq_int @ X @ Y ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_eq_refl
% 0.32/0.59 thf(fact_4323_linorder__linear,axiom,
% 0.32/0.59 ! [X: rat,Y: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X @ Y )
% 0.32/0.59 | ( ord_less_eq_rat @ Y @ X ) ) ).
% 0.32/0.59
% 0.32/0.59 % linorder_linear
% 0.32/0.59 thf(fact_4324_linorder__linear,axiom,
% 0.32/0.59 ! [X: num,Y: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X @ Y )
% 0.32/0.59 | ( ord_less_eq_num @ Y @ X ) ) ).
% 0.32/0.59
% 0.32/0.59 % linorder_linear
% 0.32/0.59 thf(fact_4325_linorder__linear,axiom,
% 0.32/0.59 ! [X: nat,Y: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ X @ Y )
% 0.32/0.59 | ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.32/0.59
% 0.32/0.59 % linorder_linear
% 0.32/0.59 thf(fact_4326_linorder__linear,axiom,
% 0.32/0.59 ! [X: int,Y: int] :
% 0.32/0.59 ( ( ord_less_eq_int @ X @ Y )
% 0.32/0.59 | ( ord_less_eq_int @ Y @ X ) ) ).
% 0.32/0.59
% 0.32/0.59 % linorder_linear
% 0.32/0.59 thf(fact_4327_ord__eq__le__subst,axiom,
% 0.32/0.59 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.32/0.59 ( ( A
% 0.32/0.59 = ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.59 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_eq_le_subst
% 0.32/0.59 thf(fact_4328_ord__eq__le__subst,axiom,
% 0.32/0.59 ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.32/0.59 ( ( A
% 0.32/0.59 = ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.59 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_eq_le_subst
% 0.32/0.59 thf(fact_4329_ord__eq__le__subst,axiom,
% 0.32/0.59 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 0.32/0.59 ( ( A
% 0.32/0.59 = ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.59 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_eq_le_subst
% 0.32/0.59 thf(fact_4330_ord__eq__le__subst,axiom,
% 0.32/0.59 ! [A: int,F: rat > int,B: rat,C: rat] :
% 0.32/0.59 ( ( A
% 0.32/0.59 = ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.59 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_eq_le_subst
% 0.32/0.59 thf(fact_4331_ord__eq__le__subst,axiom,
% 0.32/0.59 ! [A: rat,F: num > rat,B: num,C: num] :
% 0.32/0.59 ( ( A
% 0.32/0.59 = ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.59 => ( ! [X4: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_eq_le_subst
% 0.32/0.59 thf(fact_4332_ord__eq__le__subst,axiom,
% 0.32/0.59 ! [A: num,F: num > num,B: num,C: num] :
% 0.32/0.59 ( ( A
% 0.32/0.59 = ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.59 => ( ! [X4: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_eq_le_subst
% 0.32/0.59 thf(fact_4333_ord__eq__le__subst,axiom,
% 0.32/0.59 ! [A: nat,F: num > nat,B: num,C: num] :
% 0.32/0.59 ( ( A
% 0.32/0.59 = ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.59 => ( ! [X4: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_eq_le_subst
% 0.32/0.59 thf(fact_4334_ord__eq__le__subst,axiom,
% 0.32/0.59 ! [A: int,F: num > int,B: num,C: num] :
% 0.32/0.59 ( ( A
% 0.32/0.59 = ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.59 => ( ! [X4: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_eq_le_subst
% 0.32/0.59 thf(fact_4335_ord__eq__le__subst,axiom,
% 0.32/0.59 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 0.32/0.59 ( ( A
% 0.32/0.59 = ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/0.59 => ( ! [X4: nat,Y4: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_eq_le_subst
% 0.32/0.59 thf(fact_4336_ord__eq__le__subst,axiom,
% 0.32/0.59 ! [A: num,F: nat > num,B: nat,C: nat] :
% 0.32/0.59 ( ( A
% 0.32/0.59 = ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/0.59 => ( ! [X4: nat,Y4: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_eq_le_subst
% 0.32/0.59 thf(fact_4337_ord__le__eq__subst,axiom,
% 0.32/0.59 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.59 => ( ( ( F @ B )
% 0.32/0.59 = C )
% 0.32/0.59 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_le_eq_subst
% 0.32/0.59 thf(fact_4338_ord__le__eq__subst,axiom,
% 0.32/0.59 ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.59 => ( ( ( F @ B )
% 0.32/0.59 = C )
% 0.32/0.59 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_le_eq_subst
% 0.32/0.59 thf(fact_4339_ord__le__eq__subst,axiom,
% 0.32/0.59 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.59 => ( ( ( F @ B )
% 0.32/0.59 = C )
% 0.32/0.59 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_le_eq_subst
% 0.32/0.59 thf(fact_4340_ord__le__eq__subst,axiom,
% 0.32/0.59 ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.59 => ( ( ( F @ B )
% 0.32/0.59 = C )
% 0.32/0.59 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_le_eq_subst
% 0.32/0.59 thf(fact_4341_ord__le__eq__subst,axiom,
% 0.32/0.59 ! [A: num,B: num,F: num > rat,C: rat] :
% 0.32/0.59 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.59 => ( ( ( F @ B )
% 0.32/0.59 = C )
% 0.32/0.59 => ( ! [X4: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_le_eq_subst
% 0.32/0.59 thf(fact_4342_ord__le__eq__subst,axiom,
% 0.32/0.59 ! [A: num,B: num,F: num > num,C: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.59 => ( ( ( F @ B )
% 0.32/0.59 = C )
% 0.32/0.59 => ( ! [X4: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_le_eq_subst
% 0.32/0.59 thf(fact_4343_ord__le__eq__subst,axiom,
% 0.32/0.59 ! [A: num,B: num,F: num > nat,C: nat] :
% 0.32/0.59 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.59 => ( ( ( F @ B )
% 0.32/0.59 = C )
% 0.32/0.59 => ( ! [X4: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_le_eq_subst
% 0.32/0.59 thf(fact_4344_ord__le__eq__subst,axiom,
% 0.32/0.59 ! [A: num,B: num,F: num > int,C: int] :
% 0.32/0.59 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.59 => ( ( ( F @ B )
% 0.32/0.59 = C )
% 0.32/0.59 => ( ! [X4: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_le_eq_subst
% 0.32/0.59 thf(fact_4345_ord__le__eq__subst,axiom,
% 0.32/0.59 ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.59 => ( ( ( F @ B )
% 0.32/0.59 = C )
% 0.32/0.59 => ( ! [X4: nat,Y4: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_le_eq_subst
% 0.32/0.59 thf(fact_4346_ord__le__eq__subst,axiom,
% 0.32/0.59 ! [A: nat,B: nat,F: nat > num,C: num] :
% 0.32/0.59 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.59 => ( ( ( F @ B )
% 0.32/0.59 = C )
% 0.32/0.59 => ( ! [X4: nat,Y4: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ X4 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ord_le_eq_subst
% 0.32/0.59 thf(fact_4347_linorder__le__cases,axiom,
% 0.32/0.59 ! [X: rat,Y: rat] :
% 0.32/0.59 ( ~ ( ord_less_eq_rat @ X @ Y )
% 0.32/0.59 => ( ord_less_eq_rat @ Y @ X ) ) ).
% 0.32/0.59
% 0.32/0.59 % linorder_le_cases
% 0.32/0.59 thf(fact_4348_linorder__le__cases,axiom,
% 0.32/0.59 ! [X: num,Y: num] :
% 0.32/0.59 ( ~ ( ord_less_eq_num @ X @ Y )
% 0.32/0.59 => ( ord_less_eq_num @ Y @ X ) ) ).
% 0.32/0.59
% 0.32/0.59 % linorder_le_cases
% 0.32/0.60 thf(fact_4349_linorder__le__cases,axiom,
% 0.32/0.60 ! [X: nat,Y: nat] :
% 0.32/0.60 ( ~ ( ord_less_eq_nat @ X @ Y )
% 0.32/0.60 => ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_le_cases
% 0.32/0.60 thf(fact_4350_linorder__le__cases,axiom,
% 0.32/0.60 ! [X: int,Y: int] :
% 0.32/0.60 ( ~ ( ord_less_eq_int @ X @ Y )
% 0.32/0.60 => ( ord_less_eq_int @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_le_cases
% 0.32/0.60 thf(fact_4351_order__antisym__conv,axiom,
% 0.32/0.60 ! [Y: set_nat,X: set_nat] :
% 0.32/0.60 ( ( ord_less_eq_set_nat @ Y @ X )
% 0.32/0.60 => ( ( ord_less_eq_set_nat @ X @ Y )
% 0.32/0.60 = ( X = Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_antisym_conv
% 0.32/0.60 thf(fact_4352_order__antisym__conv,axiom,
% 0.32/0.60 ! [Y: rat,X: rat] :
% 0.32/0.60 ( ( ord_less_eq_rat @ Y @ X )
% 0.32/0.60 => ( ( ord_less_eq_rat @ X @ Y )
% 0.32/0.60 = ( X = Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_antisym_conv
% 0.32/0.60 thf(fact_4353_order__antisym__conv,axiom,
% 0.32/0.60 ! [Y: num,X: num] :
% 0.32/0.60 ( ( ord_less_eq_num @ Y @ X )
% 0.32/0.60 => ( ( ord_less_eq_num @ X @ Y )
% 0.32/0.60 = ( X = Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_antisym_conv
% 0.32/0.60 thf(fact_4354_order__antisym__conv,axiom,
% 0.32/0.60 ! [Y: nat,X: nat] :
% 0.32/0.60 ( ( ord_less_eq_nat @ Y @ X )
% 0.32/0.60 => ( ( ord_less_eq_nat @ X @ Y )
% 0.32/0.60 = ( X = Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_antisym_conv
% 0.32/0.60 thf(fact_4355_order__antisym__conv,axiom,
% 0.32/0.60 ! [Y: int,X: int] :
% 0.32/0.60 ( ( ord_less_eq_int @ Y @ X )
% 0.32/0.60 => ( ( ord_less_eq_int @ X @ Y )
% 0.32/0.60 = ( X = Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_antisym_conv
% 0.32/0.60 thf(fact_4356_mod__greater__zero__iff__not__dvd,axiom,
% 0.32/0.60 ! [M: nat,N2: nat] :
% 0.32/0.60 ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N2 ) )
% 0.32/0.60 = ( ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mod_greater_zero_iff_not_dvd
% 0.32/0.60 thf(fact_4357_lt__ex,axiom,
% 0.32/0.60 ! [X: real] :
% 0.32/0.60 ? [Y4: real] : ( ord_less_real @ Y4 @ X ) ).
% 0.32/0.60
% 0.32/0.60 % lt_ex
% 0.32/0.60 thf(fact_4358_lt__ex,axiom,
% 0.32/0.60 ! [X: rat] :
% 0.32/0.60 ? [Y4: rat] : ( ord_less_rat @ Y4 @ X ) ).
% 0.32/0.60
% 0.32/0.60 % lt_ex
% 0.32/0.60 thf(fact_4359_lt__ex,axiom,
% 0.32/0.60 ! [X: int] :
% 0.32/0.60 ? [Y4: int] : ( ord_less_int @ Y4 @ X ) ).
% 0.32/0.60
% 0.32/0.60 % lt_ex
% 0.32/0.60 thf(fact_4360_gt__ex,axiom,
% 0.32/0.60 ! [X: real] :
% 0.32/0.60 ? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% 0.32/0.60
% 0.32/0.60 % gt_ex
% 0.32/0.60 thf(fact_4361_gt__ex,axiom,
% 0.32/0.60 ! [X: rat] :
% 0.32/0.60 ? [X_1: rat] : ( ord_less_rat @ X @ X_1 ) ).
% 0.32/0.60
% 0.32/0.60 % gt_ex
% 0.32/0.60 thf(fact_4362_gt__ex,axiom,
% 0.32/0.60 ! [X: nat] :
% 0.32/0.60 ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% 0.32/0.60
% 0.32/0.60 % gt_ex
% 0.32/0.60 thf(fact_4363_gt__ex,axiom,
% 0.32/0.60 ! [X: int] :
% 0.32/0.60 ? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% 0.32/0.60
% 0.32/0.60 % gt_ex
% 0.32/0.60 thf(fact_4364_dense,axiom,
% 0.32/0.60 ! [X: real,Y: real] :
% 0.32/0.60 ( ( ord_less_real @ X @ Y )
% 0.32/0.60 => ? [Z5: real] :
% 0.32/0.60 ( ( ord_less_real @ X @ Z5 )
% 0.32/0.60 & ( ord_less_real @ Z5 @ Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dense
% 0.32/0.60 thf(fact_4365_dense,axiom,
% 0.32/0.60 ! [X: rat,Y: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X @ Y )
% 0.32/0.60 => ? [Z5: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X @ Z5 )
% 0.32/0.60 & ( ord_less_rat @ Z5 @ Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dense
% 0.32/0.60 thf(fact_4366_less__imp__neq,axiom,
% 0.32/0.60 ! [X: real,Y: real] :
% 0.32/0.60 ( ( ord_less_real @ X @ Y )
% 0.32/0.60 => ( X != Y ) ) ).
% 0.32/0.60
% 0.32/0.60 % less_imp_neq
% 0.32/0.60 thf(fact_4367_less__imp__neq,axiom,
% 0.32/0.60 ! [X: rat,Y: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X @ Y )
% 0.32/0.60 => ( X != Y ) ) ).
% 0.32/0.60
% 0.32/0.60 % less_imp_neq
% 0.32/0.60 thf(fact_4368_less__imp__neq,axiom,
% 0.32/0.60 ! [X: num,Y: num] :
% 0.32/0.60 ( ( ord_less_num @ X @ Y )
% 0.32/0.60 => ( X != Y ) ) ).
% 0.32/0.60
% 0.32/0.60 % less_imp_neq
% 0.32/0.60 thf(fact_4369_less__imp__neq,axiom,
% 0.32/0.60 ! [X: nat,Y: nat] :
% 0.32/0.60 ( ( ord_less_nat @ X @ Y )
% 0.32/0.60 => ( X != Y ) ) ).
% 0.32/0.60
% 0.32/0.60 % less_imp_neq
% 0.32/0.60 thf(fact_4370_less__imp__neq,axiom,
% 0.32/0.60 ! [X: int,Y: int] :
% 0.32/0.60 ( ( ord_less_int @ X @ Y )
% 0.32/0.60 => ( X != Y ) ) ).
% 0.32/0.60
% 0.32/0.60 % less_imp_neq
% 0.32/0.60 thf(fact_4371_order_Oasym,axiom,
% 0.32/0.60 ! [A: real,B: real] :
% 0.32/0.60 ( ( ord_less_real @ A @ B )
% 0.32/0.60 => ~ ( ord_less_real @ B @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.asym
% 0.32/0.60 thf(fact_4372_order_Oasym,axiom,
% 0.32/0.60 ! [A: rat,B: rat] :
% 0.32/0.60 ( ( ord_less_rat @ A @ B )
% 0.32/0.60 => ~ ( ord_less_rat @ B @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.asym
% 0.32/0.60 thf(fact_4373_order_Oasym,axiom,
% 0.32/0.60 ! [A: num,B: num] :
% 0.32/0.60 ( ( ord_less_num @ A @ B )
% 0.32/0.60 => ~ ( ord_less_num @ B @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.asym
% 0.32/0.60 thf(fact_4374_order_Oasym,axiom,
% 0.32/0.60 ! [A: nat,B: nat] :
% 0.32/0.60 ( ( ord_less_nat @ A @ B )
% 0.32/0.60 => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.asym
% 0.32/0.60 thf(fact_4375_order_Oasym,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ( ord_less_int @ A @ B )
% 0.32/0.60 => ~ ( ord_less_int @ B @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.asym
% 0.32/0.60 thf(fact_4376_ord__eq__less__trans,axiom,
% 0.32/0.60 ! [A: real,B: real,C: real] :
% 0.32/0.60 ( ( A = B )
% 0.32/0.60 => ( ( ord_less_real @ B @ C )
% 0.32/0.60 => ( ord_less_real @ A @ C ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_eq_less_trans
% 0.32/0.60 thf(fact_4377_ord__eq__less__trans,axiom,
% 0.32/0.60 ! [A: rat,B: rat,C: rat] :
% 0.32/0.60 ( ( A = B )
% 0.32/0.60 => ( ( ord_less_rat @ B @ C )
% 0.32/0.60 => ( ord_less_rat @ A @ C ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_eq_less_trans
% 0.32/0.60 thf(fact_4378_ord__eq__less__trans,axiom,
% 0.32/0.60 ! [A: num,B: num,C: num] :
% 0.32/0.60 ( ( A = B )
% 0.32/0.60 => ( ( ord_less_num @ B @ C )
% 0.32/0.60 => ( ord_less_num @ A @ C ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_eq_less_trans
% 0.32/0.60 thf(fact_4379_ord__eq__less__trans,axiom,
% 0.32/0.60 ! [A: nat,B: nat,C: nat] :
% 0.32/0.60 ( ( A = B )
% 0.32/0.60 => ( ( ord_less_nat @ B @ C )
% 0.32/0.60 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_eq_less_trans
% 0.32/0.60 thf(fact_4380_ord__eq__less__trans,axiom,
% 0.32/0.60 ! [A: int,B: int,C: int] :
% 0.32/0.60 ( ( A = B )
% 0.32/0.60 => ( ( ord_less_int @ B @ C )
% 0.32/0.60 => ( ord_less_int @ A @ C ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_eq_less_trans
% 0.32/0.60 thf(fact_4381_ord__less__eq__trans,axiom,
% 0.32/0.60 ! [A: real,B: real,C: real] :
% 0.32/0.60 ( ( ord_less_real @ A @ B )
% 0.32/0.60 => ( ( B = C )
% 0.32/0.60 => ( ord_less_real @ A @ C ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_less_eq_trans
% 0.32/0.60 thf(fact_4382_ord__less__eq__trans,axiom,
% 0.32/0.60 ! [A: rat,B: rat,C: rat] :
% 0.32/0.60 ( ( ord_less_rat @ A @ B )
% 0.32/0.60 => ( ( B = C )
% 0.32/0.60 => ( ord_less_rat @ A @ C ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_less_eq_trans
% 0.32/0.60 thf(fact_4383_ord__less__eq__trans,axiom,
% 0.32/0.60 ! [A: num,B: num,C: num] :
% 0.32/0.60 ( ( ord_less_num @ A @ B )
% 0.32/0.60 => ( ( B = C )
% 0.32/0.60 => ( ord_less_num @ A @ C ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_less_eq_trans
% 0.32/0.60 thf(fact_4384_ord__less__eq__trans,axiom,
% 0.32/0.60 ! [A: nat,B: nat,C: nat] :
% 0.32/0.60 ( ( ord_less_nat @ A @ B )
% 0.32/0.60 => ( ( B = C )
% 0.32/0.60 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_less_eq_trans
% 0.32/0.60 thf(fact_4385_ord__less__eq__trans,axiom,
% 0.32/0.60 ! [A: int,B: int,C: int] :
% 0.32/0.60 ( ( ord_less_int @ A @ B )
% 0.32/0.60 => ( ( B = C )
% 0.32/0.60 => ( ord_less_int @ A @ C ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_less_eq_trans
% 0.32/0.60 thf(fact_4386_less__induct,axiom,
% 0.32/0.60 ! [P: nat > $o,A: nat] :
% 0.32/0.60 ( ! [X4: nat] :
% 0.32/0.60 ( ! [Y5: nat] :
% 0.32/0.60 ( ( ord_less_nat @ Y5 @ X4 )
% 0.32/0.60 => ( P @ Y5 ) )
% 0.32/0.60 => ( P @ X4 ) )
% 0.32/0.60 => ( P @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % less_induct
% 0.32/0.60 thf(fact_4387_antisym__conv3,axiom,
% 0.32/0.60 ! [Y: real,X: real] :
% 0.32/0.60 ( ~ ( ord_less_real @ Y @ X )
% 0.32/0.60 => ( ( ~ ( ord_less_real @ X @ Y ) )
% 0.32/0.60 = ( X = Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % antisym_conv3
% 0.32/0.60 thf(fact_4388_antisym__conv3,axiom,
% 0.32/0.60 ! [Y: rat,X: rat] :
% 0.32/0.60 ( ~ ( ord_less_rat @ Y @ X )
% 0.32/0.60 => ( ( ~ ( ord_less_rat @ X @ Y ) )
% 0.32/0.60 = ( X = Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % antisym_conv3
% 0.32/0.60 thf(fact_4389_antisym__conv3,axiom,
% 0.32/0.60 ! [Y: num,X: num] :
% 0.32/0.60 ( ~ ( ord_less_num @ Y @ X )
% 0.32/0.60 => ( ( ~ ( ord_less_num @ X @ Y ) )
% 0.32/0.60 = ( X = Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % antisym_conv3
% 0.32/0.60 thf(fact_4390_antisym__conv3,axiom,
% 0.32/0.60 ! [Y: nat,X: nat] :
% 0.32/0.60 ( ~ ( ord_less_nat @ Y @ X )
% 0.32/0.60 => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.32/0.60 = ( X = Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % antisym_conv3
% 0.32/0.60 thf(fact_4391_antisym__conv3,axiom,
% 0.32/0.60 ! [Y: int,X: int] :
% 0.32/0.60 ( ~ ( ord_less_int @ Y @ X )
% 0.32/0.60 => ( ( ~ ( ord_less_int @ X @ Y ) )
% 0.32/0.60 = ( X = Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % antisym_conv3
% 0.32/0.60 thf(fact_4392_linorder__cases,axiom,
% 0.32/0.60 ! [X: real,Y: real] :
% 0.32/0.60 ( ~ ( ord_less_real @ X @ Y )
% 0.32/0.60 => ( ( X != Y )
% 0.32/0.60 => ( ord_less_real @ Y @ X ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_cases
% 0.32/0.60 thf(fact_4393_linorder__cases,axiom,
% 0.32/0.60 ! [X: rat,Y: rat] :
% 0.32/0.60 ( ~ ( ord_less_rat @ X @ Y )
% 0.32/0.60 => ( ( X != Y )
% 0.32/0.60 => ( ord_less_rat @ Y @ X ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_cases
% 0.32/0.60 thf(fact_4394_linorder__cases,axiom,
% 0.32/0.60 ! [X: num,Y: num] :
% 0.32/0.60 ( ~ ( ord_less_num @ X @ Y )
% 0.32/0.60 => ( ( X != Y )
% 0.32/0.60 => ( ord_less_num @ Y @ X ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_cases
% 0.32/0.60 thf(fact_4395_linorder__cases,axiom,
% 0.32/0.60 ! [X: nat,Y: nat] :
% 0.32/0.60 ( ~ ( ord_less_nat @ X @ Y )
% 0.32/0.60 => ( ( X != Y )
% 0.32/0.60 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_cases
% 0.32/0.60 thf(fact_4396_linorder__cases,axiom,
% 0.32/0.60 ! [X: int,Y: int] :
% 0.32/0.60 ( ~ ( ord_less_int @ X @ Y )
% 0.32/0.60 => ( ( X != Y )
% 0.32/0.60 => ( ord_less_int @ Y @ X ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_cases
% 0.32/0.60 thf(fact_4397_dual__order_Oasym,axiom,
% 0.32/0.60 ! [B: real,A: real] :
% 0.32/0.60 ( ( ord_less_real @ B @ A )
% 0.32/0.60 => ~ ( ord_less_real @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.asym
% 0.32/0.60 thf(fact_4398_dual__order_Oasym,axiom,
% 0.32/0.60 ! [B: rat,A: rat] :
% 0.32/0.60 ( ( ord_less_rat @ B @ A )
% 0.32/0.60 => ~ ( ord_less_rat @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.asym
% 0.32/0.60 thf(fact_4399_dual__order_Oasym,axiom,
% 0.32/0.60 ! [B: num,A: num] :
% 0.32/0.60 ( ( ord_less_num @ B @ A )
% 0.32/0.60 => ~ ( ord_less_num @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.asym
% 0.32/0.60 thf(fact_4400_dual__order_Oasym,axiom,
% 0.32/0.60 ! [B: nat,A: nat] :
% 0.32/0.60 ( ( ord_less_nat @ B @ A )
% 0.32/0.60 => ~ ( ord_less_nat @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.asym
% 0.32/0.60 thf(fact_4401_dual__order_Oasym,axiom,
% 0.32/0.60 ! [B: int,A: int] :
% 0.32/0.60 ( ( ord_less_int @ B @ A )
% 0.32/0.60 => ~ ( ord_less_int @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.asym
% 0.32/0.60 thf(fact_4402_dual__order_Oirrefl,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ~ ( ord_less_real @ A @ A ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.irrefl
% 0.32/0.60 thf(fact_4403_dual__order_Oirrefl,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ~ ( ord_less_rat @ A @ A ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.irrefl
% 0.32/0.60 thf(fact_4404_dual__order_Oirrefl,axiom,
% 0.32/0.60 ! [A: num] :
% 0.32/0.60 ~ ( ord_less_num @ A @ A ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.irrefl
% 0.32/0.60 thf(fact_4405_dual__order_Oirrefl,axiom,
% 0.32/0.60 ! [A: nat] :
% 0.32/0.60 ~ ( ord_less_nat @ A @ A ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.irrefl
% 0.32/0.60 thf(fact_4406_dual__order_Oirrefl,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ~ ( ord_less_int @ A @ A ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.irrefl
% 0.32/0.60 thf(fact_4407_exists__least__iff,axiom,
% 0.32/0.60 ( ( ^ [P2: nat > $o] :
% 0.32/0.60 ? [X7: nat] : ( P2 @ X7 ) )
% 0.32/0.60 = ( ^ [P3: nat > $o] :
% 0.32/0.60 ? [N: nat] :
% 0.32/0.60 ( ( P3 @ N )
% 0.32/0.60 & ! [M6: nat] :
% 0.32/0.60 ( ( ord_less_nat @ M6 @ N )
% 0.32/0.60 => ~ ( P3 @ M6 ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % exists_least_iff
% 0.32/0.60 thf(fact_4408_linorder__less__wlog,axiom,
% 0.32/0.60 ! [P: real > real > $o,A: real,B: real] :
% 0.32/0.60 ( ! [A3: real,B2: real] :
% 0.32/0.60 ( ( ord_less_real @ A3 @ B2 )
% 0.32/0.60 => ( P @ A3 @ B2 ) )
% 0.32/0.60 => ( ! [A3: real] : ( P @ A3 @ A3 )
% 0.32/0.60 => ( ! [A3: real,B2: real] :
% 0.32/0.60 ( ( P @ B2 @ A3 )
% 0.32/0.60 => ( P @ A3 @ B2 ) )
% 0.32/0.60 => ( P @ A @ B ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_less_wlog
% 0.32/0.60 thf(fact_4409_linorder__less__wlog,axiom,
% 0.32/0.60 ! [P: rat > rat > $o,A: rat,B: rat] :
% 0.32/0.60 ( ! [A3: rat,B2: rat] :
% 0.32/0.60 ( ( ord_less_rat @ A3 @ B2 )
% 0.32/0.60 => ( P @ A3 @ B2 ) )
% 0.32/0.60 => ( ! [A3: rat] : ( P @ A3 @ A3 )
% 0.32/0.60 => ( ! [A3: rat,B2: rat] :
% 0.32/0.60 ( ( P @ B2 @ A3 )
% 0.32/0.60 => ( P @ A3 @ B2 ) )
% 0.32/0.60 => ( P @ A @ B ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_less_wlog
% 0.32/0.60 thf(fact_4410_linorder__less__wlog,axiom,
% 0.32/0.60 ! [P: num > num > $o,A: num,B: num] :
% 0.32/0.60 ( ! [A3: num,B2: num] :
% 0.32/0.60 ( ( ord_less_num @ A3 @ B2 )
% 0.32/0.60 => ( P @ A3 @ B2 ) )
% 0.32/0.60 => ( ! [A3: num] : ( P @ A3 @ A3 )
% 0.32/0.60 => ( ! [A3: num,B2: num] :
% 0.32/0.60 ( ( P @ B2 @ A3 )
% 0.32/0.60 => ( P @ A3 @ B2 ) )
% 0.32/0.60 => ( P @ A @ B ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_less_wlog
% 0.32/0.60 thf(fact_4411_linorder__less__wlog,axiom,
% 0.32/0.60 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.32/0.60 ( ! [A3: nat,B2: nat] :
% 0.32/0.60 ( ( ord_less_nat @ A3 @ B2 )
% 0.32/0.60 => ( P @ A3 @ B2 ) )
% 0.32/0.60 => ( ! [A3: nat] : ( P @ A3 @ A3 )
% 0.32/0.60 => ( ! [A3: nat,B2: nat] :
% 0.32/0.60 ( ( P @ B2 @ A3 )
% 0.32/0.60 => ( P @ A3 @ B2 ) )
% 0.32/0.60 => ( P @ A @ B ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_less_wlog
% 0.32/0.60 thf(fact_4412_linorder__less__wlog,axiom,
% 0.32/0.60 ! [P: int > int > $o,A: int,B: int] :
% 0.32/0.60 ( ! [A3: int,B2: int] :
% 0.32/0.60 ( ( ord_less_int @ A3 @ B2 )
% 0.32/0.60 => ( P @ A3 @ B2 ) )
% 0.32/0.60 => ( ! [A3: int] : ( P @ A3 @ A3 )
% 0.32/0.60 => ( ! [A3: int,B2: int] :
% 0.32/0.60 ( ( P @ B2 @ A3 )
% 0.32/0.60 => ( P @ A3 @ B2 ) )
% 0.32/0.60 => ( P @ A @ B ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_less_wlog
% 0.32/0.60 thf(fact_4413_order_Ostrict__trans,axiom,
% 0.32/0.60 ! [A: real,B: real,C: real] :
% 0.32/0.60 ( ( ord_less_real @ A @ B )
% 0.32/0.60 => ( ( ord_less_real @ B @ C )
% 0.32/0.60 => ( ord_less_real @ A @ C ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_trans
% 0.32/0.60 thf(fact_4414_order_Ostrict__trans,axiom,
% 0.32/0.60 ! [A: rat,B: rat,C: rat] :
% 0.32/0.60 ( ( ord_less_rat @ A @ B )
% 0.32/0.60 => ( ( ord_less_rat @ B @ C )
% 0.32/0.60 => ( ord_less_rat @ A @ C ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_trans
% 0.32/0.60 thf(fact_4415_order_Ostrict__trans,axiom,
% 0.32/0.60 ! [A: num,B: num,C: num] :
% 0.32/0.60 ( ( ord_less_num @ A @ B )
% 0.32/0.60 => ( ( ord_less_num @ B @ C )
% 0.32/0.60 => ( ord_less_num @ A @ C ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_trans
% 0.32/0.60 thf(fact_4416_order_Ostrict__trans,axiom,
% 0.32/0.60 ! [A: nat,B: nat,C: nat] :
% 0.32/0.60 ( ( ord_less_nat @ A @ B )
% 0.32/0.60 => ( ( ord_less_nat @ B @ C )
% 0.32/0.60 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_trans
% 0.32/0.60 thf(fact_4417_order_Ostrict__trans,axiom,
% 0.32/0.60 ! [A: int,B: int,C: int] :
% 0.32/0.60 ( ( ord_less_int @ A @ B )
% 0.32/0.60 => ( ( ord_less_int @ B @ C )
% 0.32/0.60 => ( ord_less_int @ A @ C ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_trans
% 0.32/0.60 thf(fact_4418_not__less__iff__gr__or__eq,axiom,
% 0.32/0.60 ! [X: real,Y: real] :
% 0.32/0.60 ( ( ~ ( ord_less_real @ X @ Y ) )
% 0.32/0.60 = ( ( ord_less_real @ Y @ X )
% 0.32/0.60 | ( X = Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % not_less_iff_gr_or_eq
% 0.32/0.60 thf(fact_4419_not__less__iff__gr__or__eq,axiom,
% 0.32/0.60 ! [X: rat,Y: rat] :
% 0.32/0.60 ( ( ~ ( ord_less_rat @ X @ Y ) )
% 0.32/0.60 = ( ( ord_less_rat @ Y @ X )
% 0.32/0.60 | ( X = Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % not_less_iff_gr_or_eq
% 0.32/0.60 thf(fact_4420_not__less__iff__gr__or__eq,axiom,
% 0.32/0.60 ! [X: num,Y: num] :
% 0.32/0.60 ( ( ~ ( ord_less_num @ X @ Y ) )
% 0.32/0.60 = ( ( ord_less_num @ Y @ X )
% 0.32/0.60 | ( X = Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % not_less_iff_gr_or_eq
% 0.32/0.60 thf(fact_4421_not__less__iff__gr__or__eq,axiom,
% 0.32/0.60 ! [X: nat,Y: nat] :
% 0.32/0.60 ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.32/0.60 = ( ( ord_less_nat @ Y @ X )
% 0.32/0.60 | ( X = Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % not_less_iff_gr_or_eq
% 0.32/0.60 thf(fact_4422_not__less__iff__gr__or__eq,axiom,
% 0.32/0.60 ! [X: int,Y: int] :
% 0.32/0.60 ( ( ~ ( ord_less_int @ X @ Y ) )
% 0.32/0.60 = ( ( ord_less_int @ Y @ X )
% 0.32/0.60 | ( X = Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % not_less_iff_gr_or_eq
% 0.32/0.60 thf(fact_4423_dual__order_Ostrict__trans,axiom,
% 0.32/0.60 ! [B: real,A: real,C: real] :
% 0.32/0.60 ( ( ord_less_real @ B @ A )
% 0.32/0.60 => ( ( ord_less_real @ C @ B )
% 0.32/0.60 => ( ord_less_real @ C @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_trans
% 0.32/0.60 thf(fact_4424_dual__order_Ostrict__trans,axiom,
% 0.32/0.60 ! [B: rat,A: rat,C: rat] :
% 0.32/0.60 ( ( ord_less_rat @ B @ A )
% 0.32/0.60 => ( ( ord_less_rat @ C @ B )
% 0.32/0.60 => ( ord_less_rat @ C @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_trans
% 0.32/0.60 thf(fact_4425_dual__order_Ostrict__trans,axiom,
% 0.32/0.60 ! [B: num,A: num,C: num] :
% 0.32/0.60 ( ( ord_less_num @ B @ A )
% 0.32/0.60 => ( ( ord_less_num @ C @ B )
% 0.32/0.60 => ( ord_less_num @ C @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_trans
% 0.32/0.60 thf(fact_4426_dual__order_Ostrict__trans,axiom,
% 0.32/0.60 ! [B: nat,A: nat,C: nat] :
% 0.32/0.60 ( ( ord_less_nat @ B @ A )
% 0.32/0.60 => ( ( ord_less_nat @ C @ B )
% 0.32/0.60 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_trans
% 0.32/0.60 thf(fact_4427_dual__order_Ostrict__trans,axiom,
% 0.32/0.60 ! [B: int,A: int,C: int] :
% 0.32/0.60 ( ( ord_less_int @ B @ A )
% 0.32/0.60 => ( ( ord_less_int @ C @ B )
% 0.32/0.60 => ( ord_less_int @ C @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_trans
% 0.32/0.60 thf(fact_4428_order_Ostrict__implies__not__eq,axiom,
% 0.32/0.60 ! [A: real,B: real] :
% 0.32/0.60 ( ( ord_less_real @ A @ B )
% 0.32/0.60 => ( A != B ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_implies_not_eq
% 0.32/0.60 thf(fact_4429_order_Ostrict__implies__not__eq,axiom,
% 0.32/0.60 ! [A: rat,B: rat] :
% 0.32/0.60 ( ( ord_less_rat @ A @ B )
% 0.32/0.60 => ( A != B ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_implies_not_eq
% 0.32/0.60 thf(fact_4430_order_Ostrict__implies__not__eq,axiom,
% 0.32/0.60 ! [A: num,B: num] :
% 0.32/0.60 ( ( ord_less_num @ A @ B )
% 0.32/0.60 => ( A != B ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_implies_not_eq
% 0.32/0.60 thf(fact_4431_order_Ostrict__implies__not__eq,axiom,
% 0.32/0.60 ! [A: nat,B: nat] :
% 0.32/0.60 ( ( ord_less_nat @ A @ B )
% 0.32/0.60 => ( A != B ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_implies_not_eq
% 0.32/0.60 thf(fact_4432_order_Ostrict__implies__not__eq,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ( ord_less_int @ A @ B )
% 0.32/0.60 => ( A != B ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_implies_not_eq
% 0.32/0.60 thf(fact_4433_dual__order_Ostrict__implies__not__eq,axiom,
% 0.32/0.60 ! [B: real,A: real] :
% 0.32/0.60 ( ( ord_less_real @ B @ A )
% 0.32/0.60 => ( A != B ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_implies_not_eq
% 0.32/0.60 thf(fact_4434_dual__order_Ostrict__implies__not__eq,axiom,
% 0.32/0.60 ! [B: rat,A: rat] :
% 0.32/0.60 ( ( ord_less_rat @ B @ A )
% 0.32/0.60 => ( A != B ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_implies_not_eq
% 0.32/0.60 thf(fact_4435_dual__order_Ostrict__implies__not__eq,axiom,
% 0.32/0.60 ! [B: num,A: num] :
% 0.32/0.60 ( ( ord_less_num @ B @ A )
% 0.32/0.60 => ( A != B ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_implies_not_eq
% 0.32/0.60 thf(fact_4436_dual__order_Ostrict__implies__not__eq,axiom,
% 0.32/0.60 ! [B: nat,A: nat] :
% 0.32/0.60 ( ( ord_less_nat @ B @ A )
% 0.32/0.60 => ( A != B ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_implies_not_eq
% 0.32/0.60 thf(fact_4437_dual__order_Ostrict__implies__not__eq,axiom,
% 0.32/0.60 ! [B: int,A: int] :
% 0.32/0.60 ( ( ord_less_int @ B @ A )
% 0.32/0.60 => ( A != B ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_implies_not_eq
% 0.32/0.60 thf(fact_4438_linorder__neqE,axiom,
% 0.32/0.60 ! [X: real,Y: real] :
% 0.32/0.60 ( ( X != Y )
% 0.32/0.60 => ( ~ ( ord_less_real @ X @ Y )
% 0.32/0.60 => ( ord_less_real @ Y @ X ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_neqE
% 0.32/0.60 thf(fact_4439_linorder__neqE,axiom,
% 0.32/0.60 ! [X: rat,Y: rat] :
% 0.32/0.60 ( ( X != Y )
% 0.32/0.60 => ( ~ ( ord_less_rat @ X @ Y )
% 0.32/0.60 => ( ord_less_rat @ Y @ X ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_neqE
% 0.32/0.60 thf(fact_4440_linorder__neqE,axiom,
% 0.32/0.60 ! [X: num,Y: num] :
% 0.32/0.60 ( ( X != Y )
% 0.32/0.60 => ( ~ ( ord_less_num @ X @ Y )
% 0.32/0.60 => ( ord_less_num @ Y @ X ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_neqE
% 0.32/0.60 thf(fact_4441_linorder__neqE,axiom,
% 0.32/0.60 ! [X: nat,Y: nat] :
% 0.32/0.60 ( ( X != Y )
% 0.32/0.60 => ( ~ ( ord_less_nat @ X @ Y )
% 0.32/0.60 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_neqE
% 0.32/0.60 thf(fact_4442_linorder__neqE,axiom,
% 0.32/0.60 ! [X: int,Y: int] :
% 0.32/0.60 ( ( X != Y )
% 0.32/0.60 => ( ~ ( ord_less_int @ X @ Y )
% 0.32/0.60 => ( ord_less_int @ Y @ X ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_neqE
% 0.32/0.60 thf(fact_4443_order__less__asym,axiom,
% 0.32/0.60 ! [X: real,Y: real] :
% 0.32/0.60 ( ( ord_less_real @ X @ Y )
% 0.32/0.60 => ~ ( ord_less_real @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_asym
% 0.32/0.60 thf(fact_4444_order__less__asym,axiom,
% 0.32/0.60 ! [X: rat,Y: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X @ Y )
% 0.32/0.60 => ~ ( ord_less_rat @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_asym
% 0.32/0.60 thf(fact_4445_order__less__asym,axiom,
% 0.32/0.60 ! [X: num,Y: num] :
% 0.32/0.60 ( ( ord_less_num @ X @ Y )
% 0.32/0.60 => ~ ( ord_less_num @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_asym
% 0.32/0.60 thf(fact_4446_order__less__asym,axiom,
% 0.32/0.60 ! [X: nat,Y: nat] :
% 0.32/0.60 ( ( ord_less_nat @ X @ Y )
% 0.32/0.60 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_asym
% 0.32/0.60 thf(fact_4447_order__less__asym,axiom,
% 0.32/0.60 ! [X: int,Y: int] :
% 0.32/0.60 ( ( ord_less_int @ X @ Y )
% 0.32/0.60 => ~ ( ord_less_int @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_asym
% 0.32/0.60 thf(fact_4448_linorder__neq__iff,axiom,
% 0.32/0.60 ! [X: real,Y: real] :
% 0.32/0.60 ( ( X != Y )
% 0.32/0.60 = ( ( ord_less_real @ X @ Y )
% 0.32/0.60 | ( ord_less_real @ Y @ X ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_neq_iff
% 0.32/0.60 thf(fact_4449_linorder__neq__iff,axiom,
% 0.32/0.60 ! [X: rat,Y: rat] :
% 0.32/0.60 ( ( X != Y )
% 0.32/0.60 = ( ( ord_less_rat @ X @ Y )
% 0.32/0.60 | ( ord_less_rat @ Y @ X ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_neq_iff
% 0.32/0.60 thf(fact_4450_linorder__neq__iff,axiom,
% 0.32/0.60 ! [X: num,Y: num] :
% 0.32/0.60 ( ( X != Y )
% 0.32/0.60 = ( ( ord_less_num @ X @ Y )
% 0.32/0.60 | ( ord_less_num @ Y @ X ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_neq_iff
% 0.32/0.60 thf(fact_4451_linorder__neq__iff,axiom,
% 0.32/0.60 ! [X: nat,Y: nat] :
% 0.32/0.60 ( ( X != Y )
% 0.32/0.60 = ( ( ord_less_nat @ X @ Y )
% 0.32/0.60 | ( ord_less_nat @ Y @ X ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_neq_iff
% 0.32/0.60 thf(fact_4452_linorder__neq__iff,axiom,
% 0.32/0.60 ! [X: int,Y: int] :
% 0.32/0.60 ( ( X != Y )
% 0.32/0.60 = ( ( ord_less_int @ X @ Y )
% 0.32/0.60 | ( ord_less_int @ Y @ X ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_neq_iff
% 0.32/0.60 thf(fact_4453_order__less__asym_H,axiom,
% 0.32/0.60 ! [A: real,B: real] :
% 0.32/0.60 ( ( ord_less_real @ A @ B )
% 0.32/0.60 => ~ ( ord_less_real @ B @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_asym'
% 0.32/0.60 thf(fact_4454_order__less__asym_H,axiom,
% 0.32/0.60 ! [A: rat,B: rat] :
% 0.32/0.60 ( ( ord_less_rat @ A @ B )
% 0.32/0.60 => ~ ( ord_less_rat @ B @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_asym'
% 0.32/0.60 thf(fact_4455_order__less__asym_H,axiom,
% 0.32/0.60 ! [A: num,B: num] :
% 0.32/0.60 ( ( ord_less_num @ A @ B )
% 0.32/0.60 => ~ ( ord_less_num @ B @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_asym'
% 0.32/0.60 thf(fact_4456_order__less__asym_H,axiom,
% 0.32/0.60 ! [A: nat,B: nat] :
% 0.32/0.60 ( ( ord_less_nat @ A @ B )
% 0.32/0.60 => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_asym'
% 0.32/0.60 thf(fact_4457_order__less__asym_H,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ( ord_less_int @ A @ B )
% 0.32/0.60 => ~ ( ord_less_int @ B @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_asym'
% 0.32/0.60 thf(fact_4458_order__less__trans,axiom,
% 0.32/0.60 ! [X: real,Y: real,Z: real] :
% 0.32/0.60 ( ( ord_less_real @ X @ Y )
% 0.32/0.60 => ( ( ord_less_real @ Y @ Z )
% 0.32/0.60 => ( ord_less_real @ X @ Z ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_trans
% 0.32/0.60 thf(fact_4459_order__less__trans,axiom,
% 0.32/0.60 ! [X: rat,Y: rat,Z: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X @ Y )
% 0.32/0.60 => ( ( ord_less_rat @ Y @ Z )
% 0.32/0.60 => ( ord_less_rat @ X @ Z ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_trans
% 0.32/0.60 thf(fact_4460_order__less__trans,axiom,
% 0.32/0.60 ! [X: num,Y: num,Z: num] :
% 0.32/0.60 ( ( ord_less_num @ X @ Y )
% 0.32/0.60 => ( ( ord_less_num @ Y @ Z )
% 0.32/0.60 => ( ord_less_num @ X @ Z ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_trans
% 0.32/0.60 thf(fact_4461_order__less__trans,axiom,
% 0.32/0.60 ! [X: nat,Y: nat,Z: nat] :
% 0.32/0.60 ( ( ord_less_nat @ X @ Y )
% 0.32/0.60 => ( ( ord_less_nat @ Y @ Z )
% 0.32/0.60 => ( ord_less_nat @ X @ Z ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_trans
% 0.32/0.60 thf(fact_4462_order__less__trans,axiom,
% 0.32/0.60 ! [X: int,Y: int,Z: int] :
% 0.32/0.60 ( ( ord_less_int @ X @ Y )
% 0.32/0.60 => ( ( ord_less_int @ Y @ Z )
% 0.32/0.60 => ( ord_less_int @ X @ Z ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_trans
% 0.32/0.60 thf(fact_4463_ord__eq__less__subst,axiom,
% 0.32/0.60 ! [A: real,F: real > real,B: real,C: real] :
% 0.32/0.60 ( ( A
% 0.32/0.60 = ( F @ B ) )
% 0.32/0.60 => ( ( ord_less_real @ B @ C )
% 0.32/0.60 => ( ! [X4: real,Y4: real] :
% 0.32/0.60 ( ( ord_less_real @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_eq_less_subst
% 0.32/0.60 thf(fact_4464_ord__eq__less__subst,axiom,
% 0.32/0.60 ! [A: rat,F: real > rat,B: real,C: real] :
% 0.32/0.60 ( ( A
% 0.32/0.60 = ( F @ B ) )
% 0.32/0.60 => ( ( ord_less_real @ B @ C )
% 0.32/0.60 => ( ! [X4: real,Y4: real] :
% 0.32/0.60 ( ( ord_less_real @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_eq_less_subst
% 0.32/0.60 thf(fact_4465_ord__eq__less__subst,axiom,
% 0.32/0.60 ! [A: num,F: real > num,B: real,C: real] :
% 0.32/0.60 ( ( A
% 0.32/0.60 = ( F @ B ) )
% 0.32/0.60 => ( ( ord_less_real @ B @ C )
% 0.32/0.60 => ( ! [X4: real,Y4: real] :
% 0.32/0.60 ( ( ord_less_real @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_eq_less_subst
% 0.32/0.60 thf(fact_4466_ord__eq__less__subst,axiom,
% 0.32/0.60 ! [A: nat,F: real > nat,B: real,C: real] :
% 0.32/0.60 ( ( A
% 0.32/0.60 = ( F @ B ) )
% 0.32/0.60 => ( ( ord_less_real @ B @ C )
% 0.32/0.60 => ( ! [X4: real,Y4: real] :
% 0.32/0.60 ( ( ord_less_real @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_eq_less_subst
% 0.32/0.60 thf(fact_4467_ord__eq__less__subst,axiom,
% 0.32/0.60 ! [A: int,F: real > int,B: real,C: real] :
% 0.32/0.60 ( ( A
% 0.32/0.60 = ( F @ B ) )
% 0.32/0.60 => ( ( ord_less_real @ B @ C )
% 0.32/0.60 => ( ! [X4: real,Y4: real] :
% 0.32/0.60 ( ( ord_less_real @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_eq_less_subst
% 0.32/0.60 thf(fact_4468_ord__eq__less__subst,axiom,
% 0.32/0.60 ! [A: real,F: rat > real,B: rat,C: rat] :
% 0.32/0.60 ( ( A
% 0.32/0.60 = ( F @ B ) )
% 0.32/0.60 => ( ( ord_less_rat @ B @ C )
% 0.32/0.60 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_eq_less_subst
% 0.32/0.60 thf(fact_4469_ord__eq__less__subst,axiom,
% 0.32/0.60 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.32/0.60 ( ( A
% 0.32/0.60 = ( F @ B ) )
% 0.32/0.60 => ( ( ord_less_rat @ B @ C )
% 0.32/0.60 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_eq_less_subst
% 0.32/0.60 thf(fact_4470_ord__eq__less__subst,axiom,
% 0.32/0.60 ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.32/0.60 ( ( A
% 0.32/0.60 = ( F @ B ) )
% 0.32/0.60 => ( ( ord_less_rat @ B @ C )
% 0.32/0.60 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_eq_less_subst
% 0.32/0.60 thf(fact_4471_ord__eq__less__subst,axiom,
% 0.32/0.60 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 0.32/0.60 ( ( A
% 0.32/0.60 = ( F @ B ) )
% 0.32/0.60 => ( ( ord_less_rat @ B @ C )
% 0.32/0.60 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_eq_less_subst
% 0.32/0.60 thf(fact_4472_ord__eq__less__subst,axiom,
% 0.32/0.60 ! [A: int,F: rat > int,B: rat,C: rat] :
% 0.32/0.60 ( ( A
% 0.32/0.60 = ( F @ B ) )
% 0.32/0.60 => ( ( ord_less_rat @ B @ C )
% 0.32/0.60 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_eq_less_subst
% 0.32/0.60 thf(fact_4473_ord__less__eq__subst,axiom,
% 0.32/0.60 ! [A: real,B: real,F: real > real,C: real] :
% 0.32/0.60 ( ( ord_less_real @ A @ B )
% 0.32/0.60 => ( ( ( F @ B )
% 0.32/0.60 = C )
% 0.32/0.60 => ( ! [X4: real,Y4: real] :
% 0.32/0.60 ( ( ord_less_real @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_less_eq_subst
% 0.32/0.60 thf(fact_4474_ord__less__eq__subst,axiom,
% 0.32/0.60 ! [A: real,B: real,F: real > rat,C: rat] :
% 0.32/0.60 ( ( ord_less_real @ A @ B )
% 0.32/0.60 => ( ( ( F @ B )
% 0.32/0.60 = C )
% 0.32/0.60 => ( ! [X4: real,Y4: real] :
% 0.32/0.60 ( ( ord_less_real @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_less_eq_subst
% 0.32/0.60 thf(fact_4475_ord__less__eq__subst,axiom,
% 0.32/0.60 ! [A: real,B: real,F: real > num,C: num] :
% 0.32/0.60 ( ( ord_less_real @ A @ B )
% 0.32/0.60 => ( ( ( F @ B )
% 0.32/0.60 = C )
% 0.32/0.60 => ( ! [X4: real,Y4: real] :
% 0.32/0.60 ( ( ord_less_real @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_less_eq_subst
% 0.32/0.60 thf(fact_4476_ord__less__eq__subst,axiom,
% 0.32/0.60 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.32/0.60 ( ( ord_less_real @ A @ B )
% 0.32/0.60 => ( ( ( F @ B )
% 0.32/0.60 = C )
% 0.32/0.60 => ( ! [X4: real,Y4: real] :
% 0.32/0.60 ( ( ord_less_real @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_less_eq_subst
% 0.32/0.60 thf(fact_4477_ord__less__eq__subst,axiom,
% 0.32/0.60 ! [A: real,B: real,F: real > int,C: int] :
% 0.32/0.60 ( ( ord_less_real @ A @ B )
% 0.32/0.60 => ( ( ( F @ B )
% 0.32/0.60 = C )
% 0.32/0.60 => ( ! [X4: real,Y4: real] :
% 0.32/0.60 ( ( ord_less_real @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_less_eq_subst
% 0.32/0.60 thf(fact_4478_ord__less__eq__subst,axiom,
% 0.32/0.60 ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.32/0.60 ( ( ord_less_rat @ A @ B )
% 0.32/0.60 => ( ( ( F @ B )
% 0.32/0.60 = C )
% 0.32/0.60 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_less_eq_subst
% 0.32/0.60 thf(fact_4479_ord__less__eq__subst,axiom,
% 0.32/0.60 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.32/0.60 ( ( ord_less_rat @ A @ B )
% 0.32/0.60 => ( ( ( F @ B )
% 0.32/0.60 = C )
% 0.32/0.60 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_less_eq_subst
% 0.32/0.60 thf(fact_4480_ord__less__eq__subst,axiom,
% 0.32/0.60 ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.32/0.60 ( ( ord_less_rat @ A @ B )
% 0.32/0.60 => ( ( ( F @ B )
% 0.32/0.60 = C )
% 0.32/0.60 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_less_eq_subst
% 0.32/0.60 thf(fact_4481_ord__less__eq__subst,axiom,
% 0.32/0.60 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.32/0.60 ( ( ord_less_rat @ A @ B )
% 0.32/0.60 => ( ( ( F @ B )
% 0.32/0.60 = C )
% 0.32/0.60 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_less_eq_subst
% 0.32/0.60 thf(fact_4482_ord__less__eq__subst,axiom,
% 0.32/0.60 ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.32/0.60 ( ( ord_less_rat @ A @ B )
% 0.32/0.60 => ( ( ( F @ B )
% 0.32/0.60 = C )
% 0.32/0.60 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % ord_less_eq_subst
% 0.32/0.60 thf(fact_4483_order__less__irrefl,axiom,
% 0.32/0.60 ! [X: real] :
% 0.32/0.60 ~ ( ord_less_real @ X @ X ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_irrefl
% 0.32/0.60 thf(fact_4484_order__less__irrefl,axiom,
% 0.32/0.60 ! [X: rat] :
% 0.32/0.60 ~ ( ord_less_rat @ X @ X ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_irrefl
% 0.32/0.60 thf(fact_4485_order__less__irrefl,axiom,
% 0.32/0.60 ! [X: num] :
% 0.32/0.60 ~ ( ord_less_num @ X @ X ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_irrefl
% 0.32/0.60 thf(fact_4486_order__less__irrefl,axiom,
% 0.32/0.60 ! [X: nat] :
% 0.32/0.60 ~ ( ord_less_nat @ X @ X ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_irrefl
% 0.32/0.60 thf(fact_4487_order__less__irrefl,axiom,
% 0.32/0.60 ! [X: int] :
% 0.32/0.60 ~ ( ord_less_int @ X @ X ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_irrefl
% 0.32/0.60 thf(fact_4488_order__less__subst1,axiom,
% 0.32/0.60 ! [A: real,F: real > real,B: real,C: real] :
% 0.32/0.60 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.32/0.60 => ( ( ord_less_real @ B @ C )
% 0.32/0.60 => ( ! [X4: real,Y4: real] :
% 0.32/0.60 ( ( ord_less_real @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_subst1
% 0.32/0.60 thf(fact_4489_order__less__subst1,axiom,
% 0.32/0.60 ! [A: real,F: rat > real,B: rat,C: rat] :
% 0.32/0.60 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.32/0.60 => ( ( ord_less_rat @ B @ C )
% 0.32/0.60 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_subst1
% 0.32/0.60 thf(fact_4490_order__less__subst1,axiom,
% 0.32/0.60 ! [A: real,F: num > real,B: num,C: num] :
% 0.32/0.60 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.32/0.60 => ( ( ord_less_num @ B @ C )
% 0.32/0.60 => ( ! [X4: num,Y4: num] :
% 0.32/0.60 ( ( ord_less_num @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_subst1
% 0.32/0.60 thf(fact_4491_order__less__subst1,axiom,
% 0.32/0.60 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.32/0.60 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.32/0.60 => ( ( ord_less_nat @ B @ C )
% 0.32/0.60 => ( ! [X4: nat,Y4: nat] :
% 0.32/0.60 ( ( ord_less_nat @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_subst1
% 0.32/0.60 thf(fact_4492_order__less__subst1,axiom,
% 0.32/0.60 ! [A: real,F: int > real,B: int,C: int] :
% 0.32/0.60 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.32/0.60 => ( ( ord_less_int @ B @ C )
% 0.32/0.60 => ( ! [X4: int,Y4: int] :
% 0.32/0.60 ( ( ord_less_int @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_subst1
% 0.32/0.60 thf(fact_4493_order__less__subst1,axiom,
% 0.32/0.60 ! [A: rat,F: real > rat,B: real,C: real] :
% 0.32/0.60 ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.32/0.60 => ( ( ord_less_real @ B @ C )
% 0.32/0.60 => ( ! [X4: real,Y4: real] :
% 0.32/0.60 ( ( ord_less_real @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_subst1
% 0.32/0.60 thf(fact_4494_order__less__subst1,axiom,
% 0.32/0.60 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.32/0.60 ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.32/0.60 => ( ( ord_less_rat @ B @ C )
% 0.32/0.60 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_subst1
% 0.32/0.60 thf(fact_4495_order__less__subst1,axiom,
% 0.32/0.60 ! [A: rat,F: num > rat,B: num,C: num] :
% 0.32/0.60 ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.32/0.60 => ( ( ord_less_num @ B @ C )
% 0.32/0.60 => ( ! [X4: num,Y4: num] :
% 0.32/0.60 ( ( ord_less_num @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_subst1
% 0.32/0.60 thf(fact_4496_order__less__subst1,axiom,
% 0.32/0.60 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 0.32/0.60 ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.32/0.60 => ( ( ord_less_nat @ B @ C )
% 0.32/0.60 => ( ! [X4: nat,Y4: nat] :
% 0.32/0.60 ( ( ord_less_nat @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_subst1
% 0.32/0.60 thf(fact_4497_order__less__subst1,axiom,
% 0.32/0.60 ! [A: rat,F: int > rat,B: int,C: int] :
% 0.32/0.60 ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.32/0.60 => ( ( ord_less_int @ B @ C )
% 0.32/0.60 => ( ! [X4: int,Y4: int] :
% 0.32/0.60 ( ( ord_less_int @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_subst1
% 0.32/0.60 thf(fact_4498_order__less__subst2,axiom,
% 0.32/0.60 ! [A: real,B: real,F: real > real,C: real] :
% 0.32/0.60 ( ( ord_less_real @ A @ B )
% 0.32/0.60 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.32/0.60 => ( ! [X4: real,Y4: real] :
% 0.32/0.60 ( ( ord_less_real @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_subst2
% 0.32/0.60 thf(fact_4499_order__less__subst2,axiom,
% 0.32/0.60 ! [A: real,B: real,F: real > rat,C: rat] :
% 0.32/0.60 ( ( ord_less_real @ A @ B )
% 0.32/0.60 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.32/0.60 => ( ! [X4: real,Y4: real] :
% 0.32/0.60 ( ( ord_less_real @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_subst2
% 0.32/0.60 thf(fact_4500_order__less__subst2,axiom,
% 0.32/0.60 ! [A: real,B: real,F: real > num,C: num] :
% 0.32/0.60 ( ( ord_less_real @ A @ B )
% 0.32/0.60 => ( ( ord_less_num @ ( F @ B ) @ C )
% 0.32/0.60 => ( ! [X4: real,Y4: real] :
% 0.32/0.60 ( ( ord_less_real @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_subst2
% 0.32/0.60 thf(fact_4501_order__less__subst2,axiom,
% 0.32/0.60 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.32/0.60 ( ( ord_less_real @ A @ B )
% 0.32/0.60 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.32/0.60 => ( ! [X4: real,Y4: real] :
% 0.32/0.60 ( ( ord_less_real @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_subst2
% 0.32/0.60 thf(fact_4502_order__less__subst2,axiom,
% 0.32/0.60 ! [A: real,B: real,F: real > int,C: int] :
% 0.32/0.60 ( ( ord_less_real @ A @ B )
% 0.32/0.60 => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.32/0.60 => ( ! [X4: real,Y4: real] :
% 0.32/0.60 ( ( ord_less_real @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_subst2
% 0.32/0.60 thf(fact_4503_order__less__subst2,axiom,
% 0.32/0.60 ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.32/0.60 ( ( ord_less_rat @ A @ B )
% 0.32/0.60 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.32/0.60 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_subst2
% 0.32/0.60 thf(fact_4504_order__less__subst2,axiom,
% 0.32/0.60 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.32/0.60 ( ( ord_less_rat @ A @ B )
% 0.32/0.60 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.32/0.60 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_subst2
% 0.32/0.60 thf(fact_4505_order__less__subst2,axiom,
% 0.32/0.60 ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.32/0.60 ( ( ord_less_rat @ A @ B )
% 0.32/0.60 => ( ( ord_less_num @ ( F @ B ) @ C )
% 0.32/0.60 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_subst2
% 0.32/0.60 thf(fact_4506_order__less__subst2,axiom,
% 0.32/0.60 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.32/0.60 ( ( ord_less_rat @ A @ B )
% 0.32/0.60 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.32/0.60 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_subst2
% 0.32/0.60 thf(fact_4507_order__less__subst2,axiom,
% 0.32/0.60 ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.32/0.60 ( ( ord_less_rat @ A @ B )
% 0.32/0.60 => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.32/0.60 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X4 @ Y4 )
% 0.32/0.60 => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.60 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_subst2
% 0.32/0.60 thf(fact_4508_order__less__not__sym,axiom,
% 0.32/0.60 ! [X: real,Y: real] :
% 0.32/0.60 ( ( ord_less_real @ X @ Y )
% 0.32/0.60 => ~ ( ord_less_real @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_not_sym
% 0.32/0.60 thf(fact_4509_order__less__not__sym,axiom,
% 0.32/0.60 ! [X: rat,Y: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X @ Y )
% 0.32/0.60 => ~ ( ord_less_rat @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_not_sym
% 0.32/0.60 thf(fact_4510_order__less__not__sym,axiom,
% 0.32/0.60 ! [X: num,Y: num] :
% 0.32/0.60 ( ( ord_less_num @ X @ Y )
% 0.32/0.60 => ~ ( ord_less_num @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_not_sym
% 0.32/0.60 thf(fact_4511_order__less__not__sym,axiom,
% 0.32/0.60 ! [X: nat,Y: nat] :
% 0.32/0.60 ( ( ord_less_nat @ X @ Y )
% 0.32/0.60 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_not_sym
% 0.32/0.60 thf(fact_4512_order__less__not__sym,axiom,
% 0.32/0.60 ! [X: int,Y: int] :
% 0.32/0.60 ( ( ord_less_int @ X @ Y )
% 0.32/0.60 => ~ ( ord_less_int @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_not_sym
% 0.32/0.60 thf(fact_4513_order__less__imp__triv,axiom,
% 0.32/0.60 ! [X: real,Y: real,P: $o] :
% 0.32/0.60 ( ( ord_less_real @ X @ Y )
% 0.32/0.60 => ( ( ord_less_real @ Y @ X )
% 0.32/0.60 => P ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_imp_triv
% 0.32/0.60 thf(fact_4514_order__less__imp__triv,axiom,
% 0.32/0.60 ! [X: rat,Y: rat,P: $o] :
% 0.32/0.60 ( ( ord_less_rat @ X @ Y )
% 0.32/0.60 => ( ( ord_less_rat @ Y @ X )
% 0.32/0.60 => P ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_imp_triv
% 0.32/0.60 thf(fact_4515_order__less__imp__triv,axiom,
% 0.32/0.60 ! [X: num,Y: num,P: $o] :
% 0.32/0.60 ( ( ord_less_num @ X @ Y )
% 0.32/0.60 => ( ( ord_less_num @ Y @ X )
% 0.32/0.60 => P ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_imp_triv
% 0.32/0.60 thf(fact_4516_order__less__imp__triv,axiom,
% 0.32/0.60 ! [X: nat,Y: nat,P: $o] :
% 0.32/0.60 ( ( ord_less_nat @ X @ Y )
% 0.32/0.60 => ( ( ord_less_nat @ Y @ X )
% 0.32/0.60 => P ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_imp_triv
% 0.32/0.60 thf(fact_4517_order__less__imp__triv,axiom,
% 0.32/0.60 ! [X: int,Y: int,P: $o] :
% 0.32/0.60 ( ( ord_less_int @ X @ Y )
% 0.32/0.60 => ( ( ord_less_int @ Y @ X )
% 0.32/0.60 => P ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_imp_triv
% 0.32/0.60 thf(fact_4518_linorder__less__linear,axiom,
% 0.32/0.60 ! [X: real,Y: real] :
% 0.32/0.60 ( ( ord_less_real @ X @ Y )
% 0.32/0.60 | ( X = Y )
% 0.32/0.60 | ( ord_less_real @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_less_linear
% 0.32/0.60 thf(fact_4519_linorder__less__linear,axiom,
% 0.32/0.60 ! [X: rat,Y: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X @ Y )
% 0.32/0.60 | ( X = Y )
% 0.32/0.60 | ( ord_less_rat @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_less_linear
% 0.32/0.60 thf(fact_4520_linorder__less__linear,axiom,
% 0.32/0.60 ! [X: num,Y: num] :
% 0.32/0.60 ( ( ord_less_num @ X @ Y )
% 0.32/0.60 | ( X = Y )
% 0.32/0.60 | ( ord_less_num @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_less_linear
% 0.32/0.60 thf(fact_4521_linorder__less__linear,axiom,
% 0.32/0.60 ! [X: nat,Y: nat] :
% 0.32/0.60 ( ( ord_less_nat @ X @ Y )
% 0.32/0.60 | ( X = Y )
% 0.32/0.60 | ( ord_less_nat @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_less_linear
% 0.32/0.60 thf(fact_4522_linorder__less__linear,axiom,
% 0.32/0.60 ! [X: int,Y: int] :
% 0.32/0.60 ( ( ord_less_int @ X @ Y )
% 0.32/0.60 | ( X = Y )
% 0.32/0.60 | ( ord_less_int @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_less_linear
% 0.32/0.60 thf(fact_4523_order__less__imp__not__eq,axiom,
% 0.32/0.60 ! [X: real,Y: real] :
% 0.32/0.60 ( ( ord_less_real @ X @ Y )
% 0.32/0.60 => ( X != Y ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_imp_not_eq
% 0.32/0.60 thf(fact_4524_order__less__imp__not__eq,axiom,
% 0.32/0.60 ! [X: rat,Y: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X @ Y )
% 0.32/0.60 => ( X != Y ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_imp_not_eq
% 0.32/0.60 thf(fact_4525_order__less__imp__not__eq,axiom,
% 0.32/0.60 ! [X: num,Y: num] :
% 0.32/0.60 ( ( ord_less_num @ X @ Y )
% 0.32/0.60 => ( X != Y ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_imp_not_eq
% 0.32/0.60 thf(fact_4526_order__less__imp__not__eq,axiom,
% 0.32/0.60 ! [X: nat,Y: nat] :
% 0.32/0.60 ( ( ord_less_nat @ X @ Y )
% 0.32/0.60 => ( X != Y ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_imp_not_eq
% 0.32/0.60 thf(fact_4527_order__less__imp__not__eq,axiom,
% 0.32/0.60 ! [X: int,Y: int] :
% 0.32/0.60 ( ( ord_less_int @ X @ Y )
% 0.32/0.60 => ( X != Y ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_imp_not_eq
% 0.32/0.60 thf(fact_4528_order__less__imp__not__eq2,axiom,
% 0.32/0.60 ! [X: real,Y: real] :
% 0.32/0.60 ( ( ord_less_real @ X @ Y )
% 0.32/0.60 => ( Y != X ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_imp_not_eq2
% 0.32/0.60 thf(fact_4529_order__less__imp__not__eq2,axiom,
% 0.32/0.60 ! [X: rat,Y: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X @ Y )
% 0.32/0.60 => ( Y != X ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_imp_not_eq2
% 0.32/0.60 thf(fact_4530_order__less__imp__not__eq2,axiom,
% 0.32/0.60 ! [X: num,Y: num] :
% 0.32/0.60 ( ( ord_less_num @ X @ Y )
% 0.32/0.60 => ( Y != X ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_imp_not_eq2
% 0.32/0.60 thf(fact_4531_order__less__imp__not__eq2,axiom,
% 0.32/0.60 ! [X: nat,Y: nat] :
% 0.32/0.60 ( ( ord_less_nat @ X @ Y )
% 0.32/0.60 => ( Y != X ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_imp_not_eq2
% 0.32/0.60 thf(fact_4532_order__less__imp__not__eq2,axiom,
% 0.32/0.60 ! [X: int,Y: int] :
% 0.32/0.60 ( ( ord_less_int @ X @ Y )
% 0.32/0.60 => ( Y != X ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_imp_not_eq2
% 0.32/0.60 thf(fact_4533_order__less__imp__not__less,axiom,
% 0.32/0.60 ! [X: real,Y: real] :
% 0.32/0.60 ( ( ord_less_real @ X @ Y )
% 0.32/0.60 => ~ ( ord_less_real @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_imp_not_less
% 0.32/0.60 thf(fact_4534_order__less__imp__not__less,axiom,
% 0.32/0.60 ! [X: rat,Y: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X @ Y )
% 0.32/0.60 => ~ ( ord_less_rat @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_imp_not_less
% 0.32/0.60 thf(fact_4535_order__less__imp__not__less,axiom,
% 0.32/0.60 ! [X: num,Y: num] :
% 0.32/0.60 ( ( ord_less_num @ X @ Y )
% 0.32/0.60 => ~ ( ord_less_num @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_imp_not_less
% 0.32/0.60 thf(fact_4536_order__less__imp__not__less,axiom,
% 0.32/0.60 ! [X: nat,Y: nat] :
% 0.32/0.60 ( ( ord_less_nat @ X @ Y )
% 0.32/0.60 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_imp_not_less
% 0.32/0.60 thf(fact_4537_order__less__imp__not__less,axiom,
% 0.32/0.60 ! [X: int,Y: int] :
% 0.32/0.60 ( ( ord_less_int @ X @ Y )
% 0.32/0.60 => ~ ( ord_less_int @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_imp_not_less
% 0.32/0.60 thf(fact_4538_mod__eq__dvd__iff__nat,axiom,
% 0.32/0.60 ! [N2: nat,M: nat,Q2: nat] :
% 0.32/0.60 ( ( ord_less_eq_nat @ N2 @ M )
% 0.32/0.60 => ( ( ( modulo_modulo_nat @ M @ Q2 )
% 0.32/0.60 = ( modulo_modulo_nat @ N2 @ Q2 ) )
% 0.32/0.60 = ( dvd_dvd_nat @ Q2 @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mod_eq_dvd_iff_nat
% 0.32/0.60 thf(fact_4539_prod__decode__aux_Ocases,axiom,
% 0.32/0.60 ! [X: product_prod_nat_nat] :
% 0.32/0.60 ~ ! [K2: nat,M2: nat] :
% 0.32/0.60 ( X
% 0.32/0.60 != ( product_Pair_nat_nat @ K2 @ M2 ) ) ).
% 0.32/0.60
% 0.32/0.60 % prod_decode_aux.cases
% 0.32/0.60 thf(fact_4540_finite__divisors__nat,axiom,
% 0.32/0.60 ! [M: nat] :
% 0.32/0.60 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.60 => ( finite_finite_nat
% 0.32/0.60 @ ( collect_nat
% 0.32/0.60 @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % finite_divisors_nat
% 0.32/0.60 thf(fact_4541_even__zero,axiom,
% 0.32/0.60 dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).
% 0.32/0.60
% 0.32/0.60 % even_zero
% 0.32/0.60 thf(fact_4542_even__zero,axiom,
% 0.32/0.60 dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% 0.32/0.60
% 0.32/0.60 % even_zero
% 0.32/0.60 thf(fact_4543_even__zero,axiom,
% 0.32/0.60 dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% 0.32/0.60
% 0.32/0.60 % even_zero
% 0.32/0.60 thf(fact_4544_evenE,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 => ~ ! [B2: code_integer] :
% 0.32/0.60 ( A
% 0.32/0.60 != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % evenE
% 0.32/0.60 thf(fact_4545_evenE,axiom,
% 0.32/0.60 ! [A: nat] :
% 0.32/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 => ~ ! [B2: nat] :
% 0.32/0.60 ( A
% 0.32/0.60 != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % evenE
% 0.32/0.60 thf(fact_4546_evenE,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 => ~ ! [B2: int] :
% 0.32/0.60 ( A
% 0.32/0.60 != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % evenE
% 0.32/0.60 thf(fact_4547_is__unit__div__mult__cancel__right,axiom,
% 0.32/0.60 ! [A: code_integer,B: code_integer] :
% 0.32/0.60 ( ( A != zero_z3403309356797280102nteger )
% 0.32/0.60 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.60 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ A ) )
% 0.32/0.60 = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % is_unit_div_mult_cancel_right
% 0.32/0.60 thf(fact_4548_is__unit__div__mult__cancel__right,axiom,
% 0.32/0.60 ! [A: nat,B: nat] :
% 0.32/0.60 ( ( A != zero_zero_nat )
% 0.32/0.60 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.60 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
% 0.32/0.60 = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % is_unit_div_mult_cancel_right
% 0.32/0.60 thf(fact_4549_is__unit__div__mult__cancel__right,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ( A != zero_zero_int )
% 0.32/0.60 => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.60 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
% 0.32/0.60 = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % is_unit_div_mult_cancel_right
% 0.32/0.60 thf(fact_4550_is__unit__div__mult__cancel__left,axiom,
% 0.32/0.60 ! [A: code_integer,B: code_integer] :
% 0.32/0.60 ( ( A != zero_z3403309356797280102nteger )
% 0.32/0.60 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.60 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.32/0.60 = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % is_unit_div_mult_cancel_left
% 0.32/0.60 thf(fact_4551_is__unit__div__mult__cancel__left,axiom,
% 0.32/0.60 ! [A: nat,B: nat] :
% 0.32/0.60 ( ( A != zero_zero_nat )
% 0.32/0.60 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.60 => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
% 0.32/0.60 = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % is_unit_div_mult_cancel_left
% 0.32/0.60 thf(fact_4552_is__unit__div__mult__cancel__left,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ( A != zero_zero_int )
% 0.32/0.60 => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.60 => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
% 0.32/0.60 = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % is_unit_div_mult_cancel_left
% 0.32/0.60 thf(fact_4553_is__unitE,axiom,
% 0.32/0.60 ! [A: code_integer,C: code_integer] :
% 0.32/0.60 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.60 => ~ ( ( A != zero_z3403309356797280102nteger )
% 0.32/0.60 => ! [B2: code_integer] :
% 0.32/0.60 ( ( B2 != zero_z3403309356797280102nteger )
% 0.32/0.60 => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 0.32/0.60 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
% 0.32/0.60 = B2 )
% 0.32/0.60 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B2 )
% 0.32/0.60 = A )
% 0.32/0.60 => ( ( ( times_3573771949741848930nteger @ A @ B2 )
% 0.32/0.60 = one_one_Code_integer )
% 0.32/0.60 => ( ( divide6298287555418463151nteger @ C @ A )
% 0.32/0.60 != ( times_3573771949741848930nteger @ C @ B2 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % is_unitE
% 0.32/0.60 thf(fact_4554_is__unitE,axiom,
% 0.32/0.60 ! [A: nat,C: nat] :
% 0.32/0.60 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.60 => ~ ( ( A != zero_zero_nat )
% 0.32/0.60 => ! [B2: nat] :
% 0.32/0.60 ( ( B2 != zero_zero_nat )
% 0.32/0.60 => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 0.32/0.60 => ( ( ( divide_divide_nat @ one_one_nat @ A )
% 0.32/0.60 = B2 )
% 0.32/0.60 => ( ( ( divide_divide_nat @ one_one_nat @ B2 )
% 0.32/0.60 = A )
% 0.32/0.60 => ( ( ( times_times_nat @ A @ B2 )
% 0.32/0.60 = one_one_nat )
% 0.32/0.60 => ( ( divide_divide_nat @ C @ A )
% 0.32/0.60 != ( times_times_nat @ C @ B2 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % is_unitE
% 0.32/0.60 thf(fact_4555_is__unitE,axiom,
% 0.32/0.60 ! [A: int,C: int] :
% 0.32/0.60 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.60 => ~ ( ( A != zero_zero_int )
% 0.32/0.60 => ! [B2: int] :
% 0.32/0.60 ( ( B2 != zero_zero_int )
% 0.32/0.60 => ( ( dvd_dvd_int @ B2 @ one_one_int )
% 0.32/0.60 => ( ( ( divide_divide_int @ one_one_int @ A )
% 0.32/0.60 = B2 )
% 0.32/0.60 => ( ( ( divide_divide_int @ one_one_int @ B2 )
% 0.32/0.60 = A )
% 0.32/0.60 => ( ( ( times_times_int @ A @ B2 )
% 0.32/0.60 = one_one_int )
% 0.32/0.60 => ( ( divide_divide_int @ C @ A )
% 0.32/0.60 != ( times_times_int @ C @ B2 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % is_unitE
% 0.32/0.60 thf(fact_4556_odd__even__add,axiom,
% 0.32/0.60 ! [A: code_integer,B: code_integer] :
% 0.32/0.60 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B )
% 0.32/0.60 => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % odd_even_add
% 0.32/0.60 thf(fact_4557_odd__even__add,axiom,
% 0.32/0.60 ! [A: nat,B: nat] :
% 0.32/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.32/0.60 => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % odd_even_add
% 0.32/0.60 thf(fact_4558_odd__even__add,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
% 0.32/0.60 => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % odd_even_add
% 0.32/0.60 thf(fact_4559_odd__one,axiom,
% 0.32/0.60 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).
% 0.32/0.60
% 0.32/0.60 % odd_one
% 0.32/0.60 thf(fact_4560_odd__one,axiom,
% 0.32/0.60 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% 0.32/0.60
% 0.32/0.60 % odd_one
% 0.32/0.60 thf(fact_4561_odd__one,axiom,
% 0.32/0.60 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% 0.32/0.60
% 0.32/0.60 % odd_one
% 0.32/0.60 thf(fact_4562_bit__eq__rec,axiom,
% 0.32/0.60 ( ( ^ [Y3: code_integer,Z2: code_integer] : ( Y3 = Z2 ) )
% 0.32/0.60 = ( ^ [A5: code_integer,B5: code_integer] :
% 0.32/0.60 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 )
% 0.32/0.60 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B5 ) )
% 0.32/0.60 & ( ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.60 = ( divide6298287555418463151nteger @ B5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % bit_eq_rec
% 0.32/0.60 thf(fact_4563_bit__eq__rec,axiom,
% 0.32/0.60 ( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
% 0.32/0.60 = ( ^ [A5: nat,B5: nat] :
% 0.32/0.60 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 )
% 0.32/0.60 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B5 ) )
% 0.32/0.60 & ( ( divide_divide_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.60 = ( divide_divide_nat @ B5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % bit_eq_rec
% 0.32/0.60 thf(fact_4564_bit__eq__rec,axiom,
% 0.32/0.60 ( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
% 0.32/0.60 = ( ^ [A5: int,B5: int] :
% 0.32/0.60 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 )
% 0.32/0.60 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B5 ) )
% 0.32/0.60 & ( ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.60 = ( divide_divide_int @ B5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % bit_eq_rec
% 0.32/0.60 thf(fact_4565_dvd__power__iff,axiom,
% 0.32/0.60 ! [X: code_integer,M: nat,N2: nat] :
% 0.32/0.60 ( ( X != zero_z3403309356797280102nteger )
% 0.32/0.60 => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ N2 ) )
% 0.32/0.60 = ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
% 0.32/0.60 | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dvd_power_iff
% 0.32/0.60 thf(fact_4566_dvd__power__iff,axiom,
% 0.32/0.60 ! [X: nat,M: nat,N2: nat] :
% 0.32/0.60 ( ( X != zero_zero_nat )
% 0.32/0.60 => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N2 ) )
% 0.32/0.60 = ( ( dvd_dvd_nat @ X @ one_one_nat )
% 0.32/0.60 | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dvd_power_iff
% 0.32/0.60 thf(fact_4567_dvd__power__iff,axiom,
% 0.32/0.60 ! [X: int,M: nat,N2: nat] :
% 0.32/0.60 ( ( X != zero_zero_int )
% 0.32/0.60 => ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N2 ) )
% 0.32/0.60 = ( ( dvd_dvd_int @ X @ one_one_int )
% 0.32/0.60 | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dvd_power_iff
% 0.32/0.60 thf(fact_4568_dvd__power,axiom,
% 0.32/0.60 ! [N2: nat,X: code_integer] :
% 0.32/0.60 ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.60 | ( X = one_one_Code_integer ) )
% 0.32/0.60 => ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N2 ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dvd_power
% 0.32/0.60 thf(fact_4569_dvd__power,axiom,
% 0.32/0.60 ! [N2: nat,X: rat] :
% 0.32/0.60 ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.60 | ( X = one_one_rat ) )
% 0.32/0.60 => ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N2 ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dvd_power
% 0.32/0.60 thf(fact_4570_dvd__power,axiom,
% 0.32/0.60 ! [N2: nat,X: nat] :
% 0.32/0.60 ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.60 | ( X = one_one_nat ) )
% 0.32/0.60 => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N2 ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dvd_power
% 0.32/0.60 thf(fact_4571_dvd__power,axiom,
% 0.32/0.60 ! [N2: nat,X: real] :
% 0.32/0.60 ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.60 | ( X = one_one_real ) )
% 0.32/0.60 => ( dvd_dvd_real @ X @ ( power_power_real @ X @ N2 ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dvd_power
% 0.32/0.60 thf(fact_4572_dvd__power,axiom,
% 0.32/0.60 ! [N2: nat,X: int] :
% 0.32/0.60 ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.60 | ( X = one_one_int ) )
% 0.32/0.60 => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N2 ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dvd_power
% 0.32/0.60 thf(fact_4573_dvd__power,axiom,
% 0.32/0.60 ! [N2: nat,X: complex] :
% 0.32/0.60 ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.60 | ( X = one_one_complex ) )
% 0.32/0.60 => ( dvd_dvd_complex @ X @ ( power_power_complex @ X @ N2 ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dvd_power
% 0.32/0.60 thf(fact_4574_even__even__mod__4__iff,axiom,
% 0.32/0.60 ! [N2: nat] :
% 0.32/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.60 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % even_even_mod_4_iff
% 0.32/0.60 thf(fact_4575_dvd__mult__cancel1,axiom,
% 0.32/0.60 ! [M: nat,N2: nat] :
% 0.32/0.60 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.60 => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N2 ) @ M )
% 0.32/0.60 = ( N2 = one_one_nat ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dvd_mult_cancel1
% 0.32/0.60 thf(fact_4576_dvd__mult__cancel2,axiom,
% 0.32/0.60 ! [M: nat,N2: nat] :
% 0.32/0.60 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.60 => ( ( dvd_dvd_nat @ ( times_times_nat @ N2 @ M ) @ M )
% 0.32/0.60 = ( N2 = one_one_nat ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dvd_mult_cancel2
% 0.32/0.60 thf(fact_4577_dvd__minus__add,axiom,
% 0.32/0.60 ! [Q2: nat,N2: nat,R: nat,M: nat] :
% 0.32/0.60 ( ( ord_less_eq_nat @ Q2 @ N2 )
% 0.32/0.60 => ( ( ord_less_eq_nat @ Q2 @ ( times_times_nat @ R @ M ) )
% 0.32/0.60 => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ Q2 ) )
% 0.32/0.60 = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N2 @ ( minus_minus_nat @ ( times_times_nat @ R @ M ) @ Q2 ) ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dvd_minus_add
% 0.32/0.60 thf(fact_4578_power__dvd__imp__le,axiom,
% 0.32/0.60 ! [I2: nat,M: nat,N2: nat] :
% 0.32/0.60 ( ( dvd_dvd_nat @ ( power_power_nat @ I2 @ M ) @ ( power_power_nat @ I2 @ N2 ) )
% 0.32/0.60 => ( ( ord_less_nat @ one_one_nat @ I2 )
% 0.32/0.60 => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % power_dvd_imp_le
% 0.32/0.60 thf(fact_4579_mod__nat__eqI,axiom,
% 0.32/0.60 ! [R: nat,N2: nat,M: nat] :
% 0.32/0.60 ( ( ord_less_nat @ R @ N2 )
% 0.32/0.60 => ( ( ord_less_eq_nat @ R @ M )
% 0.32/0.60 => ( ( dvd_dvd_nat @ N2 @ ( minus_minus_nat @ M @ R ) )
% 0.32/0.60 => ( ( modulo_modulo_nat @ M @ N2 )
% 0.32/0.60 = R ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mod_nat_eqI
% 0.32/0.60 thf(fact_4580_mod__int__pos__iff,axiom,
% 0.32/0.60 ! [K: int,L2: int] :
% 0.32/0.60 ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L2 ) )
% 0.32/0.60 = ( ( dvd_dvd_int @ L2 @ K )
% 0.32/0.60 | ( ( L2 = zero_zero_int )
% 0.32/0.60 & ( ord_less_eq_int @ zero_zero_int @ K ) )
% 0.32/0.60 | ( ord_less_int @ zero_zero_int @ L2 ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mod_int_pos_iff
% 0.32/0.60 thf(fact_4581_bset_I9_J,axiom,
% 0.32/0.60 ! [D: int,D3: int,B3: set_int,T: int] :
% 0.32/0.60 ( ( dvd_dvd_int @ D @ D3 )
% 0.32/0.60 => ! [X5: int] :
% 0.32/0.60 ( ! [Xa3: int] :
% 0.32/0.60 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.60 => ! [Xb3: int] :
% 0.32/0.60 ( ( member_int @ Xb3 @ B3 )
% 0.32/0.60 => ( X5
% 0.32/0.60 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.60 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 0.32/0.60 => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % bset(9)
% 0.32/0.60 thf(fact_4582_bset_I10_J,axiom,
% 0.32/0.60 ! [D: int,D3: int,B3: set_int,T: int] :
% 0.32/0.60 ( ( dvd_dvd_int @ D @ D3 )
% 0.32/0.60 => ! [X5: int] :
% 0.32/0.60 ( ! [Xa3: int] :
% 0.32/0.60 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.60 => ! [Xb3: int] :
% 0.32/0.60 ( ( member_int @ Xb3 @ B3 )
% 0.32/0.60 => ( X5
% 0.32/0.60 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.60 => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 0.32/0.60 => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % bset(10)
% 0.32/0.60 thf(fact_4583_aset_I9_J,axiom,
% 0.32/0.60 ! [D: int,D3: int,A2: set_int,T: int] :
% 0.32/0.60 ( ( dvd_dvd_int @ D @ D3 )
% 0.32/0.60 => ! [X5: int] :
% 0.32/0.60 ( ! [Xa3: int] :
% 0.32/0.60 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.60 => ! [Xb3: int] :
% 0.32/0.60 ( ( member_int @ Xb3 @ A2 )
% 0.32/0.60 => ( X5
% 0.32/0.60 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.60 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 0.32/0.60 => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % aset(9)
% 0.32/0.60 thf(fact_4584_aset_I10_J,axiom,
% 0.32/0.60 ! [D: int,D3: int,A2: set_int,T: int] :
% 0.32/0.60 ( ( dvd_dvd_int @ D @ D3 )
% 0.32/0.60 => ! [X5: int] :
% 0.32/0.60 ( ! [Xa3: int] :
% 0.32/0.60 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 0.32/0.60 => ! [Xb3: int] :
% 0.32/0.60 ( ( member_int @ Xb3 @ A2 )
% 0.32/0.60 => ( X5
% 0.32/0.60 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.60 => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 0.32/0.60 => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % aset(10)
% 0.32/0.60 thf(fact_4585_even__two__times__div__two,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 0.32/0.60 = A ) ) ).
% 0.32/0.60
% 0.32/0.60 % even_two_times_div_two
% 0.32/0.60 thf(fact_4586_even__two__times__div__two,axiom,
% 0.32/0.60 ! [A: nat] :
% 0.32/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.60 = A ) ) ).
% 0.32/0.60
% 0.32/0.60 % even_two_times_div_two
% 0.32/0.60 thf(fact_4587_even__two__times__div__two,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.32/0.60 = A ) ) ).
% 0.32/0.60
% 0.32/0.60 % even_two_times_div_two
% 0.32/0.60 thf(fact_4588_even__iff__mod__2__eq__zero,axiom,
% 0.32/0.60 ! [A: nat] :
% 0.32/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.60 = zero_zero_nat ) ) ).
% 0.32/0.60
% 0.32/0.60 % even_iff_mod_2_eq_zero
% 0.32/0.60 thf(fact_4589_even__iff__mod__2__eq__zero,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.60 = zero_zero_int ) ) ).
% 0.32/0.60
% 0.32/0.60 % even_iff_mod_2_eq_zero
% 0.32/0.60 thf(fact_4590_even__iff__mod__2__eq__zero,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.60 = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.60
% 0.32/0.60 % even_iff_mod_2_eq_zero
% 0.32/0.60 thf(fact_4591_odd__iff__mod__2__eq__one,axiom,
% 0.32/0.60 ! [A: nat] :
% 0.32/0.60 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.60 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.60 = one_one_nat ) ) ).
% 0.32/0.60
% 0.32/0.60 % odd_iff_mod_2_eq_one
% 0.32/0.60 thf(fact_4592_odd__iff__mod__2__eq__one,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.60 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.60 = one_one_int ) ) ).
% 0.32/0.60
% 0.32/0.60 % odd_iff_mod_2_eq_one
% 0.32/0.60 thf(fact_4593_odd__iff__mod__2__eq__one,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.60 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.60 = one_one_Code_integer ) ) ).
% 0.32/0.60
% 0.32/0.60 % odd_iff_mod_2_eq_one
% 0.32/0.60 thf(fact_4594_power__mono__odd,axiom,
% 0.32/0.60 ! [N2: nat,A: real,B: real] :
% 0.32/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.60 => ( ( ord_less_eq_real @ A @ B )
% 0.32/0.60 => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % power_mono_odd
% 0.32/0.60 thf(fact_4595_power__mono__odd,axiom,
% 0.32/0.60 ! [N2: nat,A: rat,B: rat] :
% 0.32/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.60 => ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.60 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % power_mono_odd
% 0.32/0.60 thf(fact_4596_power__mono__odd,axiom,
% 0.32/0.60 ! [N2: nat,A: int,B: int] :
% 0.32/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.60 => ( ( ord_less_eq_int @ A @ B )
% 0.32/0.60 => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % power_mono_odd
% 0.32/0.60 thf(fact_4597_odd__pos,axiom,
% 0.32/0.60 ! [N2: nat] :
% 0.32/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.60 => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 0.32/0.60
% 0.32/0.60 % odd_pos
% 0.32/0.60 thf(fact_4598_dvd__power__iff__le,axiom,
% 0.32/0.60 ! [K: nat,M: nat,N2: nat] :
% 0.32/0.60 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.32/0.60 => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N2 ) )
% 0.32/0.60 = ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dvd_power_iff_le
% 0.32/0.60 thf(fact_4599_even__unset__bit__iff,axiom,
% 0.32/0.60 ! [M: nat,A: code_integer] :
% 0.32/0.60 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A ) )
% 0.32/0.60 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 | ( M = zero_zero_nat ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % even_unset_bit_iff
% 0.32/0.60 thf(fact_4600_even__unset__bit__iff,axiom,
% 0.32/0.60 ! [M: nat,A: int] :
% 0.32/0.60 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 0.32/0.60 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 | ( M = zero_zero_nat ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % even_unset_bit_iff
% 0.32/0.60 thf(fact_4601_even__unset__bit__iff,axiom,
% 0.32/0.60 ! [M: nat,A: nat] :
% 0.32/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 0.32/0.60 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 | ( M = zero_zero_nat ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % even_unset_bit_iff
% 0.32/0.60 thf(fact_4602_even__set__bit__iff,axiom,
% 0.32/0.60 ! [M: nat,A: code_integer] :
% 0.32/0.60 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A ) )
% 0.32/0.60 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 & ( M != zero_zero_nat ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % even_set_bit_iff
% 0.32/0.60 thf(fact_4603_even__set__bit__iff,axiom,
% 0.32/0.60 ! [M: nat,A: int] :
% 0.32/0.60 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 0.32/0.60 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 & ( M != zero_zero_nat ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % even_set_bit_iff
% 0.32/0.60 thf(fact_4604_even__set__bit__iff,axiom,
% 0.32/0.60 ! [M: nat,A: nat] :
% 0.32/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 0.32/0.60 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 & ( M != zero_zero_nat ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % even_set_bit_iff
% 0.32/0.60 thf(fact_4605_even__flip__bit__iff,axiom,
% 0.32/0.60 ! [M: nat,A: code_integer] :
% 0.32/0.60 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A ) )
% 0.32/0.60 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 != ( M = zero_zero_nat ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % even_flip_bit_iff
% 0.32/0.60 thf(fact_4606_even__flip__bit__iff,axiom,
% 0.32/0.60 ! [M: nat,A: int] :
% 0.32/0.60 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 0.32/0.60 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 != ( M = zero_zero_nat ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % even_flip_bit_iff
% 0.32/0.60 thf(fact_4607_even__flip__bit__iff,axiom,
% 0.32/0.60 ! [M: nat,A: nat] :
% 0.32/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 0.32/0.60 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 != ( M = zero_zero_nat ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % even_flip_bit_iff
% 0.32/0.60 thf(fact_4608_even__diff__iff,axiom,
% 0.32/0.60 ! [K: int,L2: int] :
% 0.32/0.60 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L2 ) )
% 0.32/0.60 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % even_diff_iff
% 0.32/0.60 thf(fact_4609_oddE,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 => ~ ! [B2: code_integer] :
% 0.32/0.60 ( A
% 0.32/0.60 != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) @ one_one_Code_integer ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % oddE
% 0.32/0.60 thf(fact_4610_oddE,axiom,
% 0.32/0.60 ! [A: nat] :
% 0.32/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 => ~ ! [B2: nat] :
% 0.32/0.60 ( A
% 0.32/0.60 != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) @ one_one_nat ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % oddE
% 0.32/0.60 thf(fact_4611_oddE,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 => ~ ! [B2: int] :
% 0.32/0.60 ( A
% 0.32/0.60 != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ one_one_int ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % oddE
% 0.32/0.60 thf(fact_4612_parity__cases,axiom,
% 0.32/0.60 ! [A: nat] :
% 0.32/0.60 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.60 != zero_zero_nat ) )
% 0.32/0.60 => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.60 != one_one_nat ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % parity_cases
% 0.32/0.60 thf(fact_4613_parity__cases,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.60 != zero_zero_int ) )
% 0.32/0.60 => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.60 != one_one_int ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % parity_cases
% 0.32/0.60 thf(fact_4614_parity__cases,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.60 != zero_z3403309356797280102nteger ) )
% 0.32/0.60 => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.60 != one_one_Code_integer ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % parity_cases
% 0.32/0.60 thf(fact_4615_mod2__eq__if,axiom,
% 0.32/0.60 ! [A: nat] :
% 0.32/0.60 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.60 = zero_zero_nat ) )
% 0.32/0.60 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.60 = one_one_nat ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mod2_eq_if
% 0.32/0.60 thf(fact_4616_mod2__eq__if,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.60 = zero_zero_int ) )
% 0.32/0.60 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.60 = one_one_int ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mod2_eq_if
% 0.32/0.60 thf(fact_4617_mod2__eq__if,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.60 = zero_z3403309356797280102nteger ) )
% 0.32/0.60 & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.60 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.60 = one_one_Code_integer ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mod2_eq_if
% 0.32/0.60 thf(fact_4618_zero__le__even__power,axiom,
% 0.32/0.60 ! [N2: nat,A: real] :
% 0.32/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.60 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % zero_le_even_power
% 0.32/0.60 thf(fact_4619_zero__le__even__power,axiom,
% 0.32/0.60 ! [N2: nat,A: rat] :
% 0.32/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.60 => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % zero_le_even_power
% 0.32/0.60 thf(fact_4620_zero__le__even__power,axiom,
% 0.32/0.60 ! [N2: nat,A: int] :
% 0.32/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.60 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % zero_le_even_power
% 0.32/0.60 thf(fact_4621_zero__le__odd__power,axiom,
% 0.32/0.60 ! [N2: nat,A: real] :
% 0.32/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.60 => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
% 0.32/0.60 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % zero_le_odd_power
% 0.32/0.60 thf(fact_4622_zero__le__odd__power,axiom,
% 0.32/0.60 ! [N2: nat,A: rat] :
% 0.32/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
% 0.32/0.60 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % zero_le_odd_power
% 0.32/0.60 thf(fact_4623_zero__le__odd__power,axiom,
% 0.32/0.60 ! [N2: nat,A: int] :
% 0.32/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.60 => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
% 0.32/0.60 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % zero_le_odd_power
% 0.32/0.60 thf(fact_4624_zero__le__power__eq,axiom,
% 0.32/0.60 ! [A: real,N2: nat] :
% 0.32/0.60 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
% 0.32/0.60 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.60 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.60 & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % zero_le_power_eq
% 0.32/0.60 thf(fact_4625_zero__le__power__eq,axiom,
% 0.32/0.60 ! [A: rat,N2: nat] :
% 0.32/0.60 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
% 0.32/0.60 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.60 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.60 & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % zero_le_power_eq
% 0.32/0.60 thf(fact_4626_zero__le__power__eq,axiom,
% 0.32/0.60 ! [A: int,N2: nat] :
% 0.32/0.60 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
% 0.32/0.60 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.60 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.60 & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % zero_le_power_eq
% 0.32/0.60 thf(fact_4627_leD,axiom,
% 0.32/0.60 ! [Y: real,X: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ Y @ X )
% 0.32/0.60 => ~ ( ord_less_real @ X @ Y ) ) ).
% 0.32/0.60
% 0.32/0.60 % leD
% 0.32/0.60 thf(fact_4628_leD,axiom,
% 0.32/0.60 ! [Y: set_nat,X: set_nat] :
% 0.32/0.60 ( ( ord_less_eq_set_nat @ Y @ X )
% 0.32/0.60 => ~ ( ord_less_set_nat @ X @ Y ) ) ).
% 0.32/0.60
% 0.32/0.60 % leD
% 0.32/0.60 thf(fact_4629_leD,axiom,
% 0.32/0.60 ! [Y: rat,X: rat] :
% 0.32/0.60 ( ( ord_less_eq_rat @ Y @ X )
% 0.32/0.60 => ~ ( ord_less_rat @ X @ Y ) ) ).
% 0.32/0.60
% 0.32/0.60 % leD
% 0.32/0.60 thf(fact_4630_leD,axiom,
% 0.32/0.60 ! [Y: num,X: num] :
% 0.32/0.60 ( ( ord_less_eq_num @ Y @ X )
% 0.32/0.60 => ~ ( ord_less_num @ X @ Y ) ) ).
% 0.32/0.60
% 0.32/0.60 % leD
% 0.32/0.60 thf(fact_4631_leD,axiom,
% 0.32/0.60 ! [Y: nat,X: nat] :
% 0.32/0.60 ( ( ord_less_eq_nat @ Y @ X )
% 0.32/0.60 => ~ ( ord_less_nat @ X @ Y ) ) ).
% 0.32/0.60
% 0.32/0.60 % leD
% 0.32/0.60 thf(fact_4632_leD,axiom,
% 0.32/0.60 ! [Y: int,X: int] :
% 0.32/0.60 ( ( ord_less_eq_int @ Y @ X )
% 0.32/0.60 => ~ ( ord_less_int @ X @ Y ) ) ).
% 0.32/0.60
% 0.32/0.60 % leD
% 0.32/0.60 thf(fact_4633_leI,axiom,
% 0.32/0.60 ! [X: real,Y: real] :
% 0.32/0.60 ( ~ ( ord_less_real @ X @ Y )
% 0.32/0.60 => ( ord_less_eq_real @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % leI
% 0.32/0.60 thf(fact_4634_leI,axiom,
% 0.32/0.60 ! [X: rat,Y: rat] :
% 0.32/0.60 ( ~ ( ord_less_rat @ X @ Y )
% 0.32/0.60 => ( ord_less_eq_rat @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % leI
% 0.32/0.60 thf(fact_4635_leI,axiom,
% 0.32/0.60 ! [X: num,Y: num] :
% 0.32/0.60 ( ~ ( ord_less_num @ X @ Y )
% 0.32/0.60 => ( ord_less_eq_num @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % leI
% 0.32/0.60 thf(fact_4636_leI,axiom,
% 0.32/0.60 ! [X: nat,Y: nat] :
% 0.32/0.60 ( ~ ( ord_less_nat @ X @ Y )
% 0.32/0.60 => ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % leI
% 0.32/0.60 thf(fact_4637_leI,axiom,
% 0.32/0.60 ! [X: int,Y: int] :
% 0.32/0.60 ( ~ ( ord_less_int @ X @ Y )
% 0.32/0.60 => ( ord_less_eq_int @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % leI
% 0.32/0.60 thf(fact_4638_nless__le,axiom,
% 0.32/0.60 ! [A: real,B: real] :
% 0.32/0.60 ( ( ~ ( ord_less_real @ A @ B ) )
% 0.32/0.60 = ( ~ ( ord_less_eq_real @ A @ B )
% 0.32/0.60 | ( A = B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % nless_le
% 0.32/0.60 thf(fact_4639_nless__le,axiom,
% 0.32/0.60 ! [A: set_nat,B: set_nat] :
% 0.32/0.60 ( ( ~ ( ord_less_set_nat @ A @ B ) )
% 0.32/0.60 = ( ~ ( ord_less_eq_set_nat @ A @ B )
% 0.32/0.60 | ( A = B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % nless_le
% 0.32/0.60 thf(fact_4640_nless__le,axiom,
% 0.32/0.60 ! [A: rat,B: rat] :
% 0.32/0.60 ( ( ~ ( ord_less_rat @ A @ B ) )
% 0.32/0.60 = ( ~ ( ord_less_eq_rat @ A @ B )
% 0.32/0.60 | ( A = B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % nless_le
% 0.32/0.60 thf(fact_4641_nless__le,axiom,
% 0.32/0.60 ! [A: num,B: num] :
% 0.32/0.60 ( ( ~ ( ord_less_num @ A @ B ) )
% 0.32/0.60 = ( ~ ( ord_less_eq_num @ A @ B )
% 0.32/0.60 | ( A = B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % nless_le
% 0.32/0.60 thf(fact_4642_nless__le,axiom,
% 0.32/0.60 ! [A: nat,B: nat] :
% 0.32/0.60 ( ( ~ ( ord_less_nat @ A @ B ) )
% 0.32/0.60 = ( ~ ( ord_less_eq_nat @ A @ B )
% 0.32/0.60 | ( A = B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % nless_le
% 0.32/0.60 thf(fact_4643_nless__le,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ( ~ ( ord_less_int @ A @ B ) )
% 0.32/0.60 = ( ~ ( ord_less_eq_int @ A @ B )
% 0.32/0.60 | ( A = B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % nless_le
% 0.32/0.60 thf(fact_4644_antisym__conv1,axiom,
% 0.32/0.60 ! [X: real,Y: real] :
% 0.32/0.60 ( ~ ( ord_less_real @ X @ Y )
% 0.32/0.60 => ( ( ord_less_eq_real @ X @ Y )
% 0.32/0.60 = ( X = Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % antisym_conv1
% 0.32/0.60 thf(fact_4645_antisym__conv1,axiom,
% 0.32/0.60 ! [X: set_nat,Y: set_nat] :
% 0.32/0.60 ( ~ ( ord_less_set_nat @ X @ Y )
% 0.32/0.60 => ( ( ord_less_eq_set_nat @ X @ Y )
% 0.32/0.60 = ( X = Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % antisym_conv1
% 0.32/0.60 thf(fact_4646_antisym__conv1,axiom,
% 0.32/0.60 ! [X: rat,Y: rat] :
% 0.32/0.60 ( ~ ( ord_less_rat @ X @ Y )
% 0.32/0.60 => ( ( ord_less_eq_rat @ X @ Y )
% 0.32/0.60 = ( X = Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % antisym_conv1
% 0.32/0.60 thf(fact_4647_antisym__conv1,axiom,
% 0.32/0.60 ! [X: num,Y: num] :
% 0.32/0.60 ( ~ ( ord_less_num @ X @ Y )
% 0.32/0.60 => ( ( ord_less_eq_num @ X @ Y )
% 0.32/0.60 = ( X = Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % antisym_conv1
% 0.32/0.60 thf(fact_4648_antisym__conv1,axiom,
% 0.32/0.60 ! [X: nat,Y: nat] :
% 0.32/0.60 ( ~ ( ord_less_nat @ X @ Y )
% 0.32/0.60 => ( ( ord_less_eq_nat @ X @ Y )
% 0.32/0.60 = ( X = Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % antisym_conv1
% 0.32/0.60 thf(fact_4649_antisym__conv1,axiom,
% 0.32/0.60 ! [X: int,Y: int] :
% 0.32/0.60 ( ~ ( ord_less_int @ X @ Y )
% 0.32/0.60 => ( ( ord_less_eq_int @ X @ Y )
% 0.32/0.60 = ( X = Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % antisym_conv1
% 0.32/0.60 thf(fact_4650_antisym__conv2,axiom,
% 0.32/0.60 ! [X: real,Y: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ X @ Y )
% 0.32/0.60 => ( ( ~ ( ord_less_real @ X @ Y ) )
% 0.32/0.60 = ( X = Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % antisym_conv2
% 0.32/0.60 thf(fact_4651_antisym__conv2,axiom,
% 0.32/0.60 ! [X: set_nat,Y: set_nat] :
% 0.32/0.60 ( ( ord_less_eq_set_nat @ X @ Y )
% 0.32/0.60 => ( ( ~ ( ord_less_set_nat @ X @ Y ) )
% 0.32/0.60 = ( X = Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % antisym_conv2
% 0.32/0.60 thf(fact_4652_antisym__conv2,axiom,
% 0.32/0.60 ! [X: rat,Y: rat] :
% 0.32/0.60 ( ( ord_less_eq_rat @ X @ Y )
% 0.32/0.60 => ( ( ~ ( ord_less_rat @ X @ Y ) )
% 0.32/0.60 = ( X = Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % antisym_conv2
% 0.32/0.60 thf(fact_4653_antisym__conv2,axiom,
% 0.32/0.60 ! [X: num,Y: num] :
% 0.32/0.60 ( ( ord_less_eq_num @ X @ Y )
% 0.32/0.60 => ( ( ~ ( ord_less_num @ X @ Y ) )
% 0.32/0.60 = ( X = Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % antisym_conv2
% 0.32/0.60 thf(fact_4654_antisym__conv2,axiom,
% 0.32/0.60 ! [X: nat,Y: nat] :
% 0.32/0.60 ( ( ord_less_eq_nat @ X @ Y )
% 0.32/0.60 => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.32/0.60 = ( X = Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % antisym_conv2
% 0.32/0.60 thf(fact_4655_antisym__conv2,axiom,
% 0.32/0.60 ! [X: int,Y: int] :
% 0.32/0.60 ( ( ord_less_eq_int @ X @ Y )
% 0.32/0.60 => ( ( ~ ( ord_less_int @ X @ Y ) )
% 0.32/0.60 = ( X = Y ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % antisym_conv2
% 0.32/0.60 thf(fact_4656_dense__ge,axiom,
% 0.32/0.60 ! [Z: real,Y: real] :
% 0.32/0.60 ( ! [X4: real] :
% 0.32/0.60 ( ( ord_less_real @ Z @ X4 )
% 0.32/0.60 => ( ord_less_eq_real @ Y @ X4 ) )
% 0.32/0.60 => ( ord_less_eq_real @ Y @ Z ) ) ).
% 0.32/0.60
% 0.32/0.60 % dense_ge
% 0.32/0.60 thf(fact_4657_dense__ge,axiom,
% 0.32/0.60 ! [Z: rat,Y: rat] :
% 0.32/0.60 ( ! [X4: rat] :
% 0.32/0.60 ( ( ord_less_rat @ Z @ X4 )
% 0.32/0.60 => ( ord_less_eq_rat @ Y @ X4 ) )
% 0.32/0.60 => ( ord_less_eq_rat @ Y @ Z ) ) ).
% 0.32/0.60
% 0.32/0.60 % dense_ge
% 0.32/0.60 thf(fact_4658_dense__le,axiom,
% 0.32/0.60 ! [Y: real,Z: real] :
% 0.32/0.60 ( ! [X4: real] :
% 0.32/0.60 ( ( ord_less_real @ X4 @ Y )
% 0.32/0.60 => ( ord_less_eq_real @ X4 @ Z ) )
% 0.32/0.60 => ( ord_less_eq_real @ Y @ Z ) ) ).
% 0.32/0.60
% 0.32/0.60 % dense_le
% 0.32/0.60 thf(fact_4659_dense__le,axiom,
% 0.32/0.60 ! [Y: rat,Z: rat] :
% 0.32/0.60 ( ! [X4: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X4 @ Y )
% 0.32/0.60 => ( ord_less_eq_rat @ X4 @ Z ) )
% 0.32/0.60 => ( ord_less_eq_rat @ Y @ Z ) ) ).
% 0.32/0.60
% 0.32/0.60 % dense_le
% 0.32/0.60 thf(fact_4660_less__le__not__le,axiom,
% 0.32/0.60 ( ord_less_real
% 0.32/0.60 = ( ^ [X2: real,Y2: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ X2 @ Y2 )
% 0.32/0.60 & ~ ( ord_less_eq_real @ Y2 @ X2 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % less_le_not_le
% 0.32/0.60 thf(fact_4661_less__le__not__le,axiom,
% 0.32/0.60 ( ord_less_set_nat
% 0.32/0.60 = ( ^ [X2: set_nat,Y2: set_nat] :
% 0.32/0.60 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.32/0.60 & ~ ( ord_less_eq_set_nat @ Y2 @ X2 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % less_le_not_le
% 0.32/0.60 thf(fact_4662_less__le__not__le,axiom,
% 0.32/0.60 ( ord_less_rat
% 0.32/0.60 = ( ^ [X2: rat,Y2: rat] :
% 0.32/0.60 ( ( ord_less_eq_rat @ X2 @ Y2 )
% 0.32/0.60 & ~ ( ord_less_eq_rat @ Y2 @ X2 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % less_le_not_le
% 0.32/0.60 thf(fact_4663_less__le__not__le,axiom,
% 0.32/0.60 ( ord_less_num
% 0.32/0.60 = ( ^ [X2: num,Y2: num] :
% 0.32/0.60 ( ( ord_less_eq_num @ X2 @ Y2 )
% 0.32/0.60 & ~ ( ord_less_eq_num @ Y2 @ X2 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % less_le_not_le
% 0.32/0.60 thf(fact_4664_less__le__not__le,axiom,
% 0.32/0.60 ( ord_less_nat
% 0.32/0.60 = ( ^ [X2: nat,Y2: nat] :
% 0.32/0.60 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.32/0.60 & ~ ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % less_le_not_le
% 0.32/0.60 thf(fact_4665_less__le__not__le,axiom,
% 0.32/0.60 ( ord_less_int
% 0.32/0.60 = ( ^ [X2: int,Y2: int] :
% 0.32/0.60 ( ( ord_less_eq_int @ X2 @ Y2 )
% 0.32/0.60 & ~ ( ord_less_eq_int @ Y2 @ X2 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % less_le_not_le
% 0.32/0.60 thf(fact_4666_not__le__imp__less,axiom,
% 0.32/0.60 ! [Y: real,X: real] :
% 0.32/0.60 ( ~ ( ord_less_eq_real @ Y @ X )
% 0.32/0.60 => ( ord_less_real @ X @ Y ) ) ).
% 0.32/0.60
% 0.32/0.60 % not_le_imp_less
% 0.32/0.60 thf(fact_4667_not__le__imp__less,axiom,
% 0.32/0.60 ! [Y: rat,X: rat] :
% 0.32/0.60 ( ~ ( ord_less_eq_rat @ Y @ X )
% 0.32/0.60 => ( ord_less_rat @ X @ Y ) ) ).
% 0.32/0.60
% 0.32/0.60 % not_le_imp_less
% 0.32/0.60 thf(fact_4668_not__le__imp__less,axiom,
% 0.32/0.60 ! [Y: num,X: num] :
% 0.32/0.60 ( ~ ( ord_less_eq_num @ Y @ X )
% 0.32/0.60 => ( ord_less_num @ X @ Y ) ) ).
% 0.32/0.60
% 0.32/0.60 % not_le_imp_less
% 0.32/0.60 thf(fact_4669_not__le__imp__less,axiom,
% 0.32/0.60 ! [Y: nat,X: nat] :
% 0.32/0.60 ( ~ ( ord_less_eq_nat @ Y @ X )
% 0.32/0.60 => ( ord_less_nat @ X @ Y ) ) ).
% 0.32/0.60
% 0.32/0.60 % not_le_imp_less
% 0.32/0.60 thf(fact_4670_not__le__imp__less,axiom,
% 0.32/0.60 ! [Y: int,X: int] :
% 0.32/0.60 ( ~ ( ord_less_eq_int @ Y @ X )
% 0.32/0.60 => ( ord_less_int @ X @ Y ) ) ).
% 0.32/0.60
% 0.32/0.60 % not_le_imp_less
% 0.32/0.60 thf(fact_4671_order_Oorder__iff__strict,axiom,
% 0.32/0.60 ( ord_less_eq_real
% 0.32/0.60 = ( ^ [A5: real,B5: real] :
% 0.32/0.60 ( ( ord_less_real @ A5 @ B5 )
% 0.32/0.60 | ( A5 = B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.order_iff_strict
% 0.32/0.60 thf(fact_4672_order_Oorder__iff__strict,axiom,
% 0.32/0.60 ( ord_less_eq_set_nat
% 0.32/0.60 = ( ^ [A5: set_nat,B5: set_nat] :
% 0.32/0.60 ( ( ord_less_set_nat @ A5 @ B5 )
% 0.32/0.60 | ( A5 = B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.order_iff_strict
% 0.32/0.60 thf(fact_4673_order_Oorder__iff__strict,axiom,
% 0.32/0.60 ( ord_less_eq_rat
% 0.32/0.60 = ( ^ [A5: rat,B5: rat] :
% 0.32/0.60 ( ( ord_less_rat @ A5 @ B5 )
% 0.32/0.60 | ( A5 = B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.order_iff_strict
% 0.32/0.60 thf(fact_4674_order_Oorder__iff__strict,axiom,
% 0.32/0.60 ( ord_less_eq_num
% 0.32/0.60 = ( ^ [A5: num,B5: num] :
% 0.32/0.60 ( ( ord_less_num @ A5 @ B5 )
% 0.32/0.60 | ( A5 = B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.order_iff_strict
% 0.32/0.60 thf(fact_4675_order_Oorder__iff__strict,axiom,
% 0.32/0.60 ( ord_less_eq_nat
% 0.32/0.60 = ( ^ [A5: nat,B5: nat] :
% 0.32/0.60 ( ( ord_less_nat @ A5 @ B5 )
% 0.32/0.60 | ( A5 = B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.order_iff_strict
% 0.32/0.60 thf(fact_4676_order_Oorder__iff__strict,axiom,
% 0.32/0.60 ( ord_less_eq_int
% 0.32/0.60 = ( ^ [A5: int,B5: int] :
% 0.32/0.60 ( ( ord_less_int @ A5 @ B5 )
% 0.32/0.60 | ( A5 = B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.order_iff_strict
% 0.32/0.60 thf(fact_4677_order_Ostrict__iff__order,axiom,
% 0.32/0.60 ( ord_less_real
% 0.32/0.60 = ( ^ [A5: real,B5: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ A5 @ B5 )
% 0.32/0.60 & ( A5 != B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_iff_order
% 0.32/0.60 thf(fact_4678_order_Ostrict__iff__order,axiom,
% 0.32/0.60 ( ord_less_set_nat
% 0.32/0.60 = ( ^ [A5: set_nat,B5: set_nat] :
% 0.32/0.60 ( ( ord_less_eq_set_nat @ A5 @ B5 )
% 0.32/0.60 & ( A5 != B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_iff_order
% 0.32/0.60 thf(fact_4679_order_Ostrict__iff__order,axiom,
% 0.32/0.60 ( ord_less_rat
% 0.32/0.60 = ( ^ [A5: rat,B5: rat] :
% 0.32/0.60 ( ( ord_less_eq_rat @ A5 @ B5 )
% 0.32/0.60 & ( A5 != B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_iff_order
% 0.32/0.60 thf(fact_4680_order_Ostrict__iff__order,axiom,
% 0.32/0.60 ( ord_less_num
% 0.32/0.60 = ( ^ [A5: num,B5: num] :
% 0.32/0.60 ( ( ord_less_eq_num @ A5 @ B5 )
% 0.32/0.60 & ( A5 != B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_iff_order
% 0.32/0.60 thf(fact_4681_order_Ostrict__iff__order,axiom,
% 0.32/0.60 ( ord_less_nat
% 0.32/0.60 = ( ^ [A5: nat,B5: nat] :
% 0.32/0.60 ( ( ord_less_eq_nat @ A5 @ B5 )
% 0.32/0.60 & ( A5 != B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_iff_order
% 0.32/0.60 thf(fact_4682_order_Ostrict__iff__order,axiom,
% 0.32/0.60 ( ord_less_int
% 0.32/0.60 = ( ^ [A5: int,B5: int] :
% 0.32/0.60 ( ( ord_less_eq_int @ A5 @ B5 )
% 0.32/0.60 & ( A5 != B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_iff_order
% 0.32/0.60 thf(fact_4683_order_Ostrict__trans1,axiom,
% 0.32/0.60 ! [A: real,B: real,C: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.60 => ( ( ord_less_real @ B @ C )
% 0.32/0.60 => ( ord_less_real @ A @ C ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_trans1
% 0.32/0.60 thf(fact_4684_order_Ostrict__trans1,axiom,
% 0.32/0.60 ! [A: set_nat,B: set_nat,C: set_nat] :
% 0.32/0.60 ( ( ord_less_eq_set_nat @ A @ B )
% 0.32/0.60 => ( ( ord_less_set_nat @ B @ C )
% 0.32/0.60 => ( ord_less_set_nat @ A @ C ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_trans1
% 0.32/0.60 thf(fact_4685_order_Ostrict__trans1,axiom,
% 0.32/0.60 ! [A: rat,B: rat,C: rat] :
% 0.32/0.60 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.60 => ( ( ord_less_rat @ B @ C )
% 0.32/0.60 => ( ord_less_rat @ A @ C ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_trans1
% 0.32/0.60 thf(fact_4686_order_Ostrict__trans1,axiom,
% 0.32/0.60 ! [A: num,B: num,C: num] :
% 0.32/0.60 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.60 => ( ( ord_less_num @ B @ C )
% 0.32/0.60 => ( ord_less_num @ A @ C ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_trans1
% 0.32/0.60 thf(fact_4687_order_Ostrict__trans1,axiom,
% 0.32/0.60 ! [A: nat,B: nat,C: nat] :
% 0.32/0.60 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.60 => ( ( ord_less_nat @ B @ C )
% 0.32/0.60 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_trans1
% 0.32/0.60 thf(fact_4688_order_Ostrict__trans1,axiom,
% 0.32/0.60 ! [A: int,B: int,C: int] :
% 0.32/0.60 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.60 => ( ( ord_less_int @ B @ C )
% 0.32/0.60 => ( ord_less_int @ A @ C ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_trans1
% 0.32/0.60 thf(fact_4689_order_Ostrict__trans2,axiom,
% 0.32/0.60 ! [A: real,B: real,C: real] :
% 0.32/0.60 ( ( ord_less_real @ A @ B )
% 0.32/0.60 => ( ( ord_less_eq_real @ B @ C )
% 0.32/0.60 => ( ord_less_real @ A @ C ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_trans2
% 0.32/0.60 thf(fact_4690_order_Ostrict__trans2,axiom,
% 0.32/0.60 ! [A: set_nat,B: set_nat,C: set_nat] :
% 0.32/0.60 ( ( ord_less_set_nat @ A @ B )
% 0.32/0.60 => ( ( ord_less_eq_set_nat @ B @ C )
% 0.32/0.60 => ( ord_less_set_nat @ A @ C ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_trans2
% 0.32/0.60 thf(fact_4691_order_Ostrict__trans2,axiom,
% 0.32/0.60 ! [A: rat,B: rat,C: rat] :
% 0.32/0.60 ( ( ord_less_rat @ A @ B )
% 0.32/0.60 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.60 => ( ord_less_rat @ A @ C ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_trans2
% 0.32/0.60 thf(fact_4692_order_Ostrict__trans2,axiom,
% 0.32/0.60 ! [A: num,B: num,C: num] :
% 0.32/0.60 ( ( ord_less_num @ A @ B )
% 0.32/0.60 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.60 => ( ord_less_num @ A @ C ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_trans2
% 0.32/0.60 thf(fact_4693_order_Ostrict__trans2,axiom,
% 0.32/0.60 ! [A: nat,B: nat,C: nat] :
% 0.32/0.60 ( ( ord_less_nat @ A @ B )
% 0.32/0.60 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/0.60 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_trans2
% 0.32/0.60 thf(fact_4694_order_Ostrict__trans2,axiom,
% 0.32/0.60 ! [A: int,B: int,C: int] :
% 0.32/0.60 ( ( ord_less_int @ A @ B )
% 0.32/0.60 => ( ( ord_less_eq_int @ B @ C )
% 0.32/0.60 => ( ord_less_int @ A @ C ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_trans2
% 0.32/0.60 thf(fact_4695_order_Ostrict__iff__not,axiom,
% 0.32/0.60 ( ord_less_real
% 0.32/0.60 = ( ^ [A5: real,B5: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ A5 @ B5 )
% 0.32/0.60 & ~ ( ord_less_eq_real @ B5 @ A5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_iff_not
% 0.32/0.60 thf(fact_4696_order_Ostrict__iff__not,axiom,
% 0.32/0.60 ( ord_less_set_nat
% 0.32/0.60 = ( ^ [A5: set_nat,B5: set_nat] :
% 0.32/0.60 ( ( ord_less_eq_set_nat @ A5 @ B5 )
% 0.32/0.60 & ~ ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_iff_not
% 0.32/0.60 thf(fact_4697_order_Ostrict__iff__not,axiom,
% 0.32/0.60 ( ord_less_rat
% 0.32/0.60 = ( ^ [A5: rat,B5: rat] :
% 0.32/0.60 ( ( ord_less_eq_rat @ A5 @ B5 )
% 0.32/0.60 & ~ ( ord_less_eq_rat @ B5 @ A5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_iff_not
% 0.32/0.60 thf(fact_4698_order_Ostrict__iff__not,axiom,
% 0.32/0.60 ( ord_less_num
% 0.32/0.60 = ( ^ [A5: num,B5: num] :
% 0.32/0.60 ( ( ord_less_eq_num @ A5 @ B5 )
% 0.32/0.60 & ~ ( ord_less_eq_num @ B5 @ A5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_iff_not
% 0.32/0.60 thf(fact_4699_order_Ostrict__iff__not,axiom,
% 0.32/0.60 ( ord_less_nat
% 0.32/0.60 = ( ^ [A5: nat,B5: nat] :
% 0.32/0.60 ( ( ord_less_eq_nat @ A5 @ B5 )
% 0.32/0.60 & ~ ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_iff_not
% 0.32/0.60 thf(fact_4700_order_Ostrict__iff__not,axiom,
% 0.32/0.60 ( ord_less_int
% 0.32/0.60 = ( ^ [A5: int,B5: int] :
% 0.32/0.60 ( ( ord_less_eq_int @ A5 @ B5 )
% 0.32/0.60 & ~ ( ord_less_eq_int @ B5 @ A5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_iff_not
% 0.32/0.60 thf(fact_4701_dense__ge__bounded,axiom,
% 0.32/0.60 ! [Z: real,X: real,Y: real] :
% 0.32/0.60 ( ( ord_less_real @ Z @ X )
% 0.32/0.60 => ( ! [W2: real] :
% 0.32/0.60 ( ( ord_less_real @ Z @ W2 )
% 0.32/0.60 => ( ( ord_less_real @ W2 @ X )
% 0.32/0.60 => ( ord_less_eq_real @ Y @ W2 ) ) )
% 0.32/0.60 => ( ord_less_eq_real @ Y @ Z ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dense_ge_bounded
% 0.32/0.60 thf(fact_4702_dense__ge__bounded,axiom,
% 0.32/0.60 ! [Z: rat,X: rat,Y: rat] :
% 0.32/0.60 ( ( ord_less_rat @ Z @ X )
% 0.32/0.60 => ( ! [W2: rat] :
% 0.32/0.60 ( ( ord_less_rat @ Z @ W2 )
% 0.32/0.60 => ( ( ord_less_rat @ W2 @ X )
% 0.32/0.60 => ( ord_less_eq_rat @ Y @ W2 ) ) )
% 0.32/0.60 => ( ord_less_eq_rat @ Y @ Z ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dense_ge_bounded
% 0.32/0.60 thf(fact_4703_dense__le__bounded,axiom,
% 0.32/0.60 ! [X: real,Y: real,Z: real] :
% 0.32/0.60 ( ( ord_less_real @ X @ Y )
% 0.32/0.60 => ( ! [W2: real] :
% 0.32/0.60 ( ( ord_less_real @ X @ W2 )
% 0.32/0.60 => ( ( ord_less_real @ W2 @ Y )
% 0.32/0.60 => ( ord_less_eq_real @ W2 @ Z ) ) )
% 0.32/0.60 => ( ord_less_eq_real @ Y @ Z ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dense_le_bounded
% 0.32/0.60 thf(fact_4704_dense__le__bounded,axiom,
% 0.32/0.60 ! [X: rat,Y: rat,Z: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X @ Y )
% 0.32/0.60 => ( ! [W2: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X @ W2 )
% 0.32/0.60 => ( ( ord_less_rat @ W2 @ Y )
% 0.32/0.60 => ( ord_less_eq_rat @ W2 @ Z ) ) )
% 0.32/0.60 => ( ord_less_eq_rat @ Y @ Z ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dense_le_bounded
% 0.32/0.60 thf(fact_4705_dual__order_Oorder__iff__strict,axiom,
% 0.32/0.60 ( ord_less_eq_real
% 0.32/0.60 = ( ^ [B5: real,A5: real] :
% 0.32/0.60 ( ( ord_less_real @ B5 @ A5 )
% 0.32/0.60 | ( A5 = B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.order_iff_strict
% 0.32/0.60 thf(fact_4706_dual__order_Oorder__iff__strict,axiom,
% 0.32/0.60 ( ord_less_eq_set_nat
% 0.32/0.60 = ( ^ [B5: set_nat,A5: set_nat] :
% 0.32/0.60 ( ( ord_less_set_nat @ B5 @ A5 )
% 0.32/0.60 | ( A5 = B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.order_iff_strict
% 0.32/0.60 thf(fact_4707_dual__order_Oorder__iff__strict,axiom,
% 0.32/0.60 ( ord_less_eq_rat
% 0.32/0.60 = ( ^ [B5: rat,A5: rat] :
% 0.32/0.60 ( ( ord_less_rat @ B5 @ A5 )
% 0.32/0.60 | ( A5 = B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.order_iff_strict
% 0.32/0.60 thf(fact_4708_dual__order_Oorder__iff__strict,axiom,
% 0.32/0.60 ( ord_less_eq_num
% 0.32/0.60 = ( ^ [B5: num,A5: num] :
% 0.32/0.60 ( ( ord_less_num @ B5 @ A5 )
% 0.32/0.60 | ( A5 = B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.order_iff_strict
% 0.32/0.60 thf(fact_4709_dual__order_Oorder__iff__strict,axiom,
% 0.32/0.60 ( ord_less_eq_nat
% 0.32/0.60 = ( ^ [B5: nat,A5: nat] :
% 0.32/0.60 ( ( ord_less_nat @ B5 @ A5 )
% 0.32/0.60 | ( A5 = B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.order_iff_strict
% 0.32/0.60 thf(fact_4710_dual__order_Oorder__iff__strict,axiom,
% 0.32/0.60 ( ord_less_eq_int
% 0.32/0.60 = ( ^ [B5: int,A5: int] :
% 0.32/0.60 ( ( ord_less_int @ B5 @ A5 )
% 0.32/0.60 | ( A5 = B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.order_iff_strict
% 0.32/0.60 thf(fact_4711_dual__order_Ostrict__iff__order,axiom,
% 0.32/0.60 ( ord_less_real
% 0.32/0.60 = ( ^ [B5: real,A5: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ B5 @ A5 )
% 0.32/0.60 & ( A5 != B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_iff_order
% 0.32/0.60 thf(fact_4712_dual__order_Ostrict__iff__order,axiom,
% 0.32/0.60 ( ord_less_set_nat
% 0.32/0.60 = ( ^ [B5: set_nat,A5: set_nat] :
% 0.32/0.60 ( ( ord_less_eq_set_nat @ B5 @ A5 )
% 0.32/0.60 & ( A5 != B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_iff_order
% 0.32/0.60 thf(fact_4713_dual__order_Ostrict__iff__order,axiom,
% 0.32/0.60 ( ord_less_rat
% 0.32/0.60 = ( ^ [B5: rat,A5: rat] :
% 0.32/0.60 ( ( ord_less_eq_rat @ B5 @ A5 )
% 0.32/0.60 & ( A5 != B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_iff_order
% 0.32/0.60 thf(fact_4714_dual__order_Ostrict__iff__order,axiom,
% 0.32/0.60 ( ord_less_num
% 0.32/0.60 = ( ^ [B5: num,A5: num] :
% 0.32/0.60 ( ( ord_less_eq_num @ B5 @ A5 )
% 0.32/0.60 & ( A5 != B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_iff_order
% 0.32/0.60 thf(fact_4715_dual__order_Ostrict__iff__order,axiom,
% 0.32/0.60 ( ord_less_nat
% 0.32/0.60 = ( ^ [B5: nat,A5: nat] :
% 0.32/0.60 ( ( ord_less_eq_nat @ B5 @ A5 )
% 0.32/0.60 & ( A5 != B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_iff_order
% 0.32/0.60 thf(fact_4716_dual__order_Ostrict__iff__order,axiom,
% 0.32/0.60 ( ord_less_int
% 0.32/0.60 = ( ^ [B5: int,A5: int] :
% 0.32/0.60 ( ( ord_less_eq_int @ B5 @ A5 )
% 0.32/0.60 & ( A5 != B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_iff_order
% 0.32/0.60 thf(fact_4717_dual__order_Ostrict__trans1,axiom,
% 0.32/0.60 ! [B: real,A: real,C: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ B @ A )
% 0.32/0.60 => ( ( ord_less_real @ C @ B )
% 0.32/0.60 => ( ord_less_real @ C @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_trans1
% 0.32/0.60 thf(fact_4718_dual__order_Ostrict__trans1,axiom,
% 0.32/0.60 ! [B: set_nat,A: set_nat,C: set_nat] :
% 0.32/0.60 ( ( ord_less_eq_set_nat @ B @ A )
% 0.32/0.60 => ( ( ord_less_set_nat @ C @ B )
% 0.32/0.60 => ( ord_less_set_nat @ C @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_trans1
% 0.32/0.60 thf(fact_4719_dual__order_Ostrict__trans1,axiom,
% 0.32/0.60 ! [B: rat,A: rat,C: rat] :
% 0.32/0.60 ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.60 => ( ( ord_less_rat @ C @ B )
% 0.32/0.60 => ( ord_less_rat @ C @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_trans1
% 0.32/0.60 thf(fact_4720_dual__order_Ostrict__trans1,axiom,
% 0.32/0.60 ! [B: num,A: num,C: num] :
% 0.32/0.60 ( ( ord_less_eq_num @ B @ A )
% 0.32/0.60 => ( ( ord_less_num @ C @ B )
% 0.32/0.60 => ( ord_less_num @ C @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_trans1
% 0.32/0.60 thf(fact_4721_dual__order_Ostrict__trans1,axiom,
% 0.32/0.60 ! [B: nat,A: nat,C: nat] :
% 0.32/0.60 ( ( ord_less_eq_nat @ B @ A )
% 0.32/0.60 => ( ( ord_less_nat @ C @ B )
% 0.32/0.60 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_trans1
% 0.32/0.60 thf(fact_4722_dual__order_Ostrict__trans1,axiom,
% 0.32/0.60 ! [B: int,A: int,C: int] :
% 0.32/0.60 ( ( ord_less_eq_int @ B @ A )
% 0.32/0.60 => ( ( ord_less_int @ C @ B )
% 0.32/0.60 => ( ord_less_int @ C @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_trans1
% 0.32/0.60 thf(fact_4723_dual__order_Ostrict__trans2,axiom,
% 0.32/0.60 ! [B: real,A: real,C: real] :
% 0.32/0.60 ( ( ord_less_real @ B @ A )
% 0.32/0.60 => ( ( ord_less_eq_real @ C @ B )
% 0.32/0.60 => ( ord_less_real @ C @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_trans2
% 0.32/0.60 thf(fact_4724_dual__order_Ostrict__trans2,axiom,
% 0.32/0.60 ! [B: set_nat,A: set_nat,C: set_nat] :
% 0.32/0.60 ( ( ord_less_set_nat @ B @ A )
% 0.32/0.60 => ( ( ord_less_eq_set_nat @ C @ B )
% 0.32/0.60 => ( ord_less_set_nat @ C @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_trans2
% 0.32/0.60 thf(fact_4725_dual__order_Ostrict__trans2,axiom,
% 0.32/0.60 ! [B: rat,A: rat,C: rat] :
% 0.32/0.60 ( ( ord_less_rat @ B @ A )
% 0.32/0.60 => ( ( ord_less_eq_rat @ C @ B )
% 0.32/0.60 => ( ord_less_rat @ C @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_trans2
% 0.32/0.60 thf(fact_4726_dual__order_Ostrict__trans2,axiom,
% 0.32/0.60 ! [B: num,A: num,C: num] :
% 0.32/0.60 ( ( ord_less_num @ B @ A )
% 0.32/0.60 => ( ( ord_less_eq_num @ C @ B )
% 0.32/0.60 => ( ord_less_num @ C @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_trans2
% 0.32/0.60 thf(fact_4727_dual__order_Ostrict__trans2,axiom,
% 0.32/0.60 ! [B: nat,A: nat,C: nat] :
% 0.32/0.60 ( ( ord_less_nat @ B @ A )
% 0.32/0.60 => ( ( ord_less_eq_nat @ C @ B )
% 0.32/0.60 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_trans2
% 0.32/0.60 thf(fact_4728_dual__order_Ostrict__trans2,axiom,
% 0.32/0.60 ! [B: int,A: int,C: int] :
% 0.32/0.60 ( ( ord_less_int @ B @ A )
% 0.32/0.60 => ( ( ord_less_eq_int @ C @ B )
% 0.32/0.60 => ( ord_less_int @ C @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_trans2
% 0.32/0.60 thf(fact_4729_dual__order_Ostrict__iff__not,axiom,
% 0.32/0.60 ( ord_less_real
% 0.32/0.60 = ( ^ [B5: real,A5: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ B5 @ A5 )
% 0.32/0.60 & ~ ( ord_less_eq_real @ A5 @ B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_iff_not
% 0.32/0.60 thf(fact_4730_dual__order_Ostrict__iff__not,axiom,
% 0.32/0.60 ( ord_less_set_nat
% 0.32/0.60 = ( ^ [B5: set_nat,A5: set_nat] :
% 0.32/0.60 ( ( ord_less_eq_set_nat @ B5 @ A5 )
% 0.32/0.60 & ~ ( ord_less_eq_set_nat @ A5 @ B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_iff_not
% 0.32/0.60 thf(fact_4731_dual__order_Ostrict__iff__not,axiom,
% 0.32/0.60 ( ord_less_rat
% 0.32/0.60 = ( ^ [B5: rat,A5: rat] :
% 0.32/0.60 ( ( ord_less_eq_rat @ B5 @ A5 )
% 0.32/0.60 & ~ ( ord_less_eq_rat @ A5 @ B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_iff_not
% 0.32/0.60 thf(fact_4732_dual__order_Ostrict__iff__not,axiom,
% 0.32/0.60 ( ord_less_num
% 0.32/0.60 = ( ^ [B5: num,A5: num] :
% 0.32/0.60 ( ( ord_less_eq_num @ B5 @ A5 )
% 0.32/0.60 & ~ ( ord_less_eq_num @ A5 @ B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_iff_not
% 0.32/0.60 thf(fact_4733_dual__order_Ostrict__iff__not,axiom,
% 0.32/0.60 ( ord_less_nat
% 0.32/0.60 = ( ^ [B5: nat,A5: nat] :
% 0.32/0.60 ( ( ord_less_eq_nat @ B5 @ A5 )
% 0.32/0.60 & ~ ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_iff_not
% 0.32/0.60 thf(fact_4734_dual__order_Ostrict__iff__not,axiom,
% 0.32/0.60 ( ord_less_int
% 0.32/0.60 = ( ^ [B5: int,A5: int] :
% 0.32/0.60 ( ( ord_less_eq_int @ B5 @ A5 )
% 0.32/0.60 & ~ ( ord_less_eq_int @ A5 @ B5 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_iff_not
% 0.32/0.60 thf(fact_4735_order_Ostrict__implies__order,axiom,
% 0.32/0.60 ! [A: real,B: real] :
% 0.32/0.60 ( ( ord_less_real @ A @ B )
% 0.32/0.60 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_implies_order
% 0.32/0.60 thf(fact_4736_order_Ostrict__implies__order,axiom,
% 0.32/0.60 ! [A: set_nat,B: set_nat] :
% 0.32/0.60 ( ( ord_less_set_nat @ A @ B )
% 0.32/0.60 => ( ord_less_eq_set_nat @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_implies_order
% 0.32/0.60 thf(fact_4737_order_Ostrict__implies__order,axiom,
% 0.32/0.60 ! [A: rat,B: rat] :
% 0.32/0.60 ( ( ord_less_rat @ A @ B )
% 0.32/0.60 => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_implies_order
% 0.32/0.60 thf(fact_4738_order_Ostrict__implies__order,axiom,
% 0.32/0.60 ! [A: num,B: num] :
% 0.32/0.60 ( ( ord_less_num @ A @ B )
% 0.32/0.60 => ( ord_less_eq_num @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_implies_order
% 0.32/0.60 thf(fact_4739_order_Ostrict__implies__order,axiom,
% 0.32/0.60 ! [A: nat,B: nat] :
% 0.32/0.60 ( ( ord_less_nat @ A @ B )
% 0.32/0.60 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_implies_order
% 0.32/0.60 thf(fact_4740_order_Ostrict__implies__order,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ( ord_less_int @ A @ B )
% 0.32/0.60 => ( ord_less_eq_int @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % order.strict_implies_order
% 0.32/0.60 thf(fact_4741_dual__order_Ostrict__implies__order,axiom,
% 0.32/0.60 ! [B: real,A: real] :
% 0.32/0.60 ( ( ord_less_real @ B @ A )
% 0.32/0.60 => ( ord_less_eq_real @ B @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_implies_order
% 0.32/0.60 thf(fact_4742_dual__order_Ostrict__implies__order,axiom,
% 0.32/0.60 ! [B: set_nat,A: set_nat] :
% 0.32/0.60 ( ( ord_less_set_nat @ B @ A )
% 0.32/0.60 => ( ord_less_eq_set_nat @ B @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_implies_order
% 0.32/0.60 thf(fact_4743_dual__order_Ostrict__implies__order,axiom,
% 0.32/0.60 ! [B: rat,A: rat] :
% 0.32/0.60 ( ( ord_less_rat @ B @ A )
% 0.32/0.60 => ( ord_less_eq_rat @ B @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_implies_order
% 0.32/0.60 thf(fact_4744_dual__order_Ostrict__implies__order,axiom,
% 0.32/0.60 ! [B: num,A: num] :
% 0.32/0.60 ( ( ord_less_num @ B @ A )
% 0.32/0.60 => ( ord_less_eq_num @ B @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_implies_order
% 0.32/0.60 thf(fact_4745_dual__order_Ostrict__implies__order,axiom,
% 0.32/0.60 ! [B: nat,A: nat] :
% 0.32/0.60 ( ( ord_less_nat @ B @ A )
% 0.32/0.60 => ( ord_less_eq_nat @ B @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_implies_order
% 0.32/0.60 thf(fact_4746_dual__order_Ostrict__implies__order,axiom,
% 0.32/0.60 ! [B: int,A: int] :
% 0.32/0.60 ( ( ord_less_int @ B @ A )
% 0.32/0.60 => ( ord_less_eq_int @ B @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % dual_order.strict_implies_order
% 0.32/0.60 thf(fact_4747_order__le__less,axiom,
% 0.32/0.60 ( ord_less_eq_real
% 0.32/0.60 = ( ^ [X2: real,Y2: real] :
% 0.32/0.60 ( ( ord_less_real @ X2 @ Y2 )
% 0.32/0.60 | ( X2 = Y2 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_le_less
% 0.32/0.60 thf(fact_4748_order__le__less,axiom,
% 0.32/0.60 ( ord_less_eq_set_nat
% 0.32/0.60 = ( ^ [X2: set_nat,Y2: set_nat] :
% 0.32/0.60 ( ( ord_less_set_nat @ X2 @ Y2 )
% 0.32/0.60 | ( X2 = Y2 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_le_less
% 0.32/0.60 thf(fact_4749_order__le__less,axiom,
% 0.32/0.60 ( ord_less_eq_rat
% 0.32/0.60 = ( ^ [X2: rat,Y2: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X2 @ Y2 )
% 0.32/0.60 | ( X2 = Y2 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_le_less
% 0.32/0.60 thf(fact_4750_order__le__less,axiom,
% 0.32/0.60 ( ord_less_eq_num
% 0.32/0.60 = ( ^ [X2: num,Y2: num] :
% 0.32/0.60 ( ( ord_less_num @ X2 @ Y2 )
% 0.32/0.60 | ( X2 = Y2 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_le_less
% 0.32/0.60 thf(fact_4751_order__le__less,axiom,
% 0.32/0.60 ( ord_less_eq_nat
% 0.32/0.60 = ( ^ [X2: nat,Y2: nat] :
% 0.32/0.60 ( ( ord_less_nat @ X2 @ Y2 )
% 0.32/0.60 | ( X2 = Y2 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_le_less
% 0.32/0.60 thf(fact_4752_order__le__less,axiom,
% 0.32/0.60 ( ord_less_eq_int
% 0.32/0.60 = ( ^ [X2: int,Y2: int] :
% 0.32/0.60 ( ( ord_less_int @ X2 @ Y2 )
% 0.32/0.60 | ( X2 = Y2 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_le_less
% 0.32/0.60 thf(fact_4753_order__less__le,axiom,
% 0.32/0.60 ( ord_less_real
% 0.32/0.60 = ( ^ [X2: real,Y2: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ X2 @ Y2 )
% 0.32/0.60 & ( X2 != Y2 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_le
% 0.32/0.60 thf(fact_4754_order__less__le,axiom,
% 0.32/0.60 ( ord_less_set_nat
% 0.32/0.60 = ( ^ [X2: set_nat,Y2: set_nat] :
% 0.32/0.60 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 0.32/0.60 & ( X2 != Y2 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_le
% 0.32/0.60 thf(fact_4755_order__less__le,axiom,
% 0.32/0.60 ( ord_less_rat
% 0.32/0.60 = ( ^ [X2: rat,Y2: rat] :
% 0.32/0.60 ( ( ord_less_eq_rat @ X2 @ Y2 )
% 0.32/0.60 & ( X2 != Y2 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_le
% 0.32/0.60 thf(fact_4756_order__less__le,axiom,
% 0.32/0.60 ( ord_less_num
% 0.32/0.60 = ( ^ [X2: num,Y2: num] :
% 0.32/0.60 ( ( ord_less_eq_num @ X2 @ Y2 )
% 0.32/0.60 & ( X2 != Y2 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_le
% 0.32/0.60 thf(fact_4757_order__less__le,axiom,
% 0.32/0.60 ( ord_less_nat
% 0.32/0.60 = ( ^ [X2: nat,Y2: nat] :
% 0.32/0.60 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.32/0.60 & ( X2 != Y2 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_le
% 0.32/0.60 thf(fact_4758_order__less__le,axiom,
% 0.32/0.60 ( ord_less_int
% 0.32/0.60 = ( ^ [X2: int,Y2: int] :
% 0.32/0.60 ( ( ord_less_eq_int @ X2 @ Y2 )
% 0.32/0.60 & ( X2 != Y2 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_le
% 0.32/0.60 thf(fact_4759_linorder__not__le,axiom,
% 0.32/0.60 ! [X: real,Y: real] :
% 0.32/0.60 ( ( ~ ( ord_less_eq_real @ X @ Y ) )
% 0.32/0.60 = ( ord_less_real @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_not_le
% 0.32/0.60 thf(fact_4760_linorder__not__le,axiom,
% 0.32/0.60 ! [X: rat,Y: rat] :
% 0.32/0.60 ( ( ~ ( ord_less_eq_rat @ X @ Y ) )
% 0.32/0.60 = ( ord_less_rat @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_not_le
% 0.32/0.60 thf(fact_4761_linorder__not__le,axiom,
% 0.32/0.60 ! [X: num,Y: num] :
% 0.32/0.60 ( ( ~ ( ord_less_eq_num @ X @ Y ) )
% 0.32/0.60 = ( ord_less_num @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_not_le
% 0.32/0.60 thf(fact_4762_linorder__not__le,axiom,
% 0.32/0.60 ! [X: nat,Y: nat] :
% 0.32/0.60 ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
% 0.32/0.60 = ( ord_less_nat @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_not_le
% 0.32/0.60 thf(fact_4763_linorder__not__le,axiom,
% 0.32/0.60 ! [X: int,Y: int] :
% 0.32/0.60 ( ( ~ ( ord_less_eq_int @ X @ Y ) )
% 0.32/0.60 = ( ord_less_int @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_not_le
% 0.32/0.60 thf(fact_4764_linorder__not__less,axiom,
% 0.32/0.60 ! [X: real,Y: real] :
% 0.32/0.60 ( ( ~ ( ord_less_real @ X @ Y ) )
% 0.32/0.60 = ( ord_less_eq_real @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_not_less
% 0.32/0.60 thf(fact_4765_linorder__not__less,axiom,
% 0.32/0.60 ! [X: rat,Y: rat] :
% 0.32/0.60 ( ( ~ ( ord_less_rat @ X @ Y ) )
% 0.32/0.60 = ( ord_less_eq_rat @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_not_less
% 0.32/0.60 thf(fact_4766_linorder__not__less,axiom,
% 0.32/0.60 ! [X: num,Y: num] :
% 0.32/0.60 ( ( ~ ( ord_less_num @ X @ Y ) )
% 0.32/0.60 = ( ord_less_eq_num @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_not_less
% 0.32/0.60 thf(fact_4767_linorder__not__less,axiom,
% 0.32/0.60 ! [X: nat,Y: nat] :
% 0.32/0.60 ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.32/0.60 = ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_not_less
% 0.32/0.60 thf(fact_4768_linorder__not__less,axiom,
% 0.32/0.60 ! [X: int,Y: int] :
% 0.32/0.60 ( ( ~ ( ord_less_int @ X @ Y ) )
% 0.32/0.60 = ( ord_less_eq_int @ Y @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % linorder_not_less
% 0.32/0.60 thf(fact_4769_order__less__imp__le,axiom,
% 0.32/0.60 ! [X: real,Y: real] :
% 0.32/0.60 ( ( ord_less_real @ X @ Y )
% 0.32/0.60 => ( ord_less_eq_real @ X @ Y ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_imp_le
% 0.32/0.60 thf(fact_4770_order__less__imp__le,axiom,
% 0.32/0.60 ! [X: set_nat,Y: set_nat] :
% 0.32/0.60 ( ( ord_less_set_nat @ X @ Y )
% 0.32/0.60 => ( ord_less_eq_set_nat @ X @ Y ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_imp_le
% 0.32/0.60 thf(fact_4771_order__less__imp__le,axiom,
% 0.32/0.60 ! [X: rat,Y: rat] :
% 0.32/0.60 ( ( ord_less_rat @ X @ Y )
% 0.32/0.60 => ( ord_less_eq_rat @ X @ Y ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_imp_le
% 0.32/0.60 thf(fact_4772_order__less__imp__le,axiom,
% 0.32/0.60 ! [X: num,Y: num] :
% 0.32/0.60 ( ( ord_less_num @ X @ Y )
% 0.32/0.60 => ( ord_less_eq_num @ X @ Y ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_imp_le
% 0.32/0.60 thf(fact_4773_order__less__imp__le,axiom,
% 0.32/0.60 ! [X: nat,Y: nat] :
% 0.32/0.60 ( ( ord_less_nat @ X @ Y )
% 0.32/0.60 => ( ord_less_eq_nat @ X @ Y ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_imp_le
% 0.32/0.60 thf(fact_4774_order__less__imp__le,axiom,
% 0.32/0.60 ! [X: int,Y: int] :
% 0.32/0.60 ( ( ord_less_int @ X @ Y )
% 0.32/0.60 => ( ord_less_eq_int @ X @ Y ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_less_imp_le
% 0.32/0.60 thf(fact_4775_order__le__neq__trans,axiom,
% 0.32/0.60 ! [A: real,B: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.60 => ( ( A != B )
% 0.32/0.60 => ( ord_less_real @ A @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_le_neq_trans
% 0.32/0.60 thf(fact_4776_order__le__neq__trans,axiom,
% 0.32/0.60 ! [A: set_nat,B: set_nat] :
% 0.32/0.60 ( ( ord_less_eq_set_nat @ A @ B )
% 0.32/0.60 => ( ( A != B )
% 0.32/0.60 => ( ord_less_set_nat @ A @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_le_neq_trans
% 0.32/0.60 thf(fact_4777_order__le__neq__trans,axiom,
% 0.32/0.60 ! [A: rat,B: rat] :
% 0.32/0.60 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.60 => ( ( A != B )
% 0.32/0.60 => ( ord_less_rat @ A @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_le_neq_trans
% 0.32/0.60 thf(fact_4778_order__le__neq__trans,axiom,
% 0.32/0.60 ! [A: num,B: num] :
% 0.32/0.60 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.60 => ( ( A != B )
% 0.32/0.60 => ( ord_less_num @ A @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_le_neq_trans
% 0.32/0.60 thf(fact_4779_order__le__neq__trans,axiom,
% 0.32/0.60 ! [A: nat,B: nat] :
% 0.32/0.60 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.60 => ( ( A != B )
% 0.32/0.60 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % order_le_neq_trans
% 0.32/0.60 thf(fact_4780_order__le__neq__trans,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.61 => ( ( A != B )
% 0.32/0.61 => ( ord_less_int @ A @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_neq_trans
% 0.32/0.61 thf(fact_4781_order__neq__le__trans,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( A != B )
% 0.32/0.61 => ( ( ord_less_eq_real @ A @ B )
% 0.32/0.61 => ( ord_less_real @ A @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_neq_le_trans
% 0.32/0.61 thf(fact_4782_order__neq__le__trans,axiom,
% 0.32/0.61 ! [A: set_nat,B: set_nat] :
% 0.32/0.61 ( ( A != B )
% 0.32/0.61 => ( ( ord_less_eq_set_nat @ A @ B )
% 0.32/0.61 => ( ord_less_set_nat @ A @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_neq_le_trans
% 0.32/0.61 thf(fact_4783_order__neq__le__trans,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( A != B )
% 0.32/0.61 => ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.61 => ( ord_less_rat @ A @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_neq_le_trans
% 0.32/0.61 thf(fact_4784_order__neq__le__trans,axiom,
% 0.32/0.61 ! [A: num,B: num] :
% 0.32/0.61 ( ( A != B )
% 0.32/0.61 => ( ( ord_less_eq_num @ A @ B )
% 0.32/0.61 => ( ord_less_num @ A @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_neq_le_trans
% 0.32/0.61 thf(fact_4785_order__neq__le__trans,axiom,
% 0.32/0.61 ! [A: nat,B: nat] :
% 0.32/0.61 ( ( A != B )
% 0.32/0.61 => ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.61 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_neq_le_trans
% 0.32/0.61 thf(fact_4786_order__neq__le__trans,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( A != B )
% 0.32/0.61 => ( ( ord_less_eq_int @ A @ B )
% 0.32/0.61 => ( ord_less_int @ A @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_neq_le_trans
% 0.32/0.61 thf(fact_4787_order__le__less__trans,axiom,
% 0.32/0.61 ! [X: real,Y: real,Z: real] :
% 0.32/0.61 ( ( ord_less_eq_real @ X @ Y )
% 0.32/0.61 => ( ( ord_less_real @ Y @ Z )
% 0.32/0.61 => ( ord_less_real @ X @ Z ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_less_trans
% 0.32/0.61 thf(fact_4788_order__le__less__trans,axiom,
% 0.32/0.61 ! [X: set_nat,Y: set_nat,Z: set_nat] :
% 0.32/0.61 ( ( ord_less_eq_set_nat @ X @ Y )
% 0.32/0.61 => ( ( ord_less_set_nat @ Y @ Z )
% 0.32/0.61 => ( ord_less_set_nat @ X @ Z ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_less_trans
% 0.32/0.61 thf(fact_4789_order__le__less__trans,axiom,
% 0.32/0.61 ! [X: rat,Y: rat,Z: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ X @ Y )
% 0.32/0.61 => ( ( ord_less_rat @ Y @ Z )
% 0.32/0.61 => ( ord_less_rat @ X @ Z ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_less_trans
% 0.32/0.61 thf(fact_4790_order__le__less__trans,axiom,
% 0.32/0.61 ! [X: num,Y: num,Z: num] :
% 0.32/0.61 ( ( ord_less_eq_num @ X @ Y )
% 0.32/0.61 => ( ( ord_less_num @ Y @ Z )
% 0.32/0.61 => ( ord_less_num @ X @ Z ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_less_trans
% 0.32/0.61 thf(fact_4791_order__le__less__trans,axiom,
% 0.32/0.61 ! [X: nat,Y: nat,Z: nat] :
% 0.32/0.61 ( ( ord_less_eq_nat @ X @ Y )
% 0.32/0.61 => ( ( ord_less_nat @ Y @ Z )
% 0.32/0.61 => ( ord_less_nat @ X @ Z ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_less_trans
% 0.32/0.61 thf(fact_4792_order__le__less__trans,axiom,
% 0.32/0.61 ! [X: int,Y: int,Z: int] :
% 0.32/0.61 ( ( ord_less_eq_int @ X @ Y )
% 0.32/0.61 => ( ( ord_less_int @ Y @ Z )
% 0.32/0.61 => ( ord_less_int @ X @ Z ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_less_trans
% 0.32/0.61 thf(fact_4793_order__less__le__trans,axiom,
% 0.32/0.61 ! [X: real,Y: real,Z: real] :
% 0.32/0.61 ( ( ord_less_real @ X @ Y )
% 0.32/0.61 => ( ( ord_less_eq_real @ Y @ Z )
% 0.32/0.61 => ( ord_less_real @ X @ Z ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_less_le_trans
% 0.32/0.61 thf(fact_4794_order__less__le__trans,axiom,
% 0.32/0.61 ! [X: set_nat,Y: set_nat,Z: set_nat] :
% 0.32/0.61 ( ( ord_less_set_nat @ X @ Y )
% 0.32/0.61 => ( ( ord_less_eq_set_nat @ Y @ Z )
% 0.32/0.61 => ( ord_less_set_nat @ X @ Z ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_less_le_trans
% 0.32/0.61 thf(fact_4795_order__less__le__trans,axiom,
% 0.32/0.61 ! [X: rat,Y: rat,Z: rat] :
% 0.32/0.61 ( ( ord_less_rat @ X @ Y )
% 0.32/0.61 => ( ( ord_less_eq_rat @ Y @ Z )
% 0.32/0.61 => ( ord_less_rat @ X @ Z ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_less_le_trans
% 0.32/0.61 thf(fact_4796_order__less__le__trans,axiom,
% 0.32/0.61 ! [X: num,Y: num,Z: num] :
% 0.32/0.61 ( ( ord_less_num @ X @ Y )
% 0.32/0.61 => ( ( ord_less_eq_num @ Y @ Z )
% 0.32/0.61 => ( ord_less_num @ X @ Z ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_less_le_trans
% 0.32/0.61 thf(fact_4797_order__less__le__trans,axiom,
% 0.32/0.61 ! [X: nat,Y: nat,Z: nat] :
% 0.32/0.61 ( ( ord_less_nat @ X @ Y )
% 0.32/0.61 => ( ( ord_less_eq_nat @ Y @ Z )
% 0.32/0.61 => ( ord_less_nat @ X @ Z ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_less_le_trans
% 0.32/0.61 thf(fact_4798_order__less__le__trans,axiom,
% 0.32/0.61 ! [X: int,Y: int,Z: int] :
% 0.32/0.61 ( ( ord_less_int @ X @ Y )
% 0.32/0.61 => ( ( ord_less_eq_int @ Y @ Z )
% 0.32/0.61 => ( ord_less_int @ X @ Z ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_less_le_trans
% 0.32/0.61 thf(fact_4799_order__le__less__subst1,axiom,
% 0.32/0.61 ! [A: real,F: real > real,B: real,C: real] :
% 0.32/0.61 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.32/0.61 => ( ( ord_less_real @ B @ C )
% 0.32/0.61 => ( ! [X4: real,Y4: real] :
% 0.32/0.61 ( ( ord_less_real @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_less_subst1
% 0.32/0.61 thf(fact_4800_order__le__less__subst1,axiom,
% 0.32/0.61 ! [A: real,F: rat > real,B: rat,C: rat] :
% 0.32/0.61 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.32/0.61 => ( ( ord_less_rat @ B @ C )
% 0.32/0.61 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.61 ( ( ord_less_rat @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_less_subst1
% 0.32/0.61 thf(fact_4801_order__le__less__subst1,axiom,
% 0.32/0.61 ! [A: real,F: num > real,B: num,C: num] :
% 0.32/0.61 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.32/0.61 => ( ( ord_less_num @ B @ C )
% 0.32/0.61 => ( ! [X4: num,Y4: num] :
% 0.32/0.61 ( ( ord_less_num @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_less_subst1
% 0.32/0.61 thf(fact_4802_order__le__less__subst1,axiom,
% 0.32/0.61 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.32/0.61 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.32/0.61 => ( ( ord_less_nat @ B @ C )
% 0.32/0.61 => ( ! [X4: nat,Y4: nat] :
% 0.32/0.61 ( ( ord_less_nat @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_less_subst1
% 0.32/0.61 thf(fact_4803_order__le__less__subst1,axiom,
% 0.32/0.61 ! [A: real,F: int > real,B: int,C: int] :
% 0.32/0.61 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.32/0.61 => ( ( ord_less_int @ B @ C )
% 0.32/0.61 => ( ! [X4: int,Y4: int] :
% 0.32/0.61 ( ( ord_less_int @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_less_subst1
% 0.32/0.61 thf(fact_4804_order__le__less__subst1,axiom,
% 0.32/0.61 ! [A: rat,F: real > rat,B: real,C: real] :
% 0.32/0.61 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.32/0.61 => ( ( ord_less_real @ B @ C )
% 0.32/0.61 => ( ! [X4: real,Y4: real] :
% 0.32/0.61 ( ( ord_less_real @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_less_subst1
% 0.32/0.61 thf(fact_4805_order__le__less__subst1,axiom,
% 0.32/0.61 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.32/0.61 => ( ( ord_less_rat @ B @ C )
% 0.32/0.61 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.61 ( ( ord_less_rat @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_less_subst1
% 0.32/0.61 thf(fact_4806_order__le__less__subst1,axiom,
% 0.32/0.61 ! [A: rat,F: num > rat,B: num,C: num] :
% 0.32/0.61 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.32/0.61 => ( ( ord_less_num @ B @ C )
% 0.32/0.61 => ( ! [X4: num,Y4: num] :
% 0.32/0.61 ( ( ord_less_num @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_less_subst1
% 0.32/0.61 thf(fact_4807_order__le__less__subst1,axiom,
% 0.32/0.61 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.32/0.61 => ( ( ord_less_nat @ B @ C )
% 0.32/0.61 => ( ! [X4: nat,Y4: nat] :
% 0.32/0.61 ( ( ord_less_nat @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_less_subst1
% 0.32/0.61 thf(fact_4808_order__le__less__subst1,axiom,
% 0.32/0.61 ! [A: rat,F: int > rat,B: int,C: int] :
% 0.32/0.61 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.32/0.61 => ( ( ord_less_int @ B @ C )
% 0.32/0.61 => ( ! [X4: int,Y4: int] :
% 0.32/0.61 ( ( ord_less_int @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_less_subst1
% 0.32/0.61 thf(fact_4809_order__le__less__subst2,axiom,
% 0.32/0.61 ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.32/0.61 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.61 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.32/0.61 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_less_subst2
% 0.32/0.61 thf(fact_4810_order__le__less__subst2,axiom,
% 0.32/0.61 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.61 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.32/0.61 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_less_subst2
% 0.32/0.61 thf(fact_4811_order__le__less__subst2,axiom,
% 0.32/0.61 ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.32/0.61 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.61 => ( ( ord_less_num @ ( F @ B ) @ C )
% 0.32/0.61 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_less_subst2
% 0.32/0.61 thf(fact_4812_order__le__less__subst2,axiom,
% 0.32/0.61 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.61 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.32/0.61 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_less_subst2
% 0.32/0.61 thf(fact_4813_order__le__less__subst2,axiom,
% 0.32/0.61 ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.32/0.61 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.61 => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.32/0.61 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_less_subst2
% 0.32/0.61 thf(fact_4814_order__le__less__subst2,axiom,
% 0.32/0.61 ! [A: num,B: num,F: num > real,C: real] :
% 0.32/0.61 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.61 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.32/0.61 => ( ! [X4: num,Y4: num] :
% 0.32/0.61 ( ( ord_less_eq_num @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_less_subst2
% 0.32/0.61 thf(fact_4815_order__le__less__subst2,axiom,
% 0.32/0.61 ! [A: num,B: num,F: num > rat,C: rat] :
% 0.32/0.61 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.61 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.32/0.61 => ( ! [X4: num,Y4: num] :
% 0.32/0.61 ( ( ord_less_eq_num @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_less_subst2
% 0.32/0.61 thf(fact_4816_order__le__less__subst2,axiom,
% 0.32/0.61 ! [A: num,B: num,F: num > num,C: num] :
% 0.32/0.61 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.61 => ( ( ord_less_num @ ( F @ B ) @ C )
% 0.32/0.61 => ( ! [X4: num,Y4: num] :
% 0.32/0.61 ( ( ord_less_eq_num @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_less_subst2
% 0.32/0.61 thf(fact_4817_order__le__less__subst2,axiom,
% 0.32/0.61 ! [A: num,B: num,F: num > nat,C: nat] :
% 0.32/0.61 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.61 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.32/0.61 => ( ! [X4: num,Y4: num] :
% 0.32/0.61 ( ( ord_less_eq_num @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_less_subst2
% 0.32/0.61 thf(fact_4818_order__le__less__subst2,axiom,
% 0.32/0.61 ! [A: num,B: num,F: num > int,C: int] :
% 0.32/0.61 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.61 => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.32/0.61 => ( ! [X4: num,Y4: num] :
% 0.32/0.61 ( ( ord_less_eq_num @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_less_subst2
% 0.32/0.61 thf(fact_4819_order__less__le__subst1,axiom,
% 0.32/0.61 ! [A: real,F: rat > real,B: rat,C: rat] :
% 0.32/0.61 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.32/0.61 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.61 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_less_le_subst1
% 0.32/0.61 thf(fact_4820_order__less__le__subst1,axiom,
% 0.32/0.61 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.32/0.61 ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.32/0.61 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.61 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_less_le_subst1
% 0.32/0.61 thf(fact_4821_order__less__le__subst1,axiom,
% 0.32/0.61 ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.32/0.61 ( ( ord_less_num @ A @ ( F @ B ) )
% 0.32/0.61 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.61 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_less_le_subst1
% 0.32/0.61 thf(fact_4822_order__less__le__subst1,axiom,
% 0.32/0.61 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 0.32/0.61 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.32/0.61 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.61 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_less_le_subst1
% 0.32/0.61 thf(fact_4823_order__less__le__subst1,axiom,
% 0.32/0.61 ! [A: int,F: rat > int,B: rat,C: rat] :
% 0.32/0.61 ( ( ord_less_int @ A @ ( F @ B ) )
% 0.32/0.61 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.61 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_less_le_subst1
% 0.32/0.61 thf(fact_4824_order__less__le__subst1,axiom,
% 0.32/0.61 ! [A: real,F: num > real,B: num,C: num] :
% 0.32/0.61 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.32/0.61 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.61 => ( ! [X4: num,Y4: num] :
% 0.32/0.61 ( ( ord_less_eq_num @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_less_le_subst1
% 0.32/0.61 thf(fact_4825_order__less__le__subst1,axiom,
% 0.32/0.61 ! [A: rat,F: num > rat,B: num,C: num] :
% 0.32/0.61 ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.32/0.61 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.61 => ( ! [X4: num,Y4: num] :
% 0.32/0.61 ( ( ord_less_eq_num @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_less_le_subst1
% 0.32/0.61 thf(fact_4826_order__less__le__subst1,axiom,
% 0.32/0.61 ! [A: num,F: num > num,B: num,C: num] :
% 0.32/0.61 ( ( ord_less_num @ A @ ( F @ B ) )
% 0.32/0.61 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.61 => ( ! [X4: num,Y4: num] :
% 0.32/0.61 ( ( ord_less_eq_num @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_less_le_subst1
% 0.32/0.61 thf(fact_4827_order__less__le__subst1,axiom,
% 0.32/0.61 ! [A: nat,F: num > nat,B: num,C: num] :
% 0.32/0.61 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.32/0.61 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.61 => ( ! [X4: num,Y4: num] :
% 0.32/0.61 ( ( ord_less_eq_num @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_less_le_subst1
% 0.32/0.61 thf(fact_4828_order__less__le__subst1,axiom,
% 0.32/0.61 ! [A: int,F: num > int,B: num,C: num] :
% 0.32/0.61 ( ( ord_less_int @ A @ ( F @ B ) )
% 0.32/0.61 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.61 => ( ! [X4: num,Y4: num] :
% 0.32/0.61 ( ( ord_less_eq_num @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_less_le_subst1
% 0.32/0.61 thf(fact_4829_order__less__le__subst2,axiom,
% 0.32/0.61 ! [A: real,B: real,F: real > real,C: real] :
% 0.32/0.61 ( ( ord_less_real @ A @ B )
% 0.32/0.61 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.32/0.61 => ( ! [X4: real,Y4: real] :
% 0.32/0.61 ( ( ord_less_real @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_less_le_subst2
% 0.32/0.61 thf(fact_4830_order__less__le__subst2,axiom,
% 0.32/0.61 ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.32/0.61 ( ( ord_less_rat @ A @ B )
% 0.32/0.61 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.32/0.61 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.61 ( ( ord_less_rat @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_less_le_subst2
% 0.32/0.61 thf(fact_4831_order__less__le__subst2,axiom,
% 0.32/0.61 ! [A: num,B: num,F: num > real,C: real] :
% 0.32/0.61 ( ( ord_less_num @ A @ B )
% 0.32/0.61 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.32/0.61 => ( ! [X4: num,Y4: num] :
% 0.32/0.61 ( ( ord_less_num @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_less_le_subst2
% 0.32/0.61 thf(fact_4832_order__less__le__subst2,axiom,
% 0.32/0.61 ! [A: nat,B: nat,F: nat > real,C: real] :
% 0.32/0.61 ( ( ord_less_nat @ A @ B )
% 0.32/0.61 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.32/0.61 => ( ! [X4: nat,Y4: nat] :
% 0.32/0.61 ( ( ord_less_nat @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_less_le_subst2
% 0.32/0.61 thf(fact_4833_order__less__le__subst2,axiom,
% 0.32/0.61 ! [A: int,B: int,F: int > real,C: real] :
% 0.32/0.61 ( ( ord_less_int @ A @ B )
% 0.32/0.61 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.32/0.61 => ( ! [X4: int,Y4: int] :
% 0.32/0.61 ( ( ord_less_int @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_less_le_subst2
% 0.32/0.61 thf(fact_4834_order__less__le__subst2,axiom,
% 0.32/0.61 ! [A: real,B: real,F: real > rat,C: rat] :
% 0.32/0.61 ( ( ord_less_real @ A @ B )
% 0.32/0.61 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.32/0.61 => ( ! [X4: real,Y4: real] :
% 0.32/0.61 ( ( ord_less_real @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_less_le_subst2
% 0.32/0.61 thf(fact_4835_order__less__le__subst2,axiom,
% 0.32/0.61 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.32/0.61 ( ( ord_less_rat @ A @ B )
% 0.32/0.61 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.32/0.61 => ( ! [X4: rat,Y4: rat] :
% 0.32/0.61 ( ( ord_less_rat @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_less_le_subst2
% 0.32/0.61 thf(fact_4836_order__less__le__subst2,axiom,
% 0.32/0.61 ! [A: num,B: num,F: num > rat,C: rat] :
% 0.32/0.61 ( ( ord_less_num @ A @ B )
% 0.32/0.61 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.32/0.61 => ( ! [X4: num,Y4: num] :
% 0.32/0.61 ( ( ord_less_num @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_less_le_subst2
% 0.32/0.61 thf(fact_4837_order__less__le__subst2,axiom,
% 0.32/0.61 ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 0.32/0.61 ( ( ord_less_nat @ A @ B )
% 0.32/0.61 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.32/0.61 => ( ! [X4: nat,Y4: nat] :
% 0.32/0.61 ( ( ord_less_nat @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_less_le_subst2
% 0.32/0.61 thf(fact_4838_order__less__le__subst2,axiom,
% 0.32/0.61 ! [A: int,B: int,F: int > rat,C: rat] :
% 0.32/0.61 ( ( ord_less_int @ A @ B )
% 0.32/0.61 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.32/0.61 => ( ! [X4: int,Y4: int] :
% 0.32/0.61 ( ( ord_less_int @ X4 @ Y4 )
% 0.32/0.61 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 0.32/0.61 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_less_le_subst2
% 0.32/0.61 thf(fact_4839_linorder__le__less__linear,axiom,
% 0.32/0.61 ! [X: real,Y: real] :
% 0.32/0.61 ( ( ord_less_eq_real @ X @ Y )
% 0.32/0.61 | ( ord_less_real @ Y @ X ) ) ).
% 0.32/0.61
% 0.32/0.61 % linorder_le_less_linear
% 0.32/0.61 thf(fact_4840_linorder__le__less__linear,axiom,
% 0.32/0.61 ! [X: rat,Y: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ X @ Y )
% 0.32/0.61 | ( ord_less_rat @ Y @ X ) ) ).
% 0.32/0.61
% 0.32/0.61 % linorder_le_less_linear
% 0.32/0.61 thf(fact_4841_linorder__le__less__linear,axiom,
% 0.32/0.61 ! [X: num,Y: num] :
% 0.32/0.61 ( ( ord_less_eq_num @ X @ Y )
% 0.32/0.61 | ( ord_less_num @ Y @ X ) ) ).
% 0.32/0.61
% 0.32/0.61 % linorder_le_less_linear
% 0.32/0.61 thf(fact_4842_linorder__le__less__linear,axiom,
% 0.32/0.61 ! [X: nat,Y: nat] :
% 0.32/0.61 ( ( ord_less_eq_nat @ X @ Y )
% 0.32/0.61 | ( ord_less_nat @ Y @ X ) ) ).
% 0.32/0.61
% 0.32/0.61 % linorder_le_less_linear
% 0.32/0.61 thf(fact_4843_linorder__le__less__linear,axiom,
% 0.32/0.61 ! [X: int,Y: int] :
% 0.32/0.61 ( ( ord_less_eq_int @ X @ Y )
% 0.32/0.61 | ( ord_less_int @ Y @ X ) ) ).
% 0.32/0.61
% 0.32/0.61 % linorder_le_less_linear
% 0.32/0.61 thf(fact_4844_order__le__imp__less__or__eq,axiom,
% 0.32/0.61 ! [X: real,Y: real] :
% 0.32/0.61 ( ( ord_less_eq_real @ X @ Y )
% 0.32/0.61 => ( ( ord_less_real @ X @ Y )
% 0.32/0.61 | ( X = Y ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_imp_less_or_eq
% 0.32/0.61 thf(fact_4845_order__le__imp__less__or__eq,axiom,
% 0.32/0.61 ! [X: set_nat,Y: set_nat] :
% 0.32/0.61 ( ( ord_less_eq_set_nat @ X @ Y )
% 0.32/0.61 => ( ( ord_less_set_nat @ X @ Y )
% 0.32/0.61 | ( X = Y ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_imp_less_or_eq
% 0.32/0.61 thf(fact_4846_order__le__imp__less__or__eq,axiom,
% 0.32/0.61 ! [X: rat,Y: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ X @ Y )
% 0.32/0.61 => ( ( ord_less_rat @ X @ Y )
% 0.32/0.61 | ( X = Y ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_imp_less_or_eq
% 0.32/0.61 thf(fact_4847_order__le__imp__less__or__eq,axiom,
% 0.32/0.61 ! [X: num,Y: num] :
% 0.32/0.61 ( ( ord_less_eq_num @ X @ Y )
% 0.32/0.61 => ( ( ord_less_num @ X @ Y )
% 0.32/0.61 | ( X = Y ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_imp_less_or_eq
% 0.32/0.61 thf(fact_4848_order__le__imp__less__or__eq,axiom,
% 0.32/0.61 ! [X: nat,Y: nat] :
% 0.32/0.61 ( ( ord_less_eq_nat @ X @ Y )
% 0.32/0.61 => ( ( ord_less_nat @ X @ Y )
% 0.32/0.61 | ( X = Y ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_imp_less_or_eq
% 0.32/0.61 thf(fact_4849_order__le__imp__less__or__eq,axiom,
% 0.32/0.61 ! [X: int,Y: int] :
% 0.32/0.61 ( ( ord_less_eq_int @ X @ Y )
% 0.32/0.61 => ( ( ord_less_int @ X @ Y )
% 0.32/0.61 | ( X = Y ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % order_le_imp_less_or_eq
% 0.32/0.61 thf(fact_4850_bot_Oextremum,axiom,
% 0.32/0.61 ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).
% 0.32/0.61
% 0.32/0.61 % bot.extremum
% 0.32/0.61 thf(fact_4851_bot_Oextremum,axiom,
% 0.32/0.61 ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).
% 0.32/0.61
% 0.32/0.61 % bot.extremum
% 0.32/0.61 thf(fact_4852_bot_Oextremum,axiom,
% 0.32/0.61 ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% 0.32/0.61
% 0.32/0.61 % bot.extremum
% 0.32/0.61 thf(fact_4853_bot_Oextremum,axiom,
% 0.32/0.61 ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% 0.32/0.61
% 0.32/0.61 % bot.extremum
% 0.32/0.61 thf(fact_4854_bot_Oextremum__unique,axiom,
% 0.32/0.61 ! [A: set_int] :
% 0.32/0.61 ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 0.32/0.61 = ( A = bot_bot_set_int ) ) ).
% 0.32/0.61
% 0.32/0.61 % bot.extremum_unique
% 0.32/0.61 thf(fact_4855_bot_Oextremum__unique,axiom,
% 0.32/0.61 ! [A: set_real] :
% 0.32/0.61 ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 0.32/0.61 = ( A = bot_bot_set_real ) ) ).
% 0.32/0.61
% 0.32/0.61 % bot.extremum_unique
% 0.32/0.61 thf(fact_4856_bot_Oextremum__unique,axiom,
% 0.32/0.61 ! [A: set_nat] :
% 0.32/0.61 ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 0.32/0.61 = ( A = bot_bot_set_nat ) ) ).
% 0.32/0.61
% 0.32/0.61 % bot.extremum_unique
% 0.32/0.61 thf(fact_4857_bot_Oextremum__unique,axiom,
% 0.32/0.61 ! [A: nat] :
% 0.32/0.61 ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 0.32/0.61 = ( A = bot_bot_nat ) ) ).
% 0.32/0.61
% 0.32/0.61 % bot.extremum_unique
% 0.32/0.61 thf(fact_4858_bot_Oextremum__uniqueI,axiom,
% 0.32/0.61 ! [A: set_int] :
% 0.32/0.61 ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 0.32/0.61 => ( A = bot_bot_set_int ) ) ).
% 0.32/0.61
% 0.32/0.61 % bot.extremum_uniqueI
% 0.32/0.61 thf(fact_4859_bot_Oextremum__uniqueI,axiom,
% 0.32/0.61 ! [A: set_real] :
% 0.32/0.61 ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 0.32/0.61 => ( A = bot_bot_set_real ) ) ).
% 0.32/0.61
% 0.32/0.61 % bot.extremum_uniqueI
% 0.32/0.61 thf(fact_4860_bot_Oextremum__uniqueI,axiom,
% 0.32/0.61 ! [A: set_nat] :
% 0.32/0.61 ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 0.32/0.61 => ( A = bot_bot_set_nat ) ) ).
% 0.32/0.61
% 0.32/0.61 % bot.extremum_uniqueI
% 0.32/0.61 thf(fact_4861_bot_Oextremum__uniqueI,axiom,
% 0.32/0.61 ! [A: nat] :
% 0.32/0.61 ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 0.32/0.61 => ( A = bot_bot_nat ) ) ).
% 0.32/0.61
% 0.32/0.61 % bot.extremum_uniqueI
% 0.32/0.61 thf(fact_4862_bot_Oextremum__strict,axiom,
% 0.32/0.61 ! [A: set_nat] :
% 0.32/0.61 ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% 0.32/0.61
% 0.32/0.61 % bot.extremum_strict
% 0.32/0.61 thf(fact_4863_bot_Oextremum__strict,axiom,
% 0.32/0.61 ! [A: set_int] :
% 0.32/0.61 ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).
% 0.32/0.61
% 0.32/0.61 % bot.extremum_strict
% 0.32/0.61 thf(fact_4864_bot_Oextremum__strict,axiom,
% 0.32/0.61 ! [A: set_real] :
% 0.32/0.61 ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).
% 0.32/0.61
% 0.32/0.61 % bot.extremum_strict
% 0.32/0.61 thf(fact_4865_bot_Oextremum__strict,axiom,
% 0.32/0.61 ! [A: nat] :
% 0.32/0.61 ~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% 0.32/0.61
% 0.32/0.61 % bot.extremum_strict
% 0.32/0.61 thf(fact_4866_bot_Onot__eq__extremum,axiom,
% 0.32/0.61 ! [A: set_nat] :
% 0.32/0.61 ( ( A != bot_bot_set_nat )
% 0.32/0.61 = ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % bot.not_eq_extremum
% 0.32/0.61 thf(fact_4867_bot_Onot__eq__extremum,axiom,
% 0.32/0.61 ! [A: set_int] :
% 0.32/0.61 ( ( A != bot_bot_set_int )
% 0.32/0.61 = ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % bot.not_eq_extremum
% 0.32/0.61 thf(fact_4868_bot_Onot__eq__extremum,axiom,
% 0.32/0.61 ! [A: set_real] :
% 0.32/0.61 ( ( A != bot_bot_set_real )
% 0.32/0.61 = ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % bot.not_eq_extremum
% 0.32/0.61 thf(fact_4869_bot_Onot__eq__extremum,axiom,
% 0.32/0.61 ! [A: nat] :
% 0.32/0.61 ( ( A != bot_bot_nat )
% 0.32/0.61 = ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % bot.not_eq_extremum
% 0.32/0.61 thf(fact_4870_list__decode_Ocases,axiom,
% 0.32/0.61 ! [X: nat] :
% 0.32/0.61 ( ( X != zero_zero_nat )
% 0.32/0.61 => ~ ! [N4: nat] :
% 0.32/0.61 ( X
% 0.32/0.61 != ( suc @ N4 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % list_decode.cases
% 0.32/0.61 thf(fact_4871_zero__less__power__eq,axiom,
% 0.32/0.61 ! [A: real,N2: nat] :
% 0.32/0.61 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
% 0.32/0.61 = ( ( N2 = zero_zero_nat )
% 0.32/0.61 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.61 & ( A != zero_zero_real ) )
% 0.32/0.61 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.61 & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % zero_less_power_eq
% 0.32/0.61 thf(fact_4872_zero__less__power__eq,axiom,
% 0.32/0.61 ! [A: rat,N2: nat] :
% 0.32/0.61 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
% 0.32/0.61 = ( ( N2 = zero_zero_nat )
% 0.32/0.61 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.61 & ( A != zero_zero_rat ) )
% 0.32/0.61 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.61 & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % zero_less_power_eq
% 0.32/0.61 thf(fact_4873_zero__less__power__eq,axiom,
% 0.32/0.61 ! [A: int,N2: nat] :
% 0.32/0.61 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
% 0.32/0.61 = ( ( N2 = zero_zero_nat )
% 0.32/0.61 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.61 & ( A != zero_zero_int ) )
% 0.32/0.61 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.61 & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % zero_less_power_eq
% 0.32/0.61 thf(fact_4874_max__def,axiom,
% 0.32/0.61 ( ord_max_set_nat
% 0.32/0.61 = ( ^ [A5: set_nat,B5: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A5 @ B5 ) @ B5 @ A5 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % max_def
% 0.32/0.61 thf(fact_4875_max__def,axiom,
% 0.32/0.61 ( ord_max_rat
% 0.32/0.61 = ( ^ [A5: rat,B5: rat] : ( if_rat @ ( ord_less_eq_rat @ A5 @ B5 ) @ B5 @ A5 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % max_def
% 0.32/0.61 thf(fact_4876_max__def,axiom,
% 0.32/0.61 ( ord_max_num
% 0.32/0.61 = ( ^ [A5: num,B5: num] : ( if_num @ ( ord_less_eq_num @ A5 @ B5 ) @ B5 @ A5 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % max_def
% 0.32/0.61 thf(fact_4877_max__def,axiom,
% 0.32/0.61 ( ord_max_nat
% 0.32/0.61 = ( ^ [A5: nat,B5: nat] : ( if_nat @ ( ord_less_eq_nat @ A5 @ B5 ) @ B5 @ A5 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % max_def
% 0.32/0.61 thf(fact_4878_max__def,axiom,
% 0.32/0.61 ( ord_max_int
% 0.32/0.61 = ( ^ [A5: int,B5: int] : ( if_int @ ( ord_less_eq_int @ A5 @ B5 ) @ B5 @ A5 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % max_def
% 0.32/0.61 thf(fact_4879_max__absorb1,axiom,
% 0.32/0.61 ! [Y: set_nat,X: set_nat] :
% 0.32/0.61 ( ( ord_less_eq_set_nat @ Y @ X )
% 0.32/0.61 => ( ( ord_max_set_nat @ X @ Y )
% 0.32/0.61 = X ) ) ).
% 0.32/0.61
% 0.32/0.61 % max_absorb1
% 0.32/0.61 thf(fact_4880_max__absorb1,axiom,
% 0.32/0.61 ! [Y: rat,X: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ Y @ X )
% 0.32/0.61 => ( ( ord_max_rat @ X @ Y )
% 0.32/0.61 = X ) ) ).
% 0.32/0.61
% 0.32/0.61 % max_absorb1
% 0.32/0.61 thf(fact_4881_max__absorb1,axiom,
% 0.32/0.61 ! [Y: num,X: num] :
% 0.32/0.61 ( ( ord_less_eq_num @ Y @ X )
% 0.32/0.61 => ( ( ord_max_num @ X @ Y )
% 0.32/0.61 = X ) ) ).
% 0.32/0.61
% 0.32/0.61 % max_absorb1
% 0.32/0.61 thf(fact_4882_max__absorb1,axiom,
% 0.32/0.61 ! [Y: nat,X: nat] :
% 0.32/0.61 ( ( ord_less_eq_nat @ Y @ X )
% 0.32/0.61 => ( ( ord_max_nat @ X @ Y )
% 0.32/0.61 = X ) ) ).
% 0.32/0.61
% 0.32/0.61 % max_absorb1
% 0.32/0.61 thf(fact_4883_max__absorb1,axiom,
% 0.32/0.61 ! [Y: int,X: int] :
% 0.32/0.61 ( ( ord_less_eq_int @ Y @ X )
% 0.32/0.61 => ( ( ord_max_int @ X @ Y )
% 0.32/0.61 = X ) ) ).
% 0.32/0.61
% 0.32/0.61 % max_absorb1
% 0.32/0.61 thf(fact_4884_max__absorb2,axiom,
% 0.32/0.61 ! [X: set_nat,Y: set_nat] :
% 0.32/0.61 ( ( ord_less_eq_set_nat @ X @ Y )
% 0.32/0.61 => ( ( ord_max_set_nat @ X @ Y )
% 0.32/0.61 = Y ) ) ).
% 0.32/0.61
% 0.32/0.61 % max_absorb2
% 0.32/0.61 thf(fact_4885_max__absorb2,axiom,
% 0.32/0.61 ! [X: rat,Y: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ X @ Y )
% 0.32/0.61 => ( ( ord_max_rat @ X @ Y )
% 0.32/0.61 = Y ) ) ).
% 0.32/0.61
% 0.32/0.61 % max_absorb2
% 0.32/0.61 thf(fact_4886_max__absorb2,axiom,
% 0.32/0.61 ! [X: num,Y: num] :
% 0.32/0.61 ( ( ord_less_eq_num @ X @ Y )
% 0.32/0.61 => ( ( ord_max_num @ X @ Y )
% 0.32/0.61 = Y ) ) ).
% 0.32/0.61
% 0.32/0.61 % max_absorb2
% 0.32/0.61 thf(fact_4887_max__absorb2,axiom,
% 0.32/0.61 ! [X: nat,Y: nat] :
% 0.32/0.61 ( ( ord_less_eq_nat @ X @ Y )
% 0.32/0.61 => ( ( ord_max_nat @ X @ Y )
% 0.32/0.61 = Y ) ) ).
% 0.32/0.61
% 0.32/0.61 % max_absorb2
% 0.32/0.61 thf(fact_4888_max__absorb2,axiom,
% 0.32/0.61 ! [X: int,Y: int] :
% 0.32/0.61 ( ( ord_less_eq_int @ X @ Y )
% 0.32/0.61 => ( ( ord_max_int @ X @ Y )
% 0.32/0.61 = Y ) ) ).
% 0.32/0.61
% 0.32/0.61 % max_absorb2
% 0.32/0.61 thf(fact_4889_even__mask__div__iff_H,axiom,
% 0.32/0.61 ! [M: nat,N2: nat] :
% 0.32/0.61 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.32/0.61 = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.32/0.61
% 0.32/0.61 % even_mask_div_iff'
% 0.32/0.61 thf(fact_4890_even__mask__div__iff_H,axiom,
% 0.32/0.61 ! [M: nat,N2: nat] :
% 0.32/0.61 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.32/0.61 = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.32/0.61
% 0.32/0.61 % even_mask_div_iff'
% 0.32/0.61 thf(fact_4891_even__mask__div__iff_H,axiom,
% 0.32/0.61 ! [M: nat,N2: nat] :
% 0.32/0.61 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.32/0.61 = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.32/0.61
% 0.32/0.61 % even_mask_div_iff'
% 0.32/0.61 thf(fact_4892_power__le__zero__eq,axiom,
% 0.32/0.61 ! [A: real,N2: nat] :
% 0.32/0.61 ( ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ zero_zero_real )
% 0.32/0.61 = ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.61 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.61 & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.32/0.61 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.61 & ( A = zero_zero_real ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % power_le_zero_eq
% 0.32/0.61 thf(fact_4893_power__le__zero__eq,axiom,
% 0.32/0.61 ! [A: rat,N2: nat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ zero_zero_rat )
% 0.32/0.61 = ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.61 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.61 & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.32/0.61 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.61 & ( A = zero_zero_rat ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % power_le_zero_eq
% 0.32/0.61 thf(fact_4894_power__le__zero__eq,axiom,
% 0.32/0.61 ! [A: int,N2: nat] :
% 0.32/0.61 ( ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ zero_zero_int )
% 0.32/0.61 = ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.32/0.61 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.61 & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.32/0.61 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.61 & ( A = zero_zero_int ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % power_le_zero_eq
% 0.32/0.61 thf(fact_4895_even__mod__4__div__2,axiom,
% 0.32/0.61 ! [N2: nat] :
% 0.32/0.61 ( ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.32/0.61 = ( suc @ zero_zero_nat ) )
% 0.32/0.61 => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % even_mod_4_div_2
% 0.32/0.61 thf(fact_4896_even__mask__div__iff,axiom,
% 0.32/0.61 ! [M: nat,N2: nat] :
% 0.32/0.61 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.32/0.61 = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.61 = zero_z3403309356797280102nteger )
% 0.32/0.61 | ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % even_mask_div_iff
% 0.32/0.61 thf(fact_4897_even__mask__div__iff,axiom,
% 0.32/0.61 ! [M: nat,N2: nat] :
% 0.32/0.61 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.32/0.61 = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.61 = zero_zero_nat )
% 0.32/0.61 | ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % even_mask_div_iff
% 0.32/0.61 thf(fact_4898_even__mask__div__iff,axiom,
% 0.32/0.61 ! [M: nat,N2: nat] :
% 0.32/0.61 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.32/0.61 = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.61 = zero_zero_int )
% 0.32/0.61 | ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % even_mask_div_iff
% 0.32/0.61 thf(fact_4899_even__mult__exp__div__exp__iff,axiom,
% 0.32/0.61 ! [A: code_integer,M: nat,N2: nat] :
% 0.32/0.61 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.32/0.61 = ( ( ord_less_nat @ N2 @ M )
% 0.32/0.61 | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.61 = zero_z3403309356797280102nteger )
% 0.32/0.61 | ( ( ord_less_eq_nat @ M @ N2 )
% 0.32/0.61 & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % even_mult_exp_div_exp_iff
% 0.32/0.61 thf(fact_4900_even__mult__exp__div__exp__iff,axiom,
% 0.32/0.61 ! [A: nat,M: nat,N2: nat] :
% 0.32/0.61 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.32/0.61 = ( ( ord_less_nat @ N2 @ M )
% 0.32/0.61 | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.61 = zero_zero_nat )
% 0.32/0.61 | ( ( ord_less_eq_nat @ M @ N2 )
% 0.32/0.61 & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % even_mult_exp_div_exp_iff
% 0.32/0.61 thf(fact_4901_even__mult__exp__div__exp__iff,axiom,
% 0.32/0.61 ! [A: int,M: nat,N2: nat] :
% 0.32/0.61 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.32/0.61 = ( ( ord_less_nat @ N2 @ M )
% 0.32/0.61 | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.61 = zero_zero_int )
% 0.32/0.61 | ( ( ord_less_eq_nat @ M @ N2 )
% 0.32/0.61 & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % even_mult_exp_div_exp_iff
% 0.32/0.61 thf(fact_4902_ex__min__if__finite,axiom,
% 0.32/0.61 ! [S: set_real] :
% 0.32/0.61 ( ( finite_finite_real @ S )
% 0.32/0.61 => ( ( S != bot_bot_set_real )
% 0.32/0.61 => ? [X4: real] :
% 0.32/0.61 ( ( member_real @ X4 @ S )
% 0.32/0.61 & ~ ? [Xa: real] :
% 0.32/0.61 ( ( member_real @ Xa @ S )
% 0.32/0.61 & ( ord_less_real @ Xa @ X4 ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % ex_min_if_finite
% 0.32/0.61 thf(fact_4903_ex__min__if__finite,axiom,
% 0.32/0.61 ! [S: set_rat] :
% 0.32/0.61 ( ( finite_finite_rat @ S )
% 0.32/0.61 => ( ( S != bot_bot_set_rat )
% 0.32/0.61 => ? [X4: rat] :
% 0.32/0.61 ( ( member_rat @ X4 @ S )
% 0.32/0.61 & ~ ? [Xa: rat] :
% 0.32/0.61 ( ( member_rat @ Xa @ S )
% 0.32/0.61 & ( ord_less_rat @ Xa @ X4 ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % ex_min_if_finite
% 0.32/0.61 thf(fact_4904_ex__min__if__finite,axiom,
% 0.32/0.61 ! [S: set_num] :
% 0.32/0.61 ( ( finite_finite_num @ S )
% 0.32/0.61 => ( ( S != bot_bot_set_num )
% 0.32/0.61 => ? [X4: num] :
% 0.32/0.61 ( ( member_num @ X4 @ S )
% 0.32/0.61 & ~ ? [Xa: num] :
% 0.32/0.61 ( ( member_num @ Xa @ S )
% 0.32/0.61 & ( ord_less_num @ Xa @ X4 ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % ex_min_if_finite
% 0.32/0.61 thf(fact_4905_ex__min__if__finite,axiom,
% 0.32/0.61 ! [S: set_nat] :
% 0.32/0.61 ( ( finite_finite_nat @ S )
% 0.32/0.61 => ( ( S != bot_bot_set_nat )
% 0.32/0.61 => ? [X4: nat] :
% 0.32/0.61 ( ( member_nat @ X4 @ S )
% 0.32/0.61 & ~ ? [Xa: nat] :
% 0.32/0.61 ( ( member_nat @ Xa @ S )
% 0.32/0.61 & ( ord_less_nat @ Xa @ X4 ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % ex_min_if_finite
% 0.32/0.61 thf(fact_4906_ex__min__if__finite,axiom,
% 0.32/0.61 ! [S: set_int] :
% 0.32/0.61 ( ( finite_finite_int @ S )
% 0.32/0.61 => ( ( S != bot_bot_set_int )
% 0.32/0.61 => ? [X4: int] :
% 0.32/0.61 ( ( member_int @ X4 @ S )
% 0.32/0.61 & ~ ? [Xa: int] :
% 0.32/0.61 ( ( member_int @ Xa @ S )
% 0.32/0.61 & ( ord_less_int @ Xa @ X4 ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % ex_min_if_finite
% 0.32/0.61 thf(fact_4907_infinite__growing,axiom,
% 0.32/0.61 ! [X8: set_real] :
% 0.32/0.61 ( ( X8 != bot_bot_set_real )
% 0.32/0.61 => ( ! [X4: real] :
% 0.32/0.61 ( ( member_real @ X4 @ X8 )
% 0.32/0.61 => ? [Xa: real] :
% 0.32/0.61 ( ( member_real @ Xa @ X8 )
% 0.32/0.61 & ( ord_less_real @ X4 @ Xa ) ) )
% 0.32/0.61 => ~ ( finite_finite_real @ X8 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % infinite_growing
% 0.32/0.61 thf(fact_4908_infinite__growing,axiom,
% 0.32/0.61 ! [X8: set_rat] :
% 0.32/0.61 ( ( X8 != bot_bot_set_rat )
% 0.32/0.61 => ( ! [X4: rat] :
% 0.32/0.61 ( ( member_rat @ X4 @ X8 )
% 0.32/0.61 => ? [Xa: rat] :
% 0.32/0.61 ( ( member_rat @ Xa @ X8 )
% 0.32/0.61 & ( ord_less_rat @ X4 @ Xa ) ) )
% 0.32/0.61 => ~ ( finite_finite_rat @ X8 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % infinite_growing
% 0.32/0.61 thf(fact_4909_infinite__growing,axiom,
% 0.32/0.61 ! [X8: set_num] :
% 0.32/0.61 ( ( X8 != bot_bot_set_num )
% 0.32/0.61 => ( ! [X4: num] :
% 0.32/0.61 ( ( member_num @ X4 @ X8 )
% 0.32/0.61 => ? [Xa: num] :
% 0.32/0.61 ( ( member_num @ Xa @ X8 )
% 0.32/0.61 & ( ord_less_num @ X4 @ Xa ) ) )
% 0.32/0.61 => ~ ( finite_finite_num @ X8 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % infinite_growing
% 0.32/0.61 thf(fact_4910_infinite__growing,axiom,
% 0.32/0.61 ! [X8: set_nat] :
% 0.32/0.61 ( ( X8 != bot_bot_set_nat )
% 0.32/0.61 => ( ! [X4: nat] :
% 0.32/0.61 ( ( member_nat @ X4 @ X8 )
% 0.32/0.61 => ? [Xa: nat] :
% 0.32/0.61 ( ( member_nat @ Xa @ X8 )
% 0.32/0.61 & ( ord_less_nat @ X4 @ Xa ) ) )
% 0.32/0.61 => ~ ( finite_finite_nat @ X8 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % infinite_growing
% 0.32/0.61 thf(fact_4911_infinite__growing,axiom,
% 0.32/0.61 ! [X8: set_int] :
% 0.32/0.61 ( ( X8 != bot_bot_set_int )
% 0.32/0.61 => ( ! [X4: int] :
% 0.32/0.61 ( ( member_int @ X4 @ X8 )
% 0.32/0.61 => ? [Xa: int] :
% 0.32/0.61 ( ( member_int @ Xa @ X8 )
% 0.32/0.61 & ( ord_less_int @ X4 @ Xa ) ) )
% 0.32/0.61 => ~ ( finite_finite_int @ X8 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % infinite_growing
% 0.32/0.61 thf(fact_4912_triangle__def,axiom,
% 0.32/0.61 ( nat_triangle
% 0.32/0.61 = ( ^ [N: nat] : ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % triangle_def
% 0.32/0.61 thf(fact_4913_vebt__buildup_Oelims,axiom,
% 0.32/0.61 ! [X: nat,Y: vEBT_VEBT] :
% 0.32/0.61 ( ( ( vEBT_vebt_buildup @ X )
% 0.32/0.61 = Y )
% 0.32/0.61 => ( ( ( X = zero_zero_nat )
% 0.32/0.61 => ( Y
% 0.32/0.61 != ( vEBT_Leaf @ $false @ $false ) ) )
% 0.32/0.61 => ( ( ( X
% 0.32/0.61 = ( suc @ zero_zero_nat ) )
% 0.32/0.61 => ( Y
% 0.32/0.61 != ( vEBT_Leaf @ $false @ $false ) ) )
% 0.32/0.61 => ~ ! [Va2: nat] :
% 0.32/0.61 ( ( X
% 0.32/0.61 = ( suc @ ( suc @ Va2 ) ) )
% 0.32/0.61 => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.32/0.61 => ( Y
% 0.32/0.61 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 0.32/0.61 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.32/0.61 => ( Y
% 0.32/0.61 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % vebt_buildup.elims
% 0.32/0.61 thf(fact_4914_flip__bit__0,axiom,
% 0.32/0.61 ! [A: code_integer] :
% 0.32/0.61 ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
% 0.32/0.61 = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % flip_bit_0
% 0.32/0.61 thf(fact_4915_flip__bit__0,axiom,
% 0.32/0.61 ! [A: int] :
% 0.32/0.61 ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
% 0.32/0.61 = ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % flip_bit_0
% 0.32/0.61 thf(fact_4916_flip__bit__0,axiom,
% 0.32/0.61 ! [A: nat] :
% 0.32/0.61 ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
% 0.32/0.61 = ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % flip_bit_0
% 0.32/0.61 thf(fact_4917_option_Osize__gen_I2_J,axiom,
% 0.32/0.61 ! [X: nat > nat,X22: nat] :
% 0.32/0.61 ( ( size_option_nat @ X @ ( some_nat @ X22 ) )
% 0.32/0.61 = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % option.size_gen(2)
% 0.32/0.61 thf(fact_4918_option_Osize__gen_I2_J,axiom,
% 0.32/0.61 ! [X: product_prod_nat_nat > nat,X22: product_prod_nat_nat] :
% 0.32/0.61 ( ( size_o8335143837870341156at_nat @ X @ ( some_P7363390416028606310at_nat @ X22 ) )
% 0.32/0.61 = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % option.size_gen(2)
% 0.32/0.61 thf(fact_4919_option_Osize__gen_I2_J,axiom,
% 0.32/0.61 ! [X: num > nat,X22: num] :
% 0.32/0.61 ( ( size_option_num @ X @ ( some_num @ X22 ) )
% 0.32/0.61 = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % option.size_gen(2)
% 0.32/0.61 thf(fact_4920_signed__take__bit__Suc,axiom,
% 0.32/0.61 ! [N2: nat,A: code_integer] :
% 0.32/0.61 ( ( bit_ri6519982836138164636nteger @ ( suc @ N2 ) @ A )
% 0.32/0.61 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % signed_take_bit_Suc
% 0.32/0.61 thf(fact_4921_signed__take__bit__Suc,axiom,
% 0.32/0.61 ! [N2: nat,A: int] :
% 0.32/0.61 ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ A )
% 0.32/0.61 = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % signed_take_bit_Suc
% 0.32/0.61 thf(fact_4922_set__decode__Suc,axiom,
% 0.32/0.61 ! [N2: nat,X: nat] :
% 0.32/0.61 ( ( member_nat @ ( suc @ N2 ) @ ( nat_set_decode @ X ) )
% 0.32/0.61 = ( member_nat @ N2 @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % set_decode_Suc
% 0.32/0.61 thf(fact_4923_diff__shunt__var,axiom,
% 0.32/0.61 ! [X: set_int,Y: set_int] :
% 0.32/0.61 ( ( ( minus_minus_set_int @ X @ Y )
% 0.32/0.61 = bot_bot_set_int )
% 0.32/0.61 = ( ord_less_eq_set_int @ X @ Y ) ) ).
% 0.32/0.61
% 0.32/0.61 % diff_shunt_var
% 0.32/0.61 thf(fact_4924_diff__shunt__var,axiom,
% 0.32/0.61 ! [X: set_real,Y: set_real] :
% 0.32/0.61 ( ( ( minus_minus_set_real @ X @ Y )
% 0.32/0.61 = bot_bot_set_real )
% 0.32/0.61 = ( ord_less_eq_set_real @ X @ Y ) ) ).
% 0.32/0.61
% 0.32/0.61 % diff_shunt_var
% 0.32/0.61 thf(fact_4925_diff__shunt__var,axiom,
% 0.32/0.61 ! [X: set_nat,Y: set_nat] :
% 0.32/0.61 ( ( ( minus_minus_set_nat @ X @ Y )
% 0.32/0.61 = bot_bot_set_nat )
% 0.32/0.61 = ( ord_less_eq_set_nat @ X @ Y ) ) ).
% 0.32/0.61
% 0.32/0.61 % diff_shunt_var
% 0.32/0.61 thf(fact_4926_intind,axiom,
% 0.32/0.61 ! [I2: nat,N2: nat,P: int > $o,X: int] :
% 0.32/0.61 ( ( ord_less_nat @ I2 @ N2 )
% 0.32/0.61 => ( ( P @ X )
% 0.32/0.61 => ( P @ ( nth_int @ ( replicate_int @ N2 @ X ) @ I2 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % intind
% 0.32/0.61 thf(fact_4927_intind,axiom,
% 0.32/0.61 ! [I2: nat,N2: nat,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
% 0.32/0.61 ( ( ord_less_nat @ I2 @ N2 )
% 0.32/0.61 => ( ( P @ X )
% 0.32/0.61 => ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X ) @ I2 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % intind
% 0.32/0.61 thf(fact_4928_of__bool__less__eq__iff,axiom,
% 0.32/0.61 ! [P: $o,Q: $o] :
% 0.32/0.61 ( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 0.32/0.61 = ( P
% 0.32/0.61 => Q ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_less_eq_iff
% 0.32/0.61 thf(fact_4929_of__bool__less__eq__iff,axiom,
% 0.32/0.61 ! [P: $o,Q: $o] :
% 0.32/0.61 ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 0.32/0.61 = ( P
% 0.32/0.61 => Q ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_less_eq_iff
% 0.32/0.61 thf(fact_4930_of__bool__less__eq__iff,axiom,
% 0.32/0.61 ! [P: $o,Q: $o] :
% 0.32/0.61 ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 0.32/0.61 = ( P
% 0.32/0.61 => Q ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_less_eq_iff
% 0.32/0.61 thf(fact_4931_of__bool__less__eq__iff,axiom,
% 0.32/0.61 ! [P: $o,Q: $o] :
% 0.32/0.61 ( ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 0.32/0.61 = ( P
% 0.32/0.61 => Q ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_less_eq_iff
% 0.32/0.61 thf(fact_4932_of__bool__less__iff,axiom,
% 0.32/0.61 ! [P: $o,Q: $o] :
% 0.32/0.61 ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
% 0.32/0.61 = ( ~ P
% 0.32/0.61 & Q ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_less_iff
% 0.32/0.61 thf(fact_4933_of__bool__less__iff,axiom,
% 0.32/0.61 ! [P: $o,Q: $o] :
% 0.32/0.61 ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 0.32/0.61 = ( ~ P
% 0.32/0.61 & Q ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_less_iff
% 0.32/0.61 thf(fact_4934_of__bool__less__iff,axiom,
% 0.32/0.61 ! [P: $o,Q: $o] :
% 0.32/0.61 ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 0.32/0.61 = ( ~ P
% 0.32/0.61 & Q ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_less_iff
% 0.32/0.61 thf(fact_4935_of__bool__less__iff,axiom,
% 0.32/0.61 ! [P: $o,Q: $o] :
% 0.32/0.61 ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 0.32/0.61 = ( ~ P
% 0.32/0.61 & Q ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_less_iff
% 0.32/0.61 thf(fact_4936_of__bool__less__iff,axiom,
% 0.32/0.61 ! [P: $o,Q: $o] :
% 0.32/0.61 ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 0.32/0.61 = ( ~ P
% 0.32/0.61 & Q ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_less_iff
% 0.32/0.61 thf(fact_4937_of__bool__eq__1__iff,axiom,
% 0.32/0.61 ! [P: $o] :
% 0.32/0.61 ( ( ( zero_n1201886186963655149omplex @ P )
% 0.32/0.61 = one_one_complex )
% 0.32/0.61 = P ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_eq_1_iff
% 0.32/0.61 thf(fact_4938_of__bool__eq__1__iff,axiom,
% 0.32/0.61 ! [P: $o] :
% 0.32/0.61 ( ( ( zero_n3304061248610475627l_real @ P )
% 0.32/0.61 = one_one_real )
% 0.32/0.61 = P ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_eq_1_iff
% 0.32/0.61 thf(fact_4939_of__bool__eq__1__iff,axiom,
% 0.32/0.61 ! [P: $o] :
% 0.32/0.61 ( ( ( zero_n2052037380579107095ol_rat @ P )
% 0.32/0.61 = one_one_rat )
% 0.32/0.61 = P ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_eq_1_iff
% 0.32/0.61 thf(fact_4940_of__bool__eq__1__iff,axiom,
% 0.32/0.61 ! [P: $o] :
% 0.32/0.61 ( ( ( zero_n2687167440665602831ol_nat @ P )
% 0.32/0.61 = one_one_nat )
% 0.32/0.61 = P ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_eq_1_iff
% 0.32/0.61 thf(fact_4941_of__bool__eq__1__iff,axiom,
% 0.32/0.61 ! [P: $o] :
% 0.32/0.61 ( ( ( zero_n2684676970156552555ol_int @ P )
% 0.32/0.61 = one_one_int )
% 0.32/0.61 = P ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_eq_1_iff
% 0.32/0.61 thf(fact_4942_of__bool__eq__1__iff,axiom,
% 0.32/0.61 ! [P: $o] :
% 0.32/0.61 ( ( ( zero_n356916108424825756nteger @ P )
% 0.32/0.61 = one_one_Code_integer )
% 0.32/0.61 = P ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_eq_1_iff
% 0.32/0.61 thf(fact_4943_of__bool__eq_I2_J,axiom,
% 0.32/0.61 ( ( zero_n1201886186963655149omplex @ $true )
% 0.32/0.61 = one_one_complex ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_eq(2)
% 0.32/0.61 thf(fact_4944_of__bool__eq_I2_J,axiom,
% 0.32/0.61 ( ( zero_n3304061248610475627l_real @ $true )
% 0.32/0.61 = one_one_real ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_eq(2)
% 0.32/0.61 thf(fact_4945_of__bool__eq_I2_J,axiom,
% 0.32/0.61 ( ( zero_n2052037380579107095ol_rat @ $true )
% 0.32/0.61 = one_one_rat ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_eq(2)
% 0.32/0.61 thf(fact_4946_of__bool__eq_I2_J,axiom,
% 0.32/0.61 ( ( zero_n2687167440665602831ol_nat @ $true )
% 0.32/0.61 = one_one_nat ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_eq(2)
% 0.32/0.61 thf(fact_4947_of__bool__eq_I2_J,axiom,
% 0.32/0.61 ( ( zero_n2684676970156552555ol_int @ $true )
% 0.32/0.61 = one_one_int ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_eq(2)
% 0.32/0.61 thf(fact_4948_of__bool__eq_I2_J,axiom,
% 0.32/0.61 ( ( zero_n356916108424825756nteger @ $true )
% 0.32/0.61 = one_one_Code_integer ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_eq(2)
% 0.32/0.61 thf(fact_4949_signed__take__bit__of__0,axiom,
% 0.32/0.61 ! [N2: nat] :
% 0.32/0.61 ( ( bit_ri631733984087533419it_int @ N2 @ zero_zero_int )
% 0.32/0.61 = zero_zero_int ) ).
% 0.32/0.61
% 0.32/0.61 % signed_take_bit_of_0
% 0.32/0.61 thf(fact_4950_replicate__eq__replicate,axiom,
% 0.32/0.61 ! [M: nat,X: vEBT_VEBT,N2: nat,Y: vEBT_VEBT] :
% 0.32/0.61 ( ( ( replicate_VEBT_VEBT @ M @ X )
% 0.32/0.61 = ( replicate_VEBT_VEBT @ N2 @ Y ) )
% 0.32/0.61 = ( ( M = N2 )
% 0.32/0.61 & ( ( M != zero_zero_nat )
% 0.32/0.61 => ( X = Y ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % replicate_eq_replicate
% 0.32/0.61 thf(fact_4951_length__replicate,axiom,
% 0.32/0.61 ! [N2: nat,X: vEBT_VEBT] :
% 0.32/0.61 ( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X ) )
% 0.32/0.61 = N2 ) ).
% 0.32/0.61
% 0.32/0.61 % length_replicate
% 0.32/0.61 thf(fact_4952_length__replicate,axiom,
% 0.32/0.61 ! [N2: nat,X: $o] :
% 0.32/0.61 ( ( size_size_list_o @ ( replicate_o @ N2 @ X ) )
% 0.32/0.61 = N2 ) ).
% 0.32/0.61
% 0.32/0.61 % length_replicate
% 0.32/0.61 thf(fact_4953_length__replicate,axiom,
% 0.32/0.61 ! [N2: nat,X: int] :
% 0.32/0.61 ( ( size_size_list_int @ ( replicate_int @ N2 @ X ) )
% 0.32/0.61 = N2 ) ).
% 0.32/0.61
% 0.32/0.61 % length_replicate
% 0.32/0.61 thf(fact_4954_zero__less__of__bool__iff,axiom,
% 0.32/0.61 ! [P: $o] :
% 0.32/0.61 ( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) )
% 0.32/0.61 = P ) ).
% 0.32/0.61
% 0.32/0.61 % zero_less_of_bool_iff
% 0.32/0.61 thf(fact_4955_zero__less__of__bool__iff,axiom,
% 0.32/0.61 ! [P: $o] :
% 0.32/0.61 ( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 0.32/0.61 = P ) ).
% 0.32/0.61
% 0.32/0.61 % zero_less_of_bool_iff
% 0.32/0.61 thf(fact_4956_zero__less__of__bool__iff,axiom,
% 0.32/0.61 ! [P: $o] :
% 0.32/0.61 ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.32/0.61 = P ) ).
% 0.32/0.61
% 0.32/0.61 % zero_less_of_bool_iff
% 0.32/0.61 thf(fact_4957_zero__less__of__bool__iff,axiom,
% 0.32/0.61 ! [P: $o] :
% 0.32/0.61 ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.32/0.61 = P ) ).
% 0.32/0.61
% 0.32/0.61 % zero_less_of_bool_iff
% 0.32/0.61 thf(fact_4958_zero__less__of__bool__iff,axiom,
% 0.32/0.61 ! [P: $o] :
% 0.32/0.61 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
% 0.32/0.61 = P ) ).
% 0.32/0.61
% 0.32/0.61 % zero_less_of_bool_iff
% 0.32/0.61 thf(fact_4959_of__bool__less__one__iff,axiom,
% 0.32/0.61 ! [P: $o] :
% 0.32/0.61 ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real )
% 0.32/0.61 = ( ~ P ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_less_one_iff
% 0.32/0.61 thf(fact_4960_of__bool__less__one__iff,axiom,
% 0.32/0.61 ! [P: $o] :
% 0.32/0.61 ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
% 0.32/0.61 = ( ~ P ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_less_one_iff
% 0.32/0.61 thf(fact_4961_of__bool__less__one__iff,axiom,
% 0.32/0.61 ! [P: $o] :
% 0.32/0.61 ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
% 0.32/0.61 = ( ~ P ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_less_one_iff
% 0.32/0.61 thf(fact_4962_of__bool__less__one__iff,axiom,
% 0.32/0.61 ! [P: $o] :
% 0.32/0.61 ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
% 0.32/0.61 = ( ~ P ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_less_one_iff
% 0.32/0.61 thf(fact_4963_of__bool__less__one__iff,axiom,
% 0.32/0.61 ! [P: $o] :
% 0.32/0.61 ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer )
% 0.32/0.61 = ( ~ P ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_less_one_iff
% 0.32/0.61 thf(fact_4964_of__bool__not__iff,axiom,
% 0.32/0.61 ! [P: $o] :
% 0.32/0.61 ( ( zero_n1201886186963655149omplex @ ~ P )
% 0.32/0.61 = ( minus_minus_complex @ one_one_complex @ ( zero_n1201886186963655149omplex @ P ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_not_iff
% 0.32/0.61 thf(fact_4965_of__bool__not__iff,axiom,
% 0.32/0.61 ! [P: $o] :
% 0.32/0.61 ( ( zero_n3304061248610475627l_real @ ~ P )
% 0.32/0.61 = ( minus_minus_real @ one_one_real @ ( zero_n3304061248610475627l_real @ P ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_not_iff
% 0.32/0.61 thf(fact_4966_of__bool__not__iff,axiom,
% 0.32/0.61 ! [P: $o] :
% 0.32/0.61 ( ( zero_n2052037380579107095ol_rat @ ~ P )
% 0.32/0.61 = ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_not_iff
% 0.32/0.61 thf(fact_4967_of__bool__not__iff,axiom,
% 0.32/0.61 ! [P: $o] :
% 0.32/0.61 ( ( zero_n2684676970156552555ol_int @ ~ P )
% 0.32/0.61 = ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_not_iff
% 0.32/0.61 thf(fact_4968_of__bool__not__iff,axiom,
% 0.32/0.61 ! [P: $o] :
% 0.32/0.61 ( ( zero_n356916108424825756nteger @ ~ P )
% 0.32/0.61 = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( zero_n356916108424825756nteger @ P ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_not_iff
% 0.32/0.61 thf(fact_4969_Suc__0__mod__eq,axiom,
% 0.32/0.61 ! [N2: nat] :
% 0.32/0.61 ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 0.32/0.61 = ( zero_n2687167440665602831ol_nat
% 0.32/0.61 @ ( N2
% 0.32/0.61 != ( suc @ zero_zero_nat ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % Suc_0_mod_eq
% 0.32/0.61 thf(fact_4970_signed__take__bit__Suc__1,axiom,
% 0.32/0.61 ! [N2: nat] :
% 0.32/0.61 ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ one_one_int )
% 0.32/0.61 = one_one_int ) ).
% 0.32/0.61
% 0.32/0.61 % signed_take_bit_Suc_1
% 0.32/0.61 thf(fact_4971_signed__take__bit__numeral__of__1,axiom,
% 0.32/0.61 ! [K: num] :
% 0.32/0.61 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K ) @ one_one_int )
% 0.32/0.61 = one_one_int ) ).
% 0.32/0.61
% 0.32/0.61 % signed_take_bit_numeral_of_1
% 0.32/0.61 thf(fact_4972_Ball__set__replicate,axiom,
% 0.32/0.61 ! [N2: nat,A: int,P: int > $o] :
% 0.32/0.61 ( ( ! [X2: int] :
% 0.32/0.61 ( ( member_int @ X2 @ ( set_int2 @ ( replicate_int @ N2 @ A ) ) )
% 0.32/0.61 => ( P @ X2 ) ) )
% 0.32/0.61 = ( ( P @ A )
% 0.32/0.61 | ( N2 = zero_zero_nat ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % Ball_set_replicate
% 0.32/0.61 thf(fact_4973_Ball__set__replicate,axiom,
% 0.32/0.61 ! [N2: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.32/0.61 ( ( ! [X2: vEBT_VEBT] :
% 0.32/0.61 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ A ) ) )
% 0.32/0.61 => ( P @ X2 ) ) )
% 0.32/0.61 = ( ( P @ A )
% 0.32/0.61 | ( N2 = zero_zero_nat ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % Ball_set_replicate
% 0.32/0.61 thf(fact_4974_Bex__set__replicate,axiom,
% 0.32/0.61 ! [N2: nat,A: int,P: int > $o] :
% 0.32/0.61 ( ( ? [X2: int] :
% 0.32/0.61 ( ( member_int @ X2 @ ( set_int2 @ ( replicate_int @ N2 @ A ) ) )
% 0.32/0.61 & ( P @ X2 ) ) )
% 0.32/0.61 = ( ( P @ A )
% 0.32/0.61 & ( N2 != zero_zero_nat ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % Bex_set_replicate
% 0.32/0.61 thf(fact_4975_Bex__set__replicate,axiom,
% 0.32/0.61 ! [N2: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.32/0.61 ( ( ? [X2: vEBT_VEBT] :
% 0.32/0.61 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ A ) ) )
% 0.32/0.61 & ( P @ X2 ) ) )
% 0.32/0.61 = ( ( P @ A )
% 0.32/0.61 & ( N2 != zero_zero_nat ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % Bex_set_replicate
% 0.32/0.61 thf(fact_4976_in__set__replicate,axiom,
% 0.32/0.61 ! [X: nat,N2: nat,Y: nat] :
% 0.32/0.61 ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N2 @ Y ) ) )
% 0.32/0.61 = ( ( X = Y )
% 0.32/0.61 & ( N2 != zero_zero_nat ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % in_set_replicate
% 0.32/0.61 thf(fact_4977_in__set__replicate,axiom,
% 0.32/0.61 ! [X: real,N2: nat,Y: real] :
% 0.32/0.61 ( ( member_real @ X @ ( set_real2 @ ( replicate_real @ N2 @ Y ) ) )
% 0.32/0.61 = ( ( X = Y )
% 0.32/0.61 & ( N2 != zero_zero_nat ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % in_set_replicate
% 0.32/0.61 thf(fact_4978_in__set__replicate,axiom,
% 0.32/0.61 ! [X: complex,N2: nat,Y: complex] :
% 0.32/0.61 ( ( member_complex @ X @ ( set_complex2 @ ( replicate_complex @ N2 @ Y ) ) )
% 0.32/0.61 = ( ( X = Y )
% 0.32/0.61 & ( N2 != zero_zero_nat ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % in_set_replicate
% 0.32/0.61 thf(fact_4979_in__set__replicate,axiom,
% 0.32/0.61 ! [X: set_nat,N2: nat,Y: set_nat] :
% 0.32/0.61 ( ( member_set_nat @ X @ ( set_set_nat2 @ ( replicate_set_nat @ N2 @ Y ) ) )
% 0.32/0.61 = ( ( X = Y )
% 0.32/0.61 & ( N2 != zero_zero_nat ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % in_set_replicate
% 0.32/0.61 thf(fact_4980_in__set__replicate,axiom,
% 0.32/0.61 ! [X: int,N2: nat,Y: int] :
% 0.32/0.61 ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N2 @ Y ) ) )
% 0.32/0.61 = ( ( X = Y )
% 0.32/0.61 & ( N2 != zero_zero_nat ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % in_set_replicate
% 0.32/0.61 thf(fact_4981_in__set__replicate,axiom,
% 0.32/0.61 ! [X: vEBT_VEBT,N2: nat,Y: vEBT_VEBT] :
% 0.32/0.61 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ Y ) ) )
% 0.32/0.61 = ( ( X = Y )
% 0.32/0.61 & ( N2 != zero_zero_nat ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % in_set_replicate
% 0.32/0.61 thf(fact_4982_nth__replicate,axiom,
% 0.32/0.61 ! [I2: nat,N2: nat,X: int] :
% 0.32/0.61 ( ( ord_less_nat @ I2 @ N2 )
% 0.32/0.61 => ( ( nth_int @ ( replicate_int @ N2 @ X ) @ I2 )
% 0.32/0.61 = X ) ) ).
% 0.32/0.61
% 0.32/0.61 % nth_replicate
% 0.32/0.61 thf(fact_4983_nth__replicate,axiom,
% 0.32/0.61 ! [I2: nat,N2: nat,X: vEBT_VEBT] :
% 0.32/0.61 ( ( ord_less_nat @ I2 @ N2 )
% 0.32/0.61 => ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X ) @ I2 )
% 0.32/0.61 = X ) ) ).
% 0.32/0.61
% 0.32/0.61 % nth_replicate
% 0.32/0.61 thf(fact_4984_triangle__Suc,axiom,
% 0.32/0.61 ! [N2: nat] :
% 0.32/0.61 ( ( nat_triangle @ ( suc @ N2 ) )
% 0.32/0.61 = ( plus_plus_nat @ ( nat_triangle @ N2 ) @ ( suc @ N2 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % triangle_Suc
% 0.32/0.61 thf(fact_4985_signed__take__bit__Suc__bit0,axiom,
% 0.32/0.61 ! [N2: nat,K: num] :
% 0.32/0.61 ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.32/0.61 = ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % signed_take_bit_Suc_bit0
% 0.32/0.61 thf(fact_4986_odd__of__bool__self,axiom,
% 0.32/0.61 ! [P4: $o] :
% 0.32/0.61 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P4 ) ) )
% 0.32/0.61 = P4 ) ).
% 0.32/0.61
% 0.32/0.61 % odd_of_bool_self
% 0.32/0.61 thf(fact_4987_odd__of__bool__self,axiom,
% 0.32/0.61 ! [P4: $o] :
% 0.32/0.61 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P4 ) ) )
% 0.32/0.61 = P4 ) ).
% 0.32/0.61
% 0.32/0.61 % odd_of_bool_self
% 0.32/0.61 thf(fact_4988_odd__of__bool__self,axiom,
% 0.32/0.61 ! [P4: $o] :
% 0.32/0.61 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P4 ) ) )
% 0.32/0.61 = P4 ) ).
% 0.32/0.61
% 0.32/0.61 % odd_of_bool_self
% 0.32/0.61 thf(fact_4989_of__bool__half__eq__0,axiom,
% 0.32/0.61 ! [B: $o] :
% 0.32/0.61 ( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.61 = zero_zero_nat ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_half_eq_0
% 0.32/0.61 thf(fact_4990_of__bool__half__eq__0,axiom,
% 0.32/0.61 ! [B: $o] :
% 0.32/0.61 ( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.61 = zero_zero_int ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_half_eq_0
% 0.32/0.61 thf(fact_4991_of__bool__half__eq__0,axiom,
% 0.32/0.61 ! [B: $o] :
% 0.32/0.61 ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.61 = zero_z3403309356797280102nteger ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_half_eq_0
% 0.32/0.61 thf(fact_4992_set__decode__0,axiom,
% 0.32/0.61 ! [X: nat] :
% 0.32/0.61 ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X ) )
% 0.32/0.61 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % set_decode_0
% 0.32/0.61 thf(fact_4993_bits__1__div__exp,axiom,
% 0.32/0.61 ! [N2: nat] :
% 0.32/0.61 ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.61 = ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % bits_1_div_exp
% 0.32/0.61 thf(fact_4994_bits__1__div__exp,axiom,
% 0.32/0.61 ! [N2: nat] :
% 0.32/0.61 ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.61 = ( zero_n2684676970156552555ol_int @ ( N2 = zero_zero_nat ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % bits_1_div_exp
% 0.32/0.61 thf(fact_4995_bits__1__div__exp,axiom,
% 0.32/0.61 ! [N2: nat] :
% 0.32/0.61 ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.61 = ( zero_n356916108424825756nteger @ ( N2 = zero_zero_nat ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % bits_1_div_exp
% 0.32/0.61 thf(fact_4996_one__div__2__pow__eq,axiom,
% 0.32/0.61 ! [N2: nat] :
% 0.32/0.61 ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.61 = ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % one_div_2_pow_eq
% 0.32/0.61 thf(fact_4997_one__div__2__pow__eq,axiom,
% 0.32/0.61 ! [N2: nat] :
% 0.32/0.61 ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.61 = ( zero_n2684676970156552555ol_int @ ( N2 = zero_zero_nat ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % one_div_2_pow_eq
% 0.32/0.61 thf(fact_4998_one__div__2__pow__eq,axiom,
% 0.32/0.61 ! [N2: nat] :
% 0.32/0.61 ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.61 = ( zero_n356916108424825756nteger @ ( N2 = zero_zero_nat ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % one_div_2_pow_eq
% 0.32/0.61 thf(fact_4999_one__mod__2__pow__eq,axiom,
% 0.32/0.61 ! [N2: nat] :
% 0.32/0.61 ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.61 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % one_mod_2_pow_eq
% 0.32/0.61 thf(fact_5000_one__mod__2__pow__eq,axiom,
% 0.32/0.61 ! [N2: nat] :
% 0.32/0.61 ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.61 = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % one_mod_2_pow_eq
% 0.32/0.61 thf(fact_5001_one__mod__2__pow__eq,axiom,
% 0.32/0.61 ! [N2: nat] :
% 0.32/0.61 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.61 = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % one_mod_2_pow_eq
% 0.32/0.61 thf(fact_5002_dvd__antisym,axiom,
% 0.32/0.61 ! [M: nat,N2: nat] :
% 0.32/0.61 ( ( dvd_dvd_nat @ M @ N2 )
% 0.32/0.61 => ( ( dvd_dvd_nat @ N2 @ M )
% 0.32/0.61 => ( M = N2 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % dvd_antisym
% 0.32/0.61 thf(fact_5003_of__bool__conj,axiom,
% 0.32/0.61 ! [P: $o,Q: $o] :
% 0.32/0.61 ( ( zero_n3304061248610475627l_real
% 0.32/0.61 @ ( P
% 0.32/0.61 & Q ) )
% 0.32/0.61 = ( times_times_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_conj
% 0.32/0.61 thf(fact_5004_of__bool__conj,axiom,
% 0.32/0.61 ! [P: $o,Q: $o] :
% 0.32/0.61 ( ( zero_n2052037380579107095ol_rat
% 0.32/0.61 @ ( P
% 0.32/0.61 & Q ) )
% 0.32/0.61 = ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_conj
% 0.32/0.61 thf(fact_5005_of__bool__conj,axiom,
% 0.32/0.61 ! [P: $o,Q: $o] :
% 0.32/0.61 ( ( zero_n2687167440665602831ol_nat
% 0.32/0.61 @ ( P
% 0.32/0.61 & Q ) )
% 0.32/0.61 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_conj
% 0.32/0.61 thf(fact_5006_of__bool__conj,axiom,
% 0.32/0.61 ! [P: $o,Q: $o] :
% 0.32/0.61 ( ( zero_n2684676970156552555ol_int
% 0.32/0.61 @ ( P
% 0.32/0.61 & Q ) )
% 0.32/0.61 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_conj
% 0.32/0.61 thf(fact_5007_of__bool__conj,axiom,
% 0.32/0.61 ! [P: $o,Q: $o] :
% 0.32/0.61 ( ( zero_n356916108424825756nteger
% 0.32/0.61 @ ( P
% 0.32/0.61 & Q ) )
% 0.32/0.61 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_conj
% 0.32/0.61 thf(fact_5008_bot__nat__def,axiom,
% 0.32/0.61 bot_bot_nat = zero_zero_nat ).
% 0.32/0.61
% 0.32/0.61 % bot_nat_def
% 0.32/0.61 thf(fact_5009_signed__take__bit__mult,axiom,
% 0.32/0.61 ! [N2: nat,K: int,L2: int] :
% 0.32/0.61 ( ( bit_ri631733984087533419it_int @ N2 @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( bit_ri631733984087533419it_int @ N2 @ L2 ) ) )
% 0.32/0.61 = ( bit_ri631733984087533419it_int @ N2 @ ( times_times_int @ K @ L2 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % signed_take_bit_mult
% 0.32/0.61 thf(fact_5010_signed__take__bit__add,axiom,
% 0.32/0.61 ! [N2: nat,K: int,L2: int] :
% 0.32/0.61 ( ( bit_ri631733984087533419it_int @ N2 @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( bit_ri631733984087533419it_int @ N2 @ L2 ) ) )
% 0.32/0.61 = ( bit_ri631733984087533419it_int @ N2 @ ( plus_plus_int @ K @ L2 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % signed_take_bit_add
% 0.32/0.61 thf(fact_5011_signed__take__bit__diff,axiom,
% 0.32/0.61 ! [N2: nat,K: int,L2: int] :
% 0.32/0.61 ( ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( bit_ri631733984087533419it_int @ N2 @ L2 ) ) )
% 0.32/0.61 = ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ K @ L2 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % signed_take_bit_diff
% 0.32/0.61 thf(fact_5012_zero__less__eq__of__bool,axiom,
% 0.32/0.61 ! [P: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.32/0.61
% 0.32/0.61 % zero_less_eq_of_bool
% 0.32/0.61 thf(fact_5013_zero__less__eq__of__bool,axiom,
% 0.32/0.61 ! [P: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 0.32/0.61
% 0.32/0.61 % zero_less_eq_of_bool
% 0.32/0.61 thf(fact_5014_zero__less__eq__of__bool,axiom,
% 0.32/0.61 ! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 0.32/0.61
% 0.32/0.61 % zero_less_eq_of_bool
% 0.32/0.61 thf(fact_5015_zero__less__eq__of__bool,axiom,
% 0.32/0.61 ! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.32/0.61
% 0.32/0.61 % zero_less_eq_of_bool
% 0.32/0.61 thf(fact_5016_zero__less__eq__of__bool,axiom,
% 0.32/0.61 ! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).
% 0.32/0.61
% 0.32/0.61 % zero_less_eq_of_bool
% 0.32/0.61 thf(fact_5017_of__bool__less__eq__one,axiom,
% 0.32/0.61 ! [P: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_less_eq_one
% 0.32/0.61 thf(fact_5018_of__bool__less__eq__one,axiom,
% 0.32/0.61 ! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_less_eq_one
% 0.32/0.61 thf(fact_5019_of__bool__less__eq__one,axiom,
% 0.32/0.61 ! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_less_eq_one
% 0.32/0.61 thf(fact_5020_of__bool__less__eq__one,axiom,
% 0.32/0.61 ! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_less_eq_one
% 0.32/0.61 thf(fact_5021_of__bool__less__eq__one,axiom,
% 0.32/0.61 ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_less_eq_one
% 0.32/0.61 thf(fact_5022_split__of__bool__asm,axiom,
% 0.32/0.61 ! [P: complex > $o,P4: $o] :
% 0.32/0.61 ( ( P @ ( zero_n1201886186963655149omplex @ P4 ) )
% 0.32/0.61 = ( ~ ( ( P4
% 0.32/0.61 & ~ ( P @ one_one_complex ) )
% 0.32/0.61 | ( ~ P4
% 0.32/0.61 & ~ ( P @ zero_zero_complex ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % split_of_bool_asm
% 0.32/0.61 thf(fact_5023_split__of__bool__asm,axiom,
% 0.32/0.61 ! [P: real > $o,P4: $o] :
% 0.32/0.61 ( ( P @ ( zero_n3304061248610475627l_real @ P4 ) )
% 0.32/0.61 = ( ~ ( ( P4
% 0.32/0.61 & ~ ( P @ one_one_real ) )
% 0.32/0.61 | ( ~ P4
% 0.32/0.61 & ~ ( P @ zero_zero_real ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % split_of_bool_asm
% 0.32/0.61 thf(fact_5024_split__of__bool__asm,axiom,
% 0.32/0.61 ! [P: rat > $o,P4: $o] :
% 0.32/0.61 ( ( P @ ( zero_n2052037380579107095ol_rat @ P4 ) )
% 0.32/0.61 = ( ~ ( ( P4
% 0.32/0.61 & ~ ( P @ one_one_rat ) )
% 0.32/0.61 | ( ~ P4
% 0.32/0.61 & ~ ( P @ zero_zero_rat ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % split_of_bool_asm
% 0.32/0.61 thf(fact_5025_split__of__bool__asm,axiom,
% 0.32/0.61 ! [P: nat > $o,P4: $o] :
% 0.32/0.61 ( ( P @ ( zero_n2687167440665602831ol_nat @ P4 ) )
% 0.32/0.61 = ( ~ ( ( P4
% 0.32/0.61 & ~ ( P @ one_one_nat ) )
% 0.32/0.61 | ( ~ P4
% 0.32/0.61 & ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % split_of_bool_asm
% 0.32/0.61 thf(fact_5026_split__of__bool__asm,axiom,
% 0.32/0.61 ! [P: int > $o,P4: $o] :
% 0.32/0.61 ( ( P @ ( zero_n2684676970156552555ol_int @ P4 ) )
% 0.32/0.61 = ( ~ ( ( P4
% 0.32/0.61 & ~ ( P @ one_one_int ) )
% 0.32/0.61 | ( ~ P4
% 0.32/0.61 & ~ ( P @ zero_zero_int ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % split_of_bool_asm
% 0.32/0.61 thf(fact_5027_split__of__bool__asm,axiom,
% 0.32/0.61 ! [P: code_integer > $o,P4: $o] :
% 0.32/0.61 ( ( P @ ( zero_n356916108424825756nteger @ P4 ) )
% 0.32/0.61 = ( ~ ( ( P4
% 0.32/0.61 & ~ ( P @ one_one_Code_integer ) )
% 0.32/0.61 | ( ~ P4
% 0.32/0.61 & ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % split_of_bool_asm
% 0.32/0.61 thf(fact_5028_split__of__bool,axiom,
% 0.32/0.61 ! [P: complex > $o,P4: $o] :
% 0.32/0.61 ( ( P @ ( zero_n1201886186963655149omplex @ P4 ) )
% 0.32/0.61 = ( ( P4
% 0.32/0.61 => ( P @ one_one_complex ) )
% 0.32/0.61 & ( ~ P4
% 0.32/0.61 => ( P @ zero_zero_complex ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % split_of_bool
% 0.32/0.61 thf(fact_5029_split__of__bool,axiom,
% 0.32/0.61 ! [P: real > $o,P4: $o] :
% 0.32/0.61 ( ( P @ ( zero_n3304061248610475627l_real @ P4 ) )
% 0.32/0.61 = ( ( P4
% 0.32/0.61 => ( P @ one_one_real ) )
% 0.32/0.61 & ( ~ P4
% 0.32/0.61 => ( P @ zero_zero_real ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % split_of_bool
% 0.32/0.61 thf(fact_5030_split__of__bool,axiom,
% 0.32/0.61 ! [P: rat > $o,P4: $o] :
% 0.32/0.61 ( ( P @ ( zero_n2052037380579107095ol_rat @ P4 ) )
% 0.32/0.61 = ( ( P4
% 0.32/0.61 => ( P @ one_one_rat ) )
% 0.32/0.61 & ( ~ P4
% 0.32/0.61 => ( P @ zero_zero_rat ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % split_of_bool
% 0.32/0.61 thf(fact_5031_split__of__bool,axiom,
% 0.32/0.61 ! [P: nat > $o,P4: $o] :
% 0.32/0.61 ( ( P @ ( zero_n2687167440665602831ol_nat @ P4 ) )
% 0.32/0.61 = ( ( P4
% 0.32/0.61 => ( P @ one_one_nat ) )
% 0.32/0.61 & ( ~ P4
% 0.32/0.61 => ( P @ zero_zero_nat ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % split_of_bool
% 0.32/0.61 thf(fact_5032_split__of__bool,axiom,
% 0.32/0.61 ! [P: int > $o,P4: $o] :
% 0.32/0.61 ( ( P @ ( zero_n2684676970156552555ol_int @ P4 ) )
% 0.32/0.61 = ( ( P4
% 0.32/0.61 => ( P @ one_one_int ) )
% 0.32/0.61 & ( ~ P4
% 0.32/0.61 => ( P @ zero_zero_int ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % split_of_bool
% 0.32/0.61 thf(fact_5033_split__of__bool,axiom,
% 0.32/0.61 ! [P: code_integer > $o,P4: $o] :
% 0.32/0.61 ( ( P @ ( zero_n356916108424825756nteger @ P4 ) )
% 0.32/0.61 = ( ( P4
% 0.32/0.61 => ( P @ one_one_Code_integer ) )
% 0.32/0.61 & ( ~ P4
% 0.32/0.61 => ( P @ zero_z3403309356797280102nteger ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % split_of_bool
% 0.32/0.61 thf(fact_5034_of__bool__def,axiom,
% 0.32/0.61 ( zero_n1201886186963655149omplex
% 0.32/0.61 = ( ^ [P5: $o] : ( if_complex @ P5 @ one_one_complex @ zero_zero_complex ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_def
% 0.32/0.61 thf(fact_5035_of__bool__def,axiom,
% 0.32/0.61 ( zero_n3304061248610475627l_real
% 0.32/0.61 = ( ^ [P5: $o] : ( if_real @ P5 @ one_one_real @ zero_zero_real ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_def
% 0.32/0.61 thf(fact_5036_of__bool__def,axiom,
% 0.32/0.61 ( zero_n2052037380579107095ol_rat
% 0.32/0.61 = ( ^ [P5: $o] : ( if_rat @ P5 @ one_one_rat @ zero_zero_rat ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_def
% 0.32/0.61 thf(fact_5037_of__bool__def,axiom,
% 0.32/0.61 ( zero_n2687167440665602831ol_nat
% 0.32/0.61 = ( ^ [P5: $o] : ( if_nat @ P5 @ one_one_nat @ zero_zero_nat ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_def
% 0.32/0.61 thf(fact_5038_of__bool__def,axiom,
% 0.32/0.61 ( zero_n2684676970156552555ol_int
% 0.32/0.61 = ( ^ [P5: $o] : ( if_int @ P5 @ one_one_int @ zero_zero_int ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_def
% 0.32/0.61 thf(fact_5039_of__bool__def,axiom,
% 0.32/0.61 ( zero_n356916108424825756nteger
% 0.32/0.61 = ( ^ [P5: $o] : ( if_Code_integer @ P5 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_def
% 0.32/0.61 thf(fact_5040_replicate__eqI,axiom,
% 0.32/0.61 ! [Xs: list_nat,N2: nat,X: nat] :
% 0.32/0.61 ( ( ( size_size_list_nat @ Xs )
% 0.32/0.61 = N2 )
% 0.32/0.61 => ( ! [Y4: nat] :
% 0.32/0.61 ( ( member_nat @ Y4 @ ( set_nat2 @ Xs ) )
% 0.32/0.61 => ( Y4 = X ) )
% 0.32/0.61 => ( Xs
% 0.32/0.61 = ( replicate_nat @ N2 @ X ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % replicate_eqI
% 0.32/0.61 thf(fact_5041_replicate__eqI,axiom,
% 0.32/0.61 ! [Xs: list_real,N2: nat,X: real] :
% 0.32/0.61 ( ( ( size_size_list_real @ Xs )
% 0.32/0.61 = N2 )
% 0.32/0.61 => ( ! [Y4: real] :
% 0.32/0.61 ( ( member_real @ Y4 @ ( set_real2 @ Xs ) )
% 0.32/0.61 => ( Y4 = X ) )
% 0.32/0.61 => ( Xs
% 0.32/0.61 = ( replicate_real @ N2 @ X ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % replicate_eqI
% 0.32/0.61 thf(fact_5042_replicate__eqI,axiom,
% 0.32/0.61 ! [Xs: list_complex,N2: nat,X: complex] :
% 0.32/0.61 ( ( ( size_s3451745648224563538omplex @ Xs )
% 0.32/0.61 = N2 )
% 0.32/0.61 => ( ! [Y4: complex] :
% 0.32/0.61 ( ( member_complex @ Y4 @ ( set_complex2 @ Xs ) )
% 0.32/0.61 => ( Y4 = X ) )
% 0.32/0.61 => ( Xs
% 0.32/0.61 = ( replicate_complex @ N2 @ X ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % replicate_eqI
% 0.32/0.61 thf(fact_5043_replicate__eqI,axiom,
% 0.32/0.61 ! [Xs: list_set_nat,N2: nat,X: set_nat] :
% 0.32/0.61 ( ( ( size_s3254054031482475050et_nat @ Xs )
% 0.32/0.61 = N2 )
% 0.32/0.61 => ( ! [Y4: set_nat] :
% 0.32/0.61 ( ( member_set_nat @ Y4 @ ( set_set_nat2 @ Xs ) )
% 0.32/0.61 => ( Y4 = X ) )
% 0.32/0.61 => ( Xs
% 0.32/0.61 = ( replicate_set_nat @ N2 @ X ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % replicate_eqI
% 0.32/0.61 thf(fact_5044_replicate__eqI,axiom,
% 0.32/0.61 ! [Xs: list_VEBT_VEBT,N2: nat,X: vEBT_VEBT] :
% 0.32/0.61 ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.32/0.61 = N2 )
% 0.32/0.61 => ( ! [Y4: vEBT_VEBT] :
% 0.32/0.61 ( ( member_VEBT_VEBT @ Y4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.32/0.61 => ( Y4 = X ) )
% 0.32/0.61 => ( Xs
% 0.32/0.61 = ( replicate_VEBT_VEBT @ N2 @ X ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % replicate_eqI
% 0.32/0.61 thf(fact_5045_replicate__eqI,axiom,
% 0.32/0.61 ! [Xs: list_o,N2: nat,X: $o] :
% 0.32/0.61 ( ( ( size_size_list_o @ Xs )
% 0.32/0.61 = N2 )
% 0.32/0.61 => ( ! [Y4: $o] :
% 0.32/0.61 ( ( member_o @ Y4 @ ( set_o2 @ Xs ) )
% 0.32/0.61 => ( Y4 = X ) )
% 0.32/0.61 => ( Xs
% 0.32/0.61 = ( replicate_o @ N2 @ X ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % replicate_eqI
% 0.32/0.61 thf(fact_5046_replicate__eqI,axiom,
% 0.32/0.61 ! [Xs: list_int,N2: nat,X: int] :
% 0.32/0.61 ( ( ( size_size_list_int @ Xs )
% 0.32/0.61 = N2 )
% 0.32/0.61 => ( ! [Y4: int] :
% 0.32/0.61 ( ( member_int @ Y4 @ ( set_int2 @ Xs ) )
% 0.32/0.61 => ( Y4 = X ) )
% 0.32/0.61 => ( Xs
% 0.32/0.61 = ( replicate_int @ N2 @ X ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % replicate_eqI
% 0.32/0.61 thf(fact_5047_replicate__length__same,axiom,
% 0.32/0.61 ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
% 0.32/0.61 ( ! [X4: vEBT_VEBT] :
% 0.32/0.61 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.32/0.61 => ( X4 = X ) )
% 0.32/0.61 => ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs ) @ X )
% 0.32/0.61 = Xs ) ) ).
% 0.32/0.61
% 0.32/0.61 % replicate_length_same
% 0.32/0.61 thf(fact_5048_replicate__length__same,axiom,
% 0.32/0.61 ! [Xs: list_o,X: $o] :
% 0.32/0.61 ( ! [X4: $o] :
% 0.32/0.61 ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
% 0.32/0.61 => ( X4 = X ) )
% 0.32/0.61 => ( ( replicate_o @ ( size_size_list_o @ Xs ) @ X )
% 0.32/0.61 = Xs ) ) ).
% 0.32/0.61
% 0.32/0.61 % replicate_length_same
% 0.32/0.61 thf(fact_5049_replicate__length__same,axiom,
% 0.32/0.61 ! [Xs: list_int,X: int] :
% 0.32/0.61 ( ! [X4: int] :
% 0.32/0.61 ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
% 0.32/0.61 => ( X4 = X ) )
% 0.32/0.61 => ( ( replicate_int @ ( size_size_list_int @ Xs ) @ X )
% 0.32/0.61 = Xs ) ) ).
% 0.32/0.61
% 0.32/0.61 % replicate_length_same
% 0.32/0.61 thf(fact_5050_subset__decode__imp__le,axiom,
% 0.32/0.61 ! [M: nat,N2: nat] :
% 0.32/0.61 ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N2 ) )
% 0.32/0.61 => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.32/0.61
% 0.32/0.61 % subset_decode_imp_le
% 0.32/0.61 thf(fact_5051_of__bool__odd__eq__mod__2,axiom,
% 0.32/0.61 ! [A: nat] :
% 0.32/0.61 ( ( zero_n2687167440665602831ol_nat
% 0.32/0.61 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.61 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_odd_eq_mod_2
% 0.32/0.61 thf(fact_5052_of__bool__odd__eq__mod__2,axiom,
% 0.32/0.61 ! [A: int] :
% 0.32/0.61 ( ( zero_n2684676970156552555ol_int
% 0.32/0.61 @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.61 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_odd_eq_mod_2
% 0.32/0.61 thf(fact_5053_of__bool__odd__eq__mod__2,axiom,
% 0.32/0.61 ! [A: code_integer] :
% 0.32/0.61 ( ( zero_n356916108424825756nteger
% 0.32/0.61 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.61 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % of_bool_odd_eq_mod_2
% 0.32/0.61 thf(fact_5054_signed__take__bit__int__less__exp,axiom,
% 0.32/0.61 ! [N2: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.32/0.61
% 0.32/0.61 % signed_take_bit_int_less_exp
% 0.32/0.61 thf(fact_5055_even__signed__take__bit__iff,axiom,
% 0.32/0.61 ! [M: nat,A: code_integer] :
% 0.32/0.61 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ M @ A ) )
% 0.32/0.61 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % even_signed_take_bit_iff
% 0.32/0.61 thf(fact_5056_even__signed__take__bit__iff,axiom,
% 0.32/0.61 ! [M: nat,A: int] :
% 0.32/0.61 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ M @ A ) )
% 0.32/0.61 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % even_signed_take_bit_iff
% 0.32/0.61 thf(fact_5057_bits__induct,axiom,
% 0.32/0.61 ! [P: nat > $o,A: nat] :
% 0.32/0.61 ( ! [A3: nat] :
% 0.32/0.61 ( ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.61 = A3 )
% 0.32/0.61 => ( P @ A3 ) )
% 0.32/0.61 => ( ! [A3: nat,B2: $o] :
% 0.32/0.61 ( ( P @ A3 )
% 0.32/0.61 => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.61 = A3 )
% 0.32/0.61 => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
% 0.32/0.61 => ( P @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % bits_induct
% 0.32/0.61 thf(fact_5058_bits__induct,axiom,
% 0.32/0.61 ! [P: int > $o,A: int] :
% 0.32/0.61 ( ! [A3: int] :
% 0.32/0.61 ( ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.61 = A3 )
% 0.32/0.61 => ( P @ A3 ) )
% 0.32/0.61 => ( ! [A3: int,B2: $o] :
% 0.32/0.61 ( ( P @ A3 )
% 0.32/0.61 => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.61 = A3 )
% 0.32/0.61 => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
% 0.32/0.61 => ( P @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % bits_induct
% 0.32/0.61 thf(fact_5059_bits__induct,axiom,
% 0.32/0.61 ! [P: code_integer > $o,A: code_integer] :
% 0.32/0.61 ( ! [A3: code_integer] :
% 0.32/0.61 ( ( ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.61 = A3 )
% 0.32/0.61 => ( P @ A3 ) )
% 0.32/0.61 => ( ! [A3: code_integer,B2: $o] :
% 0.32/0.61 ( ( P @ A3 )
% 0.32/0.61 => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B2 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.61 = A3 )
% 0.32/0.61 => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B2 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
% 0.32/0.61 => ( P @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % bits_induct
% 0.32/0.61 thf(fact_5060_signed__take__bit__int__greater__eq__self__iff,axiom,
% 0.32/0.61 ! [K: int,N2: nat] :
% 0.32/0.61 ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N2 @ K ) )
% 0.32/0.61 = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % signed_take_bit_int_greater_eq_self_iff
% 0.32/0.61 thf(fact_5061_signed__take__bit__int__less__self__iff,axiom,
% 0.32/0.61 ! [N2: nat,K: int] :
% 0.32/0.61 ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ K )
% 0.32/0.61 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ).
% 0.32/0.61
% 0.32/0.61 % signed_take_bit_int_less_self_iff
% 0.32/0.61 thf(fact_5062_exp__mod__exp,axiom,
% 0.32/0.61 ! [M: nat,N2: nat] :
% 0.32/0.61 ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.61 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % exp_mod_exp
% 0.32/0.61 thf(fact_5063_exp__mod__exp,axiom,
% 0.32/0.61 ! [M: nat,N2: nat] :
% 0.32/0.61 ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.61 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % exp_mod_exp
% 0.32/0.61 thf(fact_5064_exp__mod__exp,axiom,
% 0.32/0.61 ! [M: nat,N2: nat] :
% 0.32/0.61 ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.61 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % exp_mod_exp
% 0.32/0.61 thf(fact_5065_signed__take__bit__int__less__eq,axiom,
% 0.32/0.61 ! [N2: nat,K: int] :
% 0.32/0.61 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K )
% 0.32/0.61 => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % signed_take_bit_int_less_eq
% 0.32/0.61 thf(fact_5066_option_Osize__gen_I1_J,axiom,
% 0.32/0.61 ! [X: nat > nat] :
% 0.32/0.61 ( ( size_option_nat @ X @ none_nat )
% 0.32/0.61 = ( suc @ zero_zero_nat ) ) ).
% 0.32/0.61
% 0.32/0.61 % option.size_gen(1)
% 0.32/0.61 thf(fact_5067_option_Osize__gen_I1_J,axiom,
% 0.32/0.61 ! [X: product_prod_nat_nat > nat] :
% 0.32/0.61 ( ( size_o8335143837870341156at_nat @ X @ none_P5556105721700978146at_nat )
% 0.32/0.61 = ( suc @ zero_zero_nat ) ) ).
% 0.32/0.61
% 0.32/0.61 % option.size_gen(1)
% 0.32/0.61 thf(fact_5068_option_Osize__gen_I1_J,axiom,
% 0.32/0.61 ! [X: num > nat] :
% 0.32/0.61 ( ( size_option_num @ X @ none_num )
% 0.32/0.61 = ( suc @ zero_zero_nat ) ) ).
% 0.32/0.61
% 0.32/0.61 % option.size_gen(1)
% 0.32/0.61 thf(fact_5069_set__decode__def,axiom,
% 0.32/0.61 ( nat_set_decode
% 0.32/0.61 = ( ^ [X2: nat] :
% 0.32/0.61 ( collect_nat
% 0.32/0.61 @ ^ [N: nat] :
% 0.32/0.61 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % set_decode_def
% 0.32/0.61 thf(fact_5070_exp__div__exp__eq,axiom,
% 0.32/0.61 ! [M: nat,N2: nat] :
% 0.32/0.61 ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.61 = ( times_times_nat
% 0.32/0.61 @ ( zero_n2687167440665602831ol_nat
% 0.32/0.61 @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.32/0.61 != zero_zero_nat )
% 0.32/0.61 & ( ord_less_eq_nat @ N2 @ M ) ) )
% 0.32/0.61 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % exp_div_exp_eq
% 0.32/0.61 thf(fact_5071_exp__div__exp__eq,axiom,
% 0.32/0.61 ! [M: nat,N2: nat] :
% 0.32/0.61 ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.61 = ( times_times_int
% 0.32/0.61 @ ( zero_n2684676970156552555ol_int
% 0.32/0.61 @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 0.32/0.61 != zero_zero_int )
% 0.32/0.61 & ( ord_less_eq_nat @ N2 @ M ) ) )
% 0.32/0.61 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % exp_div_exp_eq
% 0.32/0.61 thf(fact_5072_exp__div__exp__eq,axiom,
% 0.32/0.61 ! [M: nat,N2: nat] :
% 0.32/0.61 ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.61 = ( times_3573771949741848930nteger
% 0.32/0.61 @ ( zero_n356916108424825756nteger
% 0.32/0.61 @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
% 0.32/0.61 != zero_z3403309356797280102nteger )
% 0.32/0.61 & ( ord_less_eq_nat @ N2 @ M ) ) )
% 0.32/0.61 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % exp_div_exp_eq
% 0.32/0.61 thf(fact_5073_vebt__buildup_Osimps_I3_J,axiom,
% 0.32/0.61 ! [Va: nat] :
% 0.32/0.61 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.32/0.61 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
% 0.32/0.61 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 0.32/0.61 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.32/0.61 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
% 0.32/0.61 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % vebt_buildup.simps(3)
% 0.32/0.61 thf(fact_5074_signed__take__bit__rec,axiom,
% 0.32/0.61 ( bit_ri6519982836138164636nteger
% 0.32/0.61 = ( ^ [N: nat,A5: code_integer] : ( if_Code_integer @ ( N = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % signed_take_bit_rec
% 0.32/0.61 thf(fact_5075_signed__take__bit__rec,axiom,
% 0.32/0.61 ( bit_ri631733984087533419it_int
% 0.32/0.61 = ( ^ [N: nat,A5: int] : ( if_int @ ( N = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % signed_take_bit_rec
% 0.32/0.61 thf(fact_5076_vebt__buildup_Opelims,axiom,
% 0.32/0.61 ! [X: nat,Y: vEBT_VEBT] :
% 0.32/0.61 ( ( ( vEBT_vebt_buildup @ X )
% 0.32/0.61 = Y )
% 0.32/0.61 => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X )
% 0.32/0.61 => ( ( ( X = zero_zero_nat )
% 0.32/0.61 => ( ( Y
% 0.32/0.61 = ( vEBT_Leaf @ $false @ $false ) )
% 0.32/0.61 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
% 0.32/0.61 => ( ( ( X
% 0.32/0.61 = ( suc @ zero_zero_nat ) )
% 0.32/0.61 => ( ( Y
% 0.32/0.61 = ( vEBT_Leaf @ $false @ $false ) )
% 0.32/0.61 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
% 0.32/0.61 => ~ ! [Va2: nat] :
% 0.32/0.61 ( ( X
% 0.32/0.61 = ( suc @ ( suc @ Va2 ) ) )
% 0.32/0.61 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.32/0.61 => ( Y
% 0.32/0.61 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 0.32/0.61 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.32/0.61 => ( Y
% 0.32/0.61 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 0.32/0.61 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % vebt_buildup.pelims
% 0.32/0.61 thf(fact_5077_add__scale__eq__noteq,axiom,
% 0.32/0.61 ! [R: complex,A: complex,B: complex,C: complex,D: complex] :
% 0.32/0.61 ( ( R != zero_zero_complex )
% 0.32/0.61 => ( ( ( A = B )
% 0.32/0.61 & ( C != D ) )
% 0.32/0.61 => ( ( plus_plus_complex @ A @ ( times_times_complex @ R @ C ) )
% 0.32/0.61 != ( plus_plus_complex @ B @ ( times_times_complex @ R @ D ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add_scale_eq_noteq
% 0.32/0.61 thf(fact_5078_add__scale__eq__noteq,axiom,
% 0.32/0.61 ! [R: real,A: real,B: real,C: real,D: real] :
% 0.32/0.61 ( ( R != zero_zero_real )
% 0.32/0.61 => ( ( ( A = B )
% 0.32/0.61 & ( C != D ) )
% 0.32/0.61 => ( ( plus_plus_real @ A @ ( times_times_real @ R @ C ) )
% 0.32/0.61 != ( plus_plus_real @ B @ ( times_times_real @ R @ D ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add_scale_eq_noteq
% 0.32/0.61 thf(fact_5079_add__scale__eq__noteq,axiom,
% 0.32/0.61 ! [R: rat,A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.61 ( ( R != zero_zero_rat )
% 0.32/0.61 => ( ( ( A = B )
% 0.32/0.61 & ( C != D ) )
% 0.32/0.61 => ( ( plus_plus_rat @ A @ ( times_times_rat @ R @ C ) )
% 0.32/0.61 != ( plus_plus_rat @ B @ ( times_times_rat @ R @ D ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add_scale_eq_noteq
% 0.32/0.61 thf(fact_5080_add__scale__eq__noteq,axiom,
% 0.32/0.61 ! [R: nat,A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.61 ( ( R != zero_zero_nat )
% 0.32/0.61 => ( ( ( A = B )
% 0.32/0.61 & ( C != D ) )
% 0.32/0.61 => ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C ) )
% 0.32/0.61 != ( plus_plus_nat @ B @ ( times_times_nat @ R @ D ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add_scale_eq_noteq
% 0.32/0.61 thf(fact_5081_add__scale__eq__noteq,axiom,
% 0.32/0.61 ! [R: int,A: int,B: int,C: int,D: int] :
% 0.32/0.61 ( ( R != zero_zero_int )
% 0.32/0.61 => ( ( ( A = B )
% 0.32/0.61 & ( C != D ) )
% 0.32/0.61 => ( ( plus_plus_int @ A @ ( times_times_int @ R @ C ) )
% 0.32/0.61 != ( plus_plus_int @ B @ ( times_times_int @ R @ D ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add_scale_eq_noteq
% 0.32/0.61 thf(fact_5082_artanh__def,axiom,
% 0.32/0.61 ( artanh_real
% 0.32/0.61 = ( ^ [X2: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % artanh_def
% 0.32/0.61 thf(fact_5083_Sum__Icc__int,axiom,
% 0.32/0.61 ! [M: int,N2: int] :
% 0.32/0.61 ( ( ord_less_eq_int @ M @ N2 )
% 0.32/0.61 => ( ( groups4538972089207619220nt_int
% 0.32/0.61 @ ^ [X2: int] : X2
% 0.32/0.61 @ ( set_or1266510415728281911st_int @ M @ N2 ) )
% 0.32/0.61 = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N2 @ ( plus_plus_int @ N2 @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % Sum_Icc_int
% 0.32/0.61 thf(fact_5084_divmod__step__def,axiom,
% 0.32/0.61 ( unique5026877609467782581ep_nat
% 0.32/0.61 = ( ^ [L: num] :
% 0.32/0.61 ( produc2626176000494625587at_nat
% 0.32/0.61 @ ^ [Q4: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % divmod_step_def
% 0.32/0.61 thf(fact_5085_divmod__step__def,axiom,
% 0.32/0.61 ( unique5024387138958732305ep_int
% 0.32/0.61 = ( ^ [L: num] :
% 0.32/0.61 ( produc4245557441103728435nt_int
% 0.32/0.61 @ ^ [Q4: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % divmod_step_def
% 0.32/0.61 thf(fact_5086_divmod__step__def,axiom,
% 0.32/0.61 ( unique4921790084139445826nteger
% 0.32/0.61 = ( ^ [L: num] :
% 0.32/0.61 ( produc6916734918728496179nteger
% 0.32/0.61 @ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % divmod_step_def
% 0.32/0.61 thf(fact_5087_even__set__encode__iff,axiom,
% 0.32/0.61 ! [A2: set_nat] :
% 0.32/0.61 ( ( finite_finite_nat @ A2 )
% 0.32/0.61 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A2 ) )
% 0.32/0.61 = ( ~ ( member_nat @ zero_zero_nat @ A2 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % even_set_encode_iff
% 0.32/0.61 thf(fact_5088_Compl__anti__mono,axiom,
% 0.32/0.61 ! [A2: set_nat,B3: set_nat] :
% 0.32/0.61 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.32/0.61 => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B3 ) @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % Compl_anti_mono
% 0.32/0.61 thf(fact_5089_Compl__subset__Compl__iff,axiom,
% 0.32/0.61 ! [A2: set_nat,B3: set_nat] :
% 0.32/0.61 ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( uminus5710092332889474511et_nat @ B3 ) )
% 0.32/0.61 = ( ord_less_eq_set_nat @ B3 @ A2 ) ) ).
% 0.32/0.61
% 0.32/0.61 % Compl_subset_Compl_iff
% 0.32/0.61 thf(fact_5090_neg__le__iff__le,axiom,
% 0.32/0.61 ! [B: real,A: real] :
% 0.32/0.61 ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 0.32/0.61 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_le_iff_le
% 0.32/0.61 thf(fact_5091_neg__le__iff__le,axiom,
% 0.32/0.61 ! [B: code_integer,A: code_integer] :
% 0.32/0.61 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 0.32/0.61 = ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_le_iff_le
% 0.32/0.61 thf(fact_5092_neg__le__iff__le,axiom,
% 0.32/0.61 ! [B: rat,A: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 0.32/0.61 = ( ord_less_eq_rat @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_le_iff_le
% 0.32/0.61 thf(fact_5093_neg__le__iff__le,axiom,
% 0.32/0.61 ! [B: int,A: int] :
% 0.32/0.61 ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 0.32/0.61 = ( ord_less_eq_int @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_le_iff_le
% 0.32/0.61 thf(fact_5094_compl__le__compl__iff,axiom,
% 0.32/0.61 ! [X: set_nat,Y: set_nat] :
% 0.32/0.61 ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y ) )
% 0.32/0.61 = ( ord_less_eq_set_nat @ Y @ X ) ) ).
% 0.32/0.61
% 0.32/0.61 % compl_le_compl_iff
% 0.32/0.61 thf(fact_5095_neg__less__iff__less,axiom,
% 0.32/0.61 ! [B: real,A: real] :
% 0.32/0.61 ( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 0.32/0.61 = ( ord_less_real @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_less_iff_less
% 0.32/0.61 thf(fact_5096_neg__less__iff__less,axiom,
% 0.32/0.61 ! [B: int,A: int] :
% 0.32/0.61 ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 0.32/0.61 = ( ord_less_int @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_less_iff_less
% 0.32/0.61 thf(fact_5097_neg__less__iff__less,axiom,
% 0.32/0.61 ! [B: rat,A: rat] :
% 0.32/0.61 ( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 0.32/0.61 = ( ord_less_rat @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_less_iff_less
% 0.32/0.61 thf(fact_5098_neg__less__iff__less,axiom,
% 0.32/0.61 ! [B: code_integer,A: code_integer] :
% 0.32/0.61 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 0.32/0.61 = ( ord_le6747313008572928689nteger @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_less_iff_less
% 0.32/0.61 thf(fact_5099_neg__numeral__eq__iff,axiom,
% 0.32/0.61 ! [M: num,N2: num] :
% 0.32/0.61 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 0.32/0.61 = ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 0.32/0.61 = ( M = N2 ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_eq_iff
% 0.32/0.61 thf(fact_5100_neg__numeral__eq__iff,axiom,
% 0.32/0.61 ! [M: num,N2: num] :
% 0.32/0.61 ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 0.32/0.61 = ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.32/0.61 = ( M = N2 ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_eq_iff
% 0.32/0.61 thf(fact_5101_neg__numeral__eq__iff,axiom,
% 0.32/0.61 ! [M: num,N2: num] :
% 0.32/0.61 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 0.32/0.61 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
% 0.32/0.61 = ( M = N2 ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_eq_iff
% 0.32/0.61 thf(fact_5102_neg__numeral__eq__iff,axiom,
% 0.32/0.61 ! [M: num,N2: num] :
% 0.32/0.61 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 0.32/0.61 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 0.32/0.61 = ( M = N2 ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_eq_iff
% 0.32/0.61 thf(fact_5103_neg__numeral__eq__iff,axiom,
% 0.32/0.61 ! [M: num,N2: num] :
% 0.32/0.61 ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 0.32/0.61 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
% 0.32/0.61 = ( M = N2 ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_eq_iff
% 0.32/0.61 thf(fact_5104_mult__minus__right,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
% 0.32/0.61 = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_minus_right
% 0.32/0.61 thf(fact_5105_mult__minus__right,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
% 0.32/0.61 = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_minus_right
% 0.32/0.61 thf(fact_5106_mult__minus__right,axiom,
% 0.32/0.61 ! [A: complex,B: complex] :
% 0.32/0.61 ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 0.32/0.61 = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_minus_right
% 0.32/0.61 thf(fact_5107_mult__minus__right,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.32/0.61 = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_minus_right
% 0.32/0.61 thf(fact_5108_mult__minus__right,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.32/0.61 = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_minus_right
% 0.32/0.61 thf(fact_5109_minus__mult__minus,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 0.32/0.61 = ( times_times_real @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_mult_minus
% 0.32/0.61 thf(fact_5110_minus__mult__minus,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 0.32/0.61 = ( times_times_int @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_mult_minus
% 0.32/0.61 thf(fact_5111_minus__mult__minus,axiom,
% 0.32/0.61 ! [A: complex,B: complex] :
% 0.32/0.61 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 0.32/0.61 = ( times_times_complex @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_mult_minus
% 0.32/0.61 thf(fact_5112_minus__mult__minus,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 0.32/0.61 = ( times_times_rat @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_mult_minus
% 0.32/0.61 thf(fact_5113_minus__mult__minus,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 0.32/0.61 = ( times_3573771949741848930nteger @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_mult_minus
% 0.32/0.61 thf(fact_5114_mult__minus__left,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
% 0.32/0.61 = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_minus_left
% 0.32/0.61 thf(fact_5115_mult__minus__left,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
% 0.32/0.61 = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_minus_left
% 0.32/0.61 thf(fact_5116_mult__minus__left,axiom,
% 0.32/0.61 ! [A: complex,B: complex] :
% 0.32/0.61 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 0.32/0.61 = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_minus_left
% 0.32/0.61 thf(fact_5117_mult__minus__left,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.32/0.61 = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_minus_left
% 0.32/0.61 thf(fact_5118_mult__minus__left,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.32/0.61 = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_minus_left
% 0.32/0.61 thf(fact_5119_minus__add__distrib,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 0.32/0.61 = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_add_distrib
% 0.32/0.61 thf(fact_5120_minus__add__distrib,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.32/0.61 = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_add_distrib
% 0.32/0.61 thf(fact_5121_minus__add__distrib,axiom,
% 0.32/0.61 ! [A: complex,B: complex] :
% 0.32/0.61 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 0.32/0.61 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_add_distrib
% 0.32/0.61 thf(fact_5122_minus__add__distrib,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 0.32/0.61 = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_add_distrib
% 0.32/0.61 thf(fact_5123_minus__add__distrib,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.32/0.61 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_add_distrib
% 0.32/0.61 thf(fact_5124_minus__add__cancel,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
% 0.32/0.61 = B ) ).
% 0.32/0.61
% 0.32/0.61 % minus_add_cancel
% 0.32/0.61 thf(fact_5125_minus__add__cancel,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
% 0.32/0.61 = B ) ).
% 0.32/0.61
% 0.32/0.61 % minus_add_cancel
% 0.32/0.61 thf(fact_5126_minus__add__cancel,axiom,
% 0.32/0.61 ! [A: complex,B: complex] :
% 0.32/0.61 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
% 0.32/0.61 = B ) ).
% 0.32/0.61
% 0.32/0.61 % minus_add_cancel
% 0.32/0.61 thf(fact_5127_minus__add__cancel,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
% 0.32/0.61 = B ) ).
% 0.32/0.61
% 0.32/0.61 % minus_add_cancel
% 0.32/0.61 thf(fact_5128_minus__add__cancel,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.32/0.61 = B ) ).
% 0.32/0.61
% 0.32/0.61 % minus_add_cancel
% 0.32/0.61 thf(fact_5129_add__minus__cancel,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
% 0.32/0.61 = B ) ).
% 0.32/0.61
% 0.32/0.61 % add_minus_cancel
% 0.32/0.61 thf(fact_5130_add__minus__cancel,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
% 0.32/0.61 = B ) ).
% 0.32/0.61
% 0.32/0.61 % add_minus_cancel
% 0.32/0.61 thf(fact_5131_add__minus__cancel,axiom,
% 0.32/0.61 ! [A: complex,B: complex] :
% 0.32/0.61 ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
% 0.32/0.61 = B ) ).
% 0.32/0.61
% 0.32/0.61 % add_minus_cancel
% 0.32/0.61 thf(fact_5132_add__minus__cancel,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
% 0.32/0.61 = B ) ).
% 0.32/0.61
% 0.32/0.61 % add_minus_cancel
% 0.32/0.61 thf(fact_5133_add__minus__cancel,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
% 0.32/0.61 = B ) ).
% 0.32/0.61
% 0.32/0.61 % add_minus_cancel
% 0.32/0.61 thf(fact_5134_minus__diff__eq,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
% 0.32/0.61 = ( minus_minus_real @ B @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_diff_eq
% 0.32/0.61 thf(fact_5135_minus__diff__eq,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
% 0.32/0.61 = ( minus_minus_int @ B @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_diff_eq
% 0.32/0.61 thf(fact_5136_minus__diff__eq,axiom,
% 0.32/0.61 ! [A: complex,B: complex] :
% 0.32/0.61 ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
% 0.32/0.61 = ( minus_minus_complex @ B @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_diff_eq
% 0.32/0.61 thf(fact_5137_minus__diff__eq,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
% 0.32/0.61 = ( minus_minus_rat @ B @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_diff_eq
% 0.32/0.61 thf(fact_5138_minus__diff__eq,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
% 0.32/0.61 = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_diff_eq
% 0.32/0.61 thf(fact_5139_div__minus__minus,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 0.32/0.61 = ( divide_divide_int @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % div_minus_minus
% 0.32/0.61 thf(fact_5140_div__minus__minus,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 0.32/0.61 = ( divide6298287555418463151nteger @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % div_minus_minus
% 0.32/0.61 thf(fact_5141_ln__less__cancel__iff,axiom,
% 0.32/0.61 ! [X: real,Y: real] :
% 0.32/0.61 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.61 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.32/0.61 => ( ( ord_less_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
% 0.32/0.61 = ( ord_less_real @ X @ Y ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % ln_less_cancel_iff
% 0.32/0.61 thf(fact_5142_ln__inj__iff,axiom,
% 0.32/0.61 ! [X: real,Y: real] :
% 0.32/0.61 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.61 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.32/0.61 => ( ( ( ln_ln_real @ X )
% 0.32/0.61 = ( ln_ln_real @ Y ) )
% 0.32/0.61 = ( X = Y ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % ln_inj_iff
% 0.32/0.61 thf(fact_5143_mod__minus__minus,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 0.32/0.61 = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % mod_minus_minus
% 0.32/0.61 thf(fact_5144_mod__minus__minus,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 0.32/0.61 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % mod_minus_minus
% 0.32/0.61 thf(fact_5145_real__add__minus__iff,axiom,
% 0.32/0.61 ! [X: real,A: real] :
% 0.32/0.61 ( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
% 0.32/0.61 = zero_zero_real )
% 0.32/0.61 = ( X = A ) ) ).
% 0.32/0.61
% 0.32/0.61 % real_add_minus_iff
% 0.32/0.61 thf(fact_5146_sum_Oneutral__const,axiom,
% 0.32/0.61 ! [A2: set_int] :
% 0.32/0.61 ( ( groups4538972089207619220nt_int
% 0.32/0.61 @ ^ [Uu3: int] : zero_zero_int
% 0.32/0.61 @ A2 )
% 0.32/0.61 = zero_zero_int ) ).
% 0.32/0.61
% 0.32/0.61 % sum.neutral_const
% 0.32/0.61 thf(fact_5147_sum_Oneutral__const,axiom,
% 0.32/0.61 ! [A2: set_complex] :
% 0.32/0.61 ( ( groups7754918857620584856omplex
% 0.32/0.61 @ ^ [Uu3: complex] : zero_zero_complex
% 0.32/0.61 @ A2 )
% 0.32/0.61 = zero_zero_complex ) ).
% 0.32/0.61
% 0.32/0.61 % sum.neutral_const
% 0.32/0.61 thf(fact_5148_sum_Oneutral__const,axiom,
% 0.32/0.61 ! [A2: set_nat] :
% 0.32/0.61 ( ( groups3542108847815614940at_nat
% 0.32/0.61 @ ^ [Uu3: nat] : zero_zero_nat
% 0.32/0.61 @ A2 )
% 0.32/0.61 = zero_zero_nat ) ).
% 0.32/0.61
% 0.32/0.61 % sum.neutral_const
% 0.32/0.61 thf(fact_5149_sum_Oneutral__const,axiom,
% 0.32/0.61 ! [A2: set_nat] :
% 0.32/0.61 ( ( groups6591440286371151544t_real
% 0.32/0.61 @ ^ [Uu3: nat] : zero_zero_real
% 0.32/0.61 @ A2 )
% 0.32/0.61 = zero_zero_real ) ).
% 0.32/0.61
% 0.32/0.61 % sum.neutral_const
% 0.32/0.61 thf(fact_5150_case__prod__conv,axiom,
% 0.32/0.61 ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,A: nat,B: nat] :
% 0.32/0.61 ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
% 0.32/0.61 = ( F @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % case_prod_conv
% 0.32/0.61 thf(fact_5151_case__prod__conv,axiom,
% 0.32/0.61 ! [F: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat] :
% 0.32/0.61 ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
% 0.32/0.61 = ( F @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % case_prod_conv
% 0.32/0.61 thf(fact_5152_case__prod__conv,axiom,
% 0.32/0.61 ! [F: int > int > product_prod_int_int,A: int,B: int] :
% 0.32/0.61 ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
% 0.32/0.61 = ( F @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % case_prod_conv
% 0.32/0.61 thf(fact_5153_case__prod__conv,axiom,
% 0.32/0.61 ! [F: int > int > $o,A: int,B: int] :
% 0.32/0.61 ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
% 0.32/0.61 = ( F @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % case_prod_conv
% 0.32/0.61 thf(fact_5154_case__prod__conv,axiom,
% 0.32/0.61 ! [F: int > int > int,A: int,B: int] :
% 0.32/0.61 ( ( produc8211389475949308722nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
% 0.32/0.61 = ( F @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % case_prod_conv
% 0.32/0.61 thf(fact_5155_neg__0__le__iff__le,axiom,
% 0.32/0.61 ! [A: real] :
% 0.32/0.61 ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.32/0.61 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_0_le_iff_le
% 0.32/0.61 thf(fact_5156_neg__0__le__iff__le,axiom,
% 0.32/0.61 ! [A: code_integer] :
% 0.32/0.61 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.32/0.61 = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_0_le_iff_le
% 0.32/0.61 thf(fact_5157_neg__0__le__iff__le,axiom,
% 0.32/0.61 ! [A: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 0.32/0.61 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_0_le_iff_le
% 0.32/0.61 thf(fact_5158_neg__0__le__iff__le,axiom,
% 0.32/0.61 ! [A: int] :
% 0.32/0.61 ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 0.32/0.61 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_0_le_iff_le
% 0.32/0.61 thf(fact_5159_neg__le__0__iff__le,axiom,
% 0.32/0.61 ! [A: real] :
% 0.32/0.61 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.32/0.61 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_le_0_iff_le
% 0.32/0.61 thf(fact_5160_neg__le__0__iff__le,axiom,
% 0.32/0.61 ! [A: code_integer] :
% 0.32/0.61 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 0.32/0.61 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_le_0_iff_le
% 0.32/0.61 thf(fact_5161_neg__le__0__iff__le,axiom,
% 0.32/0.61 ! [A: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 0.32/0.61 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_le_0_iff_le
% 0.32/0.61 thf(fact_5162_neg__le__0__iff__le,axiom,
% 0.32/0.61 ! [A: int] :
% 0.32/0.61 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 0.32/0.61 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_le_0_iff_le
% 0.32/0.61 thf(fact_5163_less__eq__neg__nonpos,axiom,
% 0.32/0.61 ! [A: real] :
% 0.32/0.61 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 0.32/0.61 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.32/0.61
% 0.32/0.61 % less_eq_neg_nonpos
% 0.32/0.61 thf(fact_5164_less__eq__neg__nonpos,axiom,
% 0.32/0.61 ! [A: code_integer] :
% 0.32/0.61 ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.32/0.61 = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.32/0.61
% 0.32/0.61 % less_eq_neg_nonpos
% 0.32/0.61 thf(fact_5165_less__eq__neg__nonpos,axiom,
% 0.32/0.61 ! [A: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.32/0.61 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.32/0.61
% 0.32/0.61 % less_eq_neg_nonpos
% 0.32/0.61 thf(fact_5166_less__eq__neg__nonpos,axiom,
% 0.32/0.61 ! [A: int] :
% 0.32/0.61 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
% 0.32/0.61 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.32/0.61
% 0.32/0.61 % less_eq_neg_nonpos
% 0.32/0.61 thf(fact_5167_neg__less__eq__nonneg,axiom,
% 0.32/0.61 ! [A: real] :
% 0.32/0.61 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 0.32/0.61 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_less_eq_nonneg
% 0.32/0.61 thf(fact_5168_neg__less__eq__nonneg,axiom,
% 0.32/0.61 ! [A: code_integer] :
% 0.32/0.61 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.32/0.61 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_less_eq_nonneg
% 0.32/0.61 thf(fact_5169_neg__less__eq__nonneg,axiom,
% 0.32/0.61 ! [A: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.32/0.61 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_less_eq_nonneg
% 0.32/0.61 thf(fact_5170_neg__less__eq__nonneg,axiom,
% 0.32/0.61 ! [A: int] :
% 0.32/0.61 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
% 0.32/0.61 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_less_eq_nonneg
% 0.32/0.61 thf(fact_5171_less__neg__neg,axiom,
% 0.32/0.61 ! [A: real] :
% 0.32/0.61 ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
% 0.32/0.61 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.32/0.61
% 0.32/0.61 % less_neg_neg
% 0.32/0.61 thf(fact_5172_less__neg__neg,axiom,
% 0.32/0.61 ! [A: int] :
% 0.32/0.61 ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
% 0.32/0.61 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.32/0.61
% 0.32/0.61 % less_neg_neg
% 0.32/0.61 thf(fact_5173_less__neg__neg,axiom,
% 0.32/0.61 ! [A: rat] :
% 0.32/0.61 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.32/0.61 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.32/0.61
% 0.32/0.61 % less_neg_neg
% 0.32/0.61 thf(fact_5174_less__neg__neg,axiom,
% 0.32/0.61 ! [A: code_integer] :
% 0.32/0.61 ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.32/0.61 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.32/0.61
% 0.32/0.61 % less_neg_neg
% 0.32/0.61 thf(fact_5175_neg__less__pos,axiom,
% 0.32/0.61 ! [A: real] :
% 0.32/0.61 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
% 0.32/0.61 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_less_pos
% 0.32/0.61 thf(fact_5176_neg__less__pos,axiom,
% 0.32/0.61 ! [A: int] :
% 0.32/0.61 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
% 0.32/0.61 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_less_pos
% 0.32/0.61 thf(fact_5177_neg__less__pos,axiom,
% 0.32/0.61 ! [A: rat] :
% 0.32/0.61 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.32/0.61 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_less_pos
% 0.32/0.61 thf(fact_5178_neg__less__pos,axiom,
% 0.32/0.61 ! [A: code_integer] :
% 0.32/0.61 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.32/0.61 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_less_pos
% 0.32/0.61 thf(fact_5179_neg__0__less__iff__less,axiom,
% 0.32/0.61 ! [A: real] :
% 0.32/0.61 ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.32/0.61 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_0_less_iff_less
% 0.32/0.61 thf(fact_5180_neg__0__less__iff__less,axiom,
% 0.32/0.61 ! [A: int] :
% 0.32/0.61 ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 0.32/0.61 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_0_less_iff_less
% 0.32/0.61 thf(fact_5181_neg__0__less__iff__less,axiom,
% 0.32/0.61 ! [A: rat] :
% 0.32/0.61 ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 0.32/0.61 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_0_less_iff_less
% 0.32/0.61 thf(fact_5182_neg__0__less__iff__less,axiom,
% 0.32/0.61 ! [A: code_integer] :
% 0.32/0.61 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.32/0.61 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_0_less_iff_less
% 0.32/0.61 thf(fact_5183_neg__less__0__iff__less,axiom,
% 0.32/0.61 ! [A: real] :
% 0.32/0.61 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.32/0.61 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_less_0_iff_less
% 0.32/0.61 thf(fact_5184_neg__less__0__iff__less,axiom,
% 0.32/0.61 ! [A: int] :
% 0.32/0.61 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 0.32/0.61 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_less_0_iff_less
% 0.32/0.61 thf(fact_5185_neg__less__0__iff__less,axiom,
% 0.32/0.61 ! [A: rat] :
% 0.32/0.61 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 0.32/0.61 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_less_0_iff_less
% 0.32/0.61 thf(fact_5186_neg__less__0__iff__less,axiom,
% 0.32/0.61 ! [A: code_integer] :
% 0.32/0.61 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 0.32/0.61 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_less_0_iff_less
% 0.32/0.61 thf(fact_5187_add_Oright__inverse,axiom,
% 0.32/0.61 ! [A: real] :
% 0.32/0.61 ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
% 0.32/0.61 = zero_zero_real ) ).
% 0.32/0.61
% 0.32/0.61 % add.right_inverse
% 0.32/0.61 thf(fact_5188_add_Oright__inverse,axiom,
% 0.32/0.61 ! [A: int] :
% 0.32/0.61 ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
% 0.32/0.61 = zero_zero_int ) ).
% 0.32/0.61
% 0.32/0.61 % add.right_inverse
% 0.32/0.61 thf(fact_5189_add_Oright__inverse,axiom,
% 0.32/0.61 ! [A: complex] :
% 0.32/0.61 ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
% 0.32/0.61 = zero_zero_complex ) ).
% 0.32/0.61
% 0.32/0.61 % add.right_inverse
% 0.32/0.61 thf(fact_5190_add_Oright__inverse,axiom,
% 0.32/0.61 ! [A: rat] :
% 0.32/0.61 ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.32/0.61 = zero_zero_rat ) ).
% 0.32/0.61
% 0.32/0.61 % add.right_inverse
% 0.32/0.61 thf(fact_5191_add_Oright__inverse,axiom,
% 0.32/0.61 ! [A: code_integer] :
% 0.32/0.61 ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.32/0.61 = zero_z3403309356797280102nteger ) ).
% 0.32/0.61
% 0.32/0.61 % add.right_inverse
% 0.32/0.61 thf(fact_5192_ab__left__minus,axiom,
% 0.32/0.61 ! [A: real] :
% 0.32/0.61 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 0.32/0.61 = zero_zero_real ) ).
% 0.32/0.61
% 0.32/0.61 % ab_left_minus
% 0.32/0.61 thf(fact_5193_ab__left__minus,axiom,
% 0.32/0.61 ! [A: int] :
% 0.32/0.61 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 0.32/0.61 = zero_zero_int ) ).
% 0.32/0.61
% 0.32/0.61 % ab_left_minus
% 0.32/0.61 thf(fact_5194_ab__left__minus,axiom,
% 0.32/0.61 ! [A: complex] :
% 0.32/0.61 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 0.32/0.61 = zero_zero_complex ) ).
% 0.32/0.61
% 0.32/0.61 % ab_left_minus
% 0.32/0.61 thf(fact_5195_ab__left__minus,axiom,
% 0.32/0.61 ! [A: rat] :
% 0.32/0.61 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.32/0.61 = zero_zero_rat ) ).
% 0.32/0.61
% 0.32/0.61 % ab_left_minus
% 0.32/0.61 thf(fact_5196_ab__left__minus,axiom,
% 0.32/0.61 ! [A: code_integer] :
% 0.32/0.61 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.32/0.61 = zero_z3403309356797280102nteger ) ).
% 0.32/0.61
% 0.32/0.61 % ab_left_minus
% 0.32/0.61 thf(fact_5197_diff__0,axiom,
% 0.32/0.61 ! [A: real] :
% 0.32/0.61 ( ( minus_minus_real @ zero_zero_real @ A )
% 0.32/0.61 = ( uminus_uminus_real @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % diff_0
% 0.32/0.61 thf(fact_5198_diff__0,axiom,
% 0.32/0.61 ! [A: int] :
% 0.32/0.61 ( ( minus_minus_int @ zero_zero_int @ A )
% 0.32/0.61 = ( uminus_uminus_int @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % diff_0
% 0.32/0.61 thf(fact_5199_diff__0,axiom,
% 0.32/0.61 ! [A: complex] :
% 0.32/0.61 ( ( minus_minus_complex @ zero_zero_complex @ A )
% 0.32/0.61 = ( uminus1482373934393186551omplex @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % diff_0
% 0.32/0.61 thf(fact_5200_diff__0,axiom,
% 0.32/0.61 ! [A: rat] :
% 0.32/0.61 ( ( minus_minus_rat @ zero_zero_rat @ A )
% 0.32/0.61 = ( uminus_uminus_rat @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % diff_0
% 0.32/0.61 thf(fact_5201_diff__0,axiom,
% 0.32/0.61 ! [A: code_integer] :
% 0.32/0.61 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
% 0.32/0.61 = ( uminus1351360451143612070nteger @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % diff_0
% 0.32/0.61 thf(fact_5202_verit__minus__simplify_I3_J,axiom,
% 0.32/0.61 ! [B: real] :
% 0.32/0.61 ( ( minus_minus_real @ zero_zero_real @ B )
% 0.32/0.61 = ( uminus_uminus_real @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % verit_minus_simplify(3)
% 0.32/0.61 thf(fact_5203_verit__minus__simplify_I3_J,axiom,
% 0.32/0.61 ! [B: int] :
% 0.32/0.61 ( ( minus_minus_int @ zero_zero_int @ B )
% 0.32/0.61 = ( uminus_uminus_int @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % verit_minus_simplify(3)
% 0.32/0.61 thf(fact_5204_verit__minus__simplify_I3_J,axiom,
% 0.32/0.61 ! [B: complex] :
% 0.32/0.61 ( ( minus_minus_complex @ zero_zero_complex @ B )
% 0.32/0.61 = ( uminus1482373934393186551omplex @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % verit_minus_simplify(3)
% 0.32/0.61 thf(fact_5205_verit__minus__simplify_I3_J,axiom,
% 0.32/0.61 ! [B: rat] :
% 0.32/0.61 ( ( minus_minus_rat @ zero_zero_rat @ B )
% 0.32/0.61 = ( uminus_uminus_rat @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % verit_minus_simplify(3)
% 0.32/0.61 thf(fact_5206_verit__minus__simplify_I3_J,axiom,
% 0.32/0.61 ! [B: code_integer] :
% 0.32/0.61 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
% 0.32/0.61 = ( uminus1351360451143612070nteger @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % verit_minus_simplify(3)
% 0.32/0.61 thf(fact_5207_add__neg__numeral__simps_I3_J,axiom,
% 0.32/0.61 ! [M: num,N2: num] :
% 0.32/0.61 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 0.32/0.61 = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add_neg_numeral_simps(3)
% 0.32/0.61 thf(fact_5208_add__neg__numeral__simps_I3_J,axiom,
% 0.32/0.61 ! [M: num,N2: num] :
% 0.32/0.61 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.32/0.61 = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add_neg_numeral_simps(3)
% 0.32/0.61 thf(fact_5209_add__neg__numeral__simps_I3_J,axiom,
% 0.32/0.61 ! [M: num,N2: num] :
% 0.32/0.61 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
% 0.32/0.61 = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N2 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add_neg_numeral_simps(3)
% 0.32/0.61 thf(fact_5210_add__neg__numeral__simps_I3_J,axiom,
% 0.32/0.61 ! [M: num,N2: num] :
% 0.32/0.61 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 0.32/0.61 = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add_neg_numeral_simps(3)
% 0.32/0.61 thf(fact_5211_add__neg__numeral__simps_I3_J,axiom,
% 0.32/0.61 ! [M: num,N2: num] :
% 0.32/0.61 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
% 0.32/0.61 = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add_neg_numeral_simps(3)
% 0.32/0.61 thf(fact_5212_mult__minus1,axiom,
% 0.32/0.61 ! [Z: real] :
% 0.32/0.61 ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
% 0.32/0.61 = ( uminus_uminus_real @ Z ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_minus1
% 0.32/0.61 thf(fact_5213_mult__minus1,axiom,
% 0.32/0.61 ! [Z: int] :
% 0.32/0.61 ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
% 0.32/0.61 = ( uminus_uminus_int @ Z ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_minus1
% 0.32/0.61 thf(fact_5214_mult__minus1,axiom,
% 0.32/0.61 ! [Z: complex] :
% 0.32/0.61 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
% 0.32/0.61 = ( uminus1482373934393186551omplex @ Z ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_minus1
% 0.32/0.61 thf(fact_5215_mult__minus1,axiom,
% 0.32/0.61 ! [Z: rat] :
% 0.32/0.61 ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
% 0.32/0.61 = ( uminus_uminus_rat @ Z ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_minus1
% 0.32/0.61 thf(fact_5216_mult__minus1,axiom,
% 0.32/0.61 ! [Z: code_integer] :
% 0.32/0.61 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
% 0.32/0.61 = ( uminus1351360451143612070nteger @ Z ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_minus1
% 0.32/0.61 thf(fact_5217_mult__minus1__right,axiom,
% 0.32/0.61 ! [Z: real] :
% 0.32/0.61 ( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
% 0.32/0.61 = ( uminus_uminus_real @ Z ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_minus1_right
% 0.32/0.61 thf(fact_5218_mult__minus1__right,axiom,
% 0.32/0.61 ! [Z: int] :
% 0.32/0.61 ( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.61 = ( uminus_uminus_int @ Z ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_minus1_right
% 0.32/0.61 thf(fact_5219_mult__minus1__right,axiom,
% 0.32/0.61 ! [Z: complex] :
% 0.32/0.61 ( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.32/0.61 = ( uminus1482373934393186551omplex @ Z ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_minus1_right
% 0.32/0.61 thf(fact_5220_mult__minus1__right,axiom,
% 0.32/0.61 ! [Z: rat] :
% 0.32/0.61 ( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.32/0.61 = ( uminus_uminus_rat @ Z ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_minus1_right
% 0.32/0.61 thf(fact_5221_mult__minus1__right,axiom,
% 0.32/0.61 ! [Z: code_integer] :
% 0.32/0.61 ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.61 = ( uminus1351360451143612070nteger @ Z ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_minus1_right
% 0.32/0.61 thf(fact_5222_uminus__add__conv__diff,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
% 0.32/0.61 = ( minus_minus_real @ B @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % uminus_add_conv_diff
% 0.32/0.61 thf(fact_5223_uminus__add__conv__diff,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
% 0.32/0.61 = ( minus_minus_int @ B @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % uminus_add_conv_diff
% 0.32/0.61 thf(fact_5224_uminus__add__conv__diff,axiom,
% 0.32/0.61 ! [A: complex,B: complex] :
% 0.32/0.61 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 0.32/0.61 = ( minus_minus_complex @ B @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % uminus_add_conv_diff
% 0.32/0.61 thf(fact_5225_uminus__add__conv__diff,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.32/0.61 = ( minus_minus_rat @ B @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % uminus_add_conv_diff
% 0.32/0.61 thf(fact_5226_uminus__add__conv__diff,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.32/0.61 = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % uminus_add_conv_diff
% 0.32/0.61 thf(fact_5227_diff__minus__eq__add,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
% 0.32/0.61 = ( plus_plus_real @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % diff_minus_eq_add
% 0.32/0.61 thf(fact_5228_diff__minus__eq__add,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
% 0.32/0.61 = ( plus_plus_int @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % diff_minus_eq_add
% 0.32/0.61 thf(fact_5229_diff__minus__eq__add,axiom,
% 0.32/0.61 ! [A: complex,B: complex] :
% 0.32/0.61 ( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 0.32/0.61 = ( plus_plus_complex @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % diff_minus_eq_add
% 0.32/0.61 thf(fact_5230_diff__minus__eq__add,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.32/0.61 = ( plus_plus_rat @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % diff_minus_eq_add
% 0.32/0.61 thf(fact_5231_diff__minus__eq__add,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.32/0.61 = ( plus_p5714425477246183910nteger @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % diff_minus_eq_add
% 0.32/0.61 thf(fact_5232_div__minus1__right,axiom,
% 0.32/0.61 ! [A: int] :
% 0.32/0.61 ( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.61 = ( uminus_uminus_int @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % div_minus1_right
% 0.32/0.61 thf(fact_5233_div__minus1__right,axiom,
% 0.32/0.61 ! [A: code_integer] :
% 0.32/0.61 ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.61 = ( uminus1351360451143612070nteger @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % div_minus1_right
% 0.32/0.61 thf(fact_5234_divide__minus1,axiom,
% 0.32/0.61 ! [X: real] :
% 0.32/0.61 ( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
% 0.32/0.61 = ( uminus_uminus_real @ X ) ) ).
% 0.32/0.61
% 0.32/0.61 % divide_minus1
% 0.32/0.61 thf(fact_5235_divide__minus1,axiom,
% 0.32/0.61 ! [X: complex] :
% 0.32/0.61 ( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.32/0.61 = ( uminus1482373934393186551omplex @ X ) ) ).
% 0.32/0.61
% 0.32/0.61 % divide_minus1
% 0.32/0.61 thf(fact_5236_divide__minus1,axiom,
% 0.32/0.61 ! [X: rat] :
% 0.32/0.61 ( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.32/0.61 = ( uminus_uminus_rat @ X ) ) ).
% 0.32/0.61
% 0.32/0.61 % divide_minus1
% 0.32/0.61 thf(fact_5237_minus__mod__self1,axiom,
% 0.32/0.61 ! [B: int,A: int] :
% 0.32/0.61 ( ( modulo_modulo_int @ ( minus_minus_int @ B @ A ) @ B )
% 0.32/0.61 = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_mod_self1
% 0.32/0.61 thf(fact_5238_minus__mod__self1,axiom,
% 0.32/0.61 ! [B: code_integer,A: code_integer] :
% 0.32/0.61 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ B )
% 0.32/0.61 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_mod_self1
% 0.32/0.61 thf(fact_5239_ln__le__cancel__iff,axiom,
% 0.32/0.61 ! [X: real,Y: real] :
% 0.32/0.61 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.61 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.32/0.61 => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
% 0.32/0.61 = ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % ln_le_cancel_iff
% 0.32/0.61 thf(fact_5240_ln__eq__zero__iff,axiom,
% 0.32/0.61 ! [X: real] :
% 0.32/0.61 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.61 => ( ( ( ln_ln_real @ X )
% 0.32/0.61 = zero_zero_real )
% 0.32/0.61 = ( X = one_one_real ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % ln_eq_zero_iff
% 0.32/0.61 thf(fact_5241_ln__gt__zero__iff,axiom,
% 0.32/0.61 ! [X: real] :
% 0.32/0.61 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.61 => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 0.32/0.61 = ( ord_less_real @ one_one_real @ X ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % ln_gt_zero_iff
% 0.32/0.61 thf(fact_5242_ln__less__zero__iff,axiom,
% 0.32/0.61 ! [X: real] :
% 0.32/0.61 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.61 => ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
% 0.32/0.61 = ( ord_less_real @ X @ one_one_real ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % ln_less_zero_iff
% 0.32/0.61 thf(fact_5243_ln__one,axiom,
% 0.32/0.61 ( ( ln_ln_real @ one_one_real )
% 0.32/0.61 = zero_zero_real ) ).
% 0.32/0.61
% 0.32/0.61 % ln_one
% 0.32/0.61 thf(fact_5244_signed__take__bit__of__minus__1,axiom,
% 0.32/0.61 ! [N2: nat] :
% 0.32/0.61 ( ( bit_ri6519982836138164636nteger @ N2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.61 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.32/0.61
% 0.32/0.61 % signed_take_bit_of_minus_1
% 0.32/0.61 thf(fact_5245_signed__take__bit__of__minus__1,axiom,
% 0.32/0.61 ! [N2: nat] :
% 0.32/0.61 ( ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.61 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.32/0.61
% 0.32/0.61 % signed_take_bit_of_minus_1
% 0.32/0.61 thf(fact_5246_sum_Odelta,axiom,
% 0.32/0.61 ! [S: set_real,A: real,B: real > complex] :
% 0.32/0.61 ( ( finite_finite_real @ S )
% 0.32/0.61 => ( ( ( member_real @ A @ S )
% 0.32/0.61 => ( ( groups5754745047067104278omplex
% 0.32/0.61 @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.32/0.61 @ S )
% 0.32/0.61 = ( B @ A ) ) )
% 0.32/0.61 & ( ~ ( member_real @ A @ S )
% 0.32/0.61 => ( ( groups5754745047067104278omplex
% 0.32/0.61 @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.32/0.61 @ S )
% 0.32/0.61 = zero_zero_complex ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % sum.delta
% 0.32/0.61 thf(fact_5247_sum_Odelta,axiom,
% 0.32/0.62 ! [S: set_nat,A: nat,B: nat > complex] :
% 0.32/0.62 ( ( finite_finite_nat @ S )
% 0.32/0.62 => ( ( ( member_nat @ A @ S )
% 0.32/0.62 => ( ( groups2073611262835488442omplex
% 0.32/0.62 @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.32/0.62 @ S )
% 0.32/0.62 = ( B @ A ) ) )
% 0.32/0.62 & ( ~ ( member_nat @ A @ S )
% 0.32/0.62 => ( ( groups2073611262835488442omplex
% 0.32/0.62 @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.32/0.62 @ S )
% 0.32/0.62 = zero_zero_complex ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.delta
% 0.32/0.62 thf(fact_5248_sum_Odelta,axiom,
% 0.32/0.62 ! [S: set_int,A: int,B: int > complex] :
% 0.32/0.62 ( ( finite_finite_int @ S )
% 0.32/0.62 => ( ( ( member_int @ A @ S )
% 0.32/0.62 => ( ( groups3049146728041665814omplex
% 0.32/0.62 @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.32/0.62 @ S )
% 0.32/0.62 = ( B @ A ) ) )
% 0.32/0.62 & ( ~ ( member_int @ A @ S )
% 0.32/0.62 => ( ( groups3049146728041665814omplex
% 0.32/0.62 @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.32/0.62 @ S )
% 0.32/0.62 = zero_zero_complex ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.delta
% 0.32/0.62 thf(fact_5249_sum_Odelta,axiom,
% 0.32/0.62 ! [S: set_real,A: real,B: real > real] :
% 0.32/0.62 ( ( finite_finite_real @ S )
% 0.32/0.62 => ( ( ( member_real @ A @ S )
% 0.32/0.62 => ( ( groups8097168146408367636l_real
% 0.32/0.62 @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 0.32/0.62 @ S )
% 0.32/0.62 = ( B @ A ) ) )
% 0.32/0.62 & ( ~ ( member_real @ A @ S )
% 0.32/0.62 => ( ( groups8097168146408367636l_real
% 0.32/0.62 @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 0.32/0.62 @ S )
% 0.32/0.62 = zero_zero_real ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.delta
% 0.32/0.62 thf(fact_5250_sum_Odelta,axiom,
% 0.32/0.62 ! [S: set_int,A: int,B: int > real] :
% 0.32/0.62 ( ( finite_finite_int @ S )
% 0.32/0.62 => ( ( ( member_int @ A @ S )
% 0.32/0.62 => ( ( groups8778361861064173332t_real
% 0.32/0.62 @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 0.32/0.62 @ S )
% 0.32/0.62 = ( B @ A ) ) )
% 0.32/0.62 & ( ~ ( member_int @ A @ S )
% 0.32/0.62 => ( ( groups8778361861064173332t_real
% 0.32/0.62 @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 0.32/0.62 @ S )
% 0.32/0.62 = zero_zero_real ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.delta
% 0.32/0.62 thf(fact_5251_sum_Odelta,axiom,
% 0.32/0.62 ! [S: set_complex,A: complex,B: complex > real] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ S )
% 0.32/0.62 => ( ( ( member_complex @ A @ S )
% 0.32/0.62 => ( ( groups5808333547571424918x_real
% 0.32/0.62 @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 0.32/0.62 @ S )
% 0.32/0.62 = ( B @ A ) ) )
% 0.32/0.62 & ( ~ ( member_complex @ A @ S )
% 0.32/0.62 => ( ( groups5808333547571424918x_real
% 0.32/0.62 @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 0.32/0.62 @ S )
% 0.32/0.62 = zero_zero_real ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.delta
% 0.32/0.62 thf(fact_5252_sum_Odelta,axiom,
% 0.32/0.62 ! [S: set_real,A: real,B: real > rat] :
% 0.32/0.62 ( ( finite_finite_real @ S )
% 0.32/0.62 => ( ( ( member_real @ A @ S )
% 0.32/0.62 => ( ( groups1300246762558778688al_rat
% 0.32/0.62 @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.32/0.62 @ S )
% 0.32/0.62 = ( B @ A ) ) )
% 0.32/0.62 & ( ~ ( member_real @ A @ S )
% 0.32/0.62 => ( ( groups1300246762558778688al_rat
% 0.32/0.62 @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.32/0.62 @ S )
% 0.32/0.62 = zero_zero_rat ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.delta
% 0.32/0.62 thf(fact_5253_sum_Odelta,axiom,
% 0.32/0.62 ! [S: set_nat,A: nat,B: nat > rat] :
% 0.32/0.62 ( ( finite_finite_nat @ S )
% 0.32/0.62 => ( ( ( member_nat @ A @ S )
% 0.32/0.62 => ( ( groups2906978787729119204at_rat
% 0.32/0.62 @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.32/0.62 @ S )
% 0.32/0.62 = ( B @ A ) ) )
% 0.32/0.62 & ( ~ ( member_nat @ A @ S )
% 0.32/0.62 => ( ( groups2906978787729119204at_rat
% 0.32/0.62 @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.32/0.62 @ S )
% 0.32/0.62 = zero_zero_rat ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.delta
% 0.32/0.62 thf(fact_5254_sum_Odelta,axiom,
% 0.32/0.62 ! [S: set_int,A: int,B: int > rat] :
% 0.32/0.62 ( ( finite_finite_int @ S )
% 0.32/0.62 => ( ( ( member_int @ A @ S )
% 0.32/0.62 => ( ( groups3906332499630173760nt_rat
% 0.32/0.62 @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.32/0.62 @ S )
% 0.32/0.62 = ( B @ A ) ) )
% 0.32/0.62 & ( ~ ( member_int @ A @ S )
% 0.32/0.62 => ( ( groups3906332499630173760nt_rat
% 0.32/0.62 @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.32/0.62 @ S )
% 0.32/0.62 = zero_zero_rat ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.delta
% 0.32/0.62 thf(fact_5255_sum_Odelta,axiom,
% 0.32/0.62 ! [S: set_complex,A: complex,B: complex > rat] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ S )
% 0.32/0.62 => ( ( ( member_complex @ A @ S )
% 0.32/0.62 => ( ( groups5058264527183730370ex_rat
% 0.32/0.62 @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.32/0.62 @ S )
% 0.32/0.62 = ( B @ A ) ) )
% 0.32/0.62 & ( ~ ( member_complex @ A @ S )
% 0.32/0.62 => ( ( groups5058264527183730370ex_rat
% 0.32/0.62 @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.32/0.62 @ S )
% 0.32/0.62 = zero_zero_rat ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.delta
% 0.32/0.62 thf(fact_5256_sum_Odelta_H,axiom,
% 0.32/0.62 ! [S: set_real,A: real,B: real > complex] :
% 0.32/0.62 ( ( finite_finite_real @ S )
% 0.32/0.62 => ( ( ( member_real @ A @ S )
% 0.32/0.62 => ( ( groups5754745047067104278omplex
% 0.32/0.62 @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.32/0.62 @ S )
% 0.32/0.62 = ( B @ A ) ) )
% 0.32/0.62 & ( ~ ( member_real @ A @ S )
% 0.32/0.62 => ( ( groups5754745047067104278omplex
% 0.32/0.62 @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.32/0.62 @ S )
% 0.32/0.62 = zero_zero_complex ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.delta'
% 0.32/0.62 thf(fact_5257_sum_Odelta_H,axiom,
% 0.32/0.62 ! [S: set_nat,A: nat,B: nat > complex] :
% 0.32/0.62 ( ( finite_finite_nat @ S )
% 0.32/0.62 => ( ( ( member_nat @ A @ S )
% 0.32/0.62 => ( ( groups2073611262835488442omplex
% 0.32/0.62 @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.32/0.62 @ S )
% 0.32/0.62 = ( B @ A ) ) )
% 0.32/0.62 & ( ~ ( member_nat @ A @ S )
% 0.32/0.62 => ( ( groups2073611262835488442omplex
% 0.32/0.62 @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.32/0.62 @ S )
% 0.32/0.62 = zero_zero_complex ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.delta'
% 0.32/0.62 thf(fact_5258_sum_Odelta_H,axiom,
% 0.32/0.62 ! [S: set_int,A: int,B: int > complex] :
% 0.32/0.62 ( ( finite_finite_int @ S )
% 0.32/0.62 => ( ( ( member_int @ A @ S )
% 0.32/0.62 => ( ( groups3049146728041665814omplex
% 0.32/0.62 @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.32/0.62 @ S )
% 0.32/0.62 = ( B @ A ) ) )
% 0.32/0.62 & ( ~ ( member_int @ A @ S )
% 0.32/0.62 => ( ( groups3049146728041665814omplex
% 0.32/0.62 @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.32/0.62 @ S )
% 0.32/0.62 = zero_zero_complex ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.delta'
% 0.32/0.62 thf(fact_5259_sum_Odelta_H,axiom,
% 0.32/0.62 ! [S: set_real,A: real,B: real > real] :
% 0.32/0.62 ( ( finite_finite_real @ S )
% 0.32/0.62 => ( ( ( member_real @ A @ S )
% 0.32/0.62 => ( ( groups8097168146408367636l_real
% 0.32/0.62 @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 0.32/0.62 @ S )
% 0.32/0.62 = ( B @ A ) ) )
% 0.32/0.62 & ( ~ ( member_real @ A @ S )
% 0.32/0.62 => ( ( groups8097168146408367636l_real
% 0.32/0.62 @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 0.32/0.62 @ S )
% 0.32/0.62 = zero_zero_real ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.delta'
% 0.32/0.62 thf(fact_5260_sum_Odelta_H,axiom,
% 0.32/0.62 ! [S: set_int,A: int,B: int > real] :
% 0.32/0.62 ( ( finite_finite_int @ S )
% 0.32/0.62 => ( ( ( member_int @ A @ S )
% 0.32/0.62 => ( ( groups8778361861064173332t_real
% 0.32/0.62 @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 0.32/0.62 @ S )
% 0.32/0.62 = ( B @ A ) ) )
% 0.32/0.62 & ( ~ ( member_int @ A @ S )
% 0.32/0.62 => ( ( groups8778361861064173332t_real
% 0.32/0.62 @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 0.32/0.62 @ S )
% 0.32/0.62 = zero_zero_real ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.delta'
% 0.32/0.62 thf(fact_5261_sum_Odelta_H,axiom,
% 0.32/0.62 ! [S: set_complex,A: complex,B: complex > real] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ S )
% 0.32/0.62 => ( ( ( member_complex @ A @ S )
% 0.32/0.62 => ( ( groups5808333547571424918x_real
% 0.32/0.62 @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 0.32/0.62 @ S )
% 0.32/0.62 = ( B @ A ) ) )
% 0.32/0.62 & ( ~ ( member_complex @ A @ S )
% 0.32/0.62 => ( ( groups5808333547571424918x_real
% 0.32/0.62 @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 0.32/0.62 @ S )
% 0.32/0.62 = zero_zero_real ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.delta'
% 0.32/0.62 thf(fact_5262_sum_Odelta_H,axiom,
% 0.32/0.62 ! [S: set_real,A: real,B: real > rat] :
% 0.32/0.62 ( ( finite_finite_real @ S )
% 0.32/0.62 => ( ( ( member_real @ A @ S )
% 0.32/0.62 => ( ( groups1300246762558778688al_rat
% 0.32/0.62 @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.32/0.62 @ S )
% 0.32/0.62 = ( B @ A ) ) )
% 0.32/0.62 & ( ~ ( member_real @ A @ S )
% 0.32/0.62 => ( ( groups1300246762558778688al_rat
% 0.32/0.62 @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.32/0.62 @ S )
% 0.32/0.62 = zero_zero_rat ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.delta'
% 0.32/0.62 thf(fact_5263_sum_Odelta_H,axiom,
% 0.32/0.62 ! [S: set_nat,A: nat,B: nat > rat] :
% 0.32/0.62 ( ( finite_finite_nat @ S )
% 0.32/0.62 => ( ( ( member_nat @ A @ S )
% 0.32/0.62 => ( ( groups2906978787729119204at_rat
% 0.32/0.62 @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.32/0.62 @ S )
% 0.32/0.62 = ( B @ A ) ) )
% 0.32/0.62 & ( ~ ( member_nat @ A @ S )
% 0.32/0.62 => ( ( groups2906978787729119204at_rat
% 0.32/0.62 @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.32/0.62 @ S )
% 0.32/0.62 = zero_zero_rat ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.delta'
% 0.32/0.62 thf(fact_5264_sum_Odelta_H,axiom,
% 0.32/0.62 ! [S: set_int,A: int,B: int > rat] :
% 0.32/0.62 ( ( finite_finite_int @ S )
% 0.32/0.62 => ( ( ( member_int @ A @ S )
% 0.32/0.62 => ( ( groups3906332499630173760nt_rat
% 0.32/0.62 @ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.32/0.62 @ S )
% 0.32/0.62 = ( B @ A ) ) )
% 0.32/0.62 & ( ~ ( member_int @ A @ S )
% 0.32/0.62 => ( ( groups3906332499630173760nt_rat
% 0.32/0.62 @ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.32/0.62 @ S )
% 0.32/0.62 = zero_zero_rat ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.delta'
% 0.32/0.62 thf(fact_5265_sum_Odelta_H,axiom,
% 0.32/0.62 ! [S: set_complex,A: complex,B: complex > rat] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ S )
% 0.32/0.62 => ( ( ( member_complex @ A @ S )
% 0.32/0.62 => ( ( groups5058264527183730370ex_rat
% 0.32/0.62 @ ^ [K3: complex] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.32/0.62 @ S )
% 0.32/0.62 = ( B @ A ) ) )
% 0.32/0.62 & ( ~ ( member_complex @ A @ S )
% 0.32/0.62 => ( ( groups5058264527183730370ex_rat
% 0.32/0.62 @ ^ [K3: complex] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.32/0.62 @ S )
% 0.32/0.62 = zero_zero_rat ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.delta'
% 0.32/0.62 thf(fact_5266_dbl__simps_I1_J,axiom,
% 0.32/0.62 ! [K: num] :
% 0.32/0.62 ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 0.32/0.62 = ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % dbl_simps(1)
% 0.32/0.62 thf(fact_5267_dbl__simps_I1_J,axiom,
% 0.32/0.62 ! [K: num] :
% 0.32/0.62 ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.32/0.62 = ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % dbl_simps(1)
% 0.32/0.62 thf(fact_5268_dbl__simps_I1_J,axiom,
% 0.32/0.62 ! [K: num] :
% 0.32/0.62 ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 0.32/0.62 = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % dbl_simps(1)
% 0.32/0.62 thf(fact_5269_dbl__simps_I1_J,axiom,
% 0.32/0.62 ! [K: num] :
% 0.32/0.62 ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.32/0.62 = ( uminus_uminus_rat @ ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % dbl_simps(1)
% 0.32/0.62 thf(fact_5270_dbl__simps_I1_J,axiom,
% 0.32/0.62 ! [K: num] :
% 0.32/0.62 ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.32/0.62 = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % dbl_simps(1)
% 0.32/0.62 thf(fact_5271_add__neg__numeral__special_I8_J,axiom,
% 0.32/0.62 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 0.32/0.62 = zero_zero_real ) ).
% 0.32/0.62
% 0.32/0.62 % add_neg_numeral_special(8)
% 0.32/0.62 thf(fact_5272_add__neg__numeral__special_I8_J,axiom,
% 0.32/0.62 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 0.32/0.62 = zero_zero_int ) ).
% 0.32/0.62
% 0.32/0.62 % add_neg_numeral_special(8)
% 0.32/0.62 thf(fact_5273_add__neg__numeral__special_I8_J,axiom,
% 0.32/0.62 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 0.32/0.62 = zero_zero_complex ) ).
% 0.32/0.62
% 0.32/0.62 % add_neg_numeral_special(8)
% 0.32/0.62 thf(fact_5274_add__neg__numeral__special_I8_J,axiom,
% 0.32/0.62 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 0.32/0.62 = zero_zero_rat ) ).
% 0.32/0.62
% 0.32/0.62 % add_neg_numeral_special(8)
% 0.32/0.62 thf(fact_5275_add__neg__numeral__special_I8_J,axiom,
% 0.32/0.62 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 0.32/0.62 = zero_z3403309356797280102nteger ) ).
% 0.32/0.62
% 0.32/0.62 % add_neg_numeral_special(8)
% 0.32/0.62 thf(fact_5276_add__neg__numeral__special_I7_J,axiom,
% 0.32/0.62 ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.32/0.62 = zero_zero_real ) ).
% 0.32/0.62
% 0.32/0.62 % add_neg_numeral_special(7)
% 0.32/0.62 thf(fact_5277_add__neg__numeral__special_I7_J,axiom,
% 0.32/0.62 ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.62 = zero_zero_int ) ).
% 0.32/0.62
% 0.32/0.62 % add_neg_numeral_special(7)
% 0.32/0.62 thf(fact_5278_add__neg__numeral__special_I7_J,axiom,
% 0.32/0.62 ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.32/0.62 = zero_zero_complex ) ).
% 0.32/0.62
% 0.32/0.62 % add_neg_numeral_special(7)
% 0.32/0.62 thf(fact_5279_add__neg__numeral__special_I7_J,axiom,
% 0.32/0.62 ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.32/0.62 = zero_zero_rat ) ).
% 0.32/0.62
% 0.32/0.62 % add_neg_numeral_special(7)
% 0.32/0.62 thf(fact_5280_add__neg__numeral__special_I7_J,axiom,
% 0.32/0.62 ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.62 = zero_z3403309356797280102nteger ) ).
% 0.32/0.62
% 0.32/0.62 % add_neg_numeral_special(7)
% 0.32/0.62 thf(fact_5281_diff__numeral__special_I12_J,axiom,
% 0.32/0.62 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.32/0.62 = zero_zero_real ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_special(12)
% 0.32/0.62 thf(fact_5282_diff__numeral__special_I12_J,axiom,
% 0.32/0.62 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.62 = zero_zero_int ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_special(12)
% 0.32/0.62 thf(fact_5283_diff__numeral__special_I12_J,axiom,
% 0.32/0.62 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.32/0.62 = zero_zero_complex ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_special(12)
% 0.32/0.62 thf(fact_5284_diff__numeral__special_I12_J,axiom,
% 0.32/0.62 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.32/0.62 = zero_zero_rat ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_special(12)
% 0.32/0.62 thf(fact_5285_diff__numeral__special_I12_J,axiom,
% 0.32/0.62 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.62 = zero_z3403309356797280102nteger ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_special(12)
% 0.32/0.62 thf(fact_5286_numeral__eq__neg__one__iff,axiom,
% 0.32/0.62 ! [N2: num] :
% 0.32/0.62 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) )
% 0.32/0.62 = ( uminus_uminus_real @ one_one_real ) )
% 0.32/0.62 = ( N2 = one ) ) ).
% 0.32/0.62
% 0.32/0.62 % numeral_eq_neg_one_iff
% 0.32/0.62 thf(fact_5287_numeral__eq__neg__one__iff,axiom,
% 0.32/0.62 ! [N2: num] :
% 0.32/0.62 ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) )
% 0.32/0.62 = ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.62 = ( N2 = one ) ) ).
% 0.32/0.62
% 0.32/0.62 % numeral_eq_neg_one_iff
% 0.32/0.62 thf(fact_5288_numeral__eq__neg__one__iff,axiom,
% 0.32/0.62 ! [N2: num] :
% 0.32/0.62 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) )
% 0.32/0.62 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.32/0.62 = ( N2 = one ) ) ).
% 0.32/0.62
% 0.32/0.62 % numeral_eq_neg_one_iff
% 0.32/0.62 thf(fact_5289_numeral__eq__neg__one__iff,axiom,
% 0.32/0.62 ! [N2: num] :
% 0.32/0.62 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) )
% 0.32/0.62 = ( uminus_uminus_rat @ one_one_rat ) )
% 0.32/0.62 = ( N2 = one ) ) ).
% 0.32/0.62
% 0.32/0.62 % numeral_eq_neg_one_iff
% 0.32/0.62 thf(fact_5290_numeral__eq__neg__one__iff,axiom,
% 0.32/0.62 ! [N2: num] :
% 0.32/0.62 ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) )
% 0.32/0.62 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.62 = ( N2 = one ) ) ).
% 0.32/0.62
% 0.32/0.62 % numeral_eq_neg_one_iff
% 0.32/0.62 thf(fact_5291_neg__one__eq__numeral__iff,axiom,
% 0.32/0.62 ! [N2: num] :
% 0.32/0.62 ( ( ( uminus_uminus_real @ one_one_real )
% 0.32/0.62 = ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 0.32/0.62 = ( N2 = one ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_one_eq_numeral_iff
% 0.32/0.62 thf(fact_5292_neg__one__eq__numeral__iff,axiom,
% 0.32/0.62 ! [N2: num] :
% 0.32/0.62 ( ( ( uminus_uminus_int @ one_one_int )
% 0.32/0.62 = ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.32/0.62 = ( N2 = one ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_one_eq_numeral_iff
% 0.32/0.62 thf(fact_5293_neg__one__eq__numeral__iff,axiom,
% 0.32/0.62 ! [N2: num] :
% 0.32/0.62 ( ( ( uminus1482373934393186551omplex @ one_one_complex )
% 0.32/0.62 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
% 0.32/0.62 = ( N2 = one ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_one_eq_numeral_iff
% 0.32/0.62 thf(fact_5294_neg__one__eq__numeral__iff,axiom,
% 0.32/0.62 ! [N2: num] :
% 0.32/0.62 ( ( ( uminus_uminus_rat @ one_one_rat )
% 0.32/0.62 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 0.32/0.62 = ( N2 = one ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_one_eq_numeral_iff
% 0.32/0.62 thf(fact_5295_neg__one__eq__numeral__iff,axiom,
% 0.32/0.62 ! [N2: num] :
% 0.32/0.62 ( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
% 0.32/0.62 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
% 0.32/0.62 = ( N2 = one ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_one_eq_numeral_iff
% 0.32/0.62 thf(fact_5296_left__minus__one__mult__self,axiom,
% 0.32/0.62 ! [N2: nat,A: real] :
% 0.32/0.62 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ A ) )
% 0.32/0.62 = A ) ).
% 0.32/0.62
% 0.32/0.62 % left_minus_one_mult_self
% 0.32/0.62 thf(fact_5297_left__minus__one__mult__self,axiom,
% 0.32/0.62 ! [N2: nat,A: int] :
% 0.32/0.62 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ A ) )
% 0.32/0.62 = A ) ).
% 0.32/0.62
% 0.32/0.62 % left_minus_one_mult_self
% 0.32/0.62 thf(fact_5298_left__minus__one__mult__self,axiom,
% 0.32/0.62 ! [N2: nat,A: complex] :
% 0.32/0.62 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ A ) )
% 0.32/0.62 = A ) ).
% 0.32/0.62
% 0.32/0.62 % left_minus_one_mult_self
% 0.32/0.62 thf(fact_5299_left__minus__one__mult__self,axiom,
% 0.32/0.62 ! [N2: nat,A: rat] :
% 0.32/0.62 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ A ) )
% 0.32/0.62 = A ) ).
% 0.32/0.62
% 0.32/0.62 % left_minus_one_mult_self
% 0.32/0.62 thf(fact_5300_left__minus__one__mult__self,axiom,
% 0.32/0.62 ! [N2: nat,A: code_integer] :
% 0.32/0.62 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ A ) )
% 0.32/0.62 = A ) ).
% 0.32/0.62
% 0.32/0.62 % left_minus_one_mult_self
% 0.32/0.62 thf(fact_5301_minus__one__mult__self,axiom,
% 0.32/0.62 ! [N2: nat] :
% 0.32/0.62 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) )
% 0.32/0.62 = one_one_real ) ).
% 0.32/0.62
% 0.32/0.62 % minus_one_mult_self
% 0.32/0.62 thf(fact_5302_minus__one__mult__self,axiom,
% 0.32/0.62 ! [N2: nat] :
% 0.32/0.62 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) )
% 0.32/0.62 = one_one_int ) ).
% 0.32/0.62
% 0.32/0.62 % minus_one_mult_self
% 0.32/0.62 thf(fact_5303_minus__one__mult__self,axiom,
% 0.32/0.62 ! [N2: nat] :
% 0.32/0.62 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) )
% 0.32/0.62 = one_one_complex ) ).
% 0.32/0.62
% 0.32/0.62 % minus_one_mult_self
% 0.32/0.62 thf(fact_5304_minus__one__mult__self,axiom,
% 0.32/0.62 ! [N2: nat] :
% 0.32/0.62 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) )
% 0.32/0.62 = one_one_rat ) ).
% 0.32/0.62
% 0.32/0.62 % minus_one_mult_self
% 0.32/0.62 thf(fact_5305_minus__one__mult__self,axiom,
% 0.32/0.62 ! [N2: nat] :
% 0.32/0.62 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) )
% 0.32/0.62 = one_one_Code_integer ) ).
% 0.32/0.62
% 0.32/0.62 % minus_one_mult_self
% 0.32/0.62 thf(fact_5306_mod__minus1__right,axiom,
% 0.32/0.62 ! [A: int] :
% 0.32/0.62 ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.62 = zero_zero_int ) ).
% 0.32/0.62
% 0.32/0.62 % mod_minus1_right
% 0.32/0.62 thf(fact_5307_mod__minus1__right,axiom,
% 0.32/0.62 ! [A: code_integer] :
% 0.32/0.62 ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.62 = zero_z3403309356797280102nteger ) ).
% 0.32/0.62
% 0.32/0.62 % mod_minus1_right
% 0.32/0.62 thf(fact_5308_max__number__of_I2_J,axiom,
% 0.32/0.62 ! [U: num,V: num] :
% 0.32/0.62 ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.32/0.62 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.32/0.62 = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 0.32/0.62 & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.32/0.62 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.32/0.62 = ( numeral_numeral_real @ U ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % max_number_of(2)
% 0.32/0.62 thf(fact_5309_max__number__of_I2_J,axiom,
% 0.32/0.62 ! [U: num,V: num] :
% 0.32/0.62 ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.32/0.62 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.32/0.62 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 0.32/0.62 & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.32/0.62 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.32/0.62 = ( numera6620942414471956472nteger @ U ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % max_number_of(2)
% 0.32/0.62 thf(fact_5310_max__number__of_I2_J,axiom,
% 0.32/0.62 ! [U: num,V: num] :
% 0.32/0.62 ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.32/0.62 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.32/0.62 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 0.32/0.62 & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.32/0.62 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.32/0.62 = ( numeral_numeral_rat @ U ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % max_number_of(2)
% 0.32/0.62 thf(fact_5311_max__number__of_I2_J,axiom,
% 0.32/0.62 ! [U: num,V: num] :
% 0.32/0.62 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.32/0.62 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.32/0.62 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 0.32/0.62 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.32/0.62 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.32/0.62 = ( numeral_numeral_int @ U ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % max_number_of(2)
% 0.32/0.62 thf(fact_5312_max__number__of_I3_J,axiom,
% 0.32/0.62 ! [U: num,V: num] :
% 0.32/0.62 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.32/0.62 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.32/0.62 = ( numeral_numeral_real @ V ) ) )
% 0.32/0.62 & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.32/0.62 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.32/0.62 = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % max_number_of(3)
% 0.32/0.62 thf(fact_5313_max__number__of_I3_J,axiom,
% 0.32/0.62 ! [U: num,V: num] :
% 0.32/0.62 ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.32/0.62 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.32/0.62 = ( numera6620942414471956472nteger @ V ) ) )
% 0.32/0.62 & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.32/0.62 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.32/0.62 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % max_number_of(3)
% 0.32/0.62 thf(fact_5314_max__number__of_I3_J,axiom,
% 0.32/0.62 ! [U: num,V: num] :
% 0.32/0.62 ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.32/0.62 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.32/0.62 = ( numeral_numeral_rat @ V ) ) )
% 0.32/0.62 & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.32/0.62 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.32/0.62 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % max_number_of(3)
% 0.32/0.62 thf(fact_5315_max__number__of_I3_J,axiom,
% 0.32/0.62 ! [U: num,V: num] :
% 0.32/0.62 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.32/0.62 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.32/0.62 = ( numeral_numeral_int @ V ) ) )
% 0.32/0.62 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.32/0.62 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.32/0.62 = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % max_number_of(3)
% 0.32/0.62 thf(fact_5316_max__number__of_I4_J,axiom,
% 0.32/0.62 ! [U: num,V: num] :
% 0.32/0.62 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.32/0.62 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.32/0.62 = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 0.32/0.62 & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.32/0.62 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.32/0.62 = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % max_number_of(4)
% 0.32/0.62 thf(fact_5317_max__number__of_I4_J,axiom,
% 0.32/0.62 ! [U: num,V: num] :
% 0.32/0.62 ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.32/0.62 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.32/0.62 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 0.32/0.62 & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.32/0.62 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.32/0.62 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % max_number_of(4)
% 0.32/0.62 thf(fact_5318_max__number__of_I4_J,axiom,
% 0.32/0.62 ! [U: num,V: num] :
% 0.32/0.62 ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.32/0.62 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.32/0.62 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 0.32/0.62 & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.32/0.62 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.32/0.62 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % max_number_of(4)
% 0.32/0.62 thf(fact_5319_max__number__of_I4_J,axiom,
% 0.32/0.62 ! [U: num,V: num] :
% 0.32/0.62 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.32/0.62 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.32/0.62 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 0.32/0.62 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.32/0.62 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.32/0.62 = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % max_number_of(4)
% 0.32/0.62 thf(fact_5320_ln__ge__zero__iff,axiom,
% 0.32/0.62 ! [X: real] :
% 0.32/0.62 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.62 => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 0.32/0.62 = ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % ln_ge_zero_iff
% 0.32/0.62 thf(fact_5321_ln__le__zero__iff,axiom,
% 0.32/0.62 ! [X: real] :
% 0.32/0.62 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.62 => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
% 0.32/0.62 = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % ln_le_zero_iff
% 0.32/0.62 thf(fact_5322_semiring__norm_I168_J,axiom,
% 0.32/0.62 ! [V: num,W: num,Y: real] :
% 0.32/0.62 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
% 0.32/0.62 = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(168)
% 0.32/0.62 thf(fact_5323_semiring__norm_I168_J,axiom,
% 0.32/0.62 ! [V: num,W: num,Y: int] :
% 0.32/0.62 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
% 0.32/0.62 = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(168)
% 0.32/0.62 thf(fact_5324_semiring__norm_I168_J,axiom,
% 0.32/0.62 ! [V: num,W: num,Y: complex] :
% 0.32/0.62 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
% 0.32/0.62 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(168)
% 0.32/0.62 thf(fact_5325_semiring__norm_I168_J,axiom,
% 0.32/0.62 ! [V: num,W: num,Y: rat] :
% 0.32/0.62 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
% 0.32/0.62 = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(168)
% 0.32/0.62 thf(fact_5326_semiring__norm_I168_J,axiom,
% 0.32/0.62 ! [V: num,W: num,Y: code_integer] :
% 0.32/0.62 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
% 0.32/0.62 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(168)
% 0.32/0.62 thf(fact_5327_diff__numeral__simps_I3_J,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) )
% 0.32/0.62 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_simps(3)
% 0.32/0.62 thf(fact_5328_diff__numeral__simps_I3_J,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
% 0.32/0.62 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_simps(3)
% 0.32/0.62 thf(fact_5329_diff__numeral__simps_I3_J,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N2 ) )
% 0.32/0.62 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_simps(3)
% 0.32/0.62 thf(fact_5330_diff__numeral__simps_I3_J,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) )
% 0.32/0.62 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_simps(3)
% 0.32/0.62 thf(fact_5331_diff__numeral__simps_I3_J,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) )
% 0.32/0.62 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_simps(3)
% 0.32/0.62 thf(fact_5332_diff__numeral__simps_I2_J,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 0.32/0.62 = ( numeral_numeral_real @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_simps(2)
% 0.32/0.62 thf(fact_5333_diff__numeral__simps_I2_J,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.32/0.62 = ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_simps(2)
% 0.32/0.62 thf(fact_5334_diff__numeral__simps_I2_J,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
% 0.32/0.62 = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_simps(2)
% 0.32/0.62 thf(fact_5335_diff__numeral__simps_I2_J,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 0.32/0.62 = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_simps(2)
% 0.32/0.62 thf(fact_5336_diff__numeral__simps_I2_J,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
% 0.32/0.62 = ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_simps(2)
% 0.32/0.62 thf(fact_5337_semiring__norm_I170_J,axiom,
% 0.32/0.62 ! [V: num,W: num,Y: real] :
% 0.32/0.62 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y ) )
% 0.32/0.62 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(170)
% 0.32/0.62 thf(fact_5338_semiring__norm_I170_J,axiom,
% 0.32/0.62 ! [V: num,W: num,Y: int] :
% 0.32/0.62 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y ) )
% 0.32/0.62 = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(170)
% 0.32/0.62 thf(fact_5339_semiring__norm_I170_J,axiom,
% 0.32/0.62 ! [V: num,W: num,Y: complex] :
% 0.32/0.62 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Y ) )
% 0.32/0.62 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(170)
% 0.32/0.62 thf(fact_5340_semiring__norm_I170_J,axiom,
% 0.32/0.62 ! [V: num,W: num,Y: rat] :
% 0.32/0.62 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y ) )
% 0.32/0.62 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(170)
% 0.32/0.62 thf(fact_5341_semiring__norm_I170_J,axiom,
% 0.32/0.62 ! [V: num,W: num,Y: code_integer] :
% 0.32/0.62 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y ) )
% 0.32/0.62 = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(170)
% 0.32/0.62 thf(fact_5342_semiring__norm_I171_J,axiom,
% 0.32/0.62 ! [V: num,W: num,Y: real] :
% 0.32/0.62 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
% 0.32/0.62 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(171)
% 0.32/0.62 thf(fact_5343_semiring__norm_I171_J,axiom,
% 0.32/0.62 ! [V: num,W: num,Y: int] :
% 0.32/0.62 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
% 0.32/0.62 = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(171)
% 0.32/0.62 thf(fact_5344_semiring__norm_I171_J,axiom,
% 0.32/0.62 ! [V: num,W: num,Y: complex] :
% 0.32/0.62 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
% 0.32/0.62 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(171)
% 0.32/0.62 thf(fact_5345_semiring__norm_I171_J,axiom,
% 0.32/0.62 ! [V: num,W: num,Y: rat] :
% 0.32/0.62 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
% 0.32/0.62 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(171)
% 0.32/0.62 thf(fact_5346_semiring__norm_I171_J,axiom,
% 0.32/0.62 ! [V: num,W: num,Y: code_integer] :
% 0.32/0.62 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
% 0.32/0.62 = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(171)
% 0.32/0.62 thf(fact_5347_semiring__norm_I172_J,axiom,
% 0.32/0.62 ! [V: num,W: num,Y: real] :
% 0.32/0.62 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
% 0.32/0.62 = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(172)
% 0.32/0.62 thf(fact_5348_semiring__norm_I172_J,axiom,
% 0.32/0.62 ! [V: num,W: num,Y: int] :
% 0.32/0.62 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
% 0.32/0.62 = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(172)
% 0.32/0.62 thf(fact_5349_semiring__norm_I172_J,axiom,
% 0.32/0.62 ! [V: num,W: num,Y: complex] :
% 0.32/0.62 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
% 0.32/0.62 = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(172)
% 0.32/0.62 thf(fact_5350_semiring__norm_I172_J,axiom,
% 0.32/0.62 ! [V: num,W: num,Y: rat] :
% 0.32/0.62 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
% 0.32/0.62 = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(172)
% 0.32/0.62 thf(fact_5351_semiring__norm_I172_J,axiom,
% 0.32/0.62 ! [V: num,W: num,Y: code_integer] :
% 0.32/0.62 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
% 0.32/0.62 = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(172)
% 0.32/0.62 thf(fact_5352_mult__neg__numeral__simps_I1_J,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 0.32/0.62 = ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % mult_neg_numeral_simps(1)
% 0.32/0.62 thf(fact_5353_mult__neg__numeral__simps_I1_J,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.32/0.62 = ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % mult_neg_numeral_simps(1)
% 0.32/0.62 thf(fact_5354_mult__neg__numeral__simps_I1_J,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
% 0.32/0.62 = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % mult_neg_numeral_simps(1)
% 0.32/0.62 thf(fact_5355_mult__neg__numeral__simps_I1_J,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 0.32/0.62 = ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % mult_neg_numeral_simps(1)
% 0.32/0.62 thf(fact_5356_mult__neg__numeral__simps_I1_J,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
% 0.32/0.62 = ( numera6620942414471956472nteger @ ( times_times_num @ M @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % mult_neg_numeral_simps(1)
% 0.32/0.62 thf(fact_5357_mult__neg__numeral__simps_I2_J,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) )
% 0.32/0.62 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % mult_neg_numeral_simps(2)
% 0.32/0.62 thf(fact_5358_mult__neg__numeral__simps_I2_J,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
% 0.32/0.62 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % mult_neg_numeral_simps(2)
% 0.32/0.62 thf(fact_5359_mult__neg__numeral__simps_I2_J,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N2 ) )
% 0.32/0.62 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % mult_neg_numeral_simps(2)
% 0.32/0.62 thf(fact_5360_mult__neg__numeral__simps_I2_J,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) )
% 0.32/0.62 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % mult_neg_numeral_simps(2)
% 0.32/0.62 thf(fact_5361_mult__neg__numeral__simps_I2_J,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) )
% 0.32/0.62 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % mult_neg_numeral_simps(2)
% 0.32/0.62 thf(fact_5362_mult__neg__numeral__simps_I3_J,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 0.32/0.62 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % mult_neg_numeral_simps(3)
% 0.32/0.62 thf(fact_5363_mult__neg__numeral__simps_I3_J,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.32/0.62 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % mult_neg_numeral_simps(3)
% 0.32/0.62 thf(fact_5364_mult__neg__numeral__simps_I3_J,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
% 0.32/0.62 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % mult_neg_numeral_simps(3)
% 0.32/0.62 thf(fact_5365_mult__neg__numeral__simps_I3_J,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 0.32/0.62 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % mult_neg_numeral_simps(3)
% 0.32/0.62 thf(fact_5366_mult__neg__numeral__simps_I3_J,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
% 0.32/0.62 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % mult_neg_numeral_simps(3)
% 0.32/0.62 thf(fact_5367_neg__numeral__le__iff,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 0.32/0.62 = ( ord_less_eq_num @ N2 @ M ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_numeral_le_iff
% 0.32/0.62 thf(fact_5368_neg__numeral__le__iff,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
% 0.32/0.62 = ( ord_less_eq_num @ N2 @ M ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_numeral_le_iff
% 0.32/0.62 thf(fact_5369_neg__numeral__le__iff,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 0.32/0.62 = ( ord_less_eq_num @ N2 @ M ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_numeral_le_iff
% 0.32/0.62 thf(fact_5370_neg__numeral__le__iff,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.32/0.62 = ( ord_less_eq_num @ N2 @ M ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_numeral_le_iff
% 0.32/0.62 thf(fact_5371_neg__numeral__less__iff,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 0.32/0.62 = ( ord_less_num @ N2 @ M ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_numeral_less_iff
% 0.32/0.62 thf(fact_5372_neg__numeral__less__iff,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.32/0.62 = ( ord_less_num @ N2 @ M ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_numeral_less_iff
% 0.32/0.62 thf(fact_5373_neg__numeral__less__iff,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 0.32/0.62 = ( ord_less_num @ N2 @ M ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_numeral_less_iff
% 0.32/0.62 thf(fact_5374_neg__numeral__less__iff,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
% 0.32/0.62 = ( ord_less_num @ N2 @ M ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_numeral_less_iff
% 0.32/0.62 thf(fact_5375_not__neg__one__le__neg__numeral__iff,axiom,
% 0.32/0.62 ! [M: num] :
% 0.32/0.62 ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
% 0.32/0.62 = ( M != one ) ) ).
% 0.32/0.62
% 0.32/0.62 % not_neg_one_le_neg_numeral_iff
% 0.32/0.62 thf(fact_5376_not__neg__one__le__neg__numeral__iff,axiom,
% 0.32/0.62 ! [M: num] :
% 0.32/0.62 ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
% 0.32/0.62 = ( M != one ) ) ).
% 0.32/0.62
% 0.32/0.62 % not_neg_one_le_neg_numeral_iff
% 0.32/0.62 thf(fact_5377_not__neg__one__le__neg__numeral__iff,axiom,
% 0.32/0.62 ! [M: num] :
% 0.32/0.62 ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
% 0.32/0.62 = ( M != one ) ) ).
% 0.32/0.62
% 0.32/0.62 % not_neg_one_le_neg_numeral_iff
% 0.32/0.62 thf(fact_5378_not__neg__one__le__neg__numeral__iff,axiom,
% 0.32/0.62 ! [M: num] :
% 0.32/0.62 ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
% 0.32/0.62 = ( M != one ) ) ).
% 0.32/0.62
% 0.32/0.62 % not_neg_one_le_neg_numeral_iff
% 0.32/0.62 thf(fact_5379_le__divide__eq__numeral1_I2_J,axiom,
% 0.32/0.62 ! [A: real,B: real,W: num] :
% 0.32/0.62 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 0.32/0.62 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % le_divide_eq_numeral1(2)
% 0.32/0.62 thf(fact_5380_le__divide__eq__numeral1_I2_J,axiom,
% 0.32/0.62 ! [A: rat,B: rat,W: num] :
% 0.32/0.62 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 0.32/0.62 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % le_divide_eq_numeral1(2)
% 0.32/0.62 thf(fact_5381_divide__le__eq__numeral1_I2_J,axiom,
% 0.32/0.62 ! [B: real,W: num,A: real] :
% 0.32/0.62 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 0.32/0.62 = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% 0.32/0.62
% 0.32/0.62 % divide_le_eq_numeral1(2)
% 0.32/0.62 thf(fact_5382_divide__le__eq__numeral1_I2_J,axiom,
% 0.32/0.62 ! [B: rat,W: num,A: rat] :
% 0.32/0.62 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 0.32/0.62 = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% 0.32/0.62
% 0.32/0.62 % divide_le_eq_numeral1(2)
% 0.32/0.62 thf(fact_5383_eq__divide__eq__numeral1_I2_J,axiom,
% 0.32/0.62 ! [A: real,B: real,W: num] :
% 0.32/0.62 ( ( A
% 0.32/0.62 = ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 0.32/0.62 = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.32/0.62 != zero_zero_real )
% 0.32/0.62 => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.32/0.62 = B ) )
% 0.32/0.62 & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.32/0.62 = zero_zero_real )
% 0.32/0.62 => ( A = zero_zero_real ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % eq_divide_eq_numeral1(2)
% 0.32/0.62 thf(fact_5384_eq__divide__eq__numeral1_I2_J,axiom,
% 0.32/0.62 ! [A: complex,B: complex,W: num] :
% 0.32/0.62 ( ( A
% 0.32/0.62 = ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) )
% 0.32/0.62 = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.32/0.62 != zero_zero_complex )
% 0.32/0.62 => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.32/0.62 = B ) )
% 0.32/0.62 & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.32/0.62 = zero_zero_complex )
% 0.32/0.62 => ( A = zero_zero_complex ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % eq_divide_eq_numeral1(2)
% 0.32/0.62 thf(fact_5385_eq__divide__eq__numeral1_I2_J,axiom,
% 0.32/0.62 ! [A: rat,B: rat,W: num] :
% 0.32/0.62 ( ( A
% 0.32/0.62 = ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 0.32/0.62 = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.32/0.62 != zero_zero_rat )
% 0.32/0.62 => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.32/0.62 = B ) )
% 0.32/0.62 & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.32/0.62 = zero_zero_rat )
% 0.32/0.62 => ( A = zero_zero_rat ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % eq_divide_eq_numeral1(2)
% 0.32/0.62 thf(fact_5386_divide__eq__eq__numeral1_I2_J,axiom,
% 0.32/0.62 ! [B: real,W: num,A: real] :
% 0.32/0.62 ( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.32/0.62 = A )
% 0.32/0.62 = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.32/0.62 != zero_zero_real )
% 0.32/0.62 => ( B
% 0.32/0.62 = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
% 0.32/0.62 & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.32/0.62 = zero_zero_real )
% 0.32/0.62 => ( A = zero_zero_real ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % divide_eq_eq_numeral1(2)
% 0.32/0.62 thf(fact_5387_divide__eq__eq__numeral1_I2_J,axiom,
% 0.32/0.62 ! [B: complex,W: num,A: complex] :
% 0.32/0.62 ( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.32/0.62 = A )
% 0.32/0.62 = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.32/0.62 != zero_zero_complex )
% 0.32/0.62 => ( B
% 0.32/0.62 = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) )
% 0.32/0.62 & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.32/0.62 = zero_zero_complex )
% 0.32/0.62 => ( A = zero_zero_complex ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % divide_eq_eq_numeral1(2)
% 0.32/0.62 thf(fact_5388_divide__eq__eq__numeral1_I2_J,axiom,
% 0.32/0.62 ! [B: rat,W: num,A: rat] :
% 0.32/0.62 ( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.32/0.62 = A )
% 0.32/0.62 = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.32/0.62 != zero_zero_rat )
% 0.32/0.62 => ( B
% 0.32/0.62 = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
% 0.32/0.62 & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.32/0.62 = zero_zero_rat )
% 0.32/0.62 => ( A = zero_zero_rat ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % divide_eq_eq_numeral1(2)
% 0.32/0.62 thf(fact_5389_neg__numeral__less__neg__one__iff,axiom,
% 0.32/0.62 ! [M: num] :
% 0.32/0.62 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.32/0.62 = ( M != one ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_numeral_less_neg_one_iff
% 0.32/0.62 thf(fact_5390_neg__numeral__less__neg__one__iff,axiom,
% 0.32/0.62 ! [M: num] :
% 0.32/0.62 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.62 = ( M != one ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_numeral_less_neg_one_iff
% 0.32/0.62 thf(fact_5391_neg__numeral__less__neg__one__iff,axiom,
% 0.32/0.62 ! [M: num] :
% 0.32/0.62 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.32/0.62 = ( M != one ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_numeral_less_neg_one_iff
% 0.32/0.62 thf(fact_5392_neg__numeral__less__neg__one__iff,axiom,
% 0.32/0.62 ! [M: num] :
% 0.32/0.62 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.62 = ( M != one ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_numeral_less_neg_one_iff
% 0.32/0.62 thf(fact_5393_less__divide__eq__numeral1_I2_J,axiom,
% 0.32/0.62 ! [A: real,B: real,W: num] :
% 0.32/0.62 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 0.32/0.62 = ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % less_divide_eq_numeral1(2)
% 0.32/0.62 thf(fact_5394_less__divide__eq__numeral1_I2_J,axiom,
% 0.32/0.62 ! [A: rat,B: rat,W: num] :
% 0.32/0.62 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 0.32/0.62 = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % less_divide_eq_numeral1(2)
% 0.32/0.62 thf(fact_5395_divide__less__eq__numeral1_I2_J,axiom,
% 0.32/0.62 ! [B: real,W: num,A: real] :
% 0.32/0.62 ( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 0.32/0.62 = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% 0.32/0.62
% 0.32/0.62 % divide_less_eq_numeral1(2)
% 0.32/0.62 thf(fact_5396_divide__less__eq__numeral1_I2_J,axiom,
% 0.32/0.62 ! [B: rat,W: num,A: rat] :
% 0.32/0.62 ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 0.32/0.62 = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% 0.32/0.62
% 0.32/0.62 % divide_less_eq_numeral1(2)
% 0.32/0.62 thf(fact_5397_power2__minus,axiom,
% 0.32/0.62 ! [A: real] :
% 0.32/0.62 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.62 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % power2_minus
% 0.32/0.62 thf(fact_5398_power2__minus,axiom,
% 0.32/0.62 ! [A: int] :
% 0.32/0.62 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.62 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % power2_minus
% 0.32/0.62 thf(fact_5399_power2__minus,axiom,
% 0.32/0.62 ! [A: complex] :
% 0.32/0.62 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.62 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % power2_minus
% 0.32/0.62 thf(fact_5400_power2__minus,axiom,
% 0.32/0.62 ! [A: rat] :
% 0.32/0.62 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.62 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % power2_minus
% 0.32/0.62 thf(fact_5401_power2__minus,axiom,
% 0.32/0.62 ! [A: code_integer] :
% 0.32/0.62 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.62 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % power2_minus
% 0.32/0.62 thf(fact_5402_add__neg__numeral__special_I9_J,axiom,
% 0.32/0.62 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.32/0.62 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % add_neg_numeral_special(9)
% 0.32/0.62 thf(fact_5403_add__neg__numeral__special_I9_J,axiom,
% 0.32/0.62 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.62 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % add_neg_numeral_special(9)
% 0.32/0.62 thf(fact_5404_add__neg__numeral__special_I9_J,axiom,
% 0.32/0.62 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.32/0.62 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % add_neg_numeral_special(9)
% 0.32/0.62 thf(fact_5405_add__neg__numeral__special_I9_J,axiom,
% 0.32/0.62 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.32/0.62 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % add_neg_numeral_special(9)
% 0.32/0.62 thf(fact_5406_add__neg__numeral__special_I9_J,axiom,
% 0.32/0.62 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.62 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % add_neg_numeral_special(9)
% 0.32/0.62 thf(fact_5407_diff__numeral__special_I10_J,axiom,
% 0.32/0.62 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 0.32/0.62 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_special(10)
% 0.32/0.62 thf(fact_5408_diff__numeral__special_I10_J,axiom,
% 0.32/0.62 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 0.32/0.62 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_special(10)
% 0.32/0.62 thf(fact_5409_diff__numeral__special_I10_J,axiom,
% 0.32/0.62 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 0.32/0.62 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_special(10)
% 0.32/0.62 thf(fact_5410_diff__numeral__special_I10_J,axiom,
% 0.32/0.62 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 0.32/0.62 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_special(10)
% 0.32/0.62 thf(fact_5411_diff__numeral__special_I10_J,axiom,
% 0.32/0.62 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 0.32/0.62 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_special(10)
% 0.32/0.62 thf(fact_5412_diff__numeral__special_I11_J,axiom,
% 0.32/0.62 ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.32/0.62 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_special(11)
% 0.32/0.62 thf(fact_5413_diff__numeral__special_I11_J,axiom,
% 0.32/0.62 ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.62 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_special(11)
% 0.32/0.62 thf(fact_5414_diff__numeral__special_I11_J,axiom,
% 0.32/0.62 ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.32/0.62 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_special(11)
% 0.32/0.62 thf(fact_5415_diff__numeral__special_I11_J,axiom,
% 0.32/0.62 ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.32/0.62 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_special(11)
% 0.32/0.62 thf(fact_5416_diff__numeral__special_I11_J,axiom,
% 0.32/0.62 ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.62 = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_special(11)
% 0.32/0.62 thf(fact_5417_minus__1__div__2__eq,axiom,
% 0.32/0.62 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.62 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_1_div_2_eq
% 0.32/0.62 thf(fact_5418_minus__1__div__2__eq,axiom,
% 0.32/0.62 ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.62 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_1_div_2_eq
% 0.32/0.62 thf(fact_5419_bits__minus__1__mod__2__eq,axiom,
% 0.32/0.62 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.62 = one_one_int ) ).
% 0.32/0.62
% 0.32/0.62 % bits_minus_1_mod_2_eq
% 0.32/0.62 thf(fact_5420_bits__minus__1__mod__2__eq,axiom,
% 0.32/0.62 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.62 = one_one_Code_integer ) ).
% 0.32/0.62
% 0.32/0.62 % bits_minus_1_mod_2_eq
% 0.32/0.62 thf(fact_5421_minus__1__mod__2__eq,axiom,
% 0.32/0.62 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.62 = one_one_int ) ).
% 0.32/0.62
% 0.32/0.62 % minus_1_mod_2_eq
% 0.32/0.62 thf(fact_5422_minus__1__mod__2__eq,axiom,
% 0.32/0.62 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.62 = one_one_Code_integer ) ).
% 0.32/0.62
% 0.32/0.62 % minus_1_mod_2_eq
% 0.32/0.62 thf(fact_5423_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.32/0.62 ! [A: real,N2: nat] :
% 0.32/0.62 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.62 = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % Power.ring_1_class.power_minus_even
% 0.32/0.62 thf(fact_5424_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.32/0.62 ! [A: int,N2: nat] :
% 0.32/0.62 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.62 = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % Power.ring_1_class.power_minus_even
% 0.32/0.62 thf(fact_5425_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.32/0.62 ! [A: complex,N2: nat] :
% 0.32/0.62 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.62 = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % Power.ring_1_class.power_minus_even
% 0.32/0.62 thf(fact_5426_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.32/0.62 ! [A: rat,N2: nat] :
% 0.32/0.62 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.62 = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % Power.ring_1_class.power_minus_even
% 0.32/0.62 thf(fact_5427_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.32/0.62 ! [A: code_integer,N2: nat] :
% 0.32/0.62 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.62 = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % Power.ring_1_class.power_minus_even
% 0.32/0.62 thf(fact_5428_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.32/0.62 ! [N2: nat,A: real] :
% 0.32/0.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.62 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 )
% 0.32/0.62 = ( power_power_real @ A @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % Parity.ring_1_class.power_minus_even
% 0.32/0.62 thf(fact_5429_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.32/0.62 ! [N2: nat,A: int] :
% 0.32/0.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.62 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
% 0.32/0.62 = ( power_power_int @ A @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % Parity.ring_1_class.power_minus_even
% 0.32/0.62 thf(fact_5430_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.32/0.62 ! [N2: nat,A: complex] :
% 0.32/0.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.62 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 )
% 0.32/0.62 = ( power_power_complex @ A @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % Parity.ring_1_class.power_minus_even
% 0.32/0.62 thf(fact_5431_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.32/0.62 ! [N2: nat,A: rat] :
% 0.32/0.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.62 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 )
% 0.32/0.62 = ( power_power_rat @ A @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % Parity.ring_1_class.power_minus_even
% 0.32/0.62 thf(fact_5432_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.32/0.62 ! [N2: nat,A: code_integer] :
% 0.32/0.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.62 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 )
% 0.32/0.62 = ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % Parity.ring_1_class.power_minus_even
% 0.32/0.62 thf(fact_5433_power__minus__odd,axiom,
% 0.32/0.62 ! [N2: nat,A: real] :
% 0.32/0.62 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.62 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 )
% 0.32/0.62 = ( uminus_uminus_real @ ( power_power_real @ A @ N2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % power_minus_odd
% 0.32/0.62 thf(fact_5434_power__minus__odd,axiom,
% 0.32/0.62 ! [N2: nat,A: int] :
% 0.32/0.62 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.62 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
% 0.32/0.62 = ( uminus_uminus_int @ ( power_power_int @ A @ N2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % power_minus_odd
% 0.32/0.62 thf(fact_5435_power__minus__odd,axiom,
% 0.32/0.62 ! [N2: nat,A: complex] :
% 0.32/0.62 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.62 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 )
% 0.32/0.62 = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % power_minus_odd
% 0.32/0.62 thf(fact_5436_power__minus__odd,axiom,
% 0.32/0.62 ! [N2: nat,A: rat] :
% 0.32/0.62 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.62 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 )
% 0.32/0.62 = ( uminus_uminus_rat @ ( power_power_rat @ A @ N2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % power_minus_odd
% 0.32/0.62 thf(fact_5437_power__minus__odd,axiom,
% 0.32/0.62 ! [N2: nat,A: code_integer] :
% 0.32/0.62 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.62 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 )
% 0.32/0.62 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % power_minus_odd
% 0.32/0.62 thf(fact_5438_diff__numeral__special_I4_J,axiom,
% 0.32/0.62 ! [M: num] :
% 0.32/0.62 ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real )
% 0.32/0.62 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_special(4)
% 0.32/0.62 thf(fact_5439_diff__numeral__special_I4_J,axiom,
% 0.32/0.62 ! [M: num] :
% 0.32/0.62 ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
% 0.32/0.62 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_special(4)
% 0.32/0.62 thf(fact_5440_diff__numeral__special_I4_J,axiom,
% 0.32/0.62 ! [M: num] :
% 0.32/0.62 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ one_one_complex )
% 0.32/0.62 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_special(4)
% 0.32/0.62 thf(fact_5441_diff__numeral__special_I4_J,axiom,
% 0.32/0.62 ! [M: num] :
% 0.32/0.62 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
% 0.32/0.62 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_special(4)
% 0.32/0.62 thf(fact_5442_diff__numeral__special_I4_J,axiom,
% 0.32/0.62 ! [M: num] :
% 0.32/0.62 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
% 0.32/0.62 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_special(4)
% 0.32/0.62 thf(fact_5443_diff__numeral__special_I3_J,axiom,
% 0.32/0.62 ! [N2: num] :
% 0.32/0.62 ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 0.32/0.62 = ( numeral_numeral_real @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_special(3)
% 0.32/0.62 thf(fact_5444_diff__numeral__special_I3_J,axiom,
% 0.32/0.62 ! [N2: num] :
% 0.32/0.62 ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.32/0.62 = ( numeral_numeral_int @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_special(3)
% 0.32/0.62 thf(fact_5445_diff__numeral__special_I3_J,axiom,
% 0.32/0.62 ! [N2: num] :
% 0.32/0.62 ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
% 0.32/0.62 = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_special(3)
% 0.32/0.62 thf(fact_5446_diff__numeral__special_I3_J,axiom,
% 0.32/0.62 ! [N2: num] :
% 0.32/0.62 ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 0.32/0.62 = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_special(3)
% 0.32/0.62 thf(fact_5447_diff__numeral__special_I3_J,axiom,
% 0.32/0.62 ! [N2: num] :
% 0.32/0.62 ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
% 0.32/0.62 = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % diff_numeral_special(3)
% 0.32/0.62 thf(fact_5448_signed__take__bit__Suc__minus__bit0,axiom,
% 0.32/0.62 ! [N2: nat,K: num] :
% 0.32/0.62 ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.32/0.62 = ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % signed_take_bit_Suc_minus_bit0
% 0.32/0.62 thf(fact_5449_dbl__simps_I4_J,axiom,
% 0.32/0.62 ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.32/0.62 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % dbl_simps(4)
% 0.32/0.62 thf(fact_5450_dbl__simps_I4_J,axiom,
% 0.32/0.62 ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.62 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % dbl_simps(4)
% 0.32/0.62 thf(fact_5451_dbl__simps_I4_J,axiom,
% 0.32/0.62 ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.32/0.62 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % dbl_simps(4)
% 0.32/0.62 thf(fact_5452_dbl__simps_I4_J,axiom,
% 0.32/0.62 ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.32/0.62 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % dbl_simps(4)
% 0.32/0.62 thf(fact_5453_dbl__simps_I4_J,axiom,
% 0.32/0.62 ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.62 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % dbl_simps(4)
% 0.32/0.62 thf(fact_5454_power__minus1__even,axiom,
% 0.32/0.62 ! [N2: nat] :
% 0.32/0.62 ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.62 = one_one_real ) ).
% 0.32/0.62
% 0.32/0.62 % power_minus1_even
% 0.32/0.62 thf(fact_5455_power__minus1__even,axiom,
% 0.32/0.62 ! [N2: nat] :
% 0.32/0.62 ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.62 = one_one_int ) ).
% 0.32/0.62
% 0.32/0.62 % power_minus1_even
% 0.32/0.62 thf(fact_5456_power__minus1__even,axiom,
% 0.32/0.62 ! [N2: nat] :
% 0.32/0.62 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.62 = one_one_complex ) ).
% 0.32/0.62
% 0.32/0.62 % power_minus1_even
% 0.32/0.62 thf(fact_5457_power__minus1__even,axiom,
% 0.32/0.62 ! [N2: nat] :
% 0.32/0.62 ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.62 = one_one_rat ) ).
% 0.32/0.62
% 0.32/0.62 % power_minus1_even
% 0.32/0.62 thf(fact_5458_power__minus1__even,axiom,
% 0.32/0.62 ! [N2: nat] :
% 0.32/0.62 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.62 = one_one_Code_integer ) ).
% 0.32/0.62
% 0.32/0.62 % power_minus1_even
% 0.32/0.62 thf(fact_5459_neg__one__even__power,axiom,
% 0.32/0.62 ! [N2: nat] :
% 0.32/0.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.62 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 )
% 0.32/0.62 = one_one_real ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_one_even_power
% 0.32/0.62 thf(fact_5460_neg__one__even__power,axiom,
% 0.32/0.62 ! [N2: nat] :
% 0.32/0.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.62 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
% 0.32/0.62 = one_one_int ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_one_even_power
% 0.32/0.62 thf(fact_5461_neg__one__even__power,axiom,
% 0.32/0.62 ! [N2: nat] :
% 0.32/0.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.62 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 )
% 0.32/0.62 = one_one_complex ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_one_even_power
% 0.32/0.62 thf(fact_5462_neg__one__even__power,axiom,
% 0.32/0.62 ! [N2: nat] :
% 0.32/0.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.62 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
% 0.32/0.62 = one_one_rat ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_one_even_power
% 0.32/0.62 thf(fact_5463_neg__one__even__power,axiom,
% 0.32/0.62 ! [N2: nat] :
% 0.32/0.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.62 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
% 0.32/0.62 = one_one_Code_integer ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_one_even_power
% 0.32/0.62 thf(fact_5464_neg__one__odd__power,axiom,
% 0.32/0.62 ! [N2: nat] :
% 0.32/0.62 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.62 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 )
% 0.32/0.62 = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_one_odd_power
% 0.32/0.62 thf(fact_5465_neg__one__odd__power,axiom,
% 0.32/0.62 ! [N2: nat] :
% 0.32/0.62 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.62 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
% 0.32/0.62 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_one_odd_power
% 0.32/0.62 thf(fact_5466_neg__one__odd__power,axiom,
% 0.32/0.62 ! [N2: nat] :
% 0.32/0.62 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.62 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 )
% 0.32/0.62 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_one_odd_power
% 0.32/0.62 thf(fact_5467_neg__one__odd__power,axiom,
% 0.32/0.62 ! [N2: nat] :
% 0.32/0.62 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.62 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
% 0.32/0.62 = ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_one_odd_power
% 0.32/0.62 thf(fact_5468_neg__one__odd__power,axiom,
% 0.32/0.62 ! [N2: nat] :
% 0.32/0.62 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.62 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
% 0.32/0.62 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_one_odd_power
% 0.32/0.62 thf(fact_5469_signed__take__bit__0,axiom,
% 0.32/0.62 ! [A: code_integer] :
% 0.32/0.62 ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
% 0.32/0.62 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % signed_take_bit_0
% 0.32/0.62 thf(fact_5470_signed__take__bit__0,axiom,
% 0.32/0.62 ! [A: int] :
% 0.32/0.62 ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
% 0.32/0.62 = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % signed_take_bit_0
% 0.32/0.62 thf(fact_5471_signed__take__bit__minus,axiom,
% 0.32/0.62 ! [N2: nat,K: int] :
% 0.32/0.62 ( ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) ) )
% 0.32/0.62 = ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ K ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % signed_take_bit_minus
% 0.32/0.62 thf(fact_5472_sum_Oswap,axiom,
% 0.32/0.62 ! [G: int > int > int,B3: set_int,A2: set_int] :
% 0.32/0.62 ( ( groups4538972089207619220nt_int
% 0.32/0.62 @ ^ [I3: int] : ( groups4538972089207619220nt_int @ ( G @ I3 ) @ B3 )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( groups4538972089207619220nt_int
% 0.32/0.62 @ ^ [J3: int] :
% 0.32/0.62 ( groups4538972089207619220nt_int
% 0.32/0.62 @ ^ [I3: int] : ( G @ I3 @ J3 )
% 0.32/0.62 @ A2 )
% 0.32/0.62 @ B3 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.swap
% 0.32/0.62 thf(fact_5473_sum_Oswap,axiom,
% 0.32/0.62 ! [G: complex > complex > complex,B3: set_complex,A2: set_complex] :
% 0.32/0.62 ( ( groups7754918857620584856omplex
% 0.32/0.62 @ ^ [I3: complex] : ( groups7754918857620584856omplex @ ( G @ I3 ) @ B3 )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( groups7754918857620584856omplex
% 0.32/0.62 @ ^ [J3: complex] :
% 0.32/0.62 ( groups7754918857620584856omplex
% 0.32/0.62 @ ^ [I3: complex] : ( G @ I3 @ J3 )
% 0.32/0.62 @ A2 )
% 0.32/0.62 @ B3 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.swap
% 0.32/0.62 thf(fact_5474_sum_Oswap,axiom,
% 0.32/0.62 ! [G: nat > nat > nat,B3: set_nat,A2: set_nat] :
% 0.32/0.62 ( ( groups3542108847815614940at_nat
% 0.32/0.62 @ ^ [I3: nat] : ( groups3542108847815614940at_nat @ ( G @ I3 ) @ B3 )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( groups3542108847815614940at_nat
% 0.32/0.62 @ ^ [J3: nat] :
% 0.32/0.62 ( groups3542108847815614940at_nat
% 0.32/0.62 @ ^ [I3: nat] : ( G @ I3 @ J3 )
% 0.32/0.62 @ A2 )
% 0.32/0.62 @ B3 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.swap
% 0.32/0.62 thf(fact_5475_sum_Oswap,axiom,
% 0.32/0.62 ! [G: nat > nat > real,B3: set_nat,A2: set_nat] :
% 0.32/0.62 ( ( groups6591440286371151544t_real
% 0.32/0.62 @ ^ [I3: nat] : ( groups6591440286371151544t_real @ ( G @ I3 ) @ B3 )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( groups6591440286371151544t_real
% 0.32/0.62 @ ^ [J3: nat] :
% 0.32/0.62 ( groups6591440286371151544t_real
% 0.32/0.62 @ ^ [I3: nat] : ( G @ I3 @ J3 )
% 0.32/0.62 @ A2 )
% 0.32/0.62 @ B3 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.swap
% 0.32/0.62 thf(fact_5476_sum__negf,axiom,
% 0.32/0.62 ! [F: int > int,A2: set_int] :
% 0.32/0.62 ( ( groups4538972089207619220nt_int
% 0.32/0.62 @ ^ [X2: int] : ( uminus_uminus_int @ ( F @ X2 ) )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( uminus_uminus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_negf
% 0.32/0.62 thf(fact_5477_sum__negf,axiom,
% 0.32/0.62 ! [F: complex > complex,A2: set_complex] :
% 0.32/0.62 ( ( groups7754918857620584856omplex
% 0.32/0.62 @ ^ [X2: complex] : ( uminus1482373934393186551omplex @ ( F @ X2 ) )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( uminus1482373934393186551omplex @ ( groups7754918857620584856omplex @ F @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_negf
% 0.32/0.62 thf(fact_5478_sum__negf,axiom,
% 0.32/0.62 ! [F: nat > real,A2: set_nat] :
% 0.32/0.62 ( ( groups6591440286371151544t_real
% 0.32/0.62 @ ^ [X2: nat] : ( uminus_uminus_real @ ( F @ X2 ) )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( uminus_uminus_real @ ( groups6591440286371151544t_real @ F @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_negf
% 0.32/0.62 thf(fact_5479_prod_Ocase__distrib,axiom,
% 0.32/0.62 ! [H2: $o > $o,F: int > int > $o,Prod: product_prod_int_int] :
% 0.32/0.62 ( ( H2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
% 0.32/0.62 = ( produc4947309494688390418_int_o
% 0.32/0.62 @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
% 0.32/0.62 @ Prod ) ) ).
% 0.32/0.62
% 0.32/0.62 % prod.case_distrib
% 0.32/0.62 thf(fact_5480_prod_Ocase__distrib,axiom,
% 0.32/0.62 ! [H2: $o > int,F: int > int > $o,Prod: product_prod_int_int] :
% 0.32/0.62 ( ( H2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
% 0.32/0.62 = ( produc8211389475949308722nt_int
% 0.32/0.62 @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
% 0.32/0.62 @ Prod ) ) ).
% 0.32/0.62
% 0.32/0.62 % prod.case_distrib
% 0.32/0.62 thf(fact_5481_prod_Ocase__distrib,axiom,
% 0.32/0.62 ! [H2: int > $o,F: int > int > int,Prod: product_prod_int_int] :
% 0.32/0.62 ( ( H2 @ ( produc8211389475949308722nt_int @ F @ Prod ) )
% 0.32/0.62 = ( produc4947309494688390418_int_o
% 0.32/0.62 @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
% 0.32/0.62 @ Prod ) ) ).
% 0.32/0.62
% 0.32/0.62 % prod.case_distrib
% 0.32/0.62 thf(fact_5482_prod_Ocase__distrib,axiom,
% 0.32/0.62 ! [H2: int > int,F: int > int > int,Prod: product_prod_int_int] :
% 0.32/0.62 ( ( H2 @ ( produc8211389475949308722nt_int @ F @ Prod ) )
% 0.32/0.62 = ( produc8211389475949308722nt_int
% 0.32/0.62 @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
% 0.32/0.62 @ Prod ) ) ).
% 0.32/0.62
% 0.32/0.62 % prod.case_distrib
% 0.32/0.62 thf(fact_5483_prod_Ocase__distrib,axiom,
% 0.32/0.62 ! [H2: product_prod_int_int > $o,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
% 0.32/0.62 ( ( H2 @ ( produc4245557441103728435nt_int @ F @ Prod ) )
% 0.32/0.62 = ( produc4947309494688390418_int_o
% 0.32/0.62 @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
% 0.32/0.62 @ Prod ) ) ).
% 0.32/0.62
% 0.32/0.62 % prod.case_distrib
% 0.32/0.62 thf(fact_5484_prod_Ocase__distrib,axiom,
% 0.32/0.62 ! [H2: product_prod_int_int > int,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
% 0.32/0.62 ( ( H2 @ ( produc4245557441103728435nt_int @ F @ Prod ) )
% 0.32/0.62 = ( produc8211389475949308722nt_int
% 0.32/0.62 @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
% 0.32/0.62 @ Prod ) ) ).
% 0.32/0.62
% 0.32/0.62 % prod.case_distrib
% 0.32/0.62 thf(fact_5485_prod_Ocase__distrib,axiom,
% 0.32/0.62 ! [H2: $o > product_prod_int_int,F: int > int > $o,Prod: product_prod_int_int] :
% 0.32/0.62 ( ( H2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
% 0.32/0.62 = ( produc4245557441103728435nt_int
% 0.32/0.62 @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
% 0.32/0.62 @ Prod ) ) ).
% 0.32/0.62
% 0.32/0.62 % prod.case_distrib
% 0.32/0.62 thf(fact_5486_prod_Ocase__distrib,axiom,
% 0.32/0.62 ! [H2: int > product_prod_int_int,F: int > int > int,Prod: product_prod_int_int] :
% 0.32/0.62 ( ( H2 @ ( produc8211389475949308722nt_int @ F @ Prod ) )
% 0.32/0.62 = ( produc4245557441103728435nt_int
% 0.32/0.62 @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
% 0.32/0.62 @ Prod ) ) ).
% 0.32/0.62
% 0.32/0.62 % prod.case_distrib
% 0.32/0.62 thf(fact_5487_prod_Ocase__distrib,axiom,
% 0.32/0.62 ! [H2: product_prod_int_int > product_prod_int_int,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
% 0.32/0.62 ( ( H2 @ ( produc4245557441103728435nt_int @ F @ Prod ) )
% 0.32/0.62 = ( produc4245557441103728435nt_int
% 0.32/0.62 @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
% 0.32/0.62 @ Prod ) ) ).
% 0.32/0.62
% 0.32/0.62 % prod.case_distrib
% 0.32/0.62 thf(fact_5488_prod_Ocase__distrib,axiom,
% 0.32/0.62 ! [H2: ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o,F: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
% 0.32/0.62 ( ( H2 @ ( produc8739625826339149834_nat_o @ F @ Prod ) )
% 0.32/0.62 = ( produc8739625826339149834_nat_o
% 0.32/0.62 @ ^ [X15: nat,X24: nat] : ( H2 @ ( F @ X15 @ X24 ) )
% 0.32/0.62 @ Prod ) ) ).
% 0.32/0.62
% 0.32/0.62 % prod.case_distrib
% 0.32/0.62 thf(fact_5489_le__minus__iff,axiom,
% 0.32/0.62 ! [A: real,B: real] :
% 0.32/0.62 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
% 0.32/0.62 = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % le_minus_iff
% 0.32/0.62 thf(fact_5490_le__minus__iff,axiom,
% 0.32/0.62 ! [A: code_integer,B: code_integer] :
% 0.32/0.62 ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.32/0.62 = ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % le_minus_iff
% 0.32/0.62 thf(fact_5491_le__minus__iff,axiom,
% 0.32/0.62 ! [A: rat,B: rat] :
% 0.32/0.62 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.32/0.62 = ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % le_minus_iff
% 0.32/0.62 thf(fact_5492_le__minus__iff,axiom,
% 0.32/0.62 ! [A: int,B: int] :
% 0.32/0.62 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
% 0.32/0.62 = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % le_minus_iff
% 0.32/0.62 thf(fact_5493_minus__le__iff,axiom,
% 0.32/0.62 ! [A: real,B: real] :
% 0.32/0.62 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 0.32/0.62 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_le_iff
% 0.32/0.62 thf(fact_5494_minus__le__iff,axiom,
% 0.32/0.62 ! [A: code_integer,B: code_integer] :
% 0.32/0.62 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.32/0.62 = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_le_iff
% 0.32/0.62 thf(fact_5495_minus__le__iff,axiom,
% 0.32/0.62 ! [A: rat,B: rat] :
% 0.32/0.62 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.32/0.62 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_le_iff
% 0.32/0.62 thf(fact_5496_minus__le__iff,axiom,
% 0.32/0.62 ! [A: int,B: int] :
% 0.32/0.62 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 0.32/0.62 = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_le_iff
% 0.32/0.62 thf(fact_5497_le__imp__neg__le,axiom,
% 0.32/0.62 ! [A: real,B: real] :
% 0.32/0.62 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.62 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % le_imp_neg_le
% 0.32/0.62 thf(fact_5498_le__imp__neg__le,axiom,
% 0.32/0.62 ! [A: code_integer,B: code_integer] :
% 0.32/0.62 ( ( ord_le3102999989581377725nteger @ A @ B )
% 0.32/0.62 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % le_imp_neg_le
% 0.32/0.62 thf(fact_5499_le__imp__neg__le,axiom,
% 0.32/0.62 ! [A: rat,B: rat] :
% 0.32/0.62 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.62 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % le_imp_neg_le
% 0.32/0.62 thf(fact_5500_le__imp__neg__le,axiom,
% 0.32/0.62 ! [A: int,B: int] :
% 0.32/0.62 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.62 => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % le_imp_neg_le
% 0.32/0.62 thf(fact_5501_compl__le__swap2,axiom,
% 0.32/0.62 ! [Y: set_nat,X: set_nat] :
% 0.32/0.62 ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ X )
% 0.32/0.62 => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) ) ).
% 0.32/0.62
% 0.32/0.62 % compl_le_swap2
% 0.32/0.62 thf(fact_5502_compl__le__swap1,axiom,
% 0.32/0.62 ! [Y: set_nat,X: set_nat] :
% 0.32/0.62 ( ( ord_less_eq_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) )
% 0.32/0.62 => ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % compl_le_swap1
% 0.32/0.62 thf(fact_5503_compl__mono,axiom,
% 0.32/0.62 ! [X: set_nat,Y: set_nat] :
% 0.32/0.62 ( ( ord_less_eq_set_nat @ X @ Y )
% 0.32/0.62 => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ ( uminus5710092332889474511et_nat @ X ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % compl_mono
% 0.32/0.62 thf(fact_5504_less__minus__iff,axiom,
% 0.32/0.62 ! [A: real,B: real] :
% 0.32/0.62 ( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
% 0.32/0.62 = ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % less_minus_iff
% 0.32/0.62 thf(fact_5505_less__minus__iff,axiom,
% 0.32/0.62 ! [A: int,B: int] :
% 0.32/0.62 ( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
% 0.32/0.62 = ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % less_minus_iff
% 0.32/0.62 thf(fact_5506_less__minus__iff,axiom,
% 0.32/0.62 ! [A: rat,B: rat] :
% 0.32/0.62 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.32/0.62 = ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % less_minus_iff
% 0.32/0.62 thf(fact_5507_less__minus__iff,axiom,
% 0.32/0.62 ! [A: code_integer,B: code_integer] :
% 0.32/0.62 ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.32/0.62 = ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % less_minus_iff
% 0.32/0.62 thf(fact_5508_minus__less__iff,axiom,
% 0.32/0.62 ! [A: real,B: real] :
% 0.32/0.62 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
% 0.32/0.62 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_less_iff
% 0.32/0.62 thf(fact_5509_minus__less__iff,axiom,
% 0.32/0.62 ! [A: int,B: int] :
% 0.32/0.62 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
% 0.32/0.62 = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_less_iff
% 0.32/0.62 thf(fact_5510_minus__less__iff,axiom,
% 0.32/0.62 ! [A: rat,B: rat] :
% 0.32/0.62 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.32/0.62 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_less_iff
% 0.32/0.62 thf(fact_5511_minus__less__iff,axiom,
% 0.32/0.62 ! [A: code_integer,B: code_integer] :
% 0.32/0.62 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.32/0.62 = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_less_iff
% 0.32/0.62 thf(fact_5512_verit__negate__coefficient_I2_J,axiom,
% 0.32/0.62 ! [A: real,B: real] :
% 0.32/0.62 ( ( ord_less_real @ A @ B )
% 0.32/0.62 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % verit_negate_coefficient(2)
% 0.32/0.62 thf(fact_5513_verit__negate__coefficient_I2_J,axiom,
% 0.32/0.62 ! [A: int,B: int] :
% 0.32/0.62 ( ( ord_less_int @ A @ B )
% 0.32/0.62 => ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % verit_negate_coefficient(2)
% 0.32/0.62 thf(fact_5514_verit__negate__coefficient_I2_J,axiom,
% 0.32/0.62 ! [A: rat,B: rat] :
% 0.32/0.62 ( ( ord_less_rat @ A @ B )
% 0.32/0.62 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % verit_negate_coefficient(2)
% 0.32/0.62 thf(fact_5515_verit__negate__coefficient_I2_J,axiom,
% 0.32/0.62 ! [A: code_integer,B: code_integer] :
% 0.32/0.62 ( ( ord_le6747313008572928689nteger @ A @ B )
% 0.32/0.62 => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % verit_negate_coefficient(2)
% 0.32/0.62 thf(fact_5516_numeral__neq__neg__numeral,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( numeral_numeral_real @ M )
% 0.32/0.62 != ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % numeral_neq_neg_numeral
% 0.32/0.62 thf(fact_5517_numeral__neq__neg__numeral,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( numeral_numeral_int @ M )
% 0.32/0.62 != ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % numeral_neq_neg_numeral
% 0.32/0.62 thf(fact_5518_numeral__neq__neg__numeral,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( numera6690914467698888265omplex @ M )
% 0.32/0.62 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % numeral_neq_neg_numeral
% 0.32/0.62 thf(fact_5519_numeral__neq__neg__numeral,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( numeral_numeral_rat @ M )
% 0.32/0.62 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % numeral_neq_neg_numeral
% 0.32/0.62 thf(fact_5520_numeral__neq__neg__numeral,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( numera6620942414471956472nteger @ M )
% 0.32/0.62 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % numeral_neq_neg_numeral
% 0.32/0.62 thf(fact_5521_neg__numeral__neq__numeral,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 0.32/0.62 != ( numeral_numeral_real @ N2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_numeral_neq_numeral
% 0.32/0.62 thf(fact_5522_neg__numeral__neq__numeral,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 0.32/0.62 != ( numeral_numeral_int @ N2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_numeral_neq_numeral
% 0.32/0.62 thf(fact_5523_neg__numeral__neq__numeral,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 0.32/0.62 != ( numera6690914467698888265omplex @ N2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_numeral_neq_numeral
% 0.32/0.62 thf(fact_5524_neg__numeral__neq__numeral,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 0.32/0.62 != ( numeral_numeral_rat @ N2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_numeral_neq_numeral
% 0.32/0.62 thf(fact_5525_neg__numeral__neq__numeral,axiom,
% 0.32/0.62 ! [M: num,N2: num] :
% 0.32/0.62 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 0.32/0.62 != ( numera6620942414471956472nteger @ N2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % neg_numeral_neq_numeral
% 0.32/0.62 thf(fact_5526_minus__mult__commute,axiom,
% 0.32/0.62 ! [A: real,B: real] :
% 0.32/0.62 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
% 0.32/0.62 = ( times_times_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_mult_commute
% 0.32/0.62 thf(fact_5527_minus__mult__commute,axiom,
% 0.32/0.62 ! [A: int,B: int] :
% 0.32/0.62 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
% 0.32/0.62 = ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_mult_commute
% 0.32/0.62 thf(fact_5528_minus__mult__commute,axiom,
% 0.32/0.62 ! [A: complex,B: complex] :
% 0.32/0.62 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 0.32/0.62 = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_mult_commute
% 0.32/0.62 thf(fact_5529_minus__mult__commute,axiom,
% 0.32/0.62 ! [A: rat,B: rat] :
% 0.32/0.62 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.32/0.62 = ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_mult_commute
% 0.32/0.62 thf(fact_5530_minus__mult__commute,axiom,
% 0.32/0.62 ! [A: code_integer,B: code_integer] :
% 0.32/0.62 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.32/0.62 = ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_mult_commute
% 0.32/0.62 thf(fact_5531_square__eq__iff,axiom,
% 0.32/0.62 ! [A: real,B: real] :
% 0.32/0.62 ( ( ( times_times_real @ A @ A )
% 0.32/0.62 = ( times_times_real @ B @ B ) )
% 0.32/0.62 = ( ( A = B )
% 0.32/0.62 | ( A
% 0.32/0.62 = ( uminus_uminus_real @ B ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % square_eq_iff
% 0.32/0.62 thf(fact_5532_square__eq__iff,axiom,
% 0.32/0.62 ! [A: int,B: int] :
% 0.32/0.62 ( ( ( times_times_int @ A @ A )
% 0.32/0.62 = ( times_times_int @ B @ B ) )
% 0.32/0.62 = ( ( A = B )
% 0.32/0.62 | ( A
% 0.32/0.62 = ( uminus_uminus_int @ B ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % square_eq_iff
% 0.32/0.62 thf(fact_5533_square__eq__iff,axiom,
% 0.32/0.62 ! [A: complex,B: complex] :
% 0.32/0.62 ( ( ( times_times_complex @ A @ A )
% 0.32/0.62 = ( times_times_complex @ B @ B ) )
% 0.32/0.62 = ( ( A = B )
% 0.32/0.62 | ( A
% 0.32/0.62 = ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % square_eq_iff
% 0.32/0.62 thf(fact_5534_square__eq__iff,axiom,
% 0.32/0.62 ! [A: rat,B: rat] :
% 0.32/0.62 ( ( ( times_times_rat @ A @ A )
% 0.32/0.62 = ( times_times_rat @ B @ B ) )
% 0.32/0.62 = ( ( A = B )
% 0.32/0.62 | ( A
% 0.32/0.62 = ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % square_eq_iff
% 0.32/0.62 thf(fact_5535_square__eq__iff,axiom,
% 0.32/0.62 ! [A: code_integer,B: code_integer] :
% 0.32/0.62 ( ( ( times_3573771949741848930nteger @ A @ A )
% 0.32/0.62 = ( times_3573771949741848930nteger @ B @ B ) )
% 0.32/0.62 = ( ( A = B )
% 0.32/0.62 | ( A
% 0.32/0.62 = ( uminus1351360451143612070nteger @ B ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % square_eq_iff
% 0.32/0.62 thf(fact_5536_is__num__normalize_I8_J,axiom,
% 0.32/0.62 ! [A: real,B: real] :
% 0.32/0.62 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 0.32/0.62 = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % is_num_normalize(8)
% 0.32/0.62 thf(fact_5537_is__num__normalize_I8_J,axiom,
% 0.32/0.62 ! [A: int,B: int] :
% 0.32/0.62 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.32/0.62 = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % is_num_normalize(8)
% 0.32/0.62 thf(fact_5538_is__num__normalize_I8_J,axiom,
% 0.32/0.62 ! [A: complex,B: complex] :
% 0.32/0.62 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 0.32/0.62 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % is_num_normalize(8)
% 0.32/0.62 thf(fact_5539_is__num__normalize_I8_J,axiom,
% 0.32/0.62 ! [A: rat,B: rat] :
% 0.32/0.62 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 0.32/0.62 = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % is_num_normalize(8)
% 0.32/0.62 thf(fact_5540_is__num__normalize_I8_J,axiom,
% 0.32/0.62 ! [A: code_integer,B: code_integer] :
% 0.32/0.62 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.32/0.62 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % is_num_normalize(8)
% 0.32/0.62 thf(fact_5541_add_Oinverse__distrib__swap,axiom,
% 0.32/0.62 ! [A: real,B: real] :
% 0.32/0.62 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 0.32/0.62 = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % add.inverse_distrib_swap
% 0.32/0.62 thf(fact_5542_add_Oinverse__distrib__swap,axiom,
% 0.32/0.62 ! [A: int,B: int] :
% 0.32/0.62 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.32/0.62 = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % add.inverse_distrib_swap
% 0.32/0.62 thf(fact_5543_add_Oinverse__distrib__swap,axiom,
% 0.32/0.62 ! [A: complex,B: complex] :
% 0.32/0.62 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 0.32/0.62 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % add.inverse_distrib_swap
% 0.32/0.62 thf(fact_5544_add_Oinverse__distrib__swap,axiom,
% 0.32/0.62 ! [A: rat,B: rat] :
% 0.32/0.62 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 0.32/0.62 = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % add.inverse_distrib_swap
% 0.32/0.62 thf(fact_5545_add_Oinverse__distrib__swap,axiom,
% 0.32/0.62 ! [A: code_integer,B: code_integer] :
% 0.32/0.62 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.32/0.62 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % add.inverse_distrib_swap
% 0.32/0.62 thf(fact_5546_group__cancel_Oneg1,axiom,
% 0.32/0.62 ! [A2: real,K: real,A: real] :
% 0.32/0.62 ( ( A2
% 0.32/0.62 = ( plus_plus_real @ K @ A ) )
% 0.32/0.62 => ( ( uminus_uminus_real @ A2 )
% 0.32/0.62 = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % group_cancel.neg1
% 0.32/0.62 thf(fact_5547_group__cancel_Oneg1,axiom,
% 0.32/0.62 ! [A2: int,K: int,A: int] :
% 0.32/0.62 ( ( A2
% 0.32/0.62 = ( plus_plus_int @ K @ A ) )
% 0.32/0.62 => ( ( uminus_uminus_int @ A2 )
% 0.32/0.62 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % group_cancel.neg1
% 0.32/0.62 thf(fact_5548_group__cancel_Oneg1,axiom,
% 0.32/0.62 ! [A2: complex,K: complex,A: complex] :
% 0.32/0.62 ( ( A2
% 0.32/0.62 = ( plus_plus_complex @ K @ A ) )
% 0.32/0.62 => ( ( uminus1482373934393186551omplex @ A2 )
% 0.32/0.62 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % group_cancel.neg1
% 0.32/0.62 thf(fact_5549_group__cancel_Oneg1,axiom,
% 0.32/0.62 ! [A2: rat,K: rat,A: rat] :
% 0.32/0.62 ( ( A2
% 0.32/0.62 = ( plus_plus_rat @ K @ A ) )
% 0.32/0.62 => ( ( uminus_uminus_rat @ A2 )
% 0.32/0.62 = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % group_cancel.neg1
% 0.32/0.62 thf(fact_5550_group__cancel_Oneg1,axiom,
% 0.32/0.62 ! [A2: code_integer,K: code_integer,A: code_integer] :
% 0.32/0.62 ( ( A2
% 0.32/0.62 = ( plus_p5714425477246183910nteger @ K @ A ) )
% 0.32/0.62 => ( ( uminus1351360451143612070nteger @ A2 )
% 0.32/0.62 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % group_cancel.neg1
% 0.32/0.62 thf(fact_5551_one__neq__neg__one,axiom,
% 0.32/0.62 ( one_one_real
% 0.32/0.62 != ( uminus_uminus_real @ one_one_real ) ) ).
% 0.32/0.62
% 0.32/0.62 % one_neq_neg_one
% 0.32/0.62 thf(fact_5552_one__neq__neg__one,axiom,
% 0.32/0.62 ( one_one_int
% 0.32/0.62 != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.32/0.62
% 0.32/0.62 % one_neq_neg_one
% 0.32/0.62 thf(fact_5553_one__neq__neg__one,axiom,
% 0.32/0.62 ( one_one_complex
% 0.32/0.62 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.32/0.62
% 0.32/0.62 % one_neq_neg_one
% 0.32/0.62 thf(fact_5554_one__neq__neg__one,axiom,
% 0.32/0.62 ( one_one_rat
% 0.32/0.62 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.32/0.62
% 0.32/0.62 % one_neq_neg_one
% 0.32/0.62 thf(fact_5555_one__neq__neg__one,axiom,
% 0.32/0.62 ( one_one_Code_integer
% 0.32/0.62 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.32/0.62
% 0.32/0.62 % one_neq_neg_one
% 0.32/0.62 thf(fact_5556_minus__diff__minus,axiom,
% 0.32/0.62 ! [A: real,B: real] :
% 0.32/0.62 ( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 0.32/0.62 = ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_diff_minus
% 0.32/0.62 thf(fact_5557_minus__diff__minus,axiom,
% 0.32/0.62 ! [A: int,B: int] :
% 0.32/0.62 ( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 0.32/0.62 = ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_diff_minus
% 0.32/0.62 thf(fact_5558_minus__diff__minus,axiom,
% 0.32/0.62 ! [A: complex,B: complex] :
% 0.32/0.62 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 0.32/0.62 = ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_diff_minus
% 0.32/0.62 thf(fact_5559_minus__diff__minus,axiom,
% 0.32/0.62 ! [A: rat,B: rat] :
% 0.32/0.62 ( ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 0.32/0.62 = ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_diff_minus
% 0.32/0.62 thf(fact_5560_minus__diff__minus,axiom,
% 0.32/0.62 ! [A: code_integer,B: code_integer] :
% 0.32/0.62 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 0.32/0.62 = ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_diff_minus
% 0.32/0.62 thf(fact_5561_minus__diff__commute,axiom,
% 0.32/0.62 ! [B: real,A: real] :
% 0.32/0.62 ( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
% 0.32/0.62 = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_diff_commute
% 0.32/0.62 thf(fact_5562_minus__diff__commute,axiom,
% 0.32/0.62 ! [B: int,A: int] :
% 0.32/0.62 ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
% 0.32/0.62 = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_diff_commute
% 0.32/0.62 thf(fact_5563_minus__diff__commute,axiom,
% 0.32/0.62 ! [B: complex,A: complex] :
% 0.32/0.62 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B ) @ A )
% 0.32/0.62 = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_diff_commute
% 0.32/0.62 thf(fact_5564_minus__diff__commute,axiom,
% 0.32/0.62 ! [B: rat,A: rat] :
% 0.32/0.62 ( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
% 0.32/0.62 = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_diff_commute
% 0.32/0.62 thf(fact_5565_minus__diff__commute,axiom,
% 0.32/0.62 ! [B: code_integer,A: code_integer] :
% 0.32/0.62 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A )
% 0.32/0.62 = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_diff_commute
% 0.32/0.62 thf(fact_5566_div__minus__right,axiom,
% 0.32/0.62 ! [A: int,B: int] :
% 0.32/0.62 ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 0.32/0.62 = ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.32/0.62
% 0.32/0.62 % div_minus_right
% 0.32/0.62 thf(fact_5567_div__minus__right,axiom,
% 0.32/0.62 ! [A: code_integer,B: code_integer] :
% 0.32/0.62 ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.32/0.62 = ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.32/0.62
% 0.32/0.62 % div_minus_right
% 0.32/0.62 thf(fact_5568_minus__divide__left,axiom,
% 0.32/0.62 ! [A: real,B: real] :
% 0.32/0.62 ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.32/0.62 = ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_divide_left
% 0.32/0.62 thf(fact_5569_minus__divide__left,axiom,
% 0.32/0.62 ! [A: complex,B: complex] :
% 0.32/0.62 ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.32/0.62 = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_divide_left
% 0.32/0.62 thf(fact_5570_minus__divide__left,axiom,
% 0.32/0.62 ! [A: rat,B: rat] :
% 0.32/0.62 ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.32/0.62 = ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_divide_left
% 0.32/0.62 thf(fact_5571_minus__divide__divide,axiom,
% 0.32/0.62 ! [A: real,B: real] :
% 0.32/0.62 ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 0.32/0.62 = ( divide_divide_real @ A @ B ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_divide_divide
% 0.32/0.62 thf(fact_5572_minus__divide__divide,axiom,
% 0.32/0.62 ! [A: complex,B: complex] :
% 0.32/0.62 ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 0.32/0.62 = ( divide1717551699836669952omplex @ A @ B ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_divide_divide
% 0.32/0.62 thf(fact_5573_minus__divide__divide,axiom,
% 0.32/0.62 ! [A: rat,B: rat] :
% 0.32/0.62 ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 0.32/0.62 = ( divide_divide_rat @ A @ B ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_divide_divide
% 0.32/0.62 thf(fact_5574_minus__divide__right,axiom,
% 0.32/0.62 ! [A: real,B: real] :
% 0.32/0.62 ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.32/0.62 = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_divide_right
% 0.32/0.62 thf(fact_5575_minus__divide__right,axiom,
% 0.32/0.62 ! [A: complex,B: complex] :
% 0.32/0.62 ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.32/0.62 = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_divide_right
% 0.32/0.62 thf(fact_5576_minus__divide__right,axiom,
% 0.32/0.62 ! [A: rat,B: rat] :
% 0.32/0.62 ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.32/0.62 = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_divide_right
% 0.32/0.62 thf(fact_5577_old_Oprod_Ocase,axiom,
% 0.32/0.62 ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X1: nat,X22: nat] :
% 0.32/0.62 ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
% 0.32/0.62 = ( F @ X1 @ X22 ) ) ).
% 0.32/0.62
% 0.32/0.62 % old.prod.case
% 0.32/0.62 thf(fact_5578_old_Oprod_Ocase,axiom,
% 0.32/0.62 ! [F: nat > nat > product_prod_nat_nat > $o,X1: nat,X22: nat] :
% 0.32/0.62 ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
% 0.32/0.62 = ( F @ X1 @ X22 ) ) ).
% 0.32/0.62
% 0.32/0.62 % old.prod.case
% 0.32/0.62 thf(fact_5579_old_Oprod_Ocase,axiom,
% 0.32/0.62 ! [F: int > int > product_prod_int_int,X1: int,X22: int] :
% 0.32/0.62 ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
% 0.32/0.62 = ( F @ X1 @ X22 ) ) ).
% 0.32/0.62
% 0.32/0.62 % old.prod.case
% 0.32/0.62 thf(fact_5580_old_Oprod_Ocase,axiom,
% 0.32/0.62 ! [F: int > int > $o,X1: int,X22: int] :
% 0.32/0.62 ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
% 0.32/0.62 = ( F @ X1 @ X22 ) ) ).
% 0.32/0.62
% 0.32/0.62 % old.prod.case
% 0.32/0.62 thf(fact_5581_old_Oprod_Ocase,axiom,
% 0.32/0.62 ! [F: int > int > int,X1: int,X22: int] :
% 0.32/0.62 ( ( produc8211389475949308722nt_int @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
% 0.32/0.62 = ( F @ X1 @ X22 ) ) ).
% 0.32/0.62
% 0.32/0.62 % old.prod.case
% 0.32/0.62 thf(fact_5582_mod__minus__right,axiom,
% 0.32/0.62 ! [A: int,B: int] :
% 0.32/0.62 ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 0.32/0.62 = ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % mod_minus_right
% 0.32/0.62 thf(fact_5583_mod__minus__right,axiom,
% 0.32/0.62 ! [A: code_integer,B: code_integer] :
% 0.32/0.62 ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.32/0.62 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % mod_minus_right
% 0.32/0.62 thf(fact_5584_mod__minus__cong,axiom,
% 0.32/0.62 ! [A: int,B: int,A6: int] :
% 0.32/0.62 ( ( ( modulo_modulo_int @ A @ B )
% 0.32/0.62 = ( modulo_modulo_int @ A6 @ B ) )
% 0.32/0.62 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 0.32/0.62 = ( modulo_modulo_int @ ( uminus_uminus_int @ A6 ) @ B ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % mod_minus_cong
% 0.32/0.62 thf(fact_5585_mod__minus__cong,axiom,
% 0.32/0.62 ! [A: code_integer,B: code_integer,A6: code_integer] :
% 0.32/0.62 ( ( ( modulo364778990260209775nteger @ A @ B )
% 0.32/0.62 = ( modulo364778990260209775nteger @ A6 @ B ) )
% 0.32/0.62 => ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.32/0.62 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A6 ) @ B ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % mod_minus_cong
% 0.32/0.62 thf(fact_5586_mod__minus__eq,axiom,
% 0.32/0.62 ! [A: int,B: int] :
% 0.32/0.62 ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) @ B )
% 0.32/0.62 = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.32/0.62
% 0.32/0.62 % mod_minus_eq
% 0.32/0.62 thf(fact_5587_mod__minus__eq,axiom,
% 0.32/0.62 ! [A: code_integer,B: code_integer] :
% 0.32/0.62 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) @ B )
% 0.32/0.62 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.32/0.62
% 0.32/0.62 % mod_minus_eq
% 0.32/0.62 thf(fact_5588_sum__mono,axiom,
% 0.32/0.62 ! [K5: set_nat,F: nat > rat,G: nat > rat] :
% 0.32/0.62 ( ! [I4: nat] :
% 0.32/0.62 ( ( member_nat @ I4 @ K5 )
% 0.32/0.62 => ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 0.32/0.62 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K5 ) @ ( groups2906978787729119204at_rat @ G @ K5 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono
% 0.32/0.62 thf(fact_5589_sum__mono,axiom,
% 0.32/0.62 ! [K5: set_real,F: real > rat,G: real > rat] :
% 0.32/0.62 ( ! [I4: real] :
% 0.32/0.62 ( ( member_real @ I4 @ K5 )
% 0.32/0.62 => ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 0.32/0.62 => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K5 ) @ ( groups1300246762558778688al_rat @ G @ K5 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono
% 0.32/0.62 thf(fact_5590_sum__mono,axiom,
% 0.32/0.62 ! [K5: set_int,F: int > rat,G: int > rat] :
% 0.32/0.62 ( ! [I4: int] :
% 0.32/0.62 ( ( member_int @ I4 @ K5 )
% 0.32/0.62 => ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 0.32/0.62 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K5 ) @ ( groups3906332499630173760nt_rat @ G @ K5 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono
% 0.32/0.62 thf(fact_5591_sum__mono,axiom,
% 0.32/0.62 ! [K5: set_complex,F: complex > rat,G: complex > rat] :
% 0.32/0.62 ( ! [I4: complex] :
% 0.32/0.62 ( ( member_complex @ I4 @ K5 )
% 0.32/0.62 => ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 0.32/0.62 => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ K5 ) @ ( groups5058264527183730370ex_rat @ G @ K5 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono
% 0.32/0.62 thf(fact_5592_sum__mono,axiom,
% 0.32/0.62 ! [K5: set_real,F: real > nat,G: real > nat] :
% 0.32/0.62 ( ! [I4: real] :
% 0.32/0.62 ( ( member_real @ I4 @ K5 )
% 0.32/0.62 => ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 0.32/0.62 => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K5 ) @ ( groups1935376822645274424al_nat @ G @ K5 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono
% 0.32/0.62 thf(fact_5593_sum__mono,axiom,
% 0.32/0.62 ! [K5: set_int,F: int > nat,G: int > nat] :
% 0.32/0.62 ( ! [I4: int] :
% 0.32/0.62 ( ( member_int @ I4 @ K5 )
% 0.32/0.62 => ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 0.32/0.62 => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K5 ) @ ( groups4541462559716669496nt_nat @ G @ K5 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono
% 0.32/0.62 thf(fact_5594_sum__mono,axiom,
% 0.32/0.62 ! [K5: set_complex,F: complex > nat,G: complex > nat] :
% 0.32/0.62 ( ! [I4: complex] :
% 0.32/0.62 ( ( member_complex @ I4 @ K5 )
% 0.32/0.62 => ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 0.32/0.62 => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ K5 ) @ ( groups5693394587270226106ex_nat @ G @ K5 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono
% 0.32/0.62 thf(fact_5595_sum__mono,axiom,
% 0.32/0.62 ! [K5: set_nat,F: nat > int,G: nat > int] :
% 0.32/0.62 ( ! [I4: nat] :
% 0.32/0.62 ( ( member_nat @ I4 @ K5 )
% 0.32/0.62 => ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 0.32/0.62 => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K5 ) @ ( groups3539618377306564664at_int @ G @ K5 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono
% 0.32/0.62 thf(fact_5596_sum__mono,axiom,
% 0.32/0.62 ! [K5: set_real,F: real > int,G: real > int] :
% 0.32/0.62 ( ! [I4: real] :
% 0.32/0.62 ( ( member_real @ I4 @ K5 )
% 0.32/0.62 => ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 0.32/0.62 => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K5 ) @ ( groups1932886352136224148al_int @ G @ K5 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono
% 0.32/0.62 thf(fact_5597_sum__mono,axiom,
% 0.32/0.62 ! [K5: set_complex,F: complex > int,G: complex > int] :
% 0.32/0.62 ( ! [I4: complex] :
% 0.32/0.62 ( ( member_complex @ I4 @ K5 )
% 0.32/0.62 => ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 0.32/0.62 => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ K5 ) @ ( groups5690904116761175830ex_int @ G @ K5 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono
% 0.32/0.62 thf(fact_5598_sum__product,axiom,
% 0.32/0.62 ! [F: int > int,A2: set_int,G: int > int,B3: set_int] :
% 0.32/0.62 ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ B3 ) )
% 0.32/0.62 = ( groups4538972089207619220nt_int
% 0.32/0.62 @ ^ [I3: int] :
% 0.32/0.62 ( groups4538972089207619220nt_int
% 0.32/0.62 @ ^ [J3: int] : ( times_times_int @ ( F @ I3 ) @ ( G @ J3 ) )
% 0.32/0.62 @ B3 )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_product
% 0.32/0.62 thf(fact_5599_sum__product,axiom,
% 0.32/0.62 ! [F: complex > complex,A2: set_complex,G: complex > complex,B3: set_complex] :
% 0.32/0.62 ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ B3 ) )
% 0.32/0.62 = ( groups7754918857620584856omplex
% 0.32/0.62 @ ^ [I3: complex] :
% 0.32/0.62 ( groups7754918857620584856omplex
% 0.32/0.62 @ ^ [J3: complex] : ( times_times_complex @ ( F @ I3 ) @ ( G @ J3 ) )
% 0.32/0.62 @ B3 )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_product
% 0.32/0.62 thf(fact_5600_sum__product,axiom,
% 0.32/0.62 ! [F: nat > nat,A2: set_nat,G: nat > nat,B3: set_nat] :
% 0.32/0.62 ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ B3 ) )
% 0.32/0.62 = ( groups3542108847815614940at_nat
% 0.32/0.62 @ ^ [I3: nat] :
% 0.32/0.62 ( groups3542108847815614940at_nat
% 0.32/0.62 @ ^ [J3: nat] : ( times_times_nat @ ( F @ I3 ) @ ( G @ J3 ) )
% 0.32/0.62 @ B3 )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_product
% 0.32/0.62 thf(fact_5601_sum__product,axiom,
% 0.32/0.62 ! [F: nat > real,A2: set_nat,G: nat > real,B3: set_nat] :
% 0.32/0.62 ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ B3 ) )
% 0.32/0.62 = ( groups6591440286371151544t_real
% 0.32/0.62 @ ^ [I3: nat] :
% 0.32/0.62 ( groups6591440286371151544t_real
% 0.32/0.62 @ ^ [J3: nat] : ( times_times_real @ ( F @ I3 ) @ ( G @ J3 ) )
% 0.32/0.62 @ B3 )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_product
% 0.32/0.62 thf(fact_5602_sum__distrib__right,axiom,
% 0.32/0.62 ! [F: int > int,A2: set_int,R: int] :
% 0.32/0.62 ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ R )
% 0.32/0.62 = ( groups4538972089207619220nt_int
% 0.32/0.62 @ ^ [N: int] : ( times_times_int @ ( F @ N ) @ R )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_distrib_right
% 0.32/0.62 thf(fact_5603_sum__distrib__right,axiom,
% 0.32/0.62 ! [F: complex > complex,A2: set_complex,R: complex] :
% 0.32/0.62 ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R )
% 0.32/0.62 = ( groups7754918857620584856omplex
% 0.32/0.62 @ ^ [N: complex] : ( times_times_complex @ ( F @ N ) @ R )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_distrib_right
% 0.32/0.62 thf(fact_5604_sum__distrib__right,axiom,
% 0.32/0.62 ! [F: nat > nat,A2: set_nat,R: nat] :
% 0.32/0.62 ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ R )
% 0.32/0.62 = ( groups3542108847815614940at_nat
% 0.32/0.62 @ ^ [N: nat] : ( times_times_nat @ ( F @ N ) @ R )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_distrib_right
% 0.32/0.62 thf(fact_5605_sum__distrib__right,axiom,
% 0.32/0.62 ! [F: nat > real,A2: set_nat,R: real] :
% 0.32/0.62 ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R )
% 0.32/0.62 = ( groups6591440286371151544t_real
% 0.32/0.62 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ R )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_distrib_right
% 0.32/0.62 thf(fact_5606_sum__distrib__left,axiom,
% 0.32/0.62 ! [R: int,F: int > int,A2: set_int] :
% 0.32/0.62 ( ( times_times_int @ R @ ( groups4538972089207619220nt_int @ F @ A2 ) )
% 0.32/0.62 = ( groups4538972089207619220nt_int
% 0.32/0.62 @ ^ [N: int] : ( times_times_int @ R @ ( F @ N ) )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_distrib_left
% 0.32/0.62 thf(fact_5607_sum__distrib__left,axiom,
% 0.32/0.62 ! [R: complex,F: complex > complex,A2: set_complex] :
% 0.32/0.62 ( ( times_times_complex @ R @ ( groups7754918857620584856omplex @ F @ A2 ) )
% 0.32/0.62 = ( groups7754918857620584856omplex
% 0.32/0.62 @ ^ [N: complex] : ( times_times_complex @ R @ ( F @ N ) )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_distrib_left
% 0.32/0.62 thf(fact_5608_sum__distrib__left,axiom,
% 0.32/0.62 ! [R: nat,F: nat > nat,A2: set_nat] :
% 0.32/0.62 ( ( times_times_nat @ R @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 0.32/0.62 = ( groups3542108847815614940at_nat
% 0.32/0.62 @ ^ [N: nat] : ( times_times_nat @ R @ ( F @ N ) )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_distrib_left
% 0.32/0.62 thf(fact_5609_sum__distrib__left,axiom,
% 0.32/0.62 ! [R: real,F: nat > real,A2: set_nat] :
% 0.32/0.62 ( ( times_times_real @ R @ ( groups6591440286371151544t_real @ F @ A2 ) )
% 0.32/0.62 = ( groups6591440286371151544t_real
% 0.32/0.62 @ ^ [N: nat] : ( times_times_real @ R @ ( F @ N ) )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_distrib_left
% 0.32/0.62 thf(fact_5610_sum_Odistrib,axiom,
% 0.32/0.62 ! [G: int > int,H2: int > int,A2: set_int] :
% 0.32/0.62 ( ( groups4538972089207619220nt_int
% 0.32/0.62 @ ^ [X2: int] : ( plus_plus_int @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A2 ) @ ( groups4538972089207619220nt_int @ H2 @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.distrib
% 0.32/0.62 thf(fact_5611_sum_Odistrib,axiom,
% 0.32/0.62 ! [G: complex > complex,H2: complex > complex,A2: set_complex] :
% 0.32/0.62 ( ( groups7754918857620584856omplex
% 0.32/0.62 @ ^ [X2: complex] : ( plus_plus_complex @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A2 ) @ ( groups7754918857620584856omplex @ H2 @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.distrib
% 0.32/0.62 thf(fact_5612_sum_Odistrib,axiom,
% 0.32/0.62 ! [G: nat > nat,H2: nat > nat,A2: set_nat] :
% 0.32/0.62 ( ( groups3542108847815614940at_nat
% 0.32/0.62 @ ^ [X2: nat] : ( plus_plus_nat @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A2 ) @ ( groups3542108847815614940at_nat @ H2 @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.distrib
% 0.32/0.62 thf(fact_5613_sum_Odistrib,axiom,
% 0.32/0.62 ! [G: nat > real,H2: nat > real,A2: set_nat] :
% 0.32/0.62 ( ( groups6591440286371151544t_real
% 0.32/0.62 @ ^ [X2: nat] : ( plus_plus_real @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A2 ) @ ( groups6591440286371151544t_real @ H2 @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.distrib
% 0.32/0.62 thf(fact_5614_sum__subtractf,axiom,
% 0.32/0.62 ! [F: int > int,G: int > int,A2: set_int] :
% 0.32/0.62 ( ( groups4538972089207619220nt_int
% 0.32/0.62 @ ^ [X2: int] : ( minus_minus_int @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_subtractf
% 0.32/0.62 thf(fact_5615_sum__subtractf,axiom,
% 0.32/0.62 ! [F: complex > complex,G: complex > complex,A2: set_complex] :
% 0.32/0.62 ( ( groups7754918857620584856omplex
% 0.32/0.62 @ ^ [X2: complex] : ( minus_minus_complex @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_subtractf
% 0.32/0.62 thf(fact_5616_sum__subtractf,axiom,
% 0.32/0.62 ! [F: nat > real,G: nat > real,A2: set_nat] :
% 0.32/0.62 ( ( groups6591440286371151544t_real
% 0.32/0.62 @ ^ [X2: nat] : ( minus_minus_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_subtractf
% 0.32/0.62 thf(fact_5617_sum__divide__distrib,axiom,
% 0.32/0.62 ! [F: complex > complex,A2: set_complex,R: complex] :
% 0.32/0.62 ( ( divide1717551699836669952omplex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R )
% 0.32/0.62 = ( groups7754918857620584856omplex
% 0.32/0.62 @ ^ [N: complex] : ( divide1717551699836669952omplex @ ( F @ N ) @ R )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_divide_distrib
% 0.32/0.62 thf(fact_5618_sum__divide__distrib,axiom,
% 0.32/0.62 ! [F: nat > real,A2: set_nat,R: real] :
% 0.32/0.62 ( ( divide_divide_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R )
% 0.32/0.62 = ( groups6591440286371151544t_real
% 0.32/0.62 @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ R )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_divide_distrib
% 0.32/0.62 thf(fact_5619_sum_Oswap__restrict,axiom,
% 0.32/0.62 ! [A2: set_real,B3: set_int,G: real > int > int,R2: real > int > $o] :
% 0.32/0.62 ( ( finite_finite_real @ A2 )
% 0.32/0.62 => ( ( finite_finite_int @ B3 )
% 0.32/0.62 => ( ( groups1932886352136224148al_int
% 0.32/0.62 @ ^ [X2: real] :
% 0.32/0.62 ( groups4538972089207619220nt_int @ ( G @ X2 )
% 0.32/0.62 @ ( collect_int
% 0.32/0.62 @ ^ [Y2: int] :
% 0.32/0.62 ( ( member_int @ Y2 @ B3 )
% 0.32/0.62 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( groups4538972089207619220nt_int
% 0.32/0.62 @ ^ [Y2: int] :
% 0.32/0.62 ( groups1932886352136224148al_int
% 0.32/0.62 @ ^ [X2: real] : ( G @ X2 @ Y2 )
% 0.32/0.62 @ ( collect_real
% 0.32/0.62 @ ^ [X2: real] :
% 0.32/0.62 ( ( member_real @ X2 @ A2 )
% 0.32/0.62 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.32/0.62 @ B3 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.swap_restrict
% 0.32/0.62 thf(fact_5620_sum_Oswap__restrict,axiom,
% 0.32/0.62 ! [A2: set_nat,B3: set_int,G: nat > int > int,R2: nat > int > $o] :
% 0.32/0.62 ( ( finite_finite_nat @ A2 )
% 0.32/0.62 => ( ( finite_finite_int @ B3 )
% 0.32/0.62 => ( ( groups3539618377306564664at_int
% 0.32/0.62 @ ^ [X2: nat] :
% 0.32/0.62 ( groups4538972089207619220nt_int @ ( G @ X2 )
% 0.32/0.62 @ ( collect_int
% 0.32/0.62 @ ^ [Y2: int] :
% 0.32/0.62 ( ( member_int @ Y2 @ B3 )
% 0.32/0.62 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( groups4538972089207619220nt_int
% 0.32/0.62 @ ^ [Y2: int] :
% 0.32/0.62 ( groups3539618377306564664at_int
% 0.32/0.62 @ ^ [X2: nat] : ( G @ X2 @ Y2 )
% 0.32/0.62 @ ( collect_nat
% 0.32/0.62 @ ^ [X2: nat] :
% 0.32/0.62 ( ( member_nat @ X2 @ A2 )
% 0.32/0.62 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.32/0.62 @ B3 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.swap_restrict
% 0.32/0.62 thf(fact_5621_sum_Oswap__restrict,axiom,
% 0.32/0.62 ! [A2: set_complex,B3: set_int,G: complex > int > int,R2: complex > int > $o] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.62 => ( ( finite_finite_int @ B3 )
% 0.32/0.62 => ( ( groups5690904116761175830ex_int
% 0.32/0.62 @ ^ [X2: complex] :
% 0.32/0.62 ( groups4538972089207619220nt_int @ ( G @ X2 )
% 0.32/0.62 @ ( collect_int
% 0.32/0.62 @ ^ [Y2: int] :
% 0.32/0.62 ( ( member_int @ Y2 @ B3 )
% 0.32/0.62 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( groups4538972089207619220nt_int
% 0.32/0.62 @ ^ [Y2: int] :
% 0.32/0.62 ( groups5690904116761175830ex_int
% 0.32/0.62 @ ^ [X2: complex] : ( G @ X2 @ Y2 )
% 0.32/0.62 @ ( collect_complex
% 0.32/0.62 @ ^ [X2: complex] :
% 0.32/0.62 ( ( member_complex @ X2 @ A2 )
% 0.32/0.62 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.32/0.62 @ B3 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.swap_restrict
% 0.32/0.62 thf(fact_5622_sum_Oswap__restrict,axiom,
% 0.32/0.62 ! [A2: set_real,B3: set_complex,G: real > complex > complex,R2: real > complex > $o] :
% 0.32/0.62 ( ( finite_finite_real @ A2 )
% 0.32/0.62 => ( ( finite3207457112153483333omplex @ B3 )
% 0.32/0.62 => ( ( groups5754745047067104278omplex
% 0.32/0.62 @ ^ [X2: real] :
% 0.32/0.62 ( groups7754918857620584856omplex @ ( G @ X2 )
% 0.32/0.62 @ ( collect_complex
% 0.32/0.62 @ ^ [Y2: complex] :
% 0.32/0.62 ( ( member_complex @ Y2 @ B3 )
% 0.32/0.62 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( groups7754918857620584856omplex
% 0.32/0.62 @ ^ [Y2: complex] :
% 0.32/0.62 ( groups5754745047067104278omplex
% 0.32/0.62 @ ^ [X2: real] : ( G @ X2 @ Y2 )
% 0.32/0.62 @ ( collect_real
% 0.32/0.62 @ ^ [X2: real] :
% 0.32/0.62 ( ( member_real @ X2 @ A2 )
% 0.32/0.62 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.32/0.62 @ B3 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.swap_restrict
% 0.32/0.62 thf(fact_5623_sum_Oswap__restrict,axiom,
% 0.32/0.62 ! [A2: set_nat,B3: set_complex,G: nat > complex > complex,R2: nat > complex > $o] :
% 0.32/0.62 ( ( finite_finite_nat @ A2 )
% 0.32/0.62 => ( ( finite3207457112153483333omplex @ B3 )
% 0.32/0.62 => ( ( groups2073611262835488442omplex
% 0.32/0.62 @ ^ [X2: nat] :
% 0.32/0.62 ( groups7754918857620584856omplex @ ( G @ X2 )
% 0.32/0.62 @ ( collect_complex
% 0.32/0.62 @ ^ [Y2: complex] :
% 0.32/0.62 ( ( member_complex @ Y2 @ B3 )
% 0.32/0.62 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( groups7754918857620584856omplex
% 0.32/0.62 @ ^ [Y2: complex] :
% 0.32/0.62 ( groups2073611262835488442omplex
% 0.32/0.62 @ ^ [X2: nat] : ( G @ X2 @ Y2 )
% 0.32/0.62 @ ( collect_nat
% 0.32/0.62 @ ^ [X2: nat] :
% 0.32/0.62 ( ( member_nat @ X2 @ A2 )
% 0.32/0.62 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.32/0.62 @ B3 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.swap_restrict
% 0.32/0.62 thf(fact_5624_sum_Oswap__restrict,axiom,
% 0.32/0.62 ! [A2: set_int,B3: set_complex,G: int > complex > complex,R2: int > complex > $o] :
% 0.32/0.62 ( ( finite_finite_int @ A2 )
% 0.32/0.62 => ( ( finite3207457112153483333omplex @ B3 )
% 0.32/0.62 => ( ( groups3049146728041665814omplex
% 0.32/0.62 @ ^ [X2: int] :
% 0.32/0.62 ( groups7754918857620584856omplex @ ( G @ X2 )
% 0.32/0.62 @ ( collect_complex
% 0.32/0.62 @ ^ [Y2: complex] :
% 0.32/0.62 ( ( member_complex @ Y2 @ B3 )
% 0.32/0.62 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( groups7754918857620584856omplex
% 0.32/0.62 @ ^ [Y2: complex] :
% 0.32/0.62 ( groups3049146728041665814omplex
% 0.32/0.62 @ ^ [X2: int] : ( G @ X2 @ Y2 )
% 0.32/0.62 @ ( collect_int
% 0.32/0.62 @ ^ [X2: int] :
% 0.32/0.62 ( ( member_int @ X2 @ A2 )
% 0.32/0.62 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.32/0.62 @ B3 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.swap_restrict
% 0.32/0.62 thf(fact_5625_sum_Oswap__restrict,axiom,
% 0.32/0.62 ! [A2: set_real,B3: set_nat,G: real > nat > nat,R2: real > nat > $o] :
% 0.32/0.62 ( ( finite_finite_real @ A2 )
% 0.32/0.62 => ( ( finite_finite_nat @ B3 )
% 0.32/0.62 => ( ( groups1935376822645274424al_nat
% 0.32/0.62 @ ^ [X2: real] :
% 0.32/0.62 ( groups3542108847815614940at_nat @ ( G @ X2 )
% 0.32/0.62 @ ( collect_nat
% 0.32/0.62 @ ^ [Y2: nat] :
% 0.32/0.62 ( ( member_nat @ Y2 @ B3 )
% 0.32/0.62 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( groups3542108847815614940at_nat
% 0.32/0.62 @ ^ [Y2: nat] :
% 0.32/0.62 ( groups1935376822645274424al_nat
% 0.32/0.62 @ ^ [X2: real] : ( G @ X2 @ Y2 )
% 0.32/0.62 @ ( collect_real
% 0.32/0.62 @ ^ [X2: real] :
% 0.32/0.62 ( ( member_real @ X2 @ A2 )
% 0.32/0.62 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.32/0.62 @ B3 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.swap_restrict
% 0.32/0.62 thf(fact_5626_sum_Oswap__restrict,axiom,
% 0.32/0.62 ! [A2: set_int,B3: set_nat,G: int > nat > nat,R2: int > nat > $o] :
% 0.32/0.62 ( ( finite_finite_int @ A2 )
% 0.32/0.62 => ( ( finite_finite_nat @ B3 )
% 0.32/0.62 => ( ( groups4541462559716669496nt_nat
% 0.32/0.62 @ ^ [X2: int] :
% 0.32/0.62 ( groups3542108847815614940at_nat @ ( G @ X2 )
% 0.32/0.62 @ ( collect_nat
% 0.32/0.62 @ ^ [Y2: nat] :
% 0.32/0.62 ( ( member_nat @ Y2 @ B3 )
% 0.32/0.62 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( groups3542108847815614940at_nat
% 0.32/0.62 @ ^ [Y2: nat] :
% 0.32/0.62 ( groups4541462559716669496nt_nat
% 0.32/0.62 @ ^ [X2: int] : ( G @ X2 @ Y2 )
% 0.32/0.62 @ ( collect_int
% 0.32/0.62 @ ^ [X2: int] :
% 0.32/0.62 ( ( member_int @ X2 @ A2 )
% 0.32/0.62 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.32/0.62 @ B3 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.swap_restrict
% 0.32/0.62 thf(fact_5627_sum_Oswap__restrict,axiom,
% 0.32/0.62 ! [A2: set_complex,B3: set_nat,G: complex > nat > nat,R2: complex > nat > $o] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.62 => ( ( finite_finite_nat @ B3 )
% 0.32/0.62 => ( ( groups5693394587270226106ex_nat
% 0.32/0.62 @ ^ [X2: complex] :
% 0.32/0.62 ( groups3542108847815614940at_nat @ ( G @ X2 )
% 0.32/0.62 @ ( collect_nat
% 0.32/0.62 @ ^ [Y2: nat] :
% 0.32/0.62 ( ( member_nat @ Y2 @ B3 )
% 0.32/0.62 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( groups3542108847815614940at_nat
% 0.32/0.62 @ ^ [Y2: nat] :
% 0.32/0.62 ( groups5693394587270226106ex_nat
% 0.32/0.62 @ ^ [X2: complex] : ( G @ X2 @ Y2 )
% 0.32/0.62 @ ( collect_complex
% 0.32/0.62 @ ^ [X2: complex] :
% 0.32/0.62 ( ( member_complex @ X2 @ A2 )
% 0.32/0.62 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.32/0.62 @ B3 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.swap_restrict
% 0.32/0.62 thf(fact_5628_sum_Oswap__restrict,axiom,
% 0.32/0.62 ! [A2: set_real,B3: set_nat,G: real > nat > real,R2: real > nat > $o] :
% 0.32/0.62 ( ( finite_finite_real @ A2 )
% 0.32/0.62 => ( ( finite_finite_nat @ B3 )
% 0.32/0.62 => ( ( groups8097168146408367636l_real
% 0.32/0.62 @ ^ [X2: real] :
% 0.32/0.62 ( groups6591440286371151544t_real @ ( G @ X2 )
% 0.32/0.62 @ ( collect_nat
% 0.32/0.62 @ ^ [Y2: nat] :
% 0.32/0.62 ( ( member_nat @ Y2 @ B3 )
% 0.32/0.62 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( groups6591440286371151544t_real
% 0.32/0.62 @ ^ [Y2: nat] :
% 0.32/0.62 ( groups8097168146408367636l_real
% 0.32/0.62 @ ^ [X2: real] : ( G @ X2 @ Y2 )
% 0.32/0.62 @ ( collect_real
% 0.32/0.62 @ ^ [X2: real] :
% 0.32/0.62 ( ( member_real @ X2 @ A2 )
% 0.32/0.62 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.32/0.62 @ B3 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.swap_restrict
% 0.32/0.62 thf(fact_5629_mod__sum__eq,axiom,
% 0.32/0.62 ! [F: int > int,A: int,A2: set_int] :
% 0.32/0.62 ( ( modulo_modulo_int
% 0.32/0.62 @ ( groups4538972089207619220nt_int
% 0.32/0.62 @ ^ [I3: int] : ( modulo_modulo_int @ ( F @ I3 ) @ A )
% 0.32/0.62 @ A2 )
% 0.32/0.62 @ A )
% 0.32/0.62 = ( modulo_modulo_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ A ) ) ).
% 0.32/0.62
% 0.32/0.62 % mod_sum_eq
% 0.32/0.62 thf(fact_5630_mod__sum__eq,axiom,
% 0.32/0.62 ! [F: nat > nat,A: nat,A2: set_nat] :
% 0.32/0.62 ( ( modulo_modulo_nat
% 0.32/0.62 @ ( groups3542108847815614940at_nat
% 0.32/0.62 @ ^ [I3: nat] : ( modulo_modulo_nat @ ( F @ I3 ) @ A )
% 0.32/0.62 @ A2 )
% 0.32/0.62 @ A )
% 0.32/0.62 = ( modulo_modulo_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ A ) ) ).
% 0.32/0.62
% 0.32/0.62 % mod_sum_eq
% 0.32/0.62 thf(fact_5631_cond__case__prod__eta,axiom,
% 0.32/0.62 ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
% 0.32/0.62 ( ! [X4: nat,Y4: nat] :
% 0.32/0.62 ( ( F @ X4 @ Y4 )
% 0.32/0.62 = ( G @ ( product_Pair_nat_nat @ X4 @ Y4 ) ) )
% 0.32/0.62 => ( ( produc27273713700761075at_nat @ F )
% 0.32/0.62 = G ) ) ).
% 0.32/0.62
% 0.32/0.62 % cond_case_prod_eta
% 0.32/0.62 thf(fact_5632_cond__case__prod__eta,axiom,
% 0.32/0.62 ! [F: nat > nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 0.32/0.62 ( ! [X4: nat,Y4: nat] :
% 0.32/0.62 ( ( F @ X4 @ Y4 )
% 0.32/0.62 = ( G @ ( product_Pair_nat_nat @ X4 @ Y4 ) ) )
% 0.32/0.62 => ( ( produc8739625826339149834_nat_o @ F )
% 0.32/0.62 = G ) ) ).
% 0.32/0.62
% 0.32/0.62 % cond_case_prod_eta
% 0.32/0.62 thf(fact_5633_cond__case__prod__eta,axiom,
% 0.32/0.62 ! [F: int > int > product_prod_int_int,G: product_prod_int_int > product_prod_int_int] :
% 0.32/0.62 ( ! [X4: int,Y4: int] :
% 0.32/0.62 ( ( F @ X4 @ Y4 )
% 0.32/0.62 = ( G @ ( product_Pair_int_int @ X4 @ Y4 ) ) )
% 0.32/0.62 => ( ( produc4245557441103728435nt_int @ F )
% 0.32/0.62 = G ) ) ).
% 0.32/0.62
% 0.32/0.62 % cond_case_prod_eta
% 0.32/0.62 thf(fact_5634_cond__case__prod__eta,axiom,
% 0.32/0.62 ! [F: int > int > $o,G: product_prod_int_int > $o] :
% 0.32/0.62 ( ! [X4: int,Y4: int] :
% 0.32/0.62 ( ( F @ X4 @ Y4 )
% 0.32/0.62 = ( G @ ( product_Pair_int_int @ X4 @ Y4 ) ) )
% 0.32/0.62 => ( ( produc4947309494688390418_int_o @ F )
% 0.32/0.62 = G ) ) ).
% 0.32/0.62
% 0.32/0.62 % cond_case_prod_eta
% 0.32/0.62 thf(fact_5635_cond__case__prod__eta,axiom,
% 0.32/0.62 ! [F: int > int > int,G: product_prod_int_int > int] :
% 0.32/0.62 ( ! [X4: int,Y4: int] :
% 0.32/0.62 ( ( F @ X4 @ Y4 )
% 0.32/0.62 = ( G @ ( product_Pair_int_int @ X4 @ Y4 ) ) )
% 0.32/0.62 => ( ( produc8211389475949308722nt_int @ F )
% 0.32/0.62 = G ) ) ).
% 0.32/0.62
% 0.32/0.62 % cond_case_prod_eta
% 0.32/0.62 thf(fact_5636_case__prod__eta,axiom,
% 0.32/0.62 ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
% 0.32/0.62 ( ( produc27273713700761075at_nat
% 0.32/0.62 @ ^ [X2: nat,Y2: nat] : ( F @ ( product_Pair_nat_nat @ X2 @ Y2 ) ) )
% 0.32/0.62 = F ) ).
% 0.32/0.62
% 0.32/0.62 % case_prod_eta
% 0.32/0.62 thf(fact_5637_case__prod__eta,axiom,
% 0.32/0.62 ! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 0.32/0.62 ( ( produc8739625826339149834_nat_o
% 0.32/0.62 @ ^ [X2: nat,Y2: nat] : ( F @ ( product_Pair_nat_nat @ X2 @ Y2 ) ) )
% 0.32/0.62 = F ) ).
% 0.32/0.62
% 0.32/0.62 % case_prod_eta
% 0.32/0.62 thf(fact_5638_case__prod__eta,axiom,
% 0.32/0.62 ! [F: product_prod_int_int > product_prod_int_int] :
% 0.32/0.62 ( ( produc4245557441103728435nt_int
% 0.32/0.62 @ ^ [X2: int,Y2: int] : ( F @ ( product_Pair_int_int @ X2 @ Y2 ) ) )
% 0.32/0.62 = F ) ).
% 0.32/0.62
% 0.32/0.62 % case_prod_eta
% 0.32/0.62 thf(fact_5639_case__prod__eta,axiom,
% 0.32/0.62 ! [F: product_prod_int_int > $o] :
% 0.32/0.62 ( ( produc4947309494688390418_int_o
% 0.32/0.62 @ ^ [X2: int,Y2: int] : ( F @ ( product_Pair_int_int @ X2 @ Y2 ) ) )
% 0.32/0.62 = F ) ).
% 0.32/0.62
% 0.32/0.62 % case_prod_eta
% 0.32/0.62 thf(fact_5640_case__prod__eta,axiom,
% 0.32/0.62 ! [F: product_prod_int_int > int] :
% 0.32/0.62 ( ( produc8211389475949308722nt_int
% 0.32/0.62 @ ^ [X2: int,Y2: int] : ( F @ ( product_Pair_int_int @ X2 @ Y2 ) ) )
% 0.32/0.62 = F ) ).
% 0.32/0.62
% 0.32/0.62 % case_prod_eta
% 0.32/0.62 thf(fact_5641_case__prodE2,axiom,
% 0.32/0.62 ! [Q: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,P: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Z: product_prod_nat_nat] :
% 0.32/0.62 ( ( Q @ ( produc27273713700761075at_nat @ P @ Z ) )
% 0.32/0.62 => ~ ! [X4: nat,Y4: nat] :
% 0.32/0.62 ( ( Z
% 0.32/0.62 = ( product_Pair_nat_nat @ X4 @ Y4 ) )
% 0.32/0.62 => ~ ( Q @ ( P @ X4 @ Y4 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % case_prodE2
% 0.32/0.62 thf(fact_5642_case__prodE2,axiom,
% 0.32/0.62 ! [Q: ( product_prod_nat_nat > $o ) > $o,P: nat > nat > product_prod_nat_nat > $o,Z: product_prod_nat_nat] :
% 0.32/0.62 ( ( Q @ ( produc8739625826339149834_nat_o @ P @ Z ) )
% 0.32/0.62 => ~ ! [X4: nat,Y4: nat] :
% 0.32/0.62 ( ( Z
% 0.32/0.62 = ( product_Pair_nat_nat @ X4 @ Y4 ) )
% 0.32/0.62 => ~ ( Q @ ( P @ X4 @ Y4 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % case_prodE2
% 0.32/0.62 thf(fact_5643_case__prodE2,axiom,
% 0.32/0.62 ! [Q: product_prod_int_int > $o,P: int > int > product_prod_int_int,Z: product_prod_int_int] :
% 0.32/0.62 ( ( Q @ ( produc4245557441103728435nt_int @ P @ Z ) )
% 0.32/0.62 => ~ ! [X4: int,Y4: int] :
% 0.32/0.62 ( ( Z
% 0.32/0.62 = ( product_Pair_int_int @ X4 @ Y4 ) )
% 0.32/0.62 => ~ ( Q @ ( P @ X4 @ Y4 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % case_prodE2
% 0.32/0.62 thf(fact_5644_case__prodE2,axiom,
% 0.32/0.62 ! [Q: $o > $o,P: int > int > $o,Z: product_prod_int_int] :
% 0.32/0.62 ( ( Q @ ( produc4947309494688390418_int_o @ P @ Z ) )
% 0.32/0.62 => ~ ! [X4: int,Y4: int] :
% 0.32/0.62 ( ( Z
% 0.32/0.62 = ( product_Pair_int_int @ X4 @ Y4 ) )
% 0.32/0.62 => ~ ( Q @ ( P @ X4 @ Y4 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % case_prodE2
% 0.32/0.62 thf(fact_5645_case__prodE2,axiom,
% 0.32/0.62 ! [Q: int > $o,P: int > int > int,Z: product_prod_int_int] :
% 0.32/0.62 ( ( Q @ ( produc8211389475949308722nt_int @ P @ Z ) )
% 0.32/0.62 => ~ ! [X4: int,Y4: int] :
% 0.32/0.62 ( ( Z
% 0.32/0.62 = ( product_Pair_int_int @ X4 @ Y4 ) )
% 0.32/0.62 => ~ ( Q @ ( P @ X4 @ Y4 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % case_prodE2
% 0.32/0.62 thf(fact_5646_sum__nonpos,axiom,
% 0.32/0.62 ! [A2: set_real,F: real > real] :
% 0.32/0.62 ( ! [X4: real] :
% 0.32/0.62 ( ( member_real @ X4 @ A2 )
% 0.32/0.62 => ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
% 0.32/0.62 => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ zero_zero_real ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonpos
% 0.32/0.62 thf(fact_5647_sum__nonpos,axiom,
% 0.32/0.62 ! [A2: set_int,F: int > real] :
% 0.32/0.62 ( ! [X4: int] :
% 0.32/0.62 ( ( member_int @ X4 @ A2 )
% 0.32/0.62 => ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
% 0.32/0.62 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ zero_zero_real ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonpos
% 0.32/0.62 thf(fact_5648_sum__nonpos,axiom,
% 0.32/0.62 ! [A2: set_complex,F: complex > real] :
% 0.32/0.62 ( ! [X4: complex] :
% 0.32/0.62 ( ( member_complex @ X4 @ A2 )
% 0.32/0.62 => ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
% 0.32/0.62 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ zero_zero_real ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonpos
% 0.32/0.62 thf(fact_5649_sum__nonpos,axiom,
% 0.32/0.62 ! [A2: set_nat,F: nat > rat] :
% 0.32/0.62 ( ! [X4: nat] :
% 0.32/0.62 ( ( member_nat @ X4 @ A2 )
% 0.32/0.62 => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
% 0.32/0.62 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonpos
% 0.32/0.62 thf(fact_5650_sum__nonpos,axiom,
% 0.32/0.62 ! [A2: set_real,F: real > rat] :
% 0.32/0.62 ( ! [X4: real] :
% 0.32/0.62 ( ( member_real @ X4 @ A2 )
% 0.32/0.62 => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
% 0.32/0.62 => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonpos
% 0.32/0.62 thf(fact_5651_sum__nonpos,axiom,
% 0.32/0.62 ! [A2: set_int,F: int > rat] :
% 0.32/0.62 ( ! [X4: int] :
% 0.32/0.62 ( ( member_int @ X4 @ A2 )
% 0.32/0.62 => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
% 0.32/0.62 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonpos
% 0.32/0.62 thf(fact_5652_sum__nonpos,axiom,
% 0.32/0.62 ! [A2: set_complex,F: complex > rat] :
% 0.32/0.62 ( ! [X4: complex] :
% 0.32/0.62 ( ( member_complex @ X4 @ A2 )
% 0.32/0.62 => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
% 0.32/0.62 => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonpos
% 0.32/0.62 thf(fact_5653_sum__nonpos,axiom,
% 0.32/0.62 ! [A2: set_real,F: real > nat] :
% 0.32/0.62 ( ! [X4: real] :
% 0.32/0.62 ( ( member_real @ X4 @ A2 )
% 0.32/0.62 => ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
% 0.32/0.62 => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonpos
% 0.32/0.62 thf(fact_5654_sum__nonpos,axiom,
% 0.32/0.62 ! [A2: set_int,F: int > nat] :
% 0.32/0.62 ( ! [X4: int] :
% 0.32/0.62 ( ( member_int @ X4 @ A2 )
% 0.32/0.62 => ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
% 0.32/0.62 => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonpos
% 0.32/0.62 thf(fact_5655_sum__nonpos,axiom,
% 0.32/0.62 ! [A2: set_complex,F: complex > nat] :
% 0.32/0.62 ( ! [X4: complex] :
% 0.32/0.62 ( ( member_complex @ X4 @ A2 )
% 0.32/0.62 => ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
% 0.32/0.62 => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonpos
% 0.32/0.62 thf(fact_5656_sum__nonneg,axiom,
% 0.32/0.62 ! [A2: set_real,F: real > real] :
% 0.32/0.62 ( ! [X4: real] :
% 0.32/0.62 ( ( member_real @ X4 @ A2 )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg
% 0.32/0.62 thf(fact_5657_sum__nonneg,axiom,
% 0.32/0.62 ! [A2: set_int,F: int > real] :
% 0.32/0.62 ( ! [X4: int] :
% 0.32/0.62 ( ( member_int @ X4 @ A2 )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg
% 0.32/0.62 thf(fact_5658_sum__nonneg,axiom,
% 0.32/0.62 ! [A2: set_complex,F: complex > real] :
% 0.32/0.62 ( ! [X4: complex] :
% 0.32/0.62 ( ( member_complex @ X4 @ A2 )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg
% 0.32/0.62 thf(fact_5659_sum__nonneg,axiom,
% 0.32/0.62 ! [A2: set_nat,F: nat > rat] :
% 0.32/0.62 ( ! [X4: nat] :
% 0.32/0.62 ( ( member_nat @ X4 @ A2 )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg
% 0.32/0.62 thf(fact_5660_sum__nonneg,axiom,
% 0.32/0.62 ! [A2: set_real,F: real > rat] :
% 0.32/0.62 ( ! [X4: real] :
% 0.32/0.62 ( ( member_real @ X4 @ A2 )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg
% 0.32/0.62 thf(fact_5661_sum__nonneg,axiom,
% 0.32/0.62 ! [A2: set_int,F: int > rat] :
% 0.32/0.62 ( ! [X4: int] :
% 0.32/0.62 ( ( member_int @ X4 @ A2 )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg
% 0.32/0.62 thf(fact_5662_sum__nonneg,axiom,
% 0.32/0.62 ! [A2: set_complex,F: complex > rat] :
% 0.32/0.62 ( ! [X4: complex] :
% 0.32/0.62 ( ( member_complex @ X4 @ A2 )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg
% 0.32/0.62 thf(fact_5663_sum__nonneg,axiom,
% 0.32/0.62 ! [A2: set_real,F: real > nat] :
% 0.32/0.62 ( ! [X4: real] :
% 0.32/0.62 ( ( member_real @ X4 @ A2 )
% 0.32/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.32/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg
% 0.32/0.62 thf(fact_5664_sum__nonneg,axiom,
% 0.32/0.62 ! [A2: set_int,F: int > nat] :
% 0.32/0.62 ( ! [X4: int] :
% 0.32/0.62 ( ( member_int @ X4 @ A2 )
% 0.32/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.32/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg
% 0.32/0.62 thf(fact_5665_sum__nonneg,axiom,
% 0.32/0.62 ! [A2: set_complex,F: complex > nat] :
% 0.32/0.62 ( ! [X4: complex] :
% 0.32/0.62 ( ( member_complex @ X4 @ A2 )
% 0.32/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.32/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg
% 0.32/0.62 thf(fact_5666_ln__add__one__self__le__self2,axiom,
% 0.32/0.62 ! [X: real] :
% 0.32/0.62 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.32/0.62 => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% 0.32/0.62
% 0.32/0.62 % ln_add_one_self_le_self2
% 0.32/0.62 thf(fact_5667_sum__mono__inv,axiom,
% 0.32/0.62 ! [F: real > rat,I6: set_real,G: real > rat,I2: real] :
% 0.32/0.62 ( ( ( groups1300246762558778688al_rat @ F @ I6 )
% 0.32/0.62 = ( groups1300246762558778688al_rat @ G @ I6 ) )
% 0.32/0.62 => ( ! [I4: real] :
% 0.32/0.62 ( ( member_real @ I4 @ I6 )
% 0.32/0.62 => ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 0.32/0.62 => ( ( member_real @ I2 @ I6 )
% 0.32/0.62 => ( ( finite_finite_real @ I6 )
% 0.32/0.62 => ( ( F @ I2 )
% 0.32/0.62 = ( G @ I2 ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono_inv
% 0.32/0.62 thf(fact_5668_sum__mono__inv,axiom,
% 0.32/0.62 ! [F: nat > rat,I6: set_nat,G: nat > rat,I2: nat] :
% 0.32/0.62 ( ( ( groups2906978787729119204at_rat @ F @ I6 )
% 0.33/0.62 = ( groups2906978787729119204at_rat @ G @ I6 ) )
% 0.33/0.62 => ( ! [I4: nat] :
% 0.33/0.62 ( ( member_nat @ I4 @ I6 )
% 0.33/0.62 => ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 0.33/0.62 => ( ( member_nat @ I2 @ I6 )
% 0.33/0.62 => ( ( finite_finite_nat @ I6 )
% 0.33/0.62 => ( ( F @ I2 )
% 0.33/0.62 = ( G @ I2 ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sum_mono_inv
% 0.33/0.62 thf(fact_5669_sum__mono__inv,axiom,
% 0.33/0.62 ! [F: int > rat,I6: set_int,G: int > rat,I2: int] :
% 0.33/0.62 ( ( ( groups3906332499630173760nt_rat @ F @ I6 )
% 0.33/0.62 = ( groups3906332499630173760nt_rat @ G @ I6 ) )
% 0.33/0.62 => ( ! [I4: int] :
% 0.33/0.62 ( ( member_int @ I4 @ I6 )
% 0.33/0.62 => ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 0.33/0.62 => ( ( member_int @ I2 @ I6 )
% 0.33/0.62 => ( ( finite_finite_int @ I6 )
% 0.33/0.62 => ( ( F @ I2 )
% 0.33/0.62 = ( G @ I2 ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sum_mono_inv
% 0.33/0.62 thf(fact_5670_sum__mono__inv,axiom,
% 0.33/0.62 ! [F: complex > rat,I6: set_complex,G: complex > rat,I2: complex] :
% 0.33/0.62 ( ( ( groups5058264527183730370ex_rat @ F @ I6 )
% 0.33/0.62 = ( groups5058264527183730370ex_rat @ G @ I6 ) )
% 0.33/0.62 => ( ! [I4: complex] :
% 0.33/0.62 ( ( member_complex @ I4 @ I6 )
% 0.33/0.62 => ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 0.33/0.62 => ( ( member_complex @ I2 @ I6 )
% 0.33/0.62 => ( ( finite3207457112153483333omplex @ I6 )
% 0.33/0.62 => ( ( F @ I2 )
% 0.33/0.62 = ( G @ I2 ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sum_mono_inv
% 0.33/0.62 thf(fact_5671_sum__mono__inv,axiom,
% 0.33/0.62 ! [F: real > nat,I6: set_real,G: real > nat,I2: real] :
% 0.33/0.62 ( ( ( groups1935376822645274424al_nat @ F @ I6 )
% 0.33/0.62 = ( groups1935376822645274424al_nat @ G @ I6 ) )
% 0.33/0.62 => ( ! [I4: real] :
% 0.33/0.62 ( ( member_real @ I4 @ I6 )
% 0.33/0.62 => ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 0.33/0.62 => ( ( member_real @ I2 @ I6 )
% 0.33/0.62 => ( ( finite_finite_real @ I6 )
% 0.33/0.62 => ( ( F @ I2 )
% 0.33/0.62 = ( G @ I2 ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sum_mono_inv
% 0.33/0.62 thf(fact_5672_sum__mono__inv,axiom,
% 0.33/0.62 ! [F: int > nat,I6: set_int,G: int > nat,I2: int] :
% 0.33/0.62 ( ( ( groups4541462559716669496nt_nat @ F @ I6 )
% 0.33/0.62 = ( groups4541462559716669496nt_nat @ G @ I6 ) )
% 0.33/0.62 => ( ! [I4: int] :
% 0.33/0.62 ( ( member_int @ I4 @ I6 )
% 0.33/0.62 => ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 0.33/0.62 => ( ( member_int @ I2 @ I6 )
% 0.33/0.62 => ( ( finite_finite_int @ I6 )
% 0.33/0.62 => ( ( F @ I2 )
% 0.33/0.62 = ( G @ I2 ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sum_mono_inv
% 0.33/0.62 thf(fact_5673_sum__mono__inv,axiom,
% 0.33/0.62 ! [F: complex > nat,I6: set_complex,G: complex > nat,I2: complex] :
% 0.33/0.62 ( ( ( groups5693394587270226106ex_nat @ F @ I6 )
% 0.33/0.62 = ( groups5693394587270226106ex_nat @ G @ I6 ) )
% 0.33/0.62 => ( ! [I4: complex] :
% 0.33/0.62 ( ( member_complex @ I4 @ I6 )
% 0.33/0.62 => ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 0.33/0.62 => ( ( member_complex @ I2 @ I6 )
% 0.33/0.62 => ( ( finite3207457112153483333omplex @ I6 )
% 0.33/0.62 => ( ( F @ I2 )
% 0.33/0.62 = ( G @ I2 ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sum_mono_inv
% 0.33/0.62 thf(fact_5674_sum__mono__inv,axiom,
% 0.33/0.62 ! [F: real > int,I6: set_real,G: real > int,I2: real] :
% 0.33/0.62 ( ( ( groups1932886352136224148al_int @ F @ I6 )
% 0.33/0.62 = ( groups1932886352136224148al_int @ G @ I6 ) )
% 0.33/0.62 => ( ! [I4: real] :
% 0.33/0.62 ( ( member_real @ I4 @ I6 )
% 0.33/0.62 => ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 0.33/0.62 => ( ( member_real @ I2 @ I6 )
% 0.33/0.62 => ( ( finite_finite_real @ I6 )
% 0.33/0.62 => ( ( F @ I2 )
% 0.33/0.62 = ( G @ I2 ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sum_mono_inv
% 0.33/0.62 thf(fact_5675_sum__mono__inv,axiom,
% 0.33/0.62 ! [F: nat > int,I6: set_nat,G: nat > int,I2: nat] :
% 0.33/0.62 ( ( ( groups3539618377306564664at_int @ F @ I6 )
% 0.33/0.62 = ( groups3539618377306564664at_int @ G @ I6 ) )
% 0.33/0.62 => ( ! [I4: nat] :
% 0.33/0.62 ( ( member_nat @ I4 @ I6 )
% 0.33/0.62 => ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 0.33/0.62 => ( ( member_nat @ I2 @ I6 )
% 0.33/0.62 => ( ( finite_finite_nat @ I6 )
% 0.33/0.62 => ( ( F @ I2 )
% 0.33/0.62 = ( G @ I2 ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sum_mono_inv
% 0.33/0.62 thf(fact_5676_sum__mono__inv,axiom,
% 0.33/0.62 ! [F: complex > int,I6: set_complex,G: complex > int,I2: complex] :
% 0.33/0.62 ( ( ( groups5690904116761175830ex_int @ F @ I6 )
% 0.33/0.62 = ( groups5690904116761175830ex_int @ G @ I6 ) )
% 0.33/0.62 => ( ! [I4: complex] :
% 0.33/0.62 ( ( member_complex @ I4 @ I6 )
% 0.33/0.62 => ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 0.33/0.62 => ( ( member_complex @ I2 @ I6 )
% 0.33/0.62 => ( ( finite3207457112153483333omplex @ I6 )
% 0.33/0.62 => ( ( F @ I2 )
% 0.33/0.62 = ( G @ I2 ) ) ) ) ) ) ).
% 0.33/0.62
% 0.33/0.62 % sum_mono_inv
% 0.33/0.62 thf(fact_5677_ln__less__self,axiom,
% 0.33/0.62 ! [X: real] :
% 0.33/0.62 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.62 => ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).
% 0.33/0.62
% 0.33/0.62 % ln_less_self
% 0.33/0.62 thf(fact_5678_neg__numeral__le__numeral,axiom,
% 0.33/0.62 ! [M: num,N2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % neg_numeral_le_numeral
% 0.33/0.62 thf(fact_5679_neg__numeral__le__numeral,axiom,
% 0.33/0.62 ! [M: num,N2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) ) ).
% 0.33/0.62
% 0.33/0.62 % neg_numeral_le_numeral
% 0.33/0.62 thf(fact_5680_neg__numeral__le__numeral,axiom,
% 0.33/0.62 ! [M: num,N2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) ) ).
% 0.33/0.63
% 0.33/0.63 % neg_numeral_le_numeral
% 0.33/0.63 thf(fact_5681_neg__numeral__le__numeral,axiom,
% 0.33/0.63 ! [M: num,N2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) ) ).
% 0.33/0.63
% 0.33/0.63 % neg_numeral_le_numeral
% 0.33/0.63 thf(fact_5682_not__numeral__le__neg__numeral,axiom,
% 0.33/0.63 ! [M: num,N2: num] :
% 0.33/0.63 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_numeral_le_neg_numeral
% 0.33/0.63 thf(fact_5683_not__numeral__le__neg__numeral,axiom,
% 0.33/0.63 ! [M: num,N2: num] :
% 0.33/0.63 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_numeral_le_neg_numeral
% 0.33/0.63 thf(fact_5684_not__numeral__le__neg__numeral,axiom,
% 0.33/0.63 ! [M: num,N2: num] :
% 0.33/0.63 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_numeral_le_neg_numeral
% 0.33/0.63 thf(fact_5685_not__numeral__le__neg__numeral,axiom,
% 0.33/0.63 ! [M: num,N2: num] :
% 0.33/0.63 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_numeral_le_neg_numeral
% 0.33/0.63 thf(fact_5686_zero__neq__neg__numeral,axiom,
% 0.33/0.63 ! [N2: num] :
% 0.33/0.63 ( zero_zero_real
% 0.33/0.63 != ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % zero_neq_neg_numeral
% 0.33/0.63 thf(fact_5687_zero__neq__neg__numeral,axiom,
% 0.33/0.63 ! [N2: num] :
% 0.33/0.63 ( zero_zero_int
% 0.33/0.63 != ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % zero_neq_neg_numeral
% 0.33/0.63 thf(fact_5688_zero__neq__neg__numeral,axiom,
% 0.33/0.63 ! [N2: num] :
% 0.33/0.63 ( zero_zero_complex
% 0.33/0.63 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % zero_neq_neg_numeral
% 0.33/0.63 thf(fact_5689_zero__neq__neg__numeral,axiom,
% 0.33/0.63 ! [N2: num] :
% 0.33/0.63 ( zero_zero_rat
% 0.33/0.63 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % zero_neq_neg_numeral
% 0.33/0.63 thf(fact_5690_zero__neq__neg__numeral,axiom,
% 0.33/0.63 ! [N2: num] :
% 0.33/0.63 ( zero_z3403309356797280102nteger
% 0.33/0.63 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % zero_neq_neg_numeral
% 0.33/0.63 thf(fact_5691_neg__numeral__less__numeral,axiom,
% 0.33/0.63 ! [M: num,N2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) ) ).
% 0.33/0.63
% 0.33/0.63 % neg_numeral_less_numeral
% 0.33/0.63 thf(fact_5692_neg__numeral__less__numeral,axiom,
% 0.33/0.63 ! [M: num,N2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) ) ).
% 0.33/0.63
% 0.33/0.63 % neg_numeral_less_numeral
% 0.33/0.63 thf(fact_5693_neg__numeral__less__numeral,axiom,
% 0.33/0.63 ! [M: num,N2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) ) ).
% 0.33/0.63
% 0.33/0.63 % neg_numeral_less_numeral
% 0.33/0.63 thf(fact_5694_neg__numeral__less__numeral,axiom,
% 0.33/0.63 ! [M: num,N2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) ) ).
% 0.33/0.63
% 0.33/0.63 % neg_numeral_less_numeral
% 0.33/0.63 thf(fact_5695_not__numeral__less__neg__numeral,axiom,
% 0.33/0.63 ! [M: num,N2: num] :
% 0.33/0.63 ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_numeral_less_neg_numeral
% 0.33/0.63 thf(fact_5696_not__numeral__less__neg__numeral,axiom,
% 0.33/0.63 ! [M: num,N2: num] :
% 0.33/0.63 ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_numeral_less_neg_numeral
% 0.33/0.63 thf(fact_5697_not__numeral__less__neg__numeral,axiom,
% 0.33/0.63 ! [M: num,N2: num] :
% 0.33/0.63 ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_numeral_less_neg_numeral
% 0.33/0.63 thf(fact_5698_not__numeral__less__neg__numeral,axiom,
% 0.33/0.63 ! [M: num,N2: num] :
% 0.33/0.63 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_numeral_less_neg_numeral
% 0.33/0.63 thf(fact_5699_neg__eq__iff__add__eq__0,axiom,
% 0.33/0.63 ! [A: real,B: real] :
% 0.33/0.63 ( ( ( uminus_uminus_real @ A )
% 0.33/0.63 = B )
% 0.33/0.63 = ( ( plus_plus_real @ A @ B )
% 0.33/0.63 = zero_zero_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % neg_eq_iff_add_eq_0
% 0.33/0.63 thf(fact_5700_neg__eq__iff__add__eq__0,axiom,
% 0.33/0.63 ! [A: int,B: int] :
% 0.33/0.63 ( ( ( uminus_uminus_int @ A )
% 0.33/0.63 = B )
% 0.33/0.63 = ( ( plus_plus_int @ A @ B )
% 0.33/0.63 = zero_zero_int ) ) ).
% 0.33/0.63
% 0.33/0.63 % neg_eq_iff_add_eq_0
% 0.33/0.63 thf(fact_5701_neg__eq__iff__add__eq__0,axiom,
% 0.33/0.63 ! [A: complex,B: complex] :
% 0.33/0.63 ( ( ( uminus1482373934393186551omplex @ A )
% 0.33/0.63 = B )
% 0.33/0.63 = ( ( plus_plus_complex @ A @ B )
% 0.33/0.63 = zero_zero_complex ) ) ).
% 0.33/0.63
% 0.33/0.63 % neg_eq_iff_add_eq_0
% 0.33/0.63 thf(fact_5702_neg__eq__iff__add__eq__0,axiom,
% 0.33/0.63 ! [A: rat,B: rat] :
% 0.33/0.63 ( ( ( uminus_uminus_rat @ A )
% 0.33/0.63 = B )
% 0.33/0.63 = ( ( plus_plus_rat @ A @ B )
% 0.33/0.63 = zero_zero_rat ) ) ).
% 0.33/0.63
% 0.33/0.63 % neg_eq_iff_add_eq_0
% 0.33/0.63 thf(fact_5703_neg__eq__iff__add__eq__0,axiom,
% 0.33/0.63 ! [A: code_integer,B: code_integer] :
% 0.33/0.63 ( ( ( uminus1351360451143612070nteger @ A )
% 0.33/0.63 = B )
% 0.33/0.63 = ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.33/0.63 = zero_z3403309356797280102nteger ) ) ).
% 0.33/0.63
% 0.33/0.63 % neg_eq_iff_add_eq_0
% 0.33/0.63 thf(fact_5704_eq__neg__iff__add__eq__0,axiom,
% 0.33/0.63 ! [A: real,B: real] :
% 0.33/0.63 ( ( A
% 0.33/0.63 = ( uminus_uminus_real @ B ) )
% 0.33/0.63 = ( ( plus_plus_real @ A @ B )
% 0.33/0.63 = zero_zero_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % eq_neg_iff_add_eq_0
% 0.33/0.63 thf(fact_5705_eq__neg__iff__add__eq__0,axiom,
% 0.33/0.63 ! [A: int,B: int] :
% 0.33/0.63 ( ( A
% 0.33/0.63 = ( uminus_uminus_int @ B ) )
% 0.33/0.63 = ( ( plus_plus_int @ A @ B )
% 0.33/0.63 = zero_zero_int ) ) ).
% 0.33/0.63
% 0.33/0.63 % eq_neg_iff_add_eq_0
% 0.33/0.63 thf(fact_5706_eq__neg__iff__add__eq__0,axiom,
% 0.33/0.63 ! [A: complex,B: complex] :
% 0.33/0.63 ( ( A
% 0.33/0.63 = ( uminus1482373934393186551omplex @ B ) )
% 0.33/0.63 = ( ( plus_plus_complex @ A @ B )
% 0.33/0.63 = zero_zero_complex ) ) ).
% 0.33/0.63
% 0.33/0.63 % eq_neg_iff_add_eq_0
% 0.33/0.63 thf(fact_5707_eq__neg__iff__add__eq__0,axiom,
% 0.33/0.63 ! [A: rat,B: rat] :
% 0.33/0.63 ( ( A
% 0.33/0.63 = ( uminus_uminus_rat @ B ) )
% 0.33/0.63 = ( ( plus_plus_rat @ A @ B )
% 0.33/0.63 = zero_zero_rat ) ) ).
% 0.33/0.63
% 0.33/0.63 % eq_neg_iff_add_eq_0
% 0.33/0.63 thf(fact_5708_eq__neg__iff__add__eq__0,axiom,
% 0.33/0.63 ! [A: code_integer,B: code_integer] :
% 0.33/0.63 ( ( A
% 0.33/0.63 = ( uminus1351360451143612070nteger @ B ) )
% 0.33/0.63 = ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.33/0.63 = zero_z3403309356797280102nteger ) ) ).
% 0.33/0.63
% 0.33/0.63 % eq_neg_iff_add_eq_0
% 0.33/0.63 thf(fact_5709_add_Oinverse__unique,axiom,
% 0.33/0.63 ! [A: real,B: real] :
% 0.33/0.63 ( ( ( plus_plus_real @ A @ B )
% 0.33/0.63 = zero_zero_real )
% 0.33/0.63 => ( ( uminus_uminus_real @ A )
% 0.33/0.63 = B ) ) ).
% 0.33/0.63
% 0.33/0.63 % add.inverse_unique
% 0.33/0.63 thf(fact_5710_add_Oinverse__unique,axiom,
% 0.33/0.63 ! [A: int,B: int] :
% 0.33/0.63 ( ( ( plus_plus_int @ A @ B )
% 0.33/0.63 = zero_zero_int )
% 0.33/0.63 => ( ( uminus_uminus_int @ A )
% 0.33/0.63 = B ) ) ).
% 0.33/0.63
% 0.33/0.63 % add.inverse_unique
% 0.33/0.63 thf(fact_5711_add_Oinverse__unique,axiom,
% 0.33/0.63 ! [A: complex,B: complex] :
% 0.33/0.63 ( ( ( plus_plus_complex @ A @ B )
% 0.33/0.63 = zero_zero_complex )
% 0.33/0.63 => ( ( uminus1482373934393186551omplex @ A )
% 0.33/0.63 = B ) ) ).
% 0.33/0.63
% 0.33/0.63 % add.inverse_unique
% 0.33/0.63 thf(fact_5712_add_Oinverse__unique,axiom,
% 0.33/0.63 ! [A: rat,B: rat] :
% 0.33/0.63 ( ( ( plus_plus_rat @ A @ B )
% 0.33/0.63 = zero_zero_rat )
% 0.33/0.63 => ( ( uminus_uminus_rat @ A )
% 0.33/0.63 = B ) ) ).
% 0.33/0.63
% 0.33/0.63 % add.inverse_unique
% 0.33/0.63 thf(fact_5713_add_Oinverse__unique,axiom,
% 0.33/0.63 ! [A: code_integer,B: code_integer] :
% 0.33/0.63 ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.33/0.63 = zero_z3403309356797280102nteger )
% 0.33/0.63 => ( ( uminus1351360451143612070nteger @ A )
% 0.33/0.63 = B ) ) ).
% 0.33/0.63
% 0.33/0.63 % add.inverse_unique
% 0.33/0.63 thf(fact_5714_ab__group__add__class_Oab__left__minus,axiom,
% 0.33/0.63 ! [A: real] :
% 0.33/0.63 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 0.33/0.63 = zero_zero_real ) ).
% 0.33/0.63
% 0.33/0.63 % ab_group_add_class.ab_left_minus
% 0.33/0.63 thf(fact_5715_ab__group__add__class_Oab__left__minus,axiom,
% 0.33/0.63 ! [A: int] :
% 0.33/0.63 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 0.33/0.63 = zero_zero_int ) ).
% 0.33/0.63
% 0.33/0.63 % ab_group_add_class.ab_left_minus
% 0.33/0.63 thf(fact_5716_ab__group__add__class_Oab__left__minus,axiom,
% 0.33/0.63 ! [A: complex] :
% 0.33/0.63 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 0.33/0.63 = zero_zero_complex ) ).
% 0.33/0.63
% 0.33/0.63 % ab_group_add_class.ab_left_minus
% 0.33/0.63 thf(fact_5717_ab__group__add__class_Oab__left__minus,axiom,
% 0.33/0.63 ! [A: rat] :
% 0.33/0.63 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.33/0.63 = zero_zero_rat ) ).
% 0.33/0.63
% 0.33/0.63 % ab_group_add_class.ab_left_minus
% 0.33/0.63 thf(fact_5718_ab__group__add__class_Oab__left__minus,axiom,
% 0.33/0.63 ! [A: code_integer] :
% 0.33/0.63 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.33/0.63 = zero_z3403309356797280102nteger ) ).
% 0.33/0.63
% 0.33/0.63 % ab_group_add_class.ab_left_minus
% 0.33/0.63 thf(fact_5719_add__eq__0__iff,axiom,
% 0.33/0.63 ! [A: real,B: real] :
% 0.33/0.63 ( ( ( plus_plus_real @ A @ B )
% 0.33/0.63 = zero_zero_real )
% 0.33/0.63 = ( B
% 0.33/0.63 = ( uminus_uminus_real @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % add_eq_0_iff
% 0.33/0.63 thf(fact_5720_add__eq__0__iff,axiom,
% 0.33/0.63 ! [A: int,B: int] :
% 0.33/0.63 ( ( ( plus_plus_int @ A @ B )
% 0.33/0.63 = zero_zero_int )
% 0.33/0.63 = ( B
% 0.33/0.63 = ( uminus_uminus_int @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % add_eq_0_iff
% 0.33/0.63 thf(fact_5721_add__eq__0__iff,axiom,
% 0.33/0.63 ! [A: complex,B: complex] :
% 0.33/0.63 ( ( ( plus_plus_complex @ A @ B )
% 0.33/0.63 = zero_zero_complex )
% 0.33/0.63 = ( B
% 0.33/0.63 = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % add_eq_0_iff
% 0.33/0.63 thf(fact_5722_add__eq__0__iff,axiom,
% 0.33/0.63 ! [A: rat,B: rat] :
% 0.33/0.63 ( ( ( plus_plus_rat @ A @ B )
% 0.33/0.63 = zero_zero_rat )
% 0.33/0.63 = ( B
% 0.33/0.63 = ( uminus_uminus_rat @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % add_eq_0_iff
% 0.33/0.63 thf(fact_5723_add__eq__0__iff,axiom,
% 0.33/0.63 ! [A: code_integer,B: code_integer] :
% 0.33/0.63 ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.33/0.63 = zero_z3403309356797280102nteger )
% 0.33/0.63 = ( B
% 0.33/0.63 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % add_eq_0_iff
% 0.33/0.63 thf(fact_5724_le__minus__one__simps_I2_J,axiom,
% 0.33/0.63 ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 0.33/0.63
% 0.33/0.63 % le_minus_one_simps(2)
% 0.33/0.63 thf(fact_5725_le__minus__one__simps_I2_J,axiom,
% 0.33/0.63 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 0.33/0.63
% 0.33/0.63 % le_minus_one_simps(2)
% 0.33/0.63 thf(fact_5726_le__minus__one__simps_I2_J,axiom,
% 0.33/0.63 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 0.33/0.63
% 0.33/0.63 % le_minus_one_simps(2)
% 0.33/0.63 thf(fact_5727_le__minus__one__simps_I2_J,axiom,
% 0.33/0.63 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 0.33/0.63
% 0.33/0.63 % le_minus_one_simps(2)
% 0.33/0.63 thf(fact_5728_le__minus__one__simps_I4_J,axiom,
% 0.33/0.63 ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % le_minus_one_simps(4)
% 0.33/0.63 thf(fact_5729_le__minus__one__simps_I4_J,axiom,
% 0.33/0.63 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.63
% 0.33/0.63 % le_minus_one_simps(4)
% 0.33/0.63 thf(fact_5730_le__minus__one__simps_I4_J,axiom,
% 0.33/0.63 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.63
% 0.33/0.63 % le_minus_one_simps(4)
% 0.33/0.63 thf(fact_5731_le__minus__one__simps_I4_J,axiom,
% 0.33/0.63 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.63
% 0.33/0.63 % le_minus_one_simps(4)
% 0.33/0.63 thf(fact_5732_zero__neq__neg__one,axiom,
% 0.33/0.63 ( zero_zero_real
% 0.33/0.63 != ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % zero_neq_neg_one
% 0.33/0.63 thf(fact_5733_zero__neq__neg__one,axiom,
% 0.33/0.63 ( zero_zero_int
% 0.33/0.63 != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.63
% 0.33/0.63 % zero_neq_neg_one
% 0.33/0.63 thf(fact_5734_zero__neq__neg__one,axiom,
% 0.33/0.63 ( zero_zero_complex
% 0.33/0.63 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.33/0.63
% 0.33/0.63 % zero_neq_neg_one
% 0.33/0.63 thf(fact_5735_zero__neq__neg__one,axiom,
% 0.33/0.63 ( zero_zero_rat
% 0.33/0.63 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.63
% 0.33/0.63 % zero_neq_neg_one
% 0.33/0.63 thf(fact_5736_zero__neq__neg__one,axiom,
% 0.33/0.63 ( zero_z3403309356797280102nteger
% 0.33/0.63 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.63
% 0.33/0.63 % zero_neq_neg_one
% 0.33/0.63 thf(fact_5737_numeral__times__minus__swap,axiom,
% 0.33/0.63 ! [W: num,X: real] :
% 0.33/0.63 ( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X ) )
% 0.33/0.63 = ( times_times_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % numeral_times_minus_swap
% 0.33/0.63 thf(fact_5738_numeral__times__minus__swap,axiom,
% 0.33/0.63 ! [W: num,X: int] :
% 0.33/0.63 ( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X ) )
% 0.33/0.63 = ( times_times_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % numeral_times_minus_swap
% 0.33/0.63 thf(fact_5739_numeral__times__minus__swap,axiom,
% 0.33/0.63 ! [W: num,X: complex] :
% 0.33/0.63 ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ ( uminus1482373934393186551omplex @ X ) )
% 0.33/0.63 = ( times_times_complex @ X @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % numeral_times_minus_swap
% 0.33/0.63 thf(fact_5740_numeral__times__minus__swap,axiom,
% 0.33/0.63 ! [W: num,X: rat] :
% 0.33/0.63 ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X ) )
% 0.33/0.63 = ( times_times_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % numeral_times_minus_swap
% 0.33/0.63 thf(fact_5741_numeral__times__minus__swap,axiom,
% 0.33/0.63 ! [W: num,X: code_integer] :
% 0.33/0.63 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X ) )
% 0.33/0.63 = ( times_3573771949741848930nteger @ X @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % numeral_times_minus_swap
% 0.33/0.63 thf(fact_5742_less__minus__one__simps_I4_J,axiom,
% 0.33/0.63 ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % less_minus_one_simps(4)
% 0.33/0.63 thf(fact_5743_less__minus__one__simps_I4_J,axiom,
% 0.33/0.63 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.63
% 0.33/0.63 % less_minus_one_simps(4)
% 0.33/0.63 thf(fact_5744_less__minus__one__simps_I4_J,axiom,
% 0.33/0.63 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.63
% 0.33/0.63 % less_minus_one_simps(4)
% 0.33/0.63 thf(fact_5745_less__minus__one__simps_I4_J,axiom,
% 0.33/0.63 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.63
% 0.33/0.63 % less_minus_one_simps(4)
% 0.33/0.63 thf(fact_5746_less__minus__one__simps_I2_J,axiom,
% 0.33/0.63 ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 0.33/0.63
% 0.33/0.63 % less_minus_one_simps(2)
% 0.33/0.63 thf(fact_5747_less__minus__one__simps_I2_J,axiom,
% 0.33/0.63 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 0.33/0.63
% 0.33/0.63 % less_minus_one_simps(2)
% 0.33/0.63 thf(fact_5748_less__minus__one__simps_I2_J,axiom,
% 0.33/0.63 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 0.33/0.63
% 0.33/0.63 % less_minus_one_simps(2)
% 0.33/0.63 thf(fact_5749_less__minus__one__simps_I2_J,axiom,
% 0.33/0.63 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 0.33/0.63
% 0.33/0.63 % less_minus_one_simps(2)
% 0.33/0.63 thf(fact_5750_nonzero__minus__divide__right,axiom,
% 0.33/0.63 ! [B: real,A: real] :
% 0.33/0.63 ( ( B != zero_zero_real )
% 0.33/0.63 => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.33/0.63 = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_minus_divide_right
% 0.33/0.63 thf(fact_5751_nonzero__minus__divide__right,axiom,
% 0.33/0.63 ! [B: complex,A: complex] :
% 0.33/0.63 ( ( B != zero_zero_complex )
% 0.33/0.63 => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.33/0.63 = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_minus_divide_right
% 0.33/0.63 thf(fact_5752_nonzero__minus__divide__right,axiom,
% 0.33/0.63 ! [B: rat,A: rat] :
% 0.33/0.63 ( ( B != zero_zero_rat )
% 0.33/0.63 => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.33/0.63 = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_minus_divide_right
% 0.33/0.63 thf(fact_5753_nonzero__minus__divide__divide,axiom,
% 0.33/0.63 ! [B: real,A: real] :
% 0.33/0.63 ( ( B != zero_zero_real )
% 0.33/0.63 => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 0.33/0.63 = ( divide_divide_real @ A @ B ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_minus_divide_divide
% 0.33/0.63 thf(fact_5754_nonzero__minus__divide__divide,axiom,
% 0.33/0.63 ! [B: complex,A: complex] :
% 0.33/0.63 ( ( B != zero_zero_complex )
% 0.33/0.63 => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 0.33/0.63 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_minus_divide_divide
% 0.33/0.63 thf(fact_5755_nonzero__minus__divide__divide,axiom,
% 0.33/0.63 ! [B: rat,A: rat] :
% 0.33/0.63 ( ( B != zero_zero_rat )
% 0.33/0.63 => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 0.33/0.63 = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_minus_divide_divide
% 0.33/0.63 thf(fact_5756_numeral__neq__neg__one,axiom,
% 0.33/0.63 ! [N2: num] :
% 0.33/0.63 ( ( numeral_numeral_real @ N2 )
% 0.33/0.63 != ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % numeral_neq_neg_one
% 0.33/0.63 thf(fact_5757_numeral__neq__neg__one,axiom,
% 0.33/0.63 ! [N2: num] :
% 0.33/0.63 ( ( numeral_numeral_int @ N2 )
% 0.33/0.63 != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.63
% 0.33/0.63 % numeral_neq_neg_one
% 0.33/0.63 thf(fact_5758_numeral__neq__neg__one,axiom,
% 0.33/0.63 ! [N2: num] :
% 0.33/0.63 ( ( numera6690914467698888265omplex @ N2 )
% 0.33/0.63 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.33/0.63
% 0.33/0.63 % numeral_neq_neg_one
% 0.33/0.63 thf(fact_5759_numeral__neq__neg__one,axiom,
% 0.33/0.63 ! [N2: num] :
% 0.33/0.63 ( ( numeral_numeral_rat @ N2 )
% 0.33/0.63 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.63
% 0.33/0.63 % numeral_neq_neg_one
% 0.33/0.63 thf(fact_5760_numeral__neq__neg__one,axiom,
% 0.33/0.63 ! [N2: num] :
% 0.33/0.63 ( ( numera6620942414471956472nteger @ N2 )
% 0.33/0.63 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.63
% 0.33/0.63 % numeral_neq_neg_one
% 0.33/0.63 thf(fact_5761_one__neq__neg__numeral,axiom,
% 0.33/0.63 ! [N2: num] :
% 0.33/0.63 ( one_one_real
% 0.33/0.63 != ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % one_neq_neg_numeral
% 0.33/0.63 thf(fact_5762_one__neq__neg__numeral,axiom,
% 0.33/0.63 ! [N2: num] :
% 0.33/0.63 ( one_one_int
% 0.33/0.63 != ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % one_neq_neg_numeral
% 0.33/0.63 thf(fact_5763_one__neq__neg__numeral,axiom,
% 0.33/0.63 ! [N2: num] :
% 0.33/0.63 ( one_one_complex
% 0.33/0.63 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % one_neq_neg_numeral
% 0.33/0.63 thf(fact_5764_one__neq__neg__numeral,axiom,
% 0.33/0.63 ! [N2: num] :
% 0.33/0.63 ( one_one_rat
% 0.33/0.63 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % one_neq_neg_numeral
% 0.33/0.63 thf(fact_5765_one__neq__neg__numeral,axiom,
% 0.33/0.63 ! [N2: num] :
% 0.33/0.63 ( one_one_Code_integer
% 0.33/0.63 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % one_neq_neg_numeral
% 0.33/0.63 thf(fact_5766_square__eq__1__iff,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ( times_times_real @ X @ X )
% 0.33/0.63 = one_one_real )
% 0.33/0.63 = ( ( X = one_one_real )
% 0.33/0.63 | ( X
% 0.33/0.63 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % square_eq_1_iff
% 0.33/0.63 thf(fact_5767_square__eq__1__iff,axiom,
% 0.33/0.63 ! [X: int] :
% 0.33/0.63 ( ( ( times_times_int @ X @ X )
% 0.33/0.63 = one_one_int )
% 0.33/0.63 = ( ( X = one_one_int )
% 0.33/0.63 | ( X
% 0.33/0.63 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % square_eq_1_iff
% 0.33/0.63 thf(fact_5768_square__eq__1__iff,axiom,
% 0.33/0.63 ! [X: complex] :
% 0.33/0.63 ( ( ( times_times_complex @ X @ X )
% 0.33/0.63 = one_one_complex )
% 0.33/0.63 = ( ( X = one_one_complex )
% 0.33/0.63 | ( X
% 0.33/0.63 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % square_eq_1_iff
% 0.33/0.63 thf(fact_5769_square__eq__1__iff,axiom,
% 0.33/0.63 ! [X: rat] :
% 0.33/0.63 ( ( ( times_times_rat @ X @ X )
% 0.33/0.63 = one_one_rat )
% 0.33/0.63 = ( ( X = one_one_rat )
% 0.33/0.63 | ( X
% 0.33/0.63 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % square_eq_1_iff
% 0.33/0.63 thf(fact_5770_square__eq__1__iff,axiom,
% 0.33/0.63 ! [X: code_integer] :
% 0.33/0.63 ( ( ( times_3573771949741848930nteger @ X @ X )
% 0.33/0.63 = one_one_Code_integer )
% 0.33/0.63 = ( ( X = one_one_Code_integer )
% 0.33/0.63 | ( X
% 0.33/0.63 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % square_eq_1_iff
% 0.33/0.63 thf(fact_5771_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.33/0.63 ( minus_minus_real
% 0.33/0.63 = ( ^ [A5: real,B5: real] : ( plus_plus_real @ A5 @ ( uminus_uminus_real @ B5 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.33/0.63 thf(fact_5772_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.33/0.63 ( minus_minus_int
% 0.33/0.63 = ( ^ [A5: int,B5: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B5 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.33/0.63 thf(fact_5773_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.33/0.63 ( minus_minus_complex
% 0.33/0.63 = ( ^ [A5: complex,B5: complex] : ( plus_plus_complex @ A5 @ ( uminus1482373934393186551omplex @ B5 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.33/0.63 thf(fact_5774_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.33/0.63 ( minus_minus_rat
% 0.33/0.63 = ( ^ [A5: rat,B5: rat] : ( plus_plus_rat @ A5 @ ( uminus_uminus_rat @ B5 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.33/0.63 thf(fact_5775_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.33/0.63 ( minus_8373710615458151222nteger
% 0.33/0.63 = ( ^ [A5: code_integer,B5: code_integer] : ( plus_p5714425477246183910nteger @ A5 @ ( uminus1351360451143612070nteger @ B5 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.33/0.63 thf(fact_5776_diff__conv__add__uminus,axiom,
% 0.33/0.63 ( minus_minus_real
% 0.33/0.63 = ( ^ [A5: real,B5: real] : ( plus_plus_real @ A5 @ ( uminus_uminus_real @ B5 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % diff_conv_add_uminus
% 0.33/0.63 thf(fact_5777_diff__conv__add__uminus,axiom,
% 0.33/0.63 ( minus_minus_int
% 0.33/0.63 = ( ^ [A5: int,B5: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B5 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % diff_conv_add_uminus
% 0.33/0.63 thf(fact_5778_diff__conv__add__uminus,axiom,
% 0.33/0.63 ( minus_minus_complex
% 0.33/0.63 = ( ^ [A5: complex,B5: complex] : ( plus_plus_complex @ A5 @ ( uminus1482373934393186551omplex @ B5 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % diff_conv_add_uminus
% 0.33/0.63 thf(fact_5779_diff__conv__add__uminus,axiom,
% 0.33/0.63 ( minus_minus_rat
% 0.33/0.63 = ( ^ [A5: rat,B5: rat] : ( plus_plus_rat @ A5 @ ( uminus_uminus_rat @ B5 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % diff_conv_add_uminus
% 0.33/0.63 thf(fact_5780_diff__conv__add__uminus,axiom,
% 0.33/0.63 ( minus_8373710615458151222nteger
% 0.33/0.63 = ( ^ [A5: code_integer,B5: code_integer] : ( plus_p5714425477246183910nteger @ A5 @ ( uminus1351360451143612070nteger @ B5 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % diff_conv_add_uminus
% 0.33/0.63 thf(fact_5781_group__cancel_Osub2,axiom,
% 0.33/0.63 ! [B3: real,K: real,B: real,A: real] :
% 0.33/0.63 ( ( B3
% 0.33/0.63 = ( plus_plus_real @ K @ B ) )
% 0.33/0.63 => ( ( minus_minus_real @ A @ B3 )
% 0.33/0.63 = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % group_cancel.sub2
% 0.33/0.63 thf(fact_5782_group__cancel_Osub2,axiom,
% 0.33/0.63 ! [B3: int,K: int,B: int,A: int] :
% 0.33/0.63 ( ( B3
% 0.33/0.63 = ( plus_plus_int @ K @ B ) )
% 0.33/0.63 => ( ( minus_minus_int @ A @ B3 )
% 0.33/0.63 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % group_cancel.sub2
% 0.33/0.63 thf(fact_5783_group__cancel_Osub2,axiom,
% 0.33/0.63 ! [B3: complex,K: complex,B: complex,A: complex] :
% 0.33/0.63 ( ( B3
% 0.33/0.63 = ( plus_plus_complex @ K @ B ) )
% 0.33/0.63 => ( ( minus_minus_complex @ A @ B3 )
% 0.33/0.63 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % group_cancel.sub2
% 0.33/0.63 thf(fact_5784_group__cancel_Osub2,axiom,
% 0.33/0.63 ! [B3: rat,K: rat,B: rat,A: rat] :
% 0.33/0.63 ( ( B3
% 0.33/0.63 = ( plus_plus_rat @ K @ B ) )
% 0.33/0.63 => ( ( minus_minus_rat @ A @ B3 )
% 0.33/0.63 = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % group_cancel.sub2
% 0.33/0.63 thf(fact_5785_group__cancel_Osub2,axiom,
% 0.33/0.63 ! [B3: code_integer,K: code_integer,B: code_integer,A: code_integer] :
% 0.33/0.63 ( ( B3
% 0.33/0.63 = ( plus_p5714425477246183910nteger @ K @ B ) )
% 0.33/0.63 => ( ( minus_8373710615458151222nteger @ A @ B3 )
% 0.33/0.63 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % group_cancel.sub2
% 0.33/0.63 thf(fact_5786_dvd__div__neg,axiom,
% 0.33/0.63 ! [B: real,A: real] :
% 0.33/0.63 ( ( dvd_dvd_real @ B @ A )
% 0.33/0.63 => ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) )
% 0.33/0.63 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % dvd_div_neg
% 0.33/0.63 thf(fact_5787_dvd__div__neg,axiom,
% 0.33/0.63 ! [B: int,A: int] :
% 0.33/0.63 ( ( dvd_dvd_int @ B @ A )
% 0.33/0.63 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 0.33/0.63 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % dvd_div_neg
% 0.33/0.63 thf(fact_5788_dvd__div__neg,axiom,
% 0.33/0.63 ! [B: complex,A: complex] :
% 0.33/0.63 ( ( dvd_dvd_complex @ B @ A )
% 0.33/0.63 => ( ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 0.33/0.63 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % dvd_div_neg
% 0.33/0.63 thf(fact_5789_dvd__div__neg,axiom,
% 0.33/0.63 ! [B: rat,A: rat] :
% 0.33/0.63 ( ( dvd_dvd_rat @ B @ A )
% 0.33/0.63 => ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.33/0.63 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % dvd_div_neg
% 0.33/0.63 thf(fact_5790_dvd__div__neg,axiom,
% 0.33/0.63 ! [B: code_integer,A: code_integer] :
% 0.33/0.63 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.33/0.63 => ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.33/0.63 = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % dvd_div_neg
% 0.33/0.63 thf(fact_5791_dvd__neg__div,axiom,
% 0.33/0.63 ! [B: real,A: real] :
% 0.33/0.63 ( ( dvd_dvd_real @ B @ A )
% 0.33/0.63 => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B )
% 0.33/0.63 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % dvd_neg_div
% 0.33/0.63 thf(fact_5792_dvd__neg__div,axiom,
% 0.33/0.63 ! [B: int,A: int] :
% 0.33/0.63 ( ( dvd_dvd_int @ B @ A )
% 0.33/0.63 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 0.33/0.63 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % dvd_neg_div
% 0.33/0.63 thf(fact_5793_dvd__neg__div,axiom,
% 0.33/0.63 ! [B: complex,A: complex] :
% 0.33/0.63 ( ( dvd_dvd_complex @ B @ A )
% 0.33/0.63 => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 0.33/0.63 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % dvd_neg_div
% 0.33/0.63 thf(fact_5794_dvd__neg__div,axiom,
% 0.33/0.63 ! [B: rat,A: rat] :
% 0.33/0.63 ( ( dvd_dvd_rat @ B @ A )
% 0.33/0.63 => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.33/0.63 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % dvd_neg_div
% 0.33/0.63 thf(fact_5795_dvd__neg__div,axiom,
% 0.33/0.63 ! [B: code_integer,A: code_integer] :
% 0.33/0.63 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.33/0.63 => ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.33/0.63 = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % dvd_neg_div
% 0.33/0.63 thf(fact_5796_subset__Compl__self__eq,axiom,
% 0.33/0.63 ! [A2: set_int] :
% 0.33/0.63 ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ A2 ) )
% 0.33/0.63 = ( A2 = bot_bot_set_int ) ) ).
% 0.33/0.63
% 0.33/0.63 % subset_Compl_self_eq
% 0.33/0.63 thf(fact_5797_subset__Compl__self__eq,axiom,
% 0.33/0.63 ! [A2: set_real] :
% 0.33/0.63 ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ A2 ) )
% 0.33/0.63 = ( A2 = bot_bot_set_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % subset_Compl_self_eq
% 0.33/0.63 thf(fact_5798_subset__Compl__self__eq,axiom,
% 0.33/0.63 ! [A2: set_nat] :
% 0.33/0.63 ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
% 0.33/0.63 = ( A2 = bot_bot_set_nat ) ) ).
% 0.33/0.63
% 0.33/0.63 % subset_Compl_self_eq
% 0.33/0.63 thf(fact_5799_real__minus__mult__self__le,axiom,
% 0.33/0.63 ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_minus_mult_self_le
% 0.33/0.63 thf(fact_5800_zmult__eq__1__iff,axiom,
% 0.33/0.63 ! [M: int,N2: int] :
% 0.33/0.63 ( ( ( times_times_int @ M @ N2 )
% 0.33/0.63 = one_one_int )
% 0.33/0.63 = ( ( ( M = one_one_int )
% 0.33/0.63 & ( N2 = one_one_int ) )
% 0.33/0.63 | ( ( M
% 0.33/0.63 = ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.63 & ( N2
% 0.33/0.63 = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % zmult_eq_1_iff
% 0.33/0.63 thf(fact_5801_pos__zmult__eq__1__iff__lemma,axiom,
% 0.33/0.63 ! [M: int,N2: int] :
% 0.33/0.63 ( ( ( times_times_int @ M @ N2 )
% 0.33/0.63 = one_one_int )
% 0.33/0.63 => ( ( M = one_one_int )
% 0.33/0.63 | ( M
% 0.33/0.63 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % pos_zmult_eq_1_iff_lemma
% 0.33/0.63 thf(fact_5802_minus__int__code_I2_J,axiom,
% 0.33/0.63 ! [L2: int] :
% 0.33/0.63 ( ( minus_minus_int @ zero_zero_int @ L2 )
% 0.33/0.63 = ( uminus_uminus_int @ L2 ) ) ).
% 0.33/0.63
% 0.33/0.63 % minus_int_code(2)
% 0.33/0.63 thf(fact_5803_sum_Ointer__filter,axiom,
% 0.33/0.63 ! [A2: set_real,G: real > complex,P: real > $o] :
% 0.33/0.63 ( ( finite_finite_real @ A2 )
% 0.33/0.63 => ( ( groups5754745047067104278omplex @ G
% 0.33/0.63 @ ( collect_real
% 0.33/0.63 @ ^ [X2: real] :
% 0.33/0.63 ( ( member_real @ X2 @ A2 )
% 0.33/0.63 & ( P @ X2 ) ) ) )
% 0.33/0.63 = ( groups5754745047067104278omplex
% 0.33/0.63 @ ^ [X2: real] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_complex )
% 0.33/0.63 @ A2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.inter_filter
% 0.33/0.63 thf(fact_5804_sum_Ointer__filter,axiom,
% 0.33/0.63 ! [A2: set_nat,G: nat > complex,P: nat > $o] :
% 0.33/0.63 ( ( finite_finite_nat @ A2 )
% 0.33/0.63 => ( ( groups2073611262835488442omplex @ G
% 0.33/0.63 @ ( collect_nat
% 0.33/0.63 @ ^ [X2: nat] :
% 0.33/0.63 ( ( member_nat @ X2 @ A2 )
% 0.33/0.63 & ( P @ X2 ) ) ) )
% 0.33/0.63 = ( groups2073611262835488442omplex
% 0.33/0.63 @ ^ [X2: nat] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_complex )
% 0.33/0.63 @ A2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.inter_filter
% 0.33/0.63 thf(fact_5805_sum_Ointer__filter,axiom,
% 0.33/0.63 ! [A2: set_int,G: int > complex,P: int > $o] :
% 0.33/0.63 ( ( finite_finite_int @ A2 )
% 0.33/0.63 => ( ( groups3049146728041665814omplex @ G
% 0.33/0.63 @ ( collect_int
% 0.33/0.63 @ ^ [X2: int] :
% 0.33/0.63 ( ( member_int @ X2 @ A2 )
% 0.33/0.63 & ( P @ X2 ) ) ) )
% 0.33/0.63 = ( groups3049146728041665814omplex
% 0.33/0.63 @ ^ [X2: int] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_complex )
% 0.33/0.63 @ A2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.inter_filter
% 0.33/0.63 thf(fact_5806_sum_Ointer__filter,axiom,
% 0.33/0.63 ! [A2: set_real,G: real > real,P: real > $o] :
% 0.33/0.63 ( ( finite_finite_real @ A2 )
% 0.33/0.63 => ( ( groups8097168146408367636l_real @ G
% 0.33/0.63 @ ( collect_real
% 0.33/0.63 @ ^ [X2: real] :
% 0.33/0.63 ( ( member_real @ X2 @ A2 )
% 0.33/0.63 & ( P @ X2 ) ) ) )
% 0.33/0.63 = ( groups8097168146408367636l_real
% 0.33/0.63 @ ^ [X2: real] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_real )
% 0.33/0.63 @ A2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.inter_filter
% 0.33/0.63 thf(fact_5807_sum_Ointer__filter,axiom,
% 0.33/0.63 ! [A2: set_int,G: int > real,P: int > $o] :
% 0.33/0.63 ( ( finite_finite_int @ A2 )
% 0.33/0.63 => ( ( groups8778361861064173332t_real @ G
% 0.33/0.63 @ ( collect_int
% 0.33/0.63 @ ^ [X2: int] :
% 0.33/0.63 ( ( member_int @ X2 @ A2 )
% 0.33/0.63 & ( P @ X2 ) ) ) )
% 0.33/0.63 = ( groups8778361861064173332t_real
% 0.33/0.63 @ ^ [X2: int] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_real )
% 0.33/0.63 @ A2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.inter_filter
% 0.33/0.63 thf(fact_5808_sum_Ointer__filter,axiom,
% 0.33/0.63 ! [A2: set_complex,G: complex > real,P: complex > $o] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.63 => ( ( groups5808333547571424918x_real @ G
% 0.33/0.63 @ ( collect_complex
% 0.33/0.63 @ ^ [X2: complex] :
% 0.33/0.63 ( ( member_complex @ X2 @ A2 )
% 0.33/0.63 & ( P @ X2 ) ) ) )
% 0.33/0.63 = ( groups5808333547571424918x_real
% 0.33/0.63 @ ^ [X2: complex] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_real )
% 0.33/0.63 @ A2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.inter_filter
% 0.33/0.63 thf(fact_5809_sum_Ointer__filter,axiom,
% 0.33/0.63 ! [A2: set_real,G: real > rat,P: real > $o] :
% 0.33/0.63 ( ( finite_finite_real @ A2 )
% 0.33/0.63 => ( ( groups1300246762558778688al_rat @ G
% 0.33/0.63 @ ( collect_real
% 0.33/0.63 @ ^ [X2: real] :
% 0.33/0.63 ( ( member_real @ X2 @ A2 )
% 0.33/0.63 & ( P @ X2 ) ) ) )
% 0.33/0.63 = ( groups1300246762558778688al_rat
% 0.33/0.63 @ ^ [X2: real] : ( if_rat @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_rat )
% 0.33/0.63 @ A2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.inter_filter
% 0.33/0.63 thf(fact_5810_sum_Ointer__filter,axiom,
% 0.33/0.63 ! [A2: set_nat,G: nat > rat,P: nat > $o] :
% 0.33/0.63 ( ( finite_finite_nat @ A2 )
% 0.33/0.63 => ( ( groups2906978787729119204at_rat @ G
% 0.33/0.63 @ ( collect_nat
% 0.33/0.63 @ ^ [X2: nat] :
% 0.33/0.63 ( ( member_nat @ X2 @ A2 )
% 0.33/0.63 & ( P @ X2 ) ) ) )
% 0.33/0.63 = ( groups2906978787729119204at_rat
% 0.33/0.63 @ ^ [X2: nat] : ( if_rat @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_rat )
% 0.33/0.63 @ A2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.inter_filter
% 0.33/0.63 thf(fact_5811_sum_Ointer__filter,axiom,
% 0.33/0.63 ! [A2: set_int,G: int > rat,P: int > $o] :
% 0.33/0.63 ( ( finite_finite_int @ A2 )
% 0.33/0.63 => ( ( groups3906332499630173760nt_rat @ G
% 0.33/0.63 @ ( collect_int
% 0.33/0.63 @ ^ [X2: int] :
% 0.33/0.63 ( ( member_int @ X2 @ A2 )
% 0.33/0.63 & ( P @ X2 ) ) ) )
% 0.33/0.63 = ( groups3906332499630173760nt_rat
% 0.33/0.63 @ ^ [X2: int] : ( if_rat @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_rat )
% 0.33/0.63 @ A2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.inter_filter
% 0.33/0.63 thf(fact_5812_sum_Ointer__filter,axiom,
% 0.33/0.63 ! [A2: set_complex,G: complex > rat,P: complex > $o] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.63 => ( ( groups5058264527183730370ex_rat @ G
% 0.33/0.63 @ ( collect_complex
% 0.33/0.63 @ ^ [X2: complex] :
% 0.33/0.63 ( ( member_complex @ X2 @ A2 )
% 0.33/0.63 & ( P @ X2 ) ) ) )
% 0.33/0.63 = ( groups5058264527183730370ex_rat
% 0.33/0.63 @ ^ [X2: complex] : ( if_rat @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_rat )
% 0.33/0.63 @ A2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.inter_filter
% 0.33/0.63 thf(fact_5813_minus__real__def,axiom,
% 0.33/0.63 ( minus_minus_real
% 0.33/0.63 = ( ^ [X2: real,Y2: real] : ( plus_plus_real @ X2 @ ( uminus_uminus_real @ Y2 ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % minus_real_def
% 0.33/0.63 thf(fact_5814_ln__one__minus__pos__upper__bound,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ( ord_less_real @ X @ one_one_real )
% 0.33/0.63 => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) @ ( uminus_uminus_real @ X ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ln_one_minus_pos_upper_bound
% 0.33/0.63 thf(fact_5815_sum__nonneg__eq__0__iff,axiom,
% 0.33/0.63 ! [A2: set_real,F: real > real] :
% 0.33/0.63 ( ( finite_finite_real @ A2 )
% 0.33/0.63 => ( ! [X4: real] :
% 0.33/0.63 ( ( member_real @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.33/0.63 => ( ( ( groups8097168146408367636l_real @ F @ A2 )
% 0.33/0.63 = zero_zero_real )
% 0.33/0.63 = ( ! [X2: real] :
% 0.33/0.63 ( ( member_real @ X2 @ A2 )
% 0.33/0.63 => ( ( F @ X2 )
% 0.33/0.63 = zero_zero_real ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_eq_0_iff
% 0.33/0.63 thf(fact_5816_sum__nonneg__eq__0__iff,axiom,
% 0.33/0.63 ! [A2: set_int,F: int > real] :
% 0.33/0.63 ( ( finite_finite_int @ A2 )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.33/0.63 => ( ( ( groups8778361861064173332t_real @ F @ A2 )
% 0.33/0.63 = zero_zero_real )
% 0.33/0.63 = ( ! [X2: int] :
% 0.33/0.63 ( ( member_int @ X2 @ A2 )
% 0.33/0.63 => ( ( F @ X2 )
% 0.33/0.63 = zero_zero_real ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_eq_0_iff
% 0.33/0.63 thf(fact_5817_sum__nonneg__eq__0__iff,axiom,
% 0.33/0.63 ! [A2: set_complex,F: complex > real] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.33/0.63 => ( ( ( groups5808333547571424918x_real @ F @ A2 )
% 0.33/0.63 = zero_zero_real )
% 0.33/0.63 = ( ! [X2: complex] :
% 0.33/0.63 ( ( member_complex @ X2 @ A2 )
% 0.33/0.63 => ( ( F @ X2 )
% 0.33/0.63 = zero_zero_real ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_eq_0_iff
% 0.33/0.63 thf(fact_5818_sum__nonneg__eq__0__iff,axiom,
% 0.33/0.63 ! [A2: set_real,F: real > rat] :
% 0.33/0.63 ( ( finite_finite_real @ A2 )
% 0.33/0.63 => ( ! [X4: real] :
% 0.33/0.63 ( ( member_real @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.33/0.63 => ( ( ( groups1300246762558778688al_rat @ F @ A2 )
% 0.33/0.63 = zero_zero_rat )
% 0.33/0.63 = ( ! [X2: real] :
% 0.33/0.63 ( ( member_real @ X2 @ A2 )
% 0.33/0.63 => ( ( F @ X2 )
% 0.33/0.63 = zero_zero_rat ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_eq_0_iff
% 0.33/0.63 thf(fact_5819_sum__nonneg__eq__0__iff,axiom,
% 0.33/0.63 ! [A2: set_nat,F: nat > rat] :
% 0.33/0.63 ( ( finite_finite_nat @ A2 )
% 0.33/0.63 => ( ! [X4: nat] :
% 0.33/0.63 ( ( member_nat @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.33/0.63 => ( ( ( groups2906978787729119204at_rat @ F @ A2 )
% 0.33/0.63 = zero_zero_rat )
% 0.33/0.63 = ( ! [X2: nat] :
% 0.33/0.63 ( ( member_nat @ X2 @ A2 )
% 0.33/0.63 => ( ( F @ X2 )
% 0.33/0.63 = zero_zero_rat ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_eq_0_iff
% 0.33/0.63 thf(fact_5820_sum__nonneg__eq__0__iff,axiom,
% 0.33/0.63 ! [A2: set_int,F: int > rat] :
% 0.33/0.63 ( ( finite_finite_int @ A2 )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.33/0.63 => ( ( ( groups3906332499630173760nt_rat @ F @ A2 )
% 0.33/0.63 = zero_zero_rat )
% 0.33/0.63 = ( ! [X2: int] :
% 0.33/0.63 ( ( member_int @ X2 @ A2 )
% 0.33/0.63 => ( ( F @ X2 )
% 0.33/0.63 = zero_zero_rat ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_eq_0_iff
% 0.33/0.63 thf(fact_5821_sum__nonneg__eq__0__iff,axiom,
% 0.33/0.63 ! [A2: set_complex,F: complex > rat] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.33/0.63 => ( ( ( groups5058264527183730370ex_rat @ F @ A2 )
% 0.33/0.63 = zero_zero_rat )
% 0.33/0.63 = ( ! [X2: complex] :
% 0.33/0.63 ( ( member_complex @ X2 @ A2 )
% 0.33/0.63 => ( ( F @ X2 )
% 0.33/0.63 = zero_zero_rat ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_eq_0_iff
% 0.33/0.63 thf(fact_5822_sum__nonneg__eq__0__iff,axiom,
% 0.33/0.63 ! [A2: set_real,F: real > nat] :
% 0.33/0.63 ( ( finite_finite_real @ A2 )
% 0.33/0.63 => ( ! [X4: real] :
% 0.33/0.63 ( ( member_real @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.33/0.63 => ( ( ( groups1935376822645274424al_nat @ F @ A2 )
% 0.33/0.63 = zero_zero_nat )
% 0.33/0.63 = ( ! [X2: real] :
% 0.33/0.63 ( ( member_real @ X2 @ A2 )
% 0.33/0.63 => ( ( F @ X2 )
% 0.33/0.63 = zero_zero_nat ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_eq_0_iff
% 0.33/0.63 thf(fact_5823_sum__nonneg__eq__0__iff,axiom,
% 0.33/0.63 ! [A2: set_int,F: int > nat] :
% 0.33/0.63 ( ( finite_finite_int @ A2 )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.33/0.63 => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
% 0.33/0.63 = zero_zero_nat )
% 0.33/0.63 = ( ! [X2: int] :
% 0.33/0.63 ( ( member_int @ X2 @ A2 )
% 0.33/0.63 => ( ( F @ X2 )
% 0.33/0.63 = zero_zero_nat ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_eq_0_iff
% 0.33/0.63 thf(fact_5824_sum__nonneg__eq__0__iff,axiom,
% 0.33/0.63 ! [A2: set_complex,F: complex > nat] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.33/0.63 => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
% 0.33/0.63 = zero_zero_nat )
% 0.33/0.63 = ( ! [X2: complex] :
% 0.33/0.63 ( ( member_complex @ X2 @ A2 )
% 0.33/0.63 => ( ( F @ X2 )
% 0.33/0.63 = zero_zero_nat ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_eq_0_iff
% 0.33/0.63 thf(fact_5825_sum__le__included,axiom,
% 0.33/0.63 ! [S3: set_int,T: set_int,G: int > real,I2: int > int,F: int > real] :
% 0.33/0.63 ( ( finite_finite_int @ S3 )
% 0.33/0.63 => ( ( finite_finite_int @ T )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ T )
% 0.33/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ S3 )
% 0.33/0.63 => ? [Xa: int] :
% 0.33/0.63 ( ( member_int @ Xa @ T )
% 0.33/0.63 & ( ( I2 @ Xa )
% 0.33/0.63 = X4 )
% 0.33/0.63 & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.33/0.63 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S3 ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_le_included
% 0.33/0.63 thf(fact_5826_sum__le__included,axiom,
% 0.33/0.63 ! [S3: set_int,T: set_complex,G: complex > real,I2: complex > int,F: int > real] :
% 0.33/0.63 ( ( finite_finite_int @ S3 )
% 0.33/0.63 => ( ( finite3207457112153483333omplex @ T )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ T )
% 0.33/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ S3 )
% 0.33/0.63 => ? [Xa: complex] :
% 0.33/0.63 ( ( member_complex @ Xa @ T )
% 0.33/0.63 & ( ( I2 @ Xa )
% 0.33/0.63 = X4 )
% 0.33/0.63 & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.33/0.63 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S3 ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_le_included
% 0.33/0.63 thf(fact_5827_sum__le__included,axiom,
% 0.33/0.63 ! [S3: set_complex,T: set_int,G: int > real,I2: int > complex,F: complex > real] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ S3 )
% 0.33/0.63 => ( ( finite_finite_int @ T )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ T )
% 0.33/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ S3 )
% 0.33/0.63 => ? [Xa: int] :
% 0.33/0.63 ( ( member_int @ Xa @ T )
% 0.33/0.63 & ( ( I2 @ Xa )
% 0.33/0.63 = X4 )
% 0.33/0.63 & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.33/0.63 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S3 ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_le_included
% 0.33/0.63 thf(fact_5828_sum__le__included,axiom,
% 0.33/0.63 ! [S3: set_complex,T: set_complex,G: complex > real,I2: complex > complex,F: complex > real] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ S3 )
% 0.33/0.63 => ( ( finite3207457112153483333omplex @ T )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ T )
% 0.33/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ S3 )
% 0.33/0.63 => ? [Xa: complex] :
% 0.33/0.63 ( ( member_complex @ Xa @ T )
% 0.33/0.63 & ( ( I2 @ Xa )
% 0.33/0.63 = X4 )
% 0.33/0.63 & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.33/0.63 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S3 ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_le_included
% 0.33/0.63 thf(fact_5829_sum__le__included,axiom,
% 0.33/0.63 ! [S3: set_nat,T: set_nat,G: nat > rat,I2: nat > nat,F: nat > rat] :
% 0.33/0.63 ( ( finite_finite_nat @ S3 )
% 0.33/0.63 => ( ( finite_finite_nat @ T )
% 0.33/0.63 => ( ! [X4: nat] :
% 0.33/0.63 ( ( member_nat @ X4 @ T )
% 0.33/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ! [X4: nat] :
% 0.33/0.63 ( ( member_nat @ X4 @ S3 )
% 0.33/0.63 => ? [Xa: nat] :
% 0.33/0.63 ( ( member_nat @ Xa @ T )
% 0.33/0.63 & ( ( I2 @ Xa )
% 0.33/0.63 = X4 )
% 0.33/0.63 & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.33/0.63 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S3 ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_le_included
% 0.33/0.63 thf(fact_5830_sum__le__included,axiom,
% 0.33/0.63 ! [S3: set_nat,T: set_int,G: int > rat,I2: int > nat,F: nat > rat] :
% 0.33/0.63 ( ( finite_finite_nat @ S3 )
% 0.33/0.63 => ( ( finite_finite_int @ T )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ T )
% 0.33/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ! [X4: nat] :
% 0.33/0.63 ( ( member_nat @ X4 @ S3 )
% 0.33/0.63 => ? [Xa: int] :
% 0.33/0.63 ( ( member_int @ Xa @ T )
% 0.33/0.63 & ( ( I2 @ Xa )
% 0.33/0.63 = X4 )
% 0.33/0.63 & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.33/0.63 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S3 ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_le_included
% 0.33/0.63 thf(fact_5831_sum__le__included,axiom,
% 0.33/0.63 ! [S3: set_nat,T: set_complex,G: complex > rat,I2: complex > nat,F: nat > rat] :
% 0.33/0.63 ( ( finite_finite_nat @ S3 )
% 0.33/0.63 => ( ( finite3207457112153483333omplex @ T )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ T )
% 0.33/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ! [X4: nat] :
% 0.33/0.63 ( ( member_nat @ X4 @ S3 )
% 0.33/0.63 => ? [Xa: complex] :
% 0.33/0.63 ( ( member_complex @ Xa @ T )
% 0.33/0.63 & ( ( I2 @ Xa )
% 0.33/0.63 = X4 )
% 0.33/0.63 & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.33/0.63 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S3 ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_le_included
% 0.33/0.63 thf(fact_5832_sum__le__included,axiom,
% 0.33/0.63 ! [S3: set_int,T: set_nat,G: nat > rat,I2: nat > int,F: int > rat] :
% 0.33/0.63 ( ( finite_finite_int @ S3 )
% 0.33/0.63 => ( ( finite_finite_nat @ T )
% 0.33/0.63 => ( ! [X4: nat] :
% 0.33/0.63 ( ( member_nat @ X4 @ T )
% 0.33/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ S3 )
% 0.33/0.63 => ? [Xa: nat] :
% 0.33/0.63 ( ( member_nat @ Xa @ T )
% 0.33/0.63 & ( ( I2 @ Xa )
% 0.33/0.63 = X4 )
% 0.33/0.63 & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.33/0.63 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S3 ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_le_included
% 0.33/0.63 thf(fact_5833_sum__le__included,axiom,
% 0.33/0.63 ! [S3: set_int,T: set_int,G: int > rat,I2: int > int,F: int > rat] :
% 0.33/0.63 ( ( finite_finite_int @ S3 )
% 0.33/0.63 => ( ( finite_finite_int @ T )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ T )
% 0.33/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ S3 )
% 0.33/0.63 => ? [Xa: int] :
% 0.33/0.63 ( ( member_int @ Xa @ T )
% 0.33/0.63 & ( ( I2 @ Xa )
% 0.33/0.63 = X4 )
% 0.33/0.63 & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.33/0.63 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S3 ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_le_included
% 0.33/0.63 thf(fact_5834_sum__le__included,axiom,
% 0.33/0.63 ! [S3: set_int,T: set_complex,G: complex > rat,I2: complex > int,F: int > rat] :
% 0.33/0.63 ( ( finite_finite_int @ S3 )
% 0.33/0.63 => ( ( finite3207457112153483333omplex @ T )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ T )
% 0.33/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ S3 )
% 0.33/0.63 => ? [Xa: complex] :
% 0.33/0.63 ( ( member_complex @ Xa @ T )
% 0.33/0.63 & ( ( I2 @ Xa )
% 0.33/0.63 = X4 )
% 0.33/0.63 & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.33/0.63 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S3 ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_le_included
% 0.33/0.63 thf(fact_5835_sum__strict__mono__ex1,axiom,
% 0.33/0.63 ! [A2: set_int,F: int > real,G: int > real] :
% 0.33/0.63 ( ( finite_finite_int @ A2 )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ? [X5: int] :
% 0.33/0.63 ( ( member_int @ X5 @ A2 )
% 0.33/0.63 & ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.33/0.63 => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_strict_mono_ex1
% 0.33/0.63 thf(fact_5836_sum__strict__mono__ex1,axiom,
% 0.33/0.63 ! [A2: set_complex,F: complex > real,G: complex > real] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ? [X5: complex] :
% 0.33/0.63 ( ( member_complex @ X5 @ A2 )
% 0.33/0.63 & ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.33/0.63 => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_strict_mono_ex1
% 0.33/0.63 thf(fact_5837_sum__strict__mono__ex1,axiom,
% 0.33/0.63 ! [A2: set_nat,F: nat > rat,G: nat > rat] :
% 0.33/0.63 ( ( finite_finite_nat @ A2 )
% 0.33/0.63 => ( ! [X4: nat] :
% 0.33/0.63 ( ( member_nat @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ? [X5: nat] :
% 0.33/0.63 ( ( member_nat @ X5 @ A2 )
% 0.33/0.63 & ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.33/0.63 => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_strict_mono_ex1
% 0.33/0.63 thf(fact_5838_sum__strict__mono__ex1,axiom,
% 0.33/0.63 ! [A2: set_int,F: int > rat,G: int > rat] :
% 0.33/0.63 ( ( finite_finite_int @ A2 )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ? [X5: int] :
% 0.33/0.63 ( ( member_int @ X5 @ A2 )
% 0.33/0.63 & ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.33/0.63 => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_strict_mono_ex1
% 0.33/0.63 thf(fact_5839_sum__strict__mono__ex1,axiom,
% 0.33/0.63 ! [A2: set_complex,F: complex > rat,G: complex > rat] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ? [X5: complex] :
% 0.33/0.63 ( ( member_complex @ X5 @ A2 )
% 0.33/0.63 & ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.33/0.63 => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_strict_mono_ex1
% 0.33/0.63 thf(fact_5840_sum__strict__mono__ex1,axiom,
% 0.33/0.63 ! [A2: set_int,F: int > nat,G: int > nat] :
% 0.33/0.63 ( ( finite_finite_int @ A2 )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ? [X5: int] :
% 0.33/0.63 ( ( member_int @ X5 @ A2 )
% 0.33/0.63 & ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.33/0.63 => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_strict_mono_ex1
% 0.33/0.63 thf(fact_5841_sum__strict__mono__ex1,axiom,
% 0.33/0.63 ! [A2: set_complex,F: complex > nat,G: complex > nat] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ? [X5: complex] :
% 0.33/0.63 ( ( member_complex @ X5 @ A2 )
% 0.33/0.63 & ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.33/0.63 => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_strict_mono_ex1
% 0.33/0.63 thf(fact_5842_sum__strict__mono__ex1,axiom,
% 0.33/0.63 ! [A2: set_nat,F: nat > int,G: nat > int] :
% 0.33/0.63 ( ( finite_finite_nat @ A2 )
% 0.33/0.63 => ( ! [X4: nat] :
% 0.33/0.63 ( ( member_nat @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ? [X5: nat] :
% 0.33/0.63 ( ( member_nat @ X5 @ A2 )
% 0.33/0.63 & ( ord_less_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.33/0.63 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ G @ A2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_strict_mono_ex1
% 0.33/0.63 thf(fact_5843_sum__strict__mono__ex1,axiom,
% 0.33/0.63 ! [A2: set_complex,F: complex > int,G: complex > int] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ? [X5: complex] :
% 0.33/0.63 ( ( member_complex @ X5 @ A2 )
% 0.33/0.63 & ( ord_less_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.33/0.63 => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ G @ A2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_strict_mono_ex1
% 0.33/0.63 thf(fact_5844_sum__strict__mono__ex1,axiom,
% 0.33/0.63 ! [A2: set_int,F: int > int,G: int > int] :
% 0.33/0.63 ( ( finite_finite_int @ A2 )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ? [X5: int] :
% 0.33/0.63 ( ( member_int @ X5 @ A2 )
% 0.33/0.63 & ( ord_less_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.33/0.63 => ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_strict_mono_ex1
% 0.33/0.63 thf(fact_5845_sum_Orelated,axiom,
% 0.33/0.63 ! [R2: complex > complex > $o,S: set_nat,H2: nat > complex,G: nat > complex] :
% 0.33/0.63 ( ( R2 @ zero_zero_complex @ zero_zero_complex )
% 0.33/0.63 => ( ! [X16: complex,Y15: complex,X23: complex,Y23: complex] :
% 0.33/0.63 ( ( ( R2 @ X16 @ X23 )
% 0.33/0.63 & ( R2 @ Y15 @ Y23 ) )
% 0.33/0.63 => ( R2 @ ( plus_plus_complex @ X16 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
% 0.33/0.63 => ( ( finite_finite_nat @ S )
% 0.33/0.63 => ( ! [X4: nat] :
% 0.33/0.63 ( ( member_nat @ X4 @ S )
% 0.33/0.63 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( R2 @ ( groups2073611262835488442omplex @ H2 @ S ) @ ( groups2073611262835488442omplex @ G @ S ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.related
% 0.33/0.63 thf(fact_5846_sum_Orelated,axiom,
% 0.33/0.63 ! [R2: complex > complex > $o,S: set_int,H2: int > complex,G: int > complex] :
% 0.33/0.63 ( ( R2 @ zero_zero_complex @ zero_zero_complex )
% 0.33/0.63 => ( ! [X16: complex,Y15: complex,X23: complex,Y23: complex] :
% 0.33/0.63 ( ( ( R2 @ X16 @ X23 )
% 0.33/0.63 & ( R2 @ Y15 @ Y23 ) )
% 0.33/0.63 => ( R2 @ ( plus_plus_complex @ X16 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
% 0.33/0.63 => ( ( finite_finite_int @ S )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ S )
% 0.33/0.63 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( R2 @ ( groups3049146728041665814omplex @ H2 @ S ) @ ( groups3049146728041665814omplex @ G @ S ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.related
% 0.33/0.63 thf(fact_5847_sum_Orelated,axiom,
% 0.33/0.63 ! [R2: real > real > $o,S: set_int,H2: int > real,G: int > real] :
% 0.33/0.63 ( ( R2 @ zero_zero_real @ zero_zero_real )
% 0.33/0.63 => ( ! [X16: real,Y15: real,X23: real,Y23: real] :
% 0.33/0.63 ( ( ( R2 @ X16 @ X23 )
% 0.33/0.63 & ( R2 @ Y15 @ Y23 ) )
% 0.33/0.63 => ( R2 @ ( plus_plus_real @ X16 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
% 0.33/0.63 => ( ( finite_finite_int @ S )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ S )
% 0.33/0.63 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( R2 @ ( groups8778361861064173332t_real @ H2 @ S ) @ ( groups8778361861064173332t_real @ G @ S ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.related
% 0.33/0.63 thf(fact_5848_sum_Orelated,axiom,
% 0.33/0.63 ! [R2: real > real > $o,S: set_complex,H2: complex > real,G: complex > real] :
% 0.33/0.63 ( ( R2 @ zero_zero_real @ zero_zero_real )
% 0.33/0.63 => ( ! [X16: real,Y15: real,X23: real,Y23: real] :
% 0.33/0.63 ( ( ( R2 @ X16 @ X23 )
% 0.33/0.63 & ( R2 @ Y15 @ Y23 ) )
% 0.33/0.63 => ( R2 @ ( plus_plus_real @ X16 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
% 0.33/0.63 => ( ( finite3207457112153483333omplex @ S )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ S )
% 0.33/0.63 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( R2 @ ( groups5808333547571424918x_real @ H2 @ S ) @ ( groups5808333547571424918x_real @ G @ S ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.related
% 0.33/0.63 thf(fact_5849_sum_Orelated,axiom,
% 0.33/0.63 ! [R2: rat > rat > $o,S: set_nat,H2: nat > rat,G: nat > rat] :
% 0.33/0.63 ( ( R2 @ zero_zero_rat @ zero_zero_rat )
% 0.33/0.63 => ( ! [X16: rat,Y15: rat,X23: rat,Y23: rat] :
% 0.33/0.63 ( ( ( R2 @ X16 @ X23 )
% 0.33/0.63 & ( R2 @ Y15 @ Y23 ) )
% 0.33/0.63 => ( R2 @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
% 0.33/0.63 => ( ( finite_finite_nat @ S )
% 0.33/0.63 => ( ! [X4: nat] :
% 0.33/0.63 ( ( member_nat @ X4 @ S )
% 0.33/0.63 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( R2 @ ( groups2906978787729119204at_rat @ H2 @ S ) @ ( groups2906978787729119204at_rat @ G @ S ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.related
% 0.33/0.63 thf(fact_5850_sum_Orelated,axiom,
% 0.33/0.63 ! [R2: rat > rat > $o,S: set_int,H2: int > rat,G: int > rat] :
% 0.33/0.63 ( ( R2 @ zero_zero_rat @ zero_zero_rat )
% 0.33/0.63 => ( ! [X16: rat,Y15: rat,X23: rat,Y23: rat] :
% 0.33/0.63 ( ( ( R2 @ X16 @ X23 )
% 0.33/0.63 & ( R2 @ Y15 @ Y23 ) )
% 0.33/0.63 => ( R2 @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
% 0.33/0.63 => ( ( finite_finite_int @ S )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ S )
% 0.33/0.63 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( R2 @ ( groups3906332499630173760nt_rat @ H2 @ S ) @ ( groups3906332499630173760nt_rat @ G @ S ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.related
% 0.33/0.63 thf(fact_5851_sum_Orelated,axiom,
% 0.33/0.63 ! [R2: rat > rat > $o,S: set_complex,H2: complex > rat,G: complex > rat] :
% 0.33/0.63 ( ( R2 @ zero_zero_rat @ zero_zero_rat )
% 0.33/0.63 => ( ! [X16: rat,Y15: rat,X23: rat,Y23: rat] :
% 0.33/0.63 ( ( ( R2 @ X16 @ X23 )
% 0.33/0.63 & ( R2 @ Y15 @ Y23 ) )
% 0.33/0.63 => ( R2 @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
% 0.33/0.63 => ( ( finite3207457112153483333omplex @ S )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ S )
% 0.33/0.63 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( R2 @ ( groups5058264527183730370ex_rat @ H2 @ S ) @ ( groups5058264527183730370ex_rat @ G @ S ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.related
% 0.33/0.63 thf(fact_5852_sum_Orelated,axiom,
% 0.33/0.63 ! [R2: nat > nat > $o,S: set_int,H2: int > nat,G: int > nat] :
% 0.33/0.63 ( ( R2 @ zero_zero_nat @ zero_zero_nat )
% 0.33/0.63 => ( ! [X16: nat,Y15: nat,X23: nat,Y23: nat] :
% 0.33/0.63 ( ( ( R2 @ X16 @ X23 )
% 0.33/0.63 & ( R2 @ Y15 @ Y23 ) )
% 0.33/0.63 => ( R2 @ ( plus_plus_nat @ X16 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
% 0.33/0.63 => ( ( finite_finite_int @ S )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ S )
% 0.33/0.63 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( R2 @ ( groups4541462559716669496nt_nat @ H2 @ S ) @ ( groups4541462559716669496nt_nat @ G @ S ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.related
% 0.33/0.63 thf(fact_5853_sum_Orelated,axiom,
% 0.33/0.63 ! [R2: nat > nat > $o,S: set_complex,H2: complex > nat,G: complex > nat] :
% 0.33/0.63 ( ( R2 @ zero_zero_nat @ zero_zero_nat )
% 0.33/0.63 => ( ! [X16: nat,Y15: nat,X23: nat,Y23: nat] :
% 0.33/0.63 ( ( ( R2 @ X16 @ X23 )
% 0.33/0.63 & ( R2 @ Y15 @ Y23 ) )
% 0.33/0.63 => ( R2 @ ( plus_plus_nat @ X16 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
% 0.33/0.63 => ( ( finite3207457112153483333omplex @ S )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ S )
% 0.33/0.63 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( R2 @ ( groups5693394587270226106ex_nat @ H2 @ S ) @ ( groups5693394587270226106ex_nat @ G @ S ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.related
% 0.33/0.63 thf(fact_5854_sum_Orelated,axiom,
% 0.33/0.63 ! [R2: int > int > $o,S: set_nat,H2: nat > int,G: nat > int] :
% 0.33/0.63 ( ( R2 @ zero_zero_int @ zero_zero_int )
% 0.33/0.63 => ( ! [X16: int,Y15: int,X23: int,Y23: int] :
% 0.33/0.63 ( ( ( R2 @ X16 @ X23 )
% 0.33/0.63 & ( R2 @ Y15 @ Y23 ) )
% 0.33/0.63 => ( R2 @ ( plus_plus_int @ X16 @ Y15 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
% 0.33/0.63 => ( ( finite_finite_nat @ S )
% 0.33/0.63 => ( ! [X4: nat] :
% 0.33/0.63 ( ( member_nat @ X4 @ S )
% 0.33/0.63 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( R2 @ ( groups3539618377306564664at_int @ H2 @ S ) @ ( groups3539618377306564664at_int @ G @ S ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.related
% 0.33/0.63 thf(fact_5855_sum__strict__mono,axiom,
% 0.33/0.63 ! [A2: set_complex,F: complex > real,G: complex > real] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.63 => ( ( A2 != bot_bot_set_complex )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_strict_mono
% 0.33/0.63 thf(fact_5856_sum__strict__mono,axiom,
% 0.33/0.63 ! [A2: set_int,F: int > real,G: int > real] :
% 0.33/0.63 ( ( finite_finite_int @ A2 )
% 0.33/0.63 => ( ( A2 != bot_bot_set_int )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_strict_mono
% 0.33/0.63 thf(fact_5857_sum__strict__mono,axiom,
% 0.33/0.63 ! [A2: set_real,F: real > real,G: real > real] :
% 0.33/0.63 ( ( finite_finite_real @ A2 )
% 0.33/0.63 => ( ( A2 != bot_bot_set_real )
% 0.33/0.63 => ( ! [X4: real] :
% 0.33/0.63 ( ( member_real @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_strict_mono
% 0.33/0.63 thf(fact_5858_sum__strict__mono,axiom,
% 0.33/0.63 ! [A2: set_complex,F: complex > rat,G: complex > rat] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.63 => ( ( A2 != bot_bot_set_complex )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_strict_mono
% 0.33/0.63 thf(fact_5859_sum__strict__mono,axiom,
% 0.33/0.63 ! [A2: set_nat,F: nat > rat,G: nat > rat] :
% 0.33/0.63 ( ( finite_finite_nat @ A2 )
% 0.33/0.63 => ( ( A2 != bot_bot_set_nat )
% 0.33/0.63 => ( ! [X4: nat] :
% 0.33/0.63 ( ( member_nat @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_strict_mono
% 0.33/0.63 thf(fact_5860_sum__strict__mono,axiom,
% 0.33/0.63 ! [A2: set_int,F: int > rat,G: int > rat] :
% 0.33/0.63 ( ( finite_finite_int @ A2 )
% 0.33/0.63 => ( ( A2 != bot_bot_set_int )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_strict_mono
% 0.33/0.63 thf(fact_5861_sum__strict__mono,axiom,
% 0.33/0.63 ! [A2: set_real,F: real > rat,G: real > rat] :
% 0.33/0.63 ( ( finite_finite_real @ A2 )
% 0.33/0.63 => ( ( A2 != bot_bot_set_real )
% 0.33/0.63 => ( ! [X4: real] :
% 0.33/0.63 ( ( member_real @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_strict_mono
% 0.33/0.63 thf(fact_5862_sum__strict__mono,axiom,
% 0.33/0.63 ! [A2: set_complex,F: complex > nat,G: complex > nat] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.63 => ( ( A2 != bot_bot_set_complex )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_strict_mono
% 0.33/0.63 thf(fact_5863_sum__strict__mono,axiom,
% 0.33/0.63 ! [A2: set_int,F: int > nat,G: int > nat] :
% 0.33/0.63 ( ( finite_finite_int @ A2 )
% 0.33/0.63 => ( ( A2 != bot_bot_set_int )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_strict_mono
% 0.33/0.63 thf(fact_5864_sum__strict__mono,axiom,
% 0.33/0.63 ! [A2: set_real,F: real > nat,G: real > nat] :
% 0.33/0.63 ( ( finite_finite_real @ A2 )
% 0.33/0.63 => ( ( A2 != bot_bot_set_real )
% 0.33/0.63 => ( ! [X4: real] :
% 0.33/0.63 ( ( member_real @ X4 @ A2 )
% 0.33/0.63 => ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.63 => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_strict_mono
% 0.33/0.63 thf(fact_5865_ln__bound,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % ln_bound
% 0.33/0.63 thf(fact_5866_ln__gt__zero,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_real @ one_one_real @ X )
% 0.33/0.63 => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ln_gt_zero
% 0.33/0.63 thf(fact_5867_ln__less__zero,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ( ord_less_real @ X @ one_one_real )
% 0.33/0.63 => ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ln_less_zero
% 0.33/0.63 thf(fact_5868_ln__gt__zero__imp__gt__one,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 0.33/0.63 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ord_less_real @ one_one_real @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ln_gt_zero_imp_gt_one
% 0.33/0.63 thf(fact_5869_ln__ge__zero,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ one_one_real @ X )
% 0.33/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ln_ge_zero
% 0.33/0.63 thf(fact_5870_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.33/0.63 ! [S4: set_real,T4: set_real,S: set_real,I2: real > real,J: real > real,T3: set_real,G: real > complex,H2: real > complex] :
% 0.33/0.63 ( ( finite_finite_real @ S4 )
% 0.33/0.63 => ( ( finite_finite_real @ T4 )
% 0.33/0.63 => ( ! [A3: real] :
% 0.33/0.63 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 0.33/0.63 => ( ( I2 @ ( J @ A3 ) )
% 0.33/0.63 = A3 ) )
% 0.33/0.63 => ( ! [A3: real] :
% 0.33/0.63 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 0.33/0.63 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 0.33/0.63 => ( ! [B2: real] :
% 0.33/0.63 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.33/0.63 => ( ( J @ ( I2 @ B2 ) )
% 0.33/0.63 = B2 ) )
% 0.33/0.63 => ( ! [B2: real] :
% 0.33/0.63 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.33/0.63 => ( member_real @ ( I2 @ B2 ) @ ( minus_minus_set_real @ S @ S4 ) ) )
% 0.33/0.63 => ( ! [A3: real] :
% 0.33/0.63 ( ( member_real @ A3 @ S4 )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_complex ) )
% 0.33/0.63 => ( ! [B2: real] :
% 0.33/0.63 ( ( member_real @ B2 @ T4 )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_complex ) )
% 0.33/0.63 => ( ! [A3: real] :
% 0.33/0.63 ( ( member_real @ A3 @ S )
% 0.33/0.63 => ( ( H2 @ ( J @ A3 ) )
% 0.33/0.63 = ( G @ A3 ) ) )
% 0.33/0.63 => ( ( groups5754745047067104278omplex @ G @ S )
% 0.33/0.63 = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.reindex_bij_witness_not_neutral
% 0.33/0.63 thf(fact_5871_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.33/0.63 ! [S4: set_real,T4: set_int,S: set_real,I2: int > real,J: real > int,T3: set_int,G: real > complex,H2: int > complex] :
% 0.33/0.63 ( ( finite_finite_real @ S4 )
% 0.33/0.63 => ( ( finite_finite_int @ T4 )
% 0.33/0.63 => ( ! [A3: real] :
% 0.33/0.63 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 0.33/0.63 => ( ( I2 @ ( J @ A3 ) )
% 0.33/0.63 = A3 ) )
% 0.33/0.63 => ( ! [A3: real] :
% 0.33/0.63 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 0.33/0.63 => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 0.33/0.63 => ( ! [B2: int] :
% 0.33/0.63 ( ( member_int @ B2 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.33/0.63 => ( ( J @ ( I2 @ B2 ) )
% 0.33/0.63 = B2 ) )
% 0.33/0.63 => ( ! [B2: int] :
% 0.33/0.63 ( ( member_int @ B2 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.33/0.63 => ( member_real @ ( I2 @ B2 ) @ ( minus_minus_set_real @ S @ S4 ) ) )
% 0.33/0.63 => ( ! [A3: real] :
% 0.33/0.63 ( ( member_real @ A3 @ S4 )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_complex ) )
% 0.33/0.63 => ( ! [B2: int] :
% 0.33/0.63 ( ( member_int @ B2 @ T4 )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_complex ) )
% 0.33/0.63 => ( ! [A3: real] :
% 0.33/0.63 ( ( member_real @ A3 @ S )
% 0.33/0.63 => ( ( H2 @ ( J @ A3 ) )
% 0.33/0.63 = ( G @ A3 ) ) )
% 0.33/0.63 => ( ( groups5754745047067104278omplex @ G @ S )
% 0.33/0.63 = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.reindex_bij_witness_not_neutral
% 0.33/0.63 thf(fact_5872_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.33/0.63 ! [S4: set_int,T4: set_real,S: set_int,I2: real > int,J: int > real,T3: set_real,G: int > complex,H2: real > complex] :
% 0.33/0.63 ( ( finite_finite_int @ S4 )
% 0.33/0.63 => ( ( finite_finite_real @ T4 )
% 0.33/0.63 => ( ! [A3: int] :
% 0.33/0.63 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 0.33/0.63 => ( ( I2 @ ( J @ A3 ) )
% 0.33/0.63 = A3 ) )
% 0.33/0.63 => ( ! [A3: int] :
% 0.33/0.63 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 0.33/0.63 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 0.33/0.63 => ( ! [B2: real] :
% 0.33/0.63 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.33/0.63 => ( ( J @ ( I2 @ B2 ) )
% 0.33/0.63 = B2 ) )
% 0.33/0.63 => ( ! [B2: real] :
% 0.33/0.63 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.33/0.63 => ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
% 0.33/0.63 => ( ! [A3: int] :
% 0.33/0.63 ( ( member_int @ A3 @ S4 )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_complex ) )
% 0.33/0.63 => ( ! [B2: real] :
% 0.33/0.63 ( ( member_real @ B2 @ T4 )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_complex ) )
% 0.33/0.63 => ( ! [A3: int] :
% 0.33/0.63 ( ( member_int @ A3 @ S )
% 0.33/0.63 => ( ( H2 @ ( J @ A3 ) )
% 0.33/0.63 = ( G @ A3 ) ) )
% 0.33/0.63 => ( ( groups3049146728041665814omplex @ G @ S )
% 0.33/0.63 = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.reindex_bij_witness_not_neutral
% 0.33/0.63 thf(fact_5873_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.33/0.63 ! [S4: set_int,T4: set_int,S: set_int,I2: int > int,J: int > int,T3: set_int,G: int > complex,H2: int > complex] :
% 0.33/0.63 ( ( finite_finite_int @ S4 )
% 0.33/0.63 => ( ( finite_finite_int @ T4 )
% 0.33/0.63 => ( ! [A3: int] :
% 0.33/0.63 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 0.33/0.63 => ( ( I2 @ ( J @ A3 ) )
% 0.33/0.63 = A3 ) )
% 0.33/0.63 => ( ! [A3: int] :
% 0.33/0.63 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 0.33/0.63 => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 0.33/0.63 => ( ! [B2: int] :
% 0.33/0.63 ( ( member_int @ B2 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.33/0.63 => ( ( J @ ( I2 @ B2 ) )
% 0.33/0.63 = B2 ) )
% 0.33/0.63 => ( ! [B2: int] :
% 0.33/0.63 ( ( member_int @ B2 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.33/0.63 => ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
% 0.33/0.63 => ( ! [A3: int] :
% 0.33/0.63 ( ( member_int @ A3 @ S4 )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_complex ) )
% 0.33/0.63 => ( ! [B2: int] :
% 0.33/0.63 ( ( member_int @ B2 @ T4 )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_complex ) )
% 0.33/0.63 => ( ! [A3: int] :
% 0.33/0.63 ( ( member_int @ A3 @ S )
% 0.33/0.63 => ( ( H2 @ ( J @ A3 ) )
% 0.33/0.63 = ( G @ A3 ) ) )
% 0.33/0.63 => ( ( groups3049146728041665814omplex @ G @ S )
% 0.33/0.63 = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.reindex_bij_witness_not_neutral
% 0.33/0.63 thf(fact_5874_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.33/0.63 ! [S4: set_real,T4: set_real,S: set_real,I2: real > real,J: real > real,T3: set_real,G: real > real,H2: real > real] :
% 0.33/0.63 ( ( finite_finite_real @ S4 )
% 0.33/0.63 => ( ( finite_finite_real @ T4 )
% 0.33/0.63 => ( ! [A3: real] :
% 0.33/0.63 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 0.33/0.63 => ( ( I2 @ ( J @ A3 ) )
% 0.33/0.63 = A3 ) )
% 0.33/0.63 => ( ! [A3: real] :
% 0.33/0.63 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 0.33/0.63 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 0.33/0.63 => ( ! [B2: real] :
% 0.33/0.63 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.33/0.63 => ( ( J @ ( I2 @ B2 ) )
% 0.33/0.63 = B2 ) )
% 0.33/0.63 => ( ! [B2: real] :
% 0.33/0.63 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.33/0.63 => ( member_real @ ( I2 @ B2 ) @ ( minus_minus_set_real @ S @ S4 ) ) )
% 0.33/0.63 => ( ! [A3: real] :
% 0.33/0.63 ( ( member_real @ A3 @ S4 )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ! [B2: real] :
% 0.33/0.63 ( ( member_real @ B2 @ T4 )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ! [A3: real] :
% 0.33/0.63 ( ( member_real @ A3 @ S )
% 0.33/0.63 => ( ( H2 @ ( J @ A3 ) )
% 0.33/0.63 = ( G @ A3 ) ) )
% 0.33/0.63 => ( ( groups8097168146408367636l_real @ G @ S )
% 0.33/0.63 = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.reindex_bij_witness_not_neutral
% 0.33/0.63 thf(fact_5875_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.33/0.63 ! [S4: set_real,T4: set_int,S: set_real,I2: int > real,J: real > int,T3: set_int,G: real > real,H2: int > real] :
% 0.33/0.63 ( ( finite_finite_real @ S4 )
% 0.33/0.63 => ( ( finite_finite_int @ T4 )
% 0.33/0.63 => ( ! [A3: real] :
% 0.33/0.63 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 0.33/0.63 => ( ( I2 @ ( J @ A3 ) )
% 0.33/0.63 = A3 ) )
% 0.33/0.63 => ( ! [A3: real] :
% 0.33/0.63 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 0.33/0.63 => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 0.33/0.63 => ( ! [B2: int] :
% 0.33/0.63 ( ( member_int @ B2 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.33/0.63 => ( ( J @ ( I2 @ B2 ) )
% 0.33/0.63 = B2 ) )
% 0.33/0.63 => ( ! [B2: int] :
% 0.33/0.63 ( ( member_int @ B2 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.33/0.63 => ( member_real @ ( I2 @ B2 ) @ ( minus_minus_set_real @ S @ S4 ) ) )
% 0.33/0.63 => ( ! [A3: real] :
% 0.33/0.63 ( ( member_real @ A3 @ S4 )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ! [B2: int] :
% 0.33/0.63 ( ( member_int @ B2 @ T4 )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ! [A3: real] :
% 0.33/0.63 ( ( member_real @ A3 @ S )
% 0.33/0.63 => ( ( H2 @ ( J @ A3 ) )
% 0.33/0.63 = ( G @ A3 ) ) )
% 0.33/0.63 => ( ( groups8097168146408367636l_real @ G @ S )
% 0.33/0.63 = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.reindex_bij_witness_not_neutral
% 0.33/0.63 thf(fact_5876_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.33/0.63 ! [S4: set_real,T4: set_complex,S: set_real,I2: complex > real,J: real > complex,T3: set_complex,G: real > real,H2: complex > real] :
% 0.33/0.63 ( ( finite_finite_real @ S4 )
% 0.33/0.63 => ( ( finite3207457112153483333omplex @ T4 )
% 0.33/0.63 => ( ! [A3: real] :
% 0.33/0.63 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 0.33/0.63 => ( ( I2 @ ( J @ A3 ) )
% 0.33/0.63 = A3 ) )
% 0.33/0.63 => ( ! [A3: real] :
% 0.33/0.63 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 0.33/0.63 => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 0.33/0.63 => ( ! [B2: complex] :
% 0.33/0.63 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.33/0.63 => ( ( J @ ( I2 @ B2 ) )
% 0.33/0.63 = B2 ) )
% 0.33/0.63 => ( ! [B2: complex] :
% 0.33/0.63 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.33/0.63 => ( member_real @ ( I2 @ B2 ) @ ( minus_minus_set_real @ S @ S4 ) ) )
% 0.33/0.63 => ( ! [A3: real] :
% 0.33/0.63 ( ( member_real @ A3 @ S4 )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ! [B2: complex] :
% 0.33/0.63 ( ( member_complex @ B2 @ T4 )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ! [A3: real] :
% 0.33/0.63 ( ( member_real @ A3 @ S )
% 0.33/0.63 => ( ( H2 @ ( J @ A3 ) )
% 0.33/0.63 = ( G @ A3 ) ) )
% 0.33/0.63 => ( ( groups8097168146408367636l_real @ G @ S )
% 0.33/0.63 = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.reindex_bij_witness_not_neutral
% 0.33/0.63 thf(fact_5877_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.33/0.63 ! [S4: set_int,T4: set_real,S: set_int,I2: real > int,J: int > real,T3: set_real,G: int > real,H2: real > real] :
% 0.33/0.63 ( ( finite_finite_int @ S4 )
% 0.33/0.63 => ( ( finite_finite_real @ T4 )
% 0.33/0.63 => ( ! [A3: int] :
% 0.33/0.63 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 0.33/0.63 => ( ( I2 @ ( J @ A3 ) )
% 0.33/0.63 = A3 ) )
% 0.33/0.63 => ( ! [A3: int] :
% 0.33/0.63 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 0.33/0.63 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 0.33/0.63 => ( ! [B2: real] :
% 0.33/0.63 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.33/0.63 => ( ( J @ ( I2 @ B2 ) )
% 0.33/0.63 = B2 ) )
% 0.33/0.63 => ( ! [B2: real] :
% 0.33/0.63 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.33/0.63 => ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
% 0.33/0.63 => ( ! [A3: int] :
% 0.33/0.63 ( ( member_int @ A3 @ S4 )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ! [B2: real] :
% 0.33/0.63 ( ( member_real @ B2 @ T4 )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ! [A3: int] :
% 0.33/0.63 ( ( member_int @ A3 @ S )
% 0.33/0.63 => ( ( H2 @ ( J @ A3 ) )
% 0.33/0.63 = ( G @ A3 ) ) )
% 0.33/0.63 => ( ( groups8778361861064173332t_real @ G @ S )
% 0.33/0.63 = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.reindex_bij_witness_not_neutral
% 0.33/0.63 thf(fact_5878_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.33/0.63 ! [S4: set_int,T4: set_int,S: set_int,I2: int > int,J: int > int,T3: set_int,G: int > real,H2: int > real] :
% 0.33/0.63 ( ( finite_finite_int @ S4 )
% 0.33/0.63 => ( ( finite_finite_int @ T4 )
% 0.33/0.63 => ( ! [A3: int] :
% 0.33/0.63 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 0.33/0.63 => ( ( I2 @ ( J @ A3 ) )
% 0.33/0.63 = A3 ) )
% 0.33/0.63 => ( ! [A3: int] :
% 0.33/0.63 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 0.33/0.63 => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 0.33/0.63 => ( ! [B2: int] :
% 0.33/0.63 ( ( member_int @ B2 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.33/0.63 => ( ( J @ ( I2 @ B2 ) )
% 0.33/0.63 = B2 ) )
% 0.33/0.63 => ( ! [B2: int] :
% 0.33/0.63 ( ( member_int @ B2 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.33/0.63 => ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
% 0.33/0.63 => ( ! [A3: int] :
% 0.33/0.63 ( ( member_int @ A3 @ S4 )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ! [B2: int] :
% 0.33/0.63 ( ( member_int @ B2 @ T4 )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ! [A3: int] :
% 0.33/0.63 ( ( member_int @ A3 @ S )
% 0.33/0.63 => ( ( H2 @ ( J @ A3 ) )
% 0.33/0.63 = ( G @ A3 ) ) )
% 0.33/0.63 => ( ( groups8778361861064173332t_real @ G @ S )
% 0.33/0.63 = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.reindex_bij_witness_not_neutral
% 0.33/0.63 thf(fact_5879_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.33/0.63 ! [S4: set_int,T4: set_complex,S: set_int,I2: complex > int,J: int > complex,T3: set_complex,G: int > real,H2: complex > real] :
% 0.33/0.63 ( ( finite_finite_int @ S4 )
% 0.33/0.63 => ( ( finite3207457112153483333omplex @ T4 )
% 0.33/0.63 => ( ! [A3: int] :
% 0.33/0.63 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 0.33/0.63 => ( ( I2 @ ( J @ A3 ) )
% 0.33/0.63 = A3 ) )
% 0.33/0.63 => ( ! [A3: int] :
% 0.33/0.63 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 0.33/0.63 => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 0.33/0.63 => ( ! [B2: complex] :
% 0.33/0.63 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.33/0.63 => ( ( J @ ( I2 @ B2 ) )
% 0.33/0.63 = B2 ) )
% 0.33/0.63 => ( ! [B2: complex] :
% 0.33/0.63 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.33/0.63 => ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
% 0.33/0.63 => ( ! [A3: int] :
% 0.33/0.63 ( ( member_int @ A3 @ S4 )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ! [B2: complex] :
% 0.33/0.63 ( ( member_complex @ B2 @ T4 )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ! [A3: int] :
% 0.33/0.63 ( ( member_int @ A3 @ S )
% 0.33/0.63 => ( ( H2 @ ( J @ A3 ) )
% 0.33/0.63 = ( G @ A3 ) ) )
% 0.33/0.63 => ( ( groups8778361861064173332t_real @ G @ S )
% 0.33/0.63 = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.reindex_bij_witness_not_neutral
% 0.33/0.63 thf(fact_5880_not__zero__le__neg__numeral,axiom,
% 0.33/0.63 ! [N2: num] :
% 0.33/0.63 ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_zero_le_neg_numeral
% 0.33/0.63 thf(fact_5881_not__zero__le__neg__numeral,axiom,
% 0.33/0.63 ! [N2: num] :
% 0.33/0.63 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_zero_le_neg_numeral
% 0.33/0.63 thf(fact_5882_not__zero__le__neg__numeral,axiom,
% 0.33/0.63 ! [N2: num] :
% 0.33/0.63 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_zero_le_neg_numeral
% 0.33/0.63 thf(fact_5883_not__zero__le__neg__numeral,axiom,
% 0.33/0.63 ! [N2: num] :
% 0.33/0.63 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_zero_le_neg_numeral
% 0.33/0.63 thf(fact_5884_neg__numeral__le__zero,axiom,
% 0.33/0.63 ! [N2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) @ zero_zero_real ) ).
% 0.33/0.63
% 0.33/0.63 % neg_numeral_le_zero
% 0.33/0.63 thf(fact_5885_neg__numeral__le__zero,axiom,
% 0.33/0.63 ! [N2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) @ zero_z3403309356797280102nteger ) ).
% 0.33/0.63
% 0.33/0.63 % neg_numeral_le_zero
% 0.33/0.63 thf(fact_5886_neg__numeral__le__zero,axiom,
% 0.33/0.63 ! [N2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) @ zero_zero_rat ) ).
% 0.33/0.63
% 0.33/0.63 % neg_numeral_le_zero
% 0.33/0.63 thf(fact_5887_neg__numeral__le__zero,axiom,
% 0.33/0.63 ! [N2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ zero_zero_int ) ).
% 0.33/0.63
% 0.33/0.63 % neg_numeral_le_zero
% 0.33/0.63 thf(fact_5888_neg__numeral__less__zero,axiom,
% 0.33/0.63 ! [N2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) @ zero_zero_real ) ).
% 0.33/0.63
% 0.33/0.63 % neg_numeral_less_zero
% 0.33/0.63 thf(fact_5889_neg__numeral__less__zero,axiom,
% 0.33/0.63 ! [N2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ zero_zero_int ) ).
% 0.33/0.63
% 0.33/0.63 % neg_numeral_less_zero
% 0.33/0.63 thf(fact_5890_neg__numeral__less__zero,axiom,
% 0.33/0.63 ! [N2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) @ zero_zero_rat ) ).
% 0.33/0.63
% 0.33/0.63 % neg_numeral_less_zero
% 0.33/0.63 thf(fact_5891_neg__numeral__less__zero,axiom,
% 0.33/0.63 ! [N2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) @ zero_z3403309356797280102nteger ) ).
% 0.33/0.63
% 0.33/0.63 % neg_numeral_less_zero
% 0.33/0.63 thf(fact_5892_not__zero__less__neg__numeral,axiom,
% 0.33/0.63 ! [N2: num] :
% 0.33/0.63 ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_zero_less_neg_numeral
% 0.33/0.63 thf(fact_5893_not__zero__less__neg__numeral,axiom,
% 0.33/0.63 ! [N2: num] :
% 0.33/0.63 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_zero_less_neg_numeral
% 0.33/0.63 thf(fact_5894_not__zero__less__neg__numeral,axiom,
% 0.33/0.63 ! [N2: num] :
% 0.33/0.63 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_zero_less_neg_numeral
% 0.33/0.63 thf(fact_5895_not__zero__less__neg__numeral,axiom,
% 0.33/0.63 ! [N2: num] :
% 0.33/0.63 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_zero_less_neg_numeral
% 0.33/0.63 thf(fact_5896_le__minus__one__simps_I3_J,axiom,
% 0.33/0.63 ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % le_minus_one_simps(3)
% 0.33/0.63 thf(fact_5897_le__minus__one__simps_I3_J,axiom,
% 0.33/0.63 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.63
% 0.33/0.63 % le_minus_one_simps(3)
% 0.33/0.63 thf(fact_5898_le__minus__one__simps_I3_J,axiom,
% 0.33/0.63 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.63
% 0.33/0.63 % le_minus_one_simps(3)
% 0.33/0.63 thf(fact_5899_le__minus__one__simps_I3_J,axiom,
% 0.33/0.63 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.63
% 0.33/0.63 % le_minus_one_simps(3)
% 0.33/0.63 thf(fact_5900_le__minus__one__simps_I1_J,axiom,
% 0.33/0.63 ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 0.33/0.63
% 0.33/0.63 % le_minus_one_simps(1)
% 0.33/0.63 thf(fact_5901_le__minus__one__simps_I1_J,axiom,
% 0.33/0.63 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 0.33/0.63
% 0.33/0.63 % le_minus_one_simps(1)
% 0.33/0.63 thf(fact_5902_le__minus__one__simps_I1_J,axiom,
% 0.33/0.63 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 0.33/0.63
% 0.33/0.63 % le_minus_one_simps(1)
% 0.33/0.63 thf(fact_5903_le__minus__one__simps_I1_J,axiom,
% 0.33/0.63 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 0.33/0.63
% 0.33/0.63 % le_minus_one_simps(1)
% 0.33/0.63 thf(fact_5904_less__minus__one__simps_I3_J,axiom,
% 0.33/0.63 ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % less_minus_one_simps(3)
% 0.33/0.63 thf(fact_5905_less__minus__one__simps_I3_J,axiom,
% 0.33/0.63 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.63
% 0.33/0.63 % less_minus_one_simps(3)
% 0.33/0.63 thf(fact_5906_less__minus__one__simps_I3_J,axiom,
% 0.33/0.63 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.63
% 0.33/0.63 % less_minus_one_simps(3)
% 0.33/0.63 thf(fact_5907_less__minus__one__simps_I3_J,axiom,
% 0.33/0.63 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.63
% 0.33/0.63 % less_minus_one_simps(3)
% 0.33/0.63 thf(fact_5908_less__minus__one__simps_I1_J,axiom,
% 0.33/0.63 ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 0.33/0.63
% 0.33/0.63 % less_minus_one_simps(1)
% 0.33/0.63 thf(fact_5909_less__minus__one__simps_I1_J,axiom,
% 0.33/0.63 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 0.33/0.63
% 0.33/0.63 % less_minus_one_simps(1)
% 0.33/0.63 thf(fact_5910_less__minus__one__simps_I1_J,axiom,
% 0.33/0.63 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 0.33/0.63
% 0.33/0.63 % less_minus_one_simps(1)
% 0.33/0.63 thf(fact_5911_less__minus__one__simps_I1_J,axiom,
% 0.33/0.63 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 0.33/0.63
% 0.33/0.63 % less_minus_one_simps(1)
% 0.33/0.63 thf(fact_5912_not__one__le__neg__numeral,axiom,
% 0.33/0.63 ! [M: num] :
% 0.33/0.63 ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_one_le_neg_numeral
% 0.33/0.63 thf(fact_5913_not__one__le__neg__numeral,axiom,
% 0.33/0.63 ! [M: num] :
% 0.33/0.63 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_one_le_neg_numeral
% 0.33/0.63 thf(fact_5914_not__one__le__neg__numeral,axiom,
% 0.33/0.63 ! [M: num] :
% 0.33/0.63 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_one_le_neg_numeral
% 0.33/0.63 thf(fact_5915_not__one__le__neg__numeral,axiom,
% 0.33/0.63 ! [M: num] :
% 0.33/0.63 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_one_le_neg_numeral
% 0.33/0.63 thf(fact_5916_not__numeral__le__neg__one,axiom,
% 0.33/0.63 ! [M: num] :
% 0.33/0.63 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_numeral_le_neg_one
% 0.33/0.63 thf(fact_5917_not__numeral__le__neg__one,axiom,
% 0.33/0.63 ! [M: num] :
% 0.33/0.63 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_numeral_le_neg_one
% 0.33/0.63 thf(fact_5918_not__numeral__le__neg__one,axiom,
% 0.33/0.63 ! [M: num] :
% 0.33/0.63 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_numeral_le_neg_one
% 0.33/0.63 thf(fact_5919_not__numeral__le__neg__one,axiom,
% 0.33/0.63 ! [M: num] :
% 0.33/0.63 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_numeral_le_neg_one
% 0.33/0.63 thf(fact_5920_neg__numeral__le__neg__one,axiom,
% 0.33/0.63 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % neg_numeral_le_neg_one
% 0.33/0.63 thf(fact_5921_neg__numeral__le__neg__one,axiom,
% 0.33/0.63 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.63
% 0.33/0.63 % neg_numeral_le_neg_one
% 0.33/0.63 thf(fact_5922_neg__numeral__le__neg__one,axiom,
% 0.33/0.63 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.63
% 0.33/0.63 % neg_numeral_le_neg_one
% 0.33/0.63 thf(fact_5923_neg__numeral__le__neg__one,axiom,
% 0.33/0.63 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.63
% 0.33/0.63 % neg_numeral_le_neg_one
% 0.33/0.63 thf(fact_5924_neg__one__le__numeral,axiom,
% 0.33/0.63 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 0.33/0.63
% 0.33/0.63 % neg_one_le_numeral
% 0.33/0.63 thf(fact_5925_neg__one__le__numeral,axiom,
% 0.33/0.63 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 0.33/0.63
% 0.33/0.63 % neg_one_le_numeral
% 0.33/0.63 thf(fact_5926_neg__one__le__numeral,axiom,
% 0.33/0.63 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 0.33/0.63
% 0.33/0.63 % neg_one_le_numeral
% 0.33/0.63 thf(fact_5927_neg__one__le__numeral,axiom,
% 0.33/0.63 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 0.33/0.63
% 0.33/0.63 % neg_one_le_numeral
% 0.33/0.63 thf(fact_5928_neg__numeral__le__one,axiom,
% 0.33/0.63 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 0.33/0.63
% 0.33/0.63 % neg_numeral_le_one
% 0.33/0.63 thf(fact_5929_neg__numeral__le__one,axiom,
% 0.33/0.63 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 0.33/0.63
% 0.33/0.63 % neg_numeral_le_one
% 0.33/0.63 thf(fact_5930_neg__numeral__le__one,axiom,
% 0.33/0.63 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 0.33/0.63
% 0.33/0.63 % neg_numeral_le_one
% 0.33/0.63 thf(fact_5931_neg__numeral__le__one,axiom,
% 0.33/0.63 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 0.33/0.63
% 0.33/0.63 % neg_numeral_le_one
% 0.33/0.63 thf(fact_5932_neg__numeral__less__one,axiom,
% 0.33/0.63 ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 0.33/0.63
% 0.33/0.63 % neg_numeral_less_one
% 0.33/0.63 thf(fact_5933_neg__numeral__less__one,axiom,
% 0.33/0.63 ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 0.33/0.63
% 0.33/0.63 % neg_numeral_less_one
% 0.33/0.63 thf(fact_5934_neg__numeral__less__one,axiom,
% 0.33/0.63 ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 0.33/0.63
% 0.33/0.63 % neg_numeral_less_one
% 0.33/0.63 thf(fact_5935_neg__numeral__less__one,axiom,
% 0.33/0.63 ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 0.33/0.63
% 0.33/0.63 % neg_numeral_less_one
% 0.33/0.63 thf(fact_5936_neg__one__less__numeral,axiom,
% 0.33/0.63 ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 0.33/0.63
% 0.33/0.63 % neg_one_less_numeral
% 0.33/0.63 thf(fact_5937_neg__one__less__numeral,axiom,
% 0.33/0.63 ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 0.33/0.63
% 0.33/0.63 % neg_one_less_numeral
% 0.33/0.63 thf(fact_5938_neg__one__less__numeral,axiom,
% 0.33/0.63 ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 0.33/0.63
% 0.33/0.63 % neg_one_less_numeral
% 0.33/0.63 thf(fact_5939_neg__one__less__numeral,axiom,
% 0.33/0.63 ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 0.33/0.63
% 0.33/0.63 % neg_one_less_numeral
% 0.33/0.63 thf(fact_5940_not__numeral__less__neg__one,axiom,
% 0.33/0.63 ! [M: num] :
% 0.33/0.63 ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_numeral_less_neg_one
% 0.33/0.63 thf(fact_5941_not__numeral__less__neg__one,axiom,
% 0.33/0.63 ! [M: num] :
% 0.33/0.63 ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_numeral_less_neg_one
% 0.33/0.63 thf(fact_5942_not__numeral__less__neg__one,axiom,
% 0.33/0.63 ! [M: num] :
% 0.33/0.63 ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_numeral_less_neg_one
% 0.33/0.63 thf(fact_5943_not__numeral__less__neg__one,axiom,
% 0.33/0.63 ! [M: num] :
% 0.33/0.63 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_numeral_less_neg_one
% 0.33/0.63 thf(fact_5944_not__one__less__neg__numeral,axiom,
% 0.33/0.63 ! [M: num] :
% 0.33/0.63 ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_one_less_neg_numeral
% 0.33/0.63 thf(fact_5945_not__one__less__neg__numeral,axiom,
% 0.33/0.63 ! [M: num] :
% 0.33/0.63 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_one_less_neg_numeral
% 0.33/0.63 thf(fact_5946_not__one__less__neg__numeral,axiom,
% 0.33/0.63 ! [M: num] :
% 0.33/0.63 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_one_less_neg_numeral
% 0.33/0.63 thf(fact_5947_not__one__less__neg__numeral,axiom,
% 0.33/0.63 ! [M: num] :
% 0.33/0.63 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_one_less_neg_numeral
% 0.33/0.63 thf(fact_5948_not__neg__one__less__neg__numeral,axiom,
% 0.33/0.63 ! [M: num] :
% 0.33/0.63 ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_neg_one_less_neg_numeral
% 0.33/0.63 thf(fact_5949_not__neg__one__less__neg__numeral,axiom,
% 0.33/0.63 ! [M: num] :
% 0.33/0.63 ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_neg_one_less_neg_numeral
% 0.33/0.63 thf(fact_5950_not__neg__one__less__neg__numeral,axiom,
% 0.33/0.63 ! [M: num] :
% 0.33/0.63 ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_neg_one_less_neg_numeral
% 0.33/0.63 thf(fact_5951_not__neg__one__less__neg__numeral,axiom,
% 0.33/0.63 ! [M: num] :
% 0.33/0.63 ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % not_neg_one_less_neg_numeral
% 0.33/0.63 thf(fact_5952_nonzero__neg__divide__eq__eq2,axiom,
% 0.33/0.63 ! [B: real,C: real,A: real] :
% 0.33/0.63 ( ( B != zero_zero_real )
% 0.33/0.63 => ( ( C
% 0.33/0.63 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
% 0.33/0.63 = ( ( times_times_real @ C @ B )
% 0.33/0.63 = ( uminus_uminus_real @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_neg_divide_eq_eq2
% 0.33/0.63 thf(fact_5953_nonzero__neg__divide__eq__eq2,axiom,
% 0.33/0.63 ! [B: complex,C: complex,A: complex] :
% 0.33/0.63 ( ( B != zero_zero_complex )
% 0.33/0.63 => ( ( C
% 0.33/0.63 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.33/0.63 = ( ( times_times_complex @ C @ B )
% 0.33/0.63 = ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_neg_divide_eq_eq2
% 0.33/0.63 thf(fact_5954_nonzero__neg__divide__eq__eq2,axiom,
% 0.33/0.63 ! [B: rat,C: rat,A: rat] :
% 0.33/0.63 ( ( B != zero_zero_rat )
% 0.33/0.63 => ( ( C
% 0.33/0.63 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
% 0.33/0.63 = ( ( times_times_rat @ C @ B )
% 0.33/0.63 = ( uminus_uminus_rat @ A ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_neg_divide_eq_eq2
% 0.33/0.63 thf(fact_5955_nonzero__neg__divide__eq__eq,axiom,
% 0.33/0.63 ! [B: real,A: real,C: real] :
% 0.33/0.63 ( ( B != zero_zero_real )
% 0.33/0.63 => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.33/0.63 = C )
% 0.33/0.63 = ( ( uminus_uminus_real @ A )
% 0.33/0.63 = ( times_times_real @ C @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_neg_divide_eq_eq
% 0.33/0.63 thf(fact_5956_nonzero__neg__divide__eq__eq,axiom,
% 0.33/0.63 ! [B: complex,A: complex,C: complex] :
% 0.33/0.63 ( ( B != zero_zero_complex )
% 0.33/0.63 => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.33/0.63 = C )
% 0.33/0.63 = ( ( uminus1482373934393186551omplex @ A )
% 0.33/0.63 = ( times_times_complex @ C @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_neg_divide_eq_eq
% 0.33/0.63 thf(fact_5957_nonzero__neg__divide__eq__eq,axiom,
% 0.33/0.63 ! [B: rat,A: rat,C: rat] :
% 0.33/0.63 ( ( B != zero_zero_rat )
% 0.33/0.63 => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.33/0.63 = C )
% 0.33/0.63 = ( ( uminus_uminus_rat @ A )
% 0.33/0.63 = ( times_times_rat @ C @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % nonzero_neg_divide_eq_eq
% 0.33/0.63 thf(fact_5958_minus__divide__eq__eq,axiom,
% 0.33/0.63 ! [B: real,C: real,A: real] :
% 0.33/0.63 ( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) )
% 0.33/0.63 = A )
% 0.33/0.63 = ( ( ( C != zero_zero_real )
% 0.33/0.63 => ( ( uminus_uminus_real @ B )
% 0.33/0.63 = ( times_times_real @ A @ C ) ) )
% 0.33/0.63 & ( ( C = zero_zero_real )
% 0.33/0.63 => ( A = zero_zero_real ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % minus_divide_eq_eq
% 0.33/0.63 thf(fact_5959_minus__divide__eq__eq,axiom,
% 0.33/0.63 ! [B: complex,C: complex,A: complex] :
% 0.33/0.63 ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) )
% 0.33/0.63 = A )
% 0.33/0.63 = ( ( ( C != zero_zero_complex )
% 0.33/0.63 => ( ( uminus1482373934393186551omplex @ B )
% 0.33/0.63 = ( times_times_complex @ A @ C ) ) )
% 0.33/0.63 & ( ( C = zero_zero_complex )
% 0.33/0.63 => ( A = zero_zero_complex ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % minus_divide_eq_eq
% 0.33/0.63 thf(fact_5960_minus__divide__eq__eq,axiom,
% 0.33/0.63 ! [B: rat,C: rat,A: rat] :
% 0.33/0.63 ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) )
% 0.33/0.63 = A )
% 0.33/0.63 = ( ( ( C != zero_zero_rat )
% 0.33/0.63 => ( ( uminus_uminus_rat @ B )
% 0.33/0.63 = ( times_times_rat @ A @ C ) ) )
% 0.33/0.63 & ( ( C = zero_zero_rat )
% 0.33/0.63 => ( A = zero_zero_rat ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % minus_divide_eq_eq
% 0.33/0.63 thf(fact_5961_eq__minus__divide__eq,axiom,
% 0.33/0.63 ! [A: real,B: real,C: real] :
% 0.33/0.63 ( ( A
% 0.33/0.63 = ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.33/0.63 = ( ( ( C != zero_zero_real )
% 0.33/0.63 => ( ( times_times_real @ A @ C )
% 0.33/0.63 = ( uminus_uminus_real @ B ) ) )
% 0.33/0.63 & ( ( C = zero_zero_real )
% 0.33/0.63 => ( A = zero_zero_real ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % eq_minus_divide_eq
% 0.33/0.63 thf(fact_5962_eq__minus__divide__eq,axiom,
% 0.33/0.63 ! [A: complex,B: complex,C: complex] :
% 0.33/0.63 ( ( A
% 0.33/0.63 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) ) )
% 0.33/0.63 = ( ( ( C != zero_zero_complex )
% 0.33/0.63 => ( ( times_times_complex @ A @ C )
% 0.33/0.63 = ( uminus1482373934393186551omplex @ B ) ) )
% 0.33/0.63 & ( ( C = zero_zero_complex )
% 0.33/0.63 => ( A = zero_zero_complex ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % eq_minus_divide_eq
% 0.33/0.63 thf(fact_5963_eq__minus__divide__eq,axiom,
% 0.33/0.63 ! [A: rat,B: rat,C: rat] :
% 0.33/0.63 ( ( A
% 0.33/0.63 = ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.33/0.63 = ( ( ( C != zero_zero_rat )
% 0.33/0.63 => ( ( times_times_rat @ A @ C )
% 0.33/0.63 = ( uminus_uminus_rat @ B ) ) )
% 0.33/0.63 & ( ( C = zero_zero_rat )
% 0.33/0.63 => ( A = zero_zero_rat ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % eq_minus_divide_eq
% 0.33/0.63 thf(fact_5964_mult__1s__ring__1_I1_J,axiom,
% 0.33/0.63 ! [B: real] :
% 0.33/0.63 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B )
% 0.33/0.63 = ( uminus_uminus_real @ B ) ) ).
% 0.33/0.63
% 0.33/0.63 % mult_1s_ring_1(1)
% 0.33/0.63 thf(fact_5965_mult__1s__ring__1_I1_J,axiom,
% 0.33/0.63 ! [B: int] :
% 0.33/0.63 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
% 0.33/0.63 = ( uminus_uminus_int @ B ) ) ).
% 0.33/0.63
% 0.33/0.63 % mult_1s_ring_1(1)
% 0.33/0.63 thf(fact_5966_mult__1s__ring__1_I1_J,axiom,
% 0.33/0.63 ! [B: complex] :
% 0.33/0.63 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B )
% 0.33/0.63 = ( uminus1482373934393186551omplex @ B ) ) ).
% 0.33/0.63
% 0.33/0.63 % mult_1s_ring_1(1)
% 0.33/0.63 thf(fact_5967_mult__1s__ring__1_I1_J,axiom,
% 0.33/0.63 ! [B: rat] :
% 0.33/0.63 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B )
% 0.33/0.63 = ( uminus_uminus_rat @ B ) ) ).
% 0.33/0.63
% 0.33/0.63 % mult_1s_ring_1(1)
% 0.33/0.63 thf(fact_5968_mult__1s__ring__1_I1_J,axiom,
% 0.33/0.63 ! [B: code_integer] :
% 0.33/0.63 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B )
% 0.33/0.63 = ( uminus1351360451143612070nteger @ B ) ) ).
% 0.33/0.63
% 0.33/0.63 % mult_1s_ring_1(1)
% 0.33/0.63 thf(fact_5969_mult__1s__ring__1_I2_J,axiom,
% 0.33/0.63 ! [B: real] :
% 0.33/0.63 ( ( times_times_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
% 0.33/0.63 = ( uminus_uminus_real @ B ) ) ).
% 0.33/0.63
% 0.33/0.63 % mult_1s_ring_1(2)
% 0.33/0.63 thf(fact_5970_mult__1s__ring__1_I2_J,axiom,
% 0.33/0.63 ! [B: int] :
% 0.33/0.63 ( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
% 0.33/0.63 = ( uminus_uminus_int @ B ) ) ).
% 0.33/0.63
% 0.33/0.63 % mult_1s_ring_1(2)
% 0.33/0.63 thf(fact_5971_mult__1s__ring__1_I2_J,axiom,
% 0.33/0.63 ! [B: complex] :
% 0.33/0.63 ( ( times_times_complex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
% 0.33/0.63 = ( uminus1482373934393186551omplex @ B ) ) ).
% 0.33/0.63
% 0.33/0.63 % mult_1s_ring_1(2)
% 0.33/0.63 thf(fact_5972_mult__1s__ring__1_I2_J,axiom,
% 0.33/0.63 ! [B: rat] :
% 0.33/0.63 ( ( times_times_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
% 0.33/0.63 = ( uminus_uminus_rat @ B ) ) ).
% 0.33/0.63
% 0.33/0.63 % mult_1s_ring_1(2)
% 0.33/0.63 thf(fact_5973_mult__1s__ring__1_I2_J,axiom,
% 0.33/0.63 ! [B: code_integer] :
% 0.33/0.63 ( ( times_3573771949741848930nteger @ B @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
% 0.33/0.63 = ( uminus1351360451143612070nteger @ B ) ) ).
% 0.33/0.63
% 0.33/0.63 % mult_1s_ring_1(2)
% 0.33/0.63 thf(fact_5974_divide__eq__minus__1__iff,axiom,
% 0.33/0.63 ! [A: real,B: real] :
% 0.33/0.63 ( ( ( divide_divide_real @ A @ B )
% 0.33/0.63 = ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.63 = ( ( B != zero_zero_real )
% 0.33/0.63 & ( A
% 0.33/0.63 = ( uminus_uminus_real @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % divide_eq_minus_1_iff
% 0.33/0.63 thf(fact_5975_divide__eq__minus__1__iff,axiom,
% 0.33/0.63 ! [A: complex,B: complex] :
% 0.33/0.63 ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.33/0.63 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.33/0.63 = ( ( B != zero_zero_complex )
% 0.33/0.63 & ( A
% 0.33/0.63 = ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % divide_eq_minus_1_iff
% 0.33/0.63 thf(fact_5976_divide__eq__minus__1__iff,axiom,
% 0.33/0.63 ! [A: rat,B: rat] :
% 0.33/0.63 ( ( ( divide_divide_rat @ A @ B )
% 0.33/0.63 = ( uminus_uminus_rat @ one_one_rat ) )
% 0.33/0.63 = ( ( B != zero_zero_rat )
% 0.33/0.63 & ( A
% 0.33/0.63 = ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % divide_eq_minus_1_iff
% 0.33/0.63 thf(fact_5977_uminus__numeral__One,axiom,
% 0.33/0.63 ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
% 0.33/0.63 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.63
% 0.33/0.63 % uminus_numeral_One
% 0.33/0.63 thf(fact_5978_uminus__numeral__One,axiom,
% 0.33/0.63 ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
% 0.33/0.63 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.63
% 0.33/0.63 % uminus_numeral_One
% 0.33/0.63 thf(fact_5979_uminus__numeral__One,axiom,
% 0.33/0.63 ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
% 0.33/0.63 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.33/0.63
% 0.33/0.63 % uminus_numeral_One
% 0.33/0.63 thf(fact_5980_uminus__numeral__One,axiom,
% 0.33/0.63 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
% 0.33/0.63 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.63
% 0.33/0.63 % uminus_numeral_One
% 0.33/0.63 thf(fact_5981_uminus__numeral__One,axiom,
% 0.33/0.63 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
% 0.33/0.63 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.63
% 0.33/0.63 % uminus_numeral_One
% 0.33/0.63 thf(fact_5982_power__minus,axiom,
% 0.33/0.63 ! [A: real,N2: nat] :
% 0.33/0.63 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 )
% 0.33/0.63 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( power_power_real @ A @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % power_minus
% 0.33/0.63 thf(fact_5983_power__minus,axiom,
% 0.33/0.63 ! [A: int,N2: nat] :
% 0.33/0.63 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
% 0.33/0.63 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ A @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % power_minus
% 0.33/0.63 thf(fact_5984_power__minus,axiom,
% 0.33/0.63 ! [A: complex,N2: nat] :
% 0.33/0.63 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 )
% 0.33/0.63 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( power_power_complex @ A @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % power_minus
% 0.33/0.63 thf(fact_5985_power__minus,axiom,
% 0.33/0.63 ! [A: rat,N2: nat] :
% 0.33/0.63 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 )
% 0.33/0.63 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( power_power_rat @ A @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % power_minus
% 0.33/0.63 thf(fact_5986_power__minus,axiom,
% 0.33/0.63 ! [A: code_integer,N2: nat] :
% 0.33/0.63 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 )
% 0.33/0.63 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % power_minus
% 0.33/0.63 thf(fact_5987_power__minus__Bit0,axiom,
% 0.33/0.63 ! [X: real,K: num] :
% 0.33/0.63 ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.33/0.63 = ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % power_minus_Bit0
% 0.33/0.63 thf(fact_5988_power__minus__Bit0,axiom,
% 0.33/0.63 ! [X: int,K: num] :
% 0.33/0.63 ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.33/0.63 = ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % power_minus_Bit0
% 0.33/0.63 thf(fact_5989_power__minus__Bit0,axiom,
% 0.33/0.63 ! [X: complex,K: num] :
% 0.33/0.63 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.33/0.63 = ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % power_minus_Bit0
% 0.33/0.63 thf(fact_5990_power__minus__Bit0,axiom,
% 0.33/0.63 ! [X: rat,K: num] :
% 0.33/0.63 ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.33/0.63 = ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % power_minus_Bit0
% 0.33/0.63 thf(fact_5991_power__minus__Bit0,axiom,
% 0.33/0.63 ! [X: code_integer,K: num] :
% 0.33/0.63 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.33/0.63 = ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % power_minus_Bit0
% 0.33/0.63 thf(fact_5992_sum__nonneg__0,axiom,
% 0.33/0.63 ! [S3: set_real,F: real > real,I2: real] :
% 0.33/0.63 ( ( finite_finite_real @ S3 )
% 0.33/0.63 => ( ! [I4: real] :
% 0.33/0.63 ( ( member_real @ I4 @ S3 )
% 0.33/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ( ( groups8097168146408367636l_real @ F @ S3 )
% 0.33/0.63 = zero_zero_real )
% 0.33/0.63 => ( ( member_real @ I2 @ S3 )
% 0.33/0.63 => ( ( F @ I2 )
% 0.33/0.63 = zero_zero_real ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_0
% 0.33/0.63 thf(fact_5993_sum__nonneg__0,axiom,
% 0.33/0.63 ! [S3: set_int,F: int > real,I2: int] :
% 0.33/0.63 ( ( finite_finite_int @ S3 )
% 0.33/0.63 => ( ! [I4: int] :
% 0.33/0.63 ( ( member_int @ I4 @ S3 )
% 0.33/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ( ( groups8778361861064173332t_real @ F @ S3 )
% 0.33/0.63 = zero_zero_real )
% 0.33/0.63 => ( ( member_int @ I2 @ S3 )
% 0.33/0.63 => ( ( F @ I2 )
% 0.33/0.63 = zero_zero_real ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_0
% 0.33/0.63 thf(fact_5994_sum__nonneg__0,axiom,
% 0.33/0.63 ! [S3: set_complex,F: complex > real,I2: complex] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ S3 )
% 0.33/0.63 => ( ! [I4: complex] :
% 0.33/0.63 ( ( member_complex @ I4 @ S3 )
% 0.33/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ( ( groups5808333547571424918x_real @ F @ S3 )
% 0.33/0.63 = zero_zero_real )
% 0.33/0.63 => ( ( member_complex @ I2 @ S3 )
% 0.33/0.63 => ( ( F @ I2 )
% 0.33/0.63 = zero_zero_real ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_0
% 0.33/0.63 thf(fact_5995_sum__nonneg__0,axiom,
% 0.33/0.63 ! [S3: set_real,F: real > rat,I2: real] :
% 0.33/0.63 ( ( finite_finite_real @ S3 )
% 0.33/0.63 => ( ! [I4: real] :
% 0.33/0.63 ( ( member_real @ I4 @ S3 )
% 0.33/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ( ( groups1300246762558778688al_rat @ F @ S3 )
% 0.33/0.63 = zero_zero_rat )
% 0.33/0.63 => ( ( member_real @ I2 @ S3 )
% 0.33/0.63 => ( ( F @ I2 )
% 0.33/0.63 = zero_zero_rat ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_0
% 0.33/0.63 thf(fact_5996_sum__nonneg__0,axiom,
% 0.33/0.63 ! [S3: set_nat,F: nat > rat,I2: nat] :
% 0.33/0.63 ( ( finite_finite_nat @ S3 )
% 0.33/0.63 => ( ! [I4: nat] :
% 0.33/0.63 ( ( member_nat @ I4 @ S3 )
% 0.33/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ( ( groups2906978787729119204at_rat @ F @ S3 )
% 0.33/0.63 = zero_zero_rat )
% 0.33/0.63 => ( ( member_nat @ I2 @ S3 )
% 0.33/0.63 => ( ( F @ I2 )
% 0.33/0.63 = zero_zero_rat ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_0
% 0.33/0.63 thf(fact_5997_sum__nonneg__0,axiom,
% 0.33/0.63 ! [S3: set_int,F: int > rat,I2: int] :
% 0.33/0.63 ( ( finite_finite_int @ S3 )
% 0.33/0.63 => ( ! [I4: int] :
% 0.33/0.63 ( ( member_int @ I4 @ S3 )
% 0.33/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ( ( groups3906332499630173760nt_rat @ F @ S3 )
% 0.33/0.63 = zero_zero_rat )
% 0.33/0.63 => ( ( member_int @ I2 @ S3 )
% 0.33/0.63 => ( ( F @ I2 )
% 0.33/0.63 = zero_zero_rat ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_0
% 0.33/0.63 thf(fact_5998_sum__nonneg__0,axiom,
% 0.33/0.63 ! [S3: set_complex,F: complex > rat,I2: complex] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ S3 )
% 0.33/0.63 => ( ! [I4: complex] :
% 0.33/0.63 ( ( member_complex @ I4 @ S3 )
% 0.33/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ( ( groups5058264527183730370ex_rat @ F @ S3 )
% 0.33/0.63 = zero_zero_rat )
% 0.33/0.63 => ( ( member_complex @ I2 @ S3 )
% 0.33/0.63 => ( ( F @ I2 )
% 0.33/0.63 = zero_zero_rat ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_0
% 0.33/0.63 thf(fact_5999_sum__nonneg__0,axiom,
% 0.33/0.63 ! [S3: set_real,F: real > nat,I2: real] :
% 0.33/0.63 ( ( finite_finite_real @ S3 )
% 0.33/0.63 => ( ! [I4: real] :
% 0.33/0.63 ( ( member_real @ I4 @ S3 )
% 0.33/0.63 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ( ( groups1935376822645274424al_nat @ F @ S3 )
% 0.33/0.63 = zero_zero_nat )
% 0.33/0.63 => ( ( member_real @ I2 @ S3 )
% 0.33/0.63 => ( ( F @ I2 )
% 0.33/0.63 = zero_zero_nat ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_0
% 0.33/0.63 thf(fact_6000_sum__nonneg__0,axiom,
% 0.33/0.63 ! [S3: set_int,F: int > nat,I2: int] :
% 0.33/0.63 ( ( finite_finite_int @ S3 )
% 0.33/0.63 => ( ! [I4: int] :
% 0.33/0.63 ( ( member_int @ I4 @ S3 )
% 0.33/0.63 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ( ( groups4541462559716669496nt_nat @ F @ S3 )
% 0.33/0.63 = zero_zero_nat )
% 0.33/0.63 => ( ( member_int @ I2 @ S3 )
% 0.33/0.63 => ( ( F @ I2 )
% 0.33/0.63 = zero_zero_nat ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_0
% 0.33/0.63 thf(fact_6001_sum__nonneg__0,axiom,
% 0.33/0.63 ! [S3: set_complex,F: complex > nat,I2: complex] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ S3 )
% 0.33/0.63 => ( ! [I4: complex] :
% 0.33/0.63 ( ( member_complex @ I4 @ S3 )
% 0.33/0.63 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ( ( groups5693394587270226106ex_nat @ F @ S3 )
% 0.33/0.63 = zero_zero_nat )
% 0.33/0.63 => ( ( member_complex @ I2 @ S3 )
% 0.33/0.63 => ( ( F @ I2 )
% 0.33/0.63 = zero_zero_nat ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_0
% 0.33/0.63 thf(fact_6002_sum__nonneg__leq__bound,axiom,
% 0.33/0.63 ! [S3: set_real,F: real > real,B3: real,I2: real] :
% 0.33/0.63 ( ( finite_finite_real @ S3 )
% 0.33/0.63 => ( ! [I4: real] :
% 0.33/0.63 ( ( member_real @ I4 @ S3 )
% 0.33/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ( ( groups8097168146408367636l_real @ F @ S3 )
% 0.33/0.63 = B3 )
% 0.33/0.63 => ( ( member_real @ I2 @ S3 )
% 0.33/0.63 => ( ord_less_eq_real @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_leq_bound
% 0.33/0.63 thf(fact_6003_sum__nonneg__leq__bound,axiom,
% 0.33/0.63 ! [S3: set_int,F: int > real,B3: real,I2: int] :
% 0.33/0.63 ( ( finite_finite_int @ S3 )
% 0.33/0.63 => ( ! [I4: int] :
% 0.33/0.63 ( ( member_int @ I4 @ S3 )
% 0.33/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ( ( groups8778361861064173332t_real @ F @ S3 )
% 0.33/0.63 = B3 )
% 0.33/0.63 => ( ( member_int @ I2 @ S3 )
% 0.33/0.63 => ( ord_less_eq_real @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_leq_bound
% 0.33/0.63 thf(fact_6004_sum__nonneg__leq__bound,axiom,
% 0.33/0.63 ! [S3: set_complex,F: complex > real,B3: real,I2: complex] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ S3 )
% 0.33/0.63 => ( ! [I4: complex] :
% 0.33/0.63 ( ( member_complex @ I4 @ S3 )
% 0.33/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ( ( groups5808333547571424918x_real @ F @ S3 )
% 0.33/0.63 = B3 )
% 0.33/0.63 => ( ( member_complex @ I2 @ S3 )
% 0.33/0.63 => ( ord_less_eq_real @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_leq_bound
% 0.33/0.63 thf(fact_6005_sum__nonneg__leq__bound,axiom,
% 0.33/0.63 ! [S3: set_real,F: real > rat,B3: rat,I2: real] :
% 0.33/0.63 ( ( finite_finite_real @ S3 )
% 0.33/0.63 => ( ! [I4: real] :
% 0.33/0.63 ( ( member_real @ I4 @ S3 )
% 0.33/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ( ( groups1300246762558778688al_rat @ F @ S3 )
% 0.33/0.63 = B3 )
% 0.33/0.63 => ( ( member_real @ I2 @ S3 )
% 0.33/0.63 => ( ord_less_eq_rat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_leq_bound
% 0.33/0.63 thf(fact_6006_sum__nonneg__leq__bound,axiom,
% 0.33/0.63 ! [S3: set_nat,F: nat > rat,B3: rat,I2: nat] :
% 0.33/0.63 ( ( finite_finite_nat @ S3 )
% 0.33/0.63 => ( ! [I4: nat] :
% 0.33/0.63 ( ( member_nat @ I4 @ S3 )
% 0.33/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ( ( groups2906978787729119204at_rat @ F @ S3 )
% 0.33/0.63 = B3 )
% 0.33/0.63 => ( ( member_nat @ I2 @ S3 )
% 0.33/0.63 => ( ord_less_eq_rat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_leq_bound
% 0.33/0.63 thf(fact_6007_sum__nonneg__leq__bound,axiom,
% 0.33/0.63 ! [S3: set_int,F: int > rat,B3: rat,I2: int] :
% 0.33/0.63 ( ( finite_finite_int @ S3 )
% 0.33/0.63 => ( ! [I4: int] :
% 0.33/0.63 ( ( member_int @ I4 @ S3 )
% 0.33/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ( ( groups3906332499630173760nt_rat @ F @ S3 )
% 0.33/0.63 = B3 )
% 0.33/0.63 => ( ( member_int @ I2 @ S3 )
% 0.33/0.63 => ( ord_less_eq_rat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_leq_bound
% 0.33/0.63 thf(fact_6008_sum__nonneg__leq__bound,axiom,
% 0.33/0.63 ! [S3: set_complex,F: complex > rat,B3: rat,I2: complex] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ S3 )
% 0.33/0.63 => ( ! [I4: complex] :
% 0.33/0.63 ( ( member_complex @ I4 @ S3 )
% 0.33/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ( ( groups5058264527183730370ex_rat @ F @ S3 )
% 0.33/0.63 = B3 )
% 0.33/0.63 => ( ( member_complex @ I2 @ S3 )
% 0.33/0.63 => ( ord_less_eq_rat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_leq_bound
% 0.33/0.63 thf(fact_6009_sum__nonneg__leq__bound,axiom,
% 0.33/0.63 ! [S3: set_real,F: real > nat,B3: nat,I2: real] :
% 0.33/0.63 ( ( finite_finite_real @ S3 )
% 0.33/0.63 => ( ! [I4: real] :
% 0.33/0.63 ( ( member_real @ I4 @ S3 )
% 0.33/0.63 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ( ( groups1935376822645274424al_nat @ F @ S3 )
% 0.33/0.63 = B3 )
% 0.33/0.63 => ( ( member_real @ I2 @ S3 )
% 0.33/0.63 => ( ord_less_eq_nat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_leq_bound
% 0.33/0.63 thf(fact_6010_sum__nonneg__leq__bound,axiom,
% 0.33/0.63 ! [S3: set_int,F: int > nat,B3: nat,I2: int] :
% 0.33/0.63 ( ( finite_finite_int @ S3 )
% 0.33/0.63 => ( ! [I4: int] :
% 0.33/0.63 ( ( member_int @ I4 @ S3 )
% 0.33/0.63 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ( ( groups4541462559716669496nt_nat @ F @ S3 )
% 0.33/0.63 = B3 )
% 0.33/0.63 => ( ( member_int @ I2 @ S3 )
% 0.33/0.63 => ( ord_less_eq_nat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_leq_bound
% 0.33/0.63 thf(fact_6011_sum__nonneg__leq__bound,axiom,
% 0.33/0.63 ! [S3: set_complex,F: complex > nat,B3: nat,I2: complex] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ S3 )
% 0.33/0.63 => ( ! [I4: complex] :
% 0.33/0.63 ( ( member_complex @ I4 @ S3 )
% 0.33/0.63 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ( ( groups5693394587270226106ex_nat @ F @ S3 )
% 0.33/0.63 = B3 )
% 0.33/0.63 => ( ( member_complex @ I2 @ S3 )
% 0.33/0.63 => ( ord_less_eq_nat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_nonneg_leq_bound
% 0.33/0.63 thf(fact_6012_sum_Osetdiff__irrelevant,axiom,
% 0.33/0.63 ! [A2: set_int,G: int > complex] :
% 0.33/0.63 ( ( finite_finite_int @ A2 )
% 0.33/0.63 => ( ( groups3049146728041665814omplex @ G
% 0.33/0.63 @ ( minus_minus_set_int @ A2
% 0.33/0.63 @ ( collect_int
% 0.33/0.63 @ ^ [X2: int] :
% 0.33/0.63 ( ( G @ X2 )
% 0.33/0.63 = zero_zero_complex ) ) ) )
% 0.33/0.63 = ( groups3049146728041665814omplex @ G @ A2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.setdiff_irrelevant
% 0.33/0.63 thf(fact_6013_sum_Osetdiff__irrelevant,axiom,
% 0.33/0.63 ! [A2: set_int,G: int > real] :
% 0.33/0.63 ( ( finite_finite_int @ A2 )
% 0.33/0.63 => ( ( groups8778361861064173332t_real @ G
% 0.33/0.63 @ ( minus_minus_set_int @ A2
% 0.33/0.63 @ ( collect_int
% 0.33/0.63 @ ^ [X2: int] :
% 0.33/0.63 ( ( G @ X2 )
% 0.33/0.63 = zero_zero_real ) ) ) )
% 0.33/0.63 = ( groups8778361861064173332t_real @ G @ A2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.setdiff_irrelevant
% 0.33/0.63 thf(fact_6014_sum_Osetdiff__irrelevant,axiom,
% 0.33/0.63 ! [A2: set_complex,G: complex > real] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.63 => ( ( groups5808333547571424918x_real @ G
% 0.33/0.63 @ ( minus_811609699411566653omplex @ A2
% 0.33/0.63 @ ( collect_complex
% 0.33/0.63 @ ^ [X2: complex] :
% 0.33/0.63 ( ( G @ X2 )
% 0.33/0.63 = zero_zero_real ) ) ) )
% 0.33/0.63 = ( groups5808333547571424918x_real @ G @ A2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.setdiff_irrelevant
% 0.33/0.63 thf(fact_6015_sum_Osetdiff__irrelevant,axiom,
% 0.33/0.63 ! [A2: set_int,G: int > rat] :
% 0.33/0.63 ( ( finite_finite_int @ A2 )
% 0.33/0.63 => ( ( groups3906332499630173760nt_rat @ G
% 0.33/0.63 @ ( minus_minus_set_int @ A2
% 0.33/0.63 @ ( collect_int
% 0.33/0.63 @ ^ [X2: int] :
% 0.33/0.63 ( ( G @ X2 )
% 0.33/0.63 = zero_zero_rat ) ) ) )
% 0.33/0.63 = ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.setdiff_irrelevant
% 0.33/0.63 thf(fact_6016_sum_Osetdiff__irrelevant,axiom,
% 0.33/0.63 ! [A2: set_complex,G: complex > rat] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.63 => ( ( groups5058264527183730370ex_rat @ G
% 0.33/0.63 @ ( minus_811609699411566653omplex @ A2
% 0.33/0.63 @ ( collect_complex
% 0.33/0.63 @ ^ [X2: complex] :
% 0.33/0.63 ( ( G @ X2 )
% 0.33/0.63 = zero_zero_rat ) ) ) )
% 0.33/0.63 = ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.setdiff_irrelevant
% 0.33/0.63 thf(fact_6017_sum_Osetdiff__irrelevant,axiom,
% 0.33/0.63 ! [A2: set_int,G: int > nat] :
% 0.33/0.63 ( ( finite_finite_int @ A2 )
% 0.33/0.63 => ( ( groups4541462559716669496nt_nat @ G
% 0.33/0.63 @ ( minus_minus_set_int @ A2
% 0.33/0.63 @ ( collect_int
% 0.33/0.63 @ ^ [X2: int] :
% 0.33/0.63 ( ( G @ X2 )
% 0.33/0.63 = zero_zero_nat ) ) ) )
% 0.33/0.63 = ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.setdiff_irrelevant
% 0.33/0.63 thf(fact_6018_sum_Osetdiff__irrelevant,axiom,
% 0.33/0.63 ! [A2: set_complex,G: complex > nat] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.63 => ( ( groups5693394587270226106ex_nat @ G
% 0.33/0.63 @ ( minus_811609699411566653omplex @ A2
% 0.33/0.63 @ ( collect_complex
% 0.33/0.63 @ ^ [X2: complex] :
% 0.33/0.63 ( ( G @ X2 )
% 0.33/0.63 = zero_zero_nat ) ) ) )
% 0.33/0.63 = ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.setdiff_irrelevant
% 0.33/0.63 thf(fact_6019_sum_Osetdiff__irrelevant,axiom,
% 0.33/0.63 ! [A2: set_complex,G: complex > int] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.63 => ( ( groups5690904116761175830ex_int @ G
% 0.33/0.63 @ ( minus_811609699411566653omplex @ A2
% 0.33/0.63 @ ( collect_complex
% 0.33/0.63 @ ^ [X2: complex] :
% 0.33/0.63 ( ( G @ X2 )
% 0.33/0.63 = zero_zero_int ) ) ) )
% 0.33/0.63 = ( groups5690904116761175830ex_int @ G @ A2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.setdiff_irrelevant
% 0.33/0.63 thf(fact_6020_sum_Osetdiff__irrelevant,axiom,
% 0.33/0.63 ! [A2: set_nat,G: nat > complex] :
% 0.33/0.63 ( ( finite_finite_nat @ A2 )
% 0.33/0.63 => ( ( groups2073611262835488442omplex @ G
% 0.33/0.63 @ ( minus_minus_set_nat @ A2
% 0.33/0.63 @ ( collect_nat
% 0.33/0.63 @ ^ [X2: nat] :
% 0.33/0.63 ( ( G @ X2 )
% 0.33/0.63 = zero_zero_complex ) ) ) )
% 0.33/0.63 = ( groups2073611262835488442omplex @ G @ A2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.setdiff_irrelevant
% 0.33/0.63 thf(fact_6021_sum_Osetdiff__irrelevant,axiom,
% 0.33/0.63 ! [A2: set_nat,G: nat > rat] :
% 0.33/0.63 ( ( finite_finite_nat @ A2 )
% 0.33/0.63 => ( ( groups2906978787729119204at_rat @ G
% 0.33/0.63 @ ( minus_minus_set_nat @ A2
% 0.33/0.63 @ ( collect_nat
% 0.33/0.63 @ ^ [X2: nat] :
% 0.33/0.63 ( ( G @ X2 )
% 0.33/0.63 = zero_zero_rat ) ) ) )
% 0.33/0.63 = ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.setdiff_irrelevant
% 0.33/0.63 thf(fact_6022_real__0__less__add__iff,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
% 0.33/0.63 = ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_0_less_add_iff
% 0.33/0.63 thf(fact_6023_real__add__less__0__iff,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
% 0.33/0.63 = ( ord_less_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_add_less_0_iff
% 0.33/0.63 thf(fact_6024_real__add__le__0__iff,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
% 0.33/0.63 = ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_add_le_0_iff
% 0.33/0.63 thf(fact_6025_real__0__le__add__iff,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
% 0.33/0.63 = ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% 0.33/0.63
% 0.33/0.63 % real_0_le_add_iff
% 0.33/0.63 thf(fact_6026_zmod__zminus1__eq__if,axiom,
% 0.33/0.63 ! [A: int,B: int] :
% 0.33/0.63 ( ( ( ( modulo_modulo_int @ A @ B )
% 0.33/0.63 = zero_zero_int )
% 0.33/0.63 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 0.33/0.63 = zero_zero_int ) )
% 0.33/0.63 & ( ( ( modulo_modulo_int @ A @ B )
% 0.33/0.63 != zero_zero_int )
% 0.33/0.63 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 0.33/0.63 = ( minus_minus_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % zmod_zminus1_eq_if
% 0.33/0.63 thf(fact_6027_zmod__zminus2__eq__if,axiom,
% 0.33/0.63 ! [A: int,B: int] :
% 0.33/0.63 ( ( ( ( modulo_modulo_int @ A @ B )
% 0.33/0.63 = zero_zero_int )
% 0.33/0.63 => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 0.33/0.63 = zero_zero_int ) )
% 0.33/0.63 & ( ( ( modulo_modulo_int @ A @ B )
% 0.33/0.63 != zero_zero_int )
% 0.33/0.63 => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 0.33/0.63 = ( minus_minus_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % zmod_zminus2_eq_if
% 0.33/0.63 thf(fact_6028_sum__pos2,axiom,
% 0.33/0.63 ! [I6: set_real,I2: real,F: real > real] :
% 0.33/0.63 ( ( finite_finite_real @ I6 )
% 0.33/0.63 => ( ( member_real @ I2 @ I6 )
% 0.33/0.63 => ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
% 0.33/0.63 => ( ! [I4: real] :
% 0.33/0.63 ( ( member_real @ I4 @ I6 )
% 0.33/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I6 ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_pos2
% 0.33/0.63 thf(fact_6029_sum__pos2,axiom,
% 0.33/0.63 ! [I6: set_int,I2: int,F: int > real] :
% 0.33/0.63 ( ( finite_finite_int @ I6 )
% 0.33/0.63 => ( ( member_int @ I2 @ I6 )
% 0.33/0.63 => ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
% 0.33/0.63 => ( ! [I4: int] :
% 0.33/0.63 ( ( member_int @ I4 @ I6 )
% 0.33/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I6 ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_pos2
% 0.33/0.63 thf(fact_6030_sum__pos2,axiom,
% 0.33/0.63 ! [I6: set_complex,I2: complex,F: complex > real] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ I6 )
% 0.33/0.63 => ( ( member_complex @ I2 @ I6 )
% 0.33/0.63 => ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
% 0.33/0.63 => ( ! [I4: complex] :
% 0.33/0.63 ( ( member_complex @ I4 @ I6 )
% 0.33/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I6 ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_pos2
% 0.33/0.63 thf(fact_6031_sum__pos2,axiom,
% 0.33/0.63 ! [I6: set_real,I2: real,F: real > rat] :
% 0.33/0.63 ( ( finite_finite_real @ I6 )
% 0.33/0.63 => ( ( member_real @ I2 @ I6 )
% 0.33/0.63 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
% 0.33/0.63 => ( ! [I4: real] :
% 0.33/0.63 ( ( member_real @ I4 @ I6 )
% 0.33/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I6 ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_pos2
% 0.33/0.63 thf(fact_6032_sum__pos2,axiom,
% 0.33/0.63 ! [I6: set_nat,I2: nat,F: nat > rat] :
% 0.33/0.63 ( ( finite_finite_nat @ I6 )
% 0.33/0.63 => ( ( member_nat @ I2 @ I6 )
% 0.33/0.63 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
% 0.33/0.63 => ( ! [I4: nat] :
% 0.33/0.63 ( ( member_nat @ I4 @ I6 )
% 0.33/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I6 ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_pos2
% 0.33/0.63 thf(fact_6033_sum__pos2,axiom,
% 0.33/0.63 ! [I6: set_int,I2: int,F: int > rat] :
% 0.33/0.63 ( ( finite_finite_int @ I6 )
% 0.33/0.63 => ( ( member_int @ I2 @ I6 )
% 0.33/0.63 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
% 0.33/0.63 => ( ! [I4: int] :
% 0.33/0.63 ( ( member_int @ I4 @ I6 )
% 0.33/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I6 ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_pos2
% 0.33/0.63 thf(fact_6034_sum__pos2,axiom,
% 0.33/0.63 ! [I6: set_complex,I2: complex,F: complex > rat] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ I6 )
% 0.33/0.63 => ( ( member_complex @ I2 @ I6 )
% 0.33/0.63 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
% 0.33/0.63 => ( ! [I4: complex] :
% 0.33/0.63 ( ( member_complex @ I4 @ I6 )
% 0.33/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I6 ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_pos2
% 0.33/0.63 thf(fact_6035_sum__pos2,axiom,
% 0.33/0.63 ! [I6: set_real,I2: real,F: real > nat] :
% 0.33/0.63 ( ( finite_finite_real @ I6 )
% 0.33/0.63 => ( ( member_real @ I2 @ I6 )
% 0.33/0.63 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
% 0.33/0.63 => ( ! [I4: real] :
% 0.33/0.63 ( ( member_real @ I4 @ I6 )
% 0.33/0.63 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I6 ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_pos2
% 0.33/0.63 thf(fact_6036_sum__pos2,axiom,
% 0.33/0.63 ! [I6: set_int,I2: int,F: int > nat] :
% 0.33/0.63 ( ( finite_finite_int @ I6 )
% 0.33/0.63 => ( ( member_int @ I2 @ I6 )
% 0.33/0.63 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
% 0.33/0.63 => ( ! [I4: int] :
% 0.33/0.63 ( ( member_int @ I4 @ I6 )
% 0.33/0.63 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I6 ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_pos2
% 0.33/0.63 thf(fact_6037_sum__pos2,axiom,
% 0.33/0.63 ! [I6: set_complex,I2: complex,F: complex > nat] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ I6 )
% 0.33/0.63 => ( ( member_complex @ I2 @ I6 )
% 0.33/0.63 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
% 0.33/0.63 => ( ! [I4: complex] :
% 0.33/0.63 ( ( member_complex @ I4 @ I6 )
% 0.33/0.63 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I6 ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_pos2
% 0.33/0.63 thf(fact_6038_sum__pos,axiom,
% 0.33/0.63 ! [I6: set_complex,F: complex > real] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ I6 )
% 0.33/0.63 => ( ( I6 != bot_bot_set_complex )
% 0.33/0.63 => ( ! [I4: complex] :
% 0.33/0.63 ( ( member_complex @ I4 @ I6 )
% 0.33/0.63 => ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I6 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_pos
% 0.33/0.63 thf(fact_6039_sum__pos,axiom,
% 0.33/0.63 ! [I6: set_int,F: int > real] :
% 0.33/0.63 ( ( finite_finite_int @ I6 )
% 0.33/0.63 => ( ( I6 != bot_bot_set_int )
% 0.33/0.63 => ( ! [I4: int] :
% 0.33/0.63 ( ( member_int @ I4 @ I6 )
% 0.33/0.63 => ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I6 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_pos
% 0.33/0.63 thf(fact_6040_sum__pos,axiom,
% 0.33/0.63 ! [I6: set_real,F: real > real] :
% 0.33/0.63 ( ( finite_finite_real @ I6 )
% 0.33/0.63 => ( ( I6 != bot_bot_set_real )
% 0.33/0.63 => ( ! [I4: real] :
% 0.33/0.63 ( ( member_real @ I4 @ I6 )
% 0.33/0.63 => ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I6 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_pos
% 0.33/0.63 thf(fact_6041_sum__pos,axiom,
% 0.33/0.63 ! [I6: set_complex,F: complex > rat] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ I6 )
% 0.33/0.63 => ( ( I6 != bot_bot_set_complex )
% 0.33/0.63 => ( ! [I4: complex] :
% 0.33/0.63 ( ( member_complex @ I4 @ I6 )
% 0.33/0.63 => ( ord_less_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I6 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_pos
% 0.33/0.63 thf(fact_6042_sum__pos,axiom,
% 0.33/0.63 ! [I6: set_nat,F: nat > rat] :
% 0.33/0.63 ( ( finite_finite_nat @ I6 )
% 0.33/0.63 => ( ( I6 != bot_bot_set_nat )
% 0.33/0.63 => ( ! [I4: nat] :
% 0.33/0.63 ( ( member_nat @ I4 @ I6 )
% 0.33/0.63 => ( ord_less_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I6 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_pos
% 0.33/0.63 thf(fact_6043_sum__pos,axiom,
% 0.33/0.63 ! [I6: set_int,F: int > rat] :
% 0.33/0.63 ( ( finite_finite_int @ I6 )
% 0.33/0.63 => ( ( I6 != bot_bot_set_int )
% 0.33/0.63 => ( ! [I4: int] :
% 0.33/0.63 ( ( member_int @ I4 @ I6 )
% 0.33/0.63 => ( ord_less_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I6 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_pos
% 0.33/0.63 thf(fact_6044_sum__pos,axiom,
% 0.33/0.63 ! [I6: set_real,F: real > rat] :
% 0.33/0.63 ( ( finite_finite_real @ I6 )
% 0.33/0.63 => ( ( I6 != bot_bot_set_real )
% 0.33/0.63 => ( ! [I4: real] :
% 0.33/0.63 ( ( member_real @ I4 @ I6 )
% 0.33/0.63 => ( ord_less_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I6 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_pos
% 0.33/0.63 thf(fact_6045_sum__pos,axiom,
% 0.33/0.63 ! [I6: set_complex,F: complex > nat] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ I6 )
% 0.33/0.63 => ( ( I6 != bot_bot_set_complex )
% 0.33/0.63 => ( ! [I4: complex] :
% 0.33/0.63 ( ( member_complex @ I4 @ I6 )
% 0.33/0.63 => ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I6 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_pos
% 0.33/0.63 thf(fact_6046_sum__pos,axiom,
% 0.33/0.63 ! [I6: set_int,F: int > nat] :
% 0.33/0.63 ( ( finite_finite_int @ I6 )
% 0.33/0.63 => ( ( I6 != bot_bot_set_int )
% 0.33/0.63 => ( ! [I4: int] :
% 0.33/0.63 ( ( member_int @ I4 @ I6 )
% 0.33/0.63 => ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I6 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_pos
% 0.33/0.63 thf(fact_6047_sum__pos,axiom,
% 0.33/0.63 ! [I6: set_real,F: real > nat] :
% 0.33/0.63 ( ( finite_finite_real @ I6 )
% 0.33/0.63 => ( ( I6 != bot_bot_set_real )
% 0.33/0.63 => ( ! [I4: real] :
% 0.33/0.63 ( ( member_real @ I4 @ I6 )
% 0.33/0.63 => ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) )
% 0.33/0.63 => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I6 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_pos
% 0.33/0.63 thf(fact_6048_ln__ge__zero__imp__ge__one,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 0.33/0.63 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ln_ge_zero_imp_ge_one
% 0.33/0.63 thf(fact_6049_sum_Omono__neutral__cong__right,axiom,
% 0.33/0.63 ! [T3: set_real,S: set_real,G: real > complex,H2: real > complex] :
% 0.33/0.63 ( ( finite_finite_real @ T3 )
% 0.33/0.63 => ( ( ord_less_eq_set_real @ S @ T3 )
% 0.33/0.63 => ( ! [X4: real] :
% 0.33/0.63 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_complex ) )
% 0.33/0.63 => ( ! [X4: real] :
% 0.33/0.63 ( ( member_real @ X4 @ S )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = ( H2 @ X4 ) ) )
% 0.33/0.63 => ( ( groups5754745047067104278omplex @ G @ T3 )
% 0.33/0.63 = ( groups5754745047067104278omplex @ H2 @ S ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_cong_right
% 0.33/0.63 thf(fact_6050_sum_Omono__neutral__cong__right,axiom,
% 0.33/0.63 ! [T3: set_int,S: set_int,G: int > complex,H2: int > complex] :
% 0.33/0.63 ( ( finite_finite_int @ T3 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_complex ) )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ S )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = ( H2 @ X4 ) ) )
% 0.33/0.63 => ( ( groups3049146728041665814omplex @ G @ T3 )
% 0.33/0.63 = ( groups3049146728041665814omplex @ H2 @ S ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_cong_right
% 0.33/0.63 thf(fact_6051_sum_Omono__neutral__cong__right,axiom,
% 0.33/0.63 ! [T3: set_real,S: set_real,G: real > real,H2: real > real] :
% 0.33/0.63 ( ( finite_finite_real @ T3 )
% 0.33/0.63 => ( ( ord_less_eq_set_real @ S @ T3 )
% 0.33/0.63 => ( ! [X4: real] :
% 0.33/0.63 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ! [X4: real] :
% 0.33/0.63 ( ( member_real @ X4 @ S )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = ( H2 @ X4 ) ) )
% 0.33/0.63 => ( ( groups8097168146408367636l_real @ G @ T3 )
% 0.33/0.63 = ( groups8097168146408367636l_real @ H2 @ S ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_cong_right
% 0.33/0.63 thf(fact_6052_sum_Omono__neutral__cong__right,axiom,
% 0.33/0.63 ! [T3: set_int,S: set_int,G: int > real,H2: int > real] :
% 0.33/0.63 ( ( finite_finite_int @ T3 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ S )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = ( H2 @ X4 ) ) )
% 0.33/0.63 => ( ( groups8778361861064173332t_real @ G @ T3 )
% 0.33/0.63 = ( groups8778361861064173332t_real @ H2 @ S ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_cong_right
% 0.33/0.63 thf(fact_6053_sum_Omono__neutral__cong__right,axiom,
% 0.33/0.63 ! [T3: set_complex,S: set_complex,G: complex > real,H2: complex > real] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ T3 )
% 0.33/0.63 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ S )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = ( H2 @ X4 ) ) )
% 0.33/0.63 => ( ( groups5808333547571424918x_real @ G @ T3 )
% 0.33/0.63 = ( groups5808333547571424918x_real @ H2 @ S ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_cong_right
% 0.33/0.63 thf(fact_6054_sum_Omono__neutral__cong__right,axiom,
% 0.33/0.63 ! [T3: set_real,S: set_real,G: real > rat,H2: real > rat] :
% 0.33/0.63 ( ( finite_finite_real @ T3 )
% 0.33/0.63 => ( ( ord_less_eq_set_real @ S @ T3 )
% 0.33/0.63 => ( ! [X4: real] :
% 0.33/0.63 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_rat ) )
% 0.33/0.63 => ( ! [X4: real] :
% 0.33/0.63 ( ( member_real @ X4 @ S )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = ( H2 @ X4 ) ) )
% 0.33/0.63 => ( ( groups1300246762558778688al_rat @ G @ T3 )
% 0.33/0.63 = ( groups1300246762558778688al_rat @ H2 @ S ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_cong_right
% 0.33/0.63 thf(fact_6055_sum_Omono__neutral__cong__right,axiom,
% 0.33/0.63 ! [T3: set_int,S: set_int,G: int > rat,H2: int > rat] :
% 0.33/0.63 ( ( finite_finite_int @ T3 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_rat ) )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ S )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = ( H2 @ X4 ) ) )
% 0.33/0.63 => ( ( groups3906332499630173760nt_rat @ G @ T3 )
% 0.33/0.63 = ( groups3906332499630173760nt_rat @ H2 @ S ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_cong_right
% 0.33/0.63 thf(fact_6056_sum_Omono__neutral__cong__right,axiom,
% 0.33/0.63 ! [T3: set_complex,S: set_complex,G: complex > rat,H2: complex > rat] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ T3 )
% 0.33/0.63 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_rat ) )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ S )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = ( H2 @ X4 ) ) )
% 0.33/0.63 => ( ( groups5058264527183730370ex_rat @ G @ T3 )
% 0.33/0.63 = ( groups5058264527183730370ex_rat @ H2 @ S ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_cong_right
% 0.33/0.63 thf(fact_6057_sum_Omono__neutral__cong__right,axiom,
% 0.33/0.63 ! [T3: set_real,S: set_real,G: real > nat,H2: real > nat] :
% 0.33/0.63 ( ( finite_finite_real @ T3 )
% 0.33/0.63 => ( ( ord_less_eq_set_real @ S @ T3 )
% 0.33/0.63 => ( ! [X4: real] :
% 0.33/0.63 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_nat ) )
% 0.33/0.63 => ( ! [X4: real] :
% 0.33/0.63 ( ( member_real @ X4 @ S )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = ( H2 @ X4 ) ) )
% 0.33/0.63 => ( ( groups1935376822645274424al_nat @ G @ T3 )
% 0.33/0.63 = ( groups1935376822645274424al_nat @ H2 @ S ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_cong_right
% 0.33/0.63 thf(fact_6058_sum_Omono__neutral__cong__right,axiom,
% 0.33/0.63 ! [T3: set_int,S: set_int,G: int > nat,H2: int > nat] :
% 0.33/0.63 ( ( finite_finite_int @ T3 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_nat ) )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ S )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = ( H2 @ X4 ) ) )
% 0.33/0.63 => ( ( groups4541462559716669496nt_nat @ G @ T3 )
% 0.33/0.63 = ( groups4541462559716669496nt_nat @ H2 @ S ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_cong_right
% 0.33/0.63 thf(fact_6059_sum_Omono__neutral__cong__left,axiom,
% 0.33/0.63 ! [T3: set_real,S: set_real,H2: real > complex,G: real > complex] :
% 0.33/0.63 ( ( finite_finite_real @ T3 )
% 0.33/0.63 => ( ( ord_less_eq_set_real @ S @ T3 )
% 0.33/0.63 => ( ! [X4: real] :
% 0.33/0.63 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 0.33/0.63 => ( ( H2 @ X4 )
% 0.33/0.63 = zero_zero_complex ) )
% 0.33/0.63 => ( ! [X4: real] :
% 0.33/0.63 ( ( member_real @ X4 @ S )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = ( H2 @ X4 ) ) )
% 0.33/0.63 => ( ( groups5754745047067104278omplex @ G @ S )
% 0.33/0.63 = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_cong_left
% 0.33/0.63 thf(fact_6060_sum_Omono__neutral__cong__left,axiom,
% 0.33/0.63 ! [T3: set_int,S: set_int,H2: int > complex,G: int > complex] :
% 0.33/0.63 ( ( finite_finite_int @ T3 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.63 => ( ( H2 @ X4 )
% 0.33/0.63 = zero_zero_complex ) )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ S )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = ( H2 @ X4 ) ) )
% 0.33/0.63 => ( ( groups3049146728041665814omplex @ G @ S )
% 0.33/0.63 = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_cong_left
% 0.33/0.63 thf(fact_6061_sum_Omono__neutral__cong__left,axiom,
% 0.33/0.63 ! [T3: set_real,S: set_real,H2: real > real,G: real > real] :
% 0.33/0.63 ( ( finite_finite_real @ T3 )
% 0.33/0.63 => ( ( ord_less_eq_set_real @ S @ T3 )
% 0.33/0.63 => ( ! [X4: real] :
% 0.33/0.63 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 0.33/0.63 => ( ( H2 @ X4 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ! [X4: real] :
% 0.33/0.63 ( ( member_real @ X4 @ S )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = ( H2 @ X4 ) ) )
% 0.33/0.63 => ( ( groups8097168146408367636l_real @ G @ S )
% 0.33/0.63 = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_cong_left
% 0.33/0.63 thf(fact_6062_sum_Omono__neutral__cong__left,axiom,
% 0.33/0.63 ! [T3: set_int,S: set_int,H2: int > real,G: int > real] :
% 0.33/0.63 ( ( finite_finite_int @ T3 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.63 => ( ( H2 @ X4 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ S )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = ( H2 @ X4 ) ) )
% 0.33/0.63 => ( ( groups8778361861064173332t_real @ G @ S )
% 0.33/0.63 = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_cong_left
% 0.33/0.63 thf(fact_6063_sum_Omono__neutral__cong__left,axiom,
% 0.33/0.63 ! [T3: set_complex,S: set_complex,H2: complex > real,G: complex > real] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ T3 )
% 0.33/0.63 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 0.33/0.63 => ( ( H2 @ X4 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ S )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = ( H2 @ X4 ) ) )
% 0.33/0.63 => ( ( groups5808333547571424918x_real @ G @ S )
% 0.33/0.63 = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_cong_left
% 0.33/0.63 thf(fact_6064_sum_Omono__neutral__cong__left,axiom,
% 0.33/0.63 ! [T3: set_real,S: set_real,H2: real > rat,G: real > rat] :
% 0.33/0.63 ( ( finite_finite_real @ T3 )
% 0.33/0.63 => ( ( ord_less_eq_set_real @ S @ T3 )
% 0.33/0.63 => ( ! [X4: real] :
% 0.33/0.63 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 0.33/0.63 => ( ( H2 @ X4 )
% 0.33/0.63 = zero_zero_rat ) )
% 0.33/0.63 => ( ! [X4: real] :
% 0.33/0.63 ( ( member_real @ X4 @ S )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = ( H2 @ X4 ) ) )
% 0.33/0.63 => ( ( groups1300246762558778688al_rat @ G @ S )
% 0.33/0.63 = ( groups1300246762558778688al_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_cong_left
% 0.33/0.63 thf(fact_6065_sum_Omono__neutral__cong__left,axiom,
% 0.33/0.63 ! [T3: set_int,S: set_int,H2: int > rat,G: int > rat] :
% 0.33/0.63 ( ( finite_finite_int @ T3 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.63 => ( ( H2 @ X4 )
% 0.33/0.63 = zero_zero_rat ) )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ S )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = ( H2 @ X4 ) ) )
% 0.33/0.63 => ( ( groups3906332499630173760nt_rat @ G @ S )
% 0.33/0.63 = ( groups3906332499630173760nt_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_cong_left
% 0.33/0.63 thf(fact_6066_sum_Omono__neutral__cong__left,axiom,
% 0.33/0.63 ! [T3: set_complex,S: set_complex,H2: complex > rat,G: complex > rat] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ T3 )
% 0.33/0.63 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 0.33/0.63 => ( ( H2 @ X4 )
% 0.33/0.63 = zero_zero_rat ) )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ S )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = ( H2 @ X4 ) ) )
% 0.33/0.63 => ( ( groups5058264527183730370ex_rat @ G @ S )
% 0.33/0.63 = ( groups5058264527183730370ex_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_cong_left
% 0.33/0.63 thf(fact_6067_sum_Omono__neutral__cong__left,axiom,
% 0.33/0.63 ! [T3: set_real,S: set_real,H2: real > nat,G: real > nat] :
% 0.33/0.63 ( ( finite_finite_real @ T3 )
% 0.33/0.63 => ( ( ord_less_eq_set_real @ S @ T3 )
% 0.33/0.63 => ( ! [X4: real] :
% 0.33/0.63 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 0.33/0.63 => ( ( H2 @ X4 )
% 0.33/0.63 = zero_zero_nat ) )
% 0.33/0.63 => ( ! [X4: real] :
% 0.33/0.63 ( ( member_real @ X4 @ S )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = ( H2 @ X4 ) ) )
% 0.33/0.63 => ( ( groups1935376822645274424al_nat @ G @ S )
% 0.33/0.63 = ( groups1935376822645274424al_nat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_cong_left
% 0.33/0.63 thf(fact_6068_sum_Omono__neutral__cong__left,axiom,
% 0.33/0.63 ! [T3: set_int,S: set_int,H2: int > nat,G: int > nat] :
% 0.33/0.63 ( ( finite_finite_int @ T3 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.63 => ( ( H2 @ X4 )
% 0.33/0.63 = zero_zero_nat ) )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ S )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = ( H2 @ X4 ) ) )
% 0.33/0.63 => ( ( groups4541462559716669496nt_nat @ G @ S )
% 0.33/0.63 = ( groups4541462559716669496nt_nat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_cong_left
% 0.33/0.63 thf(fact_6069_sum_Omono__neutral__right,axiom,
% 0.33/0.63 ! [T3: set_int,S: set_int,G: int > complex] :
% 0.33/0.63 ( ( finite_finite_int @ T3 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_complex ) )
% 0.33/0.63 => ( ( groups3049146728041665814omplex @ G @ T3 )
% 0.33/0.63 = ( groups3049146728041665814omplex @ G @ S ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_right
% 0.33/0.63 thf(fact_6070_sum_Omono__neutral__right,axiom,
% 0.33/0.63 ! [T3: set_int,S: set_int,G: int > real] :
% 0.33/0.63 ( ( finite_finite_int @ T3 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ( groups8778361861064173332t_real @ G @ T3 )
% 0.33/0.63 = ( groups8778361861064173332t_real @ G @ S ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_right
% 0.33/0.63 thf(fact_6071_sum_Omono__neutral__right,axiom,
% 0.33/0.63 ! [T3: set_complex,S: set_complex,G: complex > real] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ T3 )
% 0.33/0.63 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ( groups5808333547571424918x_real @ G @ T3 )
% 0.33/0.63 = ( groups5808333547571424918x_real @ G @ S ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_right
% 0.33/0.63 thf(fact_6072_sum_Omono__neutral__right,axiom,
% 0.33/0.63 ! [T3: set_int,S: set_int,G: int > rat] :
% 0.33/0.63 ( ( finite_finite_int @ T3 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_rat ) )
% 0.33/0.63 => ( ( groups3906332499630173760nt_rat @ G @ T3 )
% 0.33/0.63 = ( groups3906332499630173760nt_rat @ G @ S ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_right
% 0.33/0.63 thf(fact_6073_sum_Omono__neutral__right,axiom,
% 0.33/0.63 ! [T3: set_complex,S: set_complex,G: complex > rat] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ T3 )
% 0.33/0.63 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_rat ) )
% 0.33/0.63 => ( ( groups5058264527183730370ex_rat @ G @ T3 )
% 0.33/0.63 = ( groups5058264527183730370ex_rat @ G @ S ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_right
% 0.33/0.63 thf(fact_6074_sum_Omono__neutral__right,axiom,
% 0.33/0.63 ! [T3: set_int,S: set_int,G: int > nat] :
% 0.33/0.63 ( ( finite_finite_int @ T3 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_nat ) )
% 0.33/0.63 => ( ( groups4541462559716669496nt_nat @ G @ T3 )
% 0.33/0.63 = ( groups4541462559716669496nt_nat @ G @ S ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_right
% 0.33/0.63 thf(fact_6075_sum_Omono__neutral__right,axiom,
% 0.33/0.63 ! [T3: set_complex,S: set_complex,G: complex > nat] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ T3 )
% 0.33/0.63 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_nat ) )
% 0.33/0.63 => ( ( groups5693394587270226106ex_nat @ G @ T3 )
% 0.33/0.63 = ( groups5693394587270226106ex_nat @ G @ S ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_right
% 0.33/0.63 thf(fact_6076_sum_Omono__neutral__right,axiom,
% 0.33/0.63 ! [T3: set_complex,S: set_complex,G: complex > int] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ T3 )
% 0.33/0.63 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_int ) )
% 0.33/0.63 => ( ( groups5690904116761175830ex_int @ G @ T3 )
% 0.33/0.63 = ( groups5690904116761175830ex_int @ G @ S ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_right
% 0.33/0.63 thf(fact_6077_sum_Omono__neutral__right,axiom,
% 0.33/0.63 ! [T3: set_nat,S: set_nat,G: nat > complex] :
% 0.33/0.63 ( ( finite_finite_nat @ T3 )
% 0.33/0.63 => ( ( ord_less_eq_set_nat @ S @ T3 )
% 0.33/0.63 => ( ! [X4: nat] :
% 0.33/0.63 ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_complex ) )
% 0.33/0.63 => ( ( groups2073611262835488442omplex @ G @ T3 )
% 0.33/0.63 = ( groups2073611262835488442omplex @ G @ S ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_right
% 0.33/0.63 thf(fact_6078_sum_Omono__neutral__right,axiom,
% 0.33/0.63 ! [T3: set_nat,S: set_nat,G: nat > rat] :
% 0.33/0.63 ( ( finite_finite_nat @ T3 )
% 0.33/0.63 => ( ( ord_less_eq_set_nat @ S @ T3 )
% 0.33/0.63 => ( ! [X4: nat] :
% 0.33/0.63 ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_rat ) )
% 0.33/0.63 => ( ( groups2906978787729119204at_rat @ G @ T3 )
% 0.33/0.63 = ( groups2906978787729119204at_rat @ G @ S ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_right
% 0.33/0.63 thf(fact_6079_sum_Omono__neutral__left,axiom,
% 0.33/0.63 ! [T3: set_int,S: set_int,G: int > complex] :
% 0.33/0.63 ( ( finite_finite_int @ T3 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_complex ) )
% 0.33/0.63 => ( ( groups3049146728041665814omplex @ G @ S )
% 0.33/0.63 = ( groups3049146728041665814omplex @ G @ T3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_left
% 0.33/0.63 thf(fact_6080_sum_Omono__neutral__left,axiom,
% 0.33/0.63 ! [T3: set_int,S: set_int,G: int > real] :
% 0.33/0.63 ( ( finite_finite_int @ T3 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ( groups8778361861064173332t_real @ G @ S )
% 0.33/0.63 = ( groups8778361861064173332t_real @ G @ T3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_left
% 0.33/0.63 thf(fact_6081_sum_Omono__neutral__left,axiom,
% 0.33/0.63 ! [T3: set_complex,S: set_complex,G: complex > real] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ T3 )
% 0.33/0.63 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ( groups5808333547571424918x_real @ G @ S )
% 0.33/0.63 = ( groups5808333547571424918x_real @ G @ T3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_left
% 0.33/0.63 thf(fact_6082_sum_Omono__neutral__left,axiom,
% 0.33/0.63 ! [T3: set_int,S: set_int,G: int > rat] :
% 0.33/0.63 ( ( finite_finite_int @ T3 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_rat ) )
% 0.33/0.63 => ( ( groups3906332499630173760nt_rat @ G @ S )
% 0.33/0.63 = ( groups3906332499630173760nt_rat @ G @ T3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_left
% 0.33/0.63 thf(fact_6083_sum_Omono__neutral__left,axiom,
% 0.33/0.63 ! [T3: set_complex,S: set_complex,G: complex > rat] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ T3 )
% 0.33/0.63 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_rat ) )
% 0.33/0.63 => ( ( groups5058264527183730370ex_rat @ G @ S )
% 0.33/0.63 = ( groups5058264527183730370ex_rat @ G @ T3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_left
% 0.33/0.63 thf(fact_6084_sum_Omono__neutral__left,axiom,
% 0.33/0.63 ! [T3: set_int,S: set_int,G: int > nat] :
% 0.33/0.63 ( ( finite_finite_int @ T3 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.63 => ( ! [X4: int] :
% 0.33/0.63 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_nat ) )
% 0.33/0.63 => ( ( groups4541462559716669496nt_nat @ G @ S )
% 0.33/0.63 = ( groups4541462559716669496nt_nat @ G @ T3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_left
% 0.33/0.63 thf(fact_6085_sum_Omono__neutral__left,axiom,
% 0.33/0.63 ! [T3: set_complex,S: set_complex,G: complex > nat] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ T3 )
% 0.33/0.63 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_nat ) )
% 0.33/0.63 => ( ( groups5693394587270226106ex_nat @ G @ S )
% 0.33/0.63 = ( groups5693394587270226106ex_nat @ G @ T3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_left
% 0.33/0.63 thf(fact_6086_sum_Omono__neutral__left,axiom,
% 0.33/0.63 ! [T3: set_complex,S: set_complex,G: complex > int] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ T3 )
% 0.33/0.63 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.33/0.63 => ( ! [X4: complex] :
% 0.33/0.63 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_int ) )
% 0.33/0.63 => ( ( groups5690904116761175830ex_int @ G @ S )
% 0.33/0.63 = ( groups5690904116761175830ex_int @ G @ T3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_left
% 0.33/0.63 thf(fact_6087_sum_Omono__neutral__left,axiom,
% 0.33/0.63 ! [T3: set_nat,S: set_nat,G: nat > complex] :
% 0.33/0.63 ( ( finite_finite_nat @ T3 )
% 0.33/0.63 => ( ( ord_less_eq_set_nat @ S @ T3 )
% 0.33/0.63 => ( ! [X4: nat] :
% 0.33/0.63 ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_complex ) )
% 0.33/0.63 => ( ( groups2073611262835488442omplex @ G @ S )
% 0.33/0.63 = ( groups2073611262835488442omplex @ G @ T3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_left
% 0.33/0.63 thf(fact_6088_sum_Omono__neutral__left,axiom,
% 0.33/0.63 ! [T3: set_nat,S: set_nat,G: nat > rat] :
% 0.33/0.63 ( ( finite_finite_nat @ T3 )
% 0.33/0.63 => ( ( ord_less_eq_set_nat @ S @ T3 )
% 0.33/0.63 => ( ! [X4: nat] :
% 0.33/0.63 ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S ) )
% 0.33/0.63 => ( ( G @ X4 )
% 0.33/0.63 = zero_zero_rat ) )
% 0.33/0.63 => ( ( groups2906978787729119204at_rat @ G @ S )
% 0.33/0.63 = ( groups2906978787729119204at_rat @ G @ T3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.mono_neutral_left
% 0.33/0.63 thf(fact_6089_sum_Osame__carrierI,axiom,
% 0.33/0.63 ! [C2: set_real,A2: set_real,B3: set_real,G: real > complex,H2: real > complex] :
% 0.33/0.63 ( ( finite_finite_real @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 0.33/0.63 => ( ! [A3: real] :
% 0.33/0.63 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_complex ) )
% 0.33/0.63 => ( ! [B2: real] :
% 0.33/0.63 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_complex ) )
% 0.33/0.63 => ( ( ( groups5754745047067104278omplex @ G @ C2 )
% 0.33/0.63 = ( groups5754745047067104278omplex @ H2 @ C2 ) )
% 0.33/0.63 => ( ( groups5754745047067104278omplex @ G @ A2 )
% 0.33/0.63 = ( groups5754745047067104278omplex @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.same_carrierI
% 0.33/0.63 thf(fact_6090_sum_Osame__carrierI,axiom,
% 0.33/0.63 ! [C2: set_int,A2: set_int,B3: set_int,G: int > complex,H2: int > complex] :
% 0.33/0.63 ( ( finite_finite_int @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ A2 @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ B3 @ C2 )
% 0.33/0.63 => ( ! [A3: int] :
% 0.33/0.63 ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_complex ) )
% 0.33/0.63 => ( ! [B2: int] :
% 0.33/0.63 ( ( member_int @ B2 @ ( minus_minus_set_int @ C2 @ B3 ) )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_complex ) )
% 0.33/0.63 => ( ( ( groups3049146728041665814omplex @ G @ C2 )
% 0.33/0.63 = ( groups3049146728041665814omplex @ H2 @ C2 ) )
% 0.33/0.63 => ( ( groups3049146728041665814omplex @ G @ A2 )
% 0.33/0.63 = ( groups3049146728041665814omplex @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.same_carrierI
% 0.33/0.63 thf(fact_6091_sum_Osame__carrierI,axiom,
% 0.33/0.63 ! [C2: set_real,A2: set_real,B3: set_real,G: real > real,H2: real > real] :
% 0.33/0.63 ( ( finite_finite_real @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 0.33/0.63 => ( ! [A3: real] :
% 0.33/0.63 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ! [B2: real] :
% 0.33/0.63 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ( ( groups8097168146408367636l_real @ G @ C2 )
% 0.33/0.63 = ( groups8097168146408367636l_real @ H2 @ C2 ) )
% 0.33/0.63 => ( ( groups8097168146408367636l_real @ G @ A2 )
% 0.33/0.63 = ( groups8097168146408367636l_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.same_carrierI
% 0.33/0.63 thf(fact_6092_sum_Osame__carrierI,axiom,
% 0.33/0.63 ! [C2: set_int,A2: set_int,B3: set_int,G: int > real,H2: int > real] :
% 0.33/0.63 ( ( finite_finite_int @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ A2 @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ B3 @ C2 )
% 0.33/0.63 => ( ! [A3: int] :
% 0.33/0.63 ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ! [B2: int] :
% 0.33/0.63 ( ( member_int @ B2 @ ( minus_minus_set_int @ C2 @ B3 ) )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ( ( groups8778361861064173332t_real @ G @ C2 )
% 0.33/0.63 = ( groups8778361861064173332t_real @ H2 @ C2 ) )
% 0.33/0.63 => ( ( groups8778361861064173332t_real @ G @ A2 )
% 0.33/0.63 = ( groups8778361861064173332t_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.same_carrierI
% 0.33/0.63 thf(fact_6093_sum_Osame__carrierI,axiom,
% 0.33/0.63 ! [C2: set_complex,A2: set_complex,B3: set_complex,G: complex > real,H2: complex > real] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ C2 )
% 0.33/0.63 => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 0.33/0.63 => ( ( ord_le211207098394363844omplex @ B3 @ C2 )
% 0.33/0.63 => ( ! [A3: complex] :
% 0.33/0.63 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ! [B2: complex] :
% 0.33/0.63 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B3 ) )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ( ( groups5808333547571424918x_real @ G @ C2 )
% 0.33/0.63 = ( groups5808333547571424918x_real @ H2 @ C2 ) )
% 0.33/0.63 => ( ( groups5808333547571424918x_real @ G @ A2 )
% 0.33/0.63 = ( groups5808333547571424918x_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.same_carrierI
% 0.33/0.63 thf(fact_6094_sum_Osame__carrierI,axiom,
% 0.33/0.63 ! [C2: set_real,A2: set_real,B3: set_real,G: real > rat,H2: real > rat] :
% 0.33/0.63 ( ( finite_finite_real @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 0.33/0.63 => ( ! [A3: real] :
% 0.33/0.63 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_rat ) )
% 0.33/0.63 => ( ! [B2: real] :
% 0.33/0.63 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_rat ) )
% 0.33/0.63 => ( ( ( groups1300246762558778688al_rat @ G @ C2 )
% 0.33/0.63 = ( groups1300246762558778688al_rat @ H2 @ C2 ) )
% 0.33/0.63 => ( ( groups1300246762558778688al_rat @ G @ A2 )
% 0.33/0.63 = ( groups1300246762558778688al_rat @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.same_carrierI
% 0.33/0.63 thf(fact_6095_sum_Osame__carrierI,axiom,
% 0.33/0.63 ! [C2: set_int,A2: set_int,B3: set_int,G: int > rat,H2: int > rat] :
% 0.33/0.63 ( ( finite_finite_int @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ A2 @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ B3 @ C2 )
% 0.33/0.63 => ( ! [A3: int] :
% 0.33/0.63 ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_rat ) )
% 0.33/0.63 => ( ! [B2: int] :
% 0.33/0.63 ( ( member_int @ B2 @ ( minus_minus_set_int @ C2 @ B3 ) )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_rat ) )
% 0.33/0.63 => ( ( ( groups3906332499630173760nt_rat @ G @ C2 )
% 0.33/0.63 = ( groups3906332499630173760nt_rat @ H2 @ C2 ) )
% 0.33/0.63 => ( ( groups3906332499630173760nt_rat @ G @ A2 )
% 0.33/0.63 = ( groups3906332499630173760nt_rat @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.same_carrierI
% 0.33/0.63 thf(fact_6096_sum_Osame__carrierI,axiom,
% 0.33/0.63 ! [C2: set_complex,A2: set_complex,B3: set_complex,G: complex > rat,H2: complex > rat] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ C2 )
% 0.33/0.63 => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 0.33/0.63 => ( ( ord_le211207098394363844omplex @ B3 @ C2 )
% 0.33/0.63 => ( ! [A3: complex] :
% 0.33/0.63 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_rat ) )
% 0.33/0.63 => ( ! [B2: complex] :
% 0.33/0.63 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B3 ) )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_rat ) )
% 0.33/0.63 => ( ( ( groups5058264527183730370ex_rat @ G @ C2 )
% 0.33/0.63 = ( groups5058264527183730370ex_rat @ H2 @ C2 ) )
% 0.33/0.63 => ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 0.33/0.63 = ( groups5058264527183730370ex_rat @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.same_carrierI
% 0.33/0.63 thf(fact_6097_sum_Osame__carrierI,axiom,
% 0.33/0.63 ! [C2: set_real,A2: set_real,B3: set_real,G: real > nat,H2: real > nat] :
% 0.33/0.63 ( ( finite_finite_real @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 0.33/0.63 => ( ! [A3: real] :
% 0.33/0.63 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_nat ) )
% 0.33/0.63 => ( ! [B2: real] :
% 0.33/0.63 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_nat ) )
% 0.33/0.63 => ( ( ( groups1935376822645274424al_nat @ G @ C2 )
% 0.33/0.63 = ( groups1935376822645274424al_nat @ H2 @ C2 ) )
% 0.33/0.63 => ( ( groups1935376822645274424al_nat @ G @ A2 )
% 0.33/0.63 = ( groups1935376822645274424al_nat @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.same_carrierI
% 0.33/0.63 thf(fact_6098_sum_Osame__carrierI,axiom,
% 0.33/0.63 ! [C2: set_int,A2: set_int,B3: set_int,G: int > nat,H2: int > nat] :
% 0.33/0.63 ( ( finite_finite_int @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ A2 @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ B3 @ C2 )
% 0.33/0.63 => ( ! [A3: int] :
% 0.33/0.63 ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_nat ) )
% 0.33/0.63 => ( ! [B2: int] :
% 0.33/0.63 ( ( member_int @ B2 @ ( minus_minus_set_int @ C2 @ B3 ) )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_nat ) )
% 0.33/0.63 => ( ( ( groups4541462559716669496nt_nat @ G @ C2 )
% 0.33/0.63 = ( groups4541462559716669496nt_nat @ H2 @ C2 ) )
% 0.33/0.63 => ( ( groups4541462559716669496nt_nat @ G @ A2 )
% 0.33/0.63 = ( groups4541462559716669496nt_nat @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.same_carrierI
% 0.33/0.63 thf(fact_6099_sum_Osame__carrier,axiom,
% 0.33/0.63 ! [C2: set_real,A2: set_real,B3: set_real,G: real > complex,H2: real > complex] :
% 0.33/0.63 ( ( finite_finite_real @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 0.33/0.63 => ( ! [A3: real] :
% 0.33/0.63 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_complex ) )
% 0.33/0.63 => ( ! [B2: real] :
% 0.33/0.63 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_complex ) )
% 0.33/0.63 => ( ( ( groups5754745047067104278omplex @ G @ A2 )
% 0.33/0.63 = ( groups5754745047067104278omplex @ H2 @ B3 ) )
% 0.33/0.63 = ( ( groups5754745047067104278omplex @ G @ C2 )
% 0.33/0.63 = ( groups5754745047067104278omplex @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.same_carrier
% 0.33/0.63 thf(fact_6100_sum_Osame__carrier,axiom,
% 0.33/0.63 ! [C2: set_int,A2: set_int,B3: set_int,G: int > complex,H2: int > complex] :
% 0.33/0.63 ( ( finite_finite_int @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ A2 @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ B3 @ C2 )
% 0.33/0.63 => ( ! [A3: int] :
% 0.33/0.63 ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_complex ) )
% 0.33/0.63 => ( ! [B2: int] :
% 0.33/0.63 ( ( member_int @ B2 @ ( minus_minus_set_int @ C2 @ B3 ) )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_complex ) )
% 0.33/0.63 => ( ( ( groups3049146728041665814omplex @ G @ A2 )
% 0.33/0.63 = ( groups3049146728041665814omplex @ H2 @ B3 ) )
% 0.33/0.63 = ( ( groups3049146728041665814omplex @ G @ C2 )
% 0.33/0.63 = ( groups3049146728041665814omplex @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.same_carrier
% 0.33/0.63 thf(fact_6101_sum_Osame__carrier,axiom,
% 0.33/0.63 ! [C2: set_real,A2: set_real,B3: set_real,G: real > real,H2: real > real] :
% 0.33/0.63 ( ( finite_finite_real @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 0.33/0.63 => ( ! [A3: real] :
% 0.33/0.63 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ! [B2: real] :
% 0.33/0.63 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ( ( groups8097168146408367636l_real @ G @ A2 )
% 0.33/0.63 = ( groups8097168146408367636l_real @ H2 @ B3 ) )
% 0.33/0.63 = ( ( groups8097168146408367636l_real @ G @ C2 )
% 0.33/0.63 = ( groups8097168146408367636l_real @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.same_carrier
% 0.33/0.63 thf(fact_6102_sum_Osame__carrier,axiom,
% 0.33/0.63 ! [C2: set_int,A2: set_int,B3: set_int,G: int > real,H2: int > real] :
% 0.33/0.63 ( ( finite_finite_int @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ A2 @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ B3 @ C2 )
% 0.33/0.63 => ( ! [A3: int] :
% 0.33/0.63 ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ! [B2: int] :
% 0.33/0.63 ( ( member_int @ B2 @ ( minus_minus_set_int @ C2 @ B3 ) )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ( ( groups8778361861064173332t_real @ G @ A2 )
% 0.33/0.63 = ( groups8778361861064173332t_real @ H2 @ B3 ) )
% 0.33/0.63 = ( ( groups8778361861064173332t_real @ G @ C2 )
% 0.33/0.63 = ( groups8778361861064173332t_real @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.same_carrier
% 0.33/0.63 thf(fact_6103_sum_Osame__carrier,axiom,
% 0.33/0.63 ! [C2: set_complex,A2: set_complex,B3: set_complex,G: complex > real,H2: complex > real] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ C2 )
% 0.33/0.63 => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 0.33/0.63 => ( ( ord_le211207098394363844omplex @ B3 @ C2 )
% 0.33/0.63 => ( ! [A3: complex] :
% 0.33/0.63 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ! [B2: complex] :
% 0.33/0.63 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B3 ) )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_real ) )
% 0.33/0.63 => ( ( ( groups5808333547571424918x_real @ G @ A2 )
% 0.33/0.63 = ( groups5808333547571424918x_real @ H2 @ B3 ) )
% 0.33/0.63 = ( ( groups5808333547571424918x_real @ G @ C2 )
% 0.33/0.63 = ( groups5808333547571424918x_real @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.same_carrier
% 0.33/0.63 thf(fact_6104_sum_Osame__carrier,axiom,
% 0.33/0.63 ! [C2: set_real,A2: set_real,B3: set_real,G: real > rat,H2: real > rat] :
% 0.33/0.63 ( ( finite_finite_real @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 0.33/0.63 => ( ! [A3: real] :
% 0.33/0.63 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_rat ) )
% 0.33/0.63 => ( ! [B2: real] :
% 0.33/0.63 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_rat ) )
% 0.33/0.63 => ( ( ( groups1300246762558778688al_rat @ G @ A2 )
% 0.33/0.63 = ( groups1300246762558778688al_rat @ H2 @ B3 ) )
% 0.33/0.63 = ( ( groups1300246762558778688al_rat @ G @ C2 )
% 0.33/0.63 = ( groups1300246762558778688al_rat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.same_carrier
% 0.33/0.63 thf(fact_6105_sum_Osame__carrier,axiom,
% 0.33/0.63 ! [C2: set_int,A2: set_int,B3: set_int,G: int > rat,H2: int > rat] :
% 0.33/0.63 ( ( finite_finite_int @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ A2 @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ B3 @ C2 )
% 0.33/0.63 => ( ! [A3: int] :
% 0.33/0.63 ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_rat ) )
% 0.33/0.63 => ( ! [B2: int] :
% 0.33/0.63 ( ( member_int @ B2 @ ( minus_minus_set_int @ C2 @ B3 ) )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_rat ) )
% 0.33/0.63 => ( ( ( groups3906332499630173760nt_rat @ G @ A2 )
% 0.33/0.63 = ( groups3906332499630173760nt_rat @ H2 @ B3 ) )
% 0.33/0.63 = ( ( groups3906332499630173760nt_rat @ G @ C2 )
% 0.33/0.63 = ( groups3906332499630173760nt_rat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.same_carrier
% 0.33/0.63 thf(fact_6106_sum_Osame__carrier,axiom,
% 0.33/0.63 ! [C2: set_complex,A2: set_complex,B3: set_complex,G: complex > rat,H2: complex > rat] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ C2 )
% 0.33/0.63 => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 0.33/0.63 => ( ( ord_le211207098394363844omplex @ B3 @ C2 )
% 0.33/0.63 => ( ! [A3: complex] :
% 0.33/0.63 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_rat ) )
% 0.33/0.63 => ( ! [B2: complex] :
% 0.33/0.63 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B3 ) )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_rat ) )
% 0.33/0.63 => ( ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 0.33/0.63 = ( groups5058264527183730370ex_rat @ H2 @ B3 ) )
% 0.33/0.63 = ( ( groups5058264527183730370ex_rat @ G @ C2 )
% 0.33/0.63 = ( groups5058264527183730370ex_rat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.same_carrier
% 0.33/0.63 thf(fact_6107_sum_Osame__carrier,axiom,
% 0.33/0.63 ! [C2: set_real,A2: set_real,B3: set_real,G: real > nat,H2: real > nat] :
% 0.33/0.63 ( ( finite_finite_real @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 0.33/0.63 => ( ! [A3: real] :
% 0.33/0.63 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_nat ) )
% 0.33/0.63 => ( ! [B2: real] :
% 0.33/0.63 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_nat ) )
% 0.33/0.63 => ( ( ( groups1935376822645274424al_nat @ G @ A2 )
% 0.33/0.63 = ( groups1935376822645274424al_nat @ H2 @ B3 ) )
% 0.33/0.63 = ( ( groups1935376822645274424al_nat @ G @ C2 )
% 0.33/0.63 = ( groups1935376822645274424al_nat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.same_carrier
% 0.33/0.63 thf(fact_6108_sum_Osame__carrier,axiom,
% 0.33/0.63 ! [C2: set_int,A2: set_int,B3: set_int,G: int > nat,H2: int > nat] :
% 0.33/0.63 ( ( finite_finite_int @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ A2 @ C2 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ B3 @ C2 )
% 0.33/0.63 => ( ! [A3: int] :
% 0.33/0.63 ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 0.33/0.63 => ( ( G @ A3 )
% 0.33/0.63 = zero_zero_nat ) )
% 0.33/0.63 => ( ! [B2: int] :
% 0.33/0.63 ( ( member_int @ B2 @ ( minus_minus_set_int @ C2 @ B3 ) )
% 0.33/0.63 => ( ( H2 @ B2 )
% 0.33/0.63 = zero_zero_nat ) )
% 0.33/0.63 => ( ( ( groups4541462559716669496nt_nat @ G @ A2 )
% 0.33/0.63 = ( groups4541462559716669496nt_nat @ H2 @ B3 ) )
% 0.33/0.63 = ( ( groups4541462559716669496nt_nat @ G @ C2 )
% 0.33/0.63 = ( groups4541462559716669496nt_nat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.same_carrier
% 0.33/0.63 thf(fact_6109_sum_Osubset__diff,axiom,
% 0.33/0.63 ! [B3: set_int,A2: set_int,G: int > real] :
% 0.33/0.63 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.33/0.63 => ( ( finite_finite_int @ A2 )
% 0.33/0.63 => ( ( groups8778361861064173332t_real @ G @ A2 )
% 0.33/0.63 = ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups8778361861064173332t_real @ G @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.subset_diff
% 0.33/0.63 thf(fact_6110_sum_Osubset__diff,axiom,
% 0.33/0.63 ! [B3: set_complex,A2: set_complex,G: complex > real] :
% 0.33/0.63 ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 0.33/0.63 => ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.63 => ( ( groups5808333547571424918x_real @ G @ A2 )
% 0.33/0.63 = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups5808333547571424918x_real @ G @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.subset_diff
% 0.33/0.63 thf(fact_6111_sum_Osubset__diff,axiom,
% 0.33/0.63 ! [B3: set_int,A2: set_int,G: int > rat] :
% 0.33/0.63 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.33/0.63 => ( ( finite_finite_int @ A2 )
% 0.33/0.63 => ( ( groups3906332499630173760nt_rat @ G @ A2 )
% 0.33/0.63 = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups3906332499630173760nt_rat @ G @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.subset_diff
% 0.33/0.63 thf(fact_6112_sum_Osubset__diff,axiom,
% 0.33/0.63 ! [B3: set_complex,A2: set_complex,G: complex > rat] :
% 0.33/0.63 ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 0.33/0.63 => ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.63 => ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 0.33/0.63 = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups5058264527183730370ex_rat @ G @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.subset_diff
% 0.33/0.63 thf(fact_6113_sum_Osubset__diff,axiom,
% 0.33/0.63 ! [B3: set_int,A2: set_int,G: int > nat] :
% 0.33/0.63 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.33/0.63 => ( ( finite_finite_int @ A2 )
% 0.33/0.63 => ( ( groups4541462559716669496nt_nat @ G @ A2 )
% 0.33/0.63 = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups4541462559716669496nt_nat @ G @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.subset_diff
% 0.33/0.63 thf(fact_6114_sum_Osubset__diff,axiom,
% 0.33/0.63 ! [B3: set_complex,A2: set_complex,G: complex > nat] :
% 0.33/0.63 ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 0.33/0.63 => ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.63 => ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 0.33/0.63 = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups5693394587270226106ex_nat @ G @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.subset_diff
% 0.33/0.63 thf(fact_6115_sum_Osubset__diff,axiom,
% 0.33/0.63 ! [B3: set_complex,A2: set_complex,G: complex > int] :
% 0.33/0.63 ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 0.33/0.63 => ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.63 => ( ( groups5690904116761175830ex_int @ G @ A2 )
% 0.33/0.63 = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups5690904116761175830ex_int @ G @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.subset_diff
% 0.33/0.63 thf(fact_6116_sum_Osubset__diff,axiom,
% 0.33/0.63 ! [B3: set_nat,A2: set_nat,G: nat > rat] :
% 0.33/0.63 ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 0.33/0.63 => ( ( finite_finite_nat @ A2 )
% 0.33/0.63 => ( ( groups2906978787729119204at_rat @ G @ A2 )
% 0.33/0.63 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A2 @ B3 ) ) @ ( groups2906978787729119204at_rat @ G @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.subset_diff
% 0.33/0.63 thf(fact_6117_sum_Osubset__diff,axiom,
% 0.33/0.63 ! [B3: set_nat,A2: set_nat,G: nat > int] :
% 0.33/0.63 ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 0.33/0.63 => ( ( finite_finite_nat @ A2 )
% 0.33/0.63 => ( ( groups3539618377306564664at_int @ G @ A2 )
% 0.33/0.63 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A2 @ B3 ) ) @ ( groups3539618377306564664at_int @ G @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.subset_diff
% 0.33/0.63 thf(fact_6118_sum_Osubset__diff,axiom,
% 0.33/0.63 ! [B3: set_int,A2: set_int,G: int > int] :
% 0.33/0.63 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.33/0.63 => ( ( finite_finite_int @ A2 )
% 0.33/0.63 => ( ( groups4538972089207619220nt_int @ G @ A2 )
% 0.33/0.63 = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups4538972089207619220nt_int @ G @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum.subset_diff
% 0.33/0.63 thf(fact_6119_sum__diff,axiom,
% 0.33/0.63 ! [A2: set_int,B3: set_int,F: int > real] :
% 0.33/0.63 ( ( finite_finite_int @ A2 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.33/0.63 => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ B3 ) )
% 0.33/0.63 = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_diff
% 0.33/0.63 thf(fact_6120_sum__diff,axiom,
% 0.33/0.63 ! [A2: set_complex,B3: set_complex,F: complex > real] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.63 => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 0.33/0.63 => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 0.33/0.63 = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_diff
% 0.33/0.63 thf(fact_6121_sum__diff,axiom,
% 0.33/0.63 ! [A2: set_int,B3: set_int,F: int > rat] :
% 0.33/0.63 ( ( finite_finite_int @ A2 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.33/0.63 => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ B3 ) )
% 0.33/0.63 = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_diff
% 0.33/0.63 thf(fact_6122_sum__diff,axiom,
% 0.33/0.63 ! [A2: set_complex,B3: set_complex,F: complex > rat] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.63 => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 0.33/0.63 => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 0.33/0.63 = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_diff
% 0.33/0.63 thf(fact_6123_sum__diff,axiom,
% 0.33/0.63 ! [A2: set_complex,B3: set_complex,F: complex > int] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.63 => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 0.33/0.63 => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 0.33/0.63 = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_diff
% 0.33/0.63 thf(fact_6124_sum__diff,axiom,
% 0.33/0.63 ! [A2: set_nat,B3: set_nat,F: nat > rat] :
% 0.33/0.63 ( ( finite_finite_nat @ A2 )
% 0.33/0.63 => ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 0.33/0.63 => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 0.33/0.63 = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_diff
% 0.33/0.63 thf(fact_6125_sum__diff,axiom,
% 0.33/0.63 ! [A2: set_nat,B3: set_nat,F: nat > int] :
% 0.33/0.63 ( ( finite_finite_nat @ A2 )
% 0.33/0.63 => ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 0.33/0.63 => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 0.33/0.63 = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_diff
% 0.33/0.63 thf(fact_6126_sum__diff,axiom,
% 0.33/0.63 ! [A2: set_int,B3: set_int,F: int > int] :
% 0.33/0.63 ( ( finite_finite_int @ A2 )
% 0.33/0.63 => ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.33/0.63 => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ B3 ) )
% 0.33/0.63 = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ F @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_diff
% 0.33/0.63 thf(fact_6127_sum__diff,axiom,
% 0.33/0.63 ! [A2: set_complex,B3: set_complex,F: complex > complex] :
% 0.33/0.63 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.63 => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 0.33/0.63 => ( ( groups7754918857620584856omplex @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 0.33/0.63 = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ F @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_diff
% 0.33/0.63 thf(fact_6128_sum__diff,axiom,
% 0.33/0.63 ! [A2: set_nat,B3: set_nat,F: nat > real] :
% 0.33/0.63 ( ( finite_finite_nat @ A2 )
% 0.33/0.63 => ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 0.33/0.63 => ( ( groups6591440286371151544t_real @ F @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 0.33/0.63 = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ F @ B3 ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % sum_diff
% 0.33/0.63 thf(fact_6129_ln__add__one__self__le__self,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% 0.33/0.63
% 0.33/0.63 % ln_add_one_self_le_self
% 0.33/0.63 thf(fact_6130_ln__mult,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.63 => ( ( ln_ln_real @ ( times_times_real @ X @ Y ) )
% 0.33/0.63 = ( plus_plus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ln_mult
% 0.33/0.63 thf(fact_6131_ln__eq__minus__one,axiom,
% 0.33/0.63 ! [X: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ( ( ln_ln_real @ X )
% 0.33/0.63 = ( minus_minus_real @ X @ one_one_real ) )
% 0.33/0.63 => ( X = one_one_real ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ln_eq_minus_one
% 0.33/0.63 thf(fact_6132_ln__div,axiom,
% 0.33/0.63 ! [X: real,Y: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.63 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.63 => ( ( ln_ln_real @ ( divide_divide_real @ X @ Y ) )
% 0.33/0.63 = ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % ln_div
% 0.33/0.63 thf(fact_6133_pos__minus__divide__less__eq,axiom,
% 0.33/0.63 ! [C: real,B: real,A: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.63 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.33/0.63 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % pos_minus_divide_less_eq
% 0.33/0.63 thf(fact_6134_pos__minus__divide__less__eq,axiom,
% 0.33/0.63 ! [C: rat,B: rat,A: rat] :
% 0.33/0.63 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.63 => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.33/0.63 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % pos_minus_divide_less_eq
% 0.33/0.63 thf(fact_6135_pos__less__minus__divide__eq,axiom,
% 0.33/0.63 ! [C: real,A: real,B: real] :
% 0.33/0.63 ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.63 => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.33/0.63 = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % pos_less_minus_divide_eq
% 0.33/0.63 thf(fact_6136_pos__less__minus__divide__eq,axiom,
% 0.33/0.63 ! [C: rat,A: rat,B: rat] :
% 0.33/0.63 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.63 => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.33/0.63 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % pos_less_minus_divide_eq
% 0.33/0.63 thf(fact_6137_neg__minus__divide__less__eq,axiom,
% 0.33/0.63 ! [C: real,B: real,A: real] :
% 0.33/0.63 ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.63 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.33/0.63 = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % neg_minus_divide_less_eq
% 0.33/0.63 thf(fact_6138_neg__minus__divide__less__eq,axiom,
% 0.33/0.63 ! [C: rat,B: rat,A: rat] :
% 0.33/0.63 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.63 => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.33/0.63 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % neg_minus_divide_less_eq
% 0.33/0.63 thf(fact_6139_neg__less__minus__divide__eq,axiom,
% 0.33/0.63 ! [C: real,A: real,B: real] :
% 0.33/0.63 ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.63 => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.33/0.63 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % neg_less_minus_divide_eq
% 0.33/0.63 thf(fact_6140_neg__less__minus__divide__eq,axiom,
% 0.33/0.63 ! [C: rat,A: rat,B: rat] :
% 0.33/0.63 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.63 => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.33/0.63 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % neg_less_minus_divide_eq
% 0.33/0.63 thf(fact_6141_minus__divide__less__eq,axiom,
% 0.33/0.63 ! [B: real,C: real,A: real] :
% 0.33/0.63 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.33/0.63 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.63 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.33/0.63 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.63 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.63 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.33/0.63 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.63 => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % minus_divide_less_eq
% 0.33/0.63 thf(fact_6142_minus__divide__less__eq,axiom,
% 0.33/0.63 ! [B: rat,C: rat,A: rat] :
% 0.33/0.63 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.33/0.63 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.63 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.33/0.63 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.63 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.63 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.33/0.63 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.63 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % minus_divide_less_eq
% 0.33/0.63 thf(fact_6143_less__minus__divide__eq,axiom,
% 0.33/0.63 ! [A: real,B: real,C: real] :
% 0.33/0.63 ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.33/0.63 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.63 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.33/0.63 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.63 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.63 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.33/0.63 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.63 => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % less_minus_divide_eq
% 0.33/0.63 thf(fact_6144_less__minus__divide__eq,axiom,
% 0.33/0.63 ! [A: rat,B: rat,C: rat] :
% 0.33/0.63 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.33/0.63 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.63 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.33/0.63 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.63 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.63 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.33/0.63 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.63 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % less_minus_divide_eq
% 0.33/0.63 thf(fact_6145_eq__divide__eq__numeral_I2_J,axiom,
% 0.33/0.63 ! [W: num,B: real,C: real] :
% 0.33/0.63 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.33/0.63 = ( divide_divide_real @ B @ C ) )
% 0.33/0.63 = ( ( ( C != zero_zero_real )
% 0.33/0.63 => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C )
% 0.33/0.63 = B ) )
% 0.33/0.63 & ( ( C = zero_zero_real )
% 0.33/0.63 => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.33/0.63 = zero_zero_real ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % eq_divide_eq_numeral(2)
% 0.33/0.63 thf(fact_6146_eq__divide__eq__numeral_I2_J,axiom,
% 0.33/0.63 ! [W: num,B: complex,C: complex] :
% 0.33/0.63 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.33/0.63 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.33/0.63 = ( ( ( C != zero_zero_complex )
% 0.33/0.63 => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C )
% 0.33/0.63 = B ) )
% 0.33/0.63 & ( ( C = zero_zero_complex )
% 0.33/0.63 => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.33/0.63 = zero_zero_complex ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % eq_divide_eq_numeral(2)
% 0.33/0.63 thf(fact_6147_eq__divide__eq__numeral_I2_J,axiom,
% 0.33/0.63 ! [W: num,B: rat,C: rat] :
% 0.33/0.63 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.33/0.63 = ( divide_divide_rat @ B @ C ) )
% 0.33/0.63 = ( ( ( C != zero_zero_rat )
% 0.33/0.63 => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C )
% 0.33/0.63 = B ) )
% 0.33/0.63 & ( ( C = zero_zero_rat )
% 0.33/0.63 => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.33/0.63 = zero_zero_rat ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % eq_divide_eq_numeral(2)
% 0.33/0.63 thf(fact_6148_divide__eq__eq__numeral_I2_J,axiom,
% 0.33/0.63 ! [B: real,C: real,W: num] :
% 0.33/0.63 ( ( ( divide_divide_real @ B @ C )
% 0.33/0.63 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.33/0.63 = ( ( ( C != zero_zero_real )
% 0.33/0.63 => ( B
% 0.33/0.63 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.33/0.63 & ( ( C = zero_zero_real )
% 0.33/0.63 => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.33/0.63 = zero_zero_real ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % divide_eq_eq_numeral(2)
% 0.33/0.63 thf(fact_6149_divide__eq__eq__numeral_I2_J,axiom,
% 0.33/0.63 ! [B: complex,C: complex,W: num] :
% 0.33/0.63 ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.33/0.63 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.33/0.63 = ( ( ( C != zero_zero_complex )
% 0.33/0.63 => ( B
% 0.33/0.63 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C ) ) )
% 0.33/0.63 & ( ( C = zero_zero_complex )
% 0.33/0.63 => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.33/0.63 = zero_zero_complex ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % divide_eq_eq_numeral(2)
% 0.33/0.63 thf(fact_6150_divide__eq__eq__numeral_I2_J,axiom,
% 0.33/0.63 ! [B: rat,C: rat,W: num] :
% 0.33/0.63 ( ( ( divide_divide_rat @ B @ C )
% 0.33/0.63 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.33/0.63 = ( ( ( C != zero_zero_rat )
% 0.33/0.63 => ( B
% 0.33/0.63 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.33/0.63 & ( ( C = zero_zero_rat )
% 0.33/0.63 => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.33/0.63 = zero_zero_rat ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % divide_eq_eq_numeral(2)
% 0.33/0.63 thf(fact_6151_minus__divide__add__eq__iff,axiom,
% 0.33/0.63 ! [Z: real,X: real,Y: real] :
% 0.33/0.63 ( ( Z != zero_zero_real )
% 0.33/0.63 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
% 0.33/0.63 = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % minus_divide_add_eq_iff
% 0.33/0.63 thf(fact_6152_minus__divide__add__eq__iff,axiom,
% 0.33/0.63 ! [Z: complex,X: complex,Y: complex] :
% 0.33/0.63 ( ( Z != zero_zero_complex )
% 0.33/0.63 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y )
% 0.33/0.63 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % minus_divide_add_eq_iff
% 0.33/0.63 thf(fact_6153_minus__divide__add__eq__iff,axiom,
% 0.33/0.63 ! [Z: rat,X: rat,Y: rat] :
% 0.33/0.63 ( ( Z != zero_zero_rat )
% 0.33/0.63 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
% 0.33/0.63 = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % minus_divide_add_eq_iff
% 0.33/0.63 thf(fact_6154_add__divide__eq__if__simps_I3_J,axiom,
% 0.33/0.63 ! [Z: real,A: real,B: real] :
% 0.33/0.63 ( ( ( Z = zero_zero_real )
% 0.33/0.63 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 0.33/0.63 = B ) )
% 0.33/0.63 & ( ( Z != zero_zero_real )
% 0.33/0.63 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 0.33/0.63 = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.33/0.63
% 0.33/0.63 % add_divide_eq_if_simps(3)
% 0.33/0.63 thf(fact_6155_add__divide__eq__if__simps_I3_J,axiom,
% 0.33/0.63 ! [Z: complex,A: complex,B: complex] :
% 0.33/0.63 ( ( ( Z = zero_zero_complex )
% 0.33/0.64 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 0.33/0.64 = B ) )
% 0.33/0.64 & ( ( Z != zero_zero_complex )
% 0.33/0.64 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 0.33/0.64 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % add_divide_eq_if_simps(3)
% 0.33/0.64 thf(fact_6156_add__divide__eq__if__simps_I3_J,axiom,
% 0.33/0.64 ! [Z: rat,A: rat,B: rat] :
% 0.33/0.64 ( ( ( Z = zero_zero_rat )
% 0.33/0.64 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 0.33/0.64 = B ) )
% 0.33/0.64 & ( ( Z != zero_zero_rat )
% 0.33/0.64 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 0.33/0.64 = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % add_divide_eq_if_simps(3)
% 0.33/0.64 thf(fact_6157_minus__divide__diff__eq__iff,axiom,
% 0.33/0.64 ! [Z: real,X: real,Y: real] :
% 0.33/0.64 ( ( Z != zero_zero_real )
% 0.33/0.64 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
% 0.33/0.64 = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_divide_diff_eq_iff
% 0.33/0.64 thf(fact_6158_minus__divide__diff__eq__iff,axiom,
% 0.33/0.64 ! [Z: complex,X: complex,Y: complex] :
% 0.33/0.64 ( ( Z != zero_zero_complex )
% 0.33/0.64 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y )
% 0.33/0.64 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_divide_diff_eq_iff
% 0.33/0.64 thf(fact_6159_minus__divide__diff__eq__iff,axiom,
% 0.33/0.64 ! [Z: rat,X: rat,Y: rat] :
% 0.33/0.64 ( ( Z != zero_zero_rat )
% 0.33/0.64 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
% 0.33/0.64 = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_divide_diff_eq_iff
% 0.33/0.64 thf(fact_6160_add__divide__eq__if__simps_I5_J,axiom,
% 0.33/0.64 ! [Z: real,A: real,B: real] :
% 0.33/0.64 ( ( ( Z = zero_zero_real )
% 0.33/0.64 => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 0.33/0.64 = ( uminus_uminus_real @ B ) ) )
% 0.33/0.64 & ( ( Z != zero_zero_real )
% 0.33/0.64 => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 0.33/0.64 = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % add_divide_eq_if_simps(5)
% 0.33/0.64 thf(fact_6161_add__divide__eq__if__simps_I5_J,axiom,
% 0.33/0.64 ! [Z: complex,A: complex,B: complex] :
% 0.33/0.64 ( ( ( Z = zero_zero_complex )
% 0.33/0.64 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 0.33/0.64 = ( uminus1482373934393186551omplex @ B ) ) )
% 0.33/0.64 & ( ( Z != zero_zero_complex )
% 0.33/0.64 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 0.33/0.64 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % add_divide_eq_if_simps(5)
% 0.33/0.64 thf(fact_6162_add__divide__eq__if__simps_I5_J,axiom,
% 0.33/0.64 ! [Z: rat,A: rat,B: rat] :
% 0.33/0.64 ( ( ( Z = zero_zero_rat )
% 0.33/0.64 => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 0.33/0.64 = ( uminus_uminus_rat @ B ) ) )
% 0.33/0.64 & ( ( Z != zero_zero_rat )
% 0.33/0.64 => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 0.33/0.64 = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % add_divide_eq_if_simps(5)
% 0.33/0.64 thf(fact_6163_add__divide__eq__if__simps_I6_J,axiom,
% 0.33/0.64 ! [Z: real,A: real,B: real] :
% 0.33/0.64 ( ( ( Z = zero_zero_real )
% 0.33/0.64 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 0.33/0.64 = ( uminus_uminus_real @ B ) ) )
% 0.33/0.64 & ( ( Z != zero_zero_real )
% 0.33/0.64 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 0.33/0.64 = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % add_divide_eq_if_simps(6)
% 0.33/0.64 thf(fact_6164_add__divide__eq__if__simps_I6_J,axiom,
% 0.33/0.64 ! [Z: complex,A: complex,B: complex] :
% 0.33/0.64 ( ( ( Z = zero_zero_complex )
% 0.33/0.64 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 0.33/0.64 = ( uminus1482373934393186551omplex @ B ) ) )
% 0.33/0.64 & ( ( Z != zero_zero_complex )
% 0.33/0.64 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 0.33/0.64 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % add_divide_eq_if_simps(6)
% 0.33/0.64 thf(fact_6165_add__divide__eq__if__simps_I6_J,axiom,
% 0.33/0.64 ! [Z: rat,A: rat,B: rat] :
% 0.33/0.64 ( ( ( Z = zero_zero_rat )
% 0.33/0.64 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 0.33/0.64 = ( uminus_uminus_rat @ B ) ) )
% 0.33/0.64 & ( ( Z != zero_zero_rat )
% 0.33/0.64 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 0.33/0.64 = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % add_divide_eq_if_simps(6)
% 0.33/0.64 thf(fact_6166_even__minus,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A ) )
% 0.33/0.64 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % even_minus
% 0.33/0.64 thf(fact_6167_even__minus,axiom,
% 0.33/0.64 ! [A: code_integer] :
% 0.33/0.64 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( uminus1351360451143612070nteger @ A ) )
% 0.33/0.64 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % even_minus
% 0.33/0.64 thf(fact_6168_power2__eq__iff,axiom,
% 0.33/0.64 ! [X: real,Y: real] :
% 0.33/0.64 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.64 = ( ( X = Y )
% 0.33/0.64 | ( X
% 0.33/0.64 = ( uminus_uminus_real @ Y ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power2_eq_iff
% 0.33/0.64 thf(fact_6169_power2__eq__iff,axiom,
% 0.33/0.64 ! [X: int,Y: int] :
% 0.33/0.64 ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.64 = ( ( X = Y )
% 0.33/0.64 | ( X
% 0.33/0.64 = ( uminus_uminus_int @ Y ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power2_eq_iff
% 0.33/0.64 thf(fact_6170_power2__eq__iff,axiom,
% 0.33/0.64 ! [X: complex,Y: complex] :
% 0.33/0.64 ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.64 = ( ( X = Y )
% 0.33/0.64 | ( X
% 0.33/0.64 = ( uminus1482373934393186551omplex @ Y ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power2_eq_iff
% 0.33/0.64 thf(fact_6171_power2__eq__iff,axiom,
% 0.33/0.64 ! [X: rat,Y: rat] :
% 0.33/0.64 ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.64 = ( ( X = Y )
% 0.33/0.64 | ( X
% 0.33/0.64 = ( uminus_uminus_rat @ Y ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power2_eq_iff
% 0.33/0.64 thf(fact_6172_power2__eq__iff,axiom,
% 0.33/0.64 ! [X: code_integer,Y: code_integer] :
% 0.33/0.64 ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.64 = ( ( X = Y )
% 0.33/0.64 | ( X
% 0.33/0.64 = ( uminus1351360451143612070nteger @ Y ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power2_eq_iff
% 0.33/0.64 thf(fact_6173_verit__less__mono__div__int2,axiom,
% 0.33/0.64 ! [A2: int,B3: int,N2: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ A2 @ B3 )
% 0.33/0.64 => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N2 ) )
% 0.33/0.64 => ( ord_less_eq_int @ ( divide_divide_int @ B3 @ N2 ) @ ( divide_divide_int @ A2 @ N2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % verit_less_mono_div_int2
% 0.33/0.64 thf(fact_6174_div__eq__minus1,axiom,
% 0.33/0.64 ! [B: int] :
% 0.33/0.64 ( ( ord_less_int @ zero_zero_int @ B )
% 0.33/0.64 => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 0.33/0.64 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % div_eq_minus1
% 0.33/0.64 thf(fact_6175_sum__mono2,axiom,
% 0.33/0.64 ! [B3: set_real,A2: set_real,F: real > real] :
% 0.33/0.64 ( ( finite_finite_real @ B3 )
% 0.33/0.64 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.33/0.64 => ( ! [B2: real] :
% 0.33/0.64 ( ( member_real @ B2 @ ( minus_minus_set_real @ B3 @ A2 ) )
% 0.33/0.64 => ( ord_less_eq_real @ zero_zero_real @ ( F @ B2 ) ) )
% 0.33/0.64 => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B3 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_mono2
% 0.33/0.64 thf(fact_6176_sum__mono2,axiom,
% 0.33/0.64 ! [B3: set_int,A2: set_int,F: int > real] :
% 0.33/0.64 ( ( finite_finite_int @ B3 )
% 0.33/0.64 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.33/0.64 => ( ! [B2: int] :
% 0.33/0.64 ( ( member_int @ B2 @ ( minus_minus_set_int @ B3 @ A2 ) )
% 0.33/0.64 => ( ord_less_eq_real @ zero_zero_real @ ( F @ B2 ) ) )
% 0.33/0.64 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B3 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_mono2
% 0.33/0.64 thf(fact_6177_sum__mono2,axiom,
% 0.33/0.64 ! [B3: set_complex,A2: set_complex,F: complex > real] :
% 0.33/0.64 ( ( finite3207457112153483333omplex @ B3 )
% 0.33/0.64 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 0.33/0.64 => ( ! [B2: complex] :
% 0.33/0.64 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
% 0.33/0.64 => ( ord_less_eq_real @ zero_zero_real @ ( F @ B2 ) ) )
% 0.33/0.64 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B3 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_mono2
% 0.33/0.64 thf(fact_6178_sum__mono2,axiom,
% 0.33/0.64 ! [B3: set_real,A2: set_real,F: real > rat] :
% 0.33/0.64 ( ( finite_finite_real @ B3 )
% 0.33/0.64 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.33/0.64 => ( ! [B2: real] :
% 0.33/0.64 ( ( member_real @ B2 @ ( minus_minus_set_real @ B3 @ A2 ) )
% 0.33/0.64 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B2 ) ) )
% 0.33/0.64 => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B3 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_mono2
% 0.33/0.64 thf(fact_6179_sum__mono2,axiom,
% 0.33/0.64 ! [B3: set_int,A2: set_int,F: int > rat] :
% 0.33/0.64 ( ( finite_finite_int @ B3 )
% 0.33/0.64 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.33/0.64 => ( ! [B2: int] :
% 0.33/0.64 ( ( member_int @ B2 @ ( minus_minus_set_int @ B3 @ A2 ) )
% 0.33/0.64 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B2 ) ) )
% 0.33/0.64 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B3 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_mono2
% 0.33/0.64 thf(fact_6180_sum__mono2,axiom,
% 0.33/0.64 ! [B3: set_complex,A2: set_complex,F: complex > rat] :
% 0.33/0.64 ( ( finite3207457112153483333omplex @ B3 )
% 0.33/0.64 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 0.33/0.64 => ( ! [B2: complex] :
% 0.33/0.64 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
% 0.33/0.64 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B2 ) ) )
% 0.33/0.64 => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B3 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_mono2
% 0.33/0.64 thf(fact_6181_sum__mono2,axiom,
% 0.33/0.64 ! [B3: set_real,A2: set_real,F: real > nat] :
% 0.33/0.64 ( ( finite_finite_real @ B3 )
% 0.33/0.64 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.33/0.64 => ( ! [B2: real] :
% 0.33/0.64 ( ( member_real @ B2 @ ( minus_minus_set_real @ B3 @ A2 ) )
% 0.33/0.64 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B2 ) ) )
% 0.33/0.64 => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B3 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_mono2
% 0.33/0.64 thf(fact_6182_sum__mono2,axiom,
% 0.33/0.64 ! [B3: set_int,A2: set_int,F: int > nat] :
% 0.33/0.64 ( ( finite_finite_int @ B3 )
% 0.33/0.64 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.33/0.64 => ( ! [B2: int] :
% 0.33/0.64 ( ( member_int @ B2 @ ( minus_minus_set_int @ B3 @ A2 ) )
% 0.33/0.64 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B2 ) ) )
% 0.33/0.64 => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B3 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_mono2
% 0.33/0.64 thf(fact_6183_sum__mono2,axiom,
% 0.33/0.64 ! [B3: set_complex,A2: set_complex,F: complex > nat] :
% 0.33/0.64 ( ( finite3207457112153483333omplex @ B3 )
% 0.33/0.64 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 0.33/0.64 => ( ! [B2: complex] :
% 0.33/0.64 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
% 0.33/0.64 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B2 ) ) )
% 0.33/0.64 => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B3 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_mono2
% 0.33/0.64 thf(fact_6184_sum__mono2,axiom,
% 0.33/0.64 ! [B3: set_real,A2: set_real,F: real > int] :
% 0.33/0.64 ( ( finite_finite_real @ B3 )
% 0.33/0.64 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.33/0.64 => ( ! [B2: real] :
% 0.33/0.64 ( ( member_real @ B2 @ ( minus_minus_set_real @ B3 @ A2 ) )
% 0.33/0.64 => ( ord_less_eq_int @ zero_zero_int @ ( F @ B2 ) ) )
% 0.33/0.64 => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B3 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_mono2
% 0.33/0.64 thf(fact_6185_ln__le__minus__one,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.64 => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % ln_le_minus_one
% 0.33/0.64 thf(fact_6186_ln__diff__le,axiom,
% 0.33/0.64 ! [X: real,Y: real] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.64 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.64 => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) @ ( divide_divide_real @ ( minus_minus_real @ X @ Y ) @ Y ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % ln_diff_le
% 0.33/0.64 thf(fact_6187_le__minus__divide__eq,axiom,
% 0.33/0.64 ! [A: real,B: real,C: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.33/0.64 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.64 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.33/0.64 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.64 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.64 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.33/0.64 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.64 => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % le_minus_divide_eq
% 0.33/0.64 thf(fact_6188_le__minus__divide__eq,axiom,
% 0.33/0.64 ! [A: rat,B: rat,C: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.33/0.64 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.64 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.33/0.64 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.64 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.64 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.33/0.64 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.64 => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % le_minus_divide_eq
% 0.33/0.64 thf(fact_6189_minus__divide__le__eq,axiom,
% 0.33/0.64 ! [B: real,C: real,A: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.33/0.64 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.64 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.33/0.64 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.64 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.64 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.33/0.64 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.64 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_divide_le_eq
% 0.33/0.64 thf(fact_6190_minus__divide__le__eq,axiom,
% 0.33/0.64 ! [B: rat,C: rat,A: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.33/0.64 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.64 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.33/0.64 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.64 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.64 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.33/0.64 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.64 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_divide_le_eq
% 0.33/0.64 thf(fact_6191_neg__le__minus__divide__eq,axiom,
% 0.33/0.64 ! [C: real,A: real,B: real] :
% 0.33/0.64 ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.64 => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.33/0.64 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % neg_le_minus_divide_eq
% 0.33/0.64 thf(fact_6192_neg__le__minus__divide__eq,axiom,
% 0.33/0.64 ! [C: rat,A: rat,B: rat] :
% 0.33/0.64 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.64 => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.33/0.64 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % neg_le_minus_divide_eq
% 0.33/0.64 thf(fact_6193_neg__minus__divide__le__eq,axiom,
% 0.33/0.64 ! [C: real,B: real,A: real] :
% 0.33/0.64 ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.64 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.33/0.64 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % neg_minus_divide_le_eq
% 0.33/0.64 thf(fact_6194_neg__minus__divide__le__eq,axiom,
% 0.33/0.64 ! [C: rat,B: rat,A: rat] :
% 0.33/0.64 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.64 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.33/0.64 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % neg_minus_divide_le_eq
% 0.33/0.64 thf(fact_6195_pos__le__minus__divide__eq,axiom,
% 0.33/0.64 ! [C: real,A: real,B: real] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.64 => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.33/0.64 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % pos_le_minus_divide_eq
% 0.33/0.64 thf(fact_6196_pos__le__minus__divide__eq,axiom,
% 0.33/0.64 ! [C: rat,A: rat,B: rat] :
% 0.33/0.64 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.64 => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.33/0.64 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % pos_le_minus_divide_eq
% 0.33/0.64 thf(fact_6197_pos__minus__divide__le__eq,axiom,
% 0.33/0.64 ! [C: real,B: real,A: real] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.64 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.33/0.64 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % pos_minus_divide_le_eq
% 0.33/0.64 thf(fact_6198_pos__minus__divide__le__eq,axiom,
% 0.33/0.64 ! [C: rat,B: rat,A: rat] :
% 0.33/0.64 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.64 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.33/0.64 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % pos_minus_divide_le_eq
% 0.33/0.64 thf(fact_6199_less__divide__eq__numeral_I2_J,axiom,
% 0.33/0.64 ! [W: num,B: real,C: real] :
% 0.33/0.64 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
% 0.33/0.64 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.64 => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 0.33/0.64 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.64 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.64 => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.33/0.64 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.64 => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % less_divide_eq_numeral(2)
% 0.33/0.64 thf(fact_6200_less__divide__eq__numeral_I2_J,axiom,
% 0.33/0.64 ! [W: num,B: rat,C: rat] :
% 0.33/0.64 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
% 0.33/0.64 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.64 => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 0.33/0.64 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.64 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.64 => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.33/0.64 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.64 => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % less_divide_eq_numeral(2)
% 0.33/0.64 thf(fact_6201_divide__less__eq__numeral_I2_J,axiom,
% 0.33/0.64 ! [B: real,C: real,W: num] :
% 0.33/0.64 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.33/0.64 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.64 => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.33/0.64 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.64 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.64 => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 0.33/0.64 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.64 => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divide_less_eq_numeral(2)
% 0.33/0.64 thf(fact_6202_divide__less__eq__numeral_I2_J,axiom,
% 0.33/0.64 ! [B: rat,C: rat,W: num] :
% 0.33/0.64 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.33/0.64 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.64 => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.33/0.64 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.64 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.64 => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 0.33/0.64 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.64 => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divide_less_eq_numeral(2)
% 0.33/0.64 thf(fact_6203_power2__eq__1__iff,axiom,
% 0.33/0.64 ! [A: real] :
% 0.33/0.64 ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = one_one_real )
% 0.33/0.64 = ( ( A = one_one_real )
% 0.33/0.64 | ( A
% 0.33/0.64 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power2_eq_1_iff
% 0.33/0.64 thf(fact_6204_power2__eq__1__iff,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = one_one_int )
% 0.33/0.64 = ( ( A = one_one_int )
% 0.33/0.64 | ( A
% 0.33/0.64 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power2_eq_1_iff
% 0.33/0.64 thf(fact_6205_power2__eq__1__iff,axiom,
% 0.33/0.64 ! [A: complex] :
% 0.33/0.64 ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = one_one_complex )
% 0.33/0.64 = ( ( A = one_one_complex )
% 0.33/0.64 | ( A
% 0.33/0.64 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power2_eq_1_iff
% 0.33/0.64 thf(fact_6206_power2__eq__1__iff,axiom,
% 0.33/0.64 ! [A: rat] :
% 0.33/0.64 ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = one_one_rat )
% 0.33/0.64 = ( ( A = one_one_rat )
% 0.33/0.64 | ( A
% 0.33/0.64 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power2_eq_1_iff
% 0.33/0.64 thf(fact_6207_power2__eq__1__iff,axiom,
% 0.33/0.64 ! [A: code_integer] :
% 0.33/0.64 ( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = one_one_Code_integer )
% 0.33/0.64 = ( ( A = one_one_Code_integer )
% 0.33/0.64 | ( A
% 0.33/0.64 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power2_eq_1_iff
% 0.33/0.64 thf(fact_6208_uminus__power__if,axiom,
% 0.33/0.64 ! [N2: nat,A: real] :
% 0.33/0.64 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.64 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 )
% 0.33/0.64 = ( power_power_real @ A @ N2 ) ) )
% 0.33/0.64 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.64 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 )
% 0.33/0.64 = ( uminus_uminus_real @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % uminus_power_if
% 0.33/0.64 thf(fact_6209_uminus__power__if,axiom,
% 0.33/0.64 ! [N2: nat,A: int] :
% 0.33/0.64 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.64 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
% 0.33/0.64 = ( power_power_int @ A @ N2 ) ) )
% 0.33/0.64 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.64 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
% 0.33/0.64 = ( uminus_uminus_int @ ( power_power_int @ A @ N2 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % uminus_power_if
% 0.33/0.64 thf(fact_6210_uminus__power__if,axiom,
% 0.33/0.64 ! [N2: nat,A: complex] :
% 0.33/0.64 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.64 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 )
% 0.33/0.64 = ( power_power_complex @ A @ N2 ) ) )
% 0.33/0.64 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.64 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 )
% 0.33/0.64 = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N2 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % uminus_power_if
% 0.33/0.64 thf(fact_6211_uminus__power__if,axiom,
% 0.33/0.64 ! [N2: nat,A: rat] :
% 0.33/0.64 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.64 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 )
% 0.33/0.64 = ( power_power_rat @ A @ N2 ) ) )
% 0.33/0.64 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.64 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 )
% 0.33/0.64 = ( uminus_uminus_rat @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % uminus_power_if
% 0.33/0.64 thf(fact_6212_uminus__power__if,axiom,
% 0.33/0.64 ! [N2: nat,A: code_integer] :
% 0.33/0.64 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.64 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 )
% 0.33/0.64 = ( power_8256067586552552935nteger @ A @ N2 ) ) )
% 0.33/0.64 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.64 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 )
% 0.33/0.64 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % uminus_power_if
% 0.33/0.64 thf(fact_6213_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.33/0.64 ! [K: nat,N2: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ K @ N2 )
% 0.33/0.64 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N2 @ K ) )
% 0.33/0.64 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % neg_one_power_add_eq_neg_one_power_diff
% 0.33/0.64 thf(fact_6214_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.33/0.64 ! [K: nat,N2: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ K @ N2 )
% 0.33/0.64 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N2 @ K ) )
% 0.33/0.64 = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % neg_one_power_add_eq_neg_one_power_diff
% 0.33/0.64 thf(fact_6215_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.33/0.64 ! [K: nat,N2: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ K @ N2 )
% 0.33/0.64 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N2 @ K ) )
% 0.33/0.64 = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % neg_one_power_add_eq_neg_one_power_diff
% 0.33/0.64 thf(fact_6216_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.33/0.64 ! [K: nat,N2: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ K @ N2 )
% 0.33/0.64 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N2 @ K ) )
% 0.33/0.64 = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % neg_one_power_add_eq_neg_one_power_diff
% 0.33/0.64 thf(fact_6217_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.33/0.64 ! [K: nat,N2: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ K @ N2 )
% 0.33/0.64 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N2 @ K ) )
% 0.33/0.64 = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % neg_one_power_add_eq_neg_one_power_diff
% 0.33/0.64 thf(fact_6218_realpow__square__minus__le,axiom,
% 0.33/0.64 ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % realpow_square_minus_le
% 0.33/0.64 thf(fact_6219_ln__one__minus__pos__lower__bound,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.64 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.64 => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % ln_one_minus_pos_lower_bound
% 0.33/0.64 thf(fact_6220_signed__take__bit__int__less__eq__self__iff,axiom,
% 0.33/0.64 ! [N2: nat,K: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ K )
% 0.33/0.64 = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K ) ) ).
% 0.33/0.64
% 0.33/0.64 % signed_take_bit_int_less_eq_self_iff
% 0.33/0.64 thf(fact_6221_signed__take__bit__int__greater__eq__minus__exp,axiom,
% 0.33/0.64 ! [N2: nat,K: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( bit_ri631733984087533419it_int @ N2 @ K ) ) ).
% 0.33/0.64
% 0.33/0.64 % signed_take_bit_int_greater_eq_minus_exp
% 0.33/0.64 thf(fact_6222_signed__take__bit__int__greater__self__iff,axiom,
% 0.33/0.64 ! [K: int,N2: nat] :
% 0.33/0.64 ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N2 @ K ) )
% 0.33/0.64 = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % signed_take_bit_int_greater_self_iff
% 0.33/0.64 thf(fact_6223_minus__mod__int__eq,axiom,
% 0.33/0.64 ! [L2: int,K: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ zero_zero_int @ L2 )
% 0.33/0.64 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L2 )
% 0.33/0.64 = ( minus_minus_int @ ( minus_minus_int @ L2 @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_mod_int_eq
% 0.33/0.64 thf(fact_6224_zmod__minus1,axiom,
% 0.33/0.64 ! [B: int] :
% 0.33/0.64 ( ( ord_less_int @ zero_zero_int @ B )
% 0.33/0.64 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 0.33/0.64 = ( minus_minus_int @ B @ one_one_int ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % zmod_minus1
% 0.33/0.64 thf(fact_6225_sum__strict__mono2,axiom,
% 0.33/0.64 ! [B3: set_real,A2: set_real,B: real,F: real > real] :
% 0.33/0.64 ( ( finite_finite_real @ B3 )
% 0.33/0.64 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.33/0.64 => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A2 ) )
% 0.33/0.64 => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
% 0.33/0.64 => ( ! [X4: real] :
% 0.33/0.64 ( ( member_real @ X4 @ B3 )
% 0.33/0.64 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.33/0.64 => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B3 ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_strict_mono2
% 0.33/0.64 thf(fact_6226_sum__strict__mono2,axiom,
% 0.33/0.64 ! [B3: set_int,A2: set_int,B: int,F: int > real] :
% 0.33/0.64 ( ( finite_finite_int @ B3 )
% 0.33/0.64 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.33/0.64 => ( ( member_int @ B @ ( minus_minus_set_int @ B3 @ A2 ) )
% 0.33/0.64 => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
% 0.33/0.64 => ( ! [X4: int] :
% 0.33/0.64 ( ( member_int @ X4 @ B3 )
% 0.33/0.64 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.33/0.64 => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B3 ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_strict_mono2
% 0.33/0.64 thf(fact_6227_sum__strict__mono2,axiom,
% 0.33/0.64 ! [B3: set_complex,A2: set_complex,B: complex,F: complex > real] :
% 0.33/0.64 ( ( finite3207457112153483333omplex @ B3 )
% 0.33/0.64 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 0.33/0.64 => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
% 0.33/0.64 => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
% 0.33/0.64 => ( ! [X4: complex] :
% 0.33/0.64 ( ( member_complex @ X4 @ B3 )
% 0.33/0.64 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.33/0.64 => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B3 ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_strict_mono2
% 0.33/0.64 thf(fact_6228_sum__strict__mono2,axiom,
% 0.33/0.64 ! [B3: set_real,A2: set_real,B: real,F: real > rat] :
% 0.33/0.64 ( ( finite_finite_real @ B3 )
% 0.33/0.64 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.33/0.64 => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A2 ) )
% 0.33/0.64 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 0.33/0.64 => ( ! [X4: real] :
% 0.33/0.64 ( ( member_real @ X4 @ B3 )
% 0.33/0.64 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.33/0.64 => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B3 ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_strict_mono2
% 0.33/0.64 thf(fact_6229_sum__strict__mono2,axiom,
% 0.33/0.64 ! [B3: set_int,A2: set_int,B: int,F: int > rat] :
% 0.33/0.64 ( ( finite_finite_int @ B3 )
% 0.33/0.64 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.33/0.64 => ( ( member_int @ B @ ( minus_minus_set_int @ B3 @ A2 ) )
% 0.33/0.64 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 0.33/0.64 => ( ! [X4: int] :
% 0.33/0.64 ( ( member_int @ X4 @ B3 )
% 0.33/0.64 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.33/0.64 => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B3 ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_strict_mono2
% 0.33/0.64 thf(fact_6230_sum__strict__mono2,axiom,
% 0.33/0.64 ! [B3: set_complex,A2: set_complex,B: complex,F: complex > rat] :
% 0.33/0.64 ( ( finite3207457112153483333omplex @ B3 )
% 0.33/0.64 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 0.33/0.64 => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
% 0.33/0.64 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 0.33/0.64 => ( ! [X4: complex] :
% 0.33/0.64 ( ( member_complex @ X4 @ B3 )
% 0.33/0.64 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.33/0.64 => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B3 ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_strict_mono2
% 0.33/0.64 thf(fact_6231_sum__strict__mono2,axiom,
% 0.33/0.64 ! [B3: set_real,A2: set_real,B: real,F: real > nat] :
% 0.33/0.64 ( ( finite_finite_real @ B3 )
% 0.33/0.64 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.33/0.64 => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A2 ) )
% 0.33/0.64 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
% 0.33/0.64 => ( ! [X4: real] :
% 0.33/0.64 ( ( member_real @ X4 @ B3 )
% 0.33/0.64 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.33/0.64 => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B3 ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_strict_mono2
% 0.33/0.64 thf(fact_6232_sum__strict__mono2,axiom,
% 0.33/0.64 ! [B3: set_int,A2: set_int,B: int,F: int > nat] :
% 0.33/0.64 ( ( finite_finite_int @ B3 )
% 0.33/0.64 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.33/0.64 => ( ( member_int @ B @ ( minus_minus_set_int @ B3 @ A2 ) )
% 0.33/0.64 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
% 0.33/0.64 => ( ! [X4: int] :
% 0.33/0.64 ( ( member_int @ X4 @ B3 )
% 0.33/0.64 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.33/0.64 => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B3 ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_strict_mono2
% 0.33/0.64 thf(fact_6233_sum__strict__mono2,axiom,
% 0.33/0.64 ! [B3: set_complex,A2: set_complex,B: complex,F: complex > nat] :
% 0.33/0.64 ( ( finite3207457112153483333omplex @ B3 )
% 0.33/0.64 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 0.33/0.64 => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
% 0.33/0.64 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
% 0.33/0.64 => ( ! [X4: complex] :
% 0.33/0.64 ( ( member_complex @ X4 @ B3 )
% 0.33/0.64 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.33/0.64 => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B3 ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_strict_mono2
% 0.33/0.64 thf(fact_6234_sum__strict__mono2,axiom,
% 0.33/0.64 ! [B3: set_real,A2: set_real,B: real,F: real > int] :
% 0.33/0.64 ( ( finite_finite_real @ B3 )
% 0.33/0.64 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.33/0.64 => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A2 ) )
% 0.33/0.64 => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
% 0.33/0.64 => ( ! [X4: real] :
% 0.33/0.64 ( ( member_real @ X4 @ B3 )
% 0.33/0.64 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 0.33/0.64 => ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B3 ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_strict_mono2
% 0.33/0.64 thf(fact_6235_zdiv__zminus1__eq__if,axiom,
% 0.33/0.64 ! [B: int,A: int] :
% 0.33/0.64 ( ( B != zero_zero_int )
% 0.33/0.64 => ( ( ( ( modulo_modulo_int @ A @ B )
% 0.33/0.64 = zero_zero_int )
% 0.33/0.64 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 0.33/0.64 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
% 0.33/0.64 & ( ( ( modulo_modulo_int @ A @ B )
% 0.33/0.64 != zero_zero_int )
% 0.33/0.64 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 0.33/0.64 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % zdiv_zminus1_eq_if
% 0.33/0.64 thf(fact_6236_zdiv__zminus2__eq__if,axiom,
% 0.33/0.64 ! [B: int,A: int] :
% 0.33/0.64 ( ( B != zero_zero_int )
% 0.33/0.64 => ( ( ( ( modulo_modulo_int @ A @ B )
% 0.33/0.64 = zero_zero_int )
% 0.33/0.64 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 0.33/0.64 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
% 0.33/0.64 & ( ( ( modulo_modulo_int @ A @ B )
% 0.33/0.64 != zero_zero_int )
% 0.33/0.64 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 0.33/0.64 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % zdiv_zminus2_eq_if
% 0.33/0.64 thf(fact_6237_zminus1__lemma,axiom,
% 0.33/0.64 ! [A: int,B: int,Q2: int,R: int] :
% 0.33/0.64 ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.33/0.64 => ( ( B != zero_zero_int )
% 0.33/0.64 => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R = zero_zero_int ) @ ( uminus_uminus_int @ Q2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q2 ) @ one_one_int ) ) @ ( if_int @ ( R = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % zminus1_lemma
% 0.33/0.64 thf(fact_6238_le__divide__eq__numeral_I2_J,axiom,
% 0.33/0.64 ! [W: num,B: real,C: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
% 0.33/0.64 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.64 => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 0.33/0.64 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.64 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.64 => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.33/0.64 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.64 => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % le_divide_eq_numeral(2)
% 0.33/0.64 thf(fact_6239_le__divide__eq__numeral_I2_J,axiom,
% 0.33/0.64 ! [W: num,B: rat,C: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
% 0.33/0.64 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.64 => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 0.33/0.64 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.64 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.64 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.33/0.64 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.64 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % le_divide_eq_numeral(2)
% 0.33/0.64 thf(fact_6240_divide__le__eq__numeral_I2_J,axiom,
% 0.33/0.64 ! [B: real,C: real,W: num] :
% 0.33/0.64 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.33/0.64 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.64 => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.33/0.64 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.64 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.64 => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 0.33/0.64 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.33/0.64 => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divide_le_eq_numeral(2)
% 0.33/0.64 thf(fact_6241_divide__le__eq__numeral_I2_J,axiom,
% 0.33/0.64 ! [B: rat,C: rat,W: num] :
% 0.33/0.64 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.33/0.64 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.64 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.33/0.64 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.33/0.64 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.64 => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 0.33/0.64 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.33/0.64 => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divide_le_eq_numeral(2)
% 0.33/0.64 thf(fact_6242_square__le__1,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.64 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.64 => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % square_le_1
% 0.33/0.64 thf(fact_6243_square__le__1,axiom,
% 0.33/0.64 ! [X: code_integer] :
% 0.33/0.64 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
% 0.33/0.64 => ( ( ord_le3102999989581377725nteger @ X @ one_one_Code_integer )
% 0.33/0.64 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % square_le_1
% 0.33/0.64 thf(fact_6244_square__le__1,axiom,
% 0.33/0.64 ! [X: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X )
% 0.33/0.64 => ( ( ord_less_eq_rat @ X @ one_one_rat )
% 0.33/0.64 => ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % square_le_1
% 0.33/0.64 thf(fact_6245_square__le__1,axiom,
% 0.33/0.64 ! [X: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X )
% 0.33/0.64 => ( ( ord_less_eq_int @ X @ one_one_int )
% 0.33/0.64 => ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % square_le_1
% 0.33/0.64 thf(fact_6246_minus__power__mult__self,axiom,
% 0.33/0.64 ! [A: real,N2: nat] :
% 0.33/0.64 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 ) )
% 0.33/0.64 = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_power_mult_self
% 0.33/0.64 thf(fact_6247_minus__power__mult__self,axiom,
% 0.33/0.64 ! [A: int,N2: nat] :
% 0.33/0.64 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 ) )
% 0.33/0.64 = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_power_mult_self
% 0.33/0.64 thf(fact_6248_minus__power__mult__self,axiom,
% 0.33/0.64 ! [A: complex,N2: nat] :
% 0.33/0.64 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 ) )
% 0.33/0.64 = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_power_mult_self
% 0.33/0.64 thf(fact_6249_minus__power__mult__self,axiom,
% 0.33/0.64 ! [A: rat,N2: nat] :
% 0.33/0.64 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 ) )
% 0.33/0.64 = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_power_mult_self
% 0.33/0.64 thf(fact_6250_minus__power__mult__self,axiom,
% 0.33/0.64 ! [A: code_integer,N2: nat] :
% 0.33/0.64 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 ) )
% 0.33/0.64 = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_power_mult_self
% 0.33/0.64 thf(fact_6251_minus__one__power__iff,axiom,
% 0.33/0.64 ! [N2: nat] :
% 0.33/0.64 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.64 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 )
% 0.33/0.64 = one_one_real ) )
% 0.33/0.64 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.64 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 )
% 0.33/0.64 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_one_power_iff
% 0.33/0.64 thf(fact_6252_minus__one__power__iff,axiom,
% 0.33/0.64 ! [N2: nat] :
% 0.33/0.64 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.64 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
% 0.33/0.64 = one_one_int ) )
% 0.33/0.64 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.64 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
% 0.33/0.64 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_one_power_iff
% 0.33/0.64 thf(fact_6253_minus__one__power__iff,axiom,
% 0.33/0.64 ! [N2: nat] :
% 0.33/0.64 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.64 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 )
% 0.33/0.64 = one_one_complex ) )
% 0.33/0.64 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.64 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 )
% 0.33/0.64 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_one_power_iff
% 0.33/0.64 thf(fact_6254_minus__one__power__iff,axiom,
% 0.33/0.64 ! [N2: nat] :
% 0.33/0.64 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.64 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
% 0.33/0.64 = one_one_rat ) )
% 0.33/0.64 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.64 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
% 0.33/0.64 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_one_power_iff
% 0.33/0.64 thf(fact_6255_minus__one__power__iff,axiom,
% 0.33/0.64 ! [N2: nat] :
% 0.33/0.64 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.64 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
% 0.33/0.64 = one_one_Code_integer ) )
% 0.33/0.64 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.64 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
% 0.33/0.64 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_one_power_iff
% 0.33/0.64 thf(fact_6256_signed__take__bit__int__eq__self,axiom,
% 0.33/0.64 ! [N2: nat,K: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K )
% 0.33/0.64 => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.33/0.64 => ( ( bit_ri631733984087533419it_int @ N2 @ K )
% 0.33/0.64 = K ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % signed_take_bit_int_eq_self
% 0.33/0.64 thf(fact_6257_signed__take__bit__int__eq__self__iff,axiom,
% 0.33/0.64 ! [N2: nat,K: int] :
% 0.33/0.64 ( ( ( bit_ri631733984087533419it_int @ N2 @ K )
% 0.33/0.64 = K )
% 0.33/0.64 = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K )
% 0.33/0.64 & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % signed_take_bit_int_eq_self_iff
% 0.33/0.64 thf(fact_6258_minus__1__div__exp__eq__int,axiom,
% 0.33/0.64 ! [N2: nat] :
% 0.33/0.64 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.33/0.64 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_1_div_exp_eq_int
% 0.33/0.64 thf(fact_6259_div__pos__neg__trivial,axiom,
% 0.33/0.64 ! [K: int,L2: int] :
% 0.33/0.64 ( ( ord_less_int @ zero_zero_int @ K )
% 0.33/0.64 => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L2 ) @ zero_zero_int )
% 0.33/0.64 => ( ( divide_divide_int @ K @ L2 )
% 0.33/0.64 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % div_pos_neg_trivial
% 0.33/0.64 thf(fact_6260_add__0__iff,axiom,
% 0.33/0.64 ! [B: complex,A: complex] :
% 0.33/0.64 ( ( B
% 0.33/0.64 = ( plus_plus_complex @ B @ A ) )
% 0.33/0.64 = ( A = zero_zero_complex ) ) ).
% 0.33/0.64
% 0.33/0.64 % add_0_iff
% 0.33/0.64 thf(fact_6261_add__0__iff,axiom,
% 0.33/0.64 ! [B: real,A: real] :
% 0.33/0.64 ( ( B
% 0.33/0.64 = ( plus_plus_real @ B @ A ) )
% 0.33/0.64 = ( A = zero_zero_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % add_0_iff
% 0.33/0.64 thf(fact_6262_add__0__iff,axiom,
% 0.33/0.64 ! [B: rat,A: rat] :
% 0.33/0.64 ( ( B
% 0.33/0.64 = ( plus_plus_rat @ B @ A ) )
% 0.33/0.64 = ( A = zero_zero_rat ) ) ).
% 0.33/0.64
% 0.33/0.64 % add_0_iff
% 0.33/0.64 thf(fact_6263_add__0__iff,axiom,
% 0.33/0.64 ! [B: nat,A: nat] :
% 0.33/0.64 ( ( B
% 0.33/0.64 = ( plus_plus_nat @ B @ A ) )
% 0.33/0.64 = ( A = zero_zero_nat ) ) ).
% 0.33/0.64
% 0.33/0.64 % add_0_iff
% 0.33/0.64 thf(fact_6264_add__0__iff,axiom,
% 0.33/0.64 ! [B: int,A: int] :
% 0.33/0.64 ( ( B
% 0.33/0.64 = ( plus_plus_int @ B @ A ) )
% 0.33/0.64 = ( A = zero_zero_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % add_0_iff
% 0.33/0.64 thf(fact_6265_crossproduct__noteq,axiom,
% 0.33/0.64 ! [A: real,B: real,C: real,D: real] :
% 0.33/0.64 ( ( ( A != B )
% 0.33/0.64 & ( C != D ) )
% 0.33/0.64 = ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) )
% 0.33/0.64 != ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % crossproduct_noteq
% 0.33/0.64 thf(fact_6266_crossproduct__noteq,axiom,
% 0.33/0.64 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.33/0.64 ( ( ( A != B )
% 0.33/0.64 & ( C != D ) )
% 0.33/0.64 = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) )
% 0.33/0.64 != ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % crossproduct_noteq
% 0.33/0.64 thf(fact_6267_crossproduct__noteq,axiom,
% 0.33/0.64 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.33/0.64 ( ( ( A != B )
% 0.33/0.64 & ( C != D ) )
% 0.33/0.64 = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
% 0.33/0.64 != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % crossproduct_noteq
% 0.33/0.64 thf(fact_6268_crossproduct__noteq,axiom,
% 0.33/0.64 ! [A: int,B: int,C: int,D: int] :
% 0.33/0.64 ( ( ( A != B )
% 0.33/0.64 & ( C != D ) )
% 0.33/0.64 = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
% 0.33/0.64 != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % crossproduct_noteq
% 0.33/0.64 thf(fact_6269_crossproduct__eq,axiom,
% 0.33/0.64 ! [W: real,Y: real,X: real,Z: real] :
% 0.33/0.64 ( ( ( plus_plus_real @ ( times_times_real @ W @ Y ) @ ( times_times_real @ X @ Z ) )
% 0.33/0.64 = ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X @ Y ) ) )
% 0.33/0.64 = ( ( W = X )
% 0.33/0.64 | ( Y = Z ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % crossproduct_eq
% 0.33/0.64 thf(fact_6270_crossproduct__eq,axiom,
% 0.33/0.64 ! [W: rat,Y: rat,X: rat,Z: rat] :
% 0.33/0.64 ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y ) @ ( times_times_rat @ X @ Z ) )
% 0.33/0.64 = ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X @ Y ) ) )
% 0.33/0.64 = ( ( W = X )
% 0.33/0.64 | ( Y = Z ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % crossproduct_eq
% 0.33/0.64 thf(fact_6271_crossproduct__eq,axiom,
% 0.33/0.64 ! [W: nat,Y: nat,X: nat,Z: nat] :
% 0.33/0.64 ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z ) )
% 0.33/0.64 = ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y ) ) )
% 0.33/0.64 = ( ( W = X )
% 0.33/0.64 | ( Y = Z ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % crossproduct_eq
% 0.33/0.64 thf(fact_6272_crossproduct__eq,axiom,
% 0.33/0.64 ! [W: int,Y: int,X: int,Z: int] :
% 0.33/0.64 ( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z ) )
% 0.33/0.64 = ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X @ Y ) ) )
% 0.33/0.64 = ( ( W = X )
% 0.33/0.64 | ( Y = Z ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % crossproduct_eq
% 0.33/0.64 thf(fact_6273_power__minus1__odd,axiom,
% 0.33/0.64 ! [N2: nat] :
% 0.33/0.64 ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.33/0.64 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_minus1_odd
% 0.33/0.64 thf(fact_6274_power__minus1__odd,axiom,
% 0.33/0.64 ! [N2: nat] :
% 0.33/0.64 ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.33/0.64 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_minus1_odd
% 0.33/0.64 thf(fact_6275_power__minus1__odd,axiom,
% 0.33/0.64 ! [N2: nat] :
% 0.33/0.64 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.33/0.64 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_minus1_odd
% 0.33/0.64 thf(fact_6276_power__minus1__odd,axiom,
% 0.33/0.64 ! [N2: nat] :
% 0.33/0.64 ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.33/0.64 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_minus1_odd
% 0.33/0.64 thf(fact_6277_power__minus1__odd,axiom,
% 0.33/0.64 ! [N2: nat] :
% 0.33/0.64 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.33/0.64 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_minus1_odd
% 0.33/0.64 thf(fact_6278_int__bit__induct,axiom,
% 0.33/0.64 ! [P: int > $o,K: int] :
% 0.33/0.64 ( ( P @ zero_zero_int )
% 0.33/0.64 => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.64 => ( ! [K2: int] :
% 0.33/0.64 ( ( P @ K2 )
% 0.33/0.64 => ( ( K2 != zero_zero_int )
% 0.33/0.64 => ( P @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.64 => ( ! [K2: int] :
% 0.33/0.64 ( ( P @ K2 )
% 0.33/0.64 => ( ( K2
% 0.33/0.64 != ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.64 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
% 0.33/0.64 => ( P @ K ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % int_bit_induct
% 0.33/0.64 thf(fact_6279_divmod__step__nat__def,axiom,
% 0.33/0.64 ( unique5026877609467782581ep_nat
% 0.33/0.64 = ( ^ [L: num] :
% 0.33/0.64 ( produc2626176000494625587at_nat
% 0.33/0.64 @ ^ [Q4: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_step_nat_def
% 0.33/0.64 thf(fact_6280_ln__one__plus__pos__lower__bound,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.64 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.64 => ( ord_less_eq_real @ ( minus_minus_real @ X @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % ln_one_plus_pos_lower_bound
% 0.33/0.64 thf(fact_6281_signed__take__bit__int__greater__eq,axiom,
% 0.33/0.64 ! [K: int,N2: nat] :
% 0.33/0.64 ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.33/0.64 => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) ) @ ( bit_ri631733984087533419it_int @ N2 @ K ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % signed_take_bit_int_greater_eq
% 0.33/0.64 thf(fact_6282_divmod__step__int__def,axiom,
% 0.33/0.64 ( unique5024387138958732305ep_int
% 0.33/0.64 = ( ^ [L: num] :
% 0.33/0.64 ( produc4245557441103728435nt_int
% 0.33/0.64 @ ^ [Q4: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_step_int_def
% 0.33/0.64 thf(fact_6283_ln__2__less__1,axiom,
% 0.33/0.64 ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).
% 0.33/0.64
% 0.33/0.64 % ln_2_less_1
% 0.33/0.64 thf(fact_6284_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.64 => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.33/0.64 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_ln_one_plus_x_minus_x_bound_nonpos
% 0.33/0.64 thf(fact_6285_tanh__ln__real,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.64 => ( ( tanh_real @ ( ln_ln_real @ X ) )
% 0.33/0.64 = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % tanh_ln_real
% 0.33/0.64 thf(fact_6286_divmod__algorithm__code_I5_J,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
% 0.33/0.64 = ( produc4245557441103728435nt_int
% 0.33/0.64 @ ^ [Q4: int,R5: int] : ( product_Pair_int_int @ Q4 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) )
% 0.33/0.64 @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_algorithm_code(5)
% 0.33/0.64 thf(fact_6287_divmod__algorithm__code_I5_J,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
% 0.33/0.64 = ( produc2626176000494625587at_nat
% 0.33/0.64 @ ^ [Q4: nat,R5: nat] : ( product_Pair_nat_nat @ Q4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) )
% 0.33/0.64 @ ( unique5055182867167087721od_nat @ M @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_algorithm_code(5)
% 0.33/0.64 thf(fact_6288_divmod__algorithm__code_I5_J,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
% 0.33/0.64 = ( produc6916734918728496179nteger
% 0.33/0.64 @ ^ [Q4: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q4 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) )
% 0.33/0.64 @ ( unique3479559517661332726nteger @ M @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_algorithm_code(5)
% 0.33/0.64 thf(fact_6289_divmod__nat__if,axiom,
% 0.33/0.64 ( divmod_nat
% 0.33/0.64 = ( ^ [M6: nat,N: nat] :
% 0.33/0.64 ( if_Pro6206227464963214023at_nat
% 0.33/0.64 @ ( ( N = zero_zero_nat )
% 0.33/0.64 | ( ord_less_nat @ M6 @ N ) )
% 0.33/0.64 @ ( product_Pair_nat_nat @ zero_zero_nat @ M6 )
% 0.33/0.64 @ ( produc2626176000494625587at_nat
% 0.33/0.64 @ ^ [Q4: nat] : ( product_Pair_nat_nat @ ( suc @ Q4 ) )
% 0.33/0.64 @ ( divmod_nat @ ( minus_minus_nat @ M6 @ N ) @ N ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_nat_if
% 0.33/0.64 thf(fact_6290_signed__take__bit__Suc__minus__bit1,axiom,
% 0.33/0.64 ! [N2: nat,K: num] :
% 0.33/0.64 ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.33/0.64 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % signed_take_bit_Suc_minus_bit1
% 0.33/0.64 thf(fact_6291_abs__ln__one__plus__x__minus__x__bound,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.64 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_ln_one_plus_x_minus_x_bound
% 0.33/0.64 thf(fact_6292_semiring__norm_I90_J,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( ( bit1 @ M )
% 0.33/0.64 = ( bit1 @ N2 ) )
% 0.33/0.64 = ( M = N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % semiring_norm(90)
% 0.33/0.64 thf(fact_6293_case__prodI,axiom,
% 0.33/0.64 ! [F: code_integer > $o > $o,A: code_integer,B: $o] :
% 0.33/0.64 ( ( F @ A @ B )
% 0.33/0.64 => ( produc7828578312038201481er_o_o @ F @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % case_prodI
% 0.33/0.64 thf(fact_6294_case__prodI,axiom,
% 0.33/0.64 ! [F: num > num > $o,A: num,B: num] :
% 0.33/0.64 ( ( F @ A @ B )
% 0.33/0.64 => ( produc5703948589228662326_num_o @ F @ ( product_Pair_num_num @ A @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % case_prodI
% 0.33/0.64 thf(fact_6295_case__prodI,axiom,
% 0.33/0.64 ! [F: nat > num > $o,A: nat,B: num] :
% 0.33/0.64 ( ( F @ A @ B )
% 0.33/0.64 => ( produc4927758841916487424_num_o @ F @ ( product_Pair_nat_num @ A @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % case_prodI
% 0.33/0.64 thf(fact_6296_case__prodI,axiom,
% 0.33/0.64 ! [F: nat > nat > $o,A: nat,B: nat] :
% 0.33/0.64 ( ( F @ A @ B )
% 0.33/0.64 => ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % case_prodI
% 0.33/0.64 thf(fact_6297_case__prodI,axiom,
% 0.33/0.64 ! [F: int > int > $o,A: int,B: int] :
% 0.33/0.64 ( ( F @ A @ B )
% 0.33/0.64 => ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % case_prodI
% 0.33/0.64 thf(fact_6298_case__prodI2,axiom,
% 0.33/0.64 ! [P4: produc6271795597528267376eger_o,C: code_integer > $o > $o] :
% 0.33/0.64 ( ! [A3: code_integer,B2: $o] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( produc6677183202524767010eger_o @ A3 @ B2 ) )
% 0.33/0.64 => ( C @ A3 @ B2 ) )
% 0.33/0.64 => ( produc7828578312038201481er_o_o @ C @ P4 ) ) ).
% 0.33/0.64
% 0.33/0.64 % case_prodI2
% 0.33/0.64 thf(fact_6299_case__prodI2,axiom,
% 0.33/0.64 ! [P4: product_prod_num_num,C: num > num > $o] :
% 0.33/0.64 ( ! [A3: num,B2: num] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( product_Pair_num_num @ A3 @ B2 ) )
% 0.33/0.64 => ( C @ A3 @ B2 ) )
% 0.33/0.64 => ( produc5703948589228662326_num_o @ C @ P4 ) ) ).
% 0.33/0.64
% 0.33/0.64 % case_prodI2
% 0.33/0.64 thf(fact_6300_case__prodI2,axiom,
% 0.33/0.64 ! [P4: product_prod_nat_num,C: nat > num > $o] :
% 0.33/0.64 ( ! [A3: nat,B2: num] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( product_Pair_nat_num @ A3 @ B2 ) )
% 0.33/0.64 => ( C @ A3 @ B2 ) )
% 0.33/0.64 => ( produc4927758841916487424_num_o @ C @ P4 ) ) ).
% 0.33/0.64
% 0.33/0.64 % case_prodI2
% 0.33/0.64 thf(fact_6301_case__prodI2,axiom,
% 0.33/0.64 ! [P4: product_prod_nat_nat,C: nat > nat > $o] :
% 0.33/0.64 ( ! [A3: nat,B2: nat] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( product_Pair_nat_nat @ A3 @ B2 ) )
% 0.33/0.64 => ( C @ A3 @ B2 ) )
% 0.33/0.64 => ( produc6081775807080527818_nat_o @ C @ P4 ) ) ).
% 0.33/0.64
% 0.33/0.64 % case_prodI2
% 0.33/0.64 thf(fact_6302_case__prodI2,axiom,
% 0.33/0.64 ! [P4: product_prod_int_int,C: int > int > $o] :
% 0.33/0.64 ( ! [A3: int,B2: int] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( product_Pair_int_int @ A3 @ B2 ) )
% 0.33/0.64 => ( C @ A3 @ B2 ) )
% 0.33/0.64 => ( produc4947309494688390418_int_o @ C @ P4 ) ) ).
% 0.33/0.64
% 0.33/0.64 % case_prodI2
% 0.33/0.64 thf(fact_6303_mem__case__prodI,axiom,
% 0.33/0.64 ! [Z: nat,C: code_integer > $o > set_nat,A: code_integer,B: $o] :
% 0.33/0.64 ( ( member_nat @ Z @ ( C @ A @ B ) )
% 0.33/0.64 => ( member_nat @ Z @ ( produc5431169771168744661et_nat @ C @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodI
% 0.33/0.64 thf(fact_6304_mem__case__prodI,axiom,
% 0.33/0.64 ! [Z: real,C: code_integer > $o > set_real,A: code_integer,B: $o] :
% 0.33/0.64 ( ( member_real @ Z @ ( C @ A @ B ) )
% 0.33/0.64 => ( member_real @ Z @ ( produc242741666403216561t_real @ C @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodI
% 0.33/0.64 thf(fact_6305_mem__case__prodI,axiom,
% 0.33/0.64 ! [Z: int,C: code_integer > $o > set_int,A: code_integer,B: $o] :
% 0.33/0.64 ( ( member_int @ Z @ ( C @ A @ B ) )
% 0.33/0.64 => ( member_int @ Z @ ( produc1253318751659547953et_int @ C @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodI
% 0.33/0.64 thf(fact_6306_mem__case__prodI,axiom,
% 0.33/0.64 ! [Z: complex,C: code_integer > $o > set_complex,A: code_integer,B: $o] :
% 0.33/0.64 ( ( member_complex @ Z @ ( C @ A @ B ) )
% 0.33/0.64 => ( member_complex @ Z @ ( produc1043322548047392435omplex @ C @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodI
% 0.33/0.64 thf(fact_6307_mem__case__prodI,axiom,
% 0.33/0.64 ! [Z: nat,C: num > num > set_nat,A: num,B: num] :
% 0.33/0.64 ( ( member_nat @ Z @ ( C @ A @ B ) )
% 0.33/0.64 => ( member_nat @ Z @ ( produc1361121860356118632et_nat @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodI
% 0.33/0.64 thf(fact_6308_mem__case__prodI,axiom,
% 0.33/0.64 ! [Z: real,C: num > num > set_real,A: num,B: num] :
% 0.33/0.64 ( ( member_real @ Z @ ( C @ A @ B ) )
% 0.33/0.64 => ( member_real @ Z @ ( produc8296048397933160132t_real @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodI
% 0.33/0.64 thf(fact_6309_mem__case__prodI,axiom,
% 0.33/0.64 ! [Z: int,C: num > num > set_int,A: num,B: num] :
% 0.33/0.64 ( ( member_int @ Z @ ( C @ A @ B ) )
% 0.33/0.64 => ( member_int @ Z @ ( produc6406642877701697732et_int @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodI
% 0.33/0.64 thf(fact_6310_mem__case__prodI,axiom,
% 0.33/0.64 ! [Z: complex,C: num > num > set_complex,A: num,B: num] :
% 0.33/0.64 ( ( member_complex @ Z @ ( C @ A @ B ) )
% 0.33/0.64 => ( member_complex @ Z @ ( produc2866383454006189126omplex @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodI
% 0.33/0.64 thf(fact_6311_mem__case__prodI,axiom,
% 0.33/0.64 ! [Z: nat,C: nat > num > set_nat,A: nat,B: num] :
% 0.33/0.64 ( ( member_nat @ Z @ ( C @ A @ B ) )
% 0.33/0.64 => ( member_nat @ Z @ ( produc4130284055270567454et_nat @ C @ ( product_Pair_nat_num @ A @ B ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodI
% 0.33/0.64 thf(fact_6312_mem__case__prodI,axiom,
% 0.33/0.64 ! [Z: real,C: nat > num > set_real,A: nat,B: num] :
% 0.33/0.64 ( ( member_real @ Z @ ( C @ A @ B ) )
% 0.33/0.64 => ( member_real @ Z @ ( produc1435849484188172666t_real @ C @ ( product_Pair_nat_num @ A @ B ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodI
% 0.33/0.64 thf(fact_6313_mem__case__prodI2,axiom,
% 0.33/0.64 ! [P4: produc6271795597528267376eger_o,Z: nat,C: code_integer > $o > set_nat] :
% 0.33/0.64 ( ! [A3: code_integer,B2: $o] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( produc6677183202524767010eger_o @ A3 @ B2 ) )
% 0.33/0.64 => ( member_nat @ Z @ ( C @ A3 @ B2 ) ) )
% 0.33/0.64 => ( member_nat @ Z @ ( produc5431169771168744661et_nat @ C @ P4 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodI2
% 0.33/0.64 thf(fact_6314_mem__case__prodI2,axiom,
% 0.33/0.64 ! [P4: produc6271795597528267376eger_o,Z: real,C: code_integer > $o > set_real] :
% 0.33/0.64 ( ! [A3: code_integer,B2: $o] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( produc6677183202524767010eger_o @ A3 @ B2 ) )
% 0.33/0.64 => ( member_real @ Z @ ( C @ A3 @ B2 ) ) )
% 0.33/0.64 => ( member_real @ Z @ ( produc242741666403216561t_real @ C @ P4 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodI2
% 0.33/0.64 thf(fact_6315_mem__case__prodI2,axiom,
% 0.33/0.64 ! [P4: produc6271795597528267376eger_o,Z: int,C: code_integer > $o > set_int] :
% 0.33/0.64 ( ! [A3: code_integer,B2: $o] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( produc6677183202524767010eger_o @ A3 @ B2 ) )
% 0.33/0.64 => ( member_int @ Z @ ( C @ A3 @ B2 ) ) )
% 0.33/0.64 => ( member_int @ Z @ ( produc1253318751659547953et_int @ C @ P4 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodI2
% 0.33/0.64 thf(fact_6316_mem__case__prodI2,axiom,
% 0.33/0.64 ! [P4: produc6271795597528267376eger_o,Z: complex,C: code_integer > $o > set_complex] :
% 0.33/0.64 ( ! [A3: code_integer,B2: $o] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( produc6677183202524767010eger_o @ A3 @ B2 ) )
% 0.33/0.64 => ( member_complex @ Z @ ( C @ A3 @ B2 ) ) )
% 0.33/0.64 => ( member_complex @ Z @ ( produc1043322548047392435omplex @ C @ P4 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodI2
% 0.33/0.64 thf(fact_6317_mem__case__prodI2,axiom,
% 0.33/0.64 ! [P4: product_prod_num_num,Z: nat,C: num > num > set_nat] :
% 0.33/0.64 ( ! [A3: num,B2: num] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( product_Pair_num_num @ A3 @ B2 ) )
% 0.33/0.64 => ( member_nat @ Z @ ( C @ A3 @ B2 ) ) )
% 0.33/0.64 => ( member_nat @ Z @ ( produc1361121860356118632et_nat @ C @ P4 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodI2
% 0.33/0.64 thf(fact_6318_mem__case__prodI2,axiom,
% 0.33/0.64 ! [P4: product_prod_num_num,Z: real,C: num > num > set_real] :
% 0.33/0.64 ( ! [A3: num,B2: num] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( product_Pair_num_num @ A3 @ B2 ) )
% 0.33/0.64 => ( member_real @ Z @ ( C @ A3 @ B2 ) ) )
% 0.33/0.64 => ( member_real @ Z @ ( produc8296048397933160132t_real @ C @ P4 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodI2
% 0.33/0.64 thf(fact_6319_mem__case__prodI2,axiom,
% 0.33/0.64 ! [P4: product_prod_num_num,Z: int,C: num > num > set_int] :
% 0.33/0.64 ( ! [A3: num,B2: num] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( product_Pair_num_num @ A3 @ B2 ) )
% 0.33/0.64 => ( member_int @ Z @ ( C @ A3 @ B2 ) ) )
% 0.33/0.64 => ( member_int @ Z @ ( produc6406642877701697732et_int @ C @ P4 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodI2
% 0.33/0.64 thf(fact_6320_mem__case__prodI2,axiom,
% 0.33/0.64 ! [P4: product_prod_num_num,Z: complex,C: num > num > set_complex] :
% 0.33/0.64 ( ! [A3: num,B2: num] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( product_Pair_num_num @ A3 @ B2 ) )
% 0.33/0.64 => ( member_complex @ Z @ ( C @ A3 @ B2 ) ) )
% 0.33/0.64 => ( member_complex @ Z @ ( produc2866383454006189126omplex @ C @ P4 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodI2
% 0.33/0.64 thf(fact_6321_mem__case__prodI2,axiom,
% 0.33/0.64 ! [P4: product_prod_nat_num,Z: nat,C: nat > num > set_nat] :
% 0.33/0.64 ( ! [A3: nat,B2: num] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( product_Pair_nat_num @ A3 @ B2 ) )
% 0.33/0.64 => ( member_nat @ Z @ ( C @ A3 @ B2 ) ) )
% 0.33/0.64 => ( member_nat @ Z @ ( produc4130284055270567454et_nat @ C @ P4 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodI2
% 0.33/0.64 thf(fact_6322_mem__case__prodI2,axiom,
% 0.33/0.64 ! [P4: product_prod_nat_num,Z: real,C: nat > num > set_real] :
% 0.33/0.64 ( ! [A3: nat,B2: num] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( product_Pair_nat_num @ A3 @ B2 ) )
% 0.33/0.64 => ( member_real @ Z @ ( C @ A3 @ B2 ) ) )
% 0.33/0.64 => ( member_real @ Z @ ( produc1435849484188172666t_real @ C @ P4 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodI2
% 0.33/0.64 thf(fact_6323_case__prodI2_H,axiom,
% 0.33/0.64 ! [P4: product_prod_nat_nat,C: nat > nat > product_prod_nat_nat > $o,X: product_prod_nat_nat] :
% 0.33/0.64 ( ! [A3: nat,B2: nat] :
% 0.33/0.64 ( ( ( product_Pair_nat_nat @ A3 @ B2 )
% 0.33/0.64 = P4 )
% 0.33/0.64 => ( C @ A3 @ B2 @ X ) )
% 0.33/0.64 => ( produc8739625826339149834_nat_o @ C @ P4 @ X ) ) ).
% 0.33/0.64
% 0.33/0.64 % case_prodI2'
% 0.33/0.64 thf(fact_6324_semiring__norm_I89_J,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( bit1 @ M )
% 0.33/0.64 != ( bit0 @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % semiring_norm(89)
% 0.33/0.64 thf(fact_6325_semiring__norm_I88_J,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( bit0 @ M )
% 0.33/0.64 != ( bit1 @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % semiring_norm(88)
% 0.33/0.64 thf(fact_6326_semiring__norm_I86_J,axiom,
% 0.33/0.64 ! [M: num] :
% 0.33/0.64 ( ( bit1 @ M )
% 0.33/0.64 != one ) ).
% 0.33/0.64
% 0.33/0.64 % semiring_norm(86)
% 0.33/0.64 thf(fact_6327_semiring__norm_I84_J,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( one
% 0.33/0.64 != ( bit1 @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % semiring_norm(84)
% 0.33/0.64 thf(fact_6328_abs__numeral,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N2 ) )
% 0.33/0.64 = ( numera6620942414471956472nteger @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_numeral
% 0.33/0.64 thf(fact_6329_abs__numeral,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( abs_abs_real @ ( numeral_numeral_real @ N2 ) )
% 0.33/0.64 = ( numeral_numeral_real @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_numeral
% 0.33/0.64 thf(fact_6330_abs__numeral,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( abs_abs_rat @ ( numeral_numeral_rat @ N2 ) )
% 0.33/0.64 = ( numeral_numeral_rat @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_numeral
% 0.33/0.64 thf(fact_6331_abs__numeral,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( abs_abs_int @ ( numeral_numeral_int @ N2 ) )
% 0.33/0.64 = ( numeral_numeral_int @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_numeral
% 0.33/0.64 thf(fact_6332_abs__mult__self__eq,axiom,
% 0.33/0.64 ! [A: code_integer] :
% 0.33/0.64 ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
% 0.33/0.64 = ( times_3573771949741848930nteger @ A @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_mult_self_eq
% 0.33/0.64 thf(fact_6333_abs__mult__self__eq,axiom,
% 0.33/0.64 ! [A: real] :
% 0.33/0.64 ( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
% 0.33/0.64 = ( times_times_real @ A @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_mult_self_eq
% 0.33/0.64 thf(fact_6334_abs__mult__self__eq,axiom,
% 0.33/0.64 ! [A: rat] :
% 0.33/0.64 ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
% 0.33/0.64 = ( times_times_rat @ A @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_mult_self_eq
% 0.33/0.64 thf(fact_6335_abs__mult__self__eq,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
% 0.33/0.64 = ( times_times_int @ A @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_mult_self_eq
% 0.33/0.64 thf(fact_6336_abs__add__abs,axiom,
% 0.33/0.64 ! [A: code_integer,B: code_integer] :
% 0.33/0.64 ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
% 0.33/0.64 = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_add_abs
% 0.33/0.64 thf(fact_6337_abs__add__abs,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
% 0.33/0.64 = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_add_abs
% 0.33/0.64 thf(fact_6338_abs__add__abs,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
% 0.33/0.64 = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_add_abs
% 0.33/0.64 thf(fact_6339_abs__add__abs,axiom,
% 0.33/0.64 ! [A: int,B: int] :
% 0.33/0.64 ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
% 0.33/0.64 = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_add_abs
% 0.33/0.64 thf(fact_6340_abs__1,axiom,
% 0.33/0.64 ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 0.33/0.64 = one_one_Code_integer ) ).
% 0.33/0.64
% 0.33/0.64 % abs_1
% 0.33/0.64 thf(fact_6341_abs__1,axiom,
% 0.33/0.64 ( ( abs_abs_complex @ one_one_complex )
% 0.33/0.64 = one_one_complex ) ).
% 0.33/0.64
% 0.33/0.64 % abs_1
% 0.33/0.64 thf(fact_6342_abs__1,axiom,
% 0.33/0.64 ( ( abs_abs_real @ one_one_real )
% 0.33/0.64 = one_one_real ) ).
% 0.33/0.64
% 0.33/0.64 % abs_1
% 0.33/0.64 thf(fact_6343_abs__1,axiom,
% 0.33/0.64 ( ( abs_abs_rat @ one_one_rat )
% 0.33/0.64 = one_one_rat ) ).
% 0.33/0.64
% 0.33/0.64 % abs_1
% 0.33/0.64 thf(fact_6344_abs__1,axiom,
% 0.33/0.64 ( ( abs_abs_int @ one_one_int )
% 0.33/0.64 = one_one_int ) ).
% 0.33/0.64
% 0.33/0.64 % abs_1
% 0.33/0.64 thf(fact_6345_abs__divide,axiom,
% 0.33/0.64 ! [A: complex,B: complex] :
% 0.33/0.64 ( ( abs_abs_complex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.33/0.64 = ( divide1717551699836669952omplex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_divide
% 0.33/0.64 thf(fact_6346_abs__divide,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 0.33/0.64 = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_divide
% 0.33/0.64 thf(fact_6347_abs__divide,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 0.33/0.64 = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_divide
% 0.33/0.64 thf(fact_6348_tanh__real__less__iff,axiom,
% 0.33/0.64 ! [X: real,Y: real] :
% 0.33/0.64 ( ( ord_less_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) )
% 0.33/0.64 = ( ord_less_real @ X @ Y ) ) ).
% 0.33/0.64
% 0.33/0.64 % tanh_real_less_iff
% 0.33/0.64 thf(fact_6349_tanh__real__le__iff,axiom,
% 0.33/0.64 ! [X: real,Y: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) )
% 0.33/0.64 = ( ord_less_eq_real @ X @ Y ) ) ).
% 0.33/0.64
% 0.33/0.64 % tanh_real_le_iff
% 0.33/0.64 thf(fact_6350_semiring__norm_I80_J,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( ord_less_num @ M @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % semiring_norm(80)
% 0.33/0.64 thf(fact_6351_semiring__norm_I73_J,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( ord_less_eq_num @ M @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % semiring_norm(73)
% 0.33/0.64 thf(fact_6352_abs__sum__abs,axiom,
% 0.33/0.64 ! [F: int > int,A2: set_int] :
% 0.33/0.64 ( ( abs_abs_int
% 0.33/0.64 @ ( groups4538972089207619220nt_int
% 0.33/0.64 @ ^ [A5: int] : ( abs_abs_int @ ( F @ A5 ) )
% 0.33/0.64 @ A2 ) )
% 0.33/0.64 = ( groups4538972089207619220nt_int
% 0.33/0.64 @ ^ [A5: int] : ( abs_abs_int @ ( F @ A5 ) )
% 0.33/0.64 @ A2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_sum_abs
% 0.33/0.64 thf(fact_6353_abs__sum__abs,axiom,
% 0.33/0.64 ! [F: nat > real,A2: set_nat] :
% 0.33/0.64 ( ( abs_abs_real
% 0.33/0.64 @ ( groups6591440286371151544t_real
% 0.33/0.64 @ ^ [A5: nat] : ( abs_abs_real @ ( F @ A5 ) )
% 0.33/0.64 @ A2 ) )
% 0.33/0.64 = ( groups6591440286371151544t_real
% 0.33/0.64 @ ^ [A5: nat] : ( abs_abs_real @ ( F @ A5 ) )
% 0.33/0.64 @ A2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_sum_abs
% 0.33/0.64 thf(fact_6354_abs__le__zero__iff,axiom,
% 0.33/0.64 ! [A: code_integer] :
% 0.33/0.64 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 0.33/0.64 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_le_zero_iff
% 0.33/0.64 thf(fact_6355_abs__le__zero__iff,axiom,
% 0.33/0.64 ! [A: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
% 0.33/0.64 = ( A = zero_zero_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_le_zero_iff
% 0.33/0.64 thf(fact_6356_abs__le__zero__iff,axiom,
% 0.33/0.64 ! [A: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
% 0.33/0.64 = ( A = zero_zero_rat ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_le_zero_iff
% 0.33/0.64 thf(fact_6357_abs__le__zero__iff,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
% 0.33/0.64 = ( A = zero_zero_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_le_zero_iff
% 0.33/0.64 thf(fact_6358_abs__le__self__iff,axiom,
% 0.33/0.64 ! [A: code_integer] :
% 0.33/0.64 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
% 0.33/0.64 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_le_self_iff
% 0.33/0.64 thf(fact_6359_abs__le__self__iff,axiom,
% 0.33/0.64 ! [A: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
% 0.33/0.64 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_le_self_iff
% 0.33/0.64 thf(fact_6360_abs__le__self__iff,axiom,
% 0.33/0.64 ! [A: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
% 0.33/0.64 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_le_self_iff
% 0.33/0.64 thf(fact_6361_abs__le__self__iff,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
% 0.33/0.64 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_le_self_iff
% 0.33/0.64 thf(fact_6362_abs__of__nonneg,axiom,
% 0.33/0.64 ! [A: code_integer] :
% 0.33/0.64 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.33/0.64 => ( ( abs_abs_Code_integer @ A )
% 0.33/0.64 = A ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_of_nonneg
% 0.33/0.64 thf(fact_6363_abs__of__nonneg,axiom,
% 0.33/0.64 ! [A: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.64 => ( ( abs_abs_real @ A )
% 0.33/0.64 = A ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_of_nonneg
% 0.33/0.64 thf(fact_6364_abs__of__nonneg,axiom,
% 0.33/0.64 ! [A: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.64 => ( ( abs_abs_rat @ A )
% 0.33/0.64 = A ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_of_nonneg
% 0.33/0.64 thf(fact_6365_abs__of__nonneg,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.64 => ( ( abs_abs_int @ A )
% 0.33/0.64 = A ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_of_nonneg
% 0.33/0.64 thf(fact_6366_zero__less__abs__iff,axiom,
% 0.33/0.64 ! [A: code_integer] :
% 0.33/0.64 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
% 0.33/0.64 = ( A != zero_z3403309356797280102nteger ) ) ).
% 0.33/0.64
% 0.33/0.64 % zero_less_abs_iff
% 0.33/0.64 thf(fact_6367_zero__less__abs__iff,axiom,
% 0.33/0.64 ! [A: real] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
% 0.33/0.64 = ( A != zero_zero_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % zero_less_abs_iff
% 0.33/0.64 thf(fact_6368_zero__less__abs__iff,axiom,
% 0.33/0.64 ! [A: rat] :
% 0.33/0.64 ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
% 0.33/0.64 = ( A != zero_zero_rat ) ) ).
% 0.33/0.64
% 0.33/0.64 % zero_less_abs_iff
% 0.33/0.64 thf(fact_6369_zero__less__abs__iff,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
% 0.33/0.64 = ( A != zero_zero_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % zero_less_abs_iff
% 0.33/0.64 thf(fact_6370_abs__neg__numeral,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 0.33/0.64 = ( numeral_numeral_real @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_neg_numeral
% 0.33/0.64 thf(fact_6371_abs__neg__numeral,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.33/0.64 = ( numeral_numeral_int @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_neg_numeral
% 0.33/0.64 thf(fact_6372_abs__neg__numeral,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 0.33/0.64 = ( numeral_numeral_rat @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_neg_numeral
% 0.33/0.64 thf(fact_6373_abs__neg__numeral,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
% 0.33/0.64 = ( numera6620942414471956472nteger @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_neg_numeral
% 0.33/0.64 thf(fact_6374_abs__neg__one,axiom,
% 0.33/0.64 ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.64 = one_one_real ) ).
% 0.33/0.64
% 0.33/0.64 % abs_neg_one
% 0.33/0.64 thf(fact_6375_abs__neg__one,axiom,
% 0.33/0.64 ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.64 = one_one_int ) ).
% 0.33/0.64
% 0.33/0.64 % abs_neg_one
% 0.33/0.64 thf(fact_6376_abs__neg__one,axiom,
% 0.33/0.64 ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.33/0.64 = one_one_rat ) ).
% 0.33/0.64
% 0.33/0.64 % abs_neg_one
% 0.33/0.64 thf(fact_6377_abs__neg__one,axiom,
% 0.33/0.64 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.64 = one_one_Code_integer ) ).
% 0.33/0.64
% 0.33/0.64 % abs_neg_one
% 0.33/0.64 thf(fact_6378_abs__power__minus,axiom,
% 0.33/0.64 ! [A: real,N2: nat] :
% 0.33/0.64 ( ( abs_abs_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 ) )
% 0.33/0.64 = ( abs_abs_real @ ( power_power_real @ A @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_power_minus
% 0.33/0.64 thf(fact_6379_abs__power__minus,axiom,
% 0.33/0.64 ! [A: int,N2: nat] :
% 0.33/0.64 ( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 ) )
% 0.33/0.64 = ( abs_abs_int @ ( power_power_int @ A @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_power_minus
% 0.33/0.64 thf(fact_6380_abs__power__minus,axiom,
% 0.33/0.64 ! [A: rat,N2: nat] :
% 0.33/0.64 ( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 ) )
% 0.33/0.64 = ( abs_abs_rat @ ( power_power_rat @ A @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_power_minus
% 0.33/0.64 thf(fact_6381_abs__power__minus,axiom,
% 0.33/0.64 ! [A: code_integer,N2: nat] :
% 0.33/0.64 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 ) )
% 0.33/0.64 = ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_power_minus
% 0.33/0.64 thf(fact_6382_semiring__norm_I7_J,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( bit1 @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % semiring_norm(7)
% 0.33/0.64 thf(fact_6383_semiring__norm_I9_J,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 0.33/0.64 = ( bit1 @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % semiring_norm(9)
% 0.33/0.64 thf(fact_6384_semiring__norm_I14_J,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( bit0 @ ( times_times_num @ M @ ( bit1 @ N2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % semiring_norm(14)
% 0.33/0.64 thf(fact_6385_semiring__norm_I15_J,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 0.33/0.64 = ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % semiring_norm(15)
% 0.33/0.64 thf(fact_6386_semiring__norm_I81_J,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 0.33/0.64 = ( ord_less_num @ M @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % semiring_norm(81)
% 0.33/0.64 thf(fact_6387_semiring__norm_I72_J,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( ord_less_eq_num @ M @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % semiring_norm(72)
% 0.33/0.64 thf(fact_6388_semiring__norm_I77_J,axiom,
% 0.33/0.64 ! [N2: num] : ( ord_less_num @ one @ ( bit1 @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % semiring_norm(77)
% 0.33/0.64 thf(fact_6389_semiring__norm_I70_J,axiom,
% 0.33/0.64 ! [M: num] :
% 0.33/0.64 ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% 0.33/0.64
% 0.33/0.64 % semiring_norm(70)
% 0.33/0.64 thf(fact_6390_tanh__real__pos__iff,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X ) )
% 0.33/0.64 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.33/0.64
% 0.33/0.64 % tanh_real_pos_iff
% 0.33/0.64 thf(fact_6391_tanh__real__neg__iff,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ord_less_real @ ( tanh_real @ X ) @ zero_zero_real )
% 0.33/0.64 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % tanh_real_neg_iff
% 0.33/0.64 thf(fact_6392_tanh__real__nonpos__iff,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ ( tanh_real @ X ) @ zero_zero_real )
% 0.33/0.64 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % tanh_real_nonpos_iff
% 0.33/0.64 thf(fact_6393_tanh__real__nonneg__iff,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X ) )
% 0.33/0.64 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.33/0.64
% 0.33/0.64 % tanh_real_nonneg_iff
% 0.33/0.64 thf(fact_6394_sum__abs,axiom,
% 0.33/0.64 ! [F: int > int,A2: set_int] :
% 0.33/0.64 ( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F @ A2 ) )
% 0.33/0.64 @ ( groups4538972089207619220nt_int
% 0.33/0.64 @ ^ [I3: int] : ( abs_abs_int @ ( F @ I3 ) )
% 0.33/0.64 @ A2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_abs
% 0.33/0.64 thf(fact_6395_sum__abs,axiom,
% 0.33/0.64 ! [F: nat > real,A2: set_nat] :
% 0.33/0.64 ( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
% 0.33/0.64 @ ( groups6591440286371151544t_real
% 0.33/0.64 @ ^ [I3: nat] : ( abs_abs_real @ ( F @ I3 ) )
% 0.33/0.64 @ A2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_abs
% 0.33/0.64 thf(fact_6396_divide__le__0__abs__iff,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
% 0.33/0.64 = ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.33/0.64 | ( B = zero_zero_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divide_le_0_abs_iff
% 0.33/0.64 thf(fact_6397_divide__le__0__abs__iff,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) @ zero_zero_rat )
% 0.33/0.64 = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.33/0.64 | ( B = zero_zero_rat ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divide_le_0_abs_iff
% 0.33/0.64 thf(fact_6398_zero__le__divide__abs__iff,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
% 0.33/0.64 = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.64 | ( B = zero_zero_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % zero_le_divide_abs_iff
% 0.33/0.64 thf(fact_6399_zero__le__divide__abs__iff,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) )
% 0.33/0.64 = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.64 | ( B = zero_zero_rat ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % zero_le_divide_abs_iff
% 0.33/0.64 thf(fact_6400_abs__of__nonpos,axiom,
% 0.33/0.64 ! [A: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.33/0.64 => ( ( abs_abs_real @ A )
% 0.33/0.64 = ( uminus_uminus_real @ A ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_of_nonpos
% 0.33/0.64 thf(fact_6401_abs__of__nonpos,axiom,
% 0.33/0.64 ! [A: code_integer] :
% 0.33/0.64 ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
% 0.33/0.64 => ( ( abs_abs_Code_integer @ A )
% 0.33/0.64 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_of_nonpos
% 0.33/0.64 thf(fact_6402_abs__of__nonpos,axiom,
% 0.33/0.64 ! [A: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.33/0.64 => ( ( abs_abs_rat @ A )
% 0.33/0.64 = ( uminus_uminus_rat @ A ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_of_nonpos
% 0.33/0.64 thf(fact_6403_abs__of__nonpos,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.33/0.64 => ( ( abs_abs_int @ A )
% 0.33/0.64 = ( uminus_uminus_int @ A ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_of_nonpos
% 0.33/0.64 thf(fact_6404_zdiv__numeral__Bit1,axiom,
% 0.33/0.64 ! [V: num,W: num] :
% 0.33/0.64 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.33/0.64 = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % zdiv_numeral_Bit1
% 0.33/0.64 thf(fact_6405_semiring__norm_I10_J,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N2 ) @ one ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % semiring_norm(10)
% 0.33/0.64 thf(fact_6406_semiring__norm_I8_J,axiom,
% 0.33/0.64 ! [M: num] :
% 0.33/0.64 ( ( plus_plus_num @ ( bit1 @ M ) @ one )
% 0.33/0.64 = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % semiring_norm(8)
% 0.33/0.64 thf(fact_6407_semiring__norm_I5_J,axiom,
% 0.33/0.64 ! [M: num] :
% 0.33/0.64 ( ( plus_plus_num @ ( bit0 @ M ) @ one )
% 0.33/0.64 = ( bit1 @ M ) ) ).
% 0.33/0.64
% 0.33/0.64 % semiring_norm(5)
% 0.33/0.64 thf(fact_6408_semiring__norm_I4_J,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( plus_plus_num @ one @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( bit0 @ ( plus_plus_num @ N2 @ one ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % semiring_norm(4)
% 0.33/0.64 thf(fact_6409_semiring__norm_I3_J,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( plus_plus_num @ one @ ( bit0 @ N2 ) )
% 0.33/0.64 = ( bit1 @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % semiring_norm(3)
% 0.33/0.64 thf(fact_6410_artanh__minus__real,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.64 => ( ( artanh_real @ ( uminus_uminus_real @ X ) )
% 0.33/0.64 = ( uminus_uminus_real @ ( artanh_real @ X ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % artanh_minus_real
% 0.33/0.64 thf(fact_6411_sum__abs__ge__zero,axiom,
% 0.33/0.64 ! [F: int > int,A2: set_int] :
% 0.33/0.64 ( ord_less_eq_int @ zero_zero_int
% 0.33/0.64 @ ( groups4538972089207619220nt_int
% 0.33/0.64 @ ^ [I3: int] : ( abs_abs_int @ ( F @ I3 ) )
% 0.33/0.64 @ A2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_abs_ge_zero
% 0.33/0.64 thf(fact_6412_sum__abs__ge__zero,axiom,
% 0.33/0.64 ! [F: nat > real,A2: set_nat] :
% 0.33/0.64 ( ord_less_eq_real @ zero_zero_real
% 0.33/0.64 @ ( groups6591440286371151544t_real
% 0.33/0.64 @ ^ [I3: nat] : ( abs_abs_real @ ( F @ I3 ) )
% 0.33/0.64 @ A2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_abs_ge_zero
% 0.33/0.64 thf(fact_6413_semiring__norm_I16_J,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N2 ) @ ( bit0 @ ( times_times_num @ M @ N2 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % semiring_norm(16)
% 0.33/0.64 thf(fact_6414_semiring__norm_I79_J,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( ord_less_eq_num @ M @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % semiring_norm(79)
% 0.33/0.64 thf(fact_6415_semiring__norm_I74_J,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 0.33/0.64 = ( ord_less_num @ M @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % semiring_norm(74)
% 0.33/0.64 thf(fact_6416_zero__less__power__abs__iff,axiom,
% 0.33/0.64 ! [A: code_integer,N2: nat] :
% 0.33/0.64 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N2 ) )
% 0.33/0.64 = ( ( A != zero_z3403309356797280102nteger )
% 0.33/0.64 | ( N2 = zero_zero_nat ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % zero_less_power_abs_iff
% 0.33/0.64 thf(fact_6417_zero__less__power__abs__iff,axiom,
% 0.33/0.64 ! [A: real,N2: nat] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) )
% 0.33/0.64 = ( ( A != zero_zero_real )
% 0.33/0.64 | ( N2 = zero_zero_nat ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % zero_less_power_abs_iff
% 0.33/0.64 thf(fact_6418_zero__less__power__abs__iff,axiom,
% 0.33/0.64 ! [A: rat,N2: nat] :
% 0.33/0.64 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N2 ) )
% 0.33/0.64 = ( ( A != zero_zero_rat )
% 0.33/0.64 | ( N2 = zero_zero_nat ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % zero_less_power_abs_iff
% 0.33/0.64 thf(fact_6419_zero__less__power__abs__iff,axiom,
% 0.33/0.64 ! [A: int,N2: nat] :
% 0.33/0.64 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N2 ) )
% 0.33/0.64 = ( ( A != zero_zero_int )
% 0.33/0.64 | ( N2 = zero_zero_nat ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % zero_less_power_abs_iff
% 0.33/0.64 thf(fact_6420_abs__power2,axiom,
% 0.33/0.64 ! [A: rat] :
% 0.33/0.64 ( ( abs_abs_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.64 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_power2
% 0.33/0.64 thf(fact_6421_abs__power2,axiom,
% 0.33/0.64 ! [A: code_integer] :
% 0.33/0.64 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.64 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_power2
% 0.33/0.64 thf(fact_6422_abs__power2,axiom,
% 0.33/0.64 ! [A: real] :
% 0.33/0.64 ( ( abs_abs_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.64 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_power2
% 0.33/0.64 thf(fact_6423_abs__power2,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( abs_abs_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.64 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_power2
% 0.33/0.64 thf(fact_6424_power2__abs,axiom,
% 0.33/0.64 ! [A: rat] :
% 0.33/0.64 ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power2_abs
% 0.33/0.64 thf(fact_6425_power2__abs,axiom,
% 0.33/0.64 ! [A: code_integer] :
% 0.33/0.64 ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power2_abs
% 0.33/0.64 thf(fact_6426_power2__abs,axiom,
% 0.33/0.64 ! [A: real] :
% 0.33/0.64 ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power2_abs
% 0.33/0.64 thf(fact_6427_power2__abs,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power2_abs
% 0.33/0.64 thf(fact_6428_sum_Ocl__ivl__Suc,axiom,
% 0.33/0.64 ! [N2: nat,M: nat,G: nat > complex] :
% 0.33/0.64 ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.33/0.64 => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.33/0.64 = zero_zero_complex ) )
% 0.33/0.64 & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.33/0.64 => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.33/0.64 = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.cl_ivl_Suc
% 0.33/0.64 thf(fact_6429_sum_Ocl__ivl__Suc,axiom,
% 0.33/0.64 ! [N2: nat,M: nat,G: nat > rat] :
% 0.33/0.64 ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.33/0.64 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.33/0.64 = zero_zero_rat ) )
% 0.33/0.64 & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.33/0.64 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.33/0.64 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.cl_ivl_Suc
% 0.33/0.64 thf(fact_6430_sum_Ocl__ivl__Suc,axiom,
% 0.33/0.64 ! [N2: nat,M: nat,G: nat > int] :
% 0.33/0.64 ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.33/0.64 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.33/0.64 = zero_zero_int ) )
% 0.33/0.64 & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.33/0.64 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.33/0.64 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.cl_ivl_Suc
% 0.33/0.64 thf(fact_6431_sum_Ocl__ivl__Suc,axiom,
% 0.33/0.64 ! [N2: nat,M: nat,G: nat > nat] :
% 0.33/0.64 ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.33/0.64 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.33/0.64 = zero_zero_nat ) )
% 0.33/0.64 & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.33/0.64 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.33/0.64 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.cl_ivl_Suc
% 0.33/0.64 thf(fact_6432_sum_Ocl__ivl__Suc,axiom,
% 0.33/0.64 ! [N2: nat,M: nat,G: nat > real] :
% 0.33/0.64 ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.33/0.64 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.33/0.64 = zero_zero_real ) )
% 0.33/0.64 & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.33/0.64 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.33/0.64 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.cl_ivl_Suc
% 0.33/0.64 thf(fact_6433_dvd__numeral__simp,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( dvd_dvd_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 0.33/0.64 = ( unique6319869463603278526ux_int @ ( unique5052692396658037445od_int @ N2 @ M ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % dvd_numeral_simp
% 0.33/0.64 thf(fact_6434_dvd__numeral__simp,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
% 0.33/0.64 = ( unique6322359934112328802ux_nat @ ( unique5055182867167087721od_nat @ N2 @ M ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % dvd_numeral_simp
% 0.33/0.64 thf(fact_6435_dvd__numeral__simp,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) )
% 0.33/0.64 = ( unique5706413561485394159nteger @ ( unique3479559517661332726nteger @ N2 @ M ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % dvd_numeral_simp
% 0.33/0.64 thf(fact_6436_divmod__algorithm__code_I2_J,axiom,
% 0.33/0.64 ! [M: num] :
% 0.33/0.64 ( ( unique5052692396658037445od_int @ M @ one )
% 0.33/0.64 = ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_algorithm_code(2)
% 0.33/0.64 thf(fact_6437_divmod__algorithm__code_I2_J,axiom,
% 0.33/0.64 ! [M: num] :
% 0.33/0.64 ( ( unique5055182867167087721od_nat @ M @ one )
% 0.33/0.64 = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_algorithm_code(2)
% 0.33/0.64 thf(fact_6438_divmod__algorithm__code_I2_J,axiom,
% 0.33/0.64 ! [M: num] :
% 0.33/0.64 ( ( unique3479559517661332726nteger @ M @ one )
% 0.33/0.64 = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_algorithm_code(2)
% 0.33/0.64 thf(fact_6439_sum__zero__power,axiom,
% 0.33/0.64 ! [A2: set_nat,C: nat > complex] :
% 0.33/0.64 ( ( ( ( finite_finite_nat @ A2 )
% 0.33/0.64 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.33/0.64 => ( ( groups2073611262835488442omplex
% 0.33/0.64 @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) )
% 0.33/0.64 @ A2 )
% 0.33/0.64 = ( C @ zero_zero_nat ) ) )
% 0.33/0.64 & ( ~ ( ( finite_finite_nat @ A2 )
% 0.33/0.64 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.33/0.64 => ( ( groups2073611262835488442omplex
% 0.33/0.64 @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) )
% 0.33/0.64 @ A2 )
% 0.33/0.64 = zero_zero_complex ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_zero_power
% 0.33/0.64 thf(fact_6440_sum__zero__power,axiom,
% 0.33/0.64 ! [A2: set_nat,C: nat > rat] :
% 0.33/0.64 ( ( ( ( finite_finite_nat @ A2 )
% 0.33/0.64 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.33/0.64 => ( ( groups2906978787729119204at_rat
% 0.33/0.64 @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) )
% 0.33/0.64 @ A2 )
% 0.33/0.64 = ( C @ zero_zero_nat ) ) )
% 0.33/0.64 & ( ~ ( ( finite_finite_nat @ A2 )
% 0.33/0.64 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.33/0.64 => ( ( groups2906978787729119204at_rat
% 0.33/0.64 @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) )
% 0.33/0.64 @ A2 )
% 0.33/0.64 = zero_zero_rat ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_zero_power
% 0.33/0.64 thf(fact_6441_sum__zero__power,axiom,
% 0.33/0.64 ! [A2: set_nat,C: nat > real] :
% 0.33/0.64 ( ( ( ( finite_finite_nat @ A2 )
% 0.33/0.64 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.33/0.64 => ( ( groups6591440286371151544t_real
% 0.33/0.64 @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) )
% 0.33/0.64 @ A2 )
% 0.33/0.64 = ( C @ zero_zero_nat ) ) )
% 0.33/0.64 & ( ~ ( ( finite_finite_nat @ A2 )
% 0.33/0.64 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.33/0.64 => ( ( groups6591440286371151544t_real
% 0.33/0.64 @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) )
% 0.33/0.64 @ A2 )
% 0.33/0.64 = zero_zero_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_zero_power
% 0.33/0.64 thf(fact_6442_power__even__abs__numeral,axiom,
% 0.33/0.64 ! [W: num,A: rat] :
% 0.33/0.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.33/0.64 => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.33/0.64 = ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_even_abs_numeral
% 0.33/0.64 thf(fact_6443_power__even__abs__numeral,axiom,
% 0.33/0.64 ! [W: num,A: code_integer] :
% 0.33/0.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.33/0.64 => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.33/0.64 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_even_abs_numeral
% 0.33/0.64 thf(fact_6444_power__even__abs__numeral,axiom,
% 0.33/0.64 ! [W: num,A: real] :
% 0.33/0.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.33/0.64 => ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.33/0.64 = ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_even_abs_numeral
% 0.33/0.64 thf(fact_6445_power__even__abs__numeral,axiom,
% 0.33/0.64 ! [W: num,A: int] :
% 0.33/0.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.33/0.64 => ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.33/0.64 = ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_even_abs_numeral
% 0.33/0.64 thf(fact_6446_div__Suc__eq__div__add3,axiom,
% 0.33/0.64 ! [M: nat,N2: nat] :
% 0.33/0.64 ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
% 0.33/0.64 = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % div_Suc_eq_div_add3
% 0.33/0.64 thf(fact_6447_Suc__div__eq__add3__div__numeral,axiom,
% 0.33/0.64 ! [M: nat,V: num] :
% 0.33/0.64 ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 0.33/0.64 = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Suc_div_eq_add3_div_numeral
% 0.33/0.64 thf(fact_6448_divmod__algorithm__code_I3_J,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N2 ) )
% 0.33/0.64 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_algorithm_code(3)
% 0.33/0.64 thf(fact_6449_divmod__algorithm__code_I3_J,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N2 ) )
% 0.33/0.64 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_algorithm_code(3)
% 0.33/0.64 thf(fact_6450_divmod__algorithm__code_I3_J,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N2 ) )
% 0.33/0.64 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_algorithm_code(3)
% 0.33/0.64 thf(fact_6451_mod__Suc__eq__mod__add3,axiom,
% 0.33/0.64 ! [M: nat,N2: nat] :
% 0.33/0.64 ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
% 0.33/0.64 = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mod_Suc_eq_mod_add3
% 0.33/0.64 thf(fact_6452_Suc__mod__eq__add3__mod__numeral,axiom,
% 0.33/0.64 ! [M: nat,V: num] :
% 0.33/0.64 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 0.33/0.64 = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Suc_mod_eq_add3_mod_numeral
% 0.33/0.64 thf(fact_6453_divmod__algorithm__code_I4_J,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_algorithm_code(4)
% 0.33/0.64 thf(fact_6454_divmod__algorithm__code_I4_J,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_algorithm_code(4)
% 0.33/0.64 thf(fact_6455_divmod__algorithm__code_I4_J,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_algorithm_code(4)
% 0.33/0.64 thf(fact_6456_sum__zero__power_H,axiom,
% 0.33/0.64 ! [A2: set_nat,C: nat > complex,D: nat > complex] :
% 0.33/0.64 ( ( ( ( finite_finite_nat @ A2 )
% 0.33/0.64 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.33/0.64 => ( ( groups2073611262835488442omplex
% 0.33/0.64 @ ^ [I3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) ) @ ( D @ I3 ) )
% 0.33/0.64 @ A2 )
% 0.33/0.64 = ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 0.33/0.64 & ( ~ ( ( finite_finite_nat @ A2 )
% 0.33/0.64 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.33/0.64 => ( ( groups2073611262835488442omplex
% 0.33/0.64 @ ^ [I3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) ) @ ( D @ I3 ) )
% 0.33/0.64 @ A2 )
% 0.33/0.64 = zero_zero_complex ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_zero_power'
% 0.33/0.64 thf(fact_6457_sum__zero__power_H,axiom,
% 0.33/0.64 ! [A2: set_nat,C: nat > rat,D: nat > rat] :
% 0.33/0.64 ( ( ( ( finite_finite_nat @ A2 )
% 0.33/0.64 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.33/0.64 => ( ( groups2906978787729119204at_rat
% 0.33/0.64 @ ^ [I3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) ) @ ( D @ I3 ) )
% 0.33/0.64 @ A2 )
% 0.33/0.64 = ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 0.33/0.64 & ( ~ ( ( finite_finite_nat @ A2 )
% 0.33/0.64 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.33/0.64 => ( ( groups2906978787729119204at_rat
% 0.33/0.64 @ ^ [I3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) ) @ ( D @ I3 ) )
% 0.33/0.64 @ A2 )
% 0.33/0.64 = zero_zero_rat ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_zero_power'
% 0.33/0.64 thf(fact_6458_sum__zero__power_H,axiom,
% 0.33/0.64 ! [A2: set_nat,C: nat > real,D: nat > real] :
% 0.33/0.64 ( ( ( ( finite_finite_nat @ A2 )
% 0.33/0.64 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.33/0.64 => ( ( groups6591440286371151544t_real
% 0.33/0.64 @ ^ [I3: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) ) @ ( D @ I3 ) )
% 0.33/0.64 @ A2 )
% 0.33/0.64 = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 0.33/0.64 & ( ~ ( ( finite_finite_nat @ A2 )
% 0.33/0.64 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.33/0.64 => ( ( groups6591440286371151544t_real
% 0.33/0.64 @ ^ [I3: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) ) @ ( D @ I3 ) )
% 0.33/0.64 @ A2 )
% 0.33/0.64 = zero_zero_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_zero_power'
% 0.33/0.64 thf(fact_6459_zmod__numeral__Bit1,axiom,
% 0.33/0.64 ! [V: num,W: num] :
% 0.33/0.64 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.33/0.64 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) @ one_one_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % zmod_numeral_Bit1
% 0.33/0.64 thf(fact_6460_divmod__algorithm__code_I8_J,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( ( ord_less_num @ M @ N2 )
% 0.33/0.64 => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
% 0.33/0.64 & ( ~ ( ord_less_num @ M @ N2 )
% 0.33/0.64 => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( unique5026877609467782581ep_nat @ ( bit1 @ N2 ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_algorithm_code(8)
% 0.33/0.64 thf(fact_6461_divmod__algorithm__code_I8_J,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( ( ord_less_num @ M @ N2 )
% 0.33/0.64 => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
% 0.33/0.64 & ( ~ ( ord_less_num @ M @ N2 )
% 0.33/0.64 => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( unique5024387138958732305ep_int @ ( bit1 @ N2 ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_algorithm_code(8)
% 0.33/0.64 thf(fact_6462_divmod__algorithm__code_I8_J,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( ( ord_less_num @ M @ N2 )
% 0.33/0.64 => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
% 0.33/0.64 & ( ~ ( ord_less_num @ M @ N2 )
% 0.33/0.64 => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( unique4921790084139445826nteger @ ( bit1 @ N2 ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_algorithm_code(8)
% 0.33/0.64 thf(fact_6463_divmod__algorithm__code_I7_J,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( ( ord_less_eq_num @ M @ N2 )
% 0.33/0.64 => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
% 0.33/0.64 & ( ~ ( ord_less_eq_num @ M @ N2 )
% 0.33/0.64 => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( unique5026877609467782581ep_nat @ ( bit1 @ N2 ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_algorithm_code(7)
% 0.33/0.64 thf(fact_6464_divmod__algorithm__code_I7_J,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( ( ord_less_eq_num @ M @ N2 )
% 0.33/0.64 => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
% 0.33/0.64 & ( ~ ( ord_less_eq_num @ M @ N2 )
% 0.33/0.64 => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( unique5024387138958732305ep_int @ ( bit1 @ N2 ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_algorithm_code(7)
% 0.33/0.64 thf(fact_6465_divmod__algorithm__code_I7_J,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( ( ord_less_eq_num @ M @ N2 )
% 0.33/0.64 => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
% 0.33/0.64 & ( ~ ( ord_less_eq_num @ M @ N2 )
% 0.33/0.64 => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( unique4921790084139445826nteger @ ( bit1 @ N2 ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_algorithm_code(7)
% 0.33/0.64 thf(fact_6466_signed__take__bit__Suc__bit1,axiom,
% 0.33/0.64 ! [N2: nat,K: num] :
% 0.33/0.64 ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.33/0.64 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % signed_take_bit_Suc_bit1
% 0.33/0.64 thf(fact_6467_divmod__algorithm__code_I6_J,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 0.33/0.64 = ( produc4245557441103728435nt_int
% 0.33/0.64 @ ^ [Q4: int,R5: int] : ( product_Pair_int_int @ Q4 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) @ one_one_int ) )
% 0.33/0.64 @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_algorithm_code(6)
% 0.33/0.64 thf(fact_6468_divmod__algorithm__code_I6_J,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 0.33/0.64 = ( produc2626176000494625587at_nat
% 0.33/0.64 @ ^ [Q4: nat,R5: nat] : ( product_Pair_nat_nat @ Q4 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) @ one_one_nat ) )
% 0.33/0.64 @ ( unique5055182867167087721od_nat @ M @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_algorithm_code(6)
% 0.33/0.64 thf(fact_6469_divmod__algorithm__code_I6_J,axiom,
% 0.33/0.64 ! [M: num,N2: num] :
% 0.33/0.64 ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 0.33/0.64 = ( produc6916734918728496179nteger
% 0.33/0.64 @ ^ [Q4: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q4 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) @ one_one_Code_integer ) )
% 0.33/0.64 @ ( unique3479559517661332726nteger @ M @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_algorithm_code(6)
% 0.33/0.64 thf(fact_6470_Compl__eq,axiom,
% 0.33/0.64 ( uminus612125837232591019t_real
% 0.33/0.64 = ( ^ [A4: set_real] :
% 0.33/0.64 ( collect_real
% 0.33/0.64 @ ^ [X2: real] :
% 0.33/0.64 ~ ( member_real @ X2 @ A4 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Compl_eq
% 0.33/0.64 thf(fact_6471_Compl__eq,axiom,
% 0.33/0.64 ( uminus6221592323253981072nt_int
% 0.33/0.64 = ( ^ [A4: set_Pr958786334691620121nt_int] :
% 0.33/0.64 ( collec213857154873943460nt_int
% 0.33/0.64 @ ^ [X2: product_prod_int_int] :
% 0.33/0.64 ~ ( member5262025264175285858nt_int @ X2 @ A4 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Compl_eq
% 0.33/0.64 thf(fact_6472_Compl__eq,axiom,
% 0.33/0.64 ( uminus8566677241136511917omplex
% 0.33/0.64 = ( ^ [A4: set_complex] :
% 0.33/0.64 ( collect_complex
% 0.33/0.64 @ ^ [X2: complex] :
% 0.33/0.64 ~ ( member_complex @ X2 @ A4 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Compl_eq
% 0.33/0.64 thf(fact_6473_Compl__eq,axiom,
% 0.33/0.64 ( uminus613421341184616069et_nat
% 0.33/0.64 = ( ^ [A4: set_set_nat] :
% 0.33/0.64 ( collect_set_nat
% 0.33/0.64 @ ^ [X2: set_nat] :
% 0.33/0.64 ~ ( member_set_nat @ X2 @ A4 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Compl_eq
% 0.33/0.64 thf(fact_6474_Compl__eq,axiom,
% 0.33/0.64 ( uminus5710092332889474511et_nat
% 0.33/0.64 = ( ^ [A4: set_nat] :
% 0.33/0.64 ( collect_nat
% 0.33/0.64 @ ^ [X2: nat] :
% 0.33/0.64 ~ ( member_nat @ X2 @ A4 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Compl_eq
% 0.33/0.64 thf(fact_6475_Compl__eq,axiom,
% 0.33/0.64 ( uminus1532241313380277803et_int
% 0.33/0.64 = ( ^ [A4: set_int] :
% 0.33/0.64 ( collect_int
% 0.33/0.64 @ ^ [X2: int] :
% 0.33/0.64 ~ ( member_int @ X2 @ A4 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Compl_eq
% 0.33/0.64 thf(fact_6476_Collect__neg__eq,axiom,
% 0.33/0.64 ! [P: product_prod_int_int > $o] :
% 0.33/0.64 ( ( collec213857154873943460nt_int
% 0.33/0.64 @ ^ [X2: product_prod_int_int] :
% 0.33/0.64 ~ ( P @ X2 ) )
% 0.33/0.64 = ( uminus6221592323253981072nt_int @ ( collec213857154873943460nt_int @ P ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Collect_neg_eq
% 0.33/0.64 thf(fact_6477_Collect__neg__eq,axiom,
% 0.33/0.64 ! [P: complex > $o] :
% 0.33/0.64 ( ( collect_complex
% 0.33/0.64 @ ^ [X2: complex] :
% 0.33/0.64 ~ ( P @ X2 ) )
% 0.33/0.64 = ( uminus8566677241136511917omplex @ ( collect_complex @ P ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Collect_neg_eq
% 0.33/0.64 thf(fact_6478_Collect__neg__eq,axiom,
% 0.33/0.64 ! [P: set_nat > $o] :
% 0.33/0.64 ( ( collect_set_nat
% 0.33/0.64 @ ^ [X2: set_nat] :
% 0.33/0.64 ~ ( P @ X2 ) )
% 0.33/0.64 = ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Collect_neg_eq
% 0.33/0.64 thf(fact_6479_Collect__neg__eq,axiom,
% 0.33/0.64 ! [P: nat > $o] :
% 0.33/0.64 ( ( collect_nat
% 0.33/0.64 @ ^ [X2: nat] :
% 0.33/0.64 ~ ( P @ X2 ) )
% 0.33/0.64 = ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Collect_neg_eq
% 0.33/0.64 thf(fact_6480_Collect__neg__eq,axiom,
% 0.33/0.64 ! [P: int > $o] :
% 0.33/0.64 ( ( collect_int
% 0.33/0.64 @ ^ [X2: int] :
% 0.33/0.64 ~ ( P @ X2 ) )
% 0.33/0.64 = ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Collect_neg_eq
% 0.33/0.64 thf(fact_6481_uminus__set__def,axiom,
% 0.33/0.64 ( uminus612125837232591019t_real
% 0.33/0.64 = ( ^ [A4: set_real] :
% 0.33/0.64 ( collect_real
% 0.33/0.64 @ ( uminus_uminus_real_o
% 0.33/0.64 @ ^ [X2: real] : ( member_real @ X2 @ A4 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % uminus_set_def
% 0.33/0.64 thf(fact_6482_uminus__set__def,axiom,
% 0.33/0.64 ( uminus6221592323253981072nt_int
% 0.33/0.64 = ( ^ [A4: set_Pr958786334691620121nt_int] :
% 0.33/0.64 ( collec213857154873943460nt_int
% 0.33/0.64 @ ( uminus7117520113953359693_int_o
% 0.33/0.64 @ ^ [X2: product_prod_int_int] : ( member5262025264175285858nt_int @ X2 @ A4 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % uminus_set_def
% 0.33/0.64 thf(fact_6483_uminus__set__def,axiom,
% 0.33/0.64 ( uminus8566677241136511917omplex
% 0.33/0.64 = ( ^ [A4: set_complex] :
% 0.33/0.64 ( collect_complex
% 0.33/0.64 @ ( uminus1680532995456772888plex_o
% 0.33/0.64 @ ^ [X2: complex] : ( member_complex @ X2 @ A4 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % uminus_set_def
% 0.33/0.64 thf(fact_6484_uminus__set__def,axiom,
% 0.33/0.64 ( uminus613421341184616069et_nat
% 0.33/0.64 = ( ^ [A4: set_set_nat] :
% 0.33/0.64 ( collect_set_nat
% 0.33/0.64 @ ( uminus6401447641752708672_nat_o
% 0.33/0.64 @ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A4 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % uminus_set_def
% 0.33/0.64 thf(fact_6485_uminus__set__def,axiom,
% 0.33/0.64 ( uminus5710092332889474511et_nat
% 0.33/0.64 = ( ^ [A4: set_nat] :
% 0.33/0.64 ( collect_nat
% 0.33/0.64 @ ( uminus_uminus_nat_o
% 0.33/0.64 @ ^ [X2: nat] : ( member_nat @ X2 @ A4 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % uminus_set_def
% 0.33/0.64 thf(fact_6486_uminus__set__def,axiom,
% 0.33/0.64 ( uminus1532241313380277803et_int
% 0.33/0.64 = ( ^ [A4: set_int] :
% 0.33/0.64 ( collect_int
% 0.33/0.64 @ ( uminus_uminus_int_o
% 0.33/0.64 @ ^ [X2: int] : ( member_int @ X2 @ A4 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % uminus_set_def
% 0.33/0.64 thf(fact_6487_abs__ge__self,axiom,
% 0.33/0.64 ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_ge_self
% 0.33/0.64 thf(fact_6488_abs__ge__self,axiom,
% 0.33/0.64 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_ge_self
% 0.33/0.64 thf(fact_6489_abs__ge__self,axiom,
% 0.33/0.64 ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_ge_self
% 0.33/0.64 thf(fact_6490_abs__ge__self,axiom,
% 0.33/0.64 ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_ge_self
% 0.33/0.64 thf(fact_6491_abs__le__D1,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 0.33/0.64 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_le_D1
% 0.33/0.64 thf(fact_6492_abs__le__D1,axiom,
% 0.33/0.64 ! [A: code_integer,B: code_integer] :
% 0.33/0.64 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.33/0.64 => ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_le_D1
% 0.33/0.64 thf(fact_6493_abs__le__D1,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 0.33/0.64 => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_le_D1
% 0.33/0.64 thf(fact_6494_abs__le__D1,axiom,
% 0.33/0.64 ! [A: int,B: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 0.33/0.64 => ( ord_less_eq_int @ A @ B ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_le_D1
% 0.33/0.64 thf(fact_6495_abs__mult,axiom,
% 0.33/0.64 ! [A: code_integer,B: code_integer] :
% 0.33/0.64 ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.33/0.64 = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_mult
% 0.33/0.64 thf(fact_6496_abs__mult,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 0.33/0.64 = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_mult
% 0.33/0.64 thf(fact_6497_abs__mult,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
% 0.33/0.64 = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_mult
% 0.33/0.64 thf(fact_6498_abs__mult,axiom,
% 0.33/0.64 ! [A: int,B: int] :
% 0.33/0.64 ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
% 0.33/0.64 = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_mult
% 0.33/0.64 thf(fact_6499_abs__one,axiom,
% 0.33/0.64 ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 0.33/0.64 = one_one_Code_integer ) ).
% 0.33/0.64
% 0.33/0.64 % abs_one
% 0.33/0.64 thf(fact_6500_abs__one,axiom,
% 0.33/0.64 ( ( abs_abs_real @ one_one_real )
% 0.33/0.64 = one_one_real ) ).
% 0.33/0.64
% 0.33/0.64 % abs_one
% 0.33/0.64 thf(fact_6501_abs__one,axiom,
% 0.33/0.64 ( ( abs_abs_rat @ one_one_rat )
% 0.33/0.64 = one_one_rat ) ).
% 0.33/0.64
% 0.33/0.64 % abs_one
% 0.33/0.64 thf(fact_6502_abs__one,axiom,
% 0.33/0.64 ( ( abs_abs_int @ one_one_int )
% 0.33/0.64 = one_one_int ) ).
% 0.33/0.64
% 0.33/0.64 % abs_one
% 0.33/0.64 thf(fact_6503_abs__minus__commute,axiom,
% 0.33/0.64 ! [A: code_integer,B: code_integer] :
% 0.33/0.64 ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
% 0.33/0.64 = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_minus_commute
% 0.33/0.64 thf(fact_6504_abs__minus__commute,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
% 0.33/0.64 = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_minus_commute
% 0.33/0.64 thf(fact_6505_abs__minus__commute,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
% 0.33/0.64 = ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_minus_commute
% 0.33/0.64 thf(fact_6506_abs__minus__commute,axiom,
% 0.33/0.64 ! [A: int,B: int] :
% 0.33/0.64 ( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
% 0.33/0.64 = ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_minus_commute
% 0.33/0.64 thf(fact_6507_power__abs,axiom,
% 0.33/0.64 ! [A: rat,N2: nat] :
% 0.33/0.64 ( ( abs_abs_rat @ ( power_power_rat @ A @ N2 ) )
% 0.33/0.64 = ( power_power_rat @ ( abs_abs_rat @ A ) @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_abs
% 0.33/0.64 thf(fact_6508_power__abs,axiom,
% 0.33/0.64 ! [A: code_integer,N2: nat] :
% 0.33/0.64 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) )
% 0.33/0.64 = ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_abs
% 0.33/0.64 thf(fact_6509_power__abs,axiom,
% 0.33/0.64 ! [A: real,N2: nat] :
% 0.33/0.64 ( ( abs_abs_real @ ( power_power_real @ A @ N2 ) )
% 0.33/0.64 = ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_abs
% 0.33/0.64 thf(fact_6510_power__abs,axiom,
% 0.33/0.64 ! [A: int,N2: nat] :
% 0.33/0.64 ( ( abs_abs_int @ ( power_power_int @ A @ N2 ) )
% 0.33/0.64 = ( power_power_int @ ( abs_abs_int @ A ) @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_abs
% 0.33/0.64 thf(fact_6511_mem__case__prodE,axiom,
% 0.33/0.64 ! [Z: nat,C: code_integer > $o > set_nat,P4: produc6271795597528267376eger_o] :
% 0.33/0.64 ( ( member_nat @ Z @ ( produc5431169771168744661et_nat @ C @ P4 ) )
% 0.33/0.64 => ~ ! [X4: code_integer,Y4: $o] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( produc6677183202524767010eger_o @ X4 @ Y4 ) )
% 0.33/0.64 => ~ ( member_nat @ Z @ ( C @ X4 @ Y4 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodE
% 0.33/0.64 thf(fact_6512_mem__case__prodE,axiom,
% 0.33/0.64 ! [Z: real,C: code_integer > $o > set_real,P4: produc6271795597528267376eger_o] :
% 0.33/0.64 ( ( member_real @ Z @ ( produc242741666403216561t_real @ C @ P4 ) )
% 0.33/0.64 => ~ ! [X4: code_integer,Y4: $o] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( produc6677183202524767010eger_o @ X4 @ Y4 ) )
% 0.33/0.64 => ~ ( member_real @ Z @ ( C @ X4 @ Y4 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodE
% 0.33/0.64 thf(fact_6513_mem__case__prodE,axiom,
% 0.33/0.64 ! [Z: int,C: code_integer > $o > set_int,P4: produc6271795597528267376eger_o] :
% 0.33/0.64 ( ( member_int @ Z @ ( produc1253318751659547953et_int @ C @ P4 ) )
% 0.33/0.64 => ~ ! [X4: code_integer,Y4: $o] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( produc6677183202524767010eger_o @ X4 @ Y4 ) )
% 0.33/0.64 => ~ ( member_int @ Z @ ( C @ X4 @ Y4 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodE
% 0.33/0.64 thf(fact_6514_mem__case__prodE,axiom,
% 0.33/0.64 ! [Z: complex,C: code_integer > $o > set_complex,P4: produc6271795597528267376eger_o] :
% 0.33/0.64 ( ( member_complex @ Z @ ( produc1043322548047392435omplex @ C @ P4 ) )
% 0.33/0.64 => ~ ! [X4: code_integer,Y4: $o] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( produc6677183202524767010eger_o @ X4 @ Y4 ) )
% 0.33/0.64 => ~ ( member_complex @ Z @ ( C @ X4 @ Y4 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodE
% 0.33/0.64 thf(fact_6515_mem__case__prodE,axiom,
% 0.33/0.64 ! [Z: nat,C: num > num > set_nat,P4: product_prod_num_num] :
% 0.33/0.64 ( ( member_nat @ Z @ ( produc1361121860356118632et_nat @ C @ P4 ) )
% 0.33/0.64 => ~ ! [X4: num,Y4: num] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( product_Pair_num_num @ X4 @ Y4 ) )
% 0.33/0.64 => ~ ( member_nat @ Z @ ( C @ X4 @ Y4 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodE
% 0.33/0.64 thf(fact_6516_mem__case__prodE,axiom,
% 0.33/0.64 ! [Z: real,C: num > num > set_real,P4: product_prod_num_num] :
% 0.33/0.64 ( ( member_real @ Z @ ( produc8296048397933160132t_real @ C @ P4 ) )
% 0.33/0.64 => ~ ! [X4: num,Y4: num] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( product_Pair_num_num @ X4 @ Y4 ) )
% 0.33/0.64 => ~ ( member_real @ Z @ ( C @ X4 @ Y4 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodE
% 0.33/0.64 thf(fact_6517_mem__case__prodE,axiom,
% 0.33/0.64 ! [Z: int,C: num > num > set_int,P4: product_prod_num_num] :
% 0.33/0.64 ( ( member_int @ Z @ ( produc6406642877701697732et_int @ C @ P4 ) )
% 0.33/0.64 => ~ ! [X4: num,Y4: num] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( product_Pair_num_num @ X4 @ Y4 ) )
% 0.33/0.64 => ~ ( member_int @ Z @ ( C @ X4 @ Y4 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodE
% 0.33/0.64 thf(fact_6518_mem__case__prodE,axiom,
% 0.33/0.64 ! [Z: complex,C: num > num > set_complex,P4: product_prod_num_num] :
% 0.33/0.64 ( ( member_complex @ Z @ ( produc2866383454006189126omplex @ C @ P4 ) )
% 0.33/0.64 => ~ ! [X4: num,Y4: num] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( product_Pair_num_num @ X4 @ Y4 ) )
% 0.33/0.64 => ~ ( member_complex @ Z @ ( C @ X4 @ Y4 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodE
% 0.33/0.64 thf(fact_6519_mem__case__prodE,axiom,
% 0.33/0.64 ! [Z: nat,C: nat > num > set_nat,P4: product_prod_nat_num] :
% 0.33/0.64 ( ( member_nat @ Z @ ( produc4130284055270567454et_nat @ C @ P4 ) )
% 0.33/0.64 => ~ ! [X4: nat,Y4: num] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( product_Pair_nat_num @ X4 @ Y4 ) )
% 0.33/0.64 => ~ ( member_nat @ Z @ ( C @ X4 @ Y4 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodE
% 0.33/0.64 thf(fact_6520_mem__case__prodE,axiom,
% 0.33/0.64 ! [Z: real,C: nat > num > set_real,P4: product_prod_nat_num] :
% 0.33/0.64 ( ( member_real @ Z @ ( produc1435849484188172666t_real @ C @ P4 ) )
% 0.33/0.64 => ~ ! [X4: nat,Y4: num] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( product_Pair_nat_num @ X4 @ Y4 ) )
% 0.33/0.64 => ~ ( member_real @ Z @ ( C @ X4 @ Y4 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mem_case_prodE
% 0.33/0.64 thf(fact_6521_verit__eq__simplify_I14_J,axiom,
% 0.33/0.64 ! [X22: num,X32: num] :
% 0.33/0.64 ( ( bit0 @ X22 )
% 0.33/0.64 != ( bit1 @ X32 ) ) ).
% 0.33/0.64
% 0.33/0.64 % verit_eq_simplify(14)
% 0.33/0.64 thf(fact_6522_verit__eq__simplify_I12_J,axiom,
% 0.33/0.64 ! [X32: num] :
% 0.33/0.64 ( one
% 0.33/0.64 != ( bit1 @ X32 ) ) ).
% 0.33/0.64
% 0.33/0.64 % verit_eq_simplify(12)
% 0.33/0.64 thf(fact_6523_case__prodD,axiom,
% 0.33/0.64 ! [F: code_integer > $o > $o,A: code_integer,B: $o] :
% 0.33/0.64 ( ( produc7828578312038201481er_o_o @ F @ ( produc6677183202524767010eger_o @ A @ B ) )
% 0.33/0.64 => ( F @ A @ B ) ) ).
% 0.33/0.64
% 0.33/0.64 % case_prodD
% 0.33/0.64 thf(fact_6524_case__prodD,axiom,
% 0.33/0.64 ! [F: num > num > $o,A: num,B: num] :
% 0.33/0.64 ( ( produc5703948589228662326_num_o @ F @ ( product_Pair_num_num @ A @ B ) )
% 0.33/0.64 => ( F @ A @ B ) ) ).
% 0.33/0.64
% 0.33/0.64 % case_prodD
% 0.33/0.64 thf(fact_6525_case__prodD,axiom,
% 0.33/0.64 ! [F: nat > num > $o,A: nat,B: num] :
% 0.33/0.64 ( ( produc4927758841916487424_num_o @ F @ ( product_Pair_nat_num @ A @ B ) )
% 0.33/0.64 => ( F @ A @ B ) ) ).
% 0.33/0.64
% 0.33/0.64 % case_prodD
% 0.33/0.64 thf(fact_6526_case__prodD,axiom,
% 0.33/0.64 ! [F: nat > nat > $o,A: nat,B: nat] :
% 0.33/0.64 ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
% 0.33/0.64 => ( F @ A @ B ) ) ).
% 0.33/0.64
% 0.33/0.64 % case_prodD
% 0.33/0.64 thf(fact_6527_case__prodD,axiom,
% 0.33/0.64 ! [F: int > int > $o,A: int,B: int] :
% 0.33/0.64 ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
% 0.33/0.64 => ( F @ A @ B ) ) ).
% 0.33/0.64
% 0.33/0.64 % case_prodD
% 0.33/0.64 thf(fact_6528_case__prodE,axiom,
% 0.33/0.64 ! [C: code_integer > $o > $o,P4: produc6271795597528267376eger_o] :
% 0.33/0.64 ( ( produc7828578312038201481er_o_o @ C @ P4 )
% 0.33/0.64 => ~ ! [X4: code_integer,Y4: $o] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( produc6677183202524767010eger_o @ X4 @ Y4 ) )
% 0.33/0.64 => ~ ( C @ X4 @ Y4 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % case_prodE
% 0.33/0.64 thf(fact_6529_case__prodE,axiom,
% 0.33/0.64 ! [C: num > num > $o,P4: product_prod_num_num] :
% 0.33/0.64 ( ( produc5703948589228662326_num_o @ C @ P4 )
% 0.33/0.64 => ~ ! [X4: num,Y4: num] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( product_Pair_num_num @ X4 @ Y4 ) )
% 0.33/0.64 => ~ ( C @ X4 @ Y4 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % case_prodE
% 0.33/0.64 thf(fact_6530_case__prodE,axiom,
% 0.33/0.64 ! [C: nat > num > $o,P4: product_prod_nat_num] :
% 0.33/0.64 ( ( produc4927758841916487424_num_o @ C @ P4 )
% 0.33/0.64 => ~ ! [X4: nat,Y4: num] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( product_Pair_nat_num @ X4 @ Y4 ) )
% 0.33/0.64 => ~ ( C @ X4 @ Y4 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % case_prodE
% 0.33/0.64 thf(fact_6531_case__prodE,axiom,
% 0.33/0.64 ! [C: nat > nat > $o,P4: product_prod_nat_nat] :
% 0.33/0.64 ( ( produc6081775807080527818_nat_o @ C @ P4 )
% 0.33/0.64 => ~ ! [X4: nat,Y4: nat] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( product_Pair_nat_nat @ X4 @ Y4 ) )
% 0.33/0.64 => ~ ( C @ X4 @ Y4 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % case_prodE
% 0.33/0.64 thf(fact_6532_case__prodE,axiom,
% 0.33/0.64 ! [C: int > int > $o,P4: product_prod_int_int] :
% 0.33/0.64 ( ( produc4947309494688390418_int_o @ C @ P4 )
% 0.33/0.64 => ~ ! [X4: int,Y4: int] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( product_Pair_int_int @ X4 @ Y4 ) )
% 0.33/0.64 => ~ ( C @ X4 @ Y4 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % case_prodE
% 0.33/0.64 thf(fact_6533_case__prodD_H,axiom,
% 0.33/0.64 ! [R2: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat,C: product_prod_nat_nat] :
% 0.33/0.64 ( ( produc8739625826339149834_nat_o @ R2 @ ( product_Pair_nat_nat @ A @ B ) @ C )
% 0.33/0.64 => ( R2 @ A @ B @ C ) ) ).
% 0.33/0.64
% 0.33/0.64 % case_prodD'
% 0.33/0.64 thf(fact_6534_case__prodE_H,axiom,
% 0.33/0.64 ! [C: nat > nat > product_prod_nat_nat > $o,P4: product_prod_nat_nat,Z: product_prod_nat_nat] :
% 0.33/0.64 ( ( produc8739625826339149834_nat_o @ C @ P4 @ Z )
% 0.33/0.64 => ~ ! [X4: nat,Y4: nat] :
% 0.33/0.64 ( ( P4
% 0.33/0.64 = ( product_Pair_nat_nat @ X4 @ Y4 ) )
% 0.33/0.64 => ~ ( C @ X4 @ Y4 @ Z ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % case_prodE'
% 0.33/0.64 thf(fact_6535_abs__ge__zero,axiom,
% 0.33/0.64 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_ge_zero
% 0.33/0.64 thf(fact_6536_abs__ge__zero,axiom,
% 0.33/0.64 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_ge_zero
% 0.33/0.64 thf(fact_6537_abs__ge__zero,axiom,
% 0.33/0.64 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_ge_zero
% 0.33/0.64 thf(fact_6538_abs__ge__zero,axiom,
% 0.33/0.64 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_ge_zero
% 0.33/0.64 thf(fact_6539_abs__of__pos,axiom,
% 0.33/0.64 ! [A: code_integer] :
% 0.33/0.64 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 0.33/0.64 => ( ( abs_abs_Code_integer @ A )
% 0.33/0.64 = A ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_of_pos
% 0.33/0.64 thf(fact_6540_abs__of__pos,axiom,
% 0.33/0.64 ! [A: real] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.64 => ( ( abs_abs_real @ A )
% 0.33/0.64 = A ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_of_pos
% 0.33/0.64 thf(fact_6541_abs__of__pos,axiom,
% 0.33/0.64 ! [A: rat] :
% 0.33/0.64 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.33/0.64 => ( ( abs_abs_rat @ A )
% 0.33/0.64 = A ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_of_pos
% 0.33/0.64 thf(fact_6542_abs__of__pos,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( ord_less_int @ zero_zero_int @ A )
% 0.33/0.64 => ( ( abs_abs_int @ A )
% 0.33/0.64 = A ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_of_pos
% 0.33/0.64 thf(fact_6543_abs__not__less__zero,axiom,
% 0.33/0.64 ! [A: code_integer] :
% 0.33/0.64 ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).
% 0.33/0.64
% 0.33/0.64 % abs_not_less_zero
% 0.33/0.64 thf(fact_6544_abs__not__less__zero,axiom,
% 0.33/0.64 ! [A: real] :
% 0.33/0.64 ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% 0.33/0.64
% 0.33/0.64 % abs_not_less_zero
% 0.33/0.64 thf(fact_6545_abs__not__less__zero,axiom,
% 0.33/0.64 ! [A: rat] :
% 0.33/0.64 ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).
% 0.33/0.64
% 0.33/0.64 % abs_not_less_zero
% 0.33/0.64 thf(fact_6546_abs__not__less__zero,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% 0.33/0.64
% 0.33/0.64 % abs_not_less_zero
% 0.33/0.64 thf(fact_6547_abs__triangle__ineq,axiom,
% 0.33/0.64 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_triangle_ineq
% 0.33/0.64 thf(fact_6548_abs__triangle__ineq,axiom,
% 0.33/0.64 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_triangle_ineq
% 0.33/0.64 thf(fact_6549_abs__triangle__ineq,axiom,
% 0.33/0.64 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_triangle_ineq
% 0.33/0.64 thf(fact_6550_abs__triangle__ineq,axiom,
% 0.33/0.64 ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_triangle_ineq
% 0.33/0.64 thf(fact_6551_abs__mult__less,axiom,
% 0.33/0.64 ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 0.33/0.64 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
% 0.33/0.64 => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D )
% 0.33/0.64 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_mult_less
% 0.33/0.64 thf(fact_6552_abs__mult__less,axiom,
% 0.33/0.64 ! [A: real,C: real,B: real,D: real] :
% 0.33/0.64 ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
% 0.33/0.64 => ( ( ord_less_real @ ( abs_abs_real @ B ) @ D )
% 0.33/0.64 => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C @ D ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_mult_less
% 0.33/0.64 thf(fact_6553_abs__mult__less,axiom,
% 0.33/0.64 ! [A: rat,C: rat,B: rat,D: rat] :
% 0.33/0.64 ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
% 0.33/0.64 => ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D )
% 0.33/0.64 => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_mult_less
% 0.33/0.64 thf(fact_6554_abs__mult__less,axiom,
% 0.33/0.64 ! [A: int,C: int,B: int,D: int] :
% 0.33/0.64 ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
% 0.33/0.64 => ( ( ord_less_int @ ( abs_abs_int @ B ) @ D )
% 0.33/0.64 => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_mult_less
% 0.33/0.64 thf(fact_6555_abs__triangle__ineq2,axiom,
% 0.33/0.64 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_triangle_ineq2
% 0.33/0.64 thf(fact_6556_abs__triangle__ineq2,axiom,
% 0.33/0.64 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_triangle_ineq2
% 0.33/0.64 thf(fact_6557_abs__triangle__ineq2,axiom,
% 0.33/0.64 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_triangle_ineq2
% 0.33/0.64 thf(fact_6558_abs__triangle__ineq2,axiom,
% 0.33/0.64 ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_triangle_ineq2
% 0.33/0.64 thf(fact_6559_abs__triangle__ineq3,axiom,
% 0.33/0.64 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_triangle_ineq3
% 0.33/0.64 thf(fact_6560_abs__triangle__ineq3,axiom,
% 0.33/0.64 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_triangle_ineq3
% 0.33/0.64 thf(fact_6561_abs__triangle__ineq3,axiom,
% 0.33/0.64 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_triangle_ineq3
% 0.33/0.64 thf(fact_6562_abs__triangle__ineq3,axiom,
% 0.33/0.64 ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_triangle_ineq3
% 0.33/0.64 thf(fact_6563_abs__triangle__ineq2__sym,axiom,
% 0.33/0.64 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_triangle_ineq2_sym
% 0.33/0.64 thf(fact_6564_abs__triangle__ineq2__sym,axiom,
% 0.33/0.64 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_triangle_ineq2_sym
% 0.33/0.64 thf(fact_6565_abs__triangle__ineq2__sym,axiom,
% 0.33/0.64 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_triangle_ineq2_sym
% 0.33/0.64 thf(fact_6566_abs__triangle__ineq2__sym,axiom,
% 0.33/0.64 ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_triangle_ineq2_sym
% 0.33/0.64 thf(fact_6567_nonzero__abs__divide,axiom,
% 0.33/0.64 ! [B: real,A: real] :
% 0.33/0.64 ( ( B != zero_zero_real )
% 0.33/0.64 => ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 0.33/0.64 = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % nonzero_abs_divide
% 0.33/0.64 thf(fact_6568_nonzero__abs__divide,axiom,
% 0.33/0.64 ! [B: rat,A: rat] :
% 0.33/0.64 ( ( B != zero_zero_rat )
% 0.33/0.64 => ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 0.33/0.64 = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % nonzero_abs_divide
% 0.33/0.64 thf(fact_6569_abs__ge__minus__self,axiom,
% 0.33/0.64 ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_ge_minus_self
% 0.33/0.64 thf(fact_6570_abs__ge__minus__self,axiom,
% 0.33/0.64 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_ge_minus_self
% 0.33/0.64 thf(fact_6571_abs__ge__minus__self,axiom,
% 0.33/0.64 ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_ge_minus_self
% 0.33/0.64 thf(fact_6572_abs__ge__minus__self,axiom,
% 0.33/0.64 ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_ge_minus_self
% 0.33/0.64 thf(fact_6573_abs__le__iff,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 0.33/0.64 = ( ( ord_less_eq_real @ A @ B )
% 0.33/0.64 & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_le_iff
% 0.33/0.64 thf(fact_6574_abs__le__iff,axiom,
% 0.33/0.64 ! [A: code_integer,B: code_integer] :
% 0.33/0.64 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.33/0.64 = ( ( ord_le3102999989581377725nteger @ A @ B )
% 0.33/0.64 & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_le_iff
% 0.33/0.64 thf(fact_6575_abs__le__iff,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 0.33/0.64 = ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.64 & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_le_iff
% 0.33/0.64 thf(fact_6576_abs__le__iff,axiom,
% 0.33/0.64 ! [A: int,B: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 0.33/0.64 = ( ( ord_less_eq_int @ A @ B )
% 0.33/0.64 & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_le_iff
% 0.33/0.64 thf(fact_6577_abs__le__D2,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 0.33/0.64 => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_le_D2
% 0.33/0.64 thf(fact_6578_abs__le__D2,axiom,
% 0.33/0.64 ! [A: code_integer,B: code_integer] :
% 0.33/0.64 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.33/0.64 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_le_D2
% 0.33/0.64 thf(fact_6579_abs__le__D2,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 0.33/0.64 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_le_D2
% 0.33/0.64 thf(fact_6580_abs__le__D2,axiom,
% 0.33/0.64 ! [A: int,B: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 0.33/0.64 => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_le_D2
% 0.33/0.64 thf(fact_6581_abs__leI,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ A @ B )
% 0.33/0.64 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 0.33/0.64 => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_leI
% 0.33/0.64 thf(fact_6582_abs__leI,axiom,
% 0.33/0.64 ! [A: code_integer,B: code_integer] :
% 0.33/0.64 ( ( ord_le3102999989581377725nteger @ A @ B )
% 0.33/0.64 => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.33/0.64 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_leI
% 0.33/0.64 thf(fact_6583_abs__leI,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ A @ B )
% 0.33/0.64 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.33/0.64 => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_leI
% 0.33/0.64 thf(fact_6584_abs__leI,axiom,
% 0.33/0.64 ! [A: int,B: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ A @ B )
% 0.33/0.64 => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 0.33/0.64 => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_leI
% 0.33/0.64 thf(fact_6585_abs__less__iff,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
% 0.33/0.64 = ( ( ord_less_real @ A @ B )
% 0.33/0.64 & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_less_iff
% 0.33/0.64 thf(fact_6586_abs__less__iff,axiom,
% 0.33/0.64 ! [A: int,B: int] :
% 0.33/0.64 ( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
% 0.33/0.64 = ( ( ord_less_int @ A @ B )
% 0.33/0.64 & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_less_iff
% 0.33/0.64 thf(fact_6587_abs__less__iff,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B )
% 0.33/0.64 = ( ( ord_less_rat @ A @ B )
% 0.33/0.64 & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_less_iff
% 0.33/0.64 thf(fact_6588_abs__less__iff,axiom,
% 0.33/0.64 ! [A: code_integer,B: code_integer] :
% 0.33/0.64 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.33/0.64 = ( ( ord_le6747313008572928689nteger @ A @ B )
% 0.33/0.64 & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_less_iff
% 0.33/0.64 thf(fact_6589_sum__cong__Suc,axiom,
% 0.33/0.64 ! [A2: set_nat,F: nat > nat,G: nat > nat] :
% 0.33/0.64 ( ~ ( member_nat @ zero_zero_nat @ A2 )
% 0.33/0.64 => ( ! [X4: nat] :
% 0.33/0.64 ( ( member_nat @ ( suc @ X4 ) @ A2 )
% 0.33/0.64 => ( ( F @ ( suc @ X4 ) )
% 0.33/0.64 = ( G @ ( suc @ X4 ) ) ) )
% 0.33/0.64 => ( ( groups3542108847815614940at_nat @ F @ A2 )
% 0.33/0.64 = ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_cong_Suc
% 0.33/0.64 thf(fact_6590_sum__cong__Suc,axiom,
% 0.33/0.64 ! [A2: set_nat,F: nat > real,G: nat > real] :
% 0.33/0.64 ( ~ ( member_nat @ zero_zero_nat @ A2 )
% 0.33/0.64 => ( ! [X4: nat] :
% 0.33/0.64 ( ( member_nat @ ( suc @ X4 ) @ A2 )
% 0.33/0.64 => ( ( F @ ( suc @ X4 ) )
% 0.33/0.64 = ( G @ ( suc @ X4 ) ) ) )
% 0.33/0.64 => ( ( groups6591440286371151544t_real @ F @ A2 )
% 0.33/0.64 = ( groups6591440286371151544t_real @ G @ A2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_cong_Suc
% 0.33/0.64 thf(fact_6591_abs__real__def,axiom,
% 0.33/0.64 ( abs_abs_real
% 0.33/0.64 = ( ^ [A5: real] : ( if_real @ ( ord_less_real @ A5 @ zero_zero_real ) @ ( uminus_uminus_real @ A5 ) @ A5 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_real_def
% 0.33/0.64 thf(fact_6592_num_Oexhaust,axiom,
% 0.33/0.64 ! [Y: num] :
% 0.33/0.64 ( ( Y != one )
% 0.33/0.64 => ( ! [X23: num] :
% 0.33/0.64 ( Y
% 0.33/0.64 != ( bit0 @ X23 ) )
% 0.33/0.64 => ~ ! [X33: num] :
% 0.33/0.64 ( Y
% 0.33/0.64 != ( bit1 @ X33 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % num.exhaust
% 0.33/0.64 thf(fact_6593_xor__num_Ocases,axiom,
% 0.33/0.64 ! [X: product_prod_num_num] :
% 0.33/0.64 ( ( X
% 0.33/0.64 != ( product_Pair_num_num @ one @ one ) )
% 0.33/0.64 => ( ! [N4: num] :
% 0.33/0.64 ( X
% 0.33/0.64 != ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) )
% 0.33/0.64 => ( ! [N4: num] :
% 0.33/0.64 ( X
% 0.33/0.64 != ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) )
% 0.33/0.64 => ( ! [M2: num] :
% 0.33/0.64 ( X
% 0.33/0.64 != ( product_Pair_num_num @ ( bit0 @ M2 ) @ one ) )
% 0.33/0.64 => ( ! [M2: num,N4: num] :
% 0.33/0.64 ( X
% 0.33/0.64 != ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit0 @ N4 ) ) )
% 0.33/0.64 => ( ! [M2: num,N4: num] :
% 0.33/0.64 ( X
% 0.33/0.64 != ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit1 @ N4 ) ) )
% 0.33/0.64 => ( ! [M2: num] :
% 0.33/0.64 ( X
% 0.33/0.64 != ( product_Pair_num_num @ ( bit1 @ M2 ) @ one ) )
% 0.33/0.64 => ( ! [M2: num,N4: num] :
% 0.33/0.64 ( X
% 0.33/0.64 != ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit0 @ N4 ) ) )
% 0.33/0.64 => ~ ! [M2: num,N4: num] :
% 0.33/0.64 ( X
% 0.33/0.64 != ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % xor_num.cases
% 0.33/0.64 thf(fact_6594_sin__bound__lemma,axiom,
% 0.33/0.64 ! [X: real,Y: real,U: real,V: real] :
% 0.33/0.64 ( ( X = Y )
% 0.33/0.64 => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
% 0.33/0.64 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U ) @ Y ) ) @ V ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sin_bound_lemma
% 0.33/0.64 thf(fact_6595_sum__subtractf__nat,axiom,
% 0.33/0.64 ! [A2: set_real,G: real > nat,F: real > nat] :
% 0.33/0.64 ( ! [X4: real] :
% 0.33/0.64 ( ( member_real @ X4 @ A2 )
% 0.33/0.64 => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 0.33/0.64 => ( ( groups1935376822645274424al_nat
% 0.33/0.64 @ ^ [X2: real] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.33/0.64 @ A2 )
% 0.33/0.64 = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_subtractf_nat
% 0.33/0.64 thf(fact_6596_sum__subtractf__nat,axiom,
% 0.33/0.64 ! [A2: set_int,G: int > nat,F: int > nat] :
% 0.33/0.64 ( ! [X4: int] :
% 0.33/0.64 ( ( member_int @ X4 @ A2 )
% 0.33/0.64 => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 0.33/0.64 => ( ( groups4541462559716669496nt_nat
% 0.33/0.64 @ ^ [X2: int] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.33/0.64 @ A2 )
% 0.33/0.64 = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_subtractf_nat
% 0.33/0.64 thf(fact_6597_sum__subtractf__nat,axiom,
% 0.33/0.64 ! [A2: set_complex,G: complex > nat,F: complex > nat] :
% 0.33/0.64 ( ! [X4: complex] :
% 0.33/0.64 ( ( member_complex @ X4 @ A2 )
% 0.33/0.64 => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 0.33/0.64 => ( ( groups5693394587270226106ex_nat
% 0.33/0.64 @ ^ [X2: complex] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.33/0.64 @ A2 )
% 0.33/0.64 = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_subtractf_nat
% 0.33/0.64 thf(fact_6598_sum__subtractf__nat,axiom,
% 0.33/0.64 ! [A2: set_set_nat,G: set_nat > nat,F: set_nat > nat] :
% 0.33/0.64 ( ! [X4: set_nat] :
% 0.33/0.64 ( ( member_set_nat @ X4 @ A2 )
% 0.33/0.64 => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 0.33/0.64 => ( ( groups8294997508430121362at_nat
% 0.33/0.64 @ ^ [X2: set_nat] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.33/0.64 @ A2 )
% 0.33/0.64 = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A2 ) @ ( groups8294997508430121362at_nat @ G @ A2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_subtractf_nat
% 0.33/0.64 thf(fact_6599_sum__subtractf__nat,axiom,
% 0.33/0.64 ! [A2: set_nat,G: nat > nat,F: nat > nat] :
% 0.33/0.64 ( ! [X4: nat] :
% 0.33/0.64 ( ( member_nat @ X4 @ A2 )
% 0.33/0.64 => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 0.33/0.64 => ( ( groups3542108847815614940at_nat
% 0.33/0.64 @ ^ [X2: nat] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.33/0.64 @ A2 )
% 0.33/0.64 = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_subtractf_nat
% 0.33/0.64 thf(fact_6600_sum_Oshift__bounds__cl__Suc__ivl,axiom,
% 0.33/0.64 ! [G: nat > nat,M: nat,N2: nat] :
% 0.33/0.64 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
% 0.33/0.64 = ( groups3542108847815614940at_nat
% 0.33/0.64 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.shift_bounds_cl_Suc_ivl
% 0.33/0.64 thf(fact_6601_sum_Oshift__bounds__cl__Suc__ivl,axiom,
% 0.33/0.64 ! [G: nat > real,M: nat,N2: nat] :
% 0.33/0.64 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
% 0.33/0.64 = ( groups6591440286371151544t_real
% 0.33/0.64 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.shift_bounds_cl_Suc_ivl
% 0.33/0.64 thf(fact_6602_sum_Oshift__bounds__cl__nat__ivl,axiom,
% 0.33/0.64 ! [G: nat > nat,M: nat,K: nat,N2: nat] :
% 0.33/0.64 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
% 0.33/0.64 = ( groups3542108847815614940at_nat
% 0.33/0.64 @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.shift_bounds_cl_nat_ivl
% 0.33/0.64 thf(fact_6603_sum_Oshift__bounds__cl__nat__ivl,axiom,
% 0.33/0.64 ! [G: nat > real,M: nat,K: nat,N2: nat] :
% 0.33/0.64 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
% 0.33/0.64 = ( groups6591440286371151544t_real
% 0.33/0.64 @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.shift_bounds_cl_nat_ivl
% 0.33/0.64 thf(fact_6604_tanh__real__lt__1,axiom,
% 0.33/0.64 ! [X: real] : ( ord_less_real @ ( tanh_real @ X ) @ one_one_real ) ).
% 0.33/0.64
% 0.33/0.64 % tanh_real_lt_1
% 0.33/0.64 thf(fact_6605_tanh__real__gt__neg1,axiom,
% 0.33/0.64 ! [X: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X ) ) ).
% 0.33/0.64
% 0.33/0.64 % tanh_real_gt_neg1
% 0.33/0.64 thf(fact_6606_dense__eq0__I,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ! [E2: real] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.33/0.64 => ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E2 ) )
% 0.33/0.64 => ( X = zero_zero_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % dense_eq0_I
% 0.33/0.64 thf(fact_6607_dense__eq0__I,axiom,
% 0.33/0.64 ! [X: rat] :
% 0.33/0.64 ( ! [E2: rat] :
% 0.33/0.64 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 0.33/0.64 => ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E2 ) )
% 0.33/0.64 => ( X = zero_zero_rat ) ) ).
% 0.33/0.64
% 0.33/0.64 % dense_eq0_I
% 0.33/0.64 thf(fact_6608_abs__mult__pos,axiom,
% 0.33/0.64 ! [X: code_integer,Y: code_integer] :
% 0.33/0.64 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 0.33/0.64 => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y ) @ X )
% 0.33/0.64 = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y @ X ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_mult_pos
% 0.33/0.64 thf(fact_6609_abs__mult__pos,axiom,
% 0.33/0.64 ! [X: real,Y: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.64 => ( ( times_times_real @ ( abs_abs_real @ Y ) @ X )
% 0.33/0.64 = ( abs_abs_real @ ( times_times_real @ Y @ X ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_mult_pos
% 0.33/0.64 thf(fact_6610_abs__mult__pos,axiom,
% 0.33/0.64 ! [X: rat,Y: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.33/0.64 => ( ( times_times_rat @ ( abs_abs_rat @ Y ) @ X )
% 0.33/0.64 = ( abs_abs_rat @ ( times_times_rat @ Y @ X ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_mult_pos
% 0.33/0.64 thf(fact_6611_abs__mult__pos,axiom,
% 0.33/0.64 ! [X: int,Y: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.64 => ( ( times_times_int @ ( abs_abs_int @ Y ) @ X )
% 0.33/0.64 = ( abs_abs_int @ ( times_times_int @ Y @ X ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_mult_pos
% 0.33/0.64 thf(fact_6612_abs__eq__mult,axiom,
% 0.33/0.64 ! [A: code_integer,B: code_integer] :
% 0.33/0.64 ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.33/0.64 | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
% 0.33/0.64 & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 0.33/0.64 | ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
% 0.33/0.64 => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.33/0.64 = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_eq_mult
% 0.33/0.64 thf(fact_6613_abs__eq__mult,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.64 | ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.33/0.64 & ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.33/0.64 | ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 0.33/0.64 => ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 0.33/0.64 = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_eq_mult
% 0.33/0.64 thf(fact_6614_abs__eq__mult,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.64 | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.33/0.64 & ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.33/0.64 | ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 0.33/0.64 => ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
% 0.33/0.64 = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_eq_mult
% 0.33/0.64 thf(fact_6615_abs__eq__mult,axiom,
% 0.33/0.64 ! [A: int,B: int] :
% 0.33/0.64 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.64 | ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.33/0.64 & ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.33/0.64 | ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 0.33/0.64 => ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
% 0.33/0.64 = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_eq_mult
% 0.33/0.64 thf(fact_6616_abs__eq__iff_H,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( ( abs_abs_real @ A )
% 0.33/0.64 = B )
% 0.33/0.64 = ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.33/0.64 & ( ( A = B )
% 0.33/0.64 | ( A
% 0.33/0.64 = ( uminus_uminus_real @ B ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_eq_iff'
% 0.33/0.64 thf(fact_6617_abs__eq__iff_H,axiom,
% 0.33/0.64 ! [A: code_integer,B: code_integer] :
% 0.33/0.64 ( ( ( abs_abs_Code_integer @ A )
% 0.33/0.64 = B )
% 0.33/0.64 = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 0.33/0.64 & ( ( A = B )
% 0.33/0.64 | ( A
% 0.33/0.64 = ( uminus1351360451143612070nteger @ B ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_eq_iff'
% 0.33/0.64 thf(fact_6618_abs__eq__iff_H,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( ( abs_abs_rat @ A )
% 0.33/0.64 = B )
% 0.33/0.64 = ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.33/0.64 & ( ( A = B )
% 0.33/0.64 | ( A
% 0.33/0.64 = ( uminus_uminus_rat @ B ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_eq_iff'
% 0.33/0.64 thf(fact_6619_abs__eq__iff_H,axiom,
% 0.33/0.64 ! [A: int,B: int] :
% 0.33/0.64 ( ( ( abs_abs_int @ A )
% 0.33/0.64 = B )
% 0.33/0.64 = ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.33/0.64 & ( ( A = B )
% 0.33/0.64 | ( A
% 0.33/0.64 = ( uminus_uminus_int @ B ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_eq_iff'
% 0.33/0.64 thf(fact_6620_eq__abs__iff_H,axiom,
% 0.33/0.64 ! [A: real,B: real] :
% 0.33/0.64 ( ( A
% 0.33/0.64 = ( abs_abs_real @ B ) )
% 0.33/0.64 = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.64 & ( ( B = A )
% 0.33/0.64 | ( B
% 0.33/0.64 = ( uminus_uminus_real @ A ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % eq_abs_iff'
% 0.33/0.64 thf(fact_6621_eq__abs__iff_H,axiom,
% 0.33/0.64 ! [A: code_integer,B: code_integer] :
% 0.33/0.64 ( ( A
% 0.33/0.64 = ( abs_abs_Code_integer @ B ) )
% 0.33/0.64 = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.33/0.64 & ( ( B = A )
% 0.33/0.64 | ( B
% 0.33/0.64 = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % eq_abs_iff'
% 0.33/0.64 thf(fact_6622_eq__abs__iff_H,axiom,
% 0.33/0.64 ! [A: rat,B: rat] :
% 0.33/0.64 ( ( A
% 0.33/0.64 = ( abs_abs_rat @ B ) )
% 0.33/0.64 = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.33/0.64 & ( ( B = A )
% 0.33/0.64 | ( B
% 0.33/0.64 = ( uminus_uminus_rat @ A ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % eq_abs_iff'
% 0.33/0.64 thf(fact_6623_eq__abs__iff_H,axiom,
% 0.33/0.64 ! [A: int,B: int] :
% 0.33/0.64 ( ( A
% 0.33/0.64 = ( abs_abs_int @ B ) )
% 0.33/0.64 = ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.33/0.64 & ( ( B = A )
% 0.33/0.64 | ( B
% 0.33/0.64 = ( uminus_uminus_int @ A ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % eq_abs_iff'
% 0.33/0.64 thf(fact_6624_abs__minus__le__zero,axiom,
% 0.33/0.64 ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).
% 0.33/0.64
% 0.33/0.64 % abs_minus_le_zero
% 0.33/0.64 thf(fact_6625_abs__minus__le__zero,axiom,
% 0.33/0.64 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).
% 0.33/0.64
% 0.33/0.64 % abs_minus_le_zero
% 0.33/0.64 thf(fact_6626_abs__minus__le__zero,axiom,
% 0.33/0.64 ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).
% 0.33/0.64
% 0.33/0.64 % abs_minus_le_zero
% 0.33/0.64 thf(fact_6627_abs__minus__le__zero,axiom,
% 0.33/0.64 ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% 0.33/0.64
% 0.33/0.64 % abs_minus_le_zero
% 0.33/0.64 thf(fact_6628_zero__le__power__abs,axiom,
% 0.33/0.64 ! [A: code_integer,N2: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % zero_le_power_abs
% 0.33/0.64 thf(fact_6629_zero__le__power__abs,axiom,
% 0.33/0.64 ! [A: real,N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % zero_le_power_abs
% 0.33/0.64 thf(fact_6630_zero__le__power__abs,axiom,
% 0.33/0.64 ! [A: rat,N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % zero_le_power_abs
% 0.33/0.64 thf(fact_6631_zero__le__power__abs,axiom,
% 0.33/0.64 ! [A: int,N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % zero_le_power_abs
% 0.33/0.64 thf(fact_6632_abs__div__pos,axiom,
% 0.33/0.64 ! [Y: real,X: real] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.64 => ( ( divide_divide_real @ ( abs_abs_real @ X ) @ Y )
% 0.33/0.64 = ( abs_abs_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_div_pos
% 0.33/0.64 thf(fact_6633_abs__div__pos,axiom,
% 0.33/0.64 ! [Y: rat,X: rat] :
% 0.33/0.64 ( ( ord_less_rat @ zero_zero_rat @ Y )
% 0.33/0.64 => ( ( divide_divide_rat @ ( abs_abs_rat @ X ) @ Y )
% 0.33/0.64 = ( abs_abs_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_div_pos
% 0.33/0.64 thf(fact_6634_abs__if__raw,axiom,
% 0.33/0.64 ( abs_abs_real
% 0.33/0.64 = ( ^ [A5: real] : ( if_real @ ( ord_less_real @ A5 @ zero_zero_real ) @ ( uminus_uminus_real @ A5 ) @ A5 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_if_raw
% 0.33/0.64 thf(fact_6635_abs__if__raw,axiom,
% 0.33/0.64 ( abs_abs_int
% 0.33/0.64 = ( ^ [A5: int] : ( if_int @ ( ord_less_int @ A5 @ zero_zero_int ) @ ( uminus_uminus_int @ A5 ) @ A5 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_if_raw
% 0.33/0.64 thf(fact_6636_abs__if__raw,axiom,
% 0.33/0.64 ( abs_abs_rat
% 0.33/0.64 = ( ^ [A5: rat] : ( if_rat @ ( ord_less_rat @ A5 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A5 ) @ A5 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_if_raw
% 0.33/0.64 thf(fact_6637_abs__if__raw,axiom,
% 0.33/0.64 ( abs_abs_Code_integer
% 0.33/0.64 = ( ^ [A5: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A5 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A5 ) @ A5 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_if_raw
% 0.33/0.64 thf(fact_6638_abs__of__neg,axiom,
% 0.33/0.64 ! [A: real] :
% 0.33/0.64 ( ( ord_less_real @ A @ zero_zero_real )
% 0.33/0.64 => ( ( abs_abs_real @ A )
% 0.33/0.64 = ( uminus_uminus_real @ A ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_of_neg
% 0.33/0.64 thf(fact_6639_abs__of__neg,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( ord_less_int @ A @ zero_zero_int )
% 0.33/0.64 => ( ( abs_abs_int @ A )
% 0.33/0.64 = ( uminus_uminus_int @ A ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_of_neg
% 0.33/0.64 thf(fact_6640_abs__of__neg,axiom,
% 0.33/0.64 ! [A: rat] :
% 0.33/0.64 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.33/0.64 => ( ( abs_abs_rat @ A )
% 0.33/0.64 = ( uminus_uminus_rat @ A ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_of_neg
% 0.33/0.64 thf(fact_6641_abs__of__neg,axiom,
% 0.33/0.64 ! [A: code_integer] :
% 0.33/0.64 ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 0.33/0.64 => ( ( abs_abs_Code_integer @ A )
% 0.33/0.64 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_of_neg
% 0.33/0.64 thf(fact_6642_abs__if,axiom,
% 0.33/0.64 ( abs_abs_real
% 0.33/0.64 = ( ^ [A5: real] : ( if_real @ ( ord_less_real @ A5 @ zero_zero_real ) @ ( uminus_uminus_real @ A5 ) @ A5 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_if
% 0.33/0.64 thf(fact_6643_abs__if,axiom,
% 0.33/0.64 ( abs_abs_int
% 0.33/0.64 = ( ^ [A5: int] : ( if_int @ ( ord_less_int @ A5 @ zero_zero_int ) @ ( uminus_uminus_int @ A5 ) @ A5 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_if
% 0.33/0.64 thf(fact_6644_abs__if,axiom,
% 0.33/0.64 ( abs_abs_rat
% 0.33/0.64 = ( ^ [A5: rat] : ( if_rat @ ( ord_less_rat @ A5 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A5 ) @ A5 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_if
% 0.33/0.64 thf(fact_6645_abs__if,axiom,
% 0.33/0.64 ( abs_abs_Code_integer
% 0.33/0.64 = ( ^ [A5: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A5 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A5 ) @ A5 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_if
% 0.33/0.64 thf(fact_6646_abs__triangle__ineq4,axiom,
% 0.33/0.64 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_triangle_ineq4
% 0.33/0.64 thf(fact_6647_abs__triangle__ineq4,axiom,
% 0.33/0.64 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_triangle_ineq4
% 0.33/0.64 thf(fact_6648_abs__triangle__ineq4,axiom,
% 0.33/0.64 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_triangle_ineq4
% 0.33/0.64 thf(fact_6649_abs__triangle__ineq4,axiom,
% 0.33/0.64 ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_triangle_ineq4
% 0.33/0.64 thf(fact_6650_abs__diff__triangle__ineq,axiom,
% 0.33/0.64 ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ D ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_diff_triangle_ineq
% 0.33/0.64 thf(fact_6651_abs__diff__triangle__ineq,axiom,
% 0.33/0.64 ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_diff_triangle_ineq
% 0.33/0.64 thf(fact_6652_abs__diff__triangle__ineq,axiom,
% 0.33/0.64 ! [A: rat,B: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_diff_triangle_ineq
% 0.33/0.64 thf(fact_6653_abs__diff__triangle__ineq,axiom,
% 0.33/0.64 ! [A: int,B: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_diff_triangle_ineq
% 0.33/0.64 thf(fact_6654_abs__diff__le__iff,axiom,
% 0.33/0.64 ! [X: code_integer,A: code_integer,R: code_integer] :
% 0.33/0.64 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R )
% 0.33/0.64 = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R ) @ X )
% 0.33/0.64 & ( ord_le3102999989581377725nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_diff_le_iff
% 0.33/0.64 thf(fact_6655_abs__diff__le__iff,axiom,
% 0.33/0.64 ! [X: real,A: real,R: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R )
% 0.33/0.64 = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R ) @ X )
% 0.33/0.64 & ( ord_less_eq_real @ X @ ( plus_plus_real @ A @ R ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_diff_le_iff
% 0.33/0.64 thf(fact_6656_abs__diff__le__iff,axiom,
% 0.33/0.64 ! [X: rat,A: rat,R: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R )
% 0.33/0.64 = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R ) @ X )
% 0.33/0.64 & ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A @ R ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_diff_le_iff
% 0.33/0.64 thf(fact_6657_abs__diff__le__iff,axiom,
% 0.33/0.64 ! [X: int,A: int,R: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R )
% 0.33/0.64 = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R ) @ X )
% 0.33/0.64 & ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_diff_le_iff
% 0.33/0.64 thf(fact_6658_abs__diff__less__iff,axiom,
% 0.33/0.64 ! [X: code_integer,A: code_integer,R: code_integer] :
% 0.33/0.64 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R )
% 0.33/0.64 = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R ) @ X )
% 0.33/0.64 & ( ord_le6747313008572928689nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_diff_less_iff
% 0.33/0.64 thf(fact_6659_abs__diff__less__iff,axiom,
% 0.33/0.64 ! [X: real,A: real,R: real] :
% 0.33/0.64 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R )
% 0.33/0.64 = ( ( ord_less_real @ ( minus_minus_real @ A @ R ) @ X )
% 0.33/0.64 & ( ord_less_real @ X @ ( plus_plus_real @ A @ R ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_diff_less_iff
% 0.33/0.64 thf(fact_6660_abs__diff__less__iff,axiom,
% 0.33/0.64 ! [X: rat,A: rat,R: rat] :
% 0.33/0.64 ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R )
% 0.33/0.64 = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R ) @ X )
% 0.33/0.64 & ( ord_less_rat @ X @ ( plus_plus_rat @ A @ R ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_diff_less_iff
% 0.33/0.64 thf(fact_6661_abs__diff__less__iff,axiom,
% 0.33/0.64 ! [X: int,A: int,R: int] :
% 0.33/0.64 ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R )
% 0.33/0.64 = ( ( ord_less_int @ ( minus_minus_int @ A @ R ) @ X )
% 0.33/0.64 & ( ord_less_int @ X @ ( plus_plus_int @ A @ R ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_diff_less_iff
% 0.33/0.64 thf(fact_6662_sum__eq__Suc0__iff,axiom,
% 0.33/0.64 ! [A2: set_int,F: int > nat] :
% 0.33/0.64 ( ( finite_finite_int @ A2 )
% 0.33/0.64 => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
% 0.33/0.64 = ( suc @ zero_zero_nat ) )
% 0.33/0.64 = ( ? [X2: int] :
% 0.33/0.64 ( ( member_int @ X2 @ A2 )
% 0.33/0.64 & ( ( F @ X2 )
% 0.33/0.64 = ( suc @ zero_zero_nat ) )
% 0.33/0.64 & ! [Y2: int] :
% 0.33/0.64 ( ( member_int @ Y2 @ A2 )
% 0.33/0.64 => ( ( X2 != Y2 )
% 0.33/0.64 => ( ( F @ Y2 )
% 0.33/0.64 = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_eq_Suc0_iff
% 0.33/0.64 thf(fact_6663_sum__eq__Suc0__iff,axiom,
% 0.33/0.64 ! [A2: set_complex,F: complex > nat] :
% 0.33/0.64 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.64 => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
% 0.33/0.64 = ( suc @ zero_zero_nat ) )
% 0.33/0.64 = ( ? [X2: complex] :
% 0.33/0.64 ( ( member_complex @ X2 @ A2 )
% 0.33/0.64 & ( ( F @ X2 )
% 0.33/0.64 = ( suc @ zero_zero_nat ) )
% 0.33/0.64 & ! [Y2: complex] :
% 0.33/0.64 ( ( member_complex @ Y2 @ A2 )
% 0.33/0.64 => ( ( X2 != Y2 )
% 0.33/0.64 => ( ( F @ Y2 )
% 0.33/0.64 = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_eq_Suc0_iff
% 0.33/0.64 thf(fact_6664_sum__eq__Suc0__iff,axiom,
% 0.33/0.64 ! [A2: set_nat,F: nat > nat] :
% 0.33/0.64 ( ( finite_finite_nat @ A2 )
% 0.33/0.64 => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
% 0.33/0.64 = ( suc @ zero_zero_nat ) )
% 0.33/0.64 = ( ? [X2: nat] :
% 0.33/0.64 ( ( member_nat @ X2 @ A2 )
% 0.33/0.64 & ( ( F @ X2 )
% 0.33/0.64 = ( suc @ zero_zero_nat ) )
% 0.33/0.64 & ! [Y2: nat] :
% 0.33/0.64 ( ( member_nat @ Y2 @ A2 )
% 0.33/0.64 => ( ( X2 != Y2 )
% 0.33/0.64 => ( ( F @ Y2 )
% 0.33/0.64 = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_eq_Suc0_iff
% 0.33/0.64 thf(fact_6665_sum__SucD,axiom,
% 0.33/0.64 ! [F: nat > nat,A2: set_nat,N2: nat] :
% 0.33/0.64 ( ( ( groups3542108847815614940at_nat @ F @ A2 )
% 0.33/0.64 = ( suc @ N2 ) )
% 0.33/0.64 => ? [X4: nat] :
% 0.33/0.64 ( ( member_nat @ X4 @ A2 )
% 0.33/0.64 & ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_SucD
% 0.33/0.64 thf(fact_6666_sum__eq__1__iff,axiom,
% 0.33/0.64 ! [A2: set_int,F: int > nat] :
% 0.33/0.64 ( ( finite_finite_int @ A2 )
% 0.33/0.64 => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
% 0.33/0.64 = one_one_nat )
% 0.33/0.64 = ( ? [X2: int] :
% 0.33/0.64 ( ( member_int @ X2 @ A2 )
% 0.33/0.64 & ( ( F @ X2 )
% 0.33/0.64 = one_one_nat )
% 0.33/0.64 & ! [Y2: int] :
% 0.33/0.64 ( ( member_int @ Y2 @ A2 )
% 0.33/0.64 => ( ( X2 != Y2 )
% 0.33/0.64 => ( ( F @ Y2 )
% 0.33/0.64 = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_eq_1_iff
% 0.33/0.64 thf(fact_6667_sum__eq__1__iff,axiom,
% 0.33/0.64 ! [A2: set_complex,F: complex > nat] :
% 0.33/0.64 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.64 => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
% 0.33/0.64 = one_one_nat )
% 0.33/0.64 = ( ? [X2: complex] :
% 0.33/0.64 ( ( member_complex @ X2 @ A2 )
% 0.33/0.64 & ( ( F @ X2 )
% 0.33/0.64 = one_one_nat )
% 0.33/0.64 & ! [Y2: complex] :
% 0.33/0.64 ( ( member_complex @ Y2 @ A2 )
% 0.33/0.64 => ( ( X2 != Y2 )
% 0.33/0.64 => ( ( F @ Y2 )
% 0.33/0.64 = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_eq_1_iff
% 0.33/0.64 thf(fact_6668_sum__eq__1__iff,axiom,
% 0.33/0.64 ! [A2: set_nat,F: nat > nat] :
% 0.33/0.64 ( ( finite_finite_nat @ A2 )
% 0.33/0.64 => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
% 0.33/0.64 = one_one_nat )
% 0.33/0.64 = ( ? [X2: nat] :
% 0.33/0.64 ( ( member_nat @ X2 @ A2 )
% 0.33/0.64 & ( ( F @ X2 )
% 0.33/0.64 = one_one_nat )
% 0.33/0.64 & ! [Y2: nat] :
% 0.33/0.64 ( ( member_nat @ Y2 @ A2 )
% 0.33/0.64 => ( ( X2 != Y2 )
% 0.33/0.64 => ( ( F @ Y2 )
% 0.33/0.64 = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_eq_1_iff
% 0.33/0.64 thf(fact_6669_numeral__Bit1,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( numera6690914467698888265omplex @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) @ one_one_complex ) ) ).
% 0.33/0.64
% 0.33/0.64 % numeral_Bit1
% 0.33/0.64 thf(fact_6670_numeral__Bit1,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( numeral_numeral_real @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) @ one_one_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % numeral_Bit1
% 0.33/0.64 thf(fact_6671_numeral__Bit1,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( numeral_numeral_rat @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) @ one_one_rat ) ) ).
% 0.33/0.64
% 0.33/0.64 % numeral_Bit1
% 0.33/0.64 thf(fact_6672_numeral__Bit1,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) @ one_one_nat ) ) ).
% 0.33/0.64
% 0.33/0.64 % numeral_Bit1
% 0.33/0.64 thf(fact_6673_numeral__Bit1,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( numeral_numeral_int @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) @ one_one_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % numeral_Bit1
% 0.33/0.64 thf(fact_6674_eval__nat__numeral_I3_J,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % eval_nat_numeral(3)
% 0.33/0.64 thf(fact_6675_cong__exp__iff__simps_I13_J,axiom,
% 0.33/0.64 ! [M: num,Q2: num,N2: num] :
% 0.33/0.64 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.33/0.64 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 0.33/0.64 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.33/0.64 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % cong_exp_iff_simps(13)
% 0.33/0.64 thf(fact_6676_cong__exp__iff__simps_I13_J,axiom,
% 0.33/0.64 ! [M: num,Q2: num,N2: num] :
% 0.33/0.64 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.33/0.64 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 0.33/0.64 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 0.33/0.64 = ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % cong_exp_iff_simps(13)
% 0.33/0.64 thf(fact_6677_cong__exp__iff__simps_I13_J,axiom,
% 0.33/0.64 ! [M: num,Q2: num,N2: num] :
% 0.33/0.64 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.33/0.64 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 0.33/0.64 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.33/0.64 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % cong_exp_iff_simps(13)
% 0.33/0.64 thf(fact_6678_cong__exp__iff__simps_I12_J,axiom,
% 0.33/0.64 ! [M: num,Q2: num,N2: num] :
% 0.33/0.64 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.33/0.64 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % cong_exp_iff_simps(12)
% 0.33/0.64 thf(fact_6679_cong__exp__iff__simps_I12_J,axiom,
% 0.33/0.64 ! [M: num,Q2: num,N2: num] :
% 0.33/0.64 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.33/0.64 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % cong_exp_iff_simps(12)
% 0.33/0.64 thf(fact_6680_cong__exp__iff__simps_I12_J,axiom,
% 0.33/0.64 ! [M: num,Q2: num,N2: num] :
% 0.33/0.64 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.33/0.64 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % cong_exp_iff_simps(12)
% 0.33/0.64 thf(fact_6681_cong__exp__iff__simps_I10_J,axiom,
% 0.33/0.64 ! [M: num,Q2: num,N2: num] :
% 0.33/0.64 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.33/0.64 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % cong_exp_iff_simps(10)
% 0.33/0.64 thf(fact_6682_cong__exp__iff__simps_I10_J,axiom,
% 0.33/0.64 ! [M: num,Q2: num,N2: num] :
% 0.33/0.64 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.33/0.64 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % cong_exp_iff_simps(10)
% 0.33/0.64 thf(fact_6683_cong__exp__iff__simps_I10_J,axiom,
% 0.33/0.64 ! [M: num,Q2: num,N2: num] :
% 0.33/0.64 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.33/0.64 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % cong_exp_iff_simps(10)
% 0.33/0.64 thf(fact_6684_power__minus__Bit1,axiom,
% 0.33/0.64 ! [X: real,K: num] :
% 0.33/0.64 ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.33/0.64 = ( uminus_uminus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_minus_Bit1
% 0.33/0.64 thf(fact_6685_power__minus__Bit1,axiom,
% 0.33/0.64 ! [X: int,K: num] :
% 0.33/0.64 ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.33/0.64 = ( uminus_uminus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_minus_Bit1
% 0.33/0.64 thf(fact_6686_power__minus__Bit1,axiom,
% 0.33/0.64 ! [X: complex,K: num] :
% 0.33/0.64 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.33/0.64 = ( uminus1482373934393186551omplex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_minus_Bit1
% 0.33/0.64 thf(fact_6687_power__minus__Bit1,axiom,
% 0.33/0.64 ! [X: rat,K: num] :
% 0.33/0.64 ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.33/0.64 = ( uminus_uminus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_minus_Bit1
% 0.33/0.64 thf(fact_6688_power__minus__Bit1,axiom,
% 0.33/0.64 ! [X: code_integer,K: num] :
% 0.33/0.64 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.33/0.64 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_minus_Bit1
% 0.33/0.64 thf(fact_6689_lemma__interval__lt,axiom,
% 0.33/0.64 ! [A: real,X: real,B: real] :
% 0.33/0.64 ( ( ord_less_real @ A @ X )
% 0.33/0.64 => ( ( ord_less_real @ X @ B )
% 0.33/0.64 => ? [D4: real] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ D4 )
% 0.33/0.64 & ! [Y5: real] :
% 0.33/0.64 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D4 )
% 0.33/0.64 => ( ( ord_less_real @ A @ Y5 )
% 0.33/0.64 & ( ord_less_real @ Y5 @ B ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % lemma_interval_lt
% 0.33/0.64 thf(fact_6690_sum__power__add,axiom,
% 0.33/0.64 ! [X: complex,M: nat,I6: set_nat] :
% 0.33/0.64 ( ( groups2073611262835488442omplex
% 0.33/0.64 @ ^ [I3: nat] : ( power_power_complex @ X @ ( plus_plus_nat @ M @ I3 ) )
% 0.33/0.64 @ I6 )
% 0.33/0.64 = ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ I6 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_power_add
% 0.33/0.64 thf(fact_6691_sum__power__add,axiom,
% 0.33/0.64 ! [X: rat,M: nat,I6: set_nat] :
% 0.33/0.64 ( ( groups2906978787729119204at_rat
% 0.33/0.64 @ ^ [I3: nat] : ( power_power_rat @ X @ ( plus_plus_nat @ M @ I3 ) )
% 0.33/0.64 @ I6 )
% 0.33/0.64 = ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ I6 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_power_add
% 0.33/0.64 thf(fact_6692_sum__power__add,axiom,
% 0.33/0.64 ! [X: int,M: nat,I6: set_nat] :
% 0.33/0.64 ( ( groups3539618377306564664at_int
% 0.33/0.64 @ ^ [I3: nat] : ( power_power_int @ X @ ( plus_plus_nat @ M @ I3 ) )
% 0.33/0.64 @ I6 )
% 0.33/0.64 = ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ I6 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_power_add
% 0.33/0.64 thf(fact_6693_sum__power__add,axiom,
% 0.33/0.64 ! [X: real,M: nat,I6: set_nat] :
% 0.33/0.64 ( ( groups6591440286371151544t_real
% 0.33/0.64 @ ^ [I3: nat] : ( power_power_real @ X @ ( plus_plus_nat @ M @ I3 ) )
% 0.33/0.64 @ I6 )
% 0.33/0.64 = ( times_times_real @ ( power_power_real @ X @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ I6 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_power_add
% 0.33/0.64 thf(fact_6694_sum_OatLeastAtMost__rev,axiom,
% 0.33/0.64 ! [G: nat > nat,N2: nat,M: nat] :
% 0.33/0.64 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
% 0.33/0.64 = ( groups3542108847815614940at_nat
% 0.33/0.64 @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I3 ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.atLeastAtMost_rev
% 0.33/0.64 thf(fact_6695_sum_OatLeastAtMost__rev,axiom,
% 0.33/0.64 ! [G: nat > real,N2: nat,M: nat] :
% 0.33/0.64 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
% 0.33/0.64 = ( groups6591440286371151544t_real
% 0.33/0.64 @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I3 ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.atLeastAtMost_rev
% 0.33/0.64 thf(fact_6696_numeral__code_I3_J,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( numera6690914467698888265omplex @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) @ one_one_complex ) ) ).
% 0.33/0.64
% 0.33/0.64 % numeral_code(3)
% 0.33/0.64 thf(fact_6697_numeral__code_I3_J,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( numeral_numeral_real @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) @ one_one_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % numeral_code(3)
% 0.33/0.64 thf(fact_6698_numeral__code_I3_J,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( numeral_numeral_rat @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) @ one_one_rat ) ) ).
% 0.33/0.64
% 0.33/0.64 % numeral_code(3)
% 0.33/0.64 thf(fact_6699_numeral__code_I3_J,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) @ one_one_nat ) ) ).
% 0.33/0.64
% 0.33/0.64 % numeral_code(3)
% 0.33/0.64 thf(fact_6700_numeral__code_I3_J,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( numeral_numeral_int @ ( bit1 @ N2 ) )
% 0.33/0.64 = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) @ one_one_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % numeral_code(3)
% 0.33/0.64 thf(fact_6701_power__numeral__odd,axiom,
% 0.33/0.64 ! [Z: complex,W: num] :
% 0.33/0.64 ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 0.33/0.64 = ( times_times_complex @ ( times_times_complex @ Z @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_numeral_odd
% 0.33/0.64 thf(fact_6702_power__numeral__odd,axiom,
% 0.33/0.64 ! [Z: real,W: num] :
% 0.33/0.64 ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 0.33/0.64 = ( times_times_real @ ( times_times_real @ Z @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_numeral_odd
% 0.33/0.64 thf(fact_6703_power__numeral__odd,axiom,
% 0.33/0.64 ! [Z: rat,W: num] :
% 0.33/0.64 ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 0.33/0.64 = ( times_times_rat @ ( times_times_rat @ Z @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_numeral_odd
% 0.33/0.64 thf(fact_6704_power__numeral__odd,axiom,
% 0.33/0.64 ! [Z: nat,W: num] :
% 0.33/0.64 ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 0.33/0.64 = ( times_times_nat @ ( times_times_nat @ Z @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_numeral_odd
% 0.33/0.64 thf(fact_6705_power__numeral__odd,axiom,
% 0.33/0.64 ! [Z: int,W: num] :
% 0.33/0.64 ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 0.33/0.64 = ( times_times_int @ ( times_times_int @ Z @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_numeral_odd
% 0.33/0.64 thf(fact_6706_sum__roots__unity,axiom,
% 0.33/0.64 ! [N2: nat] :
% 0.33/0.64 ( ( ord_less_nat @ one_one_nat @ N2 )
% 0.33/0.64 => ( ( groups7754918857620584856omplex
% 0.33/0.64 @ ^ [X2: complex] : X2
% 0.33/0.64 @ ( collect_complex
% 0.33/0.64 @ ^ [Z3: complex] :
% 0.33/0.64 ( ( power_power_complex @ Z3 @ N2 )
% 0.33/0.64 = one_one_complex ) ) )
% 0.33/0.64 = zero_zero_complex ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_roots_unity
% 0.33/0.64 thf(fact_6707_sum__nth__roots,axiom,
% 0.33/0.64 ! [N2: nat,C: complex] :
% 0.33/0.64 ( ( ord_less_nat @ one_one_nat @ N2 )
% 0.33/0.64 => ( ( groups7754918857620584856omplex
% 0.33/0.64 @ ^ [X2: complex] : X2
% 0.33/0.64 @ ( collect_complex
% 0.33/0.64 @ ^ [Z3: complex] :
% 0.33/0.64 ( ( power_power_complex @ Z3 @ N2 )
% 0.33/0.64 = C ) ) )
% 0.33/0.64 = zero_zero_complex ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_nth_roots
% 0.33/0.64 thf(fact_6708_abs__add__one__gt__zero,axiom,
% 0.33/0.64 ! [X: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_add_one_gt_zero
% 0.33/0.64 thf(fact_6709_abs__add__one__gt__zero,axiom,
% 0.33/0.64 ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_add_one_gt_zero
% 0.33/0.64 thf(fact_6710_abs__add__one__gt__zero,axiom,
% 0.33/0.64 ! [X: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_add_one_gt_zero
% 0.33/0.64 thf(fact_6711_abs__add__one__gt__zero,axiom,
% 0.33/0.64 ! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_add_one_gt_zero
% 0.33/0.64 thf(fact_6712_sum__diff__nat,axiom,
% 0.33/0.64 ! [B3: set_int,A2: set_int,F: int > nat] :
% 0.33/0.64 ( ( finite_finite_int @ B3 )
% 0.33/0.64 => ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.33/0.64 => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ B3 ) )
% 0.33/0.64 = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B3 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_diff_nat
% 0.33/0.64 thf(fact_6713_sum__diff__nat,axiom,
% 0.33/0.64 ! [B3: set_complex,A2: set_complex,F: complex > nat] :
% 0.33/0.64 ( ( finite3207457112153483333omplex @ B3 )
% 0.33/0.64 => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 0.33/0.64 => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 0.33/0.64 = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B3 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_diff_nat
% 0.33/0.64 thf(fact_6714_sum__diff__nat,axiom,
% 0.33/0.64 ! [B3: set_nat,A2: set_nat,F: nat > nat] :
% 0.33/0.64 ( ( finite_finite_nat @ B3 )
% 0.33/0.64 => ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 0.33/0.64 => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 0.33/0.64 = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ F @ B3 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_diff_nat
% 0.33/0.64 thf(fact_6715_sum__shift__lb__Suc0__0,axiom,
% 0.33/0.64 ! [F: nat > complex,K: nat] :
% 0.33/0.64 ( ( ( F @ zero_zero_nat )
% 0.33/0.64 = zero_zero_complex )
% 0.33/0.64 => ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 0.33/0.64 = ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_shift_lb_Suc0_0
% 0.33/0.64 thf(fact_6716_sum__shift__lb__Suc0__0,axiom,
% 0.33/0.64 ! [F: nat > rat,K: nat] :
% 0.33/0.64 ( ( ( F @ zero_zero_nat )
% 0.33/0.64 = zero_zero_rat )
% 0.33/0.64 => ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 0.33/0.64 = ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_shift_lb_Suc0_0
% 0.33/0.64 thf(fact_6717_sum__shift__lb__Suc0__0,axiom,
% 0.33/0.64 ! [F: nat > int,K: nat] :
% 0.33/0.64 ( ( ( F @ zero_zero_nat )
% 0.33/0.64 = zero_zero_int )
% 0.33/0.64 => ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 0.33/0.64 = ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_shift_lb_Suc0_0
% 0.33/0.64 thf(fact_6718_sum__shift__lb__Suc0__0,axiom,
% 0.33/0.64 ! [F: nat > nat,K: nat] :
% 0.33/0.64 ( ( ( F @ zero_zero_nat )
% 0.33/0.64 = zero_zero_nat )
% 0.33/0.64 => ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 0.33/0.64 = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_shift_lb_Suc0_0
% 0.33/0.64 thf(fact_6719_sum__shift__lb__Suc0__0,axiom,
% 0.33/0.64 ! [F: nat > real,K: nat] :
% 0.33/0.64 ( ( ( F @ zero_zero_nat )
% 0.33/0.64 = zero_zero_real )
% 0.33/0.64 => ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 0.33/0.64 = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_shift_lb_Suc0_0
% 0.33/0.64 thf(fact_6720_sum_OatLeast0__atMost__Suc,axiom,
% 0.33/0.64 ! [G: nat > rat,N2: nat] :
% 0.33/0.64 ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
% 0.33/0.64 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.atLeast0_atMost_Suc
% 0.33/0.64 thf(fact_6721_sum_OatLeast0__atMost__Suc,axiom,
% 0.33/0.64 ! [G: nat > int,N2: nat] :
% 0.33/0.64 ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
% 0.33/0.64 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.atLeast0_atMost_Suc
% 0.33/0.64 thf(fact_6722_sum_OatLeast0__atMost__Suc,axiom,
% 0.33/0.64 ! [G: nat > nat,N2: nat] :
% 0.33/0.64 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
% 0.33/0.64 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.atLeast0_atMost_Suc
% 0.33/0.64 thf(fact_6723_sum_OatLeast0__atMost__Suc,axiom,
% 0.33/0.64 ! [G: nat > real,N2: nat] :
% 0.33/0.64 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
% 0.33/0.64 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.atLeast0_atMost_Suc
% 0.33/0.64 thf(fact_6724_sum_OatLeast__Suc__atMost,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,G: nat > rat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.64 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.64 = ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.atLeast_Suc_atMost
% 0.33/0.64 thf(fact_6725_sum_OatLeast__Suc__atMost,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,G: nat > int] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.64 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.64 = ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.atLeast_Suc_atMost
% 0.33/0.64 thf(fact_6726_sum_OatLeast__Suc__atMost,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,G: nat > nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.64 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.64 = ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.atLeast_Suc_atMost
% 0.33/0.64 thf(fact_6727_sum_OatLeast__Suc__atMost,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,G: nat > real] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.64 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.64 = ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.atLeast_Suc_atMost
% 0.33/0.64 thf(fact_6728_sum_Onat__ivl__Suc_H,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,G: nat > rat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 0.33/0.64 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.33/0.64 = ( plus_plus_rat @ ( G @ ( suc @ N2 ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.nat_ivl_Suc'
% 0.33/0.64 thf(fact_6729_sum_Onat__ivl__Suc_H,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,G: nat > int] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 0.33/0.64 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.33/0.64 = ( plus_plus_int @ ( G @ ( suc @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.nat_ivl_Suc'
% 0.33/0.64 thf(fact_6730_sum_Onat__ivl__Suc_H,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,G: nat > nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 0.33/0.64 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.33/0.64 = ( plus_plus_nat @ ( G @ ( suc @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.nat_ivl_Suc'
% 0.33/0.64 thf(fact_6731_sum_Onat__ivl__Suc_H,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,G: nat > real] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 0.33/0.64 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.33/0.64 = ( plus_plus_real @ ( G @ ( suc @ N2 ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.nat_ivl_Suc'
% 0.33/0.64 thf(fact_6732_numeral__Bit1__div__2,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = ( numeral_numeral_nat @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % numeral_Bit1_div_2
% 0.33/0.64 thf(fact_6733_numeral__Bit1__div__2,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.64 = ( numeral_numeral_int @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % numeral_Bit1_div_2
% 0.33/0.64 thf(fact_6734_odd__numeral,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % odd_numeral
% 0.33/0.64 thf(fact_6735_odd__numeral,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % odd_numeral
% 0.33/0.64 thf(fact_6736_odd__numeral,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % odd_numeral
% 0.33/0.64 thf(fact_6737_cong__exp__iff__simps_I3_J,axiom,
% 0.33/0.64 ! [N2: num,Q2: num] :
% 0.33/0.64 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.33/0.64 != zero_zero_nat ) ).
% 0.33/0.64
% 0.33/0.64 % cong_exp_iff_simps(3)
% 0.33/0.64 thf(fact_6738_cong__exp__iff__simps_I3_J,axiom,
% 0.33/0.64 ! [N2: num,Q2: num] :
% 0.33/0.64 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.33/0.64 != zero_zero_int ) ).
% 0.33/0.64
% 0.33/0.64 % cong_exp_iff_simps(3)
% 0.33/0.64 thf(fact_6739_cong__exp__iff__simps_I3_J,axiom,
% 0.33/0.64 ! [N2: num,Q2: num] :
% 0.33/0.64 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.33/0.64 != zero_z3403309356797280102nteger ) ).
% 0.33/0.64
% 0.33/0.64 % cong_exp_iff_simps(3)
% 0.33/0.64 thf(fact_6740_power3__eq__cube,axiom,
% 0.33/0.64 ! [A: complex] :
% 0.33/0.64 ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.33/0.64 = ( times_times_complex @ ( times_times_complex @ A @ A ) @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % power3_eq_cube
% 0.33/0.64 thf(fact_6741_power3__eq__cube,axiom,
% 0.33/0.64 ! [A: real] :
% 0.33/0.64 ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.33/0.64 = ( times_times_real @ ( times_times_real @ A @ A ) @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % power3_eq_cube
% 0.33/0.64 thf(fact_6742_power3__eq__cube,axiom,
% 0.33/0.64 ! [A: rat] :
% 0.33/0.64 ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.33/0.64 = ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % power3_eq_cube
% 0.33/0.64 thf(fact_6743_power3__eq__cube,axiom,
% 0.33/0.64 ! [A: nat] :
% 0.33/0.64 ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.33/0.64 = ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % power3_eq_cube
% 0.33/0.64 thf(fact_6744_power3__eq__cube,axiom,
% 0.33/0.64 ! [A: int] :
% 0.33/0.64 ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.33/0.64 = ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).
% 0.33/0.64
% 0.33/0.64 % power3_eq_cube
% 0.33/0.64 thf(fact_6745_numeral__3__eq__3,axiom,
% 0.33/0.64 ( ( numeral_numeral_nat @ ( bit1 @ one ) )
% 0.33/0.64 = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % numeral_3_eq_3
% 0.33/0.64 thf(fact_6746_Suc3__eq__add__3,axiom,
% 0.33/0.64 ! [N2: nat] :
% 0.33/0.64 ( ( suc @ ( suc @ ( suc @ N2 ) ) )
% 0.33/0.64 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % Suc3_eq_add_3
% 0.33/0.64 thf(fact_6747_lemma__interval,axiom,
% 0.33/0.64 ! [A: real,X: real,B: real] :
% 0.33/0.64 ( ( ord_less_real @ A @ X )
% 0.33/0.64 => ( ( ord_less_real @ X @ B )
% 0.33/0.64 => ? [D4: real] :
% 0.33/0.64 ( ( ord_less_real @ zero_zero_real @ D4 )
% 0.33/0.64 & ! [Y5: real] :
% 0.33/0.64 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D4 )
% 0.33/0.64 => ( ( ord_less_eq_real @ A @ Y5 )
% 0.33/0.64 & ( ord_less_eq_real @ Y5 @ B ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % lemma_interval
% 0.33/0.64 thf(fact_6748_sum_OSuc__reindex__ivl,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,G: nat > rat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.64 => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
% 0.33/0.64 = ( plus_plus_rat @ ( G @ M )
% 0.33/0.64 @ ( groups2906978787729119204at_rat
% 0.33/0.64 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.Suc_reindex_ivl
% 0.33/0.64 thf(fact_6749_sum_OSuc__reindex__ivl,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,G: nat > int] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.64 => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
% 0.33/0.64 = ( plus_plus_int @ ( G @ M )
% 0.33/0.64 @ ( groups3539618377306564664at_int
% 0.33/0.64 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.Suc_reindex_ivl
% 0.33/0.64 thf(fact_6750_sum_OSuc__reindex__ivl,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,G: nat > nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.64 => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
% 0.33/0.64 = ( plus_plus_nat @ ( G @ M )
% 0.33/0.64 @ ( groups3542108847815614940at_nat
% 0.33/0.64 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.Suc_reindex_ivl
% 0.33/0.64 thf(fact_6751_sum_OSuc__reindex__ivl,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,G: nat > real] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.64 => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
% 0.33/0.64 = ( plus_plus_real @ ( G @ M )
% 0.33/0.64 @ ( groups6591440286371151544t_real
% 0.33/0.64 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.Suc_reindex_ivl
% 0.33/0.64 thf(fact_6752_sum__Suc__diff,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,F: nat > rat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 0.33/0.64 => ( ( groups2906978787729119204at_rat
% 0.33/0.64 @ ^ [I3: nat] : ( minus_minus_rat @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.64 = ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_Suc_diff
% 0.33/0.64 thf(fact_6753_sum__Suc__diff,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,F: nat > int] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 0.33/0.64 => ( ( groups3539618377306564664at_int
% 0.33/0.64 @ ^ [I3: nat] : ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.64 = ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_Suc_diff
% 0.33/0.64 thf(fact_6754_sum__Suc__diff,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,F: nat > real] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 0.33/0.64 => ( ( groups6591440286371151544t_real
% 0.33/0.64 @ ^ [I3: nat] : ( minus_minus_real @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.64 = ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_Suc_diff
% 0.33/0.64 thf(fact_6755_mod__exhaust__less__4,axiom,
% 0.33/0.64 ! [M: nat] :
% 0.33/0.64 ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.33/0.64 = zero_zero_nat )
% 0.33/0.64 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.33/0.64 = one_one_nat )
% 0.33/0.64 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.33/0.64 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.33/0.64 = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mod_exhaust_less_4
% 0.33/0.64 thf(fact_6756_abs__le__square__iff,axiom,
% 0.33/0.64 ! [X: code_integer,Y: code_integer] :
% 0.33/0.64 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ ( abs_abs_Code_integer @ Y ) )
% 0.33/0.64 = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_le_square_iff
% 0.33/0.64 thf(fact_6757_abs__le__square__iff,axiom,
% 0.33/0.64 ! [X: real,Y: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) )
% 0.33/0.64 = ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_le_square_iff
% 0.33/0.64 thf(fact_6758_abs__le__square__iff,axiom,
% 0.33/0.64 ! [X: rat,Y: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ ( abs_abs_rat @ Y ) )
% 0.33/0.64 = ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_le_square_iff
% 0.33/0.64 thf(fact_6759_abs__le__square__iff,axiom,
% 0.33/0.64 ! [X: int,Y: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) )
% 0.33/0.64 = ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_le_square_iff
% 0.33/0.64 thf(fact_6760_abs__square__eq__1,axiom,
% 0.33/0.64 ! [X: code_integer] :
% 0.33/0.64 ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = one_one_Code_integer )
% 0.33/0.64 = ( ( abs_abs_Code_integer @ X )
% 0.33/0.64 = one_one_Code_integer ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_square_eq_1
% 0.33/0.64 thf(fact_6761_abs__square__eq__1,axiom,
% 0.33/0.64 ! [X: rat] :
% 0.33/0.64 ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = one_one_rat )
% 0.33/0.64 = ( ( abs_abs_rat @ X )
% 0.33/0.64 = one_one_rat ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_square_eq_1
% 0.33/0.64 thf(fact_6762_abs__square__eq__1,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = one_one_real )
% 0.33/0.64 = ( ( abs_abs_real @ X )
% 0.33/0.64 = one_one_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_square_eq_1
% 0.33/0.64 thf(fact_6763_abs__square__eq__1,axiom,
% 0.33/0.64 ! [X: int] :
% 0.33/0.64 ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 = one_one_int )
% 0.33/0.64 = ( ( abs_abs_int @ X )
% 0.33/0.64 = one_one_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_square_eq_1
% 0.33/0.64 thf(fact_6764_num_Osize_I6_J,axiom,
% 0.33/0.64 ! [X32: num] :
% 0.33/0.64 ( ( size_size_num @ ( bit1 @ X32 ) )
% 0.33/0.64 = ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % num.size(6)
% 0.33/0.64 thf(fact_6765_power__even__abs,axiom,
% 0.33/0.64 ! [N2: nat,A: rat] :
% 0.33/0.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.64 => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ N2 )
% 0.33/0.64 = ( power_power_rat @ A @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_even_abs
% 0.33/0.64 thf(fact_6766_power__even__abs,axiom,
% 0.33/0.64 ! [N2: nat,A: code_integer] :
% 0.33/0.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.64 => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N2 )
% 0.33/0.64 = ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_even_abs
% 0.33/0.64 thf(fact_6767_power__even__abs,axiom,
% 0.33/0.64 ! [N2: nat,A: real] :
% 0.33/0.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.64 => ( ( power_power_real @ ( abs_abs_real @ A ) @ N2 )
% 0.33/0.64 = ( power_power_real @ A @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_even_abs
% 0.33/0.64 thf(fact_6768_power__even__abs,axiom,
% 0.33/0.64 ! [N2: nat,A: int] :
% 0.33/0.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.64 => ( ( power_power_int @ ( abs_abs_int @ A ) @ N2 )
% 0.33/0.64 = ( power_power_int @ A @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_even_abs
% 0.33/0.64 thf(fact_6769_sum_Oub__add__nat,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,G: nat > rat,P4: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
% 0.33/0.64 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
% 0.33/0.64 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.ub_add_nat
% 0.33/0.64 thf(fact_6770_sum_Oub__add__nat,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,G: nat > int,P4: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
% 0.33/0.64 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
% 0.33/0.64 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.ub_add_nat
% 0.33/0.64 thf(fact_6771_sum_Oub__add__nat,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,G: nat > nat,P4: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
% 0.33/0.64 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
% 0.33/0.64 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.ub_add_nat
% 0.33/0.64 thf(fact_6772_sum_Oub__add__nat,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,G: nat > real,P4: nat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
% 0.33/0.64 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
% 0.33/0.64 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.ub_add_nat
% 0.33/0.64 thf(fact_6773_cong__exp__iff__simps_I11_J,axiom,
% 0.33/0.64 ! [M: num,Q2: num] :
% 0.33/0.64 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.33/0.64 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 0.33/0.64 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.33/0.64 = zero_zero_nat ) ) ).
% 0.33/0.64
% 0.33/0.64 % cong_exp_iff_simps(11)
% 0.33/0.64 thf(fact_6774_cong__exp__iff__simps_I11_J,axiom,
% 0.33/0.64 ! [M: num,Q2: num] :
% 0.33/0.64 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.33/0.64 = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 0.33/0.64 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 0.33/0.64 = zero_zero_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % cong_exp_iff_simps(11)
% 0.33/0.64 thf(fact_6775_cong__exp__iff__simps_I11_J,axiom,
% 0.33/0.64 ! [M: num,Q2: num] :
% 0.33/0.64 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.33/0.64 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 0.33/0.64 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.33/0.64 = zero_z3403309356797280102nteger ) ) ).
% 0.33/0.64
% 0.33/0.64 % cong_exp_iff_simps(11)
% 0.33/0.64 thf(fact_6776_cong__exp__iff__simps_I7_J,axiom,
% 0.33/0.64 ! [Q2: num,N2: num] :
% 0.33/0.64 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.33/0.64 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 0.33/0.64 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.33/0.64 = zero_zero_nat ) ) ).
% 0.33/0.64
% 0.33/0.64 % cong_exp_iff_simps(7)
% 0.33/0.64 thf(fact_6777_cong__exp__iff__simps_I7_J,axiom,
% 0.33/0.64 ! [Q2: num,N2: num] :
% 0.33/0.64 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.33/0.64 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 0.33/0.64 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q2 ) )
% 0.33/0.64 = zero_zero_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % cong_exp_iff_simps(7)
% 0.33/0.64 thf(fact_6778_cong__exp__iff__simps_I7_J,axiom,
% 0.33/0.64 ! [Q2: num,N2: num] :
% 0.33/0.64 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.33/0.64 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 0.33/0.64 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.33/0.64 = zero_z3403309356797280102nteger ) ) ).
% 0.33/0.64
% 0.33/0.64 % cong_exp_iff_simps(7)
% 0.33/0.64 thf(fact_6779_Suc__div__eq__add3__div,axiom,
% 0.33/0.64 ! [M: nat,N2: nat] :
% 0.33/0.64 ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 )
% 0.33/0.64 = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % Suc_div_eq_add3_div
% 0.33/0.64 thf(fact_6780_Suc__mod__eq__add3__mod,axiom,
% 0.33/0.64 ! [M: nat,N2: nat] :
% 0.33/0.64 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 )
% 0.33/0.64 = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N2 ) ) ).
% 0.33/0.64
% 0.33/0.64 % Suc_mod_eq_add3_mod
% 0.33/0.64 thf(fact_6781_set__encode__def,axiom,
% 0.33/0.64 ( nat_set_encode
% 0.33/0.64 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % set_encode_def
% 0.33/0.64 thf(fact_6782_divmod__int__def,axiom,
% 0.33/0.64 ( unique5052692396658037445od_int
% 0.33/0.64 = ( ^ [M6: num,N: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_int_def
% 0.33/0.64 thf(fact_6783_abs__sqrt__wlog,axiom,
% 0.33/0.64 ! [P: code_integer > code_integer > $o,X: code_integer] :
% 0.33/0.64 ( ! [X4: code_integer] :
% 0.33/0.64 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X4 )
% 0.33/0.64 => ( P @ X4 @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.64 => ( P @ ( abs_abs_Code_integer @ X ) @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_sqrt_wlog
% 0.33/0.64 thf(fact_6784_abs__sqrt__wlog,axiom,
% 0.33/0.64 ! [P: real > real > $o,X: real] :
% 0.33/0.64 ( ! [X4: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.33/0.64 => ( P @ X4 @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.64 => ( P @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_sqrt_wlog
% 0.33/0.64 thf(fact_6785_abs__sqrt__wlog,axiom,
% 0.33/0.64 ! [P: rat > rat > $o,X: rat] :
% 0.33/0.64 ( ! [X4: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 0.33/0.64 => ( P @ X4 @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.64 => ( P @ ( abs_abs_rat @ X ) @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_sqrt_wlog
% 0.33/0.64 thf(fact_6786_abs__sqrt__wlog,axiom,
% 0.33/0.64 ! [P: int > int > $o,X: int] :
% 0.33/0.64 ( ! [X4: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 0.33/0.64 => ( P @ X4 @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.64 => ( P @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_sqrt_wlog
% 0.33/0.64 thf(fact_6787_power2__le__iff__abs__le,axiom,
% 0.33/0.64 ! [Y: code_integer,X: code_integer] :
% 0.33/0.64 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
% 0.33/0.64 => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.64 = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power2_le_iff_abs_le
% 0.33/0.64 thf(fact_6788_power2__le__iff__abs__le,axiom,
% 0.33/0.64 ! [Y: real,X: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.64 => ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.64 = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power2_le_iff_abs_le
% 0.33/0.64 thf(fact_6789_power2__le__iff__abs__le,axiom,
% 0.33/0.64 ! [Y: rat,X: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 0.33/0.64 => ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.64 = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power2_le_iff_abs_le
% 0.33/0.64 thf(fact_6790_power2__le__iff__abs__le,axiom,
% 0.33/0.64 ! [Y: int,X: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.64 => ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.64 = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ Y ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power2_le_iff_abs_le
% 0.33/0.64 thf(fact_6791_abs__square__le__1,axiom,
% 0.33/0.64 ! [X: code_integer] :
% 0.33/0.64 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 0.33/0.64 = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_square_le_1
% 0.33/0.64 thf(fact_6792_abs__square__le__1,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 0.33/0.64 = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_square_le_1
% 0.33/0.64 thf(fact_6793_abs__square__le__1,axiom,
% 0.33/0.64 ! [X: rat] :
% 0.33/0.64 ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 0.33/0.64 = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_square_le_1
% 0.33/0.64 thf(fact_6794_abs__square__le__1,axiom,
% 0.33/0.64 ! [X: int] :
% 0.33/0.64 ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 0.33/0.64 = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_square_le_1
% 0.33/0.64 thf(fact_6795_abs__square__less__1,axiom,
% 0.33/0.64 ! [X: code_integer] :
% 0.33/0.64 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 0.33/0.64 = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_square_less_1
% 0.33/0.64 thf(fact_6796_abs__square__less__1,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 0.33/0.64 = ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_square_less_1
% 0.33/0.64 thf(fact_6797_abs__square__less__1,axiom,
% 0.33/0.64 ! [X: rat] :
% 0.33/0.64 ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 0.33/0.64 = ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_square_less_1
% 0.33/0.64 thf(fact_6798_abs__square__less__1,axiom,
% 0.33/0.64 ! [X: int] :
% 0.33/0.64 ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 0.33/0.64 = ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_square_less_1
% 0.33/0.64 thf(fact_6799_divmod__def,axiom,
% 0.33/0.64 ( unique5052692396658037445od_int
% 0.33/0.64 = ( ^ [M6: num,N: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_def
% 0.33/0.64 thf(fact_6800_divmod__def,axiom,
% 0.33/0.64 ( unique5055182867167087721od_nat
% 0.33/0.64 = ( ^ [M6: num,N: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_def
% 0.33/0.64 thf(fact_6801_divmod__def,axiom,
% 0.33/0.64 ( unique3479559517661332726nteger
% 0.33/0.64 = ( ^ [M6: num,N: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_def
% 0.33/0.64 thf(fact_6802_divmod_H__nat__def,axiom,
% 0.33/0.64 ( unique5055182867167087721od_nat
% 0.33/0.64 = ( ^ [M6: num,N: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod'_nat_def
% 0.33/0.64 thf(fact_6803_power__mono__even,axiom,
% 0.33/0.64 ! [N2: nat,A: code_integer,B: code_integer] :
% 0.33/0.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.64 => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
% 0.33/0.64 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( power_8256067586552552935nteger @ B @ N2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_mono_even
% 0.33/0.64 thf(fact_6804_power__mono__even,axiom,
% 0.33/0.64 ! [N2: nat,A: real,B: real] :
% 0.33/0.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.64 => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) )
% 0.33/0.64 => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_mono_even
% 0.33/0.64 thf(fact_6805_power__mono__even,axiom,
% 0.33/0.64 ! [N2: nat,A: rat,B: rat] :
% 0.33/0.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.64 => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
% 0.33/0.64 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_mono_even
% 0.33/0.64 thf(fact_6806_power__mono__even,axiom,
% 0.33/0.64 ! [N2: nat,A: int,B: int] :
% 0.33/0.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.64 => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
% 0.33/0.64 => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % power_mono_even
% 0.33/0.64 thf(fact_6807_convex__sum__bound__le,axiom,
% 0.33/0.64 ! [I6: set_nat,X: nat > code_integer,A: nat > code_integer,B: code_integer,Delta: code_integer] :
% 0.33/0.64 ( ! [I4: nat] :
% 0.33/0.64 ( ( member_nat @ I4 @ I6 )
% 0.33/0.64 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I4 ) ) )
% 0.33/0.64 => ( ( ( groups7501900531339628137nteger @ X @ I6 )
% 0.33/0.64 = one_one_Code_integer )
% 0.33/0.64 => ( ! [I4: nat] :
% 0.33/0.64 ( ( member_nat @ I4 @ I6 )
% 0.33/0.64 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I4 ) @ B ) ) @ Delta ) )
% 0.33/0.64 => ( ord_le3102999989581377725nteger
% 0.33/0.64 @ ( abs_abs_Code_integer
% 0.33/0.64 @ ( minus_8373710615458151222nteger
% 0.33/0.64 @ ( groups7501900531339628137nteger
% 0.33/0.64 @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
% 0.33/0.64 @ I6 )
% 0.33/0.64 @ B ) )
% 0.33/0.64 @ Delta ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % convex_sum_bound_le
% 0.33/0.64 thf(fact_6808_convex__sum__bound__le,axiom,
% 0.33/0.64 ! [I6: set_real,X: real > code_integer,A: real > code_integer,B: code_integer,Delta: code_integer] :
% 0.33/0.64 ( ! [I4: real] :
% 0.33/0.64 ( ( member_real @ I4 @ I6 )
% 0.33/0.64 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I4 ) ) )
% 0.33/0.64 => ( ( ( groups7713935264441627589nteger @ X @ I6 )
% 0.33/0.64 = one_one_Code_integer )
% 0.33/0.64 => ( ! [I4: real] :
% 0.33/0.64 ( ( member_real @ I4 @ I6 )
% 0.33/0.64 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I4 ) @ B ) ) @ Delta ) )
% 0.33/0.64 => ( ord_le3102999989581377725nteger
% 0.33/0.64 @ ( abs_abs_Code_integer
% 0.33/0.64 @ ( minus_8373710615458151222nteger
% 0.33/0.64 @ ( groups7713935264441627589nteger
% 0.33/0.64 @ ^ [I3: real] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
% 0.33/0.64 @ I6 )
% 0.33/0.64 @ B ) )
% 0.33/0.64 @ Delta ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % convex_sum_bound_le
% 0.33/0.64 thf(fact_6809_convex__sum__bound__le,axiom,
% 0.33/0.64 ! [I6: set_int,X: int > code_integer,A: int > code_integer,B: code_integer,Delta: code_integer] :
% 0.33/0.64 ( ! [I4: int] :
% 0.33/0.64 ( ( member_int @ I4 @ I6 )
% 0.33/0.64 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I4 ) ) )
% 0.33/0.64 => ( ( ( groups7873554091576472773nteger @ X @ I6 )
% 0.33/0.64 = one_one_Code_integer )
% 0.33/0.64 => ( ! [I4: int] :
% 0.33/0.64 ( ( member_int @ I4 @ I6 )
% 0.33/0.64 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I4 ) @ B ) ) @ Delta ) )
% 0.33/0.64 => ( ord_le3102999989581377725nteger
% 0.33/0.64 @ ( abs_abs_Code_integer
% 0.33/0.64 @ ( minus_8373710615458151222nteger
% 0.33/0.64 @ ( groups7873554091576472773nteger
% 0.33/0.64 @ ^ [I3: int] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
% 0.33/0.64 @ I6 )
% 0.33/0.64 @ B ) )
% 0.33/0.64 @ Delta ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % convex_sum_bound_le
% 0.33/0.64 thf(fact_6810_convex__sum__bound__le,axiom,
% 0.33/0.64 ! [I6: set_complex,X: complex > code_integer,A: complex > code_integer,B: code_integer,Delta: code_integer] :
% 0.33/0.64 ( ! [I4: complex] :
% 0.33/0.64 ( ( member_complex @ I4 @ I6 )
% 0.33/0.64 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I4 ) ) )
% 0.33/0.64 => ( ( ( groups6621422865394947399nteger @ X @ I6 )
% 0.33/0.64 = one_one_Code_integer )
% 0.33/0.64 => ( ! [I4: complex] :
% 0.33/0.64 ( ( member_complex @ I4 @ I6 )
% 0.33/0.64 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I4 ) @ B ) ) @ Delta ) )
% 0.33/0.64 => ( ord_le3102999989581377725nteger
% 0.33/0.64 @ ( abs_abs_Code_integer
% 0.33/0.64 @ ( minus_8373710615458151222nteger
% 0.33/0.64 @ ( groups6621422865394947399nteger
% 0.33/0.64 @ ^ [I3: complex] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
% 0.33/0.64 @ I6 )
% 0.33/0.64 @ B ) )
% 0.33/0.64 @ Delta ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % convex_sum_bound_le
% 0.33/0.64 thf(fact_6811_convex__sum__bound__le,axiom,
% 0.33/0.64 ! [I6: set_real,X: real > real,A: real > real,B: real,Delta: real] :
% 0.33/0.64 ( ! [I4: real] :
% 0.33/0.64 ( ( member_real @ I4 @ I6 )
% 0.33/0.64 => ( ord_less_eq_real @ zero_zero_real @ ( X @ I4 ) ) )
% 0.33/0.64 => ( ( ( groups8097168146408367636l_real @ X @ I6 )
% 0.33/0.64 = one_one_real )
% 0.33/0.64 => ( ! [I4: real] :
% 0.33/0.64 ( ( member_real @ I4 @ I6 )
% 0.33/0.64 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I4 ) @ B ) ) @ Delta ) )
% 0.33/0.64 => ( ord_less_eq_real
% 0.33/0.64 @ ( abs_abs_real
% 0.33/0.64 @ ( minus_minus_real
% 0.33/0.64 @ ( groups8097168146408367636l_real
% 0.33/0.64 @ ^ [I3: real] : ( times_times_real @ ( A @ I3 ) @ ( X @ I3 ) )
% 0.33/0.64 @ I6 )
% 0.33/0.64 @ B ) )
% 0.33/0.64 @ Delta ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % convex_sum_bound_le
% 0.33/0.64 thf(fact_6812_convex__sum__bound__le,axiom,
% 0.33/0.64 ! [I6: set_int,X: int > real,A: int > real,B: real,Delta: real] :
% 0.33/0.64 ( ! [I4: int] :
% 0.33/0.64 ( ( member_int @ I4 @ I6 )
% 0.33/0.64 => ( ord_less_eq_real @ zero_zero_real @ ( X @ I4 ) ) )
% 0.33/0.64 => ( ( ( groups8778361861064173332t_real @ X @ I6 )
% 0.33/0.64 = one_one_real )
% 0.33/0.64 => ( ! [I4: int] :
% 0.33/0.64 ( ( member_int @ I4 @ I6 )
% 0.33/0.64 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I4 ) @ B ) ) @ Delta ) )
% 0.33/0.64 => ( ord_less_eq_real
% 0.33/0.64 @ ( abs_abs_real
% 0.33/0.64 @ ( minus_minus_real
% 0.33/0.64 @ ( groups8778361861064173332t_real
% 0.33/0.64 @ ^ [I3: int] : ( times_times_real @ ( A @ I3 ) @ ( X @ I3 ) )
% 0.33/0.64 @ I6 )
% 0.33/0.64 @ B ) )
% 0.33/0.64 @ Delta ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % convex_sum_bound_le
% 0.33/0.64 thf(fact_6813_convex__sum__bound__le,axiom,
% 0.33/0.64 ! [I6: set_complex,X: complex > real,A: complex > real,B: real,Delta: real] :
% 0.33/0.64 ( ! [I4: complex] :
% 0.33/0.64 ( ( member_complex @ I4 @ I6 )
% 0.33/0.64 => ( ord_less_eq_real @ zero_zero_real @ ( X @ I4 ) ) )
% 0.33/0.64 => ( ( ( groups5808333547571424918x_real @ X @ I6 )
% 0.33/0.64 = one_one_real )
% 0.33/0.64 => ( ! [I4: complex] :
% 0.33/0.64 ( ( member_complex @ I4 @ I6 )
% 0.33/0.64 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I4 ) @ B ) ) @ Delta ) )
% 0.33/0.64 => ( ord_less_eq_real
% 0.33/0.64 @ ( abs_abs_real
% 0.33/0.64 @ ( minus_minus_real
% 0.33/0.64 @ ( groups5808333547571424918x_real
% 0.33/0.64 @ ^ [I3: complex] : ( times_times_real @ ( A @ I3 ) @ ( X @ I3 ) )
% 0.33/0.64 @ I6 )
% 0.33/0.64 @ B ) )
% 0.33/0.64 @ Delta ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % convex_sum_bound_le
% 0.33/0.64 thf(fact_6814_convex__sum__bound__le,axiom,
% 0.33/0.64 ! [I6: set_nat,X: nat > rat,A: nat > rat,B: rat,Delta: rat] :
% 0.33/0.64 ( ! [I4: nat] :
% 0.33/0.64 ( ( member_nat @ I4 @ I6 )
% 0.33/0.64 => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I4 ) ) )
% 0.33/0.64 => ( ( ( groups2906978787729119204at_rat @ X @ I6 )
% 0.33/0.64 = one_one_rat )
% 0.33/0.64 => ( ! [I4: nat] :
% 0.33/0.64 ( ( member_nat @ I4 @ I6 )
% 0.33/0.64 => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I4 ) @ B ) ) @ Delta ) )
% 0.33/0.64 => ( ord_less_eq_rat
% 0.33/0.64 @ ( abs_abs_rat
% 0.33/0.64 @ ( minus_minus_rat
% 0.33/0.64 @ ( groups2906978787729119204at_rat
% 0.33/0.64 @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( X @ I3 ) )
% 0.33/0.64 @ I6 )
% 0.33/0.64 @ B ) )
% 0.33/0.64 @ Delta ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % convex_sum_bound_le
% 0.33/0.64 thf(fact_6815_convex__sum__bound__le,axiom,
% 0.33/0.64 ! [I6: set_real,X: real > rat,A: real > rat,B: rat,Delta: rat] :
% 0.33/0.64 ( ! [I4: real] :
% 0.33/0.64 ( ( member_real @ I4 @ I6 )
% 0.33/0.64 => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I4 ) ) )
% 0.33/0.64 => ( ( ( groups1300246762558778688al_rat @ X @ I6 )
% 0.33/0.64 = one_one_rat )
% 0.33/0.64 => ( ! [I4: real] :
% 0.33/0.64 ( ( member_real @ I4 @ I6 )
% 0.33/0.64 => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I4 ) @ B ) ) @ Delta ) )
% 0.33/0.64 => ( ord_less_eq_rat
% 0.33/0.64 @ ( abs_abs_rat
% 0.33/0.64 @ ( minus_minus_rat
% 0.33/0.64 @ ( groups1300246762558778688al_rat
% 0.33/0.64 @ ^ [I3: real] : ( times_times_rat @ ( A @ I3 ) @ ( X @ I3 ) )
% 0.33/0.64 @ I6 )
% 0.33/0.64 @ B ) )
% 0.33/0.64 @ Delta ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % convex_sum_bound_le
% 0.33/0.64 thf(fact_6816_convex__sum__bound__le,axiom,
% 0.33/0.64 ! [I6: set_int,X: int > rat,A: int > rat,B: rat,Delta: rat] :
% 0.33/0.64 ( ! [I4: int] :
% 0.33/0.64 ( ( member_int @ I4 @ I6 )
% 0.33/0.64 => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I4 ) ) )
% 0.33/0.64 => ( ( ( groups3906332499630173760nt_rat @ X @ I6 )
% 0.33/0.64 = one_one_rat )
% 0.33/0.64 => ( ! [I4: int] :
% 0.33/0.64 ( ( member_int @ I4 @ I6 )
% 0.33/0.64 => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I4 ) @ B ) ) @ Delta ) )
% 0.33/0.64 => ( ord_less_eq_rat
% 0.33/0.64 @ ( abs_abs_rat
% 0.33/0.64 @ ( minus_minus_rat
% 0.33/0.64 @ ( groups3906332499630173760nt_rat
% 0.33/0.64 @ ^ [I3: int] : ( times_times_rat @ ( A @ I3 ) @ ( X @ I3 ) )
% 0.33/0.64 @ I6 )
% 0.33/0.64 @ B ) )
% 0.33/0.64 @ Delta ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % convex_sum_bound_le
% 0.33/0.64 thf(fact_6817_sum__natinterval__diff,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,F: nat > complex] :
% 0.33/0.64 ( ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.64 => ( ( groups2073611262835488442omplex
% 0.33/0.64 @ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.64 = ( minus_minus_complex @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
% 0.33/0.64 & ( ~ ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.64 => ( ( groups2073611262835488442omplex
% 0.33/0.64 @ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.64 = zero_zero_complex ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_natinterval_diff
% 0.33/0.64 thf(fact_6818_sum__natinterval__diff,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,F: nat > rat] :
% 0.33/0.64 ( ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.64 => ( ( groups2906978787729119204at_rat
% 0.33/0.64 @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.64 = ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
% 0.33/0.64 & ( ~ ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.64 => ( ( groups2906978787729119204at_rat
% 0.33/0.64 @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.64 = zero_zero_rat ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_natinterval_diff
% 0.33/0.64 thf(fact_6819_sum__natinterval__diff,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,F: nat > int] :
% 0.33/0.64 ( ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.64 => ( ( groups3539618377306564664at_int
% 0.33/0.64 @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.64 = ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
% 0.33/0.64 & ( ~ ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.64 => ( ( groups3539618377306564664at_int
% 0.33/0.64 @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.64 = zero_zero_int ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_natinterval_diff
% 0.33/0.64 thf(fact_6820_sum__natinterval__diff,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,F: nat > real] :
% 0.33/0.64 ( ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.64 => ( ( groups6591440286371151544t_real
% 0.33/0.64 @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.64 = ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
% 0.33/0.64 & ( ~ ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.64 => ( ( groups6591440286371151544t_real
% 0.33/0.64 @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.64 = zero_zero_real ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_natinterval_diff
% 0.33/0.64 thf(fact_6821_sum__telescope_H_H,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,F: nat > rat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.64 => ( ( groups2906978787729119204at_rat
% 0.33/0.64 @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
% 0.33/0.64 = ( minus_minus_rat @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_telescope''
% 0.33/0.64 thf(fact_6822_sum__telescope_H_H,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,F: nat > int] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.64 => ( ( groups3539618377306564664at_int
% 0.33/0.64 @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
% 0.33/0.64 = ( minus_minus_int @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_telescope''
% 0.33/0.64 thf(fact_6823_sum__telescope_H_H,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,F: nat > real] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.64 => ( ( groups6591440286371151544t_real
% 0.33/0.64 @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
% 0.33/0.64 = ( minus_minus_real @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_telescope''
% 0.33/0.64 thf(fact_6824_divmod__nat__def,axiom,
% 0.33/0.64 ( divmod_nat
% 0.33/0.64 = ( ^ [M6: nat,N: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M6 @ N ) @ ( modulo_modulo_nat @ M6 @ N ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_nat_def
% 0.33/0.64 thf(fact_6825_mask__eq__sum__exp,axiom,
% 0.33/0.64 ! [N2: nat] :
% 0.33/0.64 ( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int )
% 0.33/0.64 = ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.64 @ ( collect_nat
% 0.33/0.64 @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mask_eq_sum_exp
% 0.33/0.64 thf(fact_6826_mask__eq__sum__exp,axiom,
% 0.33/0.64 ! [N2: nat] :
% 0.33/0.64 ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat )
% 0.33/0.64 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 @ ( collect_nat
% 0.33/0.64 @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mask_eq_sum_exp
% 0.33/0.64 thf(fact_6827_sum__gp__multiplied,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,X: complex] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.64 => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
% 0.33/0.64 = ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_gp_multiplied
% 0.33/0.64 thf(fact_6828_sum__gp__multiplied,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,X: rat] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.64 => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
% 0.33/0.64 = ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_gp_multiplied
% 0.33/0.64 thf(fact_6829_sum__gp__multiplied,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,X: int] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.64 => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
% 0.33/0.64 = ( minus_minus_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ ( suc @ N2 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_gp_multiplied
% 0.33/0.64 thf(fact_6830_sum__gp__multiplied,axiom,
% 0.33/0.64 ! [M: nat,N2: nat,X: real] :
% 0.33/0.64 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.64 => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
% 0.33/0.64 = ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N2 ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum_gp_multiplied
% 0.33/0.64 thf(fact_6831_sum_Oin__pairs,axiom,
% 0.33/0.64 ! [G: nat > rat,M: nat,N2: nat] :
% 0.33/0.64 ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 0.33/0.64 = ( groups2906978787729119204at_rat
% 0.33/0.64 @ ^ [I3: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.in_pairs
% 0.33/0.64 thf(fact_6832_sum_Oin__pairs,axiom,
% 0.33/0.64 ! [G: nat > int,M: nat,N2: nat] :
% 0.33/0.64 ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 0.33/0.64 = ( groups3539618377306564664at_int
% 0.33/0.64 @ ^ [I3: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.in_pairs
% 0.33/0.64 thf(fact_6833_sum_Oin__pairs,axiom,
% 0.33/0.64 ! [G: nat > nat,M: nat,N2: nat] :
% 0.33/0.64 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 0.33/0.64 = ( groups3542108847815614940at_nat
% 0.33/0.64 @ ^ [I3: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.in_pairs
% 0.33/0.64 thf(fact_6834_sum_Oin__pairs,axiom,
% 0.33/0.64 ! [G: nat > real,M: nat,N2: nat] :
% 0.33/0.64 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 0.33/0.64 = ( groups6591440286371151544t_real
% 0.33/0.64 @ ^ [I3: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % sum.in_pairs
% 0.33/0.64 thf(fact_6835_eq__diff__eq_H,axiom,
% 0.33/0.64 ! [X: real,Y: real,Z: real] :
% 0.33/0.64 ( ( X
% 0.33/0.64 = ( minus_minus_real @ Y @ Z ) )
% 0.33/0.64 = ( Y
% 0.33/0.64 = ( plus_plus_real @ X @ Z ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % eq_diff_eq'
% 0.33/0.64 thf(fact_6836_mask__eq__sum__exp__nat,axiom,
% 0.33/0.64 ! [N2: nat] :
% 0.33/0.64 ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( suc @ zero_zero_nat ) )
% 0.33/0.64 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.64 @ ( collect_nat
% 0.33/0.64 @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % mask_eq_sum_exp_nat
% 0.33/0.64 thf(fact_6837_gauss__sum__nat,axiom,
% 0.33/0.64 ! [N2: nat] :
% 0.33/0.64 ( ( groups3542108847815614940at_nat
% 0.33/0.64 @ ^ [X2: nat] : X2
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 0.33/0.64 = ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % gauss_sum_nat
% 0.33/0.64 thf(fact_6838_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
% 0.33/0.64 ! [X: real] :
% 0.33/0.64 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.64 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.64 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % abs_ln_one_plus_x_minus_x_bound_nonneg
% 0.33/0.64 thf(fact_6839_arith__series__nat,axiom,
% 0.33/0.64 ! [A: nat,D: nat,N2: nat] :
% 0.33/0.64 ( ( groups3542108847815614940at_nat
% 0.33/0.64 @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I3 @ D ) )
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 0.33/0.64 = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N2 @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % arith_series_nat
% 0.33/0.64 thf(fact_6840_Sum__Icc__nat,axiom,
% 0.33/0.64 ! [M: nat,N2: nat] :
% 0.33/0.64 ( ( groups3542108847815614940at_nat
% 0.33/0.64 @ ^ [X2: nat] : X2
% 0.33/0.64 @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.64 = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( plus_plus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % Sum_Icc_nat
% 0.33/0.64 thf(fact_6841_odd__mod__4__div__2,axiom,
% 0.33/0.64 ! [N2: nat] :
% 0.33/0.64 ( ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.33/0.64 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.33/0.64 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % odd_mod_4_div_2
% 0.33/0.64 thf(fact_6842_divmod__divmod__step,axiom,
% 0.33/0.64 ( unique5055182867167087721od_nat
% 0.33/0.64 = ( ^ [M6: num,N: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M6 @ N ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M6 ) ) @ ( unique5026877609467782581ep_nat @ N @ ( unique5055182867167087721od_nat @ M6 @ ( bit0 @ N ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_divmod_step
% 0.33/0.64 thf(fact_6843_divmod__divmod__step,axiom,
% 0.33/0.64 ( unique5052692396658037445od_int
% 0.33/0.64 = ( ^ [M6: num,N: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M6 @ N ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M6 ) ) @ ( unique5024387138958732305ep_int @ N @ ( unique5052692396658037445od_int @ M6 @ ( bit0 @ N ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_divmod_step
% 0.33/0.64 thf(fact_6844_divmod__divmod__step,axiom,
% 0.33/0.64 ( unique3479559517661332726nteger
% 0.33/0.64 = ( ^ [M6: num,N: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M6 @ N ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M6 ) ) @ ( unique4921790084139445826nteger @ N @ ( unique3479559517661332726nteger @ M6 @ ( bit0 @ N ) ) ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % divmod_divmod_step
% 0.33/0.64 thf(fact_6845_one__div__minus__numeral,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.33/0.64 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % one_div_minus_numeral
% 0.33/0.64 thf(fact_6846_minus__one__div__numeral,axiom,
% 0.33/0.64 ! [N2: num] :
% 0.33/0.64 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N2 ) )
% 0.33/0.64 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % minus_one_div_numeral
% 0.33/0.64 thf(fact_6847_signed__take__bit__numeral__minus__bit1,axiom,
% 0.33/0.64 ! [L2: num,K: num] :
% 0.33/0.64 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.33/0.64 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.33/0.64
% 0.33/0.64 % signed_take_bit_numeral_minus_bit1
% 0.33/0.64 thf(fact_6848_dbl__dec__simps_I4_J,axiom,
% 0.33/0.64 ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.64 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % dbl_dec_simps(4)
% 0.33/0.64 thf(fact_6849_dbl__dec__simps_I4_J,axiom,
% 0.33/0.64 ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.64 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % dbl_dec_simps(4)
% 0.33/0.64 thf(fact_6850_dbl__dec__simps_I4_J,axiom,
% 0.33/0.64 ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.33/0.64 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).
% 0.33/0.64
% 0.33/0.64 % dbl_dec_simps(4)
% 0.33/0.64 thf(fact_6851_dbl__dec__simps_I4_J,axiom,
% 0.33/0.64 ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.33/0.64 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_dec_simps(4)
% 0.33/0.65 thf(fact_6852_dbl__dec__simps_I4_J,axiom,
% 0.33/0.65 ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.65 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_dec_simps(4)
% 0.33/0.65 thf(fact_6853_signed__take__bit__numeral__bit1,axiom,
% 0.33/0.65 ! [L2: num,K: num] :
% 0.33/0.65 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.33/0.65 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % signed_take_bit_numeral_bit1
% 0.33/0.65 thf(fact_6854_arctan__double,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.65 => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X ) )
% 0.33/0.65 = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % arctan_double
% 0.33/0.65 thf(fact_6855_split__part,axiom,
% 0.33/0.65 ! [P: $o,Q: int > int > $o] :
% 0.33/0.65 ( ( produc4947309494688390418_int_o
% 0.33/0.65 @ ^ [A5: int,B5: int] :
% 0.33/0.65 ( P
% 0.33/0.65 & ( Q @ A5 @ B5 ) ) )
% 0.33/0.65 = ( ^ [Ab: product_prod_int_int] :
% 0.33/0.65 ( P
% 0.33/0.65 & ( produc4947309494688390418_int_o @ Q @ Ab ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % split_part
% 0.33/0.65 thf(fact_6856_zdvd1__eq,axiom,
% 0.33/0.65 ! [X: int] :
% 0.33/0.65 ( ( dvd_dvd_int @ X @ one_one_int )
% 0.33/0.65 = ( ( abs_abs_int @ X )
% 0.33/0.65 = one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % zdvd1_eq
% 0.33/0.65 thf(fact_6857_dbl__dec__simps_I3_J,axiom,
% 0.33/0.65 ( ( neg_nu6511756317524482435omplex @ one_one_complex )
% 0.33/0.65 = one_one_complex ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_dec_simps(3)
% 0.33/0.65 thf(fact_6858_dbl__dec__simps_I3_J,axiom,
% 0.33/0.65 ( ( neg_nu6075765906172075777c_real @ one_one_real )
% 0.33/0.65 = one_one_real ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_dec_simps(3)
% 0.33/0.65 thf(fact_6859_dbl__dec__simps_I3_J,axiom,
% 0.33/0.65 ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
% 0.33/0.65 = one_one_rat ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_dec_simps(3)
% 0.33/0.65 thf(fact_6860_dbl__dec__simps_I3_J,axiom,
% 0.33/0.65 ( ( neg_nu3811975205180677377ec_int @ one_one_int )
% 0.33/0.65 = one_one_int ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_dec_simps(3)
% 0.33/0.65 thf(fact_6861_zabs__less__one__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
% 0.33/0.65 = ( Z = zero_zero_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % zabs_less_one_iff
% 0.33/0.65 thf(fact_6862_pred__numeral__simps_I1_J,axiom,
% 0.33/0.65 ( ( pred_numeral @ one )
% 0.33/0.65 = zero_zero_nat ) ).
% 0.33/0.65
% 0.33/0.65 % pred_numeral_simps(1)
% 0.33/0.65 thf(fact_6863_Suc__eq__numeral,axiom,
% 0.33/0.65 ! [N2: nat,K: num] :
% 0.33/0.65 ( ( ( suc @ N2 )
% 0.33/0.65 = ( numeral_numeral_nat @ K ) )
% 0.33/0.65 = ( N2
% 0.33/0.65 = ( pred_numeral @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % Suc_eq_numeral
% 0.33/0.65 thf(fact_6864_eq__numeral__Suc,axiom,
% 0.33/0.65 ! [K: num,N2: nat] :
% 0.33/0.65 ( ( ( numeral_numeral_nat @ K )
% 0.33/0.65 = ( suc @ N2 ) )
% 0.33/0.65 = ( ( pred_numeral @ K )
% 0.33/0.65 = N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % eq_numeral_Suc
% 0.33/0.65 thf(fact_6865_arctan__less__zero__iff,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ( ( ord_less_real @ ( arctan @ X ) @ zero_zero_real )
% 0.33/0.65 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.33/0.65
% 0.33/0.65 % arctan_less_zero_iff
% 0.33/0.65 thf(fact_6866_zero__less__arctan__iff,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ( ( ord_less_real @ zero_zero_real @ ( arctan @ X ) )
% 0.33/0.65 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % zero_less_arctan_iff
% 0.33/0.65 thf(fact_6867_arctan__le__zero__iff,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( arctan @ X ) @ zero_zero_real )
% 0.33/0.65 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.33/0.65
% 0.33/0.65 % arctan_le_zero_iff
% 0.33/0.65 thf(fact_6868_zero__le__arctan__iff,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X ) )
% 0.33/0.65 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % zero_le_arctan_iff
% 0.33/0.65 thf(fact_6869_less__Suc__numeral,axiom,
% 0.33/0.65 ! [N2: nat,K: num] :
% 0.33/0.65 ( ( ord_less_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
% 0.33/0.65 = ( ord_less_nat @ N2 @ ( pred_numeral @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % less_Suc_numeral
% 0.33/0.65 thf(fact_6870_less__numeral__Suc,axiom,
% 0.33/0.65 ! [K: num,N2: nat] :
% 0.33/0.65 ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
% 0.33/0.65 = ( ord_less_nat @ ( pred_numeral @ K ) @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % less_numeral_Suc
% 0.33/0.65 thf(fact_6871_pred__numeral__simps_I3_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( pred_numeral @ ( bit1 @ K ) )
% 0.33/0.65 = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % pred_numeral_simps(3)
% 0.33/0.65 thf(fact_6872_le__numeral__Suc,axiom,
% 0.33/0.65 ! [K: num,N2: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
% 0.33/0.65 = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % le_numeral_Suc
% 0.33/0.65 thf(fact_6873_le__Suc__numeral,axiom,
% 0.33/0.65 ! [N2: nat,K: num] :
% 0.33/0.65 ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
% 0.33/0.65 = ( ord_less_eq_nat @ N2 @ ( pred_numeral @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % le_Suc_numeral
% 0.33/0.65 thf(fact_6874_diff__Suc__numeral,axiom,
% 0.33/0.65 ! [N2: nat,K: num] :
% 0.33/0.65 ( ( minus_minus_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
% 0.33/0.65 = ( minus_minus_nat @ N2 @ ( pred_numeral @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_Suc_numeral
% 0.33/0.65 thf(fact_6875_diff__numeral__Suc,axiom,
% 0.33/0.65 ! [K: num,N2: nat] :
% 0.33/0.65 ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
% 0.33/0.65 = ( minus_minus_nat @ ( pred_numeral @ K ) @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_numeral_Suc
% 0.33/0.65 thf(fact_6876_max__Suc__numeral,axiom,
% 0.33/0.65 ! [N2: nat,K: num] :
% 0.33/0.65 ( ( ord_max_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
% 0.33/0.65 = ( suc @ ( ord_max_nat @ N2 @ ( pred_numeral @ K ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_Suc_numeral
% 0.33/0.65 thf(fact_6877_max__numeral__Suc,axiom,
% 0.33/0.65 ! [K: num,N2: nat] :
% 0.33/0.65 ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
% 0.33/0.65 = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % max_numeral_Suc
% 0.33/0.65 thf(fact_6878_dbl__dec__simps_I2_J,axiom,
% 0.33/0.65 ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
% 0.33/0.65 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_dec_simps(2)
% 0.33/0.65 thf(fact_6879_dbl__dec__simps_I2_J,axiom,
% 0.33/0.65 ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
% 0.33/0.65 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_dec_simps(2)
% 0.33/0.65 thf(fact_6880_dbl__dec__simps_I2_J,axiom,
% 0.33/0.65 ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
% 0.33/0.65 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_dec_simps(2)
% 0.33/0.65 thf(fact_6881_dbl__dec__simps_I2_J,axiom,
% 0.33/0.65 ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
% 0.33/0.65 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_dec_simps(2)
% 0.33/0.65 thf(fact_6882_dbl__dec__simps_I2_J,axiom,
% 0.33/0.65 ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
% 0.33/0.65 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_dec_simps(2)
% 0.33/0.65 thf(fact_6883_numeral__div__minus__numeral,axiom,
% 0.33/0.65 ! [M: num,N2: num] :
% 0.33/0.65 ( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.33/0.65 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_div_minus_numeral
% 0.33/0.65 thf(fact_6884_minus__numeral__div__numeral,axiom,
% 0.33/0.65 ! [M: num,N2: num] :
% 0.33/0.65 ( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
% 0.33/0.65 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % minus_numeral_div_numeral
% 0.33/0.65 thf(fact_6885_signed__take__bit__numeral__bit0,axiom,
% 0.33/0.65 ! [L2: num,K: num] :
% 0.33/0.65 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.33/0.65 = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % signed_take_bit_numeral_bit0
% 0.33/0.65 thf(fact_6886_signed__take__bit__numeral__minus__bit0,axiom,
% 0.33/0.65 ! [L2: num,K: num] :
% 0.33/0.65 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.33/0.65 = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % signed_take_bit_numeral_minus_bit0
% 0.33/0.65 thf(fact_6887_prod_Odisc__eq__case,axiom,
% 0.33/0.65 ! [Prod: product_prod_int_int] :
% 0.33/0.65 ( produc4947309494688390418_int_o
% 0.33/0.65 @ ^ [Uu3: int,Uv3: int] : $true
% 0.33/0.65 @ Prod ) ).
% 0.33/0.65
% 0.33/0.65 % prod.disc_eq_case
% 0.33/0.65 thf(fact_6888_arctan__less__iff,axiom,
% 0.33/0.65 ! [X: real,Y: real] :
% 0.33/0.65 ( ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) )
% 0.33/0.65 = ( ord_less_real @ X @ Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % arctan_less_iff
% 0.33/0.65 thf(fact_6889_arctan__monotone,axiom,
% 0.33/0.65 ! [X: real,Y: real] :
% 0.33/0.65 ( ( ord_less_real @ X @ Y )
% 0.33/0.65 => ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % arctan_monotone
% 0.33/0.65 thf(fact_6890_arctan__le__iff,axiom,
% 0.33/0.65 ! [X: real,Y: real] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y ) )
% 0.33/0.65 = ( ord_less_eq_real @ X @ Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % arctan_le_iff
% 0.33/0.65 thf(fact_6891_arctan__monotone_H,axiom,
% 0.33/0.65 ! [X: real,Y: real] :
% 0.33/0.65 ( ( ord_less_eq_real @ X @ Y )
% 0.33/0.65 => ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % arctan_monotone'
% 0.33/0.65 thf(fact_6892_abs__zmult__eq__1,axiom,
% 0.33/0.65 ! [M: int,N2: int] :
% 0.33/0.65 ( ( ( abs_abs_int @ ( times_times_int @ M @ N2 ) )
% 0.33/0.65 = one_one_int )
% 0.33/0.65 => ( ( abs_abs_int @ M )
% 0.33/0.65 = one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % abs_zmult_eq_1
% 0.33/0.65 thf(fact_6893_abs__div,axiom,
% 0.33/0.65 ! [Y: int,X: int] :
% 0.33/0.65 ( ( dvd_dvd_int @ Y @ X )
% 0.33/0.65 => ( ( abs_abs_int @ ( divide_divide_int @ X @ Y ) )
% 0.33/0.65 = ( divide_divide_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % abs_div
% 0.33/0.65 thf(fact_6894_numeral__eq__Suc,axiom,
% 0.33/0.65 ( numeral_numeral_nat
% 0.33/0.65 = ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_eq_Suc
% 0.33/0.65 thf(fact_6895_zabs__def,axiom,
% 0.33/0.65 ( abs_abs_int
% 0.33/0.65 = ( ^ [I3: int] : ( if_int @ ( ord_less_int @ I3 @ zero_zero_int ) @ ( uminus_uminus_int @ I3 ) @ I3 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % zabs_def
% 0.33/0.65 thf(fact_6896_abs__mod__less,axiom,
% 0.33/0.65 ! [L2: int,K: int] :
% 0.33/0.65 ( ( L2 != zero_zero_int )
% 0.33/0.65 => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L2 ) ) @ ( abs_abs_int @ L2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % abs_mod_less
% 0.33/0.65 thf(fact_6897_pred__numeral__def,axiom,
% 0.33/0.65 ( pred_numeral
% 0.33/0.65 = ( ^ [K3: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K3 ) @ one_one_nat ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % pred_numeral_def
% 0.33/0.65 thf(fact_6898_zdvd__mult__cancel1,axiom,
% 0.33/0.65 ! [M: int,N2: int] :
% 0.33/0.65 ( ( M != zero_zero_int )
% 0.33/0.65 => ( ( dvd_dvd_int @ ( times_times_int @ M @ N2 ) @ M )
% 0.33/0.65 = ( ( abs_abs_int @ N2 )
% 0.33/0.65 = one_one_int ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % zdvd_mult_cancel1
% 0.33/0.65 thf(fact_6899_even__abs__add__iff,axiom,
% 0.33/0.65 ! [K: int,L2: int] :
% 0.33/0.65 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L2 ) )
% 0.33/0.65 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % even_abs_add_iff
% 0.33/0.65 thf(fact_6900_even__add__abs__iff,axiom,
% 0.33/0.65 ! [K: int,L2: int] :
% 0.33/0.65 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L2 ) ) )
% 0.33/0.65 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % even_add_abs_iff
% 0.33/0.65 thf(fact_6901_Divides_Oadjust__div__def,axiom,
% 0.33/0.65 ( adjust_div
% 0.33/0.65 = ( produc8211389475949308722nt_int
% 0.33/0.65 @ ^ [Q4: int,R5: int] : ( plus_plus_int @ Q4 @ ( zero_n2684676970156552555ol_int @ ( R5 != zero_zero_int ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % Divides.adjust_div_def
% 0.33/0.65 thf(fact_6902_nat__intermed__int__val,axiom,
% 0.33/0.65 ! [M: nat,N2: nat,F: nat > int,K: int] :
% 0.33/0.65 ( ! [I4: nat] :
% 0.33/0.65 ( ( ( ord_less_eq_nat @ M @ I4 )
% 0.33/0.65 & ( ord_less_nat @ I4 @ N2 ) )
% 0.33/0.65 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) ) ) @ one_one_int ) )
% 0.33/0.65 => ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.65 => ( ( ord_less_eq_int @ ( F @ M ) @ K )
% 0.33/0.65 => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
% 0.33/0.65 => ? [I4: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ M @ I4 )
% 0.33/0.65 & ( ord_less_eq_nat @ I4 @ N2 )
% 0.33/0.65 & ( ( F @ I4 )
% 0.33/0.65 = K ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % nat_intermed_int_val
% 0.33/0.65 thf(fact_6903_dbl__dec__def,axiom,
% 0.33/0.65 ( neg_nu6511756317524482435omplex
% 0.33/0.65 = ( ^ [X2: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X2 @ X2 ) @ one_one_complex ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_dec_def
% 0.33/0.65 thf(fact_6904_dbl__dec__def,axiom,
% 0.33/0.65 ( neg_nu6075765906172075777c_real
% 0.33/0.65 = ( ^ [X2: real] : ( minus_minus_real @ ( plus_plus_real @ X2 @ X2 ) @ one_one_real ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_dec_def
% 0.33/0.65 thf(fact_6905_dbl__dec__def,axiom,
% 0.33/0.65 ( neg_nu3179335615603231917ec_rat
% 0.33/0.65 = ( ^ [X2: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_dec_def
% 0.33/0.65 thf(fact_6906_dbl__dec__def,axiom,
% 0.33/0.65 ( neg_nu3811975205180677377ec_int
% 0.33/0.65 = ( ^ [X2: int] : ( minus_minus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_dec_def
% 0.33/0.65 thf(fact_6907_decr__lemma,axiom,
% 0.33/0.65 ! [D: int,X: int,Z: int] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ D )
% 0.33/0.65 => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).
% 0.33/0.65
% 0.33/0.65 % decr_lemma
% 0.33/0.65 thf(fact_6908_incr__lemma,axiom,
% 0.33/0.65 ! [D: int,Z: int,X: int] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ D )
% 0.33/0.65 => ( ord_less_int @ Z @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % incr_lemma
% 0.33/0.65 thf(fact_6909_nat__ivt__aux,axiom,
% 0.33/0.65 ! [N2: nat,F: nat > int,K: int] :
% 0.33/0.65 ( ! [I4: nat] :
% 0.33/0.65 ( ( ord_less_nat @ I4 @ N2 )
% 0.33/0.65 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) ) ) @ one_one_int ) )
% 0.33/0.65 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 0.33/0.65 => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
% 0.33/0.65 => ? [I4: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ I4 @ N2 )
% 0.33/0.65 & ( ( F @ I4 )
% 0.33/0.65 = K ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % nat_ivt_aux
% 0.33/0.65 thf(fact_6910_nat0__intermed__int__val,axiom,
% 0.33/0.65 ! [N2: nat,F: nat > int,K: int] :
% 0.33/0.65 ( ! [I4: nat] :
% 0.33/0.65 ( ( ord_less_nat @ I4 @ N2 )
% 0.33/0.65 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I4 @ one_one_nat ) ) @ ( F @ I4 ) ) ) @ one_one_int ) )
% 0.33/0.65 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 0.33/0.65 => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
% 0.33/0.65 => ? [I4: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ I4 @ N2 )
% 0.33/0.65 & ( ( F @ I4 )
% 0.33/0.65 = K ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % nat0_intermed_int_val
% 0.33/0.65 thf(fact_6911_arctan__add,axiom,
% 0.33/0.65 ! [X: real,Y: real] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.65 => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 0.33/0.65 => ( ( plus_plus_real @ ( arctan @ X ) @ ( arctan @ Y ) )
% 0.33/0.65 = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X @ Y ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % arctan_add
% 0.33/0.65 thf(fact_6912_sum__gp,axiom,
% 0.33/0.65 ! [N2: nat,M: nat,X: rat] :
% 0.33/0.65 ( ( ( ord_less_nat @ N2 @ M )
% 0.33/0.65 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.65 = zero_zero_rat ) )
% 0.33/0.65 & ( ~ ( ord_less_nat @ N2 @ M )
% 0.33/0.65 => ( ( ( X = one_one_rat )
% 0.33/0.65 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.65 = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
% 0.33/0.65 & ( ( X != one_one_rat )
% 0.33/0.65 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.65 = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum_gp
% 0.33/0.65 thf(fact_6913_sum__gp,axiom,
% 0.33/0.65 ! [N2: nat,M: nat,X: complex] :
% 0.33/0.65 ( ( ( ord_less_nat @ N2 @ M )
% 0.33/0.65 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.65 = zero_zero_complex ) )
% 0.33/0.65 & ( ~ ( ord_less_nat @ N2 @ M )
% 0.33/0.65 => ( ( ( X = one_one_complex )
% 0.33/0.65 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.65 = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
% 0.33/0.65 & ( ( X != one_one_complex )
% 0.33/0.65 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.65 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum_gp
% 0.33/0.65 thf(fact_6914_sum__gp,axiom,
% 0.33/0.65 ! [N2: nat,M: nat,X: real] :
% 0.33/0.65 ( ( ( ord_less_nat @ N2 @ M )
% 0.33/0.65 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.65 = zero_zero_real ) )
% 0.33/0.65 & ( ~ ( ord_less_nat @ N2 @ M )
% 0.33/0.65 => ( ( ( X = one_one_real )
% 0.33/0.65 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.65 = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
% 0.33/0.65 & ( ( X != one_one_real )
% 0.33/0.65 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.65 = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum_gp
% 0.33/0.65 thf(fact_6915_dbl__inc__simps_I3_J,axiom,
% 0.33/0.65 ( ( neg_nu8557863876264182079omplex @ one_one_complex )
% 0.33/0.65 = ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_inc_simps(3)
% 0.33/0.65 thf(fact_6916_dbl__inc__simps_I3_J,axiom,
% 0.33/0.65 ( ( neg_nu8295874005876285629c_real @ one_one_real )
% 0.33/0.65 = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_inc_simps(3)
% 0.33/0.65 thf(fact_6917_dbl__inc__simps_I3_J,axiom,
% 0.33/0.65 ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
% 0.33/0.65 = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_inc_simps(3)
% 0.33/0.65 thf(fact_6918_dbl__inc__simps_I3_J,axiom,
% 0.33/0.65 ( ( neg_nu5851722552734809277nc_int @ one_one_int )
% 0.33/0.65 = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_inc_simps(3)
% 0.33/0.65 thf(fact_6919_divmod__BitM__2__eq,axiom,
% 0.33/0.65 ! [M: num] :
% 0.33/0.65 ( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
% 0.33/0.65 = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % divmod_BitM_2_eq
% 0.33/0.65 thf(fact_6920_gauss__sum__from__Suc__0,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
% 0.33/0.65 = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gauss_sum_from_Suc_0
% 0.33/0.65 thf(fact_6921_gauss__sum__from__Suc__0,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
% 0.33/0.65 = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gauss_sum_from_Suc_0
% 0.33/0.65 thf(fact_6922_of__int__code__if,axiom,
% 0.33/0.65 ( ring_1_of_int_real
% 0.33/0.65 = ( ^ [K3: int] :
% 0.33/0.65 ( if_real @ ( K3 = zero_zero_int ) @ zero_zero_real
% 0.33/0.65 @ ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K3 ) ) )
% 0.33/0.65 @ ( if_real
% 0.33/0.65 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.65 = zero_zero_int )
% 0.33/0.65 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.33/0.65 @ ( 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 ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_code_if
% 0.33/0.65 thf(fact_6923_of__int__code__if,axiom,
% 0.33/0.65 ( ring_1_of_int_int
% 0.33/0.65 = ( ^ [K3: int] :
% 0.33/0.65 ( if_int @ ( K3 = zero_zero_int ) @ zero_zero_int
% 0.33/0.65 @ ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K3 ) ) )
% 0.33/0.65 @ ( if_int
% 0.33/0.65 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.65 = zero_zero_int )
% 0.33/0.65 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.33/0.65 @ ( 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 ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_code_if
% 0.33/0.65 thf(fact_6924_of__int__code__if,axiom,
% 0.33/0.65 ( ring_17405671764205052669omplex
% 0.33/0.65 = ( ^ [K3: int] :
% 0.33/0.65 ( if_complex @ ( K3 = zero_zero_int ) @ zero_zero_complex
% 0.33/0.65 @ ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K3 ) ) )
% 0.33/0.65 @ ( if_complex
% 0.33/0.65 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.65 = zero_zero_int )
% 0.33/0.65 @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.33/0.65 @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_code_if
% 0.33/0.65 thf(fact_6925_of__int__code__if,axiom,
% 0.33/0.65 ( ring_1_of_int_rat
% 0.33/0.65 = ( ^ [K3: int] :
% 0.33/0.65 ( if_rat @ ( K3 = zero_zero_int ) @ zero_zero_rat
% 0.33/0.65 @ ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K3 ) ) )
% 0.33/0.65 @ ( if_rat
% 0.33/0.65 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.65 = zero_zero_int )
% 0.33/0.65 @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.33/0.65 @ ( 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 ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_code_if
% 0.33/0.65 thf(fact_6926_of__int__code__if,axiom,
% 0.33/0.65 ( ring_18347121197199848620nteger
% 0.33/0.65 = ( ^ [K3: int] :
% 0.33/0.65 ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 0.33/0.65 @ ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K3 ) ) )
% 0.33/0.65 @ ( if_Code_integer
% 0.33/0.65 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.65 = zero_zero_int )
% 0.33/0.65 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.33/0.65 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_code_if
% 0.33/0.65 thf(fact_6927_of__nat__eq__iff,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( ( semiri1314217659103216013at_int @ M )
% 0.33/0.65 = ( semiri1314217659103216013at_int @ N2 ) )
% 0.33/0.65 = ( M = N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_eq_iff
% 0.33/0.65 thf(fact_6928_of__nat__eq__iff,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( ( semiri5074537144036343181t_real @ M )
% 0.33/0.65 = ( semiri5074537144036343181t_real @ N2 ) )
% 0.33/0.65 = ( M = N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_eq_iff
% 0.33/0.65 thf(fact_6929_of__nat__eq__iff,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( ( semiri1316708129612266289at_nat @ M )
% 0.33/0.65 = ( semiri1316708129612266289at_nat @ N2 ) )
% 0.33/0.65 = ( M = N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_eq_iff
% 0.33/0.65 thf(fact_6930_of__nat__eq__iff,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( ( semiri8010041392384452111omplex @ M )
% 0.33/0.65 = ( semiri8010041392384452111omplex @ N2 ) )
% 0.33/0.65 = ( M = N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_eq_iff
% 0.33/0.65 thf(fact_6931_int__eq__iff__numeral,axiom,
% 0.33/0.65 ! [M: nat,V: num] :
% 0.33/0.65 ( ( ( semiri1314217659103216013at_int @ M )
% 0.33/0.65 = ( numeral_numeral_int @ V ) )
% 0.33/0.65 = ( M
% 0.33/0.65 = ( numeral_numeral_nat @ V ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % int_eq_iff_numeral
% 0.33/0.65 thf(fact_6932_abs__of__nat,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N2 ) )
% 0.33/0.65 = ( semiri681578069525770553at_rat @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % abs_of_nat
% 0.33/0.65 thf(fact_6933_abs__of__nat,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N2 ) )
% 0.33/0.65 = ( semiri4939895301339042750nteger @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % abs_of_nat
% 0.33/0.65 thf(fact_6934_abs__of__nat,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N2 ) )
% 0.33/0.65 = ( semiri1314217659103216013at_int @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % abs_of_nat
% 0.33/0.65 thf(fact_6935_abs__of__nat,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N2 ) )
% 0.33/0.65 = ( semiri5074537144036343181t_real @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % abs_of_nat
% 0.33/0.65 thf(fact_6936_of__nat__0,axiom,
% 0.33/0.65 ( ( semiri681578069525770553at_rat @ zero_zero_nat )
% 0.33/0.65 = zero_zero_rat ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_0
% 0.33/0.65 thf(fact_6937_of__nat__0,axiom,
% 0.33/0.65 ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 0.33/0.65 = zero_zero_int ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_0
% 0.33/0.65 thf(fact_6938_of__nat__0,axiom,
% 0.33/0.65 ( ( semiri5074537144036343181t_real @ zero_zero_nat )
% 0.33/0.65 = zero_zero_real ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_0
% 0.33/0.65 thf(fact_6939_of__nat__0,axiom,
% 0.33/0.65 ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
% 0.33/0.65 = zero_zero_nat ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_0
% 0.33/0.65 thf(fact_6940_of__nat__0,axiom,
% 0.33/0.65 ( ( semiri8010041392384452111omplex @ zero_zero_nat )
% 0.33/0.65 = zero_zero_complex ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_0
% 0.33/0.65 thf(fact_6941_of__nat__0__eq__iff,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( zero_zero_rat
% 0.33/0.65 = ( semiri681578069525770553at_rat @ N2 ) )
% 0.33/0.65 = ( zero_zero_nat = N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_0_eq_iff
% 0.33/0.65 thf(fact_6942_of__nat__0__eq__iff,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( zero_zero_int
% 0.33/0.65 = ( semiri1314217659103216013at_int @ N2 ) )
% 0.33/0.65 = ( zero_zero_nat = N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_0_eq_iff
% 0.33/0.65 thf(fact_6943_of__nat__0__eq__iff,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( zero_zero_real
% 0.33/0.65 = ( semiri5074537144036343181t_real @ N2 ) )
% 0.33/0.65 = ( zero_zero_nat = N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_0_eq_iff
% 0.33/0.65 thf(fact_6944_of__nat__0__eq__iff,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( zero_zero_nat
% 0.33/0.65 = ( semiri1316708129612266289at_nat @ N2 ) )
% 0.33/0.65 = ( zero_zero_nat = N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_0_eq_iff
% 0.33/0.65 thf(fact_6945_of__nat__0__eq__iff,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( zero_zero_complex
% 0.33/0.65 = ( semiri8010041392384452111omplex @ N2 ) )
% 0.33/0.65 = ( zero_zero_nat = N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_0_eq_iff
% 0.33/0.65 thf(fact_6946_of__nat__eq__0__iff,axiom,
% 0.33/0.65 ! [M: nat] :
% 0.33/0.65 ( ( ( semiri681578069525770553at_rat @ M )
% 0.33/0.65 = zero_zero_rat )
% 0.33/0.65 = ( M = zero_zero_nat ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_eq_0_iff
% 0.33/0.65 thf(fact_6947_of__nat__eq__0__iff,axiom,
% 0.33/0.65 ! [M: nat] :
% 0.33/0.65 ( ( ( semiri1314217659103216013at_int @ M )
% 0.33/0.65 = zero_zero_int )
% 0.33/0.65 = ( M = zero_zero_nat ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_eq_0_iff
% 0.33/0.65 thf(fact_6948_of__nat__eq__0__iff,axiom,
% 0.33/0.65 ! [M: nat] :
% 0.33/0.65 ( ( ( semiri5074537144036343181t_real @ M )
% 0.33/0.65 = zero_zero_real )
% 0.33/0.65 = ( M = zero_zero_nat ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_eq_0_iff
% 0.33/0.65 thf(fact_6949_of__nat__eq__0__iff,axiom,
% 0.33/0.65 ! [M: nat] :
% 0.33/0.65 ( ( ( semiri1316708129612266289at_nat @ M )
% 0.33/0.65 = zero_zero_nat )
% 0.33/0.65 = ( M = zero_zero_nat ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_eq_0_iff
% 0.33/0.65 thf(fact_6950_of__nat__eq__0__iff,axiom,
% 0.33/0.65 ! [M: nat] :
% 0.33/0.65 ( ( ( semiri8010041392384452111omplex @ M )
% 0.33/0.65 = zero_zero_complex )
% 0.33/0.65 = ( M = zero_zero_nat ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_eq_0_iff
% 0.33/0.65 thf(fact_6951_of__nat__numeral,axiom,
% 0.33/0.65 ! [N2: num] :
% 0.33/0.65 ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N2 ) )
% 0.33/0.65 = ( numeral_numeral_rat @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_numeral
% 0.33/0.65 thf(fact_6952_of__nat__numeral,axiom,
% 0.33/0.65 ! [N2: num] :
% 0.33/0.65 ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N2 ) )
% 0.33/0.65 = ( numeral_numeral_int @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_numeral
% 0.33/0.65 thf(fact_6953_of__nat__numeral,axiom,
% 0.33/0.65 ! [N2: num] :
% 0.33/0.65 ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N2 ) )
% 0.33/0.65 = ( numeral_numeral_real @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_numeral
% 0.33/0.65 thf(fact_6954_of__nat__numeral,axiom,
% 0.33/0.65 ! [N2: num] :
% 0.33/0.65 ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N2 ) )
% 0.33/0.65 = ( numeral_numeral_nat @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_numeral
% 0.33/0.65 thf(fact_6955_of__nat__numeral,axiom,
% 0.33/0.65 ! [N2: num] :
% 0.33/0.65 ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N2 ) )
% 0.33/0.65 = ( numera6690914467698888265omplex @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_numeral
% 0.33/0.65 thf(fact_6956_of__nat__less__iff,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
% 0.33/0.65 = ( ord_less_nat @ M @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_less_iff
% 0.33/0.65 thf(fact_6957_of__nat__less__iff,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
% 0.33/0.65 = ( ord_less_nat @ M @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_less_iff
% 0.33/0.65 thf(fact_6958_of__nat__less__iff,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
% 0.33/0.65 = ( ord_less_nat @ M @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_less_iff
% 0.33/0.65 thf(fact_6959_of__nat__less__iff,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
% 0.33/0.65 = ( ord_less_nat @ M @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_less_iff
% 0.33/0.65 thf(fact_6960_of__nat__le__iff,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
% 0.33/0.65 = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_le_iff
% 0.33/0.65 thf(fact_6961_of__nat__le__iff,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
% 0.33/0.65 = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_le_iff
% 0.33/0.65 thf(fact_6962_of__nat__le__iff,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
% 0.33/0.65 = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_le_iff
% 0.33/0.65 thf(fact_6963_of__nat__le__iff,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
% 0.33/0.65 = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_le_iff
% 0.33/0.65 thf(fact_6964_of__nat__add,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N2 ) )
% 0.33/0.65 = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_add
% 0.33/0.65 thf(fact_6965_of__nat__add,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) )
% 0.33/0.65 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_add
% 0.33/0.65 thf(fact_6966_of__nat__add,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N2 ) )
% 0.33/0.65 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_add
% 0.33/0.65 thf(fact_6967_of__nat__add,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N2 ) )
% 0.33/0.65 = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_add
% 0.33/0.65 thf(fact_6968_of__nat__add,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ N2 ) )
% 0.33/0.65 = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_add
% 0.33/0.65 thf(fact_6969_of__nat__mult,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N2 ) )
% 0.33/0.65 = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_mult
% 0.33/0.65 thf(fact_6970_of__nat__mult,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N2 ) )
% 0.33/0.65 = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_mult
% 0.33/0.65 thf(fact_6971_of__nat__mult,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N2 ) )
% 0.33/0.65 = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_mult
% 0.33/0.65 thf(fact_6972_of__nat__mult,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N2 ) )
% 0.33/0.65 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_mult
% 0.33/0.65 thf(fact_6973_of__nat__mult,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N2 ) )
% 0.33/0.65 = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_mult
% 0.33/0.65 thf(fact_6974_of__nat__eq__1__iff,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( ( semiri681578069525770553at_rat @ N2 )
% 0.33/0.65 = one_one_rat )
% 0.33/0.65 = ( N2 = one_one_nat ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_eq_1_iff
% 0.33/0.65 thf(fact_6975_of__nat__eq__1__iff,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( ( semiri1314217659103216013at_int @ N2 )
% 0.33/0.65 = one_one_int )
% 0.33/0.65 = ( N2 = one_one_nat ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_eq_1_iff
% 0.33/0.65 thf(fact_6976_of__nat__eq__1__iff,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( ( semiri5074537144036343181t_real @ N2 )
% 0.33/0.65 = one_one_real )
% 0.33/0.65 = ( N2 = one_one_nat ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_eq_1_iff
% 0.33/0.65 thf(fact_6977_of__nat__eq__1__iff,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( ( semiri1316708129612266289at_nat @ N2 )
% 0.33/0.65 = one_one_nat )
% 0.33/0.65 = ( N2 = one_one_nat ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_eq_1_iff
% 0.33/0.65 thf(fact_6978_of__nat__eq__1__iff,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( ( semiri8010041392384452111omplex @ N2 )
% 0.33/0.65 = one_one_complex )
% 0.33/0.65 = ( N2 = one_one_nat ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_eq_1_iff
% 0.33/0.65 thf(fact_6979_of__nat__1__eq__iff,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( one_one_rat
% 0.33/0.65 = ( semiri681578069525770553at_rat @ N2 ) )
% 0.33/0.65 = ( N2 = one_one_nat ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_1_eq_iff
% 0.33/0.65 thf(fact_6980_of__nat__1__eq__iff,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( one_one_int
% 0.33/0.65 = ( semiri1314217659103216013at_int @ N2 ) )
% 0.33/0.65 = ( N2 = one_one_nat ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_1_eq_iff
% 0.33/0.65 thf(fact_6981_of__nat__1__eq__iff,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( one_one_real
% 0.33/0.65 = ( semiri5074537144036343181t_real @ N2 ) )
% 0.33/0.65 = ( N2 = one_one_nat ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_1_eq_iff
% 0.33/0.65 thf(fact_6982_of__nat__1__eq__iff,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( one_one_nat
% 0.33/0.65 = ( semiri1316708129612266289at_nat @ N2 ) )
% 0.33/0.65 = ( N2 = one_one_nat ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_1_eq_iff
% 0.33/0.65 thf(fact_6983_of__nat__1__eq__iff,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( one_one_complex
% 0.33/0.65 = ( semiri8010041392384452111omplex @ N2 ) )
% 0.33/0.65 = ( N2 = one_one_nat ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_1_eq_iff
% 0.33/0.65 thf(fact_6984_of__nat__1,axiom,
% 0.33/0.65 ( ( semiri681578069525770553at_rat @ one_one_nat )
% 0.33/0.65 = one_one_rat ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_1
% 0.33/0.65 thf(fact_6985_of__nat__1,axiom,
% 0.33/0.65 ( ( semiri1314217659103216013at_int @ one_one_nat )
% 0.33/0.65 = one_one_int ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_1
% 0.33/0.65 thf(fact_6986_of__nat__1,axiom,
% 0.33/0.65 ( ( semiri5074537144036343181t_real @ one_one_nat )
% 0.33/0.65 = one_one_real ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_1
% 0.33/0.65 thf(fact_6987_of__nat__1,axiom,
% 0.33/0.65 ( ( semiri1316708129612266289at_nat @ one_one_nat )
% 0.33/0.65 = one_one_nat ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_1
% 0.33/0.65 thf(fact_6988_of__nat__1,axiom,
% 0.33/0.65 ( ( semiri8010041392384452111omplex @ one_one_nat )
% 0.33/0.65 = one_one_complex ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_1
% 0.33/0.65 thf(fact_6989_of__nat__power,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N2 ) )
% 0.33/0.65 = ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_power
% 0.33/0.65 thf(fact_6990_of__nat__power,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N2 ) )
% 0.33/0.65 = ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_power
% 0.33/0.65 thf(fact_6991_of__nat__power,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N2 ) )
% 0.33/0.65 = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_power
% 0.33/0.65 thf(fact_6992_of__nat__power,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N2 ) )
% 0.33/0.65 = ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_power
% 0.33/0.65 thf(fact_6993_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.33/0.65 ! [B: nat,W: nat,X: nat] :
% 0.33/0.65 ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
% 0.33/0.65 = ( semiri1314217659103216013at_int @ X ) )
% 0.33/0.65 = ( ( power_power_nat @ B @ W )
% 0.33/0.65 = X ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_eq_of_nat_power_cancel_iff
% 0.33/0.65 thf(fact_6994_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.33/0.65 ! [B: nat,W: nat,X: nat] :
% 0.33/0.65 ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
% 0.33/0.65 = ( semiri5074537144036343181t_real @ X ) )
% 0.33/0.65 = ( ( power_power_nat @ B @ W )
% 0.33/0.65 = X ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_eq_of_nat_power_cancel_iff
% 0.33/0.65 thf(fact_6995_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.33/0.65 ! [B: nat,W: nat,X: nat] :
% 0.33/0.65 ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
% 0.33/0.65 = ( semiri1316708129612266289at_nat @ X ) )
% 0.33/0.65 = ( ( power_power_nat @ B @ W )
% 0.33/0.65 = X ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_eq_of_nat_power_cancel_iff
% 0.33/0.65 thf(fact_6996_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.33/0.65 ! [B: nat,W: nat,X: nat] :
% 0.33/0.65 ( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W )
% 0.33/0.65 = ( semiri8010041392384452111omplex @ X ) )
% 0.33/0.65 = ( ( power_power_nat @ B @ W )
% 0.33/0.65 = X ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_eq_of_nat_power_cancel_iff
% 0.33/0.65 thf(fact_6997_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.33/0.65 ! [X: nat,B: nat,W: nat] :
% 0.33/0.65 ( ( ( semiri1314217659103216013at_int @ X )
% 0.33/0.65 = ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 0.33/0.65 = ( X
% 0.33/0.65 = ( power_power_nat @ B @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_power_eq_of_nat_cancel_iff
% 0.33/0.65 thf(fact_6998_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.33/0.65 ! [X: nat,B: nat,W: nat] :
% 0.33/0.65 ( ( ( semiri5074537144036343181t_real @ X )
% 0.33/0.65 = ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 0.33/0.65 = ( X
% 0.33/0.65 = ( power_power_nat @ B @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_power_eq_of_nat_cancel_iff
% 0.33/0.65 thf(fact_6999_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.33/0.65 ! [X: nat,B: nat,W: nat] :
% 0.33/0.65 ( ( ( semiri1316708129612266289at_nat @ X )
% 0.33/0.65 = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 0.33/0.65 = ( X
% 0.33/0.65 = ( power_power_nat @ B @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_power_eq_of_nat_cancel_iff
% 0.33/0.65 thf(fact_7000_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.33/0.65 ! [X: nat,B: nat,W: nat] :
% 0.33/0.65 ( ( ( semiri8010041392384452111omplex @ X )
% 0.33/0.65 = ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W ) )
% 0.33/0.65 = ( X
% 0.33/0.65 = ( power_power_nat @ B @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_power_eq_of_nat_cancel_iff
% 0.33/0.65 thf(fact_7001_of__int__numeral,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
% 0.33/0.65 = ( numera6690914467698888265omplex @ K ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_numeral
% 0.33/0.65 thf(fact_7002_of__int__numeral,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
% 0.33/0.65 = ( numeral_numeral_real @ K ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_numeral
% 0.33/0.65 thf(fact_7003_of__int__numeral,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
% 0.33/0.65 = ( numeral_numeral_rat @ K ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_numeral
% 0.33/0.65 thf(fact_7004_of__int__numeral,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
% 0.33/0.65 = ( numeral_numeral_int @ K ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_numeral
% 0.33/0.65 thf(fact_7005_of__int__eq__numeral__iff,axiom,
% 0.33/0.65 ! [Z: int,N2: num] :
% 0.33/0.65 ( ( ( ring_17405671764205052669omplex @ Z )
% 0.33/0.65 = ( numera6690914467698888265omplex @ N2 ) )
% 0.33/0.65 = ( Z
% 0.33/0.65 = ( numeral_numeral_int @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_eq_numeral_iff
% 0.33/0.65 thf(fact_7006_of__int__eq__numeral__iff,axiom,
% 0.33/0.65 ! [Z: int,N2: num] :
% 0.33/0.65 ( ( ( ring_1_of_int_real @ Z )
% 0.33/0.65 = ( numeral_numeral_real @ N2 ) )
% 0.33/0.65 = ( Z
% 0.33/0.65 = ( numeral_numeral_int @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_eq_numeral_iff
% 0.33/0.65 thf(fact_7007_of__int__eq__numeral__iff,axiom,
% 0.33/0.65 ! [Z: int,N2: num] :
% 0.33/0.65 ( ( ( ring_1_of_int_rat @ Z )
% 0.33/0.65 = ( numeral_numeral_rat @ N2 ) )
% 0.33/0.65 = ( Z
% 0.33/0.65 = ( numeral_numeral_int @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_eq_numeral_iff
% 0.33/0.65 thf(fact_7008_of__int__eq__numeral__iff,axiom,
% 0.33/0.65 ! [Z: int,N2: num] :
% 0.33/0.65 ( ( ( ring_1_of_int_int @ Z )
% 0.33/0.65 = ( numeral_numeral_int @ N2 ) )
% 0.33/0.65 = ( Z
% 0.33/0.65 = ( numeral_numeral_int @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_eq_numeral_iff
% 0.33/0.65 thf(fact_7009_of__int__le__iff,axiom,
% 0.33/0.65 ! [W: int,Z: int] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 0.33/0.65 = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_le_iff
% 0.33/0.65 thf(fact_7010_of__int__le__iff,axiom,
% 0.33/0.65 ! [W: int,Z: int] :
% 0.33/0.65 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
% 0.33/0.65 = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_le_iff
% 0.33/0.65 thf(fact_7011_of__int__le__iff,axiom,
% 0.33/0.65 ! [W: int,Z: int] :
% 0.33/0.65 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 0.33/0.65 = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_le_iff
% 0.33/0.65 thf(fact_7012_of__int__less__iff,axiom,
% 0.33/0.65 ! [W: int,Z: int] :
% 0.33/0.65 ( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 0.33/0.65 = ( ord_less_int @ W @ Z ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_less_iff
% 0.33/0.65 thf(fact_7013_of__int__less__iff,axiom,
% 0.33/0.65 ! [W: int,Z: int] :
% 0.33/0.65 ( ( ord_less_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
% 0.33/0.65 = ( ord_less_int @ W @ Z ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_less_iff
% 0.33/0.65 thf(fact_7014_of__int__less__iff,axiom,
% 0.33/0.65 ! [W: int,Z: int] :
% 0.33/0.65 ( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 0.33/0.65 = ( ord_less_int @ W @ Z ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_less_iff
% 0.33/0.65 thf(fact_7015_of__int__eq__1__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ( ring_17405671764205052669omplex @ Z )
% 0.33/0.65 = one_one_complex )
% 0.33/0.65 = ( Z = one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_eq_1_iff
% 0.33/0.65 thf(fact_7016_of__int__eq__1__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ( ring_1_of_int_int @ Z )
% 0.33/0.65 = one_one_int )
% 0.33/0.65 = ( Z = one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_eq_1_iff
% 0.33/0.65 thf(fact_7017_of__int__eq__1__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ( ring_1_of_int_real @ Z )
% 0.33/0.65 = one_one_real )
% 0.33/0.65 = ( Z = one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_eq_1_iff
% 0.33/0.65 thf(fact_7018_of__int__eq__1__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ( ring_1_of_int_rat @ Z )
% 0.33/0.65 = one_one_rat )
% 0.33/0.65 = ( Z = one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_eq_1_iff
% 0.33/0.65 thf(fact_7019_of__int__1,axiom,
% 0.33/0.65 ( ( ring_17405671764205052669omplex @ one_one_int )
% 0.33/0.65 = one_one_complex ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_1
% 0.33/0.65 thf(fact_7020_of__int__1,axiom,
% 0.33/0.65 ( ( ring_1_of_int_int @ one_one_int )
% 0.33/0.65 = one_one_int ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_1
% 0.33/0.65 thf(fact_7021_of__int__1,axiom,
% 0.33/0.65 ( ( ring_1_of_int_real @ one_one_int )
% 0.33/0.65 = one_one_real ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_1
% 0.33/0.65 thf(fact_7022_of__int__1,axiom,
% 0.33/0.65 ( ( ring_1_of_int_rat @ one_one_int )
% 0.33/0.65 = one_one_rat ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_1
% 0.33/0.65 thf(fact_7023_of__int__mult,axiom,
% 0.33/0.65 ! [W: int,Z: int] :
% 0.33/0.65 ( ( ring_1_of_int_real @ ( times_times_int @ W @ Z ) )
% 0.33/0.65 = ( times_times_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_mult
% 0.33/0.65 thf(fact_7024_of__int__mult,axiom,
% 0.33/0.65 ! [W: int,Z: int] :
% 0.33/0.65 ( ( ring_1_of_int_rat @ ( times_times_int @ W @ Z ) )
% 0.33/0.65 = ( times_times_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_mult
% 0.33/0.65 thf(fact_7025_of__int__mult,axiom,
% 0.33/0.65 ! [W: int,Z: int] :
% 0.33/0.65 ( ( ring_1_of_int_int @ ( times_times_int @ W @ Z ) )
% 0.33/0.65 = ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_mult
% 0.33/0.65 thf(fact_7026_of__int__add,axiom,
% 0.33/0.65 ! [W: int,Z: int] :
% 0.33/0.65 ( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z ) )
% 0.33/0.65 = ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_add
% 0.33/0.65 thf(fact_7027_of__int__add,axiom,
% 0.33/0.65 ! [W: int,Z: int] :
% 0.33/0.65 ( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z ) )
% 0.33/0.65 = ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_add
% 0.33/0.65 thf(fact_7028_of__int__add,axiom,
% 0.33/0.65 ! [W: int,Z: int] :
% 0.33/0.65 ( ( ring_1_of_int_rat @ ( plus_plus_int @ W @ Z ) )
% 0.33/0.65 = ( plus_plus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_add
% 0.33/0.65 thf(fact_7029_negative__zless,axiom,
% 0.33/0.65 ! [N2: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % negative_zless
% 0.33/0.65 thf(fact_7030_of__int__diff,axiom,
% 0.33/0.65 ! [W: int,Z: int] :
% 0.33/0.65 ( ( ring_1_of_int_real @ ( minus_minus_int @ W @ Z ) )
% 0.33/0.65 = ( minus_minus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_diff
% 0.33/0.65 thf(fact_7031_of__int__diff,axiom,
% 0.33/0.65 ! [W: int,Z: int] :
% 0.33/0.65 ( ( ring_1_of_int_rat @ ( minus_minus_int @ W @ Z ) )
% 0.33/0.65 = ( minus_minus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_diff
% 0.33/0.65 thf(fact_7032_of__int__diff,axiom,
% 0.33/0.65 ! [W: int,Z: int] :
% 0.33/0.65 ( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z ) )
% 0.33/0.65 = ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_diff
% 0.33/0.65 thf(fact_7033_of__int__power,axiom,
% 0.33/0.65 ! [Z: int,N2: nat] :
% 0.33/0.65 ( ( ring_1_of_int_rat @ ( power_power_int @ Z @ N2 ) )
% 0.33/0.65 = ( power_power_rat @ ( ring_1_of_int_rat @ Z ) @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_power
% 0.33/0.65 thf(fact_7034_of__int__power,axiom,
% 0.33/0.65 ! [Z: int,N2: nat] :
% 0.33/0.65 ( ( ring_1_of_int_real @ ( power_power_int @ Z @ N2 ) )
% 0.33/0.65 = ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_power
% 0.33/0.65 thf(fact_7035_of__int__power,axiom,
% 0.33/0.65 ! [Z: int,N2: nat] :
% 0.33/0.65 ( ( ring_1_of_int_int @ ( power_power_int @ Z @ N2 ) )
% 0.33/0.65 = ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_power
% 0.33/0.65 thf(fact_7036_of__int__power,axiom,
% 0.33/0.65 ! [Z: int,N2: nat] :
% 0.33/0.65 ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z @ N2 ) )
% 0.33/0.65 = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z ) @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_power
% 0.33/0.65 thf(fact_7037_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.33/0.65 ! [B: int,W: nat,X: int] :
% 0.33/0.65 ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W )
% 0.33/0.65 = ( ring_1_of_int_rat @ X ) )
% 0.33/0.65 = ( ( power_power_int @ B @ W )
% 0.33/0.65 = X ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_eq_of_int_power_cancel_iff
% 0.33/0.65 thf(fact_7038_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.33/0.65 ! [B: int,W: nat,X: int] :
% 0.33/0.65 ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
% 0.33/0.65 = ( ring_1_of_int_real @ X ) )
% 0.33/0.65 = ( ( power_power_int @ B @ W )
% 0.33/0.65 = X ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_eq_of_int_power_cancel_iff
% 0.33/0.65 thf(fact_7039_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.33/0.65 ! [B: int,W: nat,X: int] :
% 0.33/0.65 ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
% 0.33/0.65 = ( ring_1_of_int_int @ X ) )
% 0.33/0.65 = ( ( power_power_int @ B @ W )
% 0.33/0.65 = X ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_eq_of_int_power_cancel_iff
% 0.33/0.65 thf(fact_7040_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.33/0.65 ! [B: int,W: nat,X: int] :
% 0.33/0.65 ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W )
% 0.33/0.65 = ( ring_17405671764205052669omplex @ X ) )
% 0.33/0.65 = ( ( power_power_int @ B @ W )
% 0.33/0.65 = X ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_eq_of_int_power_cancel_iff
% 0.33/0.65 thf(fact_7041_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: int,B: int,W: nat] :
% 0.33/0.65 ( ( ( ring_1_of_int_rat @ X )
% 0.33/0.65 = ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 0.33/0.65 = ( X
% 0.33/0.65 = ( power_power_int @ B @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_power_eq_of_int_cancel_iff
% 0.33/0.65 thf(fact_7042_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: int,B: int,W: nat] :
% 0.33/0.65 ( ( ( ring_1_of_int_real @ X )
% 0.33/0.65 = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 0.33/0.65 = ( X
% 0.33/0.65 = ( power_power_int @ B @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_power_eq_of_int_cancel_iff
% 0.33/0.65 thf(fact_7043_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: int,B: int,W: nat] :
% 0.33/0.65 ( ( ( ring_1_of_int_int @ X )
% 0.33/0.65 = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 0.33/0.65 = ( X
% 0.33/0.65 = ( power_power_int @ B @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_power_eq_of_int_cancel_iff
% 0.33/0.65 thf(fact_7044_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: int,B: int,W: nat] :
% 0.33/0.65 ( ( ( ring_17405671764205052669omplex @ X )
% 0.33/0.65 = ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W ) )
% 0.33/0.65 = ( X
% 0.33/0.65 = ( power_power_int @ B @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_power_eq_of_int_cancel_iff
% 0.33/0.65 thf(fact_7045_dbl__inc__simps_I2_J,axiom,
% 0.33/0.65 ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
% 0.33/0.65 = one_one_complex ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_inc_simps(2)
% 0.33/0.65 thf(fact_7046_dbl__inc__simps_I2_J,axiom,
% 0.33/0.65 ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
% 0.33/0.65 = one_one_real ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_inc_simps(2)
% 0.33/0.65 thf(fact_7047_dbl__inc__simps_I2_J,axiom,
% 0.33/0.65 ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
% 0.33/0.65 = one_one_rat ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_inc_simps(2)
% 0.33/0.65 thf(fact_7048_dbl__inc__simps_I2_J,axiom,
% 0.33/0.65 ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
% 0.33/0.65 = one_one_int ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_inc_simps(2)
% 0.33/0.65 thf(fact_7049_of__nat__of__bool,axiom,
% 0.33/0.65 ! [P: $o] :
% 0.33/0.65 ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.33/0.65 = ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_of_bool
% 0.33/0.65 thf(fact_7050_of__nat__of__bool,axiom,
% 0.33/0.65 ! [P: $o] :
% 0.33/0.65 ( ( semiri8010041392384452111omplex @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.33/0.65 = ( zero_n1201886186963655149omplex @ P ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_of_bool
% 0.33/0.65 thf(fact_7051_of__nat__of__bool,axiom,
% 0.33/0.65 ! [P: $o] :
% 0.33/0.65 ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.33/0.65 = ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_of_bool
% 0.33/0.65 thf(fact_7052_of__nat__of__bool,axiom,
% 0.33/0.65 ! [P: $o] :
% 0.33/0.65 ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.33/0.65 = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_of_bool
% 0.33/0.65 thf(fact_7053_of__nat__of__bool,axiom,
% 0.33/0.65 ! [P: $o] :
% 0.33/0.65 ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.33/0.65 = ( zero_n356916108424825756nteger @ P ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_of_bool
% 0.33/0.65 thf(fact_7054_dbl__inc__simps_I4_J,axiom,
% 0.33/0.65 ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.65 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_inc_simps(4)
% 0.33/0.65 thf(fact_7055_dbl__inc__simps_I4_J,axiom,
% 0.33/0.65 ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.65 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_inc_simps(4)
% 0.33/0.65 thf(fact_7056_dbl__inc__simps_I4_J,axiom,
% 0.33/0.65 ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.33/0.65 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_inc_simps(4)
% 0.33/0.65 thf(fact_7057_dbl__inc__simps_I4_J,axiom,
% 0.33/0.65 ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.33/0.65 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_inc_simps(4)
% 0.33/0.65 thf(fact_7058_dbl__inc__simps_I4_J,axiom,
% 0.33/0.65 ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.33/0.65 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_inc_simps(4)
% 0.33/0.65 thf(fact_7059_dbl__inc__simps_I5_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) )
% 0.33/0.65 = ( numera6690914467698888265omplex @ ( bit1 @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_inc_simps(5)
% 0.33/0.65 thf(fact_7060_dbl__inc__simps_I5_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) )
% 0.33/0.65 = ( numeral_numeral_real @ ( bit1 @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_inc_simps(5)
% 0.33/0.65 thf(fact_7061_dbl__inc__simps_I5_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) )
% 0.33/0.65 = ( numeral_numeral_rat @ ( bit1 @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_inc_simps(5)
% 0.33/0.65 thf(fact_7062_dbl__inc__simps_I5_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
% 0.33/0.65 = ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_inc_simps(5)
% 0.33/0.65 thf(fact_7063_of__nat__sum,axiom,
% 0.33/0.65 ! [F: int > nat,A2: set_int] :
% 0.33/0.65 ( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F @ A2 ) )
% 0.33/0.65 = ( groups4538972089207619220nt_int
% 0.33/0.65 @ ^ [X2: int] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
% 0.33/0.65 @ A2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_sum
% 0.33/0.65 thf(fact_7064_of__nat__sum,axiom,
% 0.33/0.65 ! [F: complex > nat,A2: set_complex] :
% 0.33/0.65 ( ( semiri8010041392384452111omplex @ ( groups5693394587270226106ex_nat @ F @ A2 ) )
% 0.33/0.65 = ( groups7754918857620584856omplex
% 0.33/0.65 @ ^ [X2: complex] : ( semiri8010041392384452111omplex @ ( F @ X2 ) )
% 0.33/0.65 @ A2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_sum
% 0.33/0.65 thf(fact_7065_of__nat__sum,axiom,
% 0.33/0.65 ! [F: nat > nat,A2: set_nat] :
% 0.33/0.65 ( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 0.33/0.65 = ( groups3539618377306564664at_int
% 0.33/0.65 @ ^ [X2: nat] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
% 0.33/0.65 @ A2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_sum
% 0.33/0.65 thf(fact_7066_of__nat__sum,axiom,
% 0.33/0.65 ! [F: nat > nat,A2: set_nat] :
% 0.33/0.65 ( ( semiri8010041392384452111omplex @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 0.33/0.65 = ( groups2073611262835488442omplex
% 0.33/0.65 @ ^ [X2: nat] : ( semiri8010041392384452111omplex @ ( F @ X2 ) )
% 0.33/0.65 @ A2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_sum
% 0.33/0.65 thf(fact_7067_of__nat__sum,axiom,
% 0.33/0.65 ! [F: nat > nat,A2: set_nat] :
% 0.33/0.65 ( ( semiri1316708129612266289at_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 0.33/0.65 = ( groups3542108847815614940at_nat
% 0.33/0.65 @ ^ [X2: nat] : ( semiri1316708129612266289at_nat @ ( F @ X2 ) )
% 0.33/0.65 @ A2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_sum
% 0.33/0.65 thf(fact_7068_of__nat__sum,axiom,
% 0.33/0.65 ! [F: nat > nat,A2: set_nat] :
% 0.33/0.65 ( ( semiri5074537144036343181t_real @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 0.33/0.65 = ( groups6591440286371151544t_real
% 0.33/0.65 @ ^ [X2: nat] : ( semiri5074537144036343181t_real @ ( F @ X2 ) )
% 0.33/0.65 @ A2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_sum
% 0.33/0.65 thf(fact_7069_dbl__dec__simps_I5_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) )
% 0.33/0.65 = ( numera6690914467698888265omplex @ ( bitM @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_dec_simps(5)
% 0.33/0.65 thf(fact_7070_dbl__dec__simps_I5_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) )
% 0.33/0.65 = ( numeral_numeral_real @ ( bitM @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_dec_simps(5)
% 0.33/0.65 thf(fact_7071_dbl__dec__simps_I5_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) )
% 0.33/0.65 = ( numeral_numeral_rat @ ( bitM @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_dec_simps(5)
% 0.33/0.65 thf(fact_7072_dbl__dec__simps_I5_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) )
% 0.33/0.65 = ( numeral_numeral_int @ ( bitM @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_dec_simps(5)
% 0.33/0.65 thf(fact_7073_of__nat__le__0__iff,axiom,
% 0.33/0.65 ! [M: nat] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
% 0.33/0.65 = ( M = zero_zero_nat ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_le_0_iff
% 0.33/0.65 thf(fact_7074_of__nat__le__0__iff,axiom,
% 0.33/0.65 ! [M: nat] :
% 0.33/0.65 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
% 0.33/0.65 = ( M = zero_zero_nat ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_le_0_iff
% 0.33/0.65 thf(fact_7075_of__nat__le__0__iff,axiom,
% 0.33/0.65 ! [M: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
% 0.33/0.65 = ( M = zero_zero_nat ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_le_0_iff
% 0.33/0.65 thf(fact_7076_of__nat__le__0__iff,axiom,
% 0.33/0.65 ! [M: nat] :
% 0.33/0.65 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
% 0.33/0.65 = ( M = zero_zero_nat ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_le_0_iff
% 0.33/0.65 thf(fact_7077_of__nat__Suc,axiom,
% 0.33/0.65 ! [M: nat] :
% 0.33/0.65 ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
% 0.33/0.65 = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_Suc
% 0.33/0.65 thf(fact_7078_of__nat__Suc,axiom,
% 0.33/0.65 ! [M: nat] :
% 0.33/0.65 ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
% 0.33/0.65 = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_Suc
% 0.33/0.65 thf(fact_7079_of__nat__Suc,axiom,
% 0.33/0.65 ! [M: nat] :
% 0.33/0.65 ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
% 0.33/0.65 = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_Suc
% 0.33/0.65 thf(fact_7080_of__nat__Suc,axiom,
% 0.33/0.65 ! [M: nat] :
% 0.33/0.65 ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
% 0.33/0.65 = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_Suc
% 0.33/0.65 thf(fact_7081_of__nat__Suc,axiom,
% 0.33/0.65 ! [M: nat] :
% 0.33/0.65 ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
% 0.33/0.65 = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_Suc
% 0.33/0.65 thf(fact_7082_real__of__nat__less__numeral__iff,axiom,
% 0.33/0.65 ! [N2: nat,W: num] :
% 0.33/0.65 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( numeral_numeral_real @ W ) )
% 0.33/0.65 = ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_of_nat_less_numeral_iff
% 0.33/0.65 thf(fact_7083_numeral__less__real__of__nat__iff,axiom,
% 0.33/0.65 ! [W: num,N2: nat] :
% 0.33/0.65 ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N2 ) )
% 0.33/0.65 = ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_less_real_of_nat_iff
% 0.33/0.65 thf(fact_7084_numeral__le__real__of__nat__iff,axiom,
% 0.33/0.65 ! [N2: num,M: nat] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ ( semiri5074537144036343181t_real @ M ) )
% 0.33/0.65 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ M ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_le_real_of_nat_iff
% 0.33/0.65 thf(fact_7085_of__int__sum,axiom,
% 0.33/0.65 ! [F: complex > int,A2: set_complex] :
% 0.33/0.65 ( ( ring_17405671764205052669omplex @ ( groups5690904116761175830ex_int @ F @ A2 ) )
% 0.33/0.65 = ( groups7754918857620584856omplex
% 0.33/0.65 @ ^ [X2: complex] : ( ring_17405671764205052669omplex @ ( F @ X2 ) )
% 0.33/0.65 @ A2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_sum
% 0.33/0.65 thf(fact_7086_of__int__sum,axiom,
% 0.33/0.65 ! [F: nat > int,A2: set_nat] :
% 0.33/0.65 ( ( ring_1_of_int_real @ ( groups3539618377306564664at_int @ F @ A2 ) )
% 0.33/0.65 = ( groups6591440286371151544t_real
% 0.33/0.65 @ ^ [X2: nat] : ( ring_1_of_int_real @ ( F @ X2 ) )
% 0.33/0.65 @ A2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_sum
% 0.33/0.65 thf(fact_7087_of__int__sum,axiom,
% 0.33/0.65 ! [F: int > int,A2: set_int] :
% 0.33/0.65 ( ( ring_1_of_int_real @ ( groups4538972089207619220nt_int @ F @ A2 ) )
% 0.33/0.65 = ( groups8778361861064173332t_real
% 0.33/0.65 @ ^ [X2: int] : ( ring_1_of_int_real @ ( F @ X2 ) )
% 0.33/0.65 @ A2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_sum
% 0.33/0.65 thf(fact_7088_of__int__sum,axiom,
% 0.33/0.65 ! [F: int > int,A2: set_int] :
% 0.33/0.65 ( ( ring_1_of_int_rat @ ( groups4538972089207619220nt_int @ F @ A2 ) )
% 0.33/0.65 = ( groups3906332499630173760nt_rat
% 0.33/0.65 @ ^ [X2: int] : ( ring_1_of_int_rat @ ( F @ X2 ) )
% 0.33/0.65 @ A2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_sum
% 0.33/0.65 thf(fact_7089_of__int__sum,axiom,
% 0.33/0.65 ! [F: int > int,A2: set_int] :
% 0.33/0.65 ( ( ring_1_of_int_int @ ( groups4538972089207619220nt_int @ F @ A2 ) )
% 0.33/0.65 = ( groups4538972089207619220nt_int
% 0.33/0.65 @ ^ [X2: int] : ( ring_1_of_int_int @ ( F @ X2 ) )
% 0.33/0.65 @ A2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_sum
% 0.33/0.65 thf(fact_7090_pred__numeral__simps_I2_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( pred_numeral @ ( bit0 @ K ) )
% 0.33/0.65 = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % pred_numeral_simps(2)
% 0.33/0.65 thf(fact_7091_of__nat__0__less__iff,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N2 ) )
% 0.33/0.65 = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_0_less_iff
% 0.33/0.65 thf(fact_7092_of__nat__0__less__iff,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) )
% 0.33/0.65 = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_0_less_iff
% 0.33/0.65 thf(fact_7093_of__nat__0__less__iff,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) )
% 0.33/0.65 = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_0_less_iff
% 0.33/0.65 thf(fact_7094_of__nat__0__less__iff,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) )
% 0.33/0.65 = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_0_less_iff
% 0.33/0.65 thf(fact_7095_dbl__inc__simps_I1_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 0.33/0.65 = ( uminus_uminus_real @ ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_inc_simps(1)
% 0.33/0.65 thf(fact_7096_dbl__inc__simps_I1_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.33/0.65 = ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_inc_simps(1)
% 0.33/0.65 thf(fact_7097_dbl__inc__simps_I1_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 0.33/0.65 = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_inc_simps(1)
% 0.33/0.65 thf(fact_7098_dbl__inc__simps_I1_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.33/0.65 = ( uminus_uminus_rat @ ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_inc_simps(1)
% 0.33/0.65 thf(fact_7099_dbl__inc__simps_I1_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.33/0.65 = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_inc_simps(1)
% 0.33/0.65 thf(fact_7100_dbl__dec__simps_I1_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 0.33/0.65 = ( uminus_uminus_real @ ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_dec_simps(1)
% 0.33/0.65 thf(fact_7101_dbl__dec__simps_I1_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.33/0.65 = ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_dec_simps(1)
% 0.33/0.65 thf(fact_7102_dbl__dec__simps_I1_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 0.33/0.65 = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_dec_simps(1)
% 0.33/0.65 thf(fact_7103_dbl__dec__simps_I1_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.33/0.65 = ( uminus_uminus_rat @ ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_dec_simps(1)
% 0.33/0.65 thf(fact_7104_dbl__dec__simps_I1_J,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.33/0.65 = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_dec_simps(1)
% 0.33/0.65 thf(fact_7105_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.33/0.65 ! [X: num,N2: nat,Y: nat] :
% 0.33/0.65 ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 )
% 0.33/0.65 = ( semiri681578069525770553at_rat @ Y ) )
% 0.33/0.65 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
% 0.33/0.65 = Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_power_eq_of_nat_cancel_iff
% 0.33/0.65 thf(fact_7106_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.33/0.65 ! [X: num,N2: nat,Y: nat] :
% 0.33/0.65 ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
% 0.33/0.65 = ( semiri1314217659103216013at_int @ Y ) )
% 0.33/0.65 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
% 0.33/0.65 = Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_power_eq_of_nat_cancel_iff
% 0.33/0.65 thf(fact_7107_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.33/0.65 ! [X: num,N2: nat,Y: nat] :
% 0.33/0.65 ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 )
% 0.33/0.65 = ( semiri5074537144036343181t_real @ Y ) )
% 0.33/0.65 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
% 0.33/0.65 = Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_power_eq_of_nat_cancel_iff
% 0.33/0.65 thf(fact_7108_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.33/0.65 ! [X: num,N2: nat,Y: nat] :
% 0.33/0.65 ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
% 0.33/0.65 = ( semiri1316708129612266289at_nat @ Y ) )
% 0.33/0.65 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
% 0.33/0.65 = Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_power_eq_of_nat_cancel_iff
% 0.33/0.65 thf(fact_7109_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.33/0.65 ! [X: num,N2: nat,Y: nat] :
% 0.33/0.65 ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N2 )
% 0.33/0.65 = ( semiri8010041392384452111omplex @ Y ) )
% 0.33/0.65 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
% 0.33/0.65 = Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_power_eq_of_nat_cancel_iff
% 0.33/0.65 thf(fact_7110_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [Y: nat,X: num,N2: nat] :
% 0.33/0.65 ( ( ( semiri681578069525770553at_rat @ Y )
% 0.33/0.65 = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) )
% 0.33/0.65 = ( Y
% 0.33/0.65 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_of_nat_eq_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7111_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [Y: nat,X: num,N2: nat] :
% 0.33/0.65 ( ( ( semiri1314217659103216013at_int @ Y )
% 0.33/0.65 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
% 0.33/0.65 = ( Y
% 0.33/0.65 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_of_nat_eq_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7112_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [Y: nat,X: num,N2: nat] :
% 0.33/0.65 ( ( ( semiri5074537144036343181t_real @ Y )
% 0.33/0.65 = ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
% 0.33/0.65 = ( Y
% 0.33/0.65 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_of_nat_eq_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7113_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [Y: nat,X: num,N2: nat] :
% 0.33/0.65 ( ( ( semiri1316708129612266289at_nat @ Y )
% 0.33/0.65 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
% 0.33/0.65 = ( Y
% 0.33/0.65 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_of_nat_eq_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7114_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [Y: nat,X: num,N2: nat] :
% 0.33/0.65 ( ( ( semiri8010041392384452111omplex @ Y )
% 0.33/0.65 = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N2 ) )
% 0.33/0.65 = ( Y
% 0.33/0.65 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_of_nat_eq_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7115_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.33/0.65 ! [B: nat,W: nat,X: nat] :
% 0.33/0.65 ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
% 0.33/0.65 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_less_of_nat_power_cancel_iff
% 0.33/0.65 thf(fact_7116_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.33/0.65 ! [B: nat,W: nat,X: nat] :
% 0.33/0.65 ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
% 0.33/0.65 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_less_of_nat_power_cancel_iff
% 0.33/0.65 thf(fact_7117_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.33/0.65 ! [B: nat,W: nat,X: nat] :
% 0.33/0.65 ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
% 0.33/0.65 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_less_of_nat_power_cancel_iff
% 0.33/0.65 thf(fact_7118_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.33/0.65 ! [B: nat,W: nat,X: nat] :
% 0.33/0.65 ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
% 0.33/0.65 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_less_of_nat_power_cancel_iff
% 0.33/0.65 thf(fact_7119_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.33/0.65 ! [X: nat,B: nat,W: nat] :
% 0.33/0.65 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 0.33/0.65 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_power_less_of_nat_cancel_iff
% 0.33/0.65 thf(fact_7120_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.33/0.65 ! [X: nat,B: nat,W: nat] :
% 0.33/0.65 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 0.33/0.65 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_power_less_of_nat_cancel_iff
% 0.33/0.65 thf(fact_7121_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.33/0.65 ! [X: nat,B: nat,W: nat] :
% 0.33/0.65 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 0.33/0.65 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_power_less_of_nat_cancel_iff
% 0.33/0.65 thf(fact_7122_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.33/0.65 ! [X: nat,B: nat,W: nat] :
% 0.33/0.65 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 0.33/0.65 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_power_less_of_nat_cancel_iff
% 0.33/0.65 thf(fact_7123_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.33/0.65 ! [B: nat,W: nat,X: nat] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
% 0.33/0.65 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_le_of_nat_power_cancel_iff
% 0.33/0.65 thf(fact_7124_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.33/0.65 ! [B: nat,W: nat,X: nat] :
% 0.33/0.65 ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
% 0.33/0.65 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_le_of_nat_power_cancel_iff
% 0.33/0.65 thf(fact_7125_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.33/0.65 ! [B: nat,W: nat,X: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
% 0.33/0.65 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_le_of_nat_power_cancel_iff
% 0.33/0.65 thf(fact_7126_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.33/0.65 ! [B: nat,W: nat,X: nat] :
% 0.33/0.65 ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
% 0.33/0.65 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_le_of_nat_power_cancel_iff
% 0.33/0.65 thf(fact_7127_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.33/0.65 ! [X: nat,B: nat,W: nat] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 0.33/0.65 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_power_le_of_nat_cancel_iff
% 0.33/0.65 thf(fact_7128_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.33/0.65 ! [X: nat,B: nat,W: nat] :
% 0.33/0.65 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 0.33/0.65 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_power_le_of_nat_cancel_iff
% 0.33/0.65 thf(fact_7129_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.33/0.65 ! [X: nat,B: nat,W: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 0.33/0.65 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_power_le_of_nat_cancel_iff
% 0.33/0.65 thf(fact_7130_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.33/0.65 ! [X: nat,B: nat,W: nat] :
% 0.33/0.65 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 0.33/0.65 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_power_le_of_nat_cancel_iff
% 0.33/0.65 thf(fact_7131_of__int__0__le__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 0.33/0.65 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_0_le_iff
% 0.33/0.65 thf(fact_7132_of__int__0__le__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.33/0.65 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_0_le_iff
% 0.33/0.65 thf(fact_7133_of__int__0__le__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 0.33/0.65 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_0_le_iff
% 0.33/0.65 thf(fact_7134_of__int__le__0__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 0.33/0.65 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_le_0_iff
% 0.33/0.65 thf(fact_7135_of__int__le__0__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
% 0.33/0.65 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_le_0_iff
% 0.33/0.65 thf(fact_7136_of__int__le__0__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 0.33/0.65 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_le_0_iff
% 0.33/0.65 thf(fact_7137_of__int__0__less__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 0.33/0.65 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_0_less_iff
% 0.33/0.65 thf(fact_7138_of__int__0__less__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.33/0.65 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_0_less_iff
% 0.33/0.65 thf(fact_7139_of__int__0__less__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 0.33/0.65 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_0_less_iff
% 0.33/0.65 thf(fact_7140_of__int__less__0__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 0.33/0.65 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_less_0_iff
% 0.33/0.65 thf(fact_7141_of__int__less__0__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
% 0.33/0.65 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_less_0_iff
% 0.33/0.65 thf(fact_7142_of__int__less__0__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 0.33/0.65 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_less_0_iff
% 0.33/0.65 thf(fact_7143_of__int__le__numeral__iff,axiom,
% 0.33/0.65 ! [Z: int,N2: num] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N2 ) )
% 0.33/0.65 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_le_numeral_iff
% 0.33/0.65 thf(fact_7144_of__int__le__numeral__iff,axiom,
% 0.33/0.65 ! [Z: int,N2: num] :
% 0.33/0.65 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N2 ) )
% 0.33/0.65 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_le_numeral_iff
% 0.33/0.65 thf(fact_7145_of__int__le__numeral__iff,axiom,
% 0.33/0.65 ! [Z: int,N2: num] :
% 0.33/0.65 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N2 ) )
% 0.33/0.65 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_le_numeral_iff
% 0.33/0.65 thf(fact_7146_of__int__numeral__le__iff,axiom,
% 0.33/0.65 ! [N2: num,Z: int] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ ( ring_1_of_int_real @ Z ) )
% 0.33/0.65 = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_numeral_le_iff
% 0.33/0.65 thf(fact_7147_of__int__numeral__le__iff,axiom,
% 0.33/0.65 ! [N2: num,Z: int] :
% 0.33/0.65 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ ( ring_1_of_int_rat @ Z ) )
% 0.33/0.65 = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_numeral_le_iff
% 0.33/0.65 thf(fact_7148_of__int__numeral__le__iff,axiom,
% 0.33/0.65 ! [N2: num,Z: int] :
% 0.33/0.65 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ ( ring_1_of_int_int @ Z ) )
% 0.33/0.65 = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_numeral_le_iff
% 0.33/0.65 thf(fact_7149_of__int__numeral__less__iff,axiom,
% 0.33/0.65 ! [N2: num,Z: int] :
% 0.33/0.65 ( ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ ( ring_1_of_int_real @ Z ) )
% 0.33/0.65 = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_numeral_less_iff
% 0.33/0.65 thf(fact_7150_of__int__numeral__less__iff,axiom,
% 0.33/0.65 ! [N2: num,Z: int] :
% 0.33/0.65 ( ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ ( ring_1_of_int_rat @ Z ) )
% 0.33/0.65 = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_numeral_less_iff
% 0.33/0.65 thf(fact_7151_of__int__numeral__less__iff,axiom,
% 0.33/0.65 ! [N2: num,Z: int] :
% 0.33/0.65 ( ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ ( ring_1_of_int_int @ Z ) )
% 0.33/0.65 = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_numeral_less_iff
% 0.33/0.65 thf(fact_7152_of__int__less__numeral__iff,axiom,
% 0.33/0.65 ! [Z: int,N2: num] :
% 0.33/0.65 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N2 ) )
% 0.33/0.65 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_less_numeral_iff
% 0.33/0.65 thf(fact_7153_of__int__less__numeral__iff,axiom,
% 0.33/0.65 ! [Z: int,N2: num] :
% 0.33/0.65 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N2 ) )
% 0.33/0.65 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_less_numeral_iff
% 0.33/0.65 thf(fact_7154_of__int__less__numeral__iff,axiom,
% 0.33/0.65 ! [Z: int,N2: num] :
% 0.33/0.65 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N2 ) )
% 0.33/0.65 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_less_numeral_iff
% 0.33/0.65 thf(fact_7155_of__int__1__le__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 0.33/0.65 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_1_le_iff
% 0.33/0.65 thf(fact_7156_of__int__1__le__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.33/0.65 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_1_le_iff
% 0.33/0.65 thf(fact_7157_of__int__1__le__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 0.33/0.65 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_1_le_iff
% 0.33/0.65 thf(fact_7158_of__int__le__1__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 0.33/0.65 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_le_1_iff
% 0.33/0.65 thf(fact_7159_of__int__le__1__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
% 0.33/0.65 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_le_1_iff
% 0.33/0.65 thf(fact_7160_of__int__le__1__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 0.33/0.65 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_le_1_iff
% 0.33/0.65 thf(fact_7161_of__int__less__1__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 0.33/0.65 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_less_1_iff
% 0.33/0.65 thf(fact_7162_of__int__less__1__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
% 0.33/0.65 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_less_1_iff
% 0.33/0.65 thf(fact_7163_of__int__less__1__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 0.33/0.65 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_less_1_iff
% 0.33/0.65 thf(fact_7164_of__int__1__less__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 0.33/0.65 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_1_less_iff
% 0.33/0.65 thf(fact_7165_of__int__1__less__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.33/0.65 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_1_less_iff
% 0.33/0.65 thf(fact_7166_of__int__1__less__iff,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 0.33/0.65 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_1_less_iff
% 0.33/0.65 thf(fact_7167_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: num,N2: nat,Y: int] :
% 0.33/0.65 ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N2 )
% 0.33/0.65 = ( ring_17405671764205052669omplex @ Y ) )
% 0.33/0.65 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
% 0.33/0.65 = Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_power_eq_of_int_cancel_iff
% 0.33/0.65 thf(fact_7168_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: num,N2: nat,Y: int] :
% 0.33/0.65 ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 )
% 0.33/0.65 = ( ring_1_of_int_real @ Y ) )
% 0.33/0.65 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
% 0.33/0.65 = Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_power_eq_of_int_cancel_iff
% 0.33/0.65 thf(fact_7169_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: num,N2: nat,Y: int] :
% 0.33/0.65 ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 )
% 0.33/0.65 = ( ring_1_of_int_rat @ Y ) )
% 0.33/0.65 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
% 0.33/0.65 = Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_power_eq_of_int_cancel_iff
% 0.33/0.65 thf(fact_7170_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: num,N2: nat,Y: int] :
% 0.33/0.65 ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
% 0.33/0.65 = ( ring_1_of_int_int @ Y ) )
% 0.33/0.65 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
% 0.33/0.65 = Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_power_eq_of_int_cancel_iff
% 0.33/0.65 thf(fact_7171_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [Y: int,X: num,N2: nat] :
% 0.33/0.65 ( ( ( ring_17405671764205052669omplex @ Y )
% 0.33/0.65 = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N2 ) )
% 0.33/0.65 = ( Y
% 0.33/0.65 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_eq_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7172_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [Y: int,X: num,N2: nat] :
% 0.33/0.65 ( ( ( ring_1_of_int_real @ Y )
% 0.33/0.65 = ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
% 0.33/0.65 = ( Y
% 0.33/0.65 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_eq_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7173_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [Y: int,X: num,N2: nat] :
% 0.33/0.65 ( ( ( ring_1_of_int_rat @ Y )
% 0.33/0.65 = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) )
% 0.33/0.65 = ( Y
% 0.33/0.65 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_eq_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7174_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [Y: int,X: num,N2: nat] :
% 0.33/0.65 ( ( ( ring_1_of_int_int @ Y )
% 0.33/0.65 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
% 0.33/0.65 = ( Y
% 0.33/0.65 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_eq_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7175_of__int__le__of__int__power__cancel__iff,axiom,
% 0.33/0.65 ! [B: int,W: nat,X: int] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 0.33/0.65 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_le_of_int_power_cancel_iff
% 0.33/0.65 thf(fact_7176_of__int__le__of__int__power__cancel__iff,axiom,
% 0.33/0.65 ! [B: int,W: nat,X: int] :
% 0.33/0.65 ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
% 0.33/0.65 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_le_of_int_power_cancel_iff
% 0.33/0.65 thf(fact_7177_of__int__le__of__int__power__cancel__iff,axiom,
% 0.33/0.65 ! [B: int,W: nat,X: int] :
% 0.33/0.65 ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 0.33/0.65 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_le_of_int_power_cancel_iff
% 0.33/0.65 thf(fact_7178_of__int__power__le__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: int,B: int,W: nat] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 0.33/0.65 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_power_le_of_int_cancel_iff
% 0.33/0.65 thf(fact_7179_of__int__power__le__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: int,B: int,W: nat] :
% 0.33/0.65 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 0.33/0.65 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_power_le_of_int_cancel_iff
% 0.33/0.65 thf(fact_7180_of__int__power__le__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: int,B: int,W: nat] :
% 0.33/0.65 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 0.33/0.65 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_power_le_of_int_cancel_iff
% 0.33/0.65 thf(fact_7181_of__int__power__less__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: int,B: int,W: nat] :
% 0.33/0.65 ( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 0.33/0.65 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_power_less_of_int_cancel_iff
% 0.33/0.65 thf(fact_7182_of__int__power__less__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: int,B: int,W: nat] :
% 0.33/0.65 ( ( ord_less_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 0.33/0.65 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_power_less_of_int_cancel_iff
% 0.33/0.65 thf(fact_7183_of__int__power__less__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: int,B: int,W: nat] :
% 0.33/0.65 ( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 0.33/0.65 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_power_less_of_int_cancel_iff
% 0.33/0.65 thf(fact_7184_of__int__less__of__int__power__cancel__iff,axiom,
% 0.33/0.65 ! [B: int,W: nat,X: int] :
% 0.33/0.65 ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 0.33/0.65 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_less_of_int_power_cancel_iff
% 0.33/0.65 thf(fact_7185_of__int__less__of__int__power__cancel__iff,axiom,
% 0.33/0.65 ! [B: int,W: nat,X: int] :
% 0.33/0.65 ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
% 0.33/0.65 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_less_of_int_power_cancel_iff
% 0.33/0.65 thf(fact_7186_of__int__less__of__int__power__cancel__iff,axiom,
% 0.33/0.65 ! [B: int,W: nat,X: int] :
% 0.33/0.65 ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 0.33/0.65 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_less_of_int_power_cancel_iff
% 0.33/0.65 thf(fact_7187_of__nat__zero__less__power__iff,axiom,
% 0.33/0.65 ! [X: nat,N2: nat] :
% 0.33/0.65 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N2 ) )
% 0.33/0.65 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.33/0.65 | ( N2 = zero_zero_nat ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_zero_less_power_iff
% 0.33/0.65 thf(fact_7188_of__nat__zero__less__power__iff,axiom,
% 0.33/0.65 ! [X: nat,N2: nat] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N2 ) )
% 0.33/0.65 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.33/0.65 | ( N2 = zero_zero_nat ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_zero_less_power_iff
% 0.33/0.65 thf(fact_7189_of__nat__zero__less__power__iff,axiom,
% 0.33/0.65 ! [X: nat,N2: nat] :
% 0.33/0.65 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N2 ) )
% 0.33/0.65 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.33/0.65 | ( N2 = zero_zero_nat ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_zero_less_power_iff
% 0.33/0.65 thf(fact_7190_of__nat__zero__less__power__iff,axiom,
% 0.33/0.65 ! [X: nat,N2: nat] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N2 ) )
% 0.33/0.65 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.33/0.65 | ( N2 = zero_zero_nat ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_zero_less_power_iff
% 0.33/0.65 thf(fact_7191_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.33/0.65 ! [I2: num,N2: nat,X: nat] :
% 0.33/0.65 ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N2 ) @ ( semiri681578069525770553at_rat @ X ) )
% 0.33/0.65 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_power_less_of_nat_cancel_iff
% 0.33/0.65 thf(fact_7192_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.33/0.65 ! [I2: num,N2: nat,X: nat] :
% 0.33/0.65 ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N2 ) @ ( semiri1314217659103216013at_int @ X ) )
% 0.33/0.65 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_power_less_of_nat_cancel_iff
% 0.33/0.65 thf(fact_7193_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.33/0.65 ! [I2: num,N2: nat,X: nat] :
% 0.33/0.65 ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N2 ) @ ( semiri5074537144036343181t_real @ X ) )
% 0.33/0.65 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_power_less_of_nat_cancel_iff
% 0.33/0.65 thf(fact_7194_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.33/0.65 ! [I2: num,N2: nat,X: nat] :
% 0.33/0.65 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X ) )
% 0.33/0.65 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_power_less_of_nat_cancel_iff
% 0.33/0.65 thf(fact_7195_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [X: nat,I2: num,N2: nat] :
% 0.33/0.65 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N2 ) )
% 0.33/0.65 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_less_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7196_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [X: nat,I2: num,N2: nat] :
% 0.33/0.65 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N2 ) )
% 0.33/0.65 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_less_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7197_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [X: nat,I2: num,N2: nat] :
% 0.33/0.65 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N2 ) )
% 0.33/0.65 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_less_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7198_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [X: nat,I2: num,N2: nat] :
% 0.33/0.65 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) )
% 0.33/0.65 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_less_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7199_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.33/0.65 ! [I2: num,N2: nat,X: nat] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N2 ) @ ( semiri5074537144036343181t_real @ X ) )
% 0.33/0.65 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_power_le_of_nat_cancel_iff
% 0.33/0.65 thf(fact_7200_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.33/0.65 ! [I2: num,N2: nat,X: nat] :
% 0.33/0.65 ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N2 ) @ ( semiri681578069525770553at_rat @ X ) )
% 0.33/0.65 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_power_le_of_nat_cancel_iff
% 0.33/0.65 thf(fact_7201_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.33/0.65 ! [I2: num,N2: nat,X: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X ) )
% 0.33/0.65 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_power_le_of_nat_cancel_iff
% 0.33/0.65 thf(fact_7202_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.33/0.65 ! [I2: num,N2: nat,X: nat] :
% 0.33/0.65 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N2 ) @ ( semiri1314217659103216013at_int @ X ) )
% 0.33/0.65 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_power_le_of_nat_cancel_iff
% 0.33/0.65 thf(fact_7203_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [X: nat,I2: num,N2: nat] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N2 ) )
% 0.33/0.65 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_le_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7204_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [X: nat,I2: num,N2: nat] :
% 0.33/0.65 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N2 ) )
% 0.33/0.65 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_le_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7205_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [X: nat,I2: num,N2: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) )
% 0.33/0.65 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_le_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7206_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [X: nat,I2: num,N2: nat] :
% 0.33/0.65 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N2 ) )
% 0.33/0.65 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_le_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7207_even__of__nat,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ N2 ) )
% 0.33/0.65 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % even_of_nat
% 0.33/0.65 thf(fact_7208_even__of__nat,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N2 ) )
% 0.33/0.65 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % even_of_nat
% 0.33/0.65 thf(fact_7209_even__of__nat,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N2 ) )
% 0.33/0.65 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % even_of_nat
% 0.33/0.65 thf(fact_7210_numeral__power__le__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: num,N2: nat,A: int] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
% 0.33/0.65 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_power_le_of_int_cancel_iff
% 0.33/0.65 thf(fact_7211_numeral__power__le__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: num,N2: nat,A: int] :
% 0.33/0.65 ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
% 0.33/0.65 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_power_le_of_int_cancel_iff
% 0.33/0.65 thf(fact_7212_numeral__power__le__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: num,N2: nat,A: int] :
% 0.33/0.65 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
% 0.33/0.65 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_power_le_of_int_cancel_iff
% 0.33/0.65 thf(fact_7213_of__int__le__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [A: int,X: num,N2: nat] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
% 0.33/0.65 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_le_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7214_of__int__le__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [A: int,X: num,N2: nat] :
% 0.33/0.65 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) )
% 0.33/0.65 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_le_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7215_of__int__le__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [A: int,X: num,N2: nat] :
% 0.33/0.65 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
% 0.33/0.65 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_le_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7216_numeral__power__less__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: num,N2: nat,A: int] :
% 0.33/0.65 ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
% 0.33/0.65 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_power_less_of_int_cancel_iff
% 0.33/0.65 thf(fact_7217_numeral__power__less__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: num,N2: nat,A: int] :
% 0.33/0.65 ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
% 0.33/0.65 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_power_less_of_int_cancel_iff
% 0.33/0.65 thf(fact_7218_numeral__power__less__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: num,N2: nat,A: int] :
% 0.33/0.65 ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
% 0.33/0.65 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_power_less_of_int_cancel_iff
% 0.33/0.65 thf(fact_7219_of__int__less__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [A: int,X: num,N2: nat] :
% 0.33/0.65 ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
% 0.33/0.65 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_less_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7220_of__int__less__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [A: int,X: num,N2: nat] :
% 0.33/0.65 ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) )
% 0.33/0.65 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_less_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7221_of__int__less__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [A: int,X: num,N2: nat] :
% 0.33/0.65 ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
% 0.33/0.65 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_less_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7222_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: num,N2: nat,Y: int] :
% 0.33/0.65 ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 )
% 0.33/0.65 = ( ring_1_of_int_real @ Y ) )
% 0.33/0.65 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
% 0.33/0.65 = Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % neg_numeral_power_eq_of_int_cancel_iff
% 0.33/0.65 thf(fact_7223_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: num,N2: nat,Y: int] :
% 0.33/0.65 ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
% 0.33/0.65 = ( ring_1_of_int_int @ Y ) )
% 0.33/0.65 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
% 0.33/0.65 = Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % neg_numeral_power_eq_of_int_cancel_iff
% 0.33/0.65 thf(fact_7224_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: num,N2: nat,Y: int] :
% 0.33/0.65 ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N2 )
% 0.33/0.65 = ( ring_17405671764205052669omplex @ Y ) )
% 0.33/0.65 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
% 0.33/0.65 = Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % neg_numeral_power_eq_of_int_cancel_iff
% 0.33/0.65 thf(fact_7225_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: num,N2: nat,Y: int] :
% 0.33/0.65 ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 )
% 0.33/0.65 = ( ring_1_of_int_rat @ Y ) )
% 0.33/0.65 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
% 0.33/0.65 = Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % neg_numeral_power_eq_of_int_cancel_iff
% 0.33/0.65 thf(fact_7226_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: num,N2: nat,Y: int] :
% 0.33/0.65 ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 )
% 0.33/0.65 = ( ring_18347121197199848620nteger @ Y ) )
% 0.33/0.65 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
% 0.33/0.65 = Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % neg_numeral_power_eq_of_int_cancel_iff
% 0.33/0.65 thf(fact_7227_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [Y: int,X: num,N2: nat] :
% 0.33/0.65 ( ( ( ring_1_of_int_real @ Y )
% 0.33/0.65 = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) )
% 0.33/0.65 = ( Y
% 0.33/0.65 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_eq_neg_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7228_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [Y: int,X: num,N2: nat] :
% 0.33/0.65 ( ( ( ring_1_of_int_int @ Y )
% 0.33/0.65 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) )
% 0.33/0.65 = ( Y
% 0.33/0.65 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_eq_neg_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7229_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [Y: int,X: num,N2: nat] :
% 0.33/0.65 ( ( ( ring_17405671764205052669omplex @ Y )
% 0.33/0.65 = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N2 ) )
% 0.33/0.65 = ( Y
% 0.33/0.65 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_eq_neg_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7230_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [Y: int,X: num,N2: nat] :
% 0.33/0.65 ( ( ( ring_1_of_int_rat @ Y )
% 0.33/0.65 = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) )
% 0.33/0.65 = ( Y
% 0.33/0.65 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_eq_neg_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7231_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [Y: int,X: num,N2: nat] :
% 0.33/0.65 ( ( ( ring_18347121197199848620nteger @ Y )
% 0.33/0.65 = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) )
% 0.33/0.65 = ( Y
% 0.33/0.65 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_eq_neg_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7232_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: num,N2: nat,A: int] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
% 0.33/0.65 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % neg_numeral_power_le_of_int_cancel_iff
% 0.33/0.65 thf(fact_7233_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: num,N2: nat,A: int] :
% 0.33/0.65 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) @ ( ring_18347121197199848620nteger @ A ) )
% 0.33/0.65 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % neg_numeral_power_le_of_int_cancel_iff
% 0.33/0.65 thf(fact_7234_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: num,N2: nat,A: int] :
% 0.33/0.65 ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
% 0.33/0.65 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % neg_numeral_power_le_of_int_cancel_iff
% 0.33/0.65 thf(fact_7235_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: num,N2: nat,A: int] :
% 0.33/0.65 ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
% 0.33/0.65 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % neg_numeral_power_le_of_int_cancel_iff
% 0.33/0.65 thf(fact_7236_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [A: int,X: num,N2: nat] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) )
% 0.33/0.65 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_le_neg_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7237_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [A: int,X: num,N2: nat] :
% 0.33/0.65 ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) )
% 0.33/0.65 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_le_neg_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7238_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [A: int,X: num,N2: nat] :
% 0.33/0.65 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) )
% 0.33/0.65 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_le_neg_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7239_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [A: int,X: num,N2: nat] :
% 0.33/0.65 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) )
% 0.33/0.65 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_le_neg_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7240_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [A: int,X: num,N2: nat] :
% 0.33/0.65 ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) )
% 0.33/0.65 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_less_neg_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7241_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [A: int,X: num,N2: nat] :
% 0.33/0.65 ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) )
% 0.33/0.65 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_less_neg_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7242_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [A: int,X: num,N2: nat] :
% 0.33/0.65 ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) )
% 0.33/0.65 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_less_neg_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7243_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.33/0.65 ! [A: int,X: num,N2: nat] :
% 0.33/0.65 ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) )
% 0.33/0.65 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_less_neg_numeral_power_cancel_iff
% 0.33/0.65 thf(fact_7244_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: num,N2: nat,A: int] :
% 0.33/0.65 ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
% 0.33/0.65 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % neg_numeral_power_less_of_int_cancel_iff
% 0.33/0.65 thf(fact_7245_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: num,N2: nat,A: int] :
% 0.33/0.65 ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
% 0.33/0.65 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % neg_numeral_power_less_of_int_cancel_iff
% 0.33/0.65 thf(fact_7246_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: num,N2: nat,A: int] :
% 0.33/0.65 ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
% 0.33/0.65 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % neg_numeral_power_less_of_int_cancel_iff
% 0.33/0.65 thf(fact_7247_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.33/0.65 ! [X: num,N2: nat,A: int] :
% 0.33/0.65 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) @ ( ring_18347121197199848620nteger @ A ) )
% 0.33/0.65 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % neg_numeral_power_less_of_int_cancel_iff
% 0.33/0.65 thf(fact_7248_of__nat__less__of__int__iff,axiom,
% 0.33/0.65 ! [N2: nat,X: int] :
% 0.33/0.65 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( ring_1_of_int_rat @ X ) )
% 0.33/0.65 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_less_of_int_iff
% 0.33/0.65 thf(fact_7249_of__nat__less__of__int__iff,axiom,
% 0.33/0.65 ! [N2: nat,X: int] :
% 0.33/0.65 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( ring_1_of_int_int @ X ) )
% 0.33/0.65 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_less_of_int_iff
% 0.33/0.65 thf(fact_7250_of__nat__less__of__int__iff,axiom,
% 0.33/0.65 ! [N2: nat,X: int] :
% 0.33/0.65 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( ring_1_of_int_real @ X ) )
% 0.33/0.65 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_less_of_int_iff
% 0.33/0.65 thf(fact_7251_mult__of__int__commute,axiom,
% 0.33/0.65 ! [X: int,Y: real] :
% 0.33/0.65 ( ( times_times_real @ ( ring_1_of_int_real @ X ) @ Y )
% 0.33/0.65 = ( times_times_real @ Y @ ( ring_1_of_int_real @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % mult_of_int_commute
% 0.33/0.65 thf(fact_7252_mult__of__int__commute,axiom,
% 0.33/0.65 ! [X: int,Y: rat] :
% 0.33/0.65 ( ( times_times_rat @ ( ring_1_of_int_rat @ X ) @ Y )
% 0.33/0.65 = ( times_times_rat @ Y @ ( ring_1_of_int_rat @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % mult_of_int_commute
% 0.33/0.65 thf(fact_7253_mult__of__int__commute,axiom,
% 0.33/0.65 ! [X: int,Y: int] :
% 0.33/0.65 ( ( times_times_int @ ( ring_1_of_int_int @ X ) @ Y )
% 0.33/0.65 = ( times_times_int @ Y @ ( ring_1_of_int_int @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % mult_of_int_commute
% 0.33/0.65 thf(fact_7254_mult__of__nat__commute,axiom,
% 0.33/0.65 ! [X: nat,Y: rat] :
% 0.33/0.65 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y )
% 0.33/0.65 = ( times_times_rat @ Y @ ( semiri681578069525770553at_rat @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % mult_of_nat_commute
% 0.33/0.65 thf(fact_7255_mult__of__nat__commute,axiom,
% 0.33/0.65 ! [X: nat,Y: int] :
% 0.33/0.65 ( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
% 0.33/0.65 = ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % mult_of_nat_commute
% 0.33/0.65 thf(fact_7256_mult__of__nat__commute,axiom,
% 0.33/0.65 ! [X: nat,Y: real] :
% 0.33/0.65 ( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
% 0.33/0.65 = ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % mult_of_nat_commute
% 0.33/0.65 thf(fact_7257_mult__of__nat__commute,axiom,
% 0.33/0.65 ! [X: nat,Y: nat] :
% 0.33/0.65 ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
% 0.33/0.65 = ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % mult_of_nat_commute
% 0.33/0.65 thf(fact_7258_mult__of__nat__commute,axiom,
% 0.33/0.65 ! [X: nat,Y: complex] :
% 0.33/0.65 ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X ) @ Y )
% 0.33/0.65 = ( times_times_complex @ Y @ ( semiri8010041392384452111omplex @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % mult_of_nat_commute
% 0.33/0.65 thf(fact_7259_int__diff__cases,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ~ ! [M2: nat,N4: nat] :
% 0.33/0.65 ( Z
% 0.33/0.65 != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % int_diff_cases
% 0.33/0.65 thf(fact_7260_semiring__norm_I26_J,axiom,
% 0.33/0.65 ( ( bitM @ one )
% 0.33/0.65 = one ) ).
% 0.33/0.65
% 0.33/0.65 % semiring_norm(26)
% 0.33/0.65 thf(fact_7261_of__nat__0__le__iff,axiom,
% 0.33/0.65 ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_0_le_iff
% 0.33/0.65 thf(fact_7262_of__nat__0__le__iff,axiom,
% 0.33/0.65 ! [N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_0_le_iff
% 0.33/0.65 thf(fact_7263_of__nat__0__le__iff,axiom,
% 0.33/0.65 ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_0_le_iff
% 0.33/0.65 thf(fact_7264_of__nat__0__le__iff,axiom,
% 0.33/0.65 ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_0_le_iff
% 0.33/0.65 thf(fact_7265_of__nat__less__0__iff,axiom,
% 0.33/0.65 ! [M: nat] :
% 0.33/0.65 ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_less_0_iff
% 0.33/0.65 thf(fact_7266_of__nat__less__0__iff,axiom,
% 0.33/0.65 ! [M: nat] :
% 0.33/0.65 ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_less_0_iff
% 0.33/0.65 thf(fact_7267_of__nat__less__0__iff,axiom,
% 0.33/0.65 ! [M: nat] :
% 0.33/0.65 ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_less_0_iff
% 0.33/0.65 thf(fact_7268_of__nat__less__0__iff,axiom,
% 0.33/0.65 ! [M: nat] :
% 0.33/0.65 ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_less_0_iff
% 0.33/0.65 thf(fact_7269_of__nat__neq__0,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( semiri681578069525770553at_rat @ ( suc @ N2 ) )
% 0.33/0.65 != zero_zero_rat ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_neq_0
% 0.33/0.65 thf(fact_7270_of__nat__neq__0,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
% 0.33/0.65 != zero_zero_int ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_neq_0
% 0.33/0.65 thf(fact_7271_of__nat__neq__0,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( semiri5074537144036343181t_real @ ( suc @ N2 ) )
% 0.33/0.65 != zero_zero_real ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_neq_0
% 0.33/0.65 thf(fact_7272_of__nat__neq__0,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( semiri1316708129612266289at_nat @ ( suc @ N2 ) )
% 0.33/0.65 != zero_zero_nat ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_neq_0
% 0.33/0.65 thf(fact_7273_of__nat__neq__0,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( semiri8010041392384452111omplex @ ( suc @ N2 ) )
% 0.33/0.65 != zero_zero_complex ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_neq_0
% 0.33/0.65 thf(fact_7274_div__mult2__eq_H,axiom,
% 0.33/0.65 ! [A: int,M: nat,N2: nat] :
% 0.33/0.65 ( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.33/0.65 = ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % div_mult2_eq'
% 0.33/0.65 thf(fact_7275_div__mult2__eq_H,axiom,
% 0.33/0.65 ! [A: nat,M: nat,N2: nat] :
% 0.33/0.65 ( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) )
% 0.33/0.65 = ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % div_mult2_eq'
% 0.33/0.65 thf(fact_7276_of__nat__less__imp__less,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
% 0.33/0.65 => ( ord_less_nat @ M @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_less_imp_less
% 0.33/0.65 thf(fact_7277_of__nat__less__imp__less,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
% 0.33/0.65 => ( ord_less_nat @ M @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_less_imp_less
% 0.33/0.65 thf(fact_7278_of__nat__less__imp__less,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
% 0.33/0.65 => ( ord_less_nat @ M @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_less_imp_less
% 0.33/0.65 thf(fact_7279_of__nat__less__imp__less,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
% 0.33/0.65 => ( ord_less_nat @ M @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_less_imp_less
% 0.33/0.65 thf(fact_7280_less__imp__of__nat__less,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( ord_less_nat @ M @ N2 )
% 0.33/0.65 => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % less_imp_of_nat_less
% 0.33/0.65 thf(fact_7281_less__imp__of__nat__less,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( ord_less_nat @ M @ N2 )
% 0.33/0.65 => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % less_imp_of_nat_less
% 0.33/0.65 thf(fact_7282_less__imp__of__nat__less,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( ord_less_nat @ M @ N2 )
% 0.33/0.65 => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % less_imp_of_nat_less
% 0.33/0.65 thf(fact_7283_less__imp__of__nat__less,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( ord_less_nat @ M @ N2 )
% 0.33/0.65 => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % less_imp_of_nat_less
% 0.33/0.65 thf(fact_7284_of__nat__mono,axiom,
% 0.33/0.65 ! [I2: nat,J: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ I2 @ J )
% 0.33/0.65 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I2 ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_mono
% 0.33/0.65 thf(fact_7285_of__nat__mono,axiom,
% 0.33/0.65 ! [I2: nat,J: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ I2 @ J )
% 0.33/0.65 => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I2 ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_mono
% 0.33/0.65 thf(fact_7286_of__nat__mono,axiom,
% 0.33/0.65 ! [I2: nat,J: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ I2 @ J )
% 0.33/0.65 => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_mono
% 0.33/0.65 thf(fact_7287_of__nat__mono,axiom,
% 0.33/0.65 ! [I2: nat,J: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ I2 @ J )
% 0.33/0.65 => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_mono
% 0.33/0.65 thf(fact_7288_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) )
% 0.33/0.65 = ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % unique_euclidean_semiring_with_nat_class.of_nat_div
% 0.33/0.65 thf(fact_7289_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N2 ) )
% 0.33/0.65 = ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % unique_euclidean_semiring_with_nat_class.of_nat_div
% 0.33/0.65 thf(fact_7290_of__nat__dvd__iff,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( dvd_dvd_Code_integer @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) )
% 0.33/0.65 = ( dvd_dvd_nat @ M @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_dvd_iff
% 0.33/0.65 thf(fact_7291_of__nat__dvd__iff,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
% 0.33/0.65 = ( dvd_dvd_nat @ M @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_dvd_iff
% 0.33/0.65 thf(fact_7292_of__nat__dvd__iff,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
% 0.33/0.65 = ( dvd_dvd_nat @ M @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_dvd_iff
% 0.33/0.65 thf(fact_7293_int__ops_I3_J,axiom,
% 0.33/0.65 ! [N2: num] :
% 0.33/0.65 ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N2 ) )
% 0.33/0.65 = ( numeral_numeral_int @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % int_ops(3)
% 0.33/0.65 thf(fact_7294_int__of__nat__induct,axiom,
% 0.33/0.65 ! [P: int > $o,Z: int] :
% 0.33/0.65 ( ! [N4: nat] : ( P @ ( semiri1314217659103216013at_int @ N4 ) )
% 0.33/0.65 => ( ! [N4: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) )
% 0.33/0.65 => ( P @ Z ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % int_of_nat_induct
% 0.33/0.65 thf(fact_7295_int__cases,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ! [N4: nat] :
% 0.33/0.65 ( Z
% 0.33/0.65 != ( semiri1314217659103216013at_int @ N4 ) )
% 0.33/0.65 => ~ ! [N4: nat] :
% 0.33/0.65 ( Z
% 0.33/0.65 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % int_cases
% 0.33/0.65 thf(fact_7296_nat__int__comparison_I2_J,axiom,
% 0.33/0.65 ( ord_less_nat
% 0.33/0.65 = ( ^ [A5: nat,B5: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % nat_int_comparison(2)
% 0.33/0.65 thf(fact_7297_nat__int__comparison_I3_J,axiom,
% 0.33/0.65 ( ord_less_eq_nat
% 0.33/0.65 = ( ^ [A5: nat,B5: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % nat_int_comparison(3)
% 0.33/0.65 thf(fact_7298_zle__int,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
% 0.33/0.65 = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % zle_int
% 0.33/0.65 thf(fact_7299_of__nat__mod,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N2 ) )
% 0.33/0.65 = ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_mod
% 0.33/0.65 thf(fact_7300_of__nat__mod,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) )
% 0.33/0.65 = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_mod
% 0.33/0.65 thf(fact_7301_of__nat__mod,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N2 ) )
% 0.33/0.65 = ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_mod
% 0.33/0.65 thf(fact_7302_zadd__int__left,axiom,
% 0.33/0.65 ! [M: nat,N2: nat,Z: int] :
% 0.33/0.65 ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ Z ) )
% 0.33/0.65 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) ) @ Z ) ) ).
% 0.33/0.65
% 0.33/0.65 % zadd_int_left
% 0.33/0.65 thf(fact_7303_int__plus,axiom,
% 0.33/0.65 ! [N2: nat,M: nat] :
% 0.33/0.65 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N2 @ M ) )
% 0.33/0.65 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % int_plus
% 0.33/0.65 thf(fact_7304_int__ops_I5_J,axiom,
% 0.33/0.65 ! [A: nat,B: nat] :
% 0.33/0.65 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
% 0.33/0.65 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % int_ops(5)
% 0.33/0.65 thf(fact_7305_int__ops_I7_J,axiom,
% 0.33/0.65 ! [A: nat,B: nat] :
% 0.33/0.65 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
% 0.33/0.65 = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % int_ops(7)
% 0.33/0.65 thf(fact_7306_int__ops_I2_J,axiom,
% 0.33/0.65 ( ( semiri1314217659103216013at_int @ one_one_nat )
% 0.33/0.65 = one_one_int ) ).
% 0.33/0.65
% 0.33/0.65 % int_ops(2)
% 0.33/0.65 thf(fact_7307_not__int__zless__negative,axiom,
% 0.33/0.65 ! [N2: nat,M: nat] :
% 0.33/0.65 ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % not_int_zless_negative
% 0.33/0.65 thf(fact_7308_zdiv__int,axiom,
% 0.33/0.65 ! [A: nat,B: nat] :
% 0.33/0.65 ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
% 0.33/0.65 = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % zdiv_int
% 0.33/0.65 thf(fact_7309_of__nat__max,axiom,
% 0.33/0.65 ! [X: nat,Y: nat] :
% 0.33/0.65 ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X @ Y ) )
% 0.33/0.65 = ( ord_max_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_max
% 0.33/0.65 thf(fact_7310_of__nat__max,axiom,
% 0.33/0.65 ! [X: nat,Y: nat] :
% 0.33/0.65 ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X @ Y ) )
% 0.33/0.65 = ( ord_max_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_max
% 0.33/0.65 thf(fact_7311_of__nat__max,axiom,
% 0.33/0.65 ! [X: nat,Y: nat] :
% 0.33/0.65 ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X @ Y ) )
% 0.33/0.65 = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( semiri1316708129612266289at_nat @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_max
% 0.33/0.65 thf(fact_7312_semiring__norm_I28_J,axiom,
% 0.33/0.65 ! [N2: num] :
% 0.33/0.65 ( ( bitM @ ( bit1 @ N2 ) )
% 0.33/0.65 = ( bit1 @ ( bit0 @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % semiring_norm(28)
% 0.33/0.65 thf(fact_7313_semiring__norm_I27_J,axiom,
% 0.33/0.65 ! [N2: num] :
% 0.33/0.65 ( ( bitM @ ( bit0 @ N2 ) )
% 0.33/0.65 = ( bit1 @ ( bitM @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % semiring_norm(27)
% 0.33/0.65 thf(fact_7314_nat__less__as__int,axiom,
% 0.33/0.65 ( ord_less_nat
% 0.33/0.65 = ( ^ [A5: nat,B5: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % nat_less_as_int
% 0.33/0.65 thf(fact_7315_nat__leq__as__int,axiom,
% 0.33/0.65 ( ord_less_eq_nat
% 0.33/0.65 = ( ^ [A5: nat,B5: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % nat_leq_as_int
% 0.33/0.65 thf(fact_7316_of__nat__diff,axiom,
% 0.33/0.65 ! [N2: nat,M: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ N2 @ M )
% 0.33/0.65 => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N2 ) )
% 0.33/0.65 = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_diff
% 0.33/0.65 thf(fact_7317_of__nat__diff,axiom,
% 0.33/0.65 ! [N2: nat,M: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ N2 @ M )
% 0.33/0.65 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N2 ) )
% 0.33/0.65 = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_diff
% 0.33/0.65 thf(fact_7318_of__nat__diff,axiom,
% 0.33/0.65 ! [N2: nat,M: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ N2 @ M )
% 0.33/0.65 => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N2 ) )
% 0.33/0.65 = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_diff
% 0.33/0.65 thf(fact_7319_of__nat__diff,axiom,
% 0.33/0.65 ! [N2: nat,M: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ N2 @ M )
% 0.33/0.65 => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N2 ) )
% 0.33/0.65 = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_diff
% 0.33/0.65 thf(fact_7320_of__nat__diff,axiom,
% 0.33/0.65 ! [N2: nat,M: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ N2 @ M )
% 0.33/0.65 => ( ( semiri8010041392384452111omplex @ ( minus_minus_nat @ M @ N2 ) )
% 0.33/0.65 = ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_diff
% 0.33/0.65 thf(fact_7321_reals__Archimedean3,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.65 => ! [Y5: real] :
% 0.33/0.65 ? [N4: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % reals_Archimedean3
% 0.33/0.65 thf(fact_7322_real__of__int__div4,axiom,
% 0.33/0.65 ! [N2: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_of_int_div4
% 0.33/0.65 thf(fact_7323_int__cases4,axiom,
% 0.33/0.65 ! [M: int] :
% 0.33/0.65 ( ! [N4: nat] :
% 0.33/0.65 ( M
% 0.33/0.65 != ( semiri1314217659103216013at_int @ N4 ) )
% 0.33/0.65 => ~ ! [N4: nat] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 0.33/0.65 => ( M
% 0.33/0.65 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % int_cases4
% 0.33/0.65 thf(fact_7324_real__of__nat__div4,axiom,
% 0.33/0.65 ! [N2: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_of_nat_div4
% 0.33/0.65 thf(fact_7325_int__Suc,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
% 0.33/0.65 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % int_Suc
% 0.33/0.65 thf(fact_7326_int__ops_I4_J,axiom,
% 0.33/0.65 ! [A: nat] :
% 0.33/0.65 ( ( semiri1314217659103216013at_int @ ( suc @ A ) )
% 0.33/0.65 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % int_ops(4)
% 0.33/0.65 thf(fact_7327_zless__iff__Suc__zadd,axiom,
% 0.33/0.65 ( ord_less_int
% 0.33/0.65 = ( ^ [W3: int,Z3: int] :
% 0.33/0.65 ? [N: nat] :
% 0.33/0.65 ( Z3
% 0.33/0.65 = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % zless_iff_Suc_zadd
% 0.33/0.65 thf(fact_7328_real__of__nat__div,axiom,
% 0.33/0.65 ! [D: nat,N2: nat] :
% 0.33/0.65 ( ( dvd_dvd_nat @ D @ N2 )
% 0.33/0.65 => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ D ) )
% 0.33/0.65 = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_of_nat_div
% 0.33/0.65 thf(fact_7329_real__of__int__div,axiom,
% 0.33/0.65 ! [D: int,N2: int] :
% 0.33/0.65 ( ( dvd_dvd_int @ D @ N2 )
% 0.33/0.65 => ( ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ D ) )
% 0.33/0.65 = ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_of_int_div
% 0.33/0.65 thf(fact_7330_int__sum,axiom,
% 0.33/0.65 ! [F: int > nat,A2: set_int] :
% 0.33/0.65 ( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F @ A2 ) )
% 0.33/0.65 = ( groups4538972089207619220nt_int
% 0.33/0.65 @ ^ [X2: int] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
% 0.33/0.65 @ A2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % int_sum
% 0.33/0.65 thf(fact_7331_int__sum,axiom,
% 0.33/0.65 ! [F: nat > nat,A2: set_nat] :
% 0.33/0.65 ( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 0.33/0.65 = ( groups3539618377306564664at_int
% 0.33/0.65 @ ^ [X2: nat] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
% 0.33/0.65 @ A2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % int_sum
% 0.33/0.65 thf(fact_7332_eval__nat__numeral_I2_J,axiom,
% 0.33/0.65 ! [N2: num] :
% 0.33/0.65 ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
% 0.33/0.65 = ( suc @ ( numeral_numeral_nat @ ( bitM @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % eval_nat_numeral(2)
% 0.33/0.65 thf(fact_7333_BitM__plus__one,axiom,
% 0.33/0.65 ! [N2: num] :
% 0.33/0.65 ( ( plus_plus_num @ ( bitM @ N2 ) @ one )
% 0.33/0.65 = ( bit0 @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % BitM_plus_one
% 0.33/0.65 thf(fact_7334_one__plus__BitM,axiom,
% 0.33/0.65 ! [N2: num] :
% 0.33/0.65 ( ( plus_plus_num @ one @ ( bitM @ N2 ) )
% 0.33/0.65 = ( bit0 @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % one_plus_BitM
% 0.33/0.65 thf(fact_7335_of__int__nonneg,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.33/0.65 => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_nonneg
% 0.33/0.65 thf(fact_7336_of__int__nonneg,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.33/0.65 => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_nonneg
% 0.33/0.65 thf(fact_7337_of__int__nonneg,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.33/0.65 => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_nonneg
% 0.33/0.65 thf(fact_7338_of__int__leD,axiom,
% 0.33/0.65 ! [N2: int,X: code_integer] :
% 0.33/0.65 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N2 ) ) @ X )
% 0.33/0.65 => ( ( N2 = zero_zero_int )
% 0.33/0.65 | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_leD
% 0.33/0.65 thf(fact_7339_of__int__leD,axiom,
% 0.33/0.65 ! [N2: int,X: real] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N2 ) ) @ X )
% 0.33/0.65 => ( ( N2 = zero_zero_int )
% 0.33/0.65 | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_leD
% 0.33/0.65 thf(fact_7340_of__int__leD,axiom,
% 0.33/0.65 ! [N2: int,X: rat] :
% 0.33/0.65 ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N2 ) ) @ X )
% 0.33/0.65 => ( ( N2 = zero_zero_int )
% 0.33/0.65 | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_leD
% 0.33/0.65 thf(fact_7341_of__int__leD,axiom,
% 0.33/0.65 ! [N2: int,X: int] :
% 0.33/0.65 ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X )
% 0.33/0.65 => ( ( N2 = zero_zero_int )
% 0.33/0.65 | ( ord_less_eq_int @ one_one_int @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_leD
% 0.33/0.65 thf(fact_7342_of__int__pos,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.33/0.65 => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_pos
% 0.33/0.65 thf(fact_7343_of__int__pos,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.33/0.65 => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_pos
% 0.33/0.65 thf(fact_7344_of__int__pos,axiom,
% 0.33/0.65 ! [Z: int] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.33/0.65 => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_pos
% 0.33/0.65 thf(fact_7345_of__int__lessD,axiom,
% 0.33/0.65 ! [N2: int,X: code_integer] :
% 0.33/0.65 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N2 ) ) @ X )
% 0.33/0.65 => ( ( N2 = zero_zero_int )
% 0.33/0.65 | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_lessD
% 0.33/0.65 thf(fact_7346_of__int__lessD,axiom,
% 0.33/0.65 ! [N2: int,X: real] :
% 0.33/0.65 ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N2 ) ) @ X )
% 0.33/0.65 => ( ( N2 = zero_zero_int )
% 0.33/0.65 | ( ord_less_real @ one_one_real @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_lessD
% 0.33/0.65 thf(fact_7347_of__int__lessD,axiom,
% 0.33/0.65 ! [N2: int,X: rat] :
% 0.33/0.65 ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N2 ) ) @ X )
% 0.33/0.65 => ( ( N2 = zero_zero_int )
% 0.33/0.65 | ( ord_less_rat @ one_one_rat @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_lessD
% 0.33/0.65 thf(fact_7348_of__int__lessD,axiom,
% 0.33/0.65 ! [N2: int,X: int] :
% 0.33/0.65 ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X )
% 0.33/0.65 => ( ( N2 = zero_zero_int )
% 0.33/0.65 | ( ord_less_int @ one_one_int @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_lessD
% 0.33/0.65 thf(fact_7349_mod__mult2__eq_H,axiom,
% 0.33/0.65 ! [A: code_integer,M: nat,N2: nat] :
% 0.33/0.65 ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) )
% 0.33/0.65 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % mod_mult2_eq'
% 0.33/0.65 thf(fact_7350_mod__mult2__eq_H,axiom,
% 0.33/0.65 ! [A: int,M: nat,N2: nat] :
% 0.33/0.65 ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.33/0.65 = ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % mod_mult2_eq'
% 0.33/0.65 thf(fact_7351_mod__mult2__eq_H,axiom,
% 0.33/0.65 ! [A: nat,M: nat,N2: nat] :
% 0.33/0.65 ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) )
% 0.33/0.65 = ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % mod_mult2_eq'
% 0.33/0.65 thf(fact_7352_of__int__neg__numeral,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.33/0.65 = ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_neg_numeral
% 0.33/0.65 thf(fact_7353_of__int__neg__numeral,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.33/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_neg_numeral
% 0.33/0.65 thf(fact_7354_of__int__neg__numeral,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.33/0.65 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_neg_numeral
% 0.33/0.65 thf(fact_7355_of__int__neg__numeral,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.33/0.65 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_neg_numeral
% 0.33/0.65 thf(fact_7356_of__int__neg__numeral,axiom,
% 0.33/0.65 ! [K: num] :
% 0.33/0.65 ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.33/0.65 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_neg_numeral
% 0.33/0.65 thf(fact_7357_field__char__0__class_Oof__nat__div,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N2 ) )
% 0.33/0.65 = ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N2 ) ) ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % field_char_0_class.of_nat_div
% 0.33/0.65 thf(fact_7358_field__char__0__class_Oof__nat__div,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N2 ) )
% 0.33/0.65 = ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N2 ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % field_char_0_class.of_nat_div
% 0.33/0.65 thf(fact_7359_field__char__0__class_Oof__nat__div,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M @ N2 ) )
% 0.33/0.65 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M @ N2 ) ) ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % field_char_0_class.of_nat_div
% 0.33/0.65 thf(fact_7360_int__le__real__less,axiom,
% 0.33/0.65 ( ord_less_eq_int
% 0.33/0.65 = ( ^ [N: int,M6: int] : ( ord_less_real @ ( ring_1_of_int_real @ N ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M6 ) @ one_one_real ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % int_le_real_less
% 0.33/0.65 thf(fact_7361_int__less__real__le,axiom,
% 0.33/0.65 ( ord_less_int
% 0.33/0.65 = ( ^ [N: int,M6: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) @ ( ring_1_of_int_real @ M6 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % int_less_real_le
% 0.33/0.65 thf(fact_7362_pos__int__cases,axiom,
% 0.33/0.65 ! [K: int] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ K )
% 0.33/0.65 => ~ ! [N4: nat] :
% 0.33/0.65 ( ( K
% 0.33/0.65 = ( semiri1314217659103216013at_int @ N4 ) )
% 0.33/0.65 => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % pos_int_cases
% 0.33/0.65 thf(fact_7363_zero__less__imp__eq__int,axiom,
% 0.33/0.65 ! [K: int] :
% 0.33/0.65 ( ( ord_less_int @ zero_zero_int @ K )
% 0.33/0.65 => ? [N4: nat] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 0.33/0.65 & ( K
% 0.33/0.65 = ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % zero_less_imp_eq_int
% 0.33/0.65 thf(fact_7364_int__cases3,axiom,
% 0.33/0.65 ! [K: int] :
% 0.33/0.65 ( ( K != zero_zero_int )
% 0.33/0.65 => ( ! [N4: nat] :
% 0.33/0.65 ( ( K
% 0.33/0.65 = ( semiri1314217659103216013at_int @ N4 ) )
% 0.33/0.65 => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) )
% 0.33/0.65 => ~ ! [N4: nat] :
% 0.33/0.65 ( ( K
% 0.33/0.65 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
% 0.33/0.65 => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % int_cases3
% 0.33/0.65 thf(fact_7365_nat__less__real__le,axiom,
% 0.33/0.65 ( ord_less_nat
% 0.33/0.65 = ( ^ [N: nat,M6: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M6 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % nat_less_real_le
% 0.33/0.65 thf(fact_7366_nat__le__real__less,axiom,
% 0.33/0.65 ( ord_less_eq_nat
% 0.33/0.65 = ( ^ [N: nat,M6: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M6 ) @ one_one_real ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % nat_le_real_less
% 0.33/0.65 thf(fact_7367_zmult__zless__mono2__lemma,axiom,
% 0.33/0.65 ! [I2: int,J: int,K: nat] :
% 0.33/0.65 ( ( ord_less_int @ I2 @ J )
% 0.33/0.65 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.33/0.65 => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % zmult_zless_mono2_lemma
% 0.33/0.65 thf(fact_7368_not__zle__0__negative,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % not_zle_0_negative
% 0.33/0.65 thf(fact_7369_negD,axiom,
% 0.33/0.65 ! [X: int] :
% 0.33/0.65 ( ( ord_less_int @ X @ zero_zero_int )
% 0.33/0.65 => ? [N4: nat] :
% 0.33/0.65 ( X
% 0.33/0.65 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % negD
% 0.33/0.65 thf(fact_7370_negative__zless__0,axiom,
% 0.33/0.65 ! [N2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ zero_zero_int ) ).
% 0.33/0.65
% 0.33/0.65 % negative_zless_0
% 0.33/0.65 thf(fact_7371_dbl__inc__def,axiom,
% 0.33/0.65 ( neg_nu8557863876264182079omplex
% 0.33/0.65 = ( ^ [X2: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X2 @ X2 ) @ one_one_complex ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_inc_def
% 0.33/0.65 thf(fact_7372_dbl__inc__def,axiom,
% 0.33/0.65 ( neg_nu8295874005876285629c_real
% 0.33/0.65 = ( ^ [X2: real] : ( plus_plus_real @ ( plus_plus_real @ X2 @ X2 ) @ one_one_real ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_inc_def
% 0.33/0.65 thf(fact_7373_dbl__inc__def,axiom,
% 0.33/0.65 ( neg_nu5219082963157363817nc_rat
% 0.33/0.65 = ( ^ [X2: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_inc_def
% 0.33/0.65 thf(fact_7374_dbl__inc__def,axiom,
% 0.33/0.65 ( neg_nu5851722552734809277nc_int
% 0.33/0.65 = ( ^ [X2: int] : ( plus_plus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % dbl_inc_def
% 0.33/0.65 thf(fact_7375_int__ops_I6_J,axiom,
% 0.33/0.65 ! [A: nat,B: nat] :
% 0.33/0.65 ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 0.33/0.65 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 0.33/0.65 = zero_zero_int ) )
% 0.33/0.65 & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 0.33/0.65 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 0.33/0.65 = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % int_ops(6)
% 0.33/0.65 thf(fact_7376_real__of__int__div__aux,axiom,
% 0.33/0.65 ! [X: int,D: int] :
% 0.33/0.65 ( ( divide_divide_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ D ) )
% 0.33/0.65 = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_of_int_div_aux
% 0.33/0.65 thf(fact_7377_real__of__nat__div__aux,axiom,
% 0.33/0.65 ! [X: nat,D: nat] :
% 0.33/0.65 ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ D ) )
% 0.33/0.65 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_of_nat_div_aux
% 0.33/0.65 thf(fact_7378_numeral__BitM,axiom,
% 0.33/0.65 ! [N2: num] :
% 0.33/0.65 ( ( numera6690914467698888265omplex @ ( bitM @ N2 ) )
% 0.33/0.65 = ( minus_minus_complex @ ( numera6690914467698888265omplex @ ( bit0 @ N2 ) ) @ one_one_complex ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_BitM
% 0.33/0.65 thf(fact_7379_numeral__BitM,axiom,
% 0.33/0.65 ! [N2: num] :
% 0.33/0.65 ( ( numeral_numeral_real @ ( bitM @ N2 ) )
% 0.33/0.65 = ( minus_minus_real @ ( numeral_numeral_real @ ( bit0 @ N2 ) ) @ one_one_real ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_BitM
% 0.33/0.65 thf(fact_7380_numeral__BitM,axiom,
% 0.33/0.65 ! [N2: num] :
% 0.33/0.65 ( ( numeral_numeral_rat @ ( bitM @ N2 ) )
% 0.33/0.65 = ( minus_minus_rat @ ( numeral_numeral_rat @ ( bit0 @ N2 ) ) @ one_one_rat ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_BitM
% 0.33/0.65 thf(fact_7381_numeral__BitM,axiom,
% 0.33/0.65 ! [N2: num] :
% 0.33/0.65 ( ( numeral_numeral_int @ ( bitM @ N2 ) )
% 0.33/0.65 = ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ one_one_int ) ) ).
% 0.33/0.65
% 0.33/0.65 % numeral_BitM
% 0.33/0.65 thf(fact_7382_odd__numeral__BitM,axiom,
% 0.33/0.65 ! [W: num] :
% 0.33/0.65 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bitM @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % odd_numeral_BitM
% 0.33/0.65 thf(fact_7383_odd__numeral__BitM,axiom,
% 0.33/0.65 ! [W: num] :
% 0.33/0.65 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bitM @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % odd_numeral_BitM
% 0.33/0.65 thf(fact_7384_odd__numeral__BitM,axiom,
% 0.33/0.65 ! [W: num] :
% 0.33/0.65 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bitM @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % odd_numeral_BitM
% 0.33/0.65 thf(fact_7385_of__nat__less__two__power,axiom,
% 0.33/0.65 ! [N2: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_less_two_power
% 0.33/0.65 thf(fact_7386_of__nat__less__two__power,axiom,
% 0.33/0.65 ! [N2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_less_two_power
% 0.33/0.65 thf(fact_7387_of__nat__less__two__power,axiom,
% 0.33/0.65 ! [N2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_less_two_power
% 0.33/0.65 thf(fact_7388_inverse__of__nat__le,axiom,
% 0.33/0.65 ! [N2: nat,M: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ N2 @ M )
% 0.33/0.65 => ( ( N2 != zero_zero_nat )
% 0.33/0.65 => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % inverse_of_nat_le
% 0.33/0.65 thf(fact_7389_inverse__of__nat__le,axiom,
% 0.33/0.65 ! [N2: nat,M: nat] :
% 0.33/0.65 ( ( ord_less_eq_nat @ N2 @ M )
% 0.33/0.65 => ( ( N2 != zero_zero_nat )
% 0.33/0.65 => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % inverse_of_nat_le
% 0.33/0.65 thf(fact_7390_real__archimedian__rdiv__eq__0,axiom,
% 0.33/0.65 ! [X: real,C: real] :
% 0.33/0.65 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.65 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.33/0.65 => ( ! [M2: nat] :
% 0.33/0.65 ( ( ord_less_nat @ zero_zero_nat @ M2 )
% 0.33/0.65 => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ X ) @ C ) )
% 0.33/0.65 => ( X = zero_zero_real ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_archimedian_rdiv_eq_0
% 0.33/0.65 thf(fact_7391_real__of__int__div2,axiom,
% 0.33/0.65 ! [N2: int,X: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ X ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_of_int_div2
% 0.33/0.65 thf(fact_7392_real__of__int__div3,axiom,
% 0.33/0.65 ! [N2: int,X: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ X ) ) ) @ one_one_real ) ).
% 0.33/0.65
% 0.33/0.65 % real_of_int_div3
% 0.33/0.65 thf(fact_7393_neg__int__cases,axiom,
% 0.33/0.65 ! [K: int] :
% 0.33/0.65 ( ( ord_less_int @ K @ zero_zero_int )
% 0.33/0.65 => ~ ! [N4: nat] :
% 0.33/0.65 ( ( K
% 0.33/0.65 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
% 0.33/0.65 => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % neg_int_cases
% 0.33/0.65 thf(fact_7394_zdiff__int__split,axiom,
% 0.33/0.65 ! [P: int > $o,X: nat,Y: nat] :
% 0.33/0.65 ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
% 0.33/0.65 = ( ( ( ord_less_eq_nat @ Y @ X )
% 0.33/0.65 => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
% 0.33/0.65 & ( ( ord_less_nat @ X @ Y )
% 0.33/0.65 => ( P @ zero_zero_int ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % zdiff_int_split
% 0.33/0.65 thf(fact_7395_real__of__nat__div2,axiom,
% 0.33/0.65 ! [N2: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_of_nat_div2
% 0.33/0.65 thf(fact_7396_real__of__nat__div3,axiom,
% 0.33/0.65 ! [N2: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X ) ) ) @ one_one_real ) ).
% 0.33/0.65
% 0.33/0.65 % real_of_nat_div3
% 0.33/0.65 thf(fact_7397_ln__realpow,axiom,
% 0.33/0.65 ! [X: real,N2: nat] :
% 0.33/0.65 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.65 => ( ( ln_ln_real @ ( power_power_real @ X @ N2 ) )
% 0.33/0.65 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( ln_ln_real @ X ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % ln_realpow
% 0.33/0.65 thf(fact_7398_even__of__int__iff,axiom,
% 0.33/0.65 ! [K: int] :
% 0.33/0.65 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ K ) )
% 0.33/0.65 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% 0.33/0.65
% 0.33/0.65 % even_of_int_iff
% 0.33/0.65 thf(fact_7399_even__of__int__iff,axiom,
% 0.33/0.65 ! [K: int] :
% 0.33/0.65 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ K ) )
% 0.33/0.65 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% 0.33/0.65
% 0.33/0.65 % even_of_int_iff
% 0.33/0.65 thf(fact_7400_linear__plus__1__le__power,axiom,
% 0.33/0.65 ! [X: real,N2: nat] :
% 0.33/0.65 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.65 => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X @ one_one_real ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % linear_plus_1_le_power
% 0.33/0.65 thf(fact_7401_Bernoulli__inequality,axiom,
% 0.33/0.65 ! [X: real,N2: nat] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.65 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % Bernoulli_inequality
% 0.33/0.65 thf(fact_7402_Bernoulli__inequality__even,axiom,
% 0.33/0.65 ! [N2: nat,X: real] :
% 0.33/0.65 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.65 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % Bernoulli_inequality_even
% 0.33/0.65 thf(fact_7403_double__gauss__sum,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 0.33/0.65 = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % double_gauss_sum
% 0.33/0.65 thf(fact_7404_double__gauss__sum,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 0.33/0.65 = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % double_gauss_sum
% 0.33/0.65 thf(fact_7405_double__gauss__sum,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 0.33/0.65 = ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % double_gauss_sum
% 0.33/0.65 thf(fact_7406_double__gauss__sum,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 0.33/0.65 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % double_gauss_sum
% 0.33/0.65 thf(fact_7407_double__gauss__sum,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 0.33/0.65 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % double_gauss_sum
% 0.33/0.65 thf(fact_7408_double__arith__series,axiom,
% 0.33/0.65 ! [A: rat,D: rat,N2: nat] :
% 0.33/0.65 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
% 0.33/0.65 @ ( groups2906978787729119204at_rat
% 0.33/0.65 @ ^ [I3: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I3 ) @ D ) )
% 0.33/0.65 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 0.33/0.65 = ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ D ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % double_arith_series
% 0.33/0.65 thf(fact_7409_double__arith__series,axiom,
% 0.33/0.65 ! [A: int,D: int,N2: nat] :
% 0.33/0.65 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
% 0.33/0.65 @ ( groups3539618377306564664at_int
% 0.33/0.65 @ ^ [I3: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I3 ) @ D ) )
% 0.33/0.65 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 0.33/0.65 = ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ D ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % double_arith_series
% 0.33/0.65 thf(fact_7410_double__arith__series,axiom,
% 0.33/0.65 ! [A: complex,D: complex,N2: nat] :
% 0.33/0.65 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
% 0.33/0.65 @ ( groups2073611262835488442omplex
% 0.33/0.65 @ ^ [I3: nat] : ( plus_plus_complex @ A @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I3 ) @ D ) )
% 0.33/0.65 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 0.33/0.65 = ( times_times_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ D ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % double_arith_series
% 0.33/0.65 thf(fact_7411_double__arith__series,axiom,
% 0.33/0.65 ! [A: nat,D: nat,N2: nat] :
% 0.33/0.65 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
% 0.33/0.65 @ ( groups3542108847815614940at_nat
% 0.33/0.65 @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ D ) )
% 0.33/0.65 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 0.33/0.65 = ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ D ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % double_arith_series
% 0.33/0.65 thf(fact_7412_double__arith__series,axiom,
% 0.33/0.65 ! [A: real,D: real,N2: nat] :
% 0.33/0.65 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 0.33/0.65 @ ( groups6591440286371151544t_real
% 0.33/0.65 @ ^ [I3: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I3 ) @ D ) )
% 0.33/0.65 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 0.33/0.65 = ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ D ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % double_arith_series
% 0.33/0.65 thf(fact_7413_gauss__sum,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 0.33/0.65 = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gauss_sum
% 0.33/0.65 thf(fact_7414_gauss__sum,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 0.33/0.65 = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % gauss_sum
% 0.33/0.65 thf(fact_7415_arith__series,axiom,
% 0.33/0.65 ! [A: int,D: int,N2: nat] :
% 0.33/0.65 ( ( groups3539618377306564664at_int
% 0.33/0.65 @ ^ [I3: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I3 ) @ D ) )
% 0.33/0.65 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 0.33/0.65 = ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % arith_series
% 0.33/0.65 thf(fact_7416_arith__series,axiom,
% 0.33/0.65 ! [A: nat,D: nat,N2: nat] :
% 0.33/0.65 ( ( groups3542108847815614940at_nat
% 0.33/0.65 @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ D ) )
% 0.33/0.65 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 0.33/0.65 = ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % arith_series
% 0.33/0.65 thf(fact_7417_double__gauss__sum__from__Suc__0,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
% 0.33/0.65 = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % double_gauss_sum_from_Suc_0
% 0.33/0.65 thf(fact_7418_double__gauss__sum__from__Suc__0,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
% 0.33/0.65 = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % double_gauss_sum_from_Suc_0
% 0.33/0.65 thf(fact_7419_double__gauss__sum__from__Suc__0,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
% 0.33/0.65 = ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % double_gauss_sum_from_Suc_0
% 0.33/0.65 thf(fact_7420_double__gauss__sum__from__Suc__0,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
% 0.33/0.65 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % double_gauss_sum_from_Suc_0
% 0.33/0.65 thf(fact_7421_double__gauss__sum__from__Suc__0,axiom,
% 0.33/0.65 ! [N2: nat] :
% 0.33/0.65 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
% 0.33/0.65 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % double_gauss_sum_from_Suc_0
% 0.33/0.65 thf(fact_7422_sum__gp__offset,axiom,
% 0.33/0.65 ! [X: rat,M: nat,N2: nat] :
% 0.33/0.65 ( ( ( X = one_one_rat )
% 0.33/0.65 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
% 0.33/0.65 = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) )
% 0.33/0.65 & ( ( X != one_one_rat )
% 0.33/0.65 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
% 0.33/0.65 = ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum_gp_offset
% 0.33/0.65 thf(fact_7423_sum__gp__offset,axiom,
% 0.33/0.65 ! [X: complex,M: nat,N2: nat] :
% 0.33/0.65 ( ( ( X = one_one_complex )
% 0.33/0.65 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
% 0.33/0.65 = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) )
% 0.33/0.65 & ( ( X != one_one_complex )
% 0.33/0.65 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
% 0.33/0.65 = ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum_gp_offset
% 0.33/0.65 thf(fact_7424_sum__gp__offset,axiom,
% 0.33/0.65 ! [X: real,M: nat,N2: nat] :
% 0.33/0.65 ( ( ( X = one_one_real )
% 0.33/0.65 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
% 0.33/0.65 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) )
% 0.33/0.65 & ( ( X != one_one_real )
% 0.33/0.65 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
% 0.33/0.65 = ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X @ M ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N2 ) ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum_gp_offset
% 0.33/0.65 thf(fact_7425_of__nat__code__if,axiom,
% 0.33/0.65 ( semiri681578069525770553at_rat
% 0.33/0.65 = ( ^ [N: nat] :
% 0.33/0.65 ( if_rat @ ( N = zero_zero_nat ) @ zero_zero_rat
% 0.33/0.65 @ ( produc6207742614233964070at_rat
% 0.33/0.65 @ ^ [M6: nat,Q4: nat] : ( if_rat @ ( Q4 = zero_zero_nat ) @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M6 ) ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M6 ) ) @ one_one_rat ) )
% 0.33/0.65 @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_code_if
% 0.33/0.65 thf(fact_7426_of__nat__code__if,axiom,
% 0.33/0.65 ( semiri1314217659103216013at_int
% 0.33/0.65 = ( ^ [N: nat] :
% 0.33/0.65 ( if_int @ ( N = zero_zero_nat ) @ zero_zero_int
% 0.33/0.65 @ ( produc6840382203811409530at_int
% 0.33/0.65 @ ^ [M6: nat,Q4: nat] : ( if_int @ ( Q4 = zero_zero_nat ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M6 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M6 ) ) @ one_one_int ) )
% 0.33/0.65 @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_code_if
% 0.33/0.65 thf(fact_7427_of__nat__code__if,axiom,
% 0.33/0.65 ( semiri5074537144036343181t_real
% 0.33/0.65 = ( ^ [N: nat] :
% 0.33/0.65 ( if_real @ ( N = zero_zero_nat ) @ zero_zero_real
% 0.33/0.65 @ ( produc1703576794950452218t_real
% 0.33/0.65 @ ^ [M6: nat,Q4: nat] : ( if_real @ ( Q4 = zero_zero_nat ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M6 ) ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M6 ) ) @ one_one_real ) )
% 0.33/0.65 @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_code_if
% 0.33/0.65 thf(fact_7428_of__nat__code__if,axiom,
% 0.33/0.65 ( semiri1316708129612266289at_nat
% 0.33/0.65 = ( ^ [N: nat] :
% 0.33/0.65 ( if_nat @ ( N = zero_zero_nat ) @ zero_zero_nat
% 0.33/0.65 @ ( produc6842872674320459806at_nat
% 0.33/0.65 @ ^ [M6: nat,Q4: nat] : ( if_nat @ ( Q4 = zero_zero_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M6 ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M6 ) ) @ one_one_nat ) )
% 0.33/0.65 @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_code_if
% 0.33/0.65 thf(fact_7429_of__nat__code__if,axiom,
% 0.33/0.65 ( semiri8010041392384452111omplex
% 0.33/0.65 = ( ^ [N: nat] :
% 0.33/0.65 ( if_complex @ ( N = zero_zero_nat ) @ zero_zero_complex
% 0.33/0.65 @ ( produc1917071388513777916omplex
% 0.33/0.65 @ ^ [M6: nat,Q4: nat] : ( if_complex @ ( Q4 = zero_zero_nat ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M6 ) ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M6 ) ) @ one_one_complex ) )
% 0.33/0.65 @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_nat_code_if
% 0.33/0.65 thf(fact_7430_nat__approx__posE,axiom,
% 0.33/0.65 ! [E: rat] :
% 0.33/0.65 ( ( ord_less_rat @ zero_zero_rat @ E )
% 0.33/0.65 => ~ ! [N4: nat] :
% 0.33/0.65 ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N4 ) ) ) @ E ) ) ).
% 0.33/0.65
% 0.33/0.65 % nat_approx_posE
% 0.33/0.65 thf(fact_7431_nat__approx__posE,axiom,
% 0.33/0.65 ! [E: real] :
% 0.33/0.65 ( ( ord_less_real @ zero_zero_real @ E )
% 0.33/0.65 => ~ ! [N4: nat] :
% 0.33/0.65 ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ E ) ) ).
% 0.33/0.65
% 0.33/0.65 % nat_approx_posE
% 0.33/0.65 thf(fact_7432_floor__exists,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ? [Z5: int] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z5 ) @ X )
% 0.33/0.65 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % floor_exists
% 0.33/0.65 thf(fact_7433_floor__exists,axiom,
% 0.33/0.65 ! [X: rat] :
% 0.33/0.65 ? [Z5: int] :
% 0.33/0.65 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z5 ) @ X )
% 0.33/0.65 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % floor_exists
% 0.33/0.65 thf(fact_7434_floor__exists1,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ? [X4: int] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X4 ) @ X )
% 0.33/0.65 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X4 @ one_one_int ) ) )
% 0.33/0.65 & ! [Y5: int] :
% 0.33/0.65 ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y5 ) @ X )
% 0.33/0.65 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
% 0.33/0.65 => ( Y5 = X4 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % floor_exists1
% 0.33/0.65 thf(fact_7435_floor__exists1,axiom,
% 0.33/0.65 ! [X: rat] :
% 0.33/0.65 ? [X4: int] :
% 0.33/0.65 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X4 ) @ X )
% 0.33/0.65 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X4 @ one_one_int ) ) )
% 0.33/0.65 & ! [Y5: int] :
% 0.33/0.65 ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y5 ) @ X )
% 0.33/0.65 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
% 0.33/0.65 => ( Y5 = X4 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % floor_exists1
% 0.33/0.65 thf(fact_7436_monoseq__arctan__series,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.65 => ( topolo6980174941875973593q_real
% 0.33/0.65 @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % monoseq_arctan_series
% 0.33/0.65 thf(fact_7437_lemma__termdiff3,axiom,
% 0.33/0.65 ! [H2: real,Z: real,K5: real,N2: nat] :
% 0.33/0.65 ( ( H2 != zero_zero_real )
% 0.33/0.65 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K5 )
% 0.33/0.65 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H2 ) ) @ K5 )
% 0.33/0.65 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N2 ) @ ( power_power_real @ Z @ N2 ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % lemma_termdiff3
% 0.33/0.65 thf(fact_7438_lemma__termdiff3,axiom,
% 0.33/0.65 ! [H2: complex,Z: complex,K5: real,N2: nat] :
% 0.33/0.65 ( ( H2 != zero_zero_complex )
% 0.33/0.65 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K5 )
% 0.33/0.65 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H2 ) ) @ K5 )
% 0.33/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % lemma_termdiff3
% 0.33/0.65 thf(fact_7439_complex__mod__minus__le__complex__mod,axiom,
% 0.33/0.65 ! [X: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % complex_mod_minus_le_complex_mod
% 0.33/0.65 thf(fact_7440_complex__mod__triangle__ineq2,axiom,
% 0.33/0.65 ! [B: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B @ A ) ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).
% 0.33/0.65
% 0.33/0.65 % complex_mod_triangle_ineq2
% 0.33/0.65 thf(fact_7441_monoseq__realpow,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.65 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.65 => ( topolo6980174941875973593q_real @ ( power_power_real @ X ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % monoseq_realpow
% 0.33/0.65 thf(fact_7442_real__arch__simple,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ? [N4: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N4 ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_arch_simple
% 0.33/0.65 thf(fact_7443_real__arch__simple,axiom,
% 0.33/0.65 ! [X: rat] :
% 0.33/0.65 ? [N4: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N4 ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_arch_simple
% 0.33/0.65 thf(fact_7444_reals__Archimedean2,axiom,
% 0.33/0.65 ! [X: rat] :
% 0.33/0.65 ? [N4: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N4 ) ) ).
% 0.33/0.65
% 0.33/0.65 % reals_Archimedean2
% 0.33/0.65 thf(fact_7445_reals__Archimedean2,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ? [N4: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N4 ) ) ).
% 0.33/0.65
% 0.33/0.65 % reals_Archimedean2
% 0.33/0.65 thf(fact_7446_exists__least__lemma,axiom,
% 0.33/0.65 ! [P: nat > $o] :
% 0.33/0.65 ( ~ ( P @ zero_zero_nat )
% 0.33/0.65 => ( ? [X_12: nat] : ( P @ X_12 )
% 0.33/0.65 => ? [N4: nat] :
% 0.33/0.65 ( ~ ( P @ N4 )
% 0.33/0.65 & ( P @ ( suc @ N4 ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % exists_least_lemma
% 0.33/0.65 thf(fact_7447_ex__le__of__int,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ? [Z5: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z5 ) ) ).
% 0.33/0.65
% 0.33/0.65 % ex_le_of_int
% 0.33/0.65 thf(fact_7448_ex__le__of__int,axiom,
% 0.33/0.65 ! [X: rat] :
% 0.33/0.65 ? [Z5: int] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z5 ) ) ).
% 0.33/0.65
% 0.33/0.65 % ex_le_of_int
% 0.33/0.65 thf(fact_7449_ex__of__int__less,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ? [Z5: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z5 ) @ X ) ).
% 0.33/0.65
% 0.33/0.65 % ex_of_int_less
% 0.33/0.65 thf(fact_7450_ex__of__int__less,axiom,
% 0.33/0.65 ! [X: rat] :
% 0.33/0.65 ? [Z5: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z5 ) @ X ) ).
% 0.33/0.65
% 0.33/0.65 % ex_of_int_less
% 0.33/0.65 thf(fact_7451_ex__less__of__int,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ? [Z5: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z5 ) ) ).
% 0.33/0.65
% 0.33/0.65 % ex_less_of_int
% 0.33/0.65 thf(fact_7452_ex__less__of__int,axiom,
% 0.33/0.65 ! [X: rat] :
% 0.33/0.65 ? [Z5: int] : ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z5 ) ) ).
% 0.33/0.65
% 0.33/0.65 % ex_less_of_int
% 0.33/0.65 thf(fact_7453_ex__less__of__nat__mult,axiom,
% 0.33/0.65 ! [X: rat,Y: rat] :
% 0.33/0.65 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.33/0.65 => ? [N4: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N4 ) @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % ex_less_of_nat_mult
% 0.33/0.65 thf(fact_7454_ex__less__of__nat__mult,axiom,
% 0.33/0.65 ! [X: real,Y: real] :
% 0.33/0.65 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.65 => ? [N4: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % ex_less_of_nat_mult
% 0.33/0.65 thf(fact_7455_norm__divide__numeral,axiom,
% 0.33/0.65 ! [A: real,W: num] :
% 0.33/0.65 ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ W ) ) )
% 0.33/0.65 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_divide_numeral
% 0.33/0.65 thf(fact_7456_norm__divide__numeral,axiom,
% 0.33/0.65 ! [A: complex,W: num] :
% 0.33/0.65 ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 0.33/0.65 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_divide_numeral
% 0.33/0.65 thf(fact_7457_norm__mult__numeral2,axiom,
% 0.33/0.65 ! [A: real,W: num] :
% 0.33/0.65 ( ( real_V7735802525324610683m_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) )
% 0.33/0.65 = ( times_times_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_mult_numeral2
% 0.33/0.65 thf(fact_7458_norm__mult__numeral2,axiom,
% 0.33/0.65 ! [A: complex,W: num] :
% 0.33/0.65 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 0.33/0.65 = ( times_times_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_mult_numeral2
% 0.33/0.65 thf(fact_7459_norm__mult__numeral1,axiom,
% 0.33/0.65 ! [W: num,A: real] :
% 0.33/0.65 ( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
% 0.33/0.65 = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V7735802525324610683m_real @ A ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_mult_numeral1
% 0.33/0.65 thf(fact_7460_norm__mult__numeral1,axiom,
% 0.33/0.65 ! [W: num,A: complex] :
% 0.33/0.65 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
% 0.33/0.65 = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V1022390504157884413omplex @ A ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_mult_numeral1
% 0.33/0.65 thf(fact_7461_norm__neg__numeral,axiom,
% 0.33/0.65 ! [W: num] :
% 0.33/0.65 ( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.33/0.65 = ( numeral_numeral_real @ W ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_neg_numeral
% 0.33/0.65 thf(fact_7462_norm__neg__numeral,axiom,
% 0.33/0.65 ! [W: num] :
% 0.33/0.65 ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.33/0.65 = ( numeral_numeral_real @ W ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_neg_numeral
% 0.33/0.65 thf(fact_7463_norm__le__zero__iff,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
% 0.33/0.65 = ( X = zero_zero_real ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_le_zero_iff
% 0.33/0.65 thf(fact_7464_norm__le__zero__iff,axiom,
% 0.33/0.65 ! [X: complex] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real )
% 0.33/0.65 = ( X = zero_zero_complex ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_le_zero_iff
% 0.33/0.65 thf(fact_7465_norm__one,axiom,
% 0.33/0.65 ( ( real_V7735802525324610683m_real @ one_one_real )
% 0.33/0.65 = one_one_real ) ).
% 0.33/0.65
% 0.33/0.65 % norm_one
% 0.33/0.65 thf(fact_7466_norm__one,axiom,
% 0.33/0.65 ( ( real_V1022390504157884413omplex @ one_one_complex )
% 0.33/0.65 = one_one_real ) ).
% 0.33/0.65
% 0.33/0.65 % norm_one
% 0.33/0.65 thf(fact_7467_norm__numeral,axiom,
% 0.33/0.65 ! [W: num] :
% 0.33/0.65 ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
% 0.33/0.65 = ( numeral_numeral_real @ W ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_numeral
% 0.33/0.65 thf(fact_7468_norm__numeral,axiom,
% 0.33/0.65 ! [W: num] :
% 0.33/0.65 ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.33/0.65 = ( numeral_numeral_real @ W ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_numeral
% 0.33/0.65 thf(fact_7469_zero__less__norm__iff,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
% 0.33/0.65 = ( X != zero_zero_real ) ) ).
% 0.33/0.65
% 0.33/0.65 % zero_less_norm_iff
% 0.33/0.65 thf(fact_7470_zero__less__norm__iff,axiom,
% 0.33/0.65 ! [X: complex] :
% 0.33/0.65 ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) )
% 0.33/0.65 = ( X != zero_zero_complex ) ) ).
% 0.33/0.65
% 0.33/0.65 % zero_less_norm_iff
% 0.33/0.65 thf(fact_7471_norm__minus__commute,axiom,
% 0.33/0.65 ! [A: real,B: real] :
% 0.33/0.65 ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) )
% 0.33/0.65 = ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_minus_commute
% 0.33/0.65 thf(fact_7472_norm__minus__commute,axiom,
% 0.33/0.65 ! [A: complex,B: complex] :
% 0.33/0.65 ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) )
% 0.33/0.65 = ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ A ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_minus_commute
% 0.33/0.65 thf(fact_7473_norm__not__less__zero,axiom,
% 0.33/0.65 ! [X: complex] :
% 0.33/0.65 ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real ) ).
% 0.33/0.65
% 0.33/0.65 % norm_not_less_zero
% 0.33/0.65 thf(fact_7474_norm__ge__zero,axiom,
% 0.33/0.65 ! [X: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_ge_zero
% 0.33/0.65 thf(fact_7475_norm__mult,axiom,
% 0.33/0.65 ! [X: real,Y: real] :
% 0.33/0.65 ( ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) )
% 0.33/0.65 = ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_mult
% 0.33/0.65 thf(fact_7476_norm__mult,axiom,
% 0.33/0.65 ! [X: complex,Y: complex] :
% 0.33/0.65 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) )
% 0.33/0.65 = ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_mult
% 0.33/0.65 thf(fact_7477_norm__divide,axiom,
% 0.33/0.65 ! [A: real,B: real] :
% 0.33/0.65 ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
% 0.33/0.65 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_divide
% 0.33/0.65 thf(fact_7478_norm__divide,axiom,
% 0.33/0.65 ! [A: complex,B: complex] :
% 0.33/0.65 ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.33/0.65 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_divide
% 0.33/0.65 thf(fact_7479_sum__norm__le,axiom,
% 0.33/0.65 ! [S: set_real,F: real > complex,G: real > real] :
% 0.33/0.65 ( ! [X4: real] :
% 0.33/0.65 ( ( member_real @ X4 @ S )
% 0.33/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X4 ) ) @ ( G @ X4 ) ) )
% 0.33/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F @ S ) ) @ ( groups8097168146408367636l_real @ G @ S ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum_norm_le
% 0.33/0.65 thf(fact_7480_sum__norm__le,axiom,
% 0.33/0.65 ! [S: set_int,F: int > complex,G: int > real] :
% 0.33/0.65 ( ! [X4: int] :
% 0.33/0.65 ( ( member_int @ X4 @ S )
% 0.33/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X4 ) ) @ ( G @ X4 ) ) )
% 0.33/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups3049146728041665814omplex @ F @ S ) ) @ ( groups8778361861064173332t_real @ G @ S ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum_norm_le
% 0.33/0.65 thf(fact_7481_sum__norm__le,axiom,
% 0.33/0.65 ! [S: set_set_nat,F: set_nat > complex,G: set_nat > real] :
% 0.33/0.65 ( ! [X4: set_nat] :
% 0.33/0.65 ( ( member_set_nat @ X4 @ S )
% 0.33/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X4 ) ) @ ( G @ X4 ) ) )
% 0.33/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups8255218700646806128omplex @ F @ S ) ) @ ( groups5107569545109728110t_real @ G @ S ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum_norm_le
% 0.33/0.65 thf(fact_7482_sum__norm__le,axiom,
% 0.33/0.65 ! [S: set_nat,F: nat > complex,G: nat > real] :
% 0.33/0.65 ( ! [X4: nat] :
% 0.33/0.65 ( ( member_nat @ X4 @ S )
% 0.33/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X4 ) ) @ ( G @ X4 ) ) )
% 0.33/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ S ) ) @ ( groups6591440286371151544t_real @ G @ S ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum_norm_le
% 0.33/0.65 thf(fact_7483_sum__norm__le,axiom,
% 0.33/0.65 ! [S: set_complex,F: complex > complex,G: complex > real] :
% 0.33/0.65 ( ! [X4: complex] :
% 0.33/0.65 ( ( member_complex @ X4 @ S )
% 0.33/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X4 ) ) @ ( G @ X4 ) ) )
% 0.33/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ S ) ) @ ( groups5808333547571424918x_real @ G @ S ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum_norm_le
% 0.33/0.65 thf(fact_7484_sum__norm__le,axiom,
% 0.33/0.65 ! [S: set_nat,F: nat > real,G: nat > real] :
% 0.33/0.65 ( ! [X4: nat] :
% 0.33/0.65 ( ( member_nat @ X4 @ S )
% 0.33/0.65 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X4 ) ) @ ( G @ X4 ) ) )
% 0.33/0.65 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ S ) ) @ ( groups6591440286371151544t_real @ G @ S ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum_norm_le
% 0.33/0.65 thf(fact_7485_norm__power,axiom,
% 0.33/0.65 ! [X: real,N2: nat] :
% 0.33/0.65 ( ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N2 ) )
% 0.33/0.65 = ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_power
% 0.33/0.65 thf(fact_7486_norm__power,axiom,
% 0.33/0.65 ! [X: complex,N2: nat] :
% 0.33/0.65 ( ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N2 ) )
% 0.33/0.65 = ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_power
% 0.33/0.65 thf(fact_7487_norm__sum,axiom,
% 0.33/0.65 ! [F: nat > complex,A2: set_nat] :
% 0.33/0.65 ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ A2 ) )
% 0.33/0.65 @ ( groups6591440286371151544t_real
% 0.33/0.65 @ ^ [I3: nat] : ( real_V1022390504157884413omplex @ ( F @ I3 ) )
% 0.33/0.65 @ A2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_sum
% 0.33/0.65 thf(fact_7488_norm__sum,axiom,
% 0.33/0.65 ! [F: complex > complex,A2: set_complex] :
% 0.33/0.65 ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ A2 ) )
% 0.33/0.65 @ ( groups5808333547571424918x_real
% 0.33/0.65 @ ^ [I3: complex] : ( real_V1022390504157884413omplex @ ( F @ I3 ) )
% 0.33/0.65 @ A2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_sum
% 0.33/0.65 thf(fact_7489_norm__sum,axiom,
% 0.33/0.65 ! [F: nat > real,A2: set_nat] :
% 0.33/0.65 ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
% 0.33/0.65 @ ( groups6591440286371151544t_real
% 0.33/0.65 @ ^ [I3: nat] : ( real_V7735802525324610683m_real @ ( F @ I3 ) )
% 0.33/0.65 @ A2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_sum
% 0.33/0.65 thf(fact_7490_norm__uminus__minus,axiom,
% 0.33/0.65 ! [X: real,Y: real] :
% 0.33/0.65 ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ Y ) )
% 0.33/0.65 = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_uminus_minus
% 0.33/0.65 thf(fact_7491_norm__uminus__minus,axiom,
% 0.33/0.65 ! [X: complex,Y: complex] :
% 0.33/0.65 ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ Y ) )
% 0.33/0.65 = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_uminus_minus
% 0.33/0.65 thf(fact_7492_nonzero__norm__divide,axiom,
% 0.33/0.65 ! [B: real,A: real] :
% 0.33/0.65 ( ( B != zero_zero_real )
% 0.33/0.65 => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
% 0.33/0.65 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % nonzero_norm_divide
% 0.33/0.65 thf(fact_7493_nonzero__norm__divide,axiom,
% 0.33/0.65 ! [B: complex,A: complex] :
% 0.33/0.65 ( ( B != zero_zero_complex )
% 0.33/0.65 => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.33/0.65 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % nonzero_norm_divide
% 0.33/0.65 thf(fact_7494_power__eq__imp__eq__norm,axiom,
% 0.33/0.65 ! [W: real,N2: nat,Z: real] :
% 0.33/0.65 ( ( ( power_power_real @ W @ N2 )
% 0.33/0.65 = ( power_power_real @ Z @ N2 ) )
% 0.33/0.65 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.65 => ( ( real_V7735802525324610683m_real @ W )
% 0.33/0.65 = ( real_V7735802525324610683m_real @ Z ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % power_eq_imp_eq_norm
% 0.33/0.65 thf(fact_7495_power__eq__imp__eq__norm,axiom,
% 0.33/0.65 ! [W: complex,N2: nat,Z: complex] :
% 0.33/0.65 ( ( ( power_power_complex @ W @ N2 )
% 0.33/0.65 = ( power_power_complex @ Z @ N2 ) )
% 0.33/0.65 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.65 => ( ( real_V1022390504157884413omplex @ W )
% 0.33/0.65 = ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % power_eq_imp_eq_norm
% 0.33/0.65 thf(fact_7496_norm__mult__less,axiom,
% 0.33/0.65 ! [X: real,R: real,Y: real,S3: real] :
% 0.33/0.65 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R )
% 0.33/0.65 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S3 )
% 0.33/0.65 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ R @ S3 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_mult_less
% 0.33/0.65 thf(fact_7497_norm__mult__less,axiom,
% 0.33/0.65 ! [X: complex,R: real,Y: complex,S3: real] :
% 0.33/0.65 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R )
% 0.33/0.65 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S3 )
% 0.33/0.65 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ R @ S3 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_mult_less
% 0.33/0.65 thf(fact_7498_norm__mult__ineq,axiom,
% 0.33/0.65 ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_mult_ineq
% 0.33/0.65 thf(fact_7499_norm__mult__ineq,axiom,
% 0.33/0.65 ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_mult_ineq
% 0.33/0.65 thf(fact_7500_norm__add__less,axiom,
% 0.33/0.65 ! [X: real,R: real,Y: real,S3: real] :
% 0.33/0.65 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R )
% 0.33/0.65 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S3 )
% 0.33/0.65 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ R @ S3 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_add_less
% 0.33/0.65 thf(fact_7501_norm__add__less,axiom,
% 0.33/0.65 ! [X: complex,R: real,Y: complex,S3: real] :
% 0.33/0.65 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R )
% 0.33/0.65 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S3 )
% 0.33/0.65 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ R @ S3 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_add_less
% 0.33/0.65 thf(fact_7502_norm__triangle__lt,axiom,
% 0.33/0.65 ! [X: real,Y: real,E: real] :
% 0.33/0.65 ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
% 0.33/0.65 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_triangle_lt
% 0.33/0.65 thf(fact_7503_norm__triangle__lt,axiom,
% 0.33/0.65 ! [X: complex,Y: complex,E: real] :
% 0.33/0.65 ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
% 0.33/0.65 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_triangle_lt
% 0.33/0.65 thf(fact_7504_norm__power__ineq,axiom,
% 0.33/0.65 ! [X: real,N2: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N2 ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_power_ineq
% 0.33/0.65 thf(fact_7505_norm__power__ineq,axiom,
% 0.33/0.65 ! [X: complex,N2: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N2 ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_power_ineq
% 0.33/0.65 thf(fact_7506_norm__triangle__mono,axiom,
% 0.33/0.65 ! [A: real,R: real,B: real,S3: real] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R )
% 0.33/0.65 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S3 )
% 0.33/0.65 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R @ S3 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_triangle_mono
% 0.33/0.65 thf(fact_7507_norm__triangle__mono,axiom,
% 0.33/0.65 ! [A: complex,R: real,B: complex,S3: real] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R )
% 0.33/0.65 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S3 )
% 0.33/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R @ S3 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_triangle_mono
% 0.33/0.65 thf(fact_7508_norm__triangle__ineq,axiom,
% 0.33/0.65 ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_triangle_ineq
% 0.33/0.65 thf(fact_7509_norm__triangle__ineq,axiom,
% 0.33/0.65 ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_triangle_ineq
% 0.33/0.65 thf(fact_7510_norm__triangle__le,axiom,
% 0.33/0.65 ! [X: real,Y: real,E: real] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
% 0.33/0.65 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_triangle_le
% 0.33/0.65 thf(fact_7511_norm__triangle__le,axiom,
% 0.33/0.65 ! [X: complex,Y: complex,E: real] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
% 0.33/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_triangle_le
% 0.33/0.65 thf(fact_7512_norm__add__leD,axiom,
% 0.33/0.65 ! [A: real,B: real,C: real] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C )
% 0.33/0.65 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_add_leD
% 0.33/0.65 thf(fact_7513_norm__add__leD,axiom,
% 0.33/0.65 ! [A: complex,B: complex,C: real] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C )
% 0.33/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_add_leD
% 0.33/0.65 thf(fact_7514_norm__diff__triangle__less,axiom,
% 0.33/0.65 ! [X: real,Y: real,E1: real,Z: real,E22: real] :
% 0.33/0.65 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
% 0.33/0.65 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z ) ) @ E22 )
% 0.33/0.65 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_diff_triangle_less
% 0.33/0.65 thf(fact_7515_norm__diff__triangle__less,axiom,
% 0.33/0.65 ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
% 0.33/0.65 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
% 0.33/0.65 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z ) ) @ E22 )
% 0.33/0.65 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_diff_triangle_less
% 0.33/0.65 thf(fact_7516_norm__triangle__sub,axiom,
% 0.33/0.65 ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_triangle_sub
% 0.33/0.65 thf(fact_7517_norm__triangle__sub,axiom,
% 0.33/0.65 ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_triangle_sub
% 0.33/0.65 thf(fact_7518_norm__triangle__ineq4,axiom,
% 0.33/0.65 ! [A: real,B: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_triangle_ineq4
% 0.33/0.65 thf(fact_7519_norm__triangle__ineq4,axiom,
% 0.33/0.65 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_triangle_ineq4
% 0.33/0.65 thf(fact_7520_norm__diff__triangle__le,axiom,
% 0.33/0.65 ! [X: real,Y: real,E1: real,Z: real,E22: real] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
% 0.33/0.65 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z ) ) @ E22 )
% 0.33/0.65 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_diff_triangle_le
% 0.33/0.65 thf(fact_7521_norm__diff__triangle__le,axiom,
% 0.33/0.65 ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
% 0.33/0.65 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z ) ) @ E22 )
% 0.33/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_diff_triangle_le
% 0.33/0.65 thf(fact_7522_norm__triangle__le__diff,axiom,
% 0.33/0.65 ! [X: real,Y: real,E: real] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
% 0.33/0.65 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_triangle_le_diff
% 0.33/0.65 thf(fact_7523_norm__triangle__le__diff,axiom,
% 0.33/0.65 ! [X: complex,Y: complex,E: real] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
% 0.33/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_triangle_le_diff
% 0.33/0.65 thf(fact_7524_norm__diff__ineq,axiom,
% 0.33/0.65 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_diff_ineq
% 0.33/0.65 thf(fact_7525_norm__diff__ineq,axiom,
% 0.33/0.65 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_diff_ineq
% 0.33/0.65 thf(fact_7526_norm__triangle__ineq2,axiom,
% 0.33/0.65 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_triangle_ineq2
% 0.33/0.65 thf(fact_7527_norm__triangle__ineq2,axiom,
% 0.33/0.65 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_triangle_ineq2
% 0.33/0.65 thf(fact_7528_power__eq__1__iff,axiom,
% 0.33/0.65 ! [W: real,N2: nat] :
% 0.33/0.65 ( ( ( power_power_real @ W @ N2 )
% 0.33/0.65 = one_one_real )
% 0.33/0.65 => ( ( ( real_V7735802525324610683m_real @ W )
% 0.33/0.65 = one_one_real )
% 0.33/0.65 | ( N2 = zero_zero_nat ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % power_eq_1_iff
% 0.33/0.65 thf(fact_7529_power__eq__1__iff,axiom,
% 0.33/0.65 ! [W: complex,N2: nat] :
% 0.33/0.65 ( ( ( power_power_complex @ W @ N2 )
% 0.33/0.65 = one_one_complex )
% 0.33/0.65 => ( ( ( real_V1022390504157884413omplex @ W )
% 0.33/0.65 = one_one_real )
% 0.33/0.65 | ( N2 = zero_zero_nat ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % power_eq_1_iff
% 0.33/0.65 thf(fact_7530_norm__diff__triangle__ineq,axiom,
% 0.33/0.65 ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_diff_triangle_ineq
% 0.33/0.65 thf(fact_7531_norm__diff__triangle__ineq,axiom,
% 0.33/0.65 ! [A: complex,B: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ D ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_diff_triangle_ineq
% 0.33/0.65 thf(fact_7532_norm__triangle__ineq3,axiom,
% 0.33/0.65 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_triangle_ineq3
% 0.33/0.65 thf(fact_7533_norm__triangle__ineq3,axiom,
% 0.33/0.65 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_triangle_ineq3
% 0.33/0.65 thf(fact_7534_square__norm__one,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.65 = one_one_real )
% 0.33/0.65 => ( ( real_V7735802525324610683m_real @ X )
% 0.33/0.65 = one_one_real ) ) ).
% 0.33/0.65
% 0.33/0.65 % square_norm_one
% 0.33/0.65 thf(fact_7535_square__norm__one,axiom,
% 0.33/0.65 ! [X: complex] :
% 0.33/0.65 ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.65 = one_one_complex )
% 0.33/0.65 => ( ( real_V1022390504157884413omplex @ X )
% 0.33/0.65 = one_one_real ) ) ).
% 0.33/0.65
% 0.33/0.65 % square_norm_one
% 0.33/0.65 thf(fact_7536_norm__power__diff,axiom,
% 0.33/0.65 ! [Z: real,W: real,M: nat] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
% 0.33/0.65 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W ) @ one_one_real )
% 0.33/0.65 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( power_power_real @ Z @ M ) @ ( power_power_real @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Z @ W ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_power_diff
% 0.33/0.65 thf(fact_7537_norm__power__diff,axiom,
% 0.33/0.65 ! [Z: complex,W: complex,M: nat] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
% 0.33/0.65 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W ) @ one_one_real )
% 0.33/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( power_power_complex @ Z @ M ) @ ( power_power_complex @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Z @ W ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % norm_power_diff
% 0.33/0.65 thf(fact_7538_ln__series,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.65 => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.33/0.65 => ( ( ln_ln_real @ X )
% 0.33/0.65 = ( suminf_real
% 0.33/0.65 @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % ln_series
% 0.33/0.65 thf(fact_7539_arctan__series,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.65 => ( ( arctan @ X )
% 0.33/0.65 = ( suminf_real
% 0.33/0.65 @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % arctan_series
% 0.33/0.65 thf(fact_7540_round__unique,axiom,
% 0.33/0.65 ! [X: real,Y: int] :
% 0.33/0.65 ( ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y ) )
% 0.33/0.65 => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.33/0.65 => ( ( archim8280529875227126926d_real @ X )
% 0.33/0.65 = Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % round_unique
% 0.33/0.65 thf(fact_7541_round__unique,axiom,
% 0.33/0.65 ! [X: rat,Y: int] :
% 0.33/0.65 ( ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y ) )
% 0.33/0.65 => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.65 => ( ( archim7778729529865785530nd_rat @ X )
% 0.33/0.65 = Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % round_unique
% 0.33/0.65 thf(fact_7542_lemma__termdiff2,axiom,
% 0.33/0.65 ! [H2: rat,Z: rat,N2: nat] :
% 0.33/0.65 ( ( H2 != zero_zero_rat )
% 0.33/0.65 => ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ N2 ) @ ( power_power_rat @ Z @ N2 ) ) @ H2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.33/0.65 = ( times_times_rat @ H2
% 0.33/0.65 @ ( groups2906978787729119204at_rat
% 0.33/0.65 @ ^ [P5: nat] :
% 0.33/0.65 ( groups2906978787729119204at_rat
% 0.33/0.65 @ ^ [Q4: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ Q4 ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % lemma_termdiff2
% 0.33/0.65 thf(fact_7543_lemma__termdiff2,axiom,
% 0.33/0.65 ! [H2: complex,Z: complex,N2: nat] :
% 0.33/0.65 ( ( H2 != zero_zero_complex )
% 0.33/0.65 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.33/0.65 = ( times_times_complex @ H2
% 0.33/0.65 @ ( groups2073611262835488442omplex
% 0.33/0.65 @ ^ [P5: nat] :
% 0.33/0.65 ( groups2073611262835488442omplex
% 0.33/0.65 @ ^ [Q4: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ Q4 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % lemma_termdiff2
% 0.33/0.65 thf(fact_7544_lemma__termdiff2,axiom,
% 0.33/0.65 ! [H2: real,Z: real,N2: nat] :
% 0.33/0.65 ( ( H2 != zero_zero_real )
% 0.33/0.65 => ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N2 ) @ ( power_power_real @ Z @ N2 ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.33/0.65 = ( times_times_real @ H2
% 0.33/0.65 @ ( groups6591440286371151544t_real
% 0.33/0.65 @ ^ [P5: nat] :
% 0.33/0.65 ( groups6591440286371151544t_real
% 0.33/0.65 @ ^ [Q4: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ Q4 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % lemma_termdiff2
% 0.33/0.65 thf(fact_7545_summable__arctan__series,axiom,
% 0.33/0.65 ! [X: real] :
% 0.33/0.65 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.65 => ( summable_real
% 0.33/0.65 @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % summable_arctan_series
% 0.33/0.65 thf(fact_7546_lessThan__iff,axiom,
% 0.33/0.65 ! [I2: set_nat,K: set_nat] :
% 0.33/0.65 ( ( member_set_nat @ I2 @ ( set_or890127255671739683et_nat @ K ) )
% 0.33/0.65 = ( ord_less_set_nat @ I2 @ K ) ) ).
% 0.33/0.65
% 0.33/0.65 % lessThan_iff
% 0.33/0.65 thf(fact_7547_lessThan__iff,axiom,
% 0.33/0.65 ! [I2: rat,K: rat] :
% 0.33/0.65 ( ( member_rat @ I2 @ ( set_ord_lessThan_rat @ K ) )
% 0.33/0.65 = ( ord_less_rat @ I2 @ K ) ) ).
% 0.33/0.65
% 0.33/0.65 % lessThan_iff
% 0.33/0.65 thf(fact_7548_lessThan__iff,axiom,
% 0.33/0.65 ! [I2: num,K: num] :
% 0.33/0.65 ( ( member_num @ I2 @ ( set_ord_lessThan_num @ K ) )
% 0.33/0.65 = ( ord_less_num @ I2 @ K ) ) ).
% 0.33/0.65
% 0.33/0.65 % lessThan_iff
% 0.33/0.65 thf(fact_7549_lessThan__iff,axiom,
% 0.33/0.65 ! [I2: int,K: int] :
% 0.33/0.65 ( ( member_int @ I2 @ ( set_ord_lessThan_int @ K ) )
% 0.33/0.65 = ( ord_less_int @ I2 @ K ) ) ).
% 0.33/0.65
% 0.33/0.65 % lessThan_iff
% 0.33/0.65 thf(fact_7550_lessThan__iff,axiom,
% 0.33/0.65 ! [I2: nat,K: nat] :
% 0.33/0.65 ( ( member_nat @ I2 @ ( set_ord_lessThan_nat @ K ) )
% 0.33/0.65 = ( ord_less_nat @ I2 @ K ) ) ).
% 0.33/0.65
% 0.33/0.65 % lessThan_iff
% 0.33/0.65 thf(fact_7551_lessThan__iff,axiom,
% 0.33/0.65 ! [I2: real,K: real] :
% 0.33/0.65 ( ( member_real @ I2 @ ( set_or5984915006950818249n_real @ K ) )
% 0.33/0.65 = ( ord_less_real @ I2 @ K ) ) ).
% 0.33/0.65
% 0.33/0.65 % lessThan_iff
% 0.33/0.65 thf(fact_7552_lessThan__subset__iff,axiom,
% 0.33/0.65 ! [X: rat,Y: rat] :
% 0.33/0.65 ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X ) @ ( set_ord_lessThan_rat @ Y ) )
% 0.33/0.65 = ( ord_less_eq_rat @ X @ Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % lessThan_subset_iff
% 0.33/0.65 thf(fact_7553_lessThan__subset__iff,axiom,
% 0.33/0.65 ! [X: num,Y: num] :
% 0.33/0.65 ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X ) @ ( set_ord_lessThan_num @ Y ) )
% 0.33/0.65 = ( ord_less_eq_num @ X @ Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % lessThan_subset_iff
% 0.33/0.65 thf(fact_7554_lessThan__subset__iff,axiom,
% 0.33/0.65 ! [X: int,Y: int] :
% 0.33/0.65 ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y ) )
% 0.33/0.65 = ( ord_less_eq_int @ X @ Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % lessThan_subset_iff
% 0.33/0.65 thf(fact_7555_lessThan__subset__iff,axiom,
% 0.33/0.65 ! [X: nat,Y: nat] :
% 0.33/0.65 ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
% 0.33/0.65 = ( ord_less_eq_nat @ X @ Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % lessThan_subset_iff
% 0.33/0.65 thf(fact_7556_lessThan__subset__iff,axiom,
% 0.33/0.65 ! [X: real,Y: real] :
% 0.33/0.65 ( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X ) @ ( set_or5984915006950818249n_real @ Y ) )
% 0.33/0.65 = ( ord_less_eq_real @ X @ Y ) ) ).
% 0.33/0.65
% 0.33/0.65 % lessThan_subset_iff
% 0.33/0.65 thf(fact_7557_round__numeral,axiom,
% 0.33/0.65 ! [N2: num] :
% 0.33/0.65 ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N2 ) )
% 0.33/0.65 = ( numeral_numeral_int @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % round_numeral
% 0.33/0.65 thf(fact_7558_round__numeral,axiom,
% 0.33/0.65 ! [N2: num] :
% 0.33/0.65 ( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N2 ) )
% 0.33/0.65 = ( numeral_numeral_int @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % round_numeral
% 0.33/0.65 thf(fact_7559_round__1,axiom,
% 0.33/0.65 ( ( archim8280529875227126926d_real @ one_one_real )
% 0.33/0.65 = one_one_int ) ).
% 0.33/0.65
% 0.33/0.65 % round_1
% 0.33/0.65 thf(fact_7560_round__1,axiom,
% 0.33/0.65 ( ( archim7778729529865785530nd_rat @ one_one_rat )
% 0.33/0.65 = one_one_int ) ).
% 0.33/0.65
% 0.33/0.65 % round_1
% 0.33/0.65 thf(fact_7561_sum_OlessThan__Suc,axiom,
% 0.33/0.65 ! [G: nat > rat,N2: nat] :
% 0.33/0.65 ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.33/0.65 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum.lessThan_Suc
% 0.33/0.65 thf(fact_7562_sum_OlessThan__Suc,axiom,
% 0.33/0.65 ! [G: nat > int,N2: nat] :
% 0.33/0.65 ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.33/0.65 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum.lessThan_Suc
% 0.33/0.65 thf(fact_7563_sum_OlessThan__Suc,axiom,
% 0.33/0.65 ! [G: nat > nat,N2: nat] :
% 0.33/0.65 ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.33/0.65 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum.lessThan_Suc
% 0.33/0.65 thf(fact_7564_sum_OlessThan__Suc,axiom,
% 0.33/0.65 ! [G: nat > real,N2: nat] :
% 0.33/0.65 ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.33/0.65 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum.lessThan_Suc
% 0.33/0.65 thf(fact_7565_round__neg__numeral,axiom,
% 0.33/0.65 ! [N2: num] :
% 0.33/0.65 ( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 0.33/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % round_neg_numeral
% 0.33/0.65 thf(fact_7566_round__neg__numeral,axiom,
% 0.33/0.65 ! [N2: num] :
% 0.33/0.65 ( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 0.33/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % round_neg_numeral
% 0.33/0.65 thf(fact_7567_powser__zero,axiom,
% 0.33/0.65 ! [F: nat > complex] :
% 0.33/0.65 ( ( suminf_complex
% 0.33/0.65 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) ) )
% 0.33/0.65 = ( F @ zero_zero_nat ) ) ).
% 0.33/0.65
% 0.33/0.65 % powser_zero
% 0.33/0.65 thf(fact_7568_powser__zero,axiom,
% 0.33/0.65 ! [F: nat > real] :
% 0.33/0.65 ( ( suminf_real
% 0.33/0.65 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ zero_zero_real @ N ) ) )
% 0.33/0.65 = ( F @ zero_zero_nat ) ) ).
% 0.33/0.65
% 0.33/0.65 % powser_zero
% 0.33/0.65 thf(fact_7569_lessThan__def,axiom,
% 0.33/0.65 ( set_or890127255671739683et_nat
% 0.33/0.65 = ( ^ [U2: set_nat] :
% 0.33/0.65 ( collect_set_nat
% 0.33/0.65 @ ^ [X2: set_nat] : ( ord_less_set_nat @ X2 @ U2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % lessThan_def
% 0.33/0.65 thf(fact_7570_lessThan__def,axiom,
% 0.33/0.65 ( set_ord_lessThan_rat
% 0.33/0.65 = ( ^ [U2: rat] :
% 0.33/0.65 ( collect_rat
% 0.33/0.65 @ ^ [X2: rat] : ( ord_less_rat @ X2 @ U2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % lessThan_def
% 0.33/0.65 thf(fact_7571_lessThan__def,axiom,
% 0.33/0.65 ( set_ord_lessThan_num
% 0.33/0.65 = ( ^ [U2: num] :
% 0.33/0.65 ( collect_num
% 0.33/0.65 @ ^ [X2: num] : ( ord_less_num @ X2 @ U2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % lessThan_def
% 0.33/0.65 thf(fact_7572_lessThan__def,axiom,
% 0.33/0.65 ( set_ord_lessThan_int
% 0.33/0.65 = ( ^ [U2: int] :
% 0.33/0.65 ( collect_int
% 0.33/0.65 @ ^ [X2: int] : ( ord_less_int @ X2 @ U2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % lessThan_def
% 0.33/0.65 thf(fact_7573_lessThan__def,axiom,
% 0.33/0.65 ( set_ord_lessThan_nat
% 0.33/0.65 = ( ^ [U2: nat] :
% 0.33/0.65 ( collect_nat
% 0.33/0.65 @ ^ [X2: nat] : ( ord_less_nat @ X2 @ U2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % lessThan_def
% 0.33/0.65 thf(fact_7574_lessThan__def,axiom,
% 0.33/0.65 ( set_or5984915006950818249n_real
% 0.33/0.65 = ( ^ [U2: real] :
% 0.33/0.65 ( collect_real
% 0.33/0.65 @ ^ [X2: real] : ( ord_less_real @ X2 @ U2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % lessThan_def
% 0.33/0.65 thf(fact_7575_powser__insidea,axiom,
% 0.33/0.65 ! [F: nat > real,X: real,Z: real] :
% 0.33/0.65 ( ( summable_real
% 0.33/0.65 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ X @ N ) ) )
% 0.33/0.65 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X ) )
% 0.33/0.65 => ( summable_real
% 0.33/0.65 @ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % powser_insidea
% 0.33/0.65 thf(fact_7576_powser__insidea,axiom,
% 0.33/0.65 ! [F: nat > complex,X: complex,Z: complex] :
% 0.33/0.65 ( ( summable_complex
% 0.33/0.65 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ X @ N ) ) )
% 0.33/0.65 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X ) )
% 0.33/0.65 => ( summable_real
% 0.33/0.65 @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % powser_insidea
% 0.33/0.65 thf(fact_7577_lessThan__strict__subset__iff,axiom,
% 0.33/0.65 ! [M: rat,N2: rat] :
% 0.33/0.65 ( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N2 ) )
% 0.33/0.65 = ( ord_less_rat @ M @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % lessThan_strict_subset_iff
% 0.33/0.65 thf(fact_7578_lessThan__strict__subset__iff,axiom,
% 0.33/0.65 ! [M: num,N2: num] :
% 0.33/0.65 ( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N2 ) )
% 0.33/0.65 = ( ord_less_num @ M @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % lessThan_strict_subset_iff
% 0.33/0.65 thf(fact_7579_lessThan__strict__subset__iff,axiom,
% 0.33/0.65 ! [M: int,N2: int] :
% 0.33/0.65 ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N2 ) )
% 0.33/0.65 = ( ord_less_int @ M @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % lessThan_strict_subset_iff
% 0.33/0.65 thf(fact_7580_lessThan__strict__subset__iff,axiom,
% 0.33/0.65 ! [M: nat,N2: nat] :
% 0.33/0.65 ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.65 = ( ord_less_nat @ M @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % lessThan_strict_subset_iff
% 0.33/0.65 thf(fact_7581_lessThan__strict__subset__iff,axiom,
% 0.33/0.65 ! [M: real,N2: real] :
% 0.33/0.65 ( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N2 ) )
% 0.33/0.65 = ( ord_less_real @ M @ N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % lessThan_strict_subset_iff
% 0.33/0.65 thf(fact_7582_sum__pos__lt__pair,axiom,
% 0.33/0.65 ! [F: nat > real,K: nat] :
% 0.33/0.65 ( ( summable_real @ F )
% 0.33/0.65 => ( ! [D4: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D4 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D4 ) @ one_one_nat ) ) ) ) )
% 0.33/0.65 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum_pos_lt_pair
% 0.33/0.65 thf(fact_7583_round__mono,axiom,
% 0.33/0.65 ! [X: rat,Y: rat] :
% 0.33/0.65 ( ( ord_less_eq_rat @ X @ Y )
% 0.33/0.65 => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X ) @ ( archim7778729529865785530nd_rat @ Y ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % round_mono
% 0.33/0.65 thf(fact_7584_sum_Onat__diff__reindex,axiom,
% 0.33/0.65 ! [G: nat > nat,N2: nat] :
% 0.33/0.65 ( ( groups3542108847815614940at_nat
% 0.33/0.65 @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.65 = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum.nat_diff_reindex
% 0.33/0.65 thf(fact_7585_sum_Onat__diff__reindex,axiom,
% 0.33/0.65 ! [G: nat > real,N2: nat] :
% 0.33/0.65 ( ( groups6591440286371151544t_real
% 0.33/0.65 @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.65 = ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum.nat_diff_reindex
% 0.33/0.65 thf(fact_7586_sum__diff__distrib,axiom,
% 0.33/0.65 ! [Q: real > nat,P: real > nat,N2: real] :
% 0.33/0.65 ( ! [X4: real] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P @ X4 ) )
% 0.33/0.65 => ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P @ ( set_or5984915006950818249n_real @ N2 ) ) @ ( groups1935376822645274424al_nat @ Q @ ( set_or5984915006950818249n_real @ N2 ) ) )
% 0.33/0.65 = ( groups1935376822645274424al_nat
% 0.33/0.65 @ ^ [X2: real] : ( minus_minus_nat @ ( P @ X2 ) @ ( Q @ X2 ) )
% 0.33/0.65 @ ( set_or5984915006950818249n_real @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum_diff_distrib
% 0.33/0.65 thf(fact_7587_sum__diff__distrib,axiom,
% 0.33/0.65 ! [Q: nat > nat,P: nat > nat,N2: nat] :
% 0.33/0.65 ( ! [X4: nat] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P @ X4 ) )
% 0.33/0.65 => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N2 ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N2 ) ) )
% 0.33/0.65 = ( groups3542108847815614940at_nat
% 0.33/0.65 @ ^ [X2: nat] : ( minus_minus_nat @ ( P @ X2 ) @ ( Q @ X2 ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum_diff_distrib
% 0.33/0.65 thf(fact_7588_powser__inside,axiom,
% 0.33/0.65 ! [F: nat > real,X: real,Z: real] :
% 0.33/0.65 ( ( summable_real
% 0.33/0.65 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ X @ N ) ) )
% 0.33/0.65 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X ) )
% 0.33/0.65 => ( summable_real
% 0.33/0.65 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % powser_inside
% 0.33/0.65 thf(fact_7589_powser__inside,axiom,
% 0.33/0.65 ! [F: nat > complex,X: complex,Z: complex] :
% 0.33/0.65 ( ( summable_complex
% 0.33/0.65 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ X @ N ) ) )
% 0.33/0.65 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X ) )
% 0.33/0.65 => ( summable_complex
% 0.33/0.65 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % powser_inside
% 0.33/0.65 thf(fact_7590_sum_OlessThan__Suc__shift,axiom,
% 0.33/0.65 ! [G: nat > rat,N2: nat] :
% 0.33/0.65 ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.33/0.65 = ( plus_plus_rat @ ( G @ zero_zero_nat )
% 0.33/0.65 @ ( groups2906978787729119204at_rat
% 0.33/0.65 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum.lessThan_Suc_shift
% 0.33/0.65 thf(fact_7591_sum_OlessThan__Suc__shift,axiom,
% 0.33/0.65 ! [G: nat > int,N2: nat] :
% 0.33/0.65 ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.33/0.65 = ( plus_plus_int @ ( G @ zero_zero_nat )
% 0.33/0.65 @ ( groups3539618377306564664at_int
% 0.33/0.65 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum.lessThan_Suc_shift
% 0.33/0.65 thf(fact_7592_sum_OlessThan__Suc__shift,axiom,
% 0.33/0.65 ! [G: nat > nat,N2: nat] :
% 0.33/0.65 ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.33/0.65 = ( plus_plus_nat @ ( G @ zero_zero_nat )
% 0.33/0.65 @ ( groups3542108847815614940at_nat
% 0.33/0.65 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum.lessThan_Suc_shift
% 0.33/0.65 thf(fact_7593_sum_OlessThan__Suc__shift,axiom,
% 0.33/0.65 ! [G: nat > real,N2: nat] :
% 0.33/0.65 ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.33/0.65 = ( plus_plus_real @ ( G @ zero_zero_nat )
% 0.33/0.65 @ ( groups6591440286371151544t_real
% 0.33/0.65 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum.lessThan_Suc_shift
% 0.33/0.65 thf(fact_7594_sum__lessThan__telescope,axiom,
% 0.33/0.65 ! [F: nat > rat,M: nat] :
% 0.33/0.65 ( ( groups2906978787729119204at_rat
% 0.33/0.65 @ ^ [N: nat] : ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ M ) )
% 0.33/0.65 = ( minus_minus_rat @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum_lessThan_telescope
% 0.33/0.65 thf(fact_7595_sum__lessThan__telescope,axiom,
% 0.33/0.65 ! [F: nat > int,M: nat] :
% 0.33/0.65 ( ( groups3539618377306564664at_int
% 0.33/0.65 @ ^ [N: nat] : ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ M ) )
% 0.33/0.65 = ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum_lessThan_telescope
% 0.33/0.65 thf(fact_7596_sum__lessThan__telescope,axiom,
% 0.33/0.65 ! [F: nat > real,M: nat] :
% 0.33/0.65 ( ( groups6591440286371151544t_real
% 0.33/0.65 @ ^ [N: nat] : ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ M ) )
% 0.33/0.65 = ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum_lessThan_telescope
% 0.33/0.65 thf(fact_7597_sum__lessThan__telescope_H,axiom,
% 0.33/0.65 ! [F: nat > rat,M: nat] :
% 0.33/0.65 ( ( groups2906978787729119204at_rat
% 0.33/0.65 @ ^ [N: nat] : ( minus_minus_rat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ M ) )
% 0.33/0.65 = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum_lessThan_telescope'
% 0.33/0.65 thf(fact_7598_sum__lessThan__telescope_H,axiom,
% 0.33/0.65 ! [F: nat > int,M: nat] :
% 0.33/0.65 ( ( groups3539618377306564664at_int
% 0.33/0.65 @ ^ [N: nat] : ( minus_minus_int @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ M ) )
% 0.33/0.65 = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum_lessThan_telescope'
% 0.33/0.65 thf(fact_7599_sum__lessThan__telescope_H,axiom,
% 0.33/0.65 ! [F: nat > real,M: nat] :
% 0.33/0.65 ( ( groups6591440286371151544t_real
% 0.33/0.65 @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ M ) )
% 0.33/0.65 = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum_lessThan_telescope'
% 0.33/0.65 thf(fact_7600_sumr__diff__mult__const2,axiom,
% 0.33/0.65 ! [F: nat > rat,N2: nat,R: rat] :
% 0.33/0.65 ( ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ R ) )
% 0.33/0.65 = ( groups2906978787729119204at_rat
% 0.33/0.65 @ ^ [I3: nat] : ( minus_minus_rat @ ( F @ I3 ) @ R )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sumr_diff_mult_const2
% 0.33/0.65 thf(fact_7601_sumr__diff__mult__const2,axiom,
% 0.33/0.65 ! [F: nat > int,N2: nat,R: int] :
% 0.33/0.65 ( ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ R ) )
% 0.33/0.65 = ( groups3539618377306564664at_int
% 0.33/0.65 @ ^ [I3: nat] : ( minus_minus_int @ ( F @ I3 ) @ R )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sumr_diff_mult_const2
% 0.33/0.65 thf(fact_7602_sumr__diff__mult__const2,axiom,
% 0.33/0.65 ! [F: nat > complex,N2: nat,R: complex] :
% 0.33/0.65 ( ( minus_minus_complex @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ R ) )
% 0.33/0.65 = ( groups2073611262835488442omplex
% 0.33/0.65 @ ^ [I3: nat] : ( minus_minus_complex @ ( F @ I3 ) @ R )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sumr_diff_mult_const2
% 0.33/0.65 thf(fact_7603_sumr__diff__mult__const2,axiom,
% 0.33/0.65 ! [F: nat > real,N2: nat,R: real] :
% 0.33/0.65 ( ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ R ) )
% 0.33/0.65 = ( groups6591440286371151544t_real
% 0.33/0.65 @ ^ [I3: nat] : ( minus_minus_real @ ( F @ I3 ) @ R )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sumr_diff_mult_const2
% 0.33/0.65 thf(fact_7604_sum_OatLeast1__atMost__eq,axiom,
% 0.33/0.65 ! [G: nat > nat,N2: nat] :
% 0.33/0.65 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
% 0.33/0.65 = ( groups3542108847815614940at_nat
% 0.33/0.65 @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum.atLeast1_atMost_eq
% 0.33/0.65 thf(fact_7605_sum_OatLeast1__atMost__eq,axiom,
% 0.33/0.65 ! [G: nat > real,N2: nat] :
% 0.33/0.65 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
% 0.33/0.65 = ( groups6591440286371151544t_real
% 0.33/0.65 @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum.atLeast1_atMost_eq
% 0.33/0.65 thf(fact_7606_power__diff__1__eq,axiom,
% 0.33/0.65 ! [X: complex,N2: nat] :
% 0.33/0.65 ( ( minus_minus_complex @ ( power_power_complex @ X @ N2 ) @ one_one_complex )
% 0.33/0.65 = ( times_times_complex @ ( minus_minus_complex @ X @ one_one_complex ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % power_diff_1_eq
% 0.33/0.65 thf(fact_7607_power__diff__1__eq,axiom,
% 0.33/0.65 ! [X: rat,N2: nat] :
% 0.33/0.65 ( ( minus_minus_rat @ ( power_power_rat @ X @ N2 ) @ one_one_rat )
% 0.33/0.65 = ( times_times_rat @ ( minus_minus_rat @ X @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % power_diff_1_eq
% 0.33/0.65 thf(fact_7608_power__diff__1__eq,axiom,
% 0.33/0.65 ! [X: int,N2: nat] :
% 0.33/0.65 ( ( minus_minus_int @ ( power_power_int @ X @ N2 ) @ one_one_int )
% 0.33/0.65 = ( times_times_int @ ( minus_minus_int @ X @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % power_diff_1_eq
% 0.33/0.65 thf(fact_7609_power__diff__1__eq,axiom,
% 0.33/0.65 ! [X: real,N2: nat] :
% 0.33/0.65 ( ( minus_minus_real @ ( power_power_real @ X @ N2 ) @ one_one_real )
% 0.33/0.65 = ( times_times_real @ ( minus_minus_real @ X @ one_one_real ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % power_diff_1_eq
% 0.33/0.65 thf(fact_7610_one__diff__power__eq,axiom,
% 0.33/0.65 ! [X: complex,N2: nat] :
% 0.33/0.65 ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N2 ) )
% 0.33/0.65 = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % one_diff_power_eq
% 0.33/0.65 thf(fact_7611_one__diff__power__eq,axiom,
% 0.33/0.65 ! [X: rat,N2: nat] :
% 0.33/0.65 ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N2 ) )
% 0.33/0.65 = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % one_diff_power_eq
% 0.33/0.65 thf(fact_7612_one__diff__power__eq,axiom,
% 0.33/0.65 ! [X: int,N2: nat] :
% 0.33/0.65 ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N2 ) )
% 0.33/0.65 = ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % one_diff_power_eq
% 0.33/0.65 thf(fact_7613_one__diff__power__eq,axiom,
% 0.33/0.65 ! [X: real,N2: nat] :
% 0.33/0.65 ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N2 ) )
% 0.33/0.65 = ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % one_diff_power_eq
% 0.33/0.65 thf(fact_7614_geometric__sum,axiom,
% 0.33/0.65 ! [X: complex,N2: nat] :
% 0.33/0.65 ( ( X != one_one_complex )
% 0.33/0.65 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.65 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ N2 ) @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % geometric_sum
% 0.33/0.65 thf(fact_7615_geometric__sum,axiom,
% 0.33/0.65 ! [X: rat,N2: nat] :
% 0.33/0.65 ( ( X != one_one_rat )
% 0.33/0.65 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.65 = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ N2 ) @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % geometric_sum
% 0.33/0.65 thf(fact_7616_geometric__sum,axiom,
% 0.33/0.65 ! [X: real,N2: nat] :
% 0.33/0.65 ( ( X != one_one_real )
% 0.33/0.65 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.65 = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ N2 ) @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % geometric_sum
% 0.33/0.65 thf(fact_7617_round__diff__minimal,axiom,
% 0.33/0.65 ! [Z: real,M: int] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ Z ) ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ Z @ ( ring_1_of_int_real @ M ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % round_diff_minimal
% 0.33/0.65 thf(fact_7618_round__diff__minimal,axiom,
% 0.33/0.65 ! [Z: rat,M: int] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ Z @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ Z ) ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ Z @ ( ring_1_of_int_rat @ M ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % round_diff_minimal
% 0.33/0.65 thf(fact_7619_sum__gp__strict,axiom,
% 0.33/0.65 ! [X: rat,N2: nat] :
% 0.33/0.65 ( ( ( X = one_one_rat )
% 0.33/0.65 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.65 = ( semiri681578069525770553at_rat @ N2 ) ) )
% 0.33/0.65 & ( ( X != one_one_rat )
% 0.33/0.65 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.65 = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N2 ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum_gp_strict
% 0.33/0.65 thf(fact_7620_sum__gp__strict,axiom,
% 0.33/0.65 ! [X: complex,N2: nat] :
% 0.33/0.65 ( ( ( X = one_one_complex )
% 0.33/0.65 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.65 = ( semiri8010041392384452111omplex @ N2 ) ) )
% 0.33/0.65 & ( ( X != one_one_complex )
% 0.33/0.65 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.65 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N2 ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum_gp_strict
% 0.33/0.65 thf(fact_7621_sum__gp__strict,axiom,
% 0.33/0.65 ! [X: real,N2: nat] :
% 0.33/0.65 ( ( ( X = one_one_real )
% 0.33/0.65 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.65 = ( semiri5074537144036343181t_real @ N2 ) ) )
% 0.33/0.65 & ( ( X != one_one_real )
% 0.33/0.65 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.65 = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N2 ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum_gp_strict
% 0.33/0.65 thf(fact_7622_lemma__termdiff1,axiom,
% 0.33/0.65 ! [Z: complex,H2: complex,M: nat] :
% 0.33/0.65 ( ( groups2073611262835488442omplex
% 0.33/0.65 @ ^ [P5: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_complex @ Z @ P5 ) ) @ ( power_power_complex @ Z @ M ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ M ) )
% 0.33/0.65 = ( groups2073611262835488442omplex
% 0.33/0.65 @ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ Z @ P5 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % lemma_termdiff1
% 0.33/0.65 thf(fact_7623_lemma__termdiff1,axiom,
% 0.33/0.65 ! [Z: rat,H2: rat,M: nat] :
% 0.33/0.65 ( ( groups2906978787729119204at_rat
% 0.33/0.65 @ ^ [P5: nat] : ( minus_minus_rat @ ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_rat @ Z @ P5 ) ) @ ( power_power_rat @ Z @ M ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ M ) )
% 0.33/0.65 = ( groups2906978787729119204at_rat
% 0.33/0.65 @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ Z @ P5 ) @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % lemma_termdiff1
% 0.33/0.65 thf(fact_7624_lemma__termdiff1,axiom,
% 0.33/0.65 ! [Z: int,H2: int,M: nat] :
% 0.33/0.65 ( ( groups3539618377306564664at_int
% 0.33/0.65 @ ^ [P5: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_int @ Z @ P5 ) ) @ ( power_power_int @ Z @ M ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ M ) )
% 0.33/0.65 = ( groups3539618377306564664at_int
% 0.33/0.65 @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ Z @ P5 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_int @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % lemma_termdiff1
% 0.33/0.65 thf(fact_7625_lemma__termdiff1,axiom,
% 0.33/0.65 ! [Z: real,H2: real,M: nat] :
% 0.33/0.65 ( ( groups6591440286371151544t_real
% 0.33/0.65 @ ^ [P5: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_real @ Z @ P5 ) ) @ ( power_power_real @ Z @ M ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ M ) )
% 0.33/0.65 = ( groups6591440286371151544t_real
% 0.33/0.65 @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ Z @ P5 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_real @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % lemma_termdiff1
% 0.33/0.65 thf(fact_7626_diff__power__eq__sum,axiom,
% 0.33/0.65 ! [X: complex,N2: nat,Y: complex] :
% 0.33/0.65 ( ( minus_minus_complex @ ( power_power_complex @ X @ ( suc @ N2 ) ) @ ( power_power_complex @ Y @ ( suc @ N2 ) ) )
% 0.33/0.65 = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
% 0.33/0.65 @ ( groups2073611262835488442omplex
% 0.33/0.65 @ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ X @ P5 ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ N2 @ P5 ) ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_power_eq_sum
% 0.33/0.65 thf(fact_7627_diff__power__eq__sum,axiom,
% 0.33/0.65 ! [X: rat,N2: nat,Y: rat] :
% 0.33/0.65 ( ( minus_minus_rat @ ( power_power_rat @ X @ ( suc @ N2 ) ) @ ( power_power_rat @ Y @ ( suc @ N2 ) ) )
% 0.33/0.65 = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
% 0.33/0.65 @ ( groups2906978787729119204at_rat
% 0.33/0.65 @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ X @ P5 ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ N2 @ P5 ) ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_power_eq_sum
% 0.33/0.65 thf(fact_7628_diff__power__eq__sum,axiom,
% 0.33/0.65 ! [X: int,N2: nat,Y: int] :
% 0.33/0.65 ( ( minus_minus_int @ ( power_power_int @ X @ ( suc @ N2 ) ) @ ( power_power_int @ Y @ ( suc @ N2 ) ) )
% 0.33/0.65 = ( times_times_int @ ( minus_minus_int @ X @ Y )
% 0.33/0.65 @ ( groups3539618377306564664at_int
% 0.33/0.65 @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ X @ P5 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ N2 @ P5 ) ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_power_eq_sum
% 0.33/0.65 thf(fact_7629_diff__power__eq__sum,axiom,
% 0.33/0.65 ! [X: real,N2: nat,Y: real] :
% 0.33/0.65 ( ( minus_minus_real @ ( power_power_real @ X @ ( suc @ N2 ) ) @ ( power_power_real @ Y @ ( suc @ N2 ) ) )
% 0.33/0.65 = ( times_times_real @ ( minus_minus_real @ X @ Y )
% 0.33/0.65 @ ( groups6591440286371151544t_real
% 0.33/0.65 @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ X @ P5 ) @ ( power_power_real @ Y @ ( minus_minus_nat @ N2 @ P5 ) ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % diff_power_eq_sum
% 0.33/0.65 thf(fact_7630_power__diff__sumr2,axiom,
% 0.33/0.65 ! [X: complex,N2: nat,Y: complex] :
% 0.33/0.65 ( ( minus_minus_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ Y @ N2 ) )
% 0.33/0.65 = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
% 0.33/0.65 @ ( groups2073611262835488442omplex
% 0.33/0.65 @ ^ [I3: nat] : ( times_times_complex @ ( power_power_complex @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_complex @ X @ I3 ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % power_diff_sumr2
% 0.33/0.65 thf(fact_7631_power__diff__sumr2,axiom,
% 0.33/0.65 ! [X: rat,N2: nat,Y: rat] :
% 0.33/0.65 ( ( minus_minus_rat @ ( power_power_rat @ X @ N2 ) @ ( power_power_rat @ Y @ N2 ) )
% 0.33/0.65 = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
% 0.33/0.65 @ ( groups2906978787729119204at_rat
% 0.33/0.65 @ ^ [I3: nat] : ( times_times_rat @ ( power_power_rat @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_rat @ X @ I3 ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % power_diff_sumr2
% 0.33/0.65 thf(fact_7632_power__diff__sumr2,axiom,
% 0.33/0.65 ! [X: int,N2: nat,Y: int] :
% 0.33/0.65 ( ( minus_minus_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ N2 ) )
% 0.33/0.65 = ( times_times_int @ ( minus_minus_int @ X @ Y )
% 0.33/0.65 @ ( groups3539618377306564664at_int
% 0.33/0.65 @ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_int @ X @ I3 ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % power_diff_sumr2
% 0.33/0.65 thf(fact_7633_power__diff__sumr2,axiom,
% 0.33/0.65 ! [X: real,N2: nat,Y: real] :
% 0.33/0.65 ( ( minus_minus_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ N2 ) )
% 0.33/0.65 = ( times_times_real @ ( minus_minus_real @ X @ Y )
% 0.33/0.65 @ ( groups6591440286371151544t_real
% 0.33/0.65 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_real @ X @ I3 ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % power_diff_sumr2
% 0.33/0.65 thf(fact_7634_real__sum__nat__ivl__bounded2,axiom,
% 0.33/0.65 ! [N2: nat,F: nat > rat,K5: rat,K: nat] :
% 0.33/0.65 ( ! [P7: nat] :
% 0.33/0.65 ( ( ord_less_nat @ P7 @ N2 )
% 0.33/0.65 => ( ord_less_eq_rat @ ( F @ P7 ) @ K5 ) )
% 0.33/0.65 => ( ( ord_less_eq_rat @ zero_zero_rat @ K5 )
% 0.33/0.65 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ K5 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_sum_nat_ivl_bounded2
% 0.33/0.65 thf(fact_7635_real__sum__nat__ivl__bounded2,axiom,
% 0.33/0.65 ! [N2: nat,F: nat > int,K5: int,K: nat] :
% 0.33/0.65 ( ! [P7: nat] :
% 0.33/0.65 ( ( ord_less_nat @ P7 @ N2 )
% 0.33/0.65 => ( ord_less_eq_int @ ( F @ P7 ) @ K5 ) )
% 0.33/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ K5 )
% 0.33/0.65 => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ K5 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_sum_nat_ivl_bounded2
% 0.33/0.65 thf(fact_7636_real__sum__nat__ivl__bounded2,axiom,
% 0.33/0.65 ! [N2: nat,F: nat > nat,K5: nat,K: nat] :
% 0.33/0.65 ( ! [P7: nat] :
% 0.33/0.65 ( ( ord_less_nat @ P7 @ N2 )
% 0.33/0.65 => ( ord_less_eq_nat @ ( F @ P7 ) @ K5 ) )
% 0.33/0.65 => ( ( ord_less_eq_nat @ zero_zero_nat @ K5 )
% 0.33/0.65 => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ K5 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_sum_nat_ivl_bounded2
% 0.33/0.65 thf(fact_7637_real__sum__nat__ivl__bounded2,axiom,
% 0.33/0.65 ! [N2: nat,F: nat > real,K5: real,K: nat] :
% 0.33/0.65 ( ! [P7: nat] :
% 0.33/0.65 ( ( ord_less_nat @ P7 @ N2 )
% 0.33/0.65 => ( ord_less_eq_real @ ( F @ P7 ) @ K5 ) )
% 0.33/0.65 => ( ( ord_less_eq_real @ zero_zero_real @ K5 )
% 0.33/0.65 => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ K5 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % real_sum_nat_ivl_bounded2
% 0.33/0.65 thf(fact_7638_one__diff__power__eq_H,axiom,
% 0.33/0.65 ! [X: complex,N2: nat] :
% 0.33/0.65 ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N2 ) )
% 0.33/0.65 = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X )
% 0.33/0.65 @ ( groups2073611262835488442omplex
% 0.33/0.65 @ ^ [I3: nat] : ( power_power_complex @ X @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % one_diff_power_eq'
% 0.33/0.65 thf(fact_7639_one__diff__power__eq_H,axiom,
% 0.33/0.65 ! [X: rat,N2: nat] :
% 0.33/0.65 ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N2 ) )
% 0.33/0.65 = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X )
% 0.33/0.65 @ ( groups2906978787729119204at_rat
% 0.33/0.65 @ ^ [I3: nat] : ( power_power_rat @ X @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % one_diff_power_eq'
% 0.33/0.65 thf(fact_7640_one__diff__power__eq_H,axiom,
% 0.33/0.65 ! [X: int,N2: nat] :
% 0.33/0.65 ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N2 ) )
% 0.33/0.65 = ( times_times_int @ ( minus_minus_int @ one_one_int @ X )
% 0.33/0.65 @ ( groups3539618377306564664at_int
% 0.33/0.65 @ ^ [I3: nat] : ( power_power_int @ X @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % one_diff_power_eq'
% 0.33/0.65 thf(fact_7641_one__diff__power__eq_H,axiom,
% 0.33/0.65 ! [X: real,N2: nat] :
% 0.33/0.65 ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N2 ) )
% 0.33/0.65 = ( times_times_real @ ( minus_minus_real @ one_one_real @ X )
% 0.33/0.65 @ ( groups6591440286371151544t_real
% 0.33/0.65 @ ^ [I3: nat] : ( power_power_real @ X @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % one_diff_power_eq'
% 0.33/0.65 thf(fact_7642_sum__split__even__odd,axiom,
% 0.33/0.65 ! [F: nat > real,G: nat > real,N2: nat] :
% 0.33/0.65 ( ( groups6591440286371151544t_real
% 0.33/0.65 @ ^ [I3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( F @ I3 ) @ ( G @ I3 ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.33/0.65 = ( plus_plus_real
% 0.33/0.65 @ ( groups6591440286371151544t_real
% 0.33/0.65 @ ^ [I3: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.65 @ ( groups6591440286371151544t_real
% 0.33/0.65 @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ one_one_nat ) )
% 0.33/0.65 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % sum_split_even_odd
% 0.33/0.65 thf(fact_7643_of__int__round__le,axiom,
% 0.33/0.65 ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_round_le
% 0.33/0.65 thf(fact_7644_of__int__round__le,axiom,
% 0.33/0.65 ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_round_le
% 0.33/0.65 thf(fact_7645_of__int__round__ge,axiom,
% 0.33/0.65 ! [X: real] : ( ord_less_eq_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_round_ge
% 0.33/0.65 thf(fact_7646_of__int__round__ge,axiom,
% 0.33/0.65 ! [X: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_round_ge
% 0.33/0.65 thf(fact_7647_of__int__round__gt,axiom,
% 0.33/0.65 ! [X: real] : ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_round_gt
% 0.33/0.65 thf(fact_7648_of__int__round__gt,axiom,
% 0.33/0.65 ! [X: rat] : ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_round_gt
% 0.33/0.65 thf(fact_7649_of__int__round__abs__le,axiom,
% 0.33/0.65 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ X ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_round_abs_le
% 0.33/0.65 thf(fact_7650_of__int__round__abs__le,axiom,
% 0.33/0.65 ! [X: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ X ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % of_int_round_abs_le
% 0.33/0.65 thf(fact_7651_round__unique_H,axiom,
% 0.33/0.65 ! [X: real,N2: int] :
% 0.33/0.65 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ N2 ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.65 => ( ( archim8280529875227126926d_real @ X )
% 0.33/0.65 = N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % round_unique'
% 0.33/0.65 thf(fact_7652_round__unique_H,axiom,
% 0.33/0.65 ! [X: rat,N2: int] :
% 0.33/0.65 ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ N2 ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.33/0.65 => ( ( archim7778729529865785530nd_rat @ X )
% 0.33/0.65 = N2 ) ) ).
% 0.33/0.65
% 0.33/0.65 % round_unique'
% 0.33/0.65 thf(fact_7653_summable__geometric__iff,axiom,
% 0.33/0.65 ! [C: real] :
% 0.33/0.65 ( ( summable_real @ ( power_power_real @ C ) )
% 0.33/0.65 = ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real ) ) ).
% 0.33/0.65
% 0.33/0.65 % summable_geometric_iff
% 0.33/0.65 thf(fact_7654_summable__geometric__iff,axiom,
% 0.33/0.65 ! [C: complex] :
% 0.33/0.65 ( ( summable_complex @ ( power_power_complex @ C ) )
% 0.33/0.65 = ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real ) ) ).
% 0.33/0.65
% 0.33/0.65 % summable_geometric_iff
% 0.33/0.65 thf(fact_7655_summable__If__finite,axiom,
% 0.33/0.65 ! [P: nat > $o,F: nat > complex] :
% 0.33/0.65 ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.33/0.65 => ( summable_complex
% 0.33/0.65 @ ^ [R5: nat] : ( if_complex @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_complex ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % summable_If_finite
% 0.33/0.65 thf(fact_7656_summable__If__finite,axiom,
% 0.33/0.65 ! [P: nat > $o,F: nat > real] :
% 0.33/0.65 ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.33/0.65 => ( summable_real
% 0.33/0.65 @ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % summable_If_finite
% 0.33/0.65 thf(fact_7657_summable__If__finite,axiom,
% 0.33/0.65 ! [P: nat > $o,F: nat > nat] :
% 0.33/0.65 ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.33/0.65 => ( summable_nat
% 0.33/0.65 @ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % summable_If_finite
% 0.33/0.65 thf(fact_7658_summable__If__finite,axiom,
% 0.33/0.65 ! [P: nat > $o,F: nat > int] :
% 0.33/0.65 ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.33/0.65 => ( summable_int
% 0.33/0.65 @ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % summable_If_finite
% 0.33/0.65 thf(fact_7659_summable__If__finite__set,axiom,
% 0.33/0.65 ! [A2: set_nat,F: nat > complex] :
% 0.33/0.65 ( ( finite_finite_nat @ A2 )
% 0.33/0.65 => ( summable_complex
% 0.33/0.65 @ ^ [R5: nat] : ( if_complex @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_complex ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % summable_If_finite_set
% 0.33/0.65 thf(fact_7660_summable__If__finite__set,axiom,
% 0.33/0.65 ! [A2: set_nat,F: nat > real] :
% 0.33/0.65 ( ( finite_finite_nat @ A2 )
% 0.33/0.65 => ( summable_real
% 0.33/0.65 @ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % summable_If_finite_set
% 0.33/0.65 thf(fact_7661_summable__If__finite__set,axiom,
% 0.33/0.65 ! [A2: set_nat,F: nat > nat] :
% 0.33/0.65 ( ( finite_finite_nat @ A2 )
% 0.33/0.65 => ( summable_nat
% 0.33/0.65 @ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % summable_If_finite_set
% 0.33/0.65 thf(fact_7662_summable__If__finite__set,axiom,
% 0.33/0.65 ! [A2: set_nat,F: nat > int] :
% 0.33/0.65 ( ( finite_finite_nat @ A2 )
% 0.33/0.65 => ( summable_int
% 0.33/0.65 @ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % summable_If_finite_set
% 0.33/0.65 thf(fact_7663_summable__divide__iff,axiom,
% 0.33/0.65 ! [F: nat > complex,C: complex] :
% 0.33/0.65 ( ( summable_complex
% 0.33/0.65 @ ^ [N: nat] : ( divide1717551699836669952omplex @ ( F @ N ) @ C ) )
% 0.33/0.65 = ( ( C = zero_zero_complex )
% 0.33/0.65 | ( summable_complex @ F ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % summable_divide_iff
% 0.33/0.65 thf(fact_7664_summable__divide__iff,axiom,
% 0.33/0.65 ! [F: nat > real,C: real] :
% 0.33/0.65 ( ( summable_real
% 0.33/0.65 @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ C ) )
% 0.33/0.65 = ( ( C = zero_zero_real )
% 0.33/0.65 | ( summable_real @ F ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % summable_divide_iff
% 0.33/0.65 thf(fact_7665_suminf__geometric,axiom,
% 0.33/0.65 ! [C: real] :
% 0.33/0.65 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 0.33/0.65 => ( ( suminf_real @ ( power_power_real @ C ) )
% 0.33/0.65 = ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % suminf_geometric
% 0.33/0.65 thf(fact_7666_suminf__geometric,axiom,
% 0.33/0.65 ! [C: complex] :
% 0.33/0.65 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 0.33/0.65 => ( ( suminf_complex @ ( power_power_complex @ C ) )
% 0.33/0.65 = ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 0.33/0.65
% 0.33/0.65 % suminf_geometric
% 0.33/0.65 thf(fact_7667_summable__zero,axiom,
% 0.33/0.65 ( summable_complex
% 0.33/0.65 @ ^ [N: nat] : zero_zero_complex ) ).
% 0.33/0.65
% 0.33/0.65 % summable_zero
% 0.33/0.65 thf(fact_7668_summable__zero,axiom,
% 0.33/0.65 ( summable_real
% 0.33/0.65 @ ^ [N: nat] : zero_zero_real ) ).
% 0.33/0.65
% 0.33/0.65 % summable_zero
% 0.33/0.65 thf(fact_7669_summable__zero,axiom,
% 0.33/0.65 ( summable_nat
% 0.33/0.65 @ ^ [N: nat] : zero_zero_nat ) ).
% 0.33/0.66
% 0.33/0.66 % summable_zero
% 0.33/0.66 thf(fact_7670_summable__zero,axiom,
% 0.33/0.66 ( summable_int
% 0.33/0.66 @ ^ [N: nat] : zero_zero_int ) ).
% 0.33/0.66
% 0.33/0.66 % summable_zero
% 0.33/0.66 thf(fact_7671_summable__single,axiom,
% 0.33/0.66 ! [I2: nat,F: nat > complex] :
% 0.33/0.66 ( summable_complex
% 0.33/0.66 @ ^ [R5: nat] : ( if_complex @ ( R5 = I2 ) @ ( F @ R5 ) @ zero_zero_complex ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_single
% 0.33/0.66 thf(fact_7672_summable__single,axiom,
% 0.33/0.66 ! [I2: nat,F: nat > real] :
% 0.33/0.66 ( summable_real
% 0.33/0.66 @ ^ [R5: nat] : ( if_real @ ( R5 = I2 ) @ ( F @ R5 ) @ zero_zero_real ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_single
% 0.33/0.66 thf(fact_7673_summable__single,axiom,
% 0.33/0.66 ! [I2: nat,F: nat > nat] :
% 0.33/0.66 ( summable_nat
% 0.33/0.66 @ ^ [R5: nat] : ( if_nat @ ( R5 = I2 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_single
% 0.33/0.66 thf(fact_7674_summable__single,axiom,
% 0.33/0.66 ! [I2: nat,F: nat > int] :
% 0.33/0.66 ( summable_int
% 0.33/0.66 @ ^ [R5: nat] : ( if_int @ ( R5 = I2 ) @ ( F @ R5 ) @ zero_zero_int ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_single
% 0.33/0.66 thf(fact_7675_suminf__zero,axiom,
% 0.33/0.66 ( ( suminf_complex
% 0.33/0.66 @ ^ [N: nat] : zero_zero_complex )
% 0.33/0.66 = zero_zero_complex ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_zero
% 0.33/0.66 thf(fact_7676_suminf__zero,axiom,
% 0.33/0.66 ( ( suminf_real
% 0.33/0.66 @ ^ [N: nat] : zero_zero_real )
% 0.33/0.66 = zero_zero_real ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_zero
% 0.33/0.66 thf(fact_7677_suminf__zero,axiom,
% 0.33/0.66 ( ( suminf_nat
% 0.33/0.66 @ ^ [N: nat] : zero_zero_nat )
% 0.33/0.66 = zero_zero_nat ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_zero
% 0.33/0.66 thf(fact_7678_suminf__zero,axiom,
% 0.33/0.66 ( ( suminf_int
% 0.33/0.66 @ ^ [N: nat] : zero_zero_int )
% 0.33/0.66 = zero_zero_int ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_zero
% 0.33/0.66 thf(fact_7679_summable__iff__shift,axiom,
% 0.33/0.66 ! [F: nat > real,K: nat] :
% 0.33/0.66 ( ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
% 0.33/0.66 = ( summable_real @ F ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_iff_shift
% 0.33/0.66 thf(fact_7680_summable__iff__shift,axiom,
% 0.33/0.66 ! [F: nat > complex,K: nat] :
% 0.33/0.66 ( ( summable_complex
% 0.33/0.66 @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
% 0.33/0.66 = ( summable_complex @ F ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_iff_shift
% 0.33/0.66 thf(fact_7681_summable__cmult__iff,axiom,
% 0.33/0.66 ! [C: complex,F: nat > complex] :
% 0.33/0.66 ( ( summable_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) ) )
% 0.33/0.66 = ( ( C = zero_zero_complex )
% 0.33/0.66 | ( summable_complex @ F ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_cmult_iff
% 0.33/0.66 thf(fact_7682_summable__cmult__iff,axiom,
% 0.33/0.66 ! [C: real,F: nat > real] :
% 0.33/0.66 ( ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) ) )
% 0.33/0.66 = ( ( C = zero_zero_real )
% 0.33/0.66 | ( summable_real @ F ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_cmult_iff
% 0.33/0.66 thf(fact_7683_summable__const__iff,axiom,
% 0.33/0.66 ! [C: complex] :
% 0.33/0.66 ( ( summable_complex
% 0.33/0.66 @ ^ [Uu3: nat] : C )
% 0.33/0.66 = ( C = zero_zero_complex ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_const_iff
% 0.33/0.66 thf(fact_7684_summable__const__iff,axiom,
% 0.33/0.66 ! [C: real] :
% 0.33/0.66 ( ( summable_real
% 0.33/0.66 @ ^ [Uu3: nat] : C )
% 0.33/0.66 = ( C = zero_zero_real ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_const_iff
% 0.33/0.66 thf(fact_7685_summable__mult2,axiom,
% 0.33/0.66 ! [F: nat > complex,C: complex] :
% 0.33/0.66 ( ( summable_complex @ F )
% 0.33/0.66 => ( summable_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ C ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_mult2
% 0.33/0.66 thf(fact_7686_summable__mult2,axiom,
% 0.33/0.66 ! [F: nat > real,C: real] :
% 0.33/0.66 ( ( summable_real @ F )
% 0.33/0.66 => ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ C ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_mult2
% 0.33/0.66 thf(fact_7687_summable__mult,axiom,
% 0.33/0.66 ! [F: nat > complex,C: complex] :
% 0.33/0.66 ( ( summable_complex @ F )
% 0.33/0.66 => ( summable_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_mult
% 0.33/0.66 thf(fact_7688_summable__mult,axiom,
% 0.33/0.66 ! [F: nat > real,C: real] :
% 0.33/0.66 ( ( summable_real @ F )
% 0.33/0.66 => ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_mult
% 0.33/0.66 thf(fact_7689_summable__add,axiom,
% 0.33/0.66 ! [F: nat > complex,G: nat > complex] :
% 0.33/0.66 ( ( summable_complex @ F )
% 0.33/0.66 => ( ( summable_complex @ G )
% 0.33/0.66 => ( summable_complex
% 0.33/0.66 @ ^ [N: nat] : ( plus_plus_complex @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_add
% 0.33/0.66 thf(fact_7690_summable__add,axiom,
% 0.33/0.66 ! [F: nat > real,G: nat > real] :
% 0.33/0.66 ( ( summable_real @ F )
% 0.33/0.66 => ( ( summable_real @ G )
% 0.33/0.66 => ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( plus_plus_real @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_add
% 0.33/0.66 thf(fact_7691_summable__add,axiom,
% 0.33/0.66 ! [F: nat > nat,G: nat > nat] :
% 0.33/0.66 ( ( summable_nat @ F )
% 0.33/0.66 => ( ( summable_nat @ G )
% 0.33/0.66 => ( summable_nat
% 0.33/0.66 @ ^ [N: nat] : ( plus_plus_nat @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_add
% 0.33/0.66 thf(fact_7692_summable__add,axiom,
% 0.33/0.66 ! [F: nat > int,G: nat > int] :
% 0.33/0.66 ( ( summable_int @ F )
% 0.33/0.66 => ( ( summable_int @ G )
% 0.33/0.66 => ( summable_int
% 0.33/0.66 @ ^ [N: nat] : ( plus_plus_int @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_add
% 0.33/0.66 thf(fact_7693_summable__diff,axiom,
% 0.33/0.66 ! [F: nat > complex,G: nat > complex] :
% 0.33/0.66 ( ( summable_complex @ F )
% 0.33/0.66 => ( ( summable_complex @ G )
% 0.33/0.66 => ( summable_complex
% 0.33/0.66 @ ^ [N: nat] : ( minus_minus_complex @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_diff
% 0.33/0.66 thf(fact_7694_summable__diff,axiom,
% 0.33/0.66 ! [F: nat > real,G: nat > real] :
% 0.33/0.66 ( ( summable_real @ F )
% 0.33/0.66 => ( ( summable_real @ G )
% 0.33/0.66 => ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_diff
% 0.33/0.66 thf(fact_7695_summable__divide,axiom,
% 0.33/0.66 ! [F: nat > complex,C: complex] :
% 0.33/0.66 ( ( summable_complex @ F )
% 0.33/0.66 => ( summable_complex
% 0.33/0.66 @ ^ [N: nat] : ( divide1717551699836669952omplex @ ( F @ N ) @ C ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_divide
% 0.33/0.66 thf(fact_7696_summable__divide,axiom,
% 0.33/0.66 ! [F: nat > real,C: real] :
% 0.33/0.66 ( ( summable_real @ F )
% 0.33/0.66 => ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ C ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_divide
% 0.33/0.66 thf(fact_7697_summable__Suc__iff,axiom,
% 0.33/0.66 ! [F: nat > real] :
% 0.33/0.66 ( ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
% 0.33/0.66 = ( summable_real @ F ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_Suc_iff
% 0.33/0.66 thf(fact_7698_summable__Suc__iff,axiom,
% 0.33/0.66 ! [F: nat > complex] :
% 0.33/0.66 ( ( summable_complex
% 0.33/0.66 @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
% 0.33/0.66 = ( summable_complex @ F ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_Suc_iff
% 0.33/0.66 thf(fact_7699_summable__minus__iff,axiom,
% 0.33/0.66 ! [F: nat > real] :
% 0.33/0.66 ( ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( uminus_uminus_real @ ( F @ N ) ) )
% 0.33/0.66 = ( summable_real @ F ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_minus_iff
% 0.33/0.66 thf(fact_7700_summable__minus__iff,axiom,
% 0.33/0.66 ! [F: nat > complex] :
% 0.33/0.66 ( ( summable_complex
% 0.33/0.66 @ ^ [N: nat] : ( uminus1482373934393186551omplex @ ( F @ N ) ) )
% 0.33/0.66 = ( summable_complex @ F ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_minus_iff
% 0.33/0.66 thf(fact_7701_summable__minus,axiom,
% 0.33/0.66 ! [F: nat > real] :
% 0.33/0.66 ( ( summable_real @ F )
% 0.33/0.66 => ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( uminus_uminus_real @ ( F @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_minus
% 0.33/0.66 thf(fact_7702_summable__minus,axiom,
% 0.33/0.66 ! [F: nat > complex] :
% 0.33/0.66 ( ( summable_complex @ F )
% 0.33/0.66 => ( summable_complex
% 0.33/0.66 @ ^ [N: nat] : ( uminus1482373934393186551omplex @ ( F @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_minus
% 0.33/0.66 thf(fact_7703_summable__ignore__initial__segment,axiom,
% 0.33/0.66 ! [F: nat > real,K: nat] :
% 0.33/0.66 ( ( summable_real @ F )
% 0.33/0.66 => ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_ignore_initial_segment
% 0.33/0.66 thf(fact_7704_summable__ignore__initial__segment,axiom,
% 0.33/0.66 ! [F: nat > complex,K: nat] :
% 0.33/0.66 ( ( summable_complex @ F )
% 0.33/0.66 => ( summable_complex
% 0.33/0.66 @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_ignore_initial_segment
% 0.33/0.66 thf(fact_7705_summable__sum,axiom,
% 0.33/0.66 ! [I6: set_real,F: real > nat > real] :
% 0.33/0.66 ( ! [I4: real] :
% 0.33/0.66 ( ( member_real @ I4 @ I6 )
% 0.33/0.66 => ( summable_real @ ( F @ I4 ) ) )
% 0.33/0.66 => ( summable_real
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups8097168146408367636l_real
% 0.33/0.66 @ ^ [I3: real] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_sum
% 0.33/0.66 thf(fact_7706_summable__sum,axiom,
% 0.33/0.66 ! [I6: set_int,F: int > nat > real] :
% 0.33/0.66 ( ! [I4: int] :
% 0.33/0.66 ( ( member_int @ I4 @ I6 )
% 0.33/0.66 => ( summable_real @ ( F @ I4 ) ) )
% 0.33/0.66 => ( summable_real
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups8778361861064173332t_real
% 0.33/0.66 @ ^ [I3: int] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_sum
% 0.33/0.66 thf(fact_7707_summable__sum,axiom,
% 0.33/0.66 ! [I6: set_complex,F: complex > nat > real] :
% 0.33/0.66 ( ! [I4: complex] :
% 0.33/0.66 ( ( member_complex @ I4 @ I6 )
% 0.33/0.66 => ( summable_real @ ( F @ I4 ) ) )
% 0.33/0.66 => ( summable_real
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups5808333547571424918x_real
% 0.33/0.66 @ ^ [I3: complex] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_sum
% 0.33/0.66 thf(fact_7708_summable__sum,axiom,
% 0.33/0.66 ! [I6: set_nat,F: nat > nat > complex] :
% 0.33/0.66 ( ! [I4: nat] :
% 0.33/0.66 ( ( member_nat @ I4 @ I6 )
% 0.33/0.66 => ( summable_complex @ ( F @ I4 ) ) )
% 0.33/0.66 => ( summable_complex
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups2073611262835488442omplex
% 0.33/0.66 @ ^ [I3: nat] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_sum
% 0.33/0.66 thf(fact_7709_summable__sum,axiom,
% 0.33/0.66 ! [I6: set_real,F: real > nat > complex] :
% 0.33/0.66 ( ! [I4: real] :
% 0.33/0.66 ( ( member_real @ I4 @ I6 )
% 0.33/0.66 => ( summable_complex @ ( F @ I4 ) ) )
% 0.33/0.66 => ( summable_complex
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups5754745047067104278omplex
% 0.33/0.66 @ ^ [I3: real] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_sum
% 0.33/0.66 thf(fact_7710_summable__sum,axiom,
% 0.33/0.66 ! [I6: set_int,F: int > nat > complex] :
% 0.33/0.66 ( ! [I4: int] :
% 0.33/0.66 ( ( member_int @ I4 @ I6 )
% 0.33/0.66 => ( summable_complex @ ( F @ I4 ) ) )
% 0.33/0.66 => ( summable_complex
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups3049146728041665814omplex
% 0.33/0.66 @ ^ [I3: int] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_sum
% 0.33/0.66 thf(fact_7711_summable__sum,axiom,
% 0.33/0.66 ! [I6: set_int,F: int > nat > int] :
% 0.33/0.66 ( ! [I4: int] :
% 0.33/0.66 ( ( member_int @ I4 @ I6 )
% 0.33/0.66 => ( summable_int @ ( F @ I4 ) ) )
% 0.33/0.66 => ( summable_int
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups4538972089207619220nt_int
% 0.33/0.66 @ ^ [I3: int] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_sum
% 0.33/0.66 thf(fact_7712_summable__sum,axiom,
% 0.33/0.66 ! [I6: set_complex,F: complex > nat > complex] :
% 0.33/0.66 ( ! [I4: complex] :
% 0.33/0.66 ( ( member_complex @ I4 @ I6 )
% 0.33/0.66 => ( summable_complex @ ( F @ I4 ) ) )
% 0.33/0.66 => ( summable_complex
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups7754918857620584856omplex
% 0.33/0.66 @ ^ [I3: complex] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_sum
% 0.33/0.66 thf(fact_7713_summable__sum,axiom,
% 0.33/0.66 ! [I6: set_nat,F: nat > nat > nat] :
% 0.33/0.66 ( ! [I4: nat] :
% 0.33/0.66 ( ( member_nat @ I4 @ I6 )
% 0.33/0.66 => ( summable_nat @ ( F @ I4 ) ) )
% 0.33/0.66 => ( summable_nat
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups3542108847815614940at_nat
% 0.33/0.66 @ ^ [I3: nat] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_sum
% 0.33/0.66 thf(fact_7714_summable__sum,axiom,
% 0.33/0.66 ! [I6: set_nat,F: nat > nat > real] :
% 0.33/0.66 ( ! [I4: nat] :
% 0.33/0.66 ( ( member_nat @ I4 @ I6 )
% 0.33/0.66 => ( summable_real @ ( F @ I4 ) ) )
% 0.33/0.66 => ( summable_real
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [I3: nat] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_sum
% 0.33/0.66 thf(fact_7715_summable__rabs__cancel,axiom,
% 0.33/0.66 ! [F: nat > real] :
% 0.33/0.66 ( ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( abs_abs_real @ ( F @ N ) ) )
% 0.33/0.66 => ( summable_real @ F ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_rabs_cancel
% 0.33/0.66 thf(fact_7716_suminf__le,axiom,
% 0.33/0.66 ! [F: nat > real,G: nat > real] :
% 0.33/0.66 ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( G @ N4 ) )
% 0.33/0.66 => ( ( summable_real @ F )
% 0.33/0.66 => ( ( summable_real @ G )
% 0.33/0.66 => ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_le
% 0.33/0.66 thf(fact_7717_suminf__le,axiom,
% 0.33/0.66 ! [F: nat > nat,G: nat > nat] :
% 0.33/0.66 ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( G @ N4 ) )
% 0.33/0.66 => ( ( summable_nat @ F )
% 0.33/0.66 => ( ( summable_nat @ G )
% 0.33/0.66 => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_le
% 0.33/0.66 thf(fact_7718_suminf__le,axiom,
% 0.33/0.66 ! [F: nat > int,G: nat > int] :
% 0.33/0.66 ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ N4 ) @ ( G @ N4 ) )
% 0.33/0.66 => ( ( summable_int @ F )
% 0.33/0.66 => ( ( summable_int @ G )
% 0.33/0.66 => ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_le
% 0.33/0.66 thf(fact_7719_summable__mult__D,axiom,
% 0.33/0.66 ! [C: complex,F: nat > complex] :
% 0.33/0.66 ( ( summable_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) ) )
% 0.33/0.66 => ( ( C != zero_zero_complex )
% 0.33/0.66 => ( summable_complex @ F ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_mult_D
% 0.33/0.66 thf(fact_7720_summable__mult__D,axiom,
% 0.33/0.66 ! [C: real,F: nat > real] :
% 0.33/0.66 ( ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) ) )
% 0.33/0.66 => ( ( C != zero_zero_real )
% 0.33/0.66 => ( summable_real @ F ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_mult_D
% 0.33/0.66 thf(fact_7721_summable__zero__power,axiom,
% 0.33/0.66 summable_real @ ( power_power_real @ zero_zero_real ) ).
% 0.33/0.66
% 0.33/0.66 % summable_zero_power
% 0.33/0.66 thf(fact_7722_summable__zero__power,axiom,
% 0.33/0.66 summable_int @ ( power_power_int @ zero_zero_int ) ).
% 0.33/0.66
% 0.33/0.66 % summable_zero_power
% 0.33/0.66 thf(fact_7723_summable__zero__power,axiom,
% 0.33/0.66 summable_complex @ ( power_power_complex @ zero_zero_complex ) ).
% 0.33/0.66
% 0.33/0.66 % summable_zero_power
% 0.33/0.66 thf(fact_7724_suminf__mult,axiom,
% 0.33/0.66 ! [F: nat > complex,C: complex] :
% 0.33/0.66 ( ( summable_complex @ F )
% 0.33/0.66 => ( ( suminf_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) ) )
% 0.33/0.66 = ( times_times_complex @ C @ ( suminf_complex @ F ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_mult
% 0.33/0.66 thf(fact_7725_suminf__mult,axiom,
% 0.33/0.66 ! [F: nat > real,C: real] :
% 0.33/0.66 ( ( summable_real @ F )
% 0.33/0.66 => ( ( suminf_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) ) )
% 0.33/0.66 = ( times_times_real @ C @ ( suminf_real @ F ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_mult
% 0.33/0.66 thf(fact_7726_suminf__mult2,axiom,
% 0.33/0.66 ! [F: nat > complex,C: complex] :
% 0.33/0.66 ( ( summable_complex @ F )
% 0.33/0.66 => ( ( times_times_complex @ ( suminf_complex @ F ) @ C )
% 0.33/0.66 = ( suminf_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ C ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_mult2
% 0.33/0.66 thf(fact_7727_suminf__mult2,axiom,
% 0.33/0.66 ! [F: nat > real,C: real] :
% 0.33/0.66 ( ( summable_real @ F )
% 0.33/0.66 => ( ( times_times_real @ ( suminf_real @ F ) @ C )
% 0.33/0.66 = ( suminf_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ C ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_mult2
% 0.33/0.66 thf(fact_7728_suminf__add,axiom,
% 0.33/0.66 ! [F: nat > complex,G: nat > complex] :
% 0.33/0.66 ( ( summable_complex @ F )
% 0.33/0.66 => ( ( summable_complex @ G )
% 0.33/0.66 => ( ( plus_plus_complex @ ( suminf_complex @ F ) @ ( suminf_complex @ G ) )
% 0.33/0.66 = ( suminf_complex
% 0.33/0.66 @ ^ [N: nat] : ( plus_plus_complex @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_add
% 0.33/0.66 thf(fact_7729_suminf__add,axiom,
% 0.33/0.66 ! [F: nat > real,G: nat > real] :
% 0.33/0.66 ( ( summable_real @ F )
% 0.33/0.66 => ( ( summable_real @ G )
% 0.33/0.66 => ( ( plus_plus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
% 0.33/0.66 = ( suminf_real
% 0.33/0.66 @ ^ [N: nat] : ( plus_plus_real @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_add
% 0.33/0.66 thf(fact_7730_suminf__add,axiom,
% 0.33/0.66 ! [F: nat > nat,G: nat > nat] :
% 0.33/0.66 ( ( summable_nat @ F )
% 0.33/0.66 => ( ( summable_nat @ G )
% 0.33/0.66 => ( ( plus_plus_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) )
% 0.33/0.66 = ( suminf_nat
% 0.33/0.66 @ ^ [N: nat] : ( plus_plus_nat @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_add
% 0.33/0.66 thf(fact_7731_suminf__add,axiom,
% 0.33/0.66 ! [F: nat > int,G: nat > int] :
% 0.33/0.66 ( ( summable_int @ F )
% 0.33/0.66 => ( ( summable_int @ G )
% 0.33/0.66 => ( ( plus_plus_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) )
% 0.33/0.66 = ( suminf_int
% 0.33/0.66 @ ^ [N: nat] : ( plus_plus_int @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_add
% 0.33/0.66 thf(fact_7732_suminf__diff,axiom,
% 0.33/0.66 ! [F: nat > complex,G: nat > complex] :
% 0.33/0.66 ( ( summable_complex @ F )
% 0.33/0.66 => ( ( summable_complex @ G )
% 0.33/0.66 => ( ( minus_minus_complex @ ( suminf_complex @ F ) @ ( suminf_complex @ G ) )
% 0.33/0.66 = ( suminf_complex
% 0.33/0.66 @ ^ [N: nat] : ( minus_minus_complex @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_diff
% 0.33/0.66 thf(fact_7733_suminf__diff,axiom,
% 0.33/0.66 ! [F: nat > real,G: nat > real] :
% 0.33/0.66 ( ( summable_real @ F )
% 0.33/0.66 => ( ( summable_real @ G )
% 0.33/0.66 => ( ( minus_minus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
% 0.33/0.66 = ( suminf_real
% 0.33/0.66 @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_diff
% 0.33/0.66 thf(fact_7734_suminf__divide,axiom,
% 0.33/0.66 ! [F: nat > complex,C: complex] :
% 0.33/0.66 ( ( summable_complex @ F )
% 0.33/0.66 => ( ( suminf_complex
% 0.33/0.66 @ ^ [N: nat] : ( divide1717551699836669952omplex @ ( F @ N ) @ C ) )
% 0.33/0.66 = ( divide1717551699836669952omplex @ ( suminf_complex @ F ) @ C ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_divide
% 0.33/0.66 thf(fact_7735_suminf__divide,axiom,
% 0.33/0.66 ! [F: nat > real,C: real] :
% 0.33/0.66 ( ( summable_real @ F )
% 0.33/0.66 => ( ( suminf_real
% 0.33/0.66 @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ C ) )
% 0.33/0.66 = ( divide_divide_real @ ( suminf_real @ F ) @ C ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_divide
% 0.33/0.66 thf(fact_7736_suminf__minus,axiom,
% 0.33/0.66 ! [F: nat > real] :
% 0.33/0.66 ( ( summable_real @ F )
% 0.33/0.66 => ( ( suminf_real
% 0.33/0.66 @ ^ [N: nat] : ( uminus_uminus_real @ ( F @ N ) ) )
% 0.33/0.66 = ( uminus_uminus_real @ ( suminf_real @ F ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_minus
% 0.33/0.66 thf(fact_7737_suminf__minus,axiom,
% 0.33/0.66 ! [F: nat > complex] :
% 0.33/0.66 ( ( summable_complex @ F )
% 0.33/0.66 => ( ( suminf_complex
% 0.33/0.66 @ ^ [N: nat] : ( uminus1482373934393186551omplex @ ( F @ N ) ) )
% 0.33/0.66 = ( uminus1482373934393186551omplex @ ( suminf_complex @ F ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_minus
% 0.33/0.66 thf(fact_7738_suminf__sum,axiom,
% 0.33/0.66 ! [I6: set_real,F: real > nat > real] :
% 0.33/0.66 ( ! [I4: real] :
% 0.33/0.66 ( ( member_real @ I4 @ I6 )
% 0.33/0.66 => ( summable_real @ ( F @ I4 ) ) )
% 0.33/0.66 => ( ( suminf_real
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups8097168146408367636l_real
% 0.33/0.66 @ ^ [I3: real] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 ) )
% 0.33/0.66 = ( groups8097168146408367636l_real
% 0.33/0.66 @ ^ [I3: real] : ( suminf_real @ ( F @ I3 ) )
% 0.33/0.66 @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_sum
% 0.33/0.66 thf(fact_7739_suminf__sum,axiom,
% 0.33/0.66 ! [I6: set_int,F: int > nat > real] :
% 0.33/0.66 ( ! [I4: int] :
% 0.33/0.66 ( ( member_int @ I4 @ I6 )
% 0.33/0.66 => ( summable_real @ ( F @ I4 ) ) )
% 0.33/0.66 => ( ( suminf_real
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups8778361861064173332t_real
% 0.33/0.66 @ ^ [I3: int] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 ) )
% 0.33/0.66 = ( groups8778361861064173332t_real
% 0.33/0.66 @ ^ [I3: int] : ( suminf_real @ ( F @ I3 ) )
% 0.33/0.66 @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_sum
% 0.33/0.66 thf(fact_7740_suminf__sum,axiom,
% 0.33/0.66 ! [I6: set_complex,F: complex > nat > real] :
% 0.33/0.66 ( ! [I4: complex] :
% 0.33/0.66 ( ( member_complex @ I4 @ I6 )
% 0.33/0.66 => ( summable_real @ ( F @ I4 ) ) )
% 0.33/0.66 => ( ( suminf_real
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups5808333547571424918x_real
% 0.33/0.66 @ ^ [I3: complex] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 ) )
% 0.33/0.66 = ( groups5808333547571424918x_real
% 0.33/0.66 @ ^ [I3: complex] : ( suminf_real @ ( F @ I3 ) )
% 0.33/0.66 @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_sum
% 0.33/0.66 thf(fact_7741_suminf__sum,axiom,
% 0.33/0.66 ! [I6: set_nat,F: nat > nat > complex] :
% 0.33/0.66 ( ! [I4: nat] :
% 0.33/0.66 ( ( member_nat @ I4 @ I6 )
% 0.33/0.66 => ( summable_complex @ ( F @ I4 ) ) )
% 0.33/0.66 => ( ( suminf_complex
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups2073611262835488442omplex
% 0.33/0.66 @ ^ [I3: nat] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 ) )
% 0.33/0.66 = ( groups2073611262835488442omplex
% 0.33/0.66 @ ^ [I3: nat] : ( suminf_complex @ ( F @ I3 ) )
% 0.33/0.66 @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_sum
% 0.33/0.66 thf(fact_7742_suminf__sum,axiom,
% 0.33/0.66 ! [I6: set_real,F: real > nat > complex] :
% 0.33/0.66 ( ! [I4: real] :
% 0.33/0.66 ( ( member_real @ I4 @ I6 )
% 0.33/0.66 => ( summable_complex @ ( F @ I4 ) ) )
% 0.33/0.66 => ( ( suminf_complex
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups5754745047067104278omplex
% 0.33/0.66 @ ^ [I3: real] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 ) )
% 0.33/0.66 = ( groups5754745047067104278omplex
% 0.33/0.66 @ ^ [I3: real] : ( suminf_complex @ ( F @ I3 ) )
% 0.33/0.66 @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_sum
% 0.33/0.66 thf(fact_7743_suminf__sum,axiom,
% 0.33/0.66 ! [I6: set_int,F: int > nat > complex] :
% 0.33/0.66 ( ! [I4: int] :
% 0.33/0.66 ( ( member_int @ I4 @ I6 )
% 0.33/0.66 => ( summable_complex @ ( F @ I4 ) ) )
% 0.33/0.66 => ( ( suminf_complex
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups3049146728041665814omplex
% 0.33/0.66 @ ^ [I3: int] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 ) )
% 0.33/0.66 = ( groups3049146728041665814omplex
% 0.33/0.66 @ ^ [I3: int] : ( suminf_complex @ ( F @ I3 ) )
% 0.33/0.66 @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_sum
% 0.33/0.66 thf(fact_7744_suminf__sum,axiom,
% 0.33/0.66 ! [I6: set_int,F: int > nat > int] :
% 0.33/0.66 ( ! [I4: int] :
% 0.33/0.66 ( ( member_int @ I4 @ I6 )
% 0.33/0.66 => ( summable_int @ ( F @ I4 ) ) )
% 0.33/0.66 => ( ( suminf_int
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups4538972089207619220nt_int
% 0.33/0.66 @ ^ [I3: int] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 ) )
% 0.33/0.66 = ( groups4538972089207619220nt_int
% 0.33/0.66 @ ^ [I3: int] : ( suminf_int @ ( F @ I3 ) )
% 0.33/0.66 @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_sum
% 0.33/0.66 thf(fact_7745_suminf__sum,axiom,
% 0.33/0.66 ! [I6: set_complex,F: complex > nat > complex] :
% 0.33/0.66 ( ! [I4: complex] :
% 0.33/0.66 ( ( member_complex @ I4 @ I6 )
% 0.33/0.66 => ( summable_complex @ ( F @ I4 ) ) )
% 0.33/0.66 => ( ( suminf_complex
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups7754918857620584856omplex
% 0.33/0.66 @ ^ [I3: complex] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 ) )
% 0.33/0.66 = ( groups7754918857620584856omplex
% 0.33/0.66 @ ^ [I3: complex] : ( suminf_complex @ ( F @ I3 ) )
% 0.33/0.66 @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_sum
% 0.33/0.66 thf(fact_7746_suminf__sum,axiom,
% 0.33/0.66 ! [I6: set_nat,F: nat > nat > nat] :
% 0.33/0.66 ( ! [I4: nat] :
% 0.33/0.66 ( ( member_nat @ I4 @ I6 )
% 0.33/0.66 => ( summable_nat @ ( F @ I4 ) ) )
% 0.33/0.66 => ( ( suminf_nat
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups3542108847815614940at_nat
% 0.33/0.66 @ ^ [I3: nat] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 ) )
% 0.33/0.66 = ( groups3542108847815614940at_nat
% 0.33/0.66 @ ^ [I3: nat] : ( suminf_nat @ ( F @ I3 ) )
% 0.33/0.66 @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_sum
% 0.33/0.66 thf(fact_7747_suminf__sum,axiom,
% 0.33/0.66 ! [I6: set_nat,F: nat > nat > real] :
% 0.33/0.66 ( ! [I4: nat] :
% 0.33/0.66 ( ( member_nat @ I4 @ I6 )
% 0.33/0.66 => ( summable_real @ ( F @ I4 ) ) )
% 0.33/0.66 => ( ( suminf_real
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [I3: nat] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 ) )
% 0.33/0.66 = ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [I3: nat] : ( suminf_real @ ( F @ I3 ) )
% 0.33/0.66 @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_sum
% 0.33/0.66 thf(fact_7748_summable__norm__cancel,axiom,
% 0.33/0.66 ! [F: nat > real] :
% 0.33/0.66 ( ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( F @ N ) ) )
% 0.33/0.66 => ( summable_real @ F ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_norm_cancel
% 0.33/0.66 thf(fact_7749_summable__norm__cancel,axiom,
% 0.33/0.66 ! [F: nat > complex] :
% 0.33/0.66 ( ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( F @ N ) ) )
% 0.33/0.66 => ( summable_complex @ F ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_norm_cancel
% 0.33/0.66 thf(fact_7750_suminf__nonneg,axiom,
% 0.33/0.66 ! [F: nat > real] :
% 0.33/0.66 ( ( summable_real @ F )
% 0.33/0.66 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
% 0.33/0.66 => ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_nonneg
% 0.33/0.66 thf(fact_7751_suminf__nonneg,axiom,
% 0.33/0.66 ! [F: nat > nat] :
% 0.33/0.66 ( ( summable_nat @ F )
% 0.33/0.66 => ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
% 0.33/0.66 => ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_nonneg
% 0.33/0.66 thf(fact_7752_suminf__nonneg,axiom,
% 0.33/0.66 ! [F: nat > int] :
% 0.33/0.66 ( ( summable_int @ F )
% 0.33/0.66 => ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
% 0.33/0.66 => ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_nonneg
% 0.33/0.66 thf(fact_7753_suminf__eq__zero__iff,axiom,
% 0.33/0.66 ! [F: nat > real] :
% 0.33/0.66 ( ( summable_real @ F )
% 0.33/0.66 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
% 0.33/0.66 => ( ( ( suminf_real @ F )
% 0.33/0.66 = zero_zero_real )
% 0.33/0.66 = ( ! [N: nat] :
% 0.33/0.66 ( ( F @ N )
% 0.33/0.66 = zero_zero_real ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_eq_zero_iff
% 0.33/0.66 thf(fact_7754_suminf__eq__zero__iff,axiom,
% 0.33/0.66 ! [F: nat > nat] :
% 0.33/0.66 ( ( summable_nat @ F )
% 0.33/0.66 => ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
% 0.33/0.66 => ( ( ( suminf_nat @ F )
% 0.33/0.66 = zero_zero_nat )
% 0.33/0.66 = ( ! [N: nat] :
% 0.33/0.66 ( ( F @ N )
% 0.33/0.66 = zero_zero_nat ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_eq_zero_iff
% 0.33/0.66 thf(fact_7755_suminf__eq__zero__iff,axiom,
% 0.33/0.66 ! [F: nat > int] :
% 0.33/0.66 ( ( summable_int @ F )
% 0.33/0.66 => ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
% 0.33/0.66 => ( ( ( suminf_int @ F )
% 0.33/0.66 = zero_zero_int )
% 0.33/0.66 = ( ! [N: nat] :
% 0.33/0.66 ( ( F @ N )
% 0.33/0.66 = zero_zero_int ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_eq_zero_iff
% 0.33/0.66 thf(fact_7756_suminf__pos,axiom,
% 0.33/0.66 ! [F: nat > real] :
% 0.33/0.66 ( ( summable_real @ F )
% 0.33/0.66 => ( ! [N4: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N4 ) )
% 0.33/0.66 => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_pos
% 0.33/0.66 thf(fact_7757_suminf__pos,axiom,
% 0.33/0.66 ! [F: nat > nat] :
% 0.33/0.66 ( ( summable_nat @ F )
% 0.33/0.66 => ( ! [N4: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N4 ) )
% 0.33/0.66 => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_pos
% 0.33/0.66 thf(fact_7758_suminf__pos,axiom,
% 0.33/0.66 ! [F: nat > int] :
% 0.33/0.66 ( ( summable_int @ F )
% 0.33/0.66 => ( ! [N4: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N4 ) )
% 0.33/0.66 => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_pos
% 0.33/0.66 thf(fact_7759_summable__0__powser,axiom,
% 0.33/0.66 ! [F: nat > complex] :
% 0.33/0.66 ( summable_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_0_powser
% 0.33/0.66 thf(fact_7760_summable__0__powser,axiom,
% 0.33/0.66 ! [F: nat > real] :
% 0.33/0.66 ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ zero_zero_real @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_0_powser
% 0.33/0.66 thf(fact_7761_summable__zero__power_H,axiom,
% 0.33/0.66 ! [F: nat > complex] :
% 0.33/0.66 ( summable_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_zero_power'
% 0.33/0.66 thf(fact_7762_summable__zero__power_H,axiom,
% 0.33/0.66 ! [F: nat > real] :
% 0.33/0.66 ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ zero_zero_real @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_zero_power'
% 0.33/0.66 thf(fact_7763_summable__zero__power_H,axiom,
% 0.33/0.66 ! [F: nat > int] :
% 0.33/0.66 ( summable_int
% 0.33/0.66 @ ^ [N: nat] : ( times_times_int @ ( F @ N ) @ ( power_power_int @ zero_zero_int @ N ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_zero_power'
% 0.33/0.66 thf(fact_7764_powser__split__head_I3_J,axiom,
% 0.33/0.66 ! [F: nat > complex,Z: complex] :
% 0.33/0.66 ( ( summable_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
% 0.33/0.66 => ( summable_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % powser_split_head(3)
% 0.33/0.66 thf(fact_7765_powser__split__head_I3_J,axiom,
% 0.33/0.66 ! [F: nat > real,Z: real] :
% 0.33/0.66 ( ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
% 0.33/0.66 => ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % powser_split_head(3)
% 0.33/0.66 thf(fact_7766_summable__powser__split__head,axiom,
% 0.33/0.66 ! [F: nat > complex,Z: complex] :
% 0.33/0.66 ( ( summable_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) ) )
% 0.33/0.66 = ( summable_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_powser_split_head
% 0.33/0.66 thf(fact_7767_summable__powser__split__head,axiom,
% 0.33/0.66 ! [F: nat > real,Z: real] :
% 0.33/0.66 ( ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) ) )
% 0.33/0.66 = ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_powser_split_head
% 0.33/0.66 thf(fact_7768_summable__powser__ignore__initial__segment,axiom,
% 0.33/0.66 ! [F: nat > complex,M: nat,Z: complex] :
% 0.33/0.66 ( ( summable_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( F @ ( plus_plus_nat @ N @ M ) ) @ ( power_power_complex @ Z @ N ) ) )
% 0.33/0.66 = ( summable_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_powser_ignore_initial_segment
% 0.33/0.66 thf(fact_7769_summable__powser__ignore__initial__segment,axiom,
% 0.33/0.66 ! [F: nat > real,M: nat,Z: real] :
% 0.33/0.66 ( ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( F @ ( plus_plus_nat @ N @ M ) ) @ ( power_power_real @ Z @ N ) ) )
% 0.33/0.66 = ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_powser_ignore_initial_segment
% 0.33/0.66 thf(fact_7770_summable__norm__comparison__test,axiom,
% 0.33/0.66 ! [F: nat > complex,G: nat > real] :
% 0.33/0.66 ( ? [N7: nat] :
% 0.33/0.66 ! [N4: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ N7 @ N4 )
% 0.33/0.66 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
% 0.33/0.66 => ( ( summable_real @ G )
% 0.33/0.66 => ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( F @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_norm_comparison_test
% 0.33/0.66 thf(fact_7771_suminf__finite,axiom,
% 0.33/0.66 ! [N3: set_nat,F: nat > complex] :
% 0.33/0.66 ( ( finite_finite_nat @ N3 )
% 0.33/0.66 => ( ! [N4: nat] :
% 0.33/0.66 ( ~ ( member_nat @ N4 @ N3 )
% 0.33/0.66 => ( ( F @ N4 )
% 0.33/0.66 = zero_zero_complex ) )
% 0.33/0.66 => ( ( suminf_complex @ F )
% 0.33/0.66 = ( groups2073611262835488442omplex @ F @ N3 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_finite
% 0.33/0.66 thf(fact_7772_suminf__finite,axiom,
% 0.33/0.66 ! [N3: set_nat,F: nat > int] :
% 0.33/0.66 ( ( finite_finite_nat @ N3 )
% 0.33/0.66 => ( ! [N4: nat] :
% 0.33/0.66 ( ~ ( member_nat @ N4 @ N3 )
% 0.33/0.66 => ( ( F @ N4 )
% 0.33/0.66 = zero_zero_int ) )
% 0.33/0.66 => ( ( suminf_int @ F )
% 0.33/0.66 = ( groups3539618377306564664at_int @ F @ N3 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_finite
% 0.33/0.66 thf(fact_7773_suminf__finite,axiom,
% 0.33/0.66 ! [N3: set_nat,F: nat > nat] :
% 0.33/0.66 ( ( finite_finite_nat @ N3 )
% 0.33/0.66 => ( ! [N4: nat] :
% 0.33/0.66 ( ~ ( member_nat @ N4 @ N3 )
% 0.33/0.66 => ( ( F @ N4 )
% 0.33/0.66 = zero_zero_nat ) )
% 0.33/0.66 => ( ( suminf_nat @ F )
% 0.33/0.66 = ( groups3542108847815614940at_nat @ F @ N3 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_finite
% 0.33/0.66 thf(fact_7774_suminf__finite,axiom,
% 0.33/0.66 ! [N3: set_nat,F: nat > real] :
% 0.33/0.66 ( ( finite_finite_nat @ N3 )
% 0.33/0.66 => ( ! [N4: nat] :
% 0.33/0.66 ( ~ ( member_nat @ N4 @ N3 )
% 0.33/0.66 => ( ( F @ N4 )
% 0.33/0.66 = zero_zero_real ) )
% 0.33/0.66 => ( ( suminf_real @ F )
% 0.33/0.66 = ( groups6591440286371151544t_real @ F @ N3 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_finite
% 0.33/0.66 thf(fact_7775_summable__rabs__comparison__test,axiom,
% 0.33/0.66 ! [F: nat > real,G: nat > real] :
% 0.33/0.66 ( ? [N7: nat] :
% 0.33/0.66 ! [N4: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ N7 @ N4 )
% 0.33/0.66 => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
% 0.33/0.66 => ( ( summable_real @ G )
% 0.33/0.66 => ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( abs_abs_real @ ( F @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_rabs_comparison_test
% 0.33/0.66 thf(fact_7776_summable__rabs,axiom,
% 0.33/0.66 ! [F: nat > real] :
% 0.33/0.66 ( ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( abs_abs_real @ ( F @ N ) ) )
% 0.33/0.66 => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
% 0.33/0.66 @ ( suminf_real
% 0.33/0.66 @ ^ [N: nat] : ( abs_abs_real @ ( F @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_rabs
% 0.33/0.66 thf(fact_7777_suminf__pos__iff,axiom,
% 0.33/0.66 ! [F: nat > real] :
% 0.33/0.66 ( ( summable_real @ F )
% 0.33/0.66 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
% 0.33/0.66 => ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
% 0.33/0.66 = ( ? [I3: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_pos_iff
% 0.33/0.66 thf(fact_7778_suminf__pos__iff,axiom,
% 0.33/0.66 ! [F: nat > nat] :
% 0.33/0.66 ( ( summable_nat @ F )
% 0.33/0.66 => ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
% 0.33/0.66 => ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
% 0.33/0.66 = ( ? [I3: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_pos_iff
% 0.33/0.66 thf(fact_7779_suminf__pos__iff,axiom,
% 0.33/0.66 ! [F: nat > int] :
% 0.33/0.66 ( ( summable_int @ F )
% 0.33/0.66 => ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
% 0.33/0.66 => ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
% 0.33/0.66 = ( ? [I3: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I3 ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_pos_iff
% 0.33/0.66 thf(fact_7780_suminf__pos2,axiom,
% 0.33/0.66 ! [F: nat > real,I2: nat] :
% 0.33/0.66 ( ( summable_real @ F )
% 0.33/0.66 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
% 0.33/0.66 => ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
% 0.33/0.66 => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_pos2
% 0.33/0.66 thf(fact_7781_suminf__pos2,axiom,
% 0.33/0.66 ! [F: nat > nat,I2: nat] :
% 0.33/0.66 ( ( summable_nat @ F )
% 0.33/0.66 => ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
% 0.33/0.66 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
% 0.33/0.66 => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_pos2
% 0.33/0.66 thf(fact_7782_suminf__pos2,axiom,
% 0.33/0.66 ! [F: nat > int,I2: nat] :
% 0.33/0.66 ( ( summable_int @ F )
% 0.33/0.66 => ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
% 0.33/0.66 => ( ( ord_less_int @ zero_zero_int @ ( F @ I2 ) )
% 0.33/0.66 => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_pos2
% 0.33/0.66 thf(fact_7783_suminf__le__const,axiom,
% 0.33/0.66 ! [F: nat > int,X: int] :
% 0.33/0.66 ( ( summable_int @ F )
% 0.33/0.66 => ( ! [N4: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X )
% 0.33/0.66 => ( ord_less_eq_int @ ( suminf_int @ F ) @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_le_const
% 0.33/0.66 thf(fact_7784_suminf__le__const,axiom,
% 0.33/0.66 ! [F: nat > nat,X: nat] :
% 0.33/0.66 ( ( summable_nat @ F )
% 0.33/0.66 => ( ! [N4: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X )
% 0.33/0.66 => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_le_const
% 0.33/0.66 thf(fact_7785_suminf__le__const,axiom,
% 0.33/0.66 ! [F: nat > real,X: real] :
% 0.33/0.66 ( ( summable_real @ F )
% 0.33/0.66 => ( ! [N4: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X )
% 0.33/0.66 => ( ord_less_eq_real @ ( suminf_real @ F ) @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_le_const
% 0.33/0.66 thf(fact_7786_summable__comparison__test_H,axiom,
% 0.33/0.66 ! [G: nat > real,N3: nat,F: nat > real] :
% 0.33/0.66 ( ( summable_real @ G )
% 0.33/0.66 => ( ! [N4: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ N3 @ N4 )
% 0.33/0.66 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
% 0.33/0.66 => ( summable_real @ F ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_comparison_test'
% 0.33/0.66 thf(fact_7787_summable__comparison__test_H,axiom,
% 0.33/0.66 ! [G: nat > real,N3: nat,F: nat > complex] :
% 0.33/0.66 ( ( summable_real @ G )
% 0.33/0.66 => ( ! [N4: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ N3 @ N4 )
% 0.33/0.66 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
% 0.33/0.66 => ( summable_complex @ F ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_comparison_test'
% 0.33/0.66 thf(fact_7788_summable__comparison__test,axiom,
% 0.33/0.66 ! [F: nat > real,G: nat > real] :
% 0.33/0.66 ( ? [N7: nat] :
% 0.33/0.66 ! [N4: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ N7 @ N4 )
% 0.33/0.66 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
% 0.33/0.66 => ( ( summable_real @ G )
% 0.33/0.66 => ( summable_real @ F ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_comparison_test
% 0.33/0.66 thf(fact_7789_summable__comparison__test,axiom,
% 0.33/0.66 ! [F: nat > complex,G: nat > real] :
% 0.33/0.66 ( ? [N7: nat] :
% 0.33/0.66 ! [N4: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ N7 @ N4 )
% 0.33/0.66 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
% 0.33/0.66 => ( ( summable_real @ G )
% 0.33/0.66 => ( summable_complex @ F ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_comparison_test
% 0.33/0.66 thf(fact_7790_summableI__nonneg__bounded,axiom,
% 0.33/0.66 ! [F: nat > int,X: int] :
% 0.33/0.66 ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
% 0.33/0.66 => ( ! [N4: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X )
% 0.33/0.66 => ( summable_int @ F ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summableI_nonneg_bounded
% 0.33/0.66 thf(fact_7791_summableI__nonneg__bounded,axiom,
% 0.33/0.66 ! [F: nat > nat,X: nat] :
% 0.33/0.66 ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
% 0.33/0.66 => ( ! [N4: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X )
% 0.33/0.66 => ( summable_nat @ F ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summableI_nonneg_bounded
% 0.33/0.66 thf(fact_7792_summableI__nonneg__bounded,axiom,
% 0.33/0.66 ! [F: nat > real,X: real] :
% 0.33/0.66 ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
% 0.33/0.66 => ( ! [N4: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X )
% 0.33/0.66 => ( summable_real @ F ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summableI_nonneg_bounded
% 0.33/0.66 thf(fact_7793_complete__algebra__summable__geometric,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ one_one_real )
% 0.33/0.66 => ( summable_real @ ( power_power_real @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % complete_algebra_summable_geometric
% 0.33/0.66 thf(fact_7794_complete__algebra__summable__geometric,axiom,
% 0.33/0.66 ! [X: complex] :
% 0.33/0.66 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ one_one_real )
% 0.33/0.66 => ( summable_complex @ ( power_power_complex @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % complete_algebra_summable_geometric
% 0.33/0.66 thf(fact_7795_summable__geometric,axiom,
% 0.33/0.66 ! [C: real] :
% 0.33/0.66 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 0.33/0.66 => ( summable_real @ ( power_power_real @ C ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_geometric
% 0.33/0.66 thf(fact_7796_summable__geometric,axiom,
% 0.33/0.66 ! [C: complex] :
% 0.33/0.66 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 0.33/0.66 => ( summable_complex @ ( power_power_complex @ C ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_geometric
% 0.33/0.66 thf(fact_7797_suminf__split__head,axiom,
% 0.33/0.66 ! [F: nat > complex] :
% 0.33/0.66 ( ( summable_complex @ F )
% 0.33/0.66 => ( ( suminf_complex
% 0.33/0.66 @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
% 0.33/0.66 = ( minus_minus_complex @ ( suminf_complex @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_split_head
% 0.33/0.66 thf(fact_7798_suminf__split__head,axiom,
% 0.33/0.66 ! [F: nat > real] :
% 0.33/0.66 ( ( summable_real @ F )
% 0.33/0.66 => ( ( suminf_real
% 0.33/0.66 @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
% 0.33/0.66 = ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_split_head
% 0.33/0.66 thf(fact_7799_summable__norm,axiom,
% 0.33/0.66 ! [F: nat > real] :
% 0.33/0.66 ( ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( F @ N ) ) )
% 0.33/0.66 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( suminf_real @ F ) )
% 0.33/0.66 @ ( suminf_real
% 0.33/0.66 @ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( F @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_norm
% 0.33/0.66 thf(fact_7800_summable__norm,axiom,
% 0.33/0.66 ! [F: nat > complex] :
% 0.33/0.66 ( ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( F @ N ) ) )
% 0.33/0.66 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( suminf_complex @ F ) )
% 0.33/0.66 @ ( suminf_real
% 0.33/0.66 @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( F @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_norm
% 0.33/0.66 thf(fact_7801_sum__le__suminf,axiom,
% 0.33/0.66 ! [F: nat > int,I6: set_nat] :
% 0.33/0.66 ( ( summable_int @ F )
% 0.33/0.66 => ( ( finite_finite_nat @ I6 )
% 0.33/0.66 => ( ! [N4: nat] :
% 0.33/0.66 ( ( member_nat @ N4 @ ( uminus5710092332889474511et_nat @ I6 ) )
% 0.33/0.66 => ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) ) )
% 0.33/0.66 => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I6 ) @ ( suminf_int @ F ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sum_le_suminf
% 0.33/0.66 thf(fact_7802_sum__le__suminf,axiom,
% 0.33/0.66 ! [F: nat > nat,I6: set_nat] :
% 0.33/0.66 ( ( summable_nat @ F )
% 0.33/0.66 => ( ( finite_finite_nat @ I6 )
% 0.33/0.66 => ( ! [N4: nat] :
% 0.33/0.66 ( ( member_nat @ N4 @ ( uminus5710092332889474511et_nat @ I6 ) )
% 0.33/0.66 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) ) )
% 0.33/0.66 => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I6 ) @ ( suminf_nat @ F ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sum_le_suminf
% 0.33/0.66 thf(fact_7803_sum__le__suminf,axiom,
% 0.33/0.66 ! [F: nat > real,I6: set_nat] :
% 0.33/0.66 ( ( summable_real @ F )
% 0.33/0.66 => ( ( finite_finite_nat @ I6 )
% 0.33/0.66 => ( ! [N4: nat] :
% 0.33/0.66 ( ( member_nat @ N4 @ ( uminus5710092332889474511et_nat @ I6 ) )
% 0.33/0.66 => ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) ) )
% 0.33/0.66 => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I6 ) @ ( suminf_real @ F ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sum_le_suminf
% 0.33/0.66 thf(fact_7804_suminf__split__initial__segment,axiom,
% 0.33/0.66 ! [F: nat > complex,K: nat] :
% 0.33/0.66 ( ( summable_complex @ F )
% 0.33/0.66 => ( ( suminf_complex @ F )
% 0.33/0.66 = ( plus_plus_complex
% 0.33/0.66 @ ( suminf_complex
% 0.33/0.66 @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
% 0.33/0.66 @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_split_initial_segment
% 0.33/0.66 thf(fact_7805_suminf__split__initial__segment,axiom,
% 0.33/0.66 ! [F: nat > real,K: nat] :
% 0.33/0.66 ( ( summable_real @ F )
% 0.33/0.66 => ( ( suminf_real @ F )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( suminf_real
% 0.33/0.66 @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
% 0.33/0.66 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_split_initial_segment
% 0.33/0.66 thf(fact_7806_suminf__minus__initial__segment,axiom,
% 0.33/0.66 ! [F: nat > complex,K: nat] :
% 0.33/0.66 ( ( summable_complex @ F )
% 0.33/0.66 => ( ( suminf_complex
% 0.33/0.66 @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
% 0.33/0.66 = ( minus_minus_complex @ ( suminf_complex @ F ) @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_minus_initial_segment
% 0.33/0.66 thf(fact_7807_suminf__minus__initial__segment,axiom,
% 0.33/0.66 ! [F: nat > real,K: nat] :
% 0.33/0.66 ( ( summable_real @ F )
% 0.33/0.66 => ( ( suminf_real
% 0.33/0.66 @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
% 0.33/0.66 = ( minus_minus_real @ ( suminf_real @ F ) @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_minus_initial_segment
% 0.33/0.66 thf(fact_7808_sum__less__suminf,axiom,
% 0.33/0.66 ! [F: nat > int,N2: nat] :
% 0.33/0.66 ( ( summable_int @ F )
% 0.33/0.66 => ( ! [M2: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ N2 @ M2 )
% 0.33/0.66 => ( ord_less_int @ zero_zero_int @ ( F @ M2 ) ) )
% 0.33/0.66 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_int @ F ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sum_less_suminf
% 0.33/0.66 thf(fact_7809_sum__less__suminf,axiom,
% 0.33/0.66 ! [F: nat > nat,N2: nat] :
% 0.33/0.66 ( ( summable_nat @ F )
% 0.33/0.66 => ( ! [M2: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ N2 @ M2 )
% 0.33/0.66 => ( ord_less_nat @ zero_zero_nat @ ( F @ M2 ) ) )
% 0.33/0.66 => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_nat @ F ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sum_less_suminf
% 0.33/0.66 thf(fact_7810_sum__less__suminf,axiom,
% 0.33/0.66 ! [F: nat > real,N2: nat] :
% 0.33/0.66 ( ( summable_real @ F )
% 0.33/0.66 => ( ! [M2: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ N2 @ M2 )
% 0.33/0.66 => ( ord_less_real @ zero_zero_real @ ( F @ M2 ) ) )
% 0.33/0.66 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_real @ F ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sum_less_suminf
% 0.33/0.66 thf(fact_7811_powser__split__head_I1_J,axiom,
% 0.33/0.66 ! [F: nat > complex,Z: complex] :
% 0.33/0.66 ( ( summable_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
% 0.33/0.66 => ( ( suminf_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
% 0.33/0.66 = ( plus_plus_complex @ ( F @ zero_zero_nat )
% 0.33/0.66 @ ( times_times_complex
% 0.33/0.66 @ ( suminf_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) ) )
% 0.33/0.66 @ Z ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % powser_split_head(1)
% 0.33/0.66 thf(fact_7812_powser__split__head_I1_J,axiom,
% 0.33/0.66 ! [F: nat > real,Z: real] :
% 0.33/0.66 ( ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
% 0.33/0.66 => ( ( suminf_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
% 0.33/0.66 = ( plus_plus_real @ ( F @ zero_zero_nat )
% 0.33/0.66 @ ( times_times_real
% 0.33/0.66 @ ( suminf_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) ) )
% 0.33/0.66 @ Z ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % powser_split_head(1)
% 0.33/0.66 thf(fact_7813_powser__split__head_I2_J,axiom,
% 0.33/0.66 ! [F: nat > complex,Z: complex] :
% 0.33/0.66 ( ( summable_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
% 0.33/0.66 => ( ( times_times_complex
% 0.33/0.66 @ ( suminf_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) ) )
% 0.33/0.66 @ Z )
% 0.33/0.66 = ( minus_minus_complex
% 0.33/0.66 @ ( suminf_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
% 0.33/0.66 @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % powser_split_head(2)
% 0.33/0.66 thf(fact_7814_powser__split__head_I2_J,axiom,
% 0.33/0.66 ! [F: nat > real,Z: real] :
% 0.33/0.66 ( ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
% 0.33/0.66 => ( ( times_times_real
% 0.33/0.66 @ ( suminf_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) ) )
% 0.33/0.66 @ Z )
% 0.33/0.66 = ( minus_minus_real
% 0.33/0.66 @ ( suminf_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
% 0.33/0.66 @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % powser_split_head(2)
% 0.33/0.66 thf(fact_7815_suminf__exist__split,axiom,
% 0.33/0.66 ! [R: real,F: nat > real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ R )
% 0.33/0.66 => ( ( summable_real @ F )
% 0.33/0.66 => ? [N8: nat] :
% 0.33/0.66 ! [N9: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ N8 @ N9 )
% 0.33/0.66 => ( ord_less_real
% 0.33/0.66 @ ( real_V7735802525324610683m_real
% 0.33/0.66 @ ( suminf_real
% 0.33/0.66 @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N9 ) ) ) )
% 0.33/0.66 @ R ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_exist_split
% 0.33/0.66 thf(fact_7816_suminf__exist__split,axiom,
% 0.33/0.66 ! [R: real,F: nat > complex] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ R )
% 0.33/0.66 => ( ( summable_complex @ F )
% 0.33/0.66 => ? [N8: nat] :
% 0.33/0.66 ! [N9: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ N8 @ N9 )
% 0.33/0.66 => ( ord_less_real
% 0.33/0.66 @ ( real_V1022390504157884413omplex
% 0.33/0.66 @ ( suminf_complex
% 0.33/0.66 @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N9 ) ) ) )
% 0.33/0.66 @ R ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % suminf_exist_split
% 0.33/0.66 thf(fact_7817_summable__partial__sum__bound,axiom,
% 0.33/0.66 ! [F: nat > complex,E: real] :
% 0.33/0.66 ( ( summable_complex @ F )
% 0.33/0.66 => ( ( ord_less_real @ zero_zero_real @ E )
% 0.33/0.66 => ~ ! [N8: nat] :
% 0.33/0.66 ~ ! [M5: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ N8 @ M5 )
% 0.33/0.66 => ! [N9: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M5 @ N9 ) ) ) @ E ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_partial_sum_bound
% 0.33/0.66 thf(fact_7818_summable__partial__sum__bound,axiom,
% 0.33/0.66 ! [F: nat > real,E: real] :
% 0.33/0.66 ( ( summable_real @ F )
% 0.33/0.66 => ( ( ord_less_real @ zero_zero_real @ E )
% 0.33/0.66 => ~ ! [N8: nat] :
% 0.33/0.66 ~ ! [M5: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ N8 @ M5 )
% 0.33/0.66 => ! [N9: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M5 @ N9 ) ) ) @ E ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_partial_sum_bound
% 0.33/0.66 thf(fact_7819_summable__power__series,axiom,
% 0.33/0.66 ! [F: nat > real,Z: real] :
% 0.33/0.66 ( ! [I4: nat] : ( ord_less_eq_real @ ( F @ I4 ) @ one_one_real )
% 0.33/0.66 => ( ! [I4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
% 0.33/0.66 => ( ( ord_less_eq_real @ zero_zero_real @ Z )
% 0.33/0.66 => ( ( ord_less_real @ Z @ one_one_real )
% 0.33/0.66 => ( summable_real
% 0.33/0.66 @ ^ [I3: nat] : ( times_times_real @ ( F @ I3 ) @ ( power_power_real @ Z @ I3 ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_power_series
% 0.33/0.66 thf(fact_7820_Abel__lemma,axiom,
% 0.33/0.66 ! [R: real,R0: real,A: nat > complex,M7: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ R )
% 0.33/0.66 => ( ( ord_less_real @ R @ R0 )
% 0.33/0.66 => ( ! [N4: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N4 ) ) @ ( power_power_real @ R0 @ N4 ) ) @ M7 )
% 0.33/0.66 => ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N ) ) @ ( power_power_real @ R @ N ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Abel_lemma
% 0.33/0.66 thf(fact_7821_summable__ratio__test,axiom,
% 0.33/0.66 ! [C: real,N3: nat,F: nat > real] :
% 0.33/0.66 ( ( ord_less_real @ C @ one_one_real )
% 0.33/0.66 => ( ! [N4: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ N3 @ N4 )
% 0.33/0.66 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N4 ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N4 ) ) ) ) )
% 0.33/0.66 => ( summable_real @ F ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_ratio_test
% 0.33/0.66 thf(fact_7822_summable__ratio__test,axiom,
% 0.33/0.66 ! [C: real,N3: nat,F: nat > complex] :
% 0.33/0.66 ( ( ord_less_real @ C @ one_one_real )
% 0.33/0.66 => ( ! [N4: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ N3 @ N4 )
% 0.33/0.66 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N4 ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N4 ) ) ) ) )
% 0.33/0.66 => ( summable_complex @ F ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % summable_ratio_test
% 0.33/0.66 thf(fact_7823_sum__less__suminf2,axiom,
% 0.33/0.66 ! [F: nat > int,N2: nat,I2: nat] :
% 0.33/0.66 ( ( summable_int @ F )
% 0.33/0.66 => ( ! [M2: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ N2 @ M2 )
% 0.33/0.66 => ( ord_less_eq_int @ zero_zero_int @ ( F @ M2 ) ) )
% 0.33/0.66 => ( ( ord_less_eq_nat @ N2 @ I2 )
% 0.33/0.66 => ( ( ord_less_int @ zero_zero_int @ ( F @ I2 ) )
% 0.33/0.66 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_int @ F ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sum_less_suminf2
% 0.33/0.66 thf(fact_7824_sum__less__suminf2,axiom,
% 0.33/0.66 ! [F: nat > nat,N2: nat,I2: nat] :
% 0.33/0.66 ( ( summable_nat @ F )
% 0.33/0.66 => ( ! [M2: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ N2 @ M2 )
% 0.33/0.66 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M2 ) ) )
% 0.33/0.66 => ( ( ord_less_eq_nat @ N2 @ I2 )
% 0.33/0.66 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
% 0.33/0.66 => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_nat @ F ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sum_less_suminf2
% 0.33/0.66 thf(fact_7825_sum__less__suminf2,axiom,
% 0.33/0.66 ! [F: nat > real,N2: nat,I2: nat] :
% 0.33/0.66 ( ( summable_real @ F )
% 0.33/0.66 => ( ! [M2: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ N2 @ M2 )
% 0.33/0.66 => ( ord_less_eq_real @ zero_zero_real @ ( F @ M2 ) ) )
% 0.33/0.66 => ( ( ord_less_eq_nat @ N2 @ I2 )
% 0.33/0.66 => ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
% 0.33/0.66 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_real @ F ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sum_less_suminf2
% 0.33/0.66 thf(fact_7826_sum__bounds__lt__plus1,axiom,
% 0.33/0.66 ! [F: nat > nat,Mm: nat] :
% 0.33/0.66 ( ( groups3542108847815614940at_nat
% 0.33/0.66 @ ^ [K3: nat] : ( F @ ( suc @ K3 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ Mm ) )
% 0.33/0.66 = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sum_bounds_lt_plus1
% 0.33/0.66 thf(fact_7827_sum__bounds__lt__plus1,axiom,
% 0.33/0.66 ! [F: nat > real,Mm: nat] :
% 0.33/0.66 ( ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [K3: nat] : ( F @ ( suc @ K3 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ Mm ) )
% 0.33/0.66 = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sum_bounds_lt_plus1
% 0.33/0.66 thf(fact_7828_sumr__cos__zero__one,axiom,
% 0.33/0.66 ! [N2: nat] :
% 0.33/0.66 ( ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ zero_zero_real @ M6 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.33/0.66 = one_one_real ) ).
% 0.33/0.66
% 0.33/0.66 % sumr_cos_zero_one
% 0.33/0.66 thf(fact_7829_pi__series,axiom,
% 0.33/0.66 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.33/0.66 = ( suminf_real
% 0.33/0.66 @ ^ [K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % pi_series
% 0.33/0.66 thf(fact_7830_infinite__int__iff__unbounded,axiom,
% 0.33/0.66 ! [S: set_int] :
% 0.33/0.66 ( ( ~ ( finite_finite_int @ S ) )
% 0.33/0.66 = ( ! [M6: int] :
% 0.33/0.66 ? [N: int] :
% 0.33/0.66 ( ( ord_less_int @ M6 @ ( abs_abs_int @ N ) )
% 0.33/0.66 & ( member_int @ N @ S ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % infinite_int_iff_unbounded
% 0.33/0.66 thf(fact_7831_geometric__deriv__sums,axiom,
% 0.33/0.66 ! [Z: real] :
% 0.33/0.66 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
% 0.33/0.66 => ( sums_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) )
% 0.33/0.66 @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % geometric_deriv_sums
% 0.33/0.66 thf(fact_7832_geometric__deriv__sums,axiom,
% 0.33/0.66 ! [Z: complex] :
% 0.33/0.66 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
% 0.33/0.66 => ( sums_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) )
% 0.33/0.66 @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % geometric_deriv_sums
% 0.33/0.66 thf(fact_7833_sums__zero,axiom,
% 0.33/0.66 ( sums_complex
% 0.33/0.66 @ ^ [N: nat] : zero_zero_complex
% 0.33/0.66 @ zero_zero_complex ) ).
% 0.33/0.66
% 0.33/0.66 % sums_zero
% 0.33/0.66 thf(fact_7834_sums__zero,axiom,
% 0.33/0.66 ( sums_real
% 0.33/0.66 @ ^ [N: nat] : zero_zero_real
% 0.33/0.66 @ zero_zero_real ) ).
% 0.33/0.66
% 0.33/0.66 % sums_zero
% 0.33/0.66 thf(fact_7835_sums__zero,axiom,
% 0.33/0.66 ( sums_nat
% 0.33/0.66 @ ^ [N: nat] : zero_zero_nat
% 0.33/0.66 @ zero_zero_nat ) ).
% 0.33/0.66
% 0.33/0.66 % sums_zero
% 0.33/0.66 thf(fact_7836_sums__zero,axiom,
% 0.33/0.66 ( sums_int
% 0.33/0.66 @ ^ [N: nat] : zero_zero_int
% 0.33/0.66 @ zero_zero_int ) ).
% 0.33/0.66
% 0.33/0.66 % sums_zero
% 0.33/0.66 thf(fact_7837_cos__coeff__0,axiom,
% 0.33/0.66 ( ( cos_coeff @ zero_zero_nat )
% 0.33/0.66 = one_one_real ) ).
% 0.33/0.66
% 0.33/0.66 % cos_coeff_0
% 0.33/0.66 thf(fact_7838_powser__sums__zero__iff,axiom,
% 0.33/0.66 ! [A: nat > complex,X: complex] :
% 0.33/0.66 ( ( sums_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( A @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) )
% 0.33/0.66 @ X )
% 0.33/0.66 = ( ( A @ zero_zero_nat )
% 0.33/0.66 = X ) ) ).
% 0.33/0.66
% 0.33/0.66 % powser_sums_zero_iff
% 0.33/0.66 thf(fact_7839_powser__sums__zero__iff,axiom,
% 0.33/0.66 ! [A: nat > real,X: real] :
% 0.33/0.66 ( ( sums_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( A @ N ) @ ( power_power_real @ zero_zero_real @ N ) )
% 0.33/0.66 @ X )
% 0.33/0.66 = ( ( A @ zero_zero_nat )
% 0.33/0.66 = X ) ) ).
% 0.33/0.66
% 0.33/0.66 % powser_sums_zero_iff
% 0.33/0.66 thf(fact_7840_sums__le,axiom,
% 0.33/0.66 ! [F: nat > real,G: nat > real,S3: real,T: real] :
% 0.33/0.66 ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( G @ N4 ) )
% 0.33/0.66 => ( ( sums_real @ F @ S3 )
% 0.33/0.66 => ( ( sums_real @ G @ T )
% 0.33/0.66 => ( ord_less_eq_real @ S3 @ T ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_le
% 0.33/0.66 thf(fact_7841_sums__le,axiom,
% 0.33/0.66 ! [F: nat > nat,G: nat > nat,S3: nat,T: nat] :
% 0.33/0.66 ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( G @ N4 ) )
% 0.33/0.66 => ( ( sums_nat @ F @ S3 )
% 0.33/0.66 => ( ( sums_nat @ G @ T )
% 0.33/0.66 => ( ord_less_eq_nat @ S3 @ T ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_le
% 0.33/0.66 thf(fact_7842_sums__le,axiom,
% 0.33/0.66 ! [F: nat > int,G: nat > int,S3: int,T: int] :
% 0.33/0.66 ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ N4 ) @ ( G @ N4 ) )
% 0.33/0.66 => ( ( sums_int @ F @ S3 )
% 0.33/0.66 => ( ( sums_int @ G @ T )
% 0.33/0.66 => ( ord_less_eq_int @ S3 @ T ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_le
% 0.33/0.66 thf(fact_7843_sums__single,axiom,
% 0.33/0.66 ! [I2: nat,F: nat > complex] :
% 0.33/0.66 ( sums_complex
% 0.33/0.66 @ ^ [R5: nat] : ( if_complex @ ( R5 = I2 ) @ ( F @ R5 ) @ zero_zero_complex )
% 0.33/0.66 @ ( F @ I2 ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_single
% 0.33/0.66 thf(fact_7844_sums__single,axiom,
% 0.33/0.66 ! [I2: nat,F: nat > real] :
% 0.33/0.66 ( sums_real
% 0.33/0.66 @ ^ [R5: nat] : ( if_real @ ( R5 = I2 ) @ ( F @ R5 ) @ zero_zero_real )
% 0.33/0.66 @ ( F @ I2 ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_single
% 0.33/0.66 thf(fact_7845_sums__single,axiom,
% 0.33/0.66 ! [I2: nat,F: nat > nat] :
% 0.33/0.66 ( sums_nat
% 0.33/0.66 @ ^ [R5: nat] : ( if_nat @ ( R5 = I2 ) @ ( F @ R5 ) @ zero_zero_nat )
% 0.33/0.66 @ ( F @ I2 ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_single
% 0.33/0.66 thf(fact_7846_sums__single,axiom,
% 0.33/0.66 ! [I2: nat,F: nat > int] :
% 0.33/0.66 ( sums_int
% 0.33/0.66 @ ^ [R5: nat] : ( if_int @ ( R5 = I2 ) @ ( F @ R5 ) @ zero_zero_int )
% 0.33/0.66 @ ( F @ I2 ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_single
% 0.33/0.66 thf(fact_7847_sums__mult,axiom,
% 0.33/0.66 ! [F: nat > complex,A: complex,C: complex] :
% 0.33/0.66 ( ( sums_complex @ F @ A )
% 0.33/0.66 => ( sums_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) )
% 0.33/0.66 @ ( times_times_complex @ C @ A ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_mult
% 0.33/0.66 thf(fact_7848_sums__mult,axiom,
% 0.33/0.66 ! [F: nat > real,A: real,C: real] :
% 0.33/0.66 ( ( sums_real @ F @ A )
% 0.33/0.66 => ( sums_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) )
% 0.33/0.66 @ ( times_times_real @ C @ A ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_mult
% 0.33/0.66 thf(fact_7849_sums__mult2,axiom,
% 0.33/0.66 ! [F: nat > complex,A: complex,C: complex] :
% 0.33/0.66 ( ( sums_complex @ F @ A )
% 0.33/0.66 => ( sums_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ C )
% 0.33/0.66 @ ( times_times_complex @ A @ C ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_mult2
% 0.33/0.66 thf(fact_7850_sums__mult2,axiom,
% 0.33/0.66 ! [F: nat > real,A: real,C: real] :
% 0.33/0.66 ( ( sums_real @ F @ A )
% 0.33/0.66 => ( sums_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ C )
% 0.33/0.66 @ ( times_times_real @ A @ C ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_mult2
% 0.33/0.66 thf(fact_7851_sums__add,axiom,
% 0.33/0.66 ! [F: nat > complex,A: complex,G: nat > complex,B: complex] :
% 0.33/0.66 ( ( sums_complex @ F @ A )
% 0.33/0.66 => ( ( sums_complex @ G @ B )
% 0.33/0.66 => ( sums_complex
% 0.33/0.66 @ ^ [N: nat] : ( plus_plus_complex @ ( F @ N ) @ ( G @ N ) )
% 0.33/0.66 @ ( plus_plus_complex @ A @ B ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_add
% 0.33/0.66 thf(fact_7852_sums__add,axiom,
% 0.33/0.66 ! [F: nat > real,A: real,G: nat > real,B: real] :
% 0.33/0.66 ( ( sums_real @ F @ A )
% 0.33/0.66 => ( ( sums_real @ G @ B )
% 0.33/0.66 => ( sums_real
% 0.33/0.66 @ ^ [N: nat] : ( plus_plus_real @ ( F @ N ) @ ( G @ N ) )
% 0.33/0.66 @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_add
% 0.33/0.66 thf(fact_7853_sums__add,axiom,
% 0.33/0.66 ! [F: nat > nat,A: nat,G: nat > nat,B: nat] :
% 0.33/0.66 ( ( sums_nat @ F @ A )
% 0.33/0.66 => ( ( sums_nat @ G @ B )
% 0.33/0.66 => ( sums_nat
% 0.33/0.66 @ ^ [N: nat] : ( plus_plus_nat @ ( F @ N ) @ ( G @ N ) )
% 0.33/0.66 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_add
% 0.33/0.66 thf(fact_7854_sums__add,axiom,
% 0.33/0.66 ! [F: nat > int,A: int,G: nat > int,B: int] :
% 0.33/0.66 ( ( sums_int @ F @ A )
% 0.33/0.66 => ( ( sums_int @ G @ B )
% 0.33/0.66 => ( sums_int
% 0.33/0.66 @ ^ [N: nat] : ( plus_plus_int @ ( F @ N ) @ ( G @ N ) )
% 0.33/0.66 @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_add
% 0.33/0.66 thf(fact_7855_sums__diff,axiom,
% 0.33/0.66 ! [F: nat > complex,A: complex,G: nat > complex,B: complex] :
% 0.33/0.66 ( ( sums_complex @ F @ A )
% 0.33/0.66 => ( ( sums_complex @ G @ B )
% 0.33/0.66 => ( sums_complex
% 0.33/0.66 @ ^ [N: nat] : ( minus_minus_complex @ ( F @ N ) @ ( G @ N ) )
% 0.33/0.66 @ ( minus_minus_complex @ A @ B ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_diff
% 0.33/0.66 thf(fact_7856_sums__diff,axiom,
% 0.33/0.66 ! [F: nat > real,A: real,G: nat > real,B: real] :
% 0.33/0.66 ( ( sums_real @ F @ A )
% 0.33/0.66 => ( ( sums_real @ G @ B )
% 0.33/0.66 => ( sums_real
% 0.33/0.66 @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( G @ N ) )
% 0.33/0.66 @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_diff
% 0.33/0.66 thf(fact_7857_sums__divide,axiom,
% 0.33/0.66 ! [F: nat > complex,A: complex,C: complex] :
% 0.33/0.66 ( ( sums_complex @ F @ A )
% 0.33/0.66 => ( sums_complex
% 0.33/0.66 @ ^ [N: nat] : ( divide1717551699836669952omplex @ ( F @ N ) @ C )
% 0.33/0.66 @ ( divide1717551699836669952omplex @ A @ C ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_divide
% 0.33/0.66 thf(fact_7858_sums__divide,axiom,
% 0.33/0.66 ! [F: nat > real,A: real,C: real] :
% 0.33/0.66 ( ( sums_real @ F @ A )
% 0.33/0.66 => ( sums_real
% 0.33/0.66 @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ C )
% 0.33/0.66 @ ( divide_divide_real @ A @ C ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_divide
% 0.33/0.66 thf(fact_7859_sums__minus,axiom,
% 0.33/0.66 ! [F: nat > real,A: real] :
% 0.33/0.66 ( ( sums_real @ F @ A )
% 0.33/0.66 => ( sums_real
% 0.33/0.66 @ ^ [N: nat] : ( uminus_uminus_real @ ( F @ N ) )
% 0.33/0.66 @ ( uminus_uminus_real @ A ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_minus
% 0.33/0.66 thf(fact_7860_sums__minus,axiom,
% 0.33/0.66 ! [F: nat > complex,A: complex] :
% 0.33/0.66 ( ( sums_complex @ F @ A )
% 0.33/0.66 => ( sums_complex
% 0.33/0.66 @ ^ [N: nat] : ( uminus1482373934393186551omplex @ ( F @ N ) )
% 0.33/0.66 @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_minus
% 0.33/0.66 thf(fact_7861_sums__sum,axiom,
% 0.33/0.66 ! [I6: set_real,F: real > nat > real,X: real > real] :
% 0.33/0.66 ( ! [I4: real] :
% 0.33/0.66 ( ( member_real @ I4 @ I6 )
% 0.33/0.66 => ( sums_real @ ( F @ I4 ) @ ( X @ I4 ) ) )
% 0.33/0.66 => ( sums_real
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups8097168146408367636l_real
% 0.33/0.66 @ ^ [I3: real] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 )
% 0.33/0.66 @ ( groups8097168146408367636l_real @ X @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_sum
% 0.33/0.66 thf(fact_7862_sums__sum,axiom,
% 0.33/0.66 ! [I6: set_int,F: int > nat > real,X: int > real] :
% 0.33/0.66 ( ! [I4: int] :
% 0.33/0.66 ( ( member_int @ I4 @ I6 )
% 0.33/0.66 => ( sums_real @ ( F @ I4 ) @ ( X @ I4 ) ) )
% 0.33/0.66 => ( sums_real
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups8778361861064173332t_real
% 0.33/0.66 @ ^ [I3: int] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 )
% 0.33/0.66 @ ( groups8778361861064173332t_real @ X @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_sum
% 0.33/0.66 thf(fact_7863_sums__sum,axiom,
% 0.33/0.66 ! [I6: set_complex,F: complex > nat > real,X: complex > real] :
% 0.33/0.66 ( ! [I4: complex] :
% 0.33/0.66 ( ( member_complex @ I4 @ I6 )
% 0.33/0.66 => ( sums_real @ ( F @ I4 ) @ ( X @ I4 ) ) )
% 0.33/0.66 => ( sums_real
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups5808333547571424918x_real
% 0.33/0.66 @ ^ [I3: complex] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 )
% 0.33/0.66 @ ( groups5808333547571424918x_real @ X @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_sum
% 0.33/0.66 thf(fact_7864_sums__sum,axiom,
% 0.33/0.66 ! [I6: set_nat,F: nat > nat > complex,X: nat > complex] :
% 0.33/0.66 ( ! [I4: nat] :
% 0.33/0.66 ( ( member_nat @ I4 @ I6 )
% 0.33/0.66 => ( sums_complex @ ( F @ I4 ) @ ( X @ I4 ) ) )
% 0.33/0.66 => ( sums_complex
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups2073611262835488442omplex
% 0.33/0.66 @ ^ [I3: nat] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 )
% 0.33/0.66 @ ( groups2073611262835488442omplex @ X @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_sum
% 0.33/0.66 thf(fact_7865_sums__sum,axiom,
% 0.33/0.66 ! [I6: set_real,F: real > nat > complex,X: real > complex] :
% 0.33/0.66 ( ! [I4: real] :
% 0.33/0.66 ( ( member_real @ I4 @ I6 )
% 0.33/0.66 => ( sums_complex @ ( F @ I4 ) @ ( X @ I4 ) ) )
% 0.33/0.66 => ( sums_complex
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups5754745047067104278omplex
% 0.33/0.66 @ ^ [I3: real] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 )
% 0.33/0.66 @ ( groups5754745047067104278omplex @ X @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_sum
% 0.33/0.66 thf(fact_7866_sums__sum,axiom,
% 0.33/0.66 ! [I6: set_int,F: int > nat > complex,X: int > complex] :
% 0.33/0.66 ( ! [I4: int] :
% 0.33/0.66 ( ( member_int @ I4 @ I6 )
% 0.33/0.66 => ( sums_complex @ ( F @ I4 ) @ ( X @ I4 ) ) )
% 0.33/0.66 => ( sums_complex
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups3049146728041665814omplex
% 0.33/0.66 @ ^ [I3: int] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 )
% 0.33/0.66 @ ( groups3049146728041665814omplex @ X @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_sum
% 0.33/0.66 thf(fact_7867_sums__sum,axiom,
% 0.33/0.66 ! [I6: set_int,F: int > nat > int,X: int > int] :
% 0.33/0.66 ( ! [I4: int] :
% 0.33/0.66 ( ( member_int @ I4 @ I6 )
% 0.33/0.66 => ( sums_int @ ( F @ I4 ) @ ( X @ I4 ) ) )
% 0.33/0.66 => ( sums_int
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups4538972089207619220nt_int
% 0.33/0.66 @ ^ [I3: int] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 )
% 0.33/0.66 @ ( groups4538972089207619220nt_int @ X @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_sum
% 0.33/0.66 thf(fact_7868_sums__sum,axiom,
% 0.33/0.66 ! [I6: set_complex,F: complex > nat > complex,X: complex > complex] :
% 0.33/0.66 ( ! [I4: complex] :
% 0.33/0.66 ( ( member_complex @ I4 @ I6 )
% 0.33/0.66 => ( sums_complex @ ( F @ I4 ) @ ( X @ I4 ) ) )
% 0.33/0.66 => ( sums_complex
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups7754918857620584856omplex
% 0.33/0.66 @ ^ [I3: complex] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 )
% 0.33/0.66 @ ( groups7754918857620584856omplex @ X @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_sum
% 0.33/0.66 thf(fact_7869_sums__sum,axiom,
% 0.33/0.66 ! [I6: set_nat,F: nat > nat > nat,X: nat > nat] :
% 0.33/0.66 ( ! [I4: nat] :
% 0.33/0.66 ( ( member_nat @ I4 @ I6 )
% 0.33/0.66 => ( sums_nat @ ( F @ I4 ) @ ( X @ I4 ) ) )
% 0.33/0.66 => ( sums_nat
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups3542108847815614940at_nat
% 0.33/0.66 @ ^ [I3: nat] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 )
% 0.33/0.66 @ ( groups3542108847815614940at_nat @ X @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_sum
% 0.33/0.66 thf(fact_7870_sums__sum,axiom,
% 0.33/0.66 ! [I6: set_nat,F: nat > nat > real,X: nat > real] :
% 0.33/0.66 ( ! [I4: nat] :
% 0.33/0.66 ( ( member_nat @ I4 @ I6 )
% 0.33/0.66 => ( sums_real @ ( F @ I4 ) @ ( X @ I4 ) ) )
% 0.33/0.66 => ( sums_real
% 0.33/0.66 @ ^ [N: nat] :
% 0.33/0.66 ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [I3: nat] : ( F @ I3 @ N )
% 0.33/0.66 @ I6 )
% 0.33/0.66 @ ( groups6591440286371151544t_real @ X @ I6 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_sum
% 0.33/0.66 thf(fact_7871_sums__mult__iff,axiom,
% 0.33/0.66 ! [C: complex,F: nat > complex,D: complex] :
% 0.33/0.66 ( ( C != zero_zero_complex )
% 0.33/0.66 => ( ( sums_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) )
% 0.33/0.66 @ ( times_times_complex @ C @ D ) )
% 0.33/0.66 = ( sums_complex @ F @ D ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_mult_iff
% 0.33/0.66 thf(fact_7872_sums__mult__iff,axiom,
% 0.33/0.66 ! [C: real,F: nat > real,D: real] :
% 0.33/0.66 ( ( C != zero_zero_real )
% 0.33/0.66 => ( ( sums_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) )
% 0.33/0.66 @ ( times_times_real @ C @ D ) )
% 0.33/0.66 = ( sums_real @ F @ D ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_mult_iff
% 0.33/0.66 thf(fact_7873_sums__mult2__iff,axiom,
% 0.33/0.66 ! [C: complex,F: nat > complex,D: complex] :
% 0.33/0.66 ( ( C != zero_zero_complex )
% 0.33/0.66 => ( ( sums_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ C )
% 0.33/0.66 @ ( times_times_complex @ D @ C ) )
% 0.33/0.66 = ( sums_complex @ F @ D ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_mult2_iff
% 0.33/0.66 thf(fact_7874_sums__mult2__iff,axiom,
% 0.33/0.66 ! [C: real,F: nat > real,D: real] :
% 0.33/0.66 ( ( C != zero_zero_real )
% 0.33/0.66 => ( ( sums_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ C )
% 0.33/0.66 @ ( times_times_real @ D @ C ) )
% 0.33/0.66 = ( sums_real @ F @ D ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_mult2_iff
% 0.33/0.66 thf(fact_7875_pi__not__less__zero,axiom,
% 0.33/0.66 ~ ( ord_less_real @ pi @ zero_zero_real ) ).
% 0.33/0.66
% 0.33/0.66 % pi_not_less_zero
% 0.33/0.66 thf(fact_7876_pi__gt__zero,axiom,
% 0.33/0.66 ord_less_real @ zero_zero_real @ pi ).
% 0.33/0.66
% 0.33/0.66 % pi_gt_zero
% 0.33/0.66 thf(fact_7877_pi__ge__zero,axiom,
% 0.33/0.66 ord_less_eq_real @ zero_zero_real @ pi ).
% 0.33/0.66
% 0.33/0.66 % pi_ge_zero
% 0.33/0.66 thf(fact_7878_sums__mult__D,axiom,
% 0.33/0.66 ! [C: complex,F: nat > complex,A: complex] :
% 0.33/0.66 ( ( sums_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) )
% 0.33/0.66 @ A )
% 0.33/0.66 => ( ( C != zero_zero_complex )
% 0.33/0.66 => ( sums_complex @ F @ ( divide1717551699836669952omplex @ A @ C ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_mult_D
% 0.33/0.66 thf(fact_7879_sums__mult__D,axiom,
% 0.33/0.66 ! [C: real,F: nat > real,A: real] :
% 0.33/0.66 ( ( sums_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) )
% 0.33/0.66 @ A )
% 0.33/0.66 => ( ( C != zero_zero_real )
% 0.33/0.66 => ( sums_real @ F @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_mult_D
% 0.33/0.66 thf(fact_7880_sums__Suc__imp,axiom,
% 0.33/0.66 ! [F: nat > complex,S3: complex] :
% 0.33/0.66 ( ( ( F @ zero_zero_nat )
% 0.33/0.66 = zero_zero_complex )
% 0.33/0.66 => ( ( sums_complex
% 0.33/0.66 @ ^ [N: nat] : ( F @ ( suc @ N ) )
% 0.33/0.66 @ S3 )
% 0.33/0.66 => ( sums_complex @ F @ S3 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_Suc_imp
% 0.33/0.66 thf(fact_7881_sums__Suc__imp,axiom,
% 0.33/0.66 ! [F: nat > real,S3: real] :
% 0.33/0.66 ( ( ( F @ zero_zero_nat )
% 0.33/0.66 = zero_zero_real )
% 0.33/0.66 => ( ( sums_real
% 0.33/0.66 @ ^ [N: nat] : ( F @ ( suc @ N ) )
% 0.33/0.66 @ S3 )
% 0.33/0.66 => ( sums_real @ F @ S3 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_Suc_imp
% 0.33/0.66 thf(fact_7882_sums__Suc__iff,axiom,
% 0.33/0.66 ! [F: nat > complex,S3: complex] :
% 0.33/0.66 ( ( sums_complex
% 0.33/0.66 @ ^ [N: nat] : ( F @ ( suc @ N ) )
% 0.33/0.66 @ S3 )
% 0.33/0.66 = ( sums_complex @ F @ ( plus_plus_complex @ S3 @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_Suc_iff
% 0.33/0.66 thf(fact_7883_sums__Suc__iff,axiom,
% 0.33/0.66 ! [F: nat > real,S3: real] :
% 0.33/0.66 ( ( sums_real
% 0.33/0.66 @ ^ [N: nat] : ( F @ ( suc @ N ) )
% 0.33/0.66 @ S3 )
% 0.33/0.66 = ( sums_real @ F @ ( plus_plus_real @ S3 @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_Suc_iff
% 0.33/0.66 thf(fact_7884_sums__Suc,axiom,
% 0.33/0.66 ! [F: nat > complex,L2: complex] :
% 0.33/0.66 ( ( sums_complex
% 0.33/0.66 @ ^ [N: nat] : ( F @ ( suc @ N ) )
% 0.33/0.66 @ L2 )
% 0.33/0.66 => ( sums_complex @ F @ ( plus_plus_complex @ L2 @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_Suc
% 0.33/0.66 thf(fact_7885_sums__Suc,axiom,
% 0.33/0.66 ! [F: nat > real,L2: real] :
% 0.33/0.66 ( ( sums_real
% 0.33/0.66 @ ^ [N: nat] : ( F @ ( suc @ N ) )
% 0.33/0.66 @ L2 )
% 0.33/0.66 => ( sums_real @ F @ ( plus_plus_real @ L2 @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_Suc
% 0.33/0.66 thf(fact_7886_sums__Suc,axiom,
% 0.33/0.66 ! [F: nat > nat,L2: nat] :
% 0.33/0.66 ( ( sums_nat
% 0.33/0.66 @ ^ [N: nat] : ( F @ ( suc @ N ) )
% 0.33/0.66 @ L2 )
% 0.33/0.66 => ( sums_nat @ F @ ( plus_plus_nat @ L2 @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_Suc
% 0.33/0.66 thf(fact_7887_sums__Suc,axiom,
% 0.33/0.66 ! [F: nat > int,L2: int] :
% 0.33/0.66 ( ( sums_int
% 0.33/0.66 @ ^ [N: nat] : ( F @ ( suc @ N ) )
% 0.33/0.66 @ L2 )
% 0.33/0.66 => ( sums_int @ F @ ( plus_plus_int @ L2 @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_Suc
% 0.33/0.66 thf(fact_7888_sums__zero__iff__shift,axiom,
% 0.33/0.66 ! [N2: nat,F: nat > complex,S3: complex] :
% 0.33/0.66 ( ! [I4: nat] :
% 0.33/0.66 ( ( ord_less_nat @ I4 @ N2 )
% 0.33/0.66 => ( ( F @ I4 )
% 0.33/0.66 = zero_zero_complex ) )
% 0.33/0.66 => ( ( sums_complex
% 0.33/0.66 @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
% 0.33/0.66 @ S3 )
% 0.33/0.66 = ( sums_complex @ F @ S3 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_zero_iff_shift
% 0.33/0.66 thf(fact_7889_sums__zero__iff__shift,axiom,
% 0.33/0.66 ! [N2: nat,F: nat > real,S3: real] :
% 0.33/0.66 ( ! [I4: nat] :
% 0.33/0.66 ( ( ord_less_nat @ I4 @ N2 )
% 0.33/0.66 => ( ( F @ I4 )
% 0.33/0.66 = zero_zero_real ) )
% 0.33/0.66 => ( ( sums_real
% 0.33/0.66 @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
% 0.33/0.66 @ S3 )
% 0.33/0.66 = ( sums_real @ F @ S3 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_zero_iff_shift
% 0.33/0.66 thf(fact_7890_sums__finite,axiom,
% 0.33/0.66 ! [N3: set_nat,F: nat > complex] :
% 0.33/0.66 ( ( finite_finite_nat @ N3 )
% 0.33/0.66 => ( ! [N4: nat] :
% 0.33/0.66 ( ~ ( member_nat @ N4 @ N3 )
% 0.33/0.66 => ( ( F @ N4 )
% 0.33/0.66 = zero_zero_complex ) )
% 0.33/0.66 => ( sums_complex @ F @ ( groups2073611262835488442omplex @ F @ N3 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_finite
% 0.33/0.66 thf(fact_7891_sums__finite,axiom,
% 0.33/0.66 ! [N3: set_nat,F: nat > int] :
% 0.33/0.66 ( ( finite_finite_nat @ N3 )
% 0.33/0.66 => ( ! [N4: nat] :
% 0.33/0.66 ( ~ ( member_nat @ N4 @ N3 )
% 0.33/0.66 => ( ( F @ N4 )
% 0.33/0.66 = zero_zero_int ) )
% 0.33/0.66 => ( sums_int @ F @ ( groups3539618377306564664at_int @ F @ N3 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_finite
% 0.33/0.66 thf(fact_7892_sums__finite,axiom,
% 0.33/0.66 ! [N3: set_nat,F: nat > nat] :
% 0.33/0.66 ( ( finite_finite_nat @ N3 )
% 0.33/0.66 => ( ! [N4: nat] :
% 0.33/0.66 ( ~ ( member_nat @ N4 @ N3 )
% 0.33/0.66 => ( ( F @ N4 )
% 0.33/0.66 = zero_zero_nat ) )
% 0.33/0.66 => ( sums_nat @ F @ ( groups3542108847815614940at_nat @ F @ N3 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_finite
% 0.33/0.66 thf(fact_7893_sums__finite,axiom,
% 0.33/0.66 ! [N3: set_nat,F: nat > real] :
% 0.33/0.66 ( ( finite_finite_nat @ N3 )
% 0.33/0.66 => ( ! [N4: nat] :
% 0.33/0.66 ( ~ ( member_nat @ N4 @ N3 )
% 0.33/0.66 => ( ( F @ N4 )
% 0.33/0.66 = zero_zero_real ) )
% 0.33/0.66 => ( sums_real @ F @ ( groups6591440286371151544t_real @ F @ N3 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_finite
% 0.33/0.66 thf(fact_7894_sums__If__finite,axiom,
% 0.33/0.66 ! [P: nat > $o,F: nat > complex] :
% 0.33/0.66 ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.33/0.66 => ( sums_complex
% 0.33/0.66 @ ^ [R5: nat] : ( if_complex @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_complex )
% 0.33/0.66 @ ( groups2073611262835488442omplex @ F @ ( collect_nat @ P ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_If_finite
% 0.33/0.66 thf(fact_7895_sums__If__finite,axiom,
% 0.33/0.66 ! [P: nat > $o,F: nat > int] :
% 0.33/0.66 ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.33/0.66 => ( sums_int
% 0.33/0.66 @ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_int )
% 0.33/0.66 @ ( groups3539618377306564664at_int @ F @ ( collect_nat @ P ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_If_finite
% 0.33/0.66 thf(fact_7896_sums__If__finite,axiom,
% 0.33/0.66 ! [P: nat > $o,F: nat > nat] :
% 0.33/0.66 ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.33/0.66 => ( sums_nat
% 0.33/0.66 @ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_nat )
% 0.33/0.66 @ ( groups3542108847815614940at_nat @ F @ ( collect_nat @ P ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_If_finite
% 0.33/0.66 thf(fact_7897_sums__If__finite,axiom,
% 0.33/0.66 ! [P: nat > $o,F: nat > real] :
% 0.33/0.66 ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.33/0.66 => ( sums_real
% 0.33/0.66 @ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_real )
% 0.33/0.66 @ ( groups6591440286371151544t_real @ F @ ( collect_nat @ P ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_If_finite
% 0.33/0.66 thf(fact_7898_sums__If__finite__set,axiom,
% 0.33/0.66 ! [A2: set_nat,F: nat > complex] :
% 0.33/0.66 ( ( finite_finite_nat @ A2 )
% 0.33/0.66 => ( sums_complex
% 0.33/0.66 @ ^ [R5: nat] : ( if_complex @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_complex )
% 0.33/0.66 @ ( groups2073611262835488442omplex @ F @ A2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_If_finite_set
% 0.33/0.66 thf(fact_7899_sums__If__finite__set,axiom,
% 0.33/0.66 ! [A2: set_nat,F: nat > int] :
% 0.33/0.66 ( ( finite_finite_nat @ A2 )
% 0.33/0.66 => ( sums_int
% 0.33/0.66 @ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_int )
% 0.33/0.66 @ ( groups3539618377306564664at_int @ F @ A2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_If_finite_set
% 0.33/0.66 thf(fact_7900_sums__If__finite__set,axiom,
% 0.33/0.66 ! [A2: set_nat,F: nat > nat] :
% 0.33/0.66 ( ( finite_finite_nat @ A2 )
% 0.33/0.66 => ( sums_nat
% 0.33/0.66 @ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_nat )
% 0.33/0.66 @ ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_If_finite_set
% 0.33/0.66 thf(fact_7901_sums__If__finite__set,axiom,
% 0.33/0.66 ! [A2: set_nat,F: nat > real] :
% 0.33/0.66 ( ( finite_finite_nat @ A2 )
% 0.33/0.66 => ( sums_real
% 0.33/0.66 @ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_real )
% 0.33/0.66 @ ( groups6591440286371151544t_real @ F @ A2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_If_finite_set
% 0.33/0.66 thf(fact_7902_powser__sums__if,axiom,
% 0.33/0.66 ! [M: nat,Z: complex] :
% 0.33/0.66 ( sums_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( if_complex @ ( N = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z @ N ) )
% 0.33/0.66 @ ( power_power_complex @ Z @ M ) ) ).
% 0.33/0.66
% 0.33/0.66 % powser_sums_if
% 0.33/0.66 thf(fact_7903_powser__sums__if,axiom,
% 0.33/0.66 ! [M: nat,Z: real] :
% 0.33/0.66 ( sums_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( if_real @ ( N = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z @ N ) )
% 0.33/0.66 @ ( power_power_real @ Z @ M ) ) ).
% 0.33/0.66
% 0.33/0.66 % powser_sums_if
% 0.33/0.66 thf(fact_7904_powser__sums__if,axiom,
% 0.33/0.66 ! [M: nat,Z: int] :
% 0.33/0.66 ( sums_int
% 0.33/0.66 @ ^ [N: nat] : ( times_times_int @ ( if_int @ ( N = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z @ N ) )
% 0.33/0.66 @ ( power_power_int @ Z @ M ) ) ).
% 0.33/0.66
% 0.33/0.66 % powser_sums_if
% 0.33/0.66 thf(fact_7905_powser__sums__zero,axiom,
% 0.33/0.66 ! [A: nat > complex] :
% 0.33/0.66 ( sums_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( A @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) )
% 0.33/0.66 @ ( A @ zero_zero_nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % powser_sums_zero
% 0.33/0.66 thf(fact_7906_powser__sums__zero,axiom,
% 0.33/0.66 ! [A: nat > real] :
% 0.33/0.66 ( sums_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( A @ N ) @ ( power_power_real @ zero_zero_real @ N ) )
% 0.33/0.66 @ ( A @ zero_zero_nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % powser_sums_zero
% 0.33/0.66 thf(fact_7907_sums__iff__shift,axiom,
% 0.33/0.66 ! [F: nat > complex,N2: nat,S3: complex] :
% 0.33/0.66 ( ( sums_complex
% 0.33/0.66 @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
% 0.33/0.66 @ S3 )
% 0.33/0.66 = ( sums_complex @ F @ ( plus_plus_complex @ S3 @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_iff_shift
% 0.33/0.66 thf(fact_7908_sums__iff__shift,axiom,
% 0.33/0.66 ! [F: nat > real,N2: nat,S3: real] :
% 0.33/0.66 ( ( sums_real
% 0.33/0.66 @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
% 0.33/0.66 @ S3 )
% 0.33/0.66 = ( sums_real @ F @ ( plus_plus_real @ S3 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_iff_shift
% 0.33/0.66 thf(fact_7909_sums__split__initial__segment,axiom,
% 0.33/0.66 ! [F: nat > complex,S3: complex,N2: nat] :
% 0.33/0.66 ( ( sums_complex @ F @ S3 )
% 0.33/0.66 => ( sums_complex
% 0.33/0.66 @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
% 0.33/0.66 @ ( minus_minus_complex @ S3 @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_split_initial_segment
% 0.33/0.66 thf(fact_7910_sums__split__initial__segment,axiom,
% 0.33/0.66 ! [F: nat > real,S3: real,N2: nat] :
% 0.33/0.66 ( ( sums_real @ F @ S3 )
% 0.33/0.66 => ( sums_real
% 0.33/0.66 @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
% 0.33/0.66 @ ( minus_minus_real @ S3 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_split_initial_segment
% 0.33/0.66 thf(fact_7911_sums__iff__shift_H,axiom,
% 0.33/0.66 ! [F: nat > complex,N2: nat,S3: complex] :
% 0.33/0.66 ( ( sums_complex
% 0.33/0.66 @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
% 0.33/0.66 @ ( minus_minus_complex @ S3 @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) )
% 0.33/0.66 = ( sums_complex @ F @ S3 ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_iff_shift'
% 0.33/0.66 thf(fact_7912_sums__iff__shift_H,axiom,
% 0.33/0.66 ! [F: nat > real,N2: nat,S3: real] :
% 0.33/0.66 ( ( sums_real
% 0.33/0.66 @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
% 0.33/0.66 @ ( minus_minus_real @ S3 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) )
% 0.33/0.66 = ( sums_real @ F @ S3 ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_iff_shift'
% 0.33/0.66 thf(fact_7913_sums__If__finite__set_H,axiom,
% 0.33/0.66 ! [G: nat > complex,S: complex,A2: set_nat,S4: complex,F: nat > complex] :
% 0.33/0.66 ( ( sums_complex @ G @ S )
% 0.33/0.66 => ( ( finite_finite_nat @ A2 )
% 0.33/0.66 => ( ( S4
% 0.33/0.66 = ( plus_plus_complex @ S
% 0.33/0.66 @ ( groups2073611262835488442omplex
% 0.33/0.66 @ ^ [N: nat] : ( minus_minus_complex @ ( F @ N ) @ ( G @ N ) )
% 0.33/0.66 @ A2 ) ) )
% 0.33/0.66 => ( sums_complex
% 0.33/0.66 @ ^ [N: nat] : ( if_complex @ ( member_nat @ N @ A2 ) @ ( F @ N ) @ ( G @ N ) )
% 0.33/0.66 @ S4 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_If_finite_set'
% 0.33/0.66 thf(fact_7914_sums__If__finite__set_H,axiom,
% 0.33/0.66 ! [G: nat > real,S: real,A2: set_nat,S4: real,F: nat > real] :
% 0.33/0.66 ( ( sums_real @ G @ S )
% 0.33/0.66 => ( ( finite_finite_nat @ A2 )
% 0.33/0.66 => ( ( S4
% 0.33/0.66 = ( plus_plus_real @ S
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( G @ N ) )
% 0.33/0.66 @ A2 ) ) )
% 0.33/0.66 => ( sums_real
% 0.33/0.66 @ ^ [N: nat] : ( if_real @ ( member_nat @ N @ A2 ) @ ( F @ N ) @ ( G @ N ) )
% 0.33/0.66 @ S4 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_If_finite_set'
% 0.33/0.66 thf(fact_7915_pi__less__4,axiom,
% 0.33/0.66 ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % pi_less_4
% 0.33/0.66 thf(fact_7916_pi__ge__two,axiom,
% 0.33/0.66 ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).
% 0.33/0.66
% 0.33/0.66 % pi_ge_two
% 0.33/0.66 thf(fact_7917_pi__half__neq__two,axiom,
% 0.33/0.66 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.33/0.66 != ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % pi_half_neq_two
% 0.33/0.66 thf(fact_7918_pi__half__neq__zero,axiom,
% 0.33/0.66 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.33/0.66 != zero_zero_real ) ).
% 0.33/0.66
% 0.33/0.66 % pi_half_neq_zero
% 0.33/0.66 thf(fact_7919_pi__half__less__two,axiom,
% 0.33/0.66 ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.33/0.66
% 0.33/0.66 % pi_half_less_two
% 0.33/0.66 thf(fact_7920_pi__half__le__two,axiom,
% 0.33/0.66 ord_less_eq_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.33/0.66
% 0.33/0.66 % pi_half_le_two
% 0.33/0.66 thf(fact_7921_pi__half__gt__zero,axiom,
% 0.33/0.66 ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % pi_half_gt_zero
% 0.33/0.66 thf(fact_7922_pi__half__ge__zero,axiom,
% 0.33/0.66 ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % pi_half_ge_zero
% 0.33/0.66 thf(fact_7923_m2pi__less__pi,axiom,
% 0.33/0.66 ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).
% 0.33/0.66
% 0.33/0.66 % m2pi_less_pi
% 0.33/0.66 thf(fact_7924_arctan__ubound,axiom,
% 0.33/0.66 ! [Y: real] : ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % arctan_ubound
% 0.33/0.66 thf(fact_7925_arctan__one,axiom,
% 0.33/0.66 ( ( arctan @ one_one_real )
% 0.33/0.66 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % arctan_one
% 0.33/0.66 thf(fact_7926_geometric__sums,axiom,
% 0.33/0.66 ! [C: real] :
% 0.33/0.66 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 0.33/0.66 => ( sums_real @ ( power_power_real @ C ) @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % geometric_sums
% 0.33/0.66 thf(fact_7927_geometric__sums,axiom,
% 0.33/0.66 ! [C: complex] :
% 0.33/0.66 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 0.33/0.66 => ( sums_complex @ ( power_power_complex @ C ) @ ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % geometric_sums
% 0.33/0.66 thf(fact_7928_power__half__series,axiom,
% 0.33/0.66 ( sums_real
% 0.33/0.66 @ ^ [N: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N ) )
% 0.33/0.66 @ one_one_real ) ).
% 0.33/0.66
% 0.33/0.66 % power_half_series
% 0.33/0.66 thf(fact_7929_minus__pi__half__less__zero,axiom,
% 0.33/0.66 ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).
% 0.33/0.66
% 0.33/0.66 % minus_pi_half_less_zero
% 0.33/0.66 thf(fact_7930_arctan__lbound,axiom,
% 0.33/0.66 ! [Y: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) ) ).
% 0.33/0.66
% 0.33/0.66 % arctan_lbound
% 0.33/0.66 thf(fact_7931_arctan__bounded,axiom,
% 0.33/0.66 ! [Y: real] :
% 0.33/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
% 0.33/0.66 & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % arctan_bounded
% 0.33/0.66 thf(fact_7932_sums__if_H,axiom,
% 0.33/0.66 ! [G: nat > real,X: real] :
% 0.33/0.66 ( ( sums_real @ G @ X )
% 0.33/0.66 => ( sums_real
% 0.33/0.66 @ ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.66 @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_if'
% 0.33/0.66 thf(fact_7933_unbounded__k__infinite,axiom,
% 0.33/0.66 ! [K: nat,S: set_nat] :
% 0.33/0.66 ( ! [M2: nat] :
% 0.33/0.66 ( ( ord_less_nat @ K @ M2 )
% 0.33/0.66 => ? [N9: nat] :
% 0.33/0.66 ( ( ord_less_nat @ M2 @ N9 )
% 0.33/0.66 & ( member_nat @ N9 @ S ) ) )
% 0.33/0.66 => ~ ( finite_finite_nat @ S ) ) ).
% 0.33/0.66
% 0.33/0.66 % unbounded_k_infinite
% 0.33/0.66 thf(fact_7934_infinite__nat__iff__unbounded,axiom,
% 0.33/0.66 ! [S: set_nat] :
% 0.33/0.66 ( ( ~ ( finite_finite_nat @ S ) )
% 0.33/0.66 = ( ! [M6: nat] :
% 0.33/0.66 ? [N: nat] :
% 0.33/0.66 ( ( ord_less_nat @ M6 @ N )
% 0.33/0.66 & ( member_nat @ N @ S ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % infinite_nat_iff_unbounded
% 0.33/0.66 thf(fact_7935_sums__if,axiom,
% 0.33/0.66 ! [G: nat > real,X: real,F: nat > real,Y: real] :
% 0.33/0.66 ( ( sums_real @ G @ X )
% 0.33/0.66 => ( ( sums_real @ F @ Y )
% 0.33/0.66 => ( sums_real
% 0.33/0.66 @ ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( F @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.66 @ ( plus_plus_real @ X @ Y ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sums_if
% 0.33/0.66 thf(fact_7936_infinite__nat__iff__unbounded__le,axiom,
% 0.33/0.66 ! [S: set_nat] :
% 0.33/0.66 ( ( ~ ( finite_finite_nat @ S ) )
% 0.33/0.66 = ( ! [M6: nat] :
% 0.33/0.66 ? [N: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M6 @ N )
% 0.33/0.66 & ( member_nat @ N @ S ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % infinite_nat_iff_unbounded_le
% 0.33/0.66 thf(fact_7937_machin__Euler,axiom,
% 0.33/0.66 ( ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.33/0.66 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % machin_Euler
% 0.33/0.66 thf(fact_7938_machin,axiom,
% 0.33/0.66 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.33/0.66 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % machin
% 0.33/0.66 thf(fact_7939_sin__cos__npi,axiom,
% 0.33/0.66 ! [N2: nat] :
% 0.33/0.66 ( ( sin_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_cos_npi
% 0.33/0.66 thf(fact_7940_diffs__equiv,axiom,
% 0.33/0.66 ! [C: nat > real,X: real] :
% 0.33/0.66 ( ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X @ N ) ) )
% 0.33/0.66 => ( sums_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( C @ N ) ) @ ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
% 0.33/0.66 @ ( suminf_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % diffs_equiv
% 0.33/0.66 thf(fact_7941_diffs__equiv,axiom,
% 0.33/0.66 ! [C: nat > complex,X: complex] :
% 0.33/0.66 ( ( summable_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X @ N ) ) )
% 0.33/0.66 => ( sums_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( C @ N ) ) @ ( power_power_complex @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
% 0.33/0.66 @ ( suminf_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % diffs_equiv
% 0.33/0.66 thf(fact_7942_cos__pi__eq__zero,axiom,
% 0.33/0.66 ! [M: nat] :
% 0.33/0.66 ( ( cos_real @ ( divide_divide_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 = zero_zero_real ) ).
% 0.33/0.66
% 0.33/0.66 % cos_pi_eq_zero
% 0.33/0.66 thf(fact_7943_monoI1,axiom,
% 0.33/0.66 ! [X8: nat > real] :
% 0.33/0.66 ( ! [M2: nat,N4: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M2 @ N4 )
% 0.33/0.66 => ( ord_less_eq_real @ ( X8 @ M2 ) @ ( X8 @ N4 ) ) )
% 0.33/0.66 => ( topolo6980174941875973593q_real @ X8 ) ) ).
% 0.33/0.66
% 0.33/0.66 % monoI1
% 0.33/0.66 thf(fact_7944_monoI1,axiom,
% 0.33/0.66 ! [X8: nat > set_nat] :
% 0.33/0.66 ( ! [M2: nat,N4: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M2 @ N4 )
% 0.33/0.66 => ( ord_less_eq_set_nat @ ( X8 @ M2 ) @ ( X8 @ N4 ) ) )
% 0.33/0.66 => ( topolo7278393974255667507et_nat @ X8 ) ) ).
% 0.33/0.66
% 0.33/0.66 % monoI1
% 0.33/0.66 thf(fact_7945_monoI1,axiom,
% 0.33/0.66 ! [X8: nat > rat] :
% 0.33/0.66 ( ! [M2: nat,N4: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M2 @ N4 )
% 0.33/0.66 => ( ord_less_eq_rat @ ( X8 @ M2 ) @ ( X8 @ N4 ) ) )
% 0.33/0.66 => ( topolo4267028734544971653eq_rat @ X8 ) ) ).
% 0.33/0.66
% 0.33/0.66 % monoI1
% 0.33/0.66 thf(fact_7946_monoI1,axiom,
% 0.33/0.66 ! [X8: nat > num] :
% 0.33/0.66 ( ! [M2: nat,N4: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M2 @ N4 )
% 0.33/0.66 => ( ord_less_eq_num @ ( X8 @ M2 ) @ ( X8 @ N4 ) ) )
% 0.33/0.66 => ( topolo1459490580787246023eq_num @ X8 ) ) ).
% 0.33/0.66
% 0.33/0.66 % monoI1
% 0.33/0.66 thf(fact_7947_monoI1,axiom,
% 0.33/0.66 ! [X8: nat > nat] :
% 0.33/0.66 ( ! [M2: nat,N4: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M2 @ N4 )
% 0.33/0.66 => ( ord_less_eq_nat @ ( X8 @ M2 ) @ ( X8 @ N4 ) ) )
% 0.33/0.66 => ( topolo4902158794631467389eq_nat @ X8 ) ) ).
% 0.33/0.66
% 0.33/0.66 % monoI1
% 0.33/0.66 thf(fact_7948_monoI1,axiom,
% 0.33/0.66 ! [X8: nat > int] :
% 0.33/0.66 ( ! [M2: nat,N4: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M2 @ N4 )
% 0.33/0.66 => ( ord_less_eq_int @ ( X8 @ M2 ) @ ( X8 @ N4 ) ) )
% 0.33/0.66 => ( topolo4899668324122417113eq_int @ X8 ) ) ).
% 0.33/0.66
% 0.33/0.66 % monoI1
% 0.33/0.66 thf(fact_7949_monoI2,axiom,
% 0.33/0.66 ! [X8: nat > real] :
% 0.33/0.66 ( ! [M2: nat,N4: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M2 @ N4 )
% 0.33/0.66 => ( ord_less_eq_real @ ( X8 @ N4 ) @ ( X8 @ M2 ) ) )
% 0.33/0.66 => ( topolo6980174941875973593q_real @ X8 ) ) ).
% 0.33/0.66
% 0.33/0.66 % monoI2
% 0.33/0.66 thf(fact_7950_monoI2,axiom,
% 0.33/0.66 ! [X8: nat > set_nat] :
% 0.33/0.66 ( ! [M2: nat,N4: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M2 @ N4 )
% 0.33/0.66 => ( ord_less_eq_set_nat @ ( X8 @ N4 ) @ ( X8 @ M2 ) ) )
% 0.33/0.66 => ( topolo7278393974255667507et_nat @ X8 ) ) ).
% 0.33/0.66
% 0.33/0.66 % monoI2
% 0.33/0.66 thf(fact_7951_monoI2,axiom,
% 0.33/0.66 ! [X8: nat > rat] :
% 0.33/0.66 ( ! [M2: nat,N4: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M2 @ N4 )
% 0.33/0.66 => ( ord_less_eq_rat @ ( X8 @ N4 ) @ ( X8 @ M2 ) ) )
% 0.33/0.66 => ( topolo4267028734544971653eq_rat @ X8 ) ) ).
% 0.33/0.66
% 0.33/0.66 % monoI2
% 0.33/0.66 thf(fact_7952_monoI2,axiom,
% 0.33/0.66 ! [X8: nat > num] :
% 0.33/0.66 ( ! [M2: nat,N4: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M2 @ N4 )
% 0.33/0.66 => ( ord_less_eq_num @ ( X8 @ N4 ) @ ( X8 @ M2 ) ) )
% 0.33/0.66 => ( topolo1459490580787246023eq_num @ X8 ) ) ).
% 0.33/0.66
% 0.33/0.66 % monoI2
% 0.33/0.66 thf(fact_7953_monoI2,axiom,
% 0.33/0.66 ! [X8: nat > nat] :
% 0.33/0.66 ( ! [M2: nat,N4: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M2 @ N4 )
% 0.33/0.66 => ( ord_less_eq_nat @ ( X8 @ N4 ) @ ( X8 @ M2 ) ) )
% 0.33/0.66 => ( topolo4902158794631467389eq_nat @ X8 ) ) ).
% 0.33/0.66
% 0.33/0.66 % monoI2
% 0.33/0.66 thf(fact_7954_monoI2,axiom,
% 0.33/0.66 ! [X8: nat > int] :
% 0.33/0.66 ( ! [M2: nat,N4: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M2 @ N4 )
% 0.33/0.66 => ( ord_less_eq_int @ ( X8 @ N4 ) @ ( X8 @ M2 ) ) )
% 0.33/0.66 => ( topolo4899668324122417113eq_int @ X8 ) ) ).
% 0.33/0.66
% 0.33/0.66 % monoI2
% 0.33/0.66 thf(fact_7955_cos__zero,axiom,
% 0.33/0.66 ( ( cos_complex @ zero_zero_complex )
% 0.33/0.66 = one_one_complex ) ).
% 0.33/0.66
% 0.33/0.66 % cos_zero
% 0.33/0.66 thf(fact_7956_cos__zero,axiom,
% 0.33/0.66 ( ( cos_real @ zero_zero_real )
% 0.33/0.66 = one_one_real ) ).
% 0.33/0.66
% 0.33/0.66 % cos_zero
% 0.33/0.66 thf(fact_7957_sin__pi__minus,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( sin_real @ ( minus_minus_real @ pi @ X ) )
% 0.33/0.66 = ( sin_real @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_pi_minus
% 0.33/0.66 thf(fact_7958_cos__pi,axiom,
% 0.33/0.66 ( ( cos_real @ pi )
% 0.33/0.66 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_pi
% 0.33/0.66 thf(fact_7959_cos__periodic__pi,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( cos_real @ ( plus_plus_real @ X @ pi ) )
% 0.33/0.66 = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_periodic_pi
% 0.33/0.66 thf(fact_7960_cos__periodic__pi2,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( cos_real @ ( plus_plus_real @ pi @ X ) )
% 0.33/0.66 = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_periodic_pi2
% 0.33/0.66 thf(fact_7961_sin__periodic__pi,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( sin_real @ ( plus_plus_real @ X @ pi ) )
% 0.33/0.66 = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_periodic_pi
% 0.33/0.66 thf(fact_7962_sin__periodic__pi2,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( sin_real @ ( plus_plus_real @ pi @ X ) )
% 0.33/0.66 = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_periodic_pi2
% 0.33/0.66 thf(fact_7963_cos__minus__pi,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( cos_real @ ( minus_minus_real @ X @ pi ) )
% 0.33/0.66 = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_minus_pi
% 0.33/0.66 thf(fact_7964_cos__pi__minus,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( cos_real @ ( minus_minus_real @ pi @ X ) )
% 0.33/0.66 = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_pi_minus
% 0.33/0.66 thf(fact_7965_sin__minus__pi,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( sin_real @ ( minus_minus_real @ X @ pi ) )
% 0.33/0.66 = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_minus_pi
% 0.33/0.66 thf(fact_7966_sin__cos__squared__add3,axiom,
% 0.33/0.66 ! [X: complex] :
% 0.33/0.66 ( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ X ) ) @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ X ) ) )
% 0.33/0.66 = one_one_complex ) ).
% 0.33/0.66
% 0.33/0.66 % sin_cos_squared_add3
% 0.33/0.66 thf(fact_7967_sin__cos__squared__add3,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ X ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ X ) ) )
% 0.33/0.66 = one_one_real ) ).
% 0.33/0.66
% 0.33/0.66 % sin_cos_squared_add3
% 0.33/0.66 thf(fact_7968_sin__npi2,axiom,
% 0.33/0.66 ! [N2: nat] :
% 0.33/0.66 ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 0.33/0.66 = zero_zero_real ) ).
% 0.33/0.66
% 0.33/0.66 % sin_npi2
% 0.33/0.66 thf(fact_7969_sin__npi,axiom,
% 0.33/0.66 ! [N2: nat] :
% 0.33/0.66 ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) )
% 0.33/0.66 = zero_zero_real ) ).
% 0.33/0.66
% 0.33/0.66 % sin_npi
% 0.33/0.66 thf(fact_7970_sin__npi__int,axiom,
% 0.33/0.66 ! [N2: int] :
% 0.33/0.66 ( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N2 ) ) )
% 0.33/0.66 = zero_zero_real ) ).
% 0.33/0.66
% 0.33/0.66 % sin_npi_int
% 0.33/0.66 thf(fact_7971_cos__pi__half,axiom,
% 0.33/0.66 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 = zero_zero_real ) ).
% 0.33/0.66
% 0.33/0.66 % cos_pi_half
% 0.33/0.66 thf(fact_7972_sin__two__pi,axiom,
% 0.33/0.66 ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.33/0.66 = zero_zero_real ) ).
% 0.33/0.66
% 0.33/0.66 % sin_two_pi
% 0.33/0.66 thf(fact_7973_sin__pi__half,axiom,
% 0.33/0.66 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 = one_one_real ) ).
% 0.33/0.66
% 0.33/0.66 % sin_pi_half
% 0.33/0.66 thf(fact_7974_cos__two__pi,axiom,
% 0.33/0.66 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.33/0.66 = one_one_real ) ).
% 0.33/0.66
% 0.33/0.66 % cos_two_pi
% 0.33/0.66 thf(fact_7975_cos__periodic,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( cos_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.33/0.66 = ( cos_real @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_periodic
% 0.33/0.66 thf(fact_7976_sin__periodic,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( sin_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.33/0.66 = ( sin_real @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_periodic
% 0.33/0.66 thf(fact_7977_cos__2pi__minus,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
% 0.33/0.66 = ( cos_real @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_2pi_minus
% 0.33/0.66 thf(fact_7978_cos__npi,axiom,
% 0.33/0.66 ! [N2: nat] :
% 0.33/0.66 ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) )
% 0.33/0.66 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_npi
% 0.33/0.66 thf(fact_7979_cos__npi2,axiom,
% 0.33/0.66 ! [N2: nat] :
% 0.33/0.66 ( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 0.33/0.66 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_npi2
% 0.33/0.66 thf(fact_7980_sin__cos__squared__add,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( plus_plus_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.66 = one_one_real ) ).
% 0.33/0.66
% 0.33/0.66 % sin_cos_squared_add
% 0.33/0.66 thf(fact_7981_sin__cos__squared__add,axiom,
% 0.33/0.66 ! [X: complex] :
% 0.33/0.66 ( ( plus_plus_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.66 = one_one_complex ) ).
% 0.33/0.66
% 0.33/0.66 % sin_cos_squared_add
% 0.33/0.66 thf(fact_7982_sin__cos__squared__add2,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( plus_plus_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.66 = one_one_real ) ).
% 0.33/0.66
% 0.33/0.66 % sin_cos_squared_add2
% 0.33/0.66 thf(fact_7983_sin__cos__squared__add2,axiom,
% 0.33/0.66 ! [X: complex] :
% 0.33/0.66 ( ( plus_plus_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.66 = one_one_complex ) ).
% 0.33/0.66
% 0.33/0.66 % sin_cos_squared_add2
% 0.33/0.66 thf(fact_7984_sin__2npi,axiom,
% 0.33/0.66 ! [N2: nat] :
% 0.33/0.66 ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) )
% 0.33/0.66 = zero_zero_real ) ).
% 0.33/0.66
% 0.33/0.66 % sin_2npi
% 0.33/0.66 thf(fact_7985_cos__2npi,axiom,
% 0.33/0.66 ! [N2: nat] :
% 0.33/0.66 ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) )
% 0.33/0.66 = one_one_real ) ).
% 0.33/0.66
% 0.33/0.66 % cos_2npi
% 0.33/0.66 thf(fact_7986_sin__2pi__minus,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
% 0.33/0.66 = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_2pi_minus
% 0.33/0.66 thf(fact_7987_sin__int__2pin,axiom,
% 0.33/0.66 ! [N2: int] :
% 0.33/0.66 ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N2 ) ) )
% 0.33/0.66 = zero_zero_real ) ).
% 0.33/0.66
% 0.33/0.66 % sin_int_2pin
% 0.33/0.66 thf(fact_7988_cos__int__2pin,axiom,
% 0.33/0.66 ! [N2: int] :
% 0.33/0.66 ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N2 ) ) )
% 0.33/0.66 = one_one_real ) ).
% 0.33/0.66
% 0.33/0.66 % cos_int_2pin
% 0.33/0.66 thf(fact_7989_cos__3over2__pi,axiom,
% 0.33/0.66 ( ( cos_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 0.33/0.66 = zero_zero_real ) ).
% 0.33/0.66
% 0.33/0.66 % cos_3over2_pi
% 0.33/0.66 thf(fact_7990_sin__3over2__pi,axiom,
% 0.33/0.66 ( ( sin_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 0.33/0.66 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_3over2_pi
% 0.33/0.66 thf(fact_7991_cos__npi__int,axiom,
% 0.33/0.66 ! [N2: int] :
% 0.33/0.66 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.66 => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N2 ) ) )
% 0.33/0.66 = one_one_real ) )
% 0.33/0.66 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.66 => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N2 ) ) )
% 0.33/0.66 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_npi_int
% 0.33/0.66 thf(fact_7992_cos__one__sin__zero,axiom,
% 0.33/0.66 ! [X: complex] :
% 0.33/0.66 ( ( ( cos_complex @ X )
% 0.33/0.66 = one_one_complex )
% 0.33/0.66 => ( ( sin_complex @ X )
% 0.33/0.66 = zero_zero_complex ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_one_sin_zero
% 0.33/0.66 thf(fact_7993_cos__one__sin__zero,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ( cos_real @ X )
% 0.33/0.66 = one_one_real )
% 0.33/0.66 => ( ( sin_real @ X )
% 0.33/0.66 = zero_zero_real ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_one_sin_zero
% 0.33/0.66 thf(fact_7994_polar__Ex,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ? [R3: real,A3: real] :
% 0.33/0.66 ( ( X
% 0.33/0.66 = ( times_times_real @ R3 @ ( cos_real @ A3 ) ) )
% 0.33/0.66 & ( Y
% 0.33/0.66 = ( times_times_real @ R3 @ ( sin_real @ A3 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % polar_Ex
% 0.33/0.66 thf(fact_7995_sin__diff,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( sin_real @ ( minus_minus_real @ X @ Y ) )
% 0.33/0.66 = ( minus_minus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_diff
% 0.33/0.66 thf(fact_7996_sin__add,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( sin_real @ ( plus_plus_real @ X @ Y ) )
% 0.33/0.66 = ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_add
% 0.33/0.66 thf(fact_7997_cos__add,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( cos_real @ ( plus_plus_real @ X @ Y ) )
% 0.33/0.66 = ( minus_minus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_add
% 0.33/0.66 thf(fact_7998_cos__diff,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( cos_real @ ( minus_minus_real @ X @ Y ) )
% 0.33/0.66 = ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_diff
% 0.33/0.66 thf(fact_7999_sin__zero__norm__cos__one,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ( sin_real @ X )
% 0.33/0.66 = zero_zero_real )
% 0.33/0.66 => ( ( real_V7735802525324610683m_real @ ( cos_real @ X ) )
% 0.33/0.66 = one_one_real ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_zero_norm_cos_one
% 0.33/0.66 thf(fact_8000_sin__zero__norm__cos__one,axiom,
% 0.33/0.66 ! [X: complex] :
% 0.33/0.66 ( ( ( sin_complex @ X )
% 0.33/0.66 = zero_zero_complex )
% 0.33/0.66 => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X ) )
% 0.33/0.66 = one_one_real ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_zero_norm_cos_one
% 0.33/0.66 thf(fact_8001_sin__zero__abs__cos__one,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ( sin_real @ X )
% 0.33/0.66 = zero_zero_real )
% 0.33/0.66 => ( ( abs_abs_real @ ( cos_real @ X ) )
% 0.33/0.66 = one_one_real ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_zero_abs_cos_one
% 0.33/0.66 thf(fact_8002_sin__double,axiom,
% 0.33/0.66 ! [X: complex] :
% 0.33/0.66 ( ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 0.33/0.66 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ X ) ) @ ( cos_complex @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_double
% 0.33/0.66 thf(fact_8003_sin__double,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 0.33/0.66 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ X ) ) @ ( cos_real @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_double
% 0.33/0.66 thf(fact_8004_sincos__principal__value,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ? [Y4: real] :
% 0.33/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y4 )
% 0.33/0.66 & ( ord_less_eq_real @ Y4 @ pi )
% 0.33/0.66 & ( ( sin_real @ Y4 )
% 0.33/0.66 = ( sin_real @ X ) )
% 0.33/0.66 & ( ( cos_real @ Y4 )
% 0.33/0.66 = ( cos_real @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sincos_principal_value
% 0.33/0.66 thf(fact_8005_sin__x__le__x,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ord_less_eq_real @ ( sin_real @ X ) @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_x_le_x
% 0.33/0.66 thf(fact_8006_sin__le__one,axiom,
% 0.33/0.66 ! [X: real] : ( ord_less_eq_real @ ( sin_real @ X ) @ one_one_real ) ).
% 0.33/0.66
% 0.33/0.66 % sin_le_one
% 0.33/0.66 thf(fact_8007_cos__le__one,axiom,
% 0.33/0.66 ! [X: real] : ( ord_less_eq_real @ ( cos_real @ X ) @ one_one_real ) ).
% 0.33/0.66
% 0.33/0.66 % cos_le_one
% 0.33/0.66 thf(fact_8008_abs__sin__x__le__abs__x,axiom,
% 0.33/0.66 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ ( abs_abs_real @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % abs_sin_x_le_abs_x
% 0.33/0.66 thf(fact_8009_diffs__minus,axiom,
% 0.33/0.66 ! [C: nat > real] :
% 0.33/0.66 ( ( diffs_real
% 0.33/0.66 @ ^ [N: nat] : ( uminus_uminus_real @ ( C @ N ) ) )
% 0.33/0.66 = ( ^ [N: nat] : ( uminus_uminus_real @ ( diffs_real @ C @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % diffs_minus
% 0.33/0.66 thf(fact_8010_diffs__minus,axiom,
% 0.33/0.66 ! [C: nat > int] :
% 0.33/0.66 ( ( diffs_int
% 0.33/0.66 @ ^ [N: nat] : ( uminus_uminus_int @ ( C @ N ) ) )
% 0.33/0.66 = ( ^ [N: nat] : ( uminus_uminus_int @ ( diffs_int @ C @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % diffs_minus
% 0.33/0.66 thf(fact_8011_diffs__minus,axiom,
% 0.33/0.66 ! [C: nat > complex] :
% 0.33/0.66 ( ( diffs_complex
% 0.33/0.66 @ ^ [N: nat] : ( uminus1482373934393186551omplex @ ( C @ N ) ) )
% 0.33/0.66 = ( ^ [N: nat] : ( uminus1482373934393186551omplex @ ( diffs_complex @ C @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % diffs_minus
% 0.33/0.66 thf(fact_8012_diffs__minus,axiom,
% 0.33/0.66 ! [C: nat > rat] :
% 0.33/0.66 ( ( diffs_rat
% 0.33/0.66 @ ^ [N: nat] : ( uminus_uminus_rat @ ( C @ N ) ) )
% 0.33/0.66 = ( ^ [N: nat] : ( uminus_uminus_rat @ ( diffs_rat @ C @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % diffs_minus
% 0.33/0.66 thf(fact_8013_diffs__minus,axiom,
% 0.33/0.66 ! [C: nat > code_integer] :
% 0.33/0.66 ( ( diffs_Code_integer
% 0.33/0.66 @ ^ [N: nat] : ( uminus1351360451143612070nteger @ ( C @ N ) ) )
% 0.33/0.66 = ( ^ [N: nat] : ( uminus1351360451143612070nteger @ ( diffs_Code_integer @ C @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % diffs_minus
% 0.33/0.66 thf(fact_8014_sin__cos__le1,axiom,
% 0.33/0.66 ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) @ one_one_real ) ).
% 0.33/0.66
% 0.33/0.66 % sin_cos_le1
% 0.33/0.66 thf(fact_8015_cos__squared__eq,axiom,
% 0.33/0.66 ! [X: complex] :
% 0.33/0.66 ( ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.66 = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_squared_eq
% 0.33/0.66 thf(fact_8016_cos__squared__eq,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.66 = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_squared_eq
% 0.33/0.66 thf(fact_8017_sin__squared__eq,axiom,
% 0.33/0.66 ! [X: complex] :
% 0.33/0.66 ( ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.66 = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_squared_eq
% 0.33/0.66 thf(fact_8018_sin__squared__eq,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.66 = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_squared_eq
% 0.33/0.66 thf(fact_8019_sin__gt__zero,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ( ord_less_real @ X @ pi )
% 0.33/0.66 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_gt_zero
% 0.33/0.66 thf(fact_8020_sin__x__ge__neg__x,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ ( sin_real @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_x_ge_neg_x
% 0.33/0.66 thf(fact_8021_sin__ge__zero,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ( ord_less_eq_real @ X @ pi )
% 0.33/0.66 => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_ge_zero
% 0.33/0.66 thf(fact_8022_sin__ge__minus__one,axiom,
% 0.33/0.66 ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_ge_minus_one
% 0.33/0.66 thf(fact_8023_cos__inj__pi,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ( ord_less_eq_real @ X @ pi )
% 0.33/0.66 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.66 => ( ( ord_less_eq_real @ Y @ pi )
% 0.33/0.66 => ( ( ( cos_real @ X )
% 0.33/0.66 = ( cos_real @ Y ) )
% 0.33/0.66 => ( X = Y ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_inj_pi
% 0.33/0.66 thf(fact_8024_cos__mono__le__eq,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ( ord_less_eq_real @ X @ pi )
% 0.33/0.66 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.66 => ( ( ord_less_eq_real @ Y @ pi )
% 0.33/0.66 => ( ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
% 0.33/0.66 = ( ord_less_eq_real @ Y @ X ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_mono_le_eq
% 0.33/0.66 thf(fact_8025_cos__monotone__0__pi__le,axiom,
% 0.33/0.66 ! [Y: real,X: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.66 => ( ( ord_less_eq_real @ Y @ X )
% 0.33/0.66 => ( ( ord_less_eq_real @ X @ pi )
% 0.33/0.66 => ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_monotone_0_pi_le
% 0.33/0.66 thf(fact_8026_cos__ge__minus__one,axiom,
% 0.33/0.66 ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_ge_minus_one
% 0.33/0.66 thf(fact_8027_abs__sin__le__one,axiom,
% 0.33/0.66 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ one_one_real ) ).
% 0.33/0.66
% 0.33/0.66 % abs_sin_le_one
% 0.33/0.66 thf(fact_8028_abs__cos__le__one,axiom,
% 0.33/0.66 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X ) ) @ one_one_real ) ).
% 0.33/0.66
% 0.33/0.66 % abs_cos_le_one
% 0.33/0.66 thf(fact_8029_cos__diff__cos,axiom,
% 0.33/0.66 ! [W: complex,Z: complex] :
% 0.33/0.66 ( ( minus_minus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 0.33/0.66 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ Z @ W ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_diff_cos
% 0.33/0.66 thf(fact_8030_cos__diff__cos,axiom,
% 0.33/0.66 ! [W: real,Z: real] :
% 0.33/0.66 ( ( minus_minus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 0.33/0.66 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ Z @ W ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_diff_cos
% 0.33/0.66 thf(fact_8031_sin__diff__sin,axiom,
% 0.33/0.66 ! [W: complex,Z: complex] :
% 0.33/0.66 ( ( minus_minus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 0.33/0.66 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_diff_sin
% 0.33/0.66 thf(fact_8032_sin__diff__sin,axiom,
% 0.33/0.66 ! [W: real,Z: real] :
% 0.33/0.66 ( ( minus_minus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 0.33/0.66 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_diff_sin
% 0.33/0.66 thf(fact_8033_sin__plus__sin,axiom,
% 0.33/0.66 ! [W: complex,Z: complex] :
% 0.33/0.66 ( ( plus_plus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 0.33/0.66 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_plus_sin
% 0.33/0.66 thf(fact_8034_sin__plus__sin,axiom,
% 0.33/0.66 ! [W: real,Z: real] :
% 0.33/0.66 ( ( plus_plus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 0.33/0.66 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_plus_sin
% 0.33/0.66 thf(fact_8035_cos__times__sin,axiom,
% 0.33/0.66 ! [W: complex,Z: complex] :
% 0.33/0.66 ( ( times_times_complex @ ( cos_complex @ W ) @ ( sin_complex @ Z ) )
% 0.33/0.66 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_times_sin
% 0.33/0.66 thf(fact_8036_cos__times__sin,axiom,
% 0.33/0.66 ! [W: real,Z: real] :
% 0.33/0.66 ( ( times_times_real @ ( cos_real @ W ) @ ( sin_real @ Z ) )
% 0.33/0.66 = ( divide_divide_real @ ( minus_minus_real @ ( sin_real @ ( plus_plus_real @ W @ Z ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_times_sin
% 0.33/0.66 thf(fact_8037_sin__times__cos,axiom,
% 0.33/0.66 ! [W: complex,Z: complex] :
% 0.33/0.66 ( ( times_times_complex @ ( sin_complex @ W ) @ ( cos_complex @ Z ) )
% 0.33/0.66 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_times_cos
% 0.33/0.66 thf(fact_8038_sin__times__cos,axiom,
% 0.33/0.66 ! [W: real,Z: real] :
% 0.33/0.66 ( ( times_times_real @ ( sin_real @ W ) @ ( cos_real @ Z ) )
% 0.33/0.66 = ( divide_divide_real @ ( plus_plus_real @ ( sin_real @ ( plus_plus_real @ W @ Z ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_times_cos
% 0.33/0.66 thf(fact_8039_sin__times__sin,axiom,
% 0.33/0.66 ! [W: complex,Z: complex] :
% 0.33/0.66 ( ( times_times_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 0.33/0.66 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_times_sin
% 0.33/0.66 thf(fact_8040_sin__times__sin,axiom,
% 0.33/0.66 ! [W: real,Z: real] :
% 0.33/0.66 ( ( times_times_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 0.33/0.66 = ( divide_divide_real @ ( minus_minus_real @ ( cos_real @ ( minus_minus_real @ W @ Z ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_times_sin
% 0.33/0.66 thf(fact_8041_cos__double,axiom,
% 0.33/0.66 ! [X: complex] :
% 0.33/0.66 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 0.33/0.66 = ( minus_minus_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_double
% 0.33/0.66 thf(fact_8042_cos__double,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 0.33/0.66 = ( minus_minus_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_double
% 0.33/0.66 thf(fact_8043_cos__double__sin,axiom,
% 0.33/0.66 ! [W: complex] :
% 0.33/0.66 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 0.33/0.66 = ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( sin_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_double_sin
% 0.33/0.66 thf(fact_8044_cos__double__sin,axiom,
% 0.33/0.66 ! [W: real] :
% 0.33/0.66 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 0.33/0.66 = ( minus_minus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( sin_real @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_double_sin
% 0.33/0.66 thf(fact_8045_cos__two__neq__zero,axiom,
% 0.33/0.66 ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.33/0.66 != zero_zero_real ) ).
% 0.33/0.66
% 0.33/0.66 % cos_two_neq_zero
% 0.33/0.66 thf(fact_8046_cos__monotone__0__pi,axiom,
% 0.33/0.66 ! [Y: real,X: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.66 => ( ( ord_less_real @ Y @ X )
% 0.33/0.66 => ( ( ord_less_eq_real @ X @ pi )
% 0.33/0.66 => ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_monotone_0_pi
% 0.33/0.66 thf(fact_8047_cos__mono__less__eq,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ( ord_less_eq_real @ X @ pi )
% 0.33/0.66 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.66 => ( ( ord_less_eq_real @ Y @ pi )
% 0.33/0.66 => ( ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
% 0.33/0.66 = ( ord_less_real @ Y @ X ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_mono_less_eq
% 0.33/0.66 thf(fact_8048_sin__eq__0__pi,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
% 0.33/0.66 => ( ( ord_less_real @ X @ pi )
% 0.33/0.66 => ( ( ( sin_real @ X )
% 0.33/0.66 = zero_zero_real )
% 0.33/0.66 => ( X = zero_zero_real ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_eq_0_pi
% 0.33/0.66 thf(fact_8049_sin__zero__pi__iff,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ ( abs_abs_real @ X ) @ pi )
% 0.33/0.66 => ( ( ( sin_real @ X )
% 0.33/0.66 = zero_zero_real )
% 0.33/0.66 = ( X = zero_zero_real ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_zero_pi_iff
% 0.33/0.66 thf(fact_8050_cos__monotone__minus__pi__0_H,axiom,
% 0.33/0.66 ! [Y: real,X: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
% 0.33/0.66 => ( ( ord_less_eq_real @ Y @ X )
% 0.33/0.66 => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.33/0.66 => ( ord_less_eq_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_monotone_minus_pi_0'
% 0.33/0.66 thf(fact_8051_sin__zero__iff__int2,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ( sin_real @ X )
% 0.33/0.66 = zero_zero_real )
% 0.33/0.66 = ( ? [I3: int] :
% 0.33/0.66 ( X
% 0.33/0.66 = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ pi ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_zero_iff_int2
% 0.33/0.66 thf(fact_8052_diffs__def,axiom,
% 0.33/0.66 ( diffs_rat
% 0.33/0.66 = ( ^ [C4: nat > rat,N: nat] : ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( C4 @ ( suc @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % diffs_def
% 0.33/0.66 thf(fact_8053_diffs__def,axiom,
% 0.33/0.66 ( diffs_int
% 0.33/0.66 = ( ^ [C4: nat > int,N: nat] : ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( C4 @ ( suc @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % diffs_def
% 0.33/0.66 thf(fact_8054_diffs__def,axiom,
% 0.33/0.66 ( diffs_real
% 0.33/0.66 = ( ^ [C4: nat > real,N: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( C4 @ ( suc @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % diffs_def
% 0.33/0.66 thf(fact_8055_diffs__def,axiom,
% 0.33/0.66 ( diffs_complex
% 0.33/0.66 = ( ^ [C4: nat > complex,N: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( C4 @ ( suc @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % diffs_def
% 0.33/0.66 thf(fact_8056_sincos__total__pi,axiom,
% 0.33/0.66 ! [Y: real,X: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.66 => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.66 = one_one_real )
% 0.33/0.66 => ? [T5: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 0.33/0.66 & ( ord_less_eq_real @ T5 @ pi )
% 0.33/0.66 & ( X
% 0.33/0.66 = ( cos_real @ T5 ) )
% 0.33/0.66 & ( Y
% 0.33/0.66 = ( sin_real @ T5 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sincos_total_pi
% 0.33/0.66 thf(fact_8057_sin__expansion__lemma,axiom,
% 0.33/0.66 ! [X: real,M: nat] :
% 0.33/0.66 ( ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.33/0.66 = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_expansion_lemma
% 0.33/0.66 thf(fact_8058_cos__expansion__lemma,axiom,
% 0.33/0.66 ! [X: real,M: nat] :
% 0.33/0.66 ( ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.33/0.66 = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_expansion_lemma
% 0.33/0.66 thf(fact_8059_termdiff__converges__all,axiom,
% 0.33/0.66 ! [C: nat > complex,X: complex] :
% 0.33/0.66 ( ! [X4: complex] :
% 0.33/0.66 ( summable_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( C @ N ) @ ( power_power_complex @ X4 @ N ) ) )
% 0.33/0.66 => ( summable_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % termdiff_converges_all
% 0.33/0.66 thf(fact_8060_termdiff__converges__all,axiom,
% 0.33/0.66 ! [C: nat > real,X: real] :
% 0.33/0.66 ( ! [X4: real] :
% 0.33/0.66 ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( C @ N ) @ ( power_power_real @ X4 @ N ) ) )
% 0.33/0.66 => ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % termdiff_converges_all
% 0.33/0.66 thf(fact_8061_sin__gt__zero__02,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.33/0.66 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_gt_zero_02
% 0.33/0.66 thf(fact_8062_cos__two__less__zero,axiom,
% 0.33/0.66 ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 0.33/0.66
% 0.33/0.66 % cos_two_less_zero
% 0.33/0.66 thf(fact_8063_cos__two__le__zero,axiom,
% 0.33/0.66 ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 0.33/0.66
% 0.33/0.66 % cos_two_le_zero
% 0.33/0.66 thf(fact_8064_cos__is__zero,axiom,
% 0.33/0.66 ? [X4: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.33/0.66 & ( ord_less_eq_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.33/0.66 & ( ( cos_real @ X4 )
% 0.33/0.66 = zero_zero_real )
% 0.33/0.66 & ! [Y5: real] :
% 0.33/0.66 ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 0.33/0.66 & ( ord_less_eq_real @ Y5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.33/0.66 & ( ( cos_real @ Y5 )
% 0.33/0.66 = zero_zero_real ) )
% 0.33/0.66 => ( Y5 = X4 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_is_zero
% 0.33/0.66 thf(fact_8065_cos__monotone__minus__pi__0,axiom,
% 0.33/0.66 ! [Y: real,X: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
% 0.33/0.66 => ( ( ord_less_real @ Y @ X )
% 0.33/0.66 => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.33/0.66 => ( ord_less_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_monotone_minus_pi_0
% 0.33/0.66 thf(fact_8066_cos__total,axiom,
% 0.33/0.66 ! [Y: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.66 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.66 => ? [X4: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.33/0.66 & ( ord_less_eq_real @ X4 @ pi )
% 0.33/0.66 & ( ( cos_real @ X4 )
% 0.33/0.66 = Y )
% 0.33/0.66 & ! [Y5: real] :
% 0.33/0.66 ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 0.33/0.66 & ( ord_less_eq_real @ Y5 @ pi )
% 0.33/0.66 & ( ( cos_real @ Y5 )
% 0.33/0.66 = Y ) )
% 0.33/0.66 => ( Y5 = X4 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_total
% 0.33/0.66 thf(fact_8067_sincos__total__pi__half,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.66 => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.66 = one_one_real )
% 0.33/0.66 => ? [T5: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 0.33/0.66 & ( ord_less_eq_real @ T5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 & ( X
% 0.33/0.66 = ( cos_real @ T5 ) )
% 0.33/0.66 & ( Y
% 0.33/0.66 = ( sin_real @ T5 ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sincos_total_pi_half
% 0.33/0.66 thf(fact_8068_sincos__total__2pi__le,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.66 = one_one_real )
% 0.33/0.66 => ? [T5: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 0.33/0.66 & ( ord_less_eq_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.33/0.66 & ( X
% 0.33/0.66 = ( cos_real @ T5 ) )
% 0.33/0.66 & ( Y
% 0.33/0.66 = ( sin_real @ T5 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sincos_total_2pi_le
% 0.33/0.66 thf(fact_8069_sincos__total__2pi,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.66 = one_one_real )
% 0.33/0.66 => ~ ! [T5: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 0.33/0.66 => ( ( ord_less_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.33/0.66 => ( ( X
% 0.33/0.66 = ( cos_real @ T5 ) )
% 0.33/0.66 => ( Y
% 0.33/0.66 != ( sin_real @ T5 ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sincos_total_2pi
% 0.33/0.66 thf(fact_8070_sin__pi__divide__n__ge__0,axiom,
% 0.33/0.66 ! [N2: nat] :
% 0.33/0.66 ( ( N2 != zero_zero_nat )
% 0.33/0.66 => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_pi_divide_n_ge_0
% 0.33/0.66 thf(fact_8071_cos__plus__cos,axiom,
% 0.33/0.66 ! [W: complex,Z: complex] :
% 0.33/0.66 ( ( plus_plus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 0.33/0.66 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_plus_cos
% 0.33/0.66 thf(fact_8072_cos__plus__cos,axiom,
% 0.33/0.66 ! [W: real,Z: real] :
% 0.33/0.66 ( ( plus_plus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 0.33/0.66 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_plus_cos
% 0.33/0.66 thf(fact_8073_cos__times__cos,axiom,
% 0.33/0.66 ! [W: complex,Z: complex] :
% 0.33/0.66 ( ( times_times_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 0.33/0.66 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_times_cos
% 0.33/0.66 thf(fact_8074_cos__times__cos,axiom,
% 0.33/0.66 ! [W: real,Z: real] :
% 0.33/0.66 ( ( times_times_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 0.33/0.66 = ( divide_divide_real @ ( plus_plus_real @ ( cos_real @ ( minus_minus_real @ W @ Z ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_times_cos
% 0.33/0.66 thf(fact_8075_sin__gt__zero2,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_gt_zero2
% 0.33/0.66 thf(fact_8076_sin__lt__zero,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ pi @ X )
% 0.33/0.66 => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.33/0.66 => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_lt_zero
% 0.33/0.66 thf(fact_8077_cos__double__less__one,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.33/0.66 => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_double_less_one
% 0.33/0.66 thf(fact_8078_sin__30,axiom,
% 0.33/0.66 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 0.33/0.66 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_30
% 0.33/0.66 thf(fact_8079_cos__gt__zero,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_gt_zero
% 0.33/0.66 thf(fact_8080_sin__inj__pi,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.66 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.33/0.66 => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 => ( ( ( sin_real @ X )
% 0.33/0.66 = ( sin_real @ Y ) )
% 0.33/0.66 => ( X = Y ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_inj_pi
% 0.33/0.66 thf(fact_8081_sin__mono__le__eq,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.66 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.33/0.66 => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 => ( ( ord_less_eq_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
% 0.33/0.66 = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_mono_le_eq
% 0.33/0.66 thf(fact_8082_sin__monotone__2pi__le,axiom,
% 0.33/0.66 ! [Y: real,X: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.33/0.66 => ( ( ord_less_eq_real @ Y @ X )
% 0.33/0.66 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 => ( ord_less_eq_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_monotone_2pi_le
% 0.33/0.66 thf(fact_8083_cos__60,axiom,
% 0.33/0.66 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 0.33/0.66 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_60
% 0.33/0.66 thf(fact_8084_cos__one__2pi__int,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ( cos_real @ X )
% 0.33/0.66 = one_one_real )
% 0.33/0.66 = ( ? [X2: int] :
% 0.33/0.66 ( X
% 0.33/0.66 = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_one_2pi_int
% 0.33/0.66 thf(fact_8085_cos__double__cos,axiom,
% 0.33/0.66 ! [W: complex] :
% 0.33/0.66 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 0.33/0.66 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_double_cos
% 0.33/0.66 thf(fact_8086_cos__double__cos,axiom,
% 0.33/0.66 ! [W: real] :
% 0.33/0.66 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 0.33/0.66 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( cos_real @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_real ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_double_cos
% 0.33/0.66 thf(fact_8087_cos__treble__cos,axiom,
% 0.33/0.66 ! [X: complex] :
% 0.33/0.66 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ X ) )
% 0.33/0.66 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ ( cos_complex @ X ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_treble_cos
% 0.33/0.66 thf(fact_8088_cos__treble__cos,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ X ) )
% 0.33/0.66 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( cos_real @ X ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_treble_cos
% 0.33/0.66 thf(fact_8089_termdiff__converges,axiom,
% 0.33/0.66 ! [X: real,K5: real,C: nat > real] :
% 0.33/0.66 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ K5 )
% 0.33/0.66 => ( ! [X4: real] :
% 0.33/0.66 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X4 ) @ K5 )
% 0.33/0.66 => ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( C @ N ) @ ( power_power_real @ X4 @ N ) ) ) )
% 0.33/0.66 => ( summable_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % termdiff_converges
% 0.33/0.66 thf(fact_8090_termdiff__converges,axiom,
% 0.33/0.66 ! [X: complex,K5: real,C: nat > complex] :
% 0.33/0.66 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ K5 )
% 0.33/0.66 => ( ! [X4: complex] :
% 0.33/0.66 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X4 ) @ K5 )
% 0.33/0.66 => ( summable_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( C @ N ) @ ( power_power_complex @ X4 @ N ) ) ) )
% 0.33/0.66 => ( summable_complex
% 0.33/0.66 @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % termdiff_converges
% 0.33/0.66 thf(fact_8091_sin__le__zero,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ pi @ X )
% 0.33/0.66 => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.33/0.66 => ( ord_less_eq_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_le_zero
% 0.33/0.66 thf(fact_8092_sin__less__zero,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 0.33/0.66 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.33/0.66 => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_less_zero
% 0.33/0.66 thf(fact_8093_sin__mono__less__eq,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.66 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.33/0.66 => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 => ( ( ord_less_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
% 0.33/0.66 = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_mono_less_eq
% 0.33/0.66 thf(fact_8094_sin__monotone__2pi,axiom,
% 0.33/0.66 ! [Y: real,X: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.33/0.66 => ( ( ord_less_real @ Y @ X )
% 0.33/0.66 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 => ( ord_less_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_monotone_2pi
% 0.33/0.66 thf(fact_8095_sin__total,axiom,
% 0.33/0.66 ! [Y: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.66 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.66 => ? [X4: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.33/0.66 & ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 & ( ( sin_real @ X4 )
% 0.33/0.66 = Y )
% 0.33/0.66 & ! [Y5: real] :
% 0.33/0.66 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 0.33/0.66 & ( ord_less_eq_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 & ( ( sin_real @ Y5 )
% 0.33/0.66 = Y ) )
% 0.33/0.66 => ( Y5 = X4 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_total
% 0.33/0.66 thf(fact_8096_cos__gt__zero__pi,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.66 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_gt_zero_pi
% 0.33/0.66 thf(fact_8097_cos__ge__zero,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.66 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_ge_zero
% 0.33/0.66 thf(fact_8098_cos__one__2pi,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ( cos_real @ X )
% 0.33/0.66 = one_one_real )
% 0.33/0.66 = ( ? [X2: nat] :
% 0.33/0.66 ( X
% 0.33/0.66 = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 0.33/0.66 | ? [X2: nat] :
% 0.33/0.66 ( X
% 0.33/0.66 = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_one_2pi
% 0.33/0.66 thf(fact_8099_sin__pi__divide__n__gt__0,axiom,
% 0.33/0.66 ! [N2: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.66 => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_pi_divide_n_gt_0
% 0.33/0.66 thf(fact_8100_monoseq__minus,axiom,
% 0.33/0.66 ! [A: nat > int] :
% 0.33/0.66 ( ( topolo4899668324122417113eq_int @ A )
% 0.33/0.66 => ( topolo4899668324122417113eq_int
% 0.33/0.66 @ ^ [N: nat] : ( uminus_uminus_int @ ( A @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % monoseq_minus
% 0.33/0.66 thf(fact_8101_monoseq__minus,axiom,
% 0.33/0.66 ! [A: nat > rat] :
% 0.33/0.66 ( ( topolo4267028734544971653eq_rat @ A )
% 0.33/0.66 => ( topolo4267028734544971653eq_rat
% 0.33/0.66 @ ^ [N: nat] : ( uminus_uminus_rat @ ( A @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % monoseq_minus
% 0.33/0.66 thf(fact_8102_monoseq__minus,axiom,
% 0.33/0.66 ! [A: nat > code_integer] :
% 0.33/0.66 ( ( topolo2919662092509805066nteger @ A )
% 0.33/0.66 => ( topolo2919662092509805066nteger
% 0.33/0.66 @ ^ [N: nat] : ( uminus1351360451143612070nteger @ ( A @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % monoseq_minus
% 0.33/0.66 thf(fact_8103_monoseq__minus,axiom,
% 0.33/0.66 ! [A: nat > real] :
% 0.33/0.66 ( ( topolo6980174941875973593q_real @ A )
% 0.33/0.66 => ( topolo6980174941875973593q_real
% 0.33/0.66 @ ^ [N: nat] : ( uminus_uminus_real @ ( A @ N ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % monoseq_minus
% 0.33/0.66 thf(fact_8104_sin__zero__iff__int,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ( sin_real @ X )
% 0.33/0.66 = zero_zero_real )
% 0.33/0.66 = ( ? [I3: int] :
% 0.33/0.66 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I3 )
% 0.33/0.66 & ( X
% 0.33/0.66 = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_zero_iff_int
% 0.33/0.66 thf(fact_8105_cos__zero__iff__int,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ( cos_real @ X )
% 0.33/0.66 = zero_zero_real )
% 0.33/0.66 = ( ? [I3: int] :
% 0.33/0.66 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I3 )
% 0.33/0.66 & ( X
% 0.33/0.66 = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_zero_iff_int
% 0.33/0.66 thf(fact_8106_sin__zero__lemma,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ( ( sin_real @ X )
% 0.33/0.66 = zero_zero_real )
% 0.33/0.66 => ? [N4: nat] :
% 0.33/0.66 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
% 0.33/0.66 & ( X
% 0.33/0.66 = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_zero_lemma
% 0.33/0.66 thf(fact_8107_sin__zero__iff,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ( sin_real @ X )
% 0.33/0.66 = zero_zero_real )
% 0.33/0.66 = ( ? [N: nat] :
% 0.33/0.66 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.66 & ( X
% 0.33/0.66 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.66 | ? [N: nat] :
% 0.33/0.66 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.66 & ( X
% 0.33/0.66 = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_zero_iff
% 0.33/0.66 thf(fact_8108_cos__zero__lemma,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ( ( cos_real @ X )
% 0.33/0.66 = zero_zero_real )
% 0.33/0.66 => ? [N4: nat] :
% 0.33/0.66 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
% 0.33/0.66 & ( X
% 0.33/0.66 = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_zero_lemma
% 0.33/0.66 thf(fact_8109_cos__zero__iff,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ( cos_real @ X )
% 0.33/0.66 = zero_zero_real )
% 0.33/0.66 = ( ? [N: nat] :
% 0.33/0.66 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.66 & ( X
% 0.33/0.66 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.66 | ? [N: nat] :
% 0.33/0.66 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.33/0.66 & ( X
% 0.33/0.66 = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_zero_iff
% 0.33/0.66 thf(fact_8110_mono__SucI1,axiom,
% 0.33/0.66 ! [X8: nat > real] :
% 0.33/0.66 ( ! [N4: nat] : ( ord_less_eq_real @ ( X8 @ N4 ) @ ( X8 @ ( suc @ N4 ) ) )
% 0.33/0.66 => ( topolo6980174941875973593q_real @ X8 ) ) ).
% 0.33/0.66
% 0.33/0.66 % mono_SucI1
% 0.33/0.66 thf(fact_8111_mono__SucI1,axiom,
% 0.33/0.66 ! [X8: nat > set_nat] :
% 0.33/0.66 ( ! [N4: nat] : ( ord_less_eq_set_nat @ ( X8 @ N4 ) @ ( X8 @ ( suc @ N4 ) ) )
% 0.33/0.66 => ( topolo7278393974255667507et_nat @ X8 ) ) ).
% 0.33/0.66
% 0.33/0.66 % mono_SucI1
% 0.33/0.66 thf(fact_8112_mono__SucI1,axiom,
% 0.33/0.66 ! [X8: nat > rat] :
% 0.33/0.66 ( ! [N4: nat] : ( ord_less_eq_rat @ ( X8 @ N4 ) @ ( X8 @ ( suc @ N4 ) ) )
% 0.33/0.66 => ( topolo4267028734544971653eq_rat @ X8 ) ) ).
% 0.33/0.66
% 0.33/0.66 % mono_SucI1
% 0.33/0.66 thf(fact_8113_mono__SucI1,axiom,
% 0.33/0.66 ! [X8: nat > num] :
% 0.33/0.66 ( ! [N4: nat] : ( ord_less_eq_num @ ( X8 @ N4 ) @ ( X8 @ ( suc @ N4 ) ) )
% 0.33/0.66 => ( topolo1459490580787246023eq_num @ X8 ) ) ).
% 0.33/0.66
% 0.33/0.66 % mono_SucI1
% 0.33/0.66 thf(fact_8114_mono__SucI1,axiom,
% 0.33/0.66 ! [X8: nat > nat] :
% 0.33/0.66 ( ! [N4: nat] : ( ord_less_eq_nat @ ( X8 @ N4 ) @ ( X8 @ ( suc @ N4 ) ) )
% 0.33/0.66 => ( topolo4902158794631467389eq_nat @ X8 ) ) ).
% 0.33/0.66
% 0.33/0.66 % mono_SucI1
% 0.33/0.66 thf(fact_8115_mono__SucI1,axiom,
% 0.33/0.66 ! [X8: nat > int] :
% 0.33/0.66 ( ! [N4: nat] : ( ord_less_eq_int @ ( X8 @ N4 ) @ ( X8 @ ( suc @ N4 ) ) )
% 0.33/0.66 => ( topolo4899668324122417113eq_int @ X8 ) ) ).
% 0.33/0.66
% 0.33/0.66 % mono_SucI1
% 0.33/0.66 thf(fact_8116_mono__SucI2,axiom,
% 0.33/0.66 ! [X8: nat > real] :
% 0.33/0.66 ( ! [N4: nat] : ( ord_less_eq_real @ ( X8 @ ( suc @ N4 ) ) @ ( X8 @ N4 ) )
% 0.33/0.66 => ( topolo6980174941875973593q_real @ X8 ) ) ).
% 0.33/0.66
% 0.33/0.66 % mono_SucI2
% 0.33/0.66 thf(fact_8117_mono__SucI2,axiom,
% 0.33/0.66 ! [X8: nat > set_nat] :
% 0.33/0.66 ( ! [N4: nat] : ( ord_less_eq_set_nat @ ( X8 @ ( suc @ N4 ) ) @ ( X8 @ N4 ) )
% 0.33/0.66 => ( topolo7278393974255667507et_nat @ X8 ) ) ).
% 0.33/0.66
% 0.33/0.66 % mono_SucI2
% 0.33/0.66 thf(fact_8118_mono__SucI2,axiom,
% 0.33/0.66 ! [X8: nat > rat] :
% 0.33/0.66 ( ! [N4: nat] : ( ord_less_eq_rat @ ( X8 @ ( suc @ N4 ) ) @ ( X8 @ N4 ) )
% 0.33/0.66 => ( topolo4267028734544971653eq_rat @ X8 ) ) ).
% 0.33/0.66
% 0.33/0.66 % mono_SucI2
% 0.33/0.66 thf(fact_8119_mono__SucI2,axiom,
% 0.33/0.66 ! [X8: nat > num] :
% 0.33/0.66 ( ! [N4: nat] : ( ord_less_eq_num @ ( X8 @ ( suc @ N4 ) ) @ ( X8 @ N4 ) )
% 0.33/0.66 => ( topolo1459490580787246023eq_num @ X8 ) ) ).
% 0.33/0.66
% 0.33/0.66 % mono_SucI2
% 0.33/0.66 thf(fact_8120_mono__SucI2,axiom,
% 0.33/0.66 ! [X8: nat > nat] :
% 0.33/0.66 ( ! [N4: nat] : ( ord_less_eq_nat @ ( X8 @ ( suc @ N4 ) ) @ ( X8 @ N4 ) )
% 0.33/0.66 => ( topolo4902158794631467389eq_nat @ X8 ) ) ).
% 0.33/0.66
% 0.33/0.66 % mono_SucI2
% 0.33/0.66 thf(fact_8121_mono__SucI2,axiom,
% 0.33/0.66 ! [X8: nat > int] :
% 0.33/0.66 ( ! [N4: nat] : ( ord_less_eq_int @ ( X8 @ ( suc @ N4 ) ) @ ( X8 @ N4 ) )
% 0.33/0.66 => ( topolo4899668324122417113eq_int @ X8 ) ) ).
% 0.33/0.66
% 0.33/0.66 % mono_SucI2
% 0.33/0.66 thf(fact_8122_monoseq__Suc,axiom,
% 0.33/0.66 ( topolo6980174941875973593q_real
% 0.33/0.66 = ( ^ [X3: nat > real] :
% 0.33/0.66 ( ! [N: nat] : ( ord_less_eq_real @ ( X3 @ N ) @ ( X3 @ ( suc @ N ) ) )
% 0.33/0.66 | ! [N: nat] : ( ord_less_eq_real @ ( X3 @ ( suc @ N ) ) @ ( X3 @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % monoseq_Suc
% 0.33/0.66 thf(fact_8123_monoseq__Suc,axiom,
% 0.33/0.66 ( topolo7278393974255667507et_nat
% 0.33/0.66 = ( ^ [X3: nat > set_nat] :
% 0.33/0.66 ( ! [N: nat] : ( ord_less_eq_set_nat @ ( X3 @ N ) @ ( X3 @ ( suc @ N ) ) )
% 0.33/0.66 | ! [N: nat] : ( ord_less_eq_set_nat @ ( X3 @ ( suc @ N ) ) @ ( X3 @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % monoseq_Suc
% 0.33/0.66 thf(fact_8124_monoseq__Suc,axiom,
% 0.33/0.66 ( topolo4267028734544971653eq_rat
% 0.33/0.66 = ( ^ [X3: nat > rat] :
% 0.33/0.66 ( ! [N: nat] : ( ord_less_eq_rat @ ( X3 @ N ) @ ( X3 @ ( suc @ N ) ) )
% 0.33/0.66 | ! [N: nat] : ( ord_less_eq_rat @ ( X3 @ ( suc @ N ) ) @ ( X3 @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % monoseq_Suc
% 0.33/0.66 thf(fact_8125_monoseq__Suc,axiom,
% 0.33/0.66 ( topolo1459490580787246023eq_num
% 0.33/0.66 = ( ^ [X3: nat > num] :
% 0.33/0.66 ( ! [N: nat] : ( ord_less_eq_num @ ( X3 @ N ) @ ( X3 @ ( suc @ N ) ) )
% 0.33/0.66 | ! [N: nat] : ( ord_less_eq_num @ ( X3 @ ( suc @ N ) ) @ ( X3 @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % monoseq_Suc
% 0.33/0.66 thf(fact_8126_monoseq__Suc,axiom,
% 0.33/0.66 ( topolo4902158794631467389eq_nat
% 0.33/0.66 = ( ^ [X3: nat > nat] :
% 0.33/0.66 ( ! [N: nat] : ( ord_less_eq_nat @ ( X3 @ N ) @ ( X3 @ ( suc @ N ) ) )
% 0.33/0.66 | ! [N: nat] : ( ord_less_eq_nat @ ( X3 @ ( suc @ N ) ) @ ( X3 @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % monoseq_Suc
% 0.33/0.66 thf(fact_8127_monoseq__Suc,axiom,
% 0.33/0.66 ( topolo4899668324122417113eq_int
% 0.33/0.66 = ( ^ [X3: nat > int] :
% 0.33/0.66 ( ! [N: nat] : ( ord_less_eq_int @ ( X3 @ N ) @ ( X3 @ ( suc @ N ) ) )
% 0.33/0.66 | ! [N: nat] : ( ord_less_eq_int @ ( X3 @ ( suc @ N ) ) @ ( X3 @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % monoseq_Suc
% 0.33/0.66 thf(fact_8128_monoseq__def,axiom,
% 0.33/0.66 ( topolo6980174941875973593q_real
% 0.33/0.66 = ( ^ [X3: nat > real] :
% 0.33/0.66 ( ! [M6: nat,N: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M6 @ N )
% 0.33/0.66 => ( ord_less_eq_real @ ( X3 @ M6 ) @ ( X3 @ N ) ) )
% 0.33/0.66 | ! [M6: nat,N: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M6 @ N )
% 0.33/0.66 => ( ord_less_eq_real @ ( X3 @ N ) @ ( X3 @ M6 ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % monoseq_def
% 0.33/0.66 thf(fact_8129_monoseq__def,axiom,
% 0.33/0.66 ( topolo7278393974255667507et_nat
% 0.33/0.66 = ( ^ [X3: nat > set_nat] :
% 0.33/0.66 ( ! [M6: nat,N: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M6 @ N )
% 0.33/0.66 => ( ord_less_eq_set_nat @ ( X3 @ M6 ) @ ( X3 @ N ) ) )
% 0.33/0.66 | ! [M6: nat,N: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M6 @ N )
% 0.33/0.66 => ( ord_less_eq_set_nat @ ( X3 @ N ) @ ( X3 @ M6 ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % monoseq_def
% 0.33/0.66 thf(fact_8130_monoseq__def,axiom,
% 0.33/0.66 ( topolo4267028734544971653eq_rat
% 0.33/0.66 = ( ^ [X3: nat > rat] :
% 0.33/0.66 ( ! [M6: nat,N: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M6 @ N )
% 0.33/0.66 => ( ord_less_eq_rat @ ( X3 @ M6 ) @ ( X3 @ N ) ) )
% 0.33/0.66 | ! [M6: nat,N: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M6 @ N )
% 0.33/0.66 => ( ord_less_eq_rat @ ( X3 @ N ) @ ( X3 @ M6 ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % monoseq_def
% 0.33/0.66 thf(fact_8131_monoseq__def,axiom,
% 0.33/0.66 ( topolo1459490580787246023eq_num
% 0.33/0.66 = ( ^ [X3: nat > num] :
% 0.33/0.66 ( ! [M6: nat,N: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M6 @ N )
% 0.33/0.66 => ( ord_less_eq_num @ ( X3 @ M6 ) @ ( X3 @ N ) ) )
% 0.33/0.66 | ! [M6: nat,N: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M6 @ N )
% 0.33/0.66 => ( ord_less_eq_num @ ( X3 @ N ) @ ( X3 @ M6 ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % monoseq_def
% 0.33/0.66 thf(fact_8132_monoseq__def,axiom,
% 0.33/0.66 ( topolo4902158794631467389eq_nat
% 0.33/0.66 = ( ^ [X3: nat > nat] :
% 0.33/0.66 ( ! [M6: nat,N: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M6 @ N )
% 0.33/0.66 => ( ord_less_eq_nat @ ( X3 @ M6 ) @ ( X3 @ N ) ) )
% 0.33/0.66 | ! [M6: nat,N: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M6 @ N )
% 0.33/0.66 => ( ord_less_eq_nat @ ( X3 @ N ) @ ( X3 @ M6 ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % monoseq_def
% 0.33/0.66 thf(fact_8133_monoseq__def,axiom,
% 0.33/0.66 ( topolo4899668324122417113eq_int
% 0.33/0.66 = ( ^ [X3: nat > int] :
% 0.33/0.66 ( ! [M6: nat,N: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M6 @ N )
% 0.33/0.66 => ( ord_less_eq_int @ ( X3 @ M6 ) @ ( X3 @ N ) ) )
% 0.33/0.66 | ! [M6: nat,N: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M6 @ N )
% 0.33/0.66 => ( ord_less_eq_int @ ( X3 @ N ) @ ( X3 @ M6 ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % monoseq_def
% 0.33/0.66 thf(fact_8134_Maclaurin__cos__expansion2,axiom,
% 0.33/0.66 ! [X: real,N2: nat] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.66 => ? [T5: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ T5 )
% 0.33/0.66 & ( ord_less_real @ T5 @ X )
% 0.33/0.66 & ( ( cos_real @ X )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.66 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_cos_expansion2
% 0.33/0.66 thf(fact_8135_Maclaurin__minus__cos__expansion,axiom,
% 0.33/0.66 ! [N2: nat,X: real] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.66 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.33/0.66 => ? [T5: real] :
% 0.33/0.66 ( ( ord_less_real @ X @ T5 )
% 0.33/0.66 & ( ord_less_real @ T5 @ zero_zero_real )
% 0.33/0.66 & ( ( cos_real @ X )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.66 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_minus_cos_expansion
% 0.33/0.66 thf(fact_8136_Maclaurin__cos__expansion,axiom,
% 0.33/0.66 ! [X: real,N2: nat] :
% 0.33/0.66 ? [T5: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 0.33/0.66 & ( ( cos_real @ X )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.66 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_cos_expansion
% 0.33/0.66 thf(fact_8137_sin__paired,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( sums_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
% 0.33/0.66 @ ( sin_real @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_paired
% 0.33/0.66 thf(fact_8138_tan__double,axiom,
% 0.33/0.66 ! [X: complex] :
% 0.33/0.66 ( ( ( cos_complex @ X )
% 0.33/0.66 != zero_zero_complex )
% 0.33/0.66 => ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 0.33/0.66 != zero_zero_complex )
% 0.33/0.66 => ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 0.33/0.66 = ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_double
% 0.33/0.66 thf(fact_8139_tan__double,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ( cos_real @ X )
% 0.33/0.66 != zero_zero_real )
% 0.33/0.66 => ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 0.33/0.66 != zero_zero_real )
% 0.33/0.66 => ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 0.33/0.66 = ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_double
% 0.33/0.66 thf(fact_8140_tan__periodic__pi,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( tan_real @ ( plus_plus_real @ X @ pi ) )
% 0.33/0.66 = ( tan_real @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_periodic_pi
% 0.33/0.66 thf(fact_8141_fact__0,axiom,
% 0.33/0.66 ( ( semiri5044797733671781792omplex @ zero_zero_nat )
% 0.33/0.66 = one_one_complex ) ).
% 0.33/0.66
% 0.33/0.66 % fact_0
% 0.33/0.66 thf(fact_8142_fact__0,axiom,
% 0.33/0.66 ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
% 0.33/0.66 = one_one_rat ) ).
% 0.33/0.66
% 0.33/0.66 % fact_0
% 0.33/0.66 thf(fact_8143_fact__0,axiom,
% 0.33/0.66 ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
% 0.33/0.66 = one_one_int ) ).
% 0.33/0.66
% 0.33/0.66 % fact_0
% 0.33/0.66 thf(fact_8144_fact__0,axiom,
% 0.33/0.66 ( ( semiri2265585572941072030t_real @ zero_zero_nat )
% 0.33/0.66 = one_one_real ) ).
% 0.33/0.66
% 0.33/0.66 % fact_0
% 0.33/0.66 thf(fact_8145_fact__0,axiom,
% 0.33/0.66 ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
% 0.33/0.66 = one_one_nat ) ).
% 0.33/0.66
% 0.33/0.66 % fact_0
% 0.33/0.66 thf(fact_8146_fact__1,axiom,
% 0.33/0.66 ( ( semiri5044797733671781792omplex @ one_one_nat )
% 0.33/0.66 = one_one_complex ) ).
% 0.33/0.66
% 0.33/0.66 % fact_1
% 0.33/0.66 thf(fact_8147_fact__1,axiom,
% 0.33/0.66 ( ( semiri773545260158071498ct_rat @ one_one_nat )
% 0.33/0.66 = one_one_rat ) ).
% 0.33/0.66
% 0.33/0.66 % fact_1
% 0.33/0.66 thf(fact_8148_fact__1,axiom,
% 0.33/0.66 ( ( semiri1406184849735516958ct_int @ one_one_nat )
% 0.33/0.66 = one_one_int ) ).
% 0.33/0.66
% 0.33/0.66 % fact_1
% 0.33/0.66 thf(fact_8149_fact__1,axiom,
% 0.33/0.66 ( ( semiri2265585572941072030t_real @ one_one_nat )
% 0.33/0.66 = one_one_real ) ).
% 0.33/0.66
% 0.33/0.66 % fact_1
% 0.33/0.66 thf(fact_8150_fact__1,axiom,
% 0.33/0.66 ( ( semiri1408675320244567234ct_nat @ one_one_nat )
% 0.33/0.66 = one_one_nat ) ).
% 0.33/0.66
% 0.33/0.66 % fact_1
% 0.33/0.66 thf(fact_8151_fact__Suc__0,axiom,
% 0.33/0.66 ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
% 0.33/0.66 = one_one_complex ) ).
% 0.33/0.66
% 0.33/0.66 % fact_Suc_0
% 0.33/0.66 thf(fact_8152_fact__Suc__0,axiom,
% 0.33/0.66 ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
% 0.33/0.66 = one_one_rat ) ).
% 0.33/0.66
% 0.33/0.66 % fact_Suc_0
% 0.33/0.66 thf(fact_8153_fact__Suc__0,axiom,
% 0.33/0.66 ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
% 0.33/0.66 = one_one_int ) ).
% 0.33/0.66
% 0.33/0.66 % fact_Suc_0
% 0.33/0.66 thf(fact_8154_fact__Suc__0,axiom,
% 0.33/0.66 ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
% 0.33/0.66 = one_one_real ) ).
% 0.33/0.66
% 0.33/0.66 % fact_Suc_0
% 0.33/0.66 thf(fact_8155_fact__Suc__0,axiom,
% 0.33/0.66 ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
% 0.33/0.66 = one_one_nat ) ).
% 0.33/0.66
% 0.33/0.66 % fact_Suc_0
% 0.33/0.66 thf(fact_8156_fact__Suc,axiom,
% 0.33/0.66 ! [N2: nat] :
% 0.33/0.66 ( ( semiri773545260158071498ct_rat @ ( suc @ N2 ) )
% 0.33/0.66 = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_Suc
% 0.33/0.66 thf(fact_8157_fact__Suc,axiom,
% 0.33/0.66 ! [N2: nat] :
% 0.33/0.66 ( ( semiri1406184849735516958ct_int @ ( suc @ N2 ) )
% 0.33/0.66 = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_Suc
% 0.33/0.66 thf(fact_8158_fact__Suc,axiom,
% 0.33/0.66 ! [N2: nat] :
% 0.33/0.66 ( ( semiri5044797733671781792omplex @ ( suc @ N2 ) )
% 0.33/0.66 = ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( semiri5044797733671781792omplex @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_Suc
% 0.33/0.66 thf(fact_8159_fact__Suc,axiom,
% 0.33/0.66 ! [N2: nat] :
% 0.33/0.66 ( ( semiri2265585572941072030t_real @ ( suc @ N2 ) )
% 0.33/0.66 = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_Suc
% 0.33/0.66 thf(fact_8160_fact__Suc,axiom,
% 0.33/0.66 ! [N2: nat] :
% 0.33/0.66 ( ( semiri1408675320244567234ct_nat @ ( suc @ N2 ) )
% 0.33/0.66 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N2 ) ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_Suc
% 0.33/0.66 thf(fact_8161_tan__npi,axiom,
% 0.33/0.66 ! [N2: nat] :
% 0.33/0.66 ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) )
% 0.33/0.66 = zero_zero_real ) ).
% 0.33/0.66
% 0.33/0.66 % tan_npi
% 0.33/0.66 thf(fact_8162_tan__periodic__n,axiom,
% 0.33/0.66 ! [X: real,N2: num] :
% 0.33/0.66 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ N2 ) @ pi ) ) )
% 0.33/0.66 = ( tan_real @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_periodic_n
% 0.33/0.66 thf(fact_8163_tan__periodic__nat,axiom,
% 0.33/0.66 ! [X: real,N2: nat] :
% 0.33/0.66 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) ) )
% 0.33/0.66 = ( tan_real @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_periodic_nat
% 0.33/0.66 thf(fact_8164_tan__periodic__int,axiom,
% 0.33/0.66 ! [X: real,I2: int] :
% 0.33/0.66 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ pi ) ) )
% 0.33/0.66 = ( tan_real @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_periodic_int
% 0.33/0.66 thf(fact_8165_fact__2,axiom,
% 0.33/0.66 ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.66 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_2
% 0.33/0.66 thf(fact_8166_fact__2,axiom,
% 0.33/0.66 ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.66 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_2
% 0.33/0.66 thf(fact_8167_fact__2,axiom,
% 0.33/0.66 ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.66 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_2
% 0.33/0.66 thf(fact_8168_fact__2,axiom,
% 0.33/0.66 ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.66 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_2
% 0.33/0.66 thf(fact_8169_fact__2,axiom,
% 0.33/0.66 ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.66 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_2
% 0.33/0.66 thf(fact_8170_tan__periodic,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.33/0.66 = ( tan_real @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_periodic
% 0.33/0.66 thf(fact_8171_fact__ge__zero,axiom,
% 0.33/0.66 ! [N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_ge_zero
% 0.33/0.66 thf(fact_8172_fact__ge__zero,axiom,
% 0.33/0.66 ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_ge_zero
% 0.33/0.66 thf(fact_8173_fact__ge__zero,axiom,
% 0.33/0.66 ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_ge_zero
% 0.33/0.66 thf(fact_8174_fact__ge__zero,axiom,
% 0.33/0.66 ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_ge_zero
% 0.33/0.66 thf(fact_8175_fact__not__neg,axiom,
% 0.33/0.66 ! [N2: nat] :
% 0.33/0.66 ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ zero_zero_rat ) ).
% 0.33/0.66
% 0.33/0.66 % fact_not_neg
% 0.33/0.66 thf(fact_8176_fact__not__neg,axiom,
% 0.33/0.66 ! [N2: nat] :
% 0.33/0.66 ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N2 ) @ zero_zero_int ) ).
% 0.33/0.66
% 0.33/0.66 % fact_not_neg
% 0.33/0.66 thf(fact_8177_fact__not__neg,axiom,
% 0.33/0.66 ! [N2: nat] :
% 0.33/0.66 ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N2 ) @ zero_zero_real ) ).
% 0.33/0.66
% 0.33/0.66 % fact_not_neg
% 0.33/0.66 thf(fact_8178_fact__not__neg,axiom,
% 0.33/0.66 ! [N2: nat] :
% 0.33/0.66 ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ zero_zero_nat ) ).
% 0.33/0.66
% 0.33/0.66 % fact_not_neg
% 0.33/0.66 thf(fact_8179_fact__gt__zero,axiom,
% 0.33/0.66 ! [N2: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_gt_zero
% 0.33/0.66 thf(fact_8180_fact__gt__zero,axiom,
% 0.33/0.66 ! [N2: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_gt_zero
% 0.33/0.66 thf(fact_8181_fact__gt__zero,axiom,
% 0.33/0.66 ! [N2: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_gt_zero
% 0.33/0.66 thf(fact_8182_fact__gt__zero,axiom,
% 0.33/0.66 ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_gt_zero
% 0.33/0.66 thf(fact_8183_fact__ge__1,axiom,
% 0.33/0.66 ! [N2: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_ge_1
% 0.33/0.66 thf(fact_8184_fact__ge__1,axiom,
% 0.33/0.66 ! [N2: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_ge_1
% 0.33/0.66 thf(fact_8185_fact__ge__1,axiom,
% 0.33/0.66 ! [N2: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_ge_1
% 0.33/0.66 thf(fact_8186_fact__ge__1,axiom,
% 0.33/0.66 ! [N2: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_ge_1
% 0.33/0.66 thf(fact_8187_fact__mono,axiom,
% 0.33/0.66 ! [M: nat,N2: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.66 => ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_mono
% 0.33/0.66 thf(fact_8188_fact__mono,axiom,
% 0.33/0.66 ! [M: nat,N2: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.66 => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_mono
% 0.33/0.66 thf(fact_8189_fact__mono,axiom,
% 0.33/0.66 ! [M: nat,N2: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.66 => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_mono
% 0.33/0.66 thf(fact_8190_fact__mono,axiom,
% 0.33/0.66 ! [M: nat,N2: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.66 => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_mono
% 0.33/0.66 thf(fact_8191_fact__dvd,axiom,
% 0.33/0.66 ! [N2: nat,M: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ N2 @ M )
% 0.33/0.66 => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_dvd
% 0.33/0.66 thf(fact_8192_fact__dvd,axiom,
% 0.33/0.66 ! [N2: nat,M: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ N2 @ M )
% 0.33/0.66 => ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N2 ) @ ( semiri3624122377584611663nteger @ M ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_dvd
% 0.33/0.66 thf(fact_8193_fact__dvd,axiom,
% 0.33/0.66 ! [N2: nat,M: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ N2 @ M )
% 0.33/0.66 => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_dvd
% 0.33/0.66 thf(fact_8194_fact__dvd,axiom,
% 0.33/0.66 ! [N2: nat,M: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ N2 @ M )
% 0.33/0.66 => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_dvd
% 0.33/0.66 thf(fact_8195_fact__less__mono,axiom,
% 0.33/0.66 ! [M: nat,N2: nat] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.33/0.66 => ( ( ord_less_nat @ M @ N2 )
% 0.33/0.66 => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_less_mono
% 0.33/0.66 thf(fact_8196_fact__less__mono,axiom,
% 0.33/0.66 ! [M: nat,N2: nat] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.33/0.66 => ( ( ord_less_nat @ M @ N2 )
% 0.33/0.66 => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_less_mono
% 0.33/0.66 thf(fact_8197_fact__less__mono,axiom,
% 0.33/0.66 ! [M: nat,N2: nat] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.33/0.66 => ( ( ord_less_nat @ M @ N2 )
% 0.33/0.66 => ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_less_mono
% 0.33/0.66 thf(fact_8198_fact__less__mono,axiom,
% 0.33/0.66 ! [M: nat,N2: nat] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.33/0.66 => ( ( ord_less_nat @ M @ N2 )
% 0.33/0.66 => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_less_mono
% 0.33/0.66 thf(fact_8199_fact__fact__dvd__fact,axiom,
% 0.33/0.66 ! [K: nat,N2: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N2 ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_fact_dvd_fact
% 0.33/0.66 thf(fact_8200_fact__fact__dvd__fact,axiom,
% 0.33/0.66 ! [K: nat,N2: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N2 ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_fact_dvd_fact
% 0.33/0.66 thf(fact_8201_fact__fact__dvd__fact,axiom,
% 0.33/0.66 ! [K: nat,N2: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N2 ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_fact_dvd_fact
% 0.33/0.66 thf(fact_8202_fact__fact__dvd__fact,axiom,
% 0.33/0.66 ! [K: nat,N2: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_fact_dvd_fact
% 0.33/0.66 thf(fact_8203_fact__fact__dvd__fact,axiom,
% 0.33/0.66 ! [K: nat,N2: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_fact_dvd_fact
% 0.33/0.66 thf(fact_8204_fact__mod,axiom,
% 0.33/0.66 ! [M: nat,N2: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.66 => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1406184849735516958ct_int @ M ) )
% 0.33/0.66 = zero_zero_int ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_mod
% 0.33/0.66 thf(fact_8205_fact__mod,axiom,
% 0.33/0.66 ! [M: nat,N2: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.66 => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N2 ) @ ( semiri3624122377584611663nteger @ M ) )
% 0.33/0.66 = zero_z3403309356797280102nteger ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_mod
% 0.33/0.66 thf(fact_8206_fact__mod,axiom,
% 0.33/0.66 ! [M: nat,N2: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.66 => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ M ) )
% 0.33/0.66 = zero_zero_nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_mod
% 0.33/0.66 thf(fact_8207_fact__le__power,axiom,
% 0.33/0.66 ! [N2: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N2 @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_le_power
% 0.33/0.66 thf(fact_8208_fact__le__power,axiom,
% 0.33/0.66 ! [N2: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N2 @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_le_power
% 0.33/0.66 thf(fact_8209_fact__le__power,axiom,
% 0.33/0.66 ! [N2: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N2 @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_le_power
% 0.33/0.66 thf(fact_8210_fact__le__power,axiom,
% 0.33/0.66 ! [N2: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N2 @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_le_power
% 0.33/0.66 thf(fact_8211_tan__def,axiom,
% 0.33/0.66 ( tan_complex
% 0.33/0.66 = ( ^ [X2: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ X2 ) @ ( cos_complex @ X2 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_def
% 0.33/0.66 thf(fact_8212_tan__def,axiom,
% 0.33/0.66 ( tan_real
% 0.33/0.66 = ( ^ [X2: real] : ( divide_divide_real @ ( sin_real @ X2 ) @ ( cos_real @ X2 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_def
% 0.33/0.66 thf(fact_8213_choose__dvd,axiom,
% 0.33/0.66 ! [K: nat,N2: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ K @ N2 )
% 0.33/0.66 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri3624122377584611663nteger @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % choose_dvd
% 0.33/0.66 thf(fact_8214_choose__dvd,axiom,
% 0.33/0.66 ! [K: nat,N2: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ K @ N2 )
% 0.33/0.66 => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % choose_dvd
% 0.33/0.66 thf(fact_8215_choose__dvd,axiom,
% 0.33/0.66 ! [K: nat,N2: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ K @ N2 )
% 0.33/0.66 => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % choose_dvd
% 0.33/0.66 thf(fact_8216_choose__dvd,axiom,
% 0.33/0.66 ! [K: nat,N2: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ K @ N2 )
% 0.33/0.66 => ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % choose_dvd
% 0.33/0.66 thf(fact_8217_choose__dvd,axiom,
% 0.33/0.66 ! [K: nat,N2: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ K @ N2 )
% 0.33/0.66 => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % choose_dvd
% 0.33/0.66 thf(fact_8218_fact__numeral,axiom,
% 0.33/0.66 ! [K: num] :
% 0.33/0.66 ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ K ) )
% 0.33/0.66 = ( times_times_complex @ ( numera6690914467698888265omplex @ K ) @ ( semiri5044797733671781792omplex @ ( pred_numeral @ K ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_numeral
% 0.33/0.66 thf(fact_8219_fact__numeral,axiom,
% 0.33/0.66 ! [K: num] :
% 0.33/0.66 ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ K ) )
% 0.33/0.66 = ( times_times_rat @ ( numeral_numeral_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( pred_numeral @ K ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_numeral
% 0.33/0.66 thf(fact_8220_fact__numeral,axiom,
% 0.33/0.66 ! [K: num] :
% 0.33/0.66 ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ K ) )
% 0.33/0.66 = ( times_times_int @ ( numeral_numeral_int @ K ) @ ( semiri1406184849735516958ct_int @ ( pred_numeral @ K ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_numeral
% 0.33/0.66 thf(fact_8221_fact__numeral,axiom,
% 0.33/0.66 ! [K: num] :
% 0.33/0.66 ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ K ) )
% 0.33/0.66 = ( times_times_real @ ( numeral_numeral_real @ K ) @ ( semiri2265585572941072030t_real @ ( pred_numeral @ K ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_numeral
% 0.33/0.66 thf(fact_8222_fact__numeral,axiom,
% 0.33/0.66 ! [K: num] :
% 0.33/0.66 ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ K ) )
% 0.33/0.66 = ( times_times_nat @ ( numeral_numeral_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( pred_numeral @ K ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_numeral
% 0.33/0.66 thf(fact_8223_square__fact__le__2__fact,axiom,
% 0.33/0.66 ! [N2: nat] : ( ord_less_eq_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % square_fact_le_2_fact
% 0.33/0.66 thf(fact_8224_tan__45,axiom,
% 0.33/0.66 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.33/0.66 = one_one_real ) ).
% 0.33/0.66
% 0.33/0.66 % tan_45
% 0.33/0.66 thf(fact_8225_fact__num__eq__if,axiom,
% 0.33/0.66 ( semiri773545260158071498ct_rat
% 0.33/0.66 = ( ^ [M6: nat] : ( if_rat @ ( M6 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M6 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_num_eq_if
% 0.33/0.66 thf(fact_8226_fact__num__eq__if,axiom,
% 0.33/0.66 ( semiri1406184849735516958ct_int
% 0.33/0.66 = ( ^ [M6: nat] : ( if_int @ ( M6 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_num_eq_if
% 0.33/0.66 thf(fact_8227_fact__num__eq__if,axiom,
% 0.33/0.66 ( semiri5044797733671781792omplex
% 0.33/0.66 = ( ^ [M6: nat] : ( if_complex @ ( M6 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M6 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_num_eq_if
% 0.33/0.66 thf(fact_8228_fact__num__eq__if,axiom,
% 0.33/0.66 ( semiri2265585572941072030t_real
% 0.33/0.66 = ( ^ [M6: nat] : ( if_real @ ( M6 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M6 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_num_eq_if
% 0.33/0.66 thf(fact_8229_fact__num__eq__if,axiom,
% 0.33/0.66 ( semiri1408675320244567234ct_nat
% 0.33/0.66 = ( ^ [M6: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M6 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_num_eq_if
% 0.33/0.66 thf(fact_8230_fact__reduce,axiom,
% 0.33/0.66 ! [N2: nat] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.66 => ( ( semiri773545260158071498ct_rat @ N2 )
% 0.33/0.66 = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_reduce
% 0.33/0.66 thf(fact_8231_fact__reduce,axiom,
% 0.33/0.66 ! [N2: nat] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.66 => ( ( semiri1406184849735516958ct_int @ N2 )
% 0.33/0.66 = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_reduce
% 0.33/0.66 thf(fact_8232_fact__reduce,axiom,
% 0.33/0.66 ! [N2: nat] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.66 => ( ( semiri5044797733671781792omplex @ N2 )
% 0.33/0.66 = ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_reduce
% 0.33/0.66 thf(fact_8233_fact__reduce,axiom,
% 0.33/0.66 ! [N2: nat] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.66 => ( ( semiri2265585572941072030t_real @ N2 )
% 0.33/0.66 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_reduce
% 0.33/0.66 thf(fact_8234_fact__reduce,axiom,
% 0.33/0.66 ! [N2: nat] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.66 => ( ( semiri1408675320244567234ct_nat @ N2 )
% 0.33/0.66 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_reduce
% 0.33/0.66 thf(fact_8235_tan__gt__zero,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 => ( ord_less_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_gt_zero
% 0.33/0.66 thf(fact_8236_lemma__tan__total,axiom,
% 0.33/0.66 ! [Y: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.66 => ? [X4: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.33/0.66 & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 & ( ord_less_real @ Y @ ( tan_real @ X4 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % lemma_tan_total
% 0.33/0.66 thf(fact_8237_tan__total,axiom,
% 0.33/0.66 ! [Y: real] :
% 0.33/0.66 ? [X4: real] :
% 0.33/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.33/0.66 & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 & ( ( tan_real @ X4 )
% 0.33/0.66 = Y )
% 0.33/0.66 & ! [Y5: real] :
% 0.33/0.66 ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 0.33/0.66 & ( ord_less_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 & ( ( tan_real @ Y5 )
% 0.33/0.66 = Y ) )
% 0.33/0.66 => ( Y5 = X4 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_total
% 0.33/0.66 thf(fact_8238_tan__monotone,axiom,
% 0.33/0.66 ! [Y: real,X: real] :
% 0.33/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.33/0.66 => ( ( ord_less_real @ Y @ X )
% 0.33/0.66 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 => ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_monotone
% 0.33/0.66 thf(fact_8239_tan__monotone_H,axiom,
% 0.33/0.66 ! [Y: real,X: real] :
% 0.33/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.33/0.66 => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.66 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 => ( ( ord_less_real @ Y @ X )
% 0.33/0.66 = ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_monotone'
% 0.33/0.66 thf(fact_8240_tan__mono__lt__eq,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.66 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.33/0.66 => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 => ( ( ord_less_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
% 0.33/0.66 = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_mono_lt_eq
% 0.33/0.66 thf(fact_8241_lemma__tan__total1,axiom,
% 0.33/0.66 ! [Y: real] :
% 0.33/0.66 ? [X4: real] :
% 0.33/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.33/0.66 & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 & ( ( tan_real @ X4 )
% 0.33/0.66 = Y ) ) ).
% 0.33/0.66
% 0.33/0.66 % lemma_tan_total1
% 0.33/0.66 thf(fact_8242_tan__minus__45,axiom,
% 0.33/0.66 ( ( tan_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.66 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_minus_45
% 0.33/0.66 thf(fact_8243_tan__inverse,axiom,
% 0.33/0.66 ! [Y: real] :
% 0.33/0.66 ( ( divide_divide_real @ one_one_real @ ( tan_real @ Y ) )
% 0.33/0.66 = ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_inverse
% 0.33/0.66 thf(fact_8244_add__tan__eq,axiom,
% 0.33/0.66 ! [X: complex,Y: complex] :
% 0.33/0.66 ( ( ( cos_complex @ X )
% 0.33/0.66 != zero_zero_complex )
% 0.33/0.66 => ( ( ( cos_complex @ Y )
% 0.33/0.66 != zero_zero_complex )
% 0.33/0.66 => ( ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) )
% 0.33/0.66 = ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % add_tan_eq
% 0.33/0.66 thf(fact_8245_add__tan__eq,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ( cos_real @ X )
% 0.33/0.66 != zero_zero_real )
% 0.33/0.66 => ( ( ( cos_real @ Y )
% 0.33/0.66 != zero_zero_real )
% 0.33/0.66 => ( ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
% 0.33/0.66 = ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % add_tan_eq
% 0.33/0.66 thf(fact_8246_tan__pos__pi2__le,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_pos_pi2_le
% 0.33/0.66 thf(fact_8247_tan__total__pos,axiom,
% 0.33/0.66 ! [Y: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.66 => ? [X4: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.33/0.66 & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 & ( ( tan_real @ X4 )
% 0.33/0.66 = Y ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_total_pos
% 0.33/0.66 thf(fact_8248_tan__less__zero,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 0.33/0.66 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.33/0.66 => ( ord_less_real @ ( tan_real @ X ) @ zero_zero_real ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_less_zero
% 0.33/0.66 thf(fact_8249_tan__mono__le__eq,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.66 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 0.33/0.66 => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 => ( ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
% 0.33/0.66 = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_mono_le_eq
% 0.33/0.66 thf(fact_8250_tan__mono__le,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.66 => ( ( ord_less_eq_real @ X @ Y )
% 0.33/0.66 => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 => ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_mono_le
% 0.33/0.66 thf(fact_8251_tan__bound__pi2,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.33/0.66 => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X ) ) @ one_one_real ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_bound_pi2
% 0.33/0.66 thf(fact_8252_arctan,axiom,
% 0.33/0.66 ! [Y: real] :
% 0.33/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
% 0.33/0.66 & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 & ( ( tan_real @ ( arctan @ Y ) )
% 0.33/0.66 = Y ) ) ).
% 0.33/0.66
% 0.33/0.66 % arctan
% 0.33/0.66 thf(fact_8253_arctan__tan,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.66 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 => ( ( arctan @ ( tan_real @ X ) )
% 0.33/0.66 = X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % arctan_tan
% 0.33/0.66 thf(fact_8254_arctan__unique,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.66 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 => ( ( ( tan_real @ X )
% 0.33/0.66 = Y )
% 0.33/0.66 => ( ( arctan @ Y )
% 0.33/0.66 = X ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % arctan_unique
% 0.33/0.66 thf(fact_8255_Maclaurin__zero,axiom,
% 0.33/0.66 ! [X: real,N2: nat,Diff: nat > complex > real] :
% 0.33/0.66 ( ( X = zero_zero_real )
% 0.33/0.66 => ( ( N2 != zero_zero_nat )
% 0.33/0.66 => ( ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_complex ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.66 = ( Diff @ zero_zero_nat @ zero_zero_complex ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_zero
% 0.33/0.66 thf(fact_8256_Maclaurin__zero,axiom,
% 0.33/0.66 ! [X: real,N2: nat,Diff: nat > real > real] :
% 0.33/0.66 ( ( X = zero_zero_real )
% 0.33/0.66 => ( ( N2 != zero_zero_nat )
% 0.33/0.66 => ( ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.66 = ( Diff @ zero_zero_nat @ zero_zero_real ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_zero
% 0.33/0.66 thf(fact_8257_Maclaurin__zero,axiom,
% 0.33/0.66 ! [X: real,N2: nat,Diff: nat > rat > real] :
% 0.33/0.66 ( ( X = zero_zero_real )
% 0.33/0.66 => ( ( N2 != zero_zero_nat )
% 0.33/0.66 => ( ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_rat ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.66 = ( Diff @ zero_zero_nat @ zero_zero_rat ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_zero
% 0.33/0.66 thf(fact_8258_Maclaurin__zero,axiom,
% 0.33/0.66 ! [X: real,N2: nat,Diff: nat > nat > real] :
% 0.33/0.66 ( ( X = zero_zero_real )
% 0.33/0.66 => ( ( N2 != zero_zero_nat )
% 0.33/0.66 => ( ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_nat ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.66 = ( Diff @ zero_zero_nat @ zero_zero_nat ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_zero
% 0.33/0.66 thf(fact_8259_Maclaurin__zero,axiom,
% 0.33/0.66 ! [X: real,N2: nat,Diff: nat > int > real] :
% 0.33/0.66 ( ( X = zero_zero_real )
% 0.33/0.66 => ( ( N2 != zero_zero_nat )
% 0.33/0.66 => ( ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_int ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.66 = ( Diff @ zero_zero_nat @ zero_zero_int ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_zero
% 0.33/0.66 thf(fact_8260_Maclaurin__lemma,axiom,
% 0.33/0.66 ! [H2: real,F: real > real,J: nat > real,N2: nat] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.33/0.66 => ? [B9: real] :
% 0.33/0.66 ( ( F @ H2 )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.66 @ ( times_times_real @ B9 @ ( divide_divide_real @ ( power_power_real @ H2 @ N2 ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_lemma
% 0.33/0.66 thf(fact_8261_tan__add,axiom,
% 0.33/0.66 ! [X: complex,Y: complex] :
% 0.33/0.66 ( ( ( cos_complex @ X )
% 0.33/0.66 != zero_zero_complex )
% 0.33/0.66 => ( ( ( cos_complex @ Y )
% 0.33/0.66 != zero_zero_complex )
% 0.33/0.66 => ( ( ( cos_complex @ ( plus_plus_complex @ X @ Y ) )
% 0.33/0.66 != zero_zero_complex )
% 0.33/0.66 => ( ( tan_complex @ ( plus_plus_complex @ X @ Y ) )
% 0.33/0.66 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_add
% 0.33/0.66 thf(fact_8262_tan__add,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ( cos_real @ X )
% 0.33/0.66 != zero_zero_real )
% 0.33/0.66 => ( ( ( cos_real @ Y )
% 0.33/0.66 != zero_zero_real )
% 0.33/0.66 => ( ( ( cos_real @ ( plus_plus_real @ X @ Y ) )
% 0.33/0.66 != zero_zero_real )
% 0.33/0.66 => ( ( tan_real @ ( plus_plus_real @ X @ Y ) )
% 0.33/0.66 = ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_add
% 0.33/0.66 thf(fact_8263_tan__diff,axiom,
% 0.33/0.66 ! [X: complex,Y: complex] :
% 0.33/0.66 ( ( ( cos_complex @ X )
% 0.33/0.66 != zero_zero_complex )
% 0.33/0.66 => ( ( ( cos_complex @ Y )
% 0.33/0.66 != zero_zero_complex )
% 0.33/0.66 => ( ( ( cos_complex @ ( minus_minus_complex @ X @ Y ) )
% 0.33/0.66 != zero_zero_complex )
% 0.33/0.66 => ( ( tan_complex @ ( minus_minus_complex @ X @ Y ) )
% 0.33/0.66 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_diff
% 0.33/0.66 thf(fact_8264_tan__diff,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ( cos_real @ X )
% 0.33/0.66 != zero_zero_real )
% 0.33/0.66 => ( ( ( cos_real @ Y )
% 0.33/0.66 != zero_zero_real )
% 0.33/0.66 => ( ( ( cos_real @ ( minus_minus_real @ X @ Y ) )
% 0.33/0.66 != zero_zero_real )
% 0.33/0.66 => ( ( tan_real @ ( minus_minus_real @ X @ Y ) )
% 0.33/0.66 = ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_diff
% 0.33/0.66 thf(fact_8265_lemma__tan__add1,axiom,
% 0.33/0.66 ! [X: complex,Y: complex] :
% 0.33/0.66 ( ( ( cos_complex @ X )
% 0.33/0.66 != zero_zero_complex )
% 0.33/0.66 => ( ( ( cos_complex @ Y )
% 0.33/0.66 != zero_zero_complex )
% 0.33/0.66 => ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) )
% 0.33/0.66 = ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % lemma_tan_add1
% 0.33/0.66 thf(fact_8266_lemma__tan__add1,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ( cos_real @ X )
% 0.33/0.66 != zero_zero_real )
% 0.33/0.66 => ( ( ( cos_real @ Y )
% 0.33/0.66 != zero_zero_real )
% 0.33/0.66 => ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) )
% 0.33/0.66 = ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % lemma_tan_add1
% 0.33/0.66 thf(fact_8267_tan__total__pi4,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.66 => ? [Z5: real] :
% 0.33/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z5 )
% 0.33/0.66 & ( ord_less_real @ Z5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.33/0.66 & ( ( tan_real @ Z5 )
% 0.33/0.66 = X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_total_pi4
% 0.33/0.66 thf(fact_8268_tan__half,axiom,
% 0.33/0.66 ( tan_complex
% 0.33/0.66 = ( ^ [X2: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) ) @ one_one_complex ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_half
% 0.33/0.66 thf(fact_8269_tan__half,axiom,
% 0.33/0.66 ( tan_real
% 0.33/0.66 = ( ^ [X2: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) @ one_one_real ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % tan_half
% 0.33/0.66 thf(fact_8270_cos__coeff__def,axiom,
% 0.33/0.66 ( cos_coeff
% 0.33/0.66 = ( ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ zero_zero_real ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_coeff_def
% 0.33/0.66 thf(fact_8271_cos__paired,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( sums_real
% 0.33/0.66 @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( power_power_real @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.33/0.66 @ ( cos_real @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_paired
% 0.33/0.66 thf(fact_8272_Maclaurin__sin__expansion3,axiom,
% 0.33/0.66 ! [N2: nat,X: real] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.66 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.66 => ? [T5: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ T5 )
% 0.33/0.66 & ( ord_less_real @ T5 @ X )
% 0.33/0.66 & ( ( sin_real @ X )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.66 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_sin_expansion3
% 0.33/0.66 thf(fact_8273_Maclaurin__sin__expansion4,axiom,
% 0.33/0.66 ! [X: real,N2: nat] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.66 => ? [T5: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ T5 )
% 0.33/0.66 & ( ord_less_eq_real @ T5 @ X )
% 0.33/0.66 & ( ( sin_real @ X )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.66 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_sin_expansion4
% 0.33/0.66 thf(fact_8274_Maclaurin__sin__expansion2,axiom,
% 0.33/0.66 ! [X: real,N2: nat] :
% 0.33/0.66 ? [T5: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 0.33/0.66 & ( ( sin_real @ X )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.66 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_sin_expansion2
% 0.33/0.66 thf(fact_8275_Maclaurin__sin__expansion,axiom,
% 0.33/0.66 ! [X: real,N2: nat] :
% 0.33/0.66 ? [T5: real] :
% 0.33/0.66 ( ( sin_real @ X )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.66 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_sin_expansion
% 0.33/0.66 thf(fact_8276_sin__coeff__def,axiom,
% 0.33/0.66 ( sin_coeff
% 0.33/0.66 = ( ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_coeff_def
% 0.33/0.66 thf(fact_8277_fact__mono__nat,axiom,
% 0.33/0.66 ! [M: nat,N2: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.66 => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_mono_nat
% 0.33/0.66 thf(fact_8278_fact__ge__self,axiom,
% 0.33/0.66 ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_ge_self
% 0.33/0.66 thf(fact_8279_fact__less__mono__nat,axiom,
% 0.33/0.66 ! [M: nat,N2: nat] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.33/0.66 => ( ( ord_less_nat @ M @ N2 )
% 0.33/0.66 => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_less_mono_nat
% 0.33/0.66 thf(fact_8280_fact__ge__Suc__0__nat,axiom,
% 0.33/0.66 ! [N2: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_ge_Suc_0_nat
% 0.33/0.66 thf(fact_8281_dvd__fact,axiom,
% 0.33/0.66 ! [M: nat,N2: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ one_one_nat @ M )
% 0.33/0.66 => ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.66 => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % dvd_fact
% 0.33/0.66 thf(fact_8282_fact__diff__Suc,axiom,
% 0.33/0.66 ! [N2: nat,M: nat] :
% 0.33/0.66 ( ( ord_less_nat @ N2 @ ( suc @ M ) )
% 0.33/0.66 => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) )
% 0.33/0.66 = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N2 ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_diff_Suc
% 0.33/0.66 thf(fact_8283_fact__div__fact__le__pow,axiom,
% 0.33/0.66 ! [R: nat,N2: nat] :
% 0.33/0.66 ( ( ord_less_eq_nat @ R @ N2 )
% 0.33/0.66 => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ R ) ) ) @ ( power_power_nat @ N2 @ R ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % fact_div_fact_le_pow
% 0.33/0.66 thf(fact_8284_sin__coeff__Suc,axiom,
% 0.33/0.66 ! [N2: nat] :
% 0.33/0.66 ( ( sin_coeff @ ( suc @ N2 ) )
% 0.33/0.66 = ( divide_divide_real @ ( cos_coeff @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_coeff_Suc
% 0.33/0.66 thf(fact_8285_cos__coeff__Suc,axiom,
% 0.33/0.66 ! [N2: nat] :
% 0.33/0.66 ( ( cos_coeff @ ( suc @ N2 ) )
% 0.33/0.66 = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N2 ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_coeff_Suc
% 0.33/0.66 thf(fact_8286_sin__tan,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 => ( ( sin_real @ X )
% 0.33/0.66 = ( divide_divide_real @ ( tan_real @ X ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sin_tan
% 0.33/0.66 thf(fact_8287_cos__tan,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.66 => ( ( cos_real @ X )
% 0.33/0.66 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % cos_tan
% 0.33/0.66 thf(fact_8288_complex__unimodular__polar,axiom,
% 0.33/0.66 ! [Z: complex] :
% 0.33/0.66 ( ( ( real_V1022390504157884413omplex @ Z )
% 0.33/0.66 = one_one_real )
% 0.33/0.66 => ~ ! [T5: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 0.33/0.66 => ( ( ord_less_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.33/0.66 => ( Z
% 0.33/0.66 != ( complex2 @ ( cos_real @ T5 ) @ ( sin_real @ T5 ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % complex_unimodular_polar
% 0.33/0.66 thf(fact_8289_Maclaurin__exp__lt,axiom,
% 0.33/0.66 ! [X: real,N2: nat] :
% 0.33/0.66 ( ( X != zero_zero_real )
% 0.33/0.66 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.66 => ? [T5: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T5 ) )
% 0.33/0.66 & ( ord_less_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 0.33/0.66 & ( ( exp_real @ X )
% 0.33/0.66 = ( plus_plus_real
% 0.33/0.66 @ ( groups6591440286371151544t_real
% 0.33/0.66 @ ^ [M6: nat] : ( divide_divide_real @ ( power_power_real @ X @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) )
% 0.33/0.66 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.66 @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T5 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Maclaurin_exp_lt
% 0.33/0.66 thf(fact_8290_pochhammer__double,axiom,
% 0.33/0.66 ! [Z: rat,N2: nat] :
% 0.33/0.66 ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.33/0.66 = ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % pochhammer_double
% 0.33/0.66 thf(fact_8291_pochhammer__double,axiom,
% 0.33/0.66 ! [Z: real,N2: nat] :
% 0.33/0.66 ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.33/0.66 = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s7457072308508201937r_real @ Z @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % pochhammer_double
% 0.33/0.66 thf(fact_8292_pochhammer__double,axiom,
% 0.33/0.66 ! [Z: complex,N2: nat] :
% 0.33/0.66 ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.33/0.66 = ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s2602460028002588243omplex @ Z @ N2 ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % pochhammer_double
% 0.33/0.66 thf(fact_8293_real__sqrt__eq__iff,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ( sqrt @ X )
% 0.33/0.66 = ( sqrt @ Y ) )
% 0.33/0.66 = ( X = Y ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_eq_iff
% 0.33/0.66 thf(fact_8294_real__sqrt__eq__zero__cancel__iff,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ( sqrt @ X )
% 0.33/0.66 = zero_zero_real )
% 0.33/0.66 = ( X = zero_zero_real ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_eq_zero_cancel_iff
% 0.33/0.66 thf(fact_8295_real__sqrt__zero,axiom,
% 0.33/0.66 ( ( sqrt @ zero_zero_real )
% 0.33/0.66 = zero_zero_real ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_zero
% 0.33/0.66 thf(fact_8296_real__sqrt__less__iff,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
% 0.33/0.66 = ( ord_less_real @ X @ Y ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_less_iff
% 0.33/0.66 thf(fact_8297_real__sqrt__le__iff,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
% 0.33/0.66 = ( ord_less_eq_real @ X @ Y ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_le_iff
% 0.33/0.66 thf(fact_8298_real__sqrt__one,axiom,
% 0.33/0.66 ( ( sqrt @ one_one_real )
% 0.33/0.66 = one_one_real ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_one
% 0.33/0.66 thf(fact_8299_real__sqrt__eq__1__iff,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ( sqrt @ X )
% 0.33/0.66 = one_one_real )
% 0.33/0.66 = ( X = one_one_real ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_eq_1_iff
% 0.33/0.66 thf(fact_8300_exp__less__mono,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ord_less_real @ X @ Y )
% 0.33/0.66 => ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % exp_less_mono
% 0.33/0.66 thf(fact_8301_exp__less__cancel__iff,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
% 0.33/0.66 = ( ord_less_real @ X @ Y ) ) ).
% 0.33/0.66
% 0.33/0.66 % exp_less_cancel_iff
% 0.33/0.66 thf(fact_8302_exp__le__cancel__iff,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
% 0.33/0.66 = ( ord_less_eq_real @ X @ Y ) ) ).
% 0.33/0.66
% 0.33/0.66 % exp_le_cancel_iff
% 0.33/0.66 thf(fact_8303_exp__zero,axiom,
% 0.33/0.66 ( ( exp_complex @ zero_zero_complex )
% 0.33/0.66 = one_one_complex ) ).
% 0.33/0.66
% 0.33/0.66 % exp_zero
% 0.33/0.66 thf(fact_8304_exp__zero,axiom,
% 0.33/0.66 ( ( exp_real @ zero_zero_real )
% 0.33/0.66 = one_one_real ) ).
% 0.33/0.66
% 0.33/0.66 % exp_zero
% 0.33/0.66 thf(fact_8305_real__sqrt__lt__0__iff,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ ( sqrt @ X ) @ zero_zero_real )
% 0.33/0.66 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_lt_0_iff
% 0.33/0.66 thf(fact_8306_real__sqrt__gt__0__iff,axiom,
% 0.33/0.66 ! [Y: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y ) )
% 0.33/0.66 = ( ord_less_real @ zero_zero_real @ Y ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_gt_0_iff
% 0.33/0.66 thf(fact_8307_real__sqrt__ge__0__iff,axiom,
% 0.33/0.66 ! [Y: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y ) )
% 0.33/0.66 = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_ge_0_iff
% 0.33/0.66 thf(fact_8308_real__sqrt__le__0__iff,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( sqrt @ X ) @ zero_zero_real )
% 0.33/0.66 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_le_0_iff
% 0.33/0.66 thf(fact_8309_real__sqrt__gt__1__iff,axiom,
% 0.33/0.66 ! [Y: real] :
% 0.33/0.66 ( ( ord_less_real @ one_one_real @ ( sqrt @ Y ) )
% 0.33/0.66 = ( ord_less_real @ one_one_real @ Y ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_gt_1_iff
% 0.33/0.66 thf(fact_8310_real__sqrt__lt__1__iff,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ ( sqrt @ X ) @ one_one_real )
% 0.33/0.66 = ( ord_less_real @ X @ one_one_real ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_lt_1_iff
% 0.33/0.66 thf(fact_8311_real__sqrt__ge__1__iff,axiom,
% 0.33/0.66 ! [Y: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y ) )
% 0.33/0.66 = ( ord_less_eq_real @ one_one_real @ Y ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_ge_1_iff
% 0.33/0.66 thf(fact_8312_real__sqrt__le__1__iff,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( sqrt @ X ) @ one_one_real )
% 0.33/0.66 = ( ord_less_eq_real @ X @ one_one_real ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_le_1_iff
% 0.33/0.66 thf(fact_8313_pochhammer__0,axiom,
% 0.33/0.66 ! [A: complex] :
% 0.33/0.66 ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
% 0.33/0.66 = one_one_complex ) ).
% 0.33/0.66
% 0.33/0.66 % pochhammer_0
% 0.33/0.66 thf(fact_8314_pochhammer__0,axiom,
% 0.33/0.66 ! [A: real] :
% 0.33/0.66 ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
% 0.33/0.66 = one_one_real ) ).
% 0.33/0.66
% 0.33/0.66 % pochhammer_0
% 0.33/0.66 thf(fact_8315_pochhammer__0,axiom,
% 0.33/0.66 ! [A: rat] :
% 0.33/0.66 ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
% 0.33/0.66 = one_one_rat ) ).
% 0.33/0.66
% 0.33/0.66 % pochhammer_0
% 0.33/0.66 thf(fact_8316_pochhammer__0,axiom,
% 0.33/0.66 ! [A: nat] :
% 0.33/0.66 ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
% 0.33/0.66 = one_one_nat ) ).
% 0.33/0.66
% 0.33/0.66 % pochhammer_0
% 0.33/0.66 thf(fact_8317_pochhammer__0,axiom,
% 0.33/0.66 ! [A: int] :
% 0.33/0.66 ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
% 0.33/0.66 = one_one_int ) ).
% 0.33/0.66
% 0.33/0.66 % pochhammer_0
% 0.33/0.66 thf(fact_8318_exp__eq__one__iff,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ( exp_real @ X )
% 0.33/0.66 = one_one_real )
% 0.33/0.66 = ( X = zero_zero_real ) ) ).
% 0.33/0.66
% 0.33/0.66 % exp_eq_one_iff
% 0.33/0.66 thf(fact_8319_real__sqrt__mult__self,axiom,
% 0.33/0.66 ! [A: real] :
% 0.33/0.66 ( ( times_times_real @ ( sqrt @ A ) @ ( sqrt @ A ) )
% 0.33/0.66 = ( abs_abs_real @ A ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_mult_self
% 0.33/0.66 thf(fact_8320_real__sqrt__abs2,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( sqrt @ ( times_times_real @ X @ X ) )
% 0.33/0.66 = ( abs_abs_real @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_abs2
% 0.33/0.66 thf(fact_8321_real__sqrt__four,axiom,
% 0.33/0.66 ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.33/0.66 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_four
% 0.33/0.66 thf(fact_8322_exp__less__one__iff,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ ( exp_real @ X ) @ one_one_real )
% 0.33/0.66 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.33/0.66
% 0.33/0.66 % exp_less_one_iff
% 0.33/0.66 thf(fact_8323_one__less__exp__iff,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ one_one_real @ ( exp_real @ X ) )
% 0.33/0.66 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % one_less_exp_iff
% 0.33/0.66 thf(fact_8324_one__le__exp__iff,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X ) )
% 0.33/0.66 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % one_le_exp_iff
% 0.33/0.66 thf(fact_8325_exp__le__one__iff,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ ( exp_real @ X ) @ one_one_real )
% 0.33/0.66 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.33/0.66
% 0.33/0.66 % exp_le_one_iff
% 0.33/0.66 thf(fact_8326_exp__ln,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ( exp_real @ ( ln_ln_real @ X ) )
% 0.33/0.66 = X ) ) ).
% 0.33/0.66
% 0.33/0.66 % exp_ln
% 0.33/0.66 thf(fact_8327_exp__ln__iff,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ( exp_real @ ( ln_ln_real @ X ) )
% 0.33/0.66 = X )
% 0.33/0.66 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % exp_ln_iff
% 0.33/0.66 thf(fact_8328_norm__cos__sin,axiom,
% 0.33/0.66 ! [T: real] :
% 0.33/0.66 ( ( real_V1022390504157884413omplex @ ( complex2 @ ( cos_real @ T ) @ ( sin_real @ T ) ) )
% 0.33/0.66 = one_one_real ) ).
% 0.33/0.66
% 0.33/0.66 % norm_cos_sin
% 0.33/0.66 thf(fact_8329_real__sqrt__abs,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( sqrt @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.66 = ( abs_abs_real @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_abs
% 0.33/0.66 thf(fact_8330_real__sqrt__pow2__iff,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.66 = X )
% 0.33/0.66 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_pow2_iff
% 0.33/0.66 thf(fact_8331_real__sqrt__pow2,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.66 = X ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_pow2
% 0.33/0.66 thf(fact_8332_real__sqrt__sum__squares__mult__squared__eq,axiom,
% 0.33/0.66 ! [X: real,Y: real,Xa2: real,Ya: real] :
% 0.33/0.66 ( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.66 = ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_sum_squares_mult_squared_eq
% 0.33/0.66 thf(fact_8333_exp__times__arg__commute,axiom,
% 0.33/0.66 ! [A2: complex] :
% 0.33/0.66 ( ( times_times_complex @ ( exp_complex @ A2 ) @ A2 )
% 0.33/0.66 = ( times_times_complex @ A2 @ ( exp_complex @ A2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % exp_times_arg_commute
% 0.33/0.66 thf(fact_8334_exp__times__arg__commute,axiom,
% 0.33/0.66 ! [A2: real] :
% 0.33/0.66 ( ( times_times_real @ ( exp_real @ A2 ) @ A2 )
% 0.33/0.66 = ( times_times_real @ A2 @ ( exp_real @ A2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % exp_times_arg_commute
% 0.33/0.66 thf(fact_8335_real__sqrt__divide,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( sqrt @ ( divide_divide_real @ X @ Y ) )
% 0.33/0.66 = ( divide_divide_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_divide
% 0.33/0.66 thf(fact_8336_real__sqrt__mult,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( sqrt @ ( times_times_real @ X @ Y ) )
% 0.33/0.66 = ( times_times_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_mult
% 0.33/0.66 thf(fact_8337_norm__exp,axiom,
% 0.33/0.66 ! [X: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ X ) ) @ ( exp_real @ ( real_V7735802525324610683m_real @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % norm_exp
% 0.33/0.66 thf(fact_8338_norm__exp,axiom,
% 0.33/0.66 ! [X: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ X ) ) @ ( exp_real @ ( real_V1022390504157884413omplex @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % norm_exp
% 0.33/0.66 thf(fact_8339_real__sqrt__le__mono,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ X @ Y )
% 0.33/0.66 => ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_le_mono
% 0.33/0.66 thf(fact_8340_exp__less__cancel,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
% 0.33/0.66 => ( ord_less_real @ X @ Y ) ) ).
% 0.33/0.66
% 0.33/0.66 % exp_less_cancel
% 0.33/0.66 thf(fact_8341_real__sqrt__less__mono,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ord_less_real @ X @ Y )
% 0.33/0.66 => ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_less_mono
% 0.33/0.66 thf(fact_8342_real__sqrt__minus,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( sqrt @ ( uminus_uminus_real @ X ) )
% 0.33/0.66 = ( uminus_uminus_real @ ( sqrt @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_minus
% 0.33/0.66 thf(fact_8343_real__sqrt__power,axiom,
% 0.33/0.66 ! [X: real,K: nat] :
% 0.33/0.66 ( ( sqrt @ ( power_power_real @ X @ K ) )
% 0.33/0.66 = ( power_power_real @ ( sqrt @ X ) @ K ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_power
% 0.33/0.66 thf(fact_8344_complex__diff,axiom,
% 0.33/0.66 ! [A: real,B: real,C: real,D: real] :
% 0.33/0.66 ( ( minus_minus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
% 0.33/0.66 = ( complex2 @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % complex_diff
% 0.33/0.66 thf(fact_8345_real__sqrt__gt__zero,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ord_less_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_gt_zero
% 0.33/0.66 thf(fact_8346_not__exp__less__zero,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ~ ( ord_less_real @ ( exp_real @ X ) @ zero_zero_real ) ).
% 0.33/0.66
% 0.33/0.66 % not_exp_less_zero
% 0.33/0.66 thf(fact_8347_exp__gt__zero,axiom,
% 0.33/0.66 ! [X: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % exp_gt_zero
% 0.33/0.66 thf(fact_8348_exp__total,axiom,
% 0.33/0.66 ! [Y: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.66 => ? [X4: real] :
% 0.33/0.66 ( ( exp_real @ X4 )
% 0.33/0.66 = Y ) ) ).
% 0.33/0.66
% 0.33/0.66 % exp_total
% 0.33/0.66 thf(fact_8349_real__sqrt__ge__zero,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_ge_zero
% 0.33/0.66 thf(fact_8350_real__sqrt__eq__zero__cancel,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ( ( sqrt @ X )
% 0.33/0.66 = zero_zero_real )
% 0.33/0.66 => ( X = zero_zero_real ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_eq_zero_cancel
% 0.33/0.66 thf(fact_8351_not__exp__le__zero,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ~ ( ord_less_eq_real @ ( exp_real @ X ) @ zero_zero_real ) ).
% 0.33/0.66
% 0.33/0.66 % not_exp_le_zero
% 0.33/0.66 thf(fact_8352_exp__ge__zero,axiom,
% 0.33/0.66 ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % exp_ge_zero
% 0.33/0.66 thf(fact_8353_real__sqrt__ge__one,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ one_one_real @ X )
% 0.33/0.66 => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_sqrt_ge_one
% 0.33/0.66 thf(fact_8354_exp__add__commuting,axiom,
% 0.33/0.66 ! [X: complex,Y: complex] :
% 0.33/0.66 ( ( ( times_times_complex @ X @ Y )
% 0.33/0.66 = ( times_times_complex @ Y @ X ) )
% 0.33/0.66 => ( ( exp_complex @ ( plus_plus_complex @ X @ Y ) )
% 0.33/0.66 = ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % exp_add_commuting
% 0.33/0.66 thf(fact_8355_exp__add__commuting,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ( times_times_real @ X @ Y )
% 0.33/0.66 = ( times_times_real @ Y @ X ) )
% 0.33/0.66 => ( ( exp_real @ ( plus_plus_real @ X @ Y ) )
% 0.33/0.66 = ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % exp_add_commuting
% 0.33/0.66 thf(fact_8356_mult__exp__exp,axiom,
% 0.33/0.66 ! [X: complex,Y: complex] :
% 0.33/0.66 ( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) )
% 0.33/0.66 = ( exp_complex @ ( plus_plus_complex @ X @ Y ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % mult_exp_exp
% 0.33/0.66 thf(fact_8357_mult__exp__exp,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
% 0.33/0.66 = ( exp_real @ ( plus_plus_real @ X @ Y ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % mult_exp_exp
% 0.33/0.66 thf(fact_8358_exp__diff,axiom,
% 0.33/0.66 ! [X: complex,Y: complex] :
% 0.33/0.66 ( ( exp_complex @ ( minus_minus_complex @ X @ Y ) )
% 0.33/0.66 = ( divide1717551699836669952omplex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % exp_diff
% 0.33/0.66 thf(fact_8359_exp__diff,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( exp_real @ ( minus_minus_real @ X @ Y ) )
% 0.33/0.66 = ( divide_divide_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % exp_diff
% 0.33/0.66 thf(fact_8360_Complex__eq__numeral,axiom,
% 0.33/0.66 ! [A: real,B: real,W: num] :
% 0.33/0.66 ( ( ( complex2 @ A @ B )
% 0.33/0.66 = ( numera6690914467698888265omplex @ W ) )
% 0.33/0.66 = ( ( A
% 0.33/0.66 = ( numeral_numeral_real @ W ) )
% 0.33/0.66 & ( B = zero_zero_real ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Complex_eq_numeral
% 0.33/0.66 thf(fact_8361_pochhammer__pos,axiom,
% 0.33/0.66 ! [X: real,N2: nat] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % pochhammer_pos
% 0.33/0.66 thf(fact_8362_pochhammer__pos,axiom,
% 0.33/0.66 ! [X: rat,N2: nat] :
% 0.33/0.66 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.33/0.66 => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % pochhammer_pos
% 0.33/0.66 thf(fact_8363_pochhammer__pos,axiom,
% 0.33/0.66 ! [X: nat,N2: nat] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.33/0.66 => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % pochhammer_pos
% 0.33/0.66 thf(fact_8364_pochhammer__pos,axiom,
% 0.33/0.66 ! [X: int,N2: nat] :
% 0.33/0.66 ( ( ord_less_int @ zero_zero_int @ X )
% 0.33/0.66 => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % pochhammer_pos
% 0.33/0.66 thf(fact_8365_pochhammer__neq__0__mono,axiom,
% 0.33/0.66 ! [A: complex,M: nat,N2: nat] :
% 0.33/0.66 ( ( ( comm_s2602460028002588243omplex @ A @ M )
% 0.33/0.66 != zero_zero_complex )
% 0.33/0.66 => ( ( ord_less_eq_nat @ N2 @ M )
% 0.33/0.66 => ( ( comm_s2602460028002588243omplex @ A @ N2 )
% 0.33/0.66 != zero_zero_complex ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % pochhammer_neq_0_mono
% 0.33/0.66 thf(fact_8366_pochhammer__neq__0__mono,axiom,
% 0.33/0.66 ! [A: real,M: nat,N2: nat] :
% 0.33/0.66 ( ( ( comm_s7457072308508201937r_real @ A @ M )
% 0.33/0.66 != zero_zero_real )
% 0.33/0.66 => ( ( ord_less_eq_nat @ N2 @ M )
% 0.33/0.66 => ( ( comm_s7457072308508201937r_real @ A @ N2 )
% 0.33/0.66 != zero_zero_real ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % pochhammer_neq_0_mono
% 0.33/0.66 thf(fact_8367_pochhammer__neq__0__mono,axiom,
% 0.33/0.66 ! [A: rat,M: nat,N2: nat] :
% 0.33/0.66 ( ( ( comm_s4028243227959126397er_rat @ A @ M )
% 0.33/0.66 != zero_zero_rat )
% 0.33/0.66 => ( ( ord_less_eq_nat @ N2 @ M )
% 0.33/0.66 => ( ( comm_s4028243227959126397er_rat @ A @ N2 )
% 0.33/0.66 != zero_zero_rat ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % pochhammer_neq_0_mono
% 0.33/0.66 thf(fact_8368_pochhammer__eq__0__mono,axiom,
% 0.33/0.66 ! [A: complex,N2: nat,M: nat] :
% 0.33/0.66 ( ( ( comm_s2602460028002588243omplex @ A @ N2 )
% 0.33/0.66 = zero_zero_complex )
% 0.33/0.66 => ( ( ord_less_eq_nat @ N2 @ M )
% 0.33/0.66 => ( ( comm_s2602460028002588243omplex @ A @ M )
% 0.33/0.66 = zero_zero_complex ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % pochhammer_eq_0_mono
% 0.33/0.66 thf(fact_8369_pochhammer__eq__0__mono,axiom,
% 0.33/0.66 ! [A: real,N2: nat,M: nat] :
% 0.33/0.66 ( ( ( comm_s7457072308508201937r_real @ A @ N2 )
% 0.33/0.66 = zero_zero_real )
% 0.33/0.66 => ( ( ord_less_eq_nat @ N2 @ M )
% 0.33/0.66 => ( ( comm_s7457072308508201937r_real @ A @ M )
% 0.33/0.66 = zero_zero_real ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % pochhammer_eq_0_mono
% 0.33/0.66 thf(fact_8370_pochhammer__eq__0__mono,axiom,
% 0.33/0.66 ! [A: rat,N2: nat,M: nat] :
% 0.33/0.66 ( ( ( comm_s4028243227959126397er_rat @ A @ N2 )
% 0.33/0.66 = zero_zero_rat )
% 0.33/0.66 => ( ( ord_less_eq_nat @ N2 @ M )
% 0.33/0.66 => ( ( comm_s4028243227959126397er_rat @ A @ M )
% 0.33/0.66 = zero_zero_rat ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % pochhammer_eq_0_mono
% 0.33/0.66 thf(fact_8371_pochhammer__fact,axiom,
% 0.33/0.66 ( semiri5044797733671781792omplex
% 0.33/0.66 = ( comm_s2602460028002588243omplex @ one_one_complex ) ) ).
% 0.33/0.66
% 0.33/0.66 % pochhammer_fact
% 0.33/0.66 thf(fact_8372_pochhammer__fact,axiom,
% 0.33/0.66 ( semiri773545260158071498ct_rat
% 0.33/0.66 = ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).
% 0.33/0.66
% 0.33/0.66 % pochhammer_fact
% 0.33/0.66 thf(fact_8373_pochhammer__fact,axiom,
% 0.33/0.66 ( semiri1406184849735516958ct_int
% 0.33/0.66 = ( comm_s4660882817536571857er_int @ one_one_int ) ) ).
% 0.33/0.66
% 0.33/0.66 % pochhammer_fact
% 0.33/0.66 thf(fact_8374_pochhammer__fact,axiom,
% 0.33/0.66 ( semiri2265585572941072030t_real
% 0.33/0.66 = ( comm_s7457072308508201937r_real @ one_one_real ) ) ).
% 0.33/0.66
% 0.33/0.66 % pochhammer_fact
% 0.33/0.66 thf(fact_8375_pochhammer__fact,axiom,
% 0.33/0.66 ( semiri1408675320244567234ct_nat
% 0.33/0.66 = ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).
% 0.33/0.66
% 0.33/0.66 % pochhammer_fact
% 0.33/0.66 thf(fact_8376_complex__add,axiom,
% 0.33/0.66 ! [A: real,B: real,C: real,D: real] :
% 0.33/0.66 ( ( plus_plus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
% 0.33/0.66 = ( complex2 @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % complex_add
% 0.33/0.66 thf(fact_8377_complex__norm,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( real_V1022390504157884413omplex @ ( complex2 @ X @ Y ) )
% 0.33/0.66 = ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % complex_norm
% 0.33/0.66 thf(fact_8378_exp__gt__one,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ord_less_real @ one_one_real @ ( exp_real @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % exp_gt_one
% 0.33/0.66 thf(fact_8379_real__div__sqrt,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ( divide_divide_real @ X @ ( sqrt @ X ) )
% 0.33/0.66 = ( sqrt @ X ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % real_div_sqrt
% 0.33/0.66 thf(fact_8380_sqrt__add__le__add__sqrt,axiom,
% 0.33/0.66 ! [X: real,Y: real] :
% 0.33/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.66 => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % sqrt_add_le_add_sqrt
% 0.33/0.66 thf(fact_8381_exp__ge__add__one__self,axiom,
% 0.33/0.66 ! [X: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ).
% 0.33/0.66
% 0.33/0.66 % exp_ge_add_one_self
% 0.33/0.66 thf(fact_8382_le__real__sqrt__sumsq,axiom,
% 0.33/0.66 ! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % le_real_sqrt_sumsq
% 0.33/0.66 thf(fact_8383_exp__minus__inverse,axiom,
% 0.33/0.66 ! [X: real] :
% 0.33/0.66 ( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) )
% 0.33/0.66 = one_one_real ) ).
% 0.33/0.66
% 0.33/0.66 % exp_minus_inverse
% 0.33/0.66 thf(fact_8384_exp__minus__inverse,axiom,
% 0.33/0.66 ! [X: complex] :
% 0.33/0.66 ( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) )
% 0.33/0.66 = one_one_complex ) ).
% 0.33/0.66
% 0.33/0.66 % exp_minus_inverse
% 0.33/0.66 thf(fact_8385_exp__of__nat__mult,axiom,
% 0.33/0.66 ! [N2: nat,X: real] :
% 0.33/0.66 ( ( exp_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) )
% 0.33/0.66 = ( power_power_real @ ( exp_real @ X ) @ N2 ) ) ).
% 0.33/0.66
% 0.33/0.66 % exp_of_nat_mult
% 0.33/0.66 thf(fact_8386_exp__of__nat__mult,axiom,
% 0.33/0.66 ! [N2: nat,X: complex] :
% 0.33/0.66 ( ( exp_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ X ) )
% 0.33/0.66 = ( power_power_complex @ ( exp_complex @ X ) @ N2 ) ) ).
% 0.33/0.66
% 0.33/0.66 % exp_of_nat_mult
% 0.33/0.66 thf(fact_8387_exp__of__nat2__mult,axiom,
% 0.33/0.66 ! [X: real,N2: nat] :
% 0.33/0.66 ( ( exp_real @ ( times_times_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 0.33/0.66 = ( power_power_real @ ( exp_real @ X ) @ N2 ) ) ).
% 0.33/0.66
% 0.33/0.66 % exp_of_nat2_mult
% 0.33/0.66 thf(fact_8388_exp__of__nat2__mult,axiom,
% 0.33/0.66 ! [X: complex,N2: nat] :
% 0.33/0.66 ( ( exp_complex @ ( times_times_complex @ X @ ( semiri8010041392384452111omplex @ N2 ) ) )
% 0.33/0.66 = ( power_power_complex @ ( exp_complex @ X ) @ N2 ) ) ).
% 0.33/0.66
% 0.33/0.66 % exp_of_nat2_mult
% 0.33/0.66 thf(fact_8389_Complex__eq__neg__numeral,axiom,
% 0.33/0.66 ! [A: real,B: real,W: num] :
% 0.33/0.66 ( ( ( complex2 @ A @ B )
% 0.33/0.66 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.33/0.66 = ( ( A
% 0.33/0.66 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.33/0.66 & ( B = zero_zero_real ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % Complex_eq_neg_numeral
% 0.33/0.66 thf(fact_8390_pochhammer__nonneg,axiom,
% 0.33/0.66 ! [X: real,N2: nat] :
% 0.33/0.66 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.66 => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % pochhammer_nonneg
% 0.33/0.66 thf(fact_8391_pochhammer__nonneg,axiom,
% 0.33/0.66 ! [X: rat,N2: nat] :
% 0.33/0.66 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.33/0.66 => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N2 ) ) ) ).
% 0.33/0.66
% 0.33/0.66 % pochhammer_nonneg
% 0.33/0.66 thf(fact_8392_pochhammer__nonneg,axiom,
% 0.33/0.66 ! [X: nat,N2: nat] :
% 0.33/0.66 ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.33/0.66 => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_nonneg
% 0.33/0.67 thf(fact_8393_pochhammer__nonneg,axiom,
% 0.33/0.67 ! [X: int,N2: nat] :
% 0.33/0.67 ( ( ord_less_int @ zero_zero_int @ X )
% 0.33/0.67 => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_nonneg
% 0.33/0.67 thf(fact_8394_pochhammer__0__left,axiom,
% 0.33/0.67 ! [N2: nat] :
% 0.33/0.67 ( ( ( N2 = zero_zero_nat )
% 0.33/0.67 => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N2 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 & ( ( N2 != zero_zero_nat )
% 0.33/0.67 => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N2 )
% 0.33/0.67 = zero_zero_complex ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_0_left
% 0.33/0.67 thf(fact_8395_pochhammer__0__left,axiom,
% 0.33/0.67 ! [N2: nat] :
% 0.33/0.67 ( ( ( N2 = zero_zero_nat )
% 0.33/0.67 => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N2 )
% 0.33/0.67 = one_one_real ) )
% 0.33/0.67 & ( ( N2 != zero_zero_nat )
% 0.33/0.67 => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N2 )
% 0.33/0.67 = zero_zero_real ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_0_left
% 0.33/0.67 thf(fact_8396_pochhammer__0__left,axiom,
% 0.33/0.67 ! [N2: nat] :
% 0.33/0.67 ( ( ( N2 = zero_zero_nat )
% 0.33/0.67 => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N2 )
% 0.33/0.67 = one_one_rat ) )
% 0.33/0.67 & ( ( N2 != zero_zero_nat )
% 0.33/0.67 => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N2 )
% 0.33/0.67 = zero_zero_rat ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_0_left
% 0.33/0.67 thf(fact_8397_pochhammer__0__left,axiom,
% 0.33/0.67 ! [N2: nat] :
% 0.33/0.67 ( ( ( N2 = zero_zero_nat )
% 0.33/0.67 => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N2 )
% 0.33/0.67 = one_one_nat ) )
% 0.33/0.67 & ( ( N2 != zero_zero_nat )
% 0.33/0.67 => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N2 )
% 0.33/0.67 = zero_zero_nat ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_0_left
% 0.33/0.67 thf(fact_8398_pochhammer__0__left,axiom,
% 0.33/0.67 ! [N2: nat] :
% 0.33/0.67 ( ( ( N2 = zero_zero_nat )
% 0.33/0.67 => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N2 )
% 0.33/0.67 = one_one_int ) )
% 0.33/0.67 & ( ( N2 != zero_zero_nat )
% 0.33/0.67 => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N2 )
% 0.33/0.67 = zero_zero_int ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_0_left
% 0.33/0.67 thf(fact_8399_complex__mult,axiom,
% 0.33/0.67 ! [A: real,B: real,C: real,D: real] :
% 0.33/0.67 ( ( times_times_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
% 0.33/0.67 = ( complex2 @ ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % complex_mult
% 0.33/0.67 thf(fact_8400_Complex__eq__1,axiom,
% 0.33/0.67 ! [A: real,B: real] :
% 0.33/0.67 ( ( ( complex2 @ A @ B )
% 0.33/0.67 = one_one_complex )
% 0.33/0.67 = ( ( A = one_one_real )
% 0.33/0.67 & ( B = zero_zero_real ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Complex_eq_1
% 0.33/0.67 thf(fact_8401_one__complex_Ocode,axiom,
% 0.33/0.67 ( one_one_complex
% 0.33/0.67 = ( complex2 @ one_one_real @ zero_zero_real ) ) ).
% 0.33/0.67
% 0.33/0.67 % one_complex.code
% 0.33/0.67 thf(fact_8402_sqrt2__less__2,axiom,
% 0.33/0.67 ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.33/0.67
% 0.33/0.67 % sqrt2_less_2
% 0.33/0.67 thf(fact_8403_exp__ge__add__one__self__aux,axiom,
% 0.33/0.67 ! [X: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.67 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % exp_ge_add_one_self_aux
% 0.33/0.67 thf(fact_8404_lemma__exp__total,axiom,
% 0.33/0.67 ! [Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ one_one_real @ Y )
% 0.33/0.67 => ? [X4: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.33/0.67 & ( ord_less_eq_real @ X4 @ ( minus_minus_real @ Y @ one_one_real ) )
% 0.33/0.67 & ( ( exp_real @ X4 )
% 0.33/0.67 = Y ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % lemma_exp_total
% 0.33/0.67 thf(fact_8405_ln__ge__iff,axiom,
% 0.33/0.67 ! [X: real,Y: real] :
% 0.33/0.67 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.67 => ( ( ord_less_eq_real @ Y @ ( ln_ln_real @ X ) )
% 0.33/0.67 = ( ord_less_eq_real @ ( exp_real @ Y ) @ X ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % ln_ge_iff
% 0.33/0.67 thf(fact_8406_ln__x__over__x__mono,axiom,
% 0.33/0.67 ! [X: real,Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X )
% 0.33/0.67 => ( ( ord_less_eq_real @ X @ Y )
% 0.33/0.67 => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y ) @ Y ) @ ( divide_divide_real @ ( ln_ln_real @ X ) @ X ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % ln_x_over_x_mono
% 0.33/0.67 thf(fact_8407_Complex__sum_H,axiom,
% 0.33/0.67 ! [F: nat > real,S3: set_nat] :
% 0.33/0.67 ( ( groups2073611262835488442omplex
% 0.33/0.67 @ ^ [X2: nat] : ( complex2 @ ( F @ X2 ) @ zero_zero_real )
% 0.33/0.67 @ S3 )
% 0.33/0.67 = ( complex2 @ ( groups6591440286371151544t_real @ F @ S3 ) @ zero_zero_real ) ) ).
% 0.33/0.67
% 0.33/0.67 % Complex_sum'
% 0.33/0.67 thf(fact_8408_Complex__sum_H,axiom,
% 0.33/0.67 ! [F: complex > real,S3: set_complex] :
% 0.33/0.67 ( ( groups7754918857620584856omplex
% 0.33/0.67 @ ^ [X2: complex] : ( complex2 @ ( F @ X2 ) @ zero_zero_real )
% 0.33/0.67 @ S3 )
% 0.33/0.67 = ( complex2 @ ( groups5808333547571424918x_real @ F @ S3 ) @ zero_zero_real ) ) ).
% 0.33/0.67
% 0.33/0.67 % Complex_sum'
% 0.33/0.67 thf(fact_8409_pochhammer__rec,axiom,
% 0.33/0.67 ! [A: complex,N2: nat] :
% 0.33/0.67 ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N2 ) )
% 0.33/0.67 = ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_rec
% 0.33/0.67 thf(fact_8410_pochhammer__rec,axiom,
% 0.33/0.67 ! [A: real,N2: nat] :
% 0.33/0.67 ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
% 0.33/0.67 = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_rec
% 0.33/0.67 thf(fact_8411_pochhammer__rec,axiom,
% 0.33/0.67 ! [A: rat,N2: nat] :
% 0.33/0.67 ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
% 0.33/0.67 = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_rec
% 0.33/0.67 thf(fact_8412_pochhammer__rec,axiom,
% 0.33/0.67 ! [A: nat,N2: nat] :
% 0.33/0.67 ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
% 0.33/0.67 = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_rec
% 0.33/0.67 thf(fact_8413_pochhammer__rec,axiom,
% 0.33/0.67 ! [A: int,N2: nat] :
% 0.33/0.67 ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
% 0.33/0.67 = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_rec
% 0.33/0.67 thf(fact_8414_pochhammer__rec_H,axiom,
% 0.33/0.67 ! [Z: rat,N2: nat] :
% 0.33/0.67 ( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N2 ) )
% 0.33/0.67 = ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_rec'
% 0.33/0.67 thf(fact_8415_pochhammer__rec_H,axiom,
% 0.33/0.67 ! [Z: int,N2: nat] :
% 0.33/0.67 ( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N2 ) )
% 0.33/0.67 = ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( comm_s4660882817536571857er_int @ Z @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_rec'
% 0.33/0.67 thf(fact_8416_pochhammer__rec_H,axiom,
% 0.33/0.67 ! [Z: real,N2: nat] :
% 0.33/0.67 ( ( comm_s7457072308508201937r_real @ Z @ ( suc @ N2 ) )
% 0.33/0.67 = ( times_times_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N2 ) ) @ ( comm_s7457072308508201937r_real @ Z @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_rec'
% 0.33/0.67 thf(fact_8417_pochhammer__rec_H,axiom,
% 0.33/0.67 ! [Z: nat,N2: nat] :
% 0.33/0.67 ( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N2 ) )
% 0.33/0.67 = ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N2 ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_rec'
% 0.33/0.67 thf(fact_8418_pochhammer__rec_H,axiom,
% 0.33/0.67 ! [Z: complex,N2: nat] :
% 0.33/0.67 ( ( comm_s2602460028002588243omplex @ Z @ ( suc @ N2 ) )
% 0.33/0.67 = ( times_times_complex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N2 ) ) @ ( comm_s2602460028002588243omplex @ Z @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_rec'
% 0.33/0.67 thf(fact_8419_pochhammer__Suc,axiom,
% 0.33/0.67 ! [A: rat,N2: nat] :
% 0.33/0.67 ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
% 0.33/0.67 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N2 ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_Suc
% 0.33/0.67 thf(fact_8420_pochhammer__Suc,axiom,
% 0.33/0.67 ! [A: int,N2: nat] :
% 0.33/0.67 ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
% 0.33/0.67 = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N2 ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_Suc
% 0.33/0.67 thf(fact_8421_pochhammer__Suc,axiom,
% 0.33/0.67 ! [A: real,N2: nat] :
% 0.33/0.67 ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
% 0.33/0.67 = ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N2 ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_Suc
% 0.33/0.67 thf(fact_8422_pochhammer__Suc,axiom,
% 0.33/0.67 ! [A: nat,N2: nat] :
% 0.33/0.67 ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
% 0.33/0.67 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N2 ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_Suc
% 0.33/0.67 thf(fact_8423_pochhammer__Suc,axiom,
% 0.33/0.67 ! [A: complex,N2: nat] :
% 0.33/0.67 ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N2 ) )
% 0.33/0.67 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ A @ N2 ) @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N2 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_Suc
% 0.33/0.67 thf(fact_8424_pochhammer__eq__0__iff,axiom,
% 0.33/0.67 ! [A: rat,N2: nat] :
% 0.33/0.67 ( ( ( comm_s4028243227959126397er_rat @ A @ N2 )
% 0.33/0.67 = zero_zero_rat )
% 0.33/0.67 = ( ? [K3: nat] :
% 0.33/0.67 ( ( ord_less_nat @ K3 @ N2 )
% 0.33/0.67 & ( A
% 0.33/0.67 = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K3 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_eq_0_iff
% 0.33/0.67 thf(fact_8425_pochhammer__eq__0__iff,axiom,
% 0.33/0.67 ! [A: real,N2: nat] :
% 0.33/0.67 ( ( ( comm_s7457072308508201937r_real @ A @ N2 )
% 0.33/0.67 = zero_zero_real )
% 0.33/0.67 = ( ? [K3: nat] :
% 0.33/0.67 ( ( ord_less_nat @ K3 @ N2 )
% 0.33/0.67 & ( A
% 0.33/0.67 = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K3 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_eq_0_iff
% 0.33/0.67 thf(fact_8426_pochhammer__eq__0__iff,axiom,
% 0.33/0.67 ! [A: complex,N2: nat] :
% 0.33/0.67 ( ( ( comm_s2602460028002588243omplex @ A @ N2 )
% 0.33/0.67 = zero_zero_complex )
% 0.33/0.67 = ( ? [K3: nat] :
% 0.33/0.67 ( ( ord_less_nat @ K3 @ N2 )
% 0.33/0.67 & ( A
% 0.33/0.67 = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K3 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_eq_0_iff
% 0.33/0.67 thf(fact_8427_pochhammer__of__nat__eq__0__iff,axiom,
% 0.33/0.67 ! [N2: nat,K: nat] :
% 0.33/0.67 ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K )
% 0.33/0.67 = zero_zero_rat )
% 0.33/0.67 = ( ord_less_nat @ N2 @ K ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_of_nat_eq_0_iff
% 0.33/0.67 thf(fact_8428_pochhammer__of__nat__eq__0__iff,axiom,
% 0.33/0.67 ! [N2: nat,K: nat] :
% 0.33/0.67 ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K )
% 0.33/0.67 = zero_z3403309356797280102nteger )
% 0.33/0.67 = ( ord_less_nat @ N2 @ K ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_of_nat_eq_0_iff
% 0.33/0.67 thf(fact_8429_pochhammer__of__nat__eq__0__iff,axiom,
% 0.33/0.67 ! [N2: nat,K: nat] :
% 0.33/0.67 ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
% 0.33/0.67 = zero_zero_int )
% 0.33/0.67 = ( ord_less_nat @ N2 @ K ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_of_nat_eq_0_iff
% 0.33/0.67 thf(fact_8430_pochhammer__of__nat__eq__0__iff,axiom,
% 0.33/0.67 ! [N2: nat,K: nat] :
% 0.33/0.67 ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ K )
% 0.33/0.67 = zero_zero_real )
% 0.33/0.67 = ( ord_less_nat @ N2 @ K ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_of_nat_eq_0_iff
% 0.33/0.67 thf(fact_8431_pochhammer__of__nat__eq__0__iff,axiom,
% 0.33/0.67 ! [N2: nat,K: nat] :
% 0.33/0.67 ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ K )
% 0.33/0.67 = zero_zero_complex )
% 0.33/0.67 = ( ord_less_nat @ N2 @ K ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_of_nat_eq_0_iff
% 0.33/0.67 thf(fact_8432_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.33/0.67 ! [N2: nat,K: nat] :
% 0.33/0.67 ( ( ord_less_nat @ N2 @ K )
% 0.33/0.67 => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K )
% 0.33/0.67 = zero_zero_rat ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_of_nat_eq_0_lemma
% 0.33/0.67 thf(fact_8433_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.33/0.67 ! [N2: nat,K: nat] :
% 0.33/0.67 ( ( ord_less_nat @ N2 @ K )
% 0.33/0.67 => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K )
% 0.33/0.67 = zero_z3403309356797280102nteger ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_of_nat_eq_0_lemma
% 0.33/0.67 thf(fact_8434_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.33/0.67 ! [N2: nat,K: nat] :
% 0.33/0.67 ( ( ord_less_nat @ N2 @ K )
% 0.33/0.67 => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
% 0.33/0.67 = zero_zero_int ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_of_nat_eq_0_lemma
% 0.33/0.67 thf(fact_8435_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.33/0.67 ! [N2: nat,K: nat] :
% 0.33/0.67 ( ( ord_less_nat @ N2 @ K )
% 0.33/0.67 => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ K )
% 0.33/0.67 = zero_zero_real ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_of_nat_eq_0_lemma
% 0.33/0.67 thf(fact_8436_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.33/0.67 ! [N2: nat,K: nat] :
% 0.33/0.67 ( ( ord_less_nat @ N2 @ K )
% 0.33/0.67 => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ K )
% 0.33/0.67 = zero_zero_complex ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_of_nat_eq_0_lemma
% 0.33/0.67 thf(fact_8437_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.33/0.67 ! [K: nat,N2: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ K @ N2 )
% 0.33/0.67 => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K )
% 0.33/0.67 != zero_zero_rat ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_of_nat_eq_0_lemma'
% 0.33/0.67 thf(fact_8438_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.33/0.67 ! [K: nat,N2: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ K @ N2 )
% 0.33/0.67 => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K )
% 0.33/0.67 != zero_z3403309356797280102nteger ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_of_nat_eq_0_lemma'
% 0.33/0.67 thf(fact_8439_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.33/0.67 ! [K: nat,N2: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ K @ N2 )
% 0.33/0.67 => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
% 0.33/0.67 != zero_zero_int ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_of_nat_eq_0_lemma'
% 0.33/0.67 thf(fact_8440_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.33/0.67 ! [K: nat,N2: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ K @ N2 )
% 0.33/0.67 => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ K )
% 0.33/0.67 != zero_zero_real ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_of_nat_eq_0_lemma'
% 0.33/0.67 thf(fact_8441_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.33/0.67 ! [K: nat,N2: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ K @ N2 )
% 0.33/0.67 => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ K )
% 0.33/0.67 != zero_zero_complex ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_of_nat_eq_0_lemma'
% 0.33/0.67 thf(fact_8442_pochhammer__product_H,axiom,
% 0.33/0.67 ! [Z: rat,N2: nat,M: nat] :
% 0.33/0.67 ( ( comm_s4028243227959126397er_rat @ Z @ ( plus_plus_nat @ N2 @ M ) )
% 0.33/0.67 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ N2 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N2 ) ) @ M ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_product'
% 0.33/0.67 thf(fact_8443_pochhammer__product_H,axiom,
% 0.33/0.67 ! [Z: int,N2: nat,M: nat] :
% 0.33/0.67 ( ( comm_s4660882817536571857er_int @ Z @ ( plus_plus_nat @ N2 @ M ) )
% 0.33/0.67 = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ N2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N2 ) ) @ M ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_product'
% 0.33/0.67 thf(fact_8444_pochhammer__product_H,axiom,
% 0.33/0.67 ! [Z: real,N2: nat,M: nat] :
% 0.33/0.67 ( ( comm_s7457072308508201937r_real @ Z @ ( plus_plus_nat @ N2 @ M ) )
% 0.33/0.67 = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ N2 ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N2 ) ) @ M ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_product'
% 0.33/0.67 thf(fact_8445_pochhammer__product_H,axiom,
% 0.33/0.67 ! [Z: nat,N2: nat,M: nat] :
% 0.33/0.67 ( ( comm_s4663373288045622133er_nat @ Z @ ( plus_plus_nat @ N2 @ M ) )
% 0.33/0.67 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ N2 ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N2 ) ) @ M ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_product'
% 0.33/0.67 thf(fact_8446_pochhammer__product_H,axiom,
% 0.33/0.67 ! [Z: complex,N2: nat,M: nat] :
% 0.33/0.67 ( ( comm_s2602460028002588243omplex @ Z @ ( plus_plus_nat @ N2 @ M ) )
% 0.33/0.67 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ N2 ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N2 ) ) @ M ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_product'
% 0.33/0.67 thf(fact_8447_Complex__eq__neg__1,axiom,
% 0.33/0.67 ! [A: real,B: real] :
% 0.33/0.67 ( ( ( complex2 @ A @ B )
% 0.33/0.67 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.33/0.67 = ( ( A
% 0.33/0.67 = ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.67 & ( B = zero_zero_real ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Complex_eq_neg_1
% 0.33/0.67 thf(fact_8448_real__less__rsqrt,axiom,
% 0.33/0.67 ! [X: real,Y: real] :
% 0.33/0.67 ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
% 0.33/0.67 => ( ord_less_real @ X @ ( sqrt @ Y ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % real_less_rsqrt
% 0.33/0.67 thf(fact_8449_real__le__rsqrt,axiom,
% 0.33/0.67 ! [X: real,Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
% 0.33/0.67 => ( ord_less_eq_real @ X @ ( sqrt @ Y ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % real_le_rsqrt
% 0.33/0.67 thf(fact_8450_sqrt__le__D,axiom,
% 0.33/0.67 ! [X: real,Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( sqrt @ X ) @ Y )
% 0.33/0.67 => ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % sqrt_le_D
% 0.33/0.67 thf(fact_8451_exp__le,axiom,
% 0.33/0.67 ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).
% 0.33/0.67
% 0.33/0.67 % exp_le
% 0.33/0.67 thf(fact_8452_exp__divide__power__eq,axiom,
% 0.33/0.67 ! [N2: nat,X: real] :
% 0.33/0.67 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.67 => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 )
% 0.33/0.67 = ( exp_real @ X ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % exp_divide_power_eq
% 0.33/0.67 thf(fact_8453_exp__divide__power__eq,axiom,
% 0.33/0.67 ! [N2: nat,X: complex] :
% 0.33/0.67 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.67 => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X @ ( semiri8010041392384452111omplex @ N2 ) ) ) @ N2 )
% 0.33/0.67 = ( exp_complex @ X ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % exp_divide_power_eq
% 0.33/0.67 thf(fact_8454_tanh__altdef,axiom,
% 0.33/0.67 ( tanh_real
% 0.33/0.67 = ( ^ [X2: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X2 ) @ ( exp_real @ ( uminus_uminus_real @ X2 ) ) ) @ ( plus_plus_real @ ( exp_real @ X2 ) @ ( exp_real @ ( uminus_uminus_real @ X2 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % tanh_altdef
% 0.33/0.67 thf(fact_8455_tanh__altdef,axiom,
% 0.33/0.67 ( tanh_complex
% 0.33/0.67 = ( ^ [X2: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X2 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X2 ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X2 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X2 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % tanh_altdef
% 0.33/0.67 thf(fact_8456_pochhammer__product,axiom,
% 0.33/0.67 ! [M: nat,N2: nat,Z: rat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.67 => ( ( comm_s4028243227959126397er_rat @ Z @ N2 )
% 0.33/0.67 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_product
% 0.33/0.67 thf(fact_8457_pochhammer__product,axiom,
% 0.33/0.67 ! [M: nat,N2: nat,Z: int] :
% 0.33/0.67 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.67 => ( ( comm_s4660882817536571857er_int @ Z @ N2 )
% 0.33/0.67 = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_product
% 0.33/0.67 thf(fact_8458_pochhammer__product,axiom,
% 0.33/0.67 ! [M: nat,N2: nat,Z: real] :
% 0.33/0.67 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.67 => ( ( comm_s7457072308508201937r_real @ Z @ N2 )
% 0.33/0.67 = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_product
% 0.33/0.67 thf(fact_8459_pochhammer__product,axiom,
% 0.33/0.67 ! [M: nat,N2: nat,Z: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.67 => ( ( comm_s4663373288045622133er_nat @ Z @ N2 )
% 0.33/0.67 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_product
% 0.33/0.67 thf(fact_8460_pochhammer__product,axiom,
% 0.33/0.67 ! [M: nat,N2: nat,Z: complex] :
% 0.33/0.67 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.67 => ( ( comm_s2602460028002588243omplex @ Z @ N2 )
% 0.33/0.67 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ M ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_product
% 0.33/0.67 thf(fact_8461_real__sqrt__unique,axiom,
% 0.33/0.67 ! [Y: real,X: real] :
% 0.33/0.67 ( ( ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.67 = X )
% 0.33/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.67 => ( ( sqrt @ X )
% 0.33/0.67 = Y ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % real_sqrt_unique
% 0.33/0.67 thf(fact_8462_real__le__lsqrt,axiom,
% 0.33/0.67 ! [X: real,Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.67 => ( ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.67 => ( ord_less_eq_real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % real_le_lsqrt
% 0.33/0.67 thf(fact_8463_lemma__real__divide__sqrt__less,axiom,
% 0.33/0.67 ! [U: real] :
% 0.33/0.67 ( ( ord_less_real @ zero_zero_real @ U )
% 0.33/0.67 => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).
% 0.33/0.67
% 0.33/0.67 % lemma_real_divide_sqrt_less
% 0.33/0.67 thf(fact_8464_real__sqrt__sum__squares__eq__cancel2,axiom,
% 0.33/0.67 ! [X: real,Y: real] :
% 0.33/0.67 ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.67 = Y )
% 0.33/0.67 => ( X = zero_zero_real ) ) ).
% 0.33/0.67
% 0.33/0.67 % real_sqrt_sum_squares_eq_cancel2
% 0.33/0.67 thf(fact_8465_real__sqrt__sum__squares__eq__cancel,axiom,
% 0.33/0.67 ! [X: real,Y: real] :
% 0.33/0.67 ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.67 = X )
% 0.33/0.67 => ( Y = zero_zero_real ) ) ).
% 0.33/0.67
% 0.33/0.67 % real_sqrt_sum_squares_eq_cancel
% 0.33/0.67 thf(fact_8466_real__sqrt__sum__squares__triangle__ineq,axiom,
% 0.33/0.67 ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A @ C ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B @ D ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % real_sqrt_sum_squares_triangle_ineq
% 0.33/0.67 thf(fact_8467_real__sqrt__sum__squares__ge2,axiom,
% 0.33/0.67 ! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % real_sqrt_sum_squares_ge2
% 0.33/0.67 thf(fact_8468_real__sqrt__sum__squares__ge1,axiom,
% 0.33/0.67 ! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % real_sqrt_sum_squares_ge1
% 0.33/0.67 thf(fact_8469_exp__half__le2,axiom,
% 0.33/0.67 ord_less_eq_real @ ( exp_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.33/0.67
% 0.33/0.67 % exp_half_le2
% 0.33/0.67 thf(fact_8470_sqrt__ge__absD,axiom,
% 0.33/0.67 ! [X: real,Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ Y ) )
% 0.33/0.67 => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) ).
% 0.33/0.67
% 0.33/0.67 % sqrt_ge_absD
% 0.33/0.67 thf(fact_8471_cos__45,axiom,
% 0.33/0.67 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.33/0.67 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % cos_45
% 0.33/0.67 thf(fact_8472_sin__45,axiom,
% 0.33/0.67 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.33/0.67 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % sin_45
% 0.33/0.67 thf(fact_8473_tan__60,axiom,
% 0.33/0.67 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 0.33/0.67 = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % tan_60
% 0.33/0.67 thf(fact_8474_exp__double,axiom,
% 0.33/0.67 ! [Z: complex] :
% 0.33/0.67 ( ( exp_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) )
% 0.33/0.67 = ( power_power_complex @ ( exp_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % exp_double
% 0.33/0.67 thf(fact_8475_exp__double,axiom,
% 0.33/0.67 ! [Z: real] :
% 0.33/0.67 ( ( exp_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) )
% 0.33/0.67 = ( power_power_real @ ( exp_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % exp_double
% 0.33/0.67 thf(fact_8476_pochhammer__absorb__comp,axiom,
% 0.33/0.67 ! [R: rat,K: nat] :
% 0.33/0.67 ( ( times_times_rat @ ( minus_minus_rat @ R @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R ) @ K ) )
% 0.33/0.67 = ( times_times_rat @ R @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R ) @ one_one_rat ) @ K ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_absorb_comp
% 0.33/0.67 thf(fact_8477_pochhammer__absorb__comp,axiom,
% 0.33/0.67 ! [R: code_integer,K: nat] :
% 0.33/0.67 ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R ) @ K ) )
% 0.33/0.67 = ( times_3573771949741848930nteger @ R @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R ) @ one_one_Code_integer ) @ K ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_absorb_comp
% 0.33/0.67 thf(fact_8478_pochhammer__absorb__comp,axiom,
% 0.33/0.67 ! [R: int,K: nat] :
% 0.33/0.67 ( ( times_times_int @ ( minus_minus_int @ R @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R ) @ K ) )
% 0.33/0.67 = ( times_times_int @ R @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R ) @ one_one_int ) @ K ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_absorb_comp
% 0.33/0.67 thf(fact_8479_pochhammer__absorb__comp,axiom,
% 0.33/0.67 ! [R: real,K: nat] :
% 0.33/0.67 ( ( times_times_real @ ( minus_minus_real @ R @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R ) @ K ) )
% 0.33/0.67 = ( times_times_real @ R @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R ) @ one_one_real ) @ K ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_absorb_comp
% 0.33/0.67 thf(fact_8480_pochhammer__absorb__comp,axiom,
% 0.33/0.67 ! [R: complex,K: nat] :
% 0.33/0.67 ( ( times_times_complex @ ( minus_minus_complex @ R @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R ) @ K ) )
% 0.33/0.67 = ( times_times_complex @ R @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R ) @ one_one_complex ) @ K ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_absorb_comp
% 0.33/0.67 thf(fact_8481_pochhammer__same,axiom,
% 0.33/0.67 ! [N2: nat] :
% 0.33/0.67 ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ N2 )
% 0.33/0.67 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_same
% 0.33/0.67 thf(fact_8482_pochhammer__same,axiom,
% 0.33/0.67 ! [N2: nat] :
% 0.33/0.67 ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ N2 )
% 0.33/0.67 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( semiri3624122377584611663nteger @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_same
% 0.33/0.67 thf(fact_8483_pochhammer__same,axiom,
% 0.33/0.67 ! [N2: nat] :
% 0.33/0.67 ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ N2 )
% 0.33/0.67 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_same
% 0.33/0.67 thf(fact_8484_pochhammer__same,axiom,
% 0.33/0.67 ! [N2: nat] :
% 0.33/0.67 ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ N2 )
% 0.33/0.67 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( semiri5044797733671781792omplex @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_same
% 0.33/0.67 thf(fact_8485_pochhammer__same,axiom,
% 0.33/0.67 ! [N2: nat] :
% 0.33/0.67 ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ N2 )
% 0.33/0.67 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_same
% 0.33/0.67 thf(fact_8486_real__less__lsqrt,axiom,
% 0.33/0.67 ! [X: real,Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.67 => ( ( ord_less_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.67 => ( ord_less_real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % real_less_lsqrt
% 0.33/0.67 thf(fact_8487_sqrt__sum__squares__le__sum,axiom,
% 0.33/0.67 ! [X: real,Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.67 => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X @ Y ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % sqrt_sum_squares_le_sum
% 0.33/0.67 thf(fact_8488_sqrt__even__pow2,axiom,
% 0.33/0.67 ! [N2: nat] :
% 0.33/0.67 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.67 => ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) )
% 0.33/0.67 = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % sqrt_even_pow2
% 0.33/0.67 thf(fact_8489_real__sqrt__ge__abs1,axiom,
% 0.33/0.67 ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % real_sqrt_ge_abs1
% 0.33/0.67 thf(fact_8490_real__sqrt__ge__abs2,axiom,
% 0.33/0.67 ! [Y: real,X: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % real_sqrt_ge_abs2
% 0.33/0.67 thf(fact_8491_sqrt__sum__squares__le__sum__abs,axiom,
% 0.33/0.67 ! [X: real,Y: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % sqrt_sum_squares_le_sum_abs
% 0.33/0.67 thf(fact_8492_ln__sqrt,axiom,
% 0.33/0.67 ! [X: real] :
% 0.33/0.67 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.67 => ( ( ln_ln_real @ ( sqrt @ X ) )
% 0.33/0.67 = ( divide_divide_real @ ( ln_ln_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % ln_sqrt
% 0.33/0.67 thf(fact_8493_cos__30,axiom,
% 0.33/0.67 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 0.33/0.67 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % cos_30
% 0.33/0.67 thf(fact_8494_sin__60,axiom,
% 0.33/0.67 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 0.33/0.67 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % sin_60
% 0.33/0.67 thf(fact_8495_exp__bound__half,axiom,
% 0.33/0.67 ! [Z: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.67 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % exp_bound_half
% 0.33/0.67 thf(fact_8496_exp__bound__half,axiom,
% 0.33/0.67 ! [Z: complex] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.67 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % exp_bound_half
% 0.33/0.67 thf(fact_8497_pochhammer__minus,axiom,
% 0.33/0.67 ! [B: rat,K: nat] :
% 0.33/0.67 ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K )
% 0.33/0.67 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_minus
% 0.33/0.67 thf(fact_8498_pochhammer__minus,axiom,
% 0.33/0.67 ! [B: code_integer,K: nat] :
% 0.33/0.67 ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K )
% 0.33/0.67 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_minus
% 0.33/0.67 thf(fact_8499_pochhammer__minus,axiom,
% 0.33/0.67 ! [B: int,K: nat] :
% 0.33/0.67 ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K )
% 0.33/0.67 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_minus
% 0.33/0.67 thf(fact_8500_pochhammer__minus,axiom,
% 0.33/0.67 ! [B: real,K: nat] :
% 0.33/0.67 ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K )
% 0.33/0.67 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_minus
% 0.33/0.67 thf(fact_8501_pochhammer__minus,axiom,
% 0.33/0.67 ! [B: complex,K: nat] :
% 0.33/0.67 ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K )
% 0.33/0.67 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_minus
% 0.33/0.67 thf(fact_8502_pochhammer__minus_H,axiom,
% 0.33/0.67 ! [B: rat,K: nat] :
% 0.33/0.67 ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K )
% 0.33/0.67 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_minus'
% 0.33/0.67 thf(fact_8503_pochhammer__minus_H,axiom,
% 0.33/0.67 ! [B: code_integer,K: nat] :
% 0.33/0.67 ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
% 0.33/0.67 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_minus'
% 0.33/0.67 thf(fact_8504_pochhammer__minus_H,axiom,
% 0.33/0.67 ! [B: int,K: nat] :
% 0.33/0.67 ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
% 0.33/0.67 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_minus'
% 0.33/0.67 thf(fact_8505_pochhammer__minus_H,axiom,
% 0.33/0.67 ! [B: real,K: nat] :
% 0.33/0.67 ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K )
% 0.33/0.67 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_minus'
% 0.33/0.67 thf(fact_8506_pochhammer__minus_H,axiom,
% 0.33/0.67 ! [B: complex,K: nat] :
% 0.33/0.67 ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
% 0.33/0.67 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_minus'
% 0.33/0.67 thf(fact_8507_arsinh__real__aux,axiom,
% 0.33/0.67 ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arsinh_real_aux
% 0.33/0.67 thf(fact_8508_real__sqrt__power__even,axiom,
% 0.33/0.67 ! [N2: nat,X: real] :
% 0.33/0.67 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.67 => ( ( power_power_real @ ( sqrt @ X ) @ N2 )
% 0.33/0.67 = ( power_power_real @ X @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % real_sqrt_power_even
% 0.33/0.67 thf(fact_8509_exp__bound,axiom,
% 0.33/0.67 ! [X: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.67 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.67 => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % exp_bound
% 0.33/0.67 thf(fact_8510_real__sqrt__sum__squares__mult__ge__zero,axiom,
% 0.33/0.67 ! [X: real,Y: real,Xa2: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % real_sqrt_sum_squares_mult_ge_zero
% 0.33/0.67 thf(fact_8511_arith__geo__mean__sqrt,axiom,
% 0.33/0.67 ! [X: real,Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.67 => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X @ Y ) ) @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arith_geo_mean_sqrt
% 0.33/0.67 thf(fact_8512_tan__30,axiom,
% 0.33/0.67 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 0.33/0.67 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % tan_30
% 0.33/0.67 thf(fact_8513_real__exp__bound__lemma,axiom,
% 0.33/0.67 ! [X: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.67 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.67 => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % real_exp_bound_lemma
% 0.33/0.67 thf(fact_8514_cos__x__y__le__one,axiom,
% 0.33/0.67 ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).
% 0.33/0.67
% 0.33/0.67 % cos_x_y_le_one
% 0.33/0.67 thf(fact_8515_real__sqrt__sum__squares__less,axiom,
% 0.33/0.67 ! [X: real,U: real,Y: real] :
% 0.33/0.67 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.33/0.67 => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.33/0.67 => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % real_sqrt_sum_squares_less
% 0.33/0.67 thf(fact_8516_arcosh__real__def,axiom,
% 0.33/0.67 ! [X: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ one_one_real @ X )
% 0.33/0.67 => ( ( arcosh_real @ X )
% 0.33/0.67 = ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arcosh_real_def
% 0.33/0.67 thf(fact_8517_exp__ge__one__plus__x__over__n__power__n,axiom,
% 0.33/0.67 ! [N2: nat,X: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ X )
% 0.33/0.67 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.67 => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 ) @ ( exp_real @ X ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % exp_ge_one_plus_x_over_n_power_n
% 0.33/0.67 thf(fact_8518_exp__ge__one__minus__x__over__n__power__n,axiom,
% 0.33/0.67 ! [X: real,N2: nat] :
% 0.33/0.67 ( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N2 ) )
% 0.33/0.67 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.67 => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % exp_ge_one_minus_x_over_n_power_n
% 0.33/0.67 thf(fact_8519_cos__arctan,axiom,
% 0.33/0.67 ! [X: real] :
% 0.33/0.67 ( ( cos_real @ ( arctan @ X ) )
% 0.33/0.67 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % cos_arctan
% 0.33/0.67 thf(fact_8520_sin__arctan,axiom,
% 0.33/0.67 ! [X: real] :
% 0.33/0.67 ( ( sin_real @ ( arctan @ X ) )
% 0.33/0.67 = ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % sin_arctan
% 0.33/0.67 thf(fact_8521_exp__bound__lemma,axiom,
% 0.33/0.67 ! [Z: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.67 => ( 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 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % exp_bound_lemma
% 0.33/0.67 thf(fact_8522_exp__bound__lemma,axiom,
% 0.33/0.67 ! [Z: complex] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.67 => ( 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 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % exp_bound_lemma
% 0.33/0.67 thf(fact_8523_Maclaurin__exp__le,axiom,
% 0.33/0.67 ! [X: real,N2: nat] :
% 0.33/0.67 ? [T5: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 0.33/0.67 & ( ( exp_real @ X )
% 0.33/0.67 = ( plus_plus_real
% 0.33/0.67 @ ( groups6591440286371151544t_real
% 0.33/0.67 @ ^ [M6: nat] : ( divide_divide_real @ ( power_power_real @ X @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) )
% 0.33/0.67 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.67 @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T5 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % Maclaurin_exp_le
% 0.33/0.67 thf(fact_8524_sqrt__sum__squares__half__less,axiom,
% 0.33/0.67 ! [X: real,U: real,Y: real] :
% 0.33/0.67 ( ( ord_less_real @ X @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.67 => ( ( ord_less_real @ Y @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.67 => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % sqrt_sum_squares_half_less
% 0.33/0.67 thf(fact_8525_exp__lower__Taylor__quadratic,axiom,
% 0.33/0.67 ! [X: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.67 => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( divide_divide_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % exp_lower_Taylor_quadratic
% 0.33/0.67 thf(fact_8526_sin__cos__sqrt,axiom,
% 0.33/0.67 ! [X: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) )
% 0.33/0.67 => ( ( sin_real @ X )
% 0.33/0.67 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % sin_cos_sqrt
% 0.33/0.67 thf(fact_8527_arctan__half,axiom,
% 0.33/0.67 ( arctan
% 0.33/0.67 = ( ^ [X2: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X2 @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arctan_half
% 0.33/0.67 thf(fact_8528_tanh__real__altdef,axiom,
% 0.33/0.67 ( tanh_real
% 0.33/0.67 = ( ^ [X2: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % tanh_real_altdef
% 0.33/0.67 thf(fact_8529_fact__double,axiom,
% 0.33/0.67 ! [N2: nat] :
% 0.33/0.67 ( ( semiri5044797733671781792omplex @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.33/0.67 = ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( comm_s2602460028002588243omplex @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ N2 ) ) @ ( semiri5044797733671781792omplex @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % fact_double
% 0.33/0.67 thf(fact_8530_fact__double,axiom,
% 0.33/0.67 ! [N2: nat] :
% 0.33/0.67 ( ( semiri773545260158071498ct_rat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.33/0.67 = ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ N2 ) ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % fact_double
% 0.33/0.67 thf(fact_8531_fact__double,axiom,
% 0.33/0.67 ! [N2: nat] :
% 0.33/0.67 ( ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.33/0.67 = ( times_times_real @ ( times_times_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ N2 ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % fact_double
% 0.33/0.67 thf(fact_8532_pochhammer__times__pochhammer__half,axiom,
% 0.33/0.67 ! [Z: rat,N2: nat] :
% 0.33/0.67 ( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( groups73079841787564623at_rat
% 0.33/0.67 @ ^ [K3: nat] : ( plus_plus_rat @ Z @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.33/0.67 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_times_pochhammer_half
% 0.33/0.67 thf(fact_8533_pochhammer__times__pochhammer__half,axiom,
% 0.33/0.67 ! [Z: real,N2: nat] :
% 0.33/0.67 ( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ ( suc @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( groups129246275422532515t_real
% 0.33/0.67 @ ^ [K3: nat] : ( plus_plus_real @ Z @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.67 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_times_pochhammer_half
% 0.33/0.67 thf(fact_8534_pochhammer__times__pochhammer__half,axiom,
% 0.33/0.67 ! [Z: complex,N2: nat] :
% 0.33/0.67 ( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ ( suc @ N2 ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( groups6464643781859351333omplex
% 0.33/0.67 @ ^ [K3: nat] : ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 0.33/0.67 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_times_pochhammer_half
% 0.33/0.67 thf(fact_8535_pochhammer__code,axiom,
% 0.33/0.67 ( comm_s4028243227959126397er_rat
% 0.33/0.67 = ( ^ [A5: rat,N: nat] :
% 0.33/0.67 ( if_rat @ ( N = zero_zero_nat ) @ one_one_rat
% 0.33/0.67 @ ( set_fo1949268297981939178at_rat
% 0.33/0.67 @ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A5 @ ( semiri681578069525770553at_rat @ O ) ) )
% 0.33/0.67 @ zero_zero_nat
% 0.33/0.67 @ ( minus_minus_nat @ N @ one_one_nat )
% 0.33/0.67 @ one_one_rat ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_code
% 0.33/0.67 thf(fact_8536_pochhammer__code,axiom,
% 0.33/0.67 ( comm_s4660882817536571857er_int
% 0.33/0.67 = ( ^ [A5: int,N: nat] :
% 0.33/0.67 ( if_int @ ( N = zero_zero_nat ) @ one_one_int
% 0.33/0.67 @ ( set_fo2581907887559384638at_int
% 0.33/0.67 @ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A5 @ ( semiri1314217659103216013at_int @ O ) ) )
% 0.33/0.67 @ zero_zero_nat
% 0.33/0.67 @ ( minus_minus_nat @ N @ one_one_nat )
% 0.33/0.67 @ one_one_int ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_code
% 0.33/0.67 thf(fact_8537_pochhammer__code,axiom,
% 0.33/0.67 ( comm_s7457072308508201937r_real
% 0.33/0.67 = ( ^ [A5: real,N: nat] :
% 0.33/0.67 ( if_real @ ( N = zero_zero_nat ) @ one_one_real
% 0.33/0.67 @ ( set_fo3111899725591712190t_real
% 0.33/0.67 @ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A5 @ ( semiri5074537144036343181t_real @ O ) ) )
% 0.33/0.67 @ zero_zero_nat
% 0.33/0.67 @ ( minus_minus_nat @ N @ one_one_nat )
% 0.33/0.67 @ one_one_real ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_code
% 0.33/0.67 thf(fact_8538_pochhammer__code,axiom,
% 0.33/0.67 ( comm_s2602460028002588243omplex
% 0.33/0.67 = ( ^ [A5: complex,N: nat] :
% 0.33/0.67 ( if_complex @ ( N = zero_zero_nat ) @ one_one_complex
% 0.33/0.67 @ ( set_fo1517530859248394432omplex
% 0.33/0.67 @ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A5 @ ( semiri8010041392384452111omplex @ O ) ) )
% 0.33/0.67 @ zero_zero_nat
% 0.33/0.67 @ ( minus_minus_nat @ N @ one_one_nat )
% 0.33/0.67 @ one_one_complex ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_code
% 0.33/0.67 thf(fact_8539_pochhammer__code,axiom,
% 0.33/0.67 ( comm_s4663373288045622133er_nat
% 0.33/0.67 = ( ^ [A5: nat,N: nat] :
% 0.33/0.67 ( if_nat @ ( N = zero_zero_nat ) @ one_one_nat
% 0.33/0.67 @ ( set_fo2584398358068434914at_nat
% 0.33/0.67 @ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A5 @ ( semiri1316708129612266289at_nat @ O ) ) )
% 0.33/0.67 @ zero_zero_nat
% 0.33/0.67 @ ( minus_minus_nat @ N @ one_one_nat )
% 0.33/0.67 @ one_one_nat ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % pochhammer_code
% 0.33/0.67 thf(fact_8540_arsinh__real__def,axiom,
% 0.33/0.67 ( arsinh_real
% 0.33/0.67 = ( ^ [X2: real] : ( ln_ln_real @ ( plus_plus_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arsinh_real_def
% 0.33/0.67 thf(fact_8541_cos__arcsin,axiom,
% 0.33/0.67 ! [X: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.67 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.67 => ( ( cos_real @ ( arcsin @ X ) )
% 0.33/0.67 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % cos_arcsin
% 0.33/0.67 thf(fact_8542_sin__arccos__abs,axiom,
% 0.33/0.67 ! [Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 0.33/0.67 => ( ( sin_real @ ( arccos @ Y ) )
% 0.33/0.67 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % sin_arccos_abs
% 0.33/0.67 thf(fact_8543_prod_Oneutral__const,axiom,
% 0.33/0.67 ! [A2: set_nat] :
% 0.33/0.67 ( ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [Uu3: nat] : one_one_nat
% 0.33/0.67 @ A2 )
% 0.33/0.67 = one_one_nat ) ).
% 0.33/0.67
% 0.33/0.67 % prod.neutral_const
% 0.33/0.67 thf(fact_8544_prod_Oneutral__const,axiom,
% 0.33/0.67 ! [A2: set_nat] :
% 0.33/0.67 ( ( groups705719431365010083at_int
% 0.33/0.67 @ ^ [Uu3: nat] : one_one_int
% 0.33/0.67 @ A2 )
% 0.33/0.67 = one_one_int ) ).
% 0.33/0.67
% 0.33/0.67 % prod.neutral_const
% 0.33/0.67 thf(fact_8545_prod_Oneutral__const,axiom,
% 0.33/0.67 ! [A2: set_int] :
% 0.33/0.67 ( ( groups1705073143266064639nt_int
% 0.33/0.67 @ ^ [Uu3: int] : one_one_int
% 0.33/0.67 @ A2 )
% 0.33/0.67 = one_one_int ) ).
% 0.33/0.67
% 0.33/0.67 % prod.neutral_const
% 0.33/0.67 thf(fact_8546_of__nat__prod,axiom,
% 0.33/0.67 ! [F: int > nat,A2: set_int] :
% 0.33/0.67 ( ( semiri1314217659103216013at_int @ ( groups1707563613775114915nt_nat @ F @ A2 ) )
% 0.33/0.67 = ( groups1705073143266064639nt_int
% 0.33/0.67 @ ^ [X2: int] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
% 0.33/0.67 @ A2 ) ) ).
% 0.33/0.67
% 0.33/0.67 % of_nat_prod
% 0.33/0.67 thf(fact_8547_of__nat__prod,axiom,
% 0.33/0.67 ! [F: nat > nat,A2: set_nat] :
% 0.33/0.67 ( ( semiri5074537144036343181t_real @ ( groups708209901874060359at_nat @ F @ A2 ) )
% 0.33/0.67 = ( groups129246275422532515t_real
% 0.33/0.67 @ ^ [X2: nat] : ( semiri5074537144036343181t_real @ ( F @ X2 ) )
% 0.33/0.67 @ A2 ) ) ).
% 0.33/0.67
% 0.33/0.67 % of_nat_prod
% 0.33/0.67 thf(fact_8548_of__nat__prod,axiom,
% 0.33/0.67 ! [F: nat > nat,A2: set_nat] :
% 0.33/0.67 ( ( semiri8010041392384452111omplex @ ( groups708209901874060359at_nat @ F @ A2 ) )
% 0.33/0.67 = ( groups6464643781859351333omplex
% 0.33/0.67 @ ^ [X2: nat] : ( semiri8010041392384452111omplex @ ( F @ X2 ) )
% 0.33/0.67 @ A2 ) ) ).
% 0.33/0.67
% 0.33/0.67 % of_nat_prod
% 0.33/0.67 thf(fact_8549_of__nat__prod,axiom,
% 0.33/0.67 ! [F: nat > nat,A2: set_nat] :
% 0.33/0.67 ( ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ F @ A2 ) )
% 0.33/0.67 = ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [X2: nat] : ( semiri1316708129612266289at_nat @ ( F @ X2 ) )
% 0.33/0.67 @ A2 ) ) ).
% 0.33/0.67
% 0.33/0.67 % of_nat_prod
% 0.33/0.67 thf(fact_8550_of__nat__prod,axiom,
% 0.33/0.67 ! [F: nat > nat,A2: set_nat] :
% 0.33/0.67 ( ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ F @ A2 ) )
% 0.33/0.67 = ( groups705719431365010083at_int
% 0.33/0.67 @ ^ [X2: nat] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
% 0.33/0.67 @ A2 ) ) ).
% 0.33/0.67
% 0.33/0.67 % of_nat_prod
% 0.33/0.67 thf(fact_8551_of__int__prod,axiom,
% 0.33/0.67 ! [F: nat > int,A2: set_nat] :
% 0.33/0.67 ( ( ring_1_of_int_real @ ( groups705719431365010083at_int @ F @ A2 ) )
% 0.33/0.67 = ( groups129246275422532515t_real
% 0.33/0.67 @ ^ [X2: nat] : ( ring_1_of_int_real @ ( F @ X2 ) )
% 0.33/0.67 @ A2 ) ) ).
% 0.33/0.67
% 0.33/0.67 % of_int_prod
% 0.33/0.67 thf(fact_8552_of__int__prod,axiom,
% 0.33/0.67 ! [F: nat > int,A2: set_nat] :
% 0.33/0.67 ( ( ring_1_of_int_rat @ ( groups705719431365010083at_int @ F @ A2 ) )
% 0.33/0.67 = ( groups73079841787564623at_rat
% 0.33/0.67 @ ^ [X2: nat] : ( ring_1_of_int_rat @ ( F @ X2 ) )
% 0.33/0.67 @ A2 ) ) ).
% 0.33/0.67
% 0.33/0.67 % of_int_prod
% 0.33/0.67 thf(fact_8553_of__int__prod,axiom,
% 0.33/0.67 ! [F: nat > int,A2: set_nat] :
% 0.33/0.67 ( ( ring_1_of_int_int @ ( groups705719431365010083at_int @ F @ A2 ) )
% 0.33/0.67 = ( groups705719431365010083at_int
% 0.33/0.67 @ ^ [X2: nat] : ( ring_1_of_int_int @ ( F @ X2 ) )
% 0.33/0.67 @ A2 ) ) ).
% 0.33/0.67
% 0.33/0.67 % of_int_prod
% 0.33/0.67 thf(fact_8554_of__int__prod,axiom,
% 0.33/0.67 ! [F: int > int,A2: set_int] :
% 0.33/0.67 ( ( ring_1_of_int_real @ ( groups1705073143266064639nt_int @ F @ A2 ) )
% 0.33/0.67 = ( groups2316167850115554303t_real
% 0.33/0.67 @ ^ [X2: int] : ( ring_1_of_int_real @ ( F @ X2 ) )
% 0.33/0.67 @ A2 ) ) ).
% 0.33/0.67
% 0.33/0.67 % of_int_prod
% 0.33/0.67 thf(fact_8555_of__int__prod,axiom,
% 0.33/0.67 ! [F: int > int,A2: set_int] :
% 0.33/0.67 ( ( ring_1_of_int_rat @ ( groups1705073143266064639nt_int @ F @ A2 ) )
% 0.33/0.67 = ( groups1072433553688619179nt_rat
% 0.33/0.67 @ ^ [X2: int] : ( ring_1_of_int_rat @ ( F @ X2 ) )
% 0.33/0.67 @ A2 ) ) ).
% 0.33/0.67
% 0.33/0.67 % of_int_prod
% 0.33/0.67 thf(fact_8556_of__int__prod,axiom,
% 0.33/0.67 ! [F: int > int,A2: set_int] :
% 0.33/0.67 ( ( ring_1_of_int_int @ ( groups1705073143266064639nt_int @ F @ A2 ) )
% 0.33/0.67 = ( groups1705073143266064639nt_int
% 0.33/0.67 @ ^ [X2: int] : ( ring_1_of_int_int @ ( F @ X2 ) )
% 0.33/0.67 @ A2 ) ) ).
% 0.33/0.67
% 0.33/0.67 % of_int_prod
% 0.33/0.67 thf(fact_8557_prod_Oempty,axiom,
% 0.33/0.67 ! [G: nat > complex] :
% 0.33/0.67 ( ( groups6464643781859351333omplex @ G @ bot_bot_set_nat )
% 0.33/0.67 = one_one_complex ) ).
% 0.33/0.67
% 0.33/0.67 % prod.empty
% 0.33/0.67 thf(fact_8558_prod_Oempty,axiom,
% 0.33/0.67 ! [G: nat > real] :
% 0.33/0.67 ( ( groups129246275422532515t_real @ G @ bot_bot_set_nat )
% 0.33/0.67 = one_one_real ) ).
% 0.33/0.67
% 0.33/0.67 % prod.empty
% 0.33/0.67 thf(fact_8559_prod_Oempty,axiom,
% 0.33/0.67 ! [G: nat > rat] :
% 0.33/0.67 ( ( groups73079841787564623at_rat @ G @ bot_bot_set_nat )
% 0.33/0.67 = one_one_rat ) ).
% 0.33/0.67
% 0.33/0.67 % prod.empty
% 0.33/0.67 thf(fact_8560_prod_Oempty,axiom,
% 0.33/0.67 ! [G: int > complex] :
% 0.33/0.67 ( ( groups7440179247065528705omplex @ G @ bot_bot_set_int )
% 0.33/0.67 = one_one_complex ) ).
% 0.33/0.67
% 0.33/0.67 % prod.empty
% 0.33/0.67 thf(fact_8561_prod_Oempty,axiom,
% 0.33/0.67 ! [G: int > real] :
% 0.33/0.67 ( ( groups2316167850115554303t_real @ G @ bot_bot_set_int )
% 0.33/0.67 = one_one_real ) ).
% 0.33/0.67
% 0.33/0.67 % prod.empty
% 0.33/0.67 thf(fact_8562_prod_Oempty,axiom,
% 0.33/0.67 ! [G: int > rat] :
% 0.33/0.67 ( ( groups1072433553688619179nt_rat @ G @ bot_bot_set_int )
% 0.33/0.67 = one_one_rat ) ).
% 0.33/0.67
% 0.33/0.67 % prod.empty
% 0.33/0.67 thf(fact_8563_prod_Oempty,axiom,
% 0.33/0.67 ! [G: int > nat] :
% 0.33/0.67 ( ( groups1707563613775114915nt_nat @ G @ bot_bot_set_int )
% 0.33/0.67 = one_one_nat ) ).
% 0.33/0.67
% 0.33/0.67 % prod.empty
% 0.33/0.67 thf(fact_8564_prod_Oempty,axiom,
% 0.33/0.67 ! [G: real > complex] :
% 0.33/0.67 ( ( groups713298508707869441omplex @ G @ bot_bot_set_real )
% 0.33/0.67 = one_one_complex ) ).
% 0.33/0.67
% 0.33/0.67 % prod.empty
% 0.33/0.67 thf(fact_8565_prod_Oempty,axiom,
% 0.33/0.67 ! [G: real > real] :
% 0.33/0.67 ( ( groups1681761925125756287l_real @ G @ bot_bot_set_real )
% 0.33/0.67 = one_one_real ) ).
% 0.33/0.67
% 0.33/0.67 % prod.empty
% 0.33/0.67 thf(fact_8566_prod_Oempty,axiom,
% 0.33/0.67 ! [G: real > rat] :
% 0.33/0.67 ( ( groups4061424788464935467al_rat @ G @ bot_bot_set_real )
% 0.33/0.67 = one_one_rat ) ).
% 0.33/0.67
% 0.33/0.67 % prod.empty
% 0.33/0.67 thf(fact_8567_prod_Oinfinite,axiom,
% 0.33/0.67 ! [A2: set_nat,G: nat > complex] :
% 0.33/0.67 ( ~ ( finite_finite_nat @ A2 )
% 0.33/0.67 => ( ( groups6464643781859351333omplex @ G @ A2 )
% 0.33/0.67 = one_one_complex ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.infinite
% 0.33/0.67 thf(fact_8568_prod_Oinfinite,axiom,
% 0.33/0.67 ! [A2: set_int,G: int > complex] :
% 0.33/0.67 ( ~ ( finite_finite_int @ A2 )
% 0.33/0.67 => ( ( groups7440179247065528705omplex @ G @ A2 )
% 0.33/0.67 = one_one_complex ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.infinite
% 0.33/0.67 thf(fact_8569_prod_Oinfinite,axiom,
% 0.33/0.67 ! [A2: set_complex,G: complex > complex] :
% 0.33/0.67 ( ~ ( finite3207457112153483333omplex @ A2 )
% 0.33/0.67 => ( ( groups3708469109370488835omplex @ G @ A2 )
% 0.33/0.67 = one_one_complex ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.infinite
% 0.33/0.67 thf(fact_8570_prod_Oinfinite,axiom,
% 0.33/0.67 ! [A2: set_nat,G: nat > real] :
% 0.33/0.67 ( ~ ( finite_finite_nat @ A2 )
% 0.33/0.67 => ( ( groups129246275422532515t_real @ G @ A2 )
% 0.33/0.67 = one_one_real ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.infinite
% 0.33/0.67 thf(fact_8571_prod_Oinfinite,axiom,
% 0.33/0.67 ! [A2: set_int,G: int > real] :
% 0.33/0.67 ( ~ ( finite_finite_int @ A2 )
% 0.33/0.67 => ( ( groups2316167850115554303t_real @ G @ A2 )
% 0.33/0.67 = one_one_real ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.infinite
% 0.33/0.67 thf(fact_8572_prod_Oinfinite,axiom,
% 0.33/0.67 ! [A2: set_complex,G: complex > real] :
% 0.33/0.67 ( ~ ( finite3207457112153483333omplex @ A2 )
% 0.33/0.67 => ( ( groups766887009212190081x_real @ G @ A2 )
% 0.33/0.67 = one_one_real ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.infinite
% 0.33/0.67 thf(fact_8573_prod_Oinfinite,axiom,
% 0.33/0.67 ! [A2: set_nat,G: nat > rat] :
% 0.33/0.67 ( ~ ( finite_finite_nat @ A2 )
% 0.33/0.67 => ( ( groups73079841787564623at_rat @ G @ A2 )
% 0.33/0.67 = one_one_rat ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.infinite
% 0.33/0.67 thf(fact_8574_prod_Oinfinite,axiom,
% 0.33/0.67 ! [A2: set_int,G: int > rat] :
% 0.33/0.67 ( ~ ( finite_finite_int @ A2 )
% 0.33/0.67 => ( ( groups1072433553688619179nt_rat @ G @ A2 )
% 0.33/0.67 = one_one_rat ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.infinite
% 0.33/0.67 thf(fact_8575_prod_Oinfinite,axiom,
% 0.33/0.67 ! [A2: set_complex,G: complex > rat] :
% 0.33/0.67 ( ~ ( finite3207457112153483333omplex @ A2 )
% 0.33/0.67 => ( ( groups225925009352817453ex_rat @ G @ A2 )
% 0.33/0.67 = one_one_rat ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.infinite
% 0.33/0.67 thf(fact_8576_prod_Oinfinite,axiom,
% 0.33/0.67 ! [A2: set_int,G: int > nat] :
% 0.33/0.67 ( ~ ( finite_finite_int @ A2 )
% 0.33/0.67 => ( ( groups1707563613775114915nt_nat @ G @ A2 )
% 0.33/0.67 = one_one_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.infinite
% 0.33/0.67 thf(fact_8577_arccos__1,axiom,
% 0.33/0.67 ( ( arccos @ one_one_real )
% 0.33/0.67 = zero_zero_real ) ).
% 0.33/0.67
% 0.33/0.67 % arccos_1
% 0.33/0.67 thf(fact_8578_prod_Odelta,axiom,
% 0.33/0.67 ! [S: set_real,A: real,B: real > complex] :
% 0.33/0.67 ( ( finite_finite_real @ S )
% 0.33/0.67 => ( ( ( member_real @ A @ S )
% 0.33/0.67 => ( ( groups713298508707869441omplex
% 0.33/0.67 @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 0.33/0.67 @ S )
% 0.33/0.67 = ( B @ A ) ) )
% 0.33/0.67 & ( ~ ( member_real @ A @ S )
% 0.33/0.67 => ( ( groups713298508707869441omplex
% 0.33/0.67 @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 0.33/0.67 @ S )
% 0.33/0.67 = one_one_complex ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.delta
% 0.33/0.67 thf(fact_8579_prod_Odelta,axiom,
% 0.33/0.67 ! [S: set_nat,A: nat,B: nat > complex] :
% 0.33/0.67 ( ( finite_finite_nat @ S )
% 0.33/0.67 => ( ( ( member_nat @ A @ S )
% 0.33/0.67 => ( ( groups6464643781859351333omplex
% 0.33/0.67 @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 0.33/0.67 @ S )
% 0.33/0.67 = ( B @ A ) ) )
% 0.33/0.67 & ( ~ ( member_nat @ A @ S )
% 0.33/0.67 => ( ( groups6464643781859351333omplex
% 0.33/0.67 @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 0.33/0.67 @ S )
% 0.33/0.67 = one_one_complex ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.delta
% 0.33/0.67 thf(fact_8580_prod_Odelta,axiom,
% 0.33/0.67 ! [S: set_int,A: int,B: int > complex] :
% 0.33/0.67 ( ( finite_finite_int @ S )
% 0.33/0.67 => ( ( ( member_int @ A @ S )
% 0.33/0.67 => ( ( groups7440179247065528705omplex
% 0.33/0.67 @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 0.33/0.67 @ S )
% 0.33/0.67 = ( B @ A ) ) )
% 0.33/0.67 & ( ~ ( member_int @ A @ S )
% 0.33/0.67 => ( ( groups7440179247065528705omplex
% 0.33/0.67 @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 0.33/0.67 @ S )
% 0.33/0.67 = one_one_complex ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.delta
% 0.33/0.67 thf(fact_8581_prod_Odelta,axiom,
% 0.33/0.67 ! [S: set_complex,A: complex,B: complex > complex] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ S )
% 0.33/0.67 => ( ( ( member_complex @ A @ S )
% 0.33/0.67 => ( ( groups3708469109370488835omplex
% 0.33/0.67 @ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 0.33/0.67 @ S )
% 0.33/0.67 = ( B @ A ) ) )
% 0.33/0.67 & ( ~ ( member_complex @ A @ S )
% 0.33/0.67 => ( ( groups3708469109370488835omplex
% 0.33/0.67 @ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 0.33/0.67 @ S )
% 0.33/0.67 = one_one_complex ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.delta
% 0.33/0.67 thf(fact_8582_prod_Odelta,axiom,
% 0.33/0.67 ! [S: set_real,A: real,B: real > real] :
% 0.33/0.67 ( ( finite_finite_real @ S )
% 0.33/0.67 => ( ( ( member_real @ A @ S )
% 0.33/0.67 => ( ( groups1681761925125756287l_real
% 0.33/0.67 @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 0.33/0.67 @ S )
% 0.33/0.67 = ( B @ A ) ) )
% 0.33/0.67 & ( ~ ( member_real @ A @ S )
% 0.33/0.67 => ( ( groups1681761925125756287l_real
% 0.33/0.67 @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 0.33/0.67 @ S )
% 0.33/0.67 = one_one_real ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.delta
% 0.33/0.67 thf(fact_8583_prod_Odelta,axiom,
% 0.33/0.67 ! [S: set_nat,A: nat,B: nat > real] :
% 0.33/0.67 ( ( finite_finite_nat @ S )
% 0.33/0.67 => ( ( ( member_nat @ A @ S )
% 0.33/0.67 => ( ( groups129246275422532515t_real
% 0.33/0.67 @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 0.33/0.67 @ S )
% 0.33/0.67 = ( B @ A ) ) )
% 0.33/0.67 & ( ~ ( member_nat @ A @ S )
% 0.33/0.67 => ( ( groups129246275422532515t_real
% 0.33/0.67 @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 0.33/0.67 @ S )
% 0.33/0.67 = one_one_real ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.delta
% 0.33/0.67 thf(fact_8584_prod_Odelta,axiom,
% 0.33/0.67 ! [S: set_int,A: int,B: int > real] :
% 0.33/0.67 ( ( finite_finite_int @ S )
% 0.33/0.67 => ( ( ( member_int @ A @ S )
% 0.33/0.67 => ( ( groups2316167850115554303t_real
% 0.33/0.67 @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 0.33/0.67 @ S )
% 0.33/0.67 = ( B @ A ) ) )
% 0.33/0.67 & ( ~ ( member_int @ A @ S )
% 0.33/0.67 => ( ( groups2316167850115554303t_real
% 0.33/0.67 @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 0.33/0.67 @ S )
% 0.33/0.67 = one_one_real ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.delta
% 0.33/0.67 thf(fact_8585_prod_Odelta,axiom,
% 0.33/0.67 ! [S: set_complex,A: complex,B: complex > real] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ S )
% 0.33/0.67 => ( ( ( member_complex @ A @ S )
% 0.33/0.67 => ( ( groups766887009212190081x_real
% 0.33/0.67 @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 0.33/0.67 @ S )
% 0.33/0.67 = ( B @ A ) ) )
% 0.33/0.67 & ( ~ ( member_complex @ A @ S )
% 0.33/0.67 => ( ( groups766887009212190081x_real
% 0.33/0.67 @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 0.33/0.67 @ S )
% 0.33/0.67 = one_one_real ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.delta
% 0.33/0.67 thf(fact_8586_prod_Odelta,axiom,
% 0.33/0.67 ! [S: set_real,A: real,B: real > rat] :
% 0.33/0.67 ( ( finite_finite_real @ S )
% 0.33/0.67 => ( ( ( member_real @ A @ S )
% 0.33/0.67 => ( ( groups4061424788464935467al_rat
% 0.33/0.67 @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
% 0.33/0.67 @ S )
% 0.33/0.67 = ( B @ A ) ) )
% 0.33/0.67 & ( ~ ( member_real @ A @ S )
% 0.33/0.67 => ( ( groups4061424788464935467al_rat
% 0.33/0.67 @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
% 0.33/0.67 @ S )
% 0.33/0.67 = one_one_rat ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.delta
% 0.33/0.67 thf(fact_8587_prod_Odelta,axiom,
% 0.33/0.67 ! [S: set_nat,A: nat,B: nat > rat] :
% 0.33/0.67 ( ( finite_finite_nat @ S )
% 0.33/0.67 => ( ( ( member_nat @ A @ S )
% 0.33/0.67 => ( ( groups73079841787564623at_rat
% 0.33/0.67 @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
% 0.33/0.67 @ S )
% 0.33/0.67 = ( B @ A ) ) )
% 0.33/0.67 & ( ~ ( member_nat @ A @ S )
% 0.33/0.67 => ( ( groups73079841787564623at_rat
% 0.33/0.67 @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
% 0.33/0.67 @ S )
% 0.33/0.67 = one_one_rat ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.delta
% 0.33/0.67 thf(fact_8588_prod_Odelta_H,axiom,
% 0.33/0.67 ! [S: set_real,A: real,B: real > complex] :
% 0.33/0.67 ( ( finite_finite_real @ S )
% 0.33/0.67 => ( ( ( member_real @ A @ S )
% 0.33/0.67 => ( ( groups713298508707869441omplex
% 0.33/0.67 @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 0.33/0.67 @ S )
% 0.33/0.67 = ( B @ A ) ) )
% 0.33/0.67 & ( ~ ( member_real @ A @ S )
% 0.33/0.67 => ( ( groups713298508707869441omplex
% 0.33/0.67 @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 0.33/0.67 @ S )
% 0.33/0.67 = one_one_complex ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.delta'
% 0.33/0.67 thf(fact_8589_prod_Odelta_H,axiom,
% 0.33/0.67 ! [S: set_nat,A: nat,B: nat > complex] :
% 0.33/0.67 ( ( finite_finite_nat @ S )
% 0.33/0.67 => ( ( ( member_nat @ A @ S )
% 0.33/0.67 => ( ( groups6464643781859351333omplex
% 0.33/0.67 @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 0.33/0.67 @ S )
% 0.33/0.67 = ( B @ A ) ) )
% 0.33/0.67 & ( ~ ( member_nat @ A @ S )
% 0.33/0.67 => ( ( groups6464643781859351333omplex
% 0.33/0.67 @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 0.33/0.67 @ S )
% 0.33/0.67 = one_one_complex ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.delta'
% 0.33/0.67 thf(fact_8590_prod_Odelta_H,axiom,
% 0.33/0.67 ! [S: set_int,A: int,B: int > complex] :
% 0.33/0.67 ( ( finite_finite_int @ S )
% 0.33/0.67 => ( ( ( member_int @ A @ S )
% 0.33/0.67 => ( ( groups7440179247065528705omplex
% 0.33/0.67 @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 0.33/0.67 @ S )
% 0.33/0.67 = ( B @ A ) ) )
% 0.33/0.67 & ( ~ ( member_int @ A @ S )
% 0.33/0.67 => ( ( groups7440179247065528705omplex
% 0.33/0.67 @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 0.33/0.67 @ S )
% 0.33/0.67 = one_one_complex ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.delta'
% 0.33/0.67 thf(fact_8591_prod_Odelta_H,axiom,
% 0.33/0.67 ! [S: set_complex,A: complex,B: complex > complex] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ S )
% 0.33/0.67 => ( ( ( member_complex @ A @ S )
% 0.33/0.67 => ( ( groups3708469109370488835omplex
% 0.33/0.67 @ ^ [K3: complex] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 0.33/0.67 @ S )
% 0.33/0.67 = ( B @ A ) ) )
% 0.33/0.67 & ( ~ ( member_complex @ A @ S )
% 0.33/0.67 => ( ( groups3708469109370488835omplex
% 0.33/0.67 @ ^ [K3: complex] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 0.33/0.67 @ S )
% 0.33/0.67 = one_one_complex ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.delta'
% 0.33/0.67 thf(fact_8592_prod_Odelta_H,axiom,
% 0.33/0.67 ! [S: set_real,A: real,B: real > real] :
% 0.33/0.67 ( ( finite_finite_real @ S )
% 0.33/0.67 => ( ( ( member_real @ A @ S )
% 0.33/0.67 => ( ( groups1681761925125756287l_real
% 0.33/0.67 @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 0.33/0.67 @ S )
% 0.33/0.67 = ( B @ A ) ) )
% 0.33/0.67 & ( ~ ( member_real @ A @ S )
% 0.33/0.67 => ( ( groups1681761925125756287l_real
% 0.33/0.67 @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 0.33/0.67 @ S )
% 0.33/0.67 = one_one_real ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.delta'
% 0.33/0.67 thf(fact_8593_prod_Odelta_H,axiom,
% 0.33/0.67 ! [S: set_nat,A: nat,B: nat > real] :
% 0.33/0.67 ( ( finite_finite_nat @ S )
% 0.33/0.67 => ( ( ( member_nat @ A @ S )
% 0.33/0.67 => ( ( groups129246275422532515t_real
% 0.33/0.67 @ ^ [K3: nat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 0.33/0.67 @ S )
% 0.33/0.67 = ( B @ A ) ) )
% 0.33/0.67 & ( ~ ( member_nat @ A @ S )
% 0.33/0.67 => ( ( groups129246275422532515t_real
% 0.33/0.67 @ ^ [K3: nat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 0.33/0.67 @ S )
% 0.33/0.67 = one_one_real ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.delta'
% 0.33/0.67 thf(fact_8594_prod_Odelta_H,axiom,
% 0.33/0.67 ! [S: set_int,A: int,B: int > real] :
% 0.33/0.67 ( ( finite_finite_int @ S )
% 0.33/0.67 => ( ( ( member_int @ A @ S )
% 0.33/0.67 => ( ( groups2316167850115554303t_real
% 0.33/0.67 @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 0.33/0.67 @ S )
% 0.33/0.67 = ( B @ A ) ) )
% 0.33/0.67 & ( ~ ( member_int @ A @ S )
% 0.33/0.67 => ( ( groups2316167850115554303t_real
% 0.33/0.67 @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 0.33/0.67 @ S )
% 0.33/0.67 = one_one_real ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.delta'
% 0.33/0.67 thf(fact_8595_prod_Odelta_H,axiom,
% 0.33/0.67 ! [S: set_complex,A: complex,B: complex > real] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ S )
% 0.33/0.67 => ( ( ( member_complex @ A @ S )
% 0.33/0.67 => ( ( groups766887009212190081x_real
% 0.33/0.67 @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 0.33/0.67 @ S )
% 0.33/0.67 = ( B @ A ) ) )
% 0.33/0.67 & ( ~ ( member_complex @ A @ S )
% 0.33/0.67 => ( ( groups766887009212190081x_real
% 0.33/0.67 @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 0.33/0.67 @ S )
% 0.33/0.67 = one_one_real ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.delta'
% 0.33/0.67 thf(fact_8596_prod_Odelta_H,axiom,
% 0.33/0.67 ! [S: set_real,A: real,B: real > rat] :
% 0.33/0.67 ( ( finite_finite_real @ S )
% 0.33/0.67 => ( ( ( member_real @ A @ S )
% 0.33/0.67 => ( ( groups4061424788464935467al_rat
% 0.33/0.67 @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
% 0.33/0.67 @ S )
% 0.33/0.67 = ( B @ A ) ) )
% 0.33/0.67 & ( ~ ( member_real @ A @ S )
% 0.33/0.67 => ( ( groups4061424788464935467al_rat
% 0.33/0.67 @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
% 0.33/0.67 @ S )
% 0.33/0.67 = one_one_rat ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.delta'
% 0.33/0.67 thf(fact_8597_prod_Odelta_H,axiom,
% 0.33/0.67 ! [S: set_nat,A: nat,B: nat > rat] :
% 0.33/0.67 ( ( finite_finite_nat @ S )
% 0.33/0.67 => ( ( ( member_nat @ A @ S )
% 0.33/0.67 => ( ( groups73079841787564623at_rat
% 0.33/0.67 @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
% 0.33/0.67 @ S )
% 0.33/0.67 = ( B @ A ) ) )
% 0.33/0.67 & ( ~ ( member_nat @ A @ S )
% 0.33/0.67 => ( ( groups73079841787564623at_rat
% 0.33/0.67 @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
% 0.33/0.67 @ S )
% 0.33/0.67 = one_one_rat ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.delta'
% 0.33/0.67 thf(fact_8598_prod_OlessThan__Suc,axiom,
% 0.33/0.67 ! [G: nat > real,N2: nat] :
% 0.33/0.67 ( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.lessThan_Suc
% 0.33/0.67 thf(fact_8599_prod_OlessThan__Suc,axiom,
% 0.33/0.67 ! [G: nat > rat,N2: nat] :
% 0.33/0.67 ( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.lessThan_Suc
% 0.33/0.67 thf(fact_8600_prod_OlessThan__Suc,axiom,
% 0.33/0.67 ! [G: nat > nat,N2: nat] :
% 0.33/0.67 ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.lessThan_Suc
% 0.33/0.67 thf(fact_8601_prod_OlessThan__Suc,axiom,
% 0.33/0.67 ! [G: nat > int,N2: nat] :
% 0.33/0.67 ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.lessThan_Suc
% 0.33/0.67 thf(fact_8602_arccos__minus__1,axiom,
% 0.33/0.67 ( ( arccos @ ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.67 = pi ) ).
% 0.33/0.67
% 0.33/0.67 % arccos_minus_1
% 0.33/0.67 thf(fact_8603_cos__arccos,axiom,
% 0.33/0.67 ! [Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.67 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.67 => ( ( cos_real @ ( arccos @ Y ) )
% 0.33/0.67 = Y ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % cos_arccos
% 0.33/0.67 thf(fact_8604_sin__arcsin,axiom,
% 0.33/0.67 ! [Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.67 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.67 => ( ( sin_real @ ( arcsin @ Y ) )
% 0.33/0.67 = Y ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % sin_arcsin
% 0.33/0.67 thf(fact_8605_prod_Ocl__ivl__Suc,axiom,
% 0.33/0.67 ! [N2: nat,M: nat,G: nat > complex] :
% 0.33/0.67 ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.33/0.67 => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.33/0.67 => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.cl_ivl_Suc
% 0.33/0.67 thf(fact_8606_prod_Ocl__ivl__Suc,axiom,
% 0.33/0.67 ! [N2: nat,M: nat,G: nat > real] :
% 0.33/0.67 ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.33/0.67 => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.33/0.67 = one_one_real ) )
% 0.33/0.67 & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.33/0.67 => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.cl_ivl_Suc
% 0.33/0.67 thf(fact_8607_prod_Ocl__ivl__Suc,axiom,
% 0.33/0.67 ! [N2: nat,M: nat,G: nat > rat] :
% 0.33/0.67 ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.33/0.67 => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.33/0.67 = one_one_rat ) )
% 0.33/0.67 & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.33/0.67 => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.cl_ivl_Suc
% 0.33/0.67 thf(fact_8608_prod_Ocl__ivl__Suc,axiom,
% 0.33/0.67 ! [N2: nat,M: nat,G: nat > nat] :
% 0.33/0.67 ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.33/0.67 => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.33/0.67 = one_one_nat ) )
% 0.33/0.67 & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.33/0.67 => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.cl_ivl_Suc
% 0.33/0.67 thf(fact_8609_prod_Ocl__ivl__Suc,axiom,
% 0.33/0.67 ! [N2: nat,M: nat,G: nat > int] :
% 0.33/0.67 ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.33/0.67 => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.33/0.67 = one_one_int ) )
% 0.33/0.67 & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.33/0.67 => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.cl_ivl_Suc
% 0.33/0.67 thf(fact_8610_arccos__0,axiom,
% 0.33/0.67 ( ( arccos @ zero_zero_real )
% 0.33/0.67 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arccos_0
% 0.33/0.67 thf(fact_8611_arcsin__1,axiom,
% 0.33/0.67 ( ( arcsin @ one_one_real )
% 0.33/0.67 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arcsin_1
% 0.33/0.67 thf(fact_8612_arcsin__minus__1,axiom,
% 0.33/0.67 ( ( arcsin @ ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.67 = ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arcsin_minus_1
% 0.33/0.67 thf(fact_8613_prod_Oswap,axiom,
% 0.33/0.67 ! [G: nat > nat > nat,B3: set_nat,A2: set_nat] :
% 0.33/0.67 ( ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [I3: nat] : ( groups708209901874060359at_nat @ ( G @ I3 ) @ B3 )
% 0.33/0.67 @ A2 )
% 0.33/0.67 = ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [J3: nat] :
% 0.33/0.67 ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [I3: nat] : ( G @ I3 @ J3 )
% 0.33/0.67 @ A2 )
% 0.33/0.67 @ B3 ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.swap
% 0.33/0.67 thf(fact_8614_prod_Oswap,axiom,
% 0.33/0.67 ! [G: nat > nat > int,B3: set_nat,A2: set_nat] :
% 0.33/0.67 ( ( groups705719431365010083at_int
% 0.33/0.67 @ ^ [I3: nat] : ( groups705719431365010083at_int @ ( G @ I3 ) @ B3 )
% 0.33/0.67 @ A2 )
% 0.33/0.67 = ( groups705719431365010083at_int
% 0.33/0.67 @ ^ [J3: nat] :
% 0.33/0.67 ( groups705719431365010083at_int
% 0.33/0.67 @ ^ [I3: nat] : ( G @ I3 @ J3 )
% 0.33/0.67 @ A2 )
% 0.33/0.67 @ B3 ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.swap
% 0.33/0.67 thf(fact_8615_prod_Oswap,axiom,
% 0.33/0.67 ! [G: nat > int > int,B3: set_int,A2: set_nat] :
% 0.33/0.67 ( ( groups705719431365010083at_int
% 0.33/0.67 @ ^ [I3: nat] : ( groups1705073143266064639nt_int @ ( G @ I3 ) @ B3 )
% 0.33/0.67 @ A2 )
% 0.33/0.67 = ( groups1705073143266064639nt_int
% 0.33/0.67 @ ^ [J3: int] :
% 0.33/0.67 ( groups705719431365010083at_int
% 0.33/0.67 @ ^ [I3: nat] : ( G @ I3 @ J3 )
% 0.33/0.67 @ A2 )
% 0.33/0.67 @ B3 ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.swap
% 0.33/0.67 thf(fact_8616_prod_Oswap,axiom,
% 0.33/0.67 ! [G: int > nat > int,B3: set_nat,A2: set_int] :
% 0.33/0.67 ( ( groups1705073143266064639nt_int
% 0.33/0.67 @ ^ [I3: int] : ( groups705719431365010083at_int @ ( G @ I3 ) @ B3 )
% 0.33/0.67 @ A2 )
% 0.33/0.67 = ( groups705719431365010083at_int
% 0.33/0.67 @ ^ [J3: nat] :
% 0.33/0.67 ( groups1705073143266064639nt_int
% 0.33/0.67 @ ^ [I3: int] : ( G @ I3 @ J3 )
% 0.33/0.67 @ A2 )
% 0.33/0.67 @ B3 ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.swap
% 0.33/0.67 thf(fact_8617_prod_Oswap,axiom,
% 0.33/0.67 ! [G: int > int > int,B3: set_int,A2: set_int] :
% 0.33/0.67 ( ( groups1705073143266064639nt_int
% 0.33/0.67 @ ^ [I3: int] : ( groups1705073143266064639nt_int @ ( G @ I3 ) @ B3 )
% 0.33/0.67 @ A2 )
% 0.33/0.67 = ( groups1705073143266064639nt_int
% 0.33/0.67 @ ^ [J3: int] :
% 0.33/0.67 ( groups1705073143266064639nt_int
% 0.33/0.67 @ ^ [I3: int] : ( G @ I3 @ J3 )
% 0.33/0.67 @ A2 )
% 0.33/0.67 @ B3 ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.swap
% 0.33/0.67 thf(fact_8618_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.33/0.67 ! [G: nat > complex,A2: set_nat] :
% 0.33/0.67 ( ( ( groups6464643781859351333omplex @ G @ A2 )
% 0.33/0.67 != one_one_complex )
% 0.33/0.67 => ~ ! [A3: nat] :
% 0.33/0.67 ( ( member_nat @ A3 @ A2 )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_complex ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.not_neutral_contains_not_neutral
% 0.33/0.67 thf(fact_8619_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.33/0.67 ! [G: real > complex,A2: set_real] :
% 0.33/0.67 ( ( ( groups713298508707869441omplex @ G @ A2 )
% 0.33/0.67 != one_one_complex )
% 0.33/0.67 => ~ ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ A2 )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_complex ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.not_neutral_contains_not_neutral
% 0.33/0.67 thf(fact_8620_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.33/0.67 ! [G: int > complex,A2: set_int] :
% 0.33/0.67 ( ( ( groups7440179247065528705omplex @ G @ A2 )
% 0.33/0.67 != one_one_complex )
% 0.33/0.67 => ~ ! [A3: int] :
% 0.33/0.67 ( ( member_int @ A3 @ A2 )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_complex ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.not_neutral_contains_not_neutral
% 0.33/0.67 thf(fact_8621_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.33/0.67 ! [G: complex > complex,A2: set_complex] :
% 0.33/0.67 ( ( ( groups3708469109370488835omplex @ G @ A2 )
% 0.33/0.67 != one_one_complex )
% 0.33/0.67 => ~ ! [A3: complex] :
% 0.33/0.67 ( ( member_complex @ A3 @ A2 )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_complex ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.not_neutral_contains_not_neutral
% 0.33/0.67 thf(fact_8622_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.33/0.67 ! [G: nat > real,A2: set_nat] :
% 0.33/0.67 ( ( ( groups129246275422532515t_real @ G @ A2 )
% 0.33/0.67 != one_one_real )
% 0.33/0.67 => ~ ! [A3: nat] :
% 0.33/0.67 ( ( member_nat @ A3 @ A2 )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_real ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.not_neutral_contains_not_neutral
% 0.33/0.67 thf(fact_8623_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.33/0.67 ! [G: real > real,A2: set_real] :
% 0.33/0.67 ( ( ( groups1681761925125756287l_real @ G @ A2 )
% 0.33/0.67 != one_one_real )
% 0.33/0.67 => ~ ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ A2 )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_real ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.not_neutral_contains_not_neutral
% 0.33/0.67 thf(fact_8624_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.33/0.67 ! [G: int > real,A2: set_int] :
% 0.33/0.67 ( ( ( groups2316167850115554303t_real @ G @ A2 )
% 0.33/0.67 != one_one_real )
% 0.33/0.67 => ~ ! [A3: int] :
% 0.33/0.67 ( ( member_int @ A3 @ A2 )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_real ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.not_neutral_contains_not_neutral
% 0.33/0.67 thf(fact_8625_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.33/0.67 ! [G: complex > real,A2: set_complex] :
% 0.33/0.67 ( ( ( groups766887009212190081x_real @ G @ A2 )
% 0.33/0.67 != one_one_real )
% 0.33/0.67 => ~ ! [A3: complex] :
% 0.33/0.67 ( ( member_complex @ A3 @ A2 )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_real ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.not_neutral_contains_not_neutral
% 0.33/0.67 thf(fact_8626_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.33/0.67 ! [G: nat > rat,A2: set_nat] :
% 0.33/0.67 ( ( ( groups73079841787564623at_rat @ G @ A2 )
% 0.33/0.67 != one_one_rat )
% 0.33/0.67 => ~ ! [A3: nat] :
% 0.33/0.67 ( ( member_nat @ A3 @ A2 )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_rat ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.not_neutral_contains_not_neutral
% 0.33/0.67 thf(fact_8627_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.33/0.67 ! [G: real > rat,A2: set_real] :
% 0.33/0.67 ( ( ( groups4061424788464935467al_rat @ G @ A2 )
% 0.33/0.67 != one_one_rat )
% 0.33/0.67 => ~ ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ A2 )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_rat ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.not_neutral_contains_not_neutral
% 0.33/0.67 thf(fact_8628_prod_Oneutral,axiom,
% 0.33/0.67 ! [A2: set_nat,G: nat > nat] :
% 0.33/0.67 ( ! [X4: nat] :
% 0.33/0.67 ( ( member_nat @ X4 @ A2 )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_nat ) )
% 0.33/0.67 => ( ( groups708209901874060359at_nat @ G @ A2 )
% 0.33/0.67 = one_one_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.neutral
% 0.33/0.67 thf(fact_8629_prod_Oneutral,axiom,
% 0.33/0.67 ! [A2: set_nat,G: nat > int] :
% 0.33/0.67 ( ! [X4: nat] :
% 0.33/0.67 ( ( member_nat @ X4 @ A2 )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_int ) )
% 0.33/0.67 => ( ( groups705719431365010083at_int @ G @ A2 )
% 0.33/0.67 = one_one_int ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.neutral
% 0.33/0.67 thf(fact_8630_prod_Oneutral,axiom,
% 0.33/0.67 ! [A2: set_int,G: int > int] :
% 0.33/0.67 ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ A2 )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_int ) )
% 0.33/0.67 => ( ( groups1705073143266064639nt_int @ G @ A2 )
% 0.33/0.67 = one_one_int ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.neutral
% 0.33/0.67 thf(fact_8631_prod_Odistrib,axiom,
% 0.33/0.67 ! [G: nat > nat,H2: nat > nat,A2: set_nat] :
% 0.33/0.67 ( ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [X2: nat] : ( times_times_nat @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 0.33/0.67 @ A2 )
% 0.33/0.67 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ A2 ) @ ( groups708209901874060359at_nat @ H2 @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.distrib
% 0.33/0.67 thf(fact_8632_prod_Odistrib,axiom,
% 0.33/0.67 ! [G: nat > int,H2: nat > int,A2: set_nat] :
% 0.33/0.67 ( ( groups705719431365010083at_int
% 0.33/0.67 @ ^ [X2: nat] : ( times_times_int @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 0.33/0.67 @ A2 )
% 0.33/0.67 = ( times_times_int @ ( groups705719431365010083at_int @ G @ A2 ) @ ( groups705719431365010083at_int @ H2 @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.distrib
% 0.33/0.67 thf(fact_8633_prod_Odistrib,axiom,
% 0.33/0.67 ! [G: int > int,H2: int > int,A2: set_int] :
% 0.33/0.67 ( ( groups1705073143266064639nt_int
% 0.33/0.67 @ ^ [X2: int] : ( times_times_int @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 0.33/0.67 @ A2 )
% 0.33/0.67 = ( times_times_int @ ( groups1705073143266064639nt_int @ G @ A2 ) @ ( groups1705073143266064639nt_int @ H2 @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.distrib
% 0.33/0.67 thf(fact_8634_prod__power__distrib,axiom,
% 0.33/0.67 ! [F: nat > nat,A2: set_nat,N2: nat] :
% 0.33/0.67 ( ( power_power_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ N2 )
% 0.33/0.67 = ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [X2: nat] : ( power_power_nat @ ( F @ X2 ) @ N2 )
% 0.33/0.67 @ A2 ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_power_distrib
% 0.33/0.67 thf(fact_8635_prod__power__distrib,axiom,
% 0.33/0.67 ! [F: nat > int,A2: set_nat,N2: nat] :
% 0.33/0.67 ( ( power_power_int @ ( groups705719431365010083at_int @ F @ A2 ) @ N2 )
% 0.33/0.67 = ( groups705719431365010083at_int
% 0.33/0.67 @ ^ [X2: nat] : ( power_power_int @ ( F @ X2 ) @ N2 )
% 0.33/0.67 @ A2 ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_power_distrib
% 0.33/0.67 thf(fact_8636_prod__power__distrib,axiom,
% 0.33/0.67 ! [F: int > int,A2: set_int,N2: nat] :
% 0.33/0.67 ( ( power_power_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ N2 )
% 0.33/0.67 = ( groups1705073143266064639nt_int
% 0.33/0.67 @ ^ [X2: int] : ( power_power_int @ ( F @ X2 ) @ N2 )
% 0.33/0.67 @ A2 ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_power_distrib
% 0.33/0.67 thf(fact_8637_prod_Oswap__restrict,axiom,
% 0.33/0.67 ! [A2: set_real,B3: set_nat,G: real > nat > nat,R2: real > nat > $o] :
% 0.33/0.67 ( ( finite_finite_real @ A2 )
% 0.33/0.67 => ( ( finite_finite_nat @ B3 )
% 0.33/0.67 => ( ( groups4696554848551431203al_nat
% 0.33/0.67 @ ^ [X2: real] :
% 0.33/0.67 ( groups708209901874060359at_nat @ ( G @ X2 )
% 0.33/0.67 @ ( collect_nat
% 0.33/0.67 @ ^ [Y2: nat] :
% 0.33/0.67 ( ( member_nat @ Y2 @ B3 )
% 0.33/0.67 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.33/0.67 @ A2 )
% 0.33/0.67 = ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [Y2: nat] :
% 0.33/0.67 ( groups4696554848551431203al_nat
% 0.33/0.67 @ ^ [X2: real] : ( G @ X2 @ Y2 )
% 0.33/0.67 @ ( collect_real
% 0.33/0.67 @ ^ [X2: real] :
% 0.33/0.67 ( ( member_real @ X2 @ A2 )
% 0.33/0.67 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.33/0.67 @ B3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.swap_restrict
% 0.33/0.67 thf(fact_8638_prod_Oswap__restrict,axiom,
% 0.33/0.67 ! [A2: set_int,B3: set_nat,G: int > nat > nat,R2: int > nat > $o] :
% 0.33/0.67 ( ( finite_finite_int @ A2 )
% 0.33/0.67 => ( ( finite_finite_nat @ B3 )
% 0.33/0.67 => ( ( groups1707563613775114915nt_nat
% 0.33/0.67 @ ^ [X2: int] :
% 0.33/0.67 ( groups708209901874060359at_nat @ ( G @ X2 )
% 0.33/0.67 @ ( collect_nat
% 0.33/0.67 @ ^ [Y2: nat] :
% 0.33/0.67 ( ( member_nat @ Y2 @ B3 )
% 0.33/0.67 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.33/0.67 @ A2 )
% 0.33/0.67 = ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [Y2: nat] :
% 0.33/0.67 ( groups1707563613775114915nt_nat
% 0.33/0.67 @ ^ [X2: int] : ( G @ X2 @ Y2 )
% 0.33/0.67 @ ( collect_int
% 0.33/0.67 @ ^ [X2: int] :
% 0.33/0.67 ( ( member_int @ X2 @ A2 )
% 0.33/0.67 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.33/0.67 @ B3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.swap_restrict
% 0.33/0.67 thf(fact_8639_prod_Oswap__restrict,axiom,
% 0.33/0.67 ! [A2: set_complex,B3: set_nat,G: complex > nat > nat,R2: complex > nat > $o] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.67 => ( ( finite_finite_nat @ B3 )
% 0.33/0.67 => ( ( groups861055069439313189ex_nat
% 0.33/0.67 @ ^ [X2: complex] :
% 0.33/0.67 ( groups708209901874060359at_nat @ ( G @ X2 )
% 0.33/0.67 @ ( collect_nat
% 0.33/0.67 @ ^ [Y2: nat] :
% 0.33/0.67 ( ( member_nat @ Y2 @ B3 )
% 0.33/0.67 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.33/0.67 @ A2 )
% 0.33/0.67 = ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [Y2: nat] :
% 0.33/0.67 ( groups861055069439313189ex_nat
% 0.33/0.67 @ ^ [X2: complex] : ( G @ X2 @ Y2 )
% 0.33/0.67 @ ( collect_complex
% 0.33/0.67 @ ^ [X2: complex] :
% 0.33/0.67 ( ( member_complex @ X2 @ A2 )
% 0.33/0.67 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.33/0.67 @ B3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.swap_restrict
% 0.33/0.67 thf(fact_8640_prod_Oswap__restrict,axiom,
% 0.33/0.67 ! [A2: set_real,B3: set_nat,G: real > nat > int,R2: real > nat > $o] :
% 0.33/0.67 ( ( finite_finite_real @ A2 )
% 0.33/0.67 => ( ( finite_finite_nat @ B3 )
% 0.33/0.67 => ( ( groups4694064378042380927al_int
% 0.33/0.67 @ ^ [X2: real] :
% 0.33/0.67 ( groups705719431365010083at_int @ ( G @ X2 )
% 0.33/0.67 @ ( collect_nat
% 0.33/0.67 @ ^ [Y2: nat] :
% 0.33/0.67 ( ( member_nat @ Y2 @ B3 )
% 0.33/0.67 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.33/0.67 @ A2 )
% 0.33/0.67 = ( groups705719431365010083at_int
% 0.33/0.67 @ ^ [Y2: nat] :
% 0.33/0.67 ( groups4694064378042380927al_int
% 0.33/0.67 @ ^ [X2: real] : ( G @ X2 @ Y2 )
% 0.33/0.67 @ ( collect_real
% 0.33/0.67 @ ^ [X2: real] :
% 0.33/0.67 ( ( member_real @ X2 @ A2 )
% 0.33/0.67 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.33/0.67 @ B3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.swap_restrict
% 0.33/0.67 thf(fact_8641_prod_Oswap__restrict,axiom,
% 0.33/0.67 ! [A2: set_complex,B3: set_nat,G: complex > nat > int,R2: complex > nat > $o] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.67 => ( ( finite_finite_nat @ B3 )
% 0.33/0.67 => ( ( groups858564598930262913ex_int
% 0.33/0.67 @ ^ [X2: complex] :
% 0.33/0.67 ( groups705719431365010083at_int @ ( G @ X2 )
% 0.33/0.67 @ ( collect_nat
% 0.33/0.67 @ ^ [Y2: nat] :
% 0.33/0.67 ( ( member_nat @ Y2 @ B3 )
% 0.33/0.67 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.33/0.67 @ A2 )
% 0.33/0.67 = ( groups705719431365010083at_int
% 0.33/0.67 @ ^ [Y2: nat] :
% 0.33/0.67 ( groups858564598930262913ex_int
% 0.33/0.67 @ ^ [X2: complex] : ( G @ X2 @ Y2 )
% 0.33/0.67 @ ( collect_complex
% 0.33/0.67 @ ^ [X2: complex] :
% 0.33/0.67 ( ( member_complex @ X2 @ A2 )
% 0.33/0.67 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.33/0.67 @ B3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.swap_restrict
% 0.33/0.67 thf(fact_8642_prod_Oswap__restrict,axiom,
% 0.33/0.67 ! [A2: set_real,B3: set_int,G: real > int > int,R2: real > int > $o] :
% 0.33/0.67 ( ( finite_finite_real @ A2 )
% 0.33/0.67 => ( ( finite_finite_int @ B3 )
% 0.33/0.67 => ( ( groups4694064378042380927al_int
% 0.33/0.67 @ ^ [X2: real] :
% 0.33/0.67 ( groups1705073143266064639nt_int @ ( G @ X2 )
% 0.33/0.67 @ ( collect_int
% 0.33/0.67 @ ^ [Y2: int] :
% 0.33/0.67 ( ( member_int @ Y2 @ B3 )
% 0.33/0.67 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.33/0.67 @ A2 )
% 0.33/0.67 = ( groups1705073143266064639nt_int
% 0.33/0.67 @ ^ [Y2: int] :
% 0.33/0.67 ( groups4694064378042380927al_int
% 0.33/0.67 @ ^ [X2: real] : ( G @ X2 @ Y2 )
% 0.33/0.67 @ ( collect_real
% 0.33/0.67 @ ^ [X2: real] :
% 0.33/0.67 ( ( member_real @ X2 @ A2 )
% 0.33/0.67 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.33/0.67 @ B3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.swap_restrict
% 0.33/0.67 thf(fact_8643_prod_Oswap__restrict,axiom,
% 0.33/0.67 ! [A2: set_complex,B3: set_int,G: complex > int > int,R2: complex > int > $o] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.67 => ( ( finite_finite_int @ B3 )
% 0.33/0.67 => ( ( groups858564598930262913ex_int
% 0.33/0.67 @ ^ [X2: complex] :
% 0.33/0.67 ( groups1705073143266064639nt_int @ ( G @ X2 )
% 0.33/0.67 @ ( collect_int
% 0.33/0.67 @ ^ [Y2: int] :
% 0.33/0.67 ( ( member_int @ Y2 @ B3 )
% 0.33/0.67 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.33/0.67 @ A2 )
% 0.33/0.67 = ( groups1705073143266064639nt_int
% 0.33/0.67 @ ^ [Y2: int] :
% 0.33/0.67 ( groups858564598930262913ex_int
% 0.33/0.67 @ ^ [X2: complex] : ( G @ X2 @ Y2 )
% 0.33/0.67 @ ( collect_complex
% 0.33/0.67 @ ^ [X2: complex] :
% 0.33/0.67 ( ( member_complex @ X2 @ A2 )
% 0.33/0.67 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.33/0.67 @ B3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.swap_restrict
% 0.33/0.67 thf(fact_8644_prod_Oswap__restrict,axiom,
% 0.33/0.67 ! [A2: set_nat,B3: set_real,G: nat > real > nat,R2: nat > real > $o] :
% 0.33/0.67 ( ( finite_finite_nat @ A2 )
% 0.33/0.67 => ( ( finite_finite_real @ B3 )
% 0.33/0.67 => ( ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [X2: nat] :
% 0.33/0.67 ( groups4696554848551431203al_nat @ ( G @ X2 )
% 0.33/0.67 @ ( collect_real
% 0.33/0.67 @ ^ [Y2: real] :
% 0.33/0.67 ( ( member_real @ Y2 @ B3 )
% 0.33/0.67 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.33/0.67 @ A2 )
% 0.33/0.67 = ( groups4696554848551431203al_nat
% 0.33/0.67 @ ^ [Y2: real] :
% 0.33/0.67 ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [X2: nat] : ( G @ X2 @ Y2 )
% 0.33/0.67 @ ( collect_nat
% 0.33/0.67 @ ^ [X2: nat] :
% 0.33/0.67 ( ( member_nat @ X2 @ A2 )
% 0.33/0.67 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.33/0.67 @ B3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.swap_restrict
% 0.33/0.67 thf(fact_8645_prod_Oswap__restrict,axiom,
% 0.33/0.67 ! [A2: set_nat,B3: set_int,G: nat > int > nat,R2: nat > int > $o] :
% 0.33/0.67 ( ( finite_finite_nat @ A2 )
% 0.33/0.67 => ( ( finite_finite_int @ B3 )
% 0.33/0.67 => ( ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [X2: nat] :
% 0.33/0.67 ( groups1707563613775114915nt_nat @ ( G @ X2 )
% 0.33/0.67 @ ( collect_int
% 0.33/0.67 @ ^ [Y2: int] :
% 0.33/0.67 ( ( member_int @ Y2 @ B3 )
% 0.33/0.67 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.33/0.67 @ A2 )
% 0.33/0.67 = ( groups1707563613775114915nt_nat
% 0.33/0.67 @ ^ [Y2: int] :
% 0.33/0.67 ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [X2: nat] : ( G @ X2 @ Y2 )
% 0.33/0.67 @ ( collect_nat
% 0.33/0.67 @ ^ [X2: nat] :
% 0.33/0.67 ( ( member_nat @ X2 @ A2 )
% 0.33/0.67 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.33/0.67 @ B3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.swap_restrict
% 0.33/0.67 thf(fact_8646_prod_Oswap__restrict,axiom,
% 0.33/0.67 ! [A2: set_nat,B3: set_complex,G: nat > complex > nat,R2: nat > complex > $o] :
% 0.33/0.67 ( ( finite_finite_nat @ A2 )
% 0.33/0.67 => ( ( finite3207457112153483333omplex @ B3 )
% 0.33/0.67 => ( ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [X2: nat] :
% 0.33/0.67 ( groups861055069439313189ex_nat @ ( G @ X2 )
% 0.33/0.67 @ ( collect_complex
% 0.33/0.67 @ ^ [Y2: complex] :
% 0.33/0.67 ( ( member_complex @ Y2 @ B3 )
% 0.33/0.67 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.33/0.67 @ A2 )
% 0.33/0.67 = ( groups861055069439313189ex_nat
% 0.33/0.67 @ ^ [Y2: complex] :
% 0.33/0.67 ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [X2: nat] : ( G @ X2 @ Y2 )
% 0.33/0.67 @ ( collect_nat
% 0.33/0.67 @ ^ [X2: nat] :
% 0.33/0.67 ( ( member_nat @ X2 @ A2 )
% 0.33/0.67 & ( R2 @ X2 @ Y2 ) ) ) )
% 0.33/0.67 @ B3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.swap_restrict
% 0.33/0.67 thf(fact_8647_mod__prod__eq,axiom,
% 0.33/0.67 ! [F: nat > nat,A: nat,A2: set_nat] :
% 0.33/0.67 ( ( modulo_modulo_nat
% 0.33/0.67 @ ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [I3: nat] : ( modulo_modulo_nat @ ( F @ I3 ) @ A )
% 0.33/0.67 @ A2 )
% 0.33/0.67 @ A )
% 0.33/0.67 = ( modulo_modulo_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ A ) ) ).
% 0.33/0.67
% 0.33/0.67 % mod_prod_eq
% 0.33/0.67 thf(fact_8648_mod__prod__eq,axiom,
% 0.33/0.67 ! [F: nat > int,A: int,A2: set_nat] :
% 0.33/0.67 ( ( modulo_modulo_int
% 0.33/0.67 @ ( groups705719431365010083at_int
% 0.33/0.67 @ ^ [I3: nat] : ( modulo_modulo_int @ ( F @ I3 ) @ A )
% 0.33/0.67 @ A2 )
% 0.33/0.67 @ A )
% 0.33/0.67 = ( modulo_modulo_int @ ( groups705719431365010083at_int @ F @ A2 ) @ A ) ) ).
% 0.33/0.67
% 0.33/0.67 % mod_prod_eq
% 0.33/0.67 thf(fact_8649_mod__prod__eq,axiom,
% 0.33/0.67 ! [F: int > int,A: int,A2: set_int] :
% 0.33/0.67 ( ( modulo_modulo_int
% 0.33/0.67 @ ( groups1705073143266064639nt_int
% 0.33/0.67 @ ^ [I3: int] : ( modulo_modulo_int @ ( F @ I3 ) @ A )
% 0.33/0.67 @ A2 )
% 0.33/0.67 @ A )
% 0.33/0.67 = ( modulo_modulo_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ A ) ) ).
% 0.33/0.67
% 0.33/0.67 % mod_prod_eq
% 0.33/0.67 thf(fact_8650_prod__mono,axiom,
% 0.33/0.67 ! [A2: set_nat,F: nat > real,G: nat > real] :
% 0.33/0.67 ( ! [I4: nat] :
% 0.33/0.67 ( ( member_nat @ I4 @ A2 )
% 0.33/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
% 0.33/0.67 & ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 0.33/0.67 => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_mono
% 0.33/0.67 thf(fact_8651_prod__mono,axiom,
% 0.33/0.67 ! [A2: set_real,F: real > real,G: real > real] :
% 0.33/0.67 ( ! [I4: real] :
% 0.33/0.67 ( ( member_real @ I4 @ A2 )
% 0.33/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
% 0.33/0.67 & ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 0.33/0.67 => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_mono
% 0.33/0.67 thf(fact_8652_prod__mono,axiom,
% 0.33/0.67 ! [A2: set_int,F: int > real,G: int > real] :
% 0.33/0.67 ( ! [I4: int] :
% 0.33/0.67 ( ( member_int @ I4 @ A2 )
% 0.33/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
% 0.33/0.67 & ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 0.33/0.67 => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_mono
% 0.33/0.67 thf(fact_8653_prod__mono,axiom,
% 0.33/0.67 ! [A2: set_complex,F: complex > real,G: complex > real] :
% 0.33/0.67 ( ! [I4: complex] :
% 0.33/0.67 ( ( member_complex @ I4 @ A2 )
% 0.33/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
% 0.33/0.67 & ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 0.33/0.67 => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A2 ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_mono
% 0.33/0.67 thf(fact_8654_prod__mono,axiom,
% 0.33/0.67 ! [A2: set_nat,F: nat > rat,G: nat > rat] :
% 0.33/0.67 ( ! [I4: nat] :
% 0.33/0.67 ( ( member_nat @ I4 @ A2 )
% 0.33/0.67 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) )
% 0.33/0.67 & ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 0.33/0.67 => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_mono
% 0.33/0.67 thf(fact_8655_prod__mono,axiom,
% 0.33/0.67 ! [A2: set_real,F: real > rat,G: real > rat] :
% 0.33/0.67 ( ! [I4: real] :
% 0.33/0.67 ( ( member_real @ I4 @ A2 )
% 0.33/0.67 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) )
% 0.33/0.67 & ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 0.33/0.67 => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_mono
% 0.33/0.67 thf(fact_8656_prod__mono,axiom,
% 0.33/0.67 ! [A2: set_int,F: int > rat,G: int > rat] :
% 0.33/0.67 ( ! [I4: int] :
% 0.33/0.67 ( ( member_int @ I4 @ A2 )
% 0.33/0.67 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) )
% 0.33/0.67 & ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 0.33/0.67 => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ ( groups1072433553688619179nt_rat @ G @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_mono
% 0.33/0.67 thf(fact_8657_prod__mono,axiom,
% 0.33/0.67 ! [A2: set_complex,F: complex > rat,G: complex > rat] :
% 0.33/0.67 ( ! [I4: complex] :
% 0.33/0.67 ( ( member_complex @ I4 @ A2 )
% 0.33/0.67 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) )
% 0.33/0.67 & ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 0.33/0.67 => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ ( groups225925009352817453ex_rat @ G @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_mono
% 0.33/0.67 thf(fact_8658_prod__mono,axiom,
% 0.33/0.67 ! [A2: set_real,F: real > nat,G: real > nat] :
% 0.33/0.67 ( ! [I4: real] :
% 0.33/0.67 ( ( member_real @ I4 @ A2 )
% 0.33/0.67 => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) )
% 0.33/0.67 & ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 0.33/0.67 => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_mono
% 0.33/0.67 thf(fact_8659_prod__mono,axiom,
% 0.33/0.67 ! [A2: set_int,F: int > nat,G: int > nat] :
% 0.33/0.67 ( ! [I4: int] :
% 0.33/0.67 ( ( member_int @ I4 @ A2 )
% 0.33/0.67 => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) )
% 0.33/0.67 & ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 0.33/0.67 => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_mono
% 0.33/0.67 thf(fact_8660_prod__nonneg,axiom,
% 0.33/0.67 ! [A2: set_nat,F: nat > nat] :
% 0.33/0.67 ( ! [X4: nat] :
% 0.33/0.67 ( ( member_nat @ X4 @ A2 )
% 0.33/0.67 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.33/0.67 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_nonneg
% 0.33/0.67 thf(fact_8661_prod__nonneg,axiom,
% 0.33/0.67 ! [A2: set_nat,F: nat > int] :
% 0.33/0.67 ( ! [X4: nat] :
% 0.33/0.67 ( ( member_nat @ X4 @ A2 )
% 0.33/0.67 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 0.33/0.67 => ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_nonneg
% 0.33/0.67 thf(fact_8662_prod__nonneg,axiom,
% 0.33/0.67 ! [A2: set_int,F: int > int] :
% 0.33/0.67 ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ A2 )
% 0.33/0.67 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 0.33/0.67 => ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_nonneg
% 0.33/0.67 thf(fact_8663_prod__pos,axiom,
% 0.33/0.67 ! [A2: set_nat,F: nat > nat] :
% 0.33/0.67 ( ! [X4: nat] :
% 0.33/0.67 ( ( member_nat @ X4 @ A2 )
% 0.33/0.67 => ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.33/0.67 => ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_pos
% 0.33/0.67 thf(fact_8664_prod__pos,axiom,
% 0.33/0.67 ! [A2: set_nat,F: nat > int] :
% 0.33/0.67 ( ! [X4: nat] :
% 0.33/0.67 ( ( member_nat @ X4 @ A2 )
% 0.33/0.67 => ( ord_less_int @ zero_zero_int @ ( F @ X4 ) ) )
% 0.33/0.67 => ( ord_less_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_pos
% 0.33/0.67 thf(fact_8665_prod__pos,axiom,
% 0.33/0.67 ! [A2: set_int,F: int > int] :
% 0.33/0.67 ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ A2 )
% 0.33/0.67 => ( ord_less_int @ zero_zero_int @ ( F @ X4 ) ) )
% 0.33/0.67 => ( ord_less_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_pos
% 0.33/0.67 thf(fact_8666_prod__ge__1,axiom,
% 0.33/0.67 ! [A2: set_nat,F: nat > real] :
% 0.33/0.67 ( ! [X4: nat] :
% 0.33/0.67 ( ( member_nat @ X4 @ A2 )
% 0.33/0.67 => ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
% 0.33/0.67 => ( ord_less_eq_real @ one_one_real @ ( groups129246275422532515t_real @ F @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_ge_1
% 0.33/0.67 thf(fact_8667_prod__ge__1,axiom,
% 0.33/0.67 ! [A2: set_real,F: real > real] :
% 0.33/0.67 ( ! [X4: real] :
% 0.33/0.67 ( ( member_real @ X4 @ A2 )
% 0.33/0.67 => ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
% 0.33/0.67 => ( ord_less_eq_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_ge_1
% 0.33/0.67 thf(fact_8668_prod__ge__1,axiom,
% 0.33/0.67 ! [A2: set_int,F: int > real] :
% 0.33/0.67 ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ A2 )
% 0.33/0.67 => ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
% 0.33/0.67 => ( ord_less_eq_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_ge_1
% 0.33/0.67 thf(fact_8669_prod__ge__1,axiom,
% 0.33/0.67 ! [A2: set_complex,F: complex > real] :
% 0.33/0.67 ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ A2 )
% 0.33/0.67 => ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
% 0.33/0.67 => ( ord_less_eq_real @ one_one_real @ ( groups766887009212190081x_real @ F @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_ge_1
% 0.33/0.67 thf(fact_8670_prod__ge__1,axiom,
% 0.33/0.67 ! [A2: set_nat,F: nat > rat] :
% 0.33/0.67 ( ! [X4: nat] :
% 0.33/0.67 ( ( member_nat @ X4 @ A2 )
% 0.33/0.67 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
% 0.33/0.67 => ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_ge_1
% 0.33/0.67 thf(fact_8671_prod__ge__1,axiom,
% 0.33/0.67 ! [A2: set_real,F: real > rat] :
% 0.33/0.67 ( ! [X4: real] :
% 0.33/0.67 ( ( member_real @ X4 @ A2 )
% 0.33/0.67 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
% 0.33/0.67 => ( ord_less_eq_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_ge_1
% 0.33/0.67 thf(fact_8672_prod__ge__1,axiom,
% 0.33/0.67 ! [A2: set_int,F: int > rat] :
% 0.33/0.67 ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ A2 )
% 0.33/0.67 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
% 0.33/0.67 => ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_ge_1
% 0.33/0.67 thf(fact_8673_prod__ge__1,axiom,
% 0.33/0.67 ! [A2: set_complex,F: complex > rat] :
% 0.33/0.67 ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ A2 )
% 0.33/0.67 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
% 0.33/0.67 => ( ord_less_eq_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_ge_1
% 0.33/0.67 thf(fact_8674_prod__ge__1,axiom,
% 0.33/0.67 ! [A2: set_real,F: real > nat] :
% 0.33/0.67 ( ! [X4: real] :
% 0.33/0.67 ( ( member_real @ X4 @ A2 )
% 0.33/0.67 => ( ord_less_eq_nat @ one_one_nat @ ( F @ X4 ) ) )
% 0.33/0.67 => ( ord_less_eq_nat @ one_one_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_ge_1
% 0.33/0.67 thf(fact_8675_prod__ge__1,axiom,
% 0.33/0.67 ! [A2: set_int,F: int > nat] :
% 0.33/0.67 ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ A2 )
% 0.33/0.67 => ( ord_less_eq_nat @ one_one_nat @ ( F @ X4 ) ) )
% 0.33/0.67 => ( ord_less_eq_nat @ one_one_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_ge_1
% 0.33/0.67 thf(fact_8676_prod__atLeastAtMost__code,axiom,
% 0.33/0.67 ! [F: nat > complex,A: nat,B: nat] :
% 0.33/0.67 ( ( groups6464643781859351333omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 0.33/0.67 = ( set_fo1517530859248394432omplex
% 0.33/0.67 @ ^ [A5: nat] : ( times_times_complex @ ( F @ A5 ) )
% 0.33/0.67 @ A
% 0.33/0.67 @ B
% 0.33/0.67 @ one_one_complex ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_atLeastAtMost_code
% 0.33/0.67 thf(fact_8677_prod__atLeastAtMost__code,axiom,
% 0.33/0.67 ! [F: nat > real,A: nat,B: nat] :
% 0.33/0.67 ( ( groups129246275422532515t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 0.33/0.67 = ( set_fo3111899725591712190t_real
% 0.33/0.67 @ ^ [A5: nat] : ( times_times_real @ ( F @ A5 ) )
% 0.33/0.67 @ A
% 0.33/0.67 @ B
% 0.33/0.67 @ one_one_real ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_atLeastAtMost_code
% 0.33/0.67 thf(fact_8678_prod__atLeastAtMost__code,axiom,
% 0.33/0.67 ! [F: nat > rat,A: nat,B: nat] :
% 0.33/0.67 ( ( groups73079841787564623at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 0.33/0.67 = ( set_fo1949268297981939178at_rat
% 0.33/0.67 @ ^ [A5: nat] : ( times_times_rat @ ( F @ A5 ) )
% 0.33/0.67 @ A
% 0.33/0.67 @ B
% 0.33/0.67 @ one_one_rat ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_atLeastAtMost_code
% 0.33/0.67 thf(fact_8679_prod__atLeastAtMost__code,axiom,
% 0.33/0.67 ! [F: nat > nat,A: nat,B: nat] :
% 0.33/0.67 ( ( groups708209901874060359at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 0.33/0.67 = ( set_fo2584398358068434914at_nat
% 0.33/0.67 @ ^ [A5: nat] : ( times_times_nat @ ( F @ A5 ) )
% 0.33/0.67 @ A
% 0.33/0.67 @ B
% 0.33/0.67 @ one_one_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_atLeastAtMost_code
% 0.33/0.67 thf(fact_8680_prod__atLeastAtMost__code,axiom,
% 0.33/0.67 ! [F: nat > int,A: nat,B: nat] :
% 0.33/0.67 ( ( groups705719431365010083at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 0.33/0.67 = ( set_fo2581907887559384638at_int
% 0.33/0.67 @ ^ [A5: nat] : ( times_times_int @ ( F @ A5 ) )
% 0.33/0.67 @ A
% 0.33/0.67 @ B
% 0.33/0.67 @ one_one_int ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_atLeastAtMost_code
% 0.33/0.67 thf(fact_8681_prod_Ointer__filter,axiom,
% 0.33/0.67 ! [A2: set_real,G: real > complex,P: real > $o] :
% 0.33/0.67 ( ( finite_finite_real @ A2 )
% 0.33/0.67 => ( ( groups713298508707869441omplex @ G
% 0.33/0.67 @ ( collect_real
% 0.33/0.67 @ ^ [X2: real] :
% 0.33/0.67 ( ( member_real @ X2 @ A2 )
% 0.33/0.67 & ( P @ X2 ) ) ) )
% 0.33/0.67 = ( groups713298508707869441omplex
% 0.33/0.67 @ ^ [X2: real] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_complex )
% 0.33/0.67 @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.inter_filter
% 0.33/0.67 thf(fact_8682_prod_Ointer__filter,axiom,
% 0.33/0.67 ! [A2: set_nat,G: nat > complex,P: nat > $o] :
% 0.33/0.67 ( ( finite_finite_nat @ A2 )
% 0.33/0.67 => ( ( groups6464643781859351333omplex @ G
% 0.33/0.67 @ ( collect_nat
% 0.33/0.67 @ ^ [X2: nat] :
% 0.33/0.67 ( ( member_nat @ X2 @ A2 )
% 0.33/0.67 & ( P @ X2 ) ) ) )
% 0.33/0.67 = ( groups6464643781859351333omplex
% 0.33/0.67 @ ^ [X2: nat] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_complex )
% 0.33/0.67 @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.inter_filter
% 0.33/0.67 thf(fact_8683_prod_Ointer__filter,axiom,
% 0.33/0.67 ! [A2: set_int,G: int > complex,P: int > $o] :
% 0.33/0.67 ( ( finite_finite_int @ A2 )
% 0.33/0.67 => ( ( groups7440179247065528705omplex @ G
% 0.33/0.67 @ ( collect_int
% 0.33/0.67 @ ^ [X2: int] :
% 0.33/0.67 ( ( member_int @ X2 @ A2 )
% 0.33/0.67 & ( P @ X2 ) ) ) )
% 0.33/0.67 = ( groups7440179247065528705omplex
% 0.33/0.67 @ ^ [X2: int] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_complex )
% 0.33/0.67 @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.inter_filter
% 0.33/0.67 thf(fact_8684_prod_Ointer__filter,axiom,
% 0.33/0.67 ! [A2: set_complex,G: complex > complex,P: complex > $o] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.67 => ( ( groups3708469109370488835omplex @ G
% 0.33/0.67 @ ( collect_complex
% 0.33/0.67 @ ^ [X2: complex] :
% 0.33/0.67 ( ( member_complex @ X2 @ A2 )
% 0.33/0.67 & ( P @ X2 ) ) ) )
% 0.33/0.67 = ( groups3708469109370488835omplex
% 0.33/0.67 @ ^ [X2: complex] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_complex )
% 0.33/0.67 @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.inter_filter
% 0.33/0.67 thf(fact_8685_prod_Ointer__filter,axiom,
% 0.33/0.67 ! [A2: set_real,G: real > real,P: real > $o] :
% 0.33/0.67 ( ( finite_finite_real @ A2 )
% 0.33/0.67 => ( ( groups1681761925125756287l_real @ G
% 0.33/0.67 @ ( collect_real
% 0.33/0.67 @ ^ [X2: real] :
% 0.33/0.67 ( ( member_real @ X2 @ A2 )
% 0.33/0.67 & ( P @ X2 ) ) ) )
% 0.33/0.67 = ( groups1681761925125756287l_real
% 0.33/0.67 @ ^ [X2: real] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_real )
% 0.33/0.67 @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.inter_filter
% 0.33/0.67 thf(fact_8686_prod_Ointer__filter,axiom,
% 0.33/0.67 ! [A2: set_nat,G: nat > real,P: nat > $o] :
% 0.33/0.67 ( ( finite_finite_nat @ A2 )
% 0.33/0.67 => ( ( groups129246275422532515t_real @ G
% 0.33/0.67 @ ( collect_nat
% 0.33/0.67 @ ^ [X2: nat] :
% 0.33/0.67 ( ( member_nat @ X2 @ A2 )
% 0.33/0.67 & ( P @ X2 ) ) ) )
% 0.33/0.67 = ( groups129246275422532515t_real
% 0.33/0.67 @ ^ [X2: nat] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_real )
% 0.33/0.67 @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.inter_filter
% 0.33/0.67 thf(fact_8687_prod_Ointer__filter,axiom,
% 0.33/0.67 ! [A2: set_int,G: int > real,P: int > $o] :
% 0.33/0.67 ( ( finite_finite_int @ A2 )
% 0.33/0.67 => ( ( groups2316167850115554303t_real @ G
% 0.33/0.67 @ ( collect_int
% 0.33/0.67 @ ^ [X2: int] :
% 0.33/0.67 ( ( member_int @ X2 @ A2 )
% 0.33/0.67 & ( P @ X2 ) ) ) )
% 0.33/0.67 = ( groups2316167850115554303t_real
% 0.33/0.67 @ ^ [X2: int] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_real )
% 0.33/0.67 @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.inter_filter
% 0.33/0.67 thf(fact_8688_prod_Ointer__filter,axiom,
% 0.33/0.67 ! [A2: set_complex,G: complex > real,P: complex > $o] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.67 => ( ( groups766887009212190081x_real @ G
% 0.33/0.67 @ ( collect_complex
% 0.33/0.67 @ ^ [X2: complex] :
% 0.33/0.67 ( ( member_complex @ X2 @ A2 )
% 0.33/0.67 & ( P @ X2 ) ) ) )
% 0.33/0.67 = ( groups766887009212190081x_real
% 0.33/0.67 @ ^ [X2: complex] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_real )
% 0.33/0.67 @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.inter_filter
% 0.33/0.67 thf(fact_8689_prod_Ointer__filter,axiom,
% 0.33/0.67 ! [A2: set_real,G: real > rat,P: real > $o] :
% 0.33/0.67 ( ( finite_finite_real @ A2 )
% 0.33/0.67 => ( ( groups4061424788464935467al_rat @ G
% 0.33/0.67 @ ( collect_real
% 0.33/0.67 @ ^ [X2: real] :
% 0.33/0.67 ( ( member_real @ X2 @ A2 )
% 0.33/0.67 & ( P @ X2 ) ) ) )
% 0.33/0.67 = ( groups4061424788464935467al_rat
% 0.33/0.67 @ ^ [X2: real] : ( if_rat @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_rat )
% 0.33/0.67 @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.inter_filter
% 0.33/0.67 thf(fact_8690_prod_Ointer__filter,axiom,
% 0.33/0.67 ! [A2: set_nat,G: nat > rat,P: nat > $o] :
% 0.33/0.67 ( ( finite_finite_nat @ A2 )
% 0.33/0.67 => ( ( groups73079841787564623at_rat @ G
% 0.33/0.67 @ ( collect_nat
% 0.33/0.67 @ ^ [X2: nat] :
% 0.33/0.67 ( ( member_nat @ X2 @ A2 )
% 0.33/0.67 & ( P @ X2 ) ) ) )
% 0.33/0.67 = ( groups73079841787564623at_rat
% 0.33/0.67 @ ^ [X2: nat] : ( if_rat @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_rat )
% 0.33/0.67 @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.inter_filter
% 0.33/0.67 thf(fact_8691_prod_Oshift__bounds__cl__Suc__ivl,axiom,
% 0.33/0.67 ! [G: nat > nat,M: nat,N2: nat] :
% 0.33/0.67 ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.33/0.67 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.shift_bounds_cl_Suc_ivl
% 0.33/0.67 thf(fact_8692_prod_Oshift__bounds__cl__Suc__ivl,axiom,
% 0.33/0.67 ! [G: nat > int,M: nat,N2: nat] :
% 0.33/0.67 ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( groups705719431365010083at_int
% 0.33/0.67 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.33/0.67 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.shift_bounds_cl_Suc_ivl
% 0.33/0.67 thf(fact_8693_power__sum,axiom,
% 0.33/0.67 ! [C: real,F: nat > nat,A2: set_nat] :
% 0.33/0.67 ( ( power_power_real @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 0.33/0.67 = ( groups129246275422532515t_real
% 0.33/0.67 @ ^ [A5: nat] : ( power_power_real @ C @ ( F @ A5 ) )
% 0.33/0.67 @ A2 ) ) ).
% 0.33/0.67
% 0.33/0.67 % power_sum
% 0.33/0.67 thf(fact_8694_power__sum,axiom,
% 0.33/0.67 ! [C: complex,F: nat > nat,A2: set_nat] :
% 0.33/0.67 ( ( power_power_complex @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 0.33/0.67 = ( groups6464643781859351333omplex
% 0.33/0.67 @ ^ [A5: nat] : ( power_power_complex @ C @ ( F @ A5 ) )
% 0.33/0.67 @ A2 ) ) ).
% 0.33/0.67
% 0.33/0.67 % power_sum
% 0.33/0.67 thf(fact_8695_power__sum,axiom,
% 0.33/0.67 ! [C: nat,F: nat > nat,A2: set_nat] :
% 0.33/0.67 ( ( power_power_nat @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 0.33/0.67 = ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [A5: nat] : ( power_power_nat @ C @ ( F @ A5 ) )
% 0.33/0.67 @ A2 ) ) ).
% 0.33/0.67
% 0.33/0.67 % power_sum
% 0.33/0.67 thf(fact_8696_power__sum,axiom,
% 0.33/0.67 ! [C: int,F: nat > nat,A2: set_nat] :
% 0.33/0.67 ( ( power_power_int @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 0.33/0.67 = ( groups705719431365010083at_int
% 0.33/0.67 @ ^ [A5: nat] : ( power_power_int @ C @ ( F @ A5 ) )
% 0.33/0.67 @ A2 ) ) ).
% 0.33/0.67
% 0.33/0.67 % power_sum
% 0.33/0.67 thf(fact_8697_power__sum,axiom,
% 0.33/0.67 ! [C: int,F: int > nat,A2: set_int] :
% 0.33/0.67 ( ( power_power_int @ C @ ( groups4541462559716669496nt_nat @ F @ A2 ) )
% 0.33/0.67 = ( groups1705073143266064639nt_int
% 0.33/0.67 @ ^ [A5: int] : ( power_power_int @ C @ ( F @ A5 ) )
% 0.33/0.67 @ A2 ) ) ).
% 0.33/0.67
% 0.33/0.67 % power_sum
% 0.33/0.67 thf(fact_8698_prod_Oshift__bounds__cl__nat__ivl,axiom,
% 0.33/0.67 ! [G: nat > nat,M: nat,K: nat,N2: nat] :
% 0.33/0.67 ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
% 0.33/0.67 = ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
% 0.33/0.67 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.shift_bounds_cl_nat_ivl
% 0.33/0.67 thf(fact_8699_prod_Oshift__bounds__cl__nat__ivl,axiom,
% 0.33/0.67 ! [G: nat > int,M: nat,K: nat,N2: nat] :
% 0.33/0.67 ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
% 0.33/0.67 = ( groups705719431365010083at_int
% 0.33/0.67 @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
% 0.33/0.67 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.shift_bounds_cl_nat_ivl
% 0.33/0.67 thf(fact_8700_prod__le__1,axiom,
% 0.33/0.67 ! [A2: set_nat,F: nat > real] :
% 0.33/0.67 ( ! [X4: nat] :
% 0.33/0.67 ( ( member_nat @ X4 @ A2 )
% 0.33/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 0.33/0.67 & ( ord_less_eq_real @ ( F @ X4 ) @ one_one_real ) ) )
% 0.33/0.67 => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ one_one_real ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_le_1
% 0.33/0.67 thf(fact_8701_prod__le__1,axiom,
% 0.33/0.67 ! [A2: set_real,F: real > real] :
% 0.33/0.67 ( ! [X4: real] :
% 0.33/0.67 ( ( member_real @ X4 @ A2 )
% 0.33/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 0.33/0.67 & ( ord_less_eq_real @ ( F @ X4 ) @ one_one_real ) ) )
% 0.33/0.67 => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ one_one_real ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_le_1
% 0.33/0.67 thf(fact_8702_prod__le__1,axiom,
% 0.33/0.67 ! [A2: set_int,F: int > real] :
% 0.33/0.67 ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ A2 )
% 0.33/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 0.33/0.67 & ( ord_less_eq_real @ ( F @ X4 ) @ one_one_real ) ) )
% 0.33/0.67 => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ one_one_real ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_le_1
% 0.33/0.67 thf(fact_8703_prod__le__1,axiom,
% 0.33/0.67 ! [A2: set_complex,F: complex > real] :
% 0.33/0.67 ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ A2 )
% 0.33/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 0.33/0.67 & ( ord_less_eq_real @ ( F @ X4 ) @ one_one_real ) ) )
% 0.33/0.67 => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A2 ) @ one_one_real ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_le_1
% 0.33/0.67 thf(fact_8704_prod__le__1,axiom,
% 0.33/0.67 ! [A2: set_nat,F: nat > rat] :
% 0.33/0.67 ( ! [X4: nat] :
% 0.33/0.67 ( ( member_nat @ X4 @ A2 )
% 0.33/0.67 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
% 0.33/0.67 & ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
% 0.33/0.67 => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ one_one_rat ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_le_1
% 0.33/0.67 thf(fact_8705_prod__le__1,axiom,
% 0.33/0.67 ! [A2: set_real,F: real > rat] :
% 0.33/0.67 ( ! [X4: real] :
% 0.33/0.67 ( ( member_real @ X4 @ A2 )
% 0.33/0.67 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
% 0.33/0.67 & ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
% 0.33/0.67 => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ one_one_rat ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_le_1
% 0.33/0.67 thf(fact_8706_prod__le__1,axiom,
% 0.33/0.67 ! [A2: set_int,F: int > rat] :
% 0.33/0.67 ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ A2 )
% 0.33/0.67 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
% 0.33/0.67 & ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
% 0.33/0.67 => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ one_one_rat ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_le_1
% 0.33/0.67 thf(fact_8707_prod__le__1,axiom,
% 0.33/0.67 ! [A2: set_complex,F: complex > rat] :
% 0.33/0.67 ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ A2 )
% 0.33/0.67 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
% 0.33/0.67 & ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
% 0.33/0.67 => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ one_one_rat ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_le_1
% 0.33/0.67 thf(fact_8708_prod__le__1,axiom,
% 0.33/0.67 ! [A2: set_real,F: real > nat] :
% 0.33/0.67 ( ! [X4: real] :
% 0.33/0.67 ( ( member_real @ X4 @ A2 )
% 0.33/0.67 => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) )
% 0.33/0.67 & ( ord_less_eq_nat @ ( F @ X4 ) @ one_one_nat ) ) )
% 0.33/0.67 => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ one_one_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_le_1
% 0.33/0.67 thf(fact_8709_prod__le__1,axiom,
% 0.33/0.67 ! [A2: set_int,F: int > nat] :
% 0.33/0.67 ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ A2 )
% 0.33/0.67 => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) )
% 0.33/0.67 & ( ord_less_eq_nat @ ( F @ X4 ) @ one_one_nat ) ) )
% 0.33/0.67 => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ one_one_nat ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_le_1
% 0.33/0.67 thf(fact_8710_prod_Orelated,axiom,
% 0.33/0.67 ! [R2: complex > complex > $o,S: set_nat,H2: nat > complex,G: nat > complex] :
% 0.33/0.67 ( ( R2 @ one_one_complex @ one_one_complex )
% 0.33/0.67 => ( ! [X16: complex,Y15: complex,X23: complex,Y23: complex] :
% 0.33/0.67 ( ( ( R2 @ X16 @ X23 )
% 0.33/0.67 & ( R2 @ Y15 @ Y23 ) )
% 0.33/0.67 => ( R2 @ ( times_times_complex @ X16 @ Y15 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
% 0.33/0.67 => ( ( finite_finite_nat @ S )
% 0.33/0.67 => ( ! [X4: nat] :
% 0.33/0.67 ( ( member_nat @ X4 @ S )
% 0.33/0.67 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.67 => ( R2 @ ( groups6464643781859351333omplex @ H2 @ S ) @ ( groups6464643781859351333omplex @ G @ S ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.related
% 0.33/0.67 thf(fact_8711_prod_Orelated,axiom,
% 0.33/0.67 ! [R2: complex > complex > $o,S: set_int,H2: int > complex,G: int > complex] :
% 0.33/0.67 ( ( R2 @ one_one_complex @ one_one_complex )
% 0.33/0.67 => ( ! [X16: complex,Y15: complex,X23: complex,Y23: complex] :
% 0.33/0.67 ( ( ( R2 @ X16 @ X23 )
% 0.33/0.67 & ( R2 @ Y15 @ Y23 ) )
% 0.33/0.67 => ( R2 @ ( times_times_complex @ X16 @ Y15 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
% 0.33/0.67 => ( ( finite_finite_int @ S )
% 0.33/0.67 => ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ S )
% 0.33/0.67 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.67 => ( R2 @ ( groups7440179247065528705omplex @ H2 @ S ) @ ( groups7440179247065528705omplex @ G @ S ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.related
% 0.33/0.67 thf(fact_8712_prod_Orelated,axiom,
% 0.33/0.67 ! [R2: complex > complex > $o,S: set_complex,H2: complex > complex,G: complex > complex] :
% 0.33/0.67 ( ( R2 @ one_one_complex @ one_one_complex )
% 0.33/0.67 => ( ! [X16: complex,Y15: complex,X23: complex,Y23: complex] :
% 0.33/0.67 ( ( ( R2 @ X16 @ X23 )
% 0.33/0.67 & ( R2 @ Y15 @ Y23 ) )
% 0.33/0.67 => ( R2 @ ( times_times_complex @ X16 @ Y15 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
% 0.33/0.67 => ( ( finite3207457112153483333omplex @ S )
% 0.33/0.67 => ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ S )
% 0.33/0.67 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.67 => ( R2 @ ( groups3708469109370488835omplex @ H2 @ S ) @ ( groups3708469109370488835omplex @ G @ S ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.related
% 0.33/0.67 thf(fact_8713_prod_Orelated,axiom,
% 0.33/0.67 ! [R2: real > real > $o,S: set_nat,H2: nat > real,G: nat > real] :
% 0.33/0.67 ( ( R2 @ one_one_real @ one_one_real )
% 0.33/0.67 => ( ! [X16: real,Y15: real,X23: real,Y23: real] :
% 0.33/0.67 ( ( ( R2 @ X16 @ X23 )
% 0.33/0.67 & ( R2 @ Y15 @ Y23 ) )
% 0.33/0.67 => ( R2 @ ( times_times_real @ X16 @ Y15 ) @ ( times_times_real @ X23 @ Y23 ) ) )
% 0.33/0.67 => ( ( finite_finite_nat @ S )
% 0.33/0.67 => ( ! [X4: nat] :
% 0.33/0.67 ( ( member_nat @ X4 @ S )
% 0.33/0.67 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.67 => ( R2 @ ( groups129246275422532515t_real @ H2 @ S ) @ ( groups129246275422532515t_real @ G @ S ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.related
% 0.33/0.67 thf(fact_8714_prod_Orelated,axiom,
% 0.33/0.67 ! [R2: real > real > $o,S: set_int,H2: int > real,G: int > real] :
% 0.33/0.67 ( ( R2 @ one_one_real @ one_one_real )
% 0.33/0.67 => ( ! [X16: real,Y15: real,X23: real,Y23: real] :
% 0.33/0.67 ( ( ( R2 @ X16 @ X23 )
% 0.33/0.67 & ( R2 @ Y15 @ Y23 ) )
% 0.33/0.67 => ( R2 @ ( times_times_real @ X16 @ Y15 ) @ ( times_times_real @ X23 @ Y23 ) ) )
% 0.33/0.67 => ( ( finite_finite_int @ S )
% 0.33/0.67 => ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ S )
% 0.33/0.67 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.67 => ( R2 @ ( groups2316167850115554303t_real @ H2 @ S ) @ ( groups2316167850115554303t_real @ G @ S ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.related
% 0.33/0.67 thf(fact_8715_prod_Orelated,axiom,
% 0.33/0.67 ! [R2: real > real > $o,S: set_complex,H2: complex > real,G: complex > real] :
% 0.33/0.67 ( ( R2 @ one_one_real @ one_one_real )
% 0.33/0.67 => ( ! [X16: real,Y15: real,X23: real,Y23: real] :
% 0.33/0.67 ( ( ( R2 @ X16 @ X23 )
% 0.33/0.67 & ( R2 @ Y15 @ Y23 ) )
% 0.33/0.67 => ( R2 @ ( times_times_real @ X16 @ Y15 ) @ ( times_times_real @ X23 @ Y23 ) ) )
% 0.33/0.67 => ( ( finite3207457112153483333omplex @ S )
% 0.33/0.67 => ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ S )
% 0.33/0.67 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.67 => ( R2 @ ( groups766887009212190081x_real @ H2 @ S ) @ ( groups766887009212190081x_real @ G @ S ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.related
% 0.33/0.67 thf(fact_8716_prod_Orelated,axiom,
% 0.33/0.67 ! [R2: rat > rat > $o,S: set_nat,H2: nat > rat,G: nat > rat] :
% 0.33/0.67 ( ( R2 @ one_one_rat @ one_one_rat )
% 0.33/0.67 => ( ! [X16: rat,Y15: rat,X23: rat,Y23: rat] :
% 0.33/0.67 ( ( ( R2 @ X16 @ X23 )
% 0.33/0.67 & ( R2 @ Y15 @ Y23 ) )
% 0.33/0.67 => ( R2 @ ( times_times_rat @ X16 @ Y15 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
% 0.33/0.67 => ( ( finite_finite_nat @ S )
% 0.33/0.67 => ( ! [X4: nat] :
% 0.33/0.67 ( ( member_nat @ X4 @ S )
% 0.33/0.67 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.67 => ( R2 @ ( groups73079841787564623at_rat @ H2 @ S ) @ ( groups73079841787564623at_rat @ G @ S ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.related
% 0.33/0.67 thf(fact_8717_prod_Orelated,axiom,
% 0.33/0.67 ! [R2: rat > rat > $o,S: set_int,H2: int > rat,G: int > rat] :
% 0.33/0.67 ( ( R2 @ one_one_rat @ one_one_rat )
% 0.33/0.67 => ( ! [X16: rat,Y15: rat,X23: rat,Y23: rat] :
% 0.33/0.67 ( ( ( R2 @ X16 @ X23 )
% 0.33/0.67 & ( R2 @ Y15 @ Y23 ) )
% 0.33/0.67 => ( R2 @ ( times_times_rat @ X16 @ Y15 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
% 0.33/0.67 => ( ( finite_finite_int @ S )
% 0.33/0.67 => ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ S )
% 0.33/0.67 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.67 => ( R2 @ ( groups1072433553688619179nt_rat @ H2 @ S ) @ ( groups1072433553688619179nt_rat @ G @ S ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.related
% 0.33/0.67 thf(fact_8718_prod_Orelated,axiom,
% 0.33/0.67 ! [R2: rat > rat > $o,S: set_complex,H2: complex > rat,G: complex > rat] :
% 0.33/0.67 ( ( R2 @ one_one_rat @ one_one_rat )
% 0.33/0.67 => ( ! [X16: rat,Y15: rat,X23: rat,Y23: rat] :
% 0.33/0.67 ( ( ( R2 @ X16 @ X23 )
% 0.33/0.67 & ( R2 @ Y15 @ Y23 ) )
% 0.33/0.67 => ( R2 @ ( times_times_rat @ X16 @ Y15 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
% 0.33/0.67 => ( ( finite3207457112153483333omplex @ S )
% 0.33/0.67 => ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ S )
% 0.33/0.67 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.67 => ( R2 @ ( groups225925009352817453ex_rat @ H2 @ S ) @ ( groups225925009352817453ex_rat @ G @ S ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.related
% 0.33/0.67 thf(fact_8719_prod_Orelated,axiom,
% 0.33/0.67 ! [R2: nat > nat > $o,S: set_int,H2: int > nat,G: int > nat] :
% 0.33/0.67 ( ( R2 @ one_one_nat @ one_one_nat )
% 0.33/0.67 => ( ! [X16: nat,Y15: nat,X23: nat,Y23: nat] :
% 0.33/0.67 ( ( ( R2 @ X16 @ X23 )
% 0.33/0.67 & ( R2 @ Y15 @ Y23 ) )
% 0.33/0.67 => ( R2 @ ( times_times_nat @ X16 @ Y15 ) @ ( times_times_nat @ X23 @ Y23 ) ) )
% 0.33/0.67 => ( ( finite_finite_int @ S )
% 0.33/0.67 => ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ S )
% 0.33/0.67 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.33/0.67 => ( R2 @ ( groups1707563613775114915nt_nat @ H2 @ S ) @ ( groups1707563613775114915nt_nat @ G @ S ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.related
% 0.33/0.67 thf(fact_8720_prod__dvd__prod__subset,axiom,
% 0.33/0.67 ! [B3: set_int,A2: set_int,F: int > nat] :
% 0.33/0.67 ( ( finite_finite_int @ B3 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.33/0.67 => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ F @ B3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_dvd_prod_subset
% 0.33/0.67 thf(fact_8721_prod__dvd__prod__subset,axiom,
% 0.33/0.67 ! [B3: set_complex,A2: set_complex,F: complex > nat] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ B3 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 0.33/0.67 => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ F @ B3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_dvd_prod_subset
% 0.33/0.67 thf(fact_8722_prod__dvd__prod__subset,axiom,
% 0.33/0.67 ! [B3: set_complex,A2: set_complex,F: complex > int] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ B3 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 0.33/0.67 => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ F @ B3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_dvd_prod_subset
% 0.33/0.67 thf(fact_8723_prod__dvd__prod__subset,axiom,
% 0.33/0.67 ! [B3: set_int,A2: set_int,F: int > code_integer] :
% 0.33/0.67 ( ( finite_finite_int @ B3 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.33/0.67 => ( dvd_dvd_Code_integer @ ( groups3827104343326376752nteger @ F @ A2 ) @ ( groups3827104343326376752nteger @ F @ B3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_dvd_prod_subset
% 0.33/0.67 thf(fact_8724_prod__dvd__prod__subset,axiom,
% 0.33/0.67 ! [B3: set_complex,A2: set_complex,F: complex > code_integer] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ B3 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 0.33/0.67 => ( dvd_dvd_Code_integer @ ( groups8682486955453173170nteger @ F @ A2 ) @ ( groups8682486955453173170nteger @ F @ B3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_dvd_prod_subset
% 0.33/0.67 thf(fact_8725_prod__dvd__prod__subset,axiom,
% 0.33/0.67 ! [B3: set_nat,A2: set_nat,F: nat > code_integer] :
% 0.33/0.67 ( ( finite_finite_nat @ B3 )
% 0.33/0.67 => ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.33/0.67 => ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A2 ) @ ( groups3455450783089532116nteger @ F @ B3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_dvd_prod_subset
% 0.33/0.67 thf(fact_8726_prod__dvd__prod__subset,axiom,
% 0.33/0.67 ! [B3: set_nat,A2: set_nat,F: nat > nat] :
% 0.33/0.67 ( ( finite_finite_nat @ B3 )
% 0.33/0.67 => ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.33/0.67 => ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ ( groups708209901874060359at_nat @ F @ B3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_dvd_prod_subset
% 0.33/0.67 thf(fact_8727_prod__dvd__prod__subset,axiom,
% 0.33/0.67 ! [B3: set_nat,A2: set_nat,F: nat > int] :
% 0.33/0.67 ( ( finite_finite_nat @ B3 )
% 0.33/0.67 => ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.33/0.67 => ( dvd_dvd_int @ ( groups705719431365010083at_int @ F @ A2 ) @ ( groups705719431365010083at_int @ F @ B3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_dvd_prod_subset
% 0.33/0.67 thf(fact_8728_prod__dvd__prod__subset,axiom,
% 0.33/0.67 ! [B3: set_int,A2: set_int,F: int > int] :
% 0.33/0.67 ( ( finite_finite_int @ B3 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.33/0.67 => ( dvd_dvd_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ ( groups1705073143266064639nt_int @ F @ B3 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_dvd_prod_subset
% 0.33/0.67 thf(fact_8729_prod__dvd__prod__subset2,axiom,
% 0.33/0.67 ! [B3: set_real,A2: set_real,F: real > nat,G: real > nat] :
% 0.33/0.67 ( ( finite_finite_real @ B3 )
% 0.33/0.67 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.33/0.67 => ( ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ A2 )
% 0.33/0.67 => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.33/0.67 => ( dvd_dvd_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ B3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_dvd_prod_subset2
% 0.33/0.67 thf(fact_8730_prod__dvd__prod__subset2,axiom,
% 0.33/0.67 ! [B3: set_int,A2: set_int,F: int > nat,G: int > nat] :
% 0.33/0.67 ( ( finite_finite_int @ B3 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.33/0.67 => ( ! [A3: int] :
% 0.33/0.67 ( ( member_int @ A3 @ A2 )
% 0.33/0.67 => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.33/0.67 => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ B3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_dvd_prod_subset2
% 0.33/0.67 thf(fact_8731_prod__dvd__prod__subset2,axiom,
% 0.33/0.67 ! [B3: set_complex,A2: set_complex,F: complex > nat,G: complex > nat] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ B3 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 0.33/0.67 => ( ! [A3: complex] :
% 0.33/0.67 ( ( member_complex @ A3 @ A2 )
% 0.33/0.67 => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.33/0.67 => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ G @ B3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_dvd_prod_subset2
% 0.33/0.67 thf(fact_8732_prod__dvd__prod__subset2,axiom,
% 0.33/0.67 ! [B3: set_real,A2: set_real,F: real > int,G: real > int] :
% 0.33/0.67 ( ( finite_finite_real @ B3 )
% 0.33/0.67 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.33/0.67 => ( ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ A2 )
% 0.33/0.67 => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.33/0.67 => ( dvd_dvd_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ ( groups4694064378042380927al_int @ G @ B3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_dvd_prod_subset2
% 0.33/0.67 thf(fact_8733_prod__dvd__prod__subset2,axiom,
% 0.33/0.67 ! [B3: set_complex,A2: set_complex,F: complex > int,G: complex > int] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ B3 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 0.33/0.67 => ( ! [A3: complex] :
% 0.33/0.67 ( ( member_complex @ A3 @ A2 )
% 0.33/0.67 => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.33/0.67 => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ G @ B3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_dvd_prod_subset2
% 0.33/0.67 thf(fact_8734_prod__dvd__prod__subset2,axiom,
% 0.33/0.67 ! [B3: set_real,A2: set_real,F: real > code_integer,G: real > code_integer] :
% 0.33/0.67 ( ( finite_finite_real @ B3 )
% 0.33/0.67 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 0.33/0.67 => ( ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ A2 )
% 0.33/0.67 => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.33/0.67 => ( dvd_dvd_Code_integer @ ( groups6225526099057966256nteger @ F @ A2 ) @ ( groups6225526099057966256nteger @ G @ B3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_dvd_prod_subset2
% 0.33/0.67 thf(fact_8735_prod__dvd__prod__subset2,axiom,
% 0.33/0.67 ! [B3: set_int,A2: set_int,F: int > code_integer,G: int > code_integer] :
% 0.33/0.67 ( ( finite_finite_int @ B3 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.33/0.67 => ( ! [A3: int] :
% 0.33/0.67 ( ( member_int @ A3 @ A2 )
% 0.33/0.67 => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.33/0.67 => ( dvd_dvd_Code_integer @ ( groups3827104343326376752nteger @ F @ A2 ) @ ( groups3827104343326376752nteger @ G @ B3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_dvd_prod_subset2
% 0.33/0.67 thf(fact_8736_prod__dvd__prod__subset2,axiom,
% 0.33/0.67 ! [B3: set_complex,A2: set_complex,F: complex > code_integer,G: complex > code_integer] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ B3 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 0.33/0.67 => ( ! [A3: complex] :
% 0.33/0.67 ( ( member_complex @ A3 @ A2 )
% 0.33/0.67 => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.33/0.67 => ( dvd_dvd_Code_integer @ ( groups8682486955453173170nteger @ F @ A2 ) @ ( groups8682486955453173170nteger @ G @ B3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_dvd_prod_subset2
% 0.33/0.67 thf(fact_8737_prod__dvd__prod__subset2,axiom,
% 0.33/0.67 ! [B3: set_nat,A2: set_nat,F: nat > code_integer,G: nat > code_integer] :
% 0.33/0.67 ( ( finite_finite_nat @ B3 )
% 0.33/0.67 => ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.33/0.67 => ( ! [A3: nat] :
% 0.33/0.67 ( ( member_nat @ A3 @ A2 )
% 0.33/0.67 => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.33/0.67 => ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A2 ) @ ( groups3455450783089532116nteger @ G @ B3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_dvd_prod_subset2
% 0.33/0.67 thf(fact_8738_prod__dvd__prod__subset2,axiom,
% 0.33/0.67 ! [B3: set_nat,A2: set_nat,F: nat > nat,G: nat > nat] :
% 0.33/0.67 ( ( finite_finite_nat @ B3 )
% 0.33/0.67 => ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.33/0.67 => ( ! [A3: nat] :
% 0.33/0.67 ( ( member_nat @ A3 @ A2 )
% 0.33/0.67 => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 0.33/0.67 => ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ ( groups708209901874060359at_nat @ G @ B3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_dvd_prod_subset2
% 0.33/0.67 thf(fact_8739_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.33/0.67 ! [S4: set_real,T4: set_real,S: set_real,I2: real > real,J: real > real,T3: set_real,G: real > complex,H2: real > complex] :
% 0.33/0.67 ( ( finite_finite_real @ S4 )
% 0.33/0.67 => ( ( finite_finite_real @ T4 )
% 0.33/0.67 => ( ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 0.33/0.67 => ( ( I2 @ ( J @ A3 ) )
% 0.33/0.67 = A3 ) )
% 0.33/0.67 => ( ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 0.33/0.67 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 0.33/0.67 => ( ! [B2: real] :
% 0.33/0.67 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.33/0.67 => ( ( J @ ( I2 @ B2 ) )
% 0.33/0.67 = B2 ) )
% 0.33/0.67 => ( ! [B2: real] :
% 0.33/0.67 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.33/0.67 => ( member_real @ ( I2 @ B2 ) @ ( minus_minus_set_real @ S @ S4 ) ) )
% 0.33/0.67 => ( ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ S4 )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [B2: real] :
% 0.33/0.67 ( ( member_real @ B2 @ T4 )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ S )
% 0.33/0.67 => ( ( H2 @ ( J @ A3 ) )
% 0.33/0.67 = ( G @ A3 ) ) )
% 0.33/0.67 => ( ( groups713298508707869441omplex @ G @ S )
% 0.33/0.67 = ( groups713298508707869441omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.reindex_bij_witness_not_neutral
% 0.33/0.67 thf(fact_8740_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.33/0.67 ! [S4: set_real,T4: set_int,S: set_real,I2: int > real,J: real > int,T3: set_int,G: real > complex,H2: int > complex] :
% 0.33/0.67 ( ( finite_finite_real @ S4 )
% 0.33/0.67 => ( ( finite_finite_int @ T4 )
% 0.33/0.67 => ( ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 0.33/0.67 => ( ( I2 @ ( J @ A3 ) )
% 0.33/0.67 = A3 ) )
% 0.33/0.67 => ( ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 0.33/0.67 => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 0.33/0.67 => ( ! [B2: int] :
% 0.33/0.67 ( ( member_int @ B2 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.33/0.67 => ( ( J @ ( I2 @ B2 ) )
% 0.33/0.67 = B2 ) )
% 0.33/0.67 => ( ! [B2: int] :
% 0.33/0.67 ( ( member_int @ B2 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.33/0.67 => ( member_real @ ( I2 @ B2 ) @ ( minus_minus_set_real @ S @ S4 ) ) )
% 0.33/0.67 => ( ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ S4 )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [B2: int] :
% 0.33/0.67 ( ( member_int @ B2 @ T4 )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ S )
% 0.33/0.67 => ( ( H2 @ ( J @ A3 ) )
% 0.33/0.67 = ( G @ A3 ) ) )
% 0.33/0.67 => ( ( groups713298508707869441omplex @ G @ S )
% 0.33/0.67 = ( groups7440179247065528705omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.reindex_bij_witness_not_neutral
% 0.33/0.67 thf(fact_8741_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.33/0.67 ! [S4: set_real,T4: set_complex,S: set_real,I2: complex > real,J: real > complex,T3: set_complex,G: real > complex,H2: complex > complex] :
% 0.33/0.67 ( ( finite_finite_real @ S4 )
% 0.33/0.67 => ( ( finite3207457112153483333omplex @ T4 )
% 0.33/0.67 => ( ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 0.33/0.67 => ( ( I2 @ ( J @ A3 ) )
% 0.33/0.67 = A3 ) )
% 0.33/0.67 => ( ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 0.33/0.67 => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 0.33/0.67 => ( ! [B2: complex] :
% 0.33/0.67 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.33/0.67 => ( ( J @ ( I2 @ B2 ) )
% 0.33/0.67 = B2 ) )
% 0.33/0.67 => ( ! [B2: complex] :
% 0.33/0.67 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.33/0.67 => ( member_real @ ( I2 @ B2 ) @ ( minus_minus_set_real @ S @ S4 ) ) )
% 0.33/0.67 => ( ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ S4 )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [B2: complex] :
% 0.33/0.67 ( ( member_complex @ B2 @ T4 )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ S )
% 0.33/0.67 => ( ( H2 @ ( J @ A3 ) )
% 0.33/0.67 = ( G @ A3 ) ) )
% 0.33/0.67 => ( ( groups713298508707869441omplex @ G @ S )
% 0.33/0.67 = ( groups3708469109370488835omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.reindex_bij_witness_not_neutral
% 0.33/0.67 thf(fact_8742_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.33/0.67 ! [S4: set_int,T4: set_real,S: set_int,I2: real > int,J: int > real,T3: set_real,G: int > complex,H2: real > complex] :
% 0.33/0.67 ( ( finite_finite_int @ S4 )
% 0.33/0.67 => ( ( finite_finite_real @ T4 )
% 0.33/0.67 => ( ! [A3: int] :
% 0.33/0.67 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 0.33/0.67 => ( ( I2 @ ( J @ A3 ) )
% 0.33/0.67 = A3 ) )
% 0.33/0.67 => ( ! [A3: int] :
% 0.33/0.67 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 0.33/0.67 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 0.33/0.67 => ( ! [B2: real] :
% 0.33/0.67 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.33/0.67 => ( ( J @ ( I2 @ B2 ) )
% 0.33/0.67 = B2 ) )
% 0.33/0.67 => ( ! [B2: real] :
% 0.33/0.67 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.33/0.67 => ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
% 0.33/0.67 => ( ! [A3: int] :
% 0.33/0.67 ( ( member_int @ A3 @ S4 )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [B2: real] :
% 0.33/0.67 ( ( member_real @ B2 @ T4 )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [A3: int] :
% 0.33/0.67 ( ( member_int @ A3 @ S )
% 0.33/0.67 => ( ( H2 @ ( J @ A3 ) )
% 0.33/0.67 = ( G @ A3 ) ) )
% 0.33/0.67 => ( ( groups7440179247065528705omplex @ G @ S )
% 0.33/0.67 = ( groups713298508707869441omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.reindex_bij_witness_not_neutral
% 0.33/0.67 thf(fact_8743_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.33/0.67 ! [S4: set_int,T4: set_int,S: set_int,I2: int > int,J: int > int,T3: set_int,G: int > complex,H2: int > complex] :
% 0.33/0.67 ( ( finite_finite_int @ S4 )
% 0.33/0.67 => ( ( finite_finite_int @ T4 )
% 0.33/0.67 => ( ! [A3: int] :
% 0.33/0.67 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 0.33/0.67 => ( ( I2 @ ( J @ A3 ) )
% 0.33/0.67 = A3 ) )
% 0.33/0.67 => ( ! [A3: int] :
% 0.33/0.67 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 0.33/0.67 => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 0.33/0.67 => ( ! [B2: int] :
% 0.33/0.67 ( ( member_int @ B2 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.33/0.67 => ( ( J @ ( I2 @ B2 ) )
% 0.33/0.67 = B2 ) )
% 0.33/0.67 => ( ! [B2: int] :
% 0.33/0.67 ( ( member_int @ B2 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.33/0.67 => ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
% 0.33/0.67 => ( ! [A3: int] :
% 0.33/0.67 ( ( member_int @ A3 @ S4 )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [B2: int] :
% 0.33/0.67 ( ( member_int @ B2 @ T4 )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [A3: int] :
% 0.33/0.67 ( ( member_int @ A3 @ S )
% 0.33/0.67 => ( ( H2 @ ( J @ A3 ) )
% 0.33/0.67 = ( G @ A3 ) ) )
% 0.33/0.67 => ( ( groups7440179247065528705omplex @ G @ S )
% 0.33/0.67 = ( groups7440179247065528705omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.reindex_bij_witness_not_neutral
% 0.33/0.67 thf(fact_8744_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.33/0.67 ! [S4: set_int,T4: set_complex,S: set_int,I2: complex > int,J: int > complex,T3: set_complex,G: int > complex,H2: complex > complex] :
% 0.33/0.67 ( ( finite_finite_int @ S4 )
% 0.33/0.67 => ( ( finite3207457112153483333omplex @ T4 )
% 0.33/0.67 => ( ! [A3: int] :
% 0.33/0.67 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 0.33/0.67 => ( ( I2 @ ( J @ A3 ) )
% 0.33/0.67 = A3 ) )
% 0.33/0.67 => ( ! [A3: int] :
% 0.33/0.67 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 0.33/0.67 => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 0.33/0.67 => ( ! [B2: complex] :
% 0.33/0.67 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.33/0.67 => ( ( J @ ( I2 @ B2 ) )
% 0.33/0.67 = B2 ) )
% 0.33/0.67 => ( ! [B2: complex] :
% 0.33/0.67 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.33/0.67 => ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
% 0.33/0.67 => ( ! [A3: int] :
% 0.33/0.67 ( ( member_int @ A3 @ S4 )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [B2: complex] :
% 0.33/0.67 ( ( member_complex @ B2 @ T4 )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [A3: int] :
% 0.33/0.67 ( ( member_int @ A3 @ S )
% 0.33/0.67 => ( ( H2 @ ( J @ A3 ) )
% 0.33/0.67 = ( G @ A3 ) ) )
% 0.33/0.67 => ( ( groups7440179247065528705omplex @ G @ S )
% 0.33/0.67 = ( groups3708469109370488835omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.reindex_bij_witness_not_neutral
% 0.33/0.67 thf(fact_8745_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.33/0.67 ! [S4: set_complex,T4: set_real,S: set_complex,I2: real > complex,J: complex > real,T3: set_real,G: complex > complex,H2: real > complex] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ S4 )
% 0.33/0.67 => ( ( finite_finite_real @ T4 )
% 0.33/0.67 => ( ! [A3: complex] :
% 0.33/0.67 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S @ S4 ) )
% 0.33/0.67 => ( ( I2 @ ( J @ A3 ) )
% 0.33/0.67 = A3 ) )
% 0.33/0.67 => ( ! [A3: complex] :
% 0.33/0.67 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S @ S4 ) )
% 0.33/0.67 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 0.33/0.67 => ( ! [B2: real] :
% 0.33/0.67 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.33/0.67 => ( ( J @ ( I2 @ B2 ) )
% 0.33/0.67 = B2 ) )
% 0.33/0.67 => ( ! [B2: real] :
% 0.33/0.67 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.33/0.67 => ( member_complex @ ( I2 @ B2 ) @ ( minus_811609699411566653omplex @ S @ S4 ) ) )
% 0.33/0.67 => ( ! [A3: complex] :
% 0.33/0.67 ( ( member_complex @ A3 @ S4 )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [B2: real] :
% 0.33/0.67 ( ( member_real @ B2 @ T4 )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [A3: complex] :
% 0.33/0.67 ( ( member_complex @ A3 @ S )
% 0.33/0.67 => ( ( H2 @ ( J @ A3 ) )
% 0.33/0.67 = ( G @ A3 ) ) )
% 0.33/0.67 => ( ( groups3708469109370488835omplex @ G @ S )
% 0.33/0.67 = ( groups713298508707869441omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.reindex_bij_witness_not_neutral
% 0.33/0.67 thf(fact_8746_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.33/0.67 ! [S4: set_complex,T4: set_int,S: set_complex,I2: int > complex,J: complex > int,T3: set_int,G: complex > complex,H2: int > complex] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ S4 )
% 0.33/0.67 => ( ( finite_finite_int @ T4 )
% 0.33/0.67 => ( ! [A3: complex] :
% 0.33/0.67 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S @ S4 ) )
% 0.33/0.67 => ( ( I2 @ ( J @ A3 ) )
% 0.33/0.67 = A3 ) )
% 0.33/0.67 => ( ! [A3: complex] :
% 0.33/0.67 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S @ S4 ) )
% 0.33/0.67 => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 0.33/0.67 => ( ! [B2: int] :
% 0.33/0.67 ( ( member_int @ B2 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.33/0.67 => ( ( J @ ( I2 @ B2 ) )
% 0.33/0.67 = B2 ) )
% 0.33/0.67 => ( ! [B2: int] :
% 0.33/0.67 ( ( member_int @ B2 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.33/0.67 => ( member_complex @ ( I2 @ B2 ) @ ( minus_811609699411566653omplex @ S @ S4 ) ) )
% 0.33/0.67 => ( ! [A3: complex] :
% 0.33/0.67 ( ( member_complex @ A3 @ S4 )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [B2: int] :
% 0.33/0.67 ( ( member_int @ B2 @ T4 )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [A3: complex] :
% 0.33/0.67 ( ( member_complex @ A3 @ S )
% 0.33/0.67 => ( ( H2 @ ( J @ A3 ) )
% 0.33/0.67 = ( G @ A3 ) ) )
% 0.33/0.67 => ( ( groups3708469109370488835omplex @ G @ S )
% 0.33/0.67 = ( groups7440179247065528705omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.reindex_bij_witness_not_neutral
% 0.33/0.67 thf(fact_8747_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.33/0.67 ! [S4: set_complex,T4: set_complex,S: set_complex,I2: complex > complex,J: complex > complex,T3: set_complex,G: complex > complex,H2: complex > complex] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ S4 )
% 0.33/0.67 => ( ( finite3207457112153483333omplex @ T4 )
% 0.33/0.67 => ( ! [A3: complex] :
% 0.33/0.67 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S @ S4 ) )
% 0.33/0.67 => ( ( I2 @ ( J @ A3 ) )
% 0.33/0.67 = A3 ) )
% 0.33/0.67 => ( ! [A3: complex] :
% 0.33/0.67 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S @ S4 ) )
% 0.33/0.67 => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 0.33/0.67 => ( ! [B2: complex] :
% 0.33/0.67 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.33/0.67 => ( ( J @ ( I2 @ B2 ) )
% 0.33/0.67 = B2 ) )
% 0.33/0.67 => ( ! [B2: complex] :
% 0.33/0.67 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.33/0.67 => ( member_complex @ ( I2 @ B2 ) @ ( minus_811609699411566653omplex @ S @ S4 ) ) )
% 0.33/0.67 => ( ! [A3: complex] :
% 0.33/0.67 ( ( member_complex @ A3 @ S4 )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [B2: complex] :
% 0.33/0.67 ( ( member_complex @ B2 @ T4 )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [A3: complex] :
% 0.33/0.67 ( ( member_complex @ A3 @ S )
% 0.33/0.67 => ( ( H2 @ ( J @ A3 ) )
% 0.33/0.67 = ( G @ A3 ) ) )
% 0.33/0.67 => ( ( groups3708469109370488835omplex @ G @ S )
% 0.33/0.67 = ( groups3708469109370488835omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.reindex_bij_witness_not_neutral
% 0.33/0.67 thf(fact_8748_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.33/0.67 ! [S4: set_real,T4: set_real,S: set_real,I2: real > real,J: real > real,T3: set_real,G: real > real,H2: real > real] :
% 0.33/0.67 ( ( finite_finite_real @ S4 )
% 0.33/0.67 => ( ( finite_finite_real @ T4 )
% 0.33/0.67 => ( ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 0.33/0.67 => ( ( I2 @ ( J @ A3 ) )
% 0.33/0.67 = A3 ) )
% 0.33/0.67 => ( ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 0.33/0.67 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 0.33/0.67 => ( ! [B2: real] :
% 0.33/0.67 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.33/0.67 => ( ( J @ ( I2 @ B2 ) )
% 0.33/0.67 = B2 ) )
% 0.33/0.67 => ( ! [B2: real] :
% 0.33/0.67 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.33/0.67 => ( member_real @ ( I2 @ B2 ) @ ( minus_minus_set_real @ S @ S4 ) ) )
% 0.33/0.67 => ( ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ S4 )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_real ) )
% 0.33/0.67 => ( ! [B2: real] :
% 0.33/0.67 ( ( member_real @ B2 @ T4 )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_real ) )
% 0.33/0.67 => ( ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ S )
% 0.33/0.67 => ( ( H2 @ ( J @ A3 ) )
% 0.33/0.67 = ( G @ A3 ) ) )
% 0.33/0.67 => ( ( groups1681761925125756287l_real @ G @ S )
% 0.33/0.67 = ( groups1681761925125756287l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.reindex_bij_witness_not_neutral
% 0.33/0.67 thf(fact_8749_prod_Osetdiff__irrelevant,axiom,
% 0.33/0.67 ! [A2: set_int,G: int > complex] :
% 0.33/0.67 ( ( finite_finite_int @ A2 )
% 0.33/0.67 => ( ( groups7440179247065528705omplex @ G
% 0.33/0.67 @ ( minus_minus_set_int @ A2
% 0.33/0.67 @ ( collect_int
% 0.33/0.67 @ ^ [X2: int] :
% 0.33/0.67 ( ( G @ X2 )
% 0.33/0.67 = one_one_complex ) ) ) )
% 0.33/0.67 = ( groups7440179247065528705omplex @ G @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.setdiff_irrelevant
% 0.33/0.67 thf(fact_8750_prod_Osetdiff__irrelevant,axiom,
% 0.33/0.67 ! [A2: set_complex,G: complex > complex] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.67 => ( ( groups3708469109370488835omplex @ G
% 0.33/0.67 @ ( minus_811609699411566653omplex @ A2
% 0.33/0.67 @ ( collect_complex
% 0.33/0.67 @ ^ [X2: complex] :
% 0.33/0.67 ( ( G @ X2 )
% 0.33/0.67 = one_one_complex ) ) ) )
% 0.33/0.67 = ( groups3708469109370488835omplex @ G @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.setdiff_irrelevant
% 0.33/0.67 thf(fact_8751_prod_Osetdiff__irrelevant,axiom,
% 0.33/0.67 ! [A2: set_int,G: int > real] :
% 0.33/0.67 ( ( finite_finite_int @ A2 )
% 0.33/0.67 => ( ( groups2316167850115554303t_real @ G
% 0.33/0.67 @ ( minus_minus_set_int @ A2
% 0.33/0.67 @ ( collect_int
% 0.33/0.67 @ ^ [X2: int] :
% 0.33/0.67 ( ( G @ X2 )
% 0.33/0.67 = one_one_real ) ) ) )
% 0.33/0.67 = ( groups2316167850115554303t_real @ G @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.setdiff_irrelevant
% 0.33/0.67 thf(fact_8752_prod_Osetdiff__irrelevant,axiom,
% 0.33/0.67 ! [A2: set_complex,G: complex > real] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.67 => ( ( groups766887009212190081x_real @ G
% 0.33/0.67 @ ( minus_811609699411566653omplex @ A2
% 0.33/0.67 @ ( collect_complex
% 0.33/0.67 @ ^ [X2: complex] :
% 0.33/0.67 ( ( G @ X2 )
% 0.33/0.67 = one_one_real ) ) ) )
% 0.33/0.67 = ( groups766887009212190081x_real @ G @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.setdiff_irrelevant
% 0.33/0.67 thf(fact_8753_prod_Osetdiff__irrelevant,axiom,
% 0.33/0.67 ! [A2: set_int,G: int > rat] :
% 0.33/0.67 ( ( finite_finite_int @ A2 )
% 0.33/0.67 => ( ( groups1072433553688619179nt_rat @ G
% 0.33/0.67 @ ( minus_minus_set_int @ A2
% 0.33/0.67 @ ( collect_int
% 0.33/0.67 @ ^ [X2: int] :
% 0.33/0.67 ( ( G @ X2 )
% 0.33/0.67 = one_one_rat ) ) ) )
% 0.33/0.67 = ( groups1072433553688619179nt_rat @ G @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.setdiff_irrelevant
% 0.33/0.67 thf(fact_8754_prod_Osetdiff__irrelevant,axiom,
% 0.33/0.67 ! [A2: set_complex,G: complex > rat] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.67 => ( ( groups225925009352817453ex_rat @ G
% 0.33/0.67 @ ( minus_811609699411566653omplex @ A2
% 0.33/0.67 @ ( collect_complex
% 0.33/0.67 @ ^ [X2: complex] :
% 0.33/0.67 ( ( G @ X2 )
% 0.33/0.67 = one_one_rat ) ) ) )
% 0.33/0.67 = ( groups225925009352817453ex_rat @ G @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.setdiff_irrelevant
% 0.33/0.67 thf(fact_8755_prod_Osetdiff__irrelevant,axiom,
% 0.33/0.67 ! [A2: set_int,G: int > nat] :
% 0.33/0.67 ( ( finite_finite_int @ A2 )
% 0.33/0.67 => ( ( groups1707563613775114915nt_nat @ G
% 0.33/0.67 @ ( minus_minus_set_int @ A2
% 0.33/0.67 @ ( collect_int
% 0.33/0.67 @ ^ [X2: int] :
% 0.33/0.67 ( ( G @ X2 )
% 0.33/0.67 = one_one_nat ) ) ) )
% 0.33/0.67 = ( groups1707563613775114915nt_nat @ G @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.setdiff_irrelevant
% 0.33/0.67 thf(fact_8756_prod_Osetdiff__irrelevant,axiom,
% 0.33/0.67 ! [A2: set_complex,G: complex > nat] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.67 => ( ( groups861055069439313189ex_nat @ G
% 0.33/0.67 @ ( minus_811609699411566653omplex @ A2
% 0.33/0.67 @ ( collect_complex
% 0.33/0.67 @ ^ [X2: complex] :
% 0.33/0.67 ( ( G @ X2 )
% 0.33/0.67 = one_one_nat ) ) ) )
% 0.33/0.67 = ( groups861055069439313189ex_nat @ G @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.setdiff_irrelevant
% 0.33/0.67 thf(fact_8757_prod_Osetdiff__irrelevant,axiom,
% 0.33/0.67 ! [A2: set_complex,G: complex > int] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.67 => ( ( groups858564598930262913ex_int @ G
% 0.33/0.67 @ ( minus_811609699411566653omplex @ A2
% 0.33/0.67 @ ( collect_complex
% 0.33/0.67 @ ^ [X2: complex] :
% 0.33/0.67 ( ( G @ X2 )
% 0.33/0.67 = one_one_int ) ) ) )
% 0.33/0.67 = ( groups858564598930262913ex_int @ G @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.setdiff_irrelevant
% 0.33/0.67 thf(fact_8758_prod_Osetdiff__irrelevant,axiom,
% 0.33/0.67 ! [A2: set_nat,G: nat > complex] :
% 0.33/0.67 ( ( finite_finite_nat @ A2 )
% 0.33/0.67 => ( ( groups6464643781859351333omplex @ G
% 0.33/0.67 @ ( minus_minus_set_nat @ A2
% 0.33/0.67 @ ( collect_nat
% 0.33/0.67 @ ^ [X2: nat] :
% 0.33/0.67 ( ( G @ X2 )
% 0.33/0.67 = one_one_complex ) ) ) )
% 0.33/0.67 = ( groups6464643781859351333omplex @ G @ A2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.setdiff_irrelevant
% 0.33/0.67 thf(fact_8759_exp__sum,axiom,
% 0.33/0.67 ! [I6: set_int,F: int > real] :
% 0.33/0.67 ( ( finite_finite_int @ I6 )
% 0.33/0.67 => ( ( exp_real @ ( groups8778361861064173332t_real @ F @ I6 ) )
% 0.33/0.67 = ( groups2316167850115554303t_real
% 0.33/0.67 @ ^ [X2: int] : ( exp_real @ ( F @ X2 ) )
% 0.33/0.67 @ I6 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % exp_sum
% 0.33/0.67 thf(fact_8760_exp__sum,axiom,
% 0.33/0.67 ! [I6: set_complex,F: complex > real] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ I6 )
% 0.33/0.67 => ( ( exp_real @ ( groups5808333547571424918x_real @ F @ I6 ) )
% 0.33/0.67 = ( groups766887009212190081x_real
% 0.33/0.67 @ ^ [X2: complex] : ( exp_real @ ( F @ X2 ) )
% 0.33/0.67 @ I6 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % exp_sum
% 0.33/0.67 thf(fact_8761_exp__sum,axiom,
% 0.33/0.67 ! [I6: set_nat,F: nat > complex] :
% 0.33/0.67 ( ( finite_finite_nat @ I6 )
% 0.33/0.67 => ( ( exp_complex @ ( groups2073611262835488442omplex @ F @ I6 ) )
% 0.33/0.67 = ( groups6464643781859351333omplex
% 0.33/0.67 @ ^ [X2: nat] : ( exp_complex @ ( F @ X2 ) )
% 0.33/0.67 @ I6 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % exp_sum
% 0.33/0.67 thf(fact_8762_exp__sum,axiom,
% 0.33/0.67 ! [I6: set_int,F: int > complex] :
% 0.33/0.67 ( ( finite_finite_int @ I6 )
% 0.33/0.67 => ( ( exp_complex @ ( groups3049146728041665814omplex @ F @ I6 ) )
% 0.33/0.67 = ( groups7440179247065528705omplex
% 0.33/0.67 @ ^ [X2: int] : ( exp_complex @ ( F @ X2 ) )
% 0.33/0.67 @ I6 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % exp_sum
% 0.33/0.67 thf(fact_8763_exp__sum,axiom,
% 0.33/0.67 ! [I6: set_complex,F: complex > complex] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ I6 )
% 0.33/0.67 => ( ( exp_complex @ ( groups7754918857620584856omplex @ F @ I6 ) )
% 0.33/0.67 = ( groups3708469109370488835omplex
% 0.33/0.67 @ ^ [X2: complex] : ( exp_complex @ ( F @ X2 ) )
% 0.33/0.67 @ I6 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % exp_sum
% 0.33/0.67 thf(fact_8764_exp__sum,axiom,
% 0.33/0.67 ! [I6: set_nat,F: nat > real] :
% 0.33/0.67 ( ( finite_finite_nat @ I6 )
% 0.33/0.67 => ( ( exp_real @ ( groups6591440286371151544t_real @ F @ I6 ) )
% 0.33/0.67 = ( groups129246275422532515t_real
% 0.33/0.67 @ ^ [X2: nat] : ( exp_real @ ( F @ X2 ) )
% 0.33/0.67 @ I6 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % exp_sum
% 0.33/0.67 thf(fact_8765_prod_Onat__diff__reindex,axiom,
% 0.33/0.67 ! [G: nat > nat,N2: nat] :
% 0.33/0.67 ( ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
% 0.33/0.67 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.67 = ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.nat_diff_reindex
% 0.33/0.67 thf(fact_8766_prod_Onat__diff__reindex,axiom,
% 0.33/0.67 ! [G: nat > int,N2: nat] :
% 0.33/0.67 ( ( groups705719431365010083at_int
% 0.33/0.67 @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
% 0.33/0.67 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.67 = ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.nat_diff_reindex
% 0.33/0.67 thf(fact_8767_prod_OatLeastAtMost__rev,axiom,
% 0.33/0.67 ! [G: nat > nat,N2: nat,M: nat] :
% 0.33/0.67 ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
% 0.33/0.67 = ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I3 ) )
% 0.33/0.67 @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.atLeastAtMost_rev
% 0.33/0.67 thf(fact_8768_prod_OatLeastAtMost__rev,axiom,
% 0.33/0.67 ! [G: nat > int,N2: nat,M: nat] :
% 0.33/0.67 ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
% 0.33/0.67 = ( groups705719431365010083at_int
% 0.33/0.67 @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I3 ) )
% 0.33/0.67 @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.atLeastAtMost_rev
% 0.33/0.67 thf(fact_8769_less__1__prod2,axiom,
% 0.33/0.67 ! [I6: set_real,I2: real,F: real > real] :
% 0.33/0.67 ( ( finite_finite_real @ I6 )
% 0.33/0.67 => ( ( member_real @ I2 @ I6 )
% 0.33/0.67 => ( ( ord_less_real @ one_one_real @ ( F @ I2 ) )
% 0.33/0.67 => ( ! [I4: real] :
% 0.33/0.67 ( ( member_real @ I4 @ I6 )
% 0.33/0.67 => ( ord_less_eq_real @ one_one_real @ ( F @ I4 ) ) )
% 0.33/0.67 => ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I6 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_1_prod2
% 0.33/0.67 thf(fact_8770_less__1__prod2,axiom,
% 0.33/0.67 ! [I6: set_nat,I2: nat,F: nat > real] :
% 0.33/0.67 ( ( finite_finite_nat @ I6 )
% 0.33/0.67 => ( ( member_nat @ I2 @ I6 )
% 0.33/0.67 => ( ( ord_less_real @ one_one_real @ ( F @ I2 ) )
% 0.33/0.67 => ( ! [I4: nat] :
% 0.33/0.67 ( ( member_nat @ I4 @ I6 )
% 0.33/0.67 => ( ord_less_eq_real @ one_one_real @ ( F @ I4 ) ) )
% 0.33/0.67 => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I6 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_1_prod2
% 0.33/0.67 thf(fact_8771_less__1__prod2,axiom,
% 0.33/0.67 ! [I6: set_int,I2: int,F: int > real] :
% 0.33/0.67 ( ( finite_finite_int @ I6 )
% 0.33/0.67 => ( ( member_int @ I2 @ I6 )
% 0.33/0.67 => ( ( ord_less_real @ one_one_real @ ( F @ I2 ) )
% 0.33/0.67 => ( ! [I4: int] :
% 0.33/0.67 ( ( member_int @ I4 @ I6 )
% 0.33/0.67 => ( ord_less_eq_real @ one_one_real @ ( F @ I4 ) ) )
% 0.33/0.67 => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I6 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_1_prod2
% 0.33/0.67 thf(fact_8772_less__1__prod2,axiom,
% 0.33/0.67 ! [I6: set_complex,I2: complex,F: complex > real] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ I6 )
% 0.33/0.67 => ( ( member_complex @ I2 @ I6 )
% 0.33/0.67 => ( ( ord_less_real @ one_one_real @ ( F @ I2 ) )
% 0.33/0.67 => ( ! [I4: complex] :
% 0.33/0.67 ( ( member_complex @ I4 @ I6 )
% 0.33/0.67 => ( ord_less_eq_real @ one_one_real @ ( F @ I4 ) ) )
% 0.33/0.67 => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I6 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_1_prod2
% 0.33/0.67 thf(fact_8773_less__1__prod2,axiom,
% 0.33/0.67 ! [I6: set_real,I2: real,F: real > rat] :
% 0.33/0.67 ( ( finite_finite_real @ I6 )
% 0.33/0.67 => ( ( member_real @ I2 @ I6 )
% 0.33/0.67 => ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
% 0.33/0.67 => ( ! [I4: real] :
% 0.33/0.67 ( ( member_real @ I4 @ I6 )
% 0.33/0.67 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I4 ) ) )
% 0.33/0.67 => ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I6 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_1_prod2
% 0.33/0.67 thf(fact_8774_less__1__prod2,axiom,
% 0.33/0.67 ! [I6: set_nat,I2: nat,F: nat > rat] :
% 0.33/0.67 ( ( finite_finite_nat @ I6 )
% 0.33/0.67 => ( ( member_nat @ I2 @ I6 )
% 0.33/0.67 => ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
% 0.33/0.67 => ( ! [I4: nat] :
% 0.33/0.67 ( ( member_nat @ I4 @ I6 )
% 0.33/0.67 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I4 ) ) )
% 0.33/0.67 => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I6 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_1_prod2
% 0.33/0.67 thf(fact_8775_less__1__prod2,axiom,
% 0.33/0.67 ! [I6: set_int,I2: int,F: int > rat] :
% 0.33/0.67 ( ( finite_finite_int @ I6 )
% 0.33/0.67 => ( ( member_int @ I2 @ I6 )
% 0.33/0.67 => ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
% 0.33/0.67 => ( ! [I4: int] :
% 0.33/0.67 ( ( member_int @ I4 @ I6 )
% 0.33/0.67 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I4 ) ) )
% 0.33/0.67 => ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I6 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_1_prod2
% 0.33/0.67 thf(fact_8776_less__1__prod2,axiom,
% 0.33/0.67 ! [I6: set_complex,I2: complex,F: complex > rat] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ I6 )
% 0.33/0.67 => ( ( member_complex @ I2 @ I6 )
% 0.33/0.67 => ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
% 0.33/0.67 => ( ! [I4: complex] :
% 0.33/0.67 ( ( member_complex @ I4 @ I6 )
% 0.33/0.67 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I4 ) ) )
% 0.33/0.67 => ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I6 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_1_prod2
% 0.33/0.67 thf(fact_8777_less__1__prod2,axiom,
% 0.33/0.67 ! [I6: set_real,I2: real,F: real > int] :
% 0.33/0.67 ( ( finite_finite_real @ I6 )
% 0.33/0.67 => ( ( member_real @ I2 @ I6 )
% 0.33/0.67 => ( ( ord_less_int @ one_one_int @ ( F @ I2 ) )
% 0.33/0.67 => ( ! [I4: real] :
% 0.33/0.67 ( ( member_real @ I4 @ I6 )
% 0.33/0.67 => ( ord_less_eq_int @ one_one_int @ ( F @ I4 ) ) )
% 0.33/0.67 => ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I6 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_1_prod2
% 0.33/0.67 thf(fact_8778_less__1__prod2,axiom,
% 0.33/0.67 ! [I6: set_complex,I2: complex,F: complex > int] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ I6 )
% 0.33/0.67 => ( ( member_complex @ I2 @ I6 )
% 0.33/0.67 => ( ( ord_less_int @ one_one_int @ ( F @ I2 ) )
% 0.33/0.67 => ( ! [I4: complex] :
% 0.33/0.67 ( ( member_complex @ I4 @ I6 )
% 0.33/0.67 => ( ord_less_eq_int @ one_one_int @ ( F @ I4 ) ) )
% 0.33/0.67 => ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I6 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_1_prod2
% 0.33/0.67 thf(fact_8779_arccos__le__arccos,axiom,
% 0.33/0.67 ! [X: real,Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.67 => ( ( ord_less_eq_real @ X @ Y )
% 0.33/0.67 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.67 => ( ord_less_eq_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arccos_le_arccos
% 0.33/0.67 thf(fact_8780_less__1__prod,axiom,
% 0.33/0.67 ! [I6: set_complex,F: complex > real] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ I6 )
% 0.33/0.67 => ( ( I6 != bot_bot_set_complex )
% 0.33/0.67 => ( ! [I4: complex] :
% 0.33/0.67 ( ( member_complex @ I4 @ I6 )
% 0.33/0.67 => ( ord_less_real @ one_one_real @ ( F @ I4 ) ) )
% 0.33/0.67 => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I6 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_1_prod
% 0.33/0.67 thf(fact_8781_less__1__prod,axiom,
% 0.33/0.67 ! [I6: set_nat,F: nat > real] :
% 0.33/0.67 ( ( finite_finite_nat @ I6 )
% 0.33/0.67 => ( ( I6 != bot_bot_set_nat )
% 0.33/0.67 => ( ! [I4: nat] :
% 0.33/0.67 ( ( member_nat @ I4 @ I6 )
% 0.33/0.67 => ( ord_less_real @ one_one_real @ ( F @ I4 ) ) )
% 0.33/0.67 => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I6 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_1_prod
% 0.33/0.67 thf(fact_8782_less__1__prod,axiom,
% 0.33/0.67 ! [I6: set_int,F: int > real] :
% 0.33/0.67 ( ( finite_finite_int @ I6 )
% 0.33/0.67 => ( ( I6 != bot_bot_set_int )
% 0.33/0.67 => ( ! [I4: int] :
% 0.33/0.67 ( ( member_int @ I4 @ I6 )
% 0.33/0.67 => ( ord_less_real @ one_one_real @ ( F @ I4 ) ) )
% 0.33/0.67 => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I6 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_1_prod
% 0.33/0.67 thf(fact_8783_less__1__prod,axiom,
% 0.33/0.67 ! [I6: set_real,F: real > real] :
% 0.33/0.67 ( ( finite_finite_real @ I6 )
% 0.33/0.67 => ( ( I6 != bot_bot_set_real )
% 0.33/0.67 => ( ! [I4: real] :
% 0.33/0.67 ( ( member_real @ I4 @ I6 )
% 0.33/0.67 => ( ord_less_real @ one_one_real @ ( F @ I4 ) ) )
% 0.33/0.67 => ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I6 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_1_prod
% 0.33/0.67 thf(fact_8784_less__1__prod,axiom,
% 0.33/0.67 ! [I6: set_complex,F: complex > rat] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ I6 )
% 0.33/0.67 => ( ( I6 != bot_bot_set_complex )
% 0.33/0.67 => ( ! [I4: complex] :
% 0.33/0.67 ( ( member_complex @ I4 @ I6 )
% 0.33/0.67 => ( ord_less_rat @ one_one_rat @ ( F @ I4 ) ) )
% 0.33/0.67 => ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I6 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_1_prod
% 0.33/0.67 thf(fact_8785_less__1__prod,axiom,
% 0.33/0.67 ! [I6: set_nat,F: nat > rat] :
% 0.33/0.67 ( ( finite_finite_nat @ I6 )
% 0.33/0.67 => ( ( I6 != bot_bot_set_nat )
% 0.33/0.67 => ( ! [I4: nat] :
% 0.33/0.67 ( ( member_nat @ I4 @ I6 )
% 0.33/0.67 => ( ord_less_rat @ one_one_rat @ ( F @ I4 ) ) )
% 0.33/0.67 => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I6 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_1_prod
% 0.33/0.67 thf(fact_8786_less__1__prod,axiom,
% 0.33/0.67 ! [I6: set_int,F: int > rat] :
% 0.33/0.67 ( ( finite_finite_int @ I6 )
% 0.33/0.67 => ( ( I6 != bot_bot_set_int )
% 0.33/0.67 => ( ! [I4: int] :
% 0.33/0.67 ( ( member_int @ I4 @ I6 )
% 0.33/0.67 => ( ord_less_rat @ one_one_rat @ ( F @ I4 ) ) )
% 0.33/0.67 => ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I6 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_1_prod
% 0.33/0.67 thf(fact_8787_less__1__prod,axiom,
% 0.33/0.67 ! [I6: set_real,F: real > rat] :
% 0.33/0.67 ( ( finite_finite_real @ I6 )
% 0.33/0.67 => ( ( I6 != bot_bot_set_real )
% 0.33/0.67 => ( ! [I4: real] :
% 0.33/0.67 ( ( member_real @ I4 @ I6 )
% 0.33/0.67 => ( ord_less_rat @ one_one_rat @ ( F @ I4 ) ) )
% 0.33/0.67 => ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I6 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_1_prod
% 0.33/0.67 thf(fact_8788_less__1__prod,axiom,
% 0.33/0.67 ! [I6: set_complex,F: complex > int] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ I6 )
% 0.33/0.67 => ( ( I6 != bot_bot_set_complex )
% 0.33/0.67 => ( ! [I4: complex] :
% 0.33/0.67 ( ( member_complex @ I4 @ I6 )
% 0.33/0.67 => ( ord_less_int @ one_one_int @ ( F @ I4 ) ) )
% 0.33/0.67 => ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I6 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_1_prod
% 0.33/0.67 thf(fact_8789_less__1__prod,axiom,
% 0.33/0.67 ! [I6: set_real,F: real > int] :
% 0.33/0.67 ( ( finite_finite_real @ I6 )
% 0.33/0.67 => ( ( I6 != bot_bot_set_real )
% 0.33/0.67 => ( ! [I4: real] :
% 0.33/0.67 ( ( member_real @ I4 @ I6 )
% 0.33/0.67 => ( ord_less_int @ one_one_int @ ( F @ I4 ) ) )
% 0.33/0.67 => ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I6 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % less_1_prod
% 0.33/0.67 thf(fact_8790_arccos__eq__iff,axiom,
% 0.33/0.67 ! [X: real,Y: real] :
% 0.33/0.67 ( ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.67 & ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real ) )
% 0.33/0.67 => ( ( ( arccos @ X )
% 0.33/0.67 = ( arccos @ Y ) )
% 0.33/0.67 = ( X = Y ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arccos_eq_iff
% 0.33/0.67 thf(fact_8791_arccos__le__mono,axiom,
% 0.33/0.67 ! [X: real,Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.67 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 0.33/0.67 => ( ( ord_less_eq_real @ ( arccos @ X ) @ ( arccos @ Y ) )
% 0.33/0.67 = ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arccos_le_mono
% 0.33/0.67 thf(fact_8792_arcsin__le__arcsin,axiom,
% 0.33/0.67 ! [X: real,Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.67 => ( ( ord_less_eq_real @ X @ Y )
% 0.33/0.67 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.67 => ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arcsin_le_arcsin
% 0.33/0.67 thf(fact_8793_arcsin__minus,axiom,
% 0.33/0.67 ! [X: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.67 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.67 => ( ( arcsin @ ( uminus_uminus_real @ X ) )
% 0.33/0.67 = ( uminus_uminus_real @ ( arcsin @ X ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arcsin_minus
% 0.33/0.67 thf(fact_8794_prod_Osubset__diff,axiom,
% 0.33/0.67 ! [B3: set_int,A2: set_int,G: int > real] :
% 0.33/0.67 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.33/0.67 => ( ( finite_finite_int @ A2 )
% 0.33/0.67 => ( ( groups2316167850115554303t_real @ G @ A2 )
% 0.33/0.67 = ( times_times_real @ ( groups2316167850115554303t_real @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups2316167850115554303t_real @ G @ B3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.subset_diff
% 0.33/0.67 thf(fact_8795_prod_Osubset__diff,axiom,
% 0.33/0.67 ! [B3: set_complex,A2: set_complex,G: complex > real] :
% 0.33/0.67 ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 0.33/0.67 => ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.67 => ( ( groups766887009212190081x_real @ G @ A2 )
% 0.33/0.67 = ( times_times_real @ ( groups766887009212190081x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups766887009212190081x_real @ G @ B3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.subset_diff
% 0.33/0.67 thf(fact_8796_prod_Osubset__diff,axiom,
% 0.33/0.67 ! [B3: set_int,A2: set_int,G: int > rat] :
% 0.33/0.67 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.33/0.67 => ( ( finite_finite_int @ A2 )
% 0.33/0.67 => ( ( groups1072433553688619179nt_rat @ G @ A2 )
% 0.33/0.67 = ( times_times_rat @ ( groups1072433553688619179nt_rat @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups1072433553688619179nt_rat @ G @ B3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.subset_diff
% 0.33/0.67 thf(fact_8797_prod_Osubset__diff,axiom,
% 0.33/0.67 ! [B3: set_complex,A2: set_complex,G: complex > rat] :
% 0.33/0.67 ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 0.33/0.67 => ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.67 => ( ( groups225925009352817453ex_rat @ G @ A2 )
% 0.33/0.67 = ( times_times_rat @ ( groups225925009352817453ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups225925009352817453ex_rat @ G @ B3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.subset_diff
% 0.33/0.67 thf(fact_8798_prod_Osubset__diff,axiom,
% 0.33/0.67 ! [B3: set_int,A2: set_int,G: int > nat] :
% 0.33/0.67 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.33/0.67 => ( ( finite_finite_int @ A2 )
% 0.33/0.67 => ( ( groups1707563613775114915nt_nat @ G @ A2 )
% 0.33/0.67 = ( times_times_nat @ ( groups1707563613775114915nt_nat @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups1707563613775114915nt_nat @ G @ B3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.subset_diff
% 0.33/0.67 thf(fact_8799_prod_Osubset__diff,axiom,
% 0.33/0.67 ! [B3: set_complex,A2: set_complex,G: complex > nat] :
% 0.33/0.67 ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 0.33/0.67 => ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.67 => ( ( groups861055069439313189ex_nat @ G @ A2 )
% 0.33/0.67 = ( times_times_nat @ ( groups861055069439313189ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups861055069439313189ex_nat @ G @ B3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.subset_diff
% 0.33/0.67 thf(fact_8800_prod_Osubset__diff,axiom,
% 0.33/0.67 ! [B3: set_complex,A2: set_complex,G: complex > int] :
% 0.33/0.67 ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 0.33/0.67 => ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.67 => ( ( groups858564598930262913ex_int @ G @ A2 )
% 0.33/0.67 = ( times_times_int @ ( groups858564598930262913ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups858564598930262913ex_int @ G @ B3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.subset_diff
% 0.33/0.67 thf(fact_8801_prod_Osubset__diff,axiom,
% 0.33/0.67 ! [B3: set_nat,A2: set_nat,G: nat > real] :
% 0.33/0.67 ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 0.33/0.67 => ( ( finite_finite_nat @ A2 )
% 0.33/0.67 => ( ( groups129246275422532515t_real @ G @ A2 )
% 0.33/0.67 = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( minus_minus_set_nat @ A2 @ B3 ) ) @ ( groups129246275422532515t_real @ G @ B3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.subset_diff
% 0.33/0.67 thf(fact_8802_prod_Osubset__diff,axiom,
% 0.33/0.67 ! [B3: set_nat,A2: set_nat,G: nat > rat] :
% 0.33/0.67 ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 0.33/0.67 => ( ( finite_finite_nat @ A2 )
% 0.33/0.67 => ( ( groups73079841787564623at_rat @ G @ A2 )
% 0.33/0.67 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A2 @ B3 ) ) @ ( groups73079841787564623at_rat @ G @ B3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.subset_diff
% 0.33/0.67 thf(fact_8803_prod_Osubset__diff,axiom,
% 0.33/0.67 ! [B3: set_nat,A2: set_nat,G: nat > nat] :
% 0.33/0.67 ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 0.33/0.67 => ( ( finite_finite_nat @ A2 )
% 0.33/0.67 => ( ( groups708209901874060359at_nat @ G @ A2 )
% 0.33/0.67 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ A2 @ B3 ) ) @ ( groups708209901874060359at_nat @ G @ B3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.subset_diff
% 0.33/0.67 thf(fact_8804_prod_Omono__neutral__cong__right,axiom,
% 0.33/0.67 ! [T3: set_real,S: set_real,G: real > complex,H2: real > complex] :
% 0.33/0.67 ( ( finite_finite_real @ T3 )
% 0.33/0.67 => ( ( ord_less_eq_set_real @ S @ T3 )
% 0.33/0.67 => ( ! [X4: real] :
% 0.33/0.67 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [X4: real] :
% 0.33/0.67 ( ( member_real @ X4 @ S )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = ( H2 @ X4 ) ) )
% 0.33/0.67 => ( ( groups713298508707869441omplex @ G @ T3 )
% 0.33/0.67 = ( groups713298508707869441omplex @ H2 @ S ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_cong_right
% 0.33/0.67 thf(fact_8805_prod_Omono__neutral__cong__right,axiom,
% 0.33/0.67 ! [T3: set_int,S: set_int,G: int > complex,H2: int > complex] :
% 0.33/0.67 ( ( finite_finite_int @ T3 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.67 => ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ S )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = ( H2 @ X4 ) ) )
% 0.33/0.67 => ( ( groups7440179247065528705omplex @ G @ T3 )
% 0.33/0.67 = ( groups7440179247065528705omplex @ H2 @ S ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_cong_right
% 0.33/0.67 thf(fact_8806_prod_Omono__neutral__cong__right,axiom,
% 0.33/0.67 ! [T3: set_complex,S: set_complex,G: complex > complex,H2: complex > complex] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ T3 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.33/0.67 => ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ S )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = ( H2 @ X4 ) ) )
% 0.33/0.67 => ( ( groups3708469109370488835omplex @ G @ T3 )
% 0.33/0.67 = ( groups3708469109370488835omplex @ H2 @ S ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_cong_right
% 0.33/0.67 thf(fact_8807_prod_Omono__neutral__cong__right,axiom,
% 0.33/0.67 ! [T3: set_real,S: set_real,G: real > real,H2: real > real] :
% 0.33/0.67 ( ( finite_finite_real @ T3 )
% 0.33/0.67 => ( ( ord_less_eq_set_real @ S @ T3 )
% 0.33/0.67 => ( ! [X4: real] :
% 0.33/0.67 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_real ) )
% 0.33/0.67 => ( ! [X4: real] :
% 0.33/0.67 ( ( member_real @ X4 @ S )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = ( H2 @ X4 ) ) )
% 0.33/0.67 => ( ( groups1681761925125756287l_real @ G @ T3 )
% 0.33/0.67 = ( groups1681761925125756287l_real @ H2 @ S ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_cong_right
% 0.33/0.67 thf(fact_8808_prod_Omono__neutral__cong__right,axiom,
% 0.33/0.67 ! [T3: set_int,S: set_int,G: int > real,H2: int > real] :
% 0.33/0.67 ( ( finite_finite_int @ T3 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.67 => ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_real ) )
% 0.33/0.67 => ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ S )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = ( H2 @ X4 ) ) )
% 0.33/0.67 => ( ( groups2316167850115554303t_real @ G @ T3 )
% 0.33/0.67 = ( groups2316167850115554303t_real @ H2 @ S ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_cong_right
% 0.33/0.67 thf(fact_8809_prod_Omono__neutral__cong__right,axiom,
% 0.33/0.67 ! [T3: set_complex,S: set_complex,G: complex > real,H2: complex > real] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ T3 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.33/0.67 => ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_real ) )
% 0.33/0.67 => ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ S )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = ( H2 @ X4 ) ) )
% 0.33/0.67 => ( ( groups766887009212190081x_real @ G @ T3 )
% 0.33/0.67 = ( groups766887009212190081x_real @ H2 @ S ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_cong_right
% 0.33/0.67 thf(fact_8810_prod_Omono__neutral__cong__right,axiom,
% 0.33/0.67 ! [T3: set_real,S: set_real,G: real > rat,H2: real > rat] :
% 0.33/0.67 ( ( finite_finite_real @ T3 )
% 0.33/0.67 => ( ( ord_less_eq_set_real @ S @ T3 )
% 0.33/0.67 => ( ! [X4: real] :
% 0.33/0.67 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_rat ) )
% 0.33/0.67 => ( ! [X4: real] :
% 0.33/0.67 ( ( member_real @ X4 @ S )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = ( H2 @ X4 ) ) )
% 0.33/0.67 => ( ( groups4061424788464935467al_rat @ G @ T3 )
% 0.33/0.67 = ( groups4061424788464935467al_rat @ H2 @ S ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_cong_right
% 0.33/0.67 thf(fact_8811_prod_Omono__neutral__cong__right,axiom,
% 0.33/0.67 ! [T3: set_int,S: set_int,G: int > rat,H2: int > rat] :
% 0.33/0.67 ( ( finite_finite_int @ T3 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.67 => ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_rat ) )
% 0.33/0.67 => ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ S )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = ( H2 @ X4 ) ) )
% 0.33/0.67 => ( ( groups1072433553688619179nt_rat @ G @ T3 )
% 0.33/0.67 = ( groups1072433553688619179nt_rat @ H2 @ S ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_cong_right
% 0.33/0.67 thf(fact_8812_prod_Omono__neutral__cong__right,axiom,
% 0.33/0.67 ! [T3: set_complex,S: set_complex,G: complex > rat,H2: complex > rat] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ T3 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.33/0.67 => ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_rat ) )
% 0.33/0.67 => ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ S )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = ( H2 @ X4 ) ) )
% 0.33/0.67 => ( ( groups225925009352817453ex_rat @ G @ T3 )
% 0.33/0.67 = ( groups225925009352817453ex_rat @ H2 @ S ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_cong_right
% 0.33/0.67 thf(fact_8813_prod_Omono__neutral__cong__right,axiom,
% 0.33/0.67 ! [T3: set_real,S: set_real,G: real > nat,H2: real > nat] :
% 0.33/0.67 ( ( finite_finite_real @ T3 )
% 0.33/0.67 => ( ( ord_less_eq_set_real @ S @ T3 )
% 0.33/0.67 => ( ! [X4: real] :
% 0.33/0.67 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_nat ) )
% 0.33/0.67 => ( ! [X4: real] :
% 0.33/0.67 ( ( member_real @ X4 @ S )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = ( H2 @ X4 ) ) )
% 0.33/0.67 => ( ( groups4696554848551431203al_nat @ G @ T3 )
% 0.33/0.67 = ( groups4696554848551431203al_nat @ H2 @ S ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_cong_right
% 0.33/0.67 thf(fact_8814_prod_Omono__neutral__cong__left,axiom,
% 0.33/0.67 ! [T3: set_real,S: set_real,H2: real > complex,G: real > complex] :
% 0.33/0.67 ( ( finite_finite_real @ T3 )
% 0.33/0.67 => ( ( ord_less_eq_set_real @ S @ T3 )
% 0.33/0.67 => ( ! [X4: real] :
% 0.33/0.67 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 0.33/0.67 => ( ( H2 @ X4 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [X4: real] :
% 0.33/0.67 ( ( member_real @ X4 @ S )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = ( H2 @ X4 ) ) )
% 0.33/0.67 => ( ( groups713298508707869441omplex @ G @ S )
% 0.33/0.67 = ( groups713298508707869441omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_cong_left
% 0.33/0.67 thf(fact_8815_prod_Omono__neutral__cong__left,axiom,
% 0.33/0.67 ! [T3: set_int,S: set_int,H2: int > complex,G: int > complex] :
% 0.33/0.67 ( ( finite_finite_int @ T3 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.67 => ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.67 => ( ( H2 @ X4 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ S )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = ( H2 @ X4 ) ) )
% 0.33/0.67 => ( ( groups7440179247065528705omplex @ G @ S )
% 0.33/0.67 = ( groups7440179247065528705omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_cong_left
% 0.33/0.67 thf(fact_8816_prod_Omono__neutral__cong__left,axiom,
% 0.33/0.67 ! [T3: set_complex,S: set_complex,H2: complex > complex,G: complex > complex] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ T3 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.33/0.67 => ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 0.33/0.67 => ( ( H2 @ X4 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ S )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = ( H2 @ X4 ) ) )
% 0.33/0.67 => ( ( groups3708469109370488835omplex @ G @ S )
% 0.33/0.67 = ( groups3708469109370488835omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_cong_left
% 0.33/0.67 thf(fact_8817_prod_Omono__neutral__cong__left,axiom,
% 0.33/0.67 ! [T3: set_real,S: set_real,H2: real > real,G: real > real] :
% 0.33/0.67 ( ( finite_finite_real @ T3 )
% 0.33/0.67 => ( ( ord_less_eq_set_real @ S @ T3 )
% 0.33/0.67 => ( ! [X4: real] :
% 0.33/0.67 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 0.33/0.67 => ( ( H2 @ X4 )
% 0.33/0.67 = one_one_real ) )
% 0.33/0.67 => ( ! [X4: real] :
% 0.33/0.67 ( ( member_real @ X4 @ S )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = ( H2 @ X4 ) ) )
% 0.33/0.67 => ( ( groups1681761925125756287l_real @ G @ S )
% 0.33/0.67 = ( groups1681761925125756287l_real @ H2 @ T3 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_cong_left
% 0.33/0.67 thf(fact_8818_prod_Omono__neutral__cong__left,axiom,
% 0.33/0.67 ! [T3: set_int,S: set_int,H2: int > real,G: int > real] :
% 0.33/0.67 ( ( finite_finite_int @ T3 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.67 => ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.67 => ( ( H2 @ X4 )
% 0.33/0.67 = one_one_real ) )
% 0.33/0.67 => ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ S )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = ( H2 @ X4 ) ) )
% 0.33/0.67 => ( ( groups2316167850115554303t_real @ G @ S )
% 0.33/0.67 = ( groups2316167850115554303t_real @ H2 @ T3 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_cong_left
% 0.33/0.67 thf(fact_8819_prod_Omono__neutral__cong__left,axiom,
% 0.33/0.67 ! [T3: set_complex,S: set_complex,H2: complex > real,G: complex > real] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ T3 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.33/0.67 => ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 0.33/0.67 => ( ( H2 @ X4 )
% 0.33/0.67 = one_one_real ) )
% 0.33/0.67 => ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ S )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = ( H2 @ X4 ) ) )
% 0.33/0.67 => ( ( groups766887009212190081x_real @ G @ S )
% 0.33/0.67 = ( groups766887009212190081x_real @ H2 @ T3 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_cong_left
% 0.33/0.67 thf(fact_8820_prod_Omono__neutral__cong__left,axiom,
% 0.33/0.67 ! [T3: set_real,S: set_real,H2: real > rat,G: real > rat] :
% 0.33/0.67 ( ( finite_finite_real @ T3 )
% 0.33/0.67 => ( ( ord_less_eq_set_real @ S @ T3 )
% 0.33/0.67 => ( ! [X4: real] :
% 0.33/0.67 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 0.33/0.67 => ( ( H2 @ X4 )
% 0.33/0.67 = one_one_rat ) )
% 0.33/0.67 => ( ! [X4: real] :
% 0.33/0.67 ( ( member_real @ X4 @ S )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = ( H2 @ X4 ) ) )
% 0.33/0.67 => ( ( groups4061424788464935467al_rat @ G @ S )
% 0.33/0.67 = ( groups4061424788464935467al_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_cong_left
% 0.33/0.67 thf(fact_8821_prod_Omono__neutral__cong__left,axiom,
% 0.33/0.67 ! [T3: set_int,S: set_int,H2: int > rat,G: int > rat] :
% 0.33/0.67 ( ( finite_finite_int @ T3 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.67 => ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.67 => ( ( H2 @ X4 )
% 0.33/0.67 = one_one_rat ) )
% 0.33/0.67 => ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ S )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = ( H2 @ X4 ) ) )
% 0.33/0.67 => ( ( groups1072433553688619179nt_rat @ G @ S )
% 0.33/0.67 = ( groups1072433553688619179nt_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_cong_left
% 0.33/0.67 thf(fact_8822_prod_Omono__neutral__cong__left,axiom,
% 0.33/0.67 ! [T3: set_complex,S: set_complex,H2: complex > rat,G: complex > rat] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ T3 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.33/0.67 => ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 0.33/0.67 => ( ( H2 @ X4 )
% 0.33/0.67 = one_one_rat ) )
% 0.33/0.67 => ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ S )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = ( H2 @ X4 ) ) )
% 0.33/0.67 => ( ( groups225925009352817453ex_rat @ G @ S )
% 0.33/0.67 = ( groups225925009352817453ex_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_cong_left
% 0.33/0.67 thf(fact_8823_prod_Omono__neutral__cong__left,axiom,
% 0.33/0.67 ! [T3: set_real,S: set_real,H2: real > nat,G: real > nat] :
% 0.33/0.67 ( ( finite_finite_real @ T3 )
% 0.33/0.67 => ( ( ord_less_eq_set_real @ S @ T3 )
% 0.33/0.67 => ( ! [X4: real] :
% 0.33/0.67 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 0.33/0.67 => ( ( H2 @ X4 )
% 0.33/0.67 = one_one_nat ) )
% 0.33/0.67 => ( ! [X4: real] :
% 0.33/0.67 ( ( member_real @ X4 @ S )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = ( H2 @ X4 ) ) )
% 0.33/0.67 => ( ( groups4696554848551431203al_nat @ G @ S )
% 0.33/0.67 = ( groups4696554848551431203al_nat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_cong_left
% 0.33/0.67 thf(fact_8824_prod_Omono__neutral__right,axiom,
% 0.33/0.67 ! [T3: set_int,S: set_int,G: int > complex] :
% 0.33/0.67 ( ( finite_finite_int @ T3 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.67 => ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ( groups7440179247065528705omplex @ G @ T3 )
% 0.33/0.67 = ( groups7440179247065528705omplex @ G @ S ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_right
% 0.33/0.67 thf(fact_8825_prod_Omono__neutral__right,axiom,
% 0.33/0.67 ! [T3: set_complex,S: set_complex,G: complex > complex] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ T3 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.33/0.67 => ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ( groups3708469109370488835omplex @ G @ T3 )
% 0.33/0.67 = ( groups3708469109370488835omplex @ G @ S ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_right
% 0.33/0.67 thf(fact_8826_prod_Omono__neutral__right,axiom,
% 0.33/0.67 ! [T3: set_int,S: set_int,G: int > real] :
% 0.33/0.67 ( ( finite_finite_int @ T3 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.67 => ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_real ) )
% 0.33/0.67 => ( ( groups2316167850115554303t_real @ G @ T3 )
% 0.33/0.67 = ( groups2316167850115554303t_real @ G @ S ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_right
% 0.33/0.67 thf(fact_8827_prod_Omono__neutral__right,axiom,
% 0.33/0.67 ! [T3: set_complex,S: set_complex,G: complex > real] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ T3 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.33/0.67 => ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_real ) )
% 0.33/0.67 => ( ( groups766887009212190081x_real @ G @ T3 )
% 0.33/0.67 = ( groups766887009212190081x_real @ G @ S ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_right
% 0.33/0.67 thf(fact_8828_prod_Omono__neutral__right,axiom,
% 0.33/0.67 ! [T3: set_int,S: set_int,G: int > rat] :
% 0.33/0.67 ( ( finite_finite_int @ T3 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.67 => ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_rat ) )
% 0.33/0.67 => ( ( groups1072433553688619179nt_rat @ G @ T3 )
% 0.33/0.67 = ( groups1072433553688619179nt_rat @ G @ S ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_right
% 0.33/0.67 thf(fact_8829_prod_Omono__neutral__right,axiom,
% 0.33/0.67 ! [T3: set_complex,S: set_complex,G: complex > rat] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ T3 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.33/0.67 => ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_rat ) )
% 0.33/0.67 => ( ( groups225925009352817453ex_rat @ G @ T3 )
% 0.33/0.67 = ( groups225925009352817453ex_rat @ G @ S ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_right
% 0.33/0.67 thf(fact_8830_prod_Omono__neutral__right,axiom,
% 0.33/0.67 ! [T3: set_int,S: set_int,G: int > nat] :
% 0.33/0.67 ( ( finite_finite_int @ T3 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.67 => ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_nat ) )
% 0.33/0.67 => ( ( groups1707563613775114915nt_nat @ G @ T3 )
% 0.33/0.67 = ( groups1707563613775114915nt_nat @ G @ S ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_right
% 0.33/0.67 thf(fact_8831_prod_Omono__neutral__right,axiom,
% 0.33/0.67 ! [T3: set_complex,S: set_complex,G: complex > nat] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ T3 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.33/0.67 => ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_nat ) )
% 0.33/0.67 => ( ( groups861055069439313189ex_nat @ G @ T3 )
% 0.33/0.67 = ( groups861055069439313189ex_nat @ G @ S ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_right
% 0.33/0.67 thf(fact_8832_prod_Omono__neutral__right,axiom,
% 0.33/0.67 ! [T3: set_complex,S: set_complex,G: complex > int] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ T3 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.33/0.67 => ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_int ) )
% 0.33/0.67 => ( ( groups858564598930262913ex_int @ G @ T3 )
% 0.33/0.67 = ( groups858564598930262913ex_int @ G @ S ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_right
% 0.33/0.67 thf(fact_8833_prod_Omono__neutral__right,axiom,
% 0.33/0.67 ! [T3: set_nat,S: set_nat,G: nat > complex] :
% 0.33/0.67 ( ( finite_finite_nat @ T3 )
% 0.33/0.67 => ( ( ord_less_eq_set_nat @ S @ T3 )
% 0.33/0.67 => ( ! [X4: nat] :
% 0.33/0.67 ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ( groups6464643781859351333omplex @ G @ T3 )
% 0.33/0.67 = ( groups6464643781859351333omplex @ G @ S ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_right
% 0.33/0.67 thf(fact_8834_prod_Omono__neutral__left,axiom,
% 0.33/0.67 ! [T3: set_int,S: set_int,G: int > complex] :
% 0.33/0.67 ( ( finite_finite_int @ T3 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.67 => ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ( groups7440179247065528705omplex @ G @ S )
% 0.33/0.67 = ( groups7440179247065528705omplex @ G @ T3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_left
% 0.33/0.67 thf(fact_8835_prod_Omono__neutral__left,axiom,
% 0.33/0.67 ! [T3: set_complex,S: set_complex,G: complex > complex] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ T3 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.33/0.67 => ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ( groups3708469109370488835omplex @ G @ S )
% 0.33/0.67 = ( groups3708469109370488835omplex @ G @ T3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_left
% 0.33/0.67 thf(fact_8836_prod_Omono__neutral__left,axiom,
% 0.33/0.67 ! [T3: set_int,S: set_int,G: int > real] :
% 0.33/0.67 ( ( finite_finite_int @ T3 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.67 => ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_real ) )
% 0.33/0.67 => ( ( groups2316167850115554303t_real @ G @ S )
% 0.33/0.67 = ( groups2316167850115554303t_real @ G @ T3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_left
% 0.33/0.67 thf(fact_8837_prod_Omono__neutral__left,axiom,
% 0.33/0.67 ! [T3: set_complex,S: set_complex,G: complex > real] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ T3 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.33/0.67 => ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_real ) )
% 0.33/0.67 => ( ( groups766887009212190081x_real @ G @ S )
% 0.33/0.67 = ( groups766887009212190081x_real @ G @ T3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_left
% 0.33/0.67 thf(fact_8838_prod_Omono__neutral__left,axiom,
% 0.33/0.67 ! [T3: set_int,S: set_int,G: int > rat] :
% 0.33/0.67 ( ( finite_finite_int @ T3 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.67 => ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_rat ) )
% 0.33/0.67 => ( ( groups1072433553688619179nt_rat @ G @ S )
% 0.33/0.67 = ( groups1072433553688619179nt_rat @ G @ T3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_left
% 0.33/0.67 thf(fact_8839_prod_Omono__neutral__left,axiom,
% 0.33/0.67 ! [T3: set_complex,S: set_complex,G: complex > rat] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ T3 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.33/0.67 => ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_rat ) )
% 0.33/0.67 => ( ( groups225925009352817453ex_rat @ G @ S )
% 0.33/0.67 = ( groups225925009352817453ex_rat @ G @ T3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_left
% 0.33/0.67 thf(fact_8840_prod_Omono__neutral__left,axiom,
% 0.33/0.67 ! [T3: set_int,S: set_int,G: int > nat] :
% 0.33/0.67 ( ( finite_finite_int @ T3 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ S @ T3 )
% 0.33/0.67 => ( ! [X4: int] :
% 0.33/0.67 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_nat ) )
% 0.33/0.67 => ( ( groups1707563613775114915nt_nat @ G @ S )
% 0.33/0.67 = ( groups1707563613775114915nt_nat @ G @ T3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_left
% 0.33/0.67 thf(fact_8841_prod_Omono__neutral__left,axiom,
% 0.33/0.67 ! [T3: set_complex,S: set_complex,G: complex > nat] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ T3 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.33/0.67 => ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_nat ) )
% 0.33/0.67 => ( ( groups861055069439313189ex_nat @ G @ S )
% 0.33/0.67 = ( groups861055069439313189ex_nat @ G @ T3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_left
% 0.33/0.67 thf(fact_8842_prod_Omono__neutral__left,axiom,
% 0.33/0.67 ! [T3: set_complex,S: set_complex,G: complex > int] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ T3 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 0.33/0.67 => ( ! [X4: complex] :
% 0.33/0.67 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_int ) )
% 0.33/0.67 => ( ( groups858564598930262913ex_int @ G @ S )
% 0.33/0.67 = ( groups858564598930262913ex_int @ G @ T3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_left
% 0.33/0.67 thf(fact_8843_prod_Omono__neutral__left,axiom,
% 0.33/0.67 ! [T3: set_nat,S: set_nat,G: nat > complex] :
% 0.33/0.67 ( ( finite_finite_nat @ T3 )
% 0.33/0.67 => ( ( ord_less_eq_set_nat @ S @ T3 )
% 0.33/0.67 => ( ! [X4: nat] :
% 0.33/0.67 ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S ) )
% 0.33/0.67 => ( ( G @ X4 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ( groups6464643781859351333omplex @ G @ S )
% 0.33/0.67 = ( groups6464643781859351333omplex @ G @ T3 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.mono_neutral_left
% 0.33/0.67 thf(fact_8844_prod_Osame__carrierI,axiom,
% 0.33/0.67 ! [C2: set_real,A2: set_real,B3: set_real,G: real > complex,H2: real > complex] :
% 0.33/0.67 ( ( finite_finite_real @ C2 )
% 0.33/0.67 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 0.33/0.67 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 0.33/0.67 => ( ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [B2: real] :
% 0.33/0.67 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ( ( groups713298508707869441omplex @ G @ C2 )
% 0.33/0.67 = ( groups713298508707869441omplex @ H2 @ C2 ) )
% 0.33/0.67 => ( ( groups713298508707869441omplex @ G @ A2 )
% 0.33/0.67 = ( groups713298508707869441omplex @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.same_carrierI
% 0.33/0.67 thf(fact_8845_prod_Osame__carrierI,axiom,
% 0.33/0.67 ! [C2: set_int,A2: set_int,B3: set_int,G: int > complex,H2: int > complex] :
% 0.33/0.67 ( ( finite_finite_int @ C2 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ A2 @ C2 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ B3 @ C2 )
% 0.33/0.67 => ( ! [A3: int] :
% 0.33/0.67 ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [B2: int] :
% 0.33/0.67 ( ( member_int @ B2 @ ( minus_minus_set_int @ C2 @ B3 ) )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ( ( groups7440179247065528705omplex @ G @ C2 )
% 0.33/0.67 = ( groups7440179247065528705omplex @ H2 @ C2 ) )
% 0.33/0.67 => ( ( groups7440179247065528705omplex @ G @ A2 )
% 0.33/0.67 = ( groups7440179247065528705omplex @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.same_carrierI
% 0.33/0.67 thf(fact_8846_prod_Osame__carrierI,axiom,
% 0.33/0.67 ! [C2: set_complex,A2: set_complex,B3: set_complex,G: complex > complex,H2: complex > complex] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ C2 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ B3 @ C2 )
% 0.33/0.67 => ( ! [A3: complex] :
% 0.33/0.67 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [B2: complex] :
% 0.33/0.67 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B3 ) )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ( ( groups3708469109370488835omplex @ G @ C2 )
% 0.33/0.67 = ( groups3708469109370488835omplex @ H2 @ C2 ) )
% 0.33/0.67 => ( ( groups3708469109370488835omplex @ G @ A2 )
% 0.33/0.67 = ( groups3708469109370488835omplex @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.same_carrierI
% 0.33/0.67 thf(fact_8847_prod_Osame__carrierI,axiom,
% 0.33/0.67 ! [C2: set_real,A2: set_real,B3: set_real,G: real > real,H2: real > real] :
% 0.33/0.67 ( ( finite_finite_real @ C2 )
% 0.33/0.67 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 0.33/0.67 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 0.33/0.67 => ( ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_real ) )
% 0.33/0.67 => ( ! [B2: real] :
% 0.33/0.67 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_real ) )
% 0.33/0.67 => ( ( ( groups1681761925125756287l_real @ G @ C2 )
% 0.33/0.67 = ( groups1681761925125756287l_real @ H2 @ C2 ) )
% 0.33/0.67 => ( ( groups1681761925125756287l_real @ G @ A2 )
% 0.33/0.67 = ( groups1681761925125756287l_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.same_carrierI
% 0.33/0.67 thf(fact_8848_prod_Osame__carrierI,axiom,
% 0.33/0.67 ! [C2: set_int,A2: set_int,B3: set_int,G: int > real,H2: int > real] :
% 0.33/0.67 ( ( finite_finite_int @ C2 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ A2 @ C2 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ B3 @ C2 )
% 0.33/0.67 => ( ! [A3: int] :
% 0.33/0.67 ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_real ) )
% 0.33/0.67 => ( ! [B2: int] :
% 0.33/0.67 ( ( member_int @ B2 @ ( minus_minus_set_int @ C2 @ B3 ) )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_real ) )
% 0.33/0.67 => ( ( ( groups2316167850115554303t_real @ G @ C2 )
% 0.33/0.67 = ( groups2316167850115554303t_real @ H2 @ C2 ) )
% 0.33/0.67 => ( ( groups2316167850115554303t_real @ G @ A2 )
% 0.33/0.67 = ( groups2316167850115554303t_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.same_carrierI
% 0.33/0.67 thf(fact_8849_prod_Osame__carrierI,axiom,
% 0.33/0.67 ! [C2: set_complex,A2: set_complex,B3: set_complex,G: complex > real,H2: complex > real] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ C2 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ B3 @ C2 )
% 0.33/0.67 => ( ! [A3: complex] :
% 0.33/0.67 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_real ) )
% 0.33/0.67 => ( ! [B2: complex] :
% 0.33/0.67 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B3 ) )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_real ) )
% 0.33/0.67 => ( ( ( groups766887009212190081x_real @ G @ C2 )
% 0.33/0.67 = ( groups766887009212190081x_real @ H2 @ C2 ) )
% 0.33/0.67 => ( ( groups766887009212190081x_real @ G @ A2 )
% 0.33/0.67 = ( groups766887009212190081x_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.same_carrierI
% 0.33/0.67 thf(fact_8850_prod_Osame__carrierI,axiom,
% 0.33/0.67 ! [C2: set_real,A2: set_real,B3: set_real,G: real > rat,H2: real > rat] :
% 0.33/0.67 ( ( finite_finite_real @ C2 )
% 0.33/0.67 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 0.33/0.67 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 0.33/0.67 => ( ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_rat ) )
% 0.33/0.67 => ( ! [B2: real] :
% 0.33/0.67 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_rat ) )
% 0.33/0.67 => ( ( ( groups4061424788464935467al_rat @ G @ C2 )
% 0.33/0.67 = ( groups4061424788464935467al_rat @ H2 @ C2 ) )
% 0.33/0.67 => ( ( groups4061424788464935467al_rat @ G @ A2 )
% 0.33/0.67 = ( groups4061424788464935467al_rat @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.same_carrierI
% 0.33/0.67 thf(fact_8851_prod_Osame__carrierI,axiom,
% 0.33/0.67 ! [C2: set_int,A2: set_int,B3: set_int,G: int > rat,H2: int > rat] :
% 0.33/0.67 ( ( finite_finite_int @ C2 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ A2 @ C2 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ B3 @ C2 )
% 0.33/0.67 => ( ! [A3: int] :
% 0.33/0.67 ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_rat ) )
% 0.33/0.67 => ( ! [B2: int] :
% 0.33/0.67 ( ( member_int @ B2 @ ( minus_minus_set_int @ C2 @ B3 ) )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_rat ) )
% 0.33/0.67 => ( ( ( groups1072433553688619179nt_rat @ G @ C2 )
% 0.33/0.67 = ( groups1072433553688619179nt_rat @ H2 @ C2 ) )
% 0.33/0.67 => ( ( groups1072433553688619179nt_rat @ G @ A2 )
% 0.33/0.67 = ( groups1072433553688619179nt_rat @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.same_carrierI
% 0.33/0.67 thf(fact_8852_prod_Osame__carrierI,axiom,
% 0.33/0.67 ! [C2: set_complex,A2: set_complex,B3: set_complex,G: complex > rat,H2: complex > rat] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ C2 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ B3 @ C2 )
% 0.33/0.67 => ( ! [A3: complex] :
% 0.33/0.67 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_rat ) )
% 0.33/0.67 => ( ! [B2: complex] :
% 0.33/0.67 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B3 ) )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_rat ) )
% 0.33/0.67 => ( ( ( groups225925009352817453ex_rat @ G @ C2 )
% 0.33/0.67 = ( groups225925009352817453ex_rat @ H2 @ C2 ) )
% 0.33/0.67 => ( ( groups225925009352817453ex_rat @ G @ A2 )
% 0.33/0.67 = ( groups225925009352817453ex_rat @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.same_carrierI
% 0.33/0.67 thf(fact_8853_prod_Osame__carrierI,axiom,
% 0.33/0.67 ! [C2: set_real,A2: set_real,B3: set_real,G: real > nat,H2: real > nat] :
% 0.33/0.67 ( ( finite_finite_real @ C2 )
% 0.33/0.67 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 0.33/0.67 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 0.33/0.67 => ( ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_nat ) )
% 0.33/0.67 => ( ! [B2: real] :
% 0.33/0.67 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_nat ) )
% 0.33/0.67 => ( ( ( groups4696554848551431203al_nat @ G @ C2 )
% 0.33/0.67 = ( groups4696554848551431203al_nat @ H2 @ C2 ) )
% 0.33/0.67 => ( ( groups4696554848551431203al_nat @ G @ A2 )
% 0.33/0.67 = ( groups4696554848551431203al_nat @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.same_carrierI
% 0.33/0.67 thf(fact_8854_prod_Osame__carrier,axiom,
% 0.33/0.67 ! [C2: set_real,A2: set_real,B3: set_real,G: real > complex,H2: real > complex] :
% 0.33/0.67 ( ( finite_finite_real @ C2 )
% 0.33/0.67 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 0.33/0.67 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 0.33/0.67 => ( ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [B2: real] :
% 0.33/0.67 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ( ( groups713298508707869441omplex @ G @ A2 )
% 0.33/0.67 = ( groups713298508707869441omplex @ H2 @ B3 ) )
% 0.33/0.67 = ( ( groups713298508707869441omplex @ G @ C2 )
% 0.33/0.67 = ( groups713298508707869441omplex @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.same_carrier
% 0.33/0.67 thf(fact_8855_prod_Osame__carrier,axiom,
% 0.33/0.67 ! [C2: set_int,A2: set_int,B3: set_int,G: int > complex,H2: int > complex] :
% 0.33/0.67 ( ( finite_finite_int @ C2 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ A2 @ C2 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ B3 @ C2 )
% 0.33/0.67 => ( ! [A3: int] :
% 0.33/0.67 ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [B2: int] :
% 0.33/0.67 ( ( member_int @ B2 @ ( minus_minus_set_int @ C2 @ B3 ) )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ( ( groups7440179247065528705omplex @ G @ A2 )
% 0.33/0.67 = ( groups7440179247065528705omplex @ H2 @ B3 ) )
% 0.33/0.67 = ( ( groups7440179247065528705omplex @ G @ C2 )
% 0.33/0.67 = ( groups7440179247065528705omplex @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.same_carrier
% 0.33/0.67 thf(fact_8856_prod_Osame__carrier,axiom,
% 0.33/0.67 ! [C2: set_complex,A2: set_complex,B3: set_complex,G: complex > complex,H2: complex > complex] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ C2 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ B3 @ C2 )
% 0.33/0.67 => ( ! [A3: complex] :
% 0.33/0.67 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ! [B2: complex] :
% 0.33/0.67 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B3 ) )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_complex ) )
% 0.33/0.67 => ( ( ( groups3708469109370488835omplex @ G @ A2 )
% 0.33/0.67 = ( groups3708469109370488835omplex @ H2 @ B3 ) )
% 0.33/0.67 = ( ( groups3708469109370488835omplex @ G @ C2 )
% 0.33/0.67 = ( groups3708469109370488835omplex @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.same_carrier
% 0.33/0.67 thf(fact_8857_prod_Osame__carrier,axiom,
% 0.33/0.67 ! [C2: set_real,A2: set_real,B3: set_real,G: real > real,H2: real > real] :
% 0.33/0.67 ( ( finite_finite_real @ C2 )
% 0.33/0.67 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 0.33/0.67 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 0.33/0.67 => ( ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_real ) )
% 0.33/0.67 => ( ! [B2: real] :
% 0.33/0.67 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_real ) )
% 0.33/0.67 => ( ( ( groups1681761925125756287l_real @ G @ A2 )
% 0.33/0.67 = ( groups1681761925125756287l_real @ H2 @ B3 ) )
% 0.33/0.67 = ( ( groups1681761925125756287l_real @ G @ C2 )
% 0.33/0.67 = ( groups1681761925125756287l_real @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.same_carrier
% 0.33/0.67 thf(fact_8858_prod_Osame__carrier,axiom,
% 0.33/0.67 ! [C2: set_int,A2: set_int,B3: set_int,G: int > real,H2: int > real] :
% 0.33/0.67 ( ( finite_finite_int @ C2 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ A2 @ C2 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ B3 @ C2 )
% 0.33/0.67 => ( ! [A3: int] :
% 0.33/0.67 ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_real ) )
% 0.33/0.67 => ( ! [B2: int] :
% 0.33/0.67 ( ( member_int @ B2 @ ( minus_minus_set_int @ C2 @ B3 ) )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_real ) )
% 0.33/0.67 => ( ( ( groups2316167850115554303t_real @ G @ A2 )
% 0.33/0.67 = ( groups2316167850115554303t_real @ H2 @ B3 ) )
% 0.33/0.67 = ( ( groups2316167850115554303t_real @ G @ C2 )
% 0.33/0.67 = ( groups2316167850115554303t_real @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.same_carrier
% 0.33/0.67 thf(fact_8859_prod_Osame__carrier,axiom,
% 0.33/0.67 ! [C2: set_complex,A2: set_complex,B3: set_complex,G: complex > real,H2: complex > real] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ C2 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ B3 @ C2 )
% 0.33/0.67 => ( ! [A3: complex] :
% 0.33/0.67 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_real ) )
% 0.33/0.67 => ( ! [B2: complex] :
% 0.33/0.67 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B3 ) )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_real ) )
% 0.33/0.67 => ( ( ( groups766887009212190081x_real @ G @ A2 )
% 0.33/0.67 = ( groups766887009212190081x_real @ H2 @ B3 ) )
% 0.33/0.67 = ( ( groups766887009212190081x_real @ G @ C2 )
% 0.33/0.67 = ( groups766887009212190081x_real @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.same_carrier
% 0.33/0.67 thf(fact_8860_prod_Osame__carrier,axiom,
% 0.33/0.67 ! [C2: set_real,A2: set_real,B3: set_real,G: real > rat,H2: real > rat] :
% 0.33/0.67 ( ( finite_finite_real @ C2 )
% 0.33/0.67 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 0.33/0.67 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 0.33/0.67 => ( ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_rat ) )
% 0.33/0.67 => ( ! [B2: real] :
% 0.33/0.67 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_rat ) )
% 0.33/0.67 => ( ( ( groups4061424788464935467al_rat @ G @ A2 )
% 0.33/0.67 = ( groups4061424788464935467al_rat @ H2 @ B3 ) )
% 0.33/0.67 = ( ( groups4061424788464935467al_rat @ G @ C2 )
% 0.33/0.67 = ( groups4061424788464935467al_rat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.same_carrier
% 0.33/0.67 thf(fact_8861_prod_Osame__carrier,axiom,
% 0.33/0.67 ! [C2: set_int,A2: set_int,B3: set_int,G: int > rat,H2: int > rat] :
% 0.33/0.67 ( ( finite_finite_int @ C2 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ A2 @ C2 )
% 0.33/0.67 => ( ( ord_less_eq_set_int @ B3 @ C2 )
% 0.33/0.67 => ( ! [A3: int] :
% 0.33/0.67 ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_rat ) )
% 0.33/0.67 => ( ! [B2: int] :
% 0.33/0.67 ( ( member_int @ B2 @ ( minus_minus_set_int @ C2 @ B3 ) )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_rat ) )
% 0.33/0.67 => ( ( ( groups1072433553688619179nt_rat @ G @ A2 )
% 0.33/0.67 = ( groups1072433553688619179nt_rat @ H2 @ B3 ) )
% 0.33/0.67 = ( ( groups1072433553688619179nt_rat @ G @ C2 )
% 0.33/0.67 = ( groups1072433553688619179nt_rat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.same_carrier
% 0.33/0.67 thf(fact_8862_prod_Osame__carrier,axiom,
% 0.33/0.67 ! [C2: set_complex,A2: set_complex,B3: set_complex,G: complex > rat,H2: complex > rat] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ C2 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 0.33/0.67 => ( ( ord_le211207098394363844omplex @ B3 @ C2 )
% 0.33/0.67 => ( ! [A3: complex] :
% 0.33/0.67 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_rat ) )
% 0.33/0.67 => ( ! [B2: complex] :
% 0.33/0.67 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B3 ) )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_rat ) )
% 0.33/0.67 => ( ( ( groups225925009352817453ex_rat @ G @ A2 )
% 0.33/0.67 = ( groups225925009352817453ex_rat @ H2 @ B3 ) )
% 0.33/0.67 = ( ( groups225925009352817453ex_rat @ G @ C2 )
% 0.33/0.67 = ( groups225925009352817453ex_rat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.same_carrier
% 0.33/0.67 thf(fact_8863_prod_Osame__carrier,axiom,
% 0.33/0.67 ! [C2: set_real,A2: set_real,B3: set_real,G: real > nat,H2: real > nat] :
% 0.33/0.67 ( ( finite_finite_real @ C2 )
% 0.33/0.67 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 0.33/0.67 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 0.33/0.67 => ( ! [A3: real] :
% 0.33/0.67 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 0.33/0.67 => ( ( G @ A3 )
% 0.33/0.67 = one_one_nat ) )
% 0.33/0.67 => ( ! [B2: real] :
% 0.33/0.67 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 0.33/0.67 => ( ( H2 @ B2 )
% 0.33/0.67 = one_one_nat ) )
% 0.33/0.67 => ( ( ( groups4696554848551431203al_nat @ G @ A2 )
% 0.33/0.67 = ( groups4696554848551431203al_nat @ H2 @ B3 ) )
% 0.33/0.67 = ( ( groups4696554848551431203al_nat @ G @ C2 )
% 0.33/0.67 = ( groups4696554848551431203al_nat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.same_carrier
% 0.33/0.67 thf(fact_8864_arcsin__le__mono,axiom,
% 0.33/0.67 ! [X: real,Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.67 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 0.33/0.67 => ( ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
% 0.33/0.67 = ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arcsin_le_mono
% 0.33/0.67 thf(fact_8865_arcsin__eq__iff,axiom,
% 0.33/0.67 ! [X: real,Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.67 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 0.33/0.67 => ( ( ( arcsin @ X )
% 0.33/0.67 = ( arcsin @ Y ) )
% 0.33/0.67 = ( X = Y ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arcsin_eq_iff
% 0.33/0.67 thf(fact_8866_prod_OatLeast0__atMost__Suc,axiom,
% 0.33/0.67 ! [G: nat > real,N2: nat] :
% 0.33/0.67 ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.atLeast0_atMost_Suc
% 0.33/0.67 thf(fact_8867_prod_OatLeast0__atMost__Suc,axiom,
% 0.33/0.67 ! [G: nat > rat,N2: nat] :
% 0.33/0.67 ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.atLeast0_atMost_Suc
% 0.33/0.67 thf(fact_8868_prod_OatLeast0__atMost__Suc,axiom,
% 0.33/0.67 ! [G: nat > nat,N2: nat] :
% 0.33/0.67 ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.atLeast0_atMost_Suc
% 0.33/0.67 thf(fact_8869_prod_OatLeast0__atMost__Suc,axiom,
% 0.33/0.67 ! [G: nat > int,N2: nat] :
% 0.33/0.67 ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.atLeast0_atMost_Suc
% 0.33/0.67 thf(fact_8870_prod_OatLeast__Suc__atMost,axiom,
% 0.33/0.67 ! [M: nat,N2: nat,G: nat > real] :
% 0.33/0.67 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.67 => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.67 = ( times_times_real @ ( G @ M ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.atLeast_Suc_atMost
% 0.33/0.67 thf(fact_8871_prod_OatLeast__Suc__atMost,axiom,
% 0.33/0.67 ! [M: nat,N2: nat,G: nat > rat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.67 => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.67 = ( times_times_rat @ ( G @ M ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.atLeast_Suc_atMost
% 0.33/0.67 thf(fact_8872_prod_OatLeast__Suc__atMost,axiom,
% 0.33/0.67 ! [M: nat,N2: nat,G: nat > nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.67 => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.67 = ( times_times_nat @ ( G @ M ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.atLeast_Suc_atMost
% 0.33/0.67 thf(fact_8873_prod_OatLeast__Suc__atMost,axiom,
% 0.33/0.67 ! [M: nat,N2: nat,G: nat > int] :
% 0.33/0.67 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.67 => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.33/0.67 = ( times_times_int @ ( G @ M ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.atLeast_Suc_atMost
% 0.33/0.67 thf(fact_8874_prod_Onat__ivl__Suc_H,axiom,
% 0.33/0.67 ! [M: nat,N2: nat,G: nat > real] :
% 0.33/0.67 ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 0.33/0.67 => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( times_times_real @ ( G @ ( suc @ N2 ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.nat_ivl_Suc'
% 0.33/0.67 thf(fact_8875_prod_Onat__ivl__Suc_H,axiom,
% 0.33/0.67 ! [M: nat,N2: nat,G: nat > rat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 0.33/0.67 => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( times_times_rat @ ( G @ ( suc @ N2 ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.nat_ivl_Suc'
% 0.33/0.67 thf(fact_8876_prod_Onat__ivl__Suc_H,axiom,
% 0.33/0.67 ! [M: nat,N2: nat,G: nat > nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 0.33/0.67 => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( times_times_nat @ ( G @ ( suc @ N2 ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.nat_ivl_Suc'
% 0.33/0.67 thf(fact_8877_prod_Onat__ivl__Suc_H,axiom,
% 0.33/0.67 ! [M: nat,N2: nat,G: nat > int] :
% 0.33/0.67 ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 0.33/0.67 => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( times_times_int @ ( G @ ( suc @ N2 ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.nat_ivl_Suc'
% 0.33/0.67 thf(fact_8878_prod_OlessThan__Suc__shift,axiom,
% 0.33/0.67 ! [G: nat > real,N2: nat] :
% 0.33/0.67 ( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( times_times_real @ ( G @ zero_zero_nat )
% 0.33/0.67 @ ( groups129246275422532515t_real
% 0.33/0.67 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.33/0.67 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.lessThan_Suc_shift
% 0.33/0.67 thf(fact_8879_prod_OlessThan__Suc__shift,axiom,
% 0.33/0.67 ! [G: nat > rat,N2: nat] :
% 0.33/0.67 ( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( times_times_rat @ ( G @ zero_zero_nat )
% 0.33/0.67 @ ( groups73079841787564623at_rat
% 0.33/0.67 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.33/0.67 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.lessThan_Suc_shift
% 0.33/0.67 thf(fact_8880_prod_OlessThan__Suc__shift,axiom,
% 0.33/0.67 ! [G: nat > nat,N2: nat] :
% 0.33/0.67 ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( times_times_nat @ ( G @ zero_zero_nat )
% 0.33/0.67 @ ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.33/0.67 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.lessThan_Suc_shift
% 0.33/0.67 thf(fact_8881_prod_OlessThan__Suc__shift,axiom,
% 0.33/0.67 ! [G: nat > int,N2: nat] :
% 0.33/0.67 ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( times_times_int @ ( G @ zero_zero_nat )
% 0.33/0.67 @ ( groups705719431365010083at_int
% 0.33/0.67 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.33/0.67 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.lessThan_Suc_shift
% 0.33/0.67 thf(fact_8882_prod_OSuc__reindex__ivl,axiom,
% 0.33/0.67 ! [M: nat,N2: nat,G: nat > real] :
% 0.33/0.67 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.67 => ( ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( times_times_real @ ( G @ M )
% 0.33/0.67 @ ( groups129246275422532515t_real
% 0.33/0.67 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.33/0.67 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.Suc_reindex_ivl
% 0.33/0.67 thf(fact_8883_prod_OSuc__reindex__ivl,axiom,
% 0.33/0.67 ! [M: nat,N2: nat,G: nat > rat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.67 => ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( times_times_rat @ ( G @ M )
% 0.33/0.67 @ ( groups73079841787564623at_rat
% 0.33/0.67 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.33/0.67 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.Suc_reindex_ivl
% 0.33/0.67 thf(fact_8884_prod_OSuc__reindex__ivl,axiom,
% 0.33/0.67 ! [M: nat,N2: nat,G: nat > nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.67 => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( times_times_nat @ ( G @ M )
% 0.33/0.67 @ ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.33/0.67 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.Suc_reindex_ivl
% 0.33/0.67 thf(fact_8885_prod_OSuc__reindex__ivl,axiom,
% 0.33/0.67 ! [M: nat,N2: nat,G: nat > int] :
% 0.33/0.67 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.67 => ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
% 0.33/0.67 = ( times_times_int @ ( G @ M )
% 0.33/0.67 @ ( groups705719431365010083at_int
% 0.33/0.67 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.33/0.67 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.Suc_reindex_ivl
% 0.33/0.67 thf(fact_8886_prod_OatLeast1__atMost__eq,axiom,
% 0.33/0.67 ! [G: nat > nat,N2: nat] :
% 0.33/0.67 ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
% 0.33/0.67 = ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
% 0.33/0.67 @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.atLeast1_atMost_eq
% 0.33/0.67 thf(fact_8887_prod_OatLeast1__atMost__eq,axiom,
% 0.33/0.67 ! [G: nat > int,N2: nat] :
% 0.33/0.67 ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
% 0.33/0.67 = ( groups705719431365010083at_int
% 0.33/0.67 @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
% 0.33/0.67 @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod.atLeast1_atMost_eq
% 0.33/0.67 thf(fact_8888_fact__prod,axiom,
% 0.33/0.67 ( semiri1406184849735516958ct_int
% 0.33/0.67 = ( ^ [N: nat] :
% 0.33/0.67 ( semiri1314217659103216013at_int
% 0.33/0.67 @ ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [X2: nat] : X2
% 0.33/0.67 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % fact_prod
% 0.33/0.67 thf(fact_8889_fact__prod,axiom,
% 0.33/0.67 ( semiri5044797733671781792omplex
% 0.33/0.67 = ( ^ [N: nat] :
% 0.33/0.67 ( semiri8010041392384452111omplex
% 0.33/0.67 @ ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [X2: nat] : X2
% 0.33/0.67 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % fact_prod
% 0.33/0.67 thf(fact_8890_fact__prod,axiom,
% 0.33/0.67 ( semiri2265585572941072030t_real
% 0.33/0.67 = ( ^ [N: nat] :
% 0.33/0.67 ( semiri5074537144036343181t_real
% 0.33/0.67 @ ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [X2: nat] : X2
% 0.33/0.67 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % fact_prod
% 0.33/0.67 thf(fact_8891_fact__prod,axiom,
% 0.33/0.67 ( semiri1408675320244567234ct_nat
% 0.33/0.67 = ( ^ [N: nat] :
% 0.33/0.67 ( semiri1316708129612266289at_nat
% 0.33/0.67 @ ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [X2: nat] : X2
% 0.33/0.67 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % fact_prod
% 0.33/0.67 thf(fact_8892_prod__mono__strict,axiom,
% 0.33/0.67 ! [A2: set_int,F: int > nat,G: int > nat] :
% 0.33/0.67 ( ( finite_finite_int @ A2 )
% 0.33/0.67 => ( ! [I4: int] :
% 0.33/0.67 ( ( member_int @ I4 @ A2 )
% 0.33/0.67 => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) )
% 0.33/0.67 & ( ord_less_nat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 0.33/0.67 => ( ( A2 != bot_bot_set_int )
% 0.33/0.67 => ( ord_less_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ A2 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_mono_strict
% 0.33/0.67 thf(fact_8893_prod__mono__strict,axiom,
% 0.33/0.67 ! [A2: set_real,F: real > nat,G: real > nat] :
% 0.33/0.67 ( ( finite_finite_real @ A2 )
% 0.33/0.67 => ( ! [I4: real] :
% 0.33/0.67 ( ( member_real @ I4 @ A2 )
% 0.33/0.67 => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) )
% 0.33/0.67 & ( ord_less_nat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 0.33/0.67 => ( ( A2 != bot_bot_set_real )
% 0.33/0.67 => ( ord_less_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_mono_strict
% 0.33/0.67 thf(fact_8894_prod__mono__strict,axiom,
% 0.33/0.67 ! [A2: set_set_nat,F: set_nat > int,G: set_nat > int] :
% 0.33/0.67 ( ( finite1152437895449049373et_nat @ A2 )
% 0.33/0.67 => ( ! [I4: set_nat] :
% 0.33/0.67 ( ( member_set_nat @ I4 @ A2 )
% 0.33/0.67 => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) )
% 0.33/0.67 & ( ord_less_int @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 0.33/0.67 => ( ( A2 != bot_bot_set_set_nat )
% 0.33/0.67 => ( ord_less_int @ ( groups4246057289670975065at_int @ F @ A2 ) @ ( groups4246057289670975065at_int @ G @ A2 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_mono_strict
% 0.33/0.67 thf(fact_8895_prod__mono__strict,axiom,
% 0.33/0.67 ! [A2: set_complex,F: complex > int,G: complex > int] :
% 0.33/0.67 ( ( finite3207457112153483333omplex @ A2 )
% 0.33/0.67 => ( ! [I4: complex] :
% 0.33/0.67 ( ( member_complex @ I4 @ A2 )
% 0.33/0.67 => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) )
% 0.33/0.67 & ( ord_less_int @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 0.33/0.67 => ( ( A2 != bot_bot_set_complex )
% 0.33/0.67 => ( ord_less_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ G @ A2 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_mono_strict
% 0.33/0.67 thf(fact_8896_prod__mono__strict,axiom,
% 0.33/0.67 ! [A2: set_real,F: real > int,G: real > int] :
% 0.33/0.67 ( ( finite_finite_real @ A2 )
% 0.33/0.67 => ( ! [I4: real] :
% 0.33/0.67 ( ( member_real @ I4 @ A2 )
% 0.33/0.67 => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) )
% 0.33/0.67 & ( ord_less_int @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 0.33/0.67 => ( ( A2 != bot_bot_set_real )
% 0.33/0.67 => ( ord_less_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ ( groups4694064378042380927al_int @ G @ A2 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_mono_strict
% 0.33/0.67 thf(fact_8897_prod__mono__strict,axiom,
% 0.33/0.67 ! [A2: set_nat,F: nat > nat,G: nat > nat] :
% 0.33/0.67 ( ( finite_finite_nat @ A2 )
% 0.33/0.67 => ( ! [I4: nat] :
% 0.33/0.67 ( ( member_nat @ I4 @ A2 )
% 0.33/0.67 => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) )
% 0.33/0.67 & ( ord_less_nat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 0.33/0.67 => ( ( A2 != bot_bot_set_nat )
% 0.33/0.67 => ( ord_less_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ ( groups708209901874060359at_nat @ G @ A2 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_mono_strict
% 0.33/0.67 thf(fact_8898_prod__mono__strict,axiom,
% 0.33/0.67 ! [A2: set_nat,F: nat > int,G: nat > int] :
% 0.33/0.67 ( ( finite_finite_nat @ A2 )
% 0.33/0.67 => ( ! [I4: nat] :
% 0.33/0.67 ( ( member_nat @ I4 @ A2 )
% 0.33/0.67 => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) )
% 0.33/0.67 & ( ord_less_int @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 0.33/0.67 => ( ( A2 != bot_bot_set_nat )
% 0.33/0.67 => ( ord_less_int @ ( groups705719431365010083at_int @ F @ A2 ) @ ( groups705719431365010083at_int @ G @ A2 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_mono_strict
% 0.33/0.67 thf(fact_8899_prod__mono__strict,axiom,
% 0.33/0.67 ! [A2: set_int,F: int > int,G: int > int] :
% 0.33/0.67 ( ( finite_finite_int @ A2 )
% 0.33/0.67 => ( ! [I4: int] :
% 0.33/0.67 ( ( member_int @ I4 @ A2 )
% 0.33/0.67 => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) )
% 0.33/0.67 & ( ord_less_int @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 0.33/0.67 => ( ( A2 != bot_bot_set_int )
% 0.33/0.67 => ( ord_less_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ ( groups1705073143266064639nt_int @ G @ A2 ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % prod_mono_strict
% 0.33/0.67 thf(fact_8900_arccos__lbound,axiom,
% 0.33/0.67 ! [Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.67 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.67 => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arccos_lbound
% 0.33/0.67 thf(fact_8901_arccos__less__arccos,axiom,
% 0.33/0.67 ! [X: real,Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.67 => ( ( ord_less_real @ X @ Y )
% 0.33/0.67 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.67 => ( ord_less_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arccos_less_arccos
% 0.33/0.67 thf(fact_8902_arccos__less__mono,axiom,
% 0.33/0.67 ! [X: real,Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.67 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 0.33/0.67 => ( ( ord_less_real @ ( arccos @ X ) @ ( arccos @ Y ) )
% 0.33/0.67 = ( ord_less_real @ Y @ X ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arccos_less_mono
% 0.33/0.67 thf(fact_8903_arccos__ubound,axiom,
% 0.33/0.67 ! [Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.67 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.67 => ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arccos_ubound
% 0.33/0.67 thf(fact_8904_arccos__cos,axiom,
% 0.33/0.67 ! [X: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.67 => ( ( ord_less_eq_real @ X @ pi )
% 0.33/0.67 => ( ( arccos @ ( cos_real @ X ) )
% 0.33/0.67 = X ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arccos_cos
% 0.33/0.67 thf(fact_8905_arcsin__less__arcsin,axiom,
% 0.33/0.67 ! [X: real,Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.67 => ( ( ord_less_real @ X @ Y )
% 0.33/0.67 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.67 => ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arcsin_less_arcsin
% 0.33/0.67 thf(fact_8906_arcsin__less__mono,axiom,
% 0.33/0.67 ! [X: real,Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.67 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 0.33/0.67 => ( ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
% 0.33/0.67 = ( ord_less_real @ X @ Y ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arcsin_less_mono
% 0.33/0.67 thf(fact_8907_cos__arccos__abs,axiom,
% 0.33/0.67 ! [Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 0.33/0.67 => ( ( cos_real @ ( arccos @ Y ) )
% 0.33/0.67 = Y ) ) ).
% 0.33/0.67
% 0.33/0.67 % cos_arccos_abs
% 0.33/0.67 thf(fact_8908_arccos__cos__eq__abs,axiom,
% 0.33/0.67 ! [Theta: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( abs_abs_real @ Theta ) @ pi )
% 0.33/0.67 => ( ( arccos @ ( cos_real @ Theta ) )
% 0.33/0.67 = ( abs_abs_real @ Theta ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arccos_cos_eq_abs
% 0.33/0.67 thf(fact_8909_fact__eq__fact__times,axiom,
% 0.33/0.67 ! [N2: nat,M: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ N2 @ M )
% 0.33/0.67 => ( ( semiri1408675320244567234ct_nat @ M )
% 0.33/0.67 = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N2 )
% 0.33/0.67 @ ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [X2: nat] : X2
% 0.33/0.67 @ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % fact_eq_fact_times
% 0.33/0.67 thf(fact_8910_arccos__lt__bounded,axiom,
% 0.33/0.67 ! [Y: real] :
% 0.33/0.67 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.67 => ( ( ord_less_real @ Y @ one_one_real )
% 0.33/0.67 => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y ) )
% 0.33/0.67 & ( ord_less_real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arccos_lt_bounded
% 0.33/0.67 thf(fact_8911_arccos__bounded,axiom,
% 0.33/0.67 ! [Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.67 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
% 0.33/0.67 & ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arccos_bounded
% 0.33/0.67 thf(fact_8912_sin__arccos__nonzero,axiom,
% 0.33/0.67 ! [X: real] :
% 0.33/0.67 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.67 => ( ( ord_less_real @ X @ one_one_real )
% 0.33/0.67 => ( ( sin_real @ ( arccos @ X ) )
% 0.33/0.67 != zero_zero_real ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % sin_arccos_nonzero
% 0.33/0.67 thf(fact_8913_arccos__cos2,axiom,
% 0.33/0.67 ! [X: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.33/0.67 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X )
% 0.33/0.67 => ( ( arccos @ ( cos_real @ X ) )
% 0.33/0.67 = ( uminus_uminus_real @ X ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arccos_cos2
% 0.33/0.67 thf(fact_8914_arccos__minus,axiom,
% 0.33/0.67 ! [X: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.67 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.67 => ( ( arccos @ ( uminus_uminus_real @ X ) )
% 0.33/0.67 = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arccos_minus
% 0.33/0.67 thf(fact_8915_cos__arcsin__nonzero,axiom,
% 0.33/0.67 ! [X: real] :
% 0.33/0.67 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.67 => ( ( ord_less_real @ X @ one_one_real )
% 0.33/0.67 => ( ( cos_real @ ( arcsin @ X ) )
% 0.33/0.67 != zero_zero_real ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % cos_arcsin_nonzero
% 0.33/0.67 thf(fact_8916_fact__div__fact,axiom,
% 0.33/0.67 ! [N2: nat,M: nat] :
% 0.33/0.67 ( ( ord_less_eq_nat @ N2 @ M )
% 0.33/0.67 => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) )
% 0.33/0.67 = ( groups708209901874060359at_nat
% 0.33/0.67 @ ^ [X2: nat] : X2
% 0.33/0.67 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % fact_div_fact
% 0.33/0.67 thf(fact_8917_arccos,axiom,
% 0.33/0.67 ! [Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.67 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
% 0.33/0.67 & ( ord_less_eq_real @ ( arccos @ Y ) @ pi )
% 0.33/0.67 & ( ( cos_real @ ( arccos @ Y ) )
% 0.33/0.67 = Y ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arccos
% 0.33/0.67 thf(fact_8918_arccos__minus__abs,axiom,
% 0.33/0.67 ! [X: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.67 => ( ( arccos @ ( uminus_uminus_real @ X ) )
% 0.33/0.67 = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arccos_minus_abs
% 0.33/0.67 thf(fact_8919_arccos__le__pi2,axiom,
% 0.33/0.67 ! [Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.67 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.67 => ( ord_less_eq_real @ ( arccos @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arccos_le_pi2
% 0.33/0.67 thf(fact_8920_arcsin__lt__bounded,axiom,
% 0.33/0.67 ! [Y: real] :
% 0.33/0.67 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.67 => ( ( ord_less_real @ Y @ one_one_real )
% 0.33/0.67 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
% 0.33/0.67 & ( ord_less_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arcsin_lt_bounded
% 0.33/0.67 thf(fact_8921_arcsin__lbound,axiom,
% 0.33/0.67 ! [Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.67 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.67 => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arcsin_lbound
% 0.33/0.67 thf(fact_8922_arcsin__ubound,axiom,
% 0.33/0.67 ! [Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.67 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.67 => ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.67
% 0.33/0.67 % arcsin_ubound
% 0.33/0.67 thf(fact_8923_arcsin__bounded,axiom,
% 0.33/0.67 ! [Y: real] :
% 0.33/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.67 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.67 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
% 0.33/0.67 & ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % arcsin_bounded
% 0.33/0.68 thf(fact_8924_arcsin__sin,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.68 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.68 => ( ( arcsin @ ( sin_real @ X ) )
% 0.33/0.68 = X ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % arcsin_sin
% 0.33/0.68 thf(fact_8925_arcsin,axiom,
% 0.33/0.68 ! [Y: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.68 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.68 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
% 0.33/0.68 & ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.68 & ( ( sin_real @ ( arcsin @ Y ) )
% 0.33/0.68 = Y ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % arcsin
% 0.33/0.68 thf(fact_8926_arcsin__pi,axiom,
% 0.33/0.68 ! [Y: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 0.33/0.68 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 0.33/0.68 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
% 0.33/0.68 & ( ord_less_eq_real @ ( arcsin @ Y ) @ pi )
% 0.33/0.68 & ( ( sin_real @ ( arcsin @ Y ) )
% 0.33/0.68 = Y ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % arcsin_pi
% 0.33/0.68 thf(fact_8927_arcsin__le__iff,axiom,
% 0.33/0.68 ! [X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.68 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.68 => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
% 0.33/0.68 => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.68 => ( ( ord_less_eq_real @ ( arcsin @ X ) @ Y )
% 0.33/0.68 = ( ord_less_eq_real @ X @ ( sin_real @ Y ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % arcsin_le_iff
% 0.33/0.68 thf(fact_8928_le__arcsin__iff,axiom,
% 0.33/0.68 ! [X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.68 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.68 => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
% 0.33/0.68 => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.68 => ( ( ord_less_eq_real @ Y @ ( arcsin @ X ) )
% 0.33/0.68 = ( ord_less_eq_real @ ( sin_real @ Y ) @ X ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % le_arcsin_iff
% 0.33/0.68 thf(fact_8929_arccos__cos__eq__abs__2pi,axiom,
% 0.33/0.68 ! [Theta: real] :
% 0.33/0.68 ~ ! [K2: int] :
% 0.33/0.68 ( ( arccos @ ( cos_real @ Theta ) )
% 0.33/0.68 != ( abs_abs_real @ ( minus_minus_real @ Theta @ ( times_times_real @ ( ring_1_of_int_real @ K2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % arccos_cos_eq_abs_2pi
% 0.33/0.68 thf(fact_8930_sin__arccos,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.68 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.68 => ( ( sin_real @ ( arccos @ X ) )
% 0.33/0.68 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % sin_arccos
% 0.33/0.68 thf(fact_8931_Maclaurin__sin__bound,axiom,
% 0.33/0.68 ! [X: real,N2: nat] :
% 0.33/0.68 ( ord_less_eq_real
% 0.33/0.68 @ ( abs_abs_real
% 0.33/0.68 @ ( minus_minus_real @ ( sin_real @ X )
% 0.33/0.68 @ ( groups6591440286371151544t_real
% 0.33/0.68 @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 0.33/0.68 @ ( set_ord_lessThan_nat @ N2 ) ) ) )
% 0.33/0.68 @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( abs_abs_real @ X ) @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Maclaurin_sin_bound
% 0.33/0.68 thf(fact_8932_central__binomial__lower__bound,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % central_binomial_lower_bound
% 0.33/0.68 thf(fact_8933_cot__less__zero,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 0.33/0.68 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.33/0.68 => ( ord_less_real @ ( cot_real @ X ) @ zero_zero_real ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % cot_less_zero
% 0.33/0.68 thf(fact_8934_binomial__Suc__n,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( binomial @ ( suc @ N2 ) @ N2 )
% 0.33/0.68 = ( suc @ N2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % binomial_Suc_n
% 0.33/0.68 thf(fact_8935_binomial__n__n,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( binomial @ N2 @ N2 )
% 0.33/0.68 = one_one_nat ) ).
% 0.33/0.68
% 0.33/0.68 % binomial_n_n
% 0.33/0.68 thf(fact_8936_binomial__0__Suc,axiom,
% 0.33/0.68 ! [K: nat] :
% 0.33/0.68 ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
% 0.33/0.68 = zero_zero_nat ) ).
% 0.33/0.68
% 0.33/0.68 % binomial_0_Suc
% 0.33/0.68 thf(fact_8937_binomial__1,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( binomial @ N2 @ ( suc @ zero_zero_nat ) )
% 0.33/0.68 = N2 ) ).
% 0.33/0.68
% 0.33/0.68 % binomial_1
% 0.33/0.68 thf(fact_8938_binomial__eq__0__iff,axiom,
% 0.33/0.68 ! [N2: nat,K: nat] :
% 0.33/0.68 ( ( ( binomial @ N2 @ K )
% 0.33/0.68 = zero_zero_nat )
% 0.33/0.68 = ( ord_less_nat @ N2 @ K ) ) ).
% 0.33/0.68
% 0.33/0.68 % binomial_eq_0_iff
% 0.33/0.68 thf(fact_8939_binomial__Suc__Suc,axiom,
% 0.33/0.68 ! [N2: nat,K: nat] :
% 0.33/0.68 ( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) )
% 0.33/0.68 = ( plus_plus_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % binomial_Suc_Suc
% 0.33/0.68 thf(fact_8940_binomial__n__0,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( binomial @ N2 @ zero_zero_nat )
% 0.33/0.68 = one_one_nat ) ).
% 0.33/0.68
% 0.33/0.68 % binomial_n_0
% 0.33/0.68 thf(fact_8941_zero__less__binomial__iff,axiom,
% 0.33/0.68 ! [N2: nat,K: nat] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N2 @ K ) )
% 0.33/0.68 = ( ord_less_eq_nat @ K @ N2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % zero_less_binomial_iff
% 0.33/0.68 thf(fact_8942_cot__npi,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) )
% 0.33/0.68 = zero_zero_real ) ).
% 0.33/0.68
% 0.33/0.68 % cot_npi
% 0.33/0.68 thf(fact_8943_cot__periodic,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( cot_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.33/0.68 = ( cot_real @ X ) ) ).
% 0.33/0.68
% 0.33/0.68 % cot_periodic
% 0.33/0.68 thf(fact_8944_choose__one,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( binomial @ N2 @ one_one_nat )
% 0.33/0.68 = N2 ) ).
% 0.33/0.68
% 0.33/0.68 % choose_one
% 0.33/0.68 thf(fact_8945_real__sqrt__inverse,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( sqrt @ ( inverse_inverse_real @ X ) )
% 0.33/0.68 = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_sqrt_inverse
% 0.33/0.68 thf(fact_8946_binomial__eq__0,axiom,
% 0.33/0.68 ! [N2: nat,K: nat] :
% 0.33/0.68 ( ( ord_less_nat @ N2 @ K )
% 0.33/0.68 => ( ( binomial @ N2 @ K )
% 0.33/0.68 = zero_zero_nat ) ) ).
% 0.33/0.68
% 0.33/0.68 % binomial_eq_0
% 0.33/0.68 thf(fact_8947_Suc__times__binomial__eq,axiom,
% 0.33/0.68 ! [N2: nat,K: nat] :
% 0.33/0.68 ( ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) )
% 0.33/0.68 = ( times_times_nat @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Suc_times_binomial_eq
% 0.33/0.68 thf(fact_8948_Suc__times__binomial,axiom,
% 0.33/0.68 ! [K: nat,N2: nat] :
% 0.33/0.68 ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) )
% 0.33/0.68 = ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Suc_times_binomial
% 0.33/0.68 thf(fact_8949_choose__mult__lemma,axiom,
% 0.33/0.68 ! [M: nat,R: nat,K: nat] :
% 0.33/0.68 ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
% 0.33/0.68 = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R ) @ M ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % choose_mult_lemma
% 0.33/0.68 thf(fact_8950_binomial__symmetric,axiom,
% 0.33/0.68 ! [K: nat,N2: nat] :
% 0.33/0.68 ( ( ord_less_eq_nat @ K @ N2 )
% 0.33/0.68 => ( ( binomial @ N2 @ K )
% 0.33/0.68 = ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % binomial_symmetric
% 0.33/0.68 thf(fact_8951_binomial__le__pow,axiom,
% 0.33/0.68 ! [R: nat,N2: nat] :
% 0.33/0.68 ( ( ord_less_eq_nat @ R @ N2 )
% 0.33/0.68 => ( ord_less_eq_nat @ ( binomial @ N2 @ R ) @ ( power_power_nat @ N2 @ R ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % binomial_le_pow
% 0.33/0.68 thf(fact_8952_divide__real__def,axiom,
% 0.33/0.68 ( divide_divide_real
% 0.33/0.68 = ( ^ [X2: real,Y2: real] : ( times_times_real @ X2 @ ( inverse_inverse_real @ Y2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % divide_real_def
% 0.33/0.68 thf(fact_8953_zero__less__binomial,axiom,
% 0.33/0.68 ! [K: nat,N2: nat] :
% 0.33/0.68 ( ( ord_less_eq_nat @ K @ N2 )
% 0.33/0.68 => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N2 @ K ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % zero_less_binomial
% 0.33/0.68 thf(fact_8954_Suc__times__binomial__add,axiom,
% 0.33/0.68 ! [A: nat,B: nat] :
% 0.33/0.68 ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
% 0.33/0.68 = ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Suc_times_binomial_add
% 0.33/0.68 thf(fact_8955_binomial__Suc__Suc__eq__times,axiom,
% 0.33/0.68 ! [N2: nat,K: nat] :
% 0.33/0.68 ( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) )
% 0.33/0.68 = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) @ ( suc @ K ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % binomial_Suc_Suc_eq_times
% 0.33/0.68 thf(fact_8956_choose__mult,axiom,
% 0.33/0.68 ! [K: nat,M: nat,N2: nat] :
% 0.33/0.68 ( ( ord_less_eq_nat @ K @ M )
% 0.33/0.68 => ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.68 => ( ( times_times_nat @ ( binomial @ N2 @ M ) @ ( binomial @ M @ K ) )
% 0.33/0.68 = ( times_times_nat @ ( binomial @ N2 @ K ) @ ( binomial @ ( minus_minus_nat @ N2 @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % choose_mult
% 0.33/0.68 thf(fact_8957_binomial__absorb__comp,axiom,
% 0.33/0.68 ! [N2: nat,K: nat] :
% 0.33/0.68 ( ( times_times_nat @ ( minus_minus_nat @ N2 @ K ) @ ( binomial @ N2 @ K ) )
% 0.33/0.68 = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % binomial_absorb_comp
% 0.33/0.68 thf(fact_8958_prod__int__eq,axiom,
% 0.33/0.68 ! [I2: nat,J: nat] :
% 0.33/0.68 ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I2 @ J ) )
% 0.33/0.68 = ( groups1705073143266064639nt_int
% 0.33/0.68 @ ^ [X2: int] : X2
% 0.33/0.68 @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % prod_int_eq
% 0.33/0.68 thf(fact_8959_binomial__absorption,axiom,
% 0.33/0.68 ! [K: nat,N2: nat] :
% 0.33/0.68 ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) )
% 0.33/0.68 = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % binomial_absorption
% 0.33/0.68 thf(fact_8960_binomial__fact__lemma,axiom,
% 0.33/0.68 ! [K: nat,N2: nat] :
% 0.33/0.68 ( ( ord_less_eq_nat @ K @ N2 )
% 0.33/0.68 => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( binomial @ N2 @ K ) )
% 0.33/0.68 = ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % binomial_fact_lemma
% 0.33/0.68 thf(fact_8961_forall__pos__mono__1,axiom,
% 0.33/0.68 ! [P: real > $o,E: real] :
% 0.33/0.68 ( ! [D4: real,E2: real] :
% 0.33/0.68 ( ( ord_less_real @ D4 @ E2 )
% 0.33/0.68 => ( ( P @ D4 )
% 0.33/0.68 => ( P @ E2 ) ) )
% 0.33/0.68 => ( ! [N4: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ E )
% 0.33/0.68 => ( P @ E ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % forall_pos_mono_1
% 0.33/0.68 thf(fact_8962_real__arch__inverse,axiom,
% 0.33/0.68 ! [E: real] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ E )
% 0.33/0.68 = ( ? [N: nat] :
% 0.33/0.68 ( ( N != zero_zero_nat )
% 0.33/0.68 & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.33/0.68 & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) @ E ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_arch_inverse
% 0.33/0.68 thf(fact_8963_forall__pos__mono,axiom,
% 0.33/0.68 ! [P: real > $o,E: real] :
% 0.33/0.68 ( ! [D4: real,E2: real] :
% 0.33/0.68 ( ( ord_less_real @ D4 @ E2 )
% 0.33/0.68 => ( ( P @ D4 )
% 0.33/0.68 => ( P @ E2 ) ) )
% 0.33/0.68 => ( ! [N4: nat] :
% 0.33/0.68 ( ( N4 != zero_zero_nat )
% 0.33/0.68 => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) ) )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ E )
% 0.33/0.68 => ( P @ E ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % forall_pos_mono
% 0.33/0.68 thf(fact_8964_sqrt__divide__self__eq,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( divide_divide_real @ ( sqrt @ X ) @ X )
% 0.33/0.68 = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % sqrt_divide_self_eq
% 0.33/0.68 thf(fact_8965_ln__inverse,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ln_ln_real @ ( inverse_inverse_real @ X ) )
% 0.33/0.68 = ( uminus_uminus_real @ ( ln_ln_real @ X ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % ln_inverse
% 0.33/0.68 thf(fact_8966_prod__int__plus__eq,axiom,
% 0.33/0.68 ! [I2: nat,J: nat] :
% 0.33/0.68 ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I2 @ ( plus_plus_nat @ I2 @ J ) ) )
% 0.33/0.68 = ( groups1705073143266064639nt_int
% 0.33/0.68 @ ^ [X2: int] : X2
% 0.33/0.68 @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I2 @ J ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % prod_int_plus_eq
% 0.33/0.68 thf(fact_8967_binomial__maximum_H,axiom,
% 0.33/0.68 ! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % binomial_maximum'
% 0.33/0.68 thf(fact_8968_binomial__mono,axiom,
% 0.33/0.68 ! [K: nat,K6: nat,N2: nat] :
% 0.33/0.68 ( ( ord_less_eq_nat @ K @ K6 )
% 0.33/0.68 => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N2 )
% 0.33/0.68 => ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K6 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % binomial_mono
% 0.33/0.68 thf(fact_8969_binomial__antimono,axiom,
% 0.33/0.68 ! [K: nat,K6: nat,N2: nat] :
% 0.33/0.68 ( ( ord_less_eq_nat @ K @ K6 )
% 0.33/0.68 => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
% 0.33/0.68 => ( ( ord_less_eq_nat @ K6 @ N2 )
% 0.33/0.68 => ( ord_less_eq_nat @ ( binomial @ N2 @ K6 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % binomial_antimono
% 0.33/0.68 thf(fact_8970_binomial__maximum,axiom,
% 0.33/0.68 ! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % binomial_maximum
% 0.33/0.68 thf(fact_8971_binomial__le__pow2,axiom,
% 0.33/0.68 ! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % binomial_le_pow2
% 0.33/0.68 thf(fact_8972_choose__reduce__nat,axiom,
% 0.33/0.68 ! [N2: nat,K: nat] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.33/0.68 => ( ( binomial @ N2 @ K )
% 0.33/0.68 = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % choose_reduce_nat
% 0.33/0.68 thf(fact_8973_times__binomial__minus1__eq,axiom,
% 0.33/0.68 ! [K: nat,N2: nat] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.33/0.68 => ( ( times_times_nat @ K @ ( binomial @ N2 @ K ) )
% 0.33/0.68 = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % times_binomial_minus1_eq
% 0.33/0.68 thf(fact_8974_binomial__altdef__nat,axiom,
% 0.33/0.68 ! [K: nat,N2: nat] :
% 0.33/0.68 ( ( ord_less_eq_nat @ K @ N2 )
% 0.33/0.68 => ( ( binomial @ N2 @ K )
% 0.33/0.68 = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % binomial_altdef_nat
% 0.33/0.68 thf(fact_8975_binomial__less__binomial__Suc,axiom,
% 0.33/0.68 ! [K: nat,N2: nat] :
% 0.33/0.68 ( ( ord_less_nat @ K @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.68 => ( ord_less_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % binomial_less_binomial_Suc
% 0.33/0.68 thf(fact_8976_binomial__strict__antimono,axiom,
% 0.33/0.68 ! [K: nat,K6: nat,N2: nat] :
% 0.33/0.68 ( ( ord_less_nat @ K @ K6 )
% 0.33/0.68 => ( ( ord_less_eq_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
% 0.33/0.68 => ( ( ord_less_eq_nat @ K6 @ N2 )
% 0.33/0.68 => ( ord_less_nat @ ( binomial @ N2 @ K6 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % binomial_strict_antimono
% 0.33/0.68 thf(fact_8977_binomial__strict__mono,axiom,
% 0.33/0.68 ! [K: nat,K6: nat,N2: nat] :
% 0.33/0.68 ( ( ord_less_nat @ K @ K6 )
% 0.33/0.68 => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N2 )
% 0.33/0.68 => ( ord_less_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K6 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % binomial_strict_mono
% 0.33/0.68 thf(fact_8978_central__binomial__odd,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.68 => ( ( binomial @ N2 @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.68 = ( binomial @ N2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % central_binomial_odd
% 0.33/0.68 thf(fact_8979_binomial__addition__formula,axiom,
% 0.33/0.68 ! [N2: nat,K: nat] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( binomial @ N2 @ ( suc @ K ) )
% 0.33/0.68 = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % binomial_addition_formula
% 0.33/0.68 thf(fact_8980_exp__plus__inverse__exp,axiom,
% 0.33/0.68 ! [X: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % exp_plus_inverse_exp
% 0.33/0.68 thf(fact_8981_choose__two,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( binomial @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.68 = ( divide_divide_nat @ ( times_times_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % choose_two
% 0.33/0.68 thf(fact_8982_plus__inverse__ge__2,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % plus_inverse_ge_2
% 0.33/0.68 thf(fact_8983_real__inv__sqrt__pow2,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.68 = ( inverse_inverse_real @ X ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_inv_sqrt_pow2
% 0.33/0.68 thf(fact_8984_tan__cot,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
% 0.33/0.68 = ( inverse_inverse_real @ ( tan_real @ X ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % tan_cot
% 0.33/0.68 thf(fact_8985_real__le__x__sinh,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ord_less_eq_real @ X @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_le_x_sinh
% 0.33/0.68 thf(fact_8986_real__le__abs__sinh,axiom,
% 0.33/0.68 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_le_abs_sinh
% 0.33/0.68 thf(fact_8987_cot__gt__zero,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.68 => ( ord_less_real @ zero_zero_real @ ( cot_real @ X ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % cot_gt_zero
% 0.33/0.68 thf(fact_8988_tan__cot_H,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
% 0.33/0.68 = ( cot_real @ X ) ) ).
% 0.33/0.68
% 0.33/0.68 % tan_cot'
% 0.33/0.68 thf(fact_8989_binomial__code,axiom,
% 0.33/0.68 ( binomial
% 0.33/0.68 = ( ^ [N: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N @ ( minus_minus_nat @ N @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N @ K3 ) @ one_one_nat ) @ N @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % binomial_code
% 0.33/0.68 thf(fact_8990_divide__complex__def,axiom,
% 0.33/0.68 ( divide1717551699836669952omplex
% 0.33/0.68 = ( ^ [X2: complex,Y2: complex] : ( times_times_complex @ X2 @ ( invers8013647133539491842omplex @ Y2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % divide_complex_def
% 0.33/0.68 thf(fact_8991_real__scaleR__def,axiom,
% 0.33/0.68 real_V1485227260804924795R_real = times_times_real ).
% 0.33/0.68
% 0.33/0.68 % real_scaleR_def
% 0.33/0.68 thf(fact_8992_lessThan__Suc__atMost,axiom,
% 0.33/0.68 ! [K: nat] :
% 0.33/0.68 ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 0.33/0.68 = ( set_ord_atMost_nat @ K ) ) ).
% 0.33/0.68
% 0.33/0.68 % lessThan_Suc_atMost
% 0.33/0.68 thf(fact_8993_complex__scaleR,axiom,
% 0.33/0.68 ! [R: real,A: real,B: real] :
% 0.33/0.68 ( ( real_V2046097035970521341omplex @ R @ ( complex2 @ A @ B ) )
% 0.33/0.68 = ( complex2 @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ B ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % complex_scaleR
% 0.33/0.68 thf(fact_8994_sum__choose__upper,axiom,
% 0.33/0.68 ! [M: nat,N2: nat] :
% 0.33/0.68 ( ( groups3542108847815614940at_nat
% 0.33/0.68 @ ^ [K3: nat] : ( binomial @ K3 @ M )
% 0.33/0.68 @ ( set_ord_atMost_nat @ N2 ) )
% 0.33/0.68 = ( binomial @ ( suc @ N2 ) @ ( suc @ M ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % sum_choose_upper
% 0.33/0.68 thf(fact_8995_sum__choose__lower,axiom,
% 0.33/0.68 ! [R: nat,N2: nat] :
% 0.33/0.68 ( ( groups3542108847815614940at_nat
% 0.33/0.68 @ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R @ K3 ) @ K3 )
% 0.33/0.68 @ ( set_ord_atMost_nat @ N2 ) )
% 0.33/0.68 = ( binomial @ ( suc @ ( plus_plus_nat @ R @ N2 ) ) @ N2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % sum_choose_lower
% 0.33/0.68 thf(fact_8996_choose__rising__sum_I2_J,axiom,
% 0.33/0.68 ! [N2: nat,M: nat] :
% 0.33/0.68 ( ( groups3542108847815614940at_nat
% 0.33/0.68 @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N2 @ J3 ) @ N2 )
% 0.33/0.68 @ ( set_ord_atMost_nat @ M ) )
% 0.33/0.68 = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N2 @ M ) @ one_one_nat ) @ M ) ) ).
% 0.33/0.68
% 0.33/0.68 % choose_rising_sum(2)
% 0.33/0.68 thf(fact_8997_choose__rising__sum_I1_J,axiom,
% 0.33/0.68 ! [N2: nat,M: nat] :
% 0.33/0.68 ( ( groups3542108847815614940at_nat
% 0.33/0.68 @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N2 @ J3 ) @ N2 )
% 0.33/0.68 @ ( set_ord_atMost_nat @ M ) )
% 0.33/0.68 = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N2 @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % choose_rising_sum(1)
% 0.33/0.68 thf(fact_8998_sum__choose__diagonal,axiom,
% 0.33/0.68 ! [M: nat,N2: nat] :
% 0.33/0.68 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.68 => ( ( groups3542108847815614940at_nat
% 0.33/0.68 @ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N2 @ K3 ) @ ( minus_minus_nat @ M @ K3 ) )
% 0.33/0.68 @ ( set_ord_atMost_nat @ M ) )
% 0.33/0.68 = ( binomial @ ( suc @ N2 ) @ M ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % sum_choose_diagonal
% 0.33/0.68 thf(fact_8999_vandermonde,axiom,
% 0.33/0.68 ! [M: nat,N2: nat,R: nat] :
% 0.33/0.68 ( ( groups3542108847815614940at_nat
% 0.33/0.68 @ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M @ K3 ) @ ( binomial @ N2 @ ( minus_minus_nat @ R @ K3 ) ) )
% 0.33/0.68 @ ( set_ord_atMost_nat @ R ) )
% 0.33/0.68 = ( binomial @ ( plus_plus_nat @ M @ N2 ) @ R ) ) ).
% 0.33/0.68
% 0.33/0.68 % vandermonde
% 0.33/0.68 thf(fact_9000_choose__row__sum,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( groups3542108847815614940at_nat @ ( binomial @ N2 ) @ ( set_ord_atMost_nat @ N2 ) )
% 0.33/0.68 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % choose_row_sum
% 0.33/0.68 thf(fact_9001_binomial,axiom,
% 0.33/0.68 ! [A: nat,B: nat,N2: nat] :
% 0.33/0.68 ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N2 )
% 0.33/0.68 = ( groups3542108847815614940at_nat
% 0.33/0.68 @ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N2 @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
% 0.33/0.68 @ ( set_ord_atMost_nat @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % binomial
% 0.33/0.68 thf(fact_9002_polynomial__product__nat,axiom,
% 0.33/0.68 ! [M: nat,A: nat > nat,N2: nat,B: nat > nat,X: nat] :
% 0.33/0.68 ( ! [I4: nat] :
% 0.33/0.68 ( ( ord_less_nat @ M @ I4 )
% 0.33/0.68 => ( ( A @ I4 )
% 0.33/0.68 = zero_zero_nat ) )
% 0.33/0.68 => ( ! [J2: nat] :
% 0.33/0.68 ( ( ord_less_nat @ N2 @ J2 )
% 0.33/0.68 => ( ( B @ J2 )
% 0.33/0.68 = zero_zero_nat ) )
% 0.33/0.68 => ( ( times_times_nat
% 0.33/0.68 @ ( groups3542108847815614940at_nat
% 0.33/0.68 @ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( power_power_nat @ X @ I3 ) )
% 0.33/0.68 @ ( set_ord_atMost_nat @ M ) )
% 0.33/0.68 @ ( groups3542108847815614940at_nat
% 0.33/0.68 @ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X @ J3 ) )
% 0.33/0.68 @ ( set_ord_atMost_nat @ N2 ) ) )
% 0.33/0.68 = ( groups3542108847815614940at_nat
% 0.33/0.68 @ ^ [R5: nat] :
% 0.33/0.68 ( times_times_nat
% 0.33/0.68 @ ( groups3542108847815614940at_nat
% 0.33/0.68 @ ^ [K3: nat] : ( times_times_nat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
% 0.33/0.68 @ ( set_ord_atMost_nat @ R5 ) )
% 0.33/0.68 @ ( power_power_nat @ X @ R5 ) )
% 0.33/0.68 @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % polynomial_product_nat
% 0.33/0.68 thf(fact_9003_choose__square__sum,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( groups3542108847815614940at_nat
% 0.33/0.68 @ ^ [K3: nat] : ( power_power_nat @ ( binomial @ N2 @ K3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.68 @ ( set_ord_atMost_nat @ N2 ) )
% 0.33/0.68 = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % choose_square_sum
% 0.33/0.68 thf(fact_9004_complex__inverse,axiom,
% 0.33/0.68 ! [A: real,B: real] :
% 0.33/0.68 ( ( invers8013647133539491842omplex @ ( complex2 @ A @ B ) )
% 0.33/0.68 = ( complex2 @ ( divide_divide_real @ A @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ B ) @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % complex_inverse
% 0.33/0.68 thf(fact_9005_binomial__r__part__sum,axiom,
% 0.33/0.68 ! [M: nat] :
% 0.33/0.68 ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
% 0.33/0.68 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % binomial_r_part_sum
% 0.33/0.68 thf(fact_9006_choose__linear__sum,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( groups3542108847815614940at_nat
% 0.33/0.68 @ ^ [I3: nat] : ( times_times_nat @ I3 @ ( binomial @ N2 @ I3 ) )
% 0.33/0.68 @ ( set_ord_atMost_nat @ N2 ) )
% 0.33/0.68 = ( times_times_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % choose_linear_sum
% 0.33/0.68 thf(fact_9007_i__even__power,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.68 = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % i_even_power
% 0.33/0.68 thf(fact_9008_log__base__10__eq1,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.68 = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_base_10_eq1
% 0.33/0.68 thf(fact_9009_sinh__real__less__iff,axiom,
% 0.33/0.68 ! [X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) )
% 0.33/0.68 = ( ord_less_real @ X @ Y ) ) ).
% 0.33/0.68
% 0.33/0.68 % sinh_real_less_iff
% 0.33/0.68 thf(fact_9010_sinh__real__le__iff,axiom,
% 0.33/0.68 ! [X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) )
% 0.33/0.68 = ( ord_less_eq_real @ X @ Y ) ) ).
% 0.33/0.68
% 0.33/0.68 % sinh_real_le_iff
% 0.33/0.68 thf(fact_9011_of__nat__id,axiom,
% 0.33/0.68 ( semiri1316708129612266289at_nat
% 0.33/0.68 = ( ^ [N: nat] : N ) ) ).
% 0.33/0.68
% 0.33/0.68 % of_nat_id
% 0.33/0.68 thf(fact_9012_sinh__real__neg__iff,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( ord_less_real @ ( sinh_real @ X ) @ zero_zero_real )
% 0.33/0.68 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.33/0.68
% 0.33/0.68 % sinh_real_neg_iff
% 0.33/0.68 thf(fact_9013_sinh__real__pos__iff,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X ) )
% 0.33/0.68 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.33/0.68
% 0.33/0.68 % sinh_real_pos_iff
% 0.33/0.68 thf(fact_9014_sinh__real__nonneg__iff,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X ) )
% 0.33/0.68 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.33/0.68
% 0.33/0.68 % sinh_real_nonneg_iff
% 0.33/0.68 thf(fact_9015_sinh__real__nonpos__iff,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ ( sinh_real @ X ) @ zero_zero_real )
% 0.33/0.68 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.33/0.68
% 0.33/0.68 % sinh_real_nonpos_iff
% 0.33/0.68 thf(fact_9016_log__one,axiom,
% 0.33/0.68 ! [A: real] :
% 0.33/0.68 ( ( log @ A @ one_one_real )
% 0.33/0.68 = zero_zero_real ) ).
% 0.33/0.68
% 0.33/0.68 % log_one
% 0.33/0.68 thf(fact_9017_norm__ii,axiom,
% 0.33/0.68 ( ( real_V1022390504157884413omplex @ imaginary_unit )
% 0.33/0.68 = one_one_real ) ).
% 0.33/0.68
% 0.33/0.68 % norm_ii
% 0.33/0.68 thf(fact_9018_log__eq__one,axiom,
% 0.33/0.68 ! [A: real] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.68 => ( ( A != one_one_real )
% 0.33/0.68 => ( ( log @ A @ A )
% 0.33/0.68 = one_one_real ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_eq_one
% 0.33/0.68 thf(fact_9019_log__less__cancel__iff,axiom,
% 0.33/0.68 ! [A: real,X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_real @ one_one_real @ A )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.68 => ( ( ord_less_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
% 0.33/0.68 = ( ord_less_real @ X @ Y ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_less_cancel_iff
% 0.33/0.68 thf(fact_9020_log__less__one__cancel__iff,axiom,
% 0.33/0.68 ! [A: real,X: real] :
% 0.33/0.68 ( ( ord_less_real @ one_one_real @ A )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_real @ ( log @ A @ X ) @ one_one_real )
% 0.33/0.68 = ( ord_less_real @ X @ A ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_less_one_cancel_iff
% 0.33/0.68 thf(fact_9021_one__less__log__cancel__iff,axiom,
% 0.33/0.68 ! [A: real,X: real] :
% 0.33/0.68 ( ( ord_less_real @ one_one_real @ A )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_real @ one_one_real @ ( log @ A @ X ) )
% 0.33/0.68 = ( ord_less_real @ A @ X ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % one_less_log_cancel_iff
% 0.33/0.68 thf(fact_9022_log__less__zero__cancel__iff,axiom,
% 0.33/0.68 ! [A: real,X: real] :
% 0.33/0.68 ( ( ord_less_real @ one_one_real @ A )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_real @ ( log @ A @ X ) @ zero_zero_real )
% 0.33/0.68 = ( ord_less_real @ X @ one_one_real ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_less_zero_cancel_iff
% 0.33/0.68 thf(fact_9023_zero__less__log__cancel__iff,axiom,
% 0.33/0.68 ! [A: real,X: real] :
% 0.33/0.68 ( ( ord_less_real @ one_one_real @ A )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X ) )
% 0.33/0.68 = ( ord_less_real @ one_one_real @ X ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % zero_less_log_cancel_iff
% 0.33/0.68 thf(fact_9024_divide__i,axiom,
% 0.33/0.68 ! [X: complex] :
% 0.33/0.68 ( ( divide1717551699836669952omplex @ X @ imaginary_unit )
% 0.33/0.68 = ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X ) ) ).
% 0.33/0.68
% 0.33/0.68 % divide_i
% 0.33/0.68 thf(fact_9025_i__squared,axiom,
% 0.33/0.68 ( ( times_times_complex @ imaginary_unit @ imaginary_unit )
% 0.33/0.68 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.33/0.68
% 0.33/0.68 % i_squared
% 0.33/0.68 thf(fact_9026_log__le__cancel__iff,axiom,
% 0.33/0.68 ! [A: real,X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_real @ one_one_real @ A )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.68 => ( ( ord_less_eq_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
% 0.33/0.68 = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_le_cancel_iff
% 0.33/0.68 thf(fact_9027_log__le__one__cancel__iff,axiom,
% 0.33/0.68 ! [A: real,X: real] :
% 0.33/0.68 ( ( ord_less_real @ one_one_real @ A )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_eq_real @ ( log @ A @ X ) @ one_one_real )
% 0.33/0.68 = ( ord_less_eq_real @ X @ A ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_le_one_cancel_iff
% 0.33/0.68 thf(fact_9028_one__le__log__cancel__iff,axiom,
% 0.33/0.68 ! [A: real,X: real] :
% 0.33/0.68 ( ( ord_less_real @ one_one_real @ A )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X ) )
% 0.33/0.68 = ( ord_less_eq_real @ A @ X ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % one_le_log_cancel_iff
% 0.33/0.68 thf(fact_9029_log__le__zero__cancel__iff,axiom,
% 0.33/0.68 ! [A: real,X: real] :
% 0.33/0.68 ( ( ord_less_real @ one_one_real @ A )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_eq_real @ ( log @ A @ X ) @ zero_zero_real )
% 0.33/0.68 = ( ord_less_eq_real @ X @ one_one_real ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_le_zero_cancel_iff
% 0.33/0.68 thf(fact_9030_zero__le__log__cancel__iff,axiom,
% 0.33/0.68 ! [A: real,X: real] :
% 0.33/0.68 ( ( ord_less_real @ one_one_real @ A )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X ) )
% 0.33/0.68 = ( ord_less_eq_real @ one_one_real @ X ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % zero_le_log_cancel_iff
% 0.33/0.68 thf(fact_9031_divide__numeral__i,axiom,
% 0.33/0.68 ! [Z: complex,N2: num] :
% 0.33/0.68 ( ( divide1717551699836669952omplex @ Z @ ( times_times_complex @ ( numera6690914467698888265omplex @ N2 ) @ imaginary_unit ) )
% 0.33/0.68 = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) @ ( numera6690914467698888265omplex @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % divide_numeral_i
% 0.33/0.68 thf(fact_9032_log__pow__cancel,axiom,
% 0.33/0.68 ! [A: real,B: nat] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.68 => ( ( A != one_one_real )
% 0.33/0.68 => ( ( log @ A @ ( power_power_real @ A @ B ) )
% 0.33/0.68 = ( semiri5074537144036343181t_real @ B ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_pow_cancel
% 0.33/0.68 thf(fact_9033_power2__i,axiom,
% 0.33/0.68 ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.68 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.33/0.68
% 0.33/0.68 % power2_i
% 0.33/0.68 thf(fact_9034_sinh__less__cosh__real,axiom,
% 0.33/0.68 ! [X: real] : ( ord_less_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).
% 0.33/0.68
% 0.33/0.68 % sinh_less_cosh_real
% 0.33/0.68 thf(fact_9035_sinh__le__cosh__real,axiom,
% 0.33/0.68 ! [X: real] : ( ord_less_eq_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).
% 0.33/0.68
% 0.33/0.68 % sinh_le_cosh_real
% 0.33/0.68 thf(fact_9036_complex__i__not__one,axiom,
% 0.33/0.68 imaginary_unit != one_one_complex ).
% 0.33/0.68
% 0.33/0.68 % complex_i_not_one
% 0.33/0.68 thf(fact_9037_log__def,axiom,
% 0.33/0.68 ( log
% 0.33/0.68 = ( ^ [A5: real,X2: real] : ( divide_divide_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ A5 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_def
% 0.33/0.68 thf(fact_9038_cosh__real__pos,axiom,
% 0.33/0.68 ! [X: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X ) ) ).
% 0.33/0.68
% 0.33/0.68 % cosh_real_pos
% 0.33/0.68 thf(fact_9039_cosh__real__nonpos__le__iff,axiom,
% 0.33/0.68 ! [X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.33/0.68 => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 0.33/0.68 => ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
% 0.33/0.68 = ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % cosh_real_nonpos_le_iff
% 0.33/0.68 thf(fact_9040_cosh__real__nonneg__le__iff,axiom,
% 0.33/0.68 ! [X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.68 => ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
% 0.33/0.68 = ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % cosh_real_nonneg_le_iff
% 0.33/0.68 thf(fact_9041_cosh__real__nonneg,axiom,
% 0.33/0.68 ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X ) ) ).
% 0.33/0.68
% 0.33/0.68 % cosh_real_nonneg
% 0.33/0.68 thf(fact_9042_cosh__real__ge__1,axiom,
% 0.33/0.68 ! [X: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X ) ) ).
% 0.33/0.68
% 0.33/0.68 % cosh_real_ge_1
% 0.33/0.68 thf(fact_9043_cosh__real__strict__mono,axiom,
% 0.33/0.68 ! [X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_real @ X @ Y )
% 0.33/0.68 => ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % cosh_real_strict_mono
% 0.33/0.68 thf(fact_9044_cosh__real__nonneg__less__iff,axiom,
% 0.33/0.68 ! [X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.68 => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
% 0.33/0.68 = ( ord_less_real @ X @ Y ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % cosh_real_nonneg_less_iff
% 0.33/0.68 thf(fact_9045_cosh__real__nonpos__less__iff,axiom,
% 0.33/0.68 ! [X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.33/0.68 => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 0.33/0.68 => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
% 0.33/0.68 = ( ord_less_real @ Y @ X ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % cosh_real_nonpos_less_iff
% 0.33/0.68 thf(fact_9046_log__ln,axiom,
% 0.33/0.68 ( ln_ln_real
% 0.33/0.68 = ( log @ ( exp_real @ one_one_real ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_ln
% 0.33/0.68 thf(fact_9047_arcosh__cosh__real,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( arcosh_real @ ( cosh_real @ X ) )
% 0.33/0.68 = X ) ) ).
% 0.33/0.68
% 0.33/0.68 % arcosh_cosh_real
% 0.33/0.68 thf(fact_9048_imaginary__unit_Ocode,axiom,
% 0.33/0.68 ( imaginary_unit
% 0.33/0.68 = ( complex2 @ zero_zero_real @ one_one_real ) ) ).
% 0.33/0.68
% 0.33/0.68 % imaginary_unit.code
% 0.33/0.68 thf(fact_9049_Complex__eq__i,axiom,
% 0.33/0.68 ! [X: real,Y: real] :
% 0.33/0.68 ( ( ( complex2 @ X @ Y )
% 0.33/0.68 = imaginary_unit )
% 0.33/0.68 = ( ( X = zero_zero_real )
% 0.33/0.68 & ( Y = one_one_real ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Complex_eq_i
% 0.33/0.68 thf(fact_9050_log__base__change,axiom,
% 0.33/0.68 ! [A: real,B: real,X: real] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.68 => ( ( A != one_one_real )
% 0.33/0.68 => ( ( log @ B @ X )
% 0.33/0.68 = ( divide_divide_real @ ( log @ A @ X ) @ ( log @ A @ B ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_base_change
% 0.33/0.68 thf(fact_9051_log__of__power__eq,axiom,
% 0.33/0.68 ! [M: nat,B: real,N2: nat] :
% 0.33/0.68 ( ( ( semiri5074537144036343181t_real @ M )
% 0.33/0.68 = ( power_power_real @ B @ N2 ) )
% 0.33/0.68 => ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.68 => ( ( semiri5074537144036343181t_real @ N2 )
% 0.33/0.68 = ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_of_power_eq
% 0.33/0.68 thf(fact_9052_less__log__of__power,axiom,
% 0.33/0.68 ! [B: real,N2: nat,M: real] :
% 0.33/0.68 ( ( ord_less_real @ ( power_power_real @ B @ N2 ) @ M )
% 0.33/0.68 => ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.68 => ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ M ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % less_log_of_power
% 0.33/0.68 thf(fact_9053_log__mult,axiom,
% 0.33/0.68 ! [A: real,X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.68 => ( ( A != one_one_real )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.68 => ( ( log @ A @ ( times_times_real @ X @ Y ) )
% 0.33/0.68 = ( plus_plus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_mult
% 0.33/0.68 thf(fact_9054_log__divide,axiom,
% 0.33/0.68 ! [A: real,X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.68 => ( ( A != one_one_real )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.68 => ( ( log @ A @ ( divide_divide_real @ X @ Y ) )
% 0.33/0.68 = ( minus_minus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_divide
% 0.33/0.68 thf(fact_9055_le__log__of__power,axiom,
% 0.33/0.68 ! [B: real,N2: nat,M: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ ( power_power_real @ B @ N2 ) @ M )
% 0.33/0.68 => ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.68 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ M ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % le_log_of_power
% 0.33/0.68 thf(fact_9056_log__base__pow,axiom,
% 0.33/0.68 ! [A: real,N2: nat,X: real] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.68 => ( ( log @ ( power_power_real @ A @ N2 ) @ X )
% 0.33/0.68 = ( divide_divide_real @ ( log @ A @ X ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_base_pow
% 0.33/0.68 thf(fact_9057_log__nat__power,axiom,
% 0.33/0.68 ! [X: real,B: real,N2: nat] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( log @ B @ ( power_power_real @ X @ N2 ) )
% 0.33/0.68 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ X ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_nat_power
% 0.33/0.68 thf(fact_9058_log__inverse,axiom,
% 0.33/0.68 ! [A: real,X: real] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.68 => ( ( A != one_one_real )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( log @ A @ ( inverse_inverse_real @ X ) )
% 0.33/0.68 = ( uminus_uminus_real @ ( log @ A @ X ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_inverse
% 0.33/0.68 thf(fact_9059_log2__of__power__eq,axiom,
% 0.33/0.68 ! [M: nat,N2: nat] :
% 0.33/0.68 ( ( M
% 0.33/0.68 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.33/0.68 => ( ( semiri5074537144036343181t_real @ N2 )
% 0.33/0.68 = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log2_of_power_eq
% 0.33/0.68 thf(fact_9060_log__of__power__less,axiom,
% 0.33/0.68 ! [M: nat,B: real,N2: nat] :
% 0.33/0.68 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N2 ) )
% 0.33/0.68 => ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.68 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.33/0.68 => ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_of_power_less
% 0.33/0.68 thf(fact_9061_log__eq__div__ln__mult__log,axiom,
% 0.33/0.68 ! [A: real,B: real,X: real] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.68 => ( ( A != one_one_real )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.33/0.68 => ( ( B != one_one_real )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( log @ A @ X )
% 0.33/0.68 = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_eq_div_ln_mult_log
% 0.33/0.68 thf(fact_9062_log__of__power__le,axiom,
% 0.33/0.68 ! [M: nat,B: real,N2: nat] :
% 0.33/0.68 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N2 ) )
% 0.33/0.68 => ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.68 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.33/0.68 => ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_of_power_le
% 0.33/0.68 thf(fact_9063_less__log2__of__power,axiom,
% 0.33/0.68 ! [N2: nat,M: nat] :
% 0.33/0.68 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M )
% 0.33/0.68 => ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % less_log2_of_power
% 0.33/0.68 thf(fact_9064_le__log2__of__power,axiom,
% 0.33/0.68 ! [N2: nat,M: nat] :
% 0.33/0.68 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M )
% 0.33/0.68 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % le_log2_of_power
% 0.33/0.68 thf(fact_9065_log2__of__power__less,axiom,
% 0.33/0.68 ! [M: nat,N2: nat] :
% 0.33/0.68 ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.33/0.68 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.33/0.68 => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log2_of_power_less
% 0.33/0.68 thf(fact_9066_cosh__ln__real,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( cosh_real @ ( ln_ln_real @ X ) )
% 0.33/0.68 = ( divide_divide_real @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % cosh_ln_real
% 0.33/0.68 thf(fact_9067_log2__of__power__le,axiom,
% 0.33/0.68 ! [M: nat,N2: nat] :
% 0.33/0.68 ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.33/0.68 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.33/0.68 => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log2_of_power_le
% 0.33/0.68 thf(fact_9068_log__base__10__eq2,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
% 0.33/0.68 = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_base_10_eq2
% 0.33/0.68 thf(fact_9069_sinh__ln__real,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( sinh_real @ ( ln_ln_real @ X ) )
% 0.33/0.68 = ( divide_divide_real @ ( minus_minus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % sinh_ln_real
% 0.33/0.68 thf(fact_9070_Arg__minus__ii,axiom,
% 0.33/0.68 ( ( arg @ ( uminus1482373934393186551omplex @ imaginary_unit ) )
% 0.33/0.68 = ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Arg_minus_ii
% 0.33/0.68 thf(fact_9071_ceiling__log__nat__eq__powr__iff,axiom,
% 0.33/0.68 ! [B: nat,K: nat,N2: nat] :
% 0.33/0.68 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.33/0.68 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.33/0.68 => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.33/0.68 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) )
% 0.33/0.68 = ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 0.33/0.68 & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % ceiling_log_nat_eq_powr_iff
% 0.33/0.68 thf(fact_9072_Arg__ii,axiom,
% 0.33/0.68 ( ( arg @ imaginary_unit )
% 0.33/0.68 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Arg_ii
% 0.33/0.68 thf(fact_9073_ceiling__log2__div2,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.68 => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 0.33/0.68 = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % ceiling_log2_div2
% 0.33/0.68 thf(fact_9074_ceiling__divide__eq__div__numeral,axiom,
% 0.33/0.68 ! [A: num,B: num] :
% 0.33/0.68 ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
% 0.33/0.68 = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % ceiling_divide_eq_div_numeral
% 0.33/0.68 thf(fact_9075_ceiling__minus__divide__eq__div__numeral,axiom,
% 0.33/0.68 ! [A: num,B: num] :
% 0.33/0.68 ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
% 0.33/0.68 = ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % ceiling_minus_divide_eq_div_numeral
% 0.33/0.68 thf(fact_9076_Arg__bounded,axiom,
% 0.33/0.68 ! [Z: complex] :
% 0.33/0.68 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 0.33/0.68 & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ).
% 0.33/0.68
% 0.33/0.68 % Arg_bounded
% 0.33/0.68 thf(fact_9077_ceiling__log__nat__eq__if,axiom,
% 0.33/0.68 ! [B: nat,N2: nat,K: nat] :
% 0.33/0.68 ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 0.33/0.68 => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) )
% 0.33/0.68 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.33/0.68 => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.33/0.68 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % ceiling_log_nat_eq_if
% 0.33/0.68 thf(fact_9078_cis__minus__pi__half,axiom,
% 0.33/0.68 ( ( cis @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.33/0.68 = ( uminus1482373934393186551omplex @ imaginary_unit ) ) ).
% 0.33/0.68
% 0.33/0.68 % cis_minus_pi_half
% 0.33/0.68 thf(fact_9079_ceiling__log__eq__powr__iff,axiom,
% 0.33/0.68 ! [X: real,B: real,K: nat] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.68 => ( ( ( archim7802044766580827645g_real @ ( log @ B @ X ) )
% 0.33/0.68 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
% 0.33/0.68 = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X )
% 0.33/0.68 & ( ord_less_eq_real @ X @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % ceiling_log_eq_powr_iff
% 0.33/0.68 thf(fact_9080_floor__log__nat__eq__powr__iff,axiom,
% 0.33/0.68 ! [B: nat,K: nat,N2: nat] :
% 0.33/0.68 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.33/0.68 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.33/0.68 => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.33/0.68 = ( semiri1314217659103216013at_int @ N2 ) )
% 0.33/0.68 = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 0.33/0.68 & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % floor_log_nat_eq_powr_iff
% 0.33/0.68 thf(fact_9081_int__ge__less__than2__def,axiom,
% 0.33/0.68 ( int_ge_less_than2
% 0.33/0.68 = ( ^ [D2: int] :
% 0.33/0.68 ( collec213857154873943460nt_int
% 0.33/0.68 @ ( produc4947309494688390418_int_o
% 0.33/0.68 @ ^ [Z6: int,Z3: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ D2 @ Z3 )
% 0.33/0.68 & ( ord_less_int @ Z6 @ Z3 ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % int_ge_less_than2_def
% 0.33/0.68 thf(fact_9082_powr__gt__zero,axiom,
% 0.33/0.68 ! [X: real,A: real] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X @ A ) )
% 0.33/0.68 = ( X != zero_zero_real ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_gt_zero
% 0.33/0.68 thf(fact_9083_powr__nonneg__iff,axiom,
% 0.33/0.68 ! [A: real,X: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ ( powr_real @ A @ X ) @ zero_zero_real )
% 0.33/0.68 = ( A = zero_zero_real ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_nonneg_iff
% 0.33/0.68 thf(fact_9084_powr__less__cancel__iff,axiom,
% 0.33/0.68 ! [X: real,A: real,B: real] :
% 0.33/0.68 ( ( ord_less_real @ one_one_real @ X )
% 0.33/0.68 => ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 0.33/0.68 = ( ord_less_real @ A @ B ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_less_cancel_iff
% 0.33/0.68 thf(fact_9085_norm__cis,axiom,
% 0.33/0.68 ! [A: real] :
% 0.33/0.68 ( ( real_V1022390504157884413omplex @ ( cis @ A ) )
% 0.33/0.68 = one_one_real ) ).
% 0.33/0.68
% 0.33/0.68 % norm_cis
% 0.33/0.68 thf(fact_9086_cis__zero,axiom,
% 0.33/0.68 ( ( cis @ zero_zero_real )
% 0.33/0.68 = one_one_complex ) ).
% 0.33/0.68
% 0.33/0.68 % cis_zero
% 0.33/0.68 thf(fact_9087_powr__eq__one__iff,axiom,
% 0.33/0.68 ! [A: real,X: real] :
% 0.33/0.68 ( ( ord_less_real @ one_one_real @ A )
% 0.33/0.68 => ( ( ( powr_real @ A @ X )
% 0.33/0.68 = one_one_real )
% 0.33/0.68 = ( X = zero_zero_real ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_eq_one_iff
% 0.33/0.68 thf(fact_9088_powr__one__gt__zero__iff,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( ( powr_real @ X @ one_one_real )
% 0.33/0.68 = X )
% 0.33/0.68 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_one_gt_zero_iff
% 0.33/0.68 thf(fact_9089_powr__one,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( powr_real @ X @ one_one_real )
% 0.33/0.68 = X ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_one
% 0.33/0.68 thf(fact_9090_powr__le__cancel__iff,axiom,
% 0.33/0.68 ! [X: real,A: real,B: real] :
% 0.33/0.68 ( ( ord_less_real @ one_one_real @ X )
% 0.33/0.68 => ( ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 0.33/0.68 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_le_cancel_iff
% 0.33/0.68 thf(fact_9091_numeral__powr__numeral__real,axiom,
% 0.33/0.68 ! [M: num,N2: num] :
% 0.33/0.68 ( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
% 0.33/0.68 = ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % numeral_powr_numeral_real
% 0.33/0.68 thf(fact_9092_cis__pi,axiom,
% 0.33/0.68 ( ( cis @ pi )
% 0.33/0.68 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.33/0.68
% 0.33/0.68 % cis_pi
% 0.33/0.68 thf(fact_9093_powr__log__cancel,axiom,
% 0.33/0.68 ! [A: real,X: real] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.68 => ( ( A != one_one_real )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( powr_real @ A @ ( log @ A @ X ) )
% 0.33/0.68 = X ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_log_cancel
% 0.33/0.68 thf(fact_9094_log__powr__cancel,axiom,
% 0.33/0.68 ! [A: real,Y: real] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.68 => ( ( A != one_one_real )
% 0.33/0.68 => ( ( log @ A @ ( powr_real @ A @ Y ) )
% 0.33/0.68 = Y ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_powr_cancel
% 0.33/0.68 thf(fact_9095_floor__divide__eq__div__numeral,axiom,
% 0.33/0.68 ! [A: num,B: num] :
% 0.33/0.68 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
% 0.33/0.68 = ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % floor_divide_eq_div_numeral
% 0.33/0.68 thf(fact_9096_powr__numeral,axiom,
% 0.33/0.68 ! [X: real,N2: num] :
% 0.33/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( powr_real @ X @ ( numeral_numeral_real @ N2 ) )
% 0.33/0.68 = ( power_power_real @ X @ ( numeral_numeral_nat @ N2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_numeral
% 0.33/0.68 thf(fact_9097_cis__pi__half,axiom,
% 0.33/0.68 ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.68 = imaginary_unit ) ).
% 0.33/0.68
% 0.33/0.68 % cis_pi_half
% 0.33/0.68 thf(fact_9098_floor__one__divide__eq__div__numeral,axiom,
% 0.33/0.68 ! [B: num] :
% 0.33/0.68 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) )
% 0.33/0.68 = ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % floor_one_divide_eq_div_numeral
% 0.33/0.68 thf(fact_9099_cis__2pi,axiom,
% 0.33/0.68 ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.33/0.68 = one_one_complex ) ).
% 0.33/0.68
% 0.33/0.68 % cis_2pi
% 0.33/0.68 thf(fact_9100_floor__minus__divide__eq__div__numeral,axiom,
% 0.33/0.68 ! [A: num,B: num] :
% 0.33/0.68 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
% 0.33/0.68 = ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % floor_minus_divide_eq_div_numeral
% 0.33/0.68 thf(fact_9101_square__powr__half,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( powr_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.68 = ( abs_abs_real @ X ) ) ).
% 0.33/0.68
% 0.33/0.68 % square_powr_half
% 0.33/0.68 thf(fact_9102_floor__minus__one__divide__eq__div__numeral,axiom,
% 0.33/0.68 ! [B: num] :
% 0.33/0.68 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) ) )
% 0.33/0.68 = ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % floor_minus_one_divide_eq_div_numeral
% 0.33/0.68 thf(fact_9103_powr__powr,axiom,
% 0.33/0.68 ! [X: real,A: real,B: real] :
% 0.33/0.68 ( ( powr_real @ ( powr_real @ X @ A ) @ B )
% 0.33/0.68 = ( powr_real @ X @ ( times_times_real @ A @ B ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_powr
% 0.33/0.68 thf(fact_9104_powr__less__mono2__neg,axiom,
% 0.33/0.68 ! [A: real,X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_real @ A @ zero_zero_real )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_real @ X @ Y )
% 0.33/0.68 => ( ord_less_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_less_mono2_neg
% 0.33/0.68 thf(fact_9105_powr__non__neg,axiom,
% 0.33/0.68 ! [A: real,X: real] :
% 0.33/0.68 ~ ( ord_less_real @ ( powr_real @ A @ X ) @ zero_zero_real ) ).
% 0.33/0.68
% 0.33/0.68 % powr_non_neg
% 0.33/0.68 thf(fact_9106_powr__ge__pzero,axiom,
% 0.33/0.68 ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X @ Y ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_ge_pzero
% 0.33/0.68 thf(fact_9107_powr__mono2,axiom,
% 0.33/0.68 ! [A: real,X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_eq_real @ X @ Y )
% 0.33/0.68 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_mono2
% 0.33/0.68 thf(fact_9108_powr__less__mono,axiom,
% 0.33/0.68 ! [A: real,B: real,X: real] :
% 0.33/0.68 ( ( ord_less_real @ A @ B )
% 0.33/0.68 => ( ( ord_less_real @ one_one_real @ X )
% 0.33/0.68 => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_less_mono
% 0.33/0.68 thf(fact_9109_powr__less__cancel,axiom,
% 0.33/0.68 ! [X: real,A: real,B: real] :
% 0.33/0.68 ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 0.33/0.68 => ( ( ord_less_real @ one_one_real @ X )
% 0.33/0.68 => ( ord_less_real @ A @ B ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_less_cancel
% 0.33/0.68 thf(fact_9110_powr__mono,axiom,
% 0.33/0.68 ! [A: real,B: real,X: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ A @ B )
% 0.33/0.68 => ( ( ord_less_eq_real @ one_one_real @ X )
% 0.33/0.68 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_mono
% 0.33/0.68 thf(fact_9111_cis__mult,axiom,
% 0.33/0.68 ! [A: real,B: real] :
% 0.33/0.68 ( ( times_times_complex @ ( cis @ A ) @ ( cis @ B ) )
% 0.33/0.68 = ( cis @ ( plus_plus_real @ A @ B ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % cis_mult
% 0.33/0.68 thf(fact_9112_cis__divide,axiom,
% 0.33/0.68 ! [A: real,B: real] :
% 0.33/0.68 ( ( divide1717551699836669952omplex @ ( cis @ A ) @ ( cis @ B ) )
% 0.33/0.68 = ( cis @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % cis_divide
% 0.33/0.68 thf(fact_9113_powr__mono2_H,axiom,
% 0.33/0.68 ! [A: real,X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_eq_real @ X @ Y )
% 0.33/0.68 => ( ord_less_eq_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_mono2'
% 0.33/0.68 thf(fact_9114_powr__less__mono2,axiom,
% 0.33/0.68 ! [A: real,X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_real @ X @ Y )
% 0.33/0.68 => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_less_mono2
% 0.33/0.68 thf(fact_9115_powr__inj,axiom,
% 0.33/0.68 ! [A: real,X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.68 => ( ( A != one_one_real )
% 0.33/0.68 => ( ( ( powr_real @ A @ X )
% 0.33/0.68 = ( powr_real @ A @ Y ) )
% 0.33/0.68 = ( X = Y ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_inj
% 0.33/0.68 thf(fact_9116_gr__one__powr,axiom,
% 0.33/0.68 ! [X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_real @ one_one_real @ X )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ Y )
% 0.33/0.68 => ( ord_less_real @ one_one_real @ ( powr_real @ X @ Y ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % gr_one_powr
% 0.33/0.68 thf(fact_9117_ge__one__powr__ge__zero,axiom,
% 0.33/0.68 ! [X: real,A: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ one_one_real @ X )
% 0.33/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.68 => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % ge_one_powr_ge_zero
% 0.33/0.68 thf(fact_9118_powr__mono__both,axiom,
% 0.33/0.68 ! [A: real,B: real,X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.68 => ( ( ord_less_eq_real @ A @ B )
% 0.33/0.68 => ( ( ord_less_eq_real @ one_one_real @ X )
% 0.33/0.68 => ( ( ord_less_eq_real @ X @ Y )
% 0.33/0.68 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ B ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_mono_both
% 0.33/0.68 thf(fact_9119_powr__le1,axiom,
% 0.33/0.68 ! [A: real,X: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.33/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.68 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ one_one_real ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_le1
% 0.33/0.68 thf(fact_9120_powr__divide,axiom,
% 0.33/0.68 ! [X: real,Y: real,A: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.68 => ( ( powr_real @ ( divide_divide_real @ X @ Y ) @ A )
% 0.33/0.68 = ( divide_divide_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_divide
% 0.33/0.68 thf(fact_9121_powr__mult,axiom,
% 0.33/0.68 ! [X: real,Y: real,A: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.68 => ( ( powr_real @ ( times_times_real @ X @ Y ) @ A )
% 0.33/0.68 = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_mult
% 0.33/0.68 thf(fact_9122_inverse__powr,axiom,
% 0.33/0.68 ! [Y: real,A: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.68 => ( ( powr_real @ ( inverse_inverse_real @ Y ) @ A )
% 0.33/0.68 = ( inverse_inverse_real @ ( powr_real @ Y @ A ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % inverse_powr
% 0.33/0.68 thf(fact_9123_divide__powr__uminus,axiom,
% 0.33/0.68 ! [A: real,B: real,C: real] :
% 0.33/0.68 ( ( divide_divide_real @ A @ ( powr_real @ B @ C ) )
% 0.33/0.68 = ( times_times_real @ A @ ( powr_real @ B @ ( uminus_uminus_real @ C ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % divide_powr_uminus
% 0.33/0.68 thf(fact_9124_log__base__powr,axiom,
% 0.33/0.68 ! [A: real,B: real,X: real] :
% 0.33/0.68 ( ( A != zero_zero_real )
% 0.33/0.68 => ( ( log @ ( powr_real @ A @ B ) @ X )
% 0.33/0.68 = ( divide_divide_real @ ( log @ A @ X ) @ B ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_base_powr
% 0.33/0.68 thf(fact_9125_ln__powr,axiom,
% 0.33/0.68 ! [X: real,Y: real] :
% 0.33/0.68 ( ( X != zero_zero_real )
% 0.33/0.68 => ( ( ln_ln_real @ ( powr_real @ X @ Y ) )
% 0.33/0.68 = ( times_times_real @ Y @ ( ln_ln_real @ X ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % ln_powr
% 0.33/0.68 thf(fact_9126_log__powr,axiom,
% 0.33/0.68 ! [X: real,B: real,Y: real] :
% 0.33/0.68 ( ( X != zero_zero_real )
% 0.33/0.68 => ( ( log @ B @ ( powr_real @ X @ Y ) )
% 0.33/0.68 = ( times_times_real @ Y @ ( log @ B @ X ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_powr
% 0.33/0.68 thf(fact_9127_floor__log__eq__powr__iff,axiom,
% 0.33/0.68 ! [X: real,B: real,K: int] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.68 => ( ( ( archim6058952711729229775r_real @ ( log @ B @ X ) )
% 0.33/0.68 = K )
% 0.33/0.68 = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X )
% 0.33/0.68 & ( ord_less_real @ X @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % floor_log_eq_powr_iff
% 0.33/0.68 thf(fact_9128_powr__realpow,axiom,
% 0.33/0.68 ! [X: real,N2: nat] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N2 ) )
% 0.33/0.68 = ( power_power_real @ X @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_realpow
% 0.33/0.68 thf(fact_9129_less__log__iff,axiom,
% 0.33/0.68 ! [B: real,X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_real @ Y @ ( log @ B @ X ) )
% 0.33/0.68 = ( ord_less_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % less_log_iff
% 0.33/0.68 thf(fact_9130_log__less__iff,axiom,
% 0.33/0.68 ! [B: real,X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_real @ ( log @ B @ X ) @ Y )
% 0.33/0.68 = ( ord_less_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_less_iff
% 0.33/0.68 thf(fact_9131_less__powr__iff,axiom,
% 0.33/0.68 ! [B: real,X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_real @ X @ ( powr_real @ B @ Y ) )
% 0.33/0.68 = ( ord_less_real @ ( log @ B @ X ) @ Y ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % less_powr_iff
% 0.33/0.68 thf(fact_9132_powr__less__iff,axiom,
% 0.33/0.68 ! [B: real,X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_real @ ( powr_real @ B @ Y ) @ X )
% 0.33/0.68 = ( ord_less_real @ Y @ ( log @ B @ X ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_less_iff
% 0.33/0.68 thf(fact_9133_real__of__int__floor__add__one__gt,axiom,
% 0.33/0.68 ! [R: real] : ( ord_less_real @ R @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) @ one_one_real ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_of_int_floor_add_one_gt
% 0.33/0.68 thf(fact_9134_floor__eq,axiom,
% 0.33/0.68 ! [N2: int,X: real] :
% 0.33/0.68 ( ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ X )
% 0.33/0.68 => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
% 0.33/0.68 => ( ( archim6058952711729229775r_real @ X )
% 0.33/0.68 = N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % floor_eq
% 0.33/0.68 thf(fact_9135_real__of__int__floor__add__one__ge,axiom,
% 0.33/0.68 ! [R: real] : ( ord_less_eq_real @ R @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) @ one_one_real ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_of_int_floor_add_one_ge
% 0.33/0.68 thf(fact_9136_real__of__int__floor__gt__diff__one,axiom,
% 0.33/0.68 ! [R: real] : ( ord_less_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_of_int_floor_gt_diff_one
% 0.33/0.68 thf(fact_9137_real__of__int__floor__ge__diff__one,axiom,
% 0.33/0.68 ! [R: real] : ( ord_less_eq_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_of_int_floor_ge_diff_one
% 0.33/0.68 thf(fact_9138_DeMoivre,axiom,
% 0.33/0.68 ! [A: real,N2: nat] :
% 0.33/0.68 ( ( power_power_complex @ ( cis @ A ) @ N2 )
% 0.33/0.68 = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ A ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % DeMoivre
% 0.33/0.68 thf(fact_9139_powr__neg__one,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( powr_real @ X @ ( uminus_uminus_real @ one_one_real ) )
% 0.33/0.68 = ( divide_divide_real @ one_one_real @ X ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_neg_one
% 0.33/0.68 thf(fact_9140_powr__mult__base,axiom,
% 0.33/0.68 ! [X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( times_times_real @ X @ ( powr_real @ X @ Y ) )
% 0.33/0.68 = ( powr_real @ X @ ( plus_plus_real @ one_one_real @ Y ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_mult_base
% 0.33/0.68 thf(fact_9141_le__log__iff,axiom,
% 0.33/0.68 ! [B: real,X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_eq_real @ Y @ ( log @ B @ X ) )
% 0.33/0.68 = ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % le_log_iff
% 0.33/0.68 thf(fact_9142_log__le__iff,axiom,
% 0.33/0.68 ! [B: real,X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_eq_real @ ( log @ B @ X ) @ Y )
% 0.33/0.68 = ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_le_iff
% 0.33/0.68 thf(fact_9143_le__powr__iff,axiom,
% 0.33/0.68 ! [B: real,X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) )
% 0.33/0.68 = ( ord_less_eq_real @ ( log @ B @ X ) @ Y ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % le_powr_iff
% 0.33/0.68 thf(fact_9144_powr__le__iff,axiom,
% 0.33/0.68 ! [B: real,X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X )
% 0.33/0.68 = ( ord_less_eq_real @ Y @ ( log @ B @ X ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_le_iff
% 0.33/0.68 thf(fact_9145_floor__eq2,axiom,
% 0.33/0.68 ! [N2: int,X: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N2 ) @ X )
% 0.33/0.68 => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
% 0.33/0.68 => ( ( archim6058952711729229775r_real @ X )
% 0.33/0.68 = N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % floor_eq2
% 0.33/0.68 thf(fact_9146_floor__divide__real__eq__div,axiom,
% 0.33/0.68 ! [B: int,A: real] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.33/0.68 => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B ) ) )
% 0.33/0.68 = ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % floor_divide_real_eq_div
% 0.33/0.68 thf(fact_9147_ln__powr__bound,axiom,
% 0.33/0.68 ! [X: real,A: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ one_one_real @ X )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.68 => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( divide_divide_real @ ( powr_real @ X @ A ) @ A ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % ln_powr_bound
% 0.33/0.68 thf(fact_9148_ln__powr__bound2,axiom,
% 0.33/0.68 ! [X: real,A: real] :
% 0.33/0.68 ( ( ord_less_real @ one_one_real @ X )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.68 => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % ln_powr_bound2
% 0.33/0.68 thf(fact_9149_add__log__eq__powr,axiom,
% 0.33/0.68 ! [B: real,X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ B )
% 0.33/0.68 => ( ( B != one_one_real )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( plus_plus_real @ Y @ ( log @ B @ X ) )
% 0.33/0.68 = ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % add_log_eq_powr
% 0.33/0.68 thf(fact_9150_log__add__eq__powr,axiom,
% 0.33/0.68 ! [B: real,X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ B )
% 0.33/0.68 => ( ( B != one_one_real )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( plus_plus_real @ ( log @ B @ X ) @ Y )
% 0.33/0.68 = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_add_eq_powr
% 0.33/0.68 thf(fact_9151_minus__log__eq__powr,axiom,
% 0.33/0.68 ! [B: real,X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ B )
% 0.33/0.68 => ( ( B != one_one_real )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( minus_minus_real @ Y @ ( log @ B @ X ) )
% 0.33/0.68 = ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % minus_log_eq_powr
% 0.33/0.68 thf(fact_9152_log__minus__eq__powr,axiom,
% 0.33/0.68 ! [B: real,X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ B )
% 0.33/0.68 => ( ( B != one_one_real )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( minus_minus_real @ ( log @ B @ X ) @ Y )
% 0.33/0.68 = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ ( uminus_uminus_real @ Y ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_minus_eq_powr
% 0.33/0.68 thf(fact_9153_powr__half__sqrt,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.68 = ( sqrt @ X ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_half_sqrt
% 0.33/0.68 thf(fact_9154_powr__neg__numeral,axiom,
% 0.33/0.68 ! [X: real,N2: num] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( powr_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 0.33/0.68 = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_neg_numeral
% 0.33/0.68 thf(fact_9155_floor__log2__div2,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.68 => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 0.33/0.68 = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % floor_log2_div2
% 0.33/0.68 thf(fact_9156_floor__log__nat__eq__if,axiom,
% 0.33/0.68 ! [B: nat,N2: nat,K: nat] :
% 0.33/0.68 ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 0.33/0.68 => ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) )
% 0.33/0.68 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.33/0.68 => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.33/0.68 = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % floor_log_nat_eq_if
% 0.33/0.68 thf(fact_9157_int__ge__less__than__def,axiom,
% 0.33/0.68 ( int_ge_less_than
% 0.33/0.68 = ( ^ [D2: int] :
% 0.33/0.68 ( collec213857154873943460nt_int
% 0.33/0.68 @ ( produc4947309494688390418_int_o
% 0.33/0.68 @ ^ [Z6: int,Z3: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ D2 @ Z6 )
% 0.33/0.68 & ( ord_less_int @ Z6 @ Z3 ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % int_ge_less_than_def
% 0.33/0.68 thf(fact_9158_bij__betw__roots__unity,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( bij_betw_nat_complex
% 0.33/0.68 @ ^ [K3: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K3 ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 0.33/0.68 @ ( set_ord_lessThan_nat @ N2 )
% 0.33/0.68 @ ( collect_complex
% 0.33/0.68 @ ^ [Z3: complex] :
% 0.33/0.68 ( ( power_power_complex @ Z3 @ N2 )
% 0.33/0.68 = one_one_complex ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % bij_betw_roots_unity
% 0.33/0.68 thf(fact_9159_summable__complex__of__real,axiom,
% 0.33/0.68 ! [F: nat > real] :
% 0.33/0.68 ( ( summable_complex
% 0.33/0.68 @ ^ [N: nat] : ( real_V4546457046886955230omplex @ ( F @ N ) ) )
% 0.33/0.68 = ( summable_real @ F ) ) ).
% 0.33/0.68
% 0.33/0.68 % summable_complex_of_real
% 0.33/0.68 thf(fact_9160_exp__pi__i,axiom,
% 0.33/0.68 ( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
% 0.33/0.68 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.33/0.68
% 0.33/0.68 % exp_pi_i
% 0.33/0.68 thf(fact_9161_exp__pi__i_H,axiom,
% 0.33/0.68 ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
% 0.33/0.68 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.33/0.68
% 0.33/0.68 % exp_pi_i'
% 0.33/0.68 thf(fact_9162_exp__two__pi__i,axiom,
% 0.33/0.68 ( ( exp_complex @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( real_V4546457046886955230omplex @ pi ) ) @ imaginary_unit ) )
% 0.33/0.68 = one_one_complex ) ).
% 0.33/0.68
% 0.33/0.68 % exp_two_pi_i
% 0.33/0.68 thf(fact_9163_exp__two__pi__i_H,axiom,
% 0.33/0.68 ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) )
% 0.33/0.68 = one_one_complex ) ).
% 0.33/0.68
% 0.33/0.68 % exp_two_pi_i'
% 0.33/0.68 thf(fact_9164_Complex__mult__complex__of__real,axiom,
% 0.33/0.68 ! [X: real,Y: real,R: real] :
% 0.33/0.68 ( ( times_times_complex @ ( complex2 @ X @ Y ) @ ( real_V4546457046886955230omplex @ R ) )
% 0.33/0.68 = ( complex2 @ ( times_times_real @ X @ R ) @ ( times_times_real @ Y @ R ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Complex_mult_complex_of_real
% 0.33/0.68 thf(fact_9165_complex__of__real__mult__Complex,axiom,
% 0.33/0.68 ! [R: real,X: real,Y: real] :
% 0.33/0.68 ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ ( complex2 @ X @ Y ) )
% 0.33/0.68 = ( complex2 @ ( times_times_real @ R @ X ) @ ( times_times_real @ R @ Y ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % complex_of_real_mult_Complex
% 0.33/0.68 thf(fact_9166_Complex__add__complex__of__real,axiom,
% 0.33/0.68 ! [X: real,Y: real,R: real] :
% 0.33/0.68 ( ( plus_plus_complex @ ( complex2 @ X @ Y ) @ ( real_V4546457046886955230omplex @ R ) )
% 0.33/0.68 = ( complex2 @ ( plus_plus_real @ X @ R ) @ Y ) ) ).
% 0.33/0.68
% 0.33/0.68 % Complex_add_complex_of_real
% 0.33/0.68 thf(fact_9167_complex__of__real__add__Complex,axiom,
% 0.33/0.68 ! [R: real,X: real,Y: real] :
% 0.33/0.68 ( ( plus_plus_complex @ ( real_V4546457046886955230omplex @ R ) @ ( complex2 @ X @ Y ) )
% 0.33/0.68 = ( complex2 @ ( plus_plus_real @ R @ X ) @ Y ) ) ).
% 0.33/0.68
% 0.33/0.68 % complex_of_real_add_Complex
% 0.33/0.68 thf(fact_9168_cmod__unit__one,axiom,
% 0.33/0.68 ! [A: real] :
% 0.33/0.68 ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) )
% 0.33/0.68 = one_one_real ) ).
% 0.33/0.68
% 0.33/0.68 % cmod_unit_one
% 0.33/0.68 thf(fact_9169_csqrt__ii,axiom,
% 0.33/0.68 ( ( csqrt @ imaginary_unit )
% 0.33/0.68 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ one_one_complex @ imaginary_unit ) @ ( real_V4546457046886955230omplex @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % csqrt_ii
% 0.33/0.68 thf(fact_9170_arctan__def,axiom,
% 0.33/0.68 ( arctan
% 0.33/0.68 = ( ^ [Y2: real] :
% 0.33/0.68 ( the_real
% 0.33/0.68 @ ^ [X2: real] :
% 0.33/0.68 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.33/0.68 & ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.68 & ( ( tan_real @ X2 )
% 0.33/0.68 = Y2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % arctan_def
% 0.33/0.68 thf(fact_9171_arcsin__def,axiom,
% 0.33/0.68 ( arcsin
% 0.33/0.68 = ( ^ [Y2: real] :
% 0.33/0.68 ( the_real
% 0.33/0.68 @ ^ [X2: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.33/0.68 & ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.68 & ( ( sin_real @ X2 )
% 0.33/0.68 = Y2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % arcsin_def
% 0.33/0.68 thf(fact_9172_modulo__int__unfold,axiom,
% 0.33/0.68 ! [L2: int,K: int,N2: nat,M: nat] :
% 0.33/0.68 ( ( ( ( ( sgn_sgn_int @ L2 )
% 0.33/0.68 = zero_zero_int )
% 0.33/0.68 | ( ( sgn_sgn_int @ K )
% 0.33/0.68 = zero_zero_int )
% 0.33/0.68 | ( N2 = zero_zero_nat ) )
% 0.33/0.68 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.33/0.68 = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
% 0.33/0.68 & ( ~ ( ( ( sgn_sgn_int @ L2 )
% 0.33/0.68 = zero_zero_int )
% 0.33/0.68 | ( ( sgn_sgn_int @ K )
% 0.33/0.68 = zero_zero_int )
% 0.33/0.68 | ( N2 = zero_zero_nat ) )
% 0.33/0.68 => ( ( ( ( sgn_sgn_int @ K )
% 0.33/0.68 = ( sgn_sgn_int @ L2 ) )
% 0.33/0.68 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.33/0.68 = ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) )
% 0.33/0.68 & ( ( ( sgn_sgn_int @ K )
% 0.33/0.68 != ( sgn_sgn_int @ L2 ) )
% 0.33/0.68 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.33/0.68 = ( times_times_int @ ( sgn_sgn_int @ L2 )
% 0.33/0.68 @ ( minus_minus_int
% 0.33/0.68 @ ( semiri1314217659103216013at_int
% 0.33/0.68 @ ( times_times_nat @ N2
% 0.33/0.68 @ ( zero_n2687167440665602831ol_nat
% 0.33/0.68 @ ~ ( dvd_dvd_nat @ N2 @ M ) ) ) )
% 0.33/0.68 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % modulo_int_unfold
% 0.33/0.68 thf(fact_9173_csqrt__1,axiom,
% 0.33/0.68 ( ( csqrt @ one_one_complex )
% 0.33/0.68 = one_one_complex ) ).
% 0.33/0.68
% 0.33/0.68 % csqrt_1
% 0.33/0.68 thf(fact_9174_csqrt__eq__1,axiom,
% 0.33/0.68 ! [Z: complex] :
% 0.33/0.68 ( ( ( csqrt @ Z )
% 0.33/0.68 = one_one_complex )
% 0.33/0.68 = ( Z = one_one_complex ) ) ).
% 0.33/0.68
% 0.33/0.68 % csqrt_eq_1
% 0.33/0.68 thf(fact_9175_power2__csqrt,axiom,
% 0.33/0.68 ! [Z: complex] :
% 0.33/0.68 ( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.68 = Z ) ).
% 0.33/0.68
% 0.33/0.68 % power2_csqrt
% 0.33/0.68 thf(fact_9176_div__eq__sgn__abs,axiom,
% 0.33/0.68 ! [K: int,L2: int] :
% 0.33/0.68 ( ( ( sgn_sgn_int @ K )
% 0.33/0.68 = ( sgn_sgn_int @ L2 ) )
% 0.33/0.68 => ( ( divide_divide_int @ K @ L2 )
% 0.33/0.68 = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % div_eq_sgn_abs
% 0.33/0.68 thf(fact_9177_ln__real__def,axiom,
% 0.33/0.68 ( ln_ln_real
% 0.33/0.68 = ( ^ [X2: real] :
% 0.33/0.68 ( the_real
% 0.33/0.68 @ ^ [U2: real] :
% 0.33/0.68 ( ( exp_real @ U2 )
% 0.33/0.68 = X2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % ln_real_def
% 0.33/0.68 thf(fact_9178_sgn__mod,axiom,
% 0.33/0.68 ! [L2: int,K: int] :
% 0.33/0.68 ( ( L2 != zero_zero_int )
% 0.33/0.68 => ( ~ ( dvd_dvd_int @ L2 @ K )
% 0.33/0.68 => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L2 ) )
% 0.33/0.68 = ( sgn_sgn_int @ L2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % sgn_mod
% 0.33/0.68 thf(fact_9179_ln__neg__is__const,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.33/0.68 => ( ( ln_ln_real @ X )
% 0.33/0.68 = ( the_real
% 0.33/0.68 @ ^ [X2: real] : $false ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % ln_neg_is_const
% 0.33/0.68 thf(fact_9180_zsgn__def,axiom,
% 0.33/0.68 ( sgn_sgn_int
% 0.33/0.68 = ( ^ [I3: int] : ( if_int @ ( I3 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % zsgn_def
% 0.33/0.68 thf(fact_9181_div__sgn__abs__cancel,axiom,
% 0.33/0.68 ! [V: int,K: int,L2: int] :
% 0.33/0.68 ( ( V != zero_zero_int )
% 0.33/0.68 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L2 ) ) )
% 0.33/0.68 = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % div_sgn_abs_cancel
% 0.33/0.68 thf(fact_9182_div__dvd__sgn__abs,axiom,
% 0.33/0.68 ! [L2: int,K: int] :
% 0.33/0.68 ( ( dvd_dvd_int @ L2 @ K )
% 0.33/0.68 => ( ( divide_divide_int @ K @ L2 )
% 0.33/0.68 = ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( sgn_sgn_int @ L2 ) ) @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % div_dvd_sgn_abs
% 0.33/0.68 thf(fact_9183_of__real__sqrt,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( real_V4546457046886955230omplex @ ( sqrt @ X ) )
% 0.33/0.68 = ( csqrt @ ( real_V4546457046886955230omplex @ X ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % of_real_sqrt
% 0.33/0.68 thf(fact_9184_arccos__def,axiom,
% 0.33/0.68 ( arccos
% 0.33/0.68 = ( ^ [Y2: real] :
% 0.33/0.68 ( the_real
% 0.33/0.68 @ ^ [X2: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.33/0.68 & ( ord_less_eq_real @ X2 @ pi )
% 0.33/0.68 & ( ( cos_real @ X2 )
% 0.33/0.68 = Y2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % arccos_def
% 0.33/0.68 thf(fact_9185_eucl__rel__int__remainderI,axiom,
% 0.33/0.68 ! [R: int,L2: int,K: int,Q2: int] :
% 0.33/0.68 ( ( ( sgn_sgn_int @ R )
% 0.33/0.68 = ( sgn_sgn_int @ L2 ) )
% 0.33/0.68 => ( ( ord_less_int @ ( abs_abs_int @ R ) @ ( abs_abs_int @ L2 ) )
% 0.33/0.68 => ( ( K
% 0.33/0.68 = ( plus_plus_int @ ( times_times_int @ Q2 @ L2 ) @ R ) )
% 0.33/0.68 => ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % eucl_rel_int_remainderI
% 0.33/0.68 thf(fact_9186_eucl__rel__int_Ocases,axiom,
% 0.33/0.68 ! [A12: int,A23: int,A32: product_prod_int_int] :
% 0.33/0.68 ( ( eucl_rel_int @ A12 @ A23 @ A32 )
% 0.33/0.68 => ( ( ( A23 = zero_zero_int )
% 0.33/0.68 => ( A32
% 0.33/0.68 != ( product_Pair_int_int @ zero_zero_int @ A12 ) ) )
% 0.33/0.68 => ( ! [Q3: int] :
% 0.33/0.68 ( ( A32
% 0.33/0.68 = ( product_Pair_int_int @ Q3 @ zero_zero_int ) )
% 0.33/0.68 => ( ( A23 != zero_zero_int )
% 0.33/0.68 => ( A12
% 0.33/0.68 != ( times_times_int @ Q3 @ A23 ) ) ) )
% 0.33/0.68 => ~ ! [R3: int,Q3: int] :
% 0.33/0.68 ( ( A32
% 0.33/0.68 = ( product_Pair_int_int @ Q3 @ R3 ) )
% 0.33/0.68 => ( ( ( sgn_sgn_int @ R3 )
% 0.33/0.68 = ( sgn_sgn_int @ A23 ) )
% 0.33/0.68 => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A23 ) )
% 0.33/0.68 => ( A12
% 0.33/0.68 != ( plus_plus_int @ ( times_times_int @ Q3 @ A23 ) @ R3 ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % eucl_rel_int.cases
% 0.33/0.68 thf(fact_9187_eucl__rel__int_Osimps,axiom,
% 0.33/0.68 ( eucl_rel_int
% 0.33/0.68 = ( ^ [A1: int,A22: int,A33: product_prod_int_int] :
% 0.33/0.68 ( ? [K3: int] :
% 0.33/0.68 ( ( A1 = K3 )
% 0.33/0.68 & ( A22 = zero_zero_int )
% 0.33/0.68 & ( A33
% 0.33/0.68 = ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
% 0.33/0.68 | ? [L: int,K3: int,Q4: int] :
% 0.33/0.68 ( ( A1 = K3 )
% 0.33/0.68 & ( A22 = L )
% 0.33/0.68 & ( A33
% 0.33/0.68 = ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
% 0.33/0.68 & ( L != zero_zero_int )
% 0.33/0.68 & ( K3
% 0.33/0.68 = ( times_times_int @ Q4 @ L ) ) )
% 0.33/0.68 | ? [R5: int,L: int,K3: int,Q4: int] :
% 0.33/0.68 ( ( A1 = K3 )
% 0.33/0.68 & ( A22 = L )
% 0.33/0.68 & ( A33
% 0.33/0.68 = ( product_Pair_int_int @ Q4 @ R5 ) )
% 0.33/0.68 & ( ( sgn_sgn_int @ R5 )
% 0.33/0.68 = ( sgn_sgn_int @ L ) )
% 0.33/0.68 & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L ) )
% 0.33/0.68 & ( K3
% 0.33/0.68 = ( plus_plus_int @ ( times_times_int @ Q4 @ L ) @ R5 ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % eucl_rel_int.simps
% 0.33/0.68 thf(fact_9188_div__noneq__sgn__abs,axiom,
% 0.33/0.68 ! [L2: int,K: int] :
% 0.33/0.68 ( ( L2 != zero_zero_int )
% 0.33/0.68 => ( ( ( sgn_sgn_int @ K )
% 0.33/0.68 != ( sgn_sgn_int @ L2 ) )
% 0.33/0.68 => ( ( divide_divide_int @ K @ L2 )
% 0.33/0.68 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) )
% 0.33/0.68 @ ( zero_n2684676970156552555ol_int
% 0.33/0.68 @ ~ ( dvd_dvd_int @ L2 @ K ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % div_noneq_sgn_abs
% 0.33/0.68 thf(fact_9189_pi__half,axiom,
% 0.33/0.68 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.33/0.68 = ( the_real
% 0.33/0.68 @ ^ [X2: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.33/0.68 & ( ord_less_eq_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.33/0.68 & ( ( cos_real @ X2 )
% 0.33/0.68 = zero_zero_real ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % pi_half
% 0.33/0.68 thf(fact_9190_pi__def,axiom,
% 0.33/0.68 ( pi
% 0.33/0.68 = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 0.33/0.68 @ ( the_real
% 0.33/0.68 @ ^ [X2: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.33/0.68 & ( ord_less_eq_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.33/0.68 & ( ( cos_real @ X2 )
% 0.33/0.68 = zero_zero_real ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % pi_def
% 0.33/0.68 thf(fact_9191_divide__int__unfold,axiom,
% 0.33/0.68 ! [L2: int,K: int,N2: nat,M: nat] :
% 0.33/0.68 ( ( ( ( ( sgn_sgn_int @ L2 )
% 0.33/0.68 = zero_zero_int )
% 0.33/0.68 | ( ( sgn_sgn_int @ K )
% 0.33/0.68 = zero_zero_int )
% 0.33/0.68 | ( N2 = zero_zero_nat ) )
% 0.33/0.68 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.33/0.68 = zero_zero_int ) )
% 0.33/0.68 & ( ~ ( ( ( sgn_sgn_int @ L2 )
% 0.33/0.68 = zero_zero_int )
% 0.33/0.68 | ( ( sgn_sgn_int @ K )
% 0.33/0.68 = zero_zero_int )
% 0.33/0.68 | ( N2 = zero_zero_nat ) )
% 0.33/0.68 => ( ( ( ( sgn_sgn_int @ K )
% 0.33/0.68 = ( sgn_sgn_int @ L2 ) )
% 0.33/0.68 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.33/0.68 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) ) ) )
% 0.33/0.68 & ( ( ( sgn_sgn_int @ K )
% 0.33/0.68 != ( sgn_sgn_int @ L2 ) )
% 0.33/0.68 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.33/0.68 = ( uminus_uminus_int
% 0.33/0.68 @ ( semiri1314217659103216013at_int
% 0.33/0.68 @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N2 )
% 0.33/0.68 @ ( zero_n2687167440665602831ol_nat
% 0.33/0.68 @ ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % divide_int_unfold
% 0.33/0.68 thf(fact_9192_sgn__le__0__iff,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ ( sgn_sgn_real @ X ) @ zero_zero_real )
% 0.33/0.68 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.33/0.68
% 0.33/0.68 % sgn_le_0_iff
% 0.33/0.68 thf(fact_9193_zero__le__sgn__iff,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X ) )
% 0.33/0.68 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.33/0.68
% 0.33/0.68 % zero_le_sgn_iff
% 0.33/0.68 thf(fact_9194_real__sgn__eq,axiom,
% 0.33/0.68 ( sgn_sgn_real
% 0.33/0.68 = ( ^ [X2: real] : ( divide_divide_real @ X2 @ ( abs_abs_real @ X2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_sgn_eq
% 0.33/0.68 thf(fact_9195_sgn__eq,axiom,
% 0.33/0.68 ( sgn_sgn_complex
% 0.33/0.68 = ( ^ [Z3: complex] : ( divide1717551699836669952omplex @ Z3 @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ Z3 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % sgn_eq
% 0.33/0.68 thf(fact_9196_sgn__real__def,axiom,
% 0.33/0.68 ( sgn_sgn_real
% 0.33/0.68 = ( ^ [A5: real] : ( if_real @ ( A5 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A5 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % sgn_real_def
% 0.33/0.68 thf(fact_9197_sgn__power__injE,axiom,
% 0.33/0.68 ! [A: real,N2: nat,X: real,B: real] :
% 0.33/0.68 ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) )
% 0.33/0.68 = X )
% 0.33/0.68 => ( ( X
% 0.33/0.68 = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N2 ) ) )
% 0.33/0.68 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( A = B ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % sgn_power_injE
% 0.33/0.68 thf(fact_9198_cis__Arg__unique,axiom,
% 0.33/0.68 ! [Z: complex,X: real] :
% 0.33/0.68 ( ( ( sgn_sgn_complex @ Z )
% 0.33/0.68 = ( cis @ X ) )
% 0.33/0.68 => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
% 0.33/0.68 => ( ( ord_less_eq_real @ X @ pi )
% 0.33/0.68 => ( ( arg @ Z )
% 0.33/0.68 = X ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % cis_Arg_unique
% 0.33/0.68 thf(fact_9199_floor__real__def,axiom,
% 0.33/0.68 ( archim6058952711729229775r_real
% 0.33/0.68 = ( ^ [X2: real] :
% 0.33/0.68 ( the_int
% 0.33/0.68 @ ^ [Z3: int] :
% 0.33/0.68 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ X2 )
% 0.33/0.68 & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % floor_real_def
% 0.33/0.68 thf(fact_9200_Arg__correct,axiom,
% 0.33/0.68 ! [Z: complex] :
% 0.33/0.68 ( ( Z != zero_zero_complex )
% 0.33/0.68 => ( ( ( sgn_sgn_complex @ Z )
% 0.33/0.68 = ( cis @ ( arg @ Z ) ) )
% 0.33/0.68 & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 0.33/0.68 & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Arg_correct
% 0.33/0.68 thf(fact_9201_arctan__inverse,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( X != zero_zero_real )
% 0.33/0.68 => ( ( arctan @ ( divide_divide_real @ one_one_real @ X ) )
% 0.33/0.68 = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % arctan_inverse
% 0.33/0.68 thf(fact_9202_bij__betw__nth__root__unity,axiom,
% 0.33/0.68 ! [C: complex,N2: nat] :
% 0.33/0.68 ( ( C != zero_zero_complex )
% 0.33/0.68 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N2 @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) )
% 0.33/0.68 @ ( collect_complex
% 0.33/0.68 @ ^ [Z3: complex] :
% 0.33/0.68 ( ( power_power_complex @ Z3 @ N2 )
% 0.33/0.68 = one_one_complex ) )
% 0.33/0.68 @ ( collect_complex
% 0.33/0.68 @ ^ [Z3: complex] :
% 0.33/0.68 ( ( power_power_complex @ Z3 @ N2 )
% 0.33/0.68 = C ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % bij_betw_nth_root_unity
% 0.33/0.68 thf(fact_9203_modulo__int__def,axiom,
% 0.33/0.68 ( modulo_modulo_int
% 0.33/0.68 = ( ^ [K3: int,L: int] :
% 0.33/0.68 ( if_int @ ( L = zero_zero_int ) @ K3
% 0.33/0.68 @ ( if_int
% 0.33/0.68 @ ( ( sgn_sgn_int @ K3 )
% 0.33/0.68 = ( sgn_sgn_int @ L ) )
% 0.33/0.68 @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) )
% 0.33/0.68 @ ( times_times_int @ ( sgn_sgn_int @ L )
% 0.33/0.68 @ ( minus_minus_int
% 0.33/0.68 @ ( times_times_int @ ( abs_abs_int @ L )
% 0.33/0.68 @ ( zero_n2684676970156552555ol_int
% 0.33/0.68 @ ~ ( dvd_dvd_int @ L @ K3 ) ) )
% 0.33/0.68 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % modulo_int_def
% 0.33/0.68 thf(fact_9204_divide__int__def,axiom,
% 0.33/0.68 ( divide_divide_int
% 0.33/0.68 = ( ^ [K3: int,L: int] :
% 0.33/0.68 ( if_int @ ( L = zero_zero_int ) @ zero_zero_int
% 0.33/0.68 @ ( if_int
% 0.33/0.68 @ ( ( sgn_sgn_int @ K3 )
% 0.33/0.68 = ( sgn_sgn_int @ L ) )
% 0.33/0.68 @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) )
% 0.33/0.68 @ ( uminus_uminus_int
% 0.33/0.68 @ ( semiri1314217659103216013at_int
% 0.33/0.68 @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) )
% 0.33/0.68 @ ( zero_n2687167440665602831ol_nat
% 0.33/0.68 @ ~ ( dvd_dvd_int @ L @ K3 ) ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % divide_int_def
% 0.33/0.68 thf(fact_9205_mask__nat__positive__iff,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N2 ) )
% 0.33/0.68 = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % mask_nat_positive_iff
% 0.33/0.68 thf(fact_9206_real__root__zero,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( root @ N2 @ zero_zero_real )
% 0.33/0.68 = zero_zero_real ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_zero
% 0.33/0.68 thf(fact_9207_nat__numeral,axiom,
% 0.33/0.68 ! [K: num] :
% 0.33/0.68 ( ( nat2 @ ( numeral_numeral_int @ K ) )
% 0.33/0.68 = ( numeral_numeral_nat @ K ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_numeral
% 0.33/0.68 thf(fact_9208_real__root__Suc__0,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( root @ ( suc @ zero_zero_nat ) @ X )
% 0.33/0.68 = X ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_Suc_0
% 0.33/0.68 thf(fact_9209_real__root__eq__iff,axiom,
% 0.33/0.68 ! [N2: nat,X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ( root @ N2 @ X )
% 0.33/0.68 = ( root @ N2 @ Y ) )
% 0.33/0.68 = ( X = Y ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_eq_iff
% 0.33/0.68 thf(fact_9210_root__0,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( root @ zero_zero_nat @ X )
% 0.33/0.68 = zero_zero_real ) ).
% 0.33/0.68
% 0.33/0.68 % root_0
% 0.33/0.68 thf(fact_9211_nat__1,axiom,
% 0.33/0.68 ( ( nat2 @ one_one_int )
% 0.33/0.68 = ( suc @ zero_zero_nat ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_1
% 0.33/0.68 thf(fact_9212_real__root__eq__0__iff,axiom,
% 0.33/0.68 ! [N2: nat,X: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ( root @ N2 @ X )
% 0.33/0.68 = zero_zero_real )
% 0.33/0.68 = ( X = zero_zero_real ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_eq_0_iff
% 0.33/0.68 thf(fact_9213_real__root__less__iff,axiom,
% 0.33/0.68 ! [N2: nat,X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) )
% 0.33/0.68 = ( ord_less_real @ X @ Y ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_less_iff
% 0.33/0.68 thf(fact_9214_real__root__le__iff,axiom,
% 0.33/0.68 ! [N2: nat,X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) )
% 0.33/0.68 = ( ord_less_eq_real @ X @ Y ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_le_iff
% 0.33/0.68 thf(fact_9215_zless__nat__conj,axiom,
% 0.33/0.68 ! [W: int,Z: int] :
% 0.33/0.68 ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.33/0.68 = ( ( ord_less_int @ zero_zero_int @ Z )
% 0.33/0.68 & ( ord_less_int @ W @ Z ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % zless_nat_conj
% 0.33/0.68 thf(fact_9216_real__root__eq__1__iff,axiom,
% 0.33/0.68 ! [N2: nat,X: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ( root @ N2 @ X )
% 0.33/0.68 = one_one_real )
% 0.33/0.68 = ( X = one_one_real ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_eq_1_iff
% 0.33/0.68 thf(fact_9217_real__root__one,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( root @ N2 @ one_one_real )
% 0.33/0.68 = one_one_real ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_one
% 0.33/0.68 thf(fact_9218_real__root__lt__0__iff,axiom,
% 0.33/0.68 ! [N2: nat,X: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ord_less_real @ ( root @ N2 @ X ) @ zero_zero_real )
% 0.33/0.68 = ( ord_less_real @ X @ zero_zero_real ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_lt_0_iff
% 0.33/0.68 thf(fact_9219_real__root__gt__0__iff,axiom,
% 0.33/0.68 ! [N2: nat,Y: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ ( root @ N2 @ Y ) )
% 0.33/0.68 = ( ord_less_real @ zero_zero_real @ Y ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_gt_0_iff
% 0.33/0.68 thf(fact_9220_real__root__le__0__iff,axiom,
% 0.33/0.68 ! [N2: nat,X: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ zero_zero_real )
% 0.33/0.68 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_le_0_iff
% 0.33/0.68 thf(fact_9221_real__root__ge__0__iff,axiom,
% 0.33/0.68 ! [N2: nat,Y: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ Y ) )
% 0.33/0.68 = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_ge_0_iff
% 0.33/0.68 thf(fact_9222_zero__less__nat__eq,axiom,
% 0.33/0.68 ! [Z: int] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
% 0.33/0.68 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.33/0.68
% 0.33/0.68 % zero_less_nat_eq
% 0.33/0.68 thf(fact_9223_real__root__lt__1__iff,axiom,
% 0.33/0.68 ! [N2: nat,X: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ord_less_real @ ( root @ N2 @ X ) @ one_one_real )
% 0.33/0.68 = ( ord_less_real @ X @ one_one_real ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_lt_1_iff
% 0.33/0.68 thf(fact_9224_real__root__gt__1__iff,axiom,
% 0.33/0.68 ! [N2: nat,Y: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ord_less_real @ one_one_real @ ( root @ N2 @ Y ) )
% 0.33/0.68 = ( ord_less_real @ one_one_real @ Y ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_gt_1_iff
% 0.33/0.68 thf(fact_9225_real__root__le__1__iff,axiom,
% 0.33/0.68 ! [N2: nat,X: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ one_one_real )
% 0.33/0.68 = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_le_1_iff
% 0.33/0.68 thf(fact_9226_real__root__ge__1__iff,axiom,
% 0.33/0.68 ! [N2: nat,Y: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ord_less_eq_real @ one_one_real @ ( root @ N2 @ Y ) )
% 0.33/0.68 = ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_ge_1_iff
% 0.33/0.68 thf(fact_9227_diff__nat__numeral,axiom,
% 0.33/0.68 ! [V: num,V3: num] :
% 0.33/0.68 ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V3 ) )
% 0.33/0.68 = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V3 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % diff_nat_numeral
% 0.33/0.68 thf(fact_9228_numeral__power__eq__nat__cancel__iff,axiom,
% 0.33/0.68 ! [X: num,N2: nat,Y: int] :
% 0.33/0.68 ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
% 0.33/0.68 = ( nat2 @ Y ) )
% 0.33/0.68 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
% 0.33/0.68 = Y ) ) ).
% 0.33/0.68
% 0.33/0.68 % numeral_power_eq_nat_cancel_iff
% 0.33/0.68 thf(fact_9229_nat__eq__numeral__power__cancel__iff,axiom,
% 0.33/0.68 ! [Y: int,X: num,N2: nat] :
% 0.33/0.68 ( ( ( nat2 @ Y )
% 0.33/0.68 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
% 0.33/0.68 = ( Y
% 0.33/0.68 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_eq_numeral_power_cancel_iff
% 0.33/0.68 thf(fact_9230_nat__ceiling__le__eq,axiom,
% 0.33/0.68 ! [X: real,A: nat] :
% 0.33/0.68 ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
% 0.33/0.68 = ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_ceiling_le_eq
% 0.33/0.68 thf(fact_9231_one__less__nat__eq,axiom,
% 0.33/0.68 ! [Z: int] :
% 0.33/0.68 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
% 0.33/0.68 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.33/0.68
% 0.33/0.68 % one_less_nat_eq
% 0.33/0.68 thf(fact_9232_real__root__pow__pos2,axiom,
% 0.33/0.68 ! [N2: nat,X: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( power_power_real @ ( root @ N2 @ X ) @ N2 )
% 0.33/0.68 = X ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_pow_pos2
% 0.33/0.68 thf(fact_9233_nat__numeral__diff__1,axiom,
% 0.33/0.68 ! [V: num] :
% 0.33/0.68 ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
% 0.33/0.68 = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_numeral_diff_1
% 0.33/0.68 thf(fact_9234_nat__less__numeral__power__cancel__iff,axiom,
% 0.33/0.68 ! [A: int,X: num,N2: nat] :
% 0.33/0.68 ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
% 0.33/0.68 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_less_numeral_power_cancel_iff
% 0.33/0.68 thf(fact_9235_numeral__power__less__nat__cancel__iff,axiom,
% 0.33/0.68 ! [X: num,N2: nat,A: int] :
% 0.33/0.68 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) @ ( nat2 @ A ) )
% 0.33/0.68 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).
% 0.33/0.68
% 0.33/0.68 % numeral_power_less_nat_cancel_iff
% 0.33/0.68 thf(fact_9236_nat__le__numeral__power__cancel__iff,axiom,
% 0.33/0.68 ! [A: int,X: num,N2: nat] :
% 0.33/0.68 ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
% 0.33/0.68 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_le_numeral_power_cancel_iff
% 0.33/0.68 thf(fact_9237_numeral__power__le__nat__cancel__iff,axiom,
% 0.33/0.68 ! [X: num,N2: nat,A: int] :
% 0.33/0.68 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) @ ( nat2 @ A ) )
% 0.33/0.68 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).
% 0.33/0.68
% 0.33/0.68 % numeral_power_le_nat_cancel_iff
% 0.33/0.68 thf(fact_9238_less__eq__mask,axiom,
% 0.33/0.68 ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % less_eq_mask
% 0.33/0.68 thf(fact_9239_real__root__minus,axiom,
% 0.33/0.68 ! [N2: nat,X: real] :
% 0.33/0.68 ( ( root @ N2 @ ( uminus_uminus_real @ X ) )
% 0.33/0.68 = ( uminus_uminus_real @ ( root @ N2 @ X ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_minus
% 0.33/0.68 thf(fact_9240_real__root__mult,axiom,
% 0.33/0.68 ! [N2: nat,X: real,Y: real] :
% 0.33/0.68 ( ( root @ N2 @ ( times_times_real @ X @ Y ) )
% 0.33/0.68 = ( times_times_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_mult
% 0.33/0.68 thf(fact_9241_real__root__mult__exp,axiom,
% 0.33/0.68 ! [M: nat,N2: nat,X: real] :
% 0.33/0.68 ( ( root @ ( times_times_nat @ M @ N2 ) @ X )
% 0.33/0.68 = ( root @ M @ ( root @ N2 @ X ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_mult_exp
% 0.33/0.68 thf(fact_9242_nat__mask__eq,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( nat2 @ ( bit_se2000444600071755411sk_int @ N2 ) )
% 0.33/0.68 = ( bit_se2002935070580805687sk_nat @ N2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_mask_eq
% 0.33/0.68 thf(fact_9243_real__root__commute,axiom,
% 0.33/0.68 ! [M: nat,N2: nat,X: real] :
% 0.33/0.68 ( ( root @ M @ ( root @ N2 @ X ) )
% 0.33/0.68 = ( root @ N2 @ ( root @ M @ X ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_commute
% 0.33/0.68 thf(fact_9244_real__root__divide,axiom,
% 0.33/0.68 ! [N2: nat,X: real,Y: real] :
% 0.33/0.68 ( ( root @ N2 @ ( divide_divide_real @ X @ Y ) )
% 0.33/0.68 = ( divide_divide_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_divide
% 0.33/0.68 thf(fact_9245_real__root__inverse,axiom,
% 0.33/0.68 ! [N2: nat,X: real] :
% 0.33/0.68 ( ( root @ N2 @ ( inverse_inverse_real @ X ) )
% 0.33/0.68 = ( inverse_inverse_real @ ( root @ N2 @ X ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_inverse
% 0.33/0.68 thf(fact_9246_sgn__root,axiom,
% 0.33/0.68 ! [N2: nat,X: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( sgn_sgn_real @ ( root @ N2 @ X ) )
% 0.33/0.68 = ( sgn_sgn_real @ X ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % sgn_root
% 0.33/0.68 thf(fact_9247_real__root__pos__pos__le,axiom,
% 0.33/0.68 ! [X: real,N2: nat] :
% 0.33/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_pos_pos_le
% 0.33/0.68 thf(fact_9248_nat__numeral__as__int,axiom,
% 0.33/0.68 ( numeral_numeral_nat
% 0.33/0.68 = ( ^ [I3: num] : ( nat2 @ ( numeral_numeral_int @ I3 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_numeral_as_int
% 0.33/0.68 thf(fact_9249_nat__mono,axiom,
% 0.33/0.68 ! [X: int,Y: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ X @ Y )
% 0.33/0.68 => ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_mono
% 0.33/0.68 thf(fact_9250_nat__one__as__int,axiom,
% 0.33/0.68 ( one_one_nat
% 0.33/0.68 = ( nat2 @ one_one_int ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_one_as_int
% 0.33/0.68 thf(fact_9251_unset__bit__nat__def,axiom,
% 0.33/0.68 ( bit_se4205575877204974255it_nat
% 0.33/0.68 = ( ^ [M6: nat,N: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M6 @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % unset_bit_nat_def
% 0.33/0.68 thf(fact_9252_mask__nonnegative__int,axiom,
% 0.33/0.68 ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % mask_nonnegative_int
% 0.33/0.68 thf(fact_9253_not__mask__negative__int,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N2 ) @ zero_zero_int ) ).
% 0.33/0.68
% 0.33/0.68 % not_mask_negative_int
% 0.33/0.68 thf(fact_9254_real__root__less__mono,axiom,
% 0.33/0.68 ! [N2: nat,X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ord_less_real @ X @ Y )
% 0.33/0.68 => ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_less_mono
% 0.33/0.68 thf(fact_9255_real__root__le__mono,axiom,
% 0.33/0.68 ! [N2: nat,X: real,Y: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ord_less_eq_real @ X @ Y )
% 0.33/0.68 => ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_le_mono
% 0.33/0.68 thf(fact_9256_real__root__power,axiom,
% 0.33/0.68 ! [N2: nat,X: real,K: nat] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( root @ N2 @ ( power_power_real @ X @ K ) )
% 0.33/0.68 = ( power_power_real @ ( root @ N2 @ X ) @ K ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_power
% 0.33/0.68 thf(fact_9257_real__root__abs,axiom,
% 0.33/0.68 ! [N2: nat,X: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( root @ N2 @ ( abs_abs_real @ X ) )
% 0.33/0.68 = ( abs_abs_real @ ( root @ N2 @ X ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_abs
% 0.33/0.68 thf(fact_9258_nat__mono__iff,axiom,
% 0.33/0.68 ! [Z: int,W: int] :
% 0.33/0.68 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.33/0.68 => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.33/0.68 = ( ord_less_int @ W @ Z ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_mono_iff
% 0.33/0.68 thf(fact_9259_zless__nat__eq__int__zless,axiom,
% 0.33/0.68 ! [M: nat,Z: int] :
% 0.33/0.68 ( ( ord_less_nat @ M @ ( nat2 @ Z ) )
% 0.33/0.68 = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).
% 0.33/0.68
% 0.33/0.68 % zless_nat_eq_int_zless
% 0.33/0.68 thf(fact_9260_nat__le__iff,axiom,
% 0.33/0.68 ! [X: int,N2: nat] :
% 0.33/0.68 ( ( ord_less_eq_nat @ ( nat2 @ X ) @ N2 )
% 0.33/0.68 = ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_le_iff
% 0.33/0.68 thf(fact_9261_nat__int__add,axiom,
% 0.33/0.68 ! [A: nat,B: nat] :
% 0.33/0.68 ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
% 0.33/0.68 = ( plus_plus_nat @ A @ B ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_int_add
% 0.33/0.68 thf(fact_9262_nat__abs__mult__distrib,axiom,
% 0.33/0.68 ! [W: int,Z: int] :
% 0.33/0.68 ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z ) ) )
% 0.33/0.68 = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_abs_mult_distrib
% 0.33/0.68 thf(fact_9263_int__minus,axiom,
% 0.33/0.68 ! [N2: nat,M: nat] :
% 0.33/0.68 ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ M ) )
% 0.33/0.68 = ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % int_minus
% 0.33/0.68 thf(fact_9264_nat__plus__as__int,axiom,
% 0.33/0.68 ( plus_plus_nat
% 0.33/0.68 = ( ^ [A5: nat,B5: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_plus_as_int
% 0.33/0.68 thf(fact_9265_nat__times__as__int,axiom,
% 0.33/0.68 ( times_times_nat
% 0.33/0.68 = ( ^ [A5: nat,B5: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_times_as_int
% 0.33/0.68 thf(fact_9266_real__nat__ceiling__ge,axiom,
% 0.33/0.68 ! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_nat_ceiling_ge
% 0.33/0.68 thf(fact_9267_less__mask,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 0.33/0.68 => ( ord_less_nat @ N2 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % less_mask
% 0.33/0.68 thf(fact_9268_nat__minus__as__int,axiom,
% 0.33/0.68 ( minus_minus_nat
% 0.33/0.68 = ( ^ [A5: nat,B5: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_minus_as_int
% 0.33/0.68 thf(fact_9269_nat__div__as__int,axiom,
% 0.33/0.68 ( divide_divide_nat
% 0.33/0.68 = ( ^ [A5: nat,B5: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_div_as_int
% 0.33/0.68 thf(fact_9270_nat__mod__as__int,axiom,
% 0.33/0.68 ( modulo_modulo_nat
% 0.33/0.68 = ( ^ [A5: nat,B5: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_mod_as_int
% 0.33/0.68 thf(fact_9271_real__root__gt__zero,axiom,
% 0.33/0.68 ! [N2: nat,X: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ord_less_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_gt_zero
% 0.33/0.68 thf(fact_9272_real__root__strict__decreasing,axiom,
% 0.33/0.68 ! [N2: nat,N3: nat,X: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ord_less_nat @ N2 @ N3 )
% 0.33/0.68 => ( ( ord_less_real @ one_one_real @ X )
% 0.33/0.68 => ( ord_less_real @ ( root @ N3 @ X ) @ ( root @ N2 @ X ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_strict_decreasing
% 0.33/0.68 thf(fact_9273_sqrt__def,axiom,
% 0.33/0.68 ( sqrt
% 0.33/0.68 = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % sqrt_def
% 0.33/0.68 thf(fact_9274_root__abs__power,axiom,
% 0.33/0.68 ! [N2: nat,Y: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( abs_abs_real @ ( root @ N2 @ ( power_power_real @ Y @ N2 ) ) )
% 0.33/0.68 = ( abs_abs_real @ Y ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % root_abs_power
% 0.33/0.68 thf(fact_9275_nat__less__eq__zless,axiom,
% 0.33/0.68 ! [W: int,Z: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.33/0.68 => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.33/0.68 = ( ord_less_int @ W @ Z ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_less_eq_zless
% 0.33/0.68 thf(fact_9276_nat__le__eq__zle,axiom,
% 0.33/0.68 ! [W: int,Z: int] :
% 0.33/0.68 ( ( ( ord_less_int @ zero_zero_int @ W )
% 0.33/0.68 | ( ord_less_eq_int @ zero_zero_int @ Z ) )
% 0.33/0.68 => ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.33/0.68 = ( ord_less_eq_int @ W @ Z ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_le_eq_zle
% 0.33/0.68 thf(fact_9277_split__nat,axiom,
% 0.33/0.68 ! [P: nat > $o,I2: int] :
% 0.33/0.68 ( ( P @ ( nat2 @ I2 ) )
% 0.33/0.68 = ( ! [N: nat] :
% 0.33/0.68 ( ( I2
% 0.33/0.68 = ( semiri1314217659103216013at_int @ N ) )
% 0.33/0.68 => ( P @ N ) )
% 0.33/0.68 & ( ( ord_less_int @ I2 @ zero_zero_int )
% 0.33/0.68 => ( P @ zero_zero_nat ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % split_nat
% 0.33/0.68 thf(fact_9278_le__nat__iff,axiom,
% 0.33/0.68 ! [K: int,N2: nat] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.33/0.68 => ( ( ord_less_eq_nat @ N2 @ ( nat2 @ K ) )
% 0.33/0.68 = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ K ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % le_nat_iff
% 0.33/0.68 thf(fact_9279_nat__add__distrib,axiom,
% 0.33/0.68 ! [Z: int,Z7: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.33/0.68 => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
% 0.33/0.68 => ( ( nat2 @ ( plus_plus_int @ Z @ Z7 ) )
% 0.33/0.68 = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_add_distrib
% 0.33/0.68 thf(fact_9280_nat__mult__distrib,axiom,
% 0.33/0.68 ! [Z: int,Z7: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.33/0.68 => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
% 0.33/0.68 = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_mult_distrib
% 0.33/0.68 thf(fact_9281_Suc__as__int,axiom,
% 0.33/0.68 ( suc
% 0.33/0.68 = ( ^ [A5: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ one_one_int ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Suc_as_int
% 0.33/0.68 thf(fact_9282_nat__diff__distrib,axiom,
% 0.33/0.68 ! [Z7: int,Z: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
% 0.33/0.68 => ( ( ord_less_eq_int @ Z7 @ Z )
% 0.33/0.68 => ( ( nat2 @ ( minus_minus_int @ Z @ Z7 ) )
% 0.33/0.68 = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_diff_distrib
% 0.33/0.68 thf(fact_9283_nat__diff__distrib_H,axiom,
% 0.33/0.68 ! [X: int,Y: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.68 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.68 => ( ( nat2 @ ( minus_minus_int @ X @ Y ) )
% 0.33/0.68 = ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_diff_distrib'
% 0.33/0.68 thf(fact_9284_nat__abs__triangle__ineq,axiom,
% 0.33/0.68 ! [K: int,L2: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L2 ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_abs_triangle_ineq
% 0.33/0.68 thf(fact_9285_nat__div__distrib_H,axiom,
% 0.33/0.68 ! [Y: int,X: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.68 => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
% 0.33/0.68 = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_div_distrib'
% 0.33/0.68 thf(fact_9286_nat__div__distrib,axiom,
% 0.33/0.68 ! [X: int,Y: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.68 => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
% 0.33/0.68 = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_div_distrib
% 0.33/0.68 thf(fact_9287_nat__power__eq,axiom,
% 0.33/0.68 ! [Z: int,N2: nat] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.33/0.68 => ( ( nat2 @ ( power_power_int @ Z @ N2 ) )
% 0.33/0.68 = ( power_power_nat @ ( nat2 @ Z ) @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_power_eq
% 0.33/0.68 thf(fact_9288_nat__floor__neg,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.33/0.68 => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 0.33/0.68 = zero_zero_nat ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_floor_neg
% 0.33/0.68 thf(fact_9289_div__abs__eq__div__nat,axiom,
% 0.33/0.68 ! [K: int,L2: int] :
% 0.33/0.68 ( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) )
% 0.33/0.68 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % div_abs_eq_div_nat
% 0.33/0.68 thf(fact_9290_nat__mod__distrib,axiom,
% 0.33/0.68 ! [X: int,Y: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.68 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.68 => ( ( nat2 @ ( modulo_modulo_int @ X @ Y ) )
% 0.33/0.68 = ( modulo_modulo_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_mod_distrib
% 0.33/0.68 thf(fact_9291_floor__eq3,axiom,
% 0.33/0.68 ! [N2: nat,X: real] :
% 0.33/0.68 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ X )
% 0.33/0.68 => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
% 0.33/0.68 => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 0.33/0.68 = N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % floor_eq3
% 0.33/0.68 thf(fact_9292_le__nat__floor,axiom,
% 0.33/0.68 ! [X: nat,A: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A )
% 0.33/0.68 => ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % le_nat_floor
% 0.33/0.68 thf(fact_9293_real__root__pos__pos,axiom,
% 0.33/0.68 ! [N2: nat,X: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_pos_pos
% 0.33/0.68 thf(fact_9294_real__root__strict__increasing,axiom,
% 0.33/0.68 ! [N2: nat,N3: nat,X: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ord_less_nat @ N2 @ N3 )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_real @ X @ one_one_real )
% 0.33/0.68 => ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N3 @ X ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_strict_increasing
% 0.33/0.68 thf(fact_9295_real__root__decreasing,axiom,
% 0.33/0.68 ! [N2: nat,N3: nat,X: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ord_less_eq_nat @ N2 @ N3 )
% 0.33/0.68 => ( ( ord_less_eq_real @ one_one_real @ X )
% 0.33/0.68 => ( ord_less_eq_real @ ( root @ N3 @ X ) @ ( root @ N2 @ X ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_decreasing
% 0.33/0.68 thf(fact_9296_real__root__pow__pos,axiom,
% 0.33/0.68 ! [N2: nat,X: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( power_power_real @ ( root @ N2 @ X ) @ N2 )
% 0.33/0.68 = X ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_pow_pos
% 0.33/0.68 thf(fact_9297_odd__real__root__power__cancel,axiom,
% 0.33/0.68 ! [N2: nat,X: real] :
% 0.33/0.68 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.68 => ( ( root @ N2 @ ( power_power_real @ X @ N2 ) )
% 0.33/0.68 = X ) ) ).
% 0.33/0.68
% 0.33/0.68 % odd_real_root_power_cancel
% 0.33/0.68 thf(fact_9298_odd__real__root__unique,axiom,
% 0.33/0.68 ! [N2: nat,Y: real,X: real] :
% 0.33/0.68 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.68 => ( ( ( power_power_real @ Y @ N2 )
% 0.33/0.68 = X )
% 0.33/0.68 => ( ( root @ N2 @ X )
% 0.33/0.68 = Y ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % odd_real_root_unique
% 0.33/0.68 thf(fact_9299_odd__real__root__pow,axiom,
% 0.33/0.68 ! [N2: nat,X: real] :
% 0.33/0.68 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.68 => ( ( power_power_real @ ( root @ N2 @ X ) @ N2 )
% 0.33/0.68 = X ) ) ).
% 0.33/0.68
% 0.33/0.68 % odd_real_root_pow
% 0.33/0.68 thf(fact_9300_real__root__power__cancel,axiom,
% 0.33/0.68 ! [N2: nat,X: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( root @ N2 @ ( power_power_real @ X @ N2 ) )
% 0.33/0.68 = X ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_power_cancel
% 0.33/0.68 thf(fact_9301_real__root__pos__unique,axiom,
% 0.33/0.68 ! [N2: nat,Y: real,X: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 0.33/0.68 => ( ( ( power_power_real @ Y @ N2 )
% 0.33/0.68 = X )
% 0.33/0.68 => ( ( root @ N2 @ X )
% 0.33/0.68 = Y ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_pos_unique
% 0.33/0.68 thf(fact_9302_nat__2,axiom,
% 0.33/0.68 ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.68 = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_2
% 0.33/0.68 thf(fact_9303_sgn__power__root,axiom,
% 0.33/0.68 ! [N2: nat,X: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N2 @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N2 @ X ) ) @ N2 ) )
% 0.33/0.68 = X ) ) ).
% 0.33/0.68
% 0.33/0.68 % sgn_power_root
% 0.33/0.68 thf(fact_9304_root__sgn__power,axiom,
% 0.33/0.68 ! [N2: nat,Y: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( root @ N2 @ ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) ) )
% 0.33/0.68 = Y ) ) ).
% 0.33/0.68
% 0.33/0.68 % root_sgn_power
% 0.33/0.68 thf(fact_9305_Suc__nat__eq__nat__zadd1,axiom,
% 0.33/0.68 ! [Z: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.33/0.68 => ( ( suc @ ( nat2 @ Z ) )
% 0.33/0.68 = ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Suc_nat_eq_nat_zadd1
% 0.33/0.68 thf(fact_9306_nat__less__iff,axiom,
% 0.33/0.68 ! [W: int,M: nat] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.33/0.68 => ( ( ord_less_nat @ ( nat2 @ W ) @ M )
% 0.33/0.68 = ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_less_iff
% 0.33/0.68 thf(fact_9307_nat__mult__distrib__neg,axiom,
% 0.33/0.68 ! [Z: int,Z7: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 0.33/0.68 => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
% 0.33/0.68 = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z7 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_mult_distrib_neg
% 0.33/0.68 thf(fact_9308_nat__abs__int__diff,axiom,
% 0.33/0.68 ! [A: nat,B: nat] :
% 0.33/0.68 ( ( ( ord_less_eq_nat @ A @ B )
% 0.33/0.68 => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 0.33/0.68 = ( minus_minus_nat @ B @ A ) ) )
% 0.33/0.68 & ( ~ ( ord_less_eq_nat @ A @ B )
% 0.33/0.68 => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 0.33/0.68 = ( minus_minus_nat @ A @ B ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_abs_int_diff
% 0.33/0.68 thf(fact_9309_floor__eq4,axiom,
% 0.33/0.68 ! [N2: nat,X: real] :
% 0.33/0.68 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ X )
% 0.33/0.68 => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
% 0.33/0.68 => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 0.33/0.68 = N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % floor_eq4
% 0.33/0.68 thf(fact_9310_diff__nat__eq__if,axiom,
% 0.33/0.68 ! [Z7: int,Z: int] :
% 0.33/0.68 ( ( ( ord_less_int @ Z7 @ zero_zero_int )
% 0.33/0.68 => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
% 0.33/0.68 = ( nat2 @ Z ) ) )
% 0.33/0.68 & ( ~ ( ord_less_int @ Z7 @ zero_zero_int )
% 0.33/0.68 => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
% 0.33/0.68 = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z7 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z7 ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % diff_nat_eq_if
% 0.33/0.68 thf(fact_9311_real__root__increasing,axiom,
% 0.33/0.68 ! [N2: nat,N3: nat,X: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ord_less_eq_nat @ N2 @ N3 )
% 0.33/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.68 => ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N3 @ X ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % real_root_increasing
% 0.33/0.68 thf(fact_9312_Suc__mask__eq__exp,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( suc @ ( bit_se2002935070580805687sk_nat @ N2 ) )
% 0.33/0.68 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % Suc_mask_eq_exp
% 0.33/0.68 thf(fact_9313_mask__nat__less__exp,axiom,
% 0.33/0.68 ! [N2: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % mask_nat_less_exp
% 0.33/0.68 thf(fact_9314_split__root,axiom,
% 0.33/0.68 ! [P: real > $o,N2: nat,X: real] :
% 0.33/0.68 ( ( P @ ( root @ N2 @ X ) )
% 0.33/0.68 = ( ( ( N2 = zero_zero_nat )
% 0.33/0.68 => ( P @ zero_zero_real ) )
% 0.33/0.68 & ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ! [Y2: real] :
% 0.33/0.68 ( ( ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N2 ) )
% 0.33/0.68 = X )
% 0.33/0.68 => ( P @ Y2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % split_root
% 0.33/0.68 thf(fact_9315_ln__root,axiom,
% 0.33/0.68 ! [N2: nat,B: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.33/0.68 => ( ( ln_ln_real @ ( root @ N2 @ B ) )
% 0.33/0.68 = ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % ln_root
% 0.33/0.68 thf(fact_9316_log__root,axiom,
% 0.33/0.68 ! [N2: nat,A: real,B: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.33/0.68 => ( ( log @ B @ ( root @ N2 @ A ) )
% 0.33/0.68 = ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_root
% 0.33/0.68 thf(fact_9317_log__base__root,axiom,
% 0.33/0.68 ! [N2: nat,B: real,X: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.33/0.68 => ( ( log @ ( root @ N2 @ B ) @ X )
% 0.33/0.68 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ X ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % log_base_root
% 0.33/0.68 thf(fact_9318_mask__nat__def,axiom,
% 0.33/0.68 ( bit_se2002935070580805687sk_nat
% 0.33/0.68 = ( ^ [N: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % mask_nat_def
% 0.33/0.68 thf(fact_9319_mask__half__int,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.68 = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % mask_half_int
% 0.33/0.68 thf(fact_9320_mask__int__def,axiom,
% 0.33/0.68 ( bit_se2000444600071755411sk_int
% 0.33/0.68 = ( ^ [N: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % mask_int_def
% 0.33/0.68 thf(fact_9321_root__powr__inverse,axiom,
% 0.33/0.68 ! [N2: nat,X: real] :
% 0.33/0.68 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( root @ N2 @ X )
% 0.33/0.68 = ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % root_powr_inverse
% 0.33/0.68 thf(fact_9322_even__nat__iff,axiom,
% 0.33/0.68 ! [K: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.33/0.68 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K ) )
% 0.33/0.68 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % even_nat_iff
% 0.33/0.68 thf(fact_9323_powr__real__of__int,axiom,
% 0.33/0.68 ! [X: real,N2: int] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ( ord_less_eq_int @ zero_zero_int @ N2 )
% 0.33/0.68 => ( ( powr_real @ X @ ( ring_1_of_int_real @ N2 ) )
% 0.33/0.68 = ( power_power_real @ X @ ( nat2 @ N2 ) ) ) )
% 0.33/0.68 & ( ~ ( ord_less_eq_int @ zero_zero_int @ N2 )
% 0.33/0.68 => ( ( powr_real @ X @ ( ring_1_of_int_real @ N2 ) )
% 0.33/0.68 = ( inverse_inverse_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ N2 ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_real_of_int
% 0.33/0.68 thf(fact_9324_powr__int,axiom,
% 0.33/0.68 ! [X: real,I2: int] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.68 => ( ( ( ord_less_eq_int @ zero_zero_int @ I2 )
% 0.33/0.68 => ( ( powr_real @ X @ ( ring_1_of_int_real @ I2 ) )
% 0.33/0.68 = ( power_power_real @ X @ ( nat2 @ I2 ) ) ) )
% 0.33/0.68 & ( ~ ( ord_less_eq_int @ zero_zero_int @ I2 )
% 0.33/0.68 => ( ( powr_real @ X @ ( ring_1_of_int_real @ I2 ) )
% 0.33/0.68 = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ I2 ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % powr_int
% 0.33/0.68 thf(fact_9325_floor__rat__def,axiom,
% 0.33/0.68 ( archim3151403230148437115or_rat
% 0.33/0.68 = ( ^ [X2: rat] :
% 0.33/0.68 ( the_int
% 0.33/0.68 @ ^ [Z3: int] :
% 0.33/0.68 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X2 )
% 0.33/0.68 & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % floor_rat_def
% 0.33/0.68 thf(fact_9326_Arg__def,axiom,
% 0.33/0.68 ( arg
% 0.33/0.68 = ( ^ [Z3: complex] :
% 0.33/0.68 ( if_real @ ( Z3 = zero_zero_complex ) @ zero_zero_real
% 0.33/0.68 @ ( fChoice_real
% 0.33/0.68 @ ^ [A5: real] :
% 0.33/0.68 ( ( ( sgn_sgn_complex @ Z3 )
% 0.33/0.68 = ( cis @ A5 ) )
% 0.33/0.68 & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A5 )
% 0.33/0.68 & ( ord_less_eq_real @ A5 @ pi ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Arg_def
% 0.33/0.68 thf(fact_9327_num_Osize__gen_I3_J,axiom,
% 0.33/0.68 ! [X32: num] :
% 0.33/0.68 ( ( size_num @ ( bit1 @ X32 ) )
% 0.33/0.68 = ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % num.size_gen(3)
% 0.33/0.68 thf(fact_9328_concat__bit__of__zero__2,axiom,
% 0.33/0.68 ! [N2: nat,K: int] :
% 0.33/0.68 ( ( bit_concat_bit @ N2 @ K @ zero_zero_int )
% 0.33/0.68 = ( bit_se2923211474154528505it_int @ N2 @ K ) ) ).
% 0.33/0.68
% 0.33/0.68 % concat_bit_of_zero_2
% 0.33/0.68 thf(fact_9329_take__bit__of__Suc__0,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( bit_se2925701944663578781it_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 0.33/0.68 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_of_Suc_0
% 0.33/0.68 thf(fact_9330_sgn__rat__def,axiom,
% 0.33/0.68 ( sgn_sgn_rat
% 0.33/0.68 = ( ^ [A5: rat] : ( if_rat @ ( A5 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A5 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % sgn_rat_def
% 0.33/0.68 thf(fact_9331_take__bit__nat__eq,axiom,
% 0.33/0.68 ! [K: int,N2: nat] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.33/0.68 => ( ( bit_se2925701944663578781it_nat @ N2 @ ( nat2 @ K ) )
% 0.33/0.68 = ( nat2 @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_nat_eq
% 0.33/0.68 thf(fact_9332_nat__take__bit__eq,axiom,
% 0.33/0.68 ! [K: int,N2: nat] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.33/0.68 => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N2 @ K ) )
% 0.33/0.68 = ( bit_se2925701944663578781it_nat @ N2 @ ( nat2 @ K ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % nat_take_bit_eq
% 0.33/0.68 thf(fact_9333_take__bit__tightened__less__eq__nat,axiom,
% 0.33/0.68 ! [M: nat,N2: nat,Q2: nat] :
% 0.33/0.68 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.68 => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q2 ) @ ( bit_se2925701944663578781it_nat @ N2 @ Q2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_tightened_less_eq_nat
% 0.33/0.68 thf(fact_9334_take__bit__nat__less__eq__self,axiom,
% 0.33/0.68 ! [N2: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ M ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_nat_less_eq_self
% 0.33/0.68 thf(fact_9335_less__eq__rat__def,axiom,
% 0.33/0.68 ( ord_less_eq_rat
% 0.33/0.68 = ( ^ [X2: rat,Y2: rat] :
% 0.33/0.68 ( ( ord_less_rat @ X2 @ Y2 )
% 0.33/0.68 | ( X2 = Y2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % less_eq_rat_def
% 0.33/0.68 thf(fact_9336_obtain__pos__sum,axiom,
% 0.33/0.68 ! [R: rat] :
% 0.33/0.68 ( ( ord_less_rat @ zero_zero_rat @ R )
% 0.33/0.68 => ~ ! [S2: rat] :
% 0.33/0.68 ( ( ord_less_rat @ zero_zero_rat @ S2 )
% 0.33/0.68 => ! [T5: rat] :
% 0.33/0.68 ( ( ord_less_rat @ zero_zero_rat @ T5 )
% 0.33/0.68 => ( R
% 0.33/0.68 != ( plus_plus_rat @ S2 @ T5 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % obtain_pos_sum
% 0.33/0.68 thf(fact_9337_take__bit__minus,axiom,
% 0.33/0.68 ! [N2: nat,K: int] :
% 0.33/0.68 ( ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) )
% 0.33/0.68 = ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ K ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_minus
% 0.33/0.68 thf(fact_9338_take__bit__mult,axiom,
% 0.33/0.68 ! [N2: nat,K: int,L2: int] :
% 0.33/0.68 ( ( bit_se2923211474154528505it_int @ N2 @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( bit_se2923211474154528505it_int @ N2 @ L2 ) ) )
% 0.33/0.68 = ( bit_se2923211474154528505it_int @ N2 @ ( times_times_int @ K @ L2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_mult
% 0.33/0.68 thf(fact_9339_abs__rat__def,axiom,
% 0.33/0.68 ( abs_abs_rat
% 0.33/0.68 = ( ^ [A5: rat] : ( if_rat @ ( ord_less_rat @ A5 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A5 ) @ A5 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % abs_rat_def
% 0.33/0.68 thf(fact_9340_take__bit__diff,axiom,
% 0.33/0.68 ! [N2: nat,K: int,L2: int] :
% 0.33/0.68 ( ( bit_se2923211474154528505it_int @ N2 @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( bit_se2923211474154528505it_int @ N2 @ L2 ) ) )
% 0.33/0.68 = ( bit_se2923211474154528505it_int @ N2 @ ( minus_minus_int @ K @ L2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_diff
% 0.33/0.68 thf(fact_9341_concat__bit__take__bit__eq,axiom,
% 0.33/0.68 ! [N2: nat,B: int] :
% 0.33/0.68 ( ( bit_concat_bit @ N2 @ ( bit_se2923211474154528505it_int @ N2 @ B ) )
% 0.33/0.68 = ( bit_concat_bit @ N2 @ B ) ) ).
% 0.33/0.68
% 0.33/0.68 % concat_bit_take_bit_eq
% 0.33/0.68 thf(fact_9342_concat__bit__eq__iff,axiom,
% 0.33/0.68 ! [N2: nat,K: int,L2: int,R: int,S3: int] :
% 0.33/0.68 ( ( ( bit_concat_bit @ N2 @ K @ L2 )
% 0.33/0.68 = ( bit_concat_bit @ N2 @ R @ S3 ) )
% 0.33/0.68 = ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
% 0.33/0.68 = ( bit_se2923211474154528505it_int @ N2 @ R ) )
% 0.33/0.68 & ( L2 = S3 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % concat_bit_eq_iff
% 0.33/0.68 thf(fact_9343_take__bit__tightened__less__eq__int,axiom,
% 0.33/0.68 ! [M: nat,N2: nat,K: int] :
% 0.33/0.68 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.68 => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_tightened_less_eq_int
% 0.33/0.68 thf(fact_9344_take__bit__int__less__eq__self__iff,axiom,
% 0.33/0.68 ! [N2: nat,K: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ K )
% 0.33/0.68 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_int_less_eq_self_iff
% 0.33/0.68 thf(fact_9345_take__bit__nonnegative,axiom,
% 0.33/0.68 ! [N2: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_nonnegative
% 0.33/0.68 thf(fact_9346_not__take__bit__negative,axiom,
% 0.33/0.68 ! [N2: nat,K: int] :
% 0.33/0.68 ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ zero_zero_int ) ).
% 0.33/0.68
% 0.33/0.68 % not_take_bit_negative
% 0.33/0.68 thf(fact_9347_take__bit__int__greater__self__iff,axiom,
% 0.33/0.68 ! [K: int,N2: nat] :
% 0.33/0.68 ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N2 @ K ) )
% 0.33/0.68 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_int_greater_self_iff
% 0.33/0.68 thf(fact_9348_take__bit__eq__mask__iff,axiom,
% 0.33/0.68 ! [N2: nat,K: int] :
% 0.33/0.68 ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
% 0.33/0.68 = ( bit_se2000444600071755411sk_int @ N2 ) )
% 0.33/0.68 = ( ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ K @ one_one_int ) )
% 0.33/0.68 = zero_zero_int ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_eq_mask_iff
% 0.33/0.68 thf(fact_9349_take__bit__decr__eq,axiom,
% 0.33/0.68 ! [N2: nat,K: int] :
% 0.33/0.68 ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
% 0.33/0.68 != zero_zero_int )
% 0.33/0.68 => ( ( bit_se2923211474154528505it_int @ N2 @ ( minus_minus_int @ K @ one_one_int ) )
% 0.33/0.68 = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ one_one_int ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_decr_eq
% 0.33/0.68 thf(fact_9350_take__bit__nat__eq__self,axiom,
% 0.33/0.68 ! [M: nat,N2: nat] :
% 0.33/0.68 ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.33/0.68 => ( ( bit_se2925701944663578781it_nat @ N2 @ M )
% 0.33/0.68 = M ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_nat_eq_self
% 0.33/0.68 thf(fact_9351_take__bit__nat__less__exp,axiom,
% 0.33/0.68 ! [N2: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_nat_less_exp
% 0.33/0.68 thf(fact_9352_take__bit__nat__eq__self__iff,axiom,
% 0.33/0.68 ! [N2: nat,M: nat] :
% 0.33/0.68 ( ( ( bit_se2925701944663578781it_nat @ N2 @ M )
% 0.33/0.68 = M )
% 0.33/0.68 = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_nat_eq_self_iff
% 0.33/0.68 thf(fact_9353_take__bit__nat__def,axiom,
% 0.33/0.68 ( bit_se2925701944663578781it_nat
% 0.33/0.68 = ( ^ [N: nat,M6: nat] : ( modulo_modulo_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_nat_def
% 0.33/0.68 thf(fact_9354_take__bit__int__less__exp,axiom,
% 0.33/0.68 ! [N2: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_int_less_exp
% 0.33/0.68 thf(fact_9355_take__bit__int__def,axiom,
% 0.33/0.68 ( bit_se2923211474154528505it_int
% 0.33/0.68 = ( ^ [N: nat,K3: int] : ( modulo_modulo_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_int_def
% 0.33/0.68 thf(fact_9356_num_Osize__gen_I1_J,axiom,
% 0.33/0.68 ( ( size_num @ one )
% 0.33/0.68 = zero_zero_nat ) ).
% 0.33/0.68
% 0.33/0.68 % num.size_gen(1)
% 0.33/0.68 thf(fact_9357_take__bit__nat__less__self__iff,axiom,
% 0.33/0.68 ! [N2: nat,M: nat] :
% 0.33/0.68 ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ M )
% 0.33/0.68 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_nat_less_self_iff
% 0.33/0.68 thf(fact_9358_take__bit__Suc__minus__bit0,axiom,
% 0.33/0.68 ! [N2: nat,K: num] :
% 0.33/0.68 ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.33/0.68 = ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_Suc_minus_bit0
% 0.33/0.68 thf(fact_9359_take__bit__int__greater__eq__self__iff,axiom,
% 0.33/0.68 ! [K: int,N2: nat] :
% 0.33/0.68 ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N2 @ K ) )
% 0.33/0.68 = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_int_greater_eq_self_iff
% 0.33/0.68 thf(fact_9360_take__bit__int__less__self__iff,axiom,
% 0.33/0.68 ! [N2: nat,K: int] :
% 0.33/0.68 ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ K )
% 0.33/0.68 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_int_less_self_iff
% 0.33/0.68 thf(fact_9361_take__bit__int__eq__self__iff,axiom,
% 0.33/0.68 ! [N2: nat,K: int] :
% 0.33/0.68 ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
% 0.33/0.68 = K )
% 0.33/0.68 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.33/0.68 & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_int_eq_self_iff
% 0.33/0.68 thf(fact_9362_take__bit__int__eq__self,axiom,
% 0.33/0.68 ! [K: int,N2: nat] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.33/0.68 => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.33/0.68 => ( ( bit_se2923211474154528505it_int @ N2 @ K )
% 0.33/0.68 = K ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_int_eq_self
% 0.33/0.68 thf(fact_9363_take__bit__numeral__minus__bit0,axiom,
% 0.33/0.68 ! [L2: num,K: num] :
% 0.33/0.68 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.33/0.68 = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_numeral_minus_bit0
% 0.33/0.68 thf(fact_9364_take__bit__incr__eq,axiom,
% 0.33/0.68 ! [N2: nat,K: int] :
% 0.33/0.68 ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
% 0.33/0.68 != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) )
% 0.33/0.68 => ( ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ K @ one_one_int ) )
% 0.33/0.68 = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_incr_eq
% 0.33/0.68 thf(fact_9365_take__bit__eq__mask__iff__exp__dvd,axiom,
% 0.33/0.68 ! [N2: nat,K: int] :
% 0.33/0.68 ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
% 0.33/0.68 = ( bit_se2000444600071755411sk_int @ N2 ) )
% 0.33/0.68 = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_eq_mask_iff_exp_dvd
% 0.33/0.68 thf(fact_9366_take__bit__int__less__eq,axiom,
% 0.33/0.68 ! [N2: nat,K: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K )
% 0.33/0.68 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.68 => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_int_less_eq
% 0.33/0.68 thf(fact_9367_take__bit__int__greater__eq,axiom,
% 0.33/0.68 ! [K: int,N2: nat] :
% 0.33/0.68 ( ( ord_less_int @ K @ zero_zero_int )
% 0.33/0.68 => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_int_greater_eq
% 0.33/0.68 thf(fact_9368_signed__take__bit__eq__take__bit__shift,axiom,
% 0.33/0.68 ( bit_ri631733984087533419it_int
% 0.33/0.68 = ( ^ [N: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( plus_plus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % signed_take_bit_eq_take_bit_shift
% 0.33/0.68 thf(fact_9369_take__bit__minus__small__eq,axiom,
% 0.33/0.68 ! [K: int,N2: nat] :
% 0.33/0.68 ( ( ord_less_int @ zero_zero_int @ K )
% 0.33/0.68 => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.33/0.68 => ( ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ K ) )
% 0.33/0.68 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_minus_small_eq
% 0.33/0.68 thf(fact_9370_num_Osize__gen_I2_J,axiom,
% 0.33/0.68 ! [X22: num] :
% 0.33/0.68 ( ( size_num @ ( bit0 @ X22 ) )
% 0.33/0.68 = ( plus_plus_nat @ ( size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % num.size_gen(2)
% 0.33/0.68 thf(fact_9371_take__bit__numeral__minus__bit1,axiom,
% 0.33/0.68 ! [L2: num,K: num] :
% 0.33/0.68 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.33/0.68 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_numeral_minus_bit1
% 0.33/0.68 thf(fact_9372_pred__numeral__inc,axiom,
% 0.33/0.68 ! [K: num] :
% 0.33/0.68 ( ( pred_numeral @ ( inc @ K ) )
% 0.33/0.68 = ( numeral_numeral_nat @ K ) ) ).
% 0.33/0.68
% 0.33/0.68 % pred_numeral_inc
% 0.33/0.68 thf(fact_9373_diff__rat__def,axiom,
% 0.33/0.68 ( minus_minus_rat
% 0.33/0.68 = ( ^ [Q4: rat,R5: rat] : ( plus_plus_rat @ Q4 @ ( uminus_uminus_rat @ R5 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % diff_rat_def
% 0.33/0.68 thf(fact_9374_num__induct,axiom,
% 0.33/0.68 ! [P: num > $o,X: num] :
% 0.33/0.68 ( ( P @ one )
% 0.33/0.68 => ( ! [X4: num] :
% 0.33/0.68 ( ( P @ X4 )
% 0.33/0.68 => ( P @ ( inc @ X4 ) ) )
% 0.33/0.68 => ( P @ X ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % num_induct
% 0.33/0.68 thf(fact_9375_add__inc,axiom,
% 0.33/0.68 ! [X: num,Y: num] :
% 0.33/0.68 ( ( plus_plus_num @ X @ ( inc @ Y ) )
% 0.33/0.68 = ( inc @ ( plus_plus_num @ X @ Y ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % add_inc
% 0.33/0.68 thf(fact_9376_inc_Osimps_I1_J,axiom,
% 0.33/0.68 ( ( inc @ one )
% 0.33/0.68 = ( bit0 @ one ) ) ).
% 0.33/0.68
% 0.33/0.68 % inc.simps(1)
% 0.33/0.68 thf(fact_9377_inc_Osimps_I2_J,axiom,
% 0.33/0.68 ! [X: num] :
% 0.33/0.68 ( ( inc @ ( bit0 @ X ) )
% 0.33/0.68 = ( bit1 @ X ) ) ).
% 0.33/0.68
% 0.33/0.68 % inc.simps(2)
% 0.33/0.68 thf(fact_9378_inc_Osimps_I3_J,axiom,
% 0.33/0.68 ! [X: num] :
% 0.33/0.68 ( ( inc @ ( bit1 @ X ) )
% 0.33/0.68 = ( bit0 @ ( inc @ X ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % inc.simps(3)
% 0.33/0.68 thf(fact_9379_add__One,axiom,
% 0.33/0.68 ! [X: num] :
% 0.33/0.68 ( ( plus_plus_num @ X @ one )
% 0.33/0.68 = ( inc @ X ) ) ).
% 0.33/0.68
% 0.33/0.68 % add_One
% 0.33/0.68 thf(fact_9380_inc__BitM__eq,axiom,
% 0.33/0.68 ! [N2: num] :
% 0.33/0.68 ( ( inc @ ( bitM @ N2 ) )
% 0.33/0.68 = ( bit0 @ N2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % inc_BitM_eq
% 0.33/0.68 thf(fact_9381_BitM__inc__eq,axiom,
% 0.33/0.68 ! [N2: num] :
% 0.33/0.68 ( ( bitM @ ( inc @ N2 ) )
% 0.33/0.68 = ( bit1 @ N2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % BitM_inc_eq
% 0.33/0.68 thf(fact_9382_mult__inc,axiom,
% 0.33/0.68 ! [X: num,Y: num] :
% 0.33/0.68 ( ( times_times_num @ X @ ( inc @ Y ) )
% 0.33/0.68 = ( plus_plus_num @ ( times_times_num @ X @ Y ) @ X ) ) ).
% 0.33/0.68
% 0.33/0.68 % mult_inc
% 0.33/0.68 thf(fact_9383_take__bit__Suc__minus__bit1,axiom,
% 0.33/0.68 ! [N2: nat,K: num] :
% 0.33/0.68 ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.33/0.68 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_Suc_minus_bit1
% 0.33/0.68 thf(fact_9384_signed__take__bit__eq__take__bit__minus,axiom,
% 0.33/0.68 ( bit_ri631733984087533419it_int
% 0.33/0.68 = ( ^ [N: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K3 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % signed_take_bit_eq_take_bit_minus
% 0.33/0.68 thf(fact_9385_and__int__unfold,axiom,
% 0.33/0.68 ( bit_se725231765392027082nd_int
% 0.33/0.68 = ( ^ [K3: int,L: int] :
% 0.33/0.68 ( if_int
% 0.33/0.68 @ ( ( K3 = zero_zero_int )
% 0.33/0.68 | ( L = zero_zero_int ) )
% 0.33/0.68 @ zero_zero_int
% 0.33/0.68 @ ( if_int
% 0.33/0.68 @ ( K3
% 0.33/0.68 = ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.68 @ L
% 0.33/0.68 @ ( if_int
% 0.33/0.68 @ ( L
% 0.33/0.68 = ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.68 @ K3
% 0.33/0.68 @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % and_int_unfold
% 0.33/0.68 thf(fact_9386_and__nonnegative__int__iff,axiom,
% 0.33/0.68 ! [K: int,L2: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) )
% 0.33/0.68 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.33/0.68 | ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % and_nonnegative_int_iff
% 0.33/0.68 thf(fact_9387_and__negative__int__iff,axiom,
% 0.33/0.68 ! [K: int,L2: int] :
% 0.33/0.68 ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ zero_zero_int )
% 0.33/0.68 = ( ( ord_less_int @ K @ zero_zero_int )
% 0.33/0.68 & ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % and_negative_int_iff
% 0.33/0.68 thf(fact_9388_signed__take__bit__nonnegative__iff,axiom,
% 0.33/0.68 ! [N2: nat,K: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) )
% 0.33/0.68 = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % signed_take_bit_nonnegative_iff
% 0.33/0.68 thf(fact_9389_signed__take__bit__negative__iff,axiom,
% 0.33/0.68 ! [N2: nat,K: int] :
% 0.33/0.68 ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ zero_zero_int )
% 0.33/0.68 = ( bit_se1146084159140164899it_int @ K @ N2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % signed_take_bit_negative_iff
% 0.33/0.68 thf(fact_9390_bit__minus__numeral__Bit0__Suc__iff,axiom,
% 0.33/0.68 ! [W: num,N2: nat] :
% 0.33/0.68 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N2 ) )
% 0.33/0.68 = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % bit_minus_numeral_Bit0_Suc_iff
% 0.33/0.68 thf(fact_9391_bit__minus__numeral__Bit1__Suc__iff,axiom,
% 0.33/0.68 ! [W: num,N2: nat] :
% 0.33/0.68 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N2 ) )
% 0.33/0.68 = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % bit_minus_numeral_Bit1_Suc_iff
% 0.33/0.68 thf(fact_9392_and__minus__numerals_I2_J,axiom,
% 0.33/0.68 ! [N2: num] :
% 0.33/0.68 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
% 0.33/0.68 = one_one_int ) ).
% 0.33/0.68
% 0.33/0.68 % and_minus_numerals(2)
% 0.33/0.68 thf(fact_9393_and__minus__numerals_I6_J,axiom,
% 0.33/0.68 ! [N2: num] :
% 0.33/0.68 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) @ one_one_int )
% 0.33/0.68 = one_one_int ) ).
% 0.33/0.68
% 0.33/0.68 % and_minus_numerals(6)
% 0.33/0.68 thf(fact_9394_and__minus__numerals_I1_J,axiom,
% 0.33/0.68 ! [N2: num] :
% 0.33/0.68 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
% 0.33/0.68 = zero_zero_int ) ).
% 0.33/0.68
% 0.33/0.68 % and_minus_numerals(1)
% 0.33/0.68 thf(fact_9395_and__minus__numerals_I5_J,axiom,
% 0.33/0.68 ! [N2: num] :
% 0.33/0.68 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) @ one_one_int )
% 0.33/0.68 = zero_zero_int ) ).
% 0.33/0.68
% 0.33/0.68 % and_minus_numerals(5)
% 0.33/0.68 thf(fact_9396_bit__minus__numeral__int_I1_J,axiom,
% 0.33/0.68 ! [W: num,N2: num] :
% 0.33/0.68 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N2 ) )
% 0.33/0.68 = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % bit_minus_numeral_int(1)
% 0.33/0.68 thf(fact_9397_bit__minus__numeral__int_I2_J,axiom,
% 0.33/0.68 ! [W: num,N2: num] :
% 0.33/0.68 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N2 ) )
% 0.33/0.68 = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % bit_minus_numeral_int(2)
% 0.33/0.68 thf(fact_9398_bit__and__int__iff,axiom,
% 0.33/0.68 ! [K: int,L2: int,N2: nat] :
% 0.33/0.68 ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ N2 )
% 0.33/0.68 = ( ( bit_se1146084159140164899it_int @ K @ N2 )
% 0.33/0.68 & ( bit_se1146084159140164899it_int @ L2 @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % bit_and_int_iff
% 0.33/0.68 thf(fact_9399_AND__upper2_H,axiom,
% 0.33/0.68 ! [Y: int,Z: int,X: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.68 => ( ( ord_less_eq_int @ Y @ Z )
% 0.33/0.68 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % AND_upper2'
% 0.33/0.68 thf(fact_9400_AND__upper1_H,axiom,
% 0.33/0.68 ! [Y: int,Z: int,Ya: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.68 => ( ( ord_less_eq_int @ Y @ Z )
% 0.33/0.68 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % AND_upper1'
% 0.33/0.68 thf(fact_9401_AND__upper2,axiom,
% 0.33/0.68 ! [Y: int,X: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.68 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Y ) ) ).
% 0.33/0.68
% 0.33/0.68 % AND_upper2
% 0.33/0.68 thf(fact_9402_AND__upper1,axiom,
% 0.33/0.68 ! [X: int,Y: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.68 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ X ) ) ).
% 0.33/0.68
% 0.33/0.68 % AND_upper1
% 0.33/0.68 thf(fact_9403_AND__lower,axiom,
% 0.33/0.68 ! [X: int,Y: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.68 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X @ Y ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % AND_lower
% 0.33/0.68 thf(fact_9404_pow_Osimps_I1_J,axiom,
% 0.33/0.68 ! [X: num] :
% 0.33/0.68 ( ( pow @ X @ one )
% 0.33/0.68 = X ) ).
% 0.33/0.68
% 0.33/0.68 % pow.simps(1)
% 0.33/0.68 thf(fact_9405_AND__upper2_H_H,axiom,
% 0.33/0.68 ! [Y: int,Z: int,X: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.68 => ( ( ord_less_int @ Y @ Z )
% 0.33/0.68 => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % AND_upper2''
% 0.33/0.68 thf(fact_9406_AND__upper1_H_H,axiom,
% 0.33/0.68 ! [Y: int,Z: int,Ya: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.68 => ( ( ord_less_int @ Y @ Z )
% 0.33/0.68 => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % AND_upper1''
% 0.33/0.68 thf(fact_9407_and__less__eq,axiom,
% 0.33/0.68 ! [L2: int,K: int] :
% 0.33/0.68 ( ( ord_less_int @ L2 @ zero_zero_int )
% 0.33/0.68 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ K ) ) ).
% 0.33/0.68
% 0.33/0.68 % and_less_eq
% 0.33/0.68 thf(fact_9408_bit__not__int__iff_H,axiom,
% 0.33/0.68 ! [K: int,N2: nat] :
% 0.33/0.68 ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N2 )
% 0.33/0.68 = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % bit_not_int_iff'
% 0.33/0.68 thf(fact_9409_even__and__iff__int,axiom,
% 0.33/0.68 ! [K: int,L2: int] :
% 0.33/0.68 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L2 ) )
% 0.33/0.68 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.33/0.68 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % even_and_iff_int
% 0.33/0.68 thf(fact_9410_bit__imp__take__bit__positive,axiom,
% 0.33/0.68 ! [N2: nat,M: nat,K: int] :
% 0.33/0.68 ( ( ord_less_nat @ N2 @ M )
% 0.33/0.68 => ( ( bit_se1146084159140164899it_int @ K @ N2 )
% 0.33/0.68 => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % bit_imp_take_bit_positive
% 0.33/0.68 thf(fact_9411_bit__concat__bit__iff,axiom,
% 0.33/0.68 ! [M: nat,K: int,L2: int,N2: nat] :
% 0.33/0.68 ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L2 ) @ N2 )
% 0.33/0.68 = ( ( ( ord_less_nat @ N2 @ M )
% 0.33/0.68 & ( bit_se1146084159140164899it_int @ K @ N2 ) )
% 0.33/0.68 | ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.68 & ( bit_se1146084159140164899it_int @ L2 @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % bit_concat_bit_iff
% 0.33/0.68 thf(fact_9412_signed__take__bit__eq__concat__bit,axiom,
% 0.33/0.68 ( bit_ri631733984087533419it_int
% 0.33/0.68 = ( ^ [N: nat,K3: int] : ( bit_concat_bit @ N @ K3 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % signed_take_bit_eq_concat_bit
% 0.33/0.68 thf(fact_9413_int__bit__bound,axiom,
% 0.33/0.68 ! [K: int] :
% 0.33/0.68 ~ ! [N4: nat] :
% 0.33/0.68 ( ! [M5: nat] :
% 0.33/0.68 ( ( ord_less_eq_nat @ N4 @ M5 )
% 0.33/0.68 => ( ( bit_se1146084159140164899it_int @ K @ M5 )
% 0.33/0.68 = ( bit_se1146084159140164899it_int @ K @ N4 ) ) )
% 0.33/0.68 => ~ ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 0.33/0.68 => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N4 @ one_one_nat ) )
% 0.33/0.68 = ( ~ ( bit_se1146084159140164899it_int @ K @ N4 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % int_bit_bound
% 0.33/0.68 thf(fact_9414_bit__int__def,axiom,
% 0.33/0.68 ( bit_se1146084159140164899it_int
% 0.33/0.68 = ( ^ [K3: int,N: nat] :
% 0.33/0.68 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % bit_int_def
% 0.33/0.68 thf(fact_9415_and__int__rec,axiom,
% 0.33/0.68 ( bit_se725231765392027082nd_int
% 0.33/0.68 = ( ^ [K3: int,L: int] :
% 0.33/0.68 ( plus_plus_int
% 0.33/0.68 @ ( zero_n2684676970156552555ol_int
% 0.33/0.68 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 0.33/0.68 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 0.33/0.68 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % and_int_rec
% 0.33/0.68 thf(fact_9416_set__bit__eq,axiom,
% 0.33/0.68 ( bit_se7879613467334960850it_int
% 0.33/0.68 = ( ^ [N: nat,K3: int] :
% 0.33/0.68 ( plus_plus_int @ K3
% 0.33/0.68 @ ( times_times_int
% 0.33/0.68 @ ( zero_n2684676970156552555ol_int
% 0.33/0.68 @ ~ ( bit_se1146084159140164899it_int @ K3 @ N ) )
% 0.33/0.68 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % set_bit_eq
% 0.33/0.68 thf(fact_9417_unset__bit__eq,axiom,
% 0.33/0.68 ( bit_se4203085406695923979it_int
% 0.33/0.68 = ( ^ [N: nat,K3: int] : ( minus_minus_int @ K3 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % unset_bit_eq
% 0.33/0.68 thf(fact_9418_take__bit__Suc__from__most,axiom,
% 0.33/0.68 ! [N2: nat,K: int] :
% 0.33/0.68 ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ K )
% 0.33/0.68 = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % take_bit_Suc_from_most
% 0.33/0.68 thf(fact_9419_and__int_Oelims,axiom,
% 0.33/0.68 ! [X: int,Xa2: int,Y: int] :
% 0.33/0.68 ( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
% 0.33/0.68 = Y )
% 0.33/0.68 => ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.33/0.68 & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.33/0.68 => ( Y
% 0.33/0.68 = ( uminus_uminus_int
% 0.33/0.68 @ ( zero_n2684676970156552555ol_int
% 0.33/0.68 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 0.33/0.68 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
% 0.33/0.68 & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.33/0.68 & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.33/0.68 => ( Y
% 0.33/0.68 = ( plus_plus_int
% 0.33/0.68 @ ( zero_n2684676970156552555ol_int
% 0.33/0.68 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 0.33/0.68 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
% 0.33/0.68 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % and_int.elims
% 0.33/0.68 thf(fact_9420_and__int_Osimps,axiom,
% 0.33/0.68 ( bit_se725231765392027082nd_int
% 0.33/0.68 = ( ^ [K3: int,L: int] :
% 0.33/0.68 ( if_int
% 0.33/0.68 @ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.33/0.68 & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.33/0.68 @ ( uminus_uminus_int
% 0.33/0.68 @ ( zero_n2684676970156552555ol_int
% 0.33/0.68 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 0.33/0.68 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) )
% 0.33/0.68 @ ( plus_plus_int
% 0.33/0.68 @ ( zero_n2684676970156552555ol_int
% 0.33/0.68 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 0.33/0.68 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 0.33/0.68 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % and_int.simps
% 0.33/0.68 thf(fact_9421_or__int__unfold,axiom,
% 0.33/0.68 ( bit_se1409905431419307370or_int
% 0.33/0.68 = ( ^ [K3: int,L: int] :
% 0.33/0.68 ( if_int
% 0.33/0.68 @ ( ( K3
% 0.33/0.68 = ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.68 | ( L
% 0.33/0.68 = ( uminus_uminus_int @ one_one_int ) ) )
% 0.33/0.68 @ ( uminus_uminus_int @ one_one_int )
% 0.33/0.68 @ ( if_int @ ( K3 = zero_zero_int ) @ L @ ( if_int @ ( L = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_int_unfold
% 0.33/0.68 thf(fact_9422_cis__multiple__2pi,axiom,
% 0.33/0.68 ! [N2: real] :
% 0.33/0.68 ( ( member_real @ N2 @ ring_1_Ints_real )
% 0.33/0.68 => ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N2 ) )
% 0.33/0.68 = one_one_complex ) ) ).
% 0.33/0.68
% 0.33/0.68 % cis_multiple_2pi
% 0.33/0.68 thf(fact_9423_or__nonnegative__int__iff,axiom,
% 0.33/0.68 ! [K: int,L2: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) )
% 0.33/0.68 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.33/0.68 & ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_nonnegative_int_iff
% 0.33/0.68 thf(fact_9424_or__negative__int__iff,axiom,
% 0.33/0.68 ! [K: int,L2: int] :
% 0.33/0.68 ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) @ zero_zero_int )
% 0.33/0.68 = ( ( ord_less_int @ K @ zero_zero_int )
% 0.33/0.68 | ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_negative_int_iff
% 0.33/0.68 thf(fact_9425_and__nat__numerals_I1_J,axiom,
% 0.33/0.68 ! [Y: num] :
% 0.33/0.68 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 0.33/0.68 = zero_zero_nat ) ).
% 0.33/0.68
% 0.33/0.68 % and_nat_numerals(1)
% 0.33/0.68 thf(fact_9426_and__nat__numerals_I3_J,axiom,
% 0.33/0.68 ! [X: num] :
% 0.33/0.68 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.33/0.68 = zero_zero_nat ) ).
% 0.33/0.68
% 0.33/0.68 % and_nat_numerals(3)
% 0.33/0.68 thf(fact_9427_or__minus__numerals_I6_J,axiom,
% 0.33/0.68 ! [N2: num] :
% 0.33/0.68 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) @ one_one_int )
% 0.33/0.68 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_minus_numerals(6)
% 0.33/0.68 thf(fact_9428_or__minus__numerals_I2_J,axiom,
% 0.33/0.68 ! [N2: num] :
% 0.33/0.68 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
% 0.33/0.68 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_minus_numerals(2)
% 0.33/0.68 thf(fact_9429_and__nat__numerals_I2_J,axiom,
% 0.33/0.68 ! [Y: num] :
% 0.33/0.68 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 0.33/0.68 = one_one_nat ) ).
% 0.33/0.68
% 0.33/0.68 % and_nat_numerals(2)
% 0.33/0.68 thf(fact_9430_and__nat__numerals_I4_J,axiom,
% 0.33/0.68 ! [X: num] :
% 0.33/0.68 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.33/0.68 = one_one_nat ) ).
% 0.33/0.68
% 0.33/0.68 % and_nat_numerals(4)
% 0.33/0.68 thf(fact_9431_and__Suc__0__eq,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( bit_se727722235901077358nd_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 0.33/0.68 = ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % and_Suc_0_eq
% 0.33/0.68 thf(fact_9432_Suc__0__and__eq,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 0.33/0.68 = ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Suc_0_and_eq
% 0.33/0.68 thf(fact_9433_bit__or__int__iff,axiom,
% 0.33/0.68 ! [K: int,L2: int,N2: nat] :
% 0.33/0.68 ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) @ N2 )
% 0.33/0.68 = ( ( bit_se1146084159140164899it_int @ K @ N2 )
% 0.33/0.68 | ( bit_se1146084159140164899it_int @ L2 @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % bit_or_int_iff
% 0.33/0.68 thf(fact_9434_bit__Suc__0__iff,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 0.33/0.68 = ( N2 = zero_zero_nat ) ) ).
% 0.33/0.68
% 0.33/0.68 % bit_Suc_0_iff
% 0.33/0.68 thf(fact_9435_not__bit__Suc__0__Suc,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % not_bit_Suc_0_Suc
% 0.33/0.68 thf(fact_9436_OR__lower,axiom,
% 0.33/0.68 ! [X: int,Y: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.68 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.68 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X @ Y ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % OR_lower
% 0.33/0.68 thf(fact_9437_or__greater__eq,axiom,
% 0.33/0.68 ! [L2: int,K: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ L2 )
% 0.33/0.68 => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_greater_eq
% 0.33/0.68 thf(fact_9438_plus__and__or,axiom,
% 0.33/0.68 ! [X: int,Y: int] :
% 0.33/0.68 ( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ ( bit_se1409905431419307370or_int @ X @ Y ) )
% 0.33/0.68 = ( plus_plus_int @ X @ Y ) ) ).
% 0.33/0.68
% 0.33/0.68 % plus_and_or
% 0.33/0.68 thf(fact_9439_not__bit__Suc__0__numeral,axiom,
% 0.33/0.68 ! [N2: num] :
% 0.33/0.68 ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % not_bit_Suc_0_numeral
% 0.33/0.68 thf(fact_9440_and__nat__def,axiom,
% 0.33/0.68 ( bit_se727722235901077358nd_nat
% 0.33/0.68 = ( ^ [M6: nat,N: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % and_nat_def
% 0.33/0.68 thf(fact_9441_bit__nat__iff,axiom,
% 0.33/0.68 ! [K: int,N2: nat] :
% 0.33/0.68 ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N2 )
% 0.33/0.68 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.33/0.68 & ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % bit_nat_iff
% 0.33/0.68 thf(fact_9442_sin__times__pi__eq__0,axiom,
% 0.33/0.68 ! [X: real] :
% 0.33/0.68 ( ( ( sin_real @ ( times_times_real @ X @ pi ) )
% 0.33/0.68 = zero_zero_real )
% 0.33/0.68 = ( member_real @ X @ ring_1_Ints_real ) ) ).
% 0.33/0.68
% 0.33/0.68 % sin_times_pi_eq_0
% 0.33/0.68 thf(fact_9443_atLeastAtMostPlus1__int__conv,axiom,
% 0.33/0.68 ! [M: int,N2: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N2 ) )
% 0.33/0.68 => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N2 ) )
% 0.33/0.68 = ( insert_int @ ( plus_plus_int @ one_one_int @ N2 ) @ ( set_or1266510415728281911st_int @ M @ N2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % atLeastAtMostPlus1_int_conv
% 0.33/0.68 thf(fact_9444_simp__from__to,axiom,
% 0.33/0.68 ( set_or1266510415728281911st_int
% 0.33/0.68 = ( ^ [I3: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I3 ) @ bot_bot_set_int @ ( insert_int @ I3 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % simp_from_to
% 0.33/0.68 thf(fact_9445_bit__nat__def,axiom,
% 0.33/0.68 ( bit_se1148574629649215175it_nat
% 0.33/0.68 = ( ^ [M6: nat,N: nat] :
% 0.33/0.68 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % bit_nat_def
% 0.33/0.68 thf(fact_9446_OR__upper,axiom,
% 0.33/0.68 ! [X: int,N2: nat,Y: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.68 => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.33/0.68 => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.33/0.68 => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % OR_upper
% 0.33/0.68 thf(fact_9447_sin__integer__2pi,axiom,
% 0.33/0.68 ! [N2: real] :
% 0.33/0.68 ( ( member_real @ N2 @ ring_1_Ints_real )
% 0.33/0.68 => ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N2 ) )
% 0.33/0.68 = zero_zero_real ) ) ).
% 0.33/0.68
% 0.33/0.68 % sin_integer_2pi
% 0.33/0.68 thf(fact_9448_cos__integer__2pi,axiom,
% 0.33/0.68 ! [N2: real] :
% 0.33/0.68 ( ( member_real @ N2 @ ring_1_Ints_real )
% 0.33/0.68 => ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N2 ) )
% 0.33/0.68 = one_one_real ) ) ).
% 0.33/0.68
% 0.33/0.68 % cos_integer_2pi
% 0.33/0.68 thf(fact_9449_and__nat__unfold,axiom,
% 0.33/0.68 ( bit_se727722235901077358nd_nat
% 0.33/0.68 = ( ^ [M6: nat,N: nat] :
% 0.33/0.68 ( if_nat
% 0.33/0.68 @ ( ( M6 = zero_zero_nat )
% 0.33/0.68 | ( N = zero_zero_nat ) )
% 0.33/0.68 @ zero_zero_nat
% 0.33/0.68 @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % and_nat_unfold
% 0.33/0.68 thf(fact_9450_and__nat__rec,axiom,
% 0.33/0.68 ( bit_se727722235901077358nd_nat
% 0.33/0.68 = ( ^ [M6: nat,N: nat] :
% 0.33/0.68 ( plus_plus_nat
% 0.33/0.68 @ ( zero_n2687167440665602831ol_nat
% 0.33/0.68 @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 )
% 0.33/0.68 & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.33/0.68 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % and_nat_rec
% 0.33/0.68 thf(fact_9451_or__int__rec,axiom,
% 0.33/0.68 ( bit_se1409905431419307370or_int
% 0.33/0.68 = ( ^ [K3: int,L: int] :
% 0.33/0.68 ( plus_plus_int
% 0.33/0.68 @ ( zero_n2684676970156552555ol_int
% 0.33/0.68 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 0.33/0.68 | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 0.33/0.68 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_int_rec
% 0.33/0.68 thf(fact_9452_and__int_Opsimps,axiom,
% 0.33/0.68 ! [K: int,L2: int] :
% 0.33/0.68 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L2 ) )
% 0.33/0.68 => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.33/0.68 & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.33/0.68 => ( ( bit_se725231765392027082nd_int @ K @ L2 )
% 0.33/0.68 = ( uminus_uminus_int
% 0.33/0.68 @ ( zero_n2684676970156552555ol_int
% 0.33/0.68 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.33/0.68 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) ) ) )
% 0.33/0.68 & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.33/0.68 & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.33/0.68 => ( ( bit_se725231765392027082nd_int @ K @ L2 )
% 0.33/0.68 = ( plus_plus_int
% 0.33/0.68 @ ( zero_n2684676970156552555ol_int
% 0.33/0.68 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.33/0.68 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 0.33/0.68 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % and_int.psimps
% 0.33/0.68 thf(fact_9453_and__int_Opelims,axiom,
% 0.33/0.68 ! [X: int,Xa2: int,Y: int] :
% 0.33/0.68 ( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
% 0.33/0.68 = Y )
% 0.33/0.68 => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa2 ) )
% 0.33/0.68 => ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.33/0.68 & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.33/0.68 => ( Y
% 0.33/0.68 = ( uminus_uminus_int
% 0.33/0.68 @ ( zero_n2684676970156552555ol_int
% 0.33/0.68 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 0.33/0.68 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
% 0.33/0.68 & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.33/0.68 & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.33/0.68 => ( Y
% 0.33/0.68 = ( plus_plus_int
% 0.33/0.68 @ ( zero_n2684676970156552555ol_int
% 0.33/0.68 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 0.33/0.68 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
% 0.33/0.68 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 0.33/0.68 => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % and_int.pelims
% 0.33/0.68 thf(fact_9454_or__minus__numerals_I1_J,axiom,
% 0.33/0.68 ! [N2: num] :
% 0.33/0.68 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
% 0.33/0.68 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_minus_numerals(1)
% 0.33/0.68 thf(fact_9455_or__nat__numerals_I2_J,axiom,
% 0.33/0.68 ! [Y: num] :
% 0.33/0.68 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 0.33/0.68 = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_nat_numerals(2)
% 0.33/0.68 thf(fact_9456_or__nat__numerals_I4_J,axiom,
% 0.33/0.68 ! [X: num] :
% 0.33/0.68 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.33/0.68 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_nat_numerals(4)
% 0.33/0.68 thf(fact_9457_or__nat__numerals_I3_J,axiom,
% 0.33/0.68 ! [X: num] :
% 0.33/0.68 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.33/0.68 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_nat_numerals(3)
% 0.33/0.68 thf(fact_9458_or__nat__numerals_I1_J,axiom,
% 0.33/0.68 ! [Y: num] :
% 0.33/0.68 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 0.33/0.68 = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_nat_numerals(1)
% 0.33/0.68 thf(fact_9459_or__minus__numerals_I8_J,axiom,
% 0.33/0.68 ! [N2: num,M: num] :
% 0.33/0.68 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.33/0.68 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_minus_numerals(8)
% 0.33/0.68 thf(fact_9460_or__minus__numerals_I4_J,axiom,
% 0.33/0.68 ! [M: num,N2: num] :
% 0.33/0.68 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
% 0.33/0.68 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_minus_numerals(4)
% 0.33/0.68 thf(fact_9461_or__minus__numerals_I7_J,axiom,
% 0.33/0.68 ! [N2: num,M: num] :
% 0.33/0.68 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.33/0.68 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_minus_numerals(7)
% 0.33/0.68 thf(fact_9462_or__minus__numerals_I3_J,axiom,
% 0.33/0.68 ! [M: num,N2: num] :
% 0.33/0.68 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
% 0.33/0.68 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_minus_numerals(3)
% 0.33/0.68 thf(fact_9463_set__encode__insert,axiom,
% 0.33/0.68 ! [A2: set_nat,N2: nat] :
% 0.33/0.68 ( ( finite_finite_nat @ A2 )
% 0.33/0.68 => ( ~ ( member_nat @ N2 @ A2 )
% 0.33/0.68 => ( ( nat_set_encode @ ( insert_nat @ N2 @ A2 ) )
% 0.33/0.68 = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % set_encode_insert
% 0.33/0.68 thf(fact_9464_or__minus__numerals_I5_J,axiom,
% 0.33/0.68 ! [N2: num] :
% 0.33/0.68 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) @ one_one_int )
% 0.33/0.68 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_minus_numerals(5)
% 0.33/0.68 thf(fact_9465_or__not__num__neg_Osimps_I1_J,axiom,
% 0.33/0.68 ( ( bit_or_not_num_neg @ one @ one )
% 0.33/0.68 = one ) ).
% 0.33/0.68
% 0.33/0.68 % or_not_num_neg.simps(1)
% 0.33/0.68 thf(fact_9466_lessThan__Suc,axiom,
% 0.33/0.68 ! [K: nat] :
% 0.33/0.68 ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 0.33/0.68 = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % lessThan_Suc
% 0.33/0.68 thf(fact_9467_atMost__Suc,axiom,
% 0.33/0.68 ! [K: nat] :
% 0.33/0.68 ( ( set_ord_atMost_nat @ ( suc @ K ) )
% 0.33/0.68 = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % atMost_Suc
% 0.33/0.68 thf(fact_9468_or__not__num__neg_Osimps_I4_J,axiom,
% 0.33/0.68 ! [N2: num] :
% 0.33/0.68 ( ( bit_or_not_num_neg @ ( bit0 @ N2 ) @ one )
% 0.33/0.68 = ( bit0 @ one ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_not_num_neg.simps(4)
% 0.33/0.68 thf(fact_9469_or__not__num__neg_Osimps_I6_J,axiom,
% 0.33/0.68 ! [N2: num,M: num] :
% 0.33/0.68 ( ( bit_or_not_num_neg @ ( bit0 @ N2 ) @ ( bit1 @ M ) )
% 0.33/0.68 = ( bit0 @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_not_num_neg.simps(6)
% 0.33/0.68 thf(fact_9470_or__not__num__neg_Osimps_I7_J,axiom,
% 0.33/0.68 ! [N2: num] :
% 0.33/0.68 ( ( bit_or_not_num_neg @ ( bit1 @ N2 ) @ one )
% 0.33/0.68 = one ) ).
% 0.33/0.68
% 0.33/0.68 % or_not_num_neg.simps(7)
% 0.33/0.68 thf(fact_9471_or__not__num__neg_Osimps_I3_J,axiom,
% 0.33/0.68 ! [M: num] :
% 0.33/0.68 ( ( bit_or_not_num_neg @ one @ ( bit1 @ M ) )
% 0.33/0.68 = ( bit1 @ M ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_not_num_neg.simps(3)
% 0.33/0.68 thf(fact_9472_or__not__num__neg_Osimps_I5_J,axiom,
% 0.33/0.68 ! [N2: num,M: num] :
% 0.33/0.68 ( ( bit_or_not_num_neg @ ( bit0 @ N2 ) @ ( bit0 @ M ) )
% 0.33/0.68 = ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_not_num_neg.simps(5)
% 0.33/0.68 thf(fact_9473_or__not__num__neg_Osimps_I9_J,axiom,
% 0.33/0.68 ! [N2: num,M: num] :
% 0.33/0.68 ( ( bit_or_not_num_neg @ ( bit1 @ N2 ) @ ( bit1 @ M ) )
% 0.33/0.68 = ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_not_num_neg.simps(9)
% 0.33/0.68 thf(fact_9474_or__nat__def,axiom,
% 0.33/0.68 ( bit_se1412395901928357646or_nat
% 0.33/0.68 = ( ^ [M6: nat,N: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_nat_def
% 0.33/0.68 thf(fact_9475_atLeast0__atMost__Suc,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) )
% 0.33/0.68 = ( insert_nat @ ( suc @ N2 ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % atLeast0_atMost_Suc
% 0.33/0.68 thf(fact_9476_Icc__eq__insert__lb__nat,axiom,
% 0.33/0.68 ! [M: nat,N2: nat] :
% 0.33/0.68 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.68 => ( ( set_or1269000886237332187st_nat @ M @ N2 )
% 0.33/0.68 = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Icc_eq_insert_lb_nat
% 0.33/0.68 thf(fact_9477_atLeastAtMostSuc__conv,axiom,
% 0.33/0.68 ! [M: nat,N2: nat] :
% 0.33/0.68 ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 0.33/0.68 => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) )
% 0.33/0.68 = ( insert_nat @ ( suc @ N2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % atLeastAtMostSuc_conv
% 0.33/0.68 thf(fact_9478_atLeastAtMost__insertL,axiom,
% 0.33/0.68 ! [M: nat,N2: nat] :
% 0.33/0.68 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.68 => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
% 0.33/0.68 = ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % atLeastAtMost_insertL
% 0.33/0.68 thf(fact_9479_lessThan__nat__numeral,axiom,
% 0.33/0.68 ! [K: num] :
% 0.33/0.68 ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
% 0.33/0.68 = ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % lessThan_nat_numeral
% 0.33/0.68 thf(fact_9480_atMost__nat__numeral,axiom,
% 0.33/0.68 ! [K: num] :
% 0.33/0.68 ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
% 0.33/0.68 = ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % atMost_nat_numeral
% 0.33/0.68 thf(fact_9481_or__not__num__neg_Osimps_I2_J,axiom,
% 0.33/0.68 ! [M: num] :
% 0.33/0.68 ( ( bit_or_not_num_neg @ one @ ( bit0 @ M ) )
% 0.33/0.68 = ( bit1 @ M ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_not_num_neg.simps(2)
% 0.33/0.68 thf(fact_9482_or__not__num__neg_Osimps_I8_J,axiom,
% 0.33/0.68 ! [N2: num,M: num] :
% 0.33/0.68 ( ( bit_or_not_num_neg @ ( bit1 @ N2 ) @ ( bit0 @ M ) )
% 0.33/0.68 = ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_not_num_neg.simps(8)
% 0.33/0.68 thf(fact_9483_or__not__num__neg_Oelims,axiom,
% 0.33/0.68 ! [X: num,Xa2: num,Y: num] :
% 0.33/0.68 ( ( ( bit_or_not_num_neg @ X @ Xa2 )
% 0.33/0.68 = Y )
% 0.33/0.68 => ( ( ( X = one )
% 0.33/0.68 => ( ( Xa2 = one )
% 0.33/0.68 => ( Y != one ) ) )
% 0.33/0.68 => ( ( ( X = one )
% 0.33/0.68 => ! [M2: num] :
% 0.33/0.68 ( ( Xa2
% 0.33/0.68 = ( bit0 @ M2 ) )
% 0.33/0.68 => ( Y
% 0.33/0.68 != ( bit1 @ M2 ) ) ) )
% 0.33/0.68 => ( ( ( X = one )
% 0.33/0.68 => ! [M2: num] :
% 0.33/0.68 ( ( Xa2
% 0.33/0.68 = ( bit1 @ M2 ) )
% 0.33/0.68 => ( Y
% 0.33/0.68 != ( bit1 @ M2 ) ) ) )
% 0.33/0.68 => ( ( ? [N4: num] :
% 0.33/0.68 ( X
% 0.33/0.68 = ( bit0 @ N4 ) )
% 0.33/0.68 => ( ( Xa2 = one )
% 0.33/0.68 => ( Y
% 0.33/0.68 != ( bit0 @ one ) ) ) )
% 0.33/0.68 => ( ! [N4: num] :
% 0.33/0.68 ( ( X
% 0.33/0.68 = ( bit0 @ N4 ) )
% 0.33/0.68 => ! [M2: num] :
% 0.33/0.68 ( ( Xa2
% 0.33/0.68 = ( bit0 @ M2 ) )
% 0.33/0.68 => ( Y
% 0.33/0.68 != ( bitM @ ( bit_or_not_num_neg @ N4 @ M2 ) ) ) ) )
% 0.33/0.68 => ( ! [N4: num] :
% 0.33/0.68 ( ( X
% 0.33/0.68 = ( bit0 @ N4 ) )
% 0.33/0.68 => ! [M2: num] :
% 0.33/0.68 ( ( Xa2
% 0.33/0.68 = ( bit1 @ M2 ) )
% 0.33/0.68 => ( Y
% 0.33/0.68 != ( bit0 @ ( bit_or_not_num_neg @ N4 @ M2 ) ) ) ) )
% 0.33/0.68 => ( ( ? [N4: num] :
% 0.33/0.68 ( X
% 0.33/0.68 = ( bit1 @ N4 ) )
% 0.33/0.68 => ( ( Xa2 = one )
% 0.33/0.68 => ( Y != one ) ) )
% 0.33/0.68 => ( ! [N4: num] :
% 0.33/0.68 ( ( X
% 0.33/0.68 = ( bit1 @ N4 ) )
% 0.33/0.68 => ! [M2: num] :
% 0.33/0.68 ( ( Xa2
% 0.33/0.68 = ( bit0 @ M2 ) )
% 0.33/0.68 => ( Y
% 0.33/0.68 != ( bitM @ ( bit_or_not_num_neg @ N4 @ M2 ) ) ) ) )
% 0.33/0.68 => ~ ! [N4: num] :
% 0.33/0.68 ( ( X
% 0.33/0.68 = ( bit1 @ N4 ) )
% 0.33/0.68 => ! [M2: num] :
% 0.33/0.68 ( ( Xa2
% 0.33/0.68 = ( bit1 @ M2 ) )
% 0.33/0.68 => ( Y
% 0.33/0.68 != ( bitM @ ( bit_or_not_num_neg @ N4 @ M2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_not_num_neg.elims
% 0.33/0.68 thf(fact_9484_atLeast1__atMost__eq__remove0,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 0.33/0.68 = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % atLeast1_atMost_eq_remove0
% 0.33/0.68 thf(fact_9485_set__decode__plus__power__2,axiom,
% 0.33/0.68 ! [N2: nat,Z: nat] :
% 0.33/0.68 ( ~ ( member_nat @ N2 @ ( nat_set_decode @ Z ) )
% 0.33/0.68 => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ Z ) )
% 0.33/0.68 = ( insert_nat @ N2 @ ( nat_set_decode @ Z ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % set_decode_plus_power_2
% 0.33/0.68 thf(fact_9486_Suc__0__or__eq,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 0.33/0.68 = ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Suc_0_or_eq
% 0.33/0.68 thf(fact_9487_or__Suc__0__eq,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( bit_se1412395901928357646or_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 0.33/0.68 = ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_Suc_0_eq
% 0.33/0.68 thf(fact_9488_or__nat__rec,axiom,
% 0.33/0.68 ( bit_se1412395901928357646or_nat
% 0.33/0.68 = ( ^ [M6: nat,N: nat] :
% 0.33/0.68 ( plus_plus_nat
% 0.33/0.68 @ ( zero_n2687167440665602831ol_nat
% 0.33/0.68 @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 )
% 0.33/0.68 | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.33/0.68 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_nat_rec
% 0.33/0.68 thf(fact_9489_and__int_Opinduct,axiom,
% 0.33/0.68 ! [A0: int,A12: int,P: int > int > $o] :
% 0.33/0.68 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
% 0.33/0.68 => ( ! [K2: int,L4: int] :
% 0.33/0.68 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L4 ) )
% 0.33/0.68 => ( ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.33/0.68 & ( member_int @ L4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.33/0.68 => ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.33/0.68 => ( P @ K2 @ L4 ) ) )
% 0.33/0.68 => ( P @ A0 @ A12 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % and_int.pinduct
% 0.33/0.68 thf(fact_9490_or__nat__unfold,axiom,
% 0.33/0.68 ( bit_se1412395901928357646or_nat
% 0.33/0.68 = ( ^ [M6: nat,N: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N @ ( if_nat @ ( N = zero_zero_nat ) @ M6 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_nat_unfold
% 0.33/0.68 thf(fact_9491_upto_Opinduct,axiom,
% 0.33/0.68 ! [A0: int,A12: int,P: int > int > $o] :
% 0.33/0.68 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
% 0.33/0.68 => ( ! [I4: int,J2: int] :
% 0.33/0.68 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I4 @ J2 ) )
% 0.33/0.68 => ( ( ( ord_less_eq_int @ I4 @ J2 )
% 0.33/0.68 => ( P @ ( plus_plus_int @ I4 @ one_one_int ) @ J2 ) )
% 0.33/0.68 => ( P @ I4 @ J2 ) ) )
% 0.33/0.68 => ( P @ A0 @ A12 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % upto.pinduct
% 0.33/0.68 thf(fact_9492_rat__inverse__code,axiom,
% 0.33/0.68 ! [P4: rat] :
% 0.33/0.68 ( ( quotient_of @ ( inverse_inverse_rat @ P4 ) )
% 0.33/0.68 = ( produc4245557441103728435nt_int
% 0.33/0.68 @ ^ [A5: int,B5: int] : ( if_Pro3027730157355071871nt_int @ ( A5 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A5 ) @ B5 ) @ ( abs_abs_int @ A5 ) ) )
% 0.33/0.68 @ ( quotient_of @ P4 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % rat_inverse_code
% 0.33/0.68 thf(fact_9493_quotient__of__number_I3_J,axiom,
% 0.33/0.68 ! [K: num] :
% 0.33/0.68 ( ( quotient_of @ ( numeral_numeral_rat @ K ) )
% 0.33/0.68 = ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).
% 0.33/0.68
% 0.33/0.68 % quotient_of_number(3)
% 0.33/0.68 thf(fact_9494_rat__one__code,axiom,
% 0.33/0.68 ( ( quotient_of @ one_one_rat )
% 0.33/0.68 = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).
% 0.33/0.68
% 0.33/0.68 % rat_one_code
% 0.33/0.68 thf(fact_9495_rat__zero__code,axiom,
% 0.33/0.68 ( ( quotient_of @ zero_zero_rat )
% 0.33/0.68 = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 0.33/0.68
% 0.33/0.68 % rat_zero_code
% 0.33/0.68 thf(fact_9496_quotient__of__number_I5_J,axiom,
% 0.33/0.68 ! [K: num] :
% 0.33/0.68 ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.33/0.68 = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 0.33/0.68
% 0.33/0.68 % quotient_of_number(5)
% 0.33/0.68 thf(fact_9497_quotient__of__number_I4_J,axiom,
% 0.33/0.68 ( ( quotient_of @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.33/0.68 = ( product_Pair_int_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ) ) ).
% 0.33/0.68
% 0.33/0.68 % quotient_of_number(4)
% 0.33/0.68 thf(fact_9498_divide__rat__def,axiom,
% 0.33/0.68 ( divide_divide_rat
% 0.33/0.68 = ( ^ [Q4: rat,R5: rat] : ( times_times_rat @ Q4 @ ( inverse_inverse_rat @ R5 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % divide_rat_def
% 0.33/0.68 thf(fact_9499_quotient__of__div,axiom,
% 0.33/0.68 ! [R: rat,N2: int,D: int] :
% 0.33/0.68 ( ( ( quotient_of @ R )
% 0.33/0.68 = ( product_Pair_int_int @ N2 @ D ) )
% 0.33/0.68 => ( R
% 0.33/0.68 = ( divide_divide_rat @ ( ring_1_of_int_rat @ N2 ) @ ( ring_1_of_int_rat @ D ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % quotient_of_div
% 0.33/0.68 thf(fact_9500_quotient__of__denom__pos,axiom,
% 0.33/0.68 ! [R: rat,P4: int,Q2: int] :
% 0.33/0.68 ( ( ( quotient_of @ R )
% 0.33/0.68 = ( product_Pair_int_int @ P4 @ Q2 ) )
% 0.33/0.68 => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % quotient_of_denom_pos
% 0.33/0.68 thf(fact_9501_rat__uminus__code,axiom,
% 0.33/0.68 ! [P4: rat] :
% 0.33/0.68 ( ( quotient_of @ ( uminus_uminus_rat @ P4 ) )
% 0.33/0.68 = ( produc4245557441103728435nt_int
% 0.33/0.68 @ ^ [A5: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A5 ) )
% 0.33/0.68 @ ( quotient_of @ P4 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % rat_uminus_code
% 0.33/0.68 thf(fact_9502_rat__less__code,axiom,
% 0.33/0.68 ( ord_less_rat
% 0.33/0.68 = ( ^ [P5: rat,Q4: rat] :
% 0.33/0.68 ( produc4947309494688390418_int_o
% 0.33/0.68 @ ^ [A5: int,C4: int] :
% 0.33/0.68 ( produc4947309494688390418_int_o
% 0.33/0.68 @ ^ [B5: int,D2: int] : ( ord_less_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ C4 @ B5 ) )
% 0.33/0.68 @ ( quotient_of @ Q4 ) )
% 0.33/0.68 @ ( quotient_of @ P5 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % rat_less_code
% 0.33/0.68 thf(fact_9503_rat__floor__code,axiom,
% 0.33/0.68 ( archim3151403230148437115or_rat
% 0.33/0.68 = ( ^ [P5: rat] : ( produc8211389475949308722nt_int @ divide_divide_int @ ( quotient_of @ P5 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % rat_floor_code
% 0.33/0.68 thf(fact_9504_rat__abs__code,axiom,
% 0.33/0.68 ! [P4: rat] :
% 0.33/0.68 ( ( quotient_of @ ( abs_abs_rat @ P4 ) )
% 0.33/0.68 = ( produc4245557441103728435nt_int
% 0.33/0.68 @ ^ [A5: int] : ( product_Pair_int_int @ ( abs_abs_int @ A5 ) )
% 0.33/0.68 @ ( quotient_of @ P4 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % rat_abs_code
% 0.33/0.68 thf(fact_9505_rat__less__eq__code,axiom,
% 0.33/0.68 ( ord_less_eq_rat
% 0.33/0.68 = ( ^ [P5: rat,Q4: rat] :
% 0.33/0.68 ( produc4947309494688390418_int_o
% 0.33/0.68 @ ^ [A5: int,C4: int] :
% 0.33/0.68 ( produc4947309494688390418_int_o
% 0.33/0.68 @ ^ [B5: int,D2: int] : ( ord_less_eq_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ C4 @ B5 ) )
% 0.33/0.68 @ ( quotient_of @ Q4 ) )
% 0.33/0.68 @ ( quotient_of @ P5 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % rat_less_eq_code
% 0.33/0.68 thf(fact_9506_quotient__of__int,axiom,
% 0.33/0.68 ! [A: int] :
% 0.33/0.68 ( ( quotient_of @ ( of_int @ A ) )
% 0.33/0.68 = ( product_Pair_int_int @ A @ one_one_int ) ) ).
% 0.33/0.68
% 0.33/0.68 % quotient_of_int
% 0.33/0.68 thf(fact_9507_rat__minus__code,axiom,
% 0.33/0.68 ! [P4: rat,Q2: rat] :
% 0.33/0.68 ( ( quotient_of @ ( minus_minus_rat @ P4 @ Q2 ) )
% 0.33/0.68 = ( produc4245557441103728435nt_int
% 0.33/0.68 @ ^ [A5: int,C4: int] :
% 0.33/0.68 ( produc4245557441103728435nt_int
% 0.33/0.68 @ ^ [B5: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ B5 @ C4 ) ) @ ( times_times_int @ C4 @ D2 ) ) )
% 0.33/0.68 @ ( quotient_of @ Q2 ) )
% 0.33/0.68 @ ( quotient_of @ P4 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % rat_minus_code
% 0.33/0.68 thf(fact_9508_rat__plus__code,axiom,
% 0.33/0.68 ! [P4: rat,Q2: rat] :
% 0.33/0.68 ( ( quotient_of @ ( plus_plus_rat @ P4 @ Q2 ) )
% 0.33/0.68 = ( produc4245557441103728435nt_int
% 0.33/0.68 @ ^ [A5: int,C4: int] :
% 0.33/0.68 ( produc4245557441103728435nt_int
% 0.33/0.68 @ ^ [B5: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ B5 @ C4 ) ) @ ( times_times_int @ C4 @ D2 ) ) )
% 0.33/0.68 @ ( quotient_of @ Q2 ) )
% 0.33/0.68 @ ( quotient_of @ P4 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % rat_plus_code
% 0.33/0.68 thf(fact_9509_normalize__denom__zero,axiom,
% 0.33/0.68 ! [P4: int] :
% 0.33/0.68 ( ( normalize @ ( product_Pair_int_int @ P4 @ zero_zero_int ) )
% 0.33/0.68 = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 0.33/0.68
% 0.33/0.68 % normalize_denom_zero
% 0.33/0.68 thf(fact_9510_normalize__negative,axiom,
% 0.33/0.68 ! [Q2: int,P4: int] :
% 0.33/0.68 ( ( ord_less_int @ Q2 @ zero_zero_int )
% 0.33/0.68 => ( ( normalize @ ( product_Pair_int_int @ P4 @ Q2 ) )
% 0.33/0.68 = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P4 ) @ ( uminus_uminus_int @ Q2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % normalize_negative
% 0.33/0.68 thf(fact_9511_normalize__denom__pos,axiom,
% 0.33/0.68 ! [R: product_prod_int_int,P4: int,Q2: int] :
% 0.33/0.68 ( ( ( normalize @ R )
% 0.33/0.68 = ( product_Pair_int_int @ P4 @ Q2 ) )
% 0.33/0.68 => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % normalize_denom_pos
% 0.33/0.68 thf(fact_9512_rat__times__code,axiom,
% 0.33/0.68 ! [P4: rat,Q2: rat] :
% 0.33/0.68 ( ( quotient_of @ ( times_times_rat @ P4 @ Q2 ) )
% 0.33/0.68 = ( produc4245557441103728435nt_int
% 0.33/0.68 @ ^ [A5: int,C4: int] :
% 0.33/0.68 ( produc4245557441103728435nt_int
% 0.33/0.68 @ ^ [B5: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A5 @ B5 ) @ ( times_times_int @ C4 @ D2 ) ) )
% 0.33/0.68 @ ( quotient_of @ Q2 ) )
% 0.33/0.68 @ ( quotient_of @ P4 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % rat_times_code
% 0.33/0.68 thf(fact_9513_rat__divide__code,axiom,
% 0.33/0.68 ! [P4: rat,Q2: rat] :
% 0.33/0.68 ( ( quotient_of @ ( divide_divide_rat @ P4 @ Q2 ) )
% 0.33/0.68 = ( produc4245557441103728435nt_int
% 0.33/0.68 @ ^ [A5: int,C4: int] :
% 0.33/0.68 ( produc4245557441103728435nt_int
% 0.33/0.68 @ ^ [B5: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ C4 @ B5 ) ) )
% 0.33/0.68 @ ( quotient_of @ Q2 ) )
% 0.33/0.68 @ ( quotient_of @ P4 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % rat_divide_code
% 0.33/0.68 thf(fact_9514_Frct__code__post_I5_J,axiom,
% 0.33/0.68 ! [K: num] :
% 0.33/0.68 ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
% 0.33/0.68 = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Frct_code_post(5)
% 0.33/0.68 thf(fact_9515_Frct__code__post_I3_J,axiom,
% 0.33/0.68 ( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
% 0.33/0.68 = one_one_rat ) ).
% 0.33/0.68
% 0.33/0.68 % Frct_code_post(3)
% 0.33/0.68 thf(fact_9516_Frct__code__post_I4_J,axiom,
% 0.33/0.68 ! [K: num] :
% 0.33/0.68 ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
% 0.33/0.68 = ( numeral_numeral_rat @ K ) ) ).
% 0.33/0.68
% 0.33/0.68 % Frct_code_post(4)
% 0.33/0.68 thf(fact_9517_Frct__code__post_I6_J,axiom,
% 0.33/0.68 ! [K: num,L2: num] :
% 0.33/0.68 ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L2 ) ) )
% 0.33/0.68 = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Frct_code_post(6)
% 0.33/0.68 thf(fact_9518_xor__Suc__0__eq,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( bit_se6528837805403552850or_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 0.33/0.68 = ( minus_minus_nat @ ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.33/0.68 @ ( zero_n2687167440665602831ol_nat
% 0.33/0.68 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % xor_Suc_0_eq
% 0.33/0.68 thf(fact_9519_Suc__0__xor__eq,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 0.33/0.68 = ( minus_minus_nat @ ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.33/0.68 @ ( zero_n2687167440665602831ol_nat
% 0.33/0.68 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Suc_0_xor_eq
% 0.33/0.68 thf(fact_9520_horner__sum__of__bool__2__less,axiom,
% 0.33/0.68 ! [Bs: list_o] : ( ord_less_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( size_size_list_o @ Bs ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % horner_sum_of_bool_2_less
% 0.33/0.68 thf(fact_9521_xor__nat__numerals_I4_J,axiom,
% 0.33/0.68 ! [X: num] :
% 0.33/0.68 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.33/0.68 = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % xor_nat_numerals(4)
% 0.33/0.68 thf(fact_9522_xor__nat__numerals_I3_J,axiom,
% 0.33/0.68 ! [X: num] :
% 0.33/0.68 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.33/0.68 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % xor_nat_numerals(3)
% 0.33/0.68 thf(fact_9523_xor__nat__numerals_I2_J,axiom,
% 0.33/0.68 ! [Y: num] :
% 0.33/0.68 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 0.33/0.68 = ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % xor_nat_numerals(2)
% 0.33/0.68 thf(fact_9524_xor__nat__numerals_I1_J,axiom,
% 0.33/0.68 ! [Y: num] :
% 0.33/0.68 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 0.33/0.68 = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % xor_nat_numerals(1)
% 0.33/0.68 thf(fact_9525_xor__nat__unfold,axiom,
% 0.33/0.68 ( bit_se6528837805403552850or_nat
% 0.33/0.68 = ( ^ [M6: nat,N: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N @ ( if_nat @ ( N = zero_zero_nat ) @ M6 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % xor_nat_unfold
% 0.33/0.68 thf(fact_9526_xor__nat__rec,axiom,
% 0.33/0.68 ( bit_se6528837805403552850or_nat
% 0.33/0.68 = ( ^ [M6: nat,N: nat] :
% 0.33/0.68 ( plus_plus_nat
% 0.33/0.68 @ ( zero_n2687167440665602831ol_nat
% 0.33/0.68 @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 ) )
% 0.33/0.68 != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.33/0.68 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % xor_nat_rec
% 0.33/0.68 thf(fact_9527_Cauchy__iff2,axiom,
% 0.33/0.68 ( topolo4055970368930404560y_real
% 0.33/0.68 = ( ^ [X3: nat > real] :
% 0.33/0.68 ! [J3: nat] :
% 0.33/0.68 ? [M8: nat] :
% 0.33/0.68 ! [M6: nat] :
% 0.33/0.68 ( ( ord_less_eq_nat @ M8 @ M6 )
% 0.33/0.68 => ! [N: nat] :
% 0.33/0.68 ( ( ord_less_eq_nat @ M8 @ N )
% 0.33/0.68 => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X3 @ M6 ) @ ( X3 @ N ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Cauchy_iff2
% 0.33/0.68 thf(fact_9528_push__bit__nonnegative__int__iff,axiom,
% 0.33/0.68 ! [N2: nat,K: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N2 @ K ) )
% 0.33/0.68 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.33/0.68
% 0.33/0.68 % push_bit_nonnegative_int_iff
% 0.33/0.68 thf(fact_9529_push__bit__negative__int__iff,axiom,
% 0.33/0.68 ! [N2: nat,K: int] :
% 0.33/0.68 ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N2 @ K ) @ zero_zero_int )
% 0.33/0.68 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.33/0.68
% 0.33/0.68 % push_bit_negative_int_iff
% 0.33/0.68 thf(fact_9530_concat__bit__of__zero__1,axiom,
% 0.33/0.68 ! [N2: nat,L2: int] :
% 0.33/0.68 ( ( bit_concat_bit @ N2 @ zero_zero_int @ L2 )
% 0.33/0.68 = ( bit_se545348938243370406it_int @ N2 @ L2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % concat_bit_of_zero_1
% 0.33/0.68 thf(fact_9531_xor__nonnegative__int__iff,axiom,
% 0.33/0.68 ! [K: int,L2: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) )
% 0.33/0.68 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.33/0.68 = ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % xor_nonnegative_int_iff
% 0.33/0.68 thf(fact_9532_xor__negative__int__iff,axiom,
% 0.33/0.68 ! [K: int,L2: int] :
% 0.33/0.68 ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) @ zero_zero_int )
% 0.33/0.68 = ( ( ord_less_int @ K @ zero_zero_int )
% 0.33/0.68 != ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % xor_negative_int_iff
% 0.33/0.68 thf(fact_9533_push__bit__of__Suc__0,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( bit_se547839408752420682it_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 0.33/0.68 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % push_bit_of_Suc_0
% 0.33/0.68 thf(fact_9534_bit__xor__int__iff,axiom,
% 0.33/0.68 ! [K: int,L2: int,N2: nat] :
% 0.33/0.68 ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) @ N2 )
% 0.33/0.68 = ( ( bit_se1146084159140164899it_int @ K @ N2 )
% 0.33/0.68 != ( bit_se1146084159140164899it_int @ L2 @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % bit_xor_int_iff
% 0.33/0.68 thf(fact_9535_flip__bit__int__def,axiom,
% 0.33/0.68 ( bit_se2159334234014336723it_int
% 0.33/0.68 = ( ^ [N: nat,K3: int] : ( bit_se6526347334894502574or_int @ K3 @ ( bit_se545348938243370406it_int @ N @ one_one_int ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % flip_bit_int_def
% 0.33/0.68 thf(fact_9536_push__bit__nat__eq,axiom,
% 0.33/0.68 ! [N2: nat,K: int] :
% 0.33/0.68 ( ( bit_se547839408752420682it_nat @ N2 @ ( nat2 @ K ) )
% 0.33/0.68 = ( nat2 @ ( bit_se545348938243370406it_int @ N2 @ K ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % push_bit_nat_eq
% 0.33/0.68 thf(fact_9537_XOR__lower,axiom,
% 0.33/0.68 ! [X: int,Y: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.68 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 0.33/0.68 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X @ Y ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % XOR_lower
% 0.33/0.68 thf(fact_9538_set__bit__nat__def,axiom,
% 0.33/0.68 ( bit_se7882103937844011126it_nat
% 0.33/0.68 = ( ^ [M6: nat,N: nat] : ( bit_se1412395901928357646or_nat @ N @ ( bit_se547839408752420682it_nat @ M6 @ one_one_nat ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % set_bit_nat_def
% 0.33/0.68 thf(fact_9539_flip__bit__nat__def,axiom,
% 0.33/0.68 ( bit_se2161824704523386999it_nat
% 0.33/0.68 = ( ^ [M6: nat,N: nat] : ( bit_se6528837805403552850or_nat @ N @ ( bit_se547839408752420682it_nat @ M6 @ one_one_nat ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % flip_bit_nat_def
% 0.33/0.68 thf(fact_9540_bit__push__bit__iff__int,axiom,
% 0.33/0.68 ! [M: nat,K: int,N2: nat] :
% 0.33/0.68 ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N2 )
% 0.33/0.68 = ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.68 & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % bit_push_bit_iff_int
% 0.33/0.68 thf(fact_9541_xor__nat__def,axiom,
% 0.33/0.68 ( bit_se6528837805403552850or_nat
% 0.33/0.68 = ( ^ [M6: nat,N: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % xor_nat_def
% 0.33/0.68 thf(fact_9542_bit__push__bit__iff__nat,axiom,
% 0.33/0.68 ! [M: nat,Q2: nat,N2: nat] :
% 0.33/0.68 ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q2 ) @ N2 )
% 0.33/0.68 = ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.68 & ( bit_se1148574629649215175it_nat @ Q2 @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % bit_push_bit_iff_nat
% 0.33/0.68 thf(fact_9543_concat__bit__eq,axiom,
% 0.33/0.68 ( bit_concat_bit
% 0.33/0.68 = ( ^ [N: nat,K3: int,L: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N @ K3 ) @ ( bit_se545348938243370406it_int @ N @ L ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % concat_bit_eq
% 0.33/0.68 thf(fact_9544_concat__bit__def,axiom,
% 0.33/0.68 ( bit_concat_bit
% 0.33/0.68 = ( ^ [N: nat,K3: int,L: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N @ K3 ) @ ( bit_se545348938243370406it_int @ N @ L ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % concat_bit_def
% 0.33/0.68 thf(fact_9545_set__bit__int__def,axiom,
% 0.33/0.68 ( bit_se7879613467334960850it_int
% 0.33/0.68 = ( ^ [N: nat,K3: int] : ( bit_se1409905431419307370or_int @ K3 @ ( bit_se545348938243370406it_int @ N @ one_one_int ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % set_bit_int_def
% 0.33/0.68 thf(fact_9546_push__bit__nat__def,axiom,
% 0.33/0.68 ( bit_se547839408752420682it_nat
% 0.33/0.68 = ( ^ [N: nat,M6: nat] : ( times_times_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % push_bit_nat_def
% 0.33/0.68 thf(fact_9547_push__bit__int__def,axiom,
% 0.33/0.68 ( bit_se545348938243370406it_int
% 0.33/0.68 = ( ^ [N: nat,K3: int] : ( times_times_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % push_bit_int_def
% 0.33/0.68 thf(fact_9548_push__bit__minus__one,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( bit_se545348938243370406it_int @ N2 @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.68 = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % push_bit_minus_one
% 0.33/0.68 thf(fact_9549_XOR__upper,axiom,
% 0.33/0.68 ! [X: int,N2: nat,Y: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.33/0.68 => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.33/0.68 => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.33/0.68 => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % XOR_upper
% 0.33/0.68 thf(fact_9550_xor__int__rec,axiom,
% 0.33/0.68 ( bit_se6526347334894502574or_int
% 0.33/0.68 = ( ^ [K3: int,L: int] :
% 0.33/0.68 ( plus_plus_int
% 0.33/0.68 @ ( zero_n2684676970156552555ol_int
% 0.33/0.68 @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) )
% 0.33/0.68 != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) )
% 0.33/0.68 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % xor_int_rec
% 0.33/0.68 thf(fact_9551_xor__int__unfold,axiom,
% 0.33/0.68 ( bit_se6526347334894502574or_int
% 0.33/0.68 = ( ^ [K3: int,L: int] :
% 0.33/0.68 ( if_int
% 0.33/0.68 @ ( K3
% 0.33/0.68 = ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.68 @ ( bit_ri7919022796975470100ot_int @ L )
% 0.33/0.68 @ ( if_int
% 0.33/0.68 @ ( L
% 0.33/0.68 = ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.68 @ ( bit_ri7919022796975470100ot_int @ K3 )
% 0.33/0.68 @ ( if_int @ ( K3 = zero_zero_int ) @ L @ ( if_int @ ( L = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % xor_int_unfold
% 0.33/0.68 thf(fact_9552_Sum__Ico__nat,axiom,
% 0.33/0.68 ! [M: nat,N2: nat] :
% 0.33/0.68 ( ( groups3542108847815614940at_nat
% 0.33/0.68 @ ^ [X2: nat] : X2
% 0.33/0.68 @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
% 0.33/0.68 = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Sum_Ico_nat
% 0.33/0.68 thf(fact_9553_VEBT_Osize_I3_J,axiom,
% 0.33/0.68 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 0.33/0.68 ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 0.33/0.68 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % VEBT.size(3)
% 0.33/0.68 thf(fact_9554_not__nonnegative__int__iff,axiom,
% 0.33/0.68 ! [K: int] :
% 0.33/0.68 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 0.33/0.68 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.33/0.68
% 0.33/0.68 % not_nonnegative_int_iff
% 0.33/0.68 thf(fact_9555_not__negative__int__iff,axiom,
% 0.33/0.68 ! [K: int] :
% 0.33/0.68 ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
% 0.33/0.68 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.33/0.68
% 0.33/0.68 % not_negative_int_iff
% 0.33/0.68 thf(fact_9556_atLeastLessThan__singleton,axiom,
% 0.33/0.68 ! [M: nat] :
% 0.33/0.68 ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
% 0.33/0.68 = ( insert_nat @ M @ bot_bot_set_nat ) ) ).
% 0.33/0.68
% 0.33/0.68 % atLeastLessThan_singleton
% 0.33/0.68 thf(fact_9557_and__minus__minus__numerals,axiom,
% 0.33/0.68 ! [M: num,N2: num] :
% 0.33/0.68 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.33/0.68 = ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % and_minus_minus_numerals
% 0.33/0.68 thf(fact_9558_or__minus__minus__numerals,axiom,
% 0.33/0.68 ! [M: num,N2: num] :
% 0.33/0.68 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.33/0.68 = ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_minus_minus_numerals
% 0.33/0.68 thf(fact_9559_bit__not__int__iff,axiom,
% 0.33/0.68 ! [K: int,N2: nat] :
% 0.33/0.68 ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ K ) @ N2 )
% 0.33/0.68 = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % bit_not_int_iff
% 0.33/0.68 thf(fact_9560_all__nat__less__eq,axiom,
% 0.33/0.68 ! [N2: nat,P: nat > $o] :
% 0.33/0.68 ( ( ! [M6: nat] :
% 0.33/0.68 ( ( ord_less_nat @ M6 @ N2 )
% 0.33/0.68 => ( P @ M6 ) ) )
% 0.33/0.68 = ( ! [X2: nat] :
% 0.33/0.68 ( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
% 0.33/0.68 => ( P @ X2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % all_nat_less_eq
% 0.33/0.68 thf(fact_9561_ex__nat__less__eq,axiom,
% 0.33/0.68 ! [N2: nat,P: nat > $o] :
% 0.33/0.68 ( ( ? [M6: nat] :
% 0.33/0.68 ( ( ord_less_nat @ M6 @ N2 )
% 0.33/0.68 & ( P @ M6 ) ) )
% 0.33/0.68 = ( ? [X2: nat] :
% 0.33/0.68 ( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
% 0.33/0.68 & ( P @ X2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % ex_nat_less_eq
% 0.33/0.68 thf(fact_9562_atLeastLessThanSuc__atLeastAtMost,axiom,
% 0.33/0.68 ! [L2: nat,U: nat] :
% 0.33/0.68 ( ( set_or4665077453230672383an_nat @ L2 @ ( suc @ U ) )
% 0.33/0.68 = ( set_or1269000886237332187st_nat @ L2 @ U ) ) ).
% 0.33/0.68
% 0.33/0.68 % atLeastLessThanSuc_atLeastAtMost
% 0.33/0.68 thf(fact_9563_or__int__def,axiom,
% 0.33/0.68 ( bit_se1409905431419307370or_int
% 0.33/0.68 = ( ^ [K3: int,L: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K3 ) @ ( bit_ri7919022796975470100ot_int @ L ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_int_def
% 0.33/0.68 thf(fact_9564_not__int__def,axiom,
% 0.33/0.68 ( bit_ri7919022796975470100ot_int
% 0.33/0.68 = ( ^ [K3: int] : ( minus_minus_int @ ( uminus_uminus_int @ K3 ) @ one_one_int ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % not_int_def
% 0.33/0.68 thf(fact_9565_and__not__numerals_I1_J,axiom,
% 0.33/0.68 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.33/0.68 = zero_zero_int ) ).
% 0.33/0.68
% 0.33/0.68 % and_not_numerals(1)
% 0.33/0.68 thf(fact_9566_or__not__numerals_I1_J,axiom,
% 0.33/0.68 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.33/0.68 = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_not_numerals(1)
% 0.33/0.68 thf(fact_9567_atLeast0__lessThan__Suc,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) )
% 0.33/0.68 = ( insert_nat @ N2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % atLeast0_lessThan_Suc
% 0.33/0.68 thf(fact_9568_unset__bit__int__def,axiom,
% 0.33/0.68 ( bit_se4203085406695923979it_int
% 0.33/0.68 = ( ^ [N: nat,K3: int] : ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N @ one_one_int ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % unset_bit_int_def
% 0.33/0.68 thf(fact_9569_xor__int__def,axiom,
% 0.33/0.68 ( bit_se6526347334894502574or_int
% 0.33/0.68 = ( ^ [K3: int,L: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ L ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K3 ) @ L ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % xor_int_def
% 0.33/0.68 thf(fact_9570_not__int__div__2,axiom,
% 0.33/0.68 ! [K: int] :
% 0.33/0.68 ( ( divide_divide_int @ ( bit_ri7919022796975470100ot_int @ K ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.68 = ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % not_int_div_2
% 0.33/0.68 thf(fact_9571_even__not__iff__int,axiom,
% 0.33/0.68 ! [K: int] :
% 0.33/0.68 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ K ) )
% 0.33/0.68 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % even_not_iff_int
% 0.33/0.68 thf(fact_9572_and__not__numerals_I2_J,axiom,
% 0.33/0.68 ! [N2: num] :
% 0.33/0.68 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
% 0.33/0.68 = one_one_int ) ).
% 0.33/0.68
% 0.33/0.68 % and_not_numerals(2)
% 0.33/0.68 thf(fact_9573_and__not__numerals_I4_J,axiom,
% 0.33/0.68 ! [M: num] :
% 0.33/0.68 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.33/0.68 = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % and_not_numerals(4)
% 0.33/0.68 thf(fact_9574_or__not__numerals_I2_J,axiom,
% 0.33/0.68 ! [N2: num] :
% 0.33/0.68 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
% 0.33/0.68 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_not_numerals(2)
% 0.33/0.68 thf(fact_9575_or__not__numerals_I4_J,axiom,
% 0.33/0.68 ! [M: num] :
% 0.33/0.68 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.33/0.68 = ( bit_ri7919022796975470100ot_int @ one_one_int ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_not_numerals(4)
% 0.33/0.68 thf(fact_9576_bit__minus__int__iff,axiom,
% 0.33/0.68 ! [K: int,N2: nat] :
% 0.33/0.68 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N2 )
% 0.33/0.68 = ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % bit_minus_int_iff
% 0.33/0.68 thf(fact_9577_numeral__or__not__num__eq,axiom,
% 0.33/0.68 ! [M: num,N2: num] :
% 0.33/0.68 ( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N2 ) )
% 0.33/0.68 = ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % numeral_or_not_num_eq
% 0.33/0.68 thf(fact_9578_int__numeral__not__or__num__neg,axiom,
% 0.33/0.68 ! [M: num,N2: num] :
% 0.33/0.68 ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
% 0.33/0.68 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % int_numeral_not_or_num_neg
% 0.33/0.68 thf(fact_9579_int__numeral__or__not__num__neg,axiom,
% 0.33/0.68 ! [M: num,N2: num] :
% 0.33/0.68 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.33/0.68 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % int_numeral_or_not_num_neg
% 0.33/0.68 thf(fact_9580_atLeastLessThanSuc,axiom,
% 0.33/0.68 ! [M: nat,N2: nat] :
% 0.33/0.68 ( ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.68 => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) )
% 0.33/0.68 = ( insert_nat @ N2 @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) )
% 0.33/0.68 & ( ~ ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.68 => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) )
% 0.33/0.68 = bot_bot_set_nat ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % atLeastLessThanSuc
% 0.33/0.68 thf(fact_9581_prod__Suc__Suc__fact,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
% 0.33/0.68 = ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % prod_Suc_Suc_fact
% 0.33/0.68 thf(fact_9582_prod__Suc__fact,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
% 0.33/0.68 = ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % prod_Suc_fact
% 0.33/0.68 thf(fact_9583_and__not__numerals_I5_J,axiom,
% 0.33/0.68 ! [M: num,N2: num] :
% 0.33/0.68 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
% 0.33/0.68 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % and_not_numerals(5)
% 0.33/0.68 thf(fact_9584_and__not__numerals_I7_J,axiom,
% 0.33/0.68 ! [M: num] :
% 0.33/0.68 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.33/0.68 = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % and_not_numerals(7)
% 0.33/0.68 thf(fact_9585_or__not__numerals_I3_J,axiom,
% 0.33/0.68 ! [N2: num] :
% 0.33/0.68 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
% 0.33/0.68 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_not_numerals(3)
% 0.33/0.68 thf(fact_9586_and__not__numerals_I3_J,axiom,
% 0.33/0.68 ! [N2: num] :
% 0.33/0.68 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
% 0.33/0.68 = zero_zero_int ) ).
% 0.33/0.68
% 0.33/0.68 % and_not_numerals(3)
% 0.33/0.68 thf(fact_9587_or__not__numerals_I7_J,axiom,
% 0.33/0.68 ! [M: num] :
% 0.33/0.68 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.33/0.68 = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_not_numerals(7)
% 0.33/0.68 thf(fact_9588_atLeastLessThan__nat__numeral,axiom,
% 0.33/0.68 ! [M: nat,K: num] :
% 0.33/0.68 ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 0.33/0.68 => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 0.33/0.68 = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
% 0.33/0.68 & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 0.33/0.68 => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 0.33/0.68 = bot_bot_set_nat ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % atLeastLessThan_nat_numeral
% 0.33/0.68 thf(fact_9589_and__not__numerals_I6_J,axiom,
% 0.33/0.68 ! [M: num,N2: num] :
% 0.33/0.68 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
% 0.33/0.68 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % and_not_numerals(6)
% 0.33/0.68 thf(fact_9590_and__not__numerals_I9_J,axiom,
% 0.33/0.68 ! [M: num,N2: num] :
% 0.33/0.68 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
% 0.33/0.68 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % and_not_numerals(9)
% 0.33/0.68 thf(fact_9591_or__not__numerals_I6_J,axiom,
% 0.33/0.68 ! [M: num,N2: num] :
% 0.33/0.68 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
% 0.33/0.68 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_not_numerals(6)
% 0.33/0.68 thf(fact_9592_or__not__numerals_I5_J,axiom,
% 0.33/0.68 ! [M: num,N2: num] :
% 0.33/0.68 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
% 0.33/0.68 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_not_numerals(5)
% 0.33/0.68 thf(fact_9593_atLeast1__lessThan__eq__remove0,axiom,
% 0.33/0.68 ! [N2: nat] :
% 0.33/0.68 ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 0.33/0.68 = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % atLeast1_lessThan_eq_remove0
% 0.33/0.68 thf(fact_9594_and__not__numerals_I8_J,axiom,
% 0.33/0.68 ! [M: num,N2: num] :
% 0.33/0.68 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
% 0.33/0.68 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % and_not_numerals(8)
% 0.33/0.68 thf(fact_9595_or__not__numerals_I9_J,axiom,
% 0.33/0.68 ! [M: num,N2: num] :
% 0.33/0.68 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
% 0.33/0.68 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_not_numerals(9)
% 0.33/0.68 thf(fact_9596_or__not__numerals_I8_J,axiom,
% 0.33/0.68 ! [M: num,N2: num] :
% 0.33/0.68 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
% 0.33/0.68 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % or_not_numerals(8)
% 0.33/0.68 thf(fact_9597_not__int__rec,axiom,
% 0.33/0.68 ( bit_ri7919022796975470100ot_int
% 0.33/0.68 = ( ^ [K3: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % not_int_rec
% 0.33/0.68 thf(fact_9598_sum__power2,axiom,
% 0.33/0.68 ! [K: nat] :
% 0.33/0.68 ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
% 0.33/0.68 = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).
% 0.33/0.68
% 0.33/0.68 % sum_power2
% 0.33/0.68 thf(fact_9599_Chebyshev__sum__upper__nat,axiom,
% 0.33/0.68 ! [N2: nat,A: nat > nat,B: nat > nat] :
% 0.33/0.68 ( ! [I4: nat,J2: nat] :
% 0.33/0.68 ( ( ord_less_eq_nat @ I4 @ J2 )
% 0.33/0.68 => ( ( ord_less_nat @ J2 @ N2 )
% 0.33/0.68 => ( ord_less_eq_nat @ ( A @ I4 ) @ ( A @ J2 ) ) ) )
% 0.33/0.68 => ( ! [I4: nat,J2: nat] :
% 0.33/0.68 ( ( ord_less_eq_nat @ I4 @ J2 )
% 0.33/0.68 => ( ( ord_less_nat @ J2 @ N2 )
% 0.33/0.68 => ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I4 ) ) ) )
% 0.33/0.68 => ( ord_less_eq_nat
% 0.33/0.68 @ ( times_times_nat @ N2
% 0.33/0.68 @ ( groups3542108847815614940at_nat
% 0.33/0.68 @ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( B @ I3 ) )
% 0.33/0.68 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) )
% 0.33/0.68 @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Chebyshev_sum_upper_nat
% 0.33/0.68 thf(fact_9600_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
% 0.33/0.68 ! [L2: int,U: int] :
% 0.33/0.68 ( ( set_or4662586982721622107an_int @ L2 @ ( plus_plus_int @ U @ one_one_int ) )
% 0.33/0.68 = ( set_or1266510415728281911st_int @ L2 @ U ) ) ).
% 0.33/0.68
% 0.33/0.68 % atLeastLessThanPlusOne_atLeastAtMost_int
% 0.33/0.68 thf(fact_9601_VEBT_Osize__gen_I1_J,axiom,
% 0.33/0.68 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 0.33/0.68 ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 0.33/0.68 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % VEBT.size_gen(1)
% 0.33/0.68 thf(fact_9602_valid__eq,axiom,
% 0.33/0.68 vEBT_VEBT_valid = vEBT_invar_vebt ).
% 0.33/0.68
% 0.33/0.68 % valid_eq
% 0.33/0.68 thf(fact_9603_valid__eq2,axiom,
% 0.33/0.68 ! [T: vEBT_VEBT,D: nat] :
% 0.33/0.68 ( ( vEBT_VEBT_valid @ T @ D )
% 0.33/0.68 => ( vEBT_invar_vebt @ T @ D ) ) ).
% 0.33/0.68
% 0.33/0.68 % valid_eq2
% 0.33/0.68 thf(fact_9604_valid__eq1,axiom,
% 0.33/0.68 ! [T: vEBT_VEBT,D: nat] :
% 0.33/0.68 ( ( vEBT_invar_vebt @ T @ D )
% 0.33/0.68 => ( vEBT_VEBT_valid @ T @ D ) ) ).
% 0.33/0.68
% 0.33/0.68 % valid_eq1
% 0.33/0.68 thf(fact_9605_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
% 0.33/0.68 ! [Uu: $o,Uv: $o,D: nat] :
% 0.33/0.68 ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D )
% 0.33/0.68 = ( D = one_one_nat ) ) ).
% 0.33/0.68
% 0.33/0.68 % VEBT_internal.valid'.simps(1)
% 0.33/0.68 thf(fact_9606_VEBT_Osize__gen_I2_J,axiom,
% 0.33/0.68 ! [X21: $o,X222: $o] :
% 0.33/0.68 ( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 0.33/0.68 = zero_zero_nat ) ).
% 0.33/0.68
% 0.33/0.68 % VEBT.size_gen(2)
% 0.33/0.68 thf(fact_9607_csqrt_Osimps_I1_J,axiom,
% 0.33/0.68 ! [Z: complex] :
% 0.33/0.68 ( ( re @ ( csqrt @ Z ) )
% 0.33/0.68 = ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % csqrt.simps(1)
% 0.33/0.68 thf(fact_9608_divmod__step__integer__def,axiom,
% 0.33/0.68 ( unique4921790084139445826nteger
% 0.33/0.68 = ( ^ [L: num] :
% 0.33/0.68 ( produc6916734918728496179nteger
% 0.33/0.68 @ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % divmod_step_integer_def
% 0.33/0.68 thf(fact_9609_complex__Re__numeral,axiom,
% 0.33/0.68 ! [V: num] :
% 0.33/0.68 ( ( re @ ( numera6690914467698888265omplex @ V ) )
% 0.33/0.68 = ( numeral_numeral_real @ V ) ) ).
% 0.33/0.68
% 0.33/0.68 % complex_Re_numeral
% 0.33/0.68 thf(fact_9610_Re__divide__of__nat,axiom,
% 0.33/0.68 ! [Z: complex,N2: nat] :
% 0.33/0.68 ( ( re @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N2 ) ) )
% 0.33/0.68 = ( divide_divide_real @ ( re @ Z ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Re_divide_of_nat
% 0.33/0.68 thf(fact_9611_Re__divide__of__real,axiom,
% 0.33/0.68 ! [Z: complex,R: real] :
% 0.33/0.68 ( ( re @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R ) ) )
% 0.33/0.68 = ( divide_divide_real @ ( re @ Z ) @ R ) ) ).
% 0.33/0.68
% 0.33/0.68 % Re_divide_of_real
% 0.33/0.68 thf(fact_9612_Re__sgn,axiom,
% 0.33/0.68 ! [Z: complex] :
% 0.33/0.68 ( ( re @ ( sgn_sgn_complex @ Z ) )
% 0.33/0.68 = ( divide_divide_real @ ( re @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Re_sgn
% 0.33/0.68 thf(fact_9613_Re__divide__numeral,axiom,
% 0.33/0.68 ! [Z: complex,W: num] :
% 0.33/0.68 ( ( re @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
% 0.33/0.68 = ( divide_divide_real @ ( re @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Re_divide_numeral
% 0.33/0.68 thf(fact_9614_divmod__integer_H__def,axiom,
% 0.33/0.68 ( unique3479559517661332726nteger
% 0.33/0.68 = ( ^ [M6: num,N: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % divmod_integer'_def
% 0.33/0.68 thf(fact_9615_sgn__integer__code,axiom,
% 0.33/0.68 ( sgn_sgn_Code_integer
% 0.33/0.68 = ( ^ [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 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % sgn_integer_code
% 0.33/0.68 thf(fact_9616_sums__Re,axiom,
% 0.33/0.68 ! [X8: nat > complex,A: complex] :
% 0.33/0.68 ( ( sums_complex @ X8 @ A )
% 0.33/0.68 => ( sums_real
% 0.33/0.68 @ ^ [N: nat] : ( re @ ( X8 @ N ) )
% 0.33/0.68 @ ( re @ A ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % sums_Re
% 0.33/0.68 thf(fact_9617_minus__integer__code_I1_J,axiom,
% 0.33/0.68 ! [K: code_integer] :
% 0.33/0.68 ( ( minus_8373710615458151222nteger @ K @ zero_z3403309356797280102nteger )
% 0.33/0.68 = K ) ).
% 0.33/0.68
% 0.33/0.68 % minus_integer_code(1)
% 0.33/0.68 thf(fact_9618_minus__integer__code_I2_J,axiom,
% 0.33/0.68 ! [L2: code_integer] :
% 0.33/0.68 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L2 )
% 0.33/0.68 = ( uminus1351360451143612070nteger @ L2 ) ) ).
% 0.33/0.68
% 0.33/0.68 % minus_integer_code(2)
% 0.33/0.68 thf(fact_9619_Cauchy__Re,axiom,
% 0.33/0.68 ! [X8: nat > complex] :
% 0.33/0.68 ( ( topolo6517432010174082258omplex @ X8 )
% 0.33/0.68 => ( topolo4055970368930404560y_real
% 0.33/0.68 @ ^ [N: nat] : ( re @ ( X8 @ N ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Cauchy_Re
% 0.33/0.68 thf(fact_9620_complex__Re__le__cmod,axiom,
% 0.33/0.68 ! [X: complex] : ( ord_less_eq_real @ ( re @ X ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 0.33/0.68
% 0.33/0.68 % complex_Re_le_cmod
% 0.33/0.68 thf(fact_9621_one__complex_Osimps_I1_J,axiom,
% 0.33/0.68 ( ( re @ one_one_complex )
% 0.33/0.68 = one_one_real ) ).
% 0.33/0.68
% 0.33/0.68 % one_complex.simps(1)
% 0.33/0.68 thf(fact_9622_plus__complex_Osimps_I1_J,axiom,
% 0.33/0.68 ! [X: complex,Y: complex] :
% 0.33/0.68 ( ( re @ ( plus_plus_complex @ X @ Y ) )
% 0.33/0.68 = ( plus_plus_real @ ( re @ X ) @ ( re @ Y ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % plus_complex.simps(1)
% 0.33/0.68 thf(fact_9623_scaleR__complex_Osimps_I1_J,axiom,
% 0.33/0.68 ! [R: real,X: complex] :
% 0.33/0.68 ( ( re @ ( real_V2046097035970521341omplex @ R @ X ) )
% 0.33/0.68 = ( times_times_real @ R @ ( re @ X ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % scaleR_complex.simps(1)
% 0.33/0.68 thf(fact_9624_minus__complex_Osimps_I1_J,axiom,
% 0.33/0.68 ! [X: complex,Y: complex] :
% 0.33/0.68 ( ( re @ ( minus_minus_complex @ X @ Y ) )
% 0.33/0.68 = ( minus_minus_real @ ( re @ X ) @ ( re @ Y ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % minus_complex.simps(1)
% 0.33/0.68 thf(fact_9625_summable__Re,axiom,
% 0.33/0.68 ! [F: nat > complex] :
% 0.33/0.68 ( ( summable_complex @ F )
% 0.33/0.68 => ( summable_real
% 0.33/0.68 @ ^ [X2: nat] : ( re @ ( F @ X2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % summable_Re
% 0.33/0.68 thf(fact_9626_abs__Re__le__cmod,axiom,
% 0.33/0.68 ! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 0.33/0.68
% 0.33/0.68 % abs_Re_le_cmod
% 0.33/0.68 thf(fact_9627_Re__csqrt,axiom,
% 0.33/0.68 ! [Z: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Re_csqrt
% 0.33/0.68 thf(fact_9628_one__integer_Orsp,axiom,
% 0.33/0.68 one_one_int = one_one_int ).
% 0.33/0.68
% 0.33/0.68 % one_integer.rsp
% 0.33/0.68 thf(fact_9629_one__natural_Orsp,axiom,
% 0.33/0.68 one_one_nat = one_one_nat ).
% 0.33/0.68
% 0.33/0.68 % one_natural.rsp
% 0.33/0.68 thf(fact_9630_cmod__plus__Re__le__0__iff,axiom,
% 0.33/0.68 ! [Z: complex] :
% 0.33/0.68 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ zero_zero_real )
% 0.33/0.68 = ( ( re @ Z )
% 0.33/0.68 = ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % cmod_plus_Re_le_0_iff
% 0.33/0.68 thf(fact_9631_cos__n__Re__cis__pow__n,axiom,
% 0.33/0.68 ! [N2: nat,A: real] :
% 0.33/0.68 ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ A ) )
% 0.33/0.68 = ( re @ ( power_power_complex @ ( cis @ A ) @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % cos_n_Re_cis_pow_n
% 0.33/0.68 thf(fact_9632_csqrt_Ocode,axiom,
% 0.33/0.68 ( csqrt
% 0.33/0.68 = ( ^ [Z3: complex] :
% 0.33/0.68 ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z3 ) @ ( re @ Z3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.68 @ ( times_times_real
% 0.33/0.68 @ ( if_real
% 0.33/0.68 @ ( ( im @ Z3 )
% 0.33/0.68 = zero_zero_real )
% 0.33/0.68 @ one_one_real
% 0.33/0.68 @ ( sgn_sgn_real @ ( im @ Z3 ) ) )
% 0.33/0.68 @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z3 ) @ ( re @ Z3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % csqrt.code
% 0.33/0.68 thf(fact_9633_csqrt_Osimps_I2_J,axiom,
% 0.33/0.68 ! [Z: complex] :
% 0.33/0.68 ( ( im @ ( csqrt @ Z ) )
% 0.33/0.68 = ( times_times_real
% 0.33/0.68 @ ( if_real
% 0.33/0.68 @ ( ( im @ Z )
% 0.33/0.68 = zero_zero_real )
% 0.33/0.68 @ one_one_real
% 0.33/0.68 @ ( sgn_sgn_real @ ( im @ Z ) ) )
% 0.33/0.68 @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % csqrt.simps(2)
% 0.33/0.68 thf(fact_9634_integer__of__int__code,axiom,
% 0.33/0.68 ( code_integer_of_int
% 0.33/0.68 = ( ^ [K3: int] :
% 0.33/0.68 ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K3 ) ) )
% 0.33/0.68 @ ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 0.33/0.68 @ ( if_Code_integer
% 0.33/0.68 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.33/0.68 = zero_zero_int )
% 0.33/0.68 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.33/0.68 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % integer_of_int_code
% 0.33/0.68 thf(fact_9635_Im__divide__of__real,axiom,
% 0.33/0.68 ! [Z: complex,R: real] :
% 0.33/0.68 ( ( im @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R ) ) )
% 0.33/0.68 = ( divide_divide_real @ ( im @ Z ) @ R ) ) ).
% 0.33/0.68
% 0.33/0.68 % Im_divide_of_real
% 0.33/0.68 thf(fact_9636_Im__sgn,axiom,
% 0.33/0.68 ! [Z: complex] :
% 0.33/0.68 ( ( im @ ( sgn_sgn_complex @ Z ) )
% 0.33/0.68 = ( divide_divide_real @ ( im @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Im_sgn
% 0.33/0.68 thf(fact_9637_Re__power__real,axiom,
% 0.33/0.68 ! [X: complex,N2: nat] :
% 0.33/0.68 ( ( ( im @ X )
% 0.33/0.68 = zero_zero_real )
% 0.33/0.68 => ( ( re @ ( power_power_complex @ X @ N2 ) )
% 0.33/0.68 = ( power_power_real @ ( re @ X ) @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Re_power_real
% 0.33/0.68 thf(fact_9638_Im__divide__numeral,axiom,
% 0.33/0.68 ! [Z: complex,W: num] :
% 0.33/0.68 ( ( im @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
% 0.33/0.68 = ( divide_divide_real @ ( im @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Im_divide_numeral
% 0.33/0.68 thf(fact_9639_Im__divide__of__nat,axiom,
% 0.33/0.68 ! [Z: complex,N2: nat] :
% 0.33/0.68 ( ( im @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N2 ) ) )
% 0.33/0.68 = ( divide_divide_real @ ( im @ Z ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Im_divide_of_nat
% 0.33/0.68 thf(fact_9640_csqrt__of__real__nonneg,axiom,
% 0.33/0.68 ! [X: complex] :
% 0.33/0.68 ( ( ( im @ X )
% 0.33/0.68 = zero_zero_real )
% 0.33/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) )
% 0.33/0.68 => ( ( csqrt @ X )
% 0.33/0.68 = ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % csqrt_of_real_nonneg
% 0.33/0.68 thf(fact_9641_csqrt__minus,axiom,
% 0.33/0.68 ! [X: complex] :
% 0.33/0.68 ( ( ( ord_less_real @ ( im @ X ) @ zero_zero_real )
% 0.33/0.68 | ( ( ( im @ X )
% 0.33/0.68 = zero_zero_real )
% 0.33/0.68 & ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) ) ) )
% 0.33/0.68 => ( ( csqrt @ ( uminus1482373934393186551omplex @ X ) )
% 0.33/0.68 = ( times_times_complex @ imaginary_unit @ ( csqrt @ X ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % csqrt_minus
% 0.33/0.68 thf(fact_9642_csqrt__of__real__nonpos,axiom,
% 0.33/0.68 ! [X: complex] :
% 0.33/0.68 ( ( ( im @ X )
% 0.33/0.68 = zero_zero_real )
% 0.33/0.68 => ( ( ord_less_eq_real @ ( re @ X ) @ zero_zero_real )
% 0.33/0.68 => ( ( csqrt @ X )
% 0.33/0.68 = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X ) ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % csqrt_of_real_nonpos
% 0.33/0.68 thf(fact_9643_divide__integer_Oabs__eq,axiom,
% 0.33/0.68 ! [Xa2: int,X: int] :
% 0.33/0.68 ( ( divide6298287555418463151nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
% 0.33/0.68 = ( code_integer_of_int @ ( divide_divide_int @ Xa2 @ X ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % divide_integer.abs_eq
% 0.33/0.68 thf(fact_9644_less__integer_Oabs__eq,axiom,
% 0.33/0.68 ! [Xa2: int,X: int] :
% 0.33/0.68 ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
% 0.33/0.68 = ( ord_less_int @ Xa2 @ X ) ) ).
% 0.33/0.68
% 0.33/0.68 % less_integer.abs_eq
% 0.33/0.68 thf(fact_9645_abs__integer__code,axiom,
% 0.33/0.68 ( abs_abs_Code_integer
% 0.33/0.68 = ( ^ [K3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K3 ) @ K3 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % abs_integer_code
% 0.33/0.68 thf(fact_9646_less__integer__code_I1_J,axiom,
% 0.33/0.68 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).
% 0.33/0.68
% 0.33/0.68 % less_integer_code(1)
% 0.33/0.68 thf(fact_9647_sums__Im,axiom,
% 0.33/0.68 ! [X8: nat > complex,A: complex] :
% 0.33/0.68 ( ( sums_complex @ X8 @ A )
% 0.33/0.68 => ( sums_real
% 0.33/0.68 @ ^ [N: nat] : ( im @ ( X8 @ N ) )
% 0.33/0.68 @ ( im @ A ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % sums_Im
% 0.33/0.68 thf(fact_9648_Cauchy__Im,axiom,
% 0.33/0.68 ! [X8: nat > complex] :
% 0.33/0.68 ( ( topolo6517432010174082258omplex @ X8 )
% 0.33/0.68 => ( topolo4055970368930404560y_real
% 0.33/0.68 @ ^ [N: nat] : ( im @ ( X8 @ N ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Cauchy_Im
% 0.33/0.68 thf(fact_9649_imaginary__unit_Osimps_I2_J,axiom,
% 0.33/0.68 ( ( im @ imaginary_unit )
% 0.33/0.68 = one_one_real ) ).
% 0.33/0.68
% 0.33/0.68 % imaginary_unit.simps(2)
% 0.33/0.68 thf(fact_9650_one__complex_Osimps_I2_J,axiom,
% 0.33/0.68 ( ( im @ one_one_complex )
% 0.33/0.68 = zero_zero_real ) ).
% 0.33/0.68
% 0.33/0.68 % one_complex.simps(2)
% 0.33/0.68 thf(fact_9651_one__integer__def,axiom,
% 0.33/0.68 ( one_one_Code_integer
% 0.33/0.68 = ( code_integer_of_int @ one_one_int ) ) ).
% 0.33/0.68
% 0.33/0.68 % one_integer_def
% 0.33/0.68 thf(fact_9652_minus__integer_Oabs__eq,axiom,
% 0.33/0.68 ! [Xa2: int,X: int] :
% 0.33/0.68 ( ( minus_8373710615458151222nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
% 0.33/0.68 = ( code_integer_of_int @ ( minus_minus_int @ Xa2 @ X ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % minus_integer.abs_eq
% 0.33/0.68 thf(fact_9653_plus__complex_Osimps_I2_J,axiom,
% 0.33/0.68 ! [X: complex,Y: complex] :
% 0.33/0.68 ( ( im @ ( plus_plus_complex @ X @ Y ) )
% 0.33/0.68 = ( plus_plus_real @ ( im @ X ) @ ( im @ Y ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % plus_complex.simps(2)
% 0.33/0.68 thf(fact_9654_scaleR__complex_Osimps_I2_J,axiom,
% 0.33/0.68 ! [R: real,X: complex] :
% 0.33/0.68 ( ( im @ ( real_V2046097035970521341omplex @ R @ X ) )
% 0.33/0.68 = ( times_times_real @ R @ ( im @ X ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % scaleR_complex.simps(2)
% 0.33/0.68 thf(fact_9655_minus__complex_Osimps_I2_J,axiom,
% 0.33/0.68 ! [X: complex,Y: complex] :
% 0.33/0.68 ( ( im @ ( minus_minus_complex @ X @ Y ) )
% 0.33/0.68 = ( minus_minus_real @ ( im @ X ) @ ( im @ Y ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % minus_complex.simps(2)
% 0.33/0.68 thf(fact_9656_sums__complex__iff,axiom,
% 0.33/0.68 ( sums_complex
% 0.33/0.68 = ( ^ [F3: nat > complex,X2: complex] :
% 0.33/0.68 ( ( sums_real
% 0.33/0.68 @ ^ [Y2: nat] : ( re @ ( F3 @ Y2 ) )
% 0.33/0.68 @ ( re @ X2 ) )
% 0.33/0.68 & ( sums_real
% 0.33/0.68 @ ^ [Y2: nat] : ( im @ ( F3 @ Y2 ) )
% 0.33/0.68 @ ( im @ X2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % sums_complex_iff
% 0.33/0.68 thf(fact_9657_summable__Im,axiom,
% 0.33/0.68 ! [F: nat > complex] :
% 0.33/0.68 ( ( summable_complex @ F )
% 0.33/0.68 => ( summable_real
% 0.33/0.68 @ ^ [X2: nat] : ( im @ ( F @ X2 ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % summable_Im
% 0.33/0.68 thf(fact_9658_abs__Im__le__cmod,axiom,
% 0.33/0.68 ! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 0.33/0.68
% 0.33/0.68 % abs_Im_le_cmod
% 0.33/0.68 thf(fact_9659_summable__complex__iff,axiom,
% 0.33/0.68 ( summable_complex
% 0.33/0.68 = ( ^ [F3: nat > complex] :
% 0.33/0.68 ( ( summable_real
% 0.33/0.68 @ ^ [X2: nat] : ( re @ ( F3 @ X2 ) ) )
% 0.33/0.68 & ( summable_real
% 0.33/0.68 @ ^ [X2: nat] : ( im @ ( F3 @ X2 ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % summable_complex_iff
% 0.33/0.68 thf(fact_9660_times__complex_Osimps_I2_J,axiom,
% 0.33/0.68 ! [X: complex,Y: complex] :
% 0.33/0.68 ( ( im @ ( times_times_complex @ X @ Y ) )
% 0.33/0.68 = ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % times_complex.simps(2)
% 0.33/0.68 thf(fact_9661_cmod__Im__le__iff,axiom,
% 0.33/0.68 ! [X: complex,Y: complex] :
% 0.33/0.68 ( ( ( re @ X )
% 0.33/0.68 = ( re @ Y ) )
% 0.33/0.68 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) )
% 0.33/0.68 = ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( abs_abs_real @ ( im @ Y ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % cmod_Im_le_iff
% 0.33/0.68 thf(fact_9662_cmod__Re__le__iff,axiom,
% 0.33/0.68 ! [X: complex,Y: complex] :
% 0.33/0.68 ( ( ( im @ X )
% 0.33/0.68 = ( im @ Y ) )
% 0.33/0.68 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) )
% 0.33/0.68 = ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( abs_abs_real @ ( re @ Y ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % cmod_Re_le_iff
% 0.33/0.68 thf(fact_9663_times__complex_Osimps_I1_J,axiom,
% 0.33/0.68 ! [X: complex,Y: complex] :
% 0.33/0.68 ( ( re @ ( times_times_complex @ X @ Y ) )
% 0.33/0.68 = ( minus_minus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % times_complex.simps(1)
% 0.33/0.68 thf(fact_9664_plus__complex_Ocode,axiom,
% 0.33/0.68 ( plus_plus_complex
% 0.33/0.68 = ( ^ [X2: complex,Y2: complex] : ( complex2 @ ( plus_plus_real @ ( re @ X2 ) @ ( re @ Y2 ) ) @ ( plus_plus_real @ ( im @ X2 ) @ ( im @ Y2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % plus_complex.code
% 0.33/0.68 thf(fact_9665_scaleR__complex_Ocode,axiom,
% 0.33/0.68 ( real_V2046097035970521341omplex
% 0.33/0.68 = ( ^ [R5: real,X2: complex] : ( complex2 @ ( times_times_real @ R5 @ ( re @ X2 ) ) @ ( times_times_real @ R5 @ ( im @ X2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % scaleR_complex.code
% 0.33/0.68 thf(fact_9666_minus__complex_Ocode,axiom,
% 0.33/0.68 ( minus_minus_complex
% 0.33/0.68 = ( ^ [X2: complex,Y2: complex] : ( complex2 @ ( minus_minus_real @ ( re @ X2 ) @ ( re @ Y2 ) ) @ ( minus_minus_real @ ( im @ X2 ) @ ( im @ Y2 ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % minus_complex.code
% 0.33/0.68 thf(fact_9667_csqrt__principal,axiom,
% 0.33/0.68 ! [Z: complex] :
% 0.33/0.68 ( ( ord_less_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) )
% 0.33/0.68 | ( ( ( re @ ( csqrt @ Z ) )
% 0.33/0.68 = zero_zero_real )
% 0.33/0.68 & ( ord_less_eq_real @ zero_zero_real @ ( im @ ( csqrt @ Z ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % csqrt_principal
% 0.33/0.68 thf(fact_9668_cmod__le,axiom,
% 0.33/0.68 ! [Z: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % cmod_le
% 0.33/0.68 thf(fact_9669_sin__n__Im__cis__pow__n,axiom,
% 0.33/0.68 ! [N2: nat,A: real] :
% 0.33/0.68 ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ A ) )
% 0.33/0.68 = ( im @ ( power_power_complex @ ( cis @ A ) @ N2 ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % sin_n_Im_cis_pow_n
% 0.33/0.68 thf(fact_9670_Re__exp,axiom,
% 0.33/0.68 ! [Z: complex] :
% 0.33/0.68 ( ( re @ ( exp_complex @ Z ) )
% 0.33/0.68 = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( cos_real @ ( im @ Z ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Re_exp
% 0.33/0.68 thf(fact_9671_Im__exp,axiom,
% 0.33/0.68 ! [Z: complex] :
% 0.33/0.68 ( ( im @ ( exp_complex @ Z ) )
% 0.33/0.68 = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( sin_real @ ( im @ Z ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Im_exp
% 0.33/0.68 thf(fact_9672_times__complex_Ocode,axiom,
% 0.33/0.68 ( times_times_complex
% 0.33/0.68 = ( ^ [X2: complex,Y2: complex] : ( complex2 @ ( minus_minus_real @ ( times_times_real @ ( re @ X2 ) @ ( re @ Y2 ) ) @ ( times_times_real @ ( im @ X2 ) @ ( im @ Y2 ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X2 ) @ ( im @ Y2 ) ) @ ( times_times_real @ ( im @ X2 ) @ ( re @ Y2 ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % times_complex.code
% 0.33/0.68 thf(fact_9673_cmod__power2,axiom,
% 0.33/0.68 ! [Z: complex] :
% 0.33/0.68 ( ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.68 = ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % cmod_power2
% 0.33/0.68 thf(fact_9674_Im__power2,axiom,
% 0.33/0.68 ! [X: complex] :
% 0.33/0.68 ( ( im @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.68 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ X ) ) @ ( im @ X ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Im_power2
% 0.33/0.68 thf(fact_9675_Re__power2,axiom,
% 0.33/0.68 ! [X: complex] :
% 0.33/0.68 ( ( re @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.68 = ( minus_minus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Re_power2
% 0.33/0.68 thf(fact_9676_complex__eq__0,axiom,
% 0.33/0.68 ! [Z: complex] :
% 0.33/0.68 ( ( Z = zero_zero_complex )
% 0.33/0.68 = ( ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.68 = zero_zero_real ) ) ).
% 0.33/0.68
% 0.33/0.68 % complex_eq_0
% 0.33/0.68 thf(fact_9677_norm__complex__def,axiom,
% 0.33/0.68 ( real_V1022390504157884413omplex
% 0.33/0.68 = ( ^ [Z3: complex] : ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % norm_complex_def
% 0.33/0.68 thf(fact_9678_inverse__complex_Osimps_I1_J,axiom,
% 0.33/0.68 ! [X: complex] :
% 0.33/0.68 ( ( re @ ( invers8013647133539491842omplex @ X ) )
% 0.33/0.68 = ( divide_divide_real @ ( re @ X ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % inverse_complex.simps(1)
% 0.33/0.68 thf(fact_9679_complex__neq__0,axiom,
% 0.33/0.68 ! [Z: complex] :
% 0.33/0.68 ( ( Z != zero_zero_complex )
% 0.33/0.68 = ( 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 ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % complex_neq_0
% 0.33/0.68 thf(fact_9680_Re__divide,axiom,
% 0.33/0.68 ! [X: complex,Y: complex] :
% 0.33/0.68 ( ( re @ ( divide1717551699836669952omplex @ X @ Y ) )
% 0.33/0.68 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % Re_divide
% 0.33/0.68 thf(fact_9681_csqrt__unique,axiom,
% 0.33/0.68 ! [W: complex,Z: complex] :
% 0.33/0.68 ( ( ( power_power_complex @ W @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.33/0.68 = Z )
% 0.33/0.68 => ( ( ( ord_less_real @ zero_zero_real @ ( re @ W ) )
% 0.33/0.68 | ( ( ( re @ W )
% 0.33/0.68 = zero_zero_real )
% 0.33/0.68 & ( ord_less_eq_real @ zero_zero_real @ ( im @ W ) ) ) )
% 0.33/0.68 => ( ( csqrt @ Z )
% 0.33/0.68 = W ) ) ) ).
% 0.33/0.68
% 0.33/0.68 % csqrt_unique
% 0.33/0.68 thf(fact_9682_csqrt__square,axiom,
% 0.33/0.69 ! [B: complex] :
% 0.33/0.69 ( ( ( ord_less_real @ zero_zero_real @ ( re @ B ) )
% 0.33/0.69 | ( ( ( re @ B )
% 0.33/0.69 = zero_zero_real )
% 0.33/0.69 & ( ord_less_eq_real @ zero_zero_real @ ( im @ B ) ) ) )
% 0.33/0.69 => ( ( csqrt @ ( power_power_complex @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.69 = B ) ) ).
% 0.33/0.69
% 0.33/0.69 % csqrt_square
% 0.33/0.69 thf(fact_9683_inverse__complex_Osimps_I2_J,axiom,
% 0.33/0.69 ! [X: complex] :
% 0.33/0.69 ( ( im @ ( invers8013647133539491842omplex @ X ) )
% 0.33/0.69 = ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % inverse_complex.simps(2)
% 0.33/0.69 thf(fact_9684_Im__divide,axiom,
% 0.33/0.69 ! [X: complex,Y: complex] :
% 0.33/0.69 ( ( im @ ( divide1717551699836669952omplex @ X @ Y ) )
% 0.33/0.69 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Im_divide
% 0.33/0.69 thf(fact_9685_complex__abs__le__norm,axiom,
% 0.33/0.69 ! [Z: complex] : ( ord_less_eq_real @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) @ ( times_times_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % complex_abs_le_norm
% 0.33/0.69 thf(fact_9686_complex__unit__circle,axiom,
% 0.33/0.69 ! [Z: complex] :
% 0.33/0.69 ( ( Z != zero_zero_complex )
% 0.33/0.69 => ( ( plus_plus_real @ ( power_power_real @ ( divide_divide_real @ ( re @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( divide_divide_real @ ( im @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.69 = one_one_real ) ) ).
% 0.33/0.69
% 0.33/0.69 % complex_unit_circle
% 0.33/0.69 thf(fact_9687_inverse__complex_Ocode,axiom,
% 0.33/0.69 ( invers8013647133539491842omplex
% 0.33/0.69 = ( ^ [X2: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X2 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X2 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % inverse_complex.code
% 0.33/0.69 thf(fact_9688_Complex__divide,axiom,
% 0.33/0.69 ( divide1717551699836669952omplex
% 0.33/0.69 = ( ^ [X2: complex,Y2: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X2 ) @ ( re @ Y2 ) ) @ ( times_times_real @ ( im @ X2 ) @ ( im @ Y2 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X2 ) @ ( re @ Y2 ) ) @ ( times_times_real @ ( re @ X2 ) @ ( im @ Y2 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Complex_divide
% 0.33/0.69 thf(fact_9689_Im__Reals__divide,axiom,
% 0.33/0.69 ! [R: complex,Z: complex] :
% 0.33/0.69 ( ( member_complex @ R @ real_V2521375963428798218omplex )
% 0.33/0.69 => ( ( im @ ( divide1717551699836669952omplex @ R @ Z ) )
% 0.33/0.69 = ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R ) ) @ ( im @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Im_Reals_divide
% 0.33/0.69 thf(fact_9690_Re__Reals__divide,axiom,
% 0.33/0.69 ! [R: complex,Z: complex] :
% 0.33/0.69 ( ( member_complex @ R @ real_V2521375963428798218omplex )
% 0.33/0.69 => ( ( re @ ( divide1717551699836669952omplex @ R @ Z ) )
% 0.33/0.69 = ( divide_divide_real @ ( times_times_real @ ( re @ R ) @ ( re @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Re_Reals_divide
% 0.33/0.69 thf(fact_9691_Re__divide__Reals,axiom,
% 0.33/0.69 ! [R: complex,Z: complex] :
% 0.33/0.69 ( ( member_complex @ R @ real_V2521375963428798218omplex )
% 0.33/0.69 => ( ( re @ ( divide1717551699836669952omplex @ Z @ R ) )
% 0.33/0.69 = ( divide_divide_real @ ( re @ Z ) @ ( re @ R ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Re_divide_Reals
% 0.33/0.69 thf(fact_9692_Im__divide__Reals,axiom,
% 0.33/0.69 ! [R: complex,Z: complex] :
% 0.33/0.69 ( ( member_complex @ R @ real_V2521375963428798218omplex )
% 0.33/0.69 => ( ( im @ ( divide1717551699836669952omplex @ Z @ R ) )
% 0.33/0.69 = ( divide_divide_real @ ( im @ Z ) @ ( re @ R ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Im_divide_Reals
% 0.33/0.69 thf(fact_9693_complex__diff__cnj,axiom,
% 0.33/0.69 ! [Z: complex] :
% 0.33/0.69 ( ( minus_minus_complex @ Z @ ( cnj @ Z ) )
% 0.33/0.69 = ( times_times_complex @ ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( im @ Z ) ) ) @ imaginary_unit ) ) ).
% 0.33/0.69
% 0.33/0.69 % complex_diff_cnj
% 0.33/0.69 thf(fact_9694_complex__mult__cnj,axiom,
% 0.33/0.69 ! [Z: complex] :
% 0.33/0.69 ( ( times_times_complex @ Z @ ( cnj @ Z ) )
% 0.33/0.69 = ( real_V4546457046886955230omplex @ ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % complex_mult_cnj
% 0.33/0.69 thf(fact_9695_complex__cnj__divide,axiom,
% 0.33/0.69 ! [X: complex,Y: complex] :
% 0.33/0.69 ( ( cnj @ ( divide1717551699836669952omplex @ X @ Y ) )
% 0.33/0.69 = ( divide1717551699836669952omplex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % complex_cnj_divide
% 0.33/0.69 thf(fact_9696_complex__cnj__one,axiom,
% 0.33/0.69 ( ( cnj @ one_one_complex )
% 0.33/0.69 = one_one_complex ) ).
% 0.33/0.69
% 0.33/0.69 % complex_cnj_one
% 0.33/0.69 thf(fact_9697_complex__cnj__one__iff,axiom,
% 0.33/0.69 ! [Z: complex] :
% 0.33/0.69 ( ( ( cnj @ Z )
% 0.33/0.69 = one_one_complex )
% 0.33/0.69 = ( Z = one_one_complex ) ) ).
% 0.33/0.69
% 0.33/0.69 % complex_cnj_one_iff
% 0.33/0.69 thf(fact_9698_complex__cnj__diff,axiom,
% 0.33/0.69 ! [X: complex,Y: complex] :
% 0.33/0.69 ( ( cnj @ ( minus_minus_complex @ X @ Y ) )
% 0.33/0.69 = ( minus_minus_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % complex_cnj_diff
% 0.33/0.69 thf(fact_9699_sums__cnj,axiom,
% 0.33/0.69 ! [F: nat > complex,L2: complex] :
% 0.33/0.69 ( ( sums_complex
% 0.33/0.69 @ ^ [X2: nat] : ( cnj @ ( F @ X2 ) )
% 0.33/0.69 @ ( cnj @ L2 ) )
% 0.33/0.69 = ( sums_complex @ F @ L2 ) ) ).
% 0.33/0.69
% 0.33/0.69 % sums_cnj
% 0.33/0.69 thf(fact_9700_Re__complex__div__eq__0,axiom,
% 0.33/0.69 ! [A: complex,B: complex] :
% 0.33/0.69 ( ( ( re @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.33/0.69 = zero_zero_real )
% 0.33/0.69 = ( ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) )
% 0.33/0.69 = zero_zero_real ) ) ).
% 0.33/0.69
% 0.33/0.69 % Re_complex_div_eq_0
% 0.33/0.69 thf(fact_9701_Im__complex__div__eq__0,axiom,
% 0.33/0.69 ! [A: complex,B: complex] :
% 0.33/0.69 ( ( ( im @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.33/0.69 = zero_zero_real )
% 0.33/0.69 = ( ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) )
% 0.33/0.69 = zero_zero_real ) ) ).
% 0.33/0.69
% 0.33/0.69 % Im_complex_div_eq_0
% 0.33/0.69 thf(fact_9702_Re__complex__div__gt__0,axiom,
% 0.33/0.69 ! [A: complex,B: complex] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.33/0.69 = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Re_complex_div_gt_0
% 0.33/0.69 thf(fact_9703_Re__complex__div__lt__0,axiom,
% 0.33/0.69 ! [A: complex,B: complex] :
% 0.33/0.69 ( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 0.33/0.69 = ( ord_less_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 0.33/0.69
% 0.33/0.69 % Re_complex_div_lt_0
% 0.33/0.69 thf(fact_9704_Re__complex__div__le__0,axiom,
% 0.33/0.69 ! [A: complex,B: complex] :
% 0.33/0.69 ( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 0.33/0.69 = ( ord_less_eq_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 0.33/0.69
% 0.33/0.69 % Re_complex_div_le_0
% 0.33/0.69 thf(fact_9705_Re__complex__div__ge__0,axiom,
% 0.33/0.69 ! [A: complex,B: complex] :
% 0.33/0.69 ( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.33/0.69 = ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Re_complex_div_ge_0
% 0.33/0.69 thf(fact_9706_Im__complex__div__gt__0,axiom,
% 0.33/0.69 ! [A: complex,B: complex] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.33/0.69 = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Im_complex_div_gt_0
% 0.33/0.69 thf(fact_9707_Im__complex__div__lt__0,axiom,
% 0.33/0.69 ! [A: complex,B: complex] :
% 0.33/0.69 ( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 0.33/0.69 = ( ord_less_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 0.33/0.69
% 0.33/0.69 % Im_complex_div_lt_0
% 0.33/0.69 thf(fact_9708_Im__complex__div__le__0,axiom,
% 0.33/0.69 ! [A: complex,B: complex] :
% 0.33/0.69 ( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 0.33/0.69 = ( ord_less_eq_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 0.33/0.69
% 0.33/0.69 % Im_complex_div_le_0
% 0.33/0.69 thf(fact_9709_Im__complex__div__ge__0,axiom,
% 0.33/0.69 ! [A: complex,B: complex] :
% 0.33/0.69 ( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.33/0.69 = ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Im_complex_div_ge_0
% 0.33/0.69 thf(fact_9710_complex__mod__mult__cnj,axiom,
% 0.33/0.69 ! [Z: complex] :
% 0.33/0.69 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
% 0.33/0.69 = ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % complex_mod_mult_cnj
% 0.33/0.69 thf(fact_9711_complex__div__gt__0,axiom,
% 0.33/0.69 ! [A: complex,B: complex] :
% 0.33/0.69 ( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.33/0.69 = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) )
% 0.33/0.69 & ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.33/0.69 = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % complex_div_gt_0
% 0.33/0.69 thf(fact_9712_complex__norm__square,axiom,
% 0.33/0.69 ! [Z: complex] :
% 0.33/0.69 ( ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.33/0.69 = ( times_times_complex @ Z @ ( cnj @ Z ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % complex_norm_square
% 0.33/0.69 thf(fact_9713_complex__add__cnj,axiom,
% 0.33/0.69 ! [Z: complex] :
% 0.33/0.69 ( ( plus_plus_complex @ Z @ ( cnj @ Z ) )
% 0.33/0.69 = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ Z ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % complex_add_cnj
% 0.33/0.69 thf(fact_9714_complex__div__cnj,axiom,
% 0.33/0.69 ( divide1717551699836669952omplex
% 0.33/0.69 = ( ^ [A5: complex,B5: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A5 @ ( cnj @ B5 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % complex_div_cnj
% 0.33/0.69 thf(fact_9715_cnj__add__mult__eq__Re,axiom,
% 0.33/0.69 ! [Z: complex,W: complex] :
% 0.33/0.69 ( ( plus_plus_complex @ ( times_times_complex @ Z @ ( cnj @ W ) ) @ ( times_times_complex @ ( cnj @ Z ) @ W ) )
% 0.33/0.69 = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ ( times_times_complex @ Z @ ( cnj @ W ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % cnj_add_mult_eq_Re
% 0.33/0.69 thf(fact_9716_integer__of__num_I3_J,axiom,
% 0.33/0.69 ! [N2: num] :
% 0.33/0.69 ( ( code_integer_of_num @ ( bit1 @ N2 ) )
% 0.33/0.69 = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N2 ) @ ( code_integer_of_num @ N2 ) ) @ one_one_Code_integer ) ) ).
% 0.33/0.69
% 0.33/0.69 % integer_of_num(3)
% 0.33/0.69 thf(fact_9717_int__of__integer__code,axiom,
% 0.33/0.69 ( code_int_of_integer
% 0.33/0.69 = ( ^ [K3: code_integer] :
% 0.33/0.69 ( if_int @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K3 ) ) )
% 0.33/0.69 @ ( if_int @ ( K3 = zero_z3403309356797280102nteger ) @ zero_zero_int
% 0.33/0.69 @ ( produc1553301316500091796er_int
% 0.33/0.69 @ ^ [L: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L ) ) @ one_one_int ) )
% 0.33/0.69 @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % int_of_integer_code
% 0.33/0.69 thf(fact_9718_bit__cut__integer__def,axiom,
% 0.33/0.69 ( code_bit_cut_integer
% 0.33/0.69 = ( ^ [K3: code_integer] :
% 0.33/0.69 ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.33/0.69 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K3 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % bit_cut_integer_def
% 0.33/0.69 thf(fact_9719_one__integer_Orep__eq,axiom,
% 0.33/0.69 ( ( code_int_of_integer @ one_one_Code_integer )
% 0.33/0.69 = one_one_int ) ).
% 0.33/0.69
% 0.33/0.69 % one_integer.rep_eq
% 0.33/0.69 thf(fact_9720_minus__integer_Orep__eq,axiom,
% 0.33/0.69 ! [X: code_integer,Xa2: code_integer] :
% 0.33/0.69 ( ( code_int_of_integer @ ( minus_8373710615458151222nteger @ X @ Xa2 ) )
% 0.33/0.69 = ( minus_minus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % minus_integer.rep_eq
% 0.33/0.69 thf(fact_9721_divide__integer_Orep__eq,axiom,
% 0.33/0.69 ! [X: code_integer,Xa2: code_integer] :
% 0.33/0.69 ( ( code_int_of_integer @ ( divide6298287555418463151nteger @ X @ Xa2 ) )
% 0.33/0.69 = ( divide_divide_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % divide_integer.rep_eq
% 0.33/0.69 thf(fact_9722_less__integer_Orep__eq,axiom,
% 0.33/0.69 ( ord_le6747313008572928689nteger
% 0.33/0.69 = ( ^ [X2: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % less_integer.rep_eq
% 0.33/0.69 thf(fact_9723_integer__less__iff,axiom,
% 0.33/0.69 ( ord_le6747313008572928689nteger
% 0.33/0.69 = ( ^ [K3: code_integer,L: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % integer_less_iff
% 0.33/0.69 thf(fact_9724_integer__of__num__triv_I1_J,axiom,
% 0.33/0.69 ( ( code_integer_of_num @ one )
% 0.33/0.69 = one_one_Code_integer ) ).
% 0.33/0.69
% 0.33/0.69 % integer_of_num_triv(1)
% 0.33/0.69 thf(fact_9725_integer__of__num_I2_J,axiom,
% 0.33/0.69 ! [N2: num] :
% 0.33/0.69 ( ( code_integer_of_num @ ( bit0 @ N2 ) )
% 0.33/0.69 = ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N2 ) @ ( code_integer_of_num @ N2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % integer_of_num(2)
% 0.33/0.69 thf(fact_9726_integer__of__num__triv_I2_J,axiom,
% 0.33/0.69 ( ( code_integer_of_num @ ( bit0 @ one ) )
% 0.33/0.69 = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % integer_of_num_triv(2)
% 0.33/0.69 thf(fact_9727_divmod__integer__def,axiom,
% 0.33/0.69 ( code_divmod_integer
% 0.33/0.69 = ( ^ [K3: code_integer,L: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K3 @ L ) @ ( modulo364778990260209775nteger @ K3 @ L ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % divmod_integer_def
% 0.33/0.69 thf(fact_9728_num__of__integer__code,axiom,
% 0.33/0.69 ( code_num_of_integer
% 0.33/0.69 = ( ^ [K3: code_integer] :
% 0.33/0.69 ( if_num @ ( ord_le3102999989581377725nteger @ K3 @ one_one_Code_integer ) @ one
% 0.33/0.69 @ ( produc7336495610019696514er_num
% 0.33/0.69 @ ^ [L: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L ) @ ( code_num_of_integer @ L ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L ) @ ( code_num_of_integer @ L ) ) @ one ) )
% 0.33/0.69 @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % num_of_integer_code
% 0.33/0.69 thf(fact_9729_bit__cut__integer__code,axiom,
% 0.33/0.69 ( code_bit_cut_integer
% 0.33/0.69 = ( ^ [K3: code_integer] :
% 0.33/0.69 ( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
% 0.33/0.69 @ ( produc9125791028180074456eger_o
% 0.33/0.69 @ ^ [R5: code_integer,S5: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S5 ) ) @ ( S5 = one_one_Code_integer ) )
% 0.33/0.69 @ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % bit_cut_integer_code
% 0.33/0.69 thf(fact_9730_nat__of__integer__code,axiom,
% 0.33/0.69 ( code_nat_of_integer
% 0.33/0.69 = ( ^ [K3: code_integer] :
% 0.33/0.69 ( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
% 0.33/0.69 @ ( produc1555791787009142072er_nat
% 0.33/0.69 @ ^ [L: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L ) @ ( code_nat_of_integer @ L ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L ) @ ( code_nat_of_integer @ L ) ) @ one_one_nat ) )
% 0.33/0.69 @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % nat_of_integer_code
% 0.33/0.69 thf(fact_9731_nat__of__integer__code__post_I3_J,axiom,
% 0.33/0.69 ! [K: num] :
% 0.33/0.69 ( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ K ) )
% 0.33/0.69 = ( numeral_numeral_nat @ K ) ) ).
% 0.33/0.69
% 0.33/0.69 % nat_of_integer_code_post(3)
% 0.33/0.69 thf(fact_9732_nat__of__integer__code__post_I2_J,axiom,
% 0.33/0.69 ( ( code_nat_of_integer @ one_one_Code_integer )
% 0.33/0.69 = one_one_nat ) ).
% 0.33/0.69
% 0.33/0.69 % nat_of_integer_code_post(2)
% 0.33/0.69 thf(fact_9733_divmod__abs__def,axiom,
% 0.33/0.69 ( code_divmod_abs
% 0.33/0.69 = ( ^ [K3: code_integer,L: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % divmod_abs_def
% 0.33/0.69 thf(fact_9734_divmod__integer__code,axiom,
% 0.33/0.69 ( code_divmod_integer
% 0.33/0.69 = ( ^ [K3: code_integer,L: code_integer] :
% 0.33/0.69 ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 0.33/0.69 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L )
% 0.33/0.69 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L )
% 0.33/0.69 @ ( produc6916734918728496179nteger
% 0.33/0.69 @ ^ [R5: code_integer,S5: code_integer] : ( if_Pro6119634080678213985nteger @ ( S5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L @ S5 ) ) )
% 0.33/0.69 @ ( code_divmod_abs @ K3 @ L ) ) )
% 0.33/0.69 @ ( if_Pro6119634080678213985nteger @ ( L = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
% 0.33/0.69 @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
% 0.33/0.69 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L )
% 0.33/0.69 @ ( produc6916734918728496179nteger
% 0.33/0.69 @ ^ [R5: code_integer,S5: code_integer] : ( if_Pro6119634080678213985nteger @ ( S5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L ) @ S5 ) ) )
% 0.33/0.69 @ ( code_divmod_abs @ K3 @ L ) ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % divmod_integer_code
% 0.33/0.69 thf(fact_9735_card__Collect__less__nat,axiom,
% 0.33/0.69 ! [N2: nat] :
% 0.33/0.69 ( ( finite_card_nat
% 0.33/0.69 @ ( collect_nat
% 0.33/0.69 @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N2 ) ) )
% 0.33/0.69 = N2 ) ).
% 0.33/0.69
% 0.33/0.69 % card_Collect_less_nat
% 0.33/0.69 thf(fact_9736_card__atMost,axiom,
% 0.33/0.69 ! [U: nat] :
% 0.33/0.69 ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
% 0.33/0.69 = ( suc @ U ) ) ).
% 0.33/0.69
% 0.33/0.69 % card_atMost
% 0.33/0.69 thf(fact_9737_card__atLeastLessThan,axiom,
% 0.33/0.69 ! [L2: nat,U: nat] :
% 0.33/0.69 ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L2 @ U ) )
% 0.33/0.69 = ( minus_minus_nat @ U @ L2 ) ) ).
% 0.33/0.69
% 0.33/0.69 % card_atLeastLessThan
% 0.33/0.69 thf(fact_9738_card__Collect__le__nat,axiom,
% 0.33/0.69 ! [N2: nat] :
% 0.33/0.69 ( ( finite_card_nat
% 0.33/0.69 @ ( collect_nat
% 0.33/0.69 @ ^ [I3: nat] : ( ord_less_eq_nat @ I3 @ N2 ) ) )
% 0.33/0.69 = ( suc @ N2 ) ) ).
% 0.33/0.69
% 0.33/0.69 % card_Collect_le_nat
% 0.33/0.69 thf(fact_9739_card__atLeastAtMost,axiom,
% 0.33/0.69 ! [L2: nat,U: nat] :
% 0.33/0.69 ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L2 @ U ) )
% 0.33/0.69 = ( minus_minus_nat @ ( suc @ U ) @ L2 ) ) ).
% 0.33/0.69
% 0.33/0.69 % card_atLeastAtMost
% 0.33/0.69 thf(fact_9740_card__atLeastLessThan__int,axiom,
% 0.33/0.69 ! [L2: int,U: int] :
% 0.33/0.69 ( ( finite_card_int @ ( set_or4662586982721622107an_int @ L2 @ U ) )
% 0.33/0.69 = ( nat2 @ ( minus_minus_int @ U @ L2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % card_atLeastLessThan_int
% 0.33/0.69 thf(fact_9741_card__atLeastAtMost__int,axiom,
% 0.33/0.69 ! [L2: int,U: int] :
% 0.33/0.69 ( ( finite_card_int @ ( set_or1266510415728281911st_int @ L2 @ U ) )
% 0.33/0.69 = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L2 ) @ one_one_int ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % card_atLeastAtMost_int
% 0.33/0.69 thf(fact_9742_nat_Odisc__eq__case_I1_J,axiom,
% 0.33/0.69 ! [Nat: nat] :
% 0.33/0.69 ( ( Nat = zero_zero_nat )
% 0.33/0.69 = ( case_nat_o @ $true
% 0.33/0.69 @ ^ [Uu3: nat] : $false
% 0.33/0.69 @ Nat ) ) ).
% 0.33/0.69
% 0.33/0.69 % nat.disc_eq_case(1)
% 0.33/0.69 thf(fact_9743_nat_Odisc__eq__case_I2_J,axiom,
% 0.33/0.69 ! [Nat: nat] :
% 0.33/0.69 ( ( Nat != zero_zero_nat )
% 0.33/0.69 = ( case_nat_o @ $false
% 0.33/0.69 @ ^ [Uu3: nat] : $true
% 0.33/0.69 @ Nat ) ) ).
% 0.33/0.69
% 0.33/0.69 % nat.disc_eq_case(2)
% 0.33/0.69 thf(fact_9744_card__less,axiom,
% 0.33/0.69 ! [M7: set_nat,I2: nat] :
% 0.33/0.69 ( ( member_nat @ zero_zero_nat @ M7 )
% 0.33/0.69 => ( ( finite_card_nat
% 0.33/0.69 @ ( collect_nat
% 0.33/0.69 @ ^ [K3: nat] :
% 0.33/0.69 ( ( member_nat @ K3 @ M7 )
% 0.33/0.69 & ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) )
% 0.33/0.69 != zero_zero_nat ) ) ).
% 0.33/0.69
% 0.33/0.69 % card_less
% 0.33/0.69 thf(fact_9745_card__less__Suc,axiom,
% 0.33/0.69 ! [M7: set_nat,I2: nat] :
% 0.33/0.69 ( ( member_nat @ zero_zero_nat @ M7 )
% 0.33/0.69 => ( ( suc
% 0.33/0.69 @ ( finite_card_nat
% 0.33/0.69 @ ( collect_nat
% 0.33/0.69 @ ^ [K3: nat] :
% 0.33/0.69 ( ( member_nat @ ( suc @ K3 ) @ M7 )
% 0.33/0.69 & ( ord_less_nat @ K3 @ I2 ) ) ) ) )
% 0.33/0.69 = ( finite_card_nat
% 0.33/0.69 @ ( collect_nat
% 0.33/0.69 @ ^ [K3: nat] :
% 0.33/0.69 ( ( member_nat @ K3 @ M7 )
% 0.33/0.69 & ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % card_less_Suc
% 0.33/0.69 thf(fact_9746_card__less__Suc2,axiom,
% 0.33/0.69 ! [M7: set_nat,I2: nat] :
% 0.33/0.69 ( ~ ( member_nat @ zero_zero_nat @ M7 )
% 0.33/0.69 => ( ( finite_card_nat
% 0.33/0.69 @ ( collect_nat
% 0.33/0.69 @ ^ [K3: nat] :
% 0.33/0.69 ( ( member_nat @ ( suc @ K3 ) @ M7 )
% 0.33/0.69 & ( ord_less_nat @ K3 @ I2 ) ) ) )
% 0.33/0.69 = ( finite_card_nat
% 0.33/0.69 @ ( collect_nat
% 0.33/0.69 @ ^ [K3: nat] :
% 0.33/0.69 ( ( member_nat @ K3 @ M7 )
% 0.33/0.69 & ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % card_less_Suc2
% 0.33/0.69 thf(fact_9747_subset__card__intvl__is__intvl,axiom,
% 0.33/0.69 ! [A2: set_nat,K: nat] :
% 0.33/0.69 ( ( ord_less_eq_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) )
% 0.33/0.69 => ( A2
% 0.33/0.69 = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % subset_card_intvl_is_intvl
% 0.33/0.69 thf(fact_9748_less__eq__nat_Osimps_I2_J,axiom,
% 0.33/0.69 ! [M: nat,N2: nat] :
% 0.33/0.69 ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
% 0.33/0.69 = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N2 ) ) ).
% 0.33/0.69
% 0.33/0.69 % less_eq_nat.simps(2)
% 0.33/0.69 thf(fact_9749_max__Suc1,axiom,
% 0.33/0.69 ! [N2: nat,M: nat] :
% 0.33/0.69 ( ( ord_max_nat @ ( suc @ N2 ) @ M )
% 0.33/0.69 = ( case_nat_nat @ ( suc @ N2 )
% 0.33/0.69 @ ^ [M3: nat] : ( suc @ ( ord_max_nat @ N2 @ M3 ) )
% 0.33/0.69 @ M ) ) ).
% 0.33/0.69
% 0.33/0.69 % max_Suc1
% 0.33/0.69 thf(fact_9750_max__Suc2,axiom,
% 0.33/0.69 ! [M: nat,N2: nat] :
% 0.33/0.69 ( ( ord_max_nat @ M @ ( suc @ N2 ) )
% 0.33/0.69 = ( case_nat_nat @ ( suc @ N2 )
% 0.33/0.69 @ ^ [M3: nat] : ( suc @ ( ord_max_nat @ M3 @ N2 ) )
% 0.33/0.69 @ M ) ) ).
% 0.33/0.69
% 0.33/0.69 % max_Suc2
% 0.33/0.69 thf(fact_9751_subset__eq__atLeast0__lessThan__card,axiom,
% 0.33/0.69 ! [N3: set_nat,N2: nat] :
% 0.33/0.69 ( ( ord_less_eq_set_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
% 0.33/0.69 => ( ord_less_eq_nat @ ( finite_card_nat @ N3 ) @ N2 ) ) ).
% 0.33/0.69
% 0.33/0.69 % subset_eq_atLeast0_lessThan_card
% 0.33/0.69 thf(fact_9752_card__sum__le__nat__sum,axiom,
% 0.33/0.69 ! [S: set_nat] :
% 0.33/0.69 ( ord_less_eq_nat
% 0.33/0.69 @ ( groups3542108847815614940at_nat
% 0.33/0.69 @ ^ [X2: nat] : X2
% 0.33/0.69 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S ) ) )
% 0.33/0.69 @ ( groups3542108847815614940at_nat
% 0.33/0.69 @ ^ [X2: nat] : X2
% 0.33/0.69 @ S ) ) ).
% 0.33/0.69
% 0.33/0.69 % card_sum_le_nat_sum
% 0.33/0.69 thf(fact_9753_card__nth__roots,axiom,
% 0.33/0.69 ! [C: complex,N2: nat] :
% 0.33/0.69 ( ( C != zero_zero_complex )
% 0.33/0.69 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.69 => ( ( finite_card_complex
% 0.33/0.69 @ ( collect_complex
% 0.33/0.69 @ ^ [Z3: complex] :
% 0.33/0.69 ( ( power_power_complex @ Z3 @ N2 )
% 0.33/0.69 = C ) ) )
% 0.33/0.69 = N2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % card_nth_roots
% 0.33/0.69 thf(fact_9754_card__roots__unity__eq,axiom,
% 0.33/0.69 ! [N2: nat] :
% 0.33/0.69 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.69 => ( ( finite_card_complex
% 0.33/0.69 @ ( collect_complex
% 0.33/0.69 @ ^ [Z3: complex] :
% 0.33/0.69 ( ( power_power_complex @ Z3 @ N2 )
% 0.33/0.69 = one_one_complex ) ) )
% 0.33/0.69 = N2 ) ) ).
% 0.33/0.69
% 0.33/0.69 % card_roots_unity_eq
% 0.33/0.69 thf(fact_9755_diff__Suc,axiom,
% 0.33/0.69 ! [M: nat,N2: nat] :
% 0.33/0.69 ( ( minus_minus_nat @ M @ ( suc @ N2 ) )
% 0.33/0.69 = ( case_nat_nat @ zero_zero_nat
% 0.33/0.69 @ ^ [K3: nat] : K3
% 0.33/0.69 @ ( minus_minus_nat @ M @ N2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % diff_Suc
% 0.33/0.69 thf(fact_9756_pred__def,axiom,
% 0.33/0.69 ( pred
% 0.33/0.69 = ( case_nat_nat @ zero_zero_nat
% 0.33/0.69 @ ^ [X24: nat] : X24 ) ) ).
% 0.33/0.69
% 0.33/0.69 % pred_def
% 0.33/0.69 thf(fact_9757_binomial__def,axiom,
% 0.33/0.69 ( binomial
% 0.33/0.69 = ( ^ [N: nat,K3: nat] :
% 0.33/0.69 ( finite_card_set_nat
% 0.33/0.69 @ ( collect_set_nat
% 0.33/0.69 @ ^ [K7: set_nat] :
% 0.33/0.69 ( ( member_set_nat @ K7 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
% 0.33/0.69 & ( ( finite_card_nat @ K7 )
% 0.33/0.69 = K3 ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % binomial_def
% 0.33/0.69 thf(fact_9758_firstsimp,axiom,
% 0.33/0.69 ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 0.33/0.69 = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 0.33/0.69 @ ( vEBT_Node
% 0.33/0.69 @ ( some_P7363390416028606310at_nat
% 0.33/0.69 @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 0.33/0.69 @ ( if_nat
% 0.33/0.69 @ ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 0.33/0.69 = ma )
% 0.33/0.69 @ ( if_nat
% 0.33/0.69 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 0.33/0.69 = none_nat )
% 0.33/0.69 @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 0.33/0.69 @ ( 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 @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( append_VEBT_VEBT @ ( take_VEBT_VEBT @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ treeList ) @ ( append_VEBT_VEBT @ ( cons_VEBT_VEBT @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ nil_VEBT_VEBT ) @ ( drop_VEBT_VEBT @ ( plus_plus_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ one_one_nat ) @ treeList ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) ) ) )
% 0.33/0.69 @ ma ) ) )
% 0.33/0.69 @ deg
% 0.33/0.69 @ ( append_VEBT_VEBT @ ( take_VEBT_VEBT @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ treeList ) @ ( append_VEBT_VEBT @ ( cons_VEBT_VEBT @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ nil_VEBT_VEBT ) @ ( drop_VEBT_VEBT @ ( plus_plus_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ one_one_nat ) @ treeList ) ) )
% 0.33/0.69 @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 0.33/0.69 @ ( vEBT_Node
% 0.33/0.69 @ ( some_P7363390416028606310at_nat
% 0.33/0.69 @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 0.33/0.69 @ ( if_nat
% 0.33/0.69 @ ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 0.33/0.69 = ma )
% 0.33/0.69 @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( 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 @ ( append_VEBT_VEBT @ ( take_VEBT_VEBT @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ treeList ) @ ( append_VEBT_VEBT @ ( cons_VEBT_VEBT @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ nil_VEBT_VEBT ) @ ( drop_VEBT_VEBT @ ( plus_plus_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ one_one_nat ) @ treeList ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) )
% 0.33/0.69 @ ma ) ) )
% 0.33/0.69 @ deg
% 0.33/0.69 @ ( append_VEBT_VEBT @ ( take_VEBT_VEBT @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ treeList ) @ ( append_VEBT_VEBT @ ( cons_VEBT_VEBT @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ nil_VEBT_VEBT ) @ ( drop_VEBT_VEBT @ ( plus_plus_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ one_one_nat ) @ treeList ) ) )
% 0.33/0.69 @ summary ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % firstsimp
% 0.33/0.69 thf(fact_9759_bezw__0,axiom,
% 0.33/0.69 ! [X: nat] :
% 0.33/0.69 ( ( bezw @ X @ zero_zero_nat )
% 0.33/0.69 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% 0.33/0.69
% 0.33/0.69 % bezw_0
% 0.33/0.69 thf(fact_9760_drop__bit__numeral__minus__bit1,axiom,
% 0.33/0.69 ! [L2: num,K: num] :
% 0.33/0.69 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.33/0.69 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % drop_bit_numeral_minus_bit1
% 0.33/0.69 thf(fact_9761_drop__bit__nonnegative__int__iff,axiom,
% 0.33/0.69 ! [N2: nat,K: int] :
% 0.33/0.69 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N2 @ K ) )
% 0.33/0.69 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.33/0.69
% 0.33/0.69 % drop_bit_nonnegative_int_iff
% 0.33/0.69 thf(fact_9762_drop__bit__negative__int__iff,axiom,
% 0.33/0.69 ! [N2: nat,K: int] :
% 0.33/0.69 ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N2 @ K ) @ zero_zero_int )
% 0.33/0.69 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.33/0.69
% 0.33/0.69 % drop_bit_negative_int_iff
% 0.33/0.69 thf(fact_9763_drop__bit__minus__one,axiom,
% 0.33/0.69 ! [N2: nat] :
% 0.33/0.69 ( ( bit_se8568078237143864401it_int @ N2 @ ( uminus_uminus_int @ one_one_int ) )
% 0.33/0.69 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.33/0.69
% 0.33/0.69 % drop_bit_minus_one
% 0.33/0.69 thf(fact_9764_drop__bit__Suc__minus__bit0,axiom,
% 0.33/0.69 ! [N2: nat,K: num] :
% 0.33/0.69 ( ( bit_se8568078237143864401it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.33/0.69 = ( bit_se8568078237143864401it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % drop_bit_Suc_minus_bit0
% 0.33/0.69 thf(fact_9765_drop__bit__numeral__minus__bit0,axiom,
% 0.33/0.69 ! [L2: num,K: num] :
% 0.33/0.69 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.33/0.69 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % drop_bit_numeral_minus_bit0
% 0.33/0.69 thf(fact_9766_drop__bit__Suc__minus__bit1,axiom,
% 0.33/0.69 ! [N2: nat,K: num] :
% 0.33/0.69 ( ( bit_se8568078237143864401it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.33/0.69 = ( bit_se8568078237143864401it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % drop_bit_Suc_minus_bit1
% 0.33/0.69 thf(fact_9767_drop__bit__push__bit__int,axiom,
% 0.33/0.69 ! [M: nat,N2: nat,K: int] :
% 0.33/0.69 ( ( bit_se8568078237143864401it_int @ M @ ( bit_se545348938243370406it_int @ N2 @ K ) )
% 0.33/0.69 = ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ M @ N2 ) @ ( bit_se545348938243370406it_int @ ( minus_minus_nat @ N2 @ M ) @ K ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % drop_bit_push_bit_int
% 0.33/0.69 thf(fact_9768_drop__bit__int__def,axiom,
% 0.33/0.69 ( bit_se8568078237143864401it_int
% 0.33/0.69 = ( ^ [N: nat,K3: int] : ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % drop_bit_int_def
% 0.33/0.69 thf(fact_9769_Suc__0__mod__numeral,axiom,
% 0.33/0.69 ! [K: num] :
% 0.33/0.69 ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 0.33/0.69 = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Suc_0_mod_numeral
% 0.33/0.69 thf(fact_9770_drop__bit__of__Suc__0,axiom,
% 0.33/0.69 ! [N2: nat] :
% 0.33/0.69 ( ( bit_se8570568707652914677it_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 0.33/0.69 = ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % drop_bit_of_Suc_0
% 0.33/0.69 thf(fact_9771_drop__bit__nat__eq,axiom,
% 0.33/0.69 ! [N2: nat,K: int] :
% 0.33/0.69 ( ( bit_se8570568707652914677it_nat @ N2 @ ( nat2 @ K ) )
% 0.33/0.69 = ( nat2 @ ( bit_se8568078237143864401it_int @ N2 @ K ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % drop_bit_nat_eq
% 0.33/0.69 thf(fact_9772_drop__bit__nat__def,axiom,
% 0.33/0.69 ( bit_se8570568707652914677it_nat
% 0.33/0.69 = ( ^ [N: nat,M6: nat] : ( divide_divide_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % drop_bit_nat_def
% 0.33/0.69 thf(fact_9773_upto__aux__rec,axiom,
% 0.33/0.69 ( upto_aux
% 0.33/0.69 = ( ^ [I3: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % upto_aux_rec
% 0.33/0.69 thf(fact_9774_quotient__of__denom__pos_H,axiom,
% 0.33/0.69 ! [R: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % quotient_of_denom_pos'
% 0.33/0.69 thf(fact_9775_one__mod__minus__numeral,axiom,
% 0.33/0.69 ! [N2: num] :
% 0.33/0.69 ( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.33/0.69 = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N2 ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % one_mod_minus_numeral
% 0.33/0.69 thf(fact_9776_minus__one__mod__numeral,axiom,
% 0.33/0.69 ! [N2: num] :
% 0.33/0.69 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N2 ) )
% 0.33/0.69 = ( adjust_mod @ ( numeral_numeral_int @ N2 ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % minus_one_mod_numeral
% 0.33/0.69 thf(fact_9777_Divides_Oadjust__mod__def,axiom,
% 0.33/0.69 ( adjust_mod
% 0.33/0.69 = ( ^ [L: int,R5: int] : ( if_int @ ( R5 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ L @ R5 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Divides.adjust_mod_def
% 0.33/0.69 thf(fact_9778_bezw_Osimps,axiom,
% 0.33/0.69 ( bezw
% 0.33/0.69 = ( ^ [X2: nat,Y2: nat] : ( if_Pro3027730157355071871nt_int @ ( Y2 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X2 @ Y2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X2 @ Y2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X2 @ Y2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Y2 ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % bezw.simps
% 0.33/0.69 thf(fact_9779_bezw_Oelims,axiom,
% 0.33/0.69 ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
% 0.33/0.69 ( ( ( bezw @ X @ Xa2 )
% 0.33/0.69 = Y )
% 0.33/0.69 => ( ( ( Xa2 = zero_zero_nat )
% 0.33/0.69 => ( Y
% 0.33/0.69 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 0.33/0.69 & ( ( Xa2 != zero_zero_nat )
% 0.33/0.69 => ( Y
% 0.33/0.69 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa2 ) ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % bezw.elims
% 0.33/0.69 thf(fact_9780_bezw__non__0,axiom,
% 0.33/0.69 ! [Y: nat,X: nat] :
% 0.33/0.69 ( ( ord_less_nat @ zero_zero_nat @ Y )
% 0.33/0.69 => ( ( bezw @ X @ Y )
% 0.33/0.69 = ( 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 ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % bezw_non_0
% 0.33/0.69 thf(fact_9781_rat__sgn__code,axiom,
% 0.33/0.69 ! [P4: rat] :
% 0.33/0.69 ( ( quotient_of @ ( sgn_sgn_rat @ P4 ) )
% 0.33/0.69 = ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P4 ) ) ) @ one_one_int ) ) ).
% 0.33/0.69
% 0.33/0.69 % rat_sgn_code
% 0.33/0.69 thf(fact_9782_take__bit__numeral__minus__numeral__int,axiom,
% 0.33/0.69 ! [M: num,N2: num] :
% 0.33/0.69 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.33/0.69 = ( case_option_int_num @ zero_zero_int
% 0.33/0.69 @ ^ [Q4: num] : ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_int @ Q4 ) ) )
% 0.33/0.69 @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % take_bit_numeral_minus_numeral_int
% 0.33/0.69 thf(fact_9783_bezw_Opelims,axiom,
% 0.33/0.69 ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
% 0.33/0.69 ( ( ( bezw @ X @ Xa2 )
% 0.33/0.69 = Y )
% 0.33/0.69 => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
% 0.33/0.69 => ~ ( ( ( ( Xa2 = zero_zero_nat )
% 0.33/0.69 => ( Y
% 0.33/0.69 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 0.33/0.69 & ( ( Xa2 != zero_zero_nat )
% 0.33/0.69 => ( Y
% 0.33/0.69 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa2 ) ) ) ) ) ) ) )
% 0.33/0.69 => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % bezw.pelims
% 0.33/0.69 thf(fact_9784_take__bit__num__simps_I1_J,axiom,
% 0.33/0.69 ! [M: num] :
% 0.33/0.69 ( ( bit_take_bit_num @ zero_zero_nat @ M )
% 0.33/0.69 = none_num ) ).
% 0.33/0.69
% 0.33/0.69 % take_bit_num_simps(1)
% 0.33/0.69 thf(fact_9785_fst__divmod__nat,axiom,
% 0.33/0.69 ! [M: nat,N2: nat] :
% 0.33/0.69 ( ( product_fst_nat_nat @ ( divmod_nat @ M @ N2 ) )
% 0.33/0.69 = ( divide_divide_nat @ M @ N2 ) ) ).
% 0.33/0.69
% 0.33/0.69 % fst_divmod_nat
% 0.33/0.69 thf(fact_9786_fst__divmod__integer,axiom,
% 0.33/0.69 ! [K: code_integer,L2: code_integer] :
% 0.33/0.69 ( ( produc8508995932063986495nteger @ ( code_divmod_integer @ K @ L2 ) )
% 0.33/0.69 = ( divide6298287555418463151nteger @ K @ L2 ) ) ).
% 0.33/0.69
% 0.33/0.69 % fst_divmod_integer
% 0.33/0.69 thf(fact_9787_take__bit__num__simps_I2_J,axiom,
% 0.33/0.69 ! [N2: nat] :
% 0.33/0.69 ( ( bit_take_bit_num @ ( suc @ N2 ) @ one )
% 0.33/0.69 = ( some_num @ one ) ) ).
% 0.33/0.69
% 0.33/0.69 % take_bit_num_simps(2)
% 0.33/0.69 thf(fact_9788_take__bit__num__simps_I5_J,axiom,
% 0.33/0.69 ! [R: num] :
% 0.33/0.69 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ one )
% 0.33/0.69 = ( some_num @ one ) ) ).
% 0.33/0.69
% 0.33/0.69 % take_bit_num_simps(5)
% 0.33/0.69 thf(fact_9789_take__bit__num__simps_I3_J,axiom,
% 0.33/0.69 ! [N2: nat,M: num] :
% 0.33/0.69 ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit0 @ M ) )
% 0.33/0.69 = ( case_o6005452278849405969um_num @ none_num
% 0.33/0.69 @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 0.33/0.69 @ ( bit_take_bit_num @ N2 @ M ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % take_bit_num_simps(3)
% 0.33/0.69 thf(fact_9790_fst__divmod__abs,axiom,
% 0.33/0.69 ! [K: code_integer,L2: code_integer] :
% 0.33/0.69 ( ( produc8508995932063986495nteger @ ( code_divmod_abs @ K @ L2 ) )
% 0.33/0.69 = ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K ) @ ( abs_abs_Code_integer @ L2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % fst_divmod_abs
% 0.33/0.69 thf(fact_9791_take__bit__num__simps_I4_J,axiom,
% 0.33/0.69 ! [N2: nat,M: num] :
% 0.33/0.69 ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit1 @ M ) )
% 0.33/0.69 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % take_bit_num_simps(4)
% 0.33/0.69 thf(fact_9792_take__bit__num__simps_I6_J,axiom,
% 0.33/0.69 ! [R: num,M: num] :
% 0.33/0.69 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ ( bit0 @ M ) )
% 0.33/0.69 = ( case_o6005452278849405969um_num @ none_num
% 0.33/0.69 @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 0.33/0.69 @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % take_bit_num_simps(6)
% 0.33/0.69 thf(fact_9793_take__bit__num__simps_I7_J,axiom,
% 0.33/0.69 ! [R: num,M: num] :
% 0.33/0.69 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ ( bit1 @ M ) )
% 0.33/0.69 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % take_bit_num_simps(7)
% 0.33/0.69 thf(fact_9794_Suc__0__div__numeral,axiom,
% 0.33/0.69 ! [K: num] :
% 0.33/0.69 ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 0.33/0.69 = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Suc_0_div_numeral
% 0.33/0.69 thf(fact_9795_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
% 0.33/0.69 ! [N2: nat,M: num] :
% 0.33/0.69 ( ( bit_take_bit_num @ N2 @ ( bit0 @ M ) )
% 0.33/0.69 = ( case_nat_option_num @ none_num
% 0.33/0.69 @ ^ [N: nat] :
% 0.33/0.69 ( case_o6005452278849405969um_num @ none_num
% 0.33/0.69 @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 0.33/0.69 @ ( bit_take_bit_num @ N @ M ) )
% 0.33/0.69 @ N2 ) ) ).
% 0.33/0.69
% 0.33/0.69 % Code_Abstract_Nat.take_bit_num_code(2)
% 0.33/0.69 thf(fact_9796_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
% 0.33/0.69 ! [N2: nat] :
% 0.33/0.69 ( ( bit_take_bit_num @ N2 @ one )
% 0.33/0.69 = ( case_nat_option_num @ none_num
% 0.33/0.69 @ ^ [N: nat] : ( some_num @ one )
% 0.33/0.69 @ N2 ) ) ).
% 0.33/0.69
% 0.33/0.69 % Code_Abstract_Nat.take_bit_num_code(1)
% 0.33/0.69 thf(fact_9797_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
% 0.33/0.69 ! [N2: nat,M: num] :
% 0.33/0.69 ( ( bit_take_bit_num @ N2 @ ( bit1 @ M ) )
% 0.33/0.69 = ( case_nat_option_num @ none_num
% 0.33/0.69 @ ^ [N: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) )
% 0.33/0.69 @ N2 ) ) ).
% 0.33/0.69
% 0.33/0.69 % Code_Abstract_Nat.take_bit_num_code(3)
% 0.33/0.69 thf(fact_9798_and__minus__numerals_I7_J,axiom,
% 0.33/0.69 ! [N2: num,M: num] :
% 0.33/0.69 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.33/0.69 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N2 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % and_minus_numerals(7)
% 0.33/0.69 thf(fact_9799_and__minus__numerals_I3_J,axiom,
% 0.33/0.69 ! [M: num,N2: num] :
% 0.33/0.69 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
% 0.33/0.69 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N2 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % and_minus_numerals(3)
% 0.33/0.69 thf(fact_9800_and__minus__numerals_I4_J,axiom,
% 0.33/0.69 ! [M: num,N2: num] :
% 0.33/0.69 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
% 0.33/0.69 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N2 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % and_minus_numerals(4)
% 0.33/0.69 thf(fact_9801_and__minus__numerals_I8_J,axiom,
% 0.33/0.69 ! [N2: num,M: num] :
% 0.33/0.69 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.33/0.69 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N2 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % and_minus_numerals(8)
% 0.33/0.69 thf(fact_9802_and__not__num_Osimps_I1_J,axiom,
% 0.33/0.69 ( ( bit_and_not_num @ one @ one )
% 0.33/0.69 = none_num ) ).
% 0.33/0.69
% 0.33/0.69 % and_not_num.simps(1)
% 0.33/0.69 thf(fact_9803_and__not__num_Osimps_I2_J,axiom,
% 0.33/0.69 ! [N2: num] :
% 0.33/0.69 ( ( bit_and_not_num @ one @ ( bit0 @ N2 ) )
% 0.33/0.69 = ( some_num @ one ) ) ).
% 0.33/0.69
% 0.33/0.69 % and_not_num.simps(2)
% 0.33/0.69 thf(fact_9804_and__not__num_Osimps_I4_J,axiom,
% 0.33/0.69 ! [M: num] :
% 0.33/0.69 ( ( bit_and_not_num @ ( bit0 @ M ) @ one )
% 0.33/0.69 = ( some_num @ ( bit0 @ M ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % and_not_num.simps(4)
% 0.33/0.69 thf(fact_9805_and__not__num_Osimps_I3_J,axiom,
% 0.33/0.69 ! [N2: num] :
% 0.33/0.69 ( ( bit_and_not_num @ one @ ( bit1 @ N2 ) )
% 0.33/0.69 = none_num ) ).
% 0.33/0.69
% 0.33/0.69 % and_not_num.simps(3)
% 0.33/0.69 thf(fact_9806_and__not__num_Osimps_I7_J,axiom,
% 0.33/0.69 ! [M: num] :
% 0.33/0.69 ( ( bit_and_not_num @ ( bit1 @ M ) @ one )
% 0.33/0.69 = ( some_num @ ( bit0 @ M ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % and_not_num.simps(7)
% 0.33/0.69 thf(fact_9807_and__not__num__eq__Some__iff,axiom,
% 0.33/0.69 ! [M: num,N2: num,Q2: num] :
% 0.33/0.69 ( ( ( bit_and_not_num @ M @ N2 )
% 0.33/0.69 = ( some_num @ Q2 ) )
% 0.33/0.69 = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.33/0.69 = ( numeral_numeral_int @ Q2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % and_not_num_eq_Some_iff
% 0.33/0.69 thf(fact_9808_and__not__num_Osimps_I8_J,axiom,
% 0.33/0.69 ! [M: num,N2: num] :
% 0.33/0.69 ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 0.33/0.69 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.33/0.69 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.33/0.69 @ ( bit_and_not_num @ M @ N2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % and_not_num.simps(8)
% 0.33/0.69 thf(fact_9809_and__not__num__eq__None__iff,axiom,
% 0.33/0.69 ! [M: num,N2: num] :
% 0.33/0.69 ( ( ( bit_and_not_num @ M @ N2 )
% 0.33/0.69 = none_num )
% 0.33/0.69 = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.33/0.69 = zero_zero_int ) ) ).
% 0.33/0.69
% 0.33/0.69 % and_not_num_eq_None_iff
% 0.33/0.69 thf(fact_9810_int__numeral__not__and__num,axiom,
% 0.33/0.69 ! [M: num,N2: num] :
% 0.33/0.69 ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
% 0.33/0.69 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N2 @ M ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % int_numeral_not_and_num
% 0.33/0.69 thf(fact_9811_int__numeral__and__not__num,axiom,
% 0.33/0.69 ! [M: num,N2: num] :
% 0.33/0.69 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.33/0.69 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % int_numeral_and_not_num
% 0.33/0.69 thf(fact_9812_Bit__Operations_Otake__bit__num__code,axiom,
% 0.33/0.69 ( bit_take_bit_num
% 0.33/0.69 = ( ^ [N: nat,M6: num] :
% 0.33/0.69 ( produc478579273971653890on_num
% 0.33/0.69 @ ^ [A5: nat,X2: num] :
% 0.33/0.69 ( case_nat_option_num @ none_num
% 0.33/0.69 @ ^ [O: nat] :
% 0.33/0.69 ( case_num_option_num @ ( some_num @ one )
% 0.33/0.69 @ ^ [P5: num] :
% 0.33/0.69 ( case_o6005452278849405969um_num @ none_num
% 0.33/0.69 @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 0.33/0.69 @ ( bit_take_bit_num @ O @ P5 ) )
% 0.33/0.69 @ ^ [P5: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P5 ) ) )
% 0.33/0.69 @ X2 )
% 0.33/0.69 @ A5 )
% 0.33/0.69 @ ( product_Pair_nat_num @ N @ M6 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Bit_Operations.take_bit_num_code
% 0.33/0.69 thf(fact_9813_normalize__def,axiom,
% 0.33/0.69 ( normalize
% 0.33/0.69 = ( ^ [P5: product_prod_int_int] :
% 0.33/0.69 ( 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 ) ) ) )
% 0.33/0.69 @ ( if_Pro3027730157355071871nt_int
% 0.33/0.69 @ ( ( product_snd_int_int @ P5 )
% 0.33/0.69 = zero_zero_int )
% 0.33/0.69 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.33/0.69 @ ( 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 ) ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % normalize_def
% 0.33/0.69 thf(fact_9814_gcd__1__int,axiom,
% 0.33/0.69 ! [M: int] :
% 0.33/0.69 ( ( gcd_gcd_int @ M @ one_one_int )
% 0.33/0.69 = one_one_int ) ).
% 0.33/0.69
% 0.33/0.69 % gcd_1_int
% 0.33/0.69 thf(fact_9815_gcd__pos__int,axiom,
% 0.33/0.69 ! [M: int,N2: int] :
% 0.33/0.69 ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N2 ) )
% 0.33/0.69 = ( ( M != zero_zero_int )
% 0.33/0.69 | ( N2 != zero_zero_int ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % gcd_pos_int
% 0.33/0.69 thf(fact_9816_gcd__le2__int,axiom,
% 0.33/0.69 ! [B: int,A: int] :
% 0.33/0.69 ( ( ord_less_int @ zero_zero_int @ B )
% 0.33/0.69 => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ B ) ) ).
% 0.33/0.69
% 0.33/0.69 % gcd_le2_int
% 0.33/0.69 thf(fact_9817_gcd__le1__int,axiom,
% 0.33/0.69 ! [A: int,B: int] :
% 0.33/0.69 ( ( ord_less_int @ zero_zero_int @ A )
% 0.33/0.69 => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ A ) ) ).
% 0.33/0.69
% 0.33/0.69 % gcd_le1_int
% 0.33/0.69 thf(fact_9818_gcd__non__0__int,axiom,
% 0.33/0.69 ! [Y: int,X: int] :
% 0.33/0.69 ( ( ord_less_int @ zero_zero_int @ Y )
% 0.33/0.69 => ( ( gcd_gcd_int @ X @ Y )
% 0.33/0.69 = ( gcd_gcd_int @ Y @ ( modulo_modulo_int @ X @ Y ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % gcd_non_0_int
% 0.33/0.69 thf(fact_9819_upto_Opsimps,axiom,
% 0.33/0.69 ! [I2: int,J: int] :
% 0.33/0.69 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J ) )
% 0.33/0.69 => ( ( ( ord_less_eq_int @ I2 @ J )
% 0.33/0.69 => ( ( upto @ I2 @ J )
% 0.33/0.69 = ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J ) ) ) )
% 0.33/0.69 & ( ~ ( ord_less_eq_int @ I2 @ J )
% 0.33/0.69 => ( ( upto @ I2 @ J )
% 0.33/0.69 = nil_int ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % upto.psimps
% 0.33/0.69 thf(fact_9820_upto_Opelims,axiom,
% 0.33/0.69 ! [X: int,Xa2: int,Y: list_int] :
% 0.33/0.69 ( ( ( upto @ X @ Xa2 )
% 0.33/0.69 = Y )
% 0.33/0.69 => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa2 ) )
% 0.33/0.69 => ~ ( ( ( ( ord_less_eq_int @ X @ Xa2 )
% 0.33/0.69 => ( Y
% 0.33/0.69 = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa2 ) ) ) )
% 0.33/0.69 & ( ~ ( ord_less_eq_int @ X @ Xa2 )
% 0.33/0.69 => ( Y = nil_int ) ) )
% 0.33/0.69 => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa2 ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % upto.pelims
% 0.33/0.69 thf(fact_9821_take__bit__num__def,axiom,
% 0.33/0.69 ( bit_take_bit_num
% 0.33/0.69 = ( ^ [N: nat,M6: num] :
% 0.33/0.69 ( if_option_num
% 0.33/0.69 @ ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ M6 ) )
% 0.33/0.69 = zero_zero_nat )
% 0.33/0.69 @ none_num
% 0.33/0.69 @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ M6 ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % take_bit_num_def
% 0.33/0.69 thf(fact_9822_gcd__1__nat,axiom,
% 0.33/0.69 ! [M: nat] :
% 0.33/0.69 ( ( gcd_gcd_nat @ M @ one_one_nat )
% 0.33/0.69 = one_one_nat ) ).
% 0.33/0.69
% 0.33/0.69 % gcd_1_nat
% 0.33/0.69 thf(fact_9823_gcd__Suc__0,axiom,
% 0.33/0.69 ! [M: nat] :
% 0.33/0.69 ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.33/0.69 = ( suc @ zero_zero_nat ) ) ).
% 0.33/0.69
% 0.33/0.69 % gcd_Suc_0
% 0.33/0.69 thf(fact_9824_gcd__pos__nat,axiom,
% 0.33/0.69 ! [M: nat,N2: nat] :
% 0.33/0.69 ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N2 ) )
% 0.33/0.69 = ( ( M != zero_zero_nat )
% 0.33/0.69 | ( N2 != zero_zero_nat ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % gcd_pos_nat
% 0.33/0.69 thf(fact_9825_num__of__nat__numeral__eq,axiom,
% 0.33/0.69 ! [Q2: num] :
% 0.33/0.69 ( ( num_of_nat @ ( numeral_numeral_nat @ Q2 ) )
% 0.33/0.69 = Q2 ) ).
% 0.33/0.69
% 0.33/0.69 % num_of_nat_numeral_eq
% 0.33/0.69 thf(fact_9826_upto__Nil,axiom,
% 0.33/0.69 ! [I2: int,J: int] :
% 0.33/0.69 ( ( ( upto @ I2 @ J )
% 0.33/0.69 = nil_int )
% 0.33/0.69 = ( ord_less_int @ J @ I2 ) ) ).
% 0.33/0.69
% 0.33/0.69 % upto_Nil
% 0.33/0.69 thf(fact_9827_upto__Nil2,axiom,
% 0.33/0.69 ! [I2: int,J: int] :
% 0.33/0.69 ( ( nil_int
% 0.33/0.69 = ( upto @ I2 @ J ) )
% 0.33/0.69 = ( ord_less_int @ J @ I2 ) ) ).
% 0.33/0.69
% 0.33/0.69 % upto_Nil2
% 0.33/0.69 thf(fact_9828_upto__empty,axiom,
% 0.33/0.69 ! [J: int,I2: int] :
% 0.33/0.69 ( ( ord_less_int @ J @ I2 )
% 0.33/0.69 => ( ( upto @ I2 @ J )
% 0.33/0.69 = nil_int ) ) ).
% 0.33/0.69
% 0.33/0.69 % upto_empty
% 0.33/0.69 thf(fact_9829_upto__single,axiom,
% 0.33/0.69 ! [I2: int] :
% 0.33/0.69 ( ( upto @ I2 @ I2 )
% 0.33/0.69 = ( cons_int @ I2 @ nil_int ) ) ).
% 0.33/0.69
% 0.33/0.69 % upto_single
% 0.33/0.69 thf(fact_9830_nth__upto,axiom,
% 0.33/0.69 ! [I2: int,K: nat,J: int] :
% 0.33/0.69 ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ ( semiri1314217659103216013at_int @ K ) ) @ J )
% 0.33/0.69 => ( ( nth_int @ ( upto @ I2 @ J ) @ K )
% 0.33/0.69 = ( plus_plus_int @ I2 @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % nth_upto
% 0.33/0.69 thf(fact_9831_length__upto,axiom,
% 0.33/0.69 ! [I2: int,J: int] :
% 0.33/0.69 ( ( size_size_list_int @ ( upto @ I2 @ J ) )
% 0.33/0.69 = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I2 ) @ one_one_int ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % length_upto
% 0.33/0.69 thf(fact_9832_upto__rec__numeral_I1_J,axiom,
% 0.33/0.69 ! [M: num,N2: num] :
% 0.33/0.69 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 0.33/0.69 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 0.33/0.69 = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N2 ) ) ) ) )
% 0.33/0.69 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 0.33/0.69 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 0.33/0.69 = nil_int ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % upto_rec_numeral(1)
% 0.33/0.69 thf(fact_9833_upto__rec__numeral_I4_J,axiom,
% 0.33/0.69 ! [M: num,N2: num] :
% 0.33/0.69 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.33/0.69 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.33/0.69 = ( 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 @ N2 ) ) ) ) ) )
% 0.33/0.69 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.33/0.69 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.33/0.69 = nil_int ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % upto_rec_numeral(4)
% 0.33/0.69 thf(fact_9834_upto__rec__numeral_I3_J,axiom,
% 0.33/0.69 ! [M: num,N2: num] :
% 0.33/0.69 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
% 0.33/0.69 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
% 0.33/0.69 = ( 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 @ N2 ) ) ) ) )
% 0.33/0.69 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
% 0.33/0.69 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
% 0.33/0.69 = nil_int ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % upto_rec_numeral(3)
% 0.33/0.69 thf(fact_9835_upto__rec__numeral_I2_J,axiom,
% 0.33/0.69 ! [M: num,N2: num] :
% 0.33/0.69 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.33/0.69 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.33/0.69 = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) )
% 0.33/0.69 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.33/0.69 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.33/0.69 = nil_int ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % upto_rec_numeral(2)
% 0.33/0.69 thf(fact_9836_gcd__mult__distrib__nat,axiom,
% 0.33/0.69 ! [K: nat,M: nat,N2: nat] :
% 0.33/0.69 ( ( times_times_nat @ K @ ( gcd_gcd_nat @ M @ N2 ) )
% 0.33/0.69 = ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % gcd_mult_distrib_nat
% 0.33/0.69 thf(fact_9837_atLeastAtMost__upto,axiom,
% 0.33/0.69 ( set_or1266510415728281911st_int
% 0.33/0.69 = ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ I3 @ J3 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % atLeastAtMost_upto
% 0.33/0.69 thf(fact_9838_gcd__diff2__nat,axiom,
% 0.33/0.69 ! [M: nat,N2: nat] :
% 0.33/0.69 ( ( ord_less_eq_nat @ M @ N2 )
% 0.33/0.69 => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N2 @ M ) @ N2 )
% 0.33/0.69 = ( gcd_gcd_nat @ M @ N2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % gcd_diff2_nat
% 0.33/0.69 thf(fact_9839_gcd__diff1__nat,axiom,
% 0.33/0.69 ! [N2: nat,M: nat] :
% 0.33/0.69 ( ( ord_less_eq_nat @ N2 @ M )
% 0.33/0.69 => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 )
% 0.33/0.69 = ( gcd_gcd_nat @ M @ N2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % gcd_diff1_nat
% 0.33/0.69 thf(fact_9840_gcd__le2__nat,axiom,
% 0.33/0.69 ! [B: nat,A: nat] :
% 0.33/0.69 ( ( B != zero_zero_nat )
% 0.33/0.69 => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).
% 0.33/0.69
% 0.33/0.69 % gcd_le2_nat
% 0.33/0.69 thf(fact_9841_gcd__le1__nat,axiom,
% 0.33/0.69 ! [A: nat,B: nat] :
% 0.33/0.69 ( ( A != zero_zero_nat )
% 0.33/0.69 => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).
% 0.33/0.69
% 0.33/0.69 % gcd_le1_nat
% 0.33/0.69 thf(fact_9842_distinct__upto,axiom,
% 0.33/0.69 ! [I2: int,J: int] : ( distinct_int @ ( upto @ I2 @ J ) ) ).
% 0.33/0.69
% 0.33/0.69 % distinct_upto
% 0.33/0.69 thf(fact_9843_upto__code,axiom,
% 0.33/0.69 ( upto
% 0.33/0.69 = ( ^ [I3: int,J3: int] : ( upto_aux @ I3 @ J3 @ nil_int ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % upto_code
% 0.33/0.69 thf(fact_9844_upto__aux__def,axiom,
% 0.33/0.69 ( upto_aux
% 0.33/0.69 = ( ^ [I3: int,J3: int] : ( append_int @ ( upto @ I3 @ J3 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % upto_aux_def
% 0.33/0.69 thf(fact_9845_num__of__nat_Osimps_I1_J,axiom,
% 0.33/0.69 ( ( num_of_nat @ zero_zero_nat )
% 0.33/0.69 = one ) ).
% 0.33/0.69
% 0.33/0.69 % num_of_nat.simps(1)
% 0.33/0.69 thf(fact_9846_bezout__nat,axiom,
% 0.33/0.69 ! [A: nat,B: nat] :
% 0.33/0.69 ( ( A != zero_zero_nat )
% 0.33/0.69 => ? [X4: nat,Y4: nat] :
% 0.33/0.69 ( ( times_times_nat @ A @ X4 )
% 0.33/0.69 = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % bezout_nat
% 0.33/0.69 thf(fact_9847_bezout__gcd__nat_H,axiom,
% 0.33/0.69 ! [B: nat,A: nat] :
% 0.33/0.69 ? [X4: nat,Y4: nat] :
% 0.33/0.69 ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y4 ) @ ( times_times_nat @ A @ X4 ) )
% 0.33/0.69 & ( ( minus_minus_nat @ ( times_times_nat @ A @ X4 ) @ ( times_times_nat @ B @ Y4 ) )
% 0.33/0.69 = ( gcd_gcd_nat @ A @ B ) ) )
% 0.33/0.69 | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y4 ) @ ( times_times_nat @ B @ X4 ) )
% 0.33/0.69 & ( ( minus_minus_nat @ ( times_times_nat @ B @ X4 ) @ ( times_times_nat @ A @ Y4 ) )
% 0.33/0.69 = ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % bezout_gcd_nat'
% 0.33/0.69 thf(fact_9848_upto__split2,axiom,
% 0.33/0.69 ! [I2: int,J: int,K: int] :
% 0.33/0.69 ( ( ord_less_eq_int @ I2 @ J )
% 0.33/0.69 => ( ( ord_less_eq_int @ J @ K )
% 0.33/0.69 => ( ( upto @ I2 @ K )
% 0.33/0.69 = ( append_int @ ( upto @ I2 @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % upto_split2
% 0.33/0.69 thf(fact_9849_upto__split1,axiom,
% 0.33/0.69 ! [I2: int,J: int,K: int] :
% 0.33/0.69 ( ( ord_less_eq_int @ I2 @ J )
% 0.33/0.69 => ( ( ord_less_eq_int @ J @ K )
% 0.33/0.69 => ( ( upto @ I2 @ K )
% 0.33/0.69 = ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % upto_split1
% 0.33/0.69 thf(fact_9850_atLeastLessThan__upto,axiom,
% 0.33/0.69 ( set_or4662586982721622107an_int
% 0.33/0.69 = ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % atLeastLessThan_upto
% 0.33/0.69 thf(fact_9851_numeral__num__of__nat,axiom,
% 0.33/0.69 ! [N2: nat] :
% 0.33/0.69 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.69 => ( ( numeral_numeral_nat @ ( num_of_nat @ N2 ) )
% 0.33/0.69 = N2 ) ) ).
% 0.33/0.69
% 0.33/0.69 % numeral_num_of_nat
% 0.33/0.69 thf(fact_9852_num__of__nat__One,axiom,
% 0.33/0.69 ! [N2: nat] :
% 0.33/0.69 ( ( ord_less_eq_nat @ N2 @ one_one_nat )
% 0.33/0.69 => ( ( num_of_nat @ N2 )
% 0.33/0.69 = one ) ) ).
% 0.33/0.69
% 0.33/0.69 % num_of_nat_One
% 0.33/0.69 thf(fact_9853_upto__rec1,axiom,
% 0.33/0.69 ! [I2: int,J: int] :
% 0.33/0.69 ( ( ord_less_eq_int @ I2 @ J )
% 0.33/0.69 => ( ( upto @ I2 @ J )
% 0.33/0.69 = ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % upto_rec1
% 0.33/0.69 thf(fact_9854_upto_Osimps,axiom,
% 0.33/0.69 ( upto
% 0.33/0.69 = ( ^ [I3: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I3 @ J3 ) @ ( cons_int @ I3 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % upto.simps
% 0.33/0.69 thf(fact_9855_upto_Oelims,axiom,
% 0.33/0.69 ! [X: int,Xa2: int,Y: list_int] :
% 0.33/0.69 ( ( ( upto @ X @ Xa2 )
% 0.33/0.69 = Y )
% 0.33/0.69 => ( ( ( ord_less_eq_int @ X @ Xa2 )
% 0.33/0.69 => ( Y
% 0.33/0.69 = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa2 ) ) ) )
% 0.33/0.69 & ( ~ ( ord_less_eq_int @ X @ Xa2 )
% 0.33/0.69 => ( Y = nil_int ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % upto.elims
% 0.33/0.69 thf(fact_9856_upto__rec2,axiom,
% 0.33/0.69 ! [I2: int,J: int] :
% 0.33/0.69 ( ( ord_less_eq_int @ I2 @ J )
% 0.33/0.69 => ( ( upto @ I2 @ J )
% 0.33/0.69 = ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % upto_rec2
% 0.33/0.69 thf(fact_9857_num__of__nat__double,axiom,
% 0.33/0.69 ! [N2: nat] :
% 0.33/0.69 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.69 => ( ( num_of_nat @ ( plus_plus_nat @ N2 @ N2 ) )
% 0.33/0.69 = ( bit0 @ ( num_of_nat @ N2 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % num_of_nat_double
% 0.33/0.69 thf(fact_9858_num__of__nat__plus__distrib,axiom,
% 0.33/0.69 ! [M: nat,N2: nat] :
% 0.33/0.69 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.33/0.69 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.69 => ( ( num_of_nat @ ( plus_plus_nat @ M @ N2 ) )
% 0.33/0.69 = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N2 ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % num_of_nat_plus_distrib
% 0.33/0.69 thf(fact_9859_upto__split3,axiom,
% 0.33/0.69 ! [I2: int,J: int,K: int] :
% 0.33/0.69 ( ( ord_less_eq_int @ I2 @ J )
% 0.33/0.69 => ( ( ord_less_eq_int @ J @ K )
% 0.33/0.69 => ( ( upto @ I2 @ K )
% 0.33/0.69 = ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % upto_split3
% 0.33/0.69 thf(fact_9860_num__of__nat_Osimps_I2_J,axiom,
% 0.33/0.69 ! [N2: nat] :
% 0.33/0.69 ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.69 => ( ( num_of_nat @ ( suc @ N2 ) )
% 0.33/0.69 = ( inc @ ( num_of_nat @ N2 ) ) ) )
% 0.33/0.69 & ( ~ ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.69 => ( ( num_of_nat @ ( suc @ N2 ) )
% 0.33/0.69 = one ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % num_of_nat.simps(2)
% 0.33/0.69 thf(fact_9861_gcd__nat_Opelims,axiom,
% 0.33/0.69 ! [X: nat,Xa2: nat,Y: nat] :
% 0.33/0.69 ( ( ( gcd_gcd_nat @ X @ Xa2 )
% 0.33/0.69 = Y )
% 0.33/0.69 => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
% 0.33/0.69 => ~ ( ( ( ( Xa2 = zero_zero_nat )
% 0.33/0.69 => ( Y = X ) )
% 0.33/0.69 & ( ( Xa2 != zero_zero_nat )
% 0.33/0.69 => ( Y
% 0.33/0.69 = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) ) )
% 0.33/0.69 => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % gcd_nat.pelims
% 0.33/0.69 thf(fact_9862_prod__decode__aux_Oelims,axiom,
% 0.33/0.69 ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
% 0.33/0.69 ( ( ( nat_prod_decode_aux @ X @ Xa2 )
% 0.33/0.69 = Y )
% 0.33/0.69 => ( ( ( ord_less_eq_nat @ Xa2 @ X )
% 0.33/0.69 => ( Y
% 0.33/0.69 = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
% 0.33/0.69 & ( ~ ( ord_less_eq_nat @ Xa2 @ X )
% 0.33/0.69 => ( Y
% 0.33/0.69 = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % prod_decode_aux.elims
% 0.33/0.69 thf(fact_9863_nat__descend__induct,axiom,
% 0.33/0.69 ! [N2: nat,P: nat > $o,M: nat] :
% 0.33/0.69 ( ! [K2: nat] :
% 0.33/0.69 ( ( ord_less_nat @ N2 @ K2 )
% 0.33/0.69 => ( P @ K2 ) )
% 0.33/0.69 => ( ! [K2: nat] :
% 0.33/0.69 ( ( ord_less_eq_nat @ K2 @ N2 )
% 0.33/0.69 => ( ! [I: nat] :
% 0.33/0.69 ( ( ord_less_nat @ K2 @ I )
% 0.33/0.69 => ( P @ I ) )
% 0.33/0.69 => ( P @ K2 ) ) )
% 0.33/0.69 => ( P @ M ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % nat_descend_induct
% 0.33/0.69 thf(fact_9864_prod__decode__aux_Osimps,axiom,
% 0.33/0.69 ( nat_prod_decode_aux
% 0.33/0.69 = ( ^ [K3: nat,M6: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M6 @ K3 ) @ ( product_Pair_nat_nat @ M6 @ ( minus_minus_nat @ K3 @ M6 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M6 @ ( suc @ K3 ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % prod_decode_aux.simps
% 0.33/0.69 thf(fact_9865_prod__decode__aux_Opelims,axiom,
% 0.33/0.69 ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
% 0.33/0.69 ( ( ( nat_prod_decode_aux @ X @ Xa2 )
% 0.33/0.69 = Y )
% 0.33/0.69 => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
% 0.33/0.69 => ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X )
% 0.33/0.69 => ( Y
% 0.33/0.69 = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
% 0.33/0.69 & ( ~ ( ord_less_eq_nat @ Xa2 @ X )
% 0.33/0.69 => ( Y
% 0.33/0.69 = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) )
% 0.33/0.69 => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % prod_decode_aux.pelims
% 0.33/0.69 thf(fact_9866_vebt__mint_Opelims,axiom,
% 0.33/0.69 ! [X: vEBT_VEBT,Y: option_nat] :
% 0.33/0.69 ( ( ( vEBT_vebt_mint @ X )
% 0.33/0.69 = Y )
% 0.33/0.69 => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X )
% 0.33/0.69 => ( ! [A3: $o,B2: $o] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.33/0.69 => ( ( ( A3
% 0.33/0.69 => ( Y
% 0.33/0.69 = ( some_nat @ zero_zero_nat ) ) )
% 0.33/0.69 & ( ~ A3
% 0.33/0.69 => ( ( B2
% 0.33/0.69 => ( Y
% 0.33/0.69 = ( some_nat @ one_one_nat ) ) )
% 0.33/0.69 & ( ~ B2
% 0.33/0.69 => ( Y = none_nat ) ) ) ) )
% 0.33/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A3 @ B2 ) ) ) )
% 0.33/0.69 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.33/0.69 => ( ( Y = none_nat )
% 0.33/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 0.33/0.69 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.33/0.69 => ( ( Y
% 0.33/0.69 = ( some_nat @ Mi2 ) )
% 0.33/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % vebt_mint.pelims
% 0.33/0.69 thf(fact_9867_vebt__maxt_Opelims,axiom,
% 0.33/0.69 ! [X: vEBT_VEBT,Y: option_nat] :
% 0.33/0.69 ( ( ( vEBT_vebt_maxt @ X )
% 0.33/0.69 = Y )
% 0.33/0.69 => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X )
% 0.33/0.69 => ( ! [A3: $o,B2: $o] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( vEBT_Leaf @ A3 @ B2 ) )
% 0.33/0.69 => ( ( ( B2
% 0.33/0.69 => ( Y
% 0.33/0.69 = ( some_nat @ one_one_nat ) ) )
% 0.33/0.69 & ( ~ B2
% 0.33/0.69 => ( ( A3
% 0.33/0.69 => ( Y
% 0.33/0.69 = ( some_nat @ zero_zero_nat ) ) )
% 0.33/0.69 & ( ~ A3
% 0.33/0.69 => ( Y = none_nat ) ) ) ) )
% 0.33/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A3 @ B2 ) ) ) )
% 0.33/0.69 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.33/0.69 => ( ( Y = none_nat )
% 0.33/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 0.33/0.69 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.33/0.69 => ( ( Y
% 0.33/0.69 = ( some_nat @ Ma2 ) )
% 0.33/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % vebt_maxt.pelims
% 0.33/0.69 thf(fact_9868_VEBT__internal_OminNull_Opelims_I1_J,axiom,
% 0.33/0.69 ! [X: vEBT_VEBT,Y: $o] :
% 0.33/0.69 ( ( ( vEBT_VEBT_minNull @ X )
% 0.33/0.69 = Y )
% 0.33/0.69 => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
% 0.33/0.69 => ( ( ( X
% 0.33/0.69 = ( vEBT_Leaf @ $false @ $false ) )
% 0.33/0.69 => ( Y
% 0.33/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
% 0.33/0.69 => ( ! [Uv2: $o] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.33/0.69 => ( ~ Y
% 0.33/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
% 0.33/0.69 => ( ! [Uu2: $o] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.33/0.69 => ( ~ Y
% 0.33/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
% 0.33/0.69 => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 0.33/0.69 => ( Y
% 0.33/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
% 0.33/0.69 => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 0.33/0.69 => ( ~ Y
% 0.33/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % VEBT_internal.minNull.pelims(1)
% 0.33/0.69 thf(fact_9869_VEBT__internal_OminNull_Opelims_I2_J,axiom,
% 0.33/0.69 ! [X: vEBT_VEBT] :
% 0.33/0.69 ( ( vEBT_VEBT_minNull @ X )
% 0.33/0.69 => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
% 0.33/0.69 => ( ( ( X
% 0.33/0.69 = ( vEBT_Leaf @ $false @ $false ) )
% 0.33/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
% 0.33/0.69 => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 0.33/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % VEBT_internal.minNull.pelims(2)
% 0.33/0.69 thf(fact_9870_VEBT__internal_OminNull_Opelims_I3_J,axiom,
% 0.33/0.69 ! [X: vEBT_VEBT] :
% 0.33/0.69 ( ~ ( vEBT_VEBT_minNull @ X )
% 0.33/0.69 => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
% 0.33/0.69 => ( ! [Uv2: $o] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.33/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
% 0.33/0.69 => ( ! [Uu2: $o] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.33/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
% 0.33/0.69 => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 0.33/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % VEBT_internal.minNull.pelims(3)
% 0.33/0.69 thf(fact_9871_finite__enumerate,axiom,
% 0.33/0.69 ! [S: set_nat] :
% 0.33/0.69 ( ( finite_finite_nat @ S )
% 0.33/0.69 => ? [R3: nat > nat] :
% 0.33/0.69 ( ( strict1292158309912662752at_nat @ R3 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S ) ) )
% 0.33/0.69 & ! [N9: nat] :
% 0.33/0.69 ( ( ord_less_nat @ N9 @ ( finite_card_nat @ S ) )
% 0.33/0.69 => ( member_nat @ ( R3 @ N9 ) @ S ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % finite_enumerate
% 0.33/0.69 thf(fact_9872_divmod__integer__eq__cases,axiom,
% 0.33/0.69 ( code_divmod_integer
% 0.33/0.69 = ( ^ [K3: code_integer,L: code_integer] :
% 0.33/0.69 ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 0.33/0.69 @ ( if_Pro6119634080678213985nteger @ ( L = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
% 0.33/0.69 @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L
% 0.33/0.69 @ ( if_Pro6119634080678213985nteger
% 0.33/0.69 @ ( ( sgn_sgn_Code_integer @ K3 )
% 0.33/0.69 = ( sgn_sgn_Code_integer @ L ) )
% 0.33/0.69 @ ( code_divmod_abs @ K3 @ L )
% 0.33/0.69 @ ( produc6916734918728496179nteger
% 0.33/0.69 @ ^ [R5: code_integer,S5: code_integer] : ( if_Pro6119634080678213985nteger @ ( S5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L ) @ S5 ) ) )
% 0.33/0.69 @ ( code_divmod_abs @ K3 @ L ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % divmod_integer_eq_cases
% 0.33/0.69 thf(fact_9873_card_Ocomp__fun__commute__on,axiom,
% 0.33/0.69 ( ( comp_nat_nat_nat @ suc @ suc )
% 0.33/0.69 = ( comp_nat_nat_nat @ suc @ suc ) ) ).
% 0.33/0.69
% 0.33/0.69 % card.comp_fun_commute_on
% 0.33/0.69 thf(fact_9874_card__greaterThanLessThan__int,axiom,
% 0.33/0.69 ! [L2: int,U: int] :
% 0.33/0.69 ( ( finite_card_int @ ( set_or5832277885323065728an_int @ L2 @ U ) )
% 0.33/0.69 = ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L2 @ one_one_int ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % card_greaterThanLessThan_int
% 0.33/0.69 thf(fact_9875_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
% 0.33/0.69 ! [L2: int,U: int] :
% 0.33/0.69 ( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L2 @ one_one_int ) @ U )
% 0.33/0.69 = ( set_or5832277885323065728an_int @ L2 @ U ) ) ).
% 0.33/0.69
% 0.33/0.69 % atLeastPlusOneLessThan_greaterThanLessThan_int
% 0.33/0.69 thf(fact_9876_greaterThanLessThan__upto,axiom,
% 0.33/0.69 ( set_or5832277885323065728an_int
% 0.33/0.69 = ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % greaterThanLessThan_upto
% 0.33/0.69 thf(fact_9877_and__not__num_Oelims,axiom,
% 0.33/0.69 ! [X: num,Xa2: num,Y: option_num] :
% 0.33/0.69 ( ( ( bit_and_not_num @ X @ Xa2 )
% 0.33/0.69 = Y )
% 0.33/0.69 => ( ( ( X = one )
% 0.33/0.69 => ( ( Xa2 = one )
% 0.33/0.69 => ( Y != none_num ) ) )
% 0.33/0.69 => ( ( ( X = one )
% 0.33/0.69 => ( ? [N4: num] :
% 0.33/0.69 ( Xa2
% 0.33/0.69 = ( bit0 @ N4 ) )
% 0.33/0.69 => ( Y
% 0.33/0.69 != ( some_num @ one ) ) ) )
% 0.33/0.69 => ( ( ( X = one )
% 0.33/0.69 => ( ? [N4: num] :
% 0.33/0.69 ( Xa2
% 0.33/0.69 = ( bit1 @ N4 ) )
% 0.33/0.69 => ( Y != none_num ) ) )
% 0.33/0.69 => ( ! [M2: num] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( bit0 @ M2 ) )
% 0.33/0.69 => ( ( Xa2 = one )
% 0.33/0.69 => ( Y
% 0.33/0.69 != ( some_num @ ( bit0 @ M2 ) ) ) ) )
% 0.33/0.69 => ( ! [M2: num] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( bit0 @ M2 ) )
% 0.33/0.69 => ! [N4: num] :
% 0.33/0.69 ( ( Xa2
% 0.33/0.69 = ( bit0 @ N4 ) )
% 0.33/0.69 => ( Y
% 0.33/0.69 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M2 @ N4 ) ) ) ) )
% 0.33/0.69 => ( ! [M2: num] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( bit0 @ M2 ) )
% 0.33/0.69 => ! [N4: num] :
% 0.33/0.69 ( ( Xa2
% 0.33/0.69 = ( bit1 @ N4 ) )
% 0.33/0.69 => ( Y
% 0.33/0.69 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M2 @ N4 ) ) ) ) )
% 0.33/0.69 => ( ! [M2: num] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( bit1 @ M2 ) )
% 0.33/0.69 => ( ( Xa2 = one )
% 0.33/0.69 => ( Y
% 0.33/0.69 != ( some_num @ ( bit0 @ M2 ) ) ) ) )
% 0.33/0.69 => ( ! [M2: num] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( bit1 @ M2 ) )
% 0.33/0.69 => ! [N4: num] :
% 0.33/0.69 ( ( Xa2
% 0.33/0.69 = ( bit0 @ N4 ) )
% 0.33/0.69 => ( Y
% 0.33/0.69 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.33/0.69 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.33/0.69 @ ( bit_and_not_num @ M2 @ N4 ) ) ) ) )
% 0.33/0.69 => ~ ! [M2: num] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( bit1 @ M2 ) )
% 0.33/0.69 => ! [N4: num] :
% 0.33/0.69 ( ( Xa2
% 0.33/0.69 = ( bit1 @ N4 ) )
% 0.33/0.69 => ( Y
% 0.33/0.69 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M2 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % and_not_num.elims
% 0.33/0.69 thf(fact_9878_card__greaterThanLessThan,axiom,
% 0.33/0.69 ! [L2: nat,U: nat] :
% 0.33/0.69 ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L2 @ U ) )
% 0.33/0.69 = ( minus_minus_nat @ U @ ( suc @ L2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % card_greaterThanLessThan
% 0.33/0.69 thf(fact_9879_atLeastSucLessThan__greaterThanLessThan,axiom,
% 0.33/0.69 ! [L2: nat,U: nat] :
% 0.33/0.69 ( ( set_or4665077453230672383an_nat @ ( suc @ L2 ) @ U )
% 0.33/0.69 = ( set_or5834768355832116004an_nat @ L2 @ U ) ) ).
% 0.33/0.69
% 0.33/0.69 % atLeastSucLessThan_greaterThanLessThan
% 0.33/0.69 thf(fact_9880_and__not__num_Osimps_I5_J,axiom,
% 0.33/0.69 ! [M: num,N2: num] :
% 0.33/0.69 ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
% 0.33/0.69 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % and_not_num.simps(5)
% 0.33/0.69 thf(fact_9881_and__not__num_Osimps_I9_J,axiom,
% 0.33/0.69 ! [M: num,N2: num] :
% 0.33/0.69 ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 0.33/0.69 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % and_not_num.simps(9)
% 0.33/0.69 thf(fact_9882_and__not__num_Osimps_I6_J,axiom,
% 0.33/0.69 ! [M: num,N2: num] :
% 0.33/0.69 ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 0.33/0.69 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % and_not_num.simps(6)
% 0.33/0.69 thf(fact_9883_tanh__real__bounds,axiom,
% 0.33/0.69 ! [X: real] : ( member_real @ ( tanh_real @ X ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) ).
% 0.33/0.69
% 0.33/0.69 % tanh_real_bounds
% 0.33/0.69 thf(fact_9884_and__num_Oelims,axiom,
% 0.33/0.69 ! [X: num,Xa2: num,Y: option_num] :
% 0.33/0.69 ( ( ( bit_un7362597486090784418nd_num @ X @ Xa2 )
% 0.33/0.69 = Y )
% 0.33/0.69 => ( ( ( X = one )
% 0.33/0.69 => ( ( Xa2 = one )
% 0.33/0.69 => ( Y
% 0.33/0.69 != ( some_num @ one ) ) ) )
% 0.33/0.69 => ( ( ( X = one )
% 0.33/0.69 => ( ? [N4: num] :
% 0.33/0.69 ( Xa2
% 0.33/0.69 = ( bit0 @ N4 ) )
% 0.33/0.69 => ( Y != none_num ) ) )
% 0.33/0.69 => ( ( ( X = one )
% 0.33/0.69 => ( ? [N4: num] :
% 0.33/0.69 ( Xa2
% 0.33/0.69 = ( bit1 @ N4 ) )
% 0.33/0.69 => ( Y
% 0.33/0.69 != ( some_num @ one ) ) ) )
% 0.33/0.69 => ( ( ? [M2: num] :
% 0.33/0.69 ( X
% 0.33/0.69 = ( bit0 @ M2 ) )
% 0.33/0.69 => ( ( Xa2 = one )
% 0.33/0.69 => ( Y != none_num ) ) )
% 0.33/0.69 => ( ! [M2: num] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( bit0 @ M2 ) )
% 0.33/0.69 => ! [N4: num] :
% 0.33/0.69 ( ( Xa2
% 0.33/0.69 = ( bit0 @ N4 ) )
% 0.33/0.69 => ( Y
% 0.33/0.69 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M2 @ N4 ) ) ) ) )
% 0.33/0.69 => ( ! [M2: num] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( bit0 @ M2 ) )
% 0.33/0.69 => ! [N4: num] :
% 0.33/0.69 ( ( Xa2
% 0.33/0.69 = ( bit1 @ N4 ) )
% 0.33/0.69 => ( Y
% 0.33/0.69 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M2 @ N4 ) ) ) ) )
% 0.33/0.69 => ( ( ? [M2: num] :
% 0.33/0.69 ( X
% 0.33/0.69 = ( bit1 @ M2 ) )
% 0.33/0.69 => ( ( Xa2 = one )
% 0.33/0.69 => ( Y
% 0.33/0.69 != ( some_num @ one ) ) ) )
% 0.33/0.69 => ( ! [M2: num] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( bit1 @ M2 ) )
% 0.33/0.69 => ! [N4: num] :
% 0.33/0.69 ( ( Xa2
% 0.33/0.69 = ( bit0 @ N4 ) )
% 0.33/0.69 => ( Y
% 0.33/0.69 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M2 @ N4 ) ) ) ) )
% 0.33/0.69 => ~ ! [M2: num] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( bit1 @ M2 ) )
% 0.33/0.69 => ! [N4: num] :
% 0.33/0.69 ( ( Xa2
% 0.33/0.69 = ( bit1 @ N4 ) )
% 0.33/0.69 => ( Y
% 0.33/0.69 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.33/0.69 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.33/0.69 @ ( bit_un7362597486090784418nd_num @ M2 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % and_num.elims
% 0.33/0.69 thf(fact_9885_and__num_Osimps_I1_J,axiom,
% 0.33/0.69 ( ( bit_un7362597486090784418nd_num @ one @ one )
% 0.33/0.69 = ( some_num @ one ) ) ).
% 0.33/0.69
% 0.33/0.69 % and_num.simps(1)
% 0.33/0.69 thf(fact_9886_and__num_Osimps_I5_J,axiom,
% 0.33/0.69 ! [M: num,N2: num] :
% 0.33/0.69 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
% 0.33/0.69 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % and_num.simps(5)
% 0.33/0.69 thf(fact_9887_and__num_Osimps_I3_J,axiom,
% 0.33/0.69 ! [N2: num] :
% 0.33/0.69 ( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N2 ) )
% 0.33/0.69 = ( some_num @ one ) ) ).
% 0.33/0.69
% 0.33/0.69 % and_num.simps(3)
% 0.33/0.69 thf(fact_9888_and__num_Osimps_I7_J,axiom,
% 0.33/0.69 ! [M: num] :
% 0.33/0.69 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one )
% 0.33/0.69 = ( some_num @ one ) ) ).
% 0.33/0.69
% 0.33/0.69 % and_num.simps(7)
% 0.33/0.69 thf(fact_9889_and__num_Osimps_I4_J,axiom,
% 0.33/0.69 ! [M: num] :
% 0.33/0.69 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
% 0.33/0.69 = none_num ) ).
% 0.33/0.69
% 0.33/0.69 % and_num.simps(4)
% 0.33/0.69 thf(fact_9890_and__num_Osimps_I2_J,axiom,
% 0.33/0.69 ! [N2: num] :
% 0.33/0.69 ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N2 ) )
% 0.33/0.69 = none_num ) ).
% 0.33/0.69
% 0.33/0.69 % and_num.simps(2)
% 0.33/0.69 thf(fact_9891_and__num_Osimps_I6_J,axiom,
% 0.33/0.69 ! [M: num,N2: num] :
% 0.33/0.69 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 0.33/0.69 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % and_num.simps(6)
% 0.33/0.69 thf(fact_9892_and__num_Osimps_I8_J,axiom,
% 0.33/0.69 ! [M: num,N2: num] :
% 0.33/0.69 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 0.33/0.69 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % and_num.simps(8)
% 0.33/0.69 thf(fact_9893_and__num_Osimps_I9_J,axiom,
% 0.33/0.69 ! [M: num,N2: num] :
% 0.33/0.69 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 0.33/0.69 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.33/0.69 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.33/0.69 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % and_num.simps(9)
% 0.33/0.69 thf(fact_9894_xor__num_Oelims,axiom,
% 0.33/0.69 ! [X: num,Xa2: num,Y: option_num] :
% 0.33/0.69 ( ( ( bit_un2480387367778600638or_num @ X @ Xa2 )
% 0.33/0.69 = Y )
% 0.33/0.69 => ( ( ( X = one )
% 0.33/0.69 => ( ( Xa2 = one )
% 0.33/0.69 => ( Y != none_num ) ) )
% 0.33/0.69 => ( ( ( X = one )
% 0.33/0.69 => ! [N4: num] :
% 0.33/0.69 ( ( Xa2
% 0.33/0.69 = ( bit0 @ N4 ) )
% 0.33/0.69 => ( Y
% 0.33/0.69 != ( some_num @ ( bit1 @ N4 ) ) ) ) )
% 0.33/0.69 => ( ( ( X = one )
% 0.33/0.69 => ! [N4: num] :
% 0.33/0.69 ( ( Xa2
% 0.33/0.69 = ( bit1 @ N4 ) )
% 0.33/0.69 => ( Y
% 0.33/0.69 != ( some_num @ ( bit0 @ N4 ) ) ) ) )
% 0.33/0.69 => ( ! [M2: num] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( bit0 @ M2 ) )
% 0.33/0.69 => ( ( Xa2 = one )
% 0.33/0.69 => ( Y
% 0.33/0.69 != ( some_num @ ( bit1 @ M2 ) ) ) ) )
% 0.33/0.69 => ( ! [M2: num] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( bit0 @ M2 ) )
% 0.33/0.69 => ! [N4: num] :
% 0.33/0.69 ( ( Xa2
% 0.33/0.69 = ( bit0 @ N4 ) )
% 0.33/0.69 => ( Y
% 0.33/0.69 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M2 @ N4 ) ) ) ) )
% 0.33/0.69 => ( ! [M2: num] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( bit0 @ M2 ) )
% 0.33/0.69 => ! [N4: num] :
% 0.33/0.69 ( ( Xa2
% 0.33/0.69 = ( bit1 @ N4 ) )
% 0.33/0.69 => ( Y
% 0.33/0.69 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M2 @ N4 ) ) ) ) ) )
% 0.33/0.69 => ( ! [M2: num] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( bit1 @ M2 ) )
% 0.33/0.69 => ( ( Xa2 = one )
% 0.33/0.69 => ( Y
% 0.33/0.69 != ( some_num @ ( bit0 @ M2 ) ) ) ) )
% 0.33/0.69 => ( ! [M2: num] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( bit1 @ M2 ) )
% 0.33/0.69 => ! [N4: num] :
% 0.33/0.69 ( ( Xa2
% 0.33/0.69 = ( bit0 @ N4 ) )
% 0.33/0.69 => ( Y
% 0.33/0.69 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M2 @ N4 ) ) ) ) ) )
% 0.33/0.69 => ~ ! [M2: num] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( bit1 @ M2 ) )
% 0.33/0.69 => ! [N4: num] :
% 0.33/0.69 ( ( Xa2
% 0.33/0.69 = ( bit1 @ N4 ) )
% 0.33/0.69 => ( Y
% 0.33/0.69 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M2 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % xor_num.elims
% 0.33/0.69 thf(fact_9895_and__num__dict,axiom,
% 0.33/0.69 bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).
% 0.33/0.69
% 0.33/0.69 % and_num_dict
% 0.33/0.69 thf(fact_9896_xor__num_Osimps_I1_J,axiom,
% 0.33/0.69 ( ( bit_un2480387367778600638or_num @ one @ one )
% 0.33/0.69 = none_num ) ).
% 0.33/0.69
% 0.33/0.69 % xor_num.simps(1)
% 0.33/0.69 thf(fact_9897_xor__num_Osimps_I5_J,axiom,
% 0.33/0.69 ! [M: num,N2: num] :
% 0.33/0.69 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
% 0.33/0.69 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % xor_num.simps(5)
% 0.33/0.69 thf(fact_9898_xor__num_Osimps_I9_J,axiom,
% 0.33/0.69 ! [M: num,N2: num] :
% 0.33/0.69 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 0.33/0.69 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % xor_num.simps(9)
% 0.33/0.69 thf(fact_9899_xor__num_Osimps_I2_J,axiom,
% 0.33/0.69 ! [N2: num] :
% 0.33/0.69 ( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N2 ) )
% 0.33/0.69 = ( some_num @ ( bit1 @ N2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % xor_num.simps(2)
% 0.33/0.69 thf(fact_9900_xor__num_Osimps_I3_J,axiom,
% 0.33/0.69 ! [N2: num] :
% 0.33/0.69 ( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N2 ) )
% 0.33/0.69 = ( some_num @ ( bit0 @ N2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % xor_num.simps(3)
% 0.33/0.69 thf(fact_9901_xor__num_Osimps_I4_J,axiom,
% 0.33/0.69 ! [M: num] :
% 0.33/0.69 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one )
% 0.33/0.69 = ( some_num @ ( bit1 @ M ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % xor_num.simps(4)
% 0.33/0.69 thf(fact_9902_xor__num_Osimps_I7_J,axiom,
% 0.33/0.69 ! [M: num] :
% 0.33/0.69 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one )
% 0.33/0.69 = ( some_num @ ( bit0 @ M ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % xor_num.simps(7)
% 0.33/0.69 thf(fact_9903_xor__num_Osimps_I8_J,axiom,
% 0.33/0.69 ! [M: num,N2: num] :
% 0.33/0.69 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 0.33/0.69 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % xor_num.simps(8)
% 0.33/0.69 thf(fact_9904_xor__num_Osimps_I6_J,axiom,
% 0.33/0.69 ! [M: num,N2: num] :
% 0.33/0.69 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 0.33/0.69 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % xor_num.simps(6)
% 0.33/0.69 thf(fact_9905_xor__num__dict,axiom,
% 0.33/0.69 bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).
% 0.33/0.69
% 0.33/0.69 % xor_num_dict
% 0.33/0.69 thf(fact_9906_xor__minus__numerals_I2_J,axiom,
% 0.33/0.69 ! [K: int,N2: num] :
% 0.33/0.69 ( ( bit_se6526347334894502574or_int @ K @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.33/0.69 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K @ ( neg_numeral_sub_int @ N2 @ one ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % xor_minus_numerals(2)
% 0.33/0.69 thf(fact_9907_xor__minus__numerals_I1_J,axiom,
% 0.33/0.69 ! [N2: num,K: int] :
% 0.33/0.69 ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ K )
% 0.33/0.69 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N2 @ one ) @ K ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % xor_minus_numerals(1)
% 0.33/0.69 thf(fact_9908_sub__BitM__One__eq,axiom,
% 0.33/0.69 ! [N2: num] :
% 0.33/0.69 ( ( neg_numeral_sub_int @ ( bitM @ N2 ) @ one )
% 0.33/0.69 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N2 @ one ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % sub_BitM_One_eq
% 0.33/0.69 thf(fact_9909_Suc__funpow,axiom,
% 0.33/0.69 ! [N2: nat] :
% 0.33/0.69 ( ( compow_nat_nat @ N2 @ suc )
% 0.33/0.69 = ( plus_plus_nat @ N2 ) ) ).
% 0.33/0.69
% 0.33/0.69 % Suc_funpow
% 0.33/0.69 thf(fact_9910_max__nat_Osemilattice__neutr__order__axioms,axiom,
% 0.33/0.69 ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
% 0.33/0.69 @ ^ [X2: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X2 )
% 0.33/0.69 @ ^ [X2: nat,Y2: nat] : ( ord_less_nat @ Y2 @ X2 ) ) ).
% 0.33/0.69
% 0.33/0.69 % max_nat.semilattice_neutr_order_axioms
% 0.33/0.69 thf(fact_9911_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
% 0.33/0.69 ( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
% 0.33/0.69 @ ^ [M6: nat,N: nat] :
% 0.33/0.69 ( ( dvd_dvd_nat @ M6 @ N )
% 0.33/0.69 & ( M6 != N ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % gcd_nat.semilattice_neutr_order_axioms
% 0.33/0.69 thf(fact_9912_Gcd__remove0__nat,axiom,
% 0.33/0.69 ! [M7: set_nat] :
% 0.33/0.69 ( ( finite_finite_nat @ M7 )
% 0.33/0.69 => ( ( gcd_Gcd_nat @ M7 )
% 0.33/0.69 = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M7 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Gcd_remove0_nat
% 0.33/0.69 thf(fact_9913_times__int_Oabs__eq,axiom,
% 0.33/0.69 ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.33/0.69 ( ( times_times_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
% 0.33/0.69 = ( abs_Integ
% 0.33/0.69 @ ( produc27273713700761075at_nat
% 0.33/0.69 @ ^ [X2: nat,Y2: nat] :
% 0.33/0.69 ( produc2626176000494625587at_nat
% 0.33/0.69 @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U2 ) @ ( times_times_nat @ Y2 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) )
% 0.33/0.69 @ Xa2
% 0.33/0.69 @ X ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % times_int.abs_eq
% 0.33/0.69 thf(fact_9914_eq__Abs__Integ,axiom,
% 0.33/0.69 ! [Z: int] :
% 0.33/0.69 ~ ! [X4: nat,Y4: nat] :
% 0.33/0.69 ( Z
% 0.33/0.69 != ( abs_Integ @ ( product_Pair_nat_nat @ X4 @ Y4 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % eq_Abs_Integ
% 0.33/0.69 thf(fact_9915_Gcd__nat__eq__one,axiom,
% 0.33/0.69 ! [N3: set_nat] :
% 0.33/0.69 ( ( member_nat @ one_one_nat @ N3 )
% 0.33/0.69 => ( ( gcd_Gcd_nat @ N3 )
% 0.33/0.69 = one_one_nat ) ) ).
% 0.33/0.69
% 0.33/0.69 % Gcd_nat_eq_one
% 0.33/0.69 thf(fact_9916_nat_Oabs__eq,axiom,
% 0.33/0.69 ! [X: product_prod_nat_nat] :
% 0.33/0.69 ( ( nat2 @ ( abs_Integ @ X ) )
% 0.33/0.69 = ( produc6842872674320459806at_nat @ minus_minus_nat @ X ) ) ).
% 0.33/0.69
% 0.33/0.69 % nat.abs_eq
% 0.33/0.69 thf(fact_9917_zero__int__def,axiom,
% 0.33/0.69 ( zero_zero_int
% 0.33/0.69 = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % zero_int_def
% 0.33/0.69 thf(fact_9918_int__def,axiom,
% 0.33/0.69 ( semiri1314217659103216013at_int
% 0.33/0.69 = ( ^ [N: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N @ zero_zero_nat ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % int_def
% 0.33/0.69 thf(fact_9919_uminus__int_Oabs__eq,axiom,
% 0.33/0.69 ! [X: product_prod_nat_nat] :
% 0.33/0.69 ( ( uminus_uminus_int @ ( abs_Integ @ X ) )
% 0.33/0.69 = ( abs_Integ
% 0.33/0.69 @ ( produc2626176000494625587at_nat
% 0.33/0.69 @ ^ [X2: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X2 )
% 0.33/0.69 @ X ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % uminus_int.abs_eq
% 0.33/0.69 thf(fact_9920_one__int__def,axiom,
% 0.33/0.69 ( one_one_int
% 0.33/0.69 = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % one_int_def
% 0.33/0.69 thf(fact_9921_less__int_Oabs__eq,axiom,
% 0.33/0.69 ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.33/0.69 ( ( ord_less_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
% 0.33/0.69 = ( produc8739625826339149834_nat_o
% 0.33/0.69 @ ^ [X2: nat,Y2: nat] :
% 0.33/0.69 ( produc6081775807080527818_nat_o
% 0.33/0.69 @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) )
% 0.33/0.69 @ Xa2
% 0.33/0.69 @ X ) ) ).
% 0.33/0.69
% 0.33/0.69 % less_int.abs_eq
% 0.33/0.69 thf(fact_9922_less__eq__int_Oabs__eq,axiom,
% 0.33/0.69 ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.33/0.69 ( ( ord_less_eq_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
% 0.33/0.69 = ( produc8739625826339149834_nat_o
% 0.33/0.69 @ ^ [X2: nat,Y2: nat] :
% 0.33/0.69 ( produc6081775807080527818_nat_o
% 0.33/0.69 @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) )
% 0.33/0.69 @ Xa2
% 0.33/0.69 @ X ) ) ).
% 0.33/0.69
% 0.33/0.69 % less_eq_int.abs_eq
% 0.33/0.69 thf(fact_9923_plus__int_Oabs__eq,axiom,
% 0.33/0.69 ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.33/0.69 ( ( plus_plus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
% 0.33/0.69 = ( abs_Integ
% 0.33/0.69 @ ( produc27273713700761075at_nat
% 0.33/0.69 @ ^ [X2: nat,Y2: nat] :
% 0.33/0.69 ( produc2626176000494625587at_nat
% 0.33/0.69 @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U2 ) @ ( plus_plus_nat @ Y2 @ V4 ) ) )
% 0.33/0.69 @ Xa2
% 0.33/0.69 @ X ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % plus_int.abs_eq
% 0.33/0.69 thf(fact_9924_minus__int_Oabs__eq,axiom,
% 0.33/0.69 ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.33/0.69 ( ( minus_minus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
% 0.33/0.69 = ( abs_Integ
% 0.33/0.69 @ ( produc27273713700761075at_nat
% 0.33/0.69 @ ^ [X2: nat,Y2: nat] :
% 0.33/0.69 ( produc2626176000494625587at_nat
% 0.33/0.69 @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) )
% 0.33/0.69 @ Xa2
% 0.33/0.69 @ X ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % minus_int.abs_eq
% 0.33/0.69 thf(fact_9925_bij__betw__Suc,axiom,
% 0.33/0.69 ! [M7: set_nat,N3: set_nat] :
% 0.33/0.69 ( ( bij_betw_nat_nat @ suc @ M7 @ N3 )
% 0.33/0.69 = ( ( image_nat_nat @ suc @ M7 )
% 0.33/0.69 = N3 ) ) ).
% 0.33/0.69
% 0.33/0.69 % bij_betw_Suc
% 0.33/0.69 thf(fact_9926_image__Suc__atLeastAtMost,axiom,
% 0.33/0.69 ! [I2: nat,J: nat] :
% 0.33/0.69 ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I2 @ J ) )
% 0.33/0.69 = ( set_or1269000886237332187st_nat @ ( suc @ I2 ) @ ( suc @ J ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % image_Suc_atLeastAtMost
% 0.33/0.69 thf(fact_9927_image__Suc__atLeastLessThan,axiom,
% 0.33/0.69 ! [I2: nat,J: nat] :
% 0.33/0.69 ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I2 @ J ) )
% 0.33/0.69 = ( set_or4665077453230672383an_nat @ ( suc @ I2 ) @ ( suc @ J ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % image_Suc_atLeastLessThan
% 0.33/0.69 thf(fact_9928_Gcd__int__eq,axiom,
% 0.33/0.69 ! [N3: set_nat] :
% 0.33/0.69 ( ( gcd_Gcd_int @ ( image_nat_int @ semiri1314217659103216013at_int @ N3 ) )
% 0.33/0.69 = ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ N3 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Gcd_int_eq
% 0.33/0.69 thf(fact_9929_zero__notin__Suc__image,axiom,
% 0.33/0.69 ! [A2: set_nat] :
% 0.33/0.69 ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% 0.33/0.69
% 0.33/0.69 % zero_notin_Suc_image
% 0.33/0.69 thf(fact_9930_image__Suc__lessThan,axiom,
% 0.33/0.69 ! [N2: nat] :
% 0.33/0.69 ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.69 = ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ).
% 0.33/0.69
% 0.33/0.69 % image_Suc_lessThan
% 0.33/0.69 thf(fact_9931_image__Suc__atMost,axiom,
% 0.33/0.69 ! [N2: nat] :
% 0.33/0.69 ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N2 ) )
% 0.33/0.69 = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % image_Suc_atMost
% 0.33/0.69 thf(fact_9932_atLeast0__atMost__Suc__eq__insert__0,axiom,
% 0.33/0.69 ! [N2: nat] :
% 0.33/0.69 ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) )
% 0.33/0.69 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % atLeast0_atMost_Suc_eq_insert_0
% 0.33/0.69 thf(fact_9933_atLeast0__lessThan__Suc__eq__insert__0,axiom,
% 0.33/0.69 ! [N2: nat] :
% 0.33/0.69 ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) )
% 0.33/0.69 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % atLeast0_lessThan_Suc_eq_insert_0
% 0.33/0.69 thf(fact_9934_lessThan__Suc__eq__insert__0,axiom,
% 0.33/0.69 ! [N2: nat] :
% 0.33/0.69 ( ( set_ord_lessThan_nat @ ( suc @ N2 ) )
% 0.33/0.69 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % lessThan_Suc_eq_insert_0
% 0.33/0.69 thf(fact_9935_atMost__Suc__eq__insert__0,axiom,
% 0.33/0.69 ! [N2: nat] :
% 0.33/0.69 ( ( set_ord_atMost_nat @ ( suc @ N2 ) )
% 0.33/0.69 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % atMost_Suc_eq_insert_0
% 0.33/0.69 thf(fact_9936_image__minus__const__atLeastLessThan__nat,axiom,
% 0.33/0.69 ! [C: nat,Y: nat,X: nat] :
% 0.33/0.69 ( ( ( ord_less_nat @ C @ Y )
% 0.33/0.69 => ( ( image_nat_nat
% 0.33/0.69 @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
% 0.33/0.69 @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 0.33/0.69 = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y @ C ) ) ) )
% 0.33/0.69 & ( ~ ( ord_less_nat @ C @ Y )
% 0.33/0.69 => ( ( ( ord_less_nat @ X @ Y )
% 0.33/0.69 => ( ( image_nat_nat
% 0.33/0.69 @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
% 0.33/0.69 @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 0.33/0.69 = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
% 0.33/0.69 & ( ~ ( ord_less_nat @ X @ Y )
% 0.33/0.69 => ( ( image_nat_nat
% 0.33/0.69 @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
% 0.33/0.69 @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 0.33/0.69 = bot_bot_set_nat ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % image_minus_const_atLeastLessThan_nat
% 0.33/0.69 thf(fact_9937_Gcd__abs__eq,axiom,
% 0.33/0.69 ! [K5: set_int] :
% 0.33/0.69 ( ( gcd_Gcd_int @ ( image_int_int @ abs_abs_int @ K5 ) )
% 0.33/0.69 = ( gcd_Gcd_int @ K5 ) ) ).
% 0.33/0.69
% 0.33/0.69 % Gcd_abs_eq
% 0.33/0.69 thf(fact_9938_Gcd__nat__abs__eq,axiom,
% 0.33/0.69 ! [K5: set_int] :
% 0.33/0.69 ( ( gcd_Gcd_nat
% 0.33/0.69 @ ( image_int_nat
% 0.33/0.69 @ ^ [K3: int] : ( nat2 @ ( abs_abs_int @ K3 ) )
% 0.33/0.69 @ K5 ) )
% 0.33/0.69 = ( nat2 @ ( gcd_Gcd_int @ K5 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Gcd_nat_abs_eq
% 0.33/0.69 thf(fact_9939_Inf__real__def,axiom,
% 0.33/0.69 ( comple4887499456419720421f_real
% 0.33/0.69 = ( ^ [X3: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X3 ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Inf_real_def
% 0.33/0.69 thf(fact_9940_image__add__int__atLeastLessThan,axiom,
% 0.33/0.69 ! [L2: int,U: int] :
% 0.33/0.69 ( ( image_int_int
% 0.33/0.69 @ ^ [X2: int] : ( plus_plus_int @ X2 @ L2 )
% 0.33/0.69 @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L2 ) ) )
% 0.33/0.69 = ( set_or4662586982721622107an_int @ L2 @ U ) ) ).
% 0.33/0.69
% 0.33/0.69 % image_add_int_atLeastLessThan
% 0.33/0.69 thf(fact_9941_Gcd__int__def,axiom,
% 0.33/0.69 ( gcd_Gcd_int
% 0.33/0.69 = ( ^ [K7: set_int] : ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ K7 ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Gcd_int_def
% 0.33/0.69 thf(fact_9942_suminf__eq__SUP__real,axiom,
% 0.33/0.69 ! [X8: nat > real] :
% 0.33/0.69 ( ( summable_real @ X8 )
% 0.33/0.69 => ( ! [I4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X8 @ I4 ) )
% 0.33/0.69 => ( ( suminf_real @ X8 )
% 0.33/0.69 = ( comple1385675409528146559p_real
% 0.33/0.69 @ ( image_nat_real
% 0.33/0.69 @ ^ [I3: nat] : ( groups6591440286371151544t_real @ X8 @ ( set_ord_lessThan_nat @ I3 ) )
% 0.33/0.69 @ top_top_set_nat ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % suminf_eq_SUP_real
% 0.33/0.69 thf(fact_9943_UN__atMost__UNIV,axiom,
% 0.33/0.69 ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atMost_nat @ top_top_set_nat ) )
% 0.33/0.69 = top_top_set_nat ) ).
% 0.33/0.69
% 0.33/0.69 % UN_atMost_UNIV
% 0.33/0.69 thf(fact_9944_UN__lessThan__UNIV,axiom,
% 0.33/0.69 ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
% 0.33/0.69 = top_top_set_nat ) ).
% 0.33/0.69
% 0.33/0.69 % UN_lessThan_UNIV
% 0.33/0.69 thf(fact_9945_UNIV__nat__eq,axiom,
% 0.33/0.69 ( top_top_set_nat
% 0.33/0.69 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % UNIV_nat_eq
% 0.33/0.69 thf(fact_9946_range__mod,axiom,
% 0.33/0.69 ! [N2: nat] :
% 0.33/0.69 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.69 => ( ( image_nat_nat
% 0.33/0.69 @ ^ [M6: nat] : ( modulo_modulo_nat @ M6 @ N2 )
% 0.33/0.69 @ top_top_set_nat )
% 0.33/0.69 = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % range_mod
% 0.33/0.69 thf(fact_9947_card__UNIV__unit,axiom,
% 0.33/0.69 ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
% 0.33/0.69 = one_one_nat ) ).
% 0.33/0.69
% 0.33/0.69 % card_UNIV_unit
% 0.33/0.69 thf(fact_9948_card__UNIV__bool,axiom,
% 0.33/0.69 ( ( finite_card_o @ top_top_set_o )
% 0.33/0.69 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % card_UNIV_bool
% 0.33/0.69 thf(fact_9949_range__mult,axiom,
% 0.33/0.69 ! [A: real] :
% 0.33/0.69 ( ( ( A = zero_zero_real )
% 0.33/0.69 => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 0.33/0.69 = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
% 0.33/0.69 & ( ( A != zero_zero_real )
% 0.33/0.69 => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 0.33/0.69 = top_top_set_real ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % range_mult
% 0.33/0.69 thf(fact_9950_root__def,axiom,
% 0.33/0.69 ( root
% 0.33/0.69 = ( ^ [N: nat,X2: real] :
% 0.33/0.69 ( if_real @ ( N = zero_zero_nat ) @ zero_zero_real
% 0.33/0.69 @ ( the_in5290026491893676941l_real @ top_top_set_real
% 0.33/0.69 @ ^ [Y2: real] : ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N ) )
% 0.33/0.69 @ X2 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % root_def
% 0.33/0.69 thf(fact_9951_card__UNIV__char,axiom,
% 0.33/0.69 ( ( finite_card_char @ top_top_set_char )
% 0.33/0.69 = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % card_UNIV_char
% 0.33/0.69 thf(fact_9952_UNIV__char__of__nat,axiom,
% 0.33/0.69 ( top_top_set_char
% 0.33/0.69 = ( image_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % UNIV_char_of_nat
% 0.33/0.69 thf(fact_9953_nat__of__char__less__256,axiom,
% 0.33/0.69 ! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % nat_of_char_less_256
% 0.33/0.69 thf(fact_9954_range__nat__of__char,axiom,
% 0.33/0.69 ( ( image_char_nat @ comm_s629917340098488124ar_nat @ top_top_set_char )
% 0.33/0.69 = ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % range_nat_of_char
% 0.33/0.69 thf(fact_9955_integer__of__char__code,axiom,
% 0.33/0.69 ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
% 0.33/0.69 ( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
% 0.33/0.69 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ B72 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B62 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B52 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B42 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B32 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B22 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B1 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B0 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % integer_of_char_code
% 0.33/0.69 thf(fact_9956_char__of__integer__code,axiom,
% 0.33/0.69 ( char_of_integer
% 0.33/0.69 = ( ^ [K3: code_integer] :
% 0.33/0.69 ( produc4188289175737317920o_char
% 0.33/0.69 @ ^ [Q0: code_integer,B02: $o] :
% 0.33/0.69 ( produc4188289175737317920o_char
% 0.33/0.69 @ ^ [Q1: code_integer,B12: $o] :
% 0.33/0.69 ( produc4188289175737317920o_char
% 0.33/0.69 @ ^ [Q22: code_integer,B23: $o] :
% 0.33/0.69 ( produc4188289175737317920o_char
% 0.33/0.69 @ ^ [Q32: code_integer,B33: $o] :
% 0.33/0.69 ( produc4188289175737317920o_char
% 0.33/0.69 @ ^ [Q42: code_integer,B43: $o] :
% 0.33/0.69 ( produc4188289175737317920o_char
% 0.33/0.69 @ ^ [Q52: code_integer,B53: $o] :
% 0.33/0.69 ( produc4188289175737317920o_char
% 0.33/0.69 @ ^ [Q62: code_integer,B63: $o] :
% 0.33/0.69 ( produc4188289175737317920o_char
% 0.33/0.69 @ ^ [Uu3: code_integer] : ( char2 @ B02 @ B12 @ B23 @ B33 @ B43 @ B53 @ B63 )
% 0.33/0.69 @ ( code_bit_cut_integer @ Q62 ) )
% 0.33/0.69 @ ( code_bit_cut_integer @ Q52 ) )
% 0.33/0.69 @ ( code_bit_cut_integer @ Q42 ) )
% 0.33/0.69 @ ( code_bit_cut_integer @ Q32 ) )
% 0.33/0.69 @ ( code_bit_cut_integer @ Q22 ) )
% 0.33/0.69 @ ( code_bit_cut_integer @ Q1 ) )
% 0.33/0.69 @ ( code_bit_cut_integer @ Q0 ) )
% 0.33/0.69 @ ( code_bit_cut_integer @ K3 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % char_of_integer_code
% 0.33/0.69 thf(fact_9957_String_Ochar__of__ascii__of,axiom,
% 0.33/0.69 ! [C: char] :
% 0.33/0.69 ( ( comm_s629917340098488124ar_nat @ ( ascii_of @ C ) )
% 0.33/0.69 = ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( comm_s629917340098488124ar_nat @ C ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % String.char_of_ascii_of
% 0.33/0.69 thf(fact_9958_DERIV__even__real__root,axiom,
% 0.33/0.69 ! [N2: nat,X: real] :
% 0.33/0.69 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.69 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.69 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_even_real_root
% 0.33/0.69 thf(fact_9959_deriv__nonneg__imp__mono,axiom,
% 0.33/0.69 ! [A: real,B: real,G: real > real,G2: real > real] :
% 0.33/0.69 ( ! [X4: real] :
% 0.33/0.69 ( ( member_real @ X4 @ ( set_or1222579329274155063t_real @ A @ B ) )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 0.33/0.69 => ( ! [X4: real] :
% 0.33/0.69 ( ( member_real @ X4 @ ( set_or1222579329274155063t_real @ A @ B ) )
% 0.33/0.69 => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X4 ) ) )
% 0.33/0.69 => ( ( ord_less_eq_real @ A @ B )
% 0.33/0.69 => ( ord_less_eq_real @ ( G @ A ) @ ( G @ B ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % deriv_nonneg_imp_mono
% 0.33/0.69 thf(fact_9960_DERIV__nonneg__imp__nondecreasing,axiom,
% 0.33/0.69 ! [A: real,B: real,F: real > real] :
% 0.33/0.69 ( ( ord_less_eq_real @ A @ B )
% 0.33/0.69 => ( ! [X4: real] :
% 0.33/0.69 ( ( ord_less_eq_real @ A @ X4 )
% 0.33/0.69 => ( ( ord_less_eq_real @ X4 @ B )
% 0.33/0.69 => ? [Y5: real] :
% 0.33/0.69 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.33/0.69 & ( ord_less_eq_real @ zero_zero_real @ Y5 ) ) ) )
% 0.33/0.69 => ( ord_less_eq_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_nonneg_imp_nondecreasing
% 0.33/0.69 thf(fact_9961_DERIV__nonpos__imp__nonincreasing,axiom,
% 0.33/0.69 ! [A: real,B: real,F: real > real] :
% 0.33/0.69 ( ( ord_less_eq_real @ A @ B )
% 0.33/0.69 => ( ! [X4: real] :
% 0.33/0.69 ( ( ord_less_eq_real @ A @ X4 )
% 0.33/0.69 => ( ( ord_less_eq_real @ X4 @ B )
% 0.33/0.69 => ? [Y5: real] :
% 0.33/0.69 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.33/0.69 & ( ord_less_eq_real @ Y5 @ zero_zero_real ) ) ) )
% 0.33/0.69 => ( ord_less_eq_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_nonpos_imp_nonincreasing
% 0.33/0.69 thf(fact_9962_DERIV__mirror,axiom,
% 0.33/0.69 ! [F: real > real,Y: real,X: real] :
% 0.33/0.69 ( ( has_fi5821293074295781190e_real @ F @ Y @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ X ) @ top_top_set_real ) )
% 0.33/0.69 = ( has_fi5821293074295781190e_real
% 0.33/0.69 @ ^ [X2: real] : ( F @ ( uminus_uminus_real @ X2 ) )
% 0.33/0.69 @ ( uminus_uminus_real @ Y )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_mirror
% 0.33/0.69 thf(fact_9963_DERIV__const__ratio__const2,axiom,
% 0.33/0.69 ! [A: real,B: real,F: real > real,K: real] :
% 0.33/0.69 ( ( A != B )
% 0.33/0.69 => ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.33/0.69 => ( ( divide_divide_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( minus_minus_real @ B @ A ) )
% 0.33/0.69 = K ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_const_ratio_const2
% 0.33/0.69 thf(fact_9964_DERIV__const__ratio__const,axiom,
% 0.33/0.69 ! [A: real,B: real,F: real > real,K: real] :
% 0.33/0.69 ( ( A != B )
% 0.33/0.69 => ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.33/0.69 => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.33/0.69 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ K ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_const_ratio_const
% 0.33/0.69 thf(fact_9965_has__real__derivative__pos__inc__right,axiom,
% 0.33/0.69 ! [F: real > real,L2: real,X: real,S: set_real] :
% 0.33/0.69 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ S ) )
% 0.33/0.69 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 0.33/0.69 => ? [D4: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ D4 )
% 0.33/0.69 & ! [H4: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.33/0.69 => ( ( member_real @ ( plus_plus_real @ X @ H4 ) @ S )
% 0.33/0.69 => ( ( ord_less_real @ H4 @ D4 )
% 0.33/0.69 => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H4 ) ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % has_real_derivative_pos_inc_right
% 0.33/0.69 thf(fact_9966_has__real__derivative__neg__dec__right,axiom,
% 0.33/0.69 ! [F: real > real,L2: real,X: real,S: set_real] :
% 0.33/0.69 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ S ) )
% 0.33/0.69 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 0.33/0.69 => ? [D4: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ D4 )
% 0.33/0.69 & ! [H4: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.33/0.69 => ( ( member_real @ ( plus_plus_real @ X @ H4 ) @ S )
% 0.33/0.69 => ( ( ord_less_real @ H4 @ D4 )
% 0.33/0.69 => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % has_real_derivative_neg_dec_right
% 0.33/0.69 thf(fact_9967_has__real__derivative__pos__inc__left,axiom,
% 0.33/0.69 ! [F: real > real,L2: real,X: real,S: set_real] :
% 0.33/0.69 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ S ) )
% 0.33/0.69 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 0.33/0.69 => ? [D4: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ D4 )
% 0.33/0.69 & ! [H4: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.33/0.69 => ( ( member_real @ ( minus_minus_real @ X @ H4 ) @ S )
% 0.33/0.69 => ( ( ord_less_real @ H4 @ D4 )
% 0.33/0.69 => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % has_real_derivative_pos_inc_left
% 0.33/0.69 thf(fact_9968_has__real__derivative__neg__dec__left,axiom,
% 0.33/0.69 ! [F: real > real,L2: real,X: real,S: set_real] :
% 0.33/0.69 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ S ) )
% 0.33/0.69 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 0.33/0.69 => ? [D4: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ D4 )
% 0.33/0.69 & ! [H4: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.33/0.69 => ( ( member_real @ ( minus_minus_real @ X @ H4 ) @ S )
% 0.33/0.69 => ( ( ord_less_real @ H4 @ D4 )
% 0.33/0.69 => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H4 ) ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % has_real_derivative_neg_dec_left
% 0.33/0.69 thf(fact_9969_DERIV__local__const,axiom,
% 0.33/0.69 ! [F: real > real,L2: real,X: real,D: real] :
% 0.33/0.69 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.69 => ( ( ord_less_real @ zero_zero_real @ D )
% 0.33/0.69 => ( ! [Y4: real] :
% 0.33/0.69 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
% 0.33/0.69 => ( ( F @ X )
% 0.33/0.69 = ( F @ Y4 ) ) )
% 0.33/0.69 => ( L2 = zero_zero_real ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_local_const
% 0.33/0.69 thf(fact_9970_DERIV__ln,axiom,
% 0.33/0.69 ! [X: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_ln
% 0.33/0.69 thf(fact_9971_DERIV__neg__dec__left,axiom,
% 0.33/0.69 ! [F: real > real,L2: real,X: real] :
% 0.33/0.69 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.69 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 0.33/0.69 => ? [D4: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ D4 )
% 0.33/0.69 & ! [H4: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.33/0.69 => ( ( ord_less_real @ H4 @ D4 )
% 0.33/0.69 => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H4 ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_neg_dec_left
% 0.33/0.69 thf(fact_9972_DERIV__pos__inc__left,axiom,
% 0.33/0.69 ! [F: real > real,L2: real,X: real] :
% 0.33/0.69 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.69 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 0.33/0.69 => ? [D4: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ D4 )
% 0.33/0.69 & ! [H4: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.33/0.69 => ( ( ord_less_real @ H4 @ D4 )
% 0.33/0.69 => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_pos_inc_left
% 0.33/0.69 thf(fact_9973_DERIV__neg__dec__right,axiom,
% 0.33/0.69 ! [F: real > real,L2: real,X: real] :
% 0.33/0.69 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.69 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 0.33/0.69 => ? [D4: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ D4 )
% 0.33/0.69 & ! [H4: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.33/0.69 => ( ( ord_less_real @ H4 @ D4 )
% 0.33/0.69 => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_neg_dec_right
% 0.33/0.69 thf(fact_9974_DERIV__pos__inc__right,axiom,
% 0.33/0.69 ! [F: real > real,L2: real,X: real] :
% 0.33/0.69 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.69 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 0.33/0.69 => ? [D4: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ D4 )
% 0.33/0.69 & ! [H4: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.33/0.69 => ( ( ord_less_real @ H4 @ D4 )
% 0.33/0.69 => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H4 ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_pos_inc_right
% 0.33/0.69 thf(fact_9975_DERIV__isconst3,axiom,
% 0.33/0.69 ! [A: real,B: real,X: real,Y: real,F: real > real] :
% 0.33/0.69 ( ( ord_less_real @ A @ B )
% 0.33/0.69 => ( ( member_real @ X @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.33/0.69 => ( ( member_real @ Y @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.33/0.69 => ( ! [X4: real] :
% 0.33/0.69 ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 0.33/0.69 => ( ( F @ X )
% 0.33/0.69 = ( F @ Y ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_isconst3
% 0.33/0.69 thf(fact_9976_DERIV__neg__imp__decreasing,axiom,
% 0.33/0.69 ! [A: real,B: real,F: real > real] :
% 0.33/0.69 ( ( ord_less_real @ A @ B )
% 0.33/0.69 => ( ! [X4: real] :
% 0.33/0.69 ( ( ord_less_eq_real @ A @ X4 )
% 0.33/0.69 => ( ( ord_less_eq_real @ X4 @ B )
% 0.33/0.69 => ? [Y5: real] :
% 0.33/0.69 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.33/0.69 & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
% 0.33/0.69 => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_neg_imp_decreasing
% 0.33/0.69 thf(fact_9977_DERIV__pos__imp__increasing,axiom,
% 0.33/0.69 ! [A: real,B: real,F: real > real] :
% 0.33/0.69 ( ( ord_less_real @ A @ B )
% 0.33/0.69 => ( ! [X4: real] :
% 0.33/0.69 ( ( ord_less_eq_real @ A @ X4 )
% 0.33/0.69 => ( ( ord_less_eq_real @ X4 @ B )
% 0.33/0.69 => ? [Y5: real] :
% 0.33/0.69 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.33/0.69 & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
% 0.33/0.69 => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_pos_imp_increasing
% 0.33/0.69 thf(fact_9978_MVT2,axiom,
% 0.33/0.69 ! [A: real,B: real,F: real > real,F4: real > real] :
% 0.33/0.69 ( ( ord_less_real @ A @ B )
% 0.33/0.69 => ( ! [X4: real] :
% 0.33/0.69 ( ( ord_less_eq_real @ A @ X4 )
% 0.33/0.69 => ( ( ord_less_eq_real @ X4 @ B )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ F @ ( F4 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.33/0.69 => ? [Z5: real] :
% 0.33/0.69 ( ( ord_less_real @ A @ Z5 )
% 0.33/0.69 & ( ord_less_real @ Z5 @ B )
% 0.33/0.69 & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.33/0.69 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F4 @ Z5 ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % MVT2
% 0.33/0.69 thf(fact_9979_DERIV__const__average,axiom,
% 0.33/0.69 ! [A: real,B: real,V: real > real,K: real] :
% 0.33/0.69 ( ( A != B )
% 0.33/0.69 => ( ! [X4: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.33/0.69 => ( ( V @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.33/0.69 = ( divide_divide_real @ ( plus_plus_real @ ( V @ A ) @ ( V @ B ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_const_average
% 0.33/0.69 thf(fact_9980_DERIV__local__max,axiom,
% 0.33/0.69 ! [F: real > real,L2: real,X: real,D: real] :
% 0.33/0.69 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.69 => ( ( ord_less_real @ zero_zero_real @ D )
% 0.33/0.69 => ( ! [Y4: real] :
% 0.33/0.69 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
% 0.33/0.69 => ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X ) ) )
% 0.33/0.69 => ( L2 = zero_zero_real ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_local_max
% 0.33/0.69 thf(fact_9981_DERIV__local__min,axiom,
% 0.33/0.69 ! [F: real > real,L2: real,X: real,D: real] :
% 0.33/0.69 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.69 => ( ( ord_less_real @ zero_zero_real @ D )
% 0.33/0.69 => ( ! [Y4: real] :
% 0.33/0.69 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
% 0.33/0.69 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.33/0.69 => ( L2 = zero_zero_real ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_local_min
% 0.33/0.69 thf(fact_9982_DERIV__ln__divide,axiom,
% 0.33/0.69 ! [X: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_ln_divide
% 0.33/0.69 thf(fact_9983_DERIV__pow,axiom,
% 0.33/0.69 ! [N2: nat,X: real,S3: set_real] :
% 0.33/0.69 ( has_fi5821293074295781190e_real
% 0.33/0.69 @ ^ [X2: real] : ( power_power_real @ X2 @ N2 )
% 0.33/0.69 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ X @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ S3 ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_pow
% 0.33/0.69 thf(fact_9984_DERIV__fun__pow,axiom,
% 0.33/0.69 ! [G: real > real,M: real,X: real,N2: nat] :
% 0.33/0.69 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.69 => ( has_fi5821293074295781190e_real
% 0.33/0.69 @ ^ [X2: real] : ( power_power_real @ ( G @ X2 ) @ N2 )
% 0.33/0.69 @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( G @ X ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) @ M )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_fun_pow
% 0.33/0.69 thf(fact_9985_has__real__derivative__powr,axiom,
% 0.33/0.69 ! [Z: real,R: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ Z )
% 0.33/0.69 => ( has_fi5821293074295781190e_real
% 0.33/0.69 @ ^ [Z3: real] : ( powr_real @ Z3 @ R )
% 0.33/0.69 @ ( times_times_real @ R @ ( powr_real @ Z @ ( minus_minus_real @ R @ one_one_real ) ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % has_real_derivative_powr
% 0.33/0.69 thf(fact_9986_DERIV__log,axiom,
% 0.33/0.69 ! [X: real,B: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ ( log @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_log
% 0.33/0.69 thf(fact_9987_DERIV__fun__powr,axiom,
% 0.33/0.69 ! [G: real > real,M: real,X: real,R: real] :
% 0.33/0.69 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.69 => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
% 0.33/0.69 => ( has_fi5821293074295781190e_real
% 0.33/0.69 @ ^ [X2: real] : ( powr_real @ ( G @ X2 ) @ R )
% 0.33/0.69 @ ( times_times_real @ ( times_times_real @ R @ ( powr_real @ ( G @ X ) @ ( minus_minus_real @ R @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_fun_powr
% 0.33/0.69 thf(fact_9988_DERIV__powr,axiom,
% 0.33/0.69 ! [G: real > real,M: real,X: real,F: real > real,R: real] :
% 0.33/0.69 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.69 => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
% 0.33/0.69 => ( ( has_fi5821293074295781190e_real @ F @ R @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.69 => ( has_fi5821293074295781190e_real
% 0.33/0.69 @ ^ [X2: real] : ( powr_real @ ( G @ X2 ) @ ( F @ X2 ) )
% 0.33/0.69 @ ( times_times_real @ ( powr_real @ ( G @ X ) @ ( F @ X ) ) @ ( plus_plus_real @ ( times_times_real @ R @ ( ln_ln_real @ ( G @ X ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X ) ) @ ( G @ X ) ) ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_powr
% 0.33/0.69 thf(fact_9989_DERIV__real__sqrt,axiom,
% 0.33/0.69 ! [X: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_real_sqrt
% 0.33/0.69 thf(fact_9990_DERIV__series_H,axiom,
% 0.33/0.69 ! [F: real > nat > real,F4: real > nat > real,X0: real,A: real,B: real,L5: nat > real] :
% 0.33/0.69 ( ! [N4: nat] :
% 0.33/0.69 ( has_fi5821293074295781190e_real
% 0.33/0.69 @ ^ [X2: real] : ( F @ X2 @ N4 )
% 0.33/0.69 @ ( F4 @ X0 @ N4 )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
% 0.33/0.69 => ( ! [X4: real] :
% 0.33/0.69 ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.33/0.69 => ( summable_real @ ( F @ X4 ) ) )
% 0.33/0.69 => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.33/0.69 => ( ( summable_real @ ( F4 @ X0 ) )
% 0.33/0.69 => ( ( summable_real @ L5 )
% 0.33/0.69 => ( ! [N4: nat,X4: real,Y4: real] :
% 0.33/0.69 ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.33/0.69 => ( ( member_real @ Y4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.33/0.69 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X4 @ N4 ) @ ( F @ Y4 @ N4 ) ) ) @ ( times_times_real @ ( L5 @ N4 ) @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y4 ) ) ) ) ) )
% 0.33/0.69 => ( has_fi5821293074295781190e_real
% 0.33/0.69 @ ^ [X2: real] : ( suminf_real @ ( F @ X2 ) )
% 0.33/0.69 @ ( suminf_real @ ( F4 @ X0 ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_series'
% 0.33/0.69 thf(fact_9991_DERIV__arctan,axiom,
% 0.33/0.69 ! [X: real] : ( has_fi5821293074295781190e_real @ arctan @ ( inverse_inverse_real @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_arctan
% 0.33/0.69 thf(fact_9992_arsinh__real__has__field__derivative,axiom,
% 0.33/0.69 ! [X: real,A2: set_real] : ( has_fi5821293074295781190e_real @ arsinh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ).
% 0.33/0.69
% 0.33/0.69 % arsinh_real_has_field_derivative
% 0.33/0.69 thf(fact_9993_DERIV__real__sqrt__generic,axiom,
% 0.33/0.69 ! [X: real,D3: real] :
% 0.33/0.69 ( ( X != zero_zero_real )
% 0.33/0.69 => ( ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.69 => ( D3
% 0.33/0.69 = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.33/0.69 => ( ( ( ord_less_real @ X @ zero_zero_real )
% 0.33/0.69 => ( D3
% 0.33/0.69 = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ sqrt @ D3 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_real_sqrt_generic
% 0.33/0.69 thf(fact_9994_arcosh__real__has__field__derivative,axiom,
% 0.33/0.69 ! [X: real,A2: set_real] :
% 0.33/0.69 ( ( ord_less_real @ one_one_real @ X )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % arcosh_real_has_field_derivative
% 0.33/0.69 thf(fact_9995_artanh__real__has__field__derivative,axiom,
% 0.33/0.69 ! [X: real,A2: set_real] :
% 0.33/0.69 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % artanh_real_has_field_derivative
% 0.33/0.69 thf(fact_9996_DERIV__power__series_H,axiom,
% 0.33/0.69 ! [R2: real,F: nat > real,X0: real] :
% 0.33/0.69 ( ! [X4: real] :
% 0.33/0.69 ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R2 ) @ R2 ) )
% 0.33/0.69 => ( summable_real
% 0.33/0.69 @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( F @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) @ ( power_power_real @ X4 @ N ) ) ) )
% 0.33/0.69 => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R2 ) @ R2 ) )
% 0.33/0.69 => ( ( ord_less_real @ zero_zero_real @ R2 )
% 0.33/0.69 => ( has_fi5821293074295781190e_real
% 0.33/0.69 @ ^ [X2: real] :
% 0.33/0.69 ( suminf_real
% 0.33/0.69 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ X2 @ ( suc @ N ) ) ) )
% 0.33/0.69 @ ( suminf_real
% 0.33/0.69 @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( F @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) @ ( power_power_real @ X0 @ N ) ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_power_series'
% 0.33/0.69 thf(fact_9997_DERIV__real__root,axiom,
% 0.33/0.69 ! [N2: nat,X: real] :
% 0.33/0.69 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.69 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_real_root
% 0.33/0.69 thf(fact_9998_DERIV__arccos,axiom,
% 0.33/0.69 ! [X: real] :
% 0.33/0.69 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.69 => ( ( ord_less_real @ X @ one_one_real )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_arccos
% 0.33/0.69 thf(fact_9999_DERIV__arcsin,axiom,
% 0.33/0.69 ! [X: real] :
% 0.33/0.69 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.69 => ( ( ord_less_real @ X @ one_one_real )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_arcsin
% 0.33/0.69 thf(fact_10000_Maclaurin__all__le,axiom,
% 0.33/0.69 ! [Diff: nat > real > real,F: real > real,X: real,N2: nat] :
% 0.33/0.69 ( ( ( Diff @ zero_zero_nat )
% 0.33/0.69 = F )
% 0.33/0.69 => ( ! [M2: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.33/0.69 => ? [T5: real] :
% 0.33/0.69 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 0.33/0.69 & ( ( F @ X )
% 0.33/0.69 = ( plus_plus_real
% 0.33/0.69 @ ( groups6591440286371151544t_real
% 0.33/0.69 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
% 0.33/0.69 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.69 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T5 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Maclaurin_all_le
% 0.33/0.69 thf(fact_10001_Maclaurin__all__le__objl,axiom,
% 0.33/0.69 ! [Diff: nat > real > real,F: real > real,X: real,N2: nat] :
% 0.33/0.69 ( ( ( ( Diff @ zero_zero_nat )
% 0.33/0.69 = F )
% 0.33/0.69 & ! [M2: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 0.33/0.69 => ? [T5: real] :
% 0.33/0.69 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 0.33/0.69 & ( ( F @ X )
% 0.33/0.69 = ( plus_plus_real
% 0.33/0.69 @ ( groups6591440286371151544t_real
% 0.33/0.69 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
% 0.33/0.69 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.69 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T5 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Maclaurin_all_le_objl
% 0.33/0.69 thf(fact_10002_DERIV__odd__real__root,axiom,
% 0.33/0.69 ! [N2: nat,X: real] :
% 0.33/0.69 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.69 => ( ( X != zero_zero_real )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_odd_real_root
% 0.33/0.69 thf(fact_10003_Maclaurin__minus,axiom,
% 0.33/0.69 ! [H2: real,N2: nat,Diff: nat > real > real,F: real > real] :
% 0.33/0.69 ( ( ord_less_real @ H2 @ zero_zero_real )
% 0.33/0.69 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.69 => ( ( ( Diff @ zero_zero_nat )
% 0.33/0.69 = F )
% 0.33/0.69 => ( ! [M2: nat,T5: real] :
% 0.33/0.69 ( ( ( ord_less_nat @ M2 @ N2 )
% 0.33/0.69 & ( ord_less_eq_real @ H2 @ T5 )
% 0.33/0.69 & ( ord_less_eq_real @ T5 @ zero_zero_real ) )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 0.33/0.69 => ? [T5: real] :
% 0.33/0.69 ( ( ord_less_real @ H2 @ T5 )
% 0.33/0.69 & ( ord_less_real @ T5 @ zero_zero_real )
% 0.33/0.69 & ( ( F @ H2 )
% 0.33/0.69 = ( plus_plus_real
% 0.33/0.69 @ ( groups6591440286371151544t_real
% 0.33/0.69 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
% 0.33/0.69 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.69 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T5 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Maclaurin_minus
% 0.33/0.69 thf(fact_10004_Maclaurin2,axiom,
% 0.33/0.69 ! [H2: real,Diff: nat > real > real,F: real > real,N2: nat] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.33/0.69 => ( ( ( Diff @ zero_zero_nat )
% 0.33/0.69 = F )
% 0.33/0.69 => ( ! [M2: nat,T5: real] :
% 0.33/0.69 ( ( ( ord_less_nat @ M2 @ N2 )
% 0.33/0.69 & ( ord_less_eq_real @ zero_zero_real @ T5 )
% 0.33/0.69 & ( ord_less_eq_real @ T5 @ H2 ) )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 0.33/0.69 => ? [T5: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ T5 )
% 0.33/0.69 & ( ord_less_eq_real @ T5 @ H2 )
% 0.33/0.69 & ( ( F @ H2 )
% 0.33/0.69 = ( plus_plus_real
% 0.33/0.69 @ ( groups6591440286371151544t_real
% 0.33/0.69 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
% 0.33/0.69 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.69 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T5 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Maclaurin2
% 0.33/0.69 thf(fact_10005_Maclaurin,axiom,
% 0.33/0.69 ! [H2: real,N2: nat,Diff: nat > real > real,F: real > real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.33/0.69 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.69 => ( ( ( Diff @ zero_zero_nat )
% 0.33/0.69 = F )
% 0.33/0.69 => ( ! [M2: nat,T5: real] :
% 0.33/0.69 ( ( ( ord_less_nat @ M2 @ N2 )
% 0.33/0.69 & ( ord_less_eq_real @ zero_zero_real @ T5 )
% 0.33/0.69 & ( ord_less_eq_real @ T5 @ H2 ) )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 0.33/0.69 => ? [T5: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ T5 )
% 0.33/0.69 & ( ord_less_real @ T5 @ H2 )
% 0.33/0.69 & ( ( F @ H2 )
% 0.33/0.69 = ( plus_plus_real
% 0.33/0.69 @ ( groups6591440286371151544t_real
% 0.33/0.69 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
% 0.33/0.69 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.69 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T5 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Maclaurin
% 0.33/0.69 thf(fact_10006_Maclaurin__all__lt,axiom,
% 0.33/0.69 ! [Diff: nat > real > real,F: real > real,N2: nat,X: real] :
% 0.33/0.69 ( ( ( Diff @ zero_zero_nat )
% 0.33/0.69 = F )
% 0.33/0.69 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.69 => ( ( X != zero_zero_real )
% 0.33/0.69 => ( ! [M2: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.33/0.69 => ? [T5: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T5 ) )
% 0.33/0.69 & ( ord_less_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 0.33/0.69 & ( ( F @ X )
% 0.33/0.69 = ( plus_plus_real
% 0.33/0.69 @ ( groups6591440286371151544t_real
% 0.33/0.69 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
% 0.33/0.69 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.69 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T5 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Maclaurin_all_lt
% 0.33/0.69 thf(fact_10007_Maclaurin__bi__le,axiom,
% 0.33/0.69 ! [Diff: nat > real > real,F: real > real,N2: nat,X: real] :
% 0.33/0.69 ( ( ( Diff @ zero_zero_nat )
% 0.33/0.69 = F )
% 0.33/0.69 => ( ! [M2: nat,T5: real] :
% 0.33/0.69 ( ( ( ord_less_nat @ M2 @ N2 )
% 0.33/0.69 & ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) ) )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 0.33/0.69 => ? [T5: real] :
% 0.33/0.69 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 0.33/0.69 & ( ( F @ X )
% 0.33/0.69 = ( plus_plus_real
% 0.33/0.69 @ ( groups6591440286371151544t_real
% 0.33/0.69 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
% 0.33/0.69 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.69 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T5 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Maclaurin_bi_le
% 0.33/0.69 thf(fact_10008_Taylor__down,axiom,
% 0.33/0.69 ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 0.33/0.69 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.69 => ( ( ( Diff @ zero_zero_nat )
% 0.33/0.69 = F )
% 0.33/0.69 => ( ! [M2: nat,T5: real] :
% 0.33/0.69 ( ( ( ord_less_nat @ M2 @ N2 )
% 0.33/0.69 & ( ord_less_eq_real @ A @ T5 )
% 0.33/0.69 & ( ord_less_eq_real @ T5 @ B ) )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 0.33/0.69 => ( ( ord_less_real @ A @ C )
% 0.33/0.69 => ( ( ord_less_eq_real @ C @ B )
% 0.33/0.69 => ? [T5: real] :
% 0.33/0.69 ( ( ord_less_real @ A @ T5 )
% 0.33/0.69 & ( ord_less_real @ T5 @ C )
% 0.33/0.69 & ( ( F @ A )
% 0.33/0.69 = ( plus_plus_real
% 0.33/0.69 @ ( groups6591440286371151544t_real
% 0.33/0.69 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M6 ) )
% 0.33/0.69 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.69 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T5 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Taylor_down
% 0.33/0.69 thf(fact_10009_Taylor__up,axiom,
% 0.33/0.69 ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 0.33/0.69 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.69 => ( ( ( Diff @ zero_zero_nat )
% 0.33/0.69 = F )
% 0.33/0.69 => ( ! [M2: nat,T5: real] :
% 0.33/0.69 ( ( ( ord_less_nat @ M2 @ N2 )
% 0.33/0.69 & ( ord_less_eq_real @ A @ T5 )
% 0.33/0.69 & ( ord_less_eq_real @ T5 @ B ) )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 0.33/0.69 => ( ( ord_less_eq_real @ A @ C )
% 0.33/0.69 => ( ( ord_less_real @ C @ B )
% 0.33/0.69 => ? [T5: real] :
% 0.33/0.69 ( ( ord_less_real @ C @ T5 )
% 0.33/0.69 & ( ord_less_real @ T5 @ B )
% 0.33/0.69 & ( ( F @ B )
% 0.33/0.69 = ( plus_plus_real
% 0.33/0.69 @ ( groups6591440286371151544t_real
% 0.33/0.69 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M6 ) )
% 0.33/0.69 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.69 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T5 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Taylor_up
% 0.33/0.69 thf(fact_10010_Taylor,axiom,
% 0.33/0.69 ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X: real] :
% 0.33/0.69 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.69 => ( ( ( Diff @ zero_zero_nat )
% 0.33/0.69 = F )
% 0.33/0.69 => ( ! [M2: nat,T5: real] :
% 0.33/0.69 ( ( ( ord_less_nat @ M2 @ N2 )
% 0.33/0.69 & ( ord_less_eq_real @ A @ T5 )
% 0.33/0.69 & ( ord_less_eq_real @ T5 @ B ) )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 0.33/0.69 => ( ( ord_less_eq_real @ A @ C )
% 0.33/0.69 => ( ( ord_less_eq_real @ C @ B )
% 0.33/0.69 => ( ( ord_less_eq_real @ A @ X )
% 0.33/0.69 => ( ( ord_less_eq_real @ X @ B )
% 0.33/0.69 => ( ( X != C )
% 0.33/0.69 => ? [T5: real] :
% 0.33/0.69 ( ( ( ord_less_real @ X @ C )
% 0.33/0.69 => ( ( ord_less_real @ X @ T5 )
% 0.33/0.69 & ( ord_less_real @ T5 @ C ) ) )
% 0.33/0.69 & ( ~ ( ord_less_real @ X @ C )
% 0.33/0.69 => ( ( ord_less_real @ C @ T5 )
% 0.33/0.69 & ( ord_less_real @ T5 @ X ) ) )
% 0.33/0.69 & ( ( F @ X )
% 0.33/0.69 = ( plus_plus_real
% 0.33/0.69 @ ( groups6591440286371151544t_real
% 0.33/0.69 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ M6 ) )
% 0.33/0.69 @ ( set_ord_lessThan_nat @ N2 ) )
% 0.33/0.69 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T5 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Taylor
% 0.33/0.69 thf(fact_10011_Maclaurin__lemma2,axiom,
% 0.33/0.69 ! [N2: nat,H2: real,Diff: nat > real > real,K: nat,B3: real] :
% 0.33/0.69 ( ! [M2: nat,T5: real] :
% 0.33/0.69 ( ( ( ord_less_nat @ M2 @ N2 )
% 0.33/0.69 & ( ord_less_eq_real @ zero_zero_real @ T5 )
% 0.33/0.69 & ( ord_less_eq_real @ T5 @ H2 ) )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 0.33/0.69 => ( ( N2
% 0.33/0.69 = ( suc @ K ) )
% 0.33/0.69 => ! [M5: nat,T6: real] :
% 0.33/0.69 ( ( ( ord_less_nat @ M5 @ N2 )
% 0.33/0.69 & ( ord_less_eq_real @ zero_zero_real @ T6 )
% 0.33/0.69 & ( ord_less_eq_real @ T6 @ H2 ) )
% 0.33/0.69 => ( has_fi5821293074295781190e_real
% 0.33/0.69 @ ^ [U2: real] :
% 0.33/0.69 ( minus_minus_real @ ( Diff @ M5 @ U2 )
% 0.33/0.69 @ ( plus_plus_real
% 0.33/0.69 @ ( groups6591440286371151544t_real
% 0.33/0.69 @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M5 @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ U2 @ P5 ) )
% 0.33/0.69 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ M5 ) ) )
% 0.33/0.69 @ ( times_times_real @ B3 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N2 @ M5 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ M5 ) ) ) ) ) )
% 0.33/0.69 @ ( minus_minus_real @ ( Diff @ ( suc @ M5 ) @ T6 )
% 0.33/0.69 @ ( plus_plus_real
% 0.33/0.69 @ ( groups6591440286371151544t_real
% 0.33/0.69 @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M5 ) @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ T6 @ P5 ) )
% 0.33/0.69 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ M5 ) ) ) )
% 0.33/0.69 @ ( times_times_real @ B3 @ ( divide_divide_real @ ( power_power_real @ T6 @ ( minus_minus_nat @ N2 @ ( suc @ M5 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ ( suc @ M5 ) ) ) ) ) ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Maclaurin_lemma2
% 0.33/0.69 thf(fact_10012_DERIV__arctan__series,axiom,
% 0.33/0.69 ! [X: real] :
% 0.33/0.69 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.69 => ( has_fi5821293074295781190e_real
% 0.33/0.69 @ ^ [X9: real] :
% 0.33/0.69 ( suminf_real
% 0.33/0.69 @ ^ [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 ) ) ) ) )
% 0.33/0.69 @ ( suminf_real
% 0.33/0.69 @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_arctan_series
% 0.33/0.69 thf(fact_10013_DERIV__real__root__generic,axiom,
% 0.33/0.69 ! [N2: nat,X: real,D3: real] :
% 0.33/0.69 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.69 => ( ( X != zero_zero_real )
% 0.33/0.69 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.69 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.33/0.69 => ( D3
% 0.33/0.69 = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
% 0.33/0.69 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.69 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.33/0.69 => ( D3
% 0.33/0.69 = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
% 0.33/0.69 => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.69 => ( D3
% 0.33/0.69 = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ D3 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_real_root_generic
% 0.33/0.69 thf(fact_10014_isCont__Lb__Ub,axiom,
% 0.33/0.69 ! [A: real,B: real,F: real > real] :
% 0.33/0.69 ( ( ord_less_eq_real @ A @ B )
% 0.33/0.69 => ( ! [X4: real] :
% 0.33/0.69 ( ( ( ord_less_eq_real @ A @ X4 )
% 0.33/0.69 & ( ord_less_eq_real @ X4 @ B ) )
% 0.33/0.69 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F ) )
% 0.33/0.69 => ? [L6: real,M9: real] :
% 0.33/0.69 ( ! [X5: real] :
% 0.33/0.69 ( ( ( ord_less_eq_real @ A @ X5 )
% 0.33/0.69 & ( ord_less_eq_real @ X5 @ B ) )
% 0.33/0.69 => ( ( ord_less_eq_real @ L6 @ ( F @ X5 ) )
% 0.33/0.69 & ( ord_less_eq_real @ ( F @ X5 ) @ M9 ) ) )
% 0.33/0.69 & ! [Y5: real] :
% 0.33/0.69 ( ( ( ord_less_eq_real @ L6 @ Y5 )
% 0.33/0.69 & ( ord_less_eq_real @ Y5 @ M9 ) )
% 0.33/0.69 => ? [X4: real] :
% 0.33/0.69 ( ( ord_less_eq_real @ A @ X4 )
% 0.33/0.69 & ( ord_less_eq_real @ X4 @ B )
% 0.33/0.69 & ( ( F @ X4 )
% 0.33/0.69 = Y5 ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % isCont_Lb_Ub
% 0.33/0.69 thf(fact_10015_LIM__fun__less__zero,axiom,
% 0.33/0.69 ! [F: real > real,L2: real,C: real] :
% 0.33/0.69 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.33/0.69 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 0.33/0.69 => ? [R3: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.33/0.69 & ! [X5: real] :
% 0.33/0.69 ( ( ( X5 != C )
% 0.33/0.69 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
% 0.33/0.69 => ( ord_less_real @ ( F @ X5 ) @ zero_zero_real ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % LIM_fun_less_zero
% 0.33/0.69 thf(fact_10016_LIM__fun__not__zero,axiom,
% 0.33/0.69 ! [F: real > real,L2: real,C: real] :
% 0.33/0.69 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.33/0.69 => ( ( L2 != zero_zero_real )
% 0.33/0.69 => ? [R3: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.33/0.69 & ! [X5: real] :
% 0.33/0.69 ( ( ( X5 != C )
% 0.33/0.69 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
% 0.33/0.69 => ( ( F @ X5 )
% 0.33/0.69 != zero_zero_real ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % LIM_fun_not_zero
% 0.33/0.69 thf(fact_10017_LIM__fun__gt__zero,axiom,
% 0.33/0.69 ! [F: real > real,L2: real,C: real] :
% 0.33/0.69 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.33/0.69 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 0.33/0.69 => ? [R3: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.33/0.69 & ! [X5: real] :
% 0.33/0.69 ( ( ( X5 != C )
% 0.33/0.69 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
% 0.33/0.69 => ( ord_less_real @ zero_zero_real @ ( F @ X5 ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % LIM_fun_gt_zero
% 0.33/0.69 thf(fact_10018_isCont__real__sqrt,axiom,
% 0.33/0.69 ! [X: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ sqrt ) ).
% 0.33/0.69
% 0.33/0.69 % isCont_real_sqrt
% 0.33/0.69 thf(fact_10019_isCont__real__root,axiom,
% 0.33/0.69 ! [X: real,N2: nat] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ( root @ N2 ) ) ).
% 0.33/0.69
% 0.33/0.69 % isCont_real_root
% 0.33/0.69 thf(fact_10020_isCont__inverse__function2,axiom,
% 0.33/0.69 ! [A: real,X: real,B: real,G: real > real,F: real > real] :
% 0.33/0.69 ( ( ord_less_real @ A @ X )
% 0.33/0.69 => ( ( ord_less_real @ X @ B )
% 0.33/0.69 => ( ! [Z5: real] :
% 0.33/0.69 ( ( ord_less_eq_real @ A @ Z5 )
% 0.33/0.69 => ( ( ord_less_eq_real @ Z5 @ B )
% 0.33/0.69 => ( ( G @ ( F @ Z5 ) )
% 0.33/0.69 = Z5 ) ) )
% 0.33/0.69 => ( ! [Z5: real] :
% 0.33/0.69 ( ( ord_less_eq_real @ A @ Z5 )
% 0.33/0.69 => ( ( ord_less_eq_real @ Z5 @ B )
% 0.33/0.69 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) @ F ) ) )
% 0.33/0.69 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % isCont_inverse_function2
% 0.33/0.69 thf(fact_10021_isCont__arcosh,axiom,
% 0.33/0.69 ! [X: real] :
% 0.33/0.69 ( ( ord_less_real @ one_one_real @ X )
% 0.33/0.69 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcosh_real ) ) ).
% 0.33/0.69
% 0.33/0.69 % isCont_arcosh
% 0.33/0.69 thf(fact_10022_LIM__cos__div__sin,axiom,
% 0.33/0.69 ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( cos_real @ X2 ) @ ( sin_real @ X2 ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ top_top_set_real ) ) ).
% 0.33/0.69
% 0.33/0.69 % LIM_cos_div_sin
% 0.33/0.69 thf(fact_10023_DERIV__inverse__function,axiom,
% 0.33/0.69 ! [F: real > real,D3: real,G: real > real,X: real,A: real,B: real] :
% 0.33/0.69 ( ( has_fi5821293074295781190e_real @ F @ D3 @ ( topolo2177554685111907308n_real @ ( G @ X ) @ top_top_set_real ) )
% 0.33/0.69 => ( ( D3 != zero_zero_real )
% 0.33/0.69 => ( ( ord_less_real @ A @ X )
% 0.33/0.69 => ( ( ord_less_real @ X @ B )
% 0.33/0.69 => ( ! [Y4: real] :
% 0.33/0.69 ( ( ord_less_real @ A @ Y4 )
% 0.33/0.69 => ( ( ord_less_real @ Y4 @ B )
% 0.33/0.69 => ( ( F @ ( G @ Y4 ) )
% 0.33/0.69 = Y4 ) ) )
% 0.33/0.69 => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ G )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D3 ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_inverse_function
% 0.33/0.69 thf(fact_10024_isCont__arccos,axiom,
% 0.33/0.69 ! [X: real] :
% 0.33/0.69 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.69 => ( ( ord_less_real @ X @ one_one_real )
% 0.33/0.69 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arccos ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % isCont_arccos
% 0.33/0.69 thf(fact_10025_isCont__arcsin,axiom,
% 0.33/0.69 ! [X: real] :
% 0.33/0.69 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.69 => ( ( ord_less_real @ X @ one_one_real )
% 0.33/0.69 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcsin ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % isCont_arcsin
% 0.33/0.69 thf(fact_10026_LIM__less__bound,axiom,
% 0.33/0.69 ! [B: real,X: real,F: real > real] :
% 0.33/0.69 ( ( ord_less_real @ B @ X )
% 0.33/0.69 => ( ! [X4: real] :
% 0.33/0.69 ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ B @ X ) )
% 0.33/0.69 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.33/0.69 => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F )
% 0.33/0.69 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % LIM_less_bound
% 0.33/0.69 thf(fact_10027_isCont__artanh,axiom,
% 0.33/0.69 ! [X: real] :
% 0.33/0.69 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.33/0.69 => ( ( ord_less_real @ X @ one_one_real )
% 0.33/0.69 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ artanh_real ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % isCont_artanh
% 0.33/0.69 thf(fact_10028_isCont__inverse__function,axiom,
% 0.33/0.69 ! [D: real,X: real,G: real > real,F: real > real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ D )
% 0.33/0.69 => ( ! [Z5: real] :
% 0.33/0.69 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z5 @ X ) ) @ D )
% 0.33/0.69 => ( ( G @ ( F @ Z5 ) )
% 0.33/0.69 = Z5 ) )
% 0.33/0.69 => ( ! [Z5: real] :
% 0.33/0.69 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z5 @ X ) ) @ D )
% 0.33/0.69 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) @ F ) )
% 0.33/0.69 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % isCont_inverse_function
% 0.33/0.69 thf(fact_10029_GMVT_H,axiom,
% 0.33/0.69 ! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F4: real > real] :
% 0.33/0.69 ( ( ord_less_real @ A @ B )
% 0.33/0.69 => ( ! [Z5: real] :
% 0.33/0.69 ( ( ord_less_eq_real @ A @ Z5 )
% 0.33/0.69 => ( ( ord_less_eq_real @ Z5 @ B )
% 0.33/0.69 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) @ F ) ) )
% 0.33/0.69 => ( ! [Z5: real] :
% 0.33/0.69 ( ( ord_less_eq_real @ A @ Z5 )
% 0.33/0.69 => ( ( ord_less_eq_real @ Z5 @ B )
% 0.33/0.69 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) @ G ) ) )
% 0.33/0.69 => ( ! [Z5: real] :
% 0.33/0.69 ( ( ord_less_real @ A @ Z5 )
% 0.33/0.69 => ( ( ord_less_real @ Z5 @ B )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z5 ) @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) ) ) )
% 0.33/0.69 => ( ! [Z5: real] :
% 0.33/0.69 ( ( ord_less_real @ A @ Z5 )
% 0.33/0.69 => ( ( ord_less_real @ Z5 @ B )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ F @ ( F4 @ Z5 ) @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) ) ) )
% 0.33/0.69 => ? [C3: real] :
% 0.33/0.69 ( ( ord_less_real @ A @ C3 )
% 0.33/0.69 & ( ord_less_real @ C3 @ B )
% 0.33/0.69 & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C3 ) )
% 0.33/0.69 = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F4 @ C3 ) ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % GMVT'
% 0.33/0.69 thf(fact_10030_summable__Leibniz_I2_J,axiom,
% 0.33/0.69 ! [A: nat > real] :
% 0.33/0.69 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.33/0.69 => ( ( topolo6980174941875973593q_real @ A )
% 0.33/0.69 => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
% 0.33/0.69 => ! [N9: nat] :
% 0.33/0.69 ( member_real
% 0.33/0.69 @ ( suminf_real
% 0.33/0.69 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
% 0.33/0.69 @ ( set_or1222579329274155063t_real
% 0.33/0.69 @ ( groups6591440286371151544t_real
% 0.33/0.69 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 0.33/0.69 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) )
% 0.33/0.69 @ ( groups6591440286371151544t_real
% 0.33/0.69 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 0.33/0.69 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % summable_Leibniz(2)
% 0.33/0.69 thf(fact_10031_summable__Leibniz_I3_J,axiom,
% 0.33/0.69 ! [A: nat > real] :
% 0.33/0.69 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.33/0.69 => ( ( topolo6980174941875973593q_real @ A )
% 0.33/0.69 => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
% 0.33/0.69 => ! [N9: nat] :
% 0.33/0.69 ( member_real
% 0.33/0.69 @ ( suminf_real
% 0.33/0.69 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
% 0.33/0.69 @ ( set_or1222579329274155063t_real
% 0.33/0.69 @ ( groups6591440286371151544t_real
% 0.33/0.69 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 0.33/0.69 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) )
% 0.33/0.69 @ ( groups6591440286371151544t_real
% 0.33/0.69 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 0.33/0.69 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % summable_Leibniz(3)
% 0.33/0.69 thf(fact_10032_filterlim__Suc,axiom,
% 0.33/0.69 filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).
% 0.33/0.69
% 0.33/0.69 % filterlim_Suc
% 0.33/0.69 thf(fact_10033_mult__nat__right__at__top,axiom,
% 0.33/0.69 ! [C: nat] :
% 0.33/0.69 ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.33/0.69 => ( filterlim_nat_nat
% 0.33/0.69 @ ^ [X2: nat] : ( times_times_nat @ X2 @ C )
% 0.33/0.69 @ at_top_nat
% 0.33/0.69 @ at_top_nat ) ) ).
% 0.33/0.69
% 0.33/0.69 % mult_nat_right_at_top
% 0.33/0.69 thf(fact_10034_mult__nat__left__at__top,axiom,
% 0.33/0.69 ! [C: nat] :
% 0.33/0.69 ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.33/0.69 => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).
% 0.33/0.69
% 0.33/0.69 % mult_nat_left_at_top
% 0.33/0.69 thf(fact_10035_monoseq__convergent,axiom,
% 0.33/0.69 ! [X8: nat > real,B3: real] :
% 0.33/0.69 ( ( topolo6980174941875973593q_real @ X8 )
% 0.33/0.69 => ( ! [I4: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X8 @ I4 ) ) @ B3 )
% 0.33/0.69 => ~ ! [L6: real] :
% 0.33/0.69 ~ ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % monoseq_convergent
% 0.33/0.69 thf(fact_10036_LIMSEQ__root,axiom,
% 0.33/0.69 ( filterlim_nat_real
% 0.33/0.69 @ ^ [N: nat] : ( root @ N @ ( semiri5074537144036343181t_real @ N ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.33/0.69 @ at_top_nat ) ).
% 0.33/0.69
% 0.33/0.69 % LIMSEQ_root
% 0.33/0.69 thf(fact_10037_nested__sequence__unique,axiom,
% 0.33/0.69 ! [F: nat > real,G: nat > real] :
% 0.33/0.69 ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.33/0.69 => ( ! [N4: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N4 ) ) @ ( G @ N4 ) )
% 0.33/0.69 => ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( G @ N4 ) )
% 0.33/0.69 => ( ( filterlim_nat_real
% 0.33/0.69 @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( G @ N ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.33/0.69 @ at_top_nat )
% 0.33/0.69 => ? [L4: real] :
% 0.33/0.69 ( ! [N9: nat] : ( ord_less_eq_real @ ( F @ N9 ) @ L4 )
% 0.33/0.69 & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat )
% 0.33/0.69 & ! [N9: nat] : ( ord_less_eq_real @ L4 @ ( G @ N9 ) )
% 0.33/0.69 & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % nested_sequence_unique
% 0.33/0.69 thf(fact_10038_LIMSEQ__inverse__zero,axiom,
% 0.33/0.69 ! [X8: nat > real] :
% 0.33/0.69 ( ! [R3: real] :
% 0.33/0.69 ? [N7: nat] :
% 0.33/0.69 ! [N4: nat] :
% 0.33/0.69 ( ( ord_less_eq_nat @ N7 @ N4 )
% 0.33/0.69 => ( ord_less_real @ R3 @ ( X8 @ N4 ) ) )
% 0.33/0.69 => ( filterlim_nat_real
% 0.33/0.69 @ ^ [N: nat] : ( inverse_inverse_real @ ( X8 @ N ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.33/0.69 @ at_top_nat ) ) ).
% 0.33/0.69
% 0.33/0.69 % LIMSEQ_inverse_zero
% 0.33/0.69 thf(fact_10039_lim__inverse__n_H,axiom,
% 0.33/0.69 ( filterlim_nat_real
% 0.33/0.69 @ ^ [N: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.33/0.69 @ at_top_nat ) ).
% 0.33/0.69
% 0.33/0.69 % lim_inverse_n'
% 0.33/0.69 thf(fact_10040_LIMSEQ__root__const,axiom,
% 0.33/0.69 ! [C: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ C )
% 0.33/0.69 => ( filterlim_nat_real
% 0.33/0.69 @ ^ [N: nat] : ( root @ N @ C )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.33/0.69 @ at_top_nat ) ) ).
% 0.33/0.69
% 0.33/0.69 % LIMSEQ_root_const
% 0.33/0.69 thf(fact_10041_LIMSEQ__inverse__real__of__nat,axiom,
% 0.33/0.69 ( filterlim_nat_real
% 0.33/0.69 @ ^ [N: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.33/0.69 @ at_top_nat ) ).
% 0.33/0.69
% 0.33/0.69 % LIMSEQ_inverse_real_of_nat
% 0.33/0.69 thf(fact_10042_LIMSEQ__inverse__real__of__nat__add,axiom,
% 0.33/0.69 ! [R: real] :
% 0.33/0.69 ( filterlim_nat_real
% 0.33/0.69 @ ^ [N: nat] : ( plus_plus_real @ R @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ R )
% 0.33/0.69 @ at_top_nat ) ).
% 0.33/0.69
% 0.33/0.69 % LIMSEQ_inverse_real_of_nat_add
% 0.33/0.69 thf(fact_10043_increasing__LIMSEQ,axiom,
% 0.33/0.69 ! [F: nat > real,L2: real] :
% 0.33/0.69 ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.33/0.69 => ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ L2 )
% 0.33/0.69 => ( ! [E2: real] :
% 0.33/0.69 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.33/0.69 => ? [N9: nat] : ( ord_less_eq_real @ L2 @ ( plus_plus_real @ ( F @ N9 ) @ E2 ) ) )
% 0.33/0.69 => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % increasing_LIMSEQ
% 0.33/0.69 thf(fact_10044_LIMSEQ__realpow__zero,axiom,
% 0.33/0.69 ! [X: real] :
% 0.33/0.69 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.69 => ( ( ord_less_real @ X @ one_one_real )
% 0.33/0.69 => ( filterlim_nat_real @ ( power_power_real @ X ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % LIMSEQ_realpow_zero
% 0.33/0.69 thf(fact_10045_LIMSEQ__divide__realpow__zero,axiom,
% 0.33/0.69 ! [X: real,A: real] :
% 0.33/0.69 ( ( ord_less_real @ one_one_real @ X )
% 0.33/0.69 => ( filterlim_nat_real
% 0.33/0.69 @ ^ [N: nat] : ( divide_divide_real @ A @ ( power_power_real @ X @ N ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.33/0.69 @ at_top_nat ) ) ).
% 0.33/0.69
% 0.33/0.69 % LIMSEQ_divide_realpow_zero
% 0.33/0.69 thf(fact_10046_LIMSEQ__abs__realpow__zero2,axiom,
% 0.33/0.69 ! [C: real] :
% 0.33/0.69 ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 0.33/0.69 => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 0.33/0.69
% 0.33/0.69 % LIMSEQ_abs_realpow_zero2
% 0.33/0.69 thf(fact_10047_LIMSEQ__abs__realpow__zero,axiom,
% 0.33/0.69 ! [C: real] :
% 0.33/0.69 ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 0.33/0.69 => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 0.33/0.69
% 0.33/0.69 % LIMSEQ_abs_realpow_zero
% 0.33/0.69 thf(fact_10048_LIMSEQ__inverse__realpow__zero,axiom,
% 0.33/0.69 ! [X: real] :
% 0.33/0.69 ( ( ord_less_real @ one_one_real @ X )
% 0.33/0.69 => ( filterlim_nat_real
% 0.33/0.69 @ ^ [N: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.33/0.69 @ at_top_nat ) ) ).
% 0.33/0.69
% 0.33/0.69 % LIMSEQ_inverse_realpow_zero
% 0.33/0.69 thf(fact_10049_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
% 0.33/0.69 ! [R: real] :
% 0.33/0.69 ( filterlim_nat_real
% 0.33/0.69 @ ^ [N: nat] : ( plus_plus_real @ R @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ R )
% 0.33/0.69 @ at_top_nat ) ).
% 0.33/0.69
% 0.33/0.69 % LIMSEQ_inverse_real_of_nat_add_minus
% 0.33/0.69 thf(fact_10050_tendsto__exp__limit__sequentially,axiom,
% 0.33/0.69 ! [X: real] :
% 0.33/0.69 ( filterlim_nat_real
% 0.33/0.69 @ ^ [N: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 0.33/0.69 @ at_top_nat ) ).
% 0.33/0.69
% 0.33/0.69 % tendsto_exp_limit_sequentially
% 0.33/0.69 thf(fact_10051_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
% 0.33/0.69 ! [R: real] :
% 0.33/0.69 ( filterlim_nat_real
% 0.33/0.69 @ ^ [N: nat] : ( times_times_real @ R @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ R )
% 0.33/0.69 @ at_top_nat ) ).
% 0.33/0.69
% 0.33/0.69 % LIMSEQ_inverse_real_of_nat_add_minus_mult
% 0.33/0.69 thf(fact_10052_summable__Leibniz_I1_J,axiom,
% 0.33/0.69 ! [A: nat > real] :
% 0.33/0.69 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.33/0.69 => ( ( topolo6980174941875973593q_real @ A )
% 0.33/0.69 => ( summable_real
% 0.33/0.69 @ ^ [N: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( A @ N ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % summable_Leibniz(1)
% 0.33/0.69 thf(fact_10053_summable,axiom,
% 0.33/0.69 ! [A: nat > real] :
% 0.33/0.69 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.33/0.69 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 0.33/0.69 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 0.33/0.69 => ( summable_real
% 0.33/0.69 @ ^ [N: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( A @ N ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % summable
% 0.33/0.69 thf(fact_10054_cos__diff__limit__1,axiom,
% 0.33/0.69 ! [Theta: nat > real,Theta2: real] :
% 0.33/0.69 ( ( filterlim_nat_real
% 0.33/0.69 @ ^ [J3: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J3 ) @ Theta2 ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.33/0.69 @ at_top_nat )
% 0.33/0.69 => ~ ! [K2: nat > int] :
% 0.33/0.69 ~ ( filterlim_nat_real
% 0.33/0.69 @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ Theta2 )
% 0.33/0.69 @ at_top_nat ) ) ).
% 0.33/0.69
% 0.33/0.69 % cos_diff_limit_1
% 0.33/0.69 thf(fact_10055_cos__limit__1,axiom,
% 0.33/0.69 ! [Theta: nat > real] :
% 0.33/0.69 ( ( filterlim_nat_real
% 0.33/0.69 @ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.33/0.69 @ at_top_nat )
% 0.33/0.69 => ? [K2: nat > int] :
% 0.33/0.69 ( filterlim_nat_real
% 0.33/0.69 @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.33/0.69 @ at_top_nat ) ) ).
% 0.33/0.69
% 0.33/0.69 % cos_limit_1
% 0.33/0.69 thf(fact_10056_summable__Leibniz_I4_J,axiom,
% 0.33/0.69 ! [A: nat > real] :
% 0.33/0.69 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.33/0.69 => ( ( topolo6980174941875973593q_real @ A )
% 0.33/0.69 => ( filterlim_nat_real
% 0.33/0.69 @ ^ [N: nat] :
% 0.33/0.69 ( groups6591440286371151544t_real
% 0.33/0.69 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 0.33/0.69 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real
% 0.33/0.69 @ ( suminf_real
% 0.33/0.69 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
% 0.33/0.69 @ at_top_nat ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % summable_Leibniz(4)
% 0.33/0.69 thf(fact_10057_zeroseq__arctan__series,axiom,
% 0.33/0.69 ! [X: real] :
% 0.33/0.69 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.33/0.69 => ( filterlim_nat_real
% 0.33/0.69 @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.33/0.69 @ at_top_nat ) ) ).
% 0.33/0.69
% 0.33/0.69 % zeroseq_arctan_series
% 0.33/0.69 thf(fact_10058_summable__Leibniz_H_I3_J,axiom,
% 0.33/0.69 ! [A: nat > real] :
% 0.33/0.69 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.33/0.69 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 0.33/0.69 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 0.33/0.69 => ( filterlim_nat_real
% 0.33/0.69 @ ^ [N: nat] :
% 0.33/0.69 ( groups6591440286371151544t_real
% 0.33/0.69 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 0.33/0.69 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real
% 0.33/0.69 @ ( suminf_real
% 0.33/0.69 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
% 0.33/0.69 @ at_top_nat ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % summable_Leibniz'(3)
% 0.33/0.69 thf(fact_10059_summable__Leibniz_H_I2_J,axiom,
% 0.33/0.69 ! [A: nat > real,N2: nat] :
% 0.33/0.69 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.33/0.69 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 0.33/0.69 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 0.33/0.69 => ( ord_less_eq_real
% 0.33/0.69 @ ( groups6591440286371151544t_real
% 0.33/0.69 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 0.33/0.69 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.33/0.69 @ ( suminf_real
% 0.33/0.69 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % summable_Leibniz'(2)
% 0.33/0.69 thf(fact_10060_sums__alternating__upper__lower,axiom,
% 0.33/0.69 ! [A: nat > real] :
% 0.33/0.69 ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 0.33/0.69 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 0.33/0.69 => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.33/0.69 => ? [L4: real] :
% 0.33/0.69 ( ! [N9: nat] :
% 0.33/0.69 ( ord_less_eq_real
% 0.33/0.69 @ ( groups6591440286371151544t_real
% 0.33/0.69 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 0.33/0.69 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) )
% 0.33/0.69 @ L4 )
% 0.33/0.69 & ( filterlim_nat_real
% 0.33/0.69 @ ^ [N: nat] :
% 0.33/0.69 ( groups6591440286371151544t_real
% 0.33/0.69 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 0.33/0.69 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ L4 )
% 0.33/0.69 @ at_top_nat )
% 0.33/0.69 & ! [N9: nat] :
% 0.33/0.69 ( ord_less_eq_real @ L4
% 0.33/0.69 @ ( groups6591440286371151544t_real
% 0.33/0.69 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 0.33/0.69 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) ) )
% 0.33/0.69 & ( filterlim_nat_real
% 0.33/0.69 @ ^ [N: nat] :
% 0.33/0.69 ( groups6591440286371151544t_real
% 0.33/0.69 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 0.33/0.69 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ L4 )
% 0.33/0.69 @ at_top_nat ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % sums_alternating_upper_lower
% 0.33/0.69 thf(fact_10061_summable__Leibniz_I5_J,axiom,
% 0.33/0.69 ! [A: nat > real] :
% 0.33/0.69 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.33/0.69 => ( ( topolo6980174941875973593q_real @ A )
% 0.33/0.69 => ( filterlim_nat_real
% 0.33/0.69 @ ^ [N: nat] :
% 0.33/0.69 ( groups6591440286371151544t_real
% 0.33/0.69 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 0.33/0.69 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real
% 0.33/0.69 @ ( suminf_real
% 0.33/0.69 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
% 0.33/0.69 @ at_top_nat ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % summable_Leibniz(5)
% 0.33/0.69 thf(fact_10062_summable__Leibniz_H_I4_J,axiom,
% 0.33/0.69 ! [A: nat > real,N2: nat] :
% 0.33/0.69 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.33/0.69 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 0.33/0.69 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 0.33/0.69 => ( ord_less_eq_real
% 0.33/0.69 @ ( suminf_real
% 0.33/0.69 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
% 0.33/0.69 @ ( groups6591440286371151544t_real
% 0.33/0.69 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 0.33/0.69 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % summable_Leibniz'(4)
% 0.33/0.69 thf(fact_10063_summable__Leibniz_H_I5_J,axiom,
% 0.33/0.69 ! [A: nat > real] :
% 0.33/0.69 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.33/0.69 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 0.33/0.69 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 0.33/0.69 => ( filterlim_nat_real
% 0.33/0.69 @ ^ [N: nat] :
% 0.33/0.69 ( groups6591440286371151544t_real
% 0.33/0.69 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 0.33/0.69 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real
% 0.33/0.69 @ ( suminf_real
% 0.33/0.69 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
% 0.33/0.69 @ at_top_nat ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % summable_Leibniz'(5)
% 0.33/0.69 thf(fact_10064_real__bounded__linear,axiom,
% 0.33/0.69 ( real_V5970128139526366754l_real
% 0.33/0.69 = ( ^ [F3: real > real] :
% 0.33/0.69 ? [C4: real] :
% 0.33/0.69 ( F3
% 0.33/0.69 = ( ^ [X2: real] : ( times_times_real @ X2 @ C4 ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % real_bounded_linear
% 0.33/0.69 thf(fact_10065_tendsto__exp__limit__at__right,axiom,
% 0.33/0.69 ! [X: real] :
% 0.33/0.69 ( filterlim_real_real
% 0.33/0.69 @ ^ [Y2: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X @ Y2 ) ) @ ( divide_divide_real @ one_one_real @ Y2 ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % tendsto_exp_limit_at_right
% 0.33/0.69 thf(fact_10066_dist__real__def,axiom,
% 0.33/0.69 ( real_V975177566351809787t_real
% 0.33/0.69 = ( ^ [X2: real,Y2: real] : ( abs_abs_real @ ( minus_minus_real @ X2 @ Y2 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % dist_real_def
% 0.33/0.69 thf(fact_10067_dist__complex__def,axiom,
% 0.33/0.69 ( real_V3694042436643373181omplex
% 0.33/0.69 = ( ^ [X2: complex,Y2: complex] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y2 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % dist_complex_def
% 0.33/0.69 thf(fact_10068_tendsto__arcosh__at__left__1,axiom,
% 0.33/0.69 filterlim_real_real @ arcosh_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5849166863359141190n_real @ one_one_real ) ) ).
% 0.33/0.69
% 0.33/0.69 % tendsto_arcosh_at_left_1
% 0.33/0.69 thf(fact_10069_INT__greaterThan__UNIV,axiom,
% 0.33/0.69 ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ set_or1210151606488870762an_nat @ top_top_set_nat ) )
% 0.33/0.69 = bot_bot_set_nat ) ).
% 0.33/0.69
% 0.33/0.69 % INT_greaterThan_UNIV
% 0.33/0.69 thf(fact_10070_greaterThan__0,axiom,
% 0.33/0.69 ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
% 0.33/0.69 = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% 0.33/0.69
% 0.33/0.69 % greaterThan_0
% 0.33/0.69 thf(fact_10071_greaterThan__Suc,axiom,
% 0.33/0.69 ! [K: nat] :
% 0.33/0.69 ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
% 0.33/0.69 = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % greaterThan_Suc
% 0.33/0.69 thf(fact_10072_filterlim__tan__at__right,axiom,
% 0.33/0.69 filterlim_real_real @ tan_real @ at_bot_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % filterlim_tan_at_right
% 0.33/0.69 thf(fact_10073_tendsto__arctan__at__bot,axiom,
% 0.33/0.69 filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ at_bot_real ).
% 0.33/0.69
% 0.33/0.69 % tendsto_arctan_at_bot
% 0.33/0.69 thf(fact_10074_tanh__real__at__bot,axiom,
% 0.33/0.69 filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ one_one_real ) ) @ at_bot_real ).
% 0.33/0.69
% 0.33/0.69 % tanh_real_at_bot
% 0.33/0.69 thf(fact_10075_ln__at__0,axiom,
% 0.33/0.69 filterlim_real_real @ ln_ln_real @ at_bot_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ).
% 0.33/0.69
% 0.33/0.69 % ln_at_0
% 0.33/0.69 thf(fact_10076_filterlim__inverse__at__bot__neg,axiom,
% 0.33/0.69 filterlim_real_real @ inverse_inverse_real @ at_bot_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5984915006950818249n_real @ zero_zero_real ) ) ).
% 0.33/0.69
% 0.33/0.69 % filterlim_inverse_at_bot_neg
% 0.33/0.69 thf(fact_10077_artanh__real__at__right__1,axiom,
% 0.33/0.69 filterlim_real_real @ artanh_real @ at_bot_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ one_one_real ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % artanh_real_at_right_1
% 0.33/0.69 thf(fact_10078_DERIV__pos__imp__increasing__at__bot,axiom,
% 0.33/0.69 ! [B: real,F: real > real,Flim: real] :
% 0.33/0.69 ( ! [X4: real] :
% 0.33/0.69 ( ( ord_less_eq_real @ X4 @ B )
% 0.33/0.69 => ? [Y5: real] :
% 0.33/0.69 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.33/0.69 & ( ord_less_real @ zero_zero_real @ Y5 ) ) )
% 0.33/0.69 => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
% 0.33/0.69 => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_pos_imp_increasing_at_bot
% 0.33/0.69 thf(fact_10079_filterlim__pow__at__bot__odd,axiom,
% 0.33/0.69 ! [N2: nat,F: real > real,F5: filter_real] :
% 0.33/0.69 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.69 => ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
% 0.33/0.69 => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.69 => ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( power_power_real @ ( F @ X2 ) @ N2 )
% 0.33/0.69 @ at_bot_real
% 0.33/0.69 @ F5 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % filterlim_pow_at_bot_odd
% 0.33/0.69 thf(fact_10080_Gcd__eq__Max,axiom,
% 0.33/0.69 ! [M7: set_nat] :
% 0.33/0.69 ( ( finite_finite_nat @ M7 )
% 0.33/0.69 => ( ( M7 != bot_bot_set_nat )
% 0.33/0.69 => ( ~ ( member_nat @ zero_zero_nat @ M7 )
% 0.33/0.69 => ( ( gcd_Gcd_nat @ M7 )
% 0.33/0.69 = ( lattic8265883725875713057ax_nat
% 0.33/0.69 @ ( comple7806235888213564991et_nat
% 0.33/0.69 @ ( image_nat_set_nat
% 0.33/0.69 @ ^ [M6: nat] :
% 0.33/0.69 ( collect_nat
% 0.33/0.69 @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M6 ) )
% 0.33/0.69 @ M7 ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Gcd_eq_Max
% 0.33/0.69 thf(fact_10081_Max__divisors__self__nat,axiom,
% 0.33/0.69 ! [N2: nat] :
% 0.33/0.69 ( ( N2 != zero_zero_nat )
% 0.33/0.69 => ( ( lattic8265883725875713057ax_nat
% 0.33/0.69 @ ( collect_nat
% 0.33/0.69 @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ N2 ) ) )
% 0.33/0.69 = N2 ) ) ).
% 0.33/0.69
% 0.33/0.69 % Max_divisors_self_nat
% 0.33/0.69 thf(fact_10082_card__le__Suc__Max,axiom,
% 0.33/0.69 ! [S: set_nat] :
% 0.33/0.69 ( ( finite_finite_nat @ S )
% 0.33/0.69 => ( ord_less_eq_nat @ ( finite_card_nat @ S ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % card_le_Suc_Max
% 0.33/0.69 thf(fact_10083_divide__nat__def,axiom,
% 0.33/0.69 ( divide_divide_nat
% 0.33/0.69 = ( ^ [M6: nat,N: nat] :
% 0.33/0.69 ( if_nat @ ( N = zero_zero_nat ) @ zero_zero_nat
% 0.33/0.69 @ ( lattic8265883725875713057ax_nat
% 0.33/0.69 @ ( collect_nat
% 0.33/0.69 @ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N ) @ M6 ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % divide_nat_def
% 0.33/0.69 thf(fact_10084_gcd__is__Max__divisors__nat,axiom,
% 0.33/0.69 ! [N2: nat,M: nat] :
% 0.33/0.69 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.69 => ( ( gcd_gcd_nat @ M @ N2 )
% 0.33/0.69 = ( lattic8265883725875713057ax_nat
% 0.33/0.69 @ ( collect_nat
% 0.33/0.69 @ ^ [D2: nat] :
% 0.33/0.69 ( ( dvd_dvd_nat @ D2 @ M )
% 0.33/0.69 & ( dvd_dvd_nat @ D2 @ N2 ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % gcd_is_Max_divisors_nat
% 0.33/0.69 thf(fact_10085_filterlim__pow__at__bot__even,axiom,
% 0.33/0.69 ! [N2: nat,F: real > real,F5: filter_real] :
% 0.33/0.69 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.69 => ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
% 0.33/0.69 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.33/0.69 => ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( power_power_real @ ( F @ X2 ) @ N2 )
% 0.33/0.69 @ at_top_real
% 0.33/0.69 @ F5 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % filterlim_pow_at_bot_even
% 0.33/0.69 thf(fact_10086_eventually__sequentially__Suc,axiom,
% 0.33/0.69 ! [P: nat > $o] :
% 0.33/0.69 ( ( eventually_nat
% 0.33/0.69 @ ^ [I3: nat] : ( P @ ( suc @ I3 ) )
% 0.33/0.69 @ at_top_nat )
% 0.33/0.69 = ( eventually_nat @ P @ at_top_nat ) ) ).
% 0.33/0.69
% 0.33/0.69 % eventually_sequentially_Suc
% 0.33/0.69 thf(fact_10087_eventually__sequentially__seg,axiom,
% 0.33/0.69 ! [P: nat > $o,K: nat] :
% 0.33/0.69 ( ( eventually_nat
% 0.33/0.69 @ ^ [N: nat] : ( P @ ( plus_plus_nat @ N @ K ) )
% 0.33/0.69 @ at_top_nat )
% 0.33/0.69 = ( eventually_nat @ P @ at_top_nat ) ) ).
% 0.33/0.69
% 0.33/0.69 % eventually_sequentially_seg
% 0.33/0.69 thf(fact_10088_Max__divisors__self__int,axiom,
% 0.33/0.69 ! [N2: int] :
% 0.33/0.69 ( ( N2 != zero_zero_int )
% 0.33/0.69 => ( ( lattic8263393255366662781ax_int
% 0.33/0.69 @ ( collect_int
% 0.33/0.69 @ ^ [D2: int] : ( dvd_dvd_int @ D2 @ N2 ) ) )
% 0.33/0.69 = ( abs_abs_int @ N2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Max_divisors_self_int
% 0.33/0.69 thf(fact_10089_lhopital__at__top__at__top,axiom,
% 0.33/0.69 ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
% 0.33/0.69 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.33/0.69 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.33/0.69 => ( ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 0.33/0.69 @ at_top_real
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.33/0.69 => ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.33/0.69 @ at_top_real
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % lhopital_at_top_at_top
% 0.33/0.69 thf(fact_10090_ln__at__top,axiom,
% 0.33/0.69 filterlim_real_real @ ln_ln_real @ at_top_real @ at_top_real ).
% 0.33/0.69
% 0.33/0.69 % ln_at_top
% 0.33/0.69 thf(fact_10091_sqrt__at__top,axiom,
% 0.33/0.69 filterlim_real_real @ sqrt @ at_top_real @ at_top_real ).
% 0.33/0.69
% 0.33/0.69 % sqrt_at_top
% 0.33/0.69 thf(fact_10092_exp__at__top,axiom,
% 0.33/0.69 filterlim_real_real @ exp_real @ at_top_real @ at_top_real ).
% 0.33/0.69
% 0.33/0.69 % exp_at_top
% 0.33/0.69 thf(fact_10093_eventually__False__sequentially,axiom,
% 0.33/0.69 ~ ( eventually_nat
% 0.33/0.69 @ ^ [N: nat] : $false
% 0.33/0.69 @ at_top_nat ) ).
% 0.33/0.69
% 0.33/0.69 % eventually_False_sequentially
% 0.33/0.69 thf(fact_10094_sequentially__offset,axiom,
% 0.33/0.69 ! [P: nat > $o,K: nat] :
% 0.33/0.69 ( ( eventually_nat @ P @ at_top_nat )
% 0.33/0.69 => ( eventually_nat
% 0.33/0.69 @ ^ [I3: nat] : ( P @ ( plus_plus_nat @ I3 @ K ) )
% 0.33/0.69 @ at_top_nat ) ) ).
% 0.33/0.69
% 0.33/0.69 % sequentially_offset
% 0.33/0.69 thf(fact_10095_eventually__sequentiallyI,axiom,
% 0.33/0.69 ! [C: nat,P: nat > $o] :
% 0.33/0.69 ( ! [X4: nat] :
% 0.33/0.69 ( ( ord_less_eq_nat @ C @ X4 )
% 0.33/0.69 => ( P @ X4 ) )
% 0.33/0.69 => ( eventually_nat @ P @ at_top_nat ) ) ).
% 0.33/0.69
% 0.33/0.69 % eventually_sequentiallyI
% 0.33/0.69 thf(fact_10096_eventually__sequentially,axiom,
% 0.33/0.69 ! [P: nat > $o] :
% 0.33/0.69 ( ( eventually_nat @ P @ at_top_nat )
% 0.33/0.69 = ( ? [N6: nat] :
% 0.33/0.69 ! [N: nat] :
% 0.33/0.69 ( ( ord_less_eq_nat @ N6 @ N )
% 0.33/0.69 => ( P @ N ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % eventually_sequentially
% 0.33/0.69 thf(fact_10097_le__sequentially,axiom,
% 0.33/0.69 ! [F5: filter_nat] :
% 0.33/0.69 ( ( ord_le2510731241096832064er_nat @ F5 @ at_top_nat )
% 0.33/0.69 = ( ! [N6: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N6 ) @ F5 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % le_sequentially
% 0.33/0.69 thf(fact_10098_eventually__at__right__to__top,axiom,
% 0.33/0.69 ! [P: real > $o] :
% 0.33/0.69 ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.33/0.69 = ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( P @ ( inverse_inverse_real @ X2 ) )
% 0.33/0.69 @ at_top_real ) ) ).
% 0.33/0.69
% 0.33/0.69 % eventually_at_right_to_top
% 0.33/0.69 thf(fact_10099_eventually__at__top__to__right,axiom,
% 0.33/0.69 ! [P: real > $o] :
% 0.33/0.69 ( ( eventually_real @ P @ at_top_real )
% 0.33/0.69 = ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( P @ ( inverse_inverse_real @ X2 ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % eventually_at_top_to_right
% 0.33/0.69 thf(fact_10100_lhopital__right__at__top__at__top,axiom,
% 0.33/0.69 ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
% 0.33/0.69 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.33/0.69 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.33/0.69 => ( ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 0.33/0.69 @ at_top_real
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.33/0.69 => ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.33/0.69 @ at_top_real
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % lhopital_right_at_top_at_top
% 0.33/0.69 thf(fact_10101_lhopital__at__top__at__bot,axiom,
% 0.33/0.69 ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
% 0.33/0.69 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.33/0.69 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.33/0.69 => ( ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 0.33/0.69 @ at_bot_real
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.33/0.69 => ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.33/0.69 @ at_bot_real
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % lhopital_at_top_at_bot
% 0.33/0.69 thf(fact_10102_lhopital__left__at__top__at__top,axiom,
% 0.33/0.69 ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
% 0.33/0.69 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.33/0.69 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.33/0.69 => ( ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 0.33/0.69 @ at_top_real
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.33/0.69 => ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.33/0.69 @ at_top_real
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % lhopital_left_at_top_at_top
% 0.33/0.69 thf(fact_10103_lhospital__at__top__at__top,axiom,
% 0.33/0.69 ! [G: real > real,G2: real > real,F: real > real,F4: real > real,X: real] :
% 0.33/0.69 ( ( filterlim_real_real @ G @ at_top_real @ at_top_real )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] :
% 0.33/0.69 ( ( G2 @ X2 )
% 0.33/0.69 != zero_zero_real )
% 0.33/0.69 @ at_top_real )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ at_top_real )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ at_top_real )
% 0.33/0.69 => ( ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ X )
% 0.33/0.69 @ at_top_real )
% 0.33/0.69 => ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ X )
% 0.33/0.69 @ at_top_real ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % lhospital_at_top_at_top
% 0.33/0.69 thf(fact_10104_lhopital__at__top,axiom,
% 0.33/0.69 ! [G: real > real,X: real,G2: real > real,F: real > real,F4: real > real,Y: real] :
% 0.33/0.69 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] :
% 0.33/0.69 ( ( G2 @ X2 )
% 0.33/0.69 != zero_zero_real )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.69 => ( ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ Y )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.69 => ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ Y )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % lhopital_at_top
% 0.33/0.69 thf(fact_10105_filterlim__real__sequentially,axiom,
% 0.33/0.69 filterlim_nat_real @ semiri5074537144036343181t_real @ at_top_real @ at_top_nat ).
% 0.33/0.69
% 0.33/0.69 % filterlim_real_sequentially
% 0.33/0.69 thf(fact_10106_filterlim__uminus__at__top__at__bot,axiom,
% 0.33/0.69 filterlim_real_real @ uminus_uminus_real @ at_top_real @ at_bot_real ).
% 0.33/0.69
% 0.33/0.69 % filterlim_uminus_at_top_at_bot
% 0.33/0.69 thf(fact_10107_filterlim__uminus__at__bot__at__top,axiom,
% 0.33/0.69 filterlim_real_real @ uminus_uminus_real @ at_bot_real @ at_top_real ).
% 0.33/0.69
% 0.33/0.69 % filterlim_uminus_at_bot_at_top
% 0.33/0.69 thf(fact_10108_lhopital__right__at__top__at__bot,axiom,
% 0.33/0.69 ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
% 0.33/0.69 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.33/0.69 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.33/0.69 => ( ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 0.33/0.69 @ at_bot_real
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.33/0.69 => ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.33/0.69 @ at_bot_real
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % lhopital_right_at_top_at_bot
% 0.33/0.69 thf(fact_10109_lhopital__left__at__top__at__bot,axiom,
% 0.33/0.69 ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
% 0.33/0.69 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.33/0.69 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.33/0.69 => ( ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 0.33/0.69 @ at_bot_real
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.33/0.69 => ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.33/0.69 @ at_bot_real
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % lhopital_left_at_top_at_bot
% 0.33/0.69 thf(fact_10110_gcd__is__Max__divisors__int,axiom,
% 0.33/0.69 ! [N2: int,M: int] :
% 0.33/0.69 ( ( N2 != zero_zero_int )
% 0.33/0.69 => ( ( gcd_gcd_int @ M @ N2 )
% 0.33/0.69 = ( lattic8263393255366662781ax_int
% 0.33/0.69 @ ( collect_int
% 0.33/0.69 @ ^ [D2: int] :
% 0.33/0.69 ( ( dvd_dvd_int @ D2 @ M )
% 0.33/0.69 & ( dvd_dvd_int @ D2 @ N2 ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % gcd_is_Max_divisors_int
% 0.33/0.69 thf(fact_10111_lhopital__right__0__at__top,axiom,
% 0.33/0.69 ! [G: real > real,G2: real > real,F: real > real,F4: real > real,X: real] :
% 0.33/0.69 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] :
% 0.33/0.69 ( ( G2 @ X2 )
% 0.33/0.69 != zero_zero_real )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.33/0.69 => ( ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ X )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.33/0.69 => ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ X )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % lhopital_right_0_at_top
% 0.33/0.69 thf(fact_10112_lhopital__right__at__top,axiom,
% 0.33/0.69 ! [G: real > real,X: real,G2: real > real,F: real > real,F4: real > real,Y: real] :
% 0.33/0.69 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] :
% 0.33/0.69 ( ( G2 @ X2 )
% 0.33/0.69 != zero_zero_real )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.33/0.69 => ( ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ Y )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.33/0.69 => ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ Y )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % lhopital_right_at_top
% 0.33/0.69 thf(fact_10113_lhopital__left__at__top,axiom,
% 0.33/0.69 ! [G: real > real,X: real,G2: real > real,F: real > real,F4: real > real,Y: real] :
% 0.33/0.69 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] :
% 0.33/0.69 ( ( G2 @ X2 )
% 0.33/0.69 != zero_zero_real )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.33/0.69 => ( ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ Y )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.33/0.69 => ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ Y )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % lhopital_left_at_top
% 0.33/0.69 thf(fact_10114_eventually__at__right__to__0,axiom,
% 0.33/0.69 ! [P: real > $o,A: real] :
% 0.33/0.69 ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.33/0.69 = ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( P @ ( plus_plus_real @ X2 @ A ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % eventually_at_right_to_0
% 0.33/0.69 thf(fact_10115_tanh__real__at__top,axiom,
% 0.33/0.69 filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ one_one_real ) @ at_top_real ).
% 0.33/0.69
% 0.33/0.69 % tanh_real_at_top
% 0.33/0.69 thf(fact_10116_eventually__at__left__to__right,axiom,
% 0.33/0.69 ! [P: real > $o,A: real] :
% 0.33/0.69 ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.33/0.69 = ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( P @ ( uminus_uminus_real @ X2 ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ A ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ A ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % eventually_at_left_to_right
% 0.33/0.69 thf(fact_10117_eventually__at__right__real,axiom,
% 0.33/0.69 ! [A: real,B: real] :
% 0.33/0.69 ( ( ord_less_real @ A @ B )
% 0.33/0.69 => ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( member_real @ X2 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % eventually_at_right_real
% 0.33/0.69 thf(fact_10118_eventually__at__left__real,axiom,
% 0.33/0.69 ! [B: real,A: real] :
% 0.33/0.69 ( ( ord_less_real @ B @ A )
% 0.33/0.69 => ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( member_real @ X2 @ ( set_or1633881224788618240n_real @ B @ A ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % eventually_at_left_real
% 0.33/0.69 thf(fact_10119_ln__x__over__x__tendsto__0,axiom,
% 0.33/0.69 ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( ln_ln_real @ X2 ) @ X2 )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.33/0.69 @ at_top_real ) ).
% 0.33/0.69
% 0.33/0.69 % ln_x_over_x_tendsto_0
% 0.33/0.69 thf(fact_10120_artanh__real__at__left__1,axiom,
% 0.33/0.69 filterlim_real_real @ artanh_real @ at_top_real @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5984915006950818249n_real @ one_one_real ) ) ).
% 0.33/0.69
% 0.33/0.69 % artanh_real_at_left_1
% 0.33/0.69 thf(fact_10121_filterlim__inverse__at__top__right,axiom,
% 0.33/0.69 filterlim_real_real @ inverse_inverse_real @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ).
% 0.33/0.69
% 0.33/0.69 % filterlim_inverse_at_top_right
% 0.33/0.69 thf(fact_10122_filterlim__inverse__at__right__top,axiom,
% 0.33/0.69 filterlim_real_real @ inverse_inverse_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) @ at_top_real ).
% 0.33/0.69
% 0.33/0.69 % filterlim_inverse_at_right_top
% 0.33/0.69 thf(fact_10123_tendsto__power__div__exp__0,axiom,
% 0.33/0.69 ! [K: nat] :
% 0.33/0.69 ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( power_power_real @ X2 @ K ) @ ( exp_real @ X2 ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.33/0.69 @ at_top_real ) ).
% 0.33/0.69
% 0.33/0.69 % tendsto_power_div_exp_0
% 0.33/0.69 thf(fact_10124_lhopital,axiom,
% 0.33/0.69 ! [F: real > real,X: real,G: real > real,G2: real > real,F4: real > real,F5: filter_real] :
% 0.33/0.69 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.69 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] :
% 0.33/0.69 ( ( G @ X2 )
% 0.33/0.69 != zero_zero_real )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] :
% 0.33/0.69 ( ( G2 @ X2 )
% 0.33/0.69 != zero_zero_real )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.69 => ( ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 0.33/0.69 @ F5
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.33/0.69 => ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.33/0.69 @ F5
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % lhopital
% 0.33/0.69 thf(fact_10125_tendsto__exp__limit__at__top,axiom,
% 0.33/0.69 ! [X: real] :
% 0.33/0.69 ( filterlim_real_real
% 0.33/0.69 @ ^ [Y2: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ Y2 ) ) @ Y2 )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 0.33/0.69 @ at_top_real ) ).
% 0.33/0.69
% 0.33/0.69 % tendsto_exp_limit_at_top
% 0.33/0.69 thf(fact_10126_DERIV__neg__imp__decreasing__at__top,axiom,
% 0.33/0.69 ! [B: real,F: real > real,Flim: real] :
% 0.33/0.69 ( ! [X4: real] :
% 0.33/0.69 ( ( ord_less_eq_real @ B @ X4 )
% 0.33/0.69 => ? [Y5: real] :
% 0.33/0.69 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.33/0.69 & ( ord_less_real @ Y5 @ zero_zero_real ) ) )
% 0.33/0.69 => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
% 0.33/0.69 => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_neg_imp_decreasing_at_top
% 0.33/0.69 thf(fact_10127_tendsto__arctan__at__top,axiom,
% 0.33/0.69 filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ at_top_real ).
% 0.33/0.69
% 0.33/0.69 % tendsto_arctan_at_top
% 0.33/0.69 thf(fact_10128_lhopital__right__0,axiom,
% 0.33/0.69 ! [F0: real > real,G0: real > real,G2: real > real,F4: real > real,F5: filter_real] :
% 0.33/0.69 ( ( filterlim_real_real @ F0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.33/0.69 => ( ( filterlim_real_real @ G0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] :
% 0.33/0.69 ( ( G0 @ X2 )
% 0.33/0.69 != zero_zero_real )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] :
% 0.33/0.69 ( ( G2 @ X2 )
% 0.33/0.69 != zero_zero_real )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F0 @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G0 @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.33/0.69 => ( ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 0.33/0.69 @ F5
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.33/0.69 => ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F0 @ X2 ) @ ( G0 @ X2 ) )
% 0.33/0.69 @ F5
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % lhopital_right_0
% 0.33/0.69 thf(fact_10129_lhopital__right,axiom,
% 0.33/0.69 ! [F: real > real,X: real,G: real > real,G2: real > real,F4: real > real,F5: filter_real] :
% 0.33/0.69 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.33/0.69 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] :
% 0.33/0.69 ( ( G @ X2 )
% 0.33/0.69 != zero_zero_real )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] :
% 0.33/0.69 ( ( G2 @ X2 )
% 0.33/0.69 != zero_zero_real )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.33/0.69 => ( ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 0.33/0.69 @ F5
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.33/0.69 => ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.33/0.69 @ F5
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % lhopital_right
% 0.33/0.69 thf(fact_10130_lhopital__left,axiom,
% 0.33/0.69 ! [F: real > real,X: real,G: real > real,G2: real > real,F4: real > real,F5: filter_real] :
% 0.33/0.69 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.33/0.69 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] :
% 0.33/0.69 ( ( G @ X2 )
% 0.33/0.69 != zero_zero_real )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] :
% 0.33/0.69 ( ( G2 @ X2 )
% 0.33/0.69 != zero_zero_real )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.33/0.69 => ( ( eventually_real
% 0.33/0.69 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.33/0.69 => ( ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 0.33/0.69 @ F5
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.33/0.69 => ( filterlim_real_real
% 0.33/0.69 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.33/0.69 @ F5
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % lhopital_left
% 0.33/0.69 thf(fact_10131_filterlim__tan__at__left,axiom,
% 0.33/0.69 filterlim_real_real @ tan_real @ at_top_real @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( set_or5984915006950818249n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % filterlim_tan_at_left
% 0.33/0.69 thf(fact_10132_Bseq__realpow,axiom,
% 0.33/0.69 ! [X: real] :
% 0.33/0.69 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.33/0.69 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.33/0.69 => ( bfun_nat_real @ ( power_power_real @ X ) @ at_top_nat ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Bseq_realpow
% 0.33/0.69 thf(fact_10133_card__greaterThanAtMost,axiom,
% 0.33/0.69 ! [L2: nat,U: nat] :
% 0.33/0.69 ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L2 @ U ) )
% 0.33/0.69 = ( minus_minus_nat @ U @ L2 ) ) ).
% 0.33/0.69
% 0.33/0.69 % card_greaterThanAtMost
% 0.33/0.69 thf(fact_10134_GreatestI__nat,axiom,
% 0.33/0.69 ! [P: nat > $o,K: nat,B: nat] :
% 0.33/0.69 ( ( P @ K )
% 0.33/0.69 => ( ! [Y4: nat] :
% 0.33/0.69 ( ( P @ Y4 )
% 0.33/0.69 => ( ord_less_eq_nat @ Y4 @ B ) )
% 0.33/0.69 => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % GreatestI_nat
% 0.33/0.69 thf(fact_10135_Greatest__le__nat,axiom,
% 0.33/0.69 ! [P: nat > $o,K: nat,B: nat] :
% 0.33/0.69 ( ( P @ K )
% 0.33/0.69 => ( ! [Y4: nat] :
% 0.33/0.69 ( ( P @ Y4 )
% 0.33/0.69 => ( ord_less_eq_nat @ Y4 @ B ) )
% 0.33/0.69 => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Greatest_le_nat
% 0.33/0.69 thf(fact_10136_GreatestI__ex__nat,axiom,
% 0.33/0.69 ! [P: nat > $o,B: nat] :
% 0.33/0.69 ( ? [X_12: nat] : ( P @ X_12 )
% 0.33/0.69 => ( ! [Y4: nat] :
% 0.33/0.69 ( ( P @ Y4 )
% 0.33/0.69 => ( ord_less_eq_nat @ Y4 @ B ) )
% 0.33/0.69 => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % GreatestI_ex_nat
% 0.33/0.69 thf(fact_10137_atLeastSucAtMost__greaterThanAtMost,axiom,
% 0.33/0.69 ! [L2: nat,U: nat] :
% 0.33/0.69 ( ( set_or1269000886237332187st_nat @ ( suc @ L2 ) @ U )
% 0.33/0.69 = ( set_or6659071591806873216st_nat @ L2 @ U ) ) ).
% 0.33/0.69
% 0.33/0.69 % atLeastSucAtMost_greaterThanAtMost
% 0.33/0.69 thf(fact_10138_card__greaterThanAtMost__int,axiom,
% 0.33/0.69 ! [L2: int,U: int] :
% 0.33/0.69 ( ( finite_card_int @ ( set_or6656581121297822940st_int @ L2 @ U ) )
% 0.33/0.69 = ( nat2 @ ( minus_minus_int @ U @ L2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % card_greaterThanAtMost_int
% 0.33/0.69 thf(fact_10139_atLeast__Suc__greaterThan,axiom,
% 0.33/0.69 ! [K: nat] :
% 0.33/0.69 ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 0.33/0.69 = ( set_or1210151606488870762an_nat @ K ) ) ).
% 0.33/0.69
% 0.33/0.69 % atLeast_Suc_greaterThan
% 0.33/0.69 thf(fact_10140_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
% 0.33/0.69 ! [L2: int,U: int] :
% 0.33/0.69 ( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L2 @ one_one_int ) @ U )
% 0.33/0.69 = ( set_or6656581121297822940st_int @ L2 @ U ) ) ).
% 0.33/0.69
% 0.33/0.69 % atLeastPlusOneAtMost_greaterThanAtMost_int
% 0.33/0.69 thf(fact_10141_decseq__bounded,axiom,
% 0.33/0.69 ! [X8: nat > real,B3: real] :
% 0.33/0.69 ( ( order_9091379641038594480t_real @ X8 )
% 0.33/0.69 => ( ! [I4: nat] : ( ord_less_eq_real @ B3 @ ( X8 @ I4 ) )
% 0.33/0.69 => ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % decseq_bounded
% 0.33/0.69 thf(fact_10142_greaterThanAtMost__upto,axiom,
% 0.33/0.69 ( set_or6656581121297822940st_int
% 0.33/0.69 = ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % greaterThanAtMost_upto
% 0.33/0.69 thf(fact_10143_decseq__convergent,axiom,
% 0.33/0.69 ! [X8: nat > real,B3: real] :
% 0.33/0.69 ( ( order_9091379641038594480t_real @ X8 )
% 0.33/0.69 => ( ! [I4: nat] : ( ord_less_eq_real @ B3 @ ( X8 @ I4 ) )
% 0.33/0.69 => ~ ! [L6: real] :
% 0.33/0.69 ( ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
% 0.33/0.69 => ~ ! [I: nat] : ( ord_less_eq_real @ L6 @ ( X8 @ I ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % decseq_convergent
% 0.33/0.69 thf(fact_10144_UN__atLeast__UNIV,axiom,
% 0.33/0.69 ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atLeast_nat @ top_top_set_nat ) )
% 0.33/0.69 = top_top_set_nat ) ).
% 0.33/0.69
% 0.33/0.69 % UN_atLeast_UNIV
% 0.33/0.69 thf(fact_10145_atLeast__Suc,axiom,
% 0.33/0.69 ! [K: nat] :
% 0.33/0.69 ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 0.33/0.69 = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % atLeast_Suc
% 0.33/0.69 thf(fact_10146_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
% 0.33/0.69 ! [X: vEBT_VEBT,Xa2: nat] :
% 0.33/0.69 ( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
% 0.33/0.69 => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.33/0.69 ( X
% 0.33/0.69 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.33/0.69 => ( Xa2 = one_one_nat ) )
% 0.33/0.69 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.33/0.69 => ( ( Deg2 = Xa2 )
% 0.33/0.69 & ! [X4: vEBT_VEBT] :
% 0.33/0.69 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.69 => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.69 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.69 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.33/0.69 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.69 & ( case_o184042715313410164at_nat
% 0.33/0.69 @ ( ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X3 )
% 0.33/0.69 & ! [X2: vEBT_VEBT] :
% 0.33/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.69 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 0.33/0.69 @ ( produc6081775807080527818_nat_o
% 0.33/0.69 @ ^ [Mi3: nat,Ma3: nat] :
% 0.33/0.69 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.33/0.69 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.33/0.69 & ! [I3: nat] :
% 0.33/0.69 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.69 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X3 ) )
% 0.33/0.69 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 0.33/0.69 & ( ( Mi3 = Ma3 )
% 0.33/0.69 => ! [X2: vEBT_VEBT] :
% 0.33/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.69 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 0.33/0.69 & ( ( Mi3 != Ma3 )
% 0.33/0.69 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.33/0.69 & ! [X2: nat] :
% 0.33/0.69 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.33/0.69 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
% 0.33/0.69 => ( ( ord_less_nat @ Mi3 @ X2 )
% 0.33/0.69 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
% 0.33/0.69 @ Mima ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % VEBT_internal.valid'.elims(3)
% 0.33/0.69 thf(fact_10147_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
% 0.33/0.69 ! [X: vEBT_VEBT,Xa2: nat] :
% 0.33/0.69 ( ( vEBT_VEBT_valid @ X @ Xa2 )
% 0.33/0.69 => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.33/0.69 ( X
% 0.33/0.69 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.33/0.69 => ( Xa2 != one_one_nat ) )
% 0.33/0.69 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.33/0.69 => ~ ( ( Deg2 = Xa2 )
% 0.33/0.69 & ! [X5: vEBT_VEBT] :
% 0.33/0.69 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.69 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.69 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.69 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.33/0.69 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.69 & ( case_o184042715313410164at_nat
% 0.33/0.69 @ ( ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X3 )
% 0.33/0.69 & ! [X2: vEBT_VEBT] :
% 0.33/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.69 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 0.33/0.69 @ ( produc6081775807080527818_nat_o
% 0.33/0.69 @ ^ [Mi3: nat,Ma3: nat] :
% 0.33/0.69 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.33/0.69 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.33/0.69 & ! [I3: nat] :
% 0.33/0.69 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.69 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X3 ) )
% 0.33/0.69 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 0.33/0.69 & ( ( Mi3 = Ma3 )
% 0.33/0.69 => ! [X2: vEBT_VEBT] :
% 0.33/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.69 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 0.33/0.69 & ( ( Mi3 != Ma3 )
% 0.33/0.69 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.33/0.69 & ! [X2: nat] :
% 0.33/0.69 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.33/0.69 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
% 0.33/0.69 => ( ( ord_less_nat @ Mi3 @ X2 )
% 0.33/0.69 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
% 0.33/0.69 @ Mima ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % VEBT_internal.valid'.elims(2)
% 0.33/0.69 thf(fact_10148_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
% 0.33/0.69 ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
% 0.33/0.69 ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg4 )
% 0.33/0.69 = ( ( Deg = Deg4 )
% 0.33/0.69 & ! [X2: vEBT_VEBT] :
% 0.33/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.33/0.69 => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.69 & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.69 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.33/0.69 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.69 & ( case_o184042715313410164at_nat
% 0.33/0.69 @ ( ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X3 )
% 0.33/0.69 & ! [X2: vEBT_VEBT] :
% 0.33/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.33/0.69 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 0.33/0.69 @ ( produc6081775807080527818_nat_o
% 0.33/0.69 @ ^ [Mi3: nat,Ma3: nat] :
% 0.33/0.69 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.33/0.69 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.33/0.69 & ! [I3: nat] :
% 0.33/0.69 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.69 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X3 ) )
% 0.33/0.69 = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
% 0.33/0.69 & ( ( Mi3 = Ma3 )
% 0.33/0.69 => ! [X2: vEBT_VEBT] :
% 0.33/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.33/0.69 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 0.33/0.69 & ( ( Mi3 != Ma3 )
% 0.33/0.69 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 0.33/0.69 & ! [X2: nat] :
% 0.33/0.69 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.33/0.69 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X2 )
% 0.33/0.69 => ( ( ord_less_nat @ Mi3 @ X2 )
% 0.33/0.69 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
% 0.33/0.69 @ Mima2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % VEBT_internal.valid'.simps(2)
% 0.33/0.69 thf(fact_10149_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
% 0.33/0.69 ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 0.33/0.69 ( ( ( vEBT_VEBT_valid @ X @ Xa2 )
% 0.33/0.69 = Y )
% 0.33/0.69 => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.33/0.69 ( X
% 0.33/0.69 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.33/0.69 => ( Y
% 0.33/0.69 = ( Xa2 != one_one_nat ) ) )
% 0.33/0.69 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.33/0.69 => ( Y
% 0.33/0.69 = ( ~ ( ( Deg2 = Xa2 )
% 0.33/0.69 & ! [X2: vEBT_VEBT] :
% 0.33/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.69 => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.69 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.69 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.33/0.69 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.69 & ( case_o184042715313410164at_nat
% 0.33/0.69 @ ( ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X3 )
% 0.33/0.69 & ! [X2: vEBT_VEBT] :
% 0.33/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.69 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 0.33/0.69 @ ( produc6081775807080527818_nat_o
% 0.33/0.69 @ ^ [Mi3: nat,Ma3: nat] :
% 0.33/0.69 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.33/0.69 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.33/0.69 & ! [I3: nat] :
% 0.33/0.69 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.69 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X3 ) )
% 0.33/0.69 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 0.33/0.69 & ( ( Mi3 = Ma3 )
% 0.33/0.69 => ! [X2: vEBT_VEBT] :
% 0.33/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.69 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 0.33/0.69 & ( ( Mi3 != Ma3 )
% 0.33/0.69 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.33/0.69 & ! [X2: nat] :
% 0.33/0.69 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.33/0.69 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
% 0.33/0.69 => ( ( ord_less_nat @ Mi3 @ X2 )
% 0.33/0.69 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
% 0.33/0.69 @ Mima ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % VEBT_internal.valid'.elims(1)
% 0.33/0.69 thf(fact_10150_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
% 0.33/0.69 ! [X: vEBT_VEBT,Xa2: nat] :
% 0.33/0.69 ( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
% 0.33/0.69 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.33/0.69 => ( ! [Uu2: $o,Uv2: $o] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.33/0.69 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
% 0.33/0.69 => ( Xa2 = one_one_nat ) ) )
% 0.33/0.69 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.33/0.69 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
% 0.33/0.69 => ( ( Deg2 = Xa2 )
% 0.33/0.69 & ! [X4: vEBT_VEBT] :
% 0.33/0.69 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.69 => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.69 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.69 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.33/0.69 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.69 & ( case_o184042715313410164at_nat
% 0.33/0.69 @ ( ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X3 )
% 0.33/0.69 & ! [X2: vEBT_VEBT] :
% 0.33/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.69 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 0.33/0.69 @ ( produc6081775807080527818_nat_o
% 0.33/0.69 @ ^ [Mi3: nat,Ma3: nat] :
% 0.33/0.69 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.33/0.69 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.33/0.69 & ! [I3: nat] :
% 0.33/0.69 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.69 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X3 ) )
% 0.33/0.69 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 0.33/0.69 & ( ( Mi3 = Ma3 )
% 0.33/0.69 => ! [X2: vEBT_VEBT] :
% 0.33/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.69 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 0.33/0.69 & ( ( Mi3 != Ma3 )
% 0.33/0.69 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.33/0.69 & ! [X2: nat] :
% 0.33/0.69 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.33/0.69 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
% 0.33/0.69 => ( ( ord_less_nat @ Mi3 @ X2 )
% 0.33/0.69 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
% 0.33/0.69 @ Mima ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % VEBT_internal.valid'.pelims(3)
% 0.33/0.69 thf(fact_10151_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
% 0.33/0.69 ! [X: vEBT_VEBT,Xa2: nat] :
% 0.33/0.69 ( ( vEBT_VEBT_valid @ X @ Xa2 )
% 0.33/0.69 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.33/0.69 => ( ! [Uu2: $o,Uv2: $o] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.33/0.69 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
% 0.33/0.69 => ( Xa2 != one_one_nat ) ) )
% 0.33/0.69 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.33/0.69 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
% 0.33/0.69 => ~ ( ( Deg2 = Xa2 )
% 0.33/0.69 & ! [X5: vEBT_VEBT] :
% 0.33/0.69 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.69 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.69 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.69 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.33/0.69 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.69 & ( case_o184042715313410164at_nat
% 0.33/0.69 @ ( ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X3 )
% 0.33/0.69 & ! [X2: vEBT_VEBT] :
% 0.33/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.69 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 0.33/0.69 @ ( produc6081775807080527818_nat_o
% 0.33/0.69 @ ^ [Mi3: nat,Ma3: nat] :
% 0.33/0.69 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.33/0.69 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.33/0.69 & ! [I3: nat] :
% 0.33/0.69 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.69 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X3 ) )
% 0.33/0.69 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 0.33/0.69 & ( ( Mi3 = Ma3 )
% 0.33/0.69 => ! [X2: vEBT_VEBT] :
% 0.33/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.69 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 0.33/0.69 & ( ( Mi3 != Ma3 )
% 0.33/0.69 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.33/0.69 & ! [X2: nat] :
% 0.33/0.69 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.33/0.69 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
% 0.33/0.69 => ( ( ord_less_nat @ Mi3 @ X2 )
% 0.33/0.69 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
% 0.33/0.69 @ Mima ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % VEBT_internal.valid'.pelims(2)
% 0.33/0.69 thf(fact_10152_Sup__int__def,axiom,
% 0.33/0.69 ( complete_Sup_Sup_int
% 0.33/0.69 = ( ^ [X3: set_int] :
% 0.33/0.69 ( the_int
% 0.33/0.69 @ ^ [X2: int] :
% 0.33/0.69 ( ( member_int @ X2 @ X3 )
% 0.33/0.69 & ! [Y2: int] :
% 0.33/0.69 ( ( member_int @ Y2 @ X3 )
% 0.33/0.69 => ( ord_less_eq_int @ Y2 @ X2 ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Sup_int_def
% 0.33/0.69 thf(fact_10153_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
% 0.33/0.69 ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 0.33/0.69 ( ( ( vEBT_VEBT_valid @ X @ Xa2 )
% 0.33/0.69 = Y )
% 0.33/0.69 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 0.33/0.69 => ( ! [Uu2: $o,Uv2: $o] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.33/0.69 => ( ( Y
% 0.33/0.69 = ( Xa2 = one_one_nat ) )
% 0.33/0.69 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
% 0.33/0.69 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.33/0.69 ( ( X
% 0.33/0.69 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.33/0.69 => ( ( Y
% 0.33/0.69 = ( ( Deg2 = Xa2 )
% 0.33/0.69 & ! [X2: vEBT_VEBT] :
% 0.33/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.69 => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.69 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.33/0.69 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.33/0.69 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.69 & ( case_o184042715313410164at_nat
% 0.33/0.69 @ ( ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X3 )
% 0.33/0.69 & ! [X2: vEBT_VEBT] :
% 0.33/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.69 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 0.33/0.69 @ ( produc6081775807080527818_nat_o
% 0.33/0.69 @ ^ [Mi3: nat,Ma3: nat] :
% 0.33/0.69 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.33/0.69 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.33/0.69 & ! [I3: nat] :
% 0.33/0.69 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.33/0.69 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X3 ) )
% 0.33/0.69 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 0.33/0.69 & ( ( Mi3 = Ma3 )
% 0.33/0.69 => ! [X2: vEBT_VEBT] :
% 0.33/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.33/0.69 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 0.33/0.69 & ( ( Mi3 != Ma3 )
% 0.33/0.69 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.33/0.69 & ! [X2: nat] :
% 0.33/0.69 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.33/0.69 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
% 0.33/0.69 => ( ( ord_less_nat @ Mi3 @ X2 )
% 0.33/0.69 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
% 0.33/0.69 @ Mima ) ) )
% 0.33/0.69 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % VEBT_internal.valid'.pelims(1)
% 0.33/0.69 thf(fact_10154_GMVT,axiom,
% 0.33/0.69 ! [A: real,B: real,F: real > real,G: real > real] :
% 0.33/0.69 ( ( ord_less_real @ A @ B )
% 0.33/0.69 => ( ! [X4: real] :
% 0.33/0.69 ( ( ( ord_less_eq_real @ A @ X4 )
% 0.33/0.69 & ( ord_less_eq_real @ X4 @ B ) )
% 0.33/0.69 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F ) )
% 0.33/0.69 => ( ! [X4: real] :
% 0.33/0.69 ( ( ( ord_less_real @ A @ X4 )
% 0.33/0.69 & ( ord_less_real @ X4 @ B ) )
% 0.33/0.69 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 0.33/0.69 => ( ! [X4: real] :
% 0.33/0.69 ( ( ( ord_less_eq_real @ A @ X4 )
% 0.33/0.69 & ( ord_less_eq_real @ X4 @ B ) )
% 0.33/0.69 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ G ) )
% 0.33/0.69 => ( ! [X4: real] :
% 0.33/0.69 ( ( ( ord_less_real @ A @ X4 )
% 0.33/0.69 & ( ord_less_real @ X4 @ B ) )
% 0.33/0.69 => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 0.33/0.69 => ? [G_c: real,F_c: real,C3: real] :
% 0.33/0.69 ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
% 0.33/0.69 & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
% 0.33/0.69 & ( ord_less_real @ A @ C3 )
% 0.33/0.69 & ( ord_less_real @ C3 @ B )
% 0.33/0.69 & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ G_c )
% 0.33/0.69 = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % GMVT
% 0.33/0.69 thf(fact_10155_MVT,axiom,
% 0.33/0.69 ! [A: real,B: real,F: real > real] :
% 0.33/0.69 ( ( ord_less_real @ A @ B )
% 0.33/0.69 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.33/0.69 => ( ! [X4: real] :
% 0.33/0.69 ( ( ord_less_real @ A @ X4 )
% 0.33/0.69 => ( ( ord_less_real @ X4 @ B )
% 0.33/0.69 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.33/0.69 => ? [L4: real,Z5: real] :
% 0.33/0.69 ( ( ord_less_real @ A @ Z5 )
% 0.33/0.69 & ( ord_less_real @ Z5 @ B )
% 0.33/0.69 & ( has_fi5821293074295781190e_real @ F @ L4 @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) )
% 0.33/0.69 & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.33/0.69 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ L4 ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % MVT
% 0.33/0.69 thf(fact_10156_continuous__on__arsinh_H,axiom,
% 0.33/0.69 ! [A2: set_real,F: real > real] :
% 0.33/0.69 ( ( topolo5044208981011980120l_real @ A2 @ F )
% 0.33/0.69 => ( topolo5044208981011980120l_real @ A2
% 0.33/0.69 @ ^ [X2: real] : ( arsinh_real @ ( F @ X2 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % continuous_on_arsinh'
% 0.33/0.69 thf(fact_10157_continuous__on__arcosh,axiom,
% 0.33/0.69 ! [A2: set_real] :
% 0.33/0.69 ( ( ord_less_eq_set_real @ A2 @ ( set_ord_atLeast_real @ one_one_real ) )
% 0.33/0.69 => ( topolo5044208981011980120l_real @ A2 @ arcosh_real ) ) ).
% 0.33/0.69
% 0.33/0.69 % continuous_on_arcosh
% 0.33/0.69 thf(fact_10158_continuous__on__arcosh_H,axiom,
% 0.33/0.69 ! [A2: set_real,F: real > real] :
% 0.33/0.69 ( ( topolo5044208981011980120l_real @ A2 @ F )
% 0.33/0.69 => ( ! [X4: real] :
% 0.33/0.69 ( ( member_real @ X4 @ A2 )
% 0.33/0.69 => ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
% 0.33/0.69 => ( topolo5044208981011980120l_real @ A2
% 0.33/0.69 @ ^ [X2: real] : ( arcosh_real @ ( F @ X2 ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % continuous_on_arcosh'
% 0.33/0.69 thf(fact_10159_continuous__image__closed__interval,axiom,
% 0.33/0.69 ! [A: real,B: real,F: real > real] :
% 0.33/0.69 ( ( ord_less_eq_real @ A @ B )
% 0.33/0.69 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.33/0.69 => ? [C3: real,D4: real] :
% 0.33/0.69 ( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
% 0.33/0.69 = ( set_or1222579329274155063t_real @ C3 @ D4 ) )
% 0.33/0.69 & ( ord_less_eq_real @ C3 @ D4 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % continuous_image_closed_interval
% 0.33/0.69 thf(fact_10160_continuous__on__arccos_H,axiom,
% 0.33/0.69 topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arccos ).
% 0.33/0.69
% 0.33/0.69 % continuous_on_arccos'
% 0.33/0.69 thf(fact_10161_continuous__on__arcsin_H,axiom,
% 0.33/0.69 topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arcsin ).
% 0.33/0.69
% 0.33/0.69 % continuous_on_arcsin'
% 0.33/0.69 thf(fact_10162_continuous__on__artanh,axiom,
% 0.33/0.69 ! [A2: set_real] :
% 0.33/0.69 ( ( ord_less_eq_set_real @ A2 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
% 0.33/0.69 => ( topolo5044208981011980120l_real @ A2 @ artanh_real ) ) ).
% 0.33/0.69
% 0.33/0.69 % continuous_on_artanh
% 0.33/0.69 thf(fact_10163_continuous__on__artanh_H,axiom,
% 0.33/0.69 ! [A2: set_real,F: real > real] :
% 0.33/0.69 ( ( topolo5044208981011980120l_real @ A2 @ F )
% 0.33/0.69 => ( ! [X4: real] :
% 0.33/0.69 ( ( member_real @ X4 @ A2 )
% 0.33/0.69 => ( member_real @ ( F @ X4 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) )
% 0.33/0.69 => ( topolo5044208981011980120l_real @ A2
% 0.33/0.69 @ ^ [X2: real] : ( artanh_real @ ( F @ X2 ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % continuous_on_artanh'
% 0.33/0.69 thf(fact_10164_Rolle__deriv,axiom,
% 0.33/0.69 ! [A: real,B: real,F: real > real,F4: real > real > real] :
% 0.33/0.69 ( ( ord_less_real @ A @ B )
% 0.33/0.69 => ( ( ( F @ A )
% 0.33/0.69 = ( F @ B ) )
% 0.33/0.69 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.33/0.69 => ( ! [X4: real] :
% 0.33/0.69 ( ( ord_less_real @ A @ X4 )
% 0.33/0.69 => ( ( ord_less_real @ X4 @ B )
% 0.33/0.69 => ( has_de1759254742604945161l_real @ F @ ( F4 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.33/0.69 => ? [Z5: real] :
% 0.33/0.69 ( ( ord_less_real @ A @ Z5 )
% 0.33/0.69 & ( ord_less_real @ Z5 @ B )
% 0.33/0.69 & ( ( F4 @ Z5 )
% 0.33/0.69 = ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Rolle_deriv
% 0.33/0.69 thf(fact_10165_mvt,axiom,
% 0.33/0.69 ! [A: real,B: real,F: real > real,F4: real > real > real] :
% 0.33/0.69 ( ( ord_less_real @ A @ B )
% 0.33/0.69 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.33/0.69 => ( ! [X4: real] :
% 0.33/0.69 ( ( ord_less_real @ A @ X4 )
% 0.33/0.69 => ( ( ord_less_real @ X4 @ B )
% 0.33/0.69 => ( has_de1759254742604945161l_real @ F @ ( F4 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.33/0.69 => ~ ! [Xi: real] :
% 0.33/0.69 ( ( ord_less_real @ A @ Xi )
% 0.33/0.69 => ( ( ord_less_real @ Xi @ B )
% 0.33/0.69 => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.33/0.69 != ( F4 @ Xi @ ( minus_minus_real @ B @ A ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % mvt
% 0.33/0.69 thf(fact_10166_DERIV__isconst__end,axiom,
% 0.33/0.69 ! [A: real,B: real,F: real > real] :
% 0.33/0.69 ( ( ord_less_real @ A @ B )
% 0.33/0.69 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.33/0.69 => ( ! [X4: real] :
% 0.33/0.69 ( ( ord_less_real @ A @ X4 )
% 0.33/0.69 => ( ( ord_less_real @ X4 @ B )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.33/0.69 => ( ( F @ B )
% 0.33/0.69 = ( F @ A ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_isconst_end
% 0.33/0.69 thf(fact_10167_DERIV__neg__imp__decreasing__open,axiom,
% 0.33/0.69 ! [A: real,B: real,F: real > real] :
% 0.33/0.69 ( ( ord_less_real @ A @ B )
% 0.33/0.69 => ( ! [X4: real] :
% 0.33/0.69 ( ( ord_less_real @ A @ X4 )
% 0.33/0.69 => ( ( ord_less_real @ X4 @ B )
% 0.33/0.69 => ? [Y5: real] :
% 0.33/0.69 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.33/0.69 & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
% 0.33/0.69 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.33/0.69 => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_neg_imp_decreasing_open
% 0.33/0.69 thf(fact_10168_DERIV__pos__imp__increasing__open,axiom,
% 0.33/0.69 ! [A: real,B: real,F: real > real] :
% 0.33/0.69 ( ( ord_less_real @ A @ B )
% 0.33/0.69 => ( ! [X4: real] :
% 0.33/0.69 ( ( ord_less_real @ A @ X4 )
% 0.33/0.69 => ( ( ord_less_real @ X4 @ B )
% 0.33/0.69 => ? [Y5: real] :
% 0.33/0.69 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.33/0.69 & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
% 0.33/0.69 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.33/0.69 => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_pos_imp_increasing_open
% 0.33/0.69 thf(fact_10169_DERIV__isconst2,axiom,
% 0.33/0.69 ! [A: real,B: real,F: real > real,X: real] :
% 0.33/0.69 ( ( ord_less_real @ A @ B )
% 0.33/0.69 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.33/0.69 => ( ! [X4: real] :
% 0.33/0.69 ( ( ord_less_real @ A @ X4 )
% 0.33/0.69 => ( ( ord_less_real @ X4 @ B )
% 0.33/0.69 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.33/0.69 => ( ( ord_less_eq_real @ A @ X )
% 0.33/0.69 => ( ( ord_less_eq_real @ X @ B )
% 0.33/0.69 => ( ( F @ X )
% 0.33/0.69 = ( F @ A ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % DERIV_isconst2
% 0.33/0.69 thf(fact_10170_Rolle,axiom,
% 0.33/0.69 ! [A: real,B: real,F: real > real] :
% 0.33/0.69 ( ( ord_less_real @ A @ B )
% 0.33/0.69 => ( ( ( F @ A )
% 0.33/0.69 = ( F @ B ) )
% 0.33/0.69 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.33/0.69 => ( ! [X4: real] :
% 0.33/0.69 ( ( ord_less_real @ A @ X4 )
% 0.33/0.69 => ( ( ord_less_real @ X4 @ B )
% 0.33/0.69 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.33/0.69 => ? [Z5: real] :
% 0.33/0.69 ( ( ord_less_real @ A @ Z5 )
% 0.33/0.69 & ( ord_less_real @ Z5 @ B )
% 0.33/0.69 & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % Rolle
% 0.33/0.69 thf(fact_10171_less__eq__int_Orep__eq,axiom,
% 0.33/0.69 ( ord_less_eq_int
% 0.33/0.69 = ( ^ [X2: int,Xa4: int] :
% 0.33/0.69 ( produc8739625826339149834_nat_o
% 0.33/0.69 @ ^ [Y2: nat,Z3: nat] :
% 0.33/0.69 ( produc6081775807080527818_nat_o
% 0.33/0.69 @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y2 @ V4 ) @ ( plus_plus_nat @ U2 @ Z3 ) ) )
% 0.33/0.69 @ ( rep_Integ @ X2 )
% 0.33/0.69 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % less_eq_int.rep_eq
% 0.33/0.69 thf(fact_10172_less__int_Orep__eq,axiom,
% 0.33/0.69 ( ord_less_int
% 0.33/0.69 = ( ^ [X2: int,Xa4: int] :
% 0.33/0.69 ( produc8739625826339149834_nat_o
% 0.33/0.69 @ ^ [Y2: nat,Z3: nat] :
% 0.33/0.69 ( produc6081775807080527818_nat_o
% 0.33/0.69 @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y2 @ V4 ) @ ( plus_plus_nat @ U2 @ Z3 ) ) )
% 0.33/0.69 @ ( rep_Integ @ X2 )
% 0.33/0.69 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % less_int.rep_eq
% 0.33/0.69 thf(fact_10173_nat_Orep__eq,axiom,
% 0.33/0.69 ( nat2
% 0.33/0.69 = ( ^ [X2: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X2 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % nat.rep_eq
% 0.33/0.69 thf(fact_10174_uniformity__real__def,axiom,
% 0.33/0.69 ( topolo1511823702728130853y_real
% 0.33/0.69 = ( comple2936214249959783750l_real
% 0.33/0.69 @ ( image_2178119161166701260l_real
% 0.33/0.69 @ ^ [E3: real] :
% 0.33/0.69 ( princi6114159922880469582l_real
% 0.33/0.69 @ ( collec3799799289383736868l_real
% 0.33/0.69 @ ( produc5414030515140494994real_o
% 0.33/0.69 @ ^ [X2: real,Y2: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X2 @ Y2 ) @ E3 ) ) ) )
% 0.33/0.69 @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % uniformity_real_def
% 0.33/0.69 thf(fact_10175_uniformity__complex__def,axiom,
% 0.33/0.69 ( topolo896644834953643431omplex
% 0.33/0.69 = ( comple8358262395181532106omplex
% 0.33/0.69 @ ( image_5971271580939081552omplex
% 0.33/0.69 @ ^ [E3: real] :
% 0.33/0.69 ( princi3496590319149328850omplex
% 0.33/0.69 @ ( collec8663557070575231912omplex
% 0.33/0.69 @ ( produc6771430404735790350plex_o
% 0.33/0.69 @ ^ [X2: complex,Y2: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X2 @ Y2 ) @ E3 ) ) ) )
% 0.33/0.69 @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % uniformity_complex_def
% 0.33/0.69 thf(fact_10176_open__complex__def,axiom,
% 0.33/0.69 ( topolo4110288021797289639omplex
% 0.33/0.69 = ( ^ [U3: set_complex] :
% 0.33/0.69 ! [X2: complex] :
% 0.33/0.69 ( ( member_complex @ X2 @ U3 )
% 0.33/0.69 => ( eventu5826381225784669381omplex
% 0.33/0.69 @ ( produc6771430404735790350plex_o
% 0.33/0.69 @ ^ [X9: complex,Y2: complex] :
% 0.33/0.69 ( ( X9 = X2 )
% 0.33/0.69 => ( member_complex @ Y2 @ U3 ) ) )
% 0.33/0.69 @ topolo896644834953643431omplex ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % open_complex_def
% 0.33/0.69 thf(fact_10177_open__real__def,axiom,
% 0.33/0.69 ( topolo4860482606490270245n_real
% 0.33/0.69 = ( ^ [U3: set_real] :
% 0.33/0.69 ! [X2: real] :
% 0.33/0.69 ( ( member_real @ X2 @ U3 )
% 0.33/0.69 => ( eventu3244425730907250241l_real
% 0.33/0.69 @ ( produc5414030515140494994real_o
% 0.33/0.69 @ ^ [X9: real,Y2: real] :
% 0.33/0.69 ( ( X9 = X2 )
% 0.33/0.69 => ( member_real @ Y2 @ U3 ) ) )
% 0.33/0.69 @ topolo1511823702728130853y_real ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % open_real_def
% 0.33/0.69 thf(fact_10178_eventually__prod__sequentially,axiom,
% 0.33/0.69 ! [P: product_prod_nat_nat > $o] :
% 0.33/0.69 ( ( eventu1038000079068216329at_nat @ P @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
% 0.33/0.69 = ( ? [N6: nat] :
% 0.33/0.69 ! [M6: nat] :
% 0.33/0.69 ( ( ord_less_eq_nat @ N6 @ M6 )
% 0.33/0.69 => ! [N: nat] :
% 0.33/0.69 ( ( ord_less_eq_nat @ N6 @ N )
% 0.33/0.69 => ( P @ ( product_Pair_nat_nat @ N @ M6 ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % eventually_prod_sequentially
% 0.33/0.69 thf(fact_10179_incseq__bounded,axiom,
% 0.33/0.69 ! [X8: nat > real,B3: real] :
% 0.33/0.69 ( ( order_mono_nat_real @ X8 )
% 0.33/0.69 => ( ! [I4: nat] : ( ord_less_eq_real @ ( X8 @ I4 ) @ B3 )
% 0.33/0.69 => ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % incseq_bounded
% 0.33/0.69 thf(fact_10180_mono__times__nat,axiom,
% 0.33/0.69 ! [N2: nat] :
% 0.33/0.69 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.69 => ( order_mono_nat_nat @ ( times_times_nat @ N2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % mono_times_nat
% 0.33/0.69 thf(fact_10181_mono__Suc,axiom,
% 0.33/0.69 order_mono_nat_nat @ suc ).
% 0.33/0.69
% 0.33/0.69 % mono_Suc
% 0.33/0.69 thf(fact_10182_filtermap__at__right__shift,axiom,
% 0.33/0.69 ! [D: real,A: real] :
% 0.33/0.69 ( ( filtermap_real_real
% 0.33/0.69 @ ^ [X2: real] : ( minus_minus_real @ X2 @ D )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.33/0.69 = ( topolo2177554685111907308n_real @ ( minus_minus_real @ A @ D ) @ ( set_or5849166863359141190n_real @ ( minus_minus_real @ A @ D ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % filtermap_at_right_shift
% 0.33/0.69 thf(fact_10183_incseq__convergent,axiom,
% 0.33/0.69 ! [X8: nat > real,B3: real] :
% 0.33/0.69 ( ( order_mono_nat_real @ X8 )
% 0.33/0.69 => ( ! [I4: nat] : ( ord_less_eq_real @ ( X8 @ I4 ) @ B3 )
% 0.33/0.69 => ~ ! [L6: real] :
% 0.33/0.69 ( ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
% 0.33/0.69 => ~ ! [I: nat] : ( ord_less_eq_real @ ( X8 @ I ) @ L6 ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % incseq_convergent
% 0.33/0.69 thf(fact_10184_at__right__to__0,axiom,
% 0.33/0.69 ! [A: real] :
% 0.33/0.69 ( ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) )
% 0.33/0.69 = ( filtermap_real_real
% 0.33/0.69 @ ^ [X2: real] : ( plus_plus_real @ X2 @ A )
% 0.33/0.69 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % at_right_to_0
% 0.33/0.69 thf(fact_10185_at__right__minus,axiom,
% 0.33/0.69 ! [A: real] :
% 0.33/0.69 ( ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) )
% 0.33/0.69 = ( filtermap_real_real @ uminus_uminus_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ A ) @ ( set_or5984915006950818249n_real @ ( uminus_uminus_real @ A ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % at_right_minus
% 0.33/0.69 thf(fact_10186_at__left__minus,axiom,
% 0.33/0.69 ! [A: real] :
% 0.33/0.69 ( ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) )
% 0.33/0.69 = ( filtermap_real_real @ uminus_uminus_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ A ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ A ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % at_left_minus
% 0.33/0.69 thf(fact_10187_mono__ge2__power__minus__self,axiom,
% 0.33/0.69 ! [K: nat] :
% 0.33/0.69 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.33/0.69 => ( order_mono_nat_nat
% 0.33/0.69 @ ^ [M6: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M6 ) @ M6 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % mono_ge2_power_minus_self
% 0.33/0.69 thf(fact_10188_tendsto__at__topI__sequentially__real,axiom,
% 0.33/0.69 ! [F: real > real,Y: real] :
% 0.33/0.69 ( ( order_mono_real_real @ F )
% 0.33/0.69 => ( ( filterlim_nat_real
% 0.33/0.69 @ ^ [N: nat] : ( F @ ( semiri5074537144036343181t_real @ N ) )
% 0.33/0.69 @ ( topolo2815343760600316023s_real @ Y )
% 0.33/0.69 @ at_top_nat )
% 0.33/0.69 => ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Y ) @ at_top_real ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % tendsto_at_topI_sequentially_real
% 0.33/0.69 thf(fact_10189_nonneg__incseq__Bseq__subseq__iff,axiom,
% 0.33/0.69 ! [F: nat > real,G: nat > nat] :
% 0.33/0.69 ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 0.33/0.69 => ( ( order_mono_nat_real @ F )
% 0.33/0.69 => ( ( order_5726023648592871131at_nat @ G )
% 0.33/0.69 => ( ( bfun_nat_real
% 0.33/0.69 @ ^ [X2: nat] : ( F @ ( G @ X2 ) )
% 0.33/0.69 @ at_top_nat )
% 0.33/0.69 = ( bfun_nat_real @ F @ at_top_nat ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % nonneg_incseq_Bseq_subseq_iff
% 0.33/0.69 thf(fact_10190_eventually__subseq,axiom,
% 0.33/0.69 ! [R: nat > nat,P: nat > $o] :
% 0.33/0.69 ( ( order_5726023648592871131at_nat @ R )
% 0.33/0.69 => ( ( eventually_nat @ P @ at_top_nat )
% 0.33/0.69 => ( eventually_nat
% 0.33/0.69 @ ^ [N: nat] : ( P @ ( R @ N ) )
% 0.33/0.69 @ at_top_nat ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % eventually_subseq
% 0.33/0.69 thf(fact_10191_strict__mono__imp__increasing,axiom,
% 0.33/0.69 ! [F: nat > nat,N2: nat] :
% 0.33/0.69 ( ( order_5726023648592871131at_nat @ F )
% 0.33/0.69 => ( ord_less_eq_nat @ N2 @ ( F @ N2 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % strict_mono_imp_increasing
% 0.33/0.69 thf(fact_10192_inj__sgn__power,axiom,
% 0.33/0.69 ! [N2: nat] :
% 0.33/0.69 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.33/0.69 => ( inj_on_real_real
% 0.33/0.69 @ ^ [Y2: real] : ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N2 ) )
% 0.33/0.69 @ top_top_set_real ) ) ).
% 0.33/0.69
% 0.33/0.69 % inj_sgn_power
% 0.33/0.69 thf(fact_10193_log__inj,axiom,
% 0.33/0.69 ! [B: real] :
% 0.33/0.69 ( ( ord_less_real @ one_one_real @ B )
% 0.33/0.69 => ( inj_on_real_real @ ( log @ B ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % log_inj
% 0.33/0.69 thf(fact_10194_pos__deriv__imp__strict__mono,axiom,
% 0.33/0.69 ! [F: real > real,F4: real > real] :
% 0.33/0.69 ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.33/0.69 => ( ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( F4 @ X4 ) )
% 0.33/0.69 => ( order_7092887310737990675l_real @ F ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % pos_deriv_imp_strict_mono
% 0.33/0.69 thf(fact_10195_inj__on__diff__nat,axiom,
% 0.33/0.69 ! [N3: set_nat,K: nat] :
% 0.33/0.69 ( ! [N4: nat] :
% 0.33/0.69 ( ( member_nat @ N4 @ N3 )
% 0.33/0.69 => ( ord_less_eq_nat @ K @ N4 ) )
% 0.33/0.69 => ( inj_on_nat_nat
% 0.33/0.69 @ ^ [N: nat] : ( minus_minus_nat @ N @ K )
% 0.33/0.69 @ N3 ) ) ).
% 0.33/0.69
% 0.33/0.69 % inj_on_diff_nat
% 0.33/0.69 thf(fact_10196_inj__Suc,axiom,
% 0.33/0.69 ! [N3: set_nat] : ( inj_on_nat_nat @ suc @ N3 ) ).
% 0.33/0.69
% 0.33/0.69 % inj_Suc
% 0.33/0.69 thf(fact_10197_summable__reindex,axiom,
% 0.33/0.69 ! [F: nat > real,G: nat > nat] :
% 0.33/0.69 ( ( summable_real @ F )
% 0.33/0.69 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 0.33/0.69 => ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 0.33/0.69 => ( summable_real @ ( comp_nat_real_nat @ F @ G ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % summable_reindex
% 0.33/0.69 thf(fact_10198_suminf__reindex__mono,axiom,
% 0.33/0.69 ! [F: nat > real,G: nat > nat] :
% 0.33/0.69 ( ( summable_real @ F )
% 0.33/0.69 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 0.33/0.69 => ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 0.33/0.69 => ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G ) ) @ ( suminf_real @ F ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % suminf_reindex_mono
% 0.33/0.69 thf(fact_10199_inj__on__char__of__nat,axiom,
% 0.33/0.69 inj_on_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % inj_on_char_of_nat
% 0.33/0.69 thf(fact_10200_suminf__reindex,axiom,
% 0.33/0.69 ! [F: nat > real,G: nat > nat] :
% 0.33/0.69 ( ( summable_real @ F )
% 0.33/0.69 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 0.33/0.69 => ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 0.33/0.69 => ( ! [X4: nat] :
% 0.33/0.69 ( ~ ( member_nat @ X4 @ ( image_nat_nat @ G @ top_top_set_nat ) )
% 0.33/0.69 => ( ( F @ X4 )
% 0.33/0.69 = zero_zero_real ) )
% 0.33/0.69 => ( ( suminf_real @ ( comp_nat_real_nat @ F @ G ) )
% 0.33/0.69 = ( suminf_real @ F ) ) ) ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % suminf_reindex
% 0.33/0.69 thf(fact_10201_prod__encode__def,axiom,
% 0.33/0.69 ( nat_prod_encode
% 0.33/0.69 = ( produc6842872674320459806at_nat
% 0.33/0.69 @ ^ [M6: nat,N: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M6 @ N ) ) @ M6 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 % prod_encode_def
% 0.33/0.69
% 0.33/0.69 % Helper facts (36)
% 0.33/0.69 thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
% 0.33/0.69 ! [X: int,Y: int] :
% 0.33/0.69 ( ( if_int @ $false @ X @ Y )
% 0.33/0.69 = Y ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
% 0.33/0.69 ! [X: int,Y: int] :
% 0.33/0.69 ( ( if_int @ $true @ X @ Y )
% 0.33/0.69 = X ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 0.33/0.69 ! [X: nat,Y: nat] :
% 0.33/0.69 ( ( if_nat @ $false @ X @ Y )
% 0.33/0.69 = Y ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 0.33/0.69 ! [X: nat,Y: nat] :
% 0.33/0.69 ( ( if_nat @ $true @ X @ Y )
% 0.33/0.69 = X ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
% 0.33/0.69 ! [X: num,Y: num] :
% 0.33/0.69 ( ( if_num @ $false @ X @ Y )
% 0.33/0.69 = Y ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
% 0.33/0.69 ! [X: num,Y: num] :
% 0.33/0.69 ( ( if_num @ $true @ X @ Y )
% 0.33/0.69 = X ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
% 0.33/0.69 ! [X: rat,Y: rat] :
% 0.33/0.69 ( ( if_rat @ $false @ X @ Y )
% 0.33/0.69 = Y ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
% 0.33/0.69 ! [X: rat,Y: rat] :
% 0.33/0.69 ( ( if_rat @ $true @ X @ Y )
% 0.33/0.69 = X ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 0.33/0.69 ! [X: real,Y: real] :
% 0.33/0.69 ( ( if_real @ $false @ X @ Y )
% 0.33/0.69 = Y ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 0.33/0.69 ! [X: real,Y: real] :
% 0.33/0.69 ( ( if_real @ $true @ X @ Y )
% 0.33/0.69 = X ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
% 0.33/0.69 ! [P: real > $o] :
% 0.33/0.69 ( ( P @ ( fChoice_real @ P ) )
% 0.33/0.69 = ( ? [X3: real] : ( P @ X3 ) ) ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
% 0.33/0.69 ! [X: complex,Y: complex] :
% 0.33/0.69 ( ( if_complex @ $false @ X @ Y )
% 0.33/0.69 = Y ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
% 0.33/0.69 ! [X: complex,Y: complex] :
% 0.33/0.69 ( ( if_complex @ $true @ X @ Y )
% 0.33/0.69 = X ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 0.33/0.69 ! [X: code_integer,Y: code_integer] :
% 0.33/0.69 ( ( if_Code_integer @ $false @ X @ Y )
% 0.33/0.69 = Y ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 0.33/0.69 ! [X: code_integer,Y: code_integer] :
% 0.33/0.69 ( ( if_Code_integer @ $true @ X @ Y )
% 0.33/0.69 = X ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 0.33/0.69 ! [X: set_int,Y: set_int] :
% 0.33/0.69 ( ( if_set_int @ $false @ X @ Y )
% 0.33/0.69 = Y ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 0.33/0.69 ! [X: set_int,Y: set_int] :
% 0.33/0.69 ( ( if_set_int @ $true @ X @ Y )
% 0.33/0.69 = X ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
% 0.33/0.69 ! [X: set_nat,Y: set_nat] :
% 0.33/0.69 ( ( if_set_nat @ $false @ X @ Y )
% 0.33/0.69 = Y ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
% 0.33/0.69 ! [X: set_nat,Y: set_nat] :
% 0.33/0.69 ( ( if_set_nat @ $true @ X @ Y )
% 0.33/0.69 = X ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
% 0.33/0.69 ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
% 0.33/0.69 ( ( if_VEBT_VEBT @ $false @ X @ Y )
% 0.33/0.69 = Y ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
% 0.33/0.69 ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
% 0.33/0.69 ( ( if_VEBT_VEBT @ $true @ X @ Y )
% 0.33/0.69 = X ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 0.33/0.69 ! [X: list_int,Y: list_int] :
% 0.33/0.69 ( ( if_list_int @ $false @ X @ Y )
% 0.33/0.69 = Y ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 0.33/0.69 ! [X: list_int,Y: list_int] :
% 0.33/0.69 ( ( if_list_int @ $true @ X @ Y )
% 0.33/0.69 = X ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
% 0.33/0.69 ! [X: option_nat,Y: option_nat] :
% 0.33/0.69 ( ( if_option_nat @ $false @ X @ Y )
% 0.33/0.69 = Y ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
% 0.33/0.69 ! [X: option_nat,Y: option_nat] :
% 0.33/0.69 ( ( if_option_nat @ $true @ X @ Y )
% 0.33/0.69 = X ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 0.33/0.69 ! [X: option_num,Y: option_num] :
% 0.33/0.69 ( ( if_option_num @ $false @ X @ Y )
% 0.33/0.69 = Y ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 0.33/0.69 ! [X: option_num,Y: option_num] :
% 0.33/0.69 ( ( if_option_num @ $true @ X @ Y )
% 0.33/0.69 = X ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.33/0.69 ! [X: product_prod_int_int,Y: product_prod_int_int] :
% 0.33/0.69 ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
% 0.33/0.69 = Y ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.33/0.69 ! [X: product_prod_int_int,Y: product_prod_int_int] :
% 0.33/0.69 ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
% 0.33/0.69 = X ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 0.33/0.69 ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 0.33/0.69 ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
% 0.33/0.69 = Y ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 0.33/0.69 ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 0.33/0.69 ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
% 0.33/0.69 = X ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 0.33/0.69 ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
% 0.33/0.69 ( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y )
% 0.33/0.69 = Y ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 0.33/0.69 ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
% 0.33/0.69 ( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y )
% 0.33/0.69 = X ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.33/0.69 ! [P: $o] :
% 0.33/0.69 ( ( P = $true )
% 0.33/0.69 | ( P = $false ) ) ).
% 0.33/0.69
% 0.33/0.69 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.33/0.69 ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
% 0.33/0.70 ( ( if_Pro6119634080678213985nteger @ $false @ X @ Y )
% 0.33/0.70 = Y ) ).
% 0.33/0.70
% 0.33/0.70 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.33/0.70 ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
% 0.33/0.70 ( ( if_Pro6119634080678213985nteger @ $true @ X @ Y )
% 0.33/0.70 = X ) ).
% 0.33/0.70
% 0.33/0.70 % Conjectures (1)
% 0.33/0.70 thf(conj_0,conjecture,
% 0.33/0.70 ( ( 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 @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) ) )
% 0.33/0.70 = ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) @ maxs ) @ maxi ) ) ).
% 0.33/0.70
% 0.33/0.70 %------------------------------------------------------------------------------
% 0.33/0.70 ------------------------
% 0.34/4.04 ---- Embedded in THF ---
% 0.34/4.04 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.34/4.04 %%% Generated on Fri Feb 27 03:51:34 EST 2026
% 0.34/4.04 %%% using '$$dhol' embedding, version 1.3.0.
% 0.34/4.04 %%% Logic specification used:
% 0.34/4.04 %%% thf(spec, logic, $$dhol).
% 0.34/4.04
% 0.34/4.04 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.YjPWRkCLRl/DTF2DTF23017.p
% See solution above
% 0.34/4.08 ------------------------
% 16.11/8.87 ---- Cleaned THF ---
% 16.11/8.87 thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 16.11/8.87 produc5542196010084753463at_nat: $tType ).
% 16.11/8.87
% 16.11/8.87 thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 16.11/8.87 produc5491161045314408544at_nat: $tType ).
% 16.11/8.87
% 16.11/8.87 thf(ty_n_t__Product____Type__Oprod_I_062_It__Num__Onum_M_062_It__Num__Onum_Mt__Num__Onum_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J_J,type,
% 16.11/8.87 produc1193250871479095198on_num: $tType ).
% 16.11/8.87
% 16.11/8.87 thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_J,type,
% 16.11/8.87 produc8306885398267862888on_nat: $tType ).
% 16.11/8.87
% 16.11/8.87 thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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% 16.11/8.87
% 16.11/8.87 thf(sy_c_Finite__Set_Ocard_001_Eo,type,
% 16.11/8.87 finite_card_o: set_o > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
% 16.11/8.87 finite_card_complex: set_complex > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
% 16.11/8.87 finite_card_int: set_int > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
% 16.11/8.87 finite_card_nat: set_nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Ounit,type,
% 16.11/8.87 finite410649719033368117t_unit: set_Product_unit > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
% 16.11/8.87 finite_card_set_nat: set_set_nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Finite__Set_Ocard_001t__String__Ochar,type,
% 16.11/8.87 finite_card_char: set_char > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Finite__Set_Ofinite_001_Eo,type,
% 16.11/8.87 finite_finite_o: set_o > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
% 16.11/8.87 finite3207457112153483333omplex: set_complex > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
% 16.11/8.87 finite_finite_int: set_int > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_I_Eo_J,type,
% 16.11/8.87 finite_finite_list_o: set_list_o > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Complex__Ocomplex_J,type,
% 16.11/8.87 finite8712137658972009173omplex: set_list_complex > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Int__Oint_J,type,
% 16.11/8.87 finite3922522038869484883st_int: set_list_int > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
% 16.11/8.87 finite8100373058378681591st_nat: set_list_nat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 16.11/8.87 finite3004134309566078307T_VEBT: set_list_VEBT_VEBT > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
% 16.11/8.87 finite_finite_nat: set_nat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Finite__Set_Ofinite_001t__Num__Onum,type,
% 16.11/8.87 finite_finite_num: set_num > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 16.11/8.87 finite2998713641127702882nt_int: set_Pr958786334691620121nt_int > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Finite__Set_Ofinite_001t__Rat__Orat,type,
% 16.11/8.87 finite_finite_rat: set_rat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
% 16.11/8.87 finite_finite_real: set_real > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 16.11/8.87 finite6551019134538273531omplex: set_set_complex > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Int__Oint_J,type,
% 16.11/8.87 finite6197958912794628473et_int: set_set_int > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
% 16.11/8.87 finite1152437895449049373et_nat: set_set_nat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Finite__Set_Ofinite_001t__VEBT____Definitions__OVEBT,type,
% 16.11/8.87 finite5795047828879050333T_VEBT: set_VEBT_VEBT > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Fun_Obij__betw_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 16.11/8.87 bij_be1856998921033663316omplex: ( complex > complex ) > set_complex > set_complex > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Complex__Ocomplex,type,
% 16.11/8.87 bij_betw_nat_complex: ( nat > complex ) > set_nat > set_complex > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
% 16.11/8.87 bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Fun_Ocomp_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
% 16.11/8.87 comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
% 16.11/8.87 comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001t__Int__Oint,type,
% 16.11/8.87 comp_int_nat_int: ( int > nat ) > ( int > int ) > int > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 16.11/8.87 comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Real__Oreal_001t__Nat__Onat,type,
% 16.11/8.87 comp_nat_real_nat: ( nat > real ) > ( nat > nat ) > nat > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
% 16.11/8.87 inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__String__Ochar,type,
% 16.11/8.87 inj_on_nat_char: ( nat > char ) > set_nat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 16.11/8.87 inj_on_real_real: ( real > real ) > set_real > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Fun_Ostrict__mono__on_001t__Nat__Onat_001t__Nat__Onat,type,
% 16.11/8.87 strict1292158309912662752at_nat: ( nat > nat ) > set_nat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Fun_Othe__inv__into_001t__Real__Oreal_001t__Real__Oreal,type,
% 16.11/8.87 the_in5290026491893676941l_real: set_real > ( real > real ) > real > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_GCD_OGcd__class_OGcd_001t__Int__Oint,type,
% 16.11/8.87 gcd_Gcd_int: set_int > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_GCD_OGcd__class_OGcd_001t__Nat__Onat,type,
% 16.11/8.87 gcd_Gcd_nat: set_nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_GCD_Obezw,type,
% 16.11/8.87 bezw: nat > nat > product_prod_int_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_GCD_Obezw__rel,type,
% 16.11/8.87 bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Int__Oint,type,
% 16.11/8.87 gcd_gcd_int: int > int > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Nat__Onat,type,
% 16.11/8.87 gcd_gcd_nat: nat > nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_GCD_Ogcd__nat__rel,type,
% 16.11/8.87 gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Oabs__class_Oabs_001t__Code____Numeral__Ointeger,type,
% 16.11/8.87 abs_abs_Code_integer: code_integer > code_integer ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Oabs__class_Oabs_001t__Complex__Ocomplex,type,
% 16.11/8.87 abs_abs_complex: complex > complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
% 16.11/8.87 abs_abs_int: int > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Oabs__class_Oabs_001t__Rat__Orat,type,
% 16.11/8.87 abs_abs_rat: rat > rat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
% 16.11/8.87 abs_abs_real: real > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Complex__Ocomplex_M_Eo_J,type,
% 16.11/8.87 minus_8727706125548526216plex_o: ( complex > $o ) > ( complex > $o ) > complex > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Int__Oint_M_Eo_J,type,
% 16.11/8.87 minus_minus_int_o: ( int > $o ) > ( int > $o ) > int > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
% 16.11/8.87 minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
% 16.11/8.87 minus_711738161318947805_int_o: ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > product_prod_int_int > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Real__Oreal_M_Eo_J,type,
% 16.11/8.87 minus_minus_real_o: ( real > $o ) > ( real > $o ) > real > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
% 16.11/8.87 minus_6910147592129066416_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > set_nat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Ominus__class_Ominus_001t__Code____Numeral__Ointeger,type,
% 16.11/8.87 minus_8373710615458151222nteger: code_integer > code_integer > code_integer ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
% 16.11/8.87 minus_minus_complex: complex > complex > complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nat__Oenat,type,
% 16.11/8.87 minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
% 16.11/8.87 minus_minus_int: int > int > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 16.11/8.87 minus_minus_nat: nat > nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Ominus__class_Ominus_001t__Rat__Orat,type,
% 16.11/8.87 minus_minus_rat: rat > rat > rat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
% 16.11/8.87 minus_minus_real: real > real > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 16.11/8.87 minus_811609699411566653omplex: set_complex > set_complex > set_complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
% 16.11/8.87 minus_minus_set_int: set_int > set_int > set_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
% 16.11/8.87 minus_minus_set_nat: set_nat > set_nat > set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 16.11/8.87 minus_1052850069191792384nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
% 16.11/8.87 minus_minus_set_real: set_real > set_real > set_real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 16.11/8.87 minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Oone__class_Oone_001t__Code____Numeral__Ointeger,type,
% 16.11/8.87 one_one_Code_integer: code_integer ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
% 16.11/8.87 one_one_complex: complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nat__Oenat,type,
% 16.11/8.87 one_on7984719198319812577d_enat: extended_enat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
% 16.11/8.87 one_one_int: int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 16.11/8.87 one_one_nat: nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Oone__class_Oone_001t__Rat__Orat,type,
% 16.11/8.87 one_one_rat: rat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
% 16.11/8.87 one_one_real: real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Oplus__class_Oplus_001t__Code____Numeral__Ointeger,type,
% 16.11/8.87 plus_p5714425477246183910nteger: code_integer > code_integer > code_integer ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
% 16.11/8.87 plus_plus_complex: complex > complex > complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nat__Oenat,type,
% 16.11/8.87 plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
% 16.11/8.87 plus_plus_int: int > int > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 16.11/8.87 plus_plus_nat: nat > nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
% 16.11/8.87 plus_plus_num: num > num > num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Oplus__class_Oplus_001t__Rat__Orat,type,
% 16.11/8.87 plus_plus_rat: rat > rat > rat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
% 16.11/8.87 plus_plus_real: real > real > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Osgn__class_Osgn_001t__Code____Numeral__Ointeger,type,
% 16.11/8.87 sgn_sgn_Code_integer: code_integer > code_integer ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Osgn__class_Osgn_001t__Complex__Ocomplex,type,
% 16.11/8.87 sgn_sgn_complex: complex > complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Osgn__class_Osgn_001t__Int__Oint,type,
% 16.11/8.87 sgn_sgn_int: int > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Osgn__class_Osgn_001t__Rat__Orat,type,
% 16.11/8.87 sgn_sgn_rat: rat > rat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Osgn__class_Osgn_001t__Real__Oreal,type,
% 16.11/8.87 sgn_sgn_real: real > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Otimes__class_Otimes_001t__Code____Numeral__Ointeger,type,
% 16.11/8.87 times_3573771949741848930nteger: code_integer > code_integer > code_integer ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
% 16.11/8.87 times_times_complex: complex > complex > complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Nat__Oenat,type,
% 16.11/8.87 times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
% 16.11/8.87 times_times_int: int > int > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
% 16.11/8.87 times_times_nat: nat > nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
% 16.11/8.87 times_times_num: num > num > num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Otimes__class_Otimes_001t__Rat__Orat,type,
% 16.11/8.87 times_times_rat: rat > rat > rat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
% 16.11/8.87 times_times_real: real > real > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Complex__Ocomplex_M_Eo_J,type,
% 16.11/8.87 uminus1680532995456772888plex_o: ( complex > $o ) > complex > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Int__Oint_M_Eo_J,type,
% 16.11/8.87 uminus_uminus_int_o: ( int > $o ) > int > $o ).
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% 16.11/8.87 if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 16.11/8.87 if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_If_001t__Rat__Orat,type,
% 16.11/8.87 if_rat: $o > rat > rat > rat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_If_001t__Real__Oreal,type,
% 16.11/8.87 if_real: $o > real > real > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_If_001t__Set__Oset_It__Int__Oint_J,type,
% 16.11/8.87 if_set_int: $o > set_int > set_int > set_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_If_001t__Set__Oset_It__Nat__Onat_J,type,
% 16.11/8.87 if_set_nat: $o > set_nat > set_nat > set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_If_001t__VEBT____Definitions__OVEBT,type,
% 16.11/8.87 if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Int_OAbs__Integ,type,
% 16.11/8.87 abs_Integ: product_prod_nat_nat > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Int_ORep__Integ,type,
% 16.11/8.87 rep_Integ: int > product_prod_nat_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Int_Oint__ge__less__than,type,
% 16.11/8.87 int_ge_less_than: int > set_Pr958786334691620121nt_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Int_Oint__ge__less__than2,type,
% 16.11/8.87 int_ge_less_than2: int > set_Pr958786334691620121nt_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Int_Onat,type,
% 16.11/8.87 nat2: int > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
% 16.11/8.87 ring_1_Ints_real: set_real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Code____Numeral__Ointeger,type,
% 16.11/8.87 ring_18347121197199848620nteger: int > code_integer ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Complex__Ocomplex,type,
% 16.11/8.87 ring_17405671764205052669omplex: int > complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
% 16.11/8.87 ring_1_of_int_int: int > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
% 16.11/8.87 ring_1_of_int_rat: int > rat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
% 16.11/8.87 ring_1_of_int_real: int > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Nat__Onat,type,
% 16.11/8.87 semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Int__Oint,type,
% 16.11/8.87 lattic8263393255366662781ax_int: set_int > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
% 16.11/8.87 lattic8265883725875713057ax_nat: set_nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Real__Oreal,type,
% 16.11/8.87 bfun_nat_real: ( nat > real ) > filter_nat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Oappend_001t__Int__Oint,type,
% 16.11/8.87 append_int: list_int > list_int > list_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Oappend_001t__VEBT____Definitions__OVEBT,type,
% 16.11/8.87 append_VEBT_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_VEBT_VEBT ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Odistinct_001t__Int__Oint,type,
% 16.11/8.87 distinct_int: list_int > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Odrop_001t__VEBT____Definitions__OVEBT,type,
% 16.11/8.87 drop_VEBT_VEBT: nat > list_VEBT_VEBT > list_VEBT_VEBT ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
% 16.11/8.87 cons_int: int > list_int > list_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Olist_OCons_001t__VEBT____Definitions__OVEBT,type,
% 16.11/8.87 cons_VEBT_VEBT: vEBT_VEBT > list_VEBT_VEBT > list_VEBT_VEBT ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
% 16.11/8.87 nil_int: list_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Olist_ONil_001t__VEBT____Definitions__OVEBT,type,
% 16.11/8.87 nil_VEBT_VEBT: list_VEBT_VEBT ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Olist_Oset_001_Eo,type,
% 16.11/8.87 set_o2: list_o > set_o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
% 16.11/8.87 set_complex2: list_complex > set_complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
% 16.11/8.87 set_int2: list_int > set_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
% 16.11/8.87 set_nat2: list_nat > set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
% 16.11/8.87 set_real2: list_real > set_real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
% 16.11/8.87 set_set_nat2: list_set_nat > set_set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
% 16.11/8.87 set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
% 16.11/8.87 size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Olist__update_001_Eo,type,
% 16.11/8.87 list_update_o: list_o > nat > $o > list_o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Olist__update_001t__Complex__Ocomplex,type,
% 16.11/8.87 list_update_complex: list_complex > nat > complex > list_complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Olist__update_001t__Int__Oint,type,
% 16.11/8.87 list_update_int: list_int > nat > int > list_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
% 16.11/8.87 list_update_nat: list_nat > nat > nat > list_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Olist__update_001t__Real__Oreal,type,
% 16.11/8.87 list_update_real: list_real > nat > real > list_real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Olist__update_001t__Set__Oset_It__Nat__Onat_J,type,
% 16.11/8.87 list_update_set_nat: list_set_nat > nat > set_nat > list_set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Olist__update_001t__VEBT____Definitions__OVEBT,type,
% 16.11/8.87 list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Onth_001_Eo,type,
% 16.11/8.87 nth_o: list_o > nat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Onth_001t__Code____Numeral__Ointeger,type,
% 16.11/8.87 nth_Code_integer: list_Code_integer > nat > code_integer ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Onth_001t__Complex__Ocomplex,type,
% 16.11/8.87 nth_complex: list_complex > nat > complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Onth_001t__Int__Oint,type,
% 16.11/8.87 nth_int: list_int > nat > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Onth_001t__Nat__Onat,type,
% 16.11/8.87 nth_nat: list_nat > nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Onth_001t__Num__Onum,type,
% 16.11/8.87 nth_num: list_num > nat > num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
% 16.11/8.87 nth_Product_prod_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
% 16.11/8.87 nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
% 16.11/8.87 nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
% 16.11/8.87 nth_Pr8522763379788166057eger_o: list_P8526636022914148096eger_o > nat > produc6271795597528267376eger_o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 16.11/8.87 nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
% 16.11/8.87 nth_Pr8326237132889035090at_num: list_P1726324292696863441at_num > nat > product_prod_nat_num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 16.11/8.87 nth_Pr6456567536196504476um_num: list_P3744719386663036955um_num > nat > product_prod_num_num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
% 16.11/8.87 nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
% 16.11/8.87 nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 16.11/8.87 nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Onth_001t__Real__Oreal,type,
% 16.11/8.87 nth_real: list_real > nat > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
% 16.11/8.87 nth_set_nat: list_set_nat > nat > set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
% 16.11/8.87 nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Oproduct_001_Eo_001_Eo,type,
% 16.11/8.87 product_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
% 16.11/8.87 product_o_int: list_o > list_int > list_P3795440434834930179_o_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Oproduct_001_Eo_001t__VEBT____Definitions__OVEBT,type,
% 16.11/8.87 product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Oproduct_001t__Code____Numeral__Ointeger_001_Eo,type,
% 16.11/8.87 produc3607205314601156340eger_o: list_Code_integer > list_o > list_P8526636022914148096eger_o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Oproduct_001t__Int__Oint_001_Eo,type,
% 16.11/8.87 product_int_o: list_int > list_o > list_P5087981734274514673_int_o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Oproduct_001t__Int__Oint_001t__Int__Oint,type,
% 16.11/8.87 product_int_int: list_int > list_int > list_P5707943133018811711nt_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Oproduct_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
% 16.11/8.87 produc662631939642741121T_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Nat__Onat,type,
% 16.11/8.87 product_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Num__Onum,type,
% 16.11/8.87 product_nat_num: list_nat > list_num > list_P1726324292696863441at_num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Oproduct_001t__Num__Onum_001t__Num__Onum,type,
% 16.11/8.87 product_num_num: list_num > list_num > list_P3744719386663036955um_num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
% 16.11/8.87 product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
% 16.11/8.87 produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 16.11/8.87 produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Oreplicate_001_Eo,type,
% 16.11/8.87 replicate_o: nat > $o > list_o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Oreplicate_001t__Complex__Ocomplex,type,
% 16.11/8.87 replicate_complex: nat > complex > list_complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Oreplicate_001t__Int__Oint,type,
% 16.11/8.87 replicate_int: nat > int > list_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
% 16.11/8.87 replicate_nat: nat > nat > list_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Oreplicate_001t__Real__Oreal,type,
% 16.11/8.87 replicate_real: nat > real > list_real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
% 16.11/8.87 replicate_set_nat: nat > set_nat > list_set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
% 16.11/8.87 replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Otake_001t__VEBT____Definitions__OVEBT,type,
% 16.11/8.87 take_VEBT_VEBT: nat > list_VEBT_VEBT > list_VEBT_VEBT ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Oupto,type,
% 16.11/8.87 upto: int > int > list_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Oupto__aux,type,
% 16.11/8.87 upto_aux: int > int > list_int > list_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_List_Oupto__rel,type,
% 16.11/8.87 upto_rel: product_prod_int_int > product_prod_int_int > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_OSuc,type,
% 16.11/8.87 suc: nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 16.11/8.87 compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
% 16.11/8.87 case_nat_o: $o > ( nat > $o ) > nat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
% 16.11/8.87 case_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
% 16.11/8.87 case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Onat_Opred,type,
% 16.11/8.87 pred: nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
% 16.11/8.87 semiri4939895301339042750nteger: nat > code_integer ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
% 16.11/8.87 semiri8010041392384452111omplex: nat > complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 16.11/8.87 semiri1314217659103216013at_int: nat > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 16.11/8.87 semiri1316708129612266289at_nat: nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
% 16.11/8.87 semiri681578069525770553at_rat: nat > rat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
% 16.11/8.87 semiri5074537144036343181t_real: nat > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
% 16.11/8.87 size_size_list_o: list_o > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
% 16.11/8.87 size_s3445333598471063425nteger: list_Code_integer > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
% 16.11/8.87 size_s3451745648224563538omplex: list_complex > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
% 16.11/8.87 size_size_list_int: list_int > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
% 16.11/8.87 size_size_list_nat: list_nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
% 16.11/8.87 size_size_list_num: list_num > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
% 16.11/8.87 size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
% 16.11/8.87 size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 16.11/8.87 size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_M_Eo_J_J,type,
% 16.11/8.87 size_s4246224855604898693_int_o: list_P5087981734274514673_int_o > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 16.11/8.87 size_s5157815400016825771nt_int: list_P5707943133018811711nt_int > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 16.11/8.87 size_s6639371672096860321T_VEBT: list_P7524865323317820941T_VEBT > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
% 16.11/8.87 size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
% 16.11/8.87 size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 16.11/8.87 size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
% 16.11/8.87 size_size_list_real: list_real > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
% 16.11/8.87 size_s3254054031482475050et_nat: list_set_nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 16.11/8.87 size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
% 16.11/8.87 size_size_num: num > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Nat__Onat_J,type,
% 16.11/8.87 size_size_option_nat: option_nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
% 16.11/8.87 size_size_option_num: option_num > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 16.11/8.87 size_s170228958280169651at_nat: option4927543243414619207at_nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
% 16.11/8.87 size_size_VEBT_VEBT: vEBT_VEBT > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
% 16.11/8.87 nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
% 16.11/8.87 nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat__Bijection_Oprod__encode,type,
% 16.11/8.87 nat_prod_encode: product_prod_nat_nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat__Bijection_Oset__decode,type,
% 16.11/8.87 nat_set_decode: nat > set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat__Bijection_Oset__encode,type,
% 16.11/8.87 nat_set_encode: set_nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Nat__Bijection_Otriangle,type,
% 16.11/8.87 nat_triangle: nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_NthRoot_Oroot,type,
% 16.11/8.87 root: nat > real > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_NthRoot_Osqrt,type,
% 16.11/8.87 sqrt: real > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_OBitM,type,
% 16.11/8.87 bitM: num > num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Oinc,type,
% 16.11/8.87 inc: num > num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
% 16.11/8.87 neg_nu8804712462038260780nteger: code_integer > code_integer ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
% 16.11/8.87 neg_nu7009210354673126013omplex: complex > complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
% 16.11/8.87 neg_numeral_dbl_int: int > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
% 16.11/8.87 neg_numeral_dbl_rat: rat > rat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
% 16.11/8.87 neg_numeral_dbl_real: real > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
% 16.11/8.87 neg_nu7757733837767384882nteger: code_integer > code_integer ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
% 16.11/8.87 neg_nu6511756317524482435omplex: complex > complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
% 16.11/8.87 neg_nu3811975205180677377ec_int: int > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
% 16.11/8.87 neg_nu3179335615603231917ec_rat: rat > rat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
% 16.11/8.87 neg_nu6075765906172075777c_real: real > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Code____Numeral__Ointeger,type,
% 16.11/8.87 neg_nu5831290666863070958nteger: code_integer > code_integer ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
% 16.11/8.87 neg_nu8557863876264182079omplex: complex > complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
% 16.11/8.87 neg_nu5851722552734809277nc_int: int > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
% 16.11/8.87 neg_nu5219082963157363817nc_rat: rat > rat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
% 16.11/8.87 neg_nu8295874005876285629c_real: real > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
% 16.11/8.87 neg_numeral_sub_int: num > num > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Onum_OBit0,type,
% 16.11/8.87 bit0: num > num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Onum_OBit1,type,
% 16.11/8.87 bit1: num > num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Onum_OOne,type,
% 16.11/8.87 one: num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
% 16.11/8.87 case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Onum_Osize__num,type,
% 16.11/8.87 size_num: num > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Onum__of__nat,type,
% 16.11/8.87 num_of_nat: nat > num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
% 16.11/8.87 numera6620942414471956472nteger: num > code_integer ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
% 16.11/8.87 numera6690914467698888265omplex: num > complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
% 16.11/8.87 numera1916890842035813515d_enat: num > extended_enat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
% 16.11/8.87 numeral_numeral_int: num > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
% 16.11/8.87 numeral_numeral_nat: num > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
% 16.11/8.87 numeral_numeral_rat: num > rat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
% 16.11/8.87 numeral_numeral_real: num > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Opow,type,
% 16.11/8.87 pow: num > num > num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Num_Opred__numeral,type,
% 16.11/8.87 pred_numeral: num > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
% 16.11/8.87 none_nat: option_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
% 16.11/8.87 none_num: option_num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 16.11/8.87 none_P5556105721700978146at_nat: option4927543243414619207at_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
% 16.11/8.87 some_nat: nat > option_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
% 16.11/8.87 some_num: num > option_num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 16.11/8.87 some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 16.11/8.87 case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Option_Ooption_Ocase__option_001t__Int__Oint_001t__Num__Onum,type,
% 16.11/8.87 case_option_int_num: int > ( num > int ) > option_num > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
% 16.11/8.87 case_option_num_num: num > ( num > num ) > option_num > num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
% 16.11/8.87 case_o6005452278849405969um_num: option_num > ( num > option_num ) > option_num > option_num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Option_Ooption_Omap__option_001t__Num__Onum_001t__Num__Onum,type,
% 16.11/8.87 map_option_num_num: ( num > num ) > option_num > option_num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Option_Ooption_Osize__option_001t__Nat__Onat,type,
% 16.11/8.87 size_option_nat: ( nat > nat ) > option_nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Option_Ooption_Osize__option_001t__Num__Onum,type,
% 16.11/8.87 size_option_num: ( num > nat ) > option_num > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Option_Ooption_Osize__option_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 16.11/8.87 size_o8335143837870341156at_nat: ( product_prod_nat_nat > nat ) > option4927543243414619207at_nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Option_Ooption_Othe_001t__Nat__Onat,type,
% 16.11/8.87 the_nat: option_nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Option_Ooption_Othe_001t__Num__Onum,type,
% 16.11/8.87 the_num: option_num > num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Option_Ooption_Othe_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 16.11/8.87 the_Pr8591224930841456533at_nat: option4927543243414619207at_nat > product_prod_nat_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
% 16.11/8.87 bot_bot_nat: nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 16.11/8.87 bot_bot_set_complex: set_complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
% 16.11/8.87 bot_bot_set_int: set_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
% 16.11/8.87 bot_bot_set_nat: set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
% 16.11/8.87 bot_bot_set_num: set_num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 16.11/8.87 bot_bo1796632182523588997nt_int: set_Pr958786334691620121nt_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
% 16.11/8.87 bot_bot_set_rat: set_rat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
% 16.11/8.87 bot_bot_set_real: set_real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 16.11/8.87 bot_bot_set_set_nat: set_set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Complex__Ocomplex_M_Eo_J,type,
% 16.11/8.87 ord_less_complex_o: ( complex > $o ) > ( complex > $o ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Int__Oint_M_Eo_J,type,
% 16.11/8.87 ord_less_int_o: ( int > $o ) > ( int > $o ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
% 16.11/8.87 ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Real__Oreal_M_Eo_J,type,
% 16.11/8.87 ord_less_real_o: ( real > $o ) > ( real > $o ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
% 16.11/8.87 ord_less_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
% 16.11/8.87 ord_le6747313008572928689nteger: code_integer > code_integer > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
% 16.11/8.87 ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 16.11/8.87 ord_less_int: int > int > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 16.11/8.87 ord_less_nat: nat > nat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
% 16.11/8.87 ord_less_num: num > num > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
% 16.11/8.87 ord_less_rat: rat > rat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
% 16.11/8.87 ord_less_real: real > real > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
% 16.11/8.87 ord_le1307284697595431911nteger: set_Code_integer > set_Code_integer > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 16.11/8.87 ord_less_set_complex: set_complex > set_complex > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
% 16.11/8.87 ord_less_set_int: set_int > set_int > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
% 16.11/8.87 ord_less_set_nat: set_nat > set_nat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
% 16.11/8.87 ord_less_set_num: set_num > set_num > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Rat__Orat_J,type,
% 16.11/8.87 ord_less_set_rat: set_rat > set_rat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
% 16.11/8.87 ord_less_set_real: set_real > set_real > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 16.11/8.87 ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Complex__Ocomplex_M_Eo_J,type,
% 16.11/8.87 ord_le4573692005234683329plex_o: ( complex > $o ) > ( complex > $o ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Int__Oint_M_Eo_J,type,
% 16.11/8.87 ord_less_eq_int_o: ( int > $o ) > ( int > $o ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
% 16.11/8.87 ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Real__Oreal_M_Eo_J,type,
% 16.11/8.87 ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
% 16.11/8.87 ord_le3964352015994296041_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
% 16.11/8.87 ord_le3102999989581377725nteger: code_integer > code_integer > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
% 16.11/8.87 ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Filter__Ofilter_It__Nat__Onat_J,type,
% 16.11/8.87 ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 16.11/8.87 ord_less_eq_int: int > int > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 16.11/8.87 ord_less_eq_nat: nat > nat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
% 16.11/8.87 ord_less_eq_num: num > num > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
% 16.11/8.87 ord_less_eq_rat: rat > rat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
% 16.11/8.87 ord_less_eq_real: real > real > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
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% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
% 16.11/8.87 ord_le7084787975880047091nteger: set_Code_integer > set_Code_integer > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 16.11/8.87 ord_le211207098394363844omplex: set_complex > set_complex > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
% 16.11/8.87 ord_less_eq_set_int: set_int > set_int > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
% 16.11/8.87 ord_less_eq_set_nat: set_nat > set_nat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
% 16.11/8.87 ord_less_eq_set_num: set_num > set_num > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 16.11/8.87 ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Rat__Orat_J,type,
% 16.11/8.87 ord_less_eq_set_rat: set_rat > set_rat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Real__Oreal_J,type,
% 16.11/8.87 ord_less_eq_set_real: set_real > set_real > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 16.11/8.87 ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
% 16.11/8.87 ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o ).
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% 16.11/8.87 ord_max_Code_integer: code_integer > code_integer > code_integer ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
% 16.11/8.87 ord_max_int: int > int > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
% 16.11/8.87 ord_max_nat: nat > nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
% 16.11/8.87 ord_max_num: num > num > num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Omax_001t__Rat__Orat,type,
% 16.11/8.87 ord_max_rat: rat > rat > rat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Omax_001t__Real__Oreal,type,
% 16.11/8.87 ord_max_real: real > real > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Nat__Onat_J,type,
% 16.11/8.87 ord_max_set_nat: set_nat > set_nat > set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
% 16.11/8.87 order_Greatest_nat: ( nat > $o ) > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Real__Oreal,type,
% 16.11/8.87 order_9091379641038594480t_real: ( nat > real ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
% 16.11/8.87 order_mono_nat_nat: ( nat > nat ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Real__Oreal,type,
% 16.11/8.87 order_mono_nat_real: ( nat > real ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oorder__class_Omono_001t__Real__Oreal_001t__Real__Oreal,type,
% 16.11/8.87 order_mono_real_real: ( real > real ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
% 16.11/8.87 order_5726023648592871131at_nat: ( nat > nat ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Real__Oreal,type,
% 16.11/8.87 order_7092887310737990675l_real: ( real > real ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
% 16.11/8.87 top_top_set_o: set_o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
% 16.11/8.87 top_top_set_nat: set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
% 16.11/8.87 top_to1996260823553986621t_unit: set_Product_unit ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
% 16.11/8.87 top_top_set_real: set_real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
% 16.11/8.87 top_top_set_char: set_char ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
% 16.11/8.87 power_8256067586552552935nteger: code_integer > nat > code_integer ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
% 16.11/8.87 power_power_complex: complex > nat > complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
% 16.11/8.87 power_power_int: int > nat > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
% 16.11/8.87 power_power_nat: nat > nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
% 16.11/8.87 power_power_rat: rat > nat > rat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
% 16.11/8.87 power_power_real: real > nat > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
% 16.11/8.87 produc4035269172776083154on_nat: ( nat > nat > $o ) > produc4953844613479565601on_nat > produc2233624965454879586on_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 16.11/8.87 produc3209952032786966637at_nat: ( nat > nat > nat ) > produc7248412053542808358at_nat > produc4471711990508489141at_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
% 16.11/8.87 produc8929957630744042906on_nat: ( nat > nat > nat ) > produc4953844613479565601on_nat > produc8306885398267862888on_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Num__Onum_Mt__Num__Onum_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
% 16.11/8.87 produc851828971589881931at_num: ( nat > num > num ) > produc2963631642982155120at_num > produc3368934014287244435at_num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_OPair_001_062_It__Num__Onum_M_062_It__Num__Onum_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
% 16.11/8.87 produc3576312749637752826on_num: ( num > num > $o ) > produc3447558737645232053on_num > produc7036089656553540234on_num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_OPair_001_062_It__Num__Onum_M_062_It__Num__Onum_Mt__Num__Onum_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
% 16.11/8.87 produc5778274026573060048on_num: ( num > num > num ) > produc3447558737645232053on_num > produc1193250871479095198on_num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 16.11/8.87 produc3994169339658061776at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > produc6121120109295599847at_nat > produc5491161045314408544at_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 16.11/8.87 produc2899441246263362727at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > produc6121120109295599847at_nat > produc5542196010084753463at_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_OPair_001_Eo_001_Eo,type,
% 16.11/8.87 product_Pair_o_o: $o > $o > product_prod_o_o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_OPair_001_Eo_001t__Int__Oint,type,
% 16.11/8.87 product_Pair_o_int: $o > int > product_prod_o_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
% 16.11/8.87 produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
% 16.11/8.87 produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
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% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
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% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
% 16.11/8.87 product_Pair_nat_num: nat > num > product_prod_nat_num ).
% 16.11/8.87
% 16.11/8.87 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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% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
% 16.11/8.87 produc1195630363706982562at_num: nat > product_prod_nat_num > produc2963631642982155120at_num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
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% 16.11/8.87
% 16.11/8.87 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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% 16.11/8.87
% 16.11/8.87 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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% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 16.11/8.87 produc488173922507101015at_nat: option4927543243414619207at_nat > option4927543243414619207at_nat > produc6121120109295599847at_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
% 16.11/8.87 produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
% 16.11/8.87 produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 16.11/8.87 produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 16.11/8.87 produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 16.11/8.87 produc6499014454317279255nteger: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001_Eo_001_Eo,type,
% 16.11/8.87 produc7828578312038201481er_o_o: ( code_integer > $o > $o ) > produc6271795597528267376eger_o > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001_Eo_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 16.11/8.87 produc1043322548047392435omplex: ( code_integer > $o > set_complex ) > produc6271795597528267376eger_o > set_complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001_Eo_001t__Set__Oset_It__Int__Oint_J,type,
% 16.11/8.87 produc1253318751659547953et_int: ( code_integer > $o > set_int ) > produc6271795597528267376eger_o > set_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
% 16.11/8.87 produc5431169771168744661et_nat: ( code_integer > $o > set_nat ) > produc6271795597528267376eger_o > set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001_Eo_001t__Set__Oset_It__Real__Oreal_J,type,
% 16.11/8.87 produc242741666403216561t_real: ( code_integer > $o > set_real ) > produc6271795597528267376eger_o > set_real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001_Eo_001t__String__Ochar,type,
% 16.11/8.87 produc4188289175737317920o_char: ( code_integer > $o > char ) > produc6271795597528267376eger_o > char ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
% 16.11/8.87 produc1553301316500091796er_int: ( code_integer > code_integer > int ) > produc8923325533196201883nteger > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
% 16.11/8.87 produc1555791787009142072er_nat: ( code_integer > code_integer > nat ) > produc8923325533196201883nteger > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
% 16.11/8.87 produc7336495610019696514er_num: ( code_integer > code_integer > num ) > produc8923325533196201883nteger > num ).
% 16.11/8.87
% 16.11/8.87 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,
% 16.11/8.87 produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).
% 16.11/8.87
% 16.11/8.87 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,
% 16.11/8.87 produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001_Eo,type,
% 16.11/8.87 produc6771430404735790350plex_o: ( complex > complex > $o ) > produc4411394909380815293omplex > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
% 16.11/8.87 produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
% 16.11/8.87 produc8211389475949308722nt_int: ( int > int > int ) > product_prod_int_int > int ).
% 16.11/8.87
% 16.11/8.87 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,
% 16.11/8.87 produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).
% 16.11/8.87
% 16.11/8.87 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,
% 16.11/8.87 produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 16.11/8.87 produc27273713700761075at_nat: ( nat > nat > product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
% 16.11/8.87 produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Complex__Ocomplex,type,
% 16.11/8.87 produc1917071388513777916omplex: ( nat > nat > complex ) > product_prod_nat_nat > complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint,type,
% 16.11/8.87 produc6840382203811409530at_int: ( nat > nat > int ) > product_prod_nat_nat > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 16.11/8.87 produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).
% 16.11/8.87
% 16.11/8.87 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,
% 16.11/8.87 produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Rat__Orat,type,
% 16.11/8.87 produc6207742614233964070at_rat: ( nat > nat > rat ) > product_prod_nat_nat > rat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal,type,
% 16.11/8.87 produc1703576794950452218t_real: ( nat > nat > real ) > product_prod_nat_nat > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001_Eo,type,
% 16.11/8.87 produc4927758841916487424_num_o: ( nat > num > $o ) > product_prod_nat_num > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
% 16.11/8.87 produc478579273971653890on_num: ( nat > num > option_num ) > product_prod_nat_num > option_num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Set__Oset_It__Nat__Onat_J,type,
% 16.11/8.87 produc4130284055270567454et_nat: ( nat > num > set_nat ) > product_prod_nat_num > set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Set__Oset_It__Real__Oreal_J,type,
% 16.11/8.87 produc1435849484188172666t_real: ( nat > num > set_real ) > product_prod_nat_num > set_real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001_Eo,type,
% 16.11/8.87 produc5703948589228662326_num_o: ( num > num > $o ) > product_prod_num_num > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 16.11/8.87 produc2866383454006189126omplex: ( num > num > set_complex ) > product_prod_num_num > set_complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Int__Oint_J,type,
% 16.11/8.87 produc6406642877701697732et_int: ( num > num > set_int ) > product_prod_num_num > set_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Nat__Onat_J,type,
% 16.11/8.87 produc1361121860356118632et_nat: ( num > num > set_nat ) > product_prod_num_num > set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Real__Oreal_J,type,
% 16.11/8.87 produc8296048397933160132t_real: ( num > num > set_real ) > product_prod_num_num > set_real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Real__Oreal_001t__Real__Oreal_001_Eo,type,
% 16.11/8.87 produc5414030515140494994real_o: ( real > real > $o ) > produc2422161461964618553l_real > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ofst_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 16.11/8.87 produc8508995932063986495nteger: produc8923325533196201883nteger > code_integer ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
% 16.11/8.87 product_fst_int_int: product_prod_int_int > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
% 16.11/8.87 product_fst_nat_nat: product_prod_nat_nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
% 16.11/8.87 product_snd_int_int: product_prod_int_int > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
% 16.11/8.87 product_snd_nat_nat: product_prod_nat_nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Rat_OFrct,type,
% 16.11/8.87 frct: product_prod_int_int > rat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Rat_Onormalize,type,
% 16.11/8.87 normalize: product_prod_int_int > product_prod_int_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Rat_Oof__int,type,
% 16.11/8.87 of_int: int > rat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Rat_Oquotient__of,type,
% 16.11/8.87 quotient_of: rat > product_prod_int_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
% 16.11/8.87 real_V2521375963428798218omplex: set_complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Real__Vector__Spaces_Obounded__linear_001t__Real__Oreal_001t__Real__Oreal,type,
% 16.11/8.87 real_V5970128139526366754l_real: ( real > real ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Complex__Ocomplex,type,
% 16.11/8.87 real_V3694042436643373181omplex: complex > complex > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
% 16.11/8.87 real_V975177566351809787t_real: real > real > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
% 16.11/8.87 real_V1022390504157884413omplex: complex > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
% 16.11/8.87 real_V7735802525324610683m_real: real > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
% 16.11/8.87 real_V4546457046886955230omplex: real > complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
% 16.11/8.87 real_V2046097035970521341omplex: real > complex > complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
% 16.11/8.87 real_V1485227260804924795R_real: real > real > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
% 16.11/8.87 divide6298287555418463151nteger: code_integer > code_integer > code_integer ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
% 16.11/8.87 divide1717551699836669952omplex: complex > complex > complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
% 16.11/8.87 divide_divide_int: int > int > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
% 16.11/8.87 divide_divide_nat: nat > nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
% 16.11/8.87 divide_divide_rat: rat > rat > rat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
% 16.11/8.87 divide_divide_real: real > real > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
% 16.11/8.87 dvd_dvd_Code_integer: code_integer > code_integer > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
% 16.11/8.87 dvd_dvd_complex: complex > complex > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
% 16.11/8.87 dvd_dvd_int: int > int > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
% 16.11/8.87 dvd_dvd_nat: nat > nat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
% 16.11/8.87 dvd_dvd_rat: rat > rat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
% 16.11/8.87 dvd_dvd_real: real > real > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
% 16.11/8.87 modulo364778990260209775nteger: code_integer > code_integer > code_integer ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
% 16.11/8.87 modulo_modulo_int: int > int > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
% 16.11/8.87 modulo_modulo_nat: nat > nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
% 16.11/8.87 zero_n356916108424825756nteger: $o > code_integer ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
% 16.11/8.87 zero_n1201886186963655149omplex: $o > complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
% 16.11/8.87 zero_n2684676970156552555ol_int: $o > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
% 16.11/8.87 zero_n2687167440665602831ol_nat: $o > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
% 16.11/8.87 zero_n2052037380579107095ol_rat: $o > rat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
% 16.11/8.87 zero_n3304061248610475627l_real: $o > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
% 16.11/8.87 suminf_complex: ( nat > complex ) > complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Series_Osuminf_001t__Int__Oint,type,
% 16.11/8.87 suminf_int: ( nat > int ) > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Series_Osuminf_001t__Nat__Onat,type,
% 16.11/8.87 suminf_nat: ( nat > nat ) > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
% 16.11/8.87 suminf_real: ( nat > real ) > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
% 16.11/8.87 summable_complex: ( nat > complex ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Series_Osummable_001t__Int__Oint,type,
% 16.11/8.87 summable_int: ( nat > int ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Series_Osummable_001t__Nat__Onat,type,
% 16.11/8.87 summable_nat: ( nat > nat ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
% 16.11/8.87 summable_real: ( nat > real ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
% 16.11/8.87 sums_complex: ( nat > complex ) > complex > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Series_Osums_001t__Int__Oint,type,
% 16.11/8.87 sums_int: ( nat > int ) > int > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Series_Osums_001t__Nat__Onat,type,
% 16.11/8.87 sums_nat: ( nat > nat ) > nat > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Series_Osums_001t__Real__Oreal,type,
% 16.11/8.87 sums_real: ( nat > real ) > real > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_OCollect_001t__Code____Numeral__Ointeger,type,
% 16.11/8.87 collect_Code_integer: ( code_integer > $o ) > set_Code_integer ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
% 16.11/8.87 collect_complex: ( complex > $o ) > set_complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_OCollect_001t__Int__Oint,type,
% 16.11/8.87 collect_int: ( int > $o ) > set_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_OCollect_001t__List__Olist_I_Eo_J,type,
% 16.11/8.87 collect_list_o: ( list_o > $o ) > set_list_o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_OCollect_001t__List__Olist_It__Complex__Ocomplex_J,type,
% 16.11/8.87 collect_list_complex: ( list_complex > $o ) > set_list_complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_OCollect_001t__List__Olist_It__Int__Oint_J,type,
% 16.11/8.87 collect_list_int: ( list_int > $o ) > set_list_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
% 16.11/8.87 collect_list_nat: ( list_nat > $o ) > set_list_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_OCollect_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 16.11/8.87 collec5608196760682091941T_VEBT: ( list_VEBT_VEBT > $o ) > set_list_VEBT_VEBT ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 16.11/8.87 collect_nat: ( nat > $o ) > set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_OCollect_001t__Num__Onum,type,
% 16.11/8.87 collect_num: ( num > $o ) > set_num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 16.11/8.87 collec8663557070575231912omplex: ( produc4411394909380815293omplex > $o ) > set_Pr5085853215250843933omplex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 16.11/8.87 collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 16.11/8.87 collec3799799289383736868l_real: ( produc2422161461964618553l_real > $o ) > set_Pr6218003697084177305l_real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_OCollect_001t__Rat__Orat,type,
% 16.11/8.87 collect_rat: ( rat > $o ) > set_rat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
% 16.11/8.87 collect_real: ( real > $o ) > set_real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 16.11/8.87 collect_set_complex: ( set_complex > $o ) > set_set_complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
% 16.11/8.87 collect_set_int: ( set_int > $o ) > set_set_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
% 16.11/8.87 collect_set_nat: ( set_nat > $o ) > set_set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_OPow_001t__Nat__Onat,type,
% 16.11/8.87 pow_nat: set_nat > set_set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
% 16.11/8.87 image_int_int: ( int > int ) > set_int > set_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
% 16.11/8.87 image_int_nat: ( int > nat ) > set_int > set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
% 16.11/8.87 image_nat_int: ( nat > int ) > set_nat > set_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
% 16.11/8.87 image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
% 16.11/8.87 image_nat_real: ( nat > real ) > set_nat > set_real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
% 16.11/8.87 image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
% 16.11/8.87 image_nat_char: ( nat > char ) > set_nat > set_char ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 16.11/8.87 image_5971271580939081552omplex: ( real > filter6041513312241820739omplex ) > set_real > set_fi4554929511873752355omplex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 16.11/8.87 image_2178119161166701260l_real: ( real > filter2146258269922977983l_real ) > set_real > set_fi7789364187291644575l_real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
% 16.11/8.87 image_real_real: ( real > real ) > set_real > set_real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
% 16.11/8.87 image_char_nat: ( char > nat ) > set_char > set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
% 16.11/8.87 insert_int: int > set_int > set_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
% 16.11/8.87 insert_nat: nat > set_nat > set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
% 16.11/8.87 insert_real: real > set_real > set_real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Complex__Ocomplex,type,
% 16.11/8.87 set_fo1517530859248394432omplex: ( nat > complex > complex ) > nat > nat > complex > complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Int__Oint,type,
% 16.11/8.87 set_fo2581907887559384638at_int: ( nat > int > int ) > nat > nat > int > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
% 16.11/8.87 set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Rat__Orat,type,
% 16.11/8.87 set_fo1949268297981939178at_rat: ( nat > rat > rat ) > nat > nat > rat > rat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Real__Oreal,type,
% 16.11/8.87 set_fo3111899725591712190t_real: ( nat > real > real ) > nat > nat > real > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
% 16.11/8.87 set_or1266510415728281911st_int: int > int > set_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
% 16.11/8.87 set_or1269000886237332187st_nat: nat > nat > set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Num__Onum,type,
% 16.11/8.87 set_or7049704709247886629st_num: num > num > set_num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Rat__Orat,type,
% 16.11/8.87 set_or633870826150836451st_rat: rat > rat > set_rat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
% 16.11/8.87 set_or1222579329274155063t_real: real > real > set_real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
% 16.11/8.87 set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
% 16.11/8.87 set_or4662586982721622107an_int: int > int > set_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
% 16.11/8.87 set_or4665077453230672383an_nat: nat > nat > set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
% 16.11/8.87 set_ord_atLeast_nat: nat > set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
% 16.11/8.87 set_ord_atLeast_real: real > set_real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
% 16.11/8.87 set_ord_atMost_nat: nat > set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
% 16.11/8.87 set_or6656581121297822940st_int: int > int > set_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
% 16.11/8.87 set_or6659071591806873216st_nat: nat > nat > set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
% 16.11/8.87 set_or5832277885323065728an_int: int > int > set_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
% 16.11/8.87 set_or5834768355832116004an_nat: nat > nat > set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
% 16.11/8.87 set_or1633881224788618240n_real: real > real > set_real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
% 16.11/8.87 set_or1210151606488870762an_nat: nat > set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
% 16.11/8.87 set_or5849166863359141190n_real: real > set_real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
% 16.11/8.87 set_ord_lessThan_int: int > set_int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
% 16.11/8.87 set_ord_lessThan_nat: nat > set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
% 16.11/8.87 set_ord_lessThan_num: num > set_num ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Rat__Orat,type,
% 16.11/8.87 set_ord_lessThan_rat: rat > set_rat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
% 16.11/8.87 set_or5984915006950818249n_real: real > set_real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
% 16.11/8.87 set_or890127255671739683et_nat: set_nat > set_set_nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_String_Oascii__of,type,
% 16.11/8.87 ascii_of: char > char ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_String_Ochar_OChar,type,
% 16.11/8.87 char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_String_Ochar__of__integer,type,
% 16.11/8.87 char_of_integer: code_integer > char ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
% 16.11/8.87 comm_s629917340098488124ar_nat: char > nat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_String_Ointeger__of__char,type,
% 16.11/8.87 integer_of_char: char > code_integer ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
% 16.11/8.87 unique3096191561947761185of_nat: nat > char ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
% 16.11/8.87 topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 16.11/8.87 topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Topological__Spaces_Omonoseq_001t__Code____Numeral__Ointeger,type,
% 16.11/8.87 topolo2919662092509805066nteger: ( nat > code_integer ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Topological__Spaces_Omonoseq_001t__Int__Oint,type,
% 16.11/8.87 topolo4899668324122417113eq_int: ( nat > int ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
% 16.11/8.87 topolo4902158794631467389eq_nat: ( nat > nat ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Topological__Spaces_Omonoseq_001t__Num__Onum,type,
% 16.11/8.87 topolo1459490580787246023eq_num: ( nat > num ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Topological__Spaces_Omonoseq_001t__Rat__Orat,type,
% 16.11/8.87 topolo4267028734544971653eq_rat: ( nat > rat ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
% 16.11/8.87 topolo6980174941875973593q_real: ( nat > real ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Nat__Onat_J,type,
% 16.11/8.87 topolo7278393974255667507et_nat: ( nat > set_nat ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Complex__Ocomplex,type,
% 16.11/8.87 topolo4110288021797289639omplex: set_complex > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Topological__Spaces_Oopen__class_Oopen_001t__Real__Oreal,type,
% 16.11/8.87 topolo4860482606490270245n_real: set_real > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
% 16.11/8.87 topolo2177554685111907308n_real: real > set_real > filter_real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
% 16.11/8.87 topolo2815343760600316023s_real: real > filter_real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Complex__Ocomplex,type,
% 16.11/8.87 topolo6517432010174082258omplex: ( nat > complex ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
% 16.11/8.87 topolo4055970368930404560y_real: ( nat > real ) > $o ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
% 16.11/8.87 topolo896644834953643431omplex: filter6041513312241820739omplex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
% 16.11/8.87 topolo1511823702728130853y_real: filter2146258269922977983l_real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Transcendental_Oarccos,type,
% 16.11/8.87 arccos: real > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
% 16.11/8.87 arcosh_real: real > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Transcendental_Oarcsin,type,
% 16.11/8.87 arcsin: real > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Transcendental_Oarctan,type,
% 16.11/8.87 arctan: real > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
% 16.11/8.87 arsinh_real: real > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
% 16.11/8.87 artanh_real: real > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
% 16.11/8.87 cos_complex: complex > complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
% 16.11/8.87 cos_real: real > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Transcendental_Ocos__coeff,type,
% 16.11/8.87 cos_coeff: nat > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
% 16.11/8.87 cosh_real: real > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
% 16.11/8.87 cot_real: real > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Transcendental_Odiffs_001t__Code____Numeral__Ointeger,type,
% 16.11/8.87 diffs_Code_integer: ( nat > code_integer ) > nat > code_integer ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Transcendental_Odiffs_001t__Complex__Ocomplex,type,
% 16.11/8.87 diffs_complex: ( nat > complex ) > nat > complex ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Transcendental_Odiffs_001t__Int__Oint,type,
% 16.11/8.87 diffs_int: ( nat > int ) > nat > int ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Transcendental_Odiffs_001t__Rat__Orat,type,
% 16.11/8.87 diffs_rat: ( nat > rat ) > nat > rat ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Transcendental_Odiffs_001t__Real__Oreal,type,
% 16.11/8.87 diffs_real: ( nat > real ) > nat > real ).
% 16.11/8.87
% 16.11/8.87 thf(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
% 16.11/8.88 exp_complex: complex > complex ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
% 16.11/8.88 exp_real: real > real ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
% 16.11/8.88 ln_ln_real: real > real ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_Transcendental_Olog,type,
% 16.11/8.88 log: real > real > real ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_Transcendental_Opi,type,
% 16.11/8.88 pi: real ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
% 16.11/8.88 powr_real: real > real > real ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
% 16.11/8.88 sin_complex: complex > complex ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
% 16.11/8.88 sin_real: real > real ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_Transcendental_Osin__coeff,type,
% 16.11/8.88 sin_coeff: nat > real ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
% 16.11/8.88 sinh_real: real > real ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
% 16.11/8.88 tan_complex: complex > complex ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
% 16.11/8.88 tan_real: real > real ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
% 16.11/8.88 tanh_complex: complex > complex ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
% 16.11/8.88 tanh_real: real > real ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
% 16.11/8.88 vEBT_Leaf: $o > $o > vEBT_VEBT ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
% 16.11/8.88 vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
% 16.11/8.88 vEBT_size_VEBT: vEBT_VEBT > nat ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
% 16.11/8.88 vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
% 16.11/8.88 vEBT_VEBT_high: nat > nat > nat ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
% 16.11/8.88 vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
% 16.11/8.88 vEBT_VEBT_low: nat > nat > nat ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
% 16.11/8.88 vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
% 16.11/8.88 vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
% 16.11/8.88 vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
% 16.11/8.88 vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
% 16.11/8.88 vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
% 16.11/8.88 vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
% 16.11/8.88 vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Definitions_Oset__vebt,type,
% 16.11/8.88 vEBT_set_vebt: vEBT_VEBT > set_nat ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
% 16.11/8.88 vEBT_vebt_buildup: nat > vEBT_VEBT ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
% 16.11/8.88 vEBT_v4011308405150292612up_rel: nat > nat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Delete_Ovebt__delete,type,
% 16.11/8.88 vEBT_vebt_delete: vEBT_VEBT > nat > vEBT_VEBT ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
% 16.11/8.88 vEBT_vebt_delete_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Insert_Ovebt__insert,type,
% 16.11/8.88 vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
% 16.11/8.88 vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
% 16.11/8.88 vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
% 16.11/8.88 vEBT_VEBT_minNull: vEBT_VEBT > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
% 16.11/8.88 vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
% 16.11/8.88 vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Member_Ovebt__member,type,
% 16.11/8.88 vEBT_vebt_member: vEBT_VEBT > nat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
% 16.11/8.88 vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
% 16.11/8.88 vEBT_VEBT_add: option_nat > option_nat > option_nat ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
% 16.11/8.88 vEBT_VEBT_greater: option_nat > option_nat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
% 16.11/8.88 vEBT_VEBT_less: option_nat > option_nat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
% 16.11/8.88 vEBT_VEBT_lesseq: option_nat > option_nat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
% 16.11/8.88 vEBT_VEBT_max_in_set: set_nat > nat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
% 16.11/8.88 vEBT_VEBT_min_in_set: set_nat > nat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
% 16.11/8.88 vEBT_VEBT_mul: option_nat > option_nat > option_nat ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Nat__Onat,type,
% 16.11/8.88 vEBT_V4262088993061758097ft_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Num__Onum,type,
% 16.11/8.88 vEBT_V819420779217536731ft_num: ( num > num > num ) > option_num > option_num > option_num ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 16.11/8.88 vEBT_V1502963449132264192at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > option4927543243414619207at_nat > option4927543243414619207at_nat > option4927543243414619207at_nat ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
% 16.11/8.88 vEBT_VEBT_power: option_nat > option_nat > option_nat ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__MinMax_Ovebt__maxt,type,
% 16.11/8.88 vEBT_vebt_maxt: vEBT_VEBT > option_nat ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
% 16.11/8.88 vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__MinMax_Ovebt__mint,type,
% 16.11/8.88 vEBT_vebt_mint: vEBT_VEBT > option_nat ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
% 16.11/8.88 vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Pred_Ois__pred__in__set,type,
% 16.11/8.88 vEBT_is_pred_in_set: set_nat > nat > nat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Pred_Ovebt__pred,type,
% 16.11/8.88 vEBT_vebt_pred: vEBT_VEBT > nat > option_nat ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
% 16.11/8.88 vEBT_vebt_pred_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Succ_Ois__succ__in__set,type,
% 16.11/8.88 vEBT_is_succ_in_set: set_nat > nat > nat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Succ_Ovebt__succ,type,
% 16.11/8.88 vEBT_vebt_succ: vEBT_VEBT > nat > option_nat ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
% 16.11/8.88 vEBT_vebt_succ_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
% 16.11/8.88 accp_nat: ( nat > nat > $o ) > nat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 16.11/8.88 accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 16.11/8.88 accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 16.11/8.88 accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
% 16.11/8.88 accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_fChoice_001t__Real__Oreal,type,
% 16.11/8.88 fChoice_real: ( real > $o ) > real ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_member_001_Eo,type,
% 16.11/8.88 member_o: $o > set_o > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_member_001t__Complex__Ocomplex,type,
% 16.11/8.88 member_complex: complex > set_complex > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_member_001t__Int__Oint,type,
% 16.11/8.88 member_int: int > set_int > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_member_001t__List__Olist_I_Eo_J,type,
% 16.11/8.88 member_list_o: list_o > set_list_o > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
% 16.11/8.88 member_list_int: list_int > set_list_int > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_member_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 16.11/8.88 member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_member_001t__Nat__Onat,type,
% 16.11/8.88 member_nat: nat > set_nat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_member_001t__Num__Onum,type,
% 16.11/8.88 member_num: num > set_num > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 16.11/8.88 member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_member_001t__Rat__Orat,type,
% 16.11/8.88 member_rat: rat > set_rat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_member_001t__Real__Oreal,type,
% 16.11/8.88 member_real: real > set_real > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
% 16.11/8.88 member_set_nat: set_nat > set_set_nat > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
% 16.11/8.88 member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
% 16.11/8.88
% 16.11/8.88 thf(sy_v_deg____,type,
% 16.11/8.88 deg: nat ).
% 16.11/8.88
% 16.11/8.88 thf(sy_v_lx____,type,
% 16.11/8.88 lx: nat ).
% 16.11/8.88
% 16.11/8.88 thf(sy_v_m____,type,
% 16.11/8.88 m: nat ).
% 16.11/8.88
% 16.11/8.88 thf(sy_v_ma____,type,
% 16.11/8.88 ma: nat ).
% 16.11/8.88
% 16.11/8.88 thf(sy_v_maxi____,type,
% 16.11/8.88 maxi: nat ).
% 16.11/8.88
% 16.11/8.88 thf(sy_v_maxs____,type,
% 16.11/8.88 maxs: nat ).
% 16.11/8.88
% 16.11/8.88 thf(sy_v_mi____,type,
% 16.11/8.88 mi: nat ).
% 16.11/8.88
% 16.11/8.88 thf(sy_v_na____,type,
% 16.11/8.88 na: nat ).
% 16.11/8.88
% 16.11/8.88 thf(sy_v_summary____,type,
% 16.11/8.88 summary: vEBT_VEBT ).
% 16.11/8.88
% 16.11/8.88 thf(sy_v_summin____,type,
% 16.11/8.88 summin: nat ).
% 16.11/8.88
% 16.11/8.88 thf(sy_v_treeList____,type,
% 16.11/8.88 treeList: list_VEBT_VEBT ).
% 16.11/8.88
% 16.11/8.88 thf(sy_v_xa____,type,
% 16.11/8.88 xa: nat ).
% 16.11/8.88
% 16.11/8.88 thf(sy_v_ya____,type,
% 16.11/8.88 ya: nat ).
% 16.11/8.88
% 16.11/8.88 thf(fact_0__C4_Ohyps_C_I3_J,axiom,
% 16.11/8.88 m = na ).
% 16.11/8.88
% 16.11/8.88 thf(fact_1__092_060open_062both__member__options_A_ItreeList_A_B_Amaxs_J_Amaxi_092_060close_062,axiom,
% 16.11/8.88 vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ maxi ).
% 16.11/8.88
% 16.11/8.88 thf(fact_2_bit__split__inv,axiom,
% 16.11/8.88 ! [X: nat,D: nat] :
% 16.11/8.88 ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X @ D ) @ ( vEBT_VEBT_low @ X @ D ) @ D )
% 16.11/8.88 = X ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_3__C4_OIH_C_I2_J,axiom,
% 16.11/8.88 ! [X: nat,Y: nat] :
% 16.11/8.88 ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ summary @ X ) @ Y )
% 16.11/8.88 = ( ( X != Y )
% 16.11/8.88 & ( vEBT_V8194947554948674370ptions @ summary @ Y ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_4_pow__sum,axiom,
% 16.11/8.88 ! [A: nat,B: nat] :
% 16.11/8.88 ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 16.11/8.88 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_5_high__def,axiom,
% 16.11/8.88 ( vEBT_VEBT_high
% 16.11/8.88 = ( ^ [X2: nat,N: nat] : ( divide_divide_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_6__092_060open_062deg_Adiv_A2_A_061_An_092_060close_062,axiom,
% 16.11/8.88 ( ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.88 = na ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_7__C4_Ohyps_C_I4_J,axiom,
% 16.11/8.88 ( deg
% 16.11/8.88 = ( plus_plus_nat @ na @ m ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_8__092_060open_062_092_060exists_062z_O_Aboth__member__options_A_ItreeList_A_B_Asummin_J_Az_092_060close_062,axiom,
% 16.11/8.88 ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ summin ) @ X_1 ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_9__092_060open_062_092_060exists_062z_O_Aboth__member__options_A_ItreeList_A_B_Amaxs_J_Az_092_060close_062,axiom,
% 16.11/8.88 ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ X_1 ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_10_bit__concat__def,axiom,
% 16.11/8.88 ( vEBT_VEBT_bit_concat
% 16.11/8.88 = ( ^ [H: nat,L: nat,D2: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D2 ) ) @ L ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_11_hprolist,axiom,
% 16.11/8.88 ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) )
% 16.11/8.88 = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_12__092_060open_062treeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_A_B_Athe_A_Ivebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_J_A_061_AtreeList_A_B_Amaxs_092_060close_062,axiom,
% 16.11/8.88 ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) )
% 16.11/8.88 = ( nth_VEBT_VEBT @ treeList @ maxs ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_13__092_060open_062the_A_Ivebt__maxt_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_A_B_Athe_A_Ivebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_J_J_J_A_061_Athe_A_Ivebt__maxt_A_ItreeList_A_B_Amaxs_J_J_092_060close_062,axiom,
% 16.11/8.88 ( ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) )
% 16.11/8.88 = ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ maxs ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_14__092_060open_062both__member__options_Asummary_Amaxs_A_092_060and_062_Amaxs_A_092_060noteq_062_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_092_060close_062,axiom,
% 16.11/8.88 ( ( vEBT_V8194947554948674370ptions @ summary @ maxs )
% 16.11/8.88 & ( maxs
% 16.11/8.88 != ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_15__092_060open_062both__member__options_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_Amaxs_092_060close_062,axiom,
% 16.11/8.88 vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ maxs ).
% 16.11/8.88
% 16.11/8.88 thf(fact_16__092_060open_062Some_Amaxi_A_061_Avebt__maxt_A_ItreeList_A_B_Amaxs_J_092_060close_062,axiom,
% 16.11/8.88 ( ( some_nat @ maxi )
% 16.11/8.88 = ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ maxs ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_17_False,axiom,
% 16.11/8.88 ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 16.11/8.88 != none_nat ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_18__092_060open_062Some_Amaxs_A_061_Avebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_092_060close_062,axiom,
% 16.11/8.88 ( ( some_nat @ maxs )
% 16.11/8.88 = ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_19_True,axiom,
% 16.11/8.88 ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 16.11/8.88 = ma ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_20__092_060open_062length_AtreeList_A_061_Alength_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_J_092_060close_062,axiom,
% 16.11/8.88 ( ( size_s6755466524823107622T_VEBT @ treeList )
% 16.11/8.88 = ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_21_add__self__div__2,axiom,
% 16.11/8.88 ! [M: nat] :
% 16.11/8.88 ( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.88 = M ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_22__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062maxs_O_ASome_Amaxs_A_061_Avebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 16.11/8.88 ~ ! [Maxs: nat] :
% 16.11/8.88 ( ( some_nat @ Maxs )
% 16.11/8.88 != ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_23__092_060open_062summin_A_K_A2_A_094_An_A_L_Alx_A_060_A2_A_094_Adeg_092_060close_062,axiom,
% 16.11/8.88 ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_24_minxnrel,axiom,
% 16.11/8.88 ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 16.11/8.88 != mi ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_25_distrib__left__numeral,axiom,
% 16.11/8.88 ! [V: num,B: complex,C: complex] :
% 16.11/8.88 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B @ C ) )
% 16.11/8.88 = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_26_distrib__left__numeral,axiom,
% 16.11/8.88 ! [V: num,B: real,C: real] :
% 16.11/8.88 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
% 16.11/8.88 = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_27_distrib__left__numeral,axiom,
% 16.11/8.88 ! [V: num,B: rat,C: rat] :
% 16.11/8.88 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C ) )
% 16.11/8.88 = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_28_distrib__left__numeral,axiom,
% 16.11/8.88 ! [V: num,B: nat,C: nat] :
% 16.11/8.88 ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
% 16.11/8.88 = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_29_distrib__left__numeral,axiom,
% 16.11/8.88 ! [V: num,B: int,C: int] :
% 16.11/8.88 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
% 16.11/8.88 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_30_distrib__right__numeral,axiom,
% 16.11/8.88 ! [A: complex,B: complex,V: num] :
% 16.11/8.88 ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
% 16.11/8.88 = ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_31_distrib__right__numeral,axiom,
% 16.11/8.88 ! [A: real,B: real,V: num] :
% 16.11/8.88 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
% 16.11/8.88 = ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_32_distrib__right__numeral,axiom,
% 16.11/8.88 ! [A: rat,B: rat,V: num] :
% 16.11/8.88 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
% 16.11/8.88 = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_33_distrib__right__numeral,axiom,
% 16.11/8.88 ! [A: nat,B: nat,V: num] :
% 16.11/8.88 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
% 16.11/8.88 = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_34_distrib__right__numeral,axiom,
% 16.11/8.88 ! [A: int,B: int,V: num] :
% 16.11/8.88 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
% 16.11/8.88 = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_35_power2__sum,axiom,
% 16.11/8.88 ! [X: complex,Y: complex] :
% 16.11/8.88 ( ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.88 = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_36_power2__sum,axiom,
% 16.11/8.88 ! [X: real,Y: real] :
% 16.11/8.88 ( ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.88 = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_37_power2__sum,axiom,
% 16.11/8.88 ! [X: rat,Y: rat] :
% 16.11/8.88 ( ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.88 = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_38_power2__sum,axiom,
% 16.11/8.88 ! [X: nat,Y: nat] :
% 16.11/8.88 ( ( power_power_nat @ ( plus_plus_nat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.88 = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_39_power2__sum,axiom,
% 16.11/8.88 ! [X: int,Y: int] :
% 16.11/8.88 ( ( power_power_int @ ( plus_plus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.88 = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_40__092_060open_062maxi_A_060_A2_A_094_An_092_060close_062,axiom,
% 16.11/8.88 ord_less_nat @ maxi @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_41__092_060open_062vebt__member_A_ItreeList_A_B_Asummin_J_Alx_092_060close_062,axiom,
% 16.11/8.88 vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ summin ) @ lx ).
% 16.11/8.88
% 16.11/8.88 thf(fact_42_div__exp__eq,axiom,
% 16.11/8.88 ! [A: nat,M: nat,N2: nat] :
% 16.11/8.88 ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.11/8.88 = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_43_div__exp__eq,axiom,
% 16.11/8.88 ! [A: int,M: nat,N2: nat] :
% 16.11/8.88 ( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 16.11/8.88 = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_44_maxbmo,axiom,
% 16.11/8.88 ! [T: vEBT_VEBT,X: nat] :
% 16.11/8.88 ( ( ( vEBT_vebt_maxt @ T )
% 16.11/8.88 = ( some_nat @ X ) )
% 16.11/8.88 => ( vEBT_V8194947554948674370ptions @ T @ X ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_45_power__shift,axiom,
% 16.11/8.88 ! [X: nat,Y: nat,Z: nat] :
% 16.11/8.88 ( ( ( power_power_nat @ X @ Y )
% 16.11/8.88 = Z )
% 16.11/8.88 = ( ( vEBT_VEBT_power @ ( some_nat @ X ) @ ( some_nat @ Y ) )
% 16.11/8.88 = ( some_nat @ Z ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_46__C4_Ohyps_C_I8_J,axiom,
% 16.11/8.88 ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_47__C4_Ohyps_C_I2_J,axiom,
% 16.11/8.88 ( ( size_s6755466524823107622T_VEBT @ treeList )
% 16.11/8.88 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_48_numeral__eq__iff,axiom,
% 16.11/8.88 ! [M: num,N2: num] :
% 16.11/8.88 ( ( ( numera6690914467698888265omplex @ M )
% 16.11/8.88 = ( numera6690914467698888265omplex @ N2 ) )
% 16.11/8.88 = ( M = N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_49_numeral__eq__iff,axiom,
% 16.11/8.88 ! [M: num,N2: num] :
% 16.11/8.88 ( ( ( numeral_numeral_real @ M )
% 16.11/8.88 = ( numeral_numeral_real @ N2 ) )
% 16.11/8.88 = ( M = N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_50_numeral__eq__iff,axiom,
% 16.11/8.88 ! [M: num,N2: num] :
% 16.11/8.88 ( ( ( numeral_numeral_rat @ M )
% 16.11/8.88 = ( numeral_numeral_rat @ N2 ) )
% 16.11/8.88 = ( M = N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_51_numeral__eq__iff,axiom,
% 16.11/8.88 ! [M: num,N2: num] :
% 16.11/8.88 ( ( ( numeral_numeral_nat @ M )
% 16.11/8.88 = ( numeral_numeral_nat @ N2 ) )
% 16.11/8.88 = ( M = N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_52_numeral__eq__iff,axiom,
% 16.11/8.88 ! [M: num,N2: num] :
% 16.11/8.88 ( ( ( numeral_numeral_int @ M )
% 16.11/8.88 = ( numeral_numeral_int @ N2 ) )
% 16.11/8.88 = ( M = N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_53_mimapr,axiom,
% 16.11/8.88 ord_less_nat @ mi @ ma ).
% 16.11/8.88
% 16.11/8.88 thf(fact_54__C4_Ohyps_C_I5_J,axiom,
% 16.11/8.88 ! [I: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 16.11/8.88 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I ) @ X3 ) )
% 16.11/8.88 = ( vEBT_V8194947554948674370ptions @ summary @ I ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_55_high__bound__aux,axiom,
% 16.11/8.88 ! [Ma: nat,N2: nat,M: nat] :
% 16.11/8.88 ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
% 16.11/8.88 => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_56__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062maxi_O_ASome_Amaxi_A_061_Avebt__maxt_A_ItreeList_A_B_Amaxs_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 16.11/8.88 ~ ! [Maxi: nat] :
% 16.11/8.88 ( ( some_nat @ Maxi )
% 16.11/8.88 != ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ maxs ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_57__092_060open_062both__member__options_Asummary_A_Ihigh_Ama_An_J_092_060close_062,axiom,
% 16.11/8.88 vEBT_V8194947554948674370ptions @ summary @ ( vEBT_VEBT_high @ ma @ na ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_58__092_060open_062vebt__member_Asummary_A_Ihigh_Ama_An_J_092_060close_062,axiom,
% 16.11/8.88 vEBT_vebt_member @ summary @ ( vEBT_VEBT_high @ ma @ na ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_59_high__inv,axiom,
% 16.11/8.88 ! [X: nat,N2: nat,Y: nat] :
% 16.11/8.88 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.11/8.88 => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ X ) @ N2 )
% 16.11/8.88 = Y ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_60_numeral__less__iff,axiom,
% 16.11/8.88 ! [M: num,N2: num] :
% 16.11/8.88 ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
% 16.11/8.88 = ( ord_less_num @ M @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_61_numeral__less__iff,axiom,
% 16.11/8.88 ! [M: num,N2: num] :
% 16.11/8.88 ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
% 16.11/8.88 = ( ord_less_num @ M @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_62_numeral__less__iff,axiom,
% 16.11/8.88 ! [M: num,N2: num] :
% 16.11/8.88 ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
% 16.11/8.88 = ( ord_less_num @ M @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_63_numeral__less__iff,axiom,
% 16.11/8.88 ! [M: num,N2: num] :
% 16.11/8.88 ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 16.11/8.88 = ( ord_less_num @ M @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_64_mem__Collect__eq,axiom,
% 16.11/8.88 ! [A: real,P: real > $o] :
% 16.11/8.88 ( ( member_real @ A @ ( collect_real @ P ) )
% 16.11/8.88 = ( P @ A ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_65_mem__Collect__eq,axiom,
% 16.11/8.88 ! [A: product_prod_int_int,P: product_prod_int_int > $o] :
% 16.11/8.88 ( ( member5262025264175285858nt_int @ A @ ( collec213857154873943460nt_int @ P ) )
% 16.11/8.88 = ( P @ A ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_66_mem__Collect__eq,axiom,
% 16.11/8.88 ! [A: complex,P: complex > $o] :
% 16.11/8.88 ( ( member_complex @ A @ ( collect_complex @ P ) )
% 16.11/8.88 = ( P @ A ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_67_mem__Collect__eq,axiom,
% 16.11/8.88 ! [A: set_nat,P: set_nat > $o] :
% 16.11/8.88 ( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
% 16.11/8.88 = ( P @ A ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_68_mem__Collect__eq,axiom,
% 16.11/8.88 ! [A: nat,P: nat > $o] :
% 16.11/8.88 ( ( member_nat @ A @ ( collect_nat @ P ) )
% 16.11/8.88 = ( P @ A ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_69_mem__Collect__eq,axiom,
% 16.11/8.88 ! [A: int,P: int > $o] :
% 16.11/8.88 ( ( member_int @ A @ ( collect_int @ P ) )
% 16.11/8.88 = ( P @ A ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_70_Collect__mem__eq,axiom,
% 16.11/8.88 ! [A2: set_real] :
% 16.11/8.88 ( ( collect_real
% 16.11/8.88 @ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
% 16.11/8.88 = A2 ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_71_Collect__mem__eq,axiom,
% 16.11/8.88 ! [A2: set_Pr958786334691620121nt_int] :
% 16.11/8.88 ( ( collec213857154873943460nt_int
% 16.11/8.88 @ ^ [X2: product_prod_int_int] : ( member5262025264175285858nt_int @ X2 @ A2 ) )
% 16.11/8.88 = A2 ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_72_Collect__mem__eq,axiom,
% 16.11/8.88 ! [A2: set_complex] :
% 16.11/8.88 ( ( collect_complex
% 16.11/8.88 @ ^ [X2: complex] : ( member_complex @ X2 @ A2 ) )
% 16.11/8.88 = A2 ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_73_Collect__mem__eq,axiom,
% 16.11/8.88 ! [A2: set_set_nat] :
% 16.11/8.88 ( ( collect_set_nat
% 16.11/8.88 @ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A2 ) )
% 16.11/8.88 = A2 ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_74_Collect__mem__eq,axiom,
% 16.11/8.88 ! [A2: set_nat] :
% 16.11/8.88 ( ( collect_nat
% 16.11/8.88 @ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
% 16.11/8.88 = A2 ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_75_Collect__mem__eq,axiom,
% 16.11/8.88 ! [A2: set_int] :
% 16.11/8.88 ( ( collect_int
% 16.11/8.88 @ ^ [X2: int] : ( member_int @ X2 @ A2 ) )
% 16.11/8.88 = A2 ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_76_Collect__cong,axiom,
% 16.11/8.88 ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
% 16.11/8.88 ( ! [X4: product_prod_int_int] :
% 16.11/8.88 ( ( P @ X4 )
% 16.11/8.88 = ( Q @ X4 ) )
% 16.11/8.88 => ( ( collec213857154873943460nt_int @ P )
% 16.11/8.88 = ( collec213857154873943460nt_int @ Q ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_77_Collect__cong,axiom,
% 16.11/8.88 ! [P: complex > $o,Q: complex > $o] :
% 16.11/8.88 ( ! [X4: complex] :
% 16.11/8.88 ( ( P @ X4 )
% 16.11/8.88 = ( Q @ X4 ) )
% 16.11/8.88 => ( ( collect_complex @ P )
% 16.11/8.88 = ( collect_complex @ Q ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_78_Collect__cong,axiom,
% 16.11/8.88 ! [P: set_nat > $o,Q: set_nat > $o] :
% 16.11/8.88 ( ! [X4: set_nat] :
% 16.11/8.88 ( ( P @ X4 )
% 16.11/8.88 = ( Q @ X4 ) )
% 16.11/8.88 => ( ( collect_set_nat @ P )
% 16.11/8.88 = ( collect_set_nat @ Q ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_79_Collect__cong,axiom,
% 16.11/8.88 ! [P: nat > $o,Q: nat > $o] :
% 16.11/8.88 ( ! [X4: nat] :
% 16.11/8.88 ( ( P @ X4 )
% 16.11/8.88 = ( Q @ X4 ) )
% 16.11/8.88 => ( ( collect_nat @ P )
% 16.11/8.88 = ( collect_nat @ Q ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_80_Collect__cong,axiom,
% 16.11/8.88 ! [P: int > $o,Q: int > $o] :
% 16.11/8.88 ( ! [X4: int] :
% 16.11/8.88 ( ( P @ X4 )
% 16.11/8.88 = ( Q @ X4 ) )
% 16.11/8.88 => ( ( collect_int @ P )
% 16.11/8.88 = ( collect_int @ Q ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_81_low__inv,axiom,
% 16.11/8.88 ! [X: nat,N2: nat,Y: nat] :
% 16.11/8.88 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.11/8.88 => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ X ) @ N2 )
% 16.11/8.88 = X ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_82_mult__numeral__left__semiring__numeral,axiom,
% 16.11/8.88 ! [V: num,W: num,Z: complex] :
% 16.11/8.88 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
% 16.11/8.88 = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_83_mult__numeral__left__semiring__numeral,axiom,
% 16.11/8.88 ! [V: num,W: num,Z: real] :
% 16.11/8.88 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
% 16.11/8.88 = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_84_mult__numeral__left__semiring__numeral,axiom,
% 16.11/8.88 ! [V: num,W: num,Z: rat] :
% 16.11/8.88 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
% 16.11/8.88 = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_85_mult__numeral__left__semiring__numeral,axiom,
% 16.11/8.88 ! [V: num,W: num,Z: nat] :
% 16.11/8.88 ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
% 16.11/8.88 = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_86_mult__numeral__left__semiring__numeral,axiom,
% 16.11/8.88 ! [V: num,W: num,Z: int] :
% 16.11/8.88 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
% 16.11/8.88 = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_87_numeral__times__numeral,axiom,
% 16.11/8.88 ! [M: num,N2: num] :
% 16.11/8.88 ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N2 ) )
% 16.11/8.88 = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_88_numeral__times__numeral,axiom,
% 16.11/8.88 ! [M: num,N2: num] :
% 16.11/8.88 ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
% 16.11/8.88 = ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_89_numeral__times__numeral,axiom,
% 16.11/8.88 ! [M: num,N2: num] :
% 16.11/8.88 ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
% 16.11/8.88 = ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_90_numeral__times__numeral,axiom,
% 16.11/8.88 ! [M: num,N2: num] :
% 16.11/8.88 ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
% 16.11/8.88 = ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_91_numeral__times__numeral,axiom,
% 16.11/8.88 ! [M: num,N2: num] :
% 16.11/8.88 ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 16.11/8.88 = ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_92_add__numeral__left,axiom,
% 16.11/8.88 ! [V: num,W: num,Z: complex] :
% 16.11/8.88 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
% 16.11/8.88 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_93_add__numeral__left,axiom,
% 16.11/8.88 ! [V: num,W: num,Z: real] :
% 16.11/8.88 ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z ) )
% 16.11/8.88 = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_94_add__numeral__left,axiom,
% 16.11/8.88 ! [V: num,W: num,Z: rat] :
% 16.11/8.88 ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
% 16.11/8.88 = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_95_add__numeral__left,axiom,
% 16.11/8.88 ! [V: num,W: num,Z: nat] :
% 16.11/8.88 ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
% 16.11/8.88 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_96_add__numeral__left,axiom,
% 16.11/8.88 ! [V: num,W: num,Z: int] :
% 16.11/8.88 ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
% 16.11/8.88 = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_97_numeral__plus__numeral,axiom,
% 16.11/8.88 ! [M: num,N2: num] :
% 16.11/8.88 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N2 ) )
% 16.11/8.88 = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_98_numeral__plus__numeral,axiom,
% 16.11/8.88 ! [M: num,N2: num] :
% 16.11/8.88 ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
% 16.11/8.88 = ( numeral_numeral_real @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_99_numeral__plus__numeral,axiom,
% 16.11/8.88 ! [M: num,N2: num] :
% 16.11/8.88 ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
% 16.11/8.88 = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_100_numeral__plus__numeral,axiom,
% 16.11/8.88 ! [M: num,N2: num] :
% 16.11/8.88 ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
% 16.11/8.88 = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_101_numeral__plus__numeral,axiom,
% 16.11/8.88 ! [M: num,N2: num] :
% 16.11/8.88 ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 16.11/8.88 = ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_102_num__double,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( times_times_num @ ( bit0 @ one ) @ N2 )
% 16.11/8.88 = ( bit0 @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_103_power__mult__numeral,axiom,
% 16.11/8.88 ! [A: nat,M: num,N2: num] :
% 16.11/8.88 ( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
% 16.11/8.88 = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_104_power__mult__numeral,axiom,
% 16.11/8.88 ! [A: real,M: num,N2: num] :
% 16.11/8.88 ( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
% 16.11/8.88 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_105_power__mult__numeral,axiom,
% 16.11/8.88 ! [A: int,M: num,N2: num] :
% 16.11/8.88 ( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
% 16.11/8.88 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_106_power__mult__numeral,axiom,
% 16.11/8.88 ! [A: complex,M: num,N2: num] :
% 16.11/8.88 ( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
% 16.11/8.88 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_107__092_060open_062summin_A_060_A2_A_094_Am_092_060close_062,axiom,
% 16.11/8.88 ord_less_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_108__092_060open_062maxs_A_060_A2_A_094_Am_092_060close_062,axiom,
% 16.11/8.88 ord_less_nat @ maxs @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_109_power__add__numeral2,axiom,
% 16.11/8.88 ! [A: complex,M: num,N2: num,B: complex] :
% 16.11/8.88 ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
% 16.11/8.88 = ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_110_power__add__numeral2,axiom,
% 16.11/8.88 ! [A: real,M: num,N2: num,B: real] :
% 16.11/8.88 ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
% 16.11/8.88 = ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_111_power__add__numeral2,axiom,
% 16.11/8.88 ! [A: rat,M: num,N2: num,B: rat] :
% 16.11/8.88 ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
% 16.11/8.88 = ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_112_power__add__numeral2,axiom,
% 16.11/8.88 ! [A: nat,M: num,N2: num,B: nat] :
% 16.11/8.88 ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
% 16.11/8.88 = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_113_power__add__numeral2,axiom,
% 16.11/8.88 ! [A: int,M: num,N2: num,B: int] :
% 16.11/8.88 ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
% 16.11/8.88 = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_114_power__add__numeral,axiom,
% 16.11/8.88 ! [A: complex,M: num,N2: num] :
% 16.11/8.88 ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N2 ) ) )
% 16.11/8.88 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_115_power__add__numeral,axiom,
% 16.11/8.88 ! [A: real,M: num,N2: num] :
% 16.11/8.88 ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N2 ) ) )
% 16.11/8.88 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_116_power__add__numeral,axiom,
% 16.11/8.88 ! [A: rat,M: num,N2: num] :
% 16.11/8.88 ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N2 ) ) )
% 16.11/8.88 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_117_power__add__numeral,axiom,
% 16.11/8.88 ! [A: nat,M: num,N2: num] :
% 16.11/8.88 ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N2 ) ) )
% 16.11/8.88 = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_118_power__add__numeral,axiom,
% 16.11/8.88 ! [A: int,M: num,N2: num] :
% 16.11/8.88 ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N2 ) ) )
% 16.11/8.88 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_119__C4_Ohyps_C_I7_J,axiom,
% 16.11/8.88 ord_less_eq_nat @ mi @ ma ).
% 16.11/8.88
% 16.11/8.88 thf(fact_120_less__divide__eq__numeral1_I1_J,axiom,
% 16.11/8.88 ! [A: real,B: real,W: num] :
% 16.11/8.88 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 16.11/8.88 = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_121_less__divide__eq__numeral1_I1_J,axiom,
% 16.11/8.88 ! [A: rat,B: rat,W: num] :
% 16.11/8.88 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 16.11/8.88 = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_122_divide__less__eq__numeral1_I1_J,axiom,
% 16.11/8.88 ! [B: real,W: num,A: real] :
% 16.11/8.88 ( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
% 16.11/8.88 = ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_123_divide__less__eq__numeral1_I1_J,axiom,
% 16.11/8.88 ! [B: rat,W: num,A: rat] :
% 16.11/8.88 ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
% 16.11/8.88 = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_124__092_060open_062high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_060_Alength_AtreeList_092_060close_062,axiom,
% 16.11/8.88 ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_125__092_060open_062Some_Asummin_A_061_Avebt__mint_Asummary_092_060close_062,axiom,
% 16.11/8.88 ( ( some_nat @ summin )
% 16.11/8.88 = ( vEBT_vebt_mint @ summary ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_126_nothprolist,axiom,
% 16.11/8.88 ! [I2: nat] :
% 16.11/8.88 ( ( ( I2
% 16.11/8.88 != ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) )
% 16.11/8.88 & ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) )
% 16.11/8.88 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ I2 )
% 16.11/8.88 = ( nth_VEBT_VEBT @ treeList @ I2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_127__C4_Ohyps_C_I6_J,axiom,
% 16.11/8.88 ( ( mi = ma )
% 16.11/8.88 => ! [X5: vEBT_VEBT] :
% 16.11/8.88 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ treeList ) )
% 16.11/8.88 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_128__092_060open_062Some_Alx_A_061_Avebt__mint_A_ItreeList_A_B_Asummin_J_092_060close_062,axiom,
% 16.11/8.88 ( ( some_nat @ lx )
% 16.11/8.88 = ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ summin ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_129__C4_Ohyps_C_I1_J,axiom,
% 16.11/8.88 vEBT_invar_vebt @ summary @ m ).
% 16.11/8.88
% 16.11/8.88 thf(fact_130__C4_Ohyps_C_I9_J,axiom,
% 16.11/8.88 ( ( mi != ma )
% 16.11/8.88 => ! [I: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 16.11/8.88 => ( ( ( ( vEBT_VEBT_high @ ma @ na )
% 16.11/8.88 = I )
% 16.11/8.88 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
% 16.11/8.88 & ! [X5: nat] :
% 16.11/8.88 ( ( ( ( vEBT_VEBT_high @ X5 @ na )
% 16.11/8.88 = I )
% 16.11/8.88 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I ) @ ( vEBT_VEBT_low @ X5 @ na ) ) )
% 16.11/8.88 => ( ( ord_less_nat @ mi @ X5 )
% 16.11/8.88 & ( ord_less_eq_nat @ X5 @ ma ) ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_131_add__One__commute,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( plus_plus_num @ one @ N2 )
% 16.11/8.88 = ( plus_plus_num @ N2 @ one ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_132_div__mult2__numeral__eq,axiom,
% 16.11/8.88 ! [A: nat,K: num,L2: num] :
% 16.11/8.88 ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L2 ) )
% 16.11/8.88 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_133_div__mult2__numeral__eq,axiom,
% 16.11/8.88 ! [A: int,K: num,L2: num] :
% 16.11/8.88 ( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L2 ) )
% 16.11/8.88 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_134_less__mult__imp__div__less,axiom,
% 16.11/8.88 ! [M: nat,I2: nat,N2: nat] :
% 16.11/8.88 ( ( ord_less_nat @ M @ ( times_times_nat @ I2 @ N2 ) )
% 16.11/8.88 => ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ I2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_135_less__exp,axiom,
% 16.11/8.88 ! [N2: nat] : ( ord_less_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_136_is__num__normalize_I1_J,axiom,
% 16.11/8.88 ! [A: real,B: real,C: real] :
% 16.11/8.88 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 16.11/8.88 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_137_is__num__normalize_I1_J,axiom,
% 16.11/8.88 ! [A: rat,B: rat,C: rat] :
% 16.11/8.88 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 16.11/8.88 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_138_is__num__normalize_I1_J,axiom,
% 16.11/8.88 ! [A: int,B: int,C: int] :
% 16.11/8.88 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 16.11/8.88 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_139_power__commuting__commutes,axiom,
% 16.11/8.88 ! [X: complex,Y: complex,N2: nat] :
% 16.11/8.88 ( ( ( times_times_complex @ X @ Y )
% 16.11/8.88 = ( times_times_complex @ Y @ X ) )
% 16.11/8.88 => ( ( times_times_complex @ ( power_power_complex @ X @ N2 ) @ Y )
% 16.11/8.88 = ( times_times_complex @ Y @ ( power_power_complex @ X @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_140_power__commuting__commutes,axiom,
% 16.11/8.88 ! [X: real,Y: real,N2: nat] :
% 16.11/8.88 ( ( ( times_times_real @ X @ Y )
% 16.11/8.88 = ( times_times_real @ Y @ X ) )
% 16.11/8.88 => ( ( times_times_real @ ( power_power_real @ X @ N2 ) @ Y )
% 16.11/8.88 = ( times_times_real @ Y @ ( power_power_real @ X @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_141_power__commuting__commutes,axiom,
% 16.11/8.88 ! [X: rat,Y: rat,N2: nat] :
% 16.11/8.88 ( ( ( times_times_rat @ X @ Y )
% 16.11/8.88 = ( times_times_rat @ Y @ X ) )
% 16.11/8.88 => ( ( times_times_rat @ ( power_power_rat @ X @ N2 ) @ Y )
% 16.11/8.88 = ( times_times_rat @ Y @ ( power_power_rat @ X @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_142_power__commuting__commutes,axiom,
% 16.11/8.88 ! [X: nat,Y: nat,N2: nat] :
% 16.11/8.88 ( ( ( times_times_nat @ X @ Y )
% 16.11/8.88 = ( times_times_nat @ Y @ X ) )
% 16.11/8.88 => ( ( times_times_nat @ ( power_power_nat @ X @ N2 ) @ Y )
% 16.11/8.88 = ( times_times_nat @ Y @ ( power_power_nat @ X @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_143_power__commuting__commutes,axiom,
% 16.11/8.88 ! [X: int,Y: int,N2: nat] :
% 16.11/8.88 ( ( ( times_times_int @ X @ Y )
% 16.11/8.88 = ( times_times_int @ Y @ X ) )
% 16.11/8.88 => ( ( times_times_int @ ( power_power_int @ X @ N2 ) @ Y )
% 16.11/8.88 = ( times_times_int @ Y @ ( power_power_int @ X @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_144_power__mult__distrib,axiom,
% 16.11/8.88 ! [A: complex,B: complex,N2: nat] :
% 16.11/8.88 ( ( power_power_complex @ ( times_times_complex @ A @ B ) @ N2 )
% 16.11/8.88 = ( times_times_complex @ ( power_power_complex @ A @ N2 ) @ ( power_power_complex @ B @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_145_power__mult__distrib,axiom,
% 16.11/8.88 ! [A: real,B: real,N2: nat] :
% 16.11/8.88 ( ( power_power_real @ ( times_times_real @ A @ B ) @ N2 )
% 16.11/8.88 = ( times_times_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_146_power__mult__distrib,axiom,
% 16.11/8.88 ! [A: rat,B: rat,N2: nat] :
% 16.11/8.88 ( ( power_power_rat @ ( times_times_rat @ A @ B ) @ N2 )
% 16.11/8.88 = ( times_times_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_147_power__mult__distrib,axiom,
% 16.11/8.88 ! [A: nat,B: nat,N2: nat] :
% 16.11/8.88 ( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N2 )
% 16.11/8.88 = ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_148_power__mult__distrib,axiom,
% 16.11/8.88 ! [A: int,B: int,N2: nat] :
% 16.11/8.88 ( ( power_power_int @ ( times_times_int @ A @ B ) @ N2 )
% 16.11/8.88 = ( times_times_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_149_power__commutes,axiom,
% 16.11/8.88 ! [A: complex,N2: nat] :
% 16.11/8.88 ( ( times_times_complex @ ( power_power_complex @ A @ N2 ) @ A )
% 16.11/8.88 = ( times_times_complex @ A @ ( power_power_complex @ A @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_150_power__commutes,axiom,
% 16.11/8.88 ! [A: real,N2: nat] :
% 16.11/8.88 ( ( times_times_real @ ( power_power_real @ A @ N2 ) @ A )
% 16.11/8.88 = ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_151_power__commutes,axiom,
% 16.11/8.88 ! [A: rat,N2: nat] :
% 16.11/8.88 ( ( times_times_rat @ ( power_power_rat @ A @ N2 ) @ A )
% 16.11/8.88 = ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_152_power__commutes,axiom,
% 16.11/8.88 ! [A: nat,N2: nat] :
% 16.11/8.88 ( ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ A )
% 16.11/8.88 = ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_153_power__commutes,axiom,
% 16.11/8.88 ! [A: int,N2: nat] :
% 16.11/8.88 ( ( times_times_int @ ( power_power_int @ A @ N2 ) @ A )
% 16.11/8.88 = ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_154_power__divide,axiom,
% 16.11/8.88 ! [A: complex,B: complex,N2: nat] :
% 16.11/8.88 ( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B ) @ N2 )
% 16.11/8.88 = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N2 ) @ ( power_power_complex @ B @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_155_power__divide,axiom,
% 16.11/8.88 ! [A: real,B: real,N2: nat] :
% 16.11/8.88 ( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N2 )
% 16.11/8.88 = ( divide_divide_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_156_power__divide,axiom,
% 16.11/8.88 ! [A: rat,B: rat,N2: nat] :
% 16.11/8.88 ( ( power_power_rat @ ( divide_divide_rat @ A @ B ) @ N2 )
% 16.11/8.88 = ( divide_divide_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_157_power__mult,axiom,
% 16.11/8.88 ! [A: nat,M: nat,N2: nat] :
% 16.11/8.88 ( ( power_power_nat @ A @ ( times_times_nat @ M @ N2 ) )
% 16.11/8.88 = ( power_power_nat @ ( power_power_nat @ A @ M ) @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_158_power__mult,axiom,
% 16.11/8.88 ! [A: real,M: nat,N2: nat] :
% 16.11/8.88 ( ( power_power_real @ A @ ( times_times_nat @ M @ N2 ) )
% 16.11/8.88 = ( power_power_real @ ( power_power_real @ A @ M ) @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_159_power__mult,axiom,
% 16.11/8.88 ! [A: int,M: nat,N2: nat] :
% 16.11/8.88 ( ( power_power_int @ A @ ( times_times_nat @ M @ N2 ) )
% 16.11/8.88 = ( power_power_int @ ( power_power_int @ A @ M ) @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_160_power__mult,axiom,
% 16.11/8.88 ! [A: complex,M: nat,N2: nat] :
% 16.11/8.88 ( ( power_power_complex @ A @ ( times_times_nat @ M @ N2 ) )
% 16.11/8.88 = ( power_power_complex @ ( power_power_complex @ A @ M ) @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_161_div__mult2__eq,axiom,
% 16.11/8.88 ! [M: nat,N2: nat,Q2: nat] :
% 16.11/8.88 ( ( divide_divide_nat @ M @ ( times_times_nat @ N2 @ Q2 ) )
% 16.11/8.88 = ( divide_divide_nat @ ( divide_divide_nat @ M @ N2 ) @ Q2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_162_mult__numeral__1__right,axiom,
% 16.11/8.88 ! [A: complex] :
% 16.11/8.88 ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
% 16.11/8.88 = A ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_163_mult__numeral__1__right,axiom,
% 16.11/8.88 ! [A: real] :
% 16.11/8.88 ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
% 16.11/8.88 = A ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_164_mult__numeral__1__right,axiom,
% 16.11/8.88 ! [A: rat] :
% 16.11/8.88 ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
% 16.11/8.88 = A ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_165_mult__numeral__1__right,axiom,
% 16.11/8.88 ! [A: nat] :
% 16.11/8.88 ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
% 16.11/8.88 = A ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_166_mult__numeral__1__right,axiom,
% 16.11/8.88 ! [A: int] :
% 16.11/8.88 ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
% 16.11/8.88 = A ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_167_mult__numeral__1,axiom,
% 16.11/8.88 ! [A: complex] :
% 16.11/8.88 ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
% 16.11/8.88 = A ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_168_mult__numeral__1,axiom,
% 16.11/8.88 ! [A: real] :
% 16.11/8.88 ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
% 16.11/8.88 = A ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_169_mult__numeral__1,axiom,
% 16.11/8.88 ! [A: rat] :
% 16.11/8.88 ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
% 16.11/8.88 = A ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_170_mult__numeral__1,axiom,
% 16.11/8.88 ! [A: nat] :
% 16.11/8.88 ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
% 16.11/8.88 = A ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_171_mult__numeral__1,axiom,
% 16.11/8.88 ! [A: int] :
% 16.11/8.88 ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
% 16.11/8.88 = A ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_172_numeral__Bit0,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( numera6690914467698888265omplex @ ( bit0 @ N2 ) )
% 16.11/8.88 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_173_numeral__Bit0,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( numeral_numeral_real @ ( bit0 @ N2 ) )
% 16.11/8.88 = ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_174_numeral__Bit0,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( numeral_numeral_rat @ ( bit0 @ N2 ) )
% 16.11/8.88 = ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_175_numeral__Bit0,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
% 16.11/8.88 = ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_176_numeral__Bit0,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( numeral_numeral_int @ ( bit0 @ N2 ) )
% 16.11/8.88 = ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_177_divide__numeral__1,axiom,
% 16.11/8.88 ! [A: complex] :
% 16.11/8.88 ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
% 16.11/8.88 = A ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_178_divide__numeral__1,axiom,
% 16.11/8.88 ! [A: real] :
% 16.11/8.88 ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
% 16.11/8.88 = A ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_179_divide__numeral__1,axiom,
% 16.11/8.88 ! [A: rat] :
% 16.11/8.88 ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
% 16.11/8.88 = A ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_180_power__add,axiom,
% 16.11/8.88 ! [A: complex,M: nat,N2: nat] :
% 16.11/8.88 ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N2 ) )
% 16.11/8.88 = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_181_power__add,axiom,
% 16.11/8.88 ! [A: real,M: nat,N2: nat] :
% 16.11/8.88 ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N2 ) )
% 16.11/8.88 = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_182_power__add,axiom,
% 16.11/8.88 ! [A: rat,M: nat,N2: nat] :
% 16.11/8.88 ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N2 ) )
% 16.11/8.88 = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_183_power__add,axiom,
% 16.11/8.88 ! [A: nat,M: nat,N2: nat] :
% 16.11/8.88 ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N2 ) )
% 16.11/8.88 = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_184_power__add,axiom,
% 16.11/8.88 ! [A: int,M: nat,N2: nat] :
% 16.11/8.88 ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N2 ) )
% 16.11/8.88 = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_185_numeral__Bit0__div__2,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.88 = ( numeral_numeral_nat @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_186_numeral__Bit0__div__2,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.11/8.88 = ( numeral_numeral_int @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_187_left__add__twice,axiom,
% 16.11/8.88 ! [A: complex,B: complex] :
% 16.11/8.88 ( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B ) )
% 16.11/8.88 = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_188_left__add__twice,axiom,
% 16.11/8.88 ! [A: real,B: real] :
% 16.11/8.88 ( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
% 16.11/8.88 = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_189_left__add__twice,axiom,
% 16.11/8.88 ! [A: rat,B: rat] :
% 16.11/8.88 ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 16.11/8.88 = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_190_left__add__twice,axiom,
% 16.11/8.88 ! [A: nat,B: nat] :
% 16.11/8.88 ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 16.11/8.88 = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_191_left__add__twice,axiom,
% 16.11/8.88 ! [A: int,B: int] :
% 16.11/8.88 ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
% 16.11/8.88 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_192_mult__2__right,axiom,
% 16.11/8.88 ! [Z: complex] :
% 16.11/8.88 ( ( times_times_complex @ Z @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
% 16.11/8.88 = ( plus_plus_complex @ Z @ Z ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_193_mult__2__right,axiom,
% 16.11/8.88 ! [Z: real] :
% 16.11/8.88 ( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 16.11/8.88 = ( plus_plus_real @ Z @ Z ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_194_mult__2__right,axiom,
% 16.11/8.88 ! [Z: rat] :
% 16.11/8.88 ( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
% 16.11/8.88 = ( plus_plus_rat @ Z @ Z ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_195_mult__2__right,axiom,
% 16.11/8.88 ! [Z: nat] :
% 16.11/8.88 ( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.88 = ( plus_plus_nat @ Z @ Z ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_196_mult__2__right,axiom,
% 16.11/8.88 ! [Z: int] :
% 16.11/8.88 ( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.11/8.88 = ( plus_plus_int @ Z @ Z ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_197_mult__2,axiom,
% 16.11/8.88 ! [Z: complex] :
% 16.11/8.88 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z )
% 16.11/8.88 = ( plus_plus_complex @ Z @ Z ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_198_mult__2,axiom,
% 16.11/8.88 ! [Z: real] :
% 16.11/8.88 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
% 16.11/8.88 = ( plus_plus_real @ Z @ Z ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_199_mult__2,axiom,
% 16.11/8.88 ! [Z: rat] :
% 16.11/8.88 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
% 16.11/8.88 = ( plus_plus_rat @ Z @ Z ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_200_mult__2,axiom,
% 16.11/8.88 ! [Z: nat] :
% 16.11/8.88 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
% 16.11/8.88 = ( plus_plus_nat @ Z @ Z ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_201_mult__2,axiom,
% 16.11/8.88 ! [Z: int] :
% 16.11/8.88 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
% 16.11/8.88 = ( plus_plus_int @ Z @ Z ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_202_power2__eq__square,axiom,
% 16.11/8.88 ! [A: complex] :
% 16.11/8.88 ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.88 = ( times_times_complex @ A @ A ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_203_power2__eq__square,axiom,
% 16.11/8.88 ! [A: real] :
% 16.11/8.88 ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.88 = ( times_times_real @ A @ A ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_204_power2__eq__square,axiom,
% 16.11/8.88 ! [A: rat] :
% 16.11/8.88 ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.88 = ( times_times_rat @ A @ A ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_205_power2__eq__square,axiom,
% 16.11/8.88 ! [A: nat] :
% 16.11/8.88 ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.88 = ( times_times_nat @ A @ A ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_206_power2__eq__square,axiom,
% 16.11/8.88 ! [A: int] :
% 16.11/8.88 ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.88 = ( times_times_int @ A @ A ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_207_power4__eq__xxxx,axiom,
% 16.11/8.88 ! [X: complex] :
% 16.11/8.88 ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 16.11/8.88 = ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X @ X ) @ X ) @ X ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_208_power4__eq__xxxx,axiom,
% 16.11/8.88 ! [X: real] :
% 16.11/8.88 ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 16.11/8.88 = ( times_times_real @ ( times_times_real @ ( times_times_real @ X @ X ) @ X ) @ X ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_209_power4__eq__xxxx,axiom,
% 16.11/8.88 ! [X: rat] :
% 16.11/8.88 ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 16.11/8.88 = ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X @ X ) @ X ) @ X ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_210_power4__eq__xxxx,axiom,
% 16.11/8.88 ! [X: nat] :
% 16.11/8.88 ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 16.11/8.88 = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_211_power4__eq__xxxx,axiom,
% 16.11/8.88 ! [X: int] :
% 16.11/8.88 ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 16.11/8.88 = ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_212_power__even__eq,axiom,
% 16.11/8.88 ! [A: nat,N2: nat] :
% 16.11/8.88 ( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.11/8.88 = ( power_power_nat @ ( power_power_nat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_213_power__even__eq,axiom,
% 16.11/8.88 ! [A: real,N2: nat] :
% 16.11/8.88 ( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.11/8.88 = ( power_power_real @ ( power_power_real @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_214_power__even__eq,axiom,
% 16.11/8.88 ! [A: int,N2: nat] :
% 16.11/8.88 ( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.11/8.88 = ( power_power_int @ ( power_power_int @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_215_power__even__eq,axiom,
% 16.11/8.88 ! [A: complex,N2: nat] :
% 16.11/8.88 ( ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.11/8.88 = ( power_power_complex @ ( power_power_complex @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_216_in__children__def,axiom,
% 16.11/8.88 ( vEBT_V5917875025757280293ildren
% 16.11/8.88 = ( ^ [N: nat,TreeList: list_VEBT_VEBT,X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ N ) ) @ ( vEBT_VEBT_low @ X2 @ N ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_217_greater__shift,axiom,
% 16.11/8.88 ( ord_less_nat
% 16.11/8.88 = ( ^ [Y2: nat,X2: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X2 ) @ ( some_nat @ Y2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_218_less__shift,axiom,
% 16.11/8.88 ( ord_less_nat
% 16.11/8.88 = ( ^ [X2: nat,Y2: nat] : ( vEBT_VEBT_less @ ( some_nat @ X2 ) @ ( some_nat @ Y2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_219__092_060open_062x_A_060_A2_A_094_Adeg_092_060close_062,axiom,
% 16.11/8.88 ord_less_nat @ xa @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_220_nth__list__update__eq,axiom,
% 16.11/8.88 ! [I2: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
% 16.11/8.88 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 16.11/8.88 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X ) @ I2 )
% 16.11/8.88 = X ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_221_nth__list__update__eq,axiom,
% 16.11/8.88 ! [I2: nat,Xs: list_o,X: $o] :
% 16.11/8.88 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 16.11/8.88 => ( ( nth_o @ ( list_update_o @ Xs @ I2 @ X ) @ I2 )
% 16.11/8.88 = X ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_222_nth__list__update__eq,axiom,
% 16.11/8.88 ! [I2: nat,Xs: list_int,X: int] :
% 16.11/8.88 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 16.11/8.88 => ( ( nth_int @ ( list_update_int @ Xs @ I2 @ X ) @ I2 )
% 16.11/8.88 = X ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_223_option_Ocollapse,axiom,
% 16.11/8.88 ! [Option: option_nat] :
% 16.11/8.88 ( ( Option != none_nat )
% 16.11/8.88 => ( ( some_nat @ ( the_nat @ Option ) )
% 16.11/8.88 = Option ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_224_option_Ocollapse,axiom,
% 16.11/8.88 ! [Option: option4927543243414619207at_nat] :
% 16.11/8.88 ( ( Option != none_P5556105721700978146at_nat )
% 16.11/8.88 => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
% 16.11/8.88 = Option ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_225_option_Ocollapse,axiom,
% 16.11/8.88 ! [Option: option_num] :
% 16.11/8.88 ( ( Option != none_num )
% 16.11/8.88 => ( ( some_num @ ( the_num @ Option ) )
% 16.11/8.88 = Option ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_226__092_060open_062ma_A_092_060le_062_A2_A_094_Adeg_092_060close_062,axiom,
% 16.11/8.88 ord_less_eq_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_227_field__less__half__sum,axiom,
% 16.11/8.88 ! [X: real,Y: real] :
% 16.11/8.88 ( ( ord_less_real @ X @ Y )
% 16.11/8.88 => ( ord_less_real @ X @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_228_field__less__half__sum,axiom,
% 16.11/8.88 ! [X: rat,Y: rat] :
% 16.11/8.88 ( ( ord_less_rat @ X @ Y )
% 16.11/8.88 => ( ord_less_rat @ X @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_229_nth__list__update__neq,axiom,
% 16.11/8.88 ! [I2: nat,J: nat,Xs: list_int,X: int] :
% 16.11/8.88 ( ( I2 != J )
% 16.11/8.88 => ( ( nth_int @ ( list_update_int @ Xs @ I2 @ X ) @ J )
% 16.11/8.88 = ( nth_int @ Xs @ J ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_230_nth__list__update__neq,axiom,
% 16.11/8.88 ! [I2: nat,J: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
% 16.11/8.88 ( ( I2 != J )
% 16.11/8.88 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X ) @ J )
% 16.11/8.88 = ( nth_VEBT_VEBT @ Xs @ J ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_231_list__update__id,axiom,
% 16.11/8.88 ! [Xs: list_int,I2: nat] :
% 16.11/8.88 ( ( list_update_int @ Xs @ I2 @ ( nth_int @ Xs @ I2 ) )
% 16.11/8.88 = Xs ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_232_list__update__id,axiom,
% 16.11/8.88 ! [Xs: list_VEBT_VEBT,I2: nat] :
% 16.11/8.88 ( ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ ( nth_VEBT_VEBT @ Xs @ I2 ) )
% 16.11/8.88 = Xs ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_233_length__list__update,axiom,
% 16.11/8.88 ! [Xs: list_VEBT_VEBT,I2: nat,X: vEBT_VEBT] :
% 16.11/8.88 ( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X ) )
% 16.11/8.88 = ( size_s6755466524823107622T_VEBT @ Xs ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_234_length__list__update,axiom,
% 16.11/8.88 ! [Xs: list_o,I2: nat,X: $o] :
% 16.11/8.88 ( ( size_size_list_o @ ( list_update_o @ Xs @ I2 @ X ) )
% 16.11/8.88 = ( size_size_list_o @ Xs ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_235_length__list__update,axiom,
% 16.11/8.88 ! [Xs: list_int,I2: nat,X: int] :
% 16.11/8.88 ( ( size_size_list_int @ ( list_update_int @ Xs @ I2 @ X ) )
% 16.11/8.88 = ( size_size_list_int @ Xs ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_236_not__None__eq,axiom,
% 16.11/8.88 ! [X: option_nat] :
% 16.11/8.88 ( ( X != none_nat )
% 16.11/8.88 = ( ? [Y2: nat] :
% 16.11/8.88 ( X
% 16.11/8.88 = ( some_nat @ Y2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_237_not__None__eq,axiom,
% 16.11/8.88 ! [X: option4927543243414619207at_nat] :
% 16.11/8.88 ( ( X != none_P5556105721700978146at_nat )
% 16.11/8.88 = ( ? [Y2: product_prod_nat_nat] :
% 16.11/8.88 ( X
% 16.11/8.88 = ( some_P7363390416028606310at_nat @ Y2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_238_not__None__eq,axiom,
% 16.11/8.88 ! [X: option_num] :
% 16.11/8.88 ( ( X != none_num )
% 16.11/8.88 = ( ? [Y2: num] :
% 16.11/8.88 ( X
% 16.11/8.88 = ( some_num @ Y2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_239_valid__member__both__member__options,axiom,
% 16.11/8.88 ! [T: vEBT_VEBT,N2: nat,X: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.11/8.88 => ( ( vEBT_V8194947554948674370ptions @ T @ X )
% 16.11/8.88 => ( vEBT_vebt_member @ T @ X ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_240_both__member__options__equiv__member,axiom,
% 16.11/8.88 ! [T: vEBT_VEBT,N2: nat,X: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.11/8.88 => ( ( vEBT_V8194947554948674370ptions @ T @ X )
% 16.11/8.88 = ( vEBT_vebt_member @ T @ X ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_241_min__in__set__def,axiom,
% 16.11/8.88 ( vEBT_VEBT_min_in_set
% 16.11/8.88 = ( ^ [Xs2: set_nat,X2: nat] :
% 16.11/8.88 ( ( member_nat @ X2 @ Xs2 )
% 16.11/8.88 & ! [Y2: nat] :
% 16.11/8.88 ( ( member_nat @ Y2 @ Xs2 )
% 16.11/8.88 => ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_242_max__in__set__def,axiom,
% 16.11/8.88 ( vEBT_VEBT_max_in_set
% 16.11/8.88 = ( ^ [Xs2: set_nat,X2: nat] :
% 16.11/8.88 ( ( member_nat @ X2 @ Xs2 )
% 16.11/8.88 & ! [Y2: nat] :
% 16.11/8.88 ( ( member_nat @ Y2 @ Xs2 )
% 16.11/8.88 => ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_243__C4_OIH_C_I1_J,axiom,
% 16.11/8.88 ! [X5: vEBT_VEBT] :
% 16.11/8.88 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ treeList ) )
% 16.11/8.88 => ( ( vEBT_invar_vebt @ X5 @ na )
% 16.11/8.88 & ! [Xa: nat,Xb: nat] :
% 16.11/8.88 ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ X5 @ Xa ) @ Xb )
% 16.11/8.88 = ( ( Xa != Xb )
% 16.11/8.88 & ( vEBT_V8194947554948674370ptions @ X5 @ Xb ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_244__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062summin_O_ASome_Asummin_A_061_Avebt__mint_Asummary_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 16.11/8.88 ~ ! [Summin: nat] :
% 16.11/8.88 ( ( some_nat @ Summin )
% 16.11/8.88 != ( vEBT_vebt_mint @ summary ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_245_inthall,axiom,
% 16.11/8.88 ! [Xs: list_nat,P: nat > $o,N2: nat] :
% 16.11/8.88 ( ! [X4: nat] :
% 16.11/8.88 ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
% 16.11/8.88 => ( P @ X4 ) )
% 16.11/8.88 => ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
% 16.11/8.88 => ( P @ ( nth_nat @ Xs @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_246_inthall,axiom,
% 16.11/8.88 ! [Xs: list_real,P: real > $o,N2: nat] :
% 16.11/8.88 ( ! [X4: real] :
% 16.11/8.88 ( ( member_real @ X4 @ ( set_real2 @ Xs ) )
% 16.11/8.88 => ( P @ X4 ) )
% 16.11/8.88 => ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs ) )
% 16.11/8.88 => ( P @ ( nth_real @ Xs @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_247_inthall,axiom,
% 16.11/8.88 ! [Xs: list_complex,P: complex > $o,N2: nat] :
% 16.11/8.88 ( ! [X4: complex] :
% 16.11/8.88 ( ( member_complex @ X4 @ ( set_complex2 @ Xs ) )
% 16.11/8.88 => ( P @ X4 ) )
% 16.11/8.88 => ( ( ord_less_nat @ N2 @ ( size_s3451745648224563538omplex @ Xs ) )
% 16.11/8.88 => ( P @ ( nth_complex @ Xs @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_248_inthall,axiom,
% 16.11/8.88 ! [Xs: list_set_nat,P: set_nat > $o,N2: nat] :
% 16.11/8.88 ( ! [X4: set_nat] :
% 16.11/8.88 ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs ) )
% 16.11/8.88 => ( P @ X4 ) )
% 16.11/8.88 => ( ( ord_less_nat @ N2 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 16.11/8.88 => ( P @ ( nth_set_nat @ Xs @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_249_inthall,axiom,
% 16.11/8.88 ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,N2: nat] :
% 16.11/8.88 ( ! [X4: vEBT_VEBT] :
% 16.11/8.88 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 16.11/8.88 => ( P @ X4 ) )
% 16.11/8.88 => ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 16.11/8.88 => ( P @ ( nth_VEBT_VEBT @ Xs @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_250_inthall,axiom,
% 16.11/8.88 ! [Xs: list_o,P: $o > $o,N2: nat] :
% 16.11/8.88 ( ! [X4: $o] :
% 16.11/8.88 ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
% 16.11/8.88 => ( P @ X4 ) )
% 16.11/8.88 => ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
% 16.11/8.88 => ( P @ ( nth_o @ Xs @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_251_inthall,axiom,
% 16.11/8.88 ! [Xs: list_int,P: int > $o,N2: nat] :
% 16.11/8.88 ( ! [X4: int] :
% 16.11/8.88 ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
% 16.11/8.88 => ( P @ X4 ) )
% 16.11/8.88 => ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
% 16.11/8.88 => ( P @ ( nth_int @ Xs @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_252__092_060open_062x_A_061_Ami_092_060close_062,axiom,
% 16.11/8.88 xa = mi ).
% 16.11/8.88
% 16.11/8.88 thf(fact_253_maxt__member,axiom,
% 16.11/8.88 ! [T: vEBT_VEBT,N2: nat,Maxi2: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.11/8.88 => ( ( ( vEBT_vebt_maxt @ T )
% 16.11/8.88 = ( some_nat @ Maxi2 ) )
% 16.11/8.88 => ( vEBT_vebt_member @ T @ Maxi2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_254_mint__member,axiom,
% 16.11/8.88 ! [T: vEBT_VEBT,N2: nat,Maxi2: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.11/8.88 => ( ( ( vEBT_vebt_mint @ T )
% 16.11/8.88 = ( some_nat @ Maxi2 ) )
% 16.11/8.88 => ( vEBT_vebt_member @ T @ Maxi2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_255_option_Oinject,axiom,
% 16.11/8.88 ! [X22: nat,Y22: nat] :
% 16.11/8.88 ( ( ( some_nat @ X22 )
% 16.11/8.88 = ( some_nat @ Y22 ) )
% 16.11/8.88 = ( X22 = Y22 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_256_option_Oinject,axiom,
% 16.11/8.88 ! [X22: product_prod_nat_nat,Y22: product_prod_nat_nat] :
% 16.11/8.88 ( ( ( some_P7363390416028606310at_nat @ X22 )
% 16.11/8.88 = ( some_P7363390416028606310at_nat @ Y22 ) )
% 16.11/8.88 = ( X22 = Y22 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_257_option_Oinject,axiom,
% 16.11/8.88 ! [X22: num,Y22: num] :
% 16.11/8.88 ( ( ( some_num @ X22 )
% 16.11/8.88 = ( some_num @ Y22 ) )
% 16.11/8.88 = ( X22 = Y22 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_258_list__update__overwrite,axiom,
% 16.11/8.88 ! [Xs: list_VEBT_VEBT,I2: nat,X: vEBT_VEBT,Y: vEBT_VEBT] :
% 16.11/8.88 ( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X ) @ I2 @ Y )
% 16.11/8.88 = ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ Y ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_259_maxt__corr__help,axiom,
% 16.11/8.88 ! [T: vEBT_VEBT,N2: nat,Maxi2: nat,X: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.11/8.88 => ( ( ( vEBT_vebt_maxt @ T )
% 16.11/8.88 = ( some_nat @ Maxi2 ) )
% 16.11/8.88 => ( ( vEBT_vebt_member @ T @ X )
% 16.11/8.88 => ( ord_less_eq_nat @ X @ Maxi2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_260_mint__corr__help,axiom,
% 16.11/8.88 ! [T: vEBT_VEBT,N2: nat,Mini: nat,X: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.11/8.88 => ( ( ( vEBT_vebt_mint @ T )
% 16.11/8.88 = ( some_nat @ Mini ) )
% 16.11/8.88 => ( ( vEBT_vebt_member @ T @ X )
% 16.11/8.88 => ( ord_less_eq_nat @ Mini @ X ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_261_dp,axiom,
% 16.11/8.88 ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ).
% 16.11/8.88
% 16.11/8.88 thf(fact_262__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062lx_O_ASome_Alx_A_061_Avebt__mint_A_ItreeList_A_B_Asummin_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 16.11/8.88 ~ ! [Lx: nat] :
% 16.11/8.88 ( ( some_nat @ Lx )
% 16.11/8.88 != ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ summin ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_263__092_060open_062mi_A_092_060le_062_Ax_A_092_060and_062_Ax_A_092_060le_062_Ama_092_060close_062,axiom,
% 16.11/8.88 ( ( ord_less_eq_nat @ mi @ xa )
% 16.11/8.88 & ( ord_less_eq_nat @ xa @ ma ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_264__092_060open_062invar__vebt_A_ItreeList_A_B_Asummin_J_An_092_060close_062,axiom,
% 16.11/8.88 vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ summin ) @ na ).
% 16.11/8.88
% 16.11/8.88 thf(fact_265_member__bound,axiom,
% 16.11/8.88 ! [Tree: vEBT_VEBT,X: nat,N2: nat] :
% 16.11/8.88 ( ( vEBT_vebt_member @ Tree @ X )
% 16.11/8.88 => ( ( vEBT_invar_vebt @ Tree @ N2 )
% 16.11/8.88 => ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_266__092_060open_062invar__vebt_A_ItreeList_A_B_Amaxs_J_An_092_060close_062,axiom,
% 16.11/8.88 vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ na ).
% 16.11/8.88
% 16.11/8.88 thf(fact_267_numeral__le__iff,axiom,
% 16.11/8.88 ! [M: num,N2: num] :
% 16.11/8.88 ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
% 16.11/8.88 = ( ord_less_eq_num @ M @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_268_numeral__le__iff,axiom,
% 16.11/8.88 ! [M: num,N2: num] :
% 16.11/8.88 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
% 16.11/8.88 = ( ord_less_eq_num @ M @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_269_numeral__le__iff,axiom,
% 16.11/8.88 ! [M: num,N2: num] :
% 16.11/8.88 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
% 16.11/8.88 = ( ord_less_eq_num @ M @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_270_numeral__le__iff,axiom,
% 16.11/8.88 ! [M: num,N2: num] :
% 16.11/8.88 ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 16.11/8.88 = ( ord_less_eq_num @ M @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_271_misiz,axiom,
% 16.11/8.88 ! [T: vEBT_VEBT,N2: nat,M: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.11/8.88 => ( ( ( some_nat @ M )
% 16.11/8.88 = ( vEBT_vebt_mint @ T ) )
% 16.11/8.88 => ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_272_not__Some__eq,axiom,
% 16.11/8.88 ! [X: option_nat] :
% 16.11/8.88 ( ( ! [Y2: nat] :
% 16.11/8.88 ( X
% 16.11/8.88 != ( some_nat @ Y2 ) ) )
% 16.11/8.88 = ( X = none_nat ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_273_not__Some__eq,axiom,
% 16.11/8.88 ! [X: option4927543243414619207at_nat] :
% 16.11/8.88 ( ( ! [Y2: product_prod_nat_nat] :
% 16.11/8.88 ( X
% 16.11/8.88 != ( some_P7363390416028606310at_nat @ Y2 ) ) )
% 16.11/8.88 = ( X = none_P5556105721700978146at_nat ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_274_not__Some__eq,axiom,
% 16.11/8.88 ! [X: option_num] :
% 16.11/8.88 ( ( ! [Y2: num] :
% 16.11/8.88 ( X
% 16.11/8.88 != ( some_num @ Y2 ) ) )
% 16.11/8.88 = ( X = none_num ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_275_member__correct,axiom,
% 16.11/8.88 ! [T: vEBT_VEBT,N2: nat,X: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.11/8.88 => ( ( vEBT_vebt_member @ T @ X )
% 16.11/8.88 = ( member_nat @ X @ ( vEBT_set_vebt @ T ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_276_list__update__beyond,axiom,
% 16.11/8.88 ! [Xs: list_VEBT_VEBT,I2: nat,X: vEBT_VEBT] :
% 16.11/8.88 ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ I2 )
% 16.11/8.88 => ( ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X )
% 16.11/8.88 = Xs ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_277_list__update__beyond,axiom,
% 16.11/8.88 ! [Xs: list_o,I2: nat,X: $o] :
% 16.11/8.88 ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ I2 )
% 16.11/8.88 => ( ( list_update_o @ Xs @ I2 @ X )
% 16.11/8.88 = Xs ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_278_list__update__beyond,axiom,
% 16.11/8.88 ! [Xs: list_int,I2: nat,X: int] :
% 16.11/8.88 ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ I2 )
% 16.11/8.88 => ( ( list_update_int @ Xs @ I2 @ X )
% 16.11/8.88 = Xs ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_279_divide__le__eq__numeral1_I1_J,axiom,
% 16.11/8.88 ! [B: real,W: num,A: real] :
% 16.11/8.88 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
% 16.11/8.88 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_280_divide__le__eq__numeral1_I1_J,axiom,
% 16.11/8.88 ! [B: rat,W: num,A: rat] :
% 16.11/8.88 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
% 16.11/8.88 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_281_le__divide__eq__numeral1_I1_J,axiom,
% 16.11/8.88 ! [A: real,B: real,W: num] :
% 16.11/8.88 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 16.11/8.88 = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_282_le__divide__eq__numeral1_I1_J,axiom,
% 16.11/8.88 ! [A: rat,B: rat,W: num] :
% 16.11/8.88 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 16.11/8.88 = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_283_lesseq__shift,axiom,
% 16.11/8.88 ( ord_less_eq_nat
% 16.11/8.88 = ( ^ [X2: nat,Y2: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X2 ) @ ( some_nat @ Y2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_284_set__swap,axiom,
% 16.11/8.88 ! [I2: nat,Xs: list_VEBT_VEBT,J: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 16.11/8.88 => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 16.11/8.88 => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ ( nth_VEBT_VEBT @ Xs @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs @ I2 ) ) )
% 16.11/8.88 = ( set_VEBT_VEBT2 @ Xs ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_285_set__swap,axiom,
% 16.11/8.88 ! [I2: nat,Xs: list_o,J: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 16.11/8.88 => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs ) )
% 16.11/8.88 => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs @ I2 @ ( nth_o @ Xs @ J ) ) @ J @ ( nth_o @ Xs @ I2 ) ) )
% 16.11/8.88 = ( set_o2 @ Xs ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_286_set__swap,axiom,
% 16.11/8.88 ! [I2: nat,Xs: list_int,J: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 16.11/8.88 => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
% 16.11/8.88 => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs @ I2 @ ( nth_int @ Xs @ J ) ) @ J @ ( nth_int @ Xs @ I2 ) ) )
% 16.11/8.88 = ( set_int2 @ Xs ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_287__092_060open_062summin_A_K_A2_A_094_An_A_L_Alx_A_061_A_Iif_Ax_A_061_Ami_Athen_Athe_A_Ivebt__mint_Asummary_J_A_K_A2_A_094_A_Ideg_Adiv_A2_J_A_L_Athe_A_Ivebt__mint_A_ItreeList_A_B_Athe_A_Ivebt__mint_Asummary_J_J_J_Aelse_Ax_J_092_060close_062,axiom,
% 16.11/8.88 ( ( ( xa = mi )
% 16.11/8.88 => ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 16.11/8.88 = ( 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 @ treeList @ ( the_nat @ ( vEBT_vebt_mint @ summary ) ) ) ) ) ) ) )
% 16.11/8.88 & ( ( xa != mi )
% 16.11/8.88 => ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 16.11/8.88 = xa ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_288_local_Opower__def,axiom,
% 16.11/8.88 ( vEBT_VEBT_power
% 16.11/8.88 = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_289_div__le__mono,axiom,
% 16.11/8.88 ! [M: nat,N2: nat,K: nat] :
% 16.11/8.88 ( ( ord_less_eq_nat @ M @ N2 )
% 16.11/8.88 => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N2 @ K ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_290_div__le__dividend,axiom,
% 16.11/8.88 ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_291_all__set__conv__all__nth,axiom,
% 16.11/8.88 ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 16.11/8.88 ( ( ! [X2: vEBT_VEBT] :
% 16.11/8.88 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
% 16.11/8.88 => ( P @ X2 ) ) )
% 16.11/8.88 = ( ! [I3: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 16.11/8.88 => ( P @ ( nth_VEBT_VEBT @ Xs @ I3 ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_292_all__set__conv__all__nth,axiom,
% 16.11/8.88 ! [Xs: list_o,P: $o > $o] :
% 16.11/8.88 ( ( ! [X2: $o] :
% 16.11/8.88 ( ( member_o @ X2 @ ( set_o2 @ Xs ) )
% 16.11/8.88 => ( P @ X2 ) ) )
% 16.11/8.88 = ( ! [I3: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
% 16.11/8.88 => ( P @ ( nth_o @ Xs @ I3 ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_293_all__set__conv__all__nth,axiom,
% 16.11/8.88 ! [Xs: list_int,P: int > $o] :
% 16.11/8.88 ( ( ! [X2: int] :
% 16.11/8.88 ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
% 16.11/8.88 => ( P @ X2 ) ) )
% 16.11/8.88 = ( ! [I3: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
% 16.11/8.88 => ( P @ ( nth_int @ Xs @ I3 ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_294_all__nth__imp__all__set,axiom,
% 16.11/8.88 ! [Xs: list_nat,P: nat > $o,X: nat] :
% 16.11/8.88 ( ! [I4: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
% 16.11/8.88 => ( P @ ( nth_nat @ Xs @ I4 ) ) )
% 16.11/8.88 => ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 16.11/8.88 => ( P @ X ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_295_all__nth__imp__all__set,axiom,
% 16.11/8.88 ! [Xs: list_real,P: real > $o,X: real] :
% 16.11/8.88 ( ! [I4: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs ) )
% 16.11/8.88 => ( P @ ( nth_real @ Xs @ I4 ) ) )
% 16.11/8.88 => ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 16.11/8.88 => ( P @ X ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_296_all__nth__imp__all__set,axiom,
% 16.11/8.88 ! [Xs: list_complex,P: complex > $o,X: complex] :
% 16.11/8.88 ( ! [I4: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I4 @ ( size_s3451745648224563538omplex @ Xs ) )
% 16.11/8.88 => ( P @ ( nth_complex @ Xs @ I4 ) ) )
% 16.11/8.88 => ( ( member_complex @ X @ ( set_complex2 @ Xs ) )
% 16.11/8.88 => ( P @ X ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_297_all__nth__imp__all__set,axiom,
% 16.11/8.88 ! [Xs: list_set_nat,P: set_nat > $o,X: set_nat] :
% 16.11/8.88 ( ! [I4: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I4 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 16.11/8.88 => ( P @ ( nth_set_nat @ Xs @ I4 ) ) )
% 16.11/8.88 => ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 16.11/8.88 => ( P @ X ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_298_all__nth__imp__all__set,axiom,
% 16.11/8.88 ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
% 16.11/8.88 ( ! [I4: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 16.11/8.88 => ( P @ ( nth_VEBT_VEBT @ Xs @ I4 ) ) )
% 16.11/8.88 => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 16.11/8.88 => ( P @ X ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_299_all__nth__imp__all__set,axiom,
% 16.11/8.88 ! [Xs: list_o,P: $o > $o,X: $o] :
% 16.11/8.88 ( ! [I4: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
% 16.11/8.88 => ( P @ ( nth_o @ Xs @ I4 ) ) )
% 16.11/8.88 => ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 16.11/8.88 => ( P @ X ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_300_all__nth__imp__all__set,axiom,
% 16.11/8.88 ! [Xs: list_int,P: int > $o,X: int] :
% 16.11/8.88 ( ! [I4: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
% 16.11/8.88 => ( P @ ( nth_int @ Xs @ I4 ) ) )
% 16.11/8.88 => ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 16.11/8.88 => ( P @ X ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_301_in__set__conv__nth,axiom,
% 16.11/8.88 ! [X: nat,Xs: list_nat] :
% 16.11/8.88 ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 16.11/8.88 = ( ? [I3: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
% 16.11/8.88 & ( ( nth_nat @ Xs @ I3 )
% 16.11/8.88 = X ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_302_in__set__conv__nth,axiom,
% 16.11/8.88 ! [X: real,Xs: list_real] :
% 16.11/8.88 ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 16.11/8.88 = ( ? [I3: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs ) )
% 16.11/8.88 & ( ( nth_real @ Xs @ I3 )
% 16.11/8.88 = X ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_303_in__set__conv__nth,axiom,
% 16.11/8.88 ! [X: complex,Xs: list_complex] :
% 16.11/8.88 ( ( member_complex @ X @ ( set_complex2 @ Xs ) )
% 16.11/8.88 = ( ? [I3: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs ) )
% 16.11/8.88 & ( ( nth_complex @ Xs @ I3 )
% 16.11/8.88 = X ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_304_in__set__conv__nth,axiom,
% 16.11/8.88 ! [X: set_nat,Xs: list_set_nat] :
% 16.11/8.88 ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 16.11/8.88 = ( ? [I3: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I3 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 16.11/8.88 & ( ( nth_set_nat @ Xs @ I3 )
% 16.11/8.88 = X ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_305_in__set__conv__nth,axiom,
% 16.11/8.88 ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 16.11/8.88 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 16.11/8.88 = ( ? [I3: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 16.11/8.88 & ( ( nth_VEBT_VEBT @ Xs @ I3 )
% 16.11/8.88 = X ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_306_in__set__conv__nth,axiom,
% 16.11/8.88 ! [X: $o,Xs: list_o] :
% 16.11/8.88 ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 16.11/8.88 = ( ? [I3: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
% 16.11/8.88 & ( ( nth_o @ Xs @ I3 )
% 16.11/8.88 = X ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_307_in__set__conv__nth,axiom,
% 16.11/8.88 ! [X: int,Xs: list_int] :
% 16.11/8.88 ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 16.11/8.88 = ( ? [I3: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
% 16.11/8.88 & ( ( nth_int @ Xs @ I3 )
% 16.11/8.88 = X ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_308_list__ball__nth,axiom,
% 16.11/8.88 ! [N2: nat,Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 16.11/8.88 ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 16.11/8.88 => ( ! [X4: vEBT_VEBT] :
% 16.11/8.88 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 16.11/8.88 => ( P @ X4 ) )
% 16.11/8.88 => ( P @ ( nth_VEBT_VEBT @ Xs @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_309_list__ball__nth,axiom,
% 16.11/8.88 ! [N2: nat,Xs: list_o,P: $o > $o] :
% 16.11/8.88 ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
% 16.11/8.88 => ( ! [X4: $o] :
% 16.11/8.88 ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
% 16.11/8.88 => ( P @ X4 ) )
% 16.11/8.88 => ( P @ ( nth_o @ Xs @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_310_list__ball__nth,axiom,
% 16.11/8.88 ! [N2: nat,Xs: list_int,P: int > $o] :
% 16.11/8.88 ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
% 16.11/8.88 => ( ! [X4: int] :
% 16.11/8.88 ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
% 16.11/8.88 => ( P @ X4 ) )
% 16.11/8.88 => ( P @ ( nth_int @ Xs @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_311_nth__mem,axiom,
% 16.11/8.88 ! [N2: nat,Xs: list_nat] :
% 16.11/8.88 ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
% 16.11/8.88 => ( member_nat @ ( nth_nat @ Xs @ N2 ) @ ( set_nat2 @ Xs ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_312_nth__mem,axiom,
% 16.11/8.88 ! [N2: nat,Xs: list_real] :
% 16.11/8.88 ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs ) )
% 16.11/8.88 => ( member_real @ ( nth_real @ Xs @ N2 ) @ ( set_real2 @ Xs ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_313_nth__mem,axiom,
% 16.11/8.88 ! [N2: nat,Xs: list_complex] :
% 16.11/8.88 ( ( ord_less_nat @ N2 @ ( size_s3451745648224563538omplex @ Xs ) )
% 16.11/8.88 => ( member_complex @ ( nth_complex @ Xs @ N2 ) @ ( set_complex2 @ Xs ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_314_nth__mem,axiom,
% 16.11/8.88 ! [N2: nat,Xs: list_set_nat] :
% 16.11/8.88 ( ( ord_less_nat @ N2 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 16.11/8.88 => ( member_set_nat @ ( nth_set_nat @ Xs @ N2 ) @ ( set_set_nat2 @ Xs ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_315_nth__mem,axiom,
% 16.11/8.88 ! [N2: nat,Xs: list_VEBT_VEBT] :
% 16.11/8.88 ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 16.11/8.88 => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs @ N2 ) @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_316_nth__mem,axiom,
% 16.11/8.88 ! [N2: nat,Xs: list_o] :
% 16.11/8.88 ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
% 16.11/8.88 => ( member_o @ ( nth_o @ Xs @ N2 ) @ ( set_o2 @ Xs ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_317_nth__mem,axiom,
% 16.11/8.88 ! [N2: nat,Xs: list_int] :
% 16.11/8.88 ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
% 16.11/8.88 => ( member_int @ ( nth_int @ Xs @ N2 ) @ ( set_int2 @ Xs ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_318_set__update__memI,axiom,
% 16.11/8.88 ! [N2: nat,Xs: list_nat,X: nat] :
% 16.11/8.88 ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
% 16.11/8.88 => ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs @ N2 @ X ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_319_set__update__memI,axiom,
% 16.11/8.88 ! [N2: nat,Xs: list_real,X: real] :
% 16.11/8.88 ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs ) )
% 16.11/8.88 => ( member_real @ X @ ( set_real2 @ ( list_update_real @ Xs @ N2 @ X ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_320_set__update__memI,axiom,
% 16.11/8.88 ! [N2: nat,Xs: list_complex,X: complex] :
% 16.11/8.88 ( ( ord_less_nat @ N2 @ ( size_s3451745648224563538omplex @ Xs ) )
% 16.11/8.88 => ( member_complex @ X @ ( set_complex2 @ ( list_update_complex @ Xs @ N2 @ X ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_321_set__update__memI,axiom,
% 16.11/8.88 ! [N2: nat,Xs: list_set_nat,X: set_nat] :
% 16.11/8.88 ( ( ord_less_nat @ N2 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 16.11/8.88 => ( member_set_nat @ X @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ N2 @ X ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_322_set__update__memI,axiom,
% 16.11/8.88 ! [N2: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
% 16.11/8.88 ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 16.11/8.88 => ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ N2 @ X ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_323_set__update__memI,axiom,
% 16.11/8.88 ! [N2: nat,Xs: list_o,X: $o] :
% 16.11/8.88 ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
% 16.11/8.88 => ( member_o @ X @ ( set_o2 @ ( list_update_o @ Xs @ N2 @ X ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_324_set__update__memI,axiom,
% 16.11/8.88 ! [N2: nat,Xs: list_int,X: int] :
% 16.11/8.88 ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
% 16.11/8.88 => ( member_int @ X @ ( set_int2 @ ( list_update_int @ Xs @ N2 @ X ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_325_div__times__less__eq__dividend,axiom,
% 16.11/8.88 ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) @ M ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_326_times__div__less__eq__dividend,axiom,
% 16.11/8.88 ! [N2: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) @ M ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_327_neq__if__length__neq,axiom,
% 16.11/8.88 ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 16.11/8.88 ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 16.11/8.88 != ( size_s6755466524823107622T_VEBT @ Ys ) )
% 16.11/8.88 => ( Xs != Ys ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_328_neq__if__length__neq,axiom,
% 16.11/8.88 ! [Xs: list_o,Ys: list_o] :
% 16.11/8.88 ( ( ( size_size_list_o @ Xs )
% 16.11/8.88 != ( size_size_list_o @ Ys ) )
% 16.11/8.88 => ( Xs != Ys ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_329_neq__if__length__neq,axiom,
% 16.11/8.88 ! [Xs: list_int,Ys: list_int] :
% 16.11/8.88 ( ( ( size_size_list_int @ Xs )
% 16.11/8.88 != ( size_size_list_int @ Ys ) )
% 16.11/8.88 => ( Xs != Ys ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_330_Ex__list__of__length,axiom,
% 16.11/8.88 ! [N2: nat] :
% 16.11/8.88 ? [Xs3: list_VEBT_VEBT] :
% 16.11/8.88 ( ( size_s6755466524823107622T_VEBT @ Xs3 )
% 16.11/8.88 = N2 ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_331_Ex__list__of__length,axiom,
% 16.11/8.88 ! [N2: nat] :
% 16.11/8.88 ? [Xs3: list_o] :
% 16.11/8.88 ( ( size_size_list_o @ Xs3 )
% 16.11/8.88 = N2 ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_332_Ex__list__of__length,axiom,
% 16.11/8.88 ! [N2: nat] :
% 16.11/8.88 ? [Xs3: list_int] :
% 16.11/8.88 ( ( size_size_list_int @ Xs3 )
% 16.11/8.88 = N2 ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_333_list__update__swap,axiom,
% 16.11/8.88 ! [I2: nat,I5: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT,X6: vEBT_VEBT] :
% 16.11/8.88 ( ( I2 != I5 )
% 16.11/8.88 => ( ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X ) @ I5 @ X6 )
% 16.11/8.88 = ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I5 @ X6 ) @ I2 @ X ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_334_self__le__ge2__pow,axiom,
% 16.11/8.88 ! [K: nat,M: nat] :
% 16.11/8.88 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 16.11/8.88 => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_335_power2__nat__le__eq__le,axiom,
% 16.11/8.88 ! [M: nat,N2: nat] :
% 16.11/8.88 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.11/8.88 = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_336_power2__nat__le__imp__le,axiom,
% 16.11/8.88 ! [M: nat,N2: nat] :
% 16.11/8.88 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N2 )
% 16.11/8.88 => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_337_length__induct,axiom,
% 16.11/8.88 ! [P: list_VEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
% 16.11/8.88 ( ! [Xs3: list_VEBT_VEBT] :
% 16.11/8.88 ( ! [Ys2: list_VEBT_VEBT] :
% 16.11/8.88 ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys2 ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
% 16.11/8.88 => ( P @ Ys2 ) )
% 16.11/8.88 => ( P @ Xs3 ) )
% 16.11/8.88 => ( P @ Xs ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_338_length__induct,axiom,
% 16.11/8.88 ! [P: list_o > $o,Xs: list_o] :
% 16.11/8.88 ( ! [Xs3: list_o] :
% 16.11/8.88 ( ! [Ys2: list_o] :
% 16.11/8.88 ( ( ord_less_nat @ ( size_size_list_o @ Ys2 ) @ ( size_size_list_o @ Xs3 ) )
% 16.11/8.88 => ( P @ Ys2 ) )
% 16.11/8.88 => ( P @ Xs3 ) )
% 16.11/8.88 => ( P @ Xs ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_339_length__induct,axiom,
% 16.11/8.88 ! [P: list_int > $o,Xs: list_int] :
% 16.11/8.88 ( ! [Xs3: list_int] :
% 16.11/8.88 ( ! [Ys2: list_int] :
% 16.11/8.88 ( ( ord_less_nat @ ( size_size_list_int @ Ys2 ) @ ( size_size_list_int @ Xs3 ) )
% 16.11/8.88 => ( P @ Ys2 ) )
% 16.11/8.88 => ( P @ Xs3 ) )
% 16.11/8.88 => ( P @ Xs ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_340_combine__options__cases,axiom,
% 16.11/8.88 ! [X: option_nat,P: option_nat > option_nat > $o,Y: option_nat] :
% 16.11/8.88 ( ( ( X = none_nat )
% 16.11/8.88 => ( P @ X @ Y ) )
% 16.11/8.88 => ( ( ( Y = none_nat )
% 16.11/8.88 => ( P @ X @ Y ) )
% 16.11/8.88 => ( ! [A3: nat,B2: nat] :
% 16.11/8.88 ( ( X
% 16.11/8.88 = ( some_nat @ A3 ) )
% 16.11/8.88 => ( ( Y
% 16.11/8.88 = ( some_nat @ B2 ) )
% 16.11/8.88 => ( P @ X @ Y ) ) )
% 16.11/8.88 => ( P @ X @ Y ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_341_combine__options__cases,axiom,
% 16.11/8.88 ! [X: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
% 16.11/8.88 ( ( ( X = none_nat )
% 16.11/8.88 => ( P @ X @ Y ) )
% 16.11/8.88 => ( ( ( Y = none_P5556105721700978146at_nat )
% 16.11/8.88 => ( P @ X @ Y ) )
% 16.11/8.88 => ( ! [A3: nat,B2: product_prod_nat_nat] :
% 16.11/8.88 ( ( X
% 16.11/8.88 = ( some_nat @ A3 ) )
% 16.11/8.88 => ( ( Y
% 16.11/8.88 = ( some_P7363390416028606310at_nat @ B2 ) )
% 16.11/8.88 => ( P @ X @ Y ) ) )
% 16.11/8.88 => ( P @ X @ Y ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_342_combine__options__cases,axiom,
% 16.11/8.88 ! [X: option_nat,P: option_nat > option_num > $o,Y: option_num] :
% 16.11/8.88 ( ( ( X = none_nat )
% 16.11/8.88 => ( P @ X @ Y ) )
% 16.11/8.88 => ( ( ( Y = none_num )
% 16.11/8.88 => ( P @ X @ Y ) )
% 16.11/8.88 => ( ! [A3: nat,B2: num] :
% 16.11/8.88 ( ( X
% 16.11/8.88 = ( some_nat @ A3 ) )
% 16.11/8.88 => ( ( Y
% 16.11/8.88 = ( some_num @ B2 ) )
% 16.11/8.88 => ( P @ X @ Y ) ) )
% 16.11/8.88 => ( P @ X @ Y ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_343_combine__options__cases,axiom,
% 16.11/8.88 ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y: option_nat] :
% 16.11/8.88 ( ( ( X = none_P5556105721700978146at_nat )
% 16.11/8.88 => ( P @ X @ Y ) )
% 16.11/8.88 => ( ( ( Y = none_nat )
% 16.11/8.88 => ( P @ X @ Y ) )
% 16.11/8.88 => ( ! [A3: product_prod_nat_nat,B2: nat] :
% 16.11/8.88 ( ( X
% 16.11/8.88 = ( some_P7363390416028606310at_nat @ A3 ) )
% 16.11/8.88 => ( ( Y
% 16.11/8.88 = ( some_nat @ B2 ) )
% 16.11/8.88 => ( P @ X @ Y ) ) )
% 16.11/8.88 => ( P @ X @ Y ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_344_combine__options__cases,axiom,
% 16.11/8.88 ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
% 16.11/8.88 ( ( ( X = none_P5556105721700978146at_nat )
% 16.11/8.88 => ( P @ X @ Y ) )
% 16.11/8.88 => ( ( ( Y = none_P5556105721700978146at_nat )
% 16.11/8.88 => ( P @ X @ Y ) )
% 16.11/8.88 => ( ! [A3: product_prod_nat_nat,B2: product_prod_nat_nat] :
% 16.11/8.88 ( ( X
% 16.11/8.88 = ( some_P7363390416028606310at_nat @ A3 ) )
% 16.11/8.88 => ( ( Y
% 16.11/8.88 = ( some_P7363390416028606310at_nat @ B2 ) )
% 16.11/8.88 => ( P @ X @ Y ) ) )
% 16.11/8.88 => ( P @ X @ Y ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_345_combine__options__cases,axiom,
% 16.11/8.88 ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y: option_num] :
% 16.11/8.88 ( ( ( X = none_P5556105721700978146at_nat )
% 16.11/8.88 => ( P @ X @ Y ) )
% 16.11/8.88 => ( ( ( Y = none_num )
% 16.11/8.88 => ( P @ X @ Y ) )
% 16.11/8.88 => ( ! [A3: product_prod_nat_nat,B2: num] :
% 16.11/8.88 ( ( X
% 16.11/8.88 = ( some_P7363390416028606310at_nat @ A3 ) )
% 16.11/8.88 => ( ( Y
% 16.11/8.88 = ( some_num @ B2 ) )
% 16.11/8.88 => ( P @ X @ Y ) ) )
% 16.11/8.88 => ( P @ X @ Y ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_346_combine__options__cases,axiom,
% 16.11/8.88 ! [X: option_num,P: option_num > option_nat > $o,Y: option_nat] :
% 16.11/8.88 ( ( ( X = none_num )
% 16.11/8.88 => ( P @ X @ Y ) )
% 16.11/8.88 => ( ( ( Y = none_nat )
% 16.11/8.88 => ( P @ X @ Y ) )
% 16.11/8.88 => ( ! [A3: num,B2: nat] :
% 16.11/8.88 ( ( X
% 16.11/8.88 = ( some_num @ A3 ) )
% 16.11/8.88 => ( ( Y
% 16.11/8.88 = ( some_nat @ B2 ) )
% 16.11/8.88 => ( P @ X @ Y ) ) )
% 16.11/8.88 => ( P @ X @ Y ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_347_combine__options__cases,axiom,
% 16.11/8.88 ! [X: option_num,P: option_num > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
% 16.11/8.88 ( ( ( X = none_num )
% 16.11/8.88 => ( P @ X @ Y ) )
% 16.11/8.88 => ( ( ( Y = none_P5556105721700978146at_nat )
% 16.11/8.88 => ( P @ X @ Y ) )
% 16.11/8.88 => ( ! [A3: num,B2: product_prod_nat_nat] :
% 16.11/8.88 ( ( X
% 16.11/8.88 = ( some_num @ A3 ) )
% 16.11/8.88 => ( ( Y
% 16.11/8.88 = ( some_P7363390416028606310at_nat @ B2 ) )
% 16.11/8.88 => ( P @ X @ Y ) ) )
% 16.11/8.88 => ( P @ X @ Y ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_348_combine__options__cases,axiom,
% 16.11/8.88 ! [X: option_num,P: option_num > option_num > $o,Y: option_num] :
% 16.11/8.88 ( ( ( X = none_num )
% 16.11/8.88 => ( P @ X @ Y ) )
% 16.11/8.88 => ( ( ( Y = none_num )
% 16.11/8.88 => ( P @ X @ Y ) )
% 16.11/8.88 => ( ! [A3: num,B2: num] :
% 16.11/8.88 ( ( X
% 16.11/8.88 = ( some_num @ A3 ) )
% 16.11/8.88 => ( ( Y
% 16.11/8.88 = ( some_num @ B2 ) )
% 16.11/8.88 => ( P @ X @ Y ) ) )
% 16.11/8.88 => ( P @ X @ Y ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_349_split__option__all,axiom,
% 16.11/8.88 ( ( ^ [P2: option_nat > $o] :
% 16.11/8.88 ! [X7: option_nat] : ( P2 @ X7 ) )
% 16.11/8.88 = ( ^ [P3: option_nat > $o] :
% 16.11/8.88 ( ( P3 @ none_nat )
% 16.11/8.88 & ! [X2: nat] : ( P3 @ ( some_nat @ X2 ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_350_split__option__all,axiom,
% 16.11/8.88 ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 16.11/8.88 ! [X7: option4927543243414619207at_nat] : ( P2 @ X7 ) )
% 16.11/8.88 = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 16.11/8.88 ( ( P3 @ none_P5556105721700978146at_nat )
% 16.11/8.88 & ! [X2: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X2 ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_351_split__option__all,axiom,
% 16.11/8.88 ( ( ^ [P2: option_num > $o] :
% 16.11/8.88 ! [X7: option_num] : ( P2 @ X7 ) )
% 16.11/8.88 = ( ^ [P3: option_num > $o] :
% 16.11/8.88 ( ( P3 @ none_num )
% 16.11/8.88 & ! [X2: num] : ( P3 @ ( some_num @ X2 ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_352_split__option__ex,axiom,
% 16.11/8.88 ( ( ^ [P2: option_nat > $o] :
% 16.11/8.88 ? [X7: option_nat] : ( P2 @ X7 ) )
% 16.11/8.88 = ( ^ [P3: option_nat > $o] :
% 16.11/8.88 ( ( P3 @ none_nat )
% 16.11/8.88 | ? [X2: nat] : ( P3 @ ( some_nat @ X2 ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_353_split__option__ex,axiom,
% 16.11/8.88 ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 16.11/8.88 ? [X7: option4927543243414619207at_nat] : ( P2 @ X7 ) )
% 16.11/8.88 = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 16.11/8.88 ( ( P3 @ none_P5556105721700978146at_nat )
% 16.11/8.88 | ? [X2: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X2 ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_354_split__option__ex,axiom,
% 16.11/8.88 ( ( ^ [P2: option_num > $o] :
% 16.11/8.88 ? [X7: option_num] : ( P2 @ X7 ) )
% 16.11/8.88 = ( ^ [P3: option_num > $o] :
% 16.11/8.88 ( ( P3 @ none_num )
% 16.11/8.88 | ? [X2: num] : ( P3 @ ( some_num @ X2 ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_355_option_Oexhaust,axiom,
% 16.11/8.88 ! [Y: option_nat] :
% 16.11/8.88 ( ( Y != none_nat )
% 16.11/8.88 => ~ ! [X23: nat] :
% 16.11/8.88 ( Y
% 16.11/8.88 != ( some_nat @ X23 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_356_option_Oexhaust,axiom,
% 16.11/8.88 ! [Y: option4927543243414619207at_nat] :
% 16.11/8.88 ( ( Y != none_P5556105721700978146at_nat )
% 16.11/8.88 => ~ ! [X23: product_prod_nat_nat] :
% 16.11/8.88 ( Y
% 16.11/8.88 != ( some_P7363390416028606310at_nat @ X23 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_357_option_Oexhaust,axiom,
% 16.11/8.88 ! [Y: option_num] :
% 16.11/8.88 ( ( Y != none_num )
% 16.11/8.88 => ~ ! [X23: num] :
% 16.11/8.88 ( Y
% 16.11/8.88 != ( some_num @ X23 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_358_option_OdiscI,axiom,
% 16.11/8.88 ! [Option: option_nat,X22: nat] :
% 16.11/8.88 ( ( Option
% 16.11/8.88 = ( some_nat @ X22 ) )
% 16.11/8.88 => ( Option != none_nat ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_359_option_OdiscI,axiom,
% 16.11/8.88 ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
% 16.11/8.88 ( ( Option
% 16.11/8.88 = ( some_P7363390416028606310at_nat @ X22 ) )
% 16.11/8.88 => ( Option != none_P5556105721700978146at_nat ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_360_option_OdiscI,axiom,
% 16.11/8.88 ! [Option: option_num,X22: num] :
% 16.11/8.88 ( ( Option
% 16.11/8.88 = ( some_num @ X22 ) )
% 16.11/8.88 => ( Option != none_num ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_361_option_Odistinct_I1_J,axiom,
% 16.11/8.88 ! [X22: nat] :
% 16.11/8.88 ( none_nat
% 16.11/8.88 != ( some_nat @ X22 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_362_option_Odistinct_I1_J,axiom,
% 16.11/8.88 ! [X22: product_prod_nat_nat] :
% 16.11/8.88 ( none_P5556105721700978146at_nat
% 16.11/8.88 != ( some_P7363390416028606310at_nat @ X22 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_363_option_Odistinct_I1_J,axiom,
% 16.11/8.88 ! [X22: num] :
% 16.11/8.88 ( none_num
% 16.11/8.88 != ( some_num @ X22 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_364_option_Osel,axiom,
% 16.11/8.88 ! [X22: nat] :
% 16.11/8.88 ( ( the_nat @ ( some_nat @ X22 ) )
% 16.11/8.88 = X22 ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_365_option_Osel,axiom,
% 16.11/8.88 ! [X22: product_prod_nat_nat] :
% 16.11/8.88 ( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
% 16.11/8.88 = X22 ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_366_option_Osel,axiom,
% 16.11/8.88 ! [X22: num] :
% 16.11/8.88 ( ( the_num @ ( some_num @ X22 ) )
% 16.11/8.88 = X22 ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_367_option_Oexpand,axiom,
% 16.11/8.88 ! [Option: option_nat,Option2: option_nat] :
% 16.11/8.88 ( ( ( Option = none_nat )
% 16.11/8.88 = ( Option2 = none_nat ) )
% 16.11/8.88 => ( ( ( Option != none_nat )
% 16.11/8.88 => ( ( Option2 != none_nat )
% 16.11/8.88 => ( ( the_nat @ Option )
% 16.11/8.88 = ( the_nat @ Option2 ) ) ) )
% 16.11/8.88 => ( Option = Option2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_368_option_Oexpand,axiom,
% 16.11/8.88 ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
% 16.11/8.88 ( ( ( Option = none_P5556105721700978146at_nat )
% 16.11/8.88 = ( Option2 = none_P5556105721700978146at_nat ) )
% 16.11/8.88 => ( ( ( Option != none_P5556105721700978146at_nat )
% 16.11/8.88 => ( ( Option2 != none_P5556105721700978146at_nat )
% 16.11/8.88 => ( ( the_Pr8591224930841456533at_nat @ Option )
% 16.11/8.88 = ( the_Pr8591224930841456533at_nat @ Option2 ) ) ) )
% 16.11/8.88 => ( Option = Option2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_369_option_Oexpand,axiom,
% 16.11/8.88 ! [Option: option_num,Option2: option_num] :
% 16.11/8.88 ( ( ( Option = none_num )
% 16.11/8.88 = ( Option2 = none_num ) )
% 16.11/8.88 => ( ( ( Option != none_num )
% 16.11/8.88 => ( ( Option2 != none_num )
% 16.11/8.88 => ( ( the_num @ Option )
% 16.11/8.88 = ( the_num @ Option2 ) ) ) )
% 16.11/8.88 => ( Option = Option2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_370_list__eq__iff__nth__eq,axiom,
% 16.11/8.88 ( ( ^ [Y3: list_VEBT_VEBT,Z2: list_VEBT_VEBT] : ( Y3 = Z2 ) )
% 16.11/8.88 = ( ^ [Xs2: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
% 16.11/8.88 ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 16.11/8.88 = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
% 16.11/8.88 & ! [I3: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 16.11/8.88 => ( ( nth_VEBT_VEBT @ Xs2 @ I3 )
% 16.11/8.88 = ( nth_VEBT_VEBT @ Ys3 @ I3 ) ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_371_list__eq__iff__nth__eq,axiom,
% 16.11/8.88 ( ( ^ [Y3: list_o,Z2: list_o] : ( Y3 = Z2 ) )
% 16.11/8.88 = ( ^ [Xs2: list_o,Ys3: list_o] :
% 16.11/8.88 ( ( ( size_size_list_o @ Xs2 )
% 16.11/8.88 = ( size_size_list_o @ Ys3 ) )
% 16.11/8.88 & ! [I3: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
% 16.11/8.88 => ( ( nth_o @ Xs2 @ I3 )
% 16.11/8.88 = ( nth_o @ Ys3 @ I3 ) ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_372_list__eq__iff__nth__eq,axiom,
% 16.11/8.88 ( ( ^ [Y3: list_int,Z2: list_int] : ( Y3 = Z2 ) )
% 16.11/8.88 = ( ^ [Xs2: list_int,Ys3: list_int] :
% 16.11/8.88 ( ( ( size_size_list_int @ Xs2 )
% 16.11/8.88 = ( size_size_list_int @ Ys3 ) )
% 16.11/8.88 & ! [I3: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
% 16.11/8.88 => ( ( nth_int @ Xs2 @ I3 )
% 16.11/8.88 = ( nth_int @ Ys3 @ I3 ) ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_373_Skolem__list__nth,axiom,
% 16.11/8.88 ! [K: nat,P: nat > vEBT_VEBT > $o] :
% 16.11/8.88 ( ( ! [I3: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I3 @ K )
% 16.11/8.88 => ? [X3: vEBT_VEBT] : ( P @ I3 @ X3 ) ) )
% 16.11/8.88 = ( ? [Xs2: list_VEBT_VEBT] :
% 16.11/8.88 ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 16.11/8.88 = K )
% 16.11/8.88 & ! [I3: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I3 @ K )
% 16.11/8.88 => ( P @ I3 @ ( nth_VEBT_VEBT @ Xs2 @ I3 ) ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_374_Skolem__list__nth,axiom,
% 16.11/8.88 ! [K: nat,P: nat > $o > $o] :
% 16.11/8.88 ( ( ! [I3: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I3 @ K )
% 16.11/8.88 => ? [X3: $o] : ( P @ I3 @ X3 ) ) )
% 16.11/8.88 = ( ? [Xs2: list_o] :
% 16.11/8.88 ( ( ( size_size_list_o @ Xs2 )
% 16.11/8.88 = K )
% 16.11/8.88 & ! [I3: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I3 @ K )
% 16.11/8.88 => ( P @ I3 @ ( nth_o @ Xs2 @ I3 ) ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_375_Skolem__list__nth,axiom,
% 16.11/8.88 ! [K: nat,P: nat > int > $o] :
% 16.11/8.88 ( ( ! [I3: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I3 @ K )
% 16.11/8.88 => ? [X3: int] : ( P @ I3 @ X3 ) ) )
% 16.11/8.88 = ( ? [Xs2: list_int] :
% 16.11/8.88 ( ( ( size_size_list_int @ Xs2 )
% 16.11/8.88 = K )
% 16.11/8.88 & ! [I3: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I3 @ K )
% 16.11/8.88 => ( P @ I3 @ ( nth_int @ Xs2 @ I3 ) ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_376_nth__equalityI,axiom,
% 16.11/8.88 ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 16.11/8.88 ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 16.11/8.88 = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 16.11/8.88 => ( ! [I4: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 16.11/8.88 => ( ( nth_VEBT_VEBT @ Xs @ I4 )
% 16.11/8.88 = ( nth_VEBT_VEBT @ Ys @ I4 ) ) )
% 16.11/8.88 => ( Xs = Ys ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_377_nth__equalityI,axiom,
% 16.11/8.88 ! [Xs: list_o,Ys: list_o] :
% 16.11/8.88 ( ( ( size_size_list_o @ Xs )
% 16.11/8.88 = ( size_size_list_o @ Ys ) )
% 16.11/8.88 => ( ! [I4: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
% 16.11/8.88 => ( ( nth_o @ Xs @ I4 )
% 16.11/8.88 = ( nth_o @ Ys @ I4 ) ) )
% 16.11/8.88 => ( Xs = Ys ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_378_nth__equalityI,axiom,
% 16.11/8.88 ! [Xs: list_int,Ys: list_int] :
% 16.11/8.88 ( ( ( size_size_list_int @ Xs )
% 16.11/8.88 = ( size_size_list_int @ Ys ) )
% 16.11/8.88 => ( ! [I4: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
% 16.11/8.88 => ( ( nth_int @ Xs @ I4 )
% 16.11/8.88 = ( nth_int @ Ys @ I4 ) ) )
% 16.11/8.88 => ( Xs = Ys ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_379_option_Oexhaust__sel,axiom,
% 16.11/8.88 ! [Option: option_nat] :
% 16.11/8.88 ( ( Option != none_nat )
% 16.11/8.88 => ( Option
% 16.11/8.88 = ( some_nat @ ( the_nat @ Option ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_380_option_Oexhaust__sel,axiom,
% 16.11/8.88 ! [Option: option4927543243414619207at_nat] :
% 16.11/8.88 ( ( Option != none_P5556105721700978146at_nat )
% 16.11/8.88 => ( Option
% 16.11/8.88 = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_381_option_Oexhaust__sel,axiom,
% 16.11/8.88 ! [Option: option_num] :
% 16.11/8.88 ( ( Option != none_num )
% 16.11/8.88 => ( Option
% 16.11/8.88 = ( some_num @ ( the_num @ Option ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_382_nth__list__update,axiom,
% 16.11/8.88 ! [I2: nat,Xs: list_VEBT_VEBT,J: nat,X: vEBT_VEBT] :
% 16.11/8.88 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 16.11/8.88 => ( ( ( I2 = J )
% 16.11/8.88 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X ) @ J )
% 16.11/8.88 = X ) )
% 16.11/8.88 & ( ( I2 != J )
% 16.11/8.88 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X ) @ J )
% 16.11/8.88 = ( nth_VEBT_VEBT @ Xs @ J ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_383_nth__list__update,axiom,
% 16.11/8.88 ! [I2: nat,Xs: list_o,X: $o,J: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 16.11/8.88 => ( ( nth_o @ ( list_update_o @ Xs @ I2 @ X ) @ J )
% 16.11/8.88 = ( ( ( I2 = J )
% 16.11/8.88 => X )
% 16.11/8.88 & ( ( I2 != J )
% 16.11/8.88 => ( nth_o @ Xs @ J ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_384_nth__list__update,axiom,
% 16.11/8.88 ! [I2: nat,Xs: list_int,J: nat,X: int] :
% 16.11/8.88 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 16.11/8.88 => ( ( ( I2 = J )
% 16.11/8.88 => ( ( nth_int @ ( list_update_int @ Xs @ I2 @ X ) @ J )
% 16.11/8.88 = X ) )
% 16.11/8.88 & ( ( I2 != J )
% 16.11/8.88 => ( ( nth_int @ ( list_update_int @ Xs @ I2 @ X ) @ J )
% 16.11/8.88 = ( nth_int @ Xs @ J ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_385_list__update__same__conv,axiom,
% 16.11/8.88 ! [I2: nat,Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
% 16.11/8.88 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 16.11/8.88 => ( ( ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X )
% 16.11/8.88 = Xs )
% 16.11/8.88 = ( ( nth_VEBT_VEBT @ Xs @ I2 )
% 16.11/8.88 = X ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_386_list__update__same__conv,axiom,
% 16.11/8.88 ! [I2: nat,Xs: list_o,X: $o] :
% 16.11/8.88 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 16.11/8.88 => ( ( ( list_update_o @ Xs @ I2 @ X )
% 16.11/8.88 = Xs )
% 16.11/8.88 = ( ( nth_o @ Xs @ I2 )
% 16.11/8.88 = X ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_387_list__update__same__conv,axiom,
% 16.11/8.88 ! [I2: nat,Xs: list_int,X: int] :
% 16.11/8.88 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 16.11/8.88 => ( ( ( list_update_int @ Xs @ I2 @ X )
% 16.11/8.88 = Xs )
% 16.11/8.88 = ( ( nth_int @ Xs @ I2 )
% 16.11/8.88 = X ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_388_field__sum__of__halves,axiom,
% 16.11/8.88 ! [X: real] :
% 16.11/8.88 ( ( plus_plus_real @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.11/8.88 = X ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_389_field__sum__of__halves,axiom,
% 16.11/8.88 ! [X: rat] :
% 16.11/8.88 ( ( plus_plus_rat @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 16.11/8.88 = X ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_390_post__member__pre__member,axiom,
% 16.11/8.88 ! [T: vEBT_VEBT,N2: nat,X: nat,Y: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.11/8.88 => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.11/8.88 => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.11/8.88 => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X ) @ Y )
% 16.11/8.88 => ( ( vEBT_vebt_member @ T @ Y )
% 16.11/8.88 | ( X = Y ) ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_391_valid__insert__both__member__options__add,axiom,
% 16.11/8.88 ! [T: vEBT_VEBT,N2: nat,X: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.11/8.88 => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.11/8.88 => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X ) @ X ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_392_valid__insert__both__member__options__pres,axiom,
% 16.11/8.88 ! [T: vEBT_VEBT,N2: nat,X: nat,Y: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.11/8.88 => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.11/8.88 => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.11/8.88 => ( ( vEBT_V8194947554948674370ptions @ T @ X )
% 16.11/8.88 => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y ) @ X ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_393_helpyd,axiom,
% 16.11/8.88 ! [T: vEBT_VEBT,N2: nat,X: nat,Y: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.11/8.88 => ( ( ( vEBT_vebt_succ @ T @ X )
% 16.11/8.88 = ( some_nat @ Y ) )
% 16.11/8.88 => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_394_helpypredd,axiom,
% 16.11/8.88 ! [T: vEBT_VEBT,N2: nat,X: nat,Y: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.11/8.88 => ( ( ( vEBT_vebt_pred @ T @ X )
% 16.11/8.88 = ( some_nat @ Y ) )
% 16.11/8.88 => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_395_semiring__norm_I76_J,axiom,
% 16.11/8.88 ! [N2: num] : ( ord_less_num @ one @ ( bit0 @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_396_both__member__options__ding,axiom,
% 16.11/8.88 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat,X: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 )
% 16.11/8.88 => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 16.11/8.88 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.11/8.88 => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ X ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_397_sum__squares__bound,axiom,
% 16.11/8.88 ! [X: real,Y: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_398_sum__squares__bound,axiom,
% 16.11/8.88 ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_399_semiring__norm_I2_J,axiom,
% 16.11/8.88 ( ( plus_plus_num @ one @ one )
% 16.11/8.88 = ( bit0 @ one ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_400_set__n__deg__not__0,axiom,
% 16.11/8.88 ! [TreeList2: list_VEBT_VEBT,N2: nat,M: nat] :
% 16.11/8.88 ( ! [X4: vEBT_VEBT] :
% 16.11/8.88 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 16.11/8.88 => ( vEBT_invar_vebt @ X4 @ N2 ) )
% 16.11/8.88 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 16.11/8.88 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 16.11/8.88 => ( ord_less_eq_nat @ one_one_nat @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_401_semiring__norm_I75_J,axiom,
% 16.11/8.88 ! [M: num] :
% 16.11/8.88 ~ ( ord_less_num @ M @ one ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_402_semiring__norm_I78_J,axiom,
% 16.11/8.88 ! [M: num,N2: num] :
% 16.11/8.88 ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
% 16.11/8.88 = ( ord_less_num @ M @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_403_deg__deg__n,axiom,
% 16.11/8.88 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 )
% 16.11/8.88 => ( Deg = N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_404_semiring__norm_I87_J,axiom,
% 16.11/8.88 ! [M: num,N2: num] :
% 16.11/8.88 ( ( ( bit0 @ M )
% 16.11/8.88 = ( bit0 @ N2 ) )
% 16.11/8.88 = ( M = N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_405_semiring__norm_I83_J,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( one
% 16.11/8.88 != ( bit0 @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_406_semiring__norm_I85_J,axiom,
% 16.11/8.88 ! [M: num] :
% 16.11/8.88 ( ( bit0 @ M )
% 16.11/8.88 != one ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_407_bits__div__by__1,axiom,
% 16.11/8.88 ! [A: nat] :
% 16.11/8.88 ( ( divide_divide_nat @ A @ one_one_nat )
% 16.11/8.88 = A ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_408_bits__div__by__1,axiom,
% 16.11/8.88 ! [A: int] :
% 16.11/8.88 ( ( divide_divide_int @ A @ one_one_int )
% 16.11/8.88 = A ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_409_power__one,axiom,
% 16.11/8.88 ! [N2: nat] :
% 16.11/8.88 ( ( power_power_rat @ one_one_rat @ N2 )
% 16.11/8.88 = one_one_rat ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_410_power__one,axiom,
% 16.11/8.88 ! [N2: nat] :
% 16.11/8.88 ( ( power_power_nat @ one_one_nat @ N2 )
% 16.11/8.88 = one_one_nat ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_411_power__one,axiom,
% 16.11/8.88 ! [N2: nat] :
% 16.11/8.88 ( ( power_power_real @ one_one_real @ N2 )
% 16.11/8.88 = one_one_real ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_412_power__one,axiom,
% 16.11/8.88 ! [N2: nat] :
% 16.11/8.88 ( ( power_power_int @ one_one_int @ N2 )
% 16.11/8.88 = one_one_int ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_413_power__one,axiom,
% 16.11/8.88 ! [N2: nat] :
% 16.11/8.88 ( ( power_power_complex @ one_one_complex @ N2 )
% 16.11/8.88 = one_one_complex ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_414_power__one__right,axiom,
% 16.11/8.88 ! [A: nat] :
% 16.11/8.88 ( ( power_power_nat @ A @ one_one_nat )
% 16.11/8.88 = A ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_415_power__one__right,axiom,
% 16.11/8.88 ! [A: real] :
% 16.11/8.88 ( ( power_power_real @ A @ one_one_nat )
% 16.11/8.88 = A ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_416_power__one__right,axiom,
% 16.11/8.88 ! [A: int] :
% 16.11/8.88 ( ( power_power_int @ A @ one_one_nat )
% 16.11/8.88 = A ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_417_power__one__right,axiom,
% 16.11/8.88 ! [A: complex] :
% 16.11/8.88 ( ( power_power_complex @ A @ one_one_nat )
% 16.11/8.88 = A ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_418_semiring__norm_I6_J,axiom,
% 16.11/8.88 ! [M: num,N2: num] :
% 16.11/8.88 ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
% 16.11/8.88 = ( bit0 @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_419_semiring__norm_I13_J,axiom,
% 16.11/8.88 ! [M: num,N2: num] :
% 16.11/8.88 ( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
% 16.11/8.88 = ( bit0 @ ( bit0 @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_420_semiring__norm_I11_J,axiom,
% 16.11/8.88 ! [M: num] :
% 16.11/8.88 ( ( times_times_num @ M @ one )
% 16.11/8.88 = M ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_421_semiring__norm_I12_J,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( times_times_num @ one @ N2 )
% 16.11/8.88 = N2 ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_422_semiring__norm_I71_J,axiom,
% 16.11/8.88 ! [M: num,N2: num] :
% 16.11/8.88 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
% 16.11/8.88 = ( ord_less_eq_num @ M @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_423_semiring__norm_I68_J,axiom,
% 16.11/8.88 ! [N2: num] : ( ord_less_eq_num @ one @ N2 ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_424_one__eq__numeral__iff,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( one_one_complex
% 16.11/8.88 = ( numera6690914467698888265omplex @ N2 ) )
% 16.11/8.88 = ( one = N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_425_one__eq__numeral__iff,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( one_one_real
% 16.11/8.88 = ( numeral_numeral_real @ N2 ) )
% 16.11/8.88 = ( one = N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_426_one__eq__numeral__iff,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( one_one_rat
% 16.11/8.88 = ( numeral_numeral_rat @ N2 ) )
% 16.11/8.88 = ( one = N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_427_one__eq__numeral__iff,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( one_one_nat
% 16.11/8.88 = ( numeral_numeral_nat @ N2 ) )
% 16.11/8.88 = ( one = N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_428_one__eq__numeral__iff,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( one_one_int
% 16.11/8.88 = ( numeral_numeral_int @ N2 ) )
% 16.11/8.88 = ( one = N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_429_numeral__eq__one__iff,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( ( numera6690914467698888265omplex @ N2 )
% 16.11/8.88 = one_one_complex )
% 16.11/8.88 = ( N2 = one ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_430_numeral__eq__one__iff,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( ( numeral_numeral_real @ N2 )
% 16.11/8.88 = one_one_real )
% 16.11/8.88 = ( N2 = one ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_431_numeral__eq__one__iff,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( ( numeral_numeral_rat @ N2 )
% 16.11/8.88 = one_one_rat )
% 16.11/8.88 = ( N2 = one ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_432_numeral__eq__one__iff,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( ( numeral_numeral_nat @ N2 )
% 16.11/8.88 = one_one_nat )
% 16.11/8.88 = ( N2 = one ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_433_numeral__eq__one__iff,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( ( numeral_numeral_int @ N2 )
% 16.11/8.88 = one_one_int )
% 16.11/8.88 = ( N2 = one ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_434_power__inject__exp,axiom,
% 16.11/8.88 ! [A: real,M: nat,N2: nat] :
% 16.11/8.88 ( ( ord_less_real @ one_one_real @ A )
% 16.11/8.88 => ( ( ( power_power_real @ A @ M )
% 16.11/8.88 = ( power_power_real @ A @ N2 ) )
% 16.11/8.88 = ( M = N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_435_power__inject__exp,axiom,
% 16.11/8.88 ! [A: rat,M: nat,N2: nat] :
% 16.11/8.88 ( ( ord_less_rat @ one_one_rat @ A )
% 16.11/8.88 => ( ( ( power_power_rat @ A @ M )
% 16.11/8.88 = ( power_power_rat @ A @ N2 ) )
% 16.11/8.88 = ( M = N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_436_power__inject__exp,axiom,
% 16.11/8.88 ! [A: nat,M: nat,N2: nat] :
% 16.11/8.88 ( ( ord_less_nat @ one_one_nat @ A )
% 16.11/8.88 => ( ( ( power_power_nat @ A @ M )
% 16.11/8.88 = ( power_power_nat @ A @ N2 ) )
% 16.11/8.88 = ( M = N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_437_power__inject__exp,axiom,
% 16.11/8.88 ! [A: int,M: nat,N2: nat] :
% 16.11/8.88 ( ( ord_less_int @ one_one_int @ A )
% 16.11/8.88 => ( ( ( power_power_int @ A @ M )
% 16.11/8.88 = ( power_power_int @ A @ N2 ) )
% 16.11/8.88 = ( M = N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_438_semiring__norm_I69_J,axiom,
% 16.11/8.88 ! [M: num] :
% 16.11/8.88 ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_439_power__strict__increasing__iff,axiom,
% 16.11/8.88 ! [B: real,X: nat,Y: nat] :
% 16.11/8.88 ( ( ord_less_real @ one_one_real @ B )
% 16.11/8.88 => ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
% 16.11/8.88 = ( ord_less_nat @ X @ Y ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_440_power__strict__increasing__iff,axiom,
% 16.11/8.88 ! [B: rat,X: nat,Y: nat] :
% 16.11/8.88 ( ( ord_less_rat @ one_one_rat @ B )
% 16.11/8.88 => ( ( ord_less_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
% 16.11/8.88 = ( ord_less_nat @ X @ Y ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_441_power__strict__increasing__iff,axiom,
% 16.11/8.88 ! [B: nat,X: nat,Y: nat] :
% 16.11/8.88 ( ( ord_less_nat @ one_one_nat @ B )
% 16.11/8.88 => ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
% 16.11/8.88 = ( ord_less_nat @ X @ Y ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_442_power__strict__increasing__iff,axiom,
% 16.11/8.88 ! [B: int,X: nat,Y: nat] :
% 16.11/8.88 ( ( ord_less_int @ one_one_int @ B )
% 16.11/8.88 => ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
% 16.11/8.88 = ( ord_less_nat @ X @ Y ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_443_one__add__one,axiom,
% 16.11/8.88 ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
% 16.11/8.88 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_444_one__add__one,axiom,
% 16.11/8.88 ( ( plus_plus_real @ one_one_real @ one_one_real )
% 16.11/8.88 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_445_one__add__one,axiom,
% 16.11/8.88 ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
% 16.11/8.88 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_446_one__add__one,axiom,
% 16.11/8.88 ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 16.11/8.88 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_447_one__add__one,axiom,
% 16.11/8.88 ( ( plus_plus_int @ one_one_int @ one_one_int )
% 16.11/8.88 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_448_power__increasing__iff,axiom,
% 16.11/8.88 ! [B: real,X: nat,Y: nat] :
% 16.11/8.88 ( ( ord_less_real @ one_one_real @ B )
% 16.11/8.88 => ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
% 16.11/8.88 = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_449_power__increasing__iff,axiom,
% 16.11/8.88 ! [B: rat,X: nat,Y: nat] :
% 16.11/8.88 ( ( ord_less_rat @ one_one_rat @ B )
% 16.11/8.88 => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
% 16.11/8.88 = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_450_power__increasing__iff,axiom,
% 16.11/8.88 ! [B: nat,X: nat,Y: nat] :
% 16.11/8.88 ( ( ord_less_nat @ one_one_nat @ B )
% 16.11/8.88 => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
% 16.11/8.88 = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_451_power__increasing__iff,axiom,
% 16.11/8.88 ! [B: int,X: nat,Y: nat] :
% 16.11/8.88 ( ( ord_less_int @ one_one_int @ B )
% 16.11/8.88 => ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
% 16.11/8.88 = ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_452_one__plus__numeral,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N2 ) )
% 16.11/8.88 = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_453_one__plus__numeral,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N2 ) )
% 16.11/8.88 = ( numeral_numeral_real @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_454_one__plus__numeral,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) )
% 16.11/8.88 = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_455_one__plus__numeral,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
% 16.11/8.88 = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_456_one__plus__numeral,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
% 16.11/8.88 = ( numeral_numeral_int @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_457_numeral__plus__one,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ one_one_complex )
% 16.11/8.88 = ( numera6690914467698888265omplex @ ( plus_plus_num @ N2 @ one ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_458_numeral__plus__one,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ one_one_real )
% 16.11/8.88 = ( numeral_numeral_real @ ( plus_plus_num @ N2 @ one ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_459_numeral__plus__one,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat )
% 16.11/8.88 = ( numeral_numeral_rat @ ( plus_plus_num @ N2 @ one ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_460_numeral__plus__one,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
% 16.11/8.88 = ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_461_numeral__plus__one,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
% 16.11/8.88 = ( numeral_numeral_int @ ( plus_plus_num @ N2 @ one ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_462_numeral__le__one__iff,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ one_one_real )
% 16.11/8.88 = ( ord_less_eq_num @ N2 @ one ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_463_numeral__le__one__iff,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat )
% 16.11/8.88 = ( ord_less_eq_num @ N2 @ one ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_464_numeral__le__one__iff,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
% 16.11/8.88 = ( ord_less_eq_num @ N2 @ one ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_465_numeral__le__one__iff,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
% 16.11/8.88 = ( ord_less_eq_num @ N2 @ one ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_466_one__less__numeral__iff,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N2 ) )
% 16.11/8.88 = ( ord_less_num @ one @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_467_one__less__numeral__iff,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) )
% 16.11/8.88 = ( ord_less_num @ one @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_468_one__less__numeral__iff,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
% 16.11/8.88 = ( ord_less_num @ one @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_469_one__less__numeral__iff,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
% 16.11/8.88 = ( ord_less_num @ one @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_470_pred__correct,axiom,
% 16.11/8.88 ! [T: vEBT_VEBT,N2: nat,X: nat,Sx: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.11/8.88 => ( ( ( vEBT_vebt_pred @ T @ X )
% 16.11/8.88 = ( some_nat @ Sx ) )
% 16.11/8.88 = ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_471_succ__correct,axiom,
% 16.11/8.88 ! [T: vEBT_VEBT,N2: nat,X: nat,Sx: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.11/8.88 => ( ( ( vEBT_vebt_succ @ T @ X )
% 16.11/8.88 = ( some_nat @ Sx ) )
% 16.11/8.88 = ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_472_le__num__One__iff,axiom,
% 16.11/8.88 ! [X: num] :
% 16.11/8.88 ( ( ord_less_eq_num @ X @ one )
% 16.11/8.88 = ( X = one ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_473_le__numeral__extra_I4_J,axiom,
% 16.11/8.88 ord_less_eq_real @ one_one_real @ one_one_real ).
% 16.11/8.88
% 16.11/8.88 thf(fact_474_le__numeral__extra_I4_J,axiom,
% 16.11/8.88 ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% 16.11/8.88
% 16.11/8.88 thf(fact_475_le__numeral__extra_I4_J,axiom,
% 16.11/8.88 ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 16.11/8.88
% 16.11/8.88 thf(fact_476_le__numeral__extra_I4_J,axiom,
% 16.11/8.88 ord_less_eq_int @ one_one_int @ one_one_int ).
% 16.11/8.88
% 16.11/8.88 thf(fact_477_less__numeral__extra_I4_J,axiom,
% 16.11/8.88 ~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_478_less__numeral__extra_I4_J,axiom,
% 16.11/8.88 ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_479_less__numeral__extra_I4_J,axiom,
% 16.11/8.88 ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_480_less__numeral__extra_I4_J,axiom,
% 16.11/8.88 ~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_481_subset__code_I1_J,axiom,
% 16.11/8.88 ! [Xs: list_real,B3: set_real] :
% 16.11/8.88 ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B3 )
% 16.11/8.88 = ( ! [X2: real] :
% 16.11/8.88 ( ( member_real @ X2 @ ( set_real2 @ Xs ) )
% 16.11/8.88 => ( member_real @ X2 @ B3 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_482_subset__code_I1_J,axiom,
% 16.11/8.88 ! [Xs: list_complex,B3: set_complex] :
% 16.11/8.88 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ B3 )
% 16.11/8.88 = ( ! [X2: complex] :
% 16.11/8.88 ( ( member_complex @ X2 @ ( set_complex2 @ Xs ) )
% 16.11/8.88 => ( member_complex @ X2 @ B3 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_483_subset__code_I1_J,axiom,
% 16.11/8.88 ! [Xs: list_set_nat,B3: set_set_nat] :
% 16.11/8.88 ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B3 )
% 16.11/8.88 = ( ! [X2: set_nat] :
% 16.11/8.88 ( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
% 16.11/8.88 => ( member_set_nat @ X2 @ B3 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_484_subset__code_I1_J,axiom,
% 16.11/8.88 ! [Xs: list_VEBT_VEBT,B3: set_VEBT_VEBT] :
% 16.11/8.88 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ B3 )
% 16.11/8.88 = ( ! [X2: vEBT_VEBT] :
% 16.11/8.88 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
% 16.11/8.88 => ( member_VEBT_VEBT @ X2 @ B3 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_485_subset__code_I1_J,axiom,
% 16.11/8.88 ! [Xs: list_int,B3: set_int] :
% 16.11/8.88 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B3 )
% 16.11/8.88 = ( ! [X2: int] :
% 16.11/8.88 ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
% 16.11/8.88 => ( member_int @ X2 @ B3 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_486_subset__code_I1_J,axiom,
% 16.11/8.88 ! [Xs: list_nat,B3: set_nat] :
% 16.11/8.88 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B3 )
% 16.11/8.88 = ( ! [X2: nat] :
% 16.11/8.88 ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
% 16.11/8.88 => ( member_nat @ X2 @ B3 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_487_vebt__delete_Osimps_I4_J,axiom,
% 16.11/8.88 ! [Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] :
% 16.11/8.88 ( ( vEBT_vebt_delete @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Uu )
% 16.11/8.88 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_488_L2__set__mult__ineq__lemma,axiom,
% 16.11/8.88 ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C ) ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_489_one__le__numeral,axiom,
% 16.11/8.88 ! [N2: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_490_one__le__numeral,axiom,
% 16.11/8.88 ! [N2: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_491_one__le__numeral,axiom,
% 16.11/8.88 ! [N2: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_492_one__le__numeral,axiom,
% 16.11/8.88 ! [N2: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_493_not__numeral__less__one,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ one_one_real ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_494_not__numeral__less__one,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_495_not__numeral__less__one,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_496_not__numeral__less__one,axiom,
% 16.11/8.88 ! [N2: num] :
% 16.11/8.88 ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_497_one__plus__numeral__commute,axiom,
% 16.11/8.88 ! [X: num] :
% 16.11/8.88 ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X ) )
% 16.11/8.88 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_498_one__plus__numeral__commute,axiom,
% 16.11/8.88 ! [X: num] :
% 16.11/8.88 ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X ) )
% 16.11/8.88 = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_499_one__plus__numeral__commute,axiom,
% 16.11/8.88 ! [X: num] :
% 16.11/8.88 ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
% 16.11/8.88 = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_500_one__plus__numeral__commute,axiom,
% 16.11/8.88 ! [X: num] :
% 16.11/8.88 ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
% 16.11/8.88 = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_501_one__plus__numeral__commute,axiom,
% 16.11/8.88 ! [X: num] :
% 16.11/8.88 ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
% 16.11/8.88 = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_502_numeral__One,axiom,
% 16.11/8.88 ( ( numera6690914467698888265omplex @ one )
% 16.11/8.88 = one_one_complex ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_503_numeral__One,axiom,
% 16.11/8.88 ( ( numeral_numeral_real @ one )
% 16.11/8.88 = one_one_real ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_504_numeral__One,axiom,
% 16.11/8.88 ( ( numeral_numeral_rat @ one )
% 16.11/8.88 = one_one_rat ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_505_numeral__One,axiom,
% 16.11/8.88 ( ( numeral_numeral_nat @ one )
% 16.11/8.88 = one_one_nat ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_506_numeral__One,axiom,
% 16.11/8.88 ( ( numeral_numeral_int @ one )
% 16.11/8.88 = one_one_int ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_507_one__le__power,axiom,
% 16.11/8.88 ! [A: real,N2: nat] :
% 16.11/8.88 ( ( ord_less_eq_real @ one_one_real @ A )
% 16.11/8.88 => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_508_one__le__power,axiom,
% 16.11/8.88 ! [A: rat,N2: nat] :
% 16.11/8.88 ( ( ord_less_eq_rat @ one_one_rat @ A )
% 16.11/8.88 => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_509_one__le__power,axiom,
% 16.11/8.88 ! [A: nat,N2: nat] :
% 16.11/8.88 ( ( ord_less_eq_nat @ one_one_nat @ A )
% 16.11/8.88 => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_510_one__le__power,axiom,
% 16.11/8.88 ! [A: int,N2: nat] :
% 16.11/8.88 ( ( ord_less_eq_int @ one_one_int @ A )
% 16.11/8.88 => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_511_left__right__inverse__power,axiom,
% 16.11/8.88 ! [X: complex,Y: complex,N2: nat] :
% 16.11/8.88 ( ( ( times_times_complex @ X @ Y )
% 16.11/8.88 = one_one_complex )
% 16.11/8.88 => ( ( times_times_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ Y @ N2 ) )
% 16.11/8.88 = one_one_complex ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_512_left__right__inverse__power,axiom,
% 16.11/8.88 ! [X: real,Y: real,N2: nat] :
% 16.11/8.88 ( ( ( times_times_real @ X @ Y )
% 16.11/8.88 = one_one_real )
% 16.11/8.88 => ( ( times_times_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ N2 ) )
% 16.11/8.88 = one_one_real ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_513_left__right__inverse__power,axiom,
% 16.11/8.88 ! [X: rat,Y: rat,N2: nat] :
% 16.11/8.88 ( ( ( times_times_rat @ X @ Y )
% 16.11/8.88 = one_one_rat )
% 16.11/8.88 => ( ( times_times_rat @ ( power_power_rat @ X @ N2 ) @ ( power_power_rat @ Y @ N2 ) )
% 16.11/8.88 = one_one_rat ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_514_left__right__inverse__power,axiom,
% 16.11/8.88 ! [X: nat,Y: nat,N2: nat] :
% 16.11/8.88 ( ( ( times_times_nat @ X @ Y )
% 16.11/8.88 = one_one_nat )
% 16.11/8.88 => ( ( times_times_nat @ ( power_power_nat @ X @ N2 ) @ ( power_power_nat @ Y @ N2 ) )
% 16.11/8.88 = one_one_nat ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_515_left__right__inverse__power,axiom,
% 16.11/8.88 ! [X: int,Y: int,N2: nat] :
% 16.11/8.88 ( ( ( times_times_int @ X @ Y )
% 16.11/8.88 = one_one_int )
% 16.11/8.88 => ( ( times_times_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ N2 ) )
% 16.11/8.88 = one_one_int ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_516_power__one__over,axiom,
% 16.11/8.88 ! [A: complex,N2: nat] :
% 16.11/8.88 ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N2 )
% 16.11/8.88 = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_517_power__one__over,axiom,
% 16.11/8.88 ! [A: real,N2: nat] :
% 16.11/8.88 ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N2 )
% 16.11/8.88 = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_518_power__one__over,axiom,
% 16.11/8.88 ! [A: rat,N2: nat] :
% 16.11/8.88 ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N2 )
% 16.11/8.88 = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_519_numerals_I1_J,axiom,
% 16.11/8.88 ( ( numeral_numeral_nat @ one )
% 16.11/8.88 = one_one_nat ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_520_set__update__subsetI,axiom,
% 16.11/8.88 ! [Xs: list_real,A2: set_real,X: real,I2: nat] :
% 16.11/8.88 ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A2 )
% 16.11/8.88 => ( ( member_real @ X @ A2 )
% 16.11/8.88 => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I2 @ X ) ) @ A2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_521_set__update__subsetI,axiom,
% 16.11/8.88 ! [Xs: list_complex,A2: set_complex,X: complex,I2: nat] :
% 16.11/8.88 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A2 )
% 16.11/8.88 => ( ( member_complex @ X @ A2 )
% 16.11/8.88 => ( ord_le211207098394363844omplex @ ( set_complex2 @ ( list_update_complex @ Xs @ I2 @ X ) ) @ A2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_522_set__update__subsetI,axiom,
% 16.11/8.88 ! [Xs: list_set_nat,A2: set_set_nat,X: set_nat,I2: nat] :
% 16.11/8.88 ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ A2 )
% 16.11/8.88 => ( ( member_set_nat @ X @ A2 )
% 16.11/8.88 => ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ I2 @ X ) ) @ A2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_523_set__update__subsetI,axiom,
% 16.11/8.88 ! [Xs: list_int,A2: set_int,X: int,I2: nat] :
% 16.11/8.88 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
% 16.11/8.88 => ( ( member_int @ X @ A2 )
% 16.11/8.88 => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I2 @ X ) ) @ A2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_524_set__update__subsetI,axiom,
% 16.11/8.88 ! [Xs: list_VEBT_VEBT,A2: set_VEBT_VEBT,X: vEBT_VEBT,I2: nat] :
% 16.11/8.88 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
% 16.11/8.88 => ( ( member_VEBT_VEBT @ X @ A2 )
% 16.11/8.88 => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I2 @ X ) ) @ A2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_525_set__update__subsetI,axiom,
% 16.11/8.88 ! [Xs: list_nat,A2: set_nat,X: nat,I2: nat] :
% 16.11/8.88 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
% 16.11/8.88 => ( ( member_nat @ X @ A2 )
% 16.11/8.88 => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I2 @ X ) ) @ A2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_526_power__less__power__Suc,axiom,
% 16.11/8.88 ! [A: real,N2: nat] :
% 16.11/8.88 ( ( ord_less_real @ one_one_real @ A )
% 16.11/8.88 => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_527_power__less__power__Suc,axiom,
% 16.11/8.88 ! [A: rat,N2: nat] :
% 16.11/8.88 ( ( ord_less_rat @ one_one_rat @ A )
% 16.11/8.88 => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_528_power__less__power__Suc,axiom,
% 16.11/8.88 ! [A: nat,N2: nat] :
% 16.11/8.88 ( ( ord_less_nat @ one_one_nat @ A )
% 16.11/8.88 => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_529_power__less__power__Suc,axiom,
% 16.11/8.88 ! [A: int,N2: nat] :
% 16.11/8.88 ( ( ord_less_int @ one_one_int @ A )
% 16.11/8.88 => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_530_power__gt1__lemma,axiom,
% 16.11/8.88 ! [A: real,N2: nat] :
% 16.11/8.88 ( ( ord_less_real @ one_one_real @ A )
% 16.11/8.88 => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_531_power__gt1__lemma,axiom,
% 16.11/8.88 ! [A: rat,N2: nat] :
% 16.11/8.88 ( ( ord_less_rat @ one_one_rat @ A )
% 16.11/8.88 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_532_power__gt1__lemma,axiom,
% 16.11/8.88 ! [A: nat,N2: nat] :
% 16.11/8.88 ( ( ord_less_nat @ one_one_nat @ A )
% 16.11/8.88 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_533_power__gt1__lemma,axiom,
% 16.11/8.88 ! [A: int,N2: nat] :
% 16.11/8.88 ( ( ord_less_int @ one_one_int @ A )
% 16.11/8.88 => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_534_power__less__imp__less__exp,axiom,
% 16.11/8.88 ! [A: real,M: nat,N2: nat] :
% 16.11/8.88 ( ( ord_less_real @ one_one_real @ A )
% 16.11/8.88 => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) )
% 16.11/8.88 => ( ord_less_nat @ M @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_535_power__less__imp__less__exp,axiom,
% 16.11/8.88 ! [A: rat,M: nat,N2: nat] :
% 16.11/8.88 ( ( ord_less_rat @ one_one_rat @ A )
% 16.11/8.88 => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) )
% 16.11/8.88 => ( ord_less_nat @ M @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_536_power__less__imp__less__exp,axiom,
% 16.11/8.88 ! [A: nat,M: nat,N2: nat] :
% 16.11/8.88 ( ( ord_less_nat @ one_one_nat @ A )
% 16.11/8.88 => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
% 16.11/8.88 => ( ord_less_nat @ M @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_537_power__less__imp__less__exp,axiom,
% 16.11/8.88 ! [A: int,M: nat,N2: nat] :
% 16.11/8.88 ( ( ord_less_int @ one_one_int @ A )
% 16.11/8.88 => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
% 16.11/8.88 => ( ord_less_nat @ M @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_538_power__strict__increasing,axiom,
% 16.11/8.88 ! [N2: nat,N3: nat,A: real] :
% 16.11/8.88 ( ( ord_less_nat @ N2 @ N3 )
% 16.11/8.88 => ( ( ord_less_real @ one_one_real @ A )
% 16.11/8.88 => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N3 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_539_power__strict__increasing,axiom,
% 16.11/8.88 ! [N2: nat,N3: nat,A: rat] :
% 16.11/8.88 ( ( ord_less_nat @ N2 @ N3 )
% 16.11/8.88 => ( ( ord_less_rat @ one_one_rat @ A )
% 16.11/8.88 => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ A @ N3 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_540_power__strict__increasing,axiom,
% 16.11/8.88 ! [N2: nat,N3: nat,A: nat] :
% 16.11/8.88 ( ( ord_less_nat @ N2 @ N3 )
% 16.11/8.88 => ( ( ord_less_nat @ one_one_nat @ A )
% 16.11/8.88 => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_541_power__strict__increasing,axiom,
% 16.11/8.88 ! [N2: nat,N3: nat,A: int] :
% 16.11/8.88 ( ( ord_less_nat @ N2 @ N3 )
% 16.11/8.88 => ( ( ord_less_int @ one_one_int @ A )
% 16.11/8.88 => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N3 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_542_power__increasing,axiom,
% 16.11/8.88 ! [N2: nat,N3: nat,A: real] :
% 16.11/8.88 ( ( ord_less_eq_nat @ N2 @ N3 )
% 16.11/8.88 => ( ( ord_less_eq_real @ one_one_real @ A )
% 16.11/8.88 => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N3 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_543_power__increasing,axiom,
% 16.11/8.88 ! [N2: nat,N3: nat,A: rat] :
% 16.11/8.88 ( ( ord_less_eq_nat @ N2 @ N3 )
% 16.11/8.88 => ( ( ord_less_eq_rat @ one_one_rat @ A )
% 16.11/8.88 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ A @ N3 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_544_power__increasing,axiom,
% 16.11/8.88 ! [N2: nat,N3: nat,A: nat] :
% 16.11/8.88 ( ( ord_less_eq_nat @ N2 @ N3 )
% 16.11/8.88 => ( ( ord_less_eq_nat @ one_one_nat @ A )
% 16.11/8.88 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_545_power__increasing,axiom,
% 16.11/8.88 ! [N2: nat,N3: nat,A: int] :
% 16.11/8.88 ( ( ord_less_eq_nat @ N2 @ N3 )
% 16.11/8.88 => ( ( ord_less_eq_int @ one_one_int @ A )
% 16.11/8.88 => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N3 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_546_power__le__imp__le__exp,axiom,
% 16.11/8.88 ! [A: real,M: nat,N2: nat] :
% 16.11/8.88 ( ( ord_less_real @ one_one_real @ A )
% 16.11/8.88 => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) )
% 16.11/8.88 => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_547_power__le__imp__le__exp,axiom,
% 16.11/8.88 ! [A: rat,M: nat,N2: nat] :
% 16.11/8.88 ( ( ord_less_rat @ one_one_rat @ A )
% 16.11/8.88 => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) )
% 16.11/8.88 => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_548_power__le__imp__le__exp,axiom,
% 16.11/8.88 ! [A: nat,M: nat,N2: nat] :
% 16.11/8.88 ( ( ord_less_nat @ one_one_nat @ A )
% 16.11/8.88 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
% 16.11/8.88 => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_549_power__le__imp__le__exp,axiom,
% 16.11/8.88 ! [A: int,M: nat,N2: nat] :
% 16.11/8.88 ( ( ord_less_int @ one_one_int @ A )
% 16.11/8.88 => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
% 16.11/8.88 => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_550_one__power2,axiom,
% 16.11/8.88 ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.88 = one_one_rat ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_551_one__power2,axiom,
% 16.11/8.88 ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.88 = one_one_nat ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_552_one__power2,axiom,
% 16.11/8.88 ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.88 = one_one_real ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_553_one__power2,axiom,
% 16.11/8.88 ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.88 = one_one_int ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_554_one__power2,axiom,
% 16.11/8.88 ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.88 = one_one_complex ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_555_nat__1__add__1,axiom,
% 16.11/8.88 ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 16.11/8.88 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_556_left__add__mult__distrib,axiom,
% 16.11/8.88 ! [I2: nat,U: nat,J: nat,K: nat] :
% 16.11/8.88 ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
% 16.11/8.88 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J ) @ U ) @ K ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_557_ex__power__ivl2,axiom,
% 16.11/8.88 ! [B: nat,K: nat] :
% 16.11/8.88 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 16.11/8.88 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 16.11/8.88 => ? [N4: nat] :
% 16.11/8.88 ( ( ord_less_nat @ ( power_power_nat @ B @ N4 ) @ K )
% 16.11/8.88 & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_558_ex__power__ivl1,axiom,
% 16.11/8.88 ! [B: nat,K: nat] :
% 16.11/8.88 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 16.11/8.88 => ( ( ord_less_eq_nat @ one_one_nat @ K )
% 16.11/8.88 => ? [N4: nat] :
% 16.11/8.88 ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N4 ) @ K )
% 16.11/8.88 & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_559_four__x__squared,axiom,
% 16.11/8.88 ! [X: real] :
% 16.11/8.88 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.11/8.88 = ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_560_mint__sound,axiom,
% 16.11/8.88 ! [T: vEBT_VEBT,N2: nat,X: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.11/8.88 => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
% 16.11/8.88 => ( ( vEBT_vebt_mint @ T )
% 16.11/8.88 = ( some_nat @ X ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_561_mint__corr,axiom,
% 16.11/8.88 ! [T: vEBT_VEBT,N2: nat,X: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.11/8.88 => ( ( ( vEBT_vebt_mint @ T )
% 16.11/8.88 = ( some_nat @ X ) )
% 16.11/8.88 => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_562_maxt__sound,axiom,
% 16.11/8.88 ! [T: vEBT_VEBT,N2: nat,X: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.11/8.88 => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
% 16.11/8.88 => ( ( vEBT_vebt_maxt @ T )
% 16.11/8.88 = ( some_nat @ X ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_563_maxt__corr,axiom,
% 16.11/8.88 ! [T: vEBT_VEBT,N2: nat,X: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.11/8.88 => ( ( ( vEBT_vebt_maxt @ T )
% 16.11/8.88 = ( some_nat @ X ) )
% 16.11/8.88 => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_564_xnin,axiom,
% 16.11/8.88 vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_565_set__vebt__set__vebt_H__valid,axiom,
% 16.11/8.88 ! [T: vEBT_VEBT,N2: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.11/8.88 => ( ( vEBT_set_vebt @ T )
% 16.11/8.88 = ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_566_invar__vebt_Ointros_I2_J,axiom,
% 16.11/8.88 ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 16.11/8.88 ( ! [X4: vEBT_VEBT] :
% 16.11/8.88 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 16.11/8.88 => ( vEBT_invar_vebt @ X4 @ N2 ) )
% 16.11/8.88 => ( ( vEBT_invar_vebt @ Summary @ M )
% 16.11/8.88 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 16.11/8.88 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 16.11/8.88 => ( ( M = N2 )
% 16.11/8.88 => ( ( Deg
% 16.11/8.88 = ( plus_plus_nat @ N2 @ M ) )
% 16.11/8.88 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 16.11/8.88 => ( ! [X4: vEBT_VEBT] :
% 16.11/8.88 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 16.11/8.88 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
% 16.11/8.88 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_567_both__member__options__from__chilf__to__complete__tree,axiom,
% 16.11/8.88 ! [X: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 16.11/8.88 ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 16.11/8.88 => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 16.11/8.88 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.11/8.88 => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_568_member__inv,axiom,
% 16.11/8.88 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 16.11/8.88 ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 16.11/8.88 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 16.11/8.88 & ( ( X = Mi )
% 16.11/8.88 | ( X = Ma )
% 16.11/8.88 | ( ( ord_less_nat @ X @ Ma )
% 16.11/8.88 & ( ord_less_nat @ Mi @ X )
% 16.11/8.88 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 16.11/8.88 & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_569_tvalid,axiom,
% 16.11/8.88 vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ deg ).
% 16.11/8.88
% 16.11/8.88 thf(fact_570_both__member__options__from__complete__tree__to__child,axiom,
% 16.11/8.88 ! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 16.11/8.88 ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 16.11/8.88 => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 16.11/8.88 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.11/8.88 | ( X = Mi )
% 16.11/8.88 | ( X = Ma ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_571_VEBT_Oinject_I1_J,axiom,
% 16.11/8.88 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
% 16.11/8.88 ( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 16.11/8.88 = ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
% 16.11/8.88 = ( ( X11 = Y11 )
% 16.11/8.88 & ( X12 = Y12 )
% 16.11/8.88 & ( X13 = Y13 )
% 16.11/8.88 & ( X14 = Y14 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_572_real__divide__square__eq,axiom,
% 16.11/8.88 ! [R: real,A: real] :
% 16.11/8.88 ( ( divide_divide_real @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ R ) )
% 16.11/8.88 = ( divide_divide_real @ A @ R ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_573_succ__member,axiom,
% 16.11/8.88 ! [T: vEBT_VEBT,X: nat,Y: nat] :
% 16.11/8.88 ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
% 16.11/8.88 = ( ( vEBT_vebt_member @ T @ Y )
% 16.11/8.88 & ( ord_less_nat @ X @ Y )
% 16.11/8.88 & ! [Z3: nat] :
% 16.11/8.88 ( ( ( vEBT_vebt_member @ T @ Z3 )
% 16.11/8.88 & ( ord_less_nat @ X @ Z3 ) )
% 16.11/8.88 => ( ord_less_eq_nat @ Y @ Z3 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_574_pred__member,axiom,
% 16.11/8.88 ! [T: vEBT_VEBT,X: nat,Y: nat] :
% 16.11/8.88 ( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
% 16.11/8.88 = ( ( vEBT_vebt_member @ T @ Y )
% 16.11/8.88 & ( ord_less_nat @ Y @ X )
% 16.11/8.88 & ! [Z3: nat] :
% 16.11/8.88 ( ( ( vEBT_vebt_member @ T @ Z3 )
% 16.11/8.88 & ( ord_less_nat @ Z3 @ X ) )
% 16.11/8.88 => ( ord_less_eq_nat @ Z3 @ Y ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_575_succ__corr,axiom,
% 16.11/8.88 ! [T: vEBT_VEBT,N2: nat,X: nat,Sx: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.11/8.88 => ( ( ( vEBT_vebt_succ @ T @ X )
% 16.11/8.88 = ( some_nat @ Sx ) )
% 16.11/8.88 = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_576_pred__corr,axiom,
% 16.11/8.88 ! [T: vEBT_VEBT,N2: nat,X: nat,Px: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.11/8.88 => ( ( ( vEBT_vebt_pred @ T @ X )
% 16.11/8.88 = ( some_nat @ Px ) )
% 16.11/8.88 = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Px ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_577_mi__eq__ma__no__ch,axiom,
% 16.11/8.88 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
% 16.11/8.88 => ( ( Mi = Ma )
% 16.11/8.88 => ( ! [X5: vEBT_VEBT] :
% 16.11/8.88 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 16.11/8.88 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
% 16.11/8.88 & ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_578_geqmaxNone,axiom,
% 16.11/8.88 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat,X: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N2 )
% 16.11/8.88 => ( ( ord_less_eq_nat @ Ma @ X )
% 16.11/8.88 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 16.11/8.88 = none_nat ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_579_insert__simp__mima,axiom,
% 16.11/8.88 ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 16.11/8.88 ( ( ( X = Mi )
% 16.11/8.88 | ( X = Ma ) )
% 16.11/8.88 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 16.11/8.88 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 16.11/8.88 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_580_delt__out__of__range,axiom,
% 16.11/8.88 ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 16.11/8.88 ( ( ( ord_less_nat @ X @ Mi )
% 16.11/8.88 | ( ord_less_nat @ Ma @ X ) )
% 16.11/8.88 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 16.11/8.88 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 16.11/8.88 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_581_del__single__cont,axiom,
% 16.11/8.88 ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 16.11/8.88 ( ( ( X = Mi )
% 16.11/8.88 & ( X = Ma ) )
% 16.11/8.88 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 16.11/8.88 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 16.11/8.88 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_582_mi__ma__2__deg,axiom,
% 16.11/8.88 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N2 )
% 16.11/8.88 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 16.11/8.88 & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_583_succ__min,axiom,
% 16.11/8.88 ! [Deg: nat,X: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 16.11/8.88 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 16.11/8.88 => ( ( ord_less_nat @ X @ Mi )
% 16.11/8.88 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 16.11/8.88 = ( some_nat @ Mi ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_584_pred__max,axiom,
% 16.11/8.88 ! [Deg: nat,Ma: nat,X: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 16.11/8.88 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 16.11/8.88 => ( ( ord_less_nat @ Ma @ X )
% 16.11/8.88 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 16.11/8.88 = ( some_nat @ Ma ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_585_summaxma,axiom,
% 16.11/8.88 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
% 16.11/8.88 => ( ( Mi != Ma )
% 16.11/8.88 => ( ( the_nat @ ( vEBT_vebt_maxt @ Summary ) )
% 16.11/8.88 = ( vEBT_VEBT_high @ Ma @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_586_pred__list__to__short,axiom,
% 16.11/8.88 ! [Deg: nat,X: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
% 16.11/8.88 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 16.11/8.88 => ( ( ord_less_eq_nat @ X @ Ma )
% 16.11/8.88 => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.11/8.88 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 16.11/8.88 = none_nat ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_587_succ__list__to__short,axiom,
% 16.11/8.88 ! [Deg: nat,Mi: nat,X: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
% 16.11/8.88 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 16.11/8.88 => ( ( ord_less_eq_nat @ Mi @ X )
% 16.11/8.88 => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.11/8.88 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 16.11/8.88 = none_nat ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_588_real__arch__pow,axiom,
% 16.11/8.88 ! [X: real,Y: real] :
% 16.11/8.88 ( ( ord_less_real @ one_one_real @ X )
% 16.11/8.88 => ? [N4: nat] : ( ord_less_real @ Y @ ( power_power_real @ X @ N4 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_589_less__eq__real__def,axiom,
% 16.11/8.88 ( ord_less_eq_real
% 16.11/8.88 = ( ^ [X2: real,Y2: real] :
% 16.11/8.88 ( ( ord_less_real @ X2 @ Y2 )
% 16.11/8.88 | ( X2 = Y2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_590_complete__real,axiom,
% 16.11/8.88 ! [S: set_real] :
% 16.11/8.88 ( ? [X5: real] : ( member_real @ X5 @ S )
% 16.11/8.88 => ( ? [Z4: real] :
% 16.11/8.88 ! [X4: real] :
% 16.11/8.88 ( ( member_real @ X4 @ S )
% 16.11/8.88 => ( ord_less_eq_real @ X4 @ Z4 ) )
% 16.11/8.88 => ? [Y4: real] :
% 16.11/8.88 ( ! [X5: real] :
% 16.11/8.88 ( ( member_real @ X5 @ S )
% 16.11/8.88 => ( ord_less_eq_real @ X5 @ Y4 ) )
% 16.11/8.88 & ! [Z4: real] :
% 16.11/8.88 ( ! [X4: real] :
% 16.11/8.88 ( ( member_real @ X4 @ S )
% 16.11/8.88 => ( ord_less_eq_real @ X4 @ Z4 ) )
% 16.11/8.88 => ( ord_less_eq_real @ Y4 @ Z4 ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_591_two__realpow__ge__one,axiom,
% 16.11/8.88 ! [N2: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_592_invar__vebt_Ointros_I4_J,axiom,
% 16.11/8.88 ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 16.11/8.88 ( ! [X4: vEBT_VEBT] :
% 16.11/8.88 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 16.11/8.88 => ( vEBT_invar_vebt @ X4 @ N2 ) )
% 16.11/8.88 => ( ( vEBT_invar_vebt @ Summary @ M )
% 16.11/8.88 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 16.11/8.88 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 16.11/8.88 => ( ( M = N2 )
% 16.11/8.88 => ( ( Deg
% 16.11/8.88 = ( plus_plus_nat @ N2 @ M ) )
% 16.11/8.88 => ( ! [I4: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 16.11/8.88 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X3 ) )
% 16.11/8.88 = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
% 16.11/8.88 => ( ( ( Mi = Ma )
% 16.11/8.88 => ! [X4: vEBT_VEBT] :
% 16.11/8.88 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 16.11/8.88 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
% 16.11/8.88 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 16.11/8.88 => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 16.11/8.88 => ( ( ( Mi != Ma )
% 16.11/8.88 => ! [I4: nat] :
% 16.11/8.88 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 16.11/8.88 => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
% 16.11/8.88 = I4 )
% 16.11/8.88 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
% 16.11/8.88 & ! [X4: nat] :
% 16.11/8.88 ( ( ( ( vEBT_VEBT_high @ X4 @ N2 )
% 16.11/8.88 = I4 )
% 16.11/8.88 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X4 @ N2 ) ) )
% 16.11/8.88 => ( ( ord_less_nat @ Mi @ X4 )
% 16.11/8.88 & ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
% 16.11/8.88 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_593__092_060open_062y_A_061_Asummin_A_K_A2_A_094_An_A_L_Alx_A_092_060Longrightarrow_062_Ax_A_092_060noteq_062_Ay_A_092_060and_062_Aboth__member__options_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Ay_092_060close_062,axiom,
% 16.11/8.88 ( ( ya
% 16.11/8.88 = ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) )
% 16.11/8.88 => ( ( xa != ya )
% 16.11/8.88 & ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ ya ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_594_del__x__mi__lets__in__not__minNull,axiom,
% 16.11/8.88 ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L2: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
% 16.11/8.88 ( ( ( X = Mi )
% 16.11/8.88 & ( ord_less_nat @ X @ Ma ) )
% 16.11/8.88 => ( ( Mi != Ma )
% 16.11/8.88 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 16.11/8.88 => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.11/8.88 = H2 )
% 16.11/8.88 => ( ( Xn
% 16.11/8.88 = ( 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 ) ) ) ) ) ) )
% 16.11/8.88 => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.11/8.88 = L2 )
% 16.11/8.88 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 16.11/8.88 => ( ( Newnode
% 16.11/8.88 = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
% 16.11/8.88 => ( ( Newlist
% 16.11/8.88 = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
% 16.11/8.88 => ( ~ ( vEBT_VEBT_minNull @ Newnode )
% 16.11/8.88 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 16.11/8.88 = ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_595_del__x__not__mi__newnode__not__nil,axiom,
% 16.11/8.88 ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L2: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 16.11/8.88 ( ( ( ord_less_nat @ Mi @ X )
% 16.11/8.88 & ( ord_less_eq_nat @ X @ Ma ) )
% 16.11/8.88 => ( ( Mi != Ma )
% 16.11/8.88 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 16.11/8.88 => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.11/8.88 = H2 )
% 16.11/8.88 => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.11/8.88 = L2 )
% 16.11/8.88 => ( ( Newnode
% 16.11/8.88 = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
% 16.11/8.88 => ( ~ ( vEBT_VEBT_minNull @ Newnode )
% 16.11/8.88 => ( ( Newlist
% 16.11/8.88 = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
% 16.11/8.88 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 16.11/8.88 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 16.11/8.88 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = 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 ) ) ) ) ) ) ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_596_nested__mint,axiom,
% 16.11/8.88 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat,Va: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N2 )
% 16.11/8.88 => ( ( N2
% 16.11/8.88 = ( suc @ ( suc @ Va ) ) )
% 16.11/8.88 => ( ~ ( ord_less_nat @ Ma @ Mi )
% 16.11/8.88 => ( ( Ma != Mi )
% 16.11/8.88 => ( 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 ) ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_597_mintlistlength,axiom,
% 16.11/8.88 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N2 )
% 16.11/8.88 => ( ( Mi != Ma )
% 16.11/8.88 => ( ( ord_less_nat @ Mi @ Ma )
% 16.11/8.88 & ? [M2: nat] :
% 16.11/8.88 ( ( ( some_nat @ M2 )
% 16.11/8.88 = ( vEBT_vebt_mint @ Summary ) )
% 16.11/8.88 & ( ord_less_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_598_mul__def,axiom,
% 16.11/8.88 ( vEBT_VEBT_mul
% 16.11/8.88 = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_599_add__def,axiom,
% 16.11/8.88 ( vEBT_VEBT_add
% 16.11/8.88 = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_600__092_060open_062vebt__delete_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Ax_A_061_ANode_A_ISome_A_Isummin_A_K_A2_A_094_An_A_L_Alx_M_Aif_Asummin_A_K_A2_A_094_An_A_L_Alx_A_061_Ama_Athen_Alet_Amaxs_A_061_Avebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_Ain_Aif_Amaxs_A_061_ANone_Athen_Asummin_A_K_A2_A_094_An_A_L_Alx_Aelse_A2_A_094_A_Ideg_Adiv_A2_J_A_K_Athe_Amaxs_A_L_Athe_A_Ivebt__maxt_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_A_B_Athe_Amaxs_J_J_Aelse_Ama_J_J_Adeg_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_J_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_092_060close_062,axiom,
% 16.11/8.88 ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 16.11/8.88 = ( vEBT_Node
% 16.11/8.88 @ ( some_P7363390416028606310at_nat
% 16.11/8.88 @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 16.11/8.88 @ ( if_nat
% 16.11/8.88 @ ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 16.11/8.88 = ma )
% 16.11/8.88 @ ( if_nat
% 16.11/8.88 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 16.11/8.88 = none_nat )
% 16.11/8.88 @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 16.11/8.88 @ ( 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 @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) ) ) )
% 16.11/8.88 @ ma ) ) )
% 16.11/8.88 @ deg
% 16.11/8.88 @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 16.11/8.88 @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_601_vebt__mint_Osimps_I3_J,axiom,
% 16.11/8.88 ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 16.11/8.88 ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 16.11/8.88 = ( some_nat @ Mi ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_602_vebt__maxt_Osimps_I3_J,axiom,
% 16.11/8.88 ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 16.11/8.88 ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 16.11/8.88 = ( some_nat @ Ma ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_603_nat__1__eq__mult__iff,axiom,
% 16.11/8.88 ! [M: nat,N2: nat] :
% 16.11/8.88 ( ( one_one_nat
% 16.11/8.88 = ( times_times_nat @ M @ N2 ) )
% 16.11/8.88 = ( ( M = one_one_nat )
% 16.11/8.88 & ( N2 = one_one_nat ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_604_even__odd__cases,axiom,
% 16.11/8.88 ! [X: nat] :
% 16.11/8.88 ( ! [N4: nat] :
% 16.11/8.88 ( X
% 16.11/8.88 != ( plus_plus_nat @ N4 @ N4 ) )
% 16.11/8.88 => ~ ! [N4: nat] :
% 16.11/8.88 ( X
% 16.11/8.88 != ( plus_plus_nat @ N4 @ ( suc @ N4 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_605_not__min__Null__member,axiom,
% 16.11/8.88 ! [T: vEBT_VEBT] :
% 16.11/8.88 ( ~ ( vEBT_VEBT_minNull @ T )
% 16.11/8.88 => ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_606_min__Null__member,axiom,
% 16.11/8.88 ! [T: vEBT_VEBT,X: nat] :
% 16.11/8.88 ( ( vEBT_VEBT_minNull @ T )
% 16.11/8.88 => ~ ( vEBT_vebt_member @ T @ X ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_607_set__vebt_H__def,axiom,
% 16.11/8.88 ( vEBT_VEBT_set_vebt
% 16.11/8.88 = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_608_deg__SUcn__Node,axiom,
% 16.11/8.88 ! [Tree: vEBT_VEBT,N2: nat] :
% 16.11/8.88 ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N2 ) ) )
% 16.11/8.88 => ? [Info2: option4927543243414619207at_nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 16.11/8.88 ( Tree
% 16.11/8.88 = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N2 ) ) @ TreeList3 @ S2 ) ) ) ).
% 16.11/8.88
% 16.11/8.88 thf(fact_609_add__shift,axiom,
% 16.11/8.88 ! [X: nat,Y: nat,Z: nat] :
% 16.11/8.88 ( ( ( plus_plus_nat @ X @ Y )
% 16.11/8.88 = Z )
% 16.11/8.88 = ( ( vEBT_VEBT_add @ ( some_nat @ X ) @ ( some_nat @ Y ) )
% 16.11/8.88 = ( some_nat @ Z ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_610_mul__shift,axiom,
% 16.11/8.89 ! [X: nat,Y: nat,Z: nat] :
% 16.11/8.89 ( ( ( times_times_nat @ X @ Y )
% 16.11/8.89 = Z )
% 16.11/8.89 = ( ( vEBT_VEBT_mul @ ( some_nat @ X ) @ ( some_nat @ Y ) )
% 16.11/8.89 = ( some_nat @ Z ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_611_minNullmin,axiom,
% 16.11/8.89 ! [T: vEBT_VEBT] :
% 16.11/8.89 ( ( vEBT_VEBT_minNull @ T )
% 16.11/8.89 => ( ( vEBT_vebt_mint @ T )
% 16.11/8.89 = none_nat ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_612_minminNull,axiom,
% 16.11/8.89 ! [T: vEBT_VEBT] :
% 16.11/8.89 ( ( ( vEBT_vebt_mint @ T )
% 16.11/8.89 = none_nat )
% 16.11/8.89 => ( vEBT_VEBT_minNull @ T ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_613_nat_Oinject,axiom,
% 16.11/8.89 ! [X22: nat,Y22: nat] :
% 16.11/8.89 ( ( ( suc @ X22 )
% 16.11/8.89 = ( suc @ Y22 ) )
% 16.11/8.89 = ( X22 = Y22 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_614_old_Onat_Oinject,axiom,
% 16.11/8.89 ! [Nat: nat,Nat2: nat] :
% 16.11/8.89 ( ( ( suc @ Nat )
% 16.11/8.89 = ( suc @ Nat2 ) )
% 16.11/8.89 = ( Nat = Nat2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_615_power__minus__is__div,axiom,
% 16.11/8.89 ! [B: nat,A: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ B @ A )
% 16.11/8.89 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A @ B ) )
% 16.11/8.89 = ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_616_lessI,axiom,
% 16.11/8.89 ! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_617_Suc__mono,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_nat @ M @ N2 )
% 16.11/8.89 => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_618_Suc__less__eq,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
% 16.11/8.89 = ( ord_less_nat @ M @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_619_Suc__le__mono,axiom,
% 16.11/8.89 ! [N2: nat,M: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M ) )
% 16.11/8.89 = ( ord_less_eq_nat @ N2 @ M ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_620_add__Suc__right,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( plus_plus_nat @ M @ ( suc @ N2 ) )
% 16.11/8.89 = ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_621_nat__add__left__cancel__less,axiom,
% 16.11/8.89 ! [K: nat,M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
% 16.11/8.89 = ( ord_less_nat @ M @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_622_diff__Suc__Suc,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N2 ) )
% 16.11/8.89 = ( minus_minus_nat @ M @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_623_Suc__diff__diff,axiom,
% 16.11/8.89 ! [M: nat,N2: nat,K: nat] :
% 16.11/8.89 ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) @ ( suc @ K ) )
% 16.11/8.89 = ( minus_minus_nat @ ( minus_minus_nat @ M @ N2 ) @ K ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_624_nat__add__left__cancel__le,axiom,
% 16.11/8.89 ! [K: nat,M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
% 16.11/8.89 = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_625_diff__diff__cancel,axiom,
% 16.11/8.89 ! [I2: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ I2 @ N2 )
% 16.11/8.89 => ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I2 ) )
% 16.11/8.89 = I2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_626_diff__diff__left,axiom,
% 16.11/8.89 ! [I2: nat,J: nat,K: nat] :
% 16.11/8.89 ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
% 16.11/8.89 = ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_627_nat__mult__eq__1__iff,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ( times_times_nat @ M @ N2 )
% 16.11/8.89 = one_one_nat )
% 16.11/8.89 = ( ( M = one_one_nat )
% 16.11/8.89 & ( N2 = one_one_nat ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_628_right__diff__distrib__numeral,axiom,
% 16.11/8.89 ! [V: num,B: complex,C: complex] :
% 16.11/8.89 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( minus_minus_complex @ B @ C ) )
% 16.11/8.89 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_629_right__diff__distrib__numeral,axiom,
% 16.11/8.89 ! [V: num,B: real,C: real] :
% 16.11/8.89 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C ) )
% 16.11/8.89 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_630_right__diff__distrib__numeral,axiom,
% 16.11/8.89 ! [V: num,B: rat,C: rat] :
% 16.11/8.89 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B @ C ) )
% 16.11/8.89 = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_631_right__diff__distrib__numeral,axiom,
% 16.11/8.89 ! [V: num,B: int,C: int] :
% 16.11/8.89 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
% 16.11/8.89 = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_632_left__diff__distrib__numeral,axiom,
% 16.11/8.89 ! [A: complex,B: complex,V: num] :
% 16.11/8.89 ( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
% 16.11/8.89 = ( minus_minus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_633_left__diff__distrib__numeral,axiom,
% 16.11/8.89 ! [A: real,B: real,V: num] :
% 16.11/8.89 ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
% 16.11/8.89 = ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_634_left__diff__distrib__numeral,axiom,
% 16.11/8.89 ! [A: rat,B: rat,V: num] :
% 16.11/8.89 ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
% 16.11/8.89 = ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_635_left__diff__distrib__numeral,axiom,
% 16.11/8.89 ! [A: int,B: int,V: num] :
% 16.11/8.89 ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
% 16.11/8.89 = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_636_mult__Suc__right,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( times_times_nat @ M @ ( suc @ N2 ) )
% 16.11/8.89 = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_637_Nat_Odiff__diff__right,axiom,
% 16.11/8.89 ! [K: nat,J: nat,I2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ K @ J )
% 16.11/8.89 => ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
% 16.11/8.89 = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_638_Nat_Oadd__diff__assoc2,axiom,
% 16.11/8.89 ! [K: nat,J: nat,I2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ K @ J )
% 16.11/8.89 => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
% 16.11/8.89 = ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_639_Nat_Oadd__diff__assoc,axiom,
% 16.11/8.89 ! [K: nat,J: nat,I2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ K @ J )
% 16.11/8.89 => ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
% 16.11/8.89 = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_640_diff__Suc__1,axiom,
% 16.11/8.89 ! [N2: nat] :
% 16.11/8.89 ( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
% 16.11/8.89 = N2 ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_641_diff__Suc__diff__eq2,axiom,
% 16.11/8.89 ! [K: nat,J: nat,I2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ K @ J )
% 16.11/8.89 => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I2 )
% 16.11/8.89 = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_642_diff__Suc__diff__eq1,axiom,
% 16.11/8.89 ! [K: nat,J: nat,I2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ K @ J )
% 16.11/8.89 => ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
% 16.11/8.89 = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_643_asmt,axiom,
% 16.11/8.89 ( ( ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 16.11/8.89 = ma )
% 16.11/8.89 => ( ya
% 16.11/8.89 = ( if_nat
% 16.11/8.89 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 16.11/8.89 = none_nat )
% 16.11/8.89 @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 16.11/8.89 @ ( 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 @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) ) ) ) ) )
% 16.11/8.89 & ( ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 16.11/8.89 != ma )
% 16.11/8.89 => ( ya = ma ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_644_Suc__numeral,axiom,
% 16.11/8.89 ! [N2: num] :
% 16.11/8.89 ( ( suc @ ( numeral_numeral_nat @ N2 ) )
% 16.11/8.89 = ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_645__092_060open_062both__member__options_A_INode_A_ISome_A_Isummin_A_K_A2_A_094_An_A_L_Alx_M_Aif_Asummin_A_K_A2_A_094_An_A_L_Alx_A_061_Ama_Athen_Alet_Amaxs_A_061_Avebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_Ain_Aif_Amaxs_A_061_ANone_Athen_Asummin_A_K_A2_A_094_An_A_L_Alx_Aelse_A2_A_094_A_Ideg_Adiv_A2_J_A_K_Athe_Amaxs_A_L_Athe_A_Ivebt__maxt_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_A_B_Athe_Amaxs_J_J_Aelse_Ama_J_J_Adeg_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_J_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_J_Ay_092_060close_062,axiom,
% 16.11/8.89 ( vEBT_V8194947554948674370ptions
% 16.11/8.89 @ ( vEBT_Node
% 16.11/8.89 @ ( some_P7363390416028606310at_nat
% 16.11/8.89 @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 16.11/8.89 @ ( if_nat
% 16.11/8.89 @ ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 16.11/8.89 = ma )
% 16.11/8.89 @ ( if_nat
% 16.11/8.89 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 16.11/8.89 = none_nat )
% 16.11/8.89 @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 16.11/8.89 @ ( 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 @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) ) ) )
% 16.11/8.89 @ ma ) ) )
% 16.11/8.89 @ deg
% 16.11/8.89 @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 16.11/8.89 @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 16.11/8.89 @ ya ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_646_add__2__eq__Suc_H,axiom,
% 16.11/8.89 ! [N2: nat] :
% 16.11/8.89 ( ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.89 = ( suc @ ( suc @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_647_add__2__eq__Suc,axiom,
% 16.11/8.89 ! [N2: nat] :
% 16.11/8.89 ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.11/8.89 = ( suc @ ( suc @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_648_div2__Suc__Suc,axiom,
% 16.11/8.89 ! [M: nat] :
% 16.11/8.89 ( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.89 = ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_649_Suc__1,axiom,
% 16.11/8.89 ( ( suc @ one_one_nat )
% 16.11/8.89 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_650_del__x__not__mia,axiom,
% 16.11/8.89 ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 16.11/8.89 ( ( ( ord_less_nat @ Mi @ X )
% 16.11/8.89 & ( ord_less_eq_nat @ X @ Ma ) )
% 16.11/8.89 => ( ( Mi != Ma )
% 16.11/8.89 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 16.11/8.89 => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.11/8.89 = H2 )
% 16.11/8.89 => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.11/8.89 = L2 )
% 16.11/8.89 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 16.11/8.89 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 16.11/8.89 = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
% 16.11/8.89 @ ( vEBT_Node
% 16.11/8.89 @ ( some_P7363390416028606310at_nat
% 16.11/8.89 @ ( product_Pair_nat_nat @ Mi
% 16.11/8.89 @ ( if_nat @ ( X = Ma )
% 16.11/8.89 @ ( if_nat
% 16.11/8.89 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 16.11/8.89 = none_nat )
% 16.11/8.89 @ Mi
% 16.11/8.89 @ ( 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 ) @ L2 ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
% 16.11/8.89 @ Ma ) ) )
% 16.11/8.89 @ Deg
% 16.11/8.89 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
% 16.11/8.89 @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 16.11/8.89 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = 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 ) @ L2 ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) ) @ Summary ) ) ) ) ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_651_del__x__not__mi__new__node__nil,axiom,
% 16.11/8.89 ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L2: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
% 16.11/8.89 ( ( ( ord_less_nat @ Mi @ X )
% 16.11/8.89 & ( ord_less_eq_nat @ X @ Ma ) )
% 16.11/8.89 => ( ( Mi != Ma )
% 16.11/8.89 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 16.11/8.89 => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.11/8.89 = H2 )
% 16.11/8.89 => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.11/8.89 = L2 )
% 16.11/8.89 => ( ( Newnode
% 16.11/8.89 = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
% 16.11/8.89 => ( ( vEBT_VEBT_minNull @ Newnode )
% 16.11/8.89 => ( ( Sn
% 16.11/8.89 = ( vEBT_vebt_delete @ Summary @ H2 ) )
% 16.11/8.89 => ( ( Newlist
% 16.11/8.89 = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
% 16.11/8.89 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 16.11/8.89 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 16.11/8.89 = ( vEBT_Node
% 16.11/8.89 @ ( some_P7363390416028606310at_nat
% 16.11/8.89 @ ( product_Pair_nat_nat @ Mi
% 16.11/8.89 @ ( if_nat @ ( X = Ma )
% 16.11/8.89 @ ( if_nat
% 16.11/8.89 @ ( ( vEBT_vebt_maxt @ Sn )
% 16.11/8.89 = none_nat )
% 16.11/8.89 @ Mi
% 16.11/8.89 @ ( 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 ) ) ) ) ) ) )
% 16.11/8.89 @ Ma ) ) )
% 16.11/8.89 @ Deg
% 16.11/8.89 @ Newlist
% 16.11/8.89 @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_652_del__x__not__mi,axiom,
% 16.11/8.89 ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H2: nat,L2: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 16.11/8.89 ( ( ( ord_less_nat @ Mi @ X )
% 16.11/8.89 & ( ord_less_eq_nat @ X @ Ma ) )
% 16.11/8.89 => ( ( Mi != Ma )
% 16.11/8.89 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 16.11/8.89 => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.11/8.89 = H2 )
% 16.11/8.89 => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.11/8.89 = L2 )
% 16.11/8.89 => ( ( Newnode
% 16.11/8.89 = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
% 16.11/8.89 => ( ( Newlist
% 16.11/8.89 = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
% 16.11/8.89 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 16.11/8.89 => ( ( ( vEBT_VEBT_minNull @ Newnode )
% 16.11/8.89 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 16.11/8.89 = ( vEBT_Node
% 16.11/8.89 @ ( some_P7363390416028606310at_nat
% 16.11/8.89 @ ( product_Pair_nat_nat @ Mi
% 16.11/8.89 @ ( if_nat @ ( X = Ma )
% 16.11/8.89 @ ( if_nat
% 16.11/8.89 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 16.11/8.89 = none_nat )
% 16.11/8.89 @ Mi
% 16.11/8.89 @ ( 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 ) ) ) ) ) ) ) )
% 16.11/8.89 @ Ma ) ) )
% 16.11/8.89 @ Deg
% 16.11/8.89 @ Newlist
% 16.11/8.89 @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
% 16.11/8.89 & ( ~ ( vEBT_VEBT_minNull @ Newnode )
% 16.11/8.89 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 16.11/8.89 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = 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 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_653_del__x__mia,axiom,
% 16.11/8.89 ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 16.11/8.89 ( ( ( X = Mi )
% 16.11/8.89 & ( ord_less_nat @ X @ Ma ) )
% 16.11/8.89 => ( ( Mi != Ma )
% 16.11/8.89 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 16.11/8.89 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 16.11/8.89 = ( 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 ) )
% 16.11/8.89 @ ( 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 ) ) ) ) ) )
% 16.11/8.89 @ ( vEBT_Node
% 16.11/8.89 @ ( some_P7363390416028606310at_nat
% 16.11/8.89 @ ( 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 ) ) ) ) ) )
% 16.11/8.89 @ ( if_nat
% 16.11/8.89 @ ( ( 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 ) ) ) ) ) )
% 16.11/8.89 = Ma )
% 16.11/8.89 @ ( if_nat
% 16.11/8.89 @ ( ( 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 ) ) ) ) ) )
% 16.11/8.89 = none_nat )
% 16.11/8.89 @ ( 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 ) ) ) ) ) )
% 16.11/8.89 @ ( 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 ) ) ) ) ) ) ) ) ) ) ) )
% 16.11/8.89 @ Ma ) ) )
% 16.11/8.89 @ Deg
% 16.11/8.89 @ ( 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 ) ) ) ) ) )
% 16.11/8.89 @ ( 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 ) ) ) ) ) )
% 16.11/8.89 @ ( vEBT_Node
% 16.11/8.89 @ ( some_P7363390416028606310at_nat
% 16.11/8.89 @ ( 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 ) ) ) ) ) )
% 16.11/8.89 @ ( if_nat
% 16.11/8.89 @ ( ( 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 ) ) ) ) ) )
% 16.11/8.89 = Ma )
% 16.11/8.89 @ ( 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 ) ) ) ) ) ) ) )
% 16.11/8.89 @ Ma ) ) )
% 16.11/8.89 @ Deg
% 16.11/8.89 @ ( 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 ) ) ) ) ) )
% 16.11/8.89 @ Summary ) )
% 16.11/8.89 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_654_del__x__mi__lets__in__minNull,axiom,
% 16.11/8.89 ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L2: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
% 16.11/8.89 ( ( ( X = Mi )
% 16.11/8.89 & ( ord_less_nat @ X @ Ma ) )
% 16.11/8.89 => ( ( Mi != Ma )
% 16.11/8.89 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 16.11/8.89 => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.11/8.89 = H2 )
% 16.11/8.89 => ( ( Xn
% 16.11/8.89 = ( 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 ) ) ) ) ) ) )
% 16.11/8.89 => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.11/8.89 = L2 )
% 16.11/8.89 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 16.11/8.89 => ( ( Newnode
% 16.11/8.89 = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
% 16.11/8.89 => ( ( Newlist
% 16.11/8.89 = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
% 16.11/8.89 => ( ( vEBT_VEBT_minNull @ Newnode )
% 16.11/8.89 => ( ( Sn
% 16.11/8.89 = ( vEBT_vebt_delete @ Summary @ H2 ) )
% 16.11/8.89 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 16.11/8.89 = ( vEBT_Node
% 16.11/8.89 @ ( some_P7363390416028606310at_nat
% 16.11/8.89 @ ( product_Pair_nat_nat @ Xn
% 16.11/8.89 @ ( if_nat @ ( Xn = Ma )
% 16.11/8.89 @ ( if_nat
% 16.11/8.89 @ ( ( vEBT_vebt_maxt @ Sn )
% 16.11/8.89 = none_nat )
% 16.11/8.89 @ Xn
% 16.11/8.89 @ ( 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 ) ) ) ) ) ) )
% 16.11/8.89 @ Ma ) ) )
% 16.11/8.89 @ Deg
% 16.11/8.89 @ Newlist
% 16.11/8.89 @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_655_del__x__mi__lets__in,axiom,
% 16.11/8.89 ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L2: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
% 16.11/8.89 ( ( ( X = Mi )
% 16.11/8.89 & ( ord_less_nat @ X @ Ma ) )
% 16.11/8.89 => ( ( Mi != Ma )
% 16.11/8.89 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 16.11/8.89 => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.11/8.89 = H2 )
% 16.11/8.89 => ( ( Xn
% 16.11/8.89 = ( 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 ) ) ) ) ) ) )
% 16.11/8.89 => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.11/8.89 = L2 )
% 16.11/8.89 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 16.11/8.89 => ( ( Newnode
% 16.11/8.89 = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
% 16.11/8.89 => ( ( Newlist
% 16.11/8.89 = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
% 16.11/8.89 => ( ( ( vEBT_VEBT_minNull @ Newnode )
% 16.11/8.89 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 16.11/8.89 = ( vEBT_Node
% 16.11/8.89 @ ( some_P7363390416028606310at_nat
% 16.11/8.89 @ ( product_Pair_nat_nat @ Xn
% 16.11/8.89 @ ( if_nat @ ( Xn = Ma )
% 16.11/8.89 @ ( if_nat
% 16.11/8.89 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 16.11/8.89 = none_nat )
% 16.11/8.89 @ Xn
% 16.11/8.89 @ ( 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 ) ) ) ) ) ) ) )
% 16.11/8.89 @ Ma ) ) )
% 16.11/8.89 @ Deg
% 16.11/8.89 @ Newlist
% 16.11/8.89 @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
% 16.11/8.89 & ( ~ ( vEBT_VEBT_minNull @ Newnode )
% 16.11/8.89 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 16.11/8.89 = ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_656_del__x__mi,axiom,
% 16.11/8.89 ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L2: nat] :
% 16.11/8.89 ( ( ( X = Mi )
% 16.11/8.89 & ( ord_less_nat @ X @ Ma ) )
% 16.11/8.89 => ( ( Mi != Ma )
% 16.11/8.89 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 16.11/8.89 => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.11/8.89 = H2 )
% 16.11/8.89 => ( ( Xn
% 16.11/8.89 = ( 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 ) ) ) ) ) ) )
% 16.11/8.89 => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.11/8.89 = L2 )
% 16.11/8.89 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 16.11/8.89 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 16.11/8.89 = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
% 16.11/8.89 @ ( vEBT_Node
% 16.11/8.89 @ ( some_P7363390416028606310at_nat
% 16.11/8.89 @ ( product_Pair_nat_nat @ Xn
% 16.11/8.89 @ ( if_nat @ ( Xn = Ma )
% 16.11/8.89 @ ( if_nat
% 16.11/8.89 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 16.11/8.89 = none_nat )
% 16.11/8.89 @ Xn
% 16.11/8.89 @ ( 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 ) @ L2 ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
% 16.11/8.89 @ Ma ) ) )
% 16.11/8.89 @ Deg
% 16.11/8.89 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) )
% 16.11/8.89 @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 16.11/8.89 @ ( 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 ) @ L2 ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L2 ) ) @ Summary ) ) ) ) ) ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_657_del__in__range,axiom,
% 16.11/8.89 ! [Mi: nat,X: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 16.11/8.89 ( ( ( ord_less_eq_nat @ Mi @ X )
% 16.11/8.89 & ( ord_less_eq_nat @ X @ Ma ) )
% 16.11/8.89 => ( ( Mi != Ma )
% 16.11/8.89 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 16.11/8.89 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 16.11/8.89 = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 16.11/8.89 @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.11/8.89 @ ( vEBT_Node
% 16.11/8.89 @ ( some_P7363390416028606310at_nat
% 16.11/8.89 @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ Mi )
% 16.11/8.89 @ ( if_nat
% 16.11/8.89 @ ( ( ( X = Mi )
% 16.11/8.89 => ( ( 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 ) ) ) ) ) )
% 16.11/8.89 = Ma ) )
% 16.11/8.89 & ( ( X != Mi )
% 16.11/8.89 => ( X = Ma ) ) )
% 16.11/8.89 @ ( if_nat
% 16.11/8.89 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.11/8.89 = none_nat )
% 16.11/8.89 @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ Mi )
% 16.11/8.89 @ ( 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 @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( 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 @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
% 16.11/8.89 @ Ma ) ) )
% 16.11/8.89 @ Deg
% 16.11/8.89 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.11/8.89 @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.11/8.89 @ ( vEBT_Node
% 16.11/8.89 @ ( some_P7363390416028606310at_nat
% 16.11/8.89 @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ Mi )
% 16.11/8.89 @ ( if_nat
% 16.11/8.89 @ ( ( ( X = Mi )
% 16.11/8.89 => ( ( 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 ) ) ) ) ) )
% 16.11/8.89 = Ma ) )
% 16.11/8.89 & ( ( X != Mi )
% 16.11/8.89 => ( X = Ma ) ) )
% 16.11/8.89 @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( 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 @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 16.11/8.89 @ Ma ) ) )
% 16.11/8.89 @ Deg
% 16.11/8.89 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.11/8.89 @ Summary ) )
% 16.11/8.89 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_658_succ__less__length__list,axiom,
% 16.11/8.89 ! [Deg: nat,Mi: nat,X: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
% 16.11/8.89 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 16.11/8.89 => ( ( ord_less_eq_nat @ Mi @ X )
% 16.11/8.89 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 16.11/8.89 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 16.11/8.89 = ( if_option_nat
% 16.11/8.89 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.11/8.89 != none_nat )
% 16.11/8.89 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 16.11/8.89 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.11/8.89 @ ( if_option_nat
% 16.11/8.89 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.11/8.89 = none_nat )
% 16.11/8.89 @ none_nat
% 16.11/8.89 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_659_succ__greatereq__min,axiom,
% 16.11/8.89 ! [Deg: nat,Mi: nat,X: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 16.11/8.89 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 16.11/8.89 => ( ( ord_less_eq_nat @ Mi @ X )
% 16.11/8.89 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 16.11/8.89 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 16.11/8.89 @ ( if_option_nat
% 16.11/8.89 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.11/8.89 != none_nat )
% 16.11/8.89 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 16.11/8.89 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.11/8.89 @ ( if_option_nat
% 16.11/8.89 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.11/8.89 = none_nat )
% 16.11/8.89 @ none_nat
% 16.11/8.89 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 16.11/8.89 @ none_nat ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_660_pred__lesseq__max,axiom,
% 16.11/8.89 ! [Deg: nat,X: nat,Ma: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 16.11/8.89 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 16.11/8.89 => ( ( ord_less_eq_nat @ X @ Ma )
% 16.11/8.89 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 16.11/8.89 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 16.11/8.89 @ ( if_option_nat
% 16.11/8.89 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.11/8.89 != none_nat )
% 16.11/8.89 & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 16.11/8.89 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.11/8.89 @ ( if_option_nat
% 16.11/8.89 @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.11/8.89 = none_nat )
% 16.11/8.89 @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
% 16.11/8.89 @ ( 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 @ X @ ( 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 @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 16.11/8.89 @ none_nat ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_661_pred__less__length__list,axiom,
% 16.11/8.89 ! [Deg: nat,X: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
% 16.11/8.89 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 16.11/8.89 => ( ( ord_less_eq_nat @ X @ Ma )
% 16.11/8.89 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 16.11/8.89 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 16.11/8.89 = ( if_option_nat
% 16.11/8.89 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.11/8.89 != none_nat )
% 16.11/8.89 & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 16.11/8.89 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.11/8.89 @ ( if_option_nat
% 16.11/8.89 @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.11/8.89 = none_nat )
% 16.11/8.89 @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
% 16.11/8.89 @ ( 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 @ X @ ( 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 @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_662_newmaext,axiom,
% 16.11/8.89 ( ( ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 16.11/8.89 = none_nat )
% 16.11/8.89 => ( ( ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 16.11/8.89 = ma )
% 16.11/8.89 => ( ( if_nat
% 16.11/8.89 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 16.11/8.89 = none_nat )
% 16.11/8.89 @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 16.11/8.89 @ ( 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 @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) ) ) )
% 16.11/8.89 = ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) ) )
% 16.11/8.89 & ( ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 16.11/8.89 != ma )
% 16.11/8.89 => ( ma
% 16.11/8.89 = ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) ) ) ) )
% 16.11/8.89 & ( ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 16.11/8.89 != none_nat )
% 16.11/8.89 => ( ( ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 16.11/8.89 = ma )
% 16.11/8.89 => ( ( if_nat
% 16.11/8.89 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 16.11/8.89 = none_nat )
% 16.11/8.89 @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 16.11/8.89 @ ( 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 @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) ) ) )
% 16.11/8.89 = ( 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 @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) ) ) ) )
% 16.11/8.89 & ( ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 16.11/8.89 != ma )
% 16.11/8.89 => ( ma
% 16.11/8.89 = ( 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 @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_663__092_060open_062y_A_061_Asummin_A_K_A2_A_094_An_A_L_Alx_A_092_060or_062_Ay_A_061_A_Iif_Asummin_A_K_A2_A_094_An_A_L_Alx_A_061_Ama_Athen_Alet_Amaxs_A_061_Avebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_Ain_Aif_Amaxs_A_061_ANone_Athen_Asummin_A_K_A2_A_094_An_A_L_Alx_Aelse_A2_A_094_A_Ideg_Adiv_A2_J_A_K_Athe_Amaxs_A_L_Athe_A_Ivebt__maxt_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_A_B_Athe_Amaxs_J_J_Aelse_Ama_J_A_092_060or_062_Aboth__member__options_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_A_B_Ahigh_Ay_A_Ideg_Adiv_A2_J_J_A_Ilow_Ay_A_Ideg_Adiv_A2_J_J_A_092_060and_062_Ahigh_Ay_A_Ideg_Adiv_A2_J_A_060_Alength_A_ItreeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_J_092_060close_062,axiom,
% 16.11/8.89 ( ( ya
% 16.11/8.89 = ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) )
% 16.11/8.89 | ( ( ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 16.11/8.89 = ma )
% 16.11/8.89 => ( ya
% 16.11/8.89 = ( if_nat
% 16.11/8.89 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 16.11/8.89 = none_nat )
% 16.11/8.89 @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 16.11/8.89 @ ( 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 @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) ) ) ) ) )
% 16.11/8.89 & ( ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 16.11/8.89 != ma )
% 16.11/8.89 => ( ya = ma ) ) )
% 16.11/8.89 | ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( vEBT_VEBT_high @ ya @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ya @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.11/8.89 & ( ord_less_nat @ ( vEBT_VEBT_high @ ya @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_664_Suc__diff__le,axiom,
% 16.11/8.89 ! [N2: nat,M: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ N2 @ M )
% 16.11/8.89 => ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
% 16.11/8.89 = ( suc @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_665_diff__Suc__eq__diff__pred,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( minus_minus_nat @ M @ ( suc @ N2 ) )
% 16.11/8.89 = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_666_Suc__diff__Suc,axiom,
% 16.11/8.89 ! [N2: nat,M: nat] :
% 16.11/8.89 ( ( ord_less_nat @ N2 @ M )
% 16.11/8.89 => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N2 ) ) )
% 16.11/8.89 = ( minus_minus_nat @ M @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_667_diff__less__Suc,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ ( suc @ M ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_668_Suc__inject,axiom,
% 16.11/8.89 ! [X: nat,Y: nat] :
% 16.11/8.89 ( ( ( suc @ X )
% 16.11/8.89 = ( suc @ Y ) )
% 16.11/8.89 => ( X = Y ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_669_n__not__Suc__n,axiom,
% 16.11/8.89 ! [N2: nat] :
% 16.11/8.89 ( N2
% 16.11/8.89 != ( suc @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_670_diff__commute,axiom,
% 16.11/8.89 ! [I2: nat,J: nat,K: nat] :
% 16.11/8.89 ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
% 16.11/8.89 = ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_671_zero__induct__lemma,axiom,
% 16.11/8.89 ! [P: nat > $o,K: nat,I2: nat] :
% 16.11/8.89 ( ( P @ K )
% 16.11/8.89 => ( ! [N4: nat] :
% 16.11/8.89 ( ( P @ ( suc @ N4 ) )
% 16.11/8.89 => ( P @ N4 ) )
% 16.11/8.89 => ( P @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_672_diff__less__mono2,axiom,
% 16.11/8.89 ! [M: nat,N2: nat,L2: nat] :
% 16.11/8.89 ( ( ord_less_nat @ M @ N2 )
% 16.11/8.89 => ( ( ord_less_nat @ M @ L2 )
% 16.11/8.89 => ( ord_less_nat @ ( minus_minus_nat @ L2 @ N2 ) @ ( minus_minus_nat @ L2 @ M ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_673_less__imp__diff__less,axiom,
% 16.11/8.89 ! [J: nat,K: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_nat @ J @ K )
% 16.11/8.89 => ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_674_Nat_OlessE,axiom,
% 16.11/8.89 ! [I2: nat,K: nat] :
% 16.11/8.89 ( ( ord_less_nat @ I2 @ K )
% 16.11/8.89 => ( ( K
% 16.11/8.89 != ( suc @ I2 ) )
% 16.11/8.89 => ~ ! [J2: nat] :
% 16.11/8.89 ( ( ord_less_nat @ I2 @ J2 )
% 16.11/8.89 => ( K
% 16.11/8.89 != ( suc @ J2 ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_675_Suc__lessD,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_nat @ ( suc @ M ) @ N2 )
% 16.11/8.89 => ( ord_less_nat @ M @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_676_Suc__lessE,axiom,
% 16.11/8.89 ! [I2: nat,K: nat] :
% 16.11/8.89 ( ( ord_less_nat @ ( suc @ I2 ) @ K )
% 16.11/8.89 => ~ ! [J2: nat] :
% 16.11/8.89 ( ( ord_less_nat @ I2 @ J2 )
% 16.11/8.89 => ( K
% 16.11/8.89 != ( suc @ J2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_677_Suc__lessI,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_nat @ M @ N2 )
% 16.11/8.89 => ( ( ( suc @ M )
% 16.11/8.89 != N2 )
% 16.11/8.89 => ( ord_less_nat @ ( suc @ M ) @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_678_less__SucE,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_nat @ M @ ( suc @ N2 ) )
% 16.11/8.89 => ( ~ ( ord_less_nat @ M @ N2 )
% 16.11/8.89 => ( M = N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_679_less__SucI,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_nat @ M @ N2 )
% 16.11/8.89 => ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_680_Ex__less__Suc,axiom,
% 16.11/8.89 ! [N2: nat,P: nat > $o] :
% 16.11/8.89 ( ( ? [I3: nat] :
% 16.11/8.89 ( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
% 16.11/8.89 & ( P @ I3 ) ) )
% 16.11/8.89 = ( ( P @ N2 )
% 16.11/8.89 | ? [I3: nat] :
% 16.11/8.89 ( ( ord_less_nat @ I3 @ N2 )
% 16.11/8.89 & ( P @ I3 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_681_less__Suc__eq,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_nat @ M @ ( suc @ N2 ) )
% 16.11/8.89 = ( ( ord_less_nat @ M @ N2 )
% 16.11/8.89 | ( M = N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_682_not__less__eq,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ~ ( ord_less_nat @ M @ N2 ) )
% 16.11/8.89 = ( ord_less_nat @ N2 @ ( suc @ M ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_683_All__less__Suc,axiom,
% 16.11/8.89 ! [N2: nat,P: nat > $o] :
% 16.11/8.89 ( ( ! [I3: nat] :
% 16.11/8.89 ( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
% 16.11/8.89 => ( P @ I3 ) ) )
% 16.11/8.89 = ( ( P @ N2 )
% 16.11/8.89 & ! [I3: nat] :
% 16.11/8.89 ( ( ord_less_nat @ I3 @ N2 )
% 16.11/8.89 => ( P @ I3 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_684_Suc__less__eq2,axiom,
% 16.11/8.89 ! [N2: nat,M: nat] :
% 16.11/8.89 ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 16.11/8.89 = ( ? [M3: nat] :
% 16.11/8.89 ( ( M
% 16.11/8.89 = ( suc @ M3 ) )
% 16.11/8.89 & ( ord_less_nat @ N2 @ M3 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_685_less__antisym,axiom,
% 16.11/8.89 ! [N2: nat,M: nat] :
% 16.11/8.89 ( ~ ( ord_less_nat @ N2 @ M )
% 16.11/8.89 => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
% 16.11/8.89 => ( M = N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_686_Suc__less__SucD,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
% 16.11/8.89 => ( ord_less_nat @ M @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_687_less__trans__Suc,axiom,
% 16.11/8.89 ! [I2: nat,J: nat,K: nat] :
% 16.11/8.89 ( ( ord_less_nat @ I2 @ J )
% 16.11/8.89 => ( ( ord_less_nat @ J @ K )
% 16.11/8.89 => ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_688_less__Suc__induct,axiom,
% 16.11/8.89 ! [I2: nat,J: nat,P: nat > nat > $o] :
% 16.11/8.89 ( ( ord_less_nat @ I2 @ J )
% 16.11/8.89 => ( ! [I4: nat] : ( P @ I4 @ ( suc @ I4 ) )
% 16.11/8.89 => ( ! [I4: nat,J2: nat,K2: nat] :
% 16.11/8.89 ( ( ord_less_nat @ I4 @ J2 )
% 16.11/8.89 => ( ( ord_less_nat @ J2 @ K2 )
% 16.11/8.89 => ( ( P @ I4 @ J2 )
% 16.11/8.89 => ( ( P @ J2 @ K2 )
% 16.11/8.89 => ( P @ I4 @ K2 ) ) ) ) )
% 16.11/8.89 => ( P @ I2 @ J ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_689_strict__inc__induct,axiom,
% 16.11/8.89 ! [I2: nat,J: nat,P: nat > $o] :
% 16.11/8.89 ( ( ord_less_nat @ I2 @ J )
% 16.11/8.89 => ( ! [I4: nat] :
% 16.11/8.89 ( ( J
% 16.11/8.89 = ( suc @ I4 ) )
% 16.11/8.89 => ( P @ I4 ) )
% 16.11/8.89 => ( ! [I4: nat] :
% 16.11/8.89 ( ( ord_less_nat @ I4 @ J )
% 16.11/8.89 => ( ( P @ ( suc @ I4 ) )
% 16.11/8.89 => ( P @ I4 ) ) )
% 16.11/8.89 => ( P @ I2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_690_not__less__less__Suc__eq,axiom,
% 16.11/8.89 ! [N2: nat,M: nat] :
% 16.11/8.89 ( ~ ( ord_less_nat @ N2 @ M )
% 16.11/8.89 => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
% 16.11/8.89 = ( N2 = M ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_691_eq__diff__iff,axiom,
% 16.11/8.89 ! [K: nat,M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ K @ M )
% 16.11/8.89 => ( ( ord_less_eq_nat @ K @ N2 )
% 16.11/8.89 => ( ( ( minus_minus_nat @ M @ K )
% 16.11/8.89 = ( minus_minus_nat @ N2 @ K ) )
% 16.11/8.89 = ( M = N2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_692_le__diff__iff,axiom,
% 16.11/8.89 ! [K: nat,M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ K @ M )
% 16.11/8.89 => ( ( ord_less_eq_nat @ K @ N2 )
% 16.11/8.89 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
% 16.11/8.89 = ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_693_Nat_Odiff__diff__eq,axiom,
% 16.11/8.89 ! [K: nat,M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ K @ M )
% 16.11/8.89 => ( ( ord_less_eq_nat @ K @ N2 )
% 16.11/8.89 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
% 16.11/8.89 = ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_694_diff__le__mono,axiom,
% 16.11/8.89 ! [M: nat,N2: nat,L2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ M @ N2 )
% 16.11/8.89 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L2 ) @ ( minus_minus_nat @ N2 @ L2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_695_diff__le__self,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_696_le__diff__iff_H,axiom,
% 16.11/8.89 ! [A: nat,C: nat,B: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ A @ C )
% 16.11/8.89 => ( ( ord_less_eq_nat @ B @ C )
% 16.11/8.89 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 16.11/8.89 = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_697_diff__le__mono2,axiom,
% 16.11/8.89 ! [M: nat,N2: nat,L2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ M @ N2 )
% 16.11/8.89 => ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N2 ) @ ( minus_minus_nat @ L2 @ M ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_698_diff__add__inverse2,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ N2 )
% 16.11/8.89 = M ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_699_diff__add__inverse,axiom,
% 16.11/8.89 ! [N2: nat,M: nat] :
% 16.11/8.89 ( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M ) @ N2 )
% 16.11/8.89 = M ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_700_diff__cancel2,axiom,
% 16.11/8.89 ! [M: nat,K: nat,N2: nat] :
% 16.11/8.89 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) )
% 16.11/8.89 = ( minus_minus_nat @ M @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_701_Nat_Odiff__cancel,axiom,
% 16.11/8.89 ! [K: nat,M: nat,N2: nat] :
% 16.11/8.89 ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
% 16.11/8.89 = ( minus_minus_nat @ M @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_702_Suc__leD,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
% 16.11/8.89 => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_703_le__SucE,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 16.11/8.89 => ( ~ ( ord_less_eq_nat @ M @ N2 )
% 16.11/8.89 => ( M
% 16.11/8.89 = ( suc @ N2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_704_le__SucI,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ M @ N2 )
% 16.11/8.89 => ( ord_less_eq_nat @ M @ ( suc @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_705_Suc__le__D,axiom,
% 16.11/8.89 ! [N2: nat,M4: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ ( suc @ N2 ) @ M4 )
% 16.11/8.89 => ? [M2: nat] :
% 16.11/8.89 ( M4
% 16.11/8.89 = ( suc @ M2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_706_le__Suc__eq,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 16.11/8.89 = ( ( ord_less_eq_nat @ M @ N2 )
% 16.11/8.89 | ( M
% 16.11/8.89 = ( suc @ N2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_707_Suc__n__not__le__n,axiom,
% 16.11/8.89 ! [N2: nat] :
% 16.11/8.89 ~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_708_not__less__eq__eq,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ~ ( ord_less_eq_nat @ M @ N2 ) )
% 16.11/8.89 = ( ord_less_eq_nat @ ( suc @ N2 ) @ M ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_709_full__nat__induct,axiom,
% 16.11/8.89 ! [P: nat > $o,N2: nat] :
% 16.11/8.89 ( ! [N4: nat] :
% 16.11/8.89 ( ! [M5: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ ( suc @ M5 ) @ N4 )
% 16.11/8.89 => ( P @ M5 ) )
% 16.11/8.89 => ( P @ N4 ) )
% 16.11/8.89 => ( P @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_710_nat__induct__at__least,axiom,
% 16.11/8.89 ! [M: nat,N2: nat,P: nat > $o] :
% 16.11/8.89 ( ( ord_less_eq_nat @ M @ N2 )
% 16.11/8.89 => ( ( P @ M )
% 16.11/8.89 => ( ! [N4: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ M @ N4 )
% 16.11/8.89 => ( ( P @ N4 )
% 16.11/8.89 => ( P @ ( suc @ N4 ) ) ) )
% 16.11/8.89 => ( P @ N2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_711_transitive__stepwise__le,axiom,
% 16.11/8.89 ! [M: nat,N2: nat,R2: nat > nat > $o] :
% 16.11/8.89 ( ( ord_less_eq_nat @ M @ N2 )
% 16.11/8.89 => ( ! [X4: nat] : ( R2 @ X4 @ X4 )
% 16.11/8.89 => ( ! [X4: nat,Y4: nat,Z5: nat] :
% 16.11/8.89 ( ( R2 @ X4 @ Y4 )
% 16.11/8.89 => ( ( R2 @ Y4 @ Z5 )
% 16.11/8.89 => ( R2 @ X4 @ Z5 ) ) )
% 16.11/8.89 => ( ! [N4: nat] : ( R2 @ N4 @ ( suc @ N4 ) )
% 16.11/8.89 => ( R2 @ M @ N2 ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_712_add__Suc__shift,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( plus_plus_nat @ ( suc @ M ) @ N2 )
% 16.11/8.89 = ( plus_plus_nat @ M @ ( suc @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_713_add__Suc,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( plus_plus_nat @ ( suc @ M ) @ N2 )
% 16.11/8.89 = ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_714_nat__arith_Osuc1,axiom,
% 16.11/8.89 ! [A2: nat,K: nat,A: nat] :
% 16.11/8.89 ( ( A2
% 16.11/8.89 = ( plus_plus_nat @ K @ A ) )
% 16.11/8.89 => ( ( suc @ A2 )
% 16.11/8.89 = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_715_diff__mult__distrib2,axiom,
% 16.11/8.89 ! [K: nat,M: nat,N2: nat] :
% 16.11/8.89 ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
% 16.11/8.89 = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_716_diff__mult__distrib,axiom,
% 16.11/8.89 ! [M: nat,N2: nat,K: nat] :
% 16.11/8.89 ( ( times_times_nat @ ( minus_minus_nat @ M @ N2 ) @ K )
% 16.11/8.89 = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_717_Suc__mult__cancel1,axiom,
% 16.11/8.89 ! [K: nat,M: nat,N2: nat] :
% 16.11/8.89 ( ( ( times_times_nat @ ( suc @ K ) @ M )
% 16.11/8.89 = ( times_times_nat @ ( suc @ K ) @ N2 ) )
% 16.11/8.89 = ( M = N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_718_add__diff__add,axiom,
% 16.11/8.89 ! [A: real,C: real,B: real,D: real] :
% 16.11/8.89 ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
% 16.11/8.89 = ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_719_add__diff__add,axiom,
% 16.11/8.89 ! [A: rat,C: rat,B: rat,D: rat] :
% 16.11/8.89 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) )
% 16.11/8.89 = ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ ( minus_minus_rat @ C @ D ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_720_add__diff__add,axiom,
% 16.11/8.89 ! [A: int,C: int,B: int,D: int] :
% 16.11/8.89 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
% 16.11/8.89 = ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_721_set__vebt__def,axiom,
% 16.11/8.89 ( vEBT_set_vebt
% 16.11/8.89 = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_722_numeral__code_I2_J,axiom,
% 16.11/8.89 ! [N2: num] :
% 16.11/8.89 ( ( numera6690914467698888265omplex @ ( bit0 @ N2 ) )
% 16.11/8.89 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_723_numeral__code_I2_J,axiom,
% 16.11/8.89 ! [N2: num] :
% 16.11/8.89 ( ( numeral_numeral_real @ ( bit0 @ N2 ) )
% 16.11/8.89 = ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_724_numeral__code_I2_J,axiom,
% 16.11/8.89 ! [N2: num] :
% 16.11/8.89 ( ( numeral_numeral_rat @ ( bit0 @ N2 ) )
% 16.11/8.89 = ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_725_numeral__code_I2_J,axiom,
% 16.11/8.89 ! [N2: num] :
% 16.11/8.89 ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
% 16.11/8.89 = ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_726_numeral__code_I2_J,axiom,
% 16.11/8.89 ! [N2: num] :
% 16.11/8.89 ( ( numeral_numeral_int @ ( bit0 @ N2 ) )
% 16.11/8.89 = ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_727_lift__Suc__mono__less,axiom,
% 16.11/8.89 ! [F: nat > real,N2: nat,N5: nat] :
% 16.11/8.89 ( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 16.11/8.89 => ( ( ord_less_nat @ N2 @ N5 )
% 16.11/8.89 => ( ord_less_real @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_728_lift__Suc__mono__less,axiom,
% 16.11/8.89 ! [F: nat > rat,N2: nat,N5: nat] :
% 16.11/8.89 ( ! [N4: nat] : ( ord_less_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 16.11/8.89 => ( ( ord_less_nat @ N2 @ N5 )
% 16.11/8.89 => ( ord_less_rat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_729_lift__Suc__mono__less,axiom,
% 16.11/8.89 ! [F: nat > num,N2: nat,N5: nat] :
% 16.11/8.89 ( ! [N4: nat] : ( ord_less_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 16.11/8.89 => ( ( ord_less_nat @ N2 @ N5 )
% 16.11/8.89 => ( ord_less_num @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_730_lift__Suc__mono__less,axiom,
% 16.11/8.89 ! [F: nat > nat,N2: nat,N5: nat] :
% 16.11/8.89 ( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 16.11/8.89 => ( ( ord_less_nat @ N2 @ N5 )
% 16.11/8.89 => ( ord_less_nat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_731_lift__Suc__mono__less,axiom,
% 16.11/8.89 ! [F: nat > int,N2: nat,N5: nat] :
% 16.11/8.89 ( ! [N4: nat] : ( ord_less_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 16.11/8.89 => ( ( ord_less_nat @ N2 @ N5 )
% 16.11/8.89 => ( ord_less_int @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_732_lift__Suc__mono__less__iff,axiom,
% 16.11/8.89 ! [F: nat > real,N2: nat,M: nat] :
% 16.11/8.89 ( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 16.11/8.89 => ( ( ord_less_real @ ( F @ N2 ) @ ( F @ M ) )
% 16.11/8.89 = ( ord_less_nat @ N2 @ M ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_733_lift__Suc__mono__less__iff,axiom,
% 16.11/8.89 ! [F: nat > rat,N2: nat,M: nat] :
% 16.11/8.89 ( ! [N4: nat] : ( ord_less_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 16.11/8.89 => ( ( ord_less_rat @ ( F @ N2 ) @ ( F @ M ) )
% 16.11/8.89 = ( ord_less_nat @ N2 @ M ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_734_lift__Suc__mono__less__iff,axiom,
% 16.11/8.89 ! [F: nat > num,N2: nat,M: nat] :
% 16.11/8.89 ( ! [N4: nat] : ( ord_less_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 16.11/8.89 => ( ( ord_less_num @ ( F @ N2 ) @ ( F @ M ) )
% 16.11/8.89 = ( ord_less_nat @ N2 @ M ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_735_lift__Suc__mono__less__iff,axiom,
% 16.11/8.89 ! [F: nat > nat,N2: nat,M: nat] :
% 16.11/8.89 ( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 16.11/8.89 => ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M ) )
% 16.11/8.89 = ( ord_less_nat @ N2 @ M ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_736_lift__Suc__mono__less__iff,axiom,
% 16.11/8.89 ! [F: nat > int,N2: nat,M: nat] :
% 16.11/8.89 ( ! [N4: nat] : ( ord_less_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 16.11/8.89 => ( ( ord_less_int @ ( F @ N2 ) @ ( F @ M ) )
% 16.11/8.89 = ( ord_less_nat @ N2 @ M ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_737_lift__Suc__mono__le,axiom,
% 16.11/8.89 ! [F: nat > set_nat,N2: nat,N5: nat] :
% 16.11/8.89 ( ! [N4: nat] : ( ord_less_eq_set_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 16.11/8.89 => ( ( ord_less_eq_nat @ N2 @ N5 )
% 16.11/8.89 => ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_738_lift__Suc__mono__le,axiom,
% 16.11/8.89 ! [F: nat > rat,N2: nat,N5: nat] :
% 16.11/8.89 ( ! [N4: nat] : ( ord_less_eq_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 16.11/8.89 => ( ( ord_less_eq_nat @ N2 @ N5 )
% 16.11/8.89 => ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_739_lift__Suc__mono__le,axiom,
% 16.11/8.89 ! [F: nat > num,N2: nat,N5: nat] :
% 16.11/8.89 ( ! [N4: nat] : ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 16.11/8.89 => ( ( ord_less_eq_nat @ N2 @ N5 )
% 16.11/8.89 => ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_740_lift__Suc__mono__le,axiom,
% 16.11/8.89 ! [F: nat > nat,N2: nat,N5: nat] :
% 16.11/8.89 ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 16.11/8.89 => ( ( ord_less_eq_nat @ N2 @ N5 )
% 16.11/8.89 => ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_741_lift__Suc__mono__le,axiom,
% 16.11/8.89 ! [F: nat > int,N2: nat,N5: nat] :
% 16.11/8.89 ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 16.11/8.89 => ( ( ord_less_eq_nat @ N2 @ N5 )
% 16.11/8.89 => ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_742_lift__Suc__antimono__le,axiom,
% 16.11/8.89 ! [F: nat > set_nat,N2: nat,N5: nat] :
% 16.11/8.89 ( ! [N4: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 16.11/8.89 => ( ( ord_less_eq_nat @ N2 @ N5 )
% 16.11/8.89 => ( ord_less_eq_set_nat @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_743_lift__Suc__antimono__le,axiom,
% 16.11/8.89 ! [F: nat > rat,N2: nat,N5: nat] :
% 16.11/8.89 ( ! [N4: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 16.11/8.89 => ( ( ord_less_eq_nat @ N2 @ N5 )
% 16.11/8.89 => ( ord_less_eq_rat @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_744_lift__Suc__antimono__le,axiom,
% 16.11/8.89 ! [F: nat > num,N2: nat,N5: nat] :
% 16.11/8.89 ( ! [N4: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 16.11/8.89 => ( ( ord_less_eq_nat @ N2 @ N5 )
% 16.11/8.89 => ( ord_less_eq_num @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_745_lift__Suc__antimono__le,axiom,
% 16.11/8.89 ! [F: nat > nat,N2: nat,N5: nat] :
% 16.11/8.89 ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 16.11/8.89 => ( ( ord_less_eq_nat @ N2 @ N5 )
% 16.11/8.89 => ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_746_lift__Suc__antimono__le,axiom,
% 16.11/8.89 ! [F: nat > int,N2: nat,N5: nat] :
% 16.11/8.89 ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 16.11/8.89 => ( ( ord_less_eq_nat @ N2 @ N5 )
% 16.11/8.89 => ( ord_less_eq_int @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_747_diff__less__mono,axiom,
% 16.11/8.89 ! [A: nat,B: nat,C: nat] :
% 16.11/8.89 ( ( ord_less_nat @ A @ B )
% 16.11/8.89 => ( ( ord_less_eq_nat @ C @ A )
% 16.11/8.89 => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_748_less__diff__iff,axiom,
% 16.11/8.89 ! [K: nat,M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ K @ M )
% 16.11/8.89 => ( ( ord_less_eq_nat @ K @ N2 )
% 16.11/8.89 => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
% 16.11/8.89 = ( ord_less_nat @ M @ N2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_749_less__diff__conv,axiom,
% 16.11/8.89 ! [I2: nat,J: nat,K: nat] :
% 16.11/8.89 ( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
% 16.11/8.89 = ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_750_add__diff__inverse__nat,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ~ ( ord_less_nat @ M @ N2 )
% 16.11/8.89 => ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M @ N2 ) )
% 16.11/8.89 = M ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_751_Nat_Ole__imp__diff__is__add,axiom,
% 16.11/8.89 ! [I2: nat,J: nat,K: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ I2 @ J )
% 16.11/8.89 => ( ( ( minus_minus_nat @ J @ I2 )
% 16.11/8.89 = K )
% 16.11/8.89 = ( J
% 16.11/8.89 = ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_752_Nat_Odiff__add__assoc2,axiom,
% 16.11/8.89 ! [K: nat,J: nat,I2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ K @ J )
% 16.11/8.89 => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K )
% 16.11/8.89 = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_753_Nat_Odiff__add__assoc,axiom,
% 16.11/8.89 ! [K: nat,J: nat,I2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ K @ J )
% 16.11/8.89 => ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
% 16.11/8.89 = ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_754_Nat_Ole__diff__conv2,axiom,
% 16.11/8.89 ! [K: nat,J: nat,I2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ K @ J )
% 16.11/8.89 => ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
% 16.11/8.89 = ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_755_le__diff__conv,axiom,
% 16.11/8.89 ! [J: nat,K: nat,I2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
% 16.11/8.89 = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_756_le__imp__less__Suc,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ M @ N2 )
% 16.11/8.89 => ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_757_less__eq__Suc__le,axiom,
% 16.11/8.89 ( ord_less_nat
% 16.11/8.89 = ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_758_less__Suc__eq__le,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_nat @ M @ ( suc @ N2 ) )
% 16.11/8.89 = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_759_le__less__Suc__eq,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ M @ N2 )
% 16.11/8.89 => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
% 16.11/8.89 = ( N2 = M ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_760_Suc__le__lessD,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
% 16.11/8.89 => ( ord_less_nat @ M @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_761_inc__induct,axiom,
% 16.11/8.89 ! [I2: nat,J: nat,P: nat > $o] :
% 16.11/8.89 ( ( ord_less_eq_nat @ I2 @ J )
% 16.11/8.89 => ( ( P @ J )
% 16.11/8.89 => ( ! [N4: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ I2 @ N4 )
% 16.11/8.89 => ( ( ord_less_nat @ N4 @ J )
% 16.11/8.89 => ( ( P @ ( suc @ N4 ) )
% 16.11/8.89 => ( P @ N4 ) ) ) )
% 16.11/8.89 => ( P @ I2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_762_dec__induct,axiom,
% 16.11/8.89 ! [I2: nat,J: nat,P: nat > $o] :
% 16.11/8.89 ( ( ord_less_eq_nat @ I2 @ J )
% 16.11/8.89 => ( ( P @ I2 )
% 16.11/8.89 => ( ! [N4: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ I2 @ N4 )
% 16.11/8.89 => ( ( ord_less_nat @ N4 @ J )
% 16.11/8.89 => ( ( P @ N4 )
% 16.11/8.89 => ( P @ ( suc @ N4 ) ) ) ) )
% 16.11/8.89 => ( P @ J ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_763_Suc__le__eq,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
% 16.11/8.89 = ( ord_less_nat @ M @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_764_Suc__leI,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_nat @ M @ N2 )
% 16.11/8.89 => ( ord_less_eq_nat @ ( suc @ M ) @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_765_less__natE,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_nat @ M @ N2 )
% 16.11/8.89 => ~ ! [Q3: nat] :
% 16.11/8.89 ( N2
% 16.11/8.89 != ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_766_less__add__Suc1,axiom,
% 16.11/8.89 ! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_767_less__add__Suc2,axiom,
% 16.11/8.89 ! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M @ I2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_768_less__iff__Suc__add,axiom,
% 16.11/8.89 ( ord_less_nat
% 16.11/8.89 = ( ^ [M6: nat,N: nat] :
% 16.11/8.89 ? [K3: nat] :
% 16.11/8.89 ( N
% 16.11/8.89 = ( suc @ ( plus_plus_nat @ M6 @ K3 ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_769_less__imp__Suc__add,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_nat @ M @ N2 )
% 16.11/8.89 => ? [K2: nat] :
% 16.11/8.89 ( N2
% 16.11/8.89 = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_770_Suc__mult__less__cancel1,axiom,
% 16.11/8.89 ! [K: nat,M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
% 16.11/8.89 = ( ord_less_nat @ M @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_771_Suc__mult__le__cancel1,axiom,
% 16.11/8.89 ! [K: nat,M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
% 16.11/8.89 = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_772_mult__Suc,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( times_times_nat @ ( suc @ M ) @ N2 )
% 16.11/8.89 = ( plus_plus_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_773_Suc__eq__plus1,axiom,
% 16.11/8.89 ( suc
% 16.11/8.89 = ( ^ [N: nat] : ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_774_plus__1__eq__Suc,axiom,
% 16.11/8.89 ( ( plus_plus_nat @ one_one_nat )
% 16.11/8.89 = suc ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_775_Suc__eq__plus1__left,axiom,
% 16.11/8.89 ( suc
% 16.11/8.89 = ( plus_plus_nat @ one_one_nat ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_776_power__numeral__even,axiom,
% 16.11/8.89 ! [Z: complex,W: num] :
% 16.11/8.89 ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 16.11/8.89 = ( times_times_complex @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_777_power__numeral__even,axiom,
% 16.11/8.89 ! [Z: real,W: num] :
% 16.11/8.89 ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 16.11/8.89 = ( times_times_real @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_778_power__numeral__even,axiom,
% 16.11/8.89 ! [Z: rat,W: num] :
% 16.11/8.89 ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 16.11/8.89 = ( times_times_rat @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_779_power__numeral__even,axiom,
% 16.11/8.89 ! [Z: nat,W: num] :
% 16.11/8.89 ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 16.11/8.89 = ( times_times_nat @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_780_power__numeral__even,axiom,
% 16.11/8.89 ! [Z: int,W: num] :
% 16.11/8.89 ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 16.11/8.89 = ( times_times_int @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_781_mult__diff__mult,axiom,
% 16.11/8.89 ! [X: real,Y: real,A: real,B: real] :
% 16.11/8.89 ( ( minus_minus_real @ ( times_times_real @ X @ Y ) @ ( times_times_real @ A @ B ) )
% 16.11/8.89 = ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_782_mult__diff__mult,axiom,
% 16.11/8.89 ! [X: rat,Y: rat,A: rat,B: rat] :
% 16.11/8.89 ( ( minus_minus_rat @ ( times_times_rat @ X @ Y ) @ ( times_times_rat @ A @ B ) )
% 16.11/8.89 = ( plus_plus_rat @ ( times_times_rat @ X @ ( minus_minus_rat @ Y @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X @ A ) @ B ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_783_mult__diff__mult,axiom,
% 16.11/8.89 ! [X: int,Y: int,A: int,B: int] :
% 16.11/8.89 ( ( minus_minus_int @ ( times_times_int @ X @ Y ) @ ( times_times_int @ A @ B ) )
% 16.11/8.89 = ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A ) @ B ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_784_less__diff__conv2,axiom,
% 16.11/8.89 ! [K: nat,J: nat,I2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ K @ J )
% 16.11/8.89 => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
% 16.11/8.89 = ( ord_less_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_785_power__Suc2,axiom,
% 16.11/8.89 ! [A: complex,N2: nat] :
% 16.11/8.89 ( ( power_power_complex @ A @ ( suc @ N2 ) )
% 16.11/8.89 = ( times_times_complex @ ( power_power_complex @ A @ N2 ) @ A ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_786_power__Suc2,axiom,
% 16.11/8.89 ! [A: real,N2: nat] :
% 16.11/8.89 ( ( power_power_real @ A @ ( suc @ N2 ) )
% 16.11/8.89 = ( times_times_real @ ( power_power_real @ A @ N2 ) @ A ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_787_power__Suc2,axiom,
% 16.11/8.89 ! [A: rat,N2: nat] :
% 16.11/8.89 ( ( power_power_rat @ A @ ( suc @ N2 ) )
% 16.11/8.89 = ( times_times_rat @ ( power_power_rat @ A @ N2 ) @ A ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_788_power__Suc2,axiom,
% 16.11/8.89 ! [A: nat,N2: nat] :
% 16.11/8.89 ( ( power_power_nat @ A @ ( suc @ N2 ) )
% 16.11/8.89 = ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ A ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_789_power__Suc2,axiom,
% 16.11/8.89 ! [A: int,N2: nat] :
% 16.11/8.89 ( ( power_power_int @ A @ ( suc @ N2 ) )
% 16.11/8.89 = ( times_times_int @ ( power_power_int @ A @ N2 ) @ A ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_790_power__Suc,axiom,
% 16.11/8.89 ! [A: complex,N2: nat] :
% 16.11/8.89 ( ( power_power_complex @ A @ ( suc @ N2 ) )
% 16.11/8.89 = ( times_times_complex @ A @ ( power_power_complex @ A @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_791_power__Suc,axiom,
% 16.11/8.89 ! [A: real,N2: nat] :
% 16.11/8.89 ( ( power_power_real @ A @ ( suc @ N2 ) )
% 16.11/8.89 = ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_792_power__Suc,axiom,
% 16.11/8.89 ! [A: rat,N2: nat] :
% 16.11/8.89 ( ( power_power_rat @ A @ ( suc @ N2 ) )
% 16.11/8.89 = ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_793_power__Suc,axiom,
% 16.11/8.89 ! [A: nat,N2: nat] :
% 16.11/8.89 ( ( power_power_nat @ A @ ( suc @ N2 ) )
% 16.11/8.89 = ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_794_power__Suc,axiom,
% 16.11/8.89 ! [A: int,N2: nat] :
% 16.11/8.89 ( ( power_power_int @ A @ ( suc @ N2 ) )
% 16.11/8.89 = ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_795_Suc__div__le__mono,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ ( divide_divide_nat @ ( suc @ M ) @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_796_power__gt1,axiom,
% 16.11/8.89 ! [A: real,N2: nat] :
% 16.11/8.89 ( ( ord_less_real @ one_one_real @ A )
% 16.11/8.89 => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_797_power__gt1,axiom,
% 16.11/8.89 ! [A: rat,N2: nat] :
% 16.11/8.89 ( ( ord_less_rat @ one_one_rat @ A )
% 16.11/8.89 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_798_power__gt1,axiom,
% 16.11/8.89 ! [A: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_nat @ one_one_nat @ A )
% 16.11/8.89 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_799_power__gt1,axiom,
% 16.11/8.89 ! [A: int,N2: nat] :
% 16.11/8.89 ( ( ord_less_int @ one_one_int @ A )
% 16.11/8.89 => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_800_nat__diff__add__eq2,axiom,
% 16.11/8.89 ! [I2: nat,J: nat,U: nat,M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ I2 @ J )
% 16.11/8.89 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 16.11/8.89 = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_801_nat__diff__add__eq1,axiom,
% 16.11/8.89 ! [J: nat,I2: nat,U: nat,M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ J @ I2 )
% 16.11/8.89 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 16.11/8.89 = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_802_nat__le__add__iff2,axiom,
% 16.11/8.89 ! [I2: nat,J: nat,U: nat,M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ I2 @ J )
% 16.11/8.89 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 16.11/8.89 = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_803_nat__le__add__iff1,axiom,
% 16.11/8.89 ! [J: nat,I2: nat,U: nat,M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ J @ I2 )
% 16.11/8.89 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 16.11/8.89 = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_804_nat__eq__add__iff2,axiom,
% 16.11/8.89 ! [I2: nat,J: nat,U: nat,M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ I2 @ J )
% 16.11/8.89 => ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M )
% 16.11/8.89 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 16.11/8.89 = ( M
% 16.11/8.89 = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_805_nat__eq__add__iff1,axiom,
% 16.11/8.89 ! [J: nat,I2: nat,U: nat,M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ J @ I2 )
% 16.11/8.89 => ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M )
% 16.11/8.89 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 16.11/8.89 = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M )
% 16.11/8.89 = N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_806_nat__neq__iff,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( M != N2 )
% 16.11/8.89 = ( ( ord_less_nat @ M @ N2 )
% 16.11/8.89 | ( ord_less_nat @ N2 @ M ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_807_less__not__refl,axiom,
% 16.11/8.89 ! [N2: nat] :
% 16.11/8.89 ~ ( ord_less_nat @ N2 @ N2 ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_808_less__not__refl2,axiom,
% 16.11/8.89 ! [N2: nat,M: nat] :
% 16.11/8.89 ( ( ord_less_nat @ N2 @ M )
% 16.11/8.89 => ( M != N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_809_less__not__refl3,axiom,
% 16.11/8.89 ! [S3: nat,T: nat] :
% 16.11/8.89 ( ( ord_less_nat @ S3 @ T )
% 16.11/8.89 => ( S3 != T ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_810_less__irrefl__nat,axiom,
% 16.11/8.89 ! [N2: nat] :
% 16.11/8.89 ~ ( ord_less_nat @ N2 @ N2 ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_811_nat__less__induct,axiom,
% 16.11/8.89 ! [P: nat > $o,N2: nat] :
% 16.11/8.89 ( ! [N4: nat] :
% 16.11/8.89 ( ! [M5: nat] :
% 16.11/8.89 ( ( ord_less_nat @ M5 @ N4 )
% 16.11/8.89 => ( P @ M5 ) )
% 16.11/8.89 => ( P @ N4 ) )
% 16.11/8.89 => ( P @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_812_infinite__descent,axiom,
% 16.11/8.89 ! [P: nat > $o,N2: nat] :
% 16.11/8.89 ( ! [N4: nat] :
% 16.11/8.89 ( ~ ( P @ N4 )
% 16.11/8.89 => ? [M5: nat] :
% 16.11/8.89 ( ( ord_less_nat @ M5 @ N4 )
% 16.11/8.89 & ~ ( P @ M5 ) ) )
% 16.11/8.89 => ( P @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_813_linorder__neqE__nat,axiom,
% 16.11/8.89 ! [X: nat,Y: nat] :
% 16.11/8.89 ( ( X != Y )
% 16.11/8.89 => ( ~ ( ord_less_nat @ X @ Y )
% 16.11/8.89 => ( ord_less_nat @ Y @ X ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_814_le__refl,axiom,
% 16.11/8.89 ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_815_le__trans,axiom,
% 16.11/8.89 ! [I2: nat,J: nat,K: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ I2 @ J )
% 16.11/8.89 => ( ( ord_less_eq_nat @ J @ K )
% 16.11/8.89 => ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_816_eq__imp__le,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( M = N2 )
% 16.11/8.89 => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_817_le__antisym,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ M @ N2 )
% 16.11/8.89 => ( ( ord_less_eq_nat @ N2 @ M )
% 16.11/8.89 => ( M = N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_818_nat__le__linear,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ M @ N2 )
% 16.11/8.89 | ( ord_less_eq_nat @ N2 @ M ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_819_Nat_Oex__has__greatest__nat,axiom,
% 16.11/8.89 ! [P: nat > $o,K: nat,B: nat] :
% 16.11/8.89 ( ( P @ K )
% 16.11/8.89 => ( ! [Y4: nat] :
% 16.11/8.89 ( ( P @ Y4 )
% 16.11/8.89 => ( ord_less_eq_nat @ Y4 @ B ) )
% 16.11/8.89 => ? [X4: nat] :
% 16.11/8.89 ( ( P @ X4 )
% 16.11/8.89 & ! [Y5: nat] :
% 16.11/8.89 ( ( P @ Y5 )
% 16.11/8.89 => ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_820_size__neq__size__imp__neq,axiom,
% 16.11/8.89 ! [X: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
% 16.11/8.89 ( ( ( size_s6755466524823107622T_VEBT @ X )
% 16.11/8.89 != ( size_s6755466524823107622T_VEBT @ Y ) )
% 16.11/8.89 => ( X != Y ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_821_size__neq__size__imp__neq,axiom,
% 16.11/8.89 ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
% 16.11/8.89 ( ( ( size_size_VEBT_VEBT @ X )
% 16.11/8.89 != ( size_size_VEBT_VEBT @ Y ) )
% 16.11/8.89 => ( X != Y ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_822_size__neq__size__imp__neq,axiom,
% 16.11/8.89 ! [X: num,Y: num] :
% 16.11/8.89 ( ( ( size_size_num @ X )
% 16.11/8.89 != ( size_size_num @ Y ) )
% 16.11/8.89 => ( X != Y ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_823_size__neq__size__imp__neq,axiom,
% 16.11/8.89 ! [X: list_o,Y: list_o] :
% 16.11/8.89 ( ( ( size_size_list_o @ X )
% 16.11/8.89 != ( size_size_list_o @ Y ) )
% 16.11/8.89 => ( X != Y ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_824_size__neq__size__imp__neq,axiom,
% 16.11/8.89 ! [X: list_int,Y: list_int] :
% 16.11/8.89 ( ( ( size_size_list_int @ X )
% 16.11/8.89 != ( size_size_list_int @ Y ) )
% 16.11/8.89 => ( X != Y ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_825_power2__commute,axiom,
% 16.11/8.89 ! [X: complex,Y: complex] :
% 16.11/8.89 ( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.89 = ( power_power_complex @ ( minus_minus_complex @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_826_power2__commute,axiom,
% 16.11/8.89 ! [X: real,Y: real] :
% 16.11/8.89 ( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.89 = ( power_power_real @ ( minus_minus_real @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_827_power2__commute,axiom,
% 16.11/8.89 ! [X: rat,Y: rat] :
% 16.11/8.89 ( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.89 = ( power_power_rat @ ( minus_minus_rat @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_828_power2__commute,axiom,
% 16.11/8.89 ! [X: int,Y: int] :
% 16.11/8.89 ( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.89 = ( power_power_int @ ( minus_minus_int @ Y @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_829_nat__less__add__iff2,axiom,
% 16.11/8.89 ! [I2: nat,J: nat,U: nat,M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ I2 @ J )
% 16.11/8.89 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 16.11/8.89 = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_830_nat__less__add__iff1,axiom,
% 16.11/8.89 ! [J: nat,I2: nat,U: nat,M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ J @ I2 )
% 16.11/8.89 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 16.11/8.89 = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_831_div__nat__eqI,axiom,
% 16.11/8.89 ! [N2: nat,Q2: nat,M: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q2 ) @ M )
% 16.11/8.89 => ( ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q2 ) ) )
% 16.11/8.89 => ( ( divide_divide_nat @ M @ N2 )
% 16.11/8.89 = Q2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_832_Suc__nat__number__of__add,axiom,
% 16.11/8.89 ! [V: num,N2: nat] :
% 16.11/8.89 ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N2 ) )
% 16.11/8.89 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_833_diff__le__diff__pow,axiom,
% 16.11/8.89 ! [K: nat,M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 16.11/8.89 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_834_vebt__delete_Osimps_I7_J,axiom,
% 16.11/8.89 ! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 16.11/8.89 ( ( ( ( ord_less_nat @ X @ Mi )
% 16.11/8.89 | ( ord_less_nat @ Ma @ X ) )
% 16.11/8.89 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
% 16.11/8.89 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) )
% 16.11/8.89 & ( ~ ( ( ord_less_nat @ X @ Mi )
% 16.11/8.89 | ( ord_less_nat @ Ma @ X ) )
% 16.11/8.89 => ( ( ( ( X = Mi )
% 16.11/8.89 & ( X = Ma ) )
% 16.11/8.89 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
% 16.11/8.89 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) )
% 16.11/8.89 & ( ~ ( ( X = Mi )
% 16.11/8.89 & ( X = Ma ) )
% 16.11/8.89 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
% 16.11/8.89 = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 16.11/8.89 @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.11/8.89 @ ( vEBT_Node
% 16.11/8.89 @ ( some_P7363390416028606310at_nat
% 16.11/8.89 @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ Mi )
% 16.11/8.89 @ ( if_nat
% 16.11/8.89 @ ( ( ( X = Mi )
% 16.11/8.89 => ( ( 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 ) ) ) ) ) )
% 16.11/8.89 = Ma ) )
% 16.11/8.89 & ( ( X != Mi )
% 16.11/8.89 => ( X = Ma ) ) )
% 16.11/8.89 @ ( if_nat
% 16.11/8.89 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.11/8.89 = none_nat )
% 16.11/8.89 @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ Mi )
% 16.11/8.89 @ ( 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 @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( 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 @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( 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 @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
% 16.11/8.89 @ Ma ) ) )
% 16.11/8.89 @ ( suc @ ( suc @ Va ) )
% 16.11/8.89 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.11/8.89 @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.11/8.89 @ ( vEBT_Node
% 16.11/8.89 @ ( some_P7363390416028606310at_nat
% 16.11/8.89 @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ Mi )
% 16.11/8.89 @ ( if_nat
% 16.11/8.89 @ ( ( ( X = Mi )
% 16.11/8.89 => ( ( 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 ) ) ) ) ) )
% 16.11/8.89 = Ma ) )
% 16.11/8.89 & ( ( X != Mi )
% 16.11/8.89 => ( X = Ma ) ) )
% 16.11/8.89 @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( 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 @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 16.11/8.89 @ Ma ) ) )
% 16.11/8.89 @ ( suc @ ( suc @ Va ) )
% 16.11/8.89 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = 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 ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.11/8.89 @ Summary ) )
% 16.11/8.89 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_835_power__odd__eq,axiom,
% 16.11/8.89 ! [A: complex,N2: nat] :
% 16.11/8.89 ( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 16.11/8.89 = ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_836_power__odd__eq,axiom,
% 16.11/8.89 ! [A: real,N2: nat] :
% 16.11/8.89 ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 16.11/8.89 = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_837_power__odd__eq,axiom,
% 16.11/8.89 ! [A: rat,N2: nat] :
% 16.11/8.89 ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 16.11/8.89 = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_838_power__odd__eq,axiom,
% 16.11/8.89 ! [A: nat,N2: nat] :
% 16.11/8.89 ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 16.11/8.89 = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_839_power__odd__eq,axiom,
% 16.11/8.89 ! [A: int,N2: nat] :
% 16.11/8.89 ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 16.11/8.89 = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_840_power2__diff,axiom,
% 16.11/8.89 ! [X: complex,Y: complex] :
% 16.11/8.89 ( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.89 = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_841_power2__diff,axiom,
% 16.11/8.89 ! [X: real,Y: real] :
% 16.11/8.89 ( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.89 = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_842_power2__diff,axiom,
% 16.11/8.89 ! [X: rat,Y: rat] :
% 16.11/8.89 ( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.89 = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_843_power2__diff,axiom,
% 16.11/8.89 ! [X: int,Y: int] :
% 16.11/8.89 ( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.11/8.89 = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_844_nat__less__le,axiom,
% 16.11/8.89 ( ord_less_nat
% 16.11/8.89 = ( ^ [M6: nat,N: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ M6 @ N )
% 16.11/8.89 & ( M6 != N ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_845_less__imp__le__nat,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_nat @ M @ N2 )
% 16.11/8.89 => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_846_le__eq__less__or__eq,axiom,
% 16.11/8.89 ( ord_less_eq_nat
% 16.11/8.89 = ( ^ [M6: nat,N: nat] :
% 16.11/8.89 ( ( ord_less_nat @ M6 @ N )
% 16.11/8.89 | ( M6 = N ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_847_less__or__eq__imp__le,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ( ord_less_nat @ M @ N2 )
% 16.11/8.89 | ( M = N2 ) )
% 16.11/8.89 => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_848_le__neq__implies__less,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ M @ N2 )
% 16.11/8.89 => ( ( M != N2 )
% 16.11/8.89 => ( ord_less_nat @ M @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_849_less__mono__imp__le__mono,axiom,
% 16.11/8.89 ! [F: nat > nat,I2: nat,J: nat] :
% 16.11/8.89 ( ! [I4: nat,J2: nat] :
% 16.11/8.89 ( ( ord_less_nat @ I4 @ J2 )
% 16.11/8.89 => ( ord_less_nat @ ( F @ I4 ) @ ( F @ J2 ) ) )
% 16.11/8.89 => ( ( ord_less_eq_nat @ I2 @ J )
% 16.11/8.89 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_850_less__add__eq__less,axiom,
% 16.11/8.89 ! [K: nat,L2: nat,M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_nat @ K @ L2 )
% 16.11/8.89 => ( ( ( plus_plus_nat @ M @ L2 )
% 16.11/8.89 = ( plus_plus_nat @ K @ N2 ) )
% 16.11/8.89 => ( ord_less_nat @ M @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_851_trans__less__add2,axiom,
% 16.11/8.89 ! [I2: nat,J: nat,M: nat] :
% 16.11/8.89 ( ( ord_less_nat @ I2 @ J )
% 16.11/8.89 => ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_852_trans__less__add1,axiom,
% 16.11/8.89 ! [I2: nat,J: nat,M: nat] :
% 16.11/8.89 ( ( ord_less_nat @ I2 @ J )
% 16.11/8.89 => ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_853_add__less__mono1,axiom,
% 16.11/8.89 ! [I2: nat,J: nat,K: nat] :
% 16.11/8.89 ( ( ord_less_nat @ I2 @ J )
% 16.11/8.89 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_854_not__add__less2,axiom,
% 16.11/8.89 ! [J: nat,I2: nat] :
% 16.11/8.89 ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_855_not__add__less1,axiom,
% 16.11/8.89 ! [I2: nat,J: nat] :
% 16.11/8.89 ~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_856_add__less__mono,axiom,
% 16.11/8.89 ! [I2: nat,J: nat,K: nat,L2: nat] :
% 16.11/8.89 ( ( ord_less_nat @ I2 @ J )
% 16.11/8.89 => ( ( ord_less_nat @ K @ L2 )
% 16.11/8.89 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_857_add__lessD1,axiom,
% 16.11/8.89 ! [I2: nat,J: nat,K: nat] :
% 16.11/8.89 ( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
% 16.11/8.89 => ( ord_less_nat @ I2 @ K ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_858_nat__le__iff__add,axiom,
% 16.11/8.89 ( ord_less_eq_nat
% 16.11/8.89 = ( ^ [M6: nat,N: nat] :
% 16.11/8.89 ? [K3: nat] :
% 16.11/8.89 ( N
% 16.11/8.89 = ( plus_plus_nat @ M6 @ K3 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_859_trans__le__add2,axiom,
% 16.11/8.89 ! [I2: nat,J: nat,M: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ I2 @ J )
% 16.11/8.89 => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_860_trans__le__add1,axiom,
% 16.11/8.89 ! [I2: nat,J: nat,M: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ I2 @ J )
% 16.11/8.89 => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_861_add__le__mono1,axiom,
% 16.11/8.89 ! [I2: nat,J: nat,K: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ I2 @ J )
% 16.11/8.89 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_862_add__le__mono,axiom,
% 16.11/8.89 ! [I2: nat,J: nat,K: nat,L2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ I2 @ J )
% 16.11/8.89 => ( ( ord_less_eq_nat @ K @ L2 )
% 16.11/8.89 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_863_le__Suc__ex,axiom,
% 16.11/8.89 ! [K: nat,L2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ K @ L2 )
% 16.11/8.89 => ? [N4: nat] :
% 16.11/8.89 ( L2
% 16.11/8.89 = ( plus_plus_nat @ K @ N4 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_864_add__leD2,axiom,
% 16.11/8.89 ! [M: nat,K: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
% 16.11/8.89 => ( ord_less_eq_nat @ K @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_865_add__leD1,axiom,
% 16.11/8.89 ! [M: nat,K: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
% 16.11/8.89 => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_866_le__add2,axiom,
% 16.11/8.89 ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_867_le__add1,axiom,
% 16.11/8.89 ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_868_add__leE,axiom,
% 16.11/8.89 ! [M: nat,K: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
% 16.11/8.89 => ~ ( ( ord_less_eq_nat @ M @ N2 )
% 16.11/8.89 => ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_869_mult__le__mono2,axiom,
% 16.11/8.89 ! [I2: nat,J: nat,K: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ I2 @ J )
% 16.11/8.89 => ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_870_mult__le__mono1,axiom,
% 16.11/8.89 ! [I2: nat,J: nat,K: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ I2 @ J )
% 16.11/8.89 => ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_871_mult__le__mono,axiom,
% 16.11/8.89 ! [I2: nat,J: nat,K: nat,L2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ I2 @ J )
% 16.11/8.89 => ( ( ord_less_eq_nat @ K @ L2 )
% 16.11/8.89 => ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ L2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_872_le__square,axiom,
% 16.11/8.89 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_873_le__cube,axiom,
% 16.11/8.89 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_874_add__mult__distrib2,axiom,
% 16.11/8.89 ! [K: nat,M: nat,N2: nat] :
% 16.11/8.89 ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N2 ) )
% 16.11/8.89 = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_875_add__mult__distrib,axiom,
% 16.11/8.89 ! [M: nat,N2: nat,K: nat] :
% 16.11/8.89 ( ( times_times_nat @ ( plus_plus_nat @ M @ N2 ) @ K )
% 16.11/8.89 = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_876_nat__mult__1__right,axiom,
% 16.11/8.89 ! [N2: nat] :
% 16.11/8.89 ( ( times_times_nat @ N2 @ one_one_nat )
% 16.11/8.89 = N2 ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_877_nat__mult__1,axiom,
% 16.11/8.89 ! [N2: nat] :
% 16.11/8.89 ( ( times_times_nat @ one_one_nat @ N2 )
% 16.11/8.89 = N2 ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_878_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 16.11/8.89 ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
% 16.11/8.89 ( ( vEBT_V1502963449132264192at_nat @ F @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B ) )
% 16.11/8.89 = ( some_P7363390416028606310at_nat @ ( F @ A @ B ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_879_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 16.11/8.89 ! [F: num > num > num,A: num,B: num] :
% 16.11/8.89 ( ( vEBT_V819420779217536731ft_num @ F @ ( some_num @ A ) @ ( some_num @ B ) )
% 16.11/8.89 = ( some_num @ ( F @ A @ B ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_880_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 16.11/8.89 ! [F: nat > nat > nat,A: nat,B: nat] :
% 16.11/8.89 ( ( vEBT_V4262088993061758097ft_nat @ F @ ( some_nat @ A ) @ ( some_nat @ B ) )
% 16.11/8.89 = ( some_nat @ ( F @ A @ B ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_881_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 16.11/8.89 ! [Uu: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv: option4927543243414619207at_nat] :
% 16.11/8.89 ( ( vEBT_V1502963449132264192at_nat @ Uu @ none_P5556105721700978146at_nat @ Uv )
% 16.11/8.89 = none_P5556105721700978146at_nat ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_882_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 16.11/8.89 ! [Uu: num > num > num,Uv: option_num] :
% 16.11/8.89 ( ( vEBT_V819420779217536731ft_num @ Uu @ none_num @ Uv )
% 16.11/8.89 = none_num ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_883_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 16.11/8.89 ! [Uu: nat > nat > nat,Uv: option_nat] :
% 16.11/8.89 ( ( vEBT_V4262088993061758097ft_nat @ Uu @ none_nat @ Uv )
% 16.11/8.89 = none_nat ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_884_mono__nat__linear__lb,axiom,
% 16.11/8.89 ! [F: nat > nat,M: nat,K: nat] :
% 16.11/8.89 ( ! [M2: nat,N4: nat] :
% 16.11/8.89 ( ( ord_less_nat @ M2 @ N4 )
% 16.11/8.89 => ( ord_less_nat @ ( F @ M2 ) @ ( F @ N4 ) ) )
% 16.11/8.89 => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_885_invar__vebt_Ointros_I3_J,axiom,
% 16.11/8.89 ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 16.11/8.89 ( ! [X4: vEBT_VEBT] :
% 16.11/8.89 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 16.11/8.89 => ( vEBT_invar_vebt @ X4 @ N2 ) )
% 16.11/8.89 => ( ( vEBT_invar_vebt @ Summary @ M )
% 16.11/8.89 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 16.11/8.89 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 16.11/8.89 => ( ( M
% 16.11/8.89 = ( suc @ N2 ) )
% 16.11/8.89 => ( ( Deg
% 16.11/8.89 = ( plus_plus_nat @ N2 @ M ) )
% 16.11/8.89 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 16.11/8.89 => ( ! [X4: vEBT_VEBT] :
% 16.11/8.89 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 16.11/8.89 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
% 16.11/8.89 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_886_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 16.11/8.89 ! [Uw: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V: product_prod_nat_nat] :
% 16.11/8.89 ( ( vEBT_V1502963449132264192at_nat @ Uw @ ( some_P7363390416028606310at_nat @ V ) @ none_P5556105721700978146at_nat )
% 16.11/8.89 = none_P5556105721700978146at_nat ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_887_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 16.11/8.89 ! [Uw: num > num > num,V: num] :
% 16.11/8.89 ( ( vEBT_V819420779217536731ft_num @ Uw @ ( some_num @ V ) @ none_num )
% 16.11/8.89 = none_num ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_888_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 16.11/8.89 ! [Uw: nat > nat > nat,V: nat] :
% 16.11/8.89 ( ( vEBT_V4262088993061758097ft_nat @ Uw @ ( some_nat @ V ) @ none_nat )
% 16.11/8.89 = none_nat ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_889_VEBT__internal_Ooption__shift_Oelims,axiom,
% 16.11/8.89 ! [X: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb2: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
% 16.11/8.89 ( ( ( vEBT_V1502963449132264192at_nat @ X @ Xa2 @ Xb2 )
% 16.11/8.89 = Y )
% 16.11/8.89 => ( ( ( Xa2 = none_P5556105721700978146at_nat )
% 16.11/8.89 => ( Y != none_P5556105721700978146at_nat ) )
% 16.11/8.89 => ( ( ? [V2: product_prod_nat_nat] :
% 16.11/8.89 ( Xa2
% 16.11/8.89 = ( some_P7363390416028606310at_nat @ V2 ) )
% 16.11/8.89 => ( ( Xb2 = none_P5556105721700978146at_nat )
% 16.11/8.89 => ( Y != none_P5556105721700978146at_nat ) ) )
% 16.11/8.89 => ~ ! [A3: product_prod_nat_nat] :
% 16.11/8.89 ( ( Xa2
% 16.11/8.89 = ( some_P7363390416028606310at_nat @ A3 ) )
% 16.11/8.89 => ! [B2: product_prod_nat_nat] :
% 16.11/8.89 ( ( Xb2
% 16.11/8.89 = ( some_P7363390416028606310at_nat @ B2 ) )
% 16.11/8.89 => ( Y
% 16.11/8.89 != ( some_P7363390416028606310at_nat @ ( X @ A3 @ B2 ) ) ) ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_890_VEBT__internal_Ooption__shift_Oelims,axiom,
% 16.11/8.89 ! [X: num > num > num,Xa2: option_num,Xb2: option_num,Y: option_num] :
% 16.11/8.89 ( ( ( vEBT_V819420779217536731ft_num @ X @ Xa2 @ Xb2 )
% 16.11/8.89 = Y )
% 16.11/8.89 => ( ( ( Xa2 = none_num )
% 16.11/8.89 => ( Y != none_num ) )
% 16.11/8.89 => ( ( ? [V2: num] :
% 16.11/8.89 ( Xa2
% 16.11/8.89 = ( some_num @ V2 ) )
% 16.11/8.89 => ( ( Xb2 = none_num )
% 16.11/8.89 => ( Y != none_num ) ) )
% 16.11/8.89 => ~ ! [A3: num] :
% 16.11/8.89 ( ( Xa2
% 16.11/8.89 = ( some_num @ A3 ) )
% 16.11/8.89 => ! [B2: num] :
% 16.11/8.89 ( ( Xb2
% 16.11/8.89 = ( some_num @ B2 ) )
% 16.11/8.89 => ( Y
% 16.11/8.89 != ( some_num @ ( X @ A3 @ B2 ) ) ) ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_891_VEBT__internal_Ooption__shift_Oelims,axiom,
% 16.11/8.89 ! [X: nat > nat > nat,Xa2: option_nat,Xb2: option_nat,Y: option_nat] :
% 16.11/8.89 ( ( ( vEBT_V4262088993061758097ft_nat @ X @ Xa2 @ Xb2 )
% 16.11/8.89 = Y )
% 16.11/8.89 => ( ( ( Xa2 = none_nat )
% 16.11/8.89 => ( Y != none_nat ) )
% 16.11/8.89 => ( ( ? [V2: nat] :
% 16.11/8.89 ( Xa2
% 16.11/8.89 = ( some_nat @ V2 ) )
% 16.11/8.89 => ( ( Xb2 = none_nat )
% 16.11/8.89 => ( Y != none_nat ) ) )
% 16.11/8.89 => ~ ! [A3: nat] :
% 16.11/8.89 ( ( Xa2
% 16.11/8.89 = ( some_nat @ A3 ) )
% 16.11/8.89 => ! [B2: nat] :
% 16.11/8.89 ( ( Xb2
% 16.11/8.89 = ( some_nat @ B2 ) )
% 16.11/8.89 => ( Y
% 16.11/8.89 != ( some_nat @ ( X @ A3 @ B2 ) ) ) ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_892_vebt__maxt_Osimps_I2_J,axiom,
% 16.11/8.89 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 16.11/8.89 ( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 16.11/8.89 = none_nat ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_893_vebt__mint_Osimps_I2_J,axiom,
% 16.11/8.89 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 16.11/8.89 ( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 16.11/8.89 = none_nat ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_894_invar__vebt_Ointros_I5_J,axiom,
% 16.11/8.89 ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 16.11/8.89 ( ! [X4: vEBT_VEBT] :
% 16.11/8.89 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 16.11/8.89 => ( vEBT_invar_vebt @ X4 @ N2 ) )
% 16.11/8.89 => ( ( vEBT_invar_vebt @ Summary @ M )
% 16.11/8.89 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 16.11/8.89 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 16.11/8.89 => ( ( M
% 16.11/8.89 = ( suc @ N2 ) )
% 16.11/8.89 => ( ( Deg
% 16.11/8.89 = ( plus_plus_nat @ N2 @ M ) )
% 16.11/8.89 => ( ! [I4: nat] :
% 16.11/8.89 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 16.11/8.89 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X3 ) )
% 16.11/8.89 = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
% 16.11/8.89 => ( ( ( Mi = Ma )
% 16.11/8.89 => ! [X4: vEBT_VEBT] :
% 16.11/8.89 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 16.11/8.89 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
% 16.11/8.89 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 16.11/8.89 => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 16.11/8.89 => ( ( ( Mi != Ma )
% 16.11/8.89 => ! [I4: nat] :
% 16.11/8.89 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 16.11/8.89 => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
% 16.11/8.89 = I4 )
% 16.11/8.89 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
% 16.11/8.89 & ! [X4: nat] :
% 16.11/8.89 ( ( ( ( vEBT_VEBT_high @ X4 @ N2 )
% 16.11/8.89 = I4 )
% 16.11/8.89 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X4 @ N2 ) ) )
% 16.11/8.89 => ( ( ord_less_nat @ Mi @ X4 )
% 16.11/8.89 & ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
% 16.11/8.89 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_895_vebt__succ_Osimps_I6_J,axiom,
% 16.11/8.89 ! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 16.11/8.89 ( ( ( ord_less_nat @ X @ Mi )
% 16.11/8.89 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
% 16.11/8.89 = ( some_nat @ Mi ) ) )
% 16.11/8.89 & ( ~ ( ord_less_nat @ X @ Mi )
% 16.11/8.89 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
% 16.11/8.89 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 16.11/8.89 @ ( if_option_nat
% 16.11/8.89 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.11/8.89 != none_nat )
% 16.11/8.89 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 16.11/8.89 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.11/8.89 @ ( if_option_nat
% 16.11/8.89 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.11/8.89 = none_nat )
% 16.11/8.89 @ none_nat
% 16.11/8.89 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 16.11/8.89 @ none_nat ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_896_vebt__pred_Osimps_I7_J,axiom,
% 16.11/8.89 ! [Ma: nat,X: nat,Mi: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 16.11/8.89 ( ( ( ord_less_nat @ Ma @ X )
% 16.11/8.89 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
% 16.11/8.89 = ( some_nat @ Ma ) ) )
% 16.11/8.89 & ( ~ ( ord_less_nat @ Ma @ X )
% 16.11/8.89 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
% 16.11/8.89 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 16.11/8.89 @ ( if_option_nat
% 16.11/8.89 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.11/8.89 != none_nat )
% 16.11/8.89 & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 16.11/8.89 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.11/8.89 @ ( if_option_nat
% 16.11/8.89 @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.11/8.89 = none_nat )
% 16.11/8.89 @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
% 16.11/8.89 @ ( 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 @ X @ ( 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 @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 16.11/8.89 @ none_nat ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_897_vebt__member_Osimps_I5_J,axiom,
% 16.11/8.89 ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 16.11/8.89 ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
% 16.11/8.89 = ( ( X != Mi )
% 16.11/8.89 => ( ( X != Ma )
% 16.11/8.89 => ( ~ ( ord_less_nat @ X @ Mi )
% 16.11/8.89 & ( ~ ( ord_less_nat @ X @ Mi )
% 16.11/8.89 => ( ~ ( ord_less_nat @ Ma @ X )
% 16.11/8.89 & ( ~ ( ord_less_nat @ Ma @ X )
% 16.11/8.89 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 16.11/8.89 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.11/8.89 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_898_insert__simp__norm,axiom,
% 16.11/8.89 ! [X: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 16.11/8.89 ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 16.11/8.89 => ( ( ord_less_nat @ Mi @ X )
% 16.11/8.89 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 16.11/8.89 => ( ( X != Ma )
% 16.11/8.89 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 16.11/8.89 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_899_insert__simp__excp,axiom,
% 16.11/8.89 ! [Mi: nat,Deg: nat,TreeList2: list_VEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
% 16.11/8.89 ( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 16.11/8.89 => ( ( ord_less_nat @ X @ Mi )
% 16.11/8.89 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 16.11/8.89 => ( ( X != Ma )
% 16.11/8.89 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 16.11/8.89 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_900_le__add__diff__inverse2,axiom,
% 16.11/8.89 ! [B: real,A: real] :
% 16.11/8.89 ( ( ord_less_eq_real @ B @ A )
% 16.11/8.89 => ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 16.11/8.89 = A ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_901_le__add__diff__inverse2,axiom,
% 16.11/8.89 ! [B: rat,A: rat] :
% 16.11/8.89 ( ( ord_less_eq_rat @ B @ A )
% 16.11/8.89 => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 16.11/8.89 = A ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_902_le__add__diff__inverse2,axiom,
% 16.11/8.89 ! [B: nat,A: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ B @ A )
% 16.11/8.89 => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
% 16.11/8.89 = A ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_903_le__add__diff__inverse2,axiom,
% 16.11/8.89 ! [B: int,A: int] :
% 16.11/8.89 ( ( ord_less_eq_int @ B @ A )
% 16.11/8.89 => ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 16.11/8.89 = A ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_904_le__add__diff__inverse,axiom,
% 16.11/8.89 ! [B: real,A: real] :
% 16.11/8.89 ( ( ord_less_eq_real @ B @ A )
% 16.11/8.89 => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 16.11/8.89 = A ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_905_le__add__diff__inverse,axiom,
% 16.11/8.89 ! [B: rat,A: rat] :
% 16.11/8.89 ( ( ord_less_eq_rat @ B @ A )
% 16.11/8.89 => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 16.11/8.89 = A ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_906_le__add__diff__inverse,axiom,
% 16.11/8.89 ! [B: nat,A: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ B @ A )
% 16.11/8.89 => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 16.11/8.89 = A ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_907_le__add__diff__inverse,axiom,
% 16.11/8.89 ! [B: int,A: int] :
% 16.11/8.89 ( ( ord_less_eq_int @ B @ A )
% 16.11/8.89 => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 16.11/8.89 = A ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_908_vebt__insert_Osimps_I4_J,axiom,
% 16.11/8.89 ! [V: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 16.11/8.89 ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) @ X )
% 16.11/8.89 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_909_pred__empty,axiom,
% 16.11/8.89 ! [T: vEBT_VEBT,N2: nat,X: nat] :
% 16.11/8.89 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.11/8.89 => ( ( ( vEBT_vebt_pred @ T @ X )
% 16.11/8.89 = none_nat )
% 16.11/8.89 = ( ( collect_nat
% 16.11/8.89 @ ^ [Y2: nat] :
% 16.11/8.89 ( ( vEBT_vebt_member @ T @ Y2 )
% 16.11/8.89 & ( ord_less_nat @ Y2 @ X ) ) )
% 16.11/8.89 = bot_bot_set_nat ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_910_succ__empty,axiom,
% 16.11/8.89 ! [T: vEBT_VEBT,N2: nat,X: nat] :
% 16.11/8.89 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.11/8.89 => ( ( ( vEBT_vebt_succ @ T @ X )
% 16.11/8.89 = none_nat )
% 16.11/8.89 = ( ( collect_nat
% 16.11/8.89 @ ^ [Y2: nat] :
% 16.11/8.89 ( ( vEBT_vebt_member @ T @ Y2 )
% 16.11/8.89 & ( ord_less_nat @ X @ Y2 ) ) )
% 16.11/8.89 = bot_bot_set_nat ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_911_enat__ord__number_I1_J,axiom,
% 16.11/8.89 ! [M: num,N2: num] :
% 16.11/8.89 ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
% 16.11/8.89 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_912_maxt__corr__help__empty,axiom,
% 16.11/8.89 ! [T: vEBT_VEBT,N2: nat] :
% 16.11/8.89 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.11/8.89 => ( ( ( vEBT_vebt_maxt @ T )
% 16.11/8.89 = none_nat )
% 16.11/8.89 => ( ( vEBT_VEBT_set_vebt @ T )
% 16.11/8.89 = bot_bot_set_nat ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_913_mint__corr__help__empty,axiom,
% 16.11/8.89 ! [T: vEBT_VEBT,N2: nat] :
% 16.11/8.89 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.11/8.89 => ( ( ( vEBT_vebt_mint @ T )
% 16.11/8.89 = none_nat )
% 16.11/8.89 => ( ( vEBT_VEBT_set_vebt @ T )
% 16.11/8.89 = bot_bot_set_nat ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_914_div__by__1,axiom,
% 16.11/8.89 ! [A: complex] :
% 16.11/8.89 ( ( divide1717551699836669952omplex @ A @ one_one_complex )
% 16.11/8.89 = A ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_915_div__by__1,axiom,
% 16.11/8.89 ! [A: real] :
% 16.11/8.89 ( ( divide_divide_real @ A @ one_one_real )
% 16.11/8.89 = A ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_916_div__by__1,axiom,
% 16.11/8.89 ! [A: rat] :
% 16.11/8.89 ( ( divide_divide_rat @ A @ one_one_rat )
% 16.11/8.89 = A ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_917_div__by__1,axiom,
% 16.11/8.89 ! [A: nat] :
% 16.11/8.89 ( ( divide_divide_nat @ A @ one_one_nat )
% 16.11/8.89 = A ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_918_div__by__1,axiom,
% 16.11/8.89 ! [A: int] :
% 16.11/8.89 ( ( divide_divide_int @ A @ one_one_int )
% 16.11/8.89 = A ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_919_max__Suc__Suc,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N2 ) )
% 16.11/8.89 = ( suc @ ( ord_max_nat @ M @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_920_max__number__of_I1_J,axiom,
% 16.11/8.89 ! [U: num,V: num] :
% 16.11/8.89 ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 16.11/8.89 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 16.11/8.89 = ( numeral_numeral_real @ V ) ) )
% 16.11/8.89 & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 16.11/8.89 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 16.11/8.89 = ( numeral_numeral_real @ U ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_921_max__number__of_I1_J,axiom,
% 16.11/8.89 ! [U: num,V: num] :
% 16.11/8.89 ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 16.11/8.89 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 16.11/8.89 = ( numeral_numeral_rat @ V ) ) )
% 16.11/8.89 & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 16.11/8.89 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 16.11/8.89 = ( numeral_numeral_rat @ U ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_922_max__number__of_I1_J,axiom,
% 16.11/8.89 ! [U: num,V: num] :
% 16.11/8.89 ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 16.11/8.89 => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 16.11/8.89 = ( numeral_numeral_nat @ V ) ) )
% 16.11/8.89 & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 16.11/8.89 => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 16.11/8.89 = ( numeral_numeral_nat @ U ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_923_max__number__of_I1_J,axiom,
% 16.11/8.89 ! [U: num,V: num] :
% 16.11/8.89 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 16.11/8.89 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 16.11/8.89 = ( numeral_numeral_int @ V ) ) )
% 16.11/8.89 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 16.11/8.89 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 16.11/8.89 = ( numeral_numeral_int @ U ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_924_max__0__1_I5_J,axiom,
% 16.11/8.89 ! [X: num] :
% 16.11/8.89 ( ( ord_max_real @ one_one_real @ ( numeral_numeral_real @ X ) )
% 16.11/8.89 = ( numeral_numeral_real @ X ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_925_max__0__1_I5_J,axiom,
% 16.11/8.89 ! [X: num] :
% 16.11/8.89 ( ( ord_max_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
% 16.11/8.89 = ( numeral_numeral_rat @ X ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_926_max__0__1_I5_J,axiom,
% 16.11/8.89 ! [X: num] :
% 16.11/8.89 ( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
% 16.11/8.89 = ( numeral_numeral_nat @ X ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_927_max__0__1_I5_J,axiom,
% 16.11/8.89 ! [X: num] :
% 16.11/8.89 ( ( ord_max_int @ one_one_int @ ( numeral_numeral_int @ X ) )
% 16.11/8.89 = ( numeral_numeral_int @ X ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_928_max__0__1_I6_J,axiom,
% 16.11/8.89 ! [X: num] :
% 16.11/8.89 ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ one_one_real )
% 16.11/8.89 = ( numeral_numeral_real @ X ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_929_max__0__1_I6_J,axiom,
% 16.11/8.89 ! [X: num] :
% 16.11/8.89 ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat )
% 16.11/8.89 = ( numeral_numeral_rat @ X ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_930_max__0__1_I6_J,axiom,
% 16.11/8.89 ! [X: num] :
% 16.11/8.89 ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
% 16.11/8.89 = ( numeral_numeral_nat @ X ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_931_max__0__1_I6_J,axiom,
% 16.11/8.89 ! [X: num] :
% 16.11/8.89 ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ one_one_int )
% 16.11/8.89 = ( numeral_numeral_int @ X ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_932_enat__ord__number_I2_J,axiom,
% 16.11/8.89 ! [M: num,N2: num] :
% 16.11/8.89 ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
% 16.11/8.89 = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_933_add__diff__assoc__enat,axiom,
% 16.11/8.89 ! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
% 16.11/8.89 ( ( ord_le2932123472753598470d_enat @ Z @ Y )
% 16.11/8.89 => ( ( plus_p3455044024723400733d_enat @ X @ ( minus_3235023915231533773d_enat @ Y @ Z ) )
% 16.11/8.89 = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ Z ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_934_nat__add__max__left,axiom,
% 16.11/8.89 ! [M: nat,N2: nat,Q2: nat] :
% 16.11/8.89 ( ( plus_plus_nat @ ( ord_max_nat @ M @ N2 ) @ Q2 )
% 16.11/8.89 = ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N2 @ Q2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_935_nat__add__max__right,axiom,
% 16.11/8.89 ! [M: nat,N2: nat,Q2: nat] :
% 16.11/8.89 ( ( plus_plus_nat @ M @ ( ord_max_nat @ N2 @ Q2 ) )
% 16.11/8.89 = ( ord_max_nat @ ( plus_plus_nat @ M @ N2 ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_936_nat__mult__max__left,axiom,
% 16.11/8.89 ! [M: nat,N2: nat,Q2: nat] :
% 16.11/8.89 ( ( times_times_nat @ ( ord_max_nat @ M @ N2 ) @ Q2 )
% 16.11/8.89 = ( ord_max_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N2 @ Q2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_937_nat__mult__max__right,axiom,
% 16.11/8.89 ! [M: nat,N2: nat,Q2: nat] :
% 16.11/8.89 ( ( times_times_nat @ M @ ( ord_max_nat @ N2 @ Q2 ) )
% 16.11/8.89 = ( ord_max_nat @ ( times_times_nat @ M @ N2 ) @ ( times_times_nat @ M @ Q2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_938_nat__minus__add__max,axiom,
% 16.11/8.89 ! [N2: nat,M: nat] :
% 16.11/8.89 ( ( plus_plus_nat @ ( minus_minus_nat @ N2 @ M ) @ M )
% 16.11/8.89 = ( ord_max_nat @ N2 @ M ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_939_linorder__neqE__linordered__idom,axiom,
% 16.11/8.89 ! [X: real,Y: real] :
% 16.11/8.89 ( ( X != Y )
% 16.11/8.89 => ( ~ ( ord_less_real @ X @ Y )
% 16.11/8.89 => ( ord_less_real @ Y @ X ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_940_linorder__neqE__linordered__idom,axiom,
% 16.11/8.89 ! [X: rat,Y: rat] :
% 16.11/8.89 ( ( X != Y )
% 16.11/8.89 => ( ~ ( ord_less_rat @ X @ Y )
% 16.11/8.89 => ( ord_less_rat @ Y @ X ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_941_linorder__neqE__linordered__idom,axiom,
% 16.11/8.89 ! [X: int,Y: int] :
% 16.11/8.89 ( ( X != Y )
% 16.11/8.89 => ( ~ ( ord_less_int @ X @ Y )
% 16.11/8.89 => ( ord_less_int @ Y @ X ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_942_ring__class_Oring__distribs_I2_J,axiom,
% 16.11/8.89 ! [A: real,B: real,C: real] :
% 16.11/8.89 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 16.11/8.89 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_943_ring__class_Oring__distribs_I2_J,axiom,
% 16.11/8.89 ! [A: rat,B: rat,C: rat] :
% 16.11/8.89 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 16.11/8.89 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_944_ring__class_Oring__distribs_I2_J,axiom,
% 16.11/8.89 ! [A: int,B: int,C: int] :
% 16.11/8.89 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 16.11/8.89 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_945_ring__class_Oring__distribs_I1_J,axiom,
% 16.11/8.89 ! [A: real,B: real,C: real] :
% 16.11/8.89 ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 16.11/8.89 = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_946_ring__class_Oring__distribs_I1_J,axiom,
% 16.11/8.89 ! [A: rat,B: rat,C: rat] :
% 16.11/8.89 ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 16.11/8.89 = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_947_ring__class_Oring__distribs_I1_J,axiom,
% 16.11/8.89 ! [A: int,B: int,C: int] :
% 16.11/8.89 ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
% 16.11/8.89 = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_948_comm__semiring__class_Odistrib,axiom,
% 16.11/8.89 ! [A: real,B: real,C: real] :
% 16.11/8.89 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 16.11/8.89 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_949_comm__semiring__class_Odistrib,axiom,
% 16.11/8.89 ! [A: rat,B: rat,C: rat] :
% 16.11/8.89 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 16.11/8.89 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_950_comm__semiring__class_Odistrib,axiom,
% 16.11/8.89 ! [A: nat,B: nat,C: nat] :
% 16.11/8.89 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 16.11/8.89 = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_951_comm__semiring__class_Odistrib,axiom,
% 16.11/8.89 ! [A: int,B: int,C: int] :
% 16.11/8.89 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 16.11/8.89 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_952_distrib__left,axiom,
% 16.11/8.89 ! [A: real,B: real,C: real] :
% 16.11/8.89 ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 16.11/8.89 = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_953_distrib__left,axiom,
% 16.11/8.89 ! [A: rat,B: rat,C: rat] :
% 16.11/8.89 ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 16.11/8.89 = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_954_distrib__left,axiom,
% 16.11/8.89 ! [A: nat,B: nat,C: nat] :
% 16.11/8.89 ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 16.11/8.89 = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_955_distrib__left,axiom,
% 16.11/8.89 ! [A: int,B: int,C: int] :
% 16.11/8.89 ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
% 16.11/8.89 = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_956_distrib__right,axiom,
% 16.11/8.89 ! [A: real,B: real,C: real] :
% 16.11/8.89 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 16.11/8.89 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_957_distrib__right,axiom,
% 16.11/8.89 ! [A: rat,B: rat,C: rat] :
% 16.11/8.89 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 16.11/8.89 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_958_distrib__right,axiom,
% 16.11/8.89 ! [A: nat,B: nat,C: nat] :
% 16.11/8.89 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 16.11/8.89 = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_959_distrib__right,axiom,
% 16.11/8.89 ! [A: int,B: int,C: int] :
% 16.11/8.89 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 16.11/8.89 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_960_combine__common__factor,axiom,
% 16.11/8.89 ! [A: real,E: real,B: real,C: real] :
% 16.11/8.89 ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
% 16.11/8.89 = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_961_combine__common__factor,axiom,
% 16.11/8.89 ! [A: rat,E: rat,B: rat,C: rat] :
% 16.11/8.89 ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ C ) )
% 16.11/8.89 = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E ) @ C ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_962_combine__common__factor,axiom,
% 16.11/8.89 ! [A: nat,E: nat,B: nat,C: nat] :
% 16.11/8.89 ( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
% 16.11/8.89 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_963_combine__common__factor,axiom,
% 16.11/8.89 ! [A: int,E: int,B: int,C: int] :
% 16.11/8.89 ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
% 16.11/8.89 = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_964_left__diff__distrib,axiom,
% 16.11/8.89 ! [A: real,B: real,C: real] :
% 16.11/8.89 ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
% 16.11/8.89 = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_965_left__diff__distrib,axiom,
% 16.11/8.89 ! [A: rat,B: rat,C: rat] :
% 16.11/8.89 ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 16.11/8.89 = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_966_left__diff__distrib,axiom,
% 16.11/8.89 ! [A: int,B: int,C: int] :
% 16.11/8.89 ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
% 16.11/8.89 = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_967_right__diff__distrib,axiom,
% 16.11/8.89 ! [A: real,B: real,C: real] :
% 16.11/8.89 ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
% 16.11/8.89 = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_968_right__diff__distrib,axiom,
% 16.11/8.89 ! [A: rat,B: rat,C: rat] :
% 16.11/8.89 ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 16.11/8.89 = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_969_right__diff__distrib,axiom,
% 16.11/8.89 ! [A: int,B: int,C: int] :
% 16.11/8.89 ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
% 16.11/8.89 = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_970_left__diff__distrib_H,axiom,
% 16.11/8.89 ! [B: real,C: real,A: real] :
% 16.11/8.89 ( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
% 16.11/8.89 = ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_971_left__diff__distrib_H,axiom,
% 16.11/8.89 ! [B: rat,C: rat,A: rat] :
% 16.11/8.89 ( ( times_times_rat @ ( minus_minus_rat @ B @ C ) @ A )
% 16.11/8.89 = ( minus_minus_rat @ ( times_times_rat @ B @ A ) @ ( times_times_rat @ C @ A ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_972_left__diff__distrib_H,axiom,
% 16.11/8.89 ! [B: nat,C: nat,A: nat] :
% 16.11/8.89 ( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
% 16.11/8.89 = ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_973_left__diff__distrib_H,axiom,
% 16.11/8.89 ! [B: int,C: int,A: int] :
% 16.11/8.89 ( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
% 16.11/8.89 = ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_974_right__diff__distrib_H,axiom,
% 16.11/8.89 ! [A: real,B: real,C: real] :
% 16.11/8.89 ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
% 16.11/8.89 = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_975_right__diff__distrib_H,axiom,
% 16.11/8.89 ! [A: rat,B: rat,C: rat] :
% 16.11/8.89 ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 16.11/8.89 = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_976_right__diff__distrib_H,axiom,
% 16.11/8.89 ! [A: nat,B: nat,C: nat] :
% 16.11/8.89 ( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
% 16.11/8.89 = ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_977_right__diff__distrib_H,axiom,
% 16.11/8.89 ! [A: int,B: int,C: int] :
% 16.11/8.89 ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
% 16.11/8.89 = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_978_lambda__one,axiom,
% 16.11/8.89 ( ( ^ [X2: complex] : X2 )
% 16.11/8.89 = ( times_times_complex @ one_one_complex ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_979_lambda__one,axiom,
% 16.11/8.89 ( ( ^ [X2: real] : X2 )
% 16.11/8.89 = ( times_times_real @ one_one_real ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_980_lambda__one,axiom,
% 16.11/8.89 ( ( ^ [X2: rat] : X2 )
% 16.11/8.89 = ( times_times_rat @ one_one_rat ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_981_lambda__one,axiom,
% 16.11/8.89 ( ( ^ [X2: nat] : X2 )
% 16.11/8.89 = ( times_times_nat @ one_one_nat ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_982_lambda__one,axiom,
% 16.11/8.89 ( ( ^ [X2: int] : X2 )
% 16.11/8.89 = ( times_times_int @ one_one_int ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_983_less__1__mult,axiom,
% 16.11/8.89 ! [M: real,N2: real] :
% 16.11/8.89 ( ( ord_less_real @ one_one_real @ M )
% 16.11/8.89 => ( ( ord_less_real @ one_one_real @ N2 )
% 16.11/8.89 => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_984_less__1__mult,axiom,
% 16.11/8.89 ! [M: rat,N2: rat] :
% 16.11/8.89 ( ( ord_less_rat @ one_one_rat @ M )
% 16.11/8.89 => ( ( ord_less_rat @ one_one_rat @ N2 )
% 16.11/8.89 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_985_less__1__mult,axiom,
% 16.11/8.89 ! [M: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_nat @ one_one_nat @ M )
% 16.11/8.89 => ( ( ord_less_nat @ one_one_nat @ N2 )
% 16.11/8.89 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_986_less__1__mult,axiom,
% 16.11/8.89 ! [M: int,N2: int] :
% 16.11/8.89 ( ( ord_less_int @ one_one_int @ M )
% 16.11/8.89 => ( ( ord_less_int @ one_one_int @ N2 )
% 16.11/8.89 => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N2 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_987_add__mono1,axiom,
% 16.11/8.89 ! [A: real,B: real] :
% 16.11/8.89 ( ( ord_less_real @ A @ B )
% 16.11/8.89 => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_988_add__mono1,axiom,
% 16.11/8.89 ! [A: rat,B: rat] :
% 16.11/8.89 ( ( ord_less_rat @ A @ B )
% 16.11/8.89 => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_989_add__mono1,axiom,
% 16.11/8.89 ! [A: nat,B: nat] :
% 16.11/8.89 ( ( ord_less_nat @ A @ B )
% 16.11/8.89 => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_990_add__mono1,axiom,
% 16.11/8.89 ! [A: int,B: int] :
% 16.11/8.89 ( ( ord_less_int @ A @ B )
% 16.11/8.89 => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_991_less__add__one,axiom,
% 16.11/8.89 ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_992_less__add__one,axiom,
% 16.11/8.89 ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_993_less__add__one,axiom,
% 16.11/8.89 ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_994_less__add__one,axiom,
% 16.11/8.89 ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_995_add__le__imp__le__diff,axiom,
% 16.11/8.89 ! [I2: real,K: real,N2: real] :
% 16.11/8.89 ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N2 )
% 16.11/8.89 => ( ord_less_eq_real @ I2 @ ( minus_minus_real @ N2 @ K ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_996_add__le__imp__le__diff,axiom,
% 16.11/8.89 ! [I2: rat,K: rat,N2: rat] :
% 16.11/8.89 ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ N2 )
% 16.11/8.89 => ( ord_less_eq_rat @ I2 @ ( minus_minus_rat @ N2 @ K ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_997_add__le__imp__le__diff,axiom,
% 16.11/8.89 ! [I2: nat,K: nat,N2: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N2 )
% 16.11/8.89 => ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N2 @ K ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_998_add__le__imp__le__diff,axiom,
% 16.11/8.89 ! [I2: int,K: int,N2: int] :
% 16.11/8.89 ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N2 )
% 16.11/8.89 => ( ord_less_eq_int @ I2 @ ( minus_minus_int @ N2 @ K ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_999_add__le__add__imp__diff__le,axiom,
% 16.11/8.89 ! [I2: real,K: real,N2: real,J: real] :
% 16.11/8.89 ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N2 )
% 16.11/8.89 => ( ( ord_less_eq_real @ N2 @ ( plus_plus_real @ J @ K ) )
% 16.11/8.89 => ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N2 )
% 16.11/8.89 => ( ( ord_less_eq_real @ N2 @ ( plus_plus_real @ J @ K ) )
% 16.11/8.89 => ( ord_less_eq_real @ ( minus_minus_real @ N2 @ K ) @ J ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1000_add__le__add__imp__diff__le,axiom,
% 16.11/8.89 ! [I2: rat,K: rat,N2: rat,J: rat] :
% 16.11/8.89 ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ N2 )
% 16.11/8.89 => ( ( ord_less_eq_rat @ N2 @ ( plus_plus_rat @ J @ K ) )
% 16.11/8.89 => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ N2 )
% 16.11/8.89 => ( ( ord_less_eq_rat @ N2 @ ( plus_plus_rat @ J @ K ) )
% 16.11/8.89 => ( ord_less_eq_rat @ ( minus_minus_rat @ N2 @ K ) @ J ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1001_add__le__add__imp__diff__le,axiom,
% 16.11/8.89 ! [I2: nat,K: nat,N2: nat,J: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N2 )
% 16.11/8.89 => ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
% 16.11/8.89 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N2 )
% 16.11/8.89 => ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
% 16.11/8.89 => ( ord_less_eq_nat @ ( minus_minus_nat @ N2 @ K ) @ J ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1002_add__le__add__imp__diff__le,axiom,
% 16.11/8.89 ! [I2: int,K: int,N2: int,J: int] :
% 16.11/8.89 ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N2 )
% 16.11/8.89 => ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K ) )
% 16.11/8.89 => ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N2 )
% 16.11/8.89 => ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K ) )
% 16.11/8.89 => ( ord_less_eq_int @ ( minus_minus_int @ N2 @ K ) @ J ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1003_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 16.11/8.89 ! [A: real,B: real] :
% 16.11/8.89 ( ~ ( ord_less_real @ A @ B )
% 16.11/8.89 => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 16.11/8.89 = A ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1004_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 16.11/8.89 ! [A: rat,B: rat] :
% 16.11/8.89 ( ~ ( ord_less_rat @ A @ B )
% 16.11/8.89 => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 16.11/8.89 = A ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1005_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 16.11/8.89 ! [A: nat,B: nat] :
% 16.11/8.89 ( ~ ( ord_less_nat @ A @ B )
% 16.11/8.89 => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 16.11/8.89 = A ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1006_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 16.11/8.89 ! [A: int,B: int] :
% 16.11/8.89 ( ~ ( ord_less_int @ A @ B )
% 16.11/8.89 => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 16.11/8.89 = A ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1007_eq__add__iff1,axiom,
% 16.11/8.89 ! [A: real,E: real,C: real,B: real,D: real] :
% 16.11/8.89 ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
% 16.11/8.89 = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 16.11/8.89 = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C )
% 16.11/8.89 = D ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1008_eq__add__iff1,axiom,
% 16.11/8.89 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 16.11/8.89 ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
% 16.11/8.89 = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 16.11/8.89 = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C )
% 16.11/8.89 = D ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1009_eq__add__iff1,axiom,
% 16.11/8.89 ! [A: int,E: int,C: int,B: int,D: int] :
% 16.11/8.89 ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
% 16.11/8.89 = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 16.11/8.89 = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
% 16.11/8.89 = D ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1010_eq__add__iff2,axiom,
% 16.11/8.89 ! [A: real,E: real,C: real,B: real,D: real] :
% 16.11/8.89 ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
% 16.11/8.89 = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 16.11/8.89 = ( C
% 16.11/8.89 = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1011_eq__add__iff2,axiom,
% 16.11/8.89 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 16.11/8.89 ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
% 16.11/8.89 = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 16.11/8.89 = ( C
% 16.11/8.89 = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1012_eq__add__iff2,axiom,
% 16.11/8.89 ! [A: int,E: int,C: int,B: int,D: int] :
% 16.11/8.89 ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
% 16.11/8.89 = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 16.11/8.89 = ( C
% 16.11/8.89 = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1013_square__diff__square__factored,axiom,
% 16.11/8.89 ! [X: real,Y: real] :
% 16.11/8.89 ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
% 16.11/8.89 = ( times_times_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ X @ Y ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1014_square__diff__square__factored,axiom,
% 16.11/8.89 ! [X: rat,Y: rat] :
% 16.11/8.89 ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
% 16.11/8.89 = ( times_times_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_rat @ X @ Y ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1015_square__diff__square__factored,axiom,
% 16.11/8.89 ! [X: int,Y: int] :
% 16.11/8.89 ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
% 16.11/8.89 = ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1016_VEBT__internal_OminNull_Osimps_I5_J,axiom,
% 16.11/8.89 ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
% 16.11/8.89 ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1017_vebt__member_Osimps_I2_J,axiom,
% 16.11/8.89 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] :
% 16.11/8.89 ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1018_VEBT__internal_OminNull_Osimps_I4_J,axiom,
% 16.11/8.89 ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1019_is__succ__in__set__def,axiom,
% 16.11/8.89 ( vEBT_is_succ_in_set
% 16.11/8.89 = ( ^ [Xs2: set_nat,X2: nat,Y2: nat] :
% 16.11/8.89 ( ( member_nat @ Y2 @ Xs2 )
% 16.11/8.89 & ( ord_less_nat @ X2 @ Y2 )
% 16.11/8.89 & ! [Z3: nat] :
% 16.11/8.89 ( ( member_nat @ Z3 @ Xs2 )
% 16.11/8.89 => ( ( ord_less_nat @ X2 @ Z3 )
% 16.11/8.89 => ( ord_less_eq_nat @ Y2 @ Z3 ) ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1020_is__pred__in__set__def,axiom,
% 16.11/8.89 ( vEBT_is_pred_in_set
% 16.11/8.89 = ( ^ [Xs2: set_nat,X2: nat,Y2: nat] :
% 16.11/8.89 ( ( member_nat @ Y2 @ Xs2 )
% 16.11/8.89 & ( ord_less_nat @ Y2 @ X2 )
% 16.11/8.89 & ! [Z3: nat] :
% 16.11/8.89 ( ( member_nat @ Z3 @ Xs2 )
% 16.11/8.89 => ( ( ord_less_nat @ Z3 @ X2 )
% 16.11/8.89 => ( ord_less_eq_nat @ Z3 @ Y2 ) ) ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1021_vebt__insert_Osimps_I5_J,axiom,
% 16.11/8.89 ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 16.11/8.89 ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
% 16.11/8.89 = ( if_VEBT_VEBT
% 16.11/8.89 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 16.11/8.89 & ~ ( ( X = Mi )
% 16.11/8.89 | ( X = Ma ) ) )
% 16.11/8.89 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ X @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
% 16.11/8.89 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1022_ordered__ring__class_Ole__add__iff1,axiom,
% 16.11/8.89 ! [A: real,E: real,C: real,B: real,D: real] :
% 16.11/8.89 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 16.11/8.89 = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1023_ordered__ring__class_Ole__add__iff1,axiom,
% 16.11/8.89 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 16.11/8.89 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 16.11/8.89 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1024_ordered__ring__class_Ole__add__iff1,axiom,
% 16.11/8.89 ! [A: int,E: int,C: int,B: int,D: int] :
% 16.11/8.89 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 16.11/8.89 = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1025_ordered__ring__class_Ole__add__iff2,axiom,
% 16.11/8.89 ! [A: real,E: real,C: real,B: real,D: real] :
% 16.11/8.89 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 16.11/8.89 = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1026_ordered__ring__class_Ole__add__iff2,axiom,
% 16.11/8.89 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 16.11/8.89 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 16.11/8.89 = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1027_ordered__ring__class_Ole__add__iff2,axiom,
% 16.11/8.89 ! [A: int,E: int,C: int,B: int,D: int] :
% 16.11/8.89 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 16.11/8.89 = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1028_less__add__iff1,axiom,
% 16.11/8.89 ! [A: real,E: real,C: real,B: real,D: real] :
% 16.11/8.89 ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 16.11/8.89 = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1029_less__add__iff1,axiom,
% 16.11/8.89 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 16.11/8.89 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 16.11/8.89 = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1030_less__add__iff1,axiom,
% 16.11/8.89 ! [A: int,E: int,C: int,B: int,D: int] :
% 16.11/8.89 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 16.11/8.89 = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1031_less__add__iff2,axiom,
% 16.11/8.89 ! [A: real,E: real,C: real,B: real,D: real] :
% 16.11/8.89 ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 16.11/8.89 = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1032_less__add__iff2,axiom,
% 16.11/8.89 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 16.11/8.89 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 16.11/8.89 = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1033_less__add__iff2,axiom,
% 16.11/8.89 ! [A: int,E: int,C: int,B: int,D: int] :
% 16.11/8.89 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 16.11/8.89 = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1034_square__diff__one__factored,axiom,
% 16.11/8.89 ! [X: complex] :
% 16.11/8.89 ( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ one_one_complex )
% 16.11/8.89 = ( times_times_complex @ ( plus_plus_complex @ X @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1035_square__diff__one__factored,axiom,
% 16.11/8.89 ! [X: real] :
% 16.11/8.89 ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
% 16.11/8.89 = ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1036_square__diff__one__factored,axiom,
% 16.11/8.89 ! [X: rat] :
% 16.11/8.89 ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ one_one_rat )
% 16.11/8.89 = ( times_times_rat @ ( plus_plus_rat @ X @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1037_square__diff__one__factored,axiom,
% 16.11/8.89 ! [X: int] :
% 16.11/8.89 ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
% 16.11/8.89 = ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1038_vebt__succ_Osimps_I3_J,axiom,
% 16.11/8.89 ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
% 16.11/8.89 ( ( vEBT_vebt_succ @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Va )
% 16.11/8.89 = none_nat ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1039_vebt__pred_Osimps_I4_J,axiom,
% 16.11/8.89 ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
% 16.11/8.89 ( ( vEBT_vebt_pred @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va ) @ Vb )
% 16.11/8.89 = none_nat ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1040_real__average__minus__second,axiom,
% 16.11/8.89 ! [B: real,A: real] :
% 16.11/8.89 ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 16.11/8.89 = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1041_real__average__minus__first,axiom,
% 16.11/8.89 ! [A: real,B: real] :
% 16.11/8.89 ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 16.11/8.89 = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1042_buildup__gives__empty,axiom,
% 16.11/8.89 ! [N2: nat] :
% 16.11/8.89 ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N2 ) )
% 16.11/8.89 = bot_bot_set_nat ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1043_max__less__iff__conj,axiom,
% 16.11/8.89 ! [X: real,Y: real,Z: real] :
% 16.11/8.89 ( ( ord_less_real @ ( ord_max_real @ X @ Y ) @ Z )
% 16.11/8.89 = ( ( ord_less_real @ X @ Z )
% 16.11/8.89 & ( ord_less_real @ Y @ Z ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1044_max__less__iff__conj,axiom,
% 16.11/8.89 ! [X: rat,Y: rat,Z: rat] :
% 16.11/8.89 ( ( ord_less_rat @ ( ord_max_rat @ X @ Y ) @ Z )
% 16.11/8.89 = ( ( ord_less_rat @ X @ Z )
% 16.11/8.89 & ( ord_less_rat @ Y @ Z ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1045_max__less__iff__conj,axiom,
% 16.11/8.89 ! [X: num,Y: num,Z: num] :
% 16.11/8.89 ( ( ord_less_num @ ( ord_max_num @ X @ Y ) @ Z )
% 16.11/8.89 = ( ( ord_less_num @ X @ Z )
% 16.11/8.89 & ( ord_less_num @ Y @ Z ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1046_max__less__iff__conj,axiom,
% 16.11/8.89 ! [X: nat,Y: nat,Z: nat] :
% 16.11/8.89 ( ( ord_less_nat @ ( ord_max_nat @ X @ Y ) @ Z )
% 16.11/8.89 = ( ( ord_less_nat @ X @ Z )
% 16.11/8.89 & ( ord_less_nat @ Y @ Z ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1047_max__less__iff__conj,axiom,
% 16.11/8.89 ! [X: int,Y: int,Z: int] :
% 16.11/8.89 ( ( ord_less_int @ ( ord_max_int @ X @ Y ) @ Z )
% 16.11/8.89 = ( ( ord_less_int @ X @ Z )
% 16.11/8.89 & ( ord_less_int @ Y @ Z ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1048_max_Oabsorb4,axiom,
% 16.11/8.89 ! [A: real,B: real] :
% 16.11/8.89 ( ( ord_less_real @ A @ B )
% 16.11/8.89 => ( ( ord_max_real @ A @ B )
% 16.11/8.89 = B ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1049_max_Oabsorb4,axiom,
% 16.11/8.89 ! [A: rat,B: rat] :
% 16.11/8.89 ( ( ord_less_rat @ A @ B )
% 16.11/8.89 => ( ( ord_max_rat @ A @ B )
% 16.11/8.89 = B ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1050_max_Oabsorb4,axiom,
% 16.11/8.89 ! [A: num,B: num] :
% 16.11/8.89 ( ( ord_less_num @ A @ B )
% 16.11/8.89 => ( ( ord_max_num @ A @ B )
% 16.11/8.89 = B ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1051_max_Oabsorb4,axiom,
% 16.11/8.89 ! [A: nat,B: nat] :
% 16.11/8.89 ( ( ord_less_nat @ A @ B )
% 16.11/8.89 => ( ( ord_max_nat @ A @ B )
% 16.11/8.89 = B ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1052_max_Oabsorb4,axiom,
% 16.11/8.89 ! [A: int,B: int] :
% 16.11/8.89 ( ( ord_less_int @ A @ B )
% 16.11/8.89 => ( ( ord_max_int @ A @ B )
% 16.11/8.89 = B ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1053_max_Oabsorb3,axiom,
% 16.11/8.89 ! [B: real,A: real] :
% 16.11/8.89 ( ( ord_less_real @ B @ A )
% 16.11/8.89 => ( ( ord_max_real @ A @ B )
% 16.11/8.89 = A ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1054_max_Oabsorb3,axiom,
% 16.11/8.89 ! [B: rat,A: rat] :
% 16.11/8.89 ( ( ord_less_rat @ B @ A )
% 16.11/8.89 => ( ( ord_max_rat @ A @ B )
% 16.11/8.89 = A ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1055_max_Oabsorb3,axiom,
% 16.11/8.89 ! [B: num,A: num] :
% 16.11/8.89 ( ( ord_less_num @ B @ A )
% 16.11/8.89 => ( ( ord_max_num @ A @ B )
% 16.11/8.89 = A ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1056_max_Oabsorb3,axiom,
% 16.11/8.89 ! [B: nat,A: nat] :
% 16.11/8.89 ( ( ord_less_nat @ B @ A )
% 16.11/8.89 => ( ( ord_max_nat @ A @ B )
% 16.11/8.89 = A ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1057_max_Oabsorb3,axiom,
% 16.11/8.89 ! [B: int,A: int] :
% 16.11/8.89 ( ( ord_less_int @ B @ A )
% 16.11/8.89 => ( ( ord_max_int @ A @ B )
% 16.11/8.89 = A ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1058_max_Obounded__iff,axiom,
% 16.11/8.89 ! [B: rat,C: rat,A: rat] :
% 16.11/8.89 ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
% 16.11/8.89 = ( ( ord_less_eq_rat @ B @ A )
% 16.11/8.89 & ( ord_less_eq_rat @ C @ A ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1059_max_Obounded__iff,axiom,
% 16.11/8.89 ! [B: num,C: num,A: num] :
% 16.11/8.89 ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
% 16.11/8.89 = ( ( ord_less_eq_num @ B @ A )
% 16.11/8.89 & ( ord_less_eq_num @ C @ A ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1060_max_Obounded__iff,axiom,
% 16.11/8.89 ! [B: nat,C: nat,A: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
% 16.11/8.89 = ( ( ord_less_eq_nat @ B @ A )
% 16.11/8.89 & ( ord_less_eq_nat @ C @ A ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1061_max_Obounded__iff,axiom,
% 16.11/8.89 ! [B: int,C: int,A: int] :
% 16.11/8.89 ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
% 16.11/8.89 = ( ( ord_less_eq_int @ B @ A )
% 16.11/8.89 & ( ord_less_eq_int @ C @ A ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1062_max_Oabsorb2,axiom,
% 16.11/8.89 ! [A: rat,B: rat] :
% 16.11/8.89 ( ( ord_less_eq_rat @ A @ B )
% 16.11/8.89 => ( ( ord_max_rat @ A @ B )
% 16.11/8.89 = B ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1063_max_Oabsorb2,axiom,
% 16.11/8.89 ! [A: num,B: num] :
% 16.11/8.89 ( ( ord_less_eq_num @ A @ B )
% 16.11/8.89 => ( ( ord_max_num @ A @ B )
% 16.11/8.89 = B ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1064_max_Oabsorb2,axiom,
% 16.11/8.89 ! [A: nat,B: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ A @ B )
% 16.11/8.89 => ( ( ord_max_nat @ A @ B )
% 16.11/8.89 = B ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1065_max_Oabsorb2,axiom,
% 16.11/8.89 ! [A: int,B: int] :
% 16.11/8.89 ( ( ord_less_eq_int @ A @ B )
% 16.11/8.89 => ( ( ord_max_int @ A @ B )
% 16.11/8.89 = B ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1066_max_Oabsorb1,axiom,
% 16.11/8.89 ! [B: rat,A: rat] :
% 16.11/8.89 ( ( ord_less_eq_rat @ B @ A )
% 16.11/8.89 => ( ( ord_max_rat @ A @ B )
% 16.11/8.89 = A ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1067_max_Oabsorb1,axiom,
% 16.11/8.89 ! [B: num,A: num] :
% 16.11/8.89 ( ( ord_less_eq_num @ B @ A )
% 16.11/8.89 => ( ( ord_max_num @ A @ B )
% 16.11/8.89 = A ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1068_max_Oabsorb1,axiom,
% 16.11/8.89 ! [B: nat,A: nat] :
% 16.11/8.89 ( ( ord_less_eq_nat @ B @ A )
% 16.11/8.89 => ( ( ord_max_nat @ A @ B )
% 16.11/8.89 = A ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1069_max_Oabsorb1,axiom,
% 16.11/8.89 ! [B: int,A: int] :
% 16.11/8.89 ( ( ord_less_eq_int @ B @ A )
% 16.11/8.89 => ( ( ord_max_int @ A @ B )
% 16.11/8.89 = A ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1070_Diff__eq__empty__iff,axiom,
% 16.11/8.89 ! [A2: set_int,B3: set_int] :
% 16.11/8.89 ( ( ( minus_minus_set_int @ A2 @ B3 )
% 16.11/8.89 = bot_bot_set_int )
% 16.11/8.89 = ( ord_less_eq_set_int @ A2 @ B3 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1071_Diff__eq__empty__iff,axiom,
% 16.11/8.89 ! [A2: set_real,B3: set_real] :
% 16.11/8.89 ( ( ( minus_minus_set_real @ A2 @ B3 )
% 16.11/8.89 = bot_bot_set_real )
% 16.11/8.89 = ( ord_less_eq_set_real @ A2 @ B3 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1072_Diff__eq__empty__iff,axiom,
% 16.11/8.89 ! [A2: set_nat,B3: set_nat] :
% 16.11/8.89 ( ( ( minus_minus_set_nat @ A2 @ B3 )
% 16.11/8.89 = bot_bot_set_nat )
% 16.11/8.89 = ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1073_subsetI,axiom,
% 16.11/8.89 ! [A2: set_real,B3: set_real] :
% 16.11/8.89 ( ! [X4: real] :
% 16.11/8.89 ( ( member_real @ X4 @ A2 )
% 16.11/8.89 => ( member_real @ X4 @ B3 ) )
% 16.11/8.89 => ( ord_less_eq_set_real @ A2 @ B3 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1074_subsetI,axiom,
% 16.11/8.89 ! [A2: set_int,B3: set_int] :
% 16.11/8.89 ( ! [X4: int] :
% 16.11/8.89 ( ( member_int @ X4 @ A2 )
% 16.11/8.89 => ( member_int @ X4 @ B3 ) )
% 16.11/8.89 => ( ord_less_eq_set_int @ A2 @ B3 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1075_subsetI,axiom,
% 16.11/8.89 ! [A2: set_complex,B3: set_complex] :
% 16.11/8.89 ( ! [X4: complex] :
% 16.11/8.89 ( ( member_complex @ X4 @ A2 )
% 16.11/8.89 => ( member_complex @ X4 @ B3 ) )
% 16.11/8.89 => ( ord_le211207098394363844omplex @ A2 @ B3 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1076_subsetI,axiom,
% 16.11/8.89 ! [A2: set_set_nat,B3: set_set_nat] :
% 16.11/8.89 ( ! [X4: set_nat] :
% 16.11/8.89 ( ( member_set_nat @ X4 @ A2 )
% 16.11/8.89 => ( member_set_nat @ X4 @ B3 ) )
% 16.11/8.89 => ( ord_le6893508408891458716et_nat @ A2 @ B3 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1077_subsetI,axiom,
% 16.11/8.89 ! [A2: set_nat,B3: set_nat] :
% 16.11/8.89 ( ! [X4: nat] :
% 16.11/8.89 ( ( member_nat @ X4 @ A2 )
% 16.11/8.89 => ( member_nat @ X4 @ B3 ) )
% 16.11/8.89 => ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1078_subset__antisym,axiom,
% 16.11/8.89 ! [A2: set_nat,B3: set_nat] :
% 16.11/8.89 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 16.11/8.89 => ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 16.11/8.89 => ( A2 = B3 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1079_psubsetI,axiom,
% 16.11/8.89 ! [A2: set_nat,B3: set_nat] :
% 16.11/8.89 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 16.11/8.89 => ( ( A2 != B3 )
% 16.11/8.89 => ( ord_less_set_nat @ A2 @ B3 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1080_DiffI,axiom,
% 16.11/8.89 ! [C: real,A2: set_real,B3: set_real] :
% 16.11/8.89 ( ( member_real @ C @ A2 )
% 16.11/8.89 => ( ~ ( member_real @ C @ B3 )
% 16.11/8.89 => ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1081_DiffI,axiom,
% 16.11/8.89 ! [C: int,A2: set_int,B3: set_int] :
% 16.11/8.89 ( ( member_int @ C @ A2 )
% 16.11/8.89 => ( ~ ( member_int @ C @ B3 )
% 16.11/8.89 => ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1082_DiffI,axiom,
% 16.11/8.89 ! [C: complex,A2: set_complex,B3: set_complex] :
% 16.11/8.89 ( ( member_complex @ C @ A2 )
% 16.11/8.89 => ( ~ ( member_complex @ C @ B3 )
% 16.11/8.89 => ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1083_DiffI,axiom,
% 16.11/8.89 ! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
% 16.11/8.89 ( ( member_set_nat @ C @ A2 )
% 16.11/8.89 => ( ~ ( member_set_nat @ C @ B3 )
% 16.11/8.89 => ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B3 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1084_DiffI,axiom,
% 16.11/8.89 ! [C: nat,A2: set_nat,B3: set_nat] :
% 16.11/8.89 ( ( member_nat @ C @ A2 )
% 16.11/8.89 => ( ~ ( member_nat @ C @ B3 )
% 16.11/8.89 => ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1085_Diff__iff,axiom,
% 16.11/8.89 ! [C: real,A2: set_real,B3: set_real] :
% 16.11/8.89 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
% 16.11/8.89 = ( ( member_real @ C @ A2 )
% 16.11/8.89 & ~ ( member_real @ C @ B3 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1086_Diff__iff,axiom,
% 16.11/8.89 ! [C: int,A2: set_int,B3: set_int] :
% 16.11/8.89 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
% 16.11/8.89 = ( ( member_int @ C @ A2 )
% 16.11/8.89 & ~ ( member_int @ C @ B3 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1087_Diff__iff,axiom,
% 16.11/8.89 ! [C: complex,A2: set_complex,B3: set_complex] :
% 16.11/8.89 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 16.11/8.89 = ( ( member_complex @ C @ A2 )
% 16.11/8.89 & ~ ( member_complex @ C @ B3 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1088_Diff__iff,axiom,
% 16.11/8.89 ! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
% 16.11/8.89 ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B3 ) )
% 16.11/8.89 = ( ( member_set_nat @ C @ A2 )
% 16.11/8.89 & ~ ( member_set_nat @ C @ B3 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1089_Diff__iff,axiom,
% 16.11/8.89 ! [C: nat,A2: set_nat,B3: set_nat] :
% 16.11/8.89 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 16.11/8.89 = ( ( member_nat @ C @ A2 )
% 16.11/8.89 & ~ ( member_nat @ C @ B3 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1090_Diff__idemp,axiom,
% 16.11/8.89 ! [A2: set_nat,B3: set_nat] :
% 16.11/8.89 ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B3 ) @ B3 )
% 16.11/8.89 = ( minus_minus_set_nat @ A2 @ B3 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1091_subset__empty,axiom,
% 16.11/8.89 ! [A2: set_int] :
% 16.11/8.89 ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
% 16.11/8.89 = ( A2 = bot_bot_set_int ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1092_subset__empty,axiom,
% 16.11/8.89 ! [A2: set_real] :
% 16.11/8.89 ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
% 16.11/8.89 = ( A2 = bot_bot_set_real ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1093_subset__empty,axiom,
% 16.11/8.89 ! [A2: set_nat] :
% 16.11/8.89 ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
% 16.11/8.89 = ( A2 = bot_bot_set_nat ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1094_empty__subsetI,axiom,
% 16.11/8.89 ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1095_empty__subsetI,axiom,
% 16.11/8.89 ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1096_empty__subsetI,axiom,
% 16.11/8.89 ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1097_Diff__cancel,axiom,
% 16.11/8.89 ! [A2: set_int] :
% 16.11/8.89 ( ( minus_minus_set_int @ A2 @ A2 )
% 16.11/8.89 = bot_bot_set_int ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1098_Diff__cancel,axiom,
% 16.11/8.89 ! [A2: set_real] :
% 16.11/8.89 ( ( minus_minus_set_real @ A2 @ A2 )
% 16.11/8.89 = bot_bot_set_real ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1099_Diff__cancel,axiom,
% 16.11/8.89 ! [A2: set_nat] :
% 16.11/8.89 ( ( minus_minus_set_nat @ A2 @ A2 )
% 16.11/8.89 = bot_bot_set_nat ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1100_empty__Diff,axiom,
% 16.11/8.89 ! [A2: set_int] :
% 16.11/8.89 ( ( minus_minus_set_int @ bot_bot_set_int @ A2 )
% 16.11/8.89 = bot_bot_set_int ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1101_empty__Diff,axiom,
% 16.11/8.89 ! [A2: set_real] :
% 16.11/8.89 ( ( minus_minus_set_real @ bot_bot_set_real @ A2 )
% 16.11/8.89 = bot_bot_set_real ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1102_empty__Diff,axiom,
% 16.11/8.89 ! [A2: set_nat] :
% 16.11/8.89 ( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
% 16.11/8.89 = bot_bot_set_nat ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1103_Diff__empty,axiom,
% 16.11/8.89 ! [A2: set_int] :
% 16.11/8.89 ( ( minus_minus_set_int @ A2 @ bot_bot_set_int )
% 16.11/8.89 = A2 ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1104_Diff__empty,axiom,
% 16.11/8.89 ! [A2: set_real] :
% 16.11/8.89 ( ( minus_minus_set_real @ A2 @ bot_bot_set_real )
% 16.11/8.89 = A2 ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1105_Diff__empty,axiom,
% 16.11/8.89 ! [A2: set_nat] :
% 16.11/8.89 ( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
% 16.11/8.89 = A2 ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1106_psubset__imp__ex__mem,axiom,
% 16.11/8.89 ! [A2: set_real,B3: set_real] :
% 16.11/8.89 ( ( ord_less_set_real @ A2 @ B3 )
% 16.11/8.89 => ? [B2: real] : ( member_real @ B2 @ ( minus_minus_set_real @ B3 @ A2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1107_psubset__imp__ex__mem,axiom,
% 16.11/8.89 ! [A2: set_int,B3: set_int] :
% 16.11/8.89 ( ( ord_less_set_int @ A2 @ B3 )
% 16.11/8.89 => ? [B2: int] : ( member_int @ B2 @ ( minus_minus_set_int @ B3 @ A2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1108_psubset__imp__ex__mem,axiom,
% 16.11/8.89 ! [A2: set_complex,B3: set_complex] :
% 16.11/8.89 ( ( ord_less_set_complex @ A2 @ B3 )
% 16.11/8.89 => ? [B2: complex] : ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B3 @ A2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1109_psubset__imp__ex__mem,axiom,
% 16.11/8.89 ! [A2: set_set_nat,B3: set_set_nat] :
% 16.11/8.89 ( ( ord_less_set_set_nat @ A2 @ B3 )
% 16.11/8.89 => ? [B2: set_nat] : ( member_set_nat @ B2 @ ( minus_2163939370556025621et_nat @ B3 @ A2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1110_psubset__imp__ex__mem,axiom,
% 16.11/8.89 ! [A2: set_nat,B3: set_nat] :
% 16.11/8.89 ( ( ord_less_set_nat @ A2 @ B3 )
% 16.11/8.89 => ? [B2: nat] : ( member_nat @ B2 @ ( minus_minus_set_nat @ B3 @ A2 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1111_enat__less__induct,axiom,
% 16.11/8.89 ! [P: extended_enat > $o,N2: extended_enat] :
% 16.11/8.89 ( ! [N4: extended_enat] :
% 16.11/8.89 ( ! [M5: extended_enat] :
% 16.11/8.89 ( ( ord_le72135733267957522d_enat @ M5 @ N4 )
% 16.11/8.89 => ( P @ M5 ) )
% 16.11/8.89 => ( P @ N4 ) )
% 16.11/8.89 => ( P @ N2 ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1112_DiffE,axiom,
% 16.11/8.89 ! [C: real,A2: set_real,B3: set_real] :
% 16.11/8.89 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
% 16.11/8.89 => ~ ( ( member_real @ C @ A2 )
% 16.11/8.89 => ( member_real @ C @ B3 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1113_DiffE,axiom,
% 16.11/8.89 ! [C: int,A2: set_int,B3: set_int] :
% 16.11/8.89 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
% 16.11/8.89 => ~ ( ( member_int @ C @ A2 )
% 16.11/8.89 => ( member_int @ C @ B3 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1114_DiffE,axiom,
% 16.11/8.89 ! [C: complex,A2: set_complex,B3: set_complex] :
% 16.11/8.89 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 16.11/8.89 => ~ ( ( member_complex @ C @ A2 )
% 16.11/8.89 => ( member_complex @ C @ B3 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1115_DiffE,axiom,
% 16.11/8.89 ! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
% 16.11/8.89 ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B3 ) )
% 16.11/8.89 => ~ ( ( member_set_nat @ C @ A2 )
% 16.11/8.89 => ( member_set_nat @ C @ B3 ) ) ) ).
% 16.11/8.89
% 16.11/8.89 thf(fact_1116_DiffE,axiom,
% 16.11/8.89 ! [C: nat,A2: set_nat,B3: set_nat] :
% 16.11/8.89 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 16.11/8.90 => ~ ( ( member_nat @ C @ A2 )
% 16.11/8.90 => ( member_nat @ C @ B3 ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1117_DiffD1,axiom,
% 16.11/8.90 ! [C: real,A2: set_real,B3: set_real] :
% 16.11/8.90 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
% 16.11/8.90 => ( member_real @ C @ A2 ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1118_DiffD1,axiom,
% 16.11/8.90 ! [C: int,A2: set_int,B3: set_int] :
% 16.11/8.90 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
% 16.11/8.90 => ( member_int @ C @ A2 ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1119_DiffD1,axiom,
% 16.11/8.90 ! [C: complex,A2: set_complex,B3: set_complex] :
% 16.11/8.90 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 16.11/8.90 => ( member_complex @ C @ A2 ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1120_DiffD1,axiom,
% 16.11/8.90 ! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
% 16.11/8.90 ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B3 ) )
% 16.11/8.90 => ( member_set_nat @ C @ A2 ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1121_DiffD1,axiom,
% 16.11/8.90 ! [C: nat,A2: set_nat,B3: set_nat] :
% 16.11/8.90 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 16.11/8.90 => ( member_nat @ C @ A2 ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1122_DiffD2,axiom,
% 16.11/8.90 ! [C: real,A2: set_real,B3: set_real] :
% 16.11/8.90 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B3 ) )
% 16.11/8.90 => ~ ( member_real @ C @ B3 ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1123_DiffD2,axiom,
% 16.11/8.90 ! [C: int,A2: set_int,B3: set_int] :
% 16.11/8.90 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B3 ) )
% 16.11/8.90 => ~ ( member_int @ C @ B3 ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1124_DiffD2,axiom,
% 16.11/8.90 ! [C: complex,A2: set_complex,B3: set_complex] :
% 16.11/8.90 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 16.11/8.90 => ~ ( member_complex @ C @ B3 ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1125_DiffD2,axiom,
% 16.11/8.90 ! [C: set_nat,A2: set_set_nat,B3: set_set_nat] :
% 16.11/8.90 ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B3 ) )
% 16.11/8.90 => ~ ( member_set_nat @ C @ B3 ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1126_DiffD2,axiom,
% 16.11/8.90 ! [C: nat,A2: set_nat,B3: set_nat] :
% 16.11/8.90 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 16.11/8.90 => ~ ( member_nat @ C @ B3 ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1127_set__diff__eq,axiom,
% 16.11/8.90 ( minus_minus_set_real
% 16.11/8.90 = ( ^ [A4: set_real,B4: set_real] :
% 16.11/8.90 ( collect_real
% 16.11/8.90 @ ^ [X2: real] :
% 16.11/8.90 ( ( member_real @ X2 @ A4 )
% 16.11/8.90 & ~ ( member_real @ X2 @ B4 ) ) ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1128_set__diff__eq,axiom,
% 16.11/8.90 ( minus_1052850069191792384nt_int
% 16.11/8.90 = ( ^ [A4: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
% 16.11/8.90 ( collec213857154873943460nt_int
% 16.11/8.90 @ ^ [X2: product_prod_int_int] :
% 16.11/8.90 ( ( member5262025264175285858nt_int @ X2 @ A4 )
% 16.11/8.90 & ~ ( member5262025264175285858nt_int @ X2 @ B4 ) ) ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1129_set__diff__eq,axiom,
% 16.11/8.90 ( minus_811609699411566653omplex
% 16.11/8.90 = ( ^ [A4: set_complex,B4: set_complex] :
% 16.11/8.90 ( collect_complex
% 16.11/8.90 @ ^ [X2: complex] :
% 16.11/8.90 ( ( member_complex @ X2 @ A4 )
% 16.11/8.90 & ~ ( member_complex @ X2 @ B4 ) ) ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1130_set__diff__eq,axiom,
% 16.11/8.90 ( minus_2163939370556025621et_nat
% 16.11/8.90 = ( ^ [A4: set_set_nat,B4: set_set_nat] :
% 16.11/8.90 ( collect_set_nat
% 16.11/8.90 @ ^ [X2: set_nat] :
% 16.11/8.90 ( ( member_set_nat @ X2 @ A4 )
% 16.11/8.90 & ~ ( member_set_nat @ X2 @ B4 ) ) ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1131_set__diff__eq,axiom,
% 16.11/8.90 ( minus_minus_set_int
% 16.11/8.90 = ( ^ [A4: set_int,B4: set_int] :
% 16.11/8.90 ( collect_int
% 16.11/8.90 @ ^ [X2: int] :
% 16.11/8.90 ( ( member_int @ X2 @ A4 )
% 16.11/8.90 & ~ ( member_int @ X2 @ B4 ) ) ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1132_set__diff__eq,axiom,
% 16.11/8.90 ( minus_minus_set_nat
% 16.11/8.90 = ( ^ [A4: set_nat,B4: set_nat] :
% 16.11/8.90 ( collect_nat
% 16.11/8.90 @ ^ [X2: nat] :
% 16.11/8.90 ( ( member_nat @ X2 @ A4 )
% 16.11/8.90 & ~ ( member_nat @ X2 @ B4 ) ) ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1133_minus__set__def,axiom,
% 16.11/8.90 ( minus_minus_set_real
% 16.11/8.90 = ( ^ [A4: set_real,B4: set_real] :
% 16.11/8.90 ( collect_real
% 16.11/8.90 @ ( minus_minus_real_o
% 16.11/8.90 @ ^ [X2: real] : ( member_real @ X2 @ A4 )
% 16.11/8.90 @ ^ [X2: real] : ( member_real @ X2 @ B4 ) ) ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1134_minus__set__def,axiom,
% 16.11/8.90 ( minus_1052850069191792384nt_int
% 16.11/8.90 = ( ^ [A4: set_Pr958786334691620121nt_int,B4: set_Pr958786334691620121nt_int] :
% 16.11/8.90 ( collec213857154873943460nt_int
% 16.11/8.90 @ ( minus_711738161318947805_int_o
% 16.11/8.90 @ ^ [X2: product_prod_int_int] : ( member5262025264175285858nt_int @ X2 @ A4 )
% 16.11/8.90 @ ^ [X2: product_prod_int_int] : ( member5262025264175285858nt_int @ X2 @ B4 ) ) ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1135_minus__set__def,axiom,
% 16.11/8.90 ( minus_811609699411566653omplex
% 16.11/8.90 = ( ^ [A4: set_complex,B4: set_complex] :
% 16.11/8.90 ( collect_complex
% 16.11/8.90 @ ( minus_8727706125548526216plex_o
% 16.11/8.90 @ ^ [X2: complex] : ( member_complex @ X2 @ A4 )
% 16.11/8.90 @ ^ [X2: complex] : ( member_complex @ X2 @ B4 ) ) ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1136_minus__set__def,axiom,
% 16.11/8.90 ( minus_2163939370556025621et_nat
% 16.11/8.90 = ( ^ [A4: set_set_nat,B4: set_set_nat] :
% 16.11/8.90 ( collect_set_nat
% 16.11/8.90 @ ( minus_6910147592129066416_nat_o
% 16.11/8.90 @ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A4 )
% 16.11/8.90 @ ^ [X2: set_nat] : ( member_set_nat @ X2 @ B4 ) ) ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1137_minus__set__def,axiom,
% 16.11/8.90 ( minus_minus_set_int
% 16.11/8.90 = ( ^ [A4: set_int,B4: set_int] :
% 16.11/8.90 ( collect_int
% 16.11/8.90 @ ( minus_minus_int_o
% 16.11/8.90 @ ^ [X2: int] : ( member_int @ X2 @ A4 )
% 16.11/8.90 @ ^ [X2: int] : ( member_int @ X2 @ B4 ) ) ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1138_minus__set__def,axiom,
% 16.11/8.90 ( minus_minus_set_nat
% 16.11/8.90 = ( ^ [A4: set_nat,B4: set_nat] :
% 16.11/8.90 ( collect_nat
% 16.11/8.90 @ ( minus_minus_nat_o
% 16.11/8.90 @ ^ [X2: nat] : ( member_nat @ X2 @ A4 )
% 16.11/8.90 @ ^ [X2: nat] : ( member_nat @ X2 @ B4 ) ) ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1139_not__psubset__empty,axiom,
% 16.11/8.90 ! [A2: set_nat] :
% 16.11/8.90 ~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1140_not__psubset__empty,axiom,
% 16.11/8.90 ! [A2: set_int] :
% 16.11/8.90 ~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1141_not__psubset__empty,axiom,
% 16.11/8.90 ! [A2: set_real] :
% 16.11/8.90 ~ ( ord_less_set_real @ A2 @ bot_bot_set_real ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1142_in__mono,axiom,
% 16.11/8.90 ! [A2: set_real,B3: set_real,X: real] :
% 16.11/8.90 ( ( ord_less_eq_set_real @ A2 @ B3 )
% 16.11/8.90 => ( ( member_real @ X @ A2 )
% 16.11/8.90 => ( member_real @ X @ B3 ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1143_in__mono,axiom,
% 16.11/8.90 ! [A2: set_int,B3: set_int,X: int] :
% 16.11/8.90 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 16.11/8.90 => ( ( member_int @ X @ A2 )
% 16.11/8.90 => ( member_int @ X @ B3 ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1144_in__mono,axiom,
% 16.11/8.90 ! [A2: set_complex,B3: set_complex,X: complex] :
% 16.11/8.90 ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 16.11/8.90 => ( ( member_complex @ X @ A2 )
% 16.11/8.90 => ( member_complex @ X @ B3 ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1145_in__mono,axiom,
% 16.11/8.90 ! [A2: set_set_nat,B3: set_set_nat,X: set_nat] :
% 16.11/8.90 ( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
% 16.11/8.90 => ( ( member_set_nat @ X @ A2 )
% 16.11/8.90 => ( member_set_nat @ X @ B3 ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1146_in__mono,axiom,
% 16.11/8.90 ! [A2: set_nat,B3: set_nat,X: nat] :
% 16.11/8.90 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 16.11/8.90 => ( ( member_nat @ X @ A2 )
% 16.11/8.90 => ( member_nat @ X @ B3 ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1147_subsetD,axiom,
% 16.11/8.90 ! [A2: set_real,B3: set_real,C: real] :
% 16.11/8.90 ( ( ord_less_eq_set_real @ A2 @ B3 )
% 16.11/8.90 => ( ( member_real @ C @ A2 )
% 16.11/8.90 => ( member_real @ C @ B3 ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1148_subsetD,axiom,
% 16.11/8.90 ! [A2: set_int,B3: set_int,C: int] :
% 16.11/8.90 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 16.11/8.90 => ( ( member_int @ C @ A2 )
% 16.11/8.90 => ( member_int @ C @ B3 ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1149_subsetD,axiom,
% 16.11/8.90 ! [A2: set_complex,B3: set_complex,C: complex] :
% 16.11/8.90 ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 16.11/8.90 => ( ( member_complex @ C @ A2 )
% 16.11/8.90 => ( member_complex @ C @ B3 ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1150_subsetD,axiom,
% 16.11/8.90 ! [A2: set_set_nat,B3: set_set_nat,C: set_nat] :
% 16.11/8.90 ( ( ord_le6893508408891458716et_nat @ A2 @ B3 )
% 16.11/8.90 => ( ( member_set_nat @ C @ A2 )
% 16.11/8.90 => ( member_set_nat @ C @ B3 ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1151_subsetD,axiom,
% 16.11/8.90 ! [A2: set_nat,B3: set_nat,C: nat] :
% 16.11/8.90 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 16.11/8.90 => ( ( member_nat @ C @ A2 )
% 16.11/8.90 => ( member_nat @ C @ B3 ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1152_equalityE,axiom,
% 16.11/8.90 ! [A2: set_nat,B3: set_nat] :
% 16.11/8.90 ( ( A2 = B3 )
% 16.11/8.90 => ~ ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 16.11/8.90 => ~ ( ord_less_eq_set_nat @ B3 @ A2 ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1153_subset__eq,axiom,
% 16.11/8.90 ( ord_less_eq_set_real
% 16.11/8.90 = ( ^ [A4: set_real,B4: set_real] :
% 16.11/8.90 ! [X2: real] :
% 16.11/8.90 ( ( member_real @ X2 @ A4 )
% 16.11/8.90 => ( member_real @ X2 @ B4 ) ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1154_subset__eq,axiom,
% 16.11/8.90 ( ord_less_eq_set_int
% 16.11/8.90 = ( ^ [A4: set_int,B4: set_int] :
% 16.11/8.90 ! [X2: int] :
% 16.11/8.90 ( ( member_int @ X2 @ A4 )
% 16.11/8.90 => ( member_int @ X2 @ B4 ) ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1155_subset__eq,axiom,
% 16.11/8.90 ( ord_le211207098394363844omplex
% 16.11/8.90 = ( ^ [A4: set_complex,B4: set_complex] :
% 16.11/8.90 ! [X2: complex] :
% 16.11/8.90 ( ( member_complex @ X2 @ A4 )
% 16.11/8.90 => ( member_complex @ X2 @ B4 ) ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1156_subset__eq,axiom,
% 16.11/8.90 ( ord_le6893508408891458716et_nat
% 16.11/8.90 = ( ^ [A4: set_set_nat,B4: set_set_nat] :
% 16.11/8.90 ! [X2: set_nat] :
% 16.11/8.90 ( ( member_set_nat @ X2 @ A4 )
% 16.11/8.90 => ( member_set_nat @ X2 @ B4 ) ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1157_subset__eq,axiom,
% 16.11/8.90 ( ord_less_eq_set_nat
% 16.11/8.90 = ( ^ [A4: set_nat,B4: set_nat] :
% 16.11/8.90 ! [X2: nat] :
% 16.11/8.90 ( ( member_nat @ X2 @ A4 )
% 16.11/8.90 => ( member_nat @ X2 @ B4 ) ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1158_equalityD1,axiom,
% 16.11/8.90 ! [A2: set_nat,B3: set_nat] :
% 16.11/8.90 ( ( A2 = B3 )
% 16.11/8.90 => ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1159_equalityD2,axiom,
% 16.11/8.90 ! [A2: set_nat,B3: set_nat] :
% 16.11/8.90 ( ( A2 = B3 )
% 16.11/8.90 => ( ord_less_eq_set_nat @ B3 @ A2 ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1160_subset__iff,axiom,
% 16.11/8.90 ( ord_less_eq_set_real
% 16.11/8.90 = ( ^ [A4: set_real,B4: set_real] :
% 16.11/8.90 ! [T2: real] :
% 16.11/8.90 ( ( member_real @ T2 @ A4 )
% 16.11/8.90 => ( member_real @ T2 @ B4 ) ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1161_subset__iff,axiom,
% 16.11/8.90 ( ord_less_eq_set_int
% 16.11/8.90 = ( ^ [A4: set_int,B4: set_int] :
% 16.11/8.90 ! [T2: int] :
% 16.11/8.90 ( ( member_int @ T2 @ A4 )
% 16.11/8.90 => ( member_int @ T2 @ B4 ) ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1162_subset__iff,axiom,
% 16.11/8.90 ( ord_le211207098394363844omplex
% 16.11/8.90 = ( ^ [A4: set_complex,B4: set_complex] :
% 16.11/8.90 ! [T2: complex] :
% 16.11/8.90 ( ( member_complex @ T2 @ A4 )
% 16.11/8.90 => ( member_complex @ T2 @ B4 ) ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1163_subset__iff,axiom,
% 16.11/8.90 ( ord_le6893508408891458716et_nat
% 16.11/8.90 = ( ^ [A4: set_set_nat,B4: set_set_nat] :
% 16.11/8.90 ! [T2: set_nat] :
% 16.11/8.90 ( ( member_set_nat @ T2 @ A4 )
% 16.11/8.90 => ( member_set_nat @ T2 @ B4 ) ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1164_subset__iff,axiom,
% 16.11/8.90 ( ord_less_eq_set_nat
% 16.11/8.90 = ( ^ [A4: set_nat,B4: set_nat] :
% 16.11/8.90 ! [T2: nat] :
% 16.11/8.90 ( ( member_nat @ T2 @ A4 )
% 16.11/8.90 => ( member_nat @ T2 @ B4 ) ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1165_subset__refl,axiom,
% 16.11/8.90 ! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1166_Collect__mono,axiom,
% 16.11/8.90 ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
% 16.11/8.90 ( ! [X4: product_prod_int_int] :
% 16.11/8.90 ( ( P @ X4 )
% 16.11/8.90 => ( Q @ X4 ) )
% 16.11/8.90 => ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1167_Collect__mono,axiom,
% 16.11/8.90 ! [P: complex > $o,Q: complex > $o] :
% 16.11/8.90 ( ! [X4: complex] :
% 16.11/8.90 ( ( P @ X4 )
% 16.11/8.90 => ( Q @ X4 ) )
% 16.11/8.90 => ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).
% 16.11/8.90
% 16.11/8.90 thf(fact_1168_Collect__mono,axiom,
% 16.11/8.90 ! [P: set_nat > $o,Q: set_nat > $o] :
% 16.19/8.90 ( ! [X4: set_nat] :
% 16.19/8.90 ( ( P @ X4 )
% 16.19/8.90 => ( Q @ X4 ) )
% 16.19/8.90 => ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1169_Collect__mono,axiom,
% 16.19/8.90 ! [P: int > $o,Q: int > $o] :
% 16.19/8.90 ( ! [X4: int] :
% 16.19/8.90 ( ( P @ X4 )
% 16.19/8.90 => ( Q @ X4 ) )
% 16.19/8.90 => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1170_Collect__mono,axiom,
% 16.19/8.90 ! [P: nat > $o,Q: nat > $o] :
% 16.19/8.90 ( ! [X4: nat] :
% 16.19/8.90 ( ( P @ X4 )
% 16.19/8.90 => ( Q @ X4 ) )
% 16.19/8.90 => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1171_subset__trans,axiom,
% 16.19/8.90 ! [A2: set_nat,B3: set_nat,C2: set_nat] :
% 16.19/8.90 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 16.19/8.90 => ( ( ord_less_eq_set_nat @ B3 @ C2 )
% 16.19/8.90 => ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1172_set__eq__subset,axiom,
% 16.19/8.90 ( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
% 16.19/8.90 = ( ^ [A4: set_nat,B4: set_nat] :
% 16.19/8.90 ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 16.19/8.90 & ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1173_Collect__mono__iff,axiom,
% 16.19/8.90 ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
% 16.19/8.90 ( ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q ) )
% 16.19/8.90 = ( ! [X2: product_prod_int_int] :
% 16.19/8.90 ( ( P @ X2 )
% 16.19/8.90 => ( Q @ X2 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1174_Collect__mono__iff,axiom,
% 16.19/8.90 ! [P: complex > $o,Q: complex > $o] :
% 16.19/8.90 ( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
% 16.19/8.90 = ( ! [X2: complex] :
% 16.19/8.90 ( ( P @ X2 )
% 16.19/8.90 => ( Q @ X2 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1175_Collect__mono__iff,axiom,
% 16.19/8.90 ! [P: set_nat > $o,Q: set_nat > $o] :
% 16.19/8.90 ( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
% 16.19/8.90 = ( ! [X2: set_nat] :
% 16.19/8.90 ( ( P @ X2 )
% 16.19/8.90 => ( Q @ X2 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1176_Collect__mono__iff,axiom,
% 16.19/8.90 ! [P: int > $o,Q: int > $o] :
% 16.19/8.90 ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
% 16.19/8.90 = ( ! [X2: int] :
% 16.19/8.90 ( ( P @ X2 )
% 16.19/8.90 => ( Q @ X2 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1177_Collect__mono__iff,axiom,
% 16.19/8.90 ! [P: nat > $o,Q: nat > $o] :
% 16.19/8.90 ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
% 16.19/8.90 = ( ! [X2: nat] :
% 16.19/8.90 ( ( P @ X2 )
% 16.19/8.90 => ( Q @ X2 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1178_subset__iff__psubset__eq,axiom,
% 16.19/8.90 ( ord_less_eq_set_nat
% 16.19/8.90 = ( ^ [A4: set_nat,B4: set_nat] :
% 16.19/8.90 ( ( ord_less_set_nat @ A4 @ B4 )
% 16.19/8.90 | ( A4 = B4 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1179_subset__psubset__trans,axiom,
% 16.19/8.90 ! [A2: set_nat,B3: set_nat,C2: set_nat] :
% 16.19/8.90 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 16.19/8.90 => ( ( ord_less_set_nat @ B3 @ C2 )
% 16.19/8.90 => ( ord_less_set_nat @ A2 @ C2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1180_subset__not__subset__eq,axiom,
% 16.19/8.90 ( ord_less_set_nat
% 16.19/8.90 = ( ^ [A4: set_nat,B4: set_nat] :
% 16.19/8.90 ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 16.19/8.90 & ~ ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1181_psubset__subset__trans,axiom,
% 16.19/8.90 ! [A2: set_nat,B3: set_nat,C2: set_nat] :
% 16.19/8.90 ( ( ord_less_set_nat @ A2 @ B3 )
% 16.19/8.90 => ( ( ord_less_eq_set_nat @ B3 @ C2 )
% 16.19/8.90 => ( ord_less_set_nat @ A2 @ C2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1182_psubset__imp__subset,axiom,
% 16.19/8.90 ! [A2: set_nat,B3: set_nat] :
% 16.19/8.90 ( ( ord_less_set_nat @ A2 @ B3 )
% 16.19/8.90 => ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1183_psubset__eq,axiom,
% 16.19/8.90 ( ord_less_set_nat
% 16.19/8.90 = ( ^ [A4: set_nat,B4: set_nat] :
% 16.19/8.90 ( ( ord_less_eq_set_nat @ A4 @ B4 )
% 16.19/8.90 & ( A4 != B4 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1184_psubsetE,axiom,
% 16.19/8.90 ! [A2: set_nat,B3: set_nat] :
% 16.19/8.90 ( ( ord_less_set_nat @ A2 @ B3 )
% 16.19/8.90 => ~ ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 16.19/8.90 => ( ord_less_eq_set_nat @ B3 @ A2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1185_Diff__mono,axiom,
% 16.19/8.90 ! [A2: set_nat,C2: set_nat,D3: set_nat,B3: set_nat] :
% 16.19/8.90 ( ( ord_less_eq_set_nat @ A2 @ C2 )
% 16.19/8.90 => ( ( ord_less_eq_set_nat @ D3 @ B3 )
% 16.19/8.90 => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B3 ) @ ( minus_minus_set_nat @ C2 @ D3 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1186_Diff__subset,axiom,
% 16.19/8.90 ! [A2: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B3 ) @ A2 ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1187_double__diff,axiom,
% 16.19/8.90 ! [A2: set_nat,B3: set_nat,C2: set_nat] :
% 16.19/8.90 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 16.19/8.90 => ( ( ord_less_eq_set_nat @ B3 @ C2 )
% 16.19/8.90 => ( ( minus_minus_set_nat @ B3 @ ( minus_minus_set_nat @ C2 @ A2 ) )
% 16.19/8.90 = A2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1188_mult__commute__abs,axiom,
% 16.19/8.90 ! [C: real] :
% 16.19/8.90 ( ( ^ [X2: real] : ( times_times_real @ X2 @ C ) )
% 16.19/8.90 = ( times_times_real @ C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1189_mult__commute__abs,axiom,
% 16.19/8.90 ! [C: rat] :
% 16.19/8.90 ( ( ^ [X2: rat] : ( times_times_rat @ X2 @ C ) )
% 16.19/8.90 = ( times_times_rat @ C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1190_mult__commute__abs,axiom,
% 16.19/8.90 ! [C: nat] :
% 16.19/8.90 ( ( ^ [X2: nat] : ( times_times_nat @ X2 @ C ) )
% 16.19/8.90 = ( times_times_nat @ C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1191_mult__commute__abs,axiom,
% 16.19/8.90 ! [C: int] :
% 16.19/8.90 ( ( ^ [X2: int] : ( times_times_int @ X2 @ C ) )
% 16.19/8.90 = ( times_times_int @ C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1192_empty__def,axiom,
% 16.19/8.90 ( bot_bo1796632182523588997nt_int
% 16.19/8.90 = ( collec213857154873943460nt_int
% 16.19/8.90 @ ^ [X2: product_prod_int_int] : $false ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1193_empty__def,axiom,
% 16.19/8.90 ( bot_bot_set_complex
% 16.19/8.90 = ( collect_complex
% 16.19/8.90 @ ^ [X2: complex] : $false ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1194_empty__def,axiom,
% 16.19/8.90 ( bot_bot_set_set_nat
% 16.19/8.90 = ( collect_set_nat
% 16.19/8.90 @ ^ [X2: set_nat] : $false ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1195_empty__def,axiom,
% 16.19/8.90 ( bot_bot_set_nat
% 16.19/8.90 = ( collect_nat
% 16.19/8.90 @ ^ [X2: nat] : $false ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1196_empty__def,axiom,
% 16.19/8.90 ( bot_bot_set_int
% 16.19/8.90 = ( collect_int
% 16.19/8.90 @ ^ [X2: int] : $false ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1197_empty__def,axiom,
% 16.19/8.90 ( bot_bot_set_real
% 16.19/8.90 = ( collect_real
% 16.19/8.90 @ ^ [X2: real] : $false ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1198_less__eq__set__def,axiom,
% 16.19/8.90 ( ord_less_eq_set_real
% 16.19/8.90 = ( ^ [A4: set_real,B4: set_real] :
% 16.19/8.90 ( ord_less_eq_real_o
% 16.19/8.90 @ ^ [X2: real] : ( member_real @ X2 @ A4 )
% 16.19/8.90 @ ^ [X2: real] : ( member_real @ X2 @ B4 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1199_less__eq__set__def,axiom,
% 16.19/8.90 ( ord_less_eq_set_int
% 16.19/8.90 = ( ^ [A4: set_int,B4: set_int] :
% 16.19/8.90 ( ord_less_eq_int_o
% 16.19/8.90 @ ^ [X2: int] : ( member_int @ X2 @ A4 )
% 16.19/8.90 @ ^ [X2: int] : ( member_int @ X2 @ B4 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1200_less__eq__set__def,axiom,
% 16.19/8.90 ( ord_le211207098394363844omplex
% 16.19/8.90 = ( ^ [A4: set_complex,B4: set_complex] :
% 16.19/8.90 ( ord_le4573692005234683329plex_o
% 16.19/8.90 @ ^ [X2: complex] : ( member_complex @ X2 @ A4 )
% 16.19/8.90 @ ^ [X2: complex] : ( member_complex @ X2 @ B4 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1201_less__eq__set__def,axiom,
% 16.19/8.90 ( ord_le6893508408891458716et_nat
% 16.19/8.90 = ( ^ [A4: set_set_nat,B4: set_set_nat] :
% 16.19/8.90 ( ord_le3964352015994296041_nat_o
% 16.19/8.90 @ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A4 )
% 16.19/8.90 @ ^ [X2: set_nat] : ( member_set_nat @ X2 @ B4 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1202_less__eq__set__def,axiom,
% 16.19/8.90 ( ord_less_eq_set_nat
% 16.19/8.90 = ( ^ [A4: set_nat,B4: set_nat] :
% 16.19/8.90 ( ord_less_eq_nat_o
% 16.19/8.90 @ ^ [X2: nat] : ( member_nat @ X2 @ A4 )
% 16.19/8.90 @ ^ [X2: nat] : ( member_nat @ X2 @ B4 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1203_Collect__subset,axiom,
% 16.19/8.90 ! [A2: set_real,P: real > $o] :
% 16.19/8.90 ( ord_less_eq_set_real
% 16.19/8.90 @ ( collect_real
% 16.19/8.90 @ ^ [X2: real] :
% 16.19/8.90 ( ( member_real @ X2 @ A2 )
% 16.19/8.90 & ( P @ X2 ) ) )
% 16.19/8.90 @ A2 ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1204_Collect__subset,axiom,
% 16.19/8.90 ! [A2: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
% 16.19/8.90 ( ord_le2843351958646193337nt_int
% 16.19/8.90 @ ( collec213857154873943460nt_int
% 16.19/8.90 @ ^ [X2: product_prod_int_int] :
% 16.19/8.90 ( ( member5262025264175285858nt_int @ X2 @ A2 )
% 16.19/8.90 & ( P @ X2 ) ) )
% 16.19/8.90 @ A2 ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1205_Collect__subset,axiom,
% 16.19/8.90 ! [A2: set_complex,P: complex > $o] :
% 16.19/8.90 ( ord_le211207098394363844omplex
% 16.19/8.90 @ ( collect_complex
% 16.19/8.90 @ ^ [X2: complex] :
% 16.19/8.90 ( ( member_complex @ X2 @ A2 )
% 16.19/8.90 & ( P @ X2 ) ) )
% 16.19/8.90 @ A2 ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1206_Collect__subset,axiom,
% 16.19/8.90 ! [A2: set_set_nat,P: set_nat > $o] :
% 16.19/8.90 ( ord_le6893508408891458716et_nat
% 16.19/8.90 @ ( collect_set_nat
% 16.19/8.90 @ ^ [X2: set_nat] :
% 16.19/8.90 ( ( member_set_nat @ X2 @ A2 )
% 16.19/8.90 & ( P @ X2 ) ) )
% 16.19/8.90 @ A2 ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1207_Collect__subset,axiom,
% 16.19/8.90 ! [A2: set_int,P: int > $o] :
% 16.19/8.90 ( ord_less_eq_set_int
% 16.19/8.90 @ ( collect_int
% 16.19/8.90 @ ^ [X2: int] :
% 16.19/8.90 ( ( member_int @ X2 @ A2 )
% 16.19/8.90 & ( P @ X2 ) ) )
% 16.19/8.90 @ A2 ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1208_Collect__subset,axiom,
% 16.19/8.90 ! [A2: set_nat,P: nat > $o] :
% 16.19/8.90 ( ord_less_eq_set_nat
% 16.19/8.90 @ ( collect_nat
% 16.19/8.90 @ ^ [X2: nat] :
% 16.19/8.90 ( ( member_nat @ X2 @ A2 )
% 16.19/8.90 & ( P @ X2 ) ) )
% 16.19/8.90 @ A2 ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1209_max_Omono,axiom,
% 16.19/8.90 ! [C: rat,A: rat,D: rat,B: rat] :
% 16.19/8.90 ( ( ord_less_eq_rat @ C @ A )
% 16.19/8.90 => ( ( ord_less_eq_rat @ D @ B )
% 16.19/8.90 => ( ord_less_eq_rat @ ( ord_max_rat @ C @ D ) @ ( ord_max_rat @ A @ B ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1210_max_Omono,axiom,
% 16.19/8.90 ! [C: num,A: num,D: num,B: num] :
% 16.19/8.90 ( ( ord_less_eq_num @ C @ A )
% 16.19/8.90 => ( ( ord_less_eq_num @ D @ B )
% 16.19/8.90 => ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A @ B ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1211_max_Omono,axiom,
% 16.19/8.90 ! [C: nat,A: nat,D: nat,B: nat] :
% 16.19/8.90 ( ( ord_less_eq_nat @ C @ A )
% 16.19/8.90 => ( ( ord_less_eq_nat @ D @ B )
% 16.19/8.90 => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1212_max_Omono,axiom,
% 16.19/8.90 ! [C: int,A: int,D: int,B: int] :
% 16.19/8.90 ( ( ord_less_eq_int @ C @ A )
% 16.19/8.90 => ( ( ord_less_eq_int @ D @ B )
% 16.19/8.90 => ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A @ B ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1213_max_OorderE,axiom,
% 16.19/8.90 ! [B: rat,A: rat] :
% 16.19/8.90 ( ( ord_less_eq_rat @ B @ A )
% 16.19/8.90 => ( A
% 16.19/8.90 = ( ord_max_rat @ A @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1214_max_OorderE,axiom,
% 16.19/8.90 ! [B: num,A: num] :
% 16.19/8.90 ( ( ord_less_eq_num @ B @ A )
% 16.19/8.90 => ( A
% 16.19/8.90 = ( ord_max_num @ A @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1215_max_OorderE,axiom,
% 16.19/8.90 ! [B: nat,A: nat] :
% 16.19/8.90 ( ( ord_less_eq_nat @ B @ A )
% 16.19/8.90 => ( A
% 16.19/8.90 = ( ord_max_nat @ A @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1216_max_OorderE,axiom,
% 16.19/8.90 ! [B: int,A: int] :
% 16.19/8.90 ( ( ord_less_eq_int @ B @ A )
% 16.19/8.90 => ( A
% 16.19/8.90 = ( ord_max_int @ A @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1217_max_OorderI,axiom,
% 16.19/8.90 ! [A: rat,B: rat] :
% 16.19/8.90 ( ( A
% 16.19/8.90 = ( ord_max_rat @ A @ B ) )
% 16.19/8.90 => ( ord_less_eq_rat @ B @ A ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1218_max_OorderI,axiom,
% 16.19/8.90 ! [A: num,B: num] :
% 16.19/8.90 ( ( A
% 16.19/8.90 = ( ord_max_num @ A @ B ) )
% 16.19/8.90 => ( ord_less_eq_num @ B @ A ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1219_max_OorderI,axiom,
% 16.19/8.90 ! [A: nat,B: nat] :
% 16.19/8.90 ( ( A
% 16.19/8.90 = ( ord_max_nat @ A @ B ) )
% 16.19/8.90 => ( ord_less_eq_nat @ B @ A ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1220_max_OorderI,axiom,
% 16.19/8.90 ! [A: int,B: int] :
% 16.19/8.90 ( ( A
% 16.19/8.90 = ( ord_max_int @ A @ B ) )
% 16.19/8.90 => ( ord_less_eq_int @ B @ A ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1221_max_OboundedE,axiom,
% 16.19/8.90 ! [B: rat,C: rat,A: rat] :
% 16.19/8.90 ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
% 16.19/8.90 => ~ ( ( ord_less_eq_rat @ B @ A )
% 16.19/8.90 => ~ ( ord_less_eq_rat @ C @ A ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1222_max_OboundedE,axiom,
% 16.19/8.90 ! [B: num,C: num,A: num] :
% 16.19/8.90 ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
% 16.19/8.90 => ~ ( ( ord_less_eq_num @ B @ A )
% 16.19/8.90 => ~ ( ord_less_eq_num @ C @ A ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1223_max_OboundedE,axiom,
% 16.19/8.90 ! [B: nat,C: nat,A: nat] :
% 16.19/8.90 ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
% 16.19/8.90 => ~ ( ( ord_less_eq_nat @ B @ A )
% 16.19/8.90 => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1224_max_OboundedE,axiom,
% 16.19/8.90 ! [B: int,C: int,A: int] :
% 16.19/8.90 ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
% 16.19/8.90 => ~ ( ( ord_less_eq_int @ B @ A )
% 16.19/8.90 => ~ ( ord_less_eq_int @ C @ A ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1225_max_OboundedI,axiom,
% 16.19/8.90 ! [B: rat,A: rat,C: rat] :
% 16.19/8.90 ( ( ord_less_eq_rat @ B @ A )
% 16.19/8.90 => ( ( ord_less_eq_rat @ C @ A )
% 16.19/8.90 => ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1226_max_OboundedI,axiom,
% 16.19/8.90 ! [B: num,A: num,C: num] :
% 16.19/8.90 ( ( ord_less_eq_num @ B @ A )
% 16.19/8.90 => ( ( ord_less_eq_num @ C @ A )
% 16.19/8.90 => ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1227_max_OboundedI,axiom,
% 16.19/8.90 ! [B: nat,A: nat,C: nat] :
% 16.19/8.90 ( ( ord_less_eq_nat @ B @ A )
% 16.19/8.90 => ( ( ord_less_eq_nat @ C @ A )
% 16.19/8.90 => ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1228_max_OboundedI,axiom,
% 16.19/8.90 ! [B: int,A: int,C: int] :
% 16.19/8.90 ( ( ord_less_eq_int @ B @ A )
% 16.19/8.90 => ( ( ord_less_eq_int @ C @ A )
% 16.19/8.90 => ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1229_max_Oorder__iff,axiom,
% 16.19/8.90 ( ord_less_eq_rat
% 16.19/8.90 = ( ^ [B5: rat,A5: rat] :
% 16.19/8.90 ( A5
% 16.19/8.90 = ( ord_max_rat @ A5 @ B5 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1230_max_Oorder__iff,axiom,
% 16.19/8.90 ( ord_less_eq_num
% 16.19/8.90 = ( ^ [B5: num,A5: num] :
% 16.19/8.90 ( A5
% 16.19/8.90 = ( ord_max_num @ A5 @ B5 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1231_max_Oorder__iff,axiom,
% 16.19/8.90 ( ord_less_eq_nat
% 16.19/8.90 = ( ^ [B5: nat,A5: nat] :
% 16.19/8.90 ( A5
% 16.19/8.90 = ( ord_max_nat @ A5 @ B5 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1232_max_Oorder__iff,axiom,
% 16.19/8.90 ( ord_less_eq_int
% 16.19/8.90 = ( ^ [B5: int,A5: int] :
% 16.19/8.90 ( A5
% 16.19/8.90 = ( ord_max_int @ A5 @ B5 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1233_max_Ocobounded1,axiom,
% 16.19/8.90 ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1234_max_Ocobounded1,axiom,
% 16.19/8.90 ! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1235_max_Ocobounded1,axiom,
% 16.19/8.90 ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1236_max_Ocobounded1,axiom,
% 16.19/8.90 ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1237_max_Ocobounded2,axiom,
% 16.19/8.90 ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1238_max_Ocobounded2,axiom,
% 16.19/8.90 ! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1239_max_Ocobounded2,axiom,
% 16.19/8.90 ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1240_max_Ocobounded2,axiom,
% 16.19/8.90 ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1241_le__max__iff__disj,axiom,
% 16.19/8.90 ! [Z: rat,X: rat,Y: rat] :
% 16.19/8.90 ( ( ord_less_eq_rat @ Z @ ( ord_max_rat @ X @ Y ) )
% 16.19/8.90 = ( ( ord_less_eq_rat @ Z @ X )
% 16.19/8.90 | ( ord_less_eq_rat @ Z @ Y ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1242_le__max__iff__disj,axiom,
% 16.19/8.90 ! [Z: num,X: num,Y: num] :
% 16.19/8.90 ( ( ord_less_eq_num @ Z @ ( ord_max_num @ X @ Y ) )
% 16.19/8.90 = ( ( ord_less_eq_num @ Z @ X )
% 16.19/8.90 | ( ord_less_eq_num @ Z @ Y ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1243_le__max__iff__disj,axiom,
% 16.19/8.90 ! [Z: nat,X: nat,Y: nat] :
% 16.19/8.90 ( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X @ Y ) )
% 16.19/8.90 = ( ( ord_less_eq_nat @ Z @ X )
% 16.19/8.90 | ( ord_less_eq_nat @ Z @ Y ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1244_le__max__iff__disj,axiom,
% 16.19/8.90 ! [Z: int,X: int,Y: int] :
% 16.19/8.90 ( ( ord_less_eq_int @ Z @ ( ord_max_int @ X @ Y ) )
% 16.19/8.90 = ( ( ord_less_eq_int @ Z @ X )
% 16.19/8.90 | ( ord_less_eq_int @ Z @ Y ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1245_max_Oabsorb__iff1,axiom,
% 16.19/8.90 ( ord_less_eq_rat
% 16.19/8.90 = ( ^ [B5: rat,A5: rat] :
% 16.19/8.90 ( ( ord_max_rat @ A5 @ B5 )
% 16.19/8.90 = A5 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1246_max_Oabsorb__iff1,axiom,
% 16.19/8.90 ( ord_less_eq_num
% 16.19/8.90 = ( ^ [B5: num,A5: num] :
% 16.19/8.90 ( ( ord_max_num @ A5 @ B5 )
% 16.19/8.90 = A5 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1247_max_Oabsorb__iff1,axiom,
% 16.19/8.90 ( ord_less_eq_nat
% 16.19/8.90 = ( ^ [B5: nat,A5: nat] :
% 16.19/8.90 ( ( ord_max_nat @ A5 @ B5 )
% 16.19/8.90 = A5 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1248_max_Oabsorb__iff1,axiom,
% 16.19/8.90 ( ord_less_eq_int
% 16.19/8.90 = ( ^ [B5: int,A5: int] :
% 16.19/8.90 ( ( ord_max_int @ A5 @ B5 )
% 16.19/8.90 = A5 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1249_max_Oabsorb__iff2,axiom,
% 16.19/8.90 ( ord_less_eq_rat
% 16.19/8.90 = ( ^ [A5: rat,B5: rat] :
% 16.19/8.90 ( ( ord_max_rat @ A5 @ B5 )
% 16.19/8.90 = B5 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1250_max_Oabsorb__iff2,axiom,
% 16.19/8.90 ( ord_less_eq_num
% 16.19/8.90 = ( ^ [A5: num,B5: num] :
% 16.19/8.90 ( ( ord_max_num @ A5 @ B5 )
% 16.19/8.90 = B5 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1251_max_Oabsorb__iff2,axiom,
% 16.19/8.90 ( ord_less_eq_nat
% 16.19/8.90 = ( ^ [A5: nat,B5: nat] :
% 16.19/8.90 ( ( ord_max_nat @ A5 @ B5 )
% 16.19/8.90 = B5 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1252_max_Oabsorb__iff2,axiom,
% 16.19/8.90 ( ord_less_eq_int
% 16.19/8.90 = ( ^ [A5: int,B5: int] :
% 16.19/8.90 ( ( ord_max_int @ A5 @ B5 )
% 16.19/8.90 = B5 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1253_max_OcoboundedI1,axiom,
% 16.19/8.90 ! [C: rat,A: rat,B: rat] :
% 16.19/8.90 ( ( ord_less_eq_rat @ C @ A )
% 16.19/8.90 => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1254_max_OcoboundedI1,axiom,
% 16.19/8.90 ! [C: num,A: num,B: num] :
% 16.19/8.90 ( ( ord_less_eq_num @ C @ A )
% 16.19/8.90 => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1255_max_OcoboundedI1,axiom,
% 16.19/8.90 ! [C: nat,A: nat,B: nat] :
% 16.19/8.90 ( ( ord_less_eq_nat @ C @ A )
% 16.19/8.90 => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1256_max_OcoboundedI1,axiom,
% 16.19/8.90 ! [C: int,A: int,B: int] :
% 16.19/8.90 ( ( ord_less_eq_int @ C @ A )
% 16.19/8.90 => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1257_max_OcoboundedI2,axiom,
% 16.19/8.90 ! [C: rat,B: rat,A: rat] :
% 16.19/8.90 ( ( ord_less_eq_rat @ C @ B )
% 16.19/8.90 => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1258_max_OcoboundedI2,axiom,
% 16.19/8.90 ! [C: num,B: num,A: num] :
% 16.19/8.90 ( ( ord_less_eq_num @ C @ B )
% 16.19/8.90 => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1259_max_OcoboundedI2,axiom,
% 16.19/8.90 ! [C: nat,B: nat,A: nat] :
% 16.19/8.90 ( ( ord_less_eq_nat @ C @ B )
% 16.19/8.90 => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1260_max_OcoboundedI2,axiom,
% 16.19/8.90 ! [C: int,B: int,A: int] :
% 16.19/8.90 ( ( ord_less_eq_int @ C @ B )
% 16.19/8.90 => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1261_less__max__iff__disj,axiom,
% 16.19/8.90 ! [Z: real,X: real,Y: real] :
% 16.19/8.90 ( ( ord_less_real @ Z @ ( ord_max_real @ X @ Y ) )
% 16.19/8.90 = ( ( ord_less_real @ Z @ X )
% 16.19/8.90 | ( ord_less_real @ Z @ Y ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1262_less__max__iff__disj,axiom,
% 16.19/8.90 ! [Z: rat,X: rat,Y: rat] :
% 16.19/8.90 ( ( ord_less_rat @ Z @ ( ord_max_rat @ X @ Y ) )
% 16.19/8.90 = ( ( ord_less_rat @ Z @ X )
% 16.19/8.90 | ( ord_less_rat @ Z @ Y ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1263_less__max__iff__disj,axiom,
% 16.19/8.90 ! [Z: num,X: num,Y: num] :
% 16.19/8.90 ( ( ord_less_num @ Z @ ( ord_max_num @ X @ Y ) )
% 16.19/8.90 = ( ( ord_less_num @ Z @ X )
% 16.19/8.90 | ( ord_less_num @ Z @ Y ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1264_less__max__iff__disj,axiom,
% 16.19/8.90 ! [Z: nat,X: nat,Y: nat] :
% 16.19/8.90 ( ( ord_less_nat @ Z @ ( ord_max_nat @ X @ Y ) )
% 16.19/8.90 = ( ( ord_less_nat @ Z @ X )
% 16.19/8.90 | ( ord_less_nat @ Z @ Y ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1265_less__max__iff__disj,axiom,
% 16.19/8.90 ! [Z: int,X: int,Y: int] :
% 16.19/8.90 ( ( ord_less_int @ Z @ ( ord_max_int @ X @ Y ) )
% 16.19/8.90 = ( ( ord_less_int @ Z @ X )
% 16.19/8.90 | ( ord_less_int @ Z @ Y ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1266_max_Ostrict__boundedE,axiom,
% 16.19/8.90 ! [B: real,C: real,A: real] :
% 16.19/8.90 ( ( ord_less_real @ ( ord_max_real @ B @ C ) @ A )
% 16.19/8.90 => ~ ( ( ord_less_real @ B @ A )
% 16.19/8.90 => ~ ( ord_less_real @ C @ A ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1267_max_Ostrict__boundedE,axiom,
% 16.19/8.90 ! [B: rat,C: rat,A: rat] :
% 16.19/8.90 ( ( ord_less_rat @ ( ord_max_rat @ B @ C ) @ A )
% 16.19/8.90 => ~ ( ( ord_less_rat @ B @ A )
% 16.19/8.90 => ~ ( ord_less_rat @ C @ A ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1268_max_Ostrict__boundedE,axiom,
% 16.19/8.90 ! [B: num,C: num,A: num] :
% 16.19/8.90 ( ( ord_less_num @ ( ord_max_num @ B @ C ) @ A )
% 16.19/8.90 => ~ ( ( ord_less_num @ B @ A )
% 16.19/8.90 => ~ ( ord_less_num @ C @ A ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1269_max_Ostrict__boundedE,axiom,
% 16.19/8.90 ! [B: nat,C: nat,A: nat] :
% 16.19/8.90 ( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
% 16.19/8.90 => ~ ( ( ord_less_nat @ B @ A )
% 16.19/8.90 => ~ ( ord_less_nat @ C @ A ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1270_max_Ostrict__boundedE,axiom,
% 16.19/8.90 ! [B: int,C: int,A: int] :
% 16.19/8.90 ( ( ord_less_int @ ( ord_max_int @ B @ C ) @ A )
% 16.19/8.90 => ~ ( ( ord_less_int @ B @ A )
% 16.19/8.90 => ~ ( ord_less_int @ C @ A ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1271_max_Ostrict__order__iff,axiom,
% 16.19/8.90 ( ord_less_real
% 16.19/8.90 = ( ^ [B5: real,A5: real] :
% 16.19/8.90 ( ( A5
% 16.19/8.90 = ( ord_max_real @ A5 @ B5 ) )
% 16.19/8.90 & ( A5 != B5 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1272_max_Ostrict__order__iff,axiom,
% 16.19/8.90 ( ord_less_rat
% 16.19/8.90 = ( ^ [B5: rat,A5: rat] :
% 16.19/8.90 ( ( A5
% 16.19/8.90 = ( ord_max_rat @ A5 @ B5 ) )
% 16.19/8.90 & ( A5 != B5 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1273_max_Ostrict__order__iff,axiom,
% 16.19/8.90 ( ord_less_num
% 16.19/8.90 = ( ^ [B5: num,A5: num] :
% 16.19/8.90 ( ( A5
% 16.19/8.90 = ( ord_max_num @ A5 @ B5 ) )
% 16.19/8.90 & ( A5 != B5 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1274_max_Ostrict__order__iff,axiom,
% 16.19/8.90 ( ord_less_nat
% 16.19/8.90 = ( ^ [B5: nat,A5: nat] :
% 16.19/8.90 ( ( A5
% 16.19/8.90 = ( ord_max_nat @ A5 @ B5 ) )
% 16.19/8.90 & ( A5 != B5 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1275_max_Ostrict__order__iff,axiom,
% 16.19/8.90 ( ord_less_int
% 16.19/8.90 = ( ^ [B5: int,A5: int] :
% 16.19/8.90 ( ( A5
% 16.19/8.90 = ( ord_max_int @ A5 @ B5 ) )
% 16.19/8.90 & ( A5 != B5 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1276_max_Ostrict__coboundedI1,axiom,
% 16.19/8.90 ! [C: real,A: real,B: real] :
% 16.19/8.90 ( ( ord_less_real @ C @ A )
% 16.19/8.90 => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1277_max_Ostrict__coboundedI1,axiom,
% 16.19/8.90 ! [C: rat,A: rat,B: rat] :
% 16.19/8.90 ( ( ord_less_rat @ C @ A )
% 16.19/8.90 => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1278_max_Ostrict__coboundedI1,axiom,
% 16.19/8.90 ! [C: num,A: num,B: num] :
% 16.19/8.90 ( ( ord_less_num @ C @ A )
% 16.19/8.90 => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1279_max_Ostrict__coboundedI1,axiom,
% 16.19/8.90 ! [C: nat,A: nat,B: nat] :
% 16.19/8.90 ( ( ord_less_nat @ C @ A )
% 16.19/8.90 => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1280_max_Ostrict__coboundedI1,axiom,
% 16.19/8.90 ! [C: int,A: int,B: int] :
% 16.19/8.90 ( ( ord_less_int @ C @ A )
% 16.19/8.90 => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1281_max_Ostrict__coboundedI2,axiom,
% 16.19/8.90 ! [C: real,B: real,A: real] :
% 16.19/8.90 ( ( ord_less_real @ C @ B )
% 16.19/8.90 => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1282_max_Ostrict__coboundedI2,axiom,
% 16.19/8.90 ! [C: rat,B: rat,A: rat] :
% 16.19/8.90 ( ( ord_less_rat @ C @ B )
% 16.19/8.90 => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1283_max_Ostrict__coboundedI2,axiom,
% 16.19/8.90 ! [C: num,B: num,A: num] :
% 16.19/8.90 ( ( ord_less_num @ C @ B )
% 16.19/8.90 => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1284_max_Ostrict__coboundedI2,axiom,
% 16.19/8.90 ! [C: nat,B: nat,A: nat] :
% 16.19/8.90 ( ( ord_less_nat @ C @ B )
% 16.19/8.90 => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1285_max_Ostrict__coboundedI2,axiom,
% 16.19/8.90 ! [C: int,B: int,A: int] :
% 16.19/8.90 ( ( ord_less_int @ C @ B )
% 16.19/8.90 => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1286_double__not__eq__Suc__double,axiom,
% 16.19/8.90 ! [M: nat,N2: nat] :
% 16.19/8.90 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 16.19/8.90 != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1287_Suc__double__not__eq__double,axiom,
% 16.19/8.90 ! [M: nat,N2: nat] :
% 16.19/8.90 ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 16.19/8.90 != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1288_add__diff__cancel__right_H,axiom,
% 16.19/8.90 ! [A: real,B: real] :
% 16.19/8.90 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1289_add__diff__cancel__right_H,axiom,
% 16.19/8.90 ! [A: rat,B: rat] :
% 16.19/8.90 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1290_add__diff__cancel__right_H,axiom,
% 16.19/8.90 ! [A: nat,B: nat] :
% 16.19/8.90 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1291_add__diff__cancel__right_H,axiom,
% 16.19/8.90 ! [A: int,B: int] :
% 16.19/8.90 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1292_add__diff__cancel__right,axiom,
% 16.19/8.90 ! [A: real,C: real,B: real] :
% 16.19/8.90 ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 16.19/8.90 = ( minus_minus_real @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1293_add__diff__cancel__right,axiom,
% 16.19/8.90 ! [A: rat,C: rat,B: rat] :
% 16.19/8.90 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 16.19/8.90 = ( minus_minus_rat @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1294_add__diff__cancel__right,axiom,
% 16.19/8.90 ! [A: nat,C: nat,B: nat] :
% 16.19/8.90 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 16.19/8.90 = ( minus_minus_nat @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1295_add__diff__cancel__right,axiom,
% 16.19/8.90 ! [A: int,C: int,B: int] :
% 16.19/8.90 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 16.19/8.90 = ( minus_minus_int @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1296_add__diff__cancel__left_H,axiom,
% 16.19/8.90 ! [A: real,B: real] :
% 16.19/8.90 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
% 16.19/8.90 = B ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1297_add__diff__cancel__left_H,axiom,
% 16.19/8.90 ! [A: rat,B: rat] :
% 16.19/8.90 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
% 16.19/8.90 = B ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1298_add__diff__cancel__left_H,axiom,
% 16.19/8.90 ! [A: nat,B: nat] :
% 16.19/8.90 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 16.19/8.90 = B ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1299_add__diff__cancel__left_H,axiom,
% 16.19/8.90 ! [A: int,B: int] :
% 16.19/8.90 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
% 16.19/8.90 = B ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1300_add__diff__cancel__left,axiom,
% 16.19/8.90 ! [C: real,A: real,B: real] :
% 16.19/8.90 ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 16.19/8.90 = ( minus_minus_real @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1301_add__diff__cancel__left,axiom,
% 16.19/8.90 ! [C: rat,A: rat,B: rat] :
% 16.19/8.90 ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 16.19/8.90 = ( minus_minus_rat @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1302_add__diff__cancel__left,axiom,
% 16.19/8.90 ! [C: nat,A: nat,B: nat] :
% 16.19/8.90 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 16.19/8.90 = ( minus_minus_nat @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1303_add__diff__cancel__left,axiom,
% 16.19/8.90 ! [C: int,A: int,B: int] :
% 16.19/8.90 ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 16.19/8.90 = ( minus_minus_int @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1304_diff__add__cancel,axiom,
% 16.19/8.90 ! [A: real,B: real] :
% 16.19/8.90 ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1305_diff__add__cancel,axiom,
% 16.19/8.90 ! [A: rat,B: rat] :
% 16.19/8.90 ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1306_diff__add__cancel,axiom,
% 16.19/8.90 ! [A: int,B: int] :
% 16.19/8.90 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1307_add__diff__cancel,axiom,
% 16.19/8.90 ! [A: real,B: real] :
% 16.19/8.90 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1308_add__diff__cancel,axiom,
% 16.19/8.90 ! [A: rat,B: rat] :
% 16.19/8.90 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1309_add__diff__cancel,axiom,
% 16.19/8.90 ! [A: int,B: int] :
% 16.19/8.90 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1310_times__divide__eq__right,axiom,
% 16.19/8.90 ! [A: complex,B: complex,C: complex] :
% 16.19/8.90 ( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
% 16.19/8.90 = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1311_times__divide__eq__right,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real] :
% 16.19/8.90 ( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
% 16.19/8.90 = ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1312_times__divide__eq__right,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat] :
% 16.19/8.90 ( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 16.19/8.90 = ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1313_divide__divide__eq__right,axiom,
% 16.19/8.90 ! [A: complex,B: complex,C: complex] :
% 16.19/8.90 ( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
% 16.19/8.90 = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1314_divide__divide__eq__right,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real] :
% 16.19/8.90 ( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
% 16.19/8.90 = ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1315_divide__divide__eq__right,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat] :
% 16.19/8.90 ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 16.19/8.90 = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1316_add__left__cancel,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real] :
% 16.19/8.90 ( ( ( plus_plus_real @ A @ B )
% 16.19/8.90 = ( plus_plus_real @ A @ C ) )
% 16.19/8.90 = ( B = C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1317_add__left__cancel,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat] :
% 16.19/8.90 ( ( ( plus_plus_rat @ A @ B )
% 16.19/8.90 = ( plus_plus_rat @ A @ C ) )
% 16.19/8.90 = ( B = C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1318_add__left__cancel,axiom,
% 16.19/8.90 ! [A: nat,B: nat,C: nat] :
% 16.19/8.90 ( ( ( plus_plus_nat @ A @ B )
% 16.19/8.90 = ( plus_plus_nat @ A @ C ) )
% 16.19/8.90 = ( B = C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1319_add__left__cancel,axiom,
% 16.19/8.90 ! [A: int,B: int,C: int] :
% 16.19/8.90 ( ( ( plus_plus_int @ A @ B )
% 16.19/8.90 = ( plus_plus_int @ A @ C ) )
% 16.19/8.90 = ( B = C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1320_add__right__cancel,axiom,
% 16.19/8.90 ! [B: real,A: real,C: real] :
% 16.19/8.90 ( ( ( plus_plus_real @ B @ A )
% 16.19/8.90 = ( plus_plus_real @ C @ A ) )
% 16.19/8.90 = ( B = C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1321_add__right__cancel,axiom,
% 16.19/8.90 ! [B: rat,A: rat,C: rat] :
% 16.19/8.90 ( ( ( plus_plus_rat @ B @ A )
% 16.19/8.90 = ( plus_plus_rat @ C @ A ) )
% 16.19/8.90 = ( B = C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1322_add__right__cancel,axiom,
% 16.19/8.90 ! [B: nat,A: nat,C: nat] :
% 16.19/8.90 ( ( ( plus_plus_nat @ B @ A )
% 16.19/8.90 = ( plus_plus_nat @ C @ A ) )
% 16.19/8.90 = ( B = C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1323_add__right__cancel,axiom,
% 16.19/8.90 ! [B: int,A: int,C: int] :
% 16.19/8.90 ( ( ( plus_plus_int @ B @ A )
% 16.19/8.90 = ( plus_plus_int @ C @ A ) )
% 16.19/8.90 = ( B = C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1324_add__le__cancel__left,axiom,
% 16.19/8.90 ! [C: real,A: real,B: real] :
% 16.19/8.90 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 16.19/8.90 = ( ord_less_eq_real @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1325_add__le__cancel__left,axiom,
% 16.19/8.90 ! [C: rat,A: rat,B: rat] :
% 16.19/8.90 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 16.19/8.90 = ( ord_less_eq_rat @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1326_add__le__cancel__left,axiom,
% 16.19/8.90 ! [C: nat,A: nat,B: nat] :
% 16.19/8.90 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 16.19/8.90 = ( ord_less_eq_nat @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1327_add__le__cancel__left,axiom,
% 16.19/8.90 ! [C: int,A: int,B: int] :
% 16.19/8.90 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 16.19/8.90 = ( ord_less_eq_int @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1328_add__le__cancel__right,axiom,
% 16.19/8.90 ! [A: real,C: real,B: real] :
% 16.19/8.90 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 16.19/8.90 = ( ord_less_eq_real @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1329_add__le__cancel__right,axiom,
% 16.19/8.90 ! [A: rat,C: rat,B: rat] :
% 16.19/8.90 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 16.19/8.90 = ( ord_less_eq_rat @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1330_add__le__cancel__right,axiom,
% 16.19/8.90 ! [A: nat,C: nat,B: nat] :
% 16.19/8.90 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 16.19/8.90 = ( ord_less_eq_nat @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1331_add__le__cancel__right,axiom,
% 16.19/8.90 ! [A: int,C: int,B: int] :
% 16.19/8.90 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 16.19/8.90 = ( ord_less_eq_int @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1332_add__less__cancel__left,axiom,
% 16.19/8.90 ! [C: real,A: real,B: real] :
% 16.19/8.90 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 16.19/8.90 = ( ord_less_real @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1333_add__less__cancel__left,axiom,
% 16.19/8.90 ! [C: rat,A: rat,B: rat] :
% 16.19/8.90 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 16.19/8.90 = ( ord_less_rat @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1334_add__less__cancel__left,axiom,
% 16.19/8.90 ! [C: nat,A: nat,B: nat] :
% 16.19/8.90 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 16.19/8.90 = ( ord_less_nat @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1335_add__less__cancel__left,axiom,
% 16.19/8.90 ! [C: int,A: int,B: int] :
% 16.19/8.90 ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 16.19/8.90 = ( ord_less_int @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1336_add__less__cancel__right,axiom,
% 16.19/8.90 ! [A: real,C: real,B: real] :
% 16.19/8.90 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 16.19/8.90 = ( ord_less_real @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1337_add__less__cancel__right,axiom,
% 16.19/8.90 ! [A: rat,C: rat,B: rat] :
% 16.19/8.90 ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 16.19/8.90 = ( ord_less_rat @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1338_add__less__cancel__right,axiom,
% 16.19/8.90 ! [A: nat,C: nat,B: nat] :
% 16.19/8.90 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 16.19/8.90 = ( ord_less_nat @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1339_add__less__cancel__right,axiom,
% 16.19/8.90 ! [A: int,C: int,B: int] :
% 16.19/8.90 ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 16.19/8.90 = ( ord_less_int @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1340_mult_Oright__neutral,axiom,
% 16.19/8.90 ! [A: complex] :
% 16.19/8.90 ( ( times_times_complex @ A @ one_one_complex )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1341_mult_Oright__neutral,axiom,
% 16.19/8.90 ! [A: real] :
% 16.19/8.90 ( ( times_times_real @ A @ one_one_real )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1342_mult_Oright__neutral,axiom,
% 16.19/8.90 ! [A: rat] :
% 16.19/8.90 ( ( times_times_rat @ A @ one_one_rat )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1343_mult_Oright__neutral,axiom,
% 16.19/8.90 ! [A: nat] :
% 16.19/8.90 ( ( times_times_nat @ A @ one_one_nat )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1344_mult_Oright__neutral,axiom,
% 16.19/8.90 ! [A: int] :
% 16.19/8.90 ( ( times_times_int @ A @ one_one_int )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1345_mult__1,axiom,
% 16.19/8.90 ! [A: complex] :
% 16.19/8.90 ( ( times_times_complex @ one_one_complex @ A )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1346_mult__1,axiom,
% 16.19/8.90 ! [A: real] :
% 16.19/8.90 ( ( times_times_real @ one_one_real @ A )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1347_mult__1,axiom,
% 16.19/8.90 ! [A: rat] :
% 16.19/8.90 ( ( times_times_rat @ one_one_rat @ A )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1348_mult__1,axiom,
% 16.19/8.90 ! [A: nat] :
% 16.19/8.90 ( ( times_times_nat @ one_one_nat @ A )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1349_mult__1,axiom,
% 16.19/8.90 ! [A: int] :
% 16.19/8.90 ( ( times_times_int @ one_one_int @ A )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1350_times__divide__eq__left,axiom,
% 16.19/8.90 ! [B: complex,C: complex,A: complex] :
% 16.19/8.90 ( ( times_times_complex @ ( divide1717551699836669952omplex @ B @ C ) @ A )
% 16.19/8.90 = ( divide1717551699836669952omplex @ ( times_times_complex @ B @ A ) @ C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1351_times__divide__eq__left,axiom,
% 16.19/8.90 ! [B: real,C: real,A: real] :
% 16.19/8.90 ( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
% 16.19/8.90 = ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1352_times__divide__eq__left,axiom,
% 16.19/8.90 ! [B: rat,C: rat,A: rat] :
% 16.19/8.90 ( ( times_times_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 16.19/8.90 = ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1353_divide__divide__eq__left,axiom,
% 16.19/8.90 ! [A: complex,B: complex,C: complex] :
% 16.19/8.90 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
% 16.19/8.90 = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1354_divide__divide__eq__left,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real] :
% 16.19/8.90 ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
% 16.19/8.90 = ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1355_divide__divide__eq__left,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat] :
% 16.19/8.90 ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
% 16.19/8.90 = ( divide_divide_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1356_less__set__def,axiom,
% 16.19/8.90 ( ord_less_set_nat
% 16.19/8.90 = ( ^ [A4: set_nat,B4: set_nat] :
% 16.19/8.90 ( ord_less_nat_o
% 16.19/8.90 @ ^ [X2: nat] : ( member_nat @ X2 @ A4 )
% 16.19/8.90 @ ^ [X2: nat] : ( member_nat @ X2 @ B4 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1357_less__set__def,axiom,
% 16.19/8.90 ( ord_less_set_real
% 16.19/8.90 = ( ^ [A4: set_real,B4: set_real] :
% 16.19/8.90 ( ord_less_real_o
% 16.19/8.90 @ ^ [X2: real] : ( member_real @ X2 @ A4 )
% 16.19/8.90 @ ^ [X2: real] : ( member_real @ X2 @ B4 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1358_less__set__def,axiom,
% 16.19/8.90 ( ord_less_set_int
% 16.19/8.90 = ( ^ [A4: set_int,B4: set_int] :
% 16.19/8.90 ( ord_less_int_o
% 16.19/8.90 @ ^ [X2: int] : ( member_int @ X2 @ A4 )
% 16.19/8.90 @ ^ [X2: int] : ( member_int @ X2 @ B4 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1359_less__set__def,axiom,
% 16.19/8.90 ( ord_less_set_complex
% 16.19/8.90 = ( ^ [A4: set_complex,B4: set_complex] :
% 16.19/8.90 ( ord_less_complex_o
% 16.19/8.90 @ ^ [X2: complex] : ( member_complex @ X2 @ A4 )
% 16.19/8.90 @ ^ [X2: complex] : ( member_complex @ X2 @ B4 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1360_less__set__def,axiom,
% 16.19/8.90 ( ord_less_set_set_nat
% 16.19/8.90 = ( ^ [A4: set_set_nat,B4: set_set_nat] :
% 16.19/8.90 ( ord_less_set_nat_o
% 16.19/8.90 @ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A4 )
% 16.19/8.90 @ ^ [X2: set_nat] : ( member_set_nat @ X2 @ B4 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1361_psubsetD,axiom,
% 16.19/8.90 ! [A2: set_nat,B3: set_nat,C: nat] :
% 16.19/8.90 ( ( ord_less_set_nat @ A2 @ B3 )
% 16.19/8.90 => ( ( member_nat @ C @ A2 )
% 16.19/8.90 => ( member_nat @ C @ B3 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1362_psubsetD,axiom,
% 16.19/8.90 ! [A2: set_real,B3: set_real,C: real] :
% 16.19/8.90 ( ( ord_less_set_real @ A2 @ B3 )
% 16.19/8.90 => ( ( member_real @ C @ A2 )
% 16.19/8.90 => ( member_real @ C @ B3 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1363_psubsetD,axiom,
% 16.19/8.90 ! [A2: set_int,B3: set_int,C: int] :
% 16.19/8.90 ( ( ord_less_set_int @ A2 @ B3 )
% 16.19/8.90 => ( ( member_int @ C @ A2 )
% 16.19/8.90 => ( member_int @ C @ B3 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1364_psubsetD,axiom,
% 16.19/8.90 ! [A2: set_complex,B3: set_complex,C: complex] :
% 16.19/8.90 ( ( ord_less_set_complex @ A2 @ B3 )
% 16.19/8.90 => ( ( member_complex @ C @ A2 )
% 16.19/8.90 => ( member_complex @ C @ B3 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1365_psubsetD,axiom,
% 16.19/8.90 ! [A2: set_set_nat,B3: set_set_nat,C: set_nat] :
% 16.19/8.90 ( ( ord_less_set_set_nat @ A2 @ B3 )
% 16.19/8.90 => ( ( member_set_nat @ C @ A2 )
% 16.19/8.90 => ( member_set_nat @ C @ B3 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1366_linordered__field__no__ub,axiom,
% 16.19/8.90 ! [X5: real] :
% 16.19/8.90 ? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1367_linordered__field__no__ub,axiom,
% 16.19/8.90 ! [X5: rat] :
% 16.19/8.90 ? [X_1: rat] : ( ord_less_rat @ X5 @ X_1 ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1368_linordered__field__no__lb,axiom,
% 16.19/8.90 ! [X5: real] :
% 16.19/8.90 ? [Y4: real] : ( ord_less_real @ Y4 @ X5 ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1369_linordered__field__no__lb,axiom,
% 16.19/8.90 ! [X5: rat] :
% 16.19/8.90 ? [Y4: rat] : ( ord_less_rat @ Y4 @ X5 ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1370_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real] :
% 16.19/8.90 ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 16.19/8.90 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1371_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat] :
% 16.19/8.90 ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 16.19/8.90 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1372_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 16.19/8.90 ! [A: nat,B: nat,C: nat] :
% 16.19/8.90 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 16.19/8.90 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1373_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 16.19/8.90 ! [A: int,B: int,C: int] :
% 16.19/8.90 ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 16.19/8.90 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1374_mult_Oassoc,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real] :
% 16.19/8.90 ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 16.19/8.90 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1375_mult_Oassoc,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat] :
% 16.19/8.90 ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 16.19/8.90 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1376_mult_Oassoc,axiom,
% 16.19/8.90 ! [A: nat,B: nat,C: nat] :
% 16.19/8.90 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 16.19/8.90 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1377_mult_Oassoc,axiom,
% 16.19/8.90 ! [A: int,B: int,C: int] :
% 16.19/8.90 ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 16.19/8.90 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1378_mult_Ocommute,axiom,
% 16.19/8.90 ( times_times_real
% 16.19/8.90 = ( ^ [A5: real,B5: real] : ( times_times_real @ B5 @ A5 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1379_mult_Ocommute,axiom,
% 16.19/8.90 ( times_times_rat
% 16.19/8.90 = ( ^ [A5: rat,B5: rat] : ( times_times_rat @ B5 @ A5 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1380_mult_Ocommute,axiom,
% 16.19/8.90 ( times_times_nat
% 16.19/8.90 = ( ^ [A5: nat,B5: nat] : ( times_times_nat @ B5 @ A5 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1381_mult_Ocommute,axiom,
% 16.19/8.90 ( times_times_int
% 16.19/8.90 = ( ^ [A5: int,B5: int] : ( times_times_int @ B5 @ A5 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1382_mult_Oleft__commute,axiom,
% 16.19/8.90 ! [B: real,A: real,C: real] :
% 16.19/8.90 ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
% 16.19/8.90 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1383_mult_Oleft__commute,axiom,
% 16.19/8.90 ! [B: rat,A: rat,C: rat] :
% 16.19/8.90 ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
% 16.19/8.90 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1384_mult_Oleft__commute,axiom,
% 16.19/8.90 ! [B: nat,A: nat,C: nat] :
% 16.19/8.90 ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
% 16.19/8.90 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1385_mult_Oleft__commute,axiom,
% 16.19/8.90 ! [B: int,A: int,C: int] :
% 16.19/8.90 ( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
% 16.19/8.90 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1386_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real] :
% 16.19/8.90 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 16.19/8.90 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1387_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat] :
% 16.19/8.90 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 16.19/8.90 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1388_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 16.19/8.90 ! [A: nat,B: nat,C: nat] :
% 16.19/8.90 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 16.19/8.90 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1389_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 16.19/8.90 ! [A: int,B: int,C: int] :
% 16.19/8.90 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 16.19/8.90 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1390_add__mono__thms__linordered__semiring_I4_J,axiom,
% 16.19/8.90 ! [I2: real,J: real,K: real,L2: real] :
% 16.19/8.90 ( ( ( I2 = J )
% 16.19/8.90 & ( K = L2 ) )
% 16.19/8.90 => ( ( plus_plus_real @ I2 @ K )
% 16.19/8.90 = ( plus_plus_real @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1391_add__mono__thms__linordered__semiring_I4_J,axiom,
% 16.19/8.90 ! [I2: rat,J: rat,K: rat,L2: rat] :
% 16.19/8.90 ( ( ( I2 = J )
% 16.19/8.90 & ( K = L2 ) )
% 16.19/8.90 => ( ( plus_plus_rat @ I2 @ K )
% 16.19/8.90 = ( plus_plus_rat @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1392_add__mono__thms__linordered__semiring_I4_J,axiom,
% 16.19/8.90 ! [I2: nat,J: nat,K: nat,L2: nat] :
% 16.19/8.90 ( ( ( I2 = J )
% 16.19/8.90 & ( K = L2 ) )
% 16.19/8.90 => ( ( plus_plus_nat @ I2 @ K )
% 16.19/8.90 = ( plus_plus_nat @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1393_add__mono__thms__linordered__semiring_I4_J,axiom,
% 16.19/8.90 ! [I2: int,J: int,K: int,L2: int] :
% 16.19/8.90 ( ( ( I2 = J )
% 16.19/8.90 & ( K = L2 ) )
% 16.19/8.90 => ( ( plus_plus_int @ I2 @ K )
% 16.19/8.90 = ( plus_plus_int @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1394_group__cancel_Oadd1,axiom,
% 16.19/8.90 ! [A2: real,K: real,A: real,B: real] :
% 16.19/8.90 ( ( A2
% 16.19/8.90 = ( plus_plus_real @ K @ A ) )
% 16.19/8.90 => ( ( plus_plus_real @ A2 @ B )
% 16.19/8.90 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1395_group__cancel_Oadd1,axiom,
% 16.19/8.90 ! [A2: rat,K: rat,A: rat,B: rat] :
% 16.19/8.90 ( ( A2
% 16.19/8.90 = ( plus_plus_rat @ K @ A ) )
% 16.19/8.90 => ( ( plus_plus_rat @ A2 @ B )
% 16.19/8.90 = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1396_group__cancel_Oadd1,axiom,
% 16.19/8.90 ! [A2: nat,K: nat,A: nat,B: nat] :
% 16.19/8.90 ( ( A2
% 16.19/8.90 = ( plus_plus_nat @ K @ A ) )
% 16.19/8.90 => ( ( plus_plus_nat @ A2 @ B )
% 16.19/8.90 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1397_group__cancel_Oadd1,axiom,
% 16.19/8.90 ! [A2: int,K: int,A: int,B: int] :
% 16.19/8.90 ( ( A2
% 16.19/8.90 = ( plus_plus_int @ K @ A ) )
% 16.19/8.90 => ( ( plus_plus_int @ A2 @ B )
% 16.19/8.90 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1398_group__cancel_Oadd2,axiom,
% 16.19/8.90 ! [B3: real,K: real,B: real,A: real] :
% 16.19/8.90 ( ( B3
% 16.19/8.90 = ( plus_plus_real @ K @ B ) )
% 16.19/8.90 => ( ( plus_plus_real @ A @ B3 )
% 16.19/8.90 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1399_group__cancel_Oadd2,axiom,
% 16.19/8.90 ! [B3: rat,K: rat,B: rat,A: rat] :
% 16.19/8.90 ( ( B3
% 16.19/8.90 = ( plus_plus_rat @ K @ B ) )
% 16.19/8.90 => ( ( plus_plus_rat @ A @ B3 )
% 16.19/8.90 = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1400_group__cancel_Oadd2,axiom,
% 16.19/8.90 ! [B3: nat,K: nat,B: nat,A: nat] :
% 16.19/8.90 ( ( B3
% 16.19/8.90 = ( plus_plus_nat @ K @ B ) )
% 16.19/8.90 => ( ( plus_plus_nat @ A @ B3 )
% 16.19/8.90 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1401_group__cancel_Oadd2,axiom,
% 16.19/8.90 ! [B3: int,K: int,B: int,A: int] :
% 16.19/8.90 ( ( B3
% 16.19/8.90 = ( plus_plus_int @ K @ B ) )
% 16.19/8.90 => ( ( plus_plus_int @ A @ B3 )
% 16.19/8.90 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1402_add_Oassoc,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real] :
% 16.19/8.90 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 16.19/8.90 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1403_add_Oassoc,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat] :
% 16.19/8.90 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 16.19/8.90 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1404_add_Oassoc,axiom,
% 16.19/8.90 ! [A: nat,B: nat,C: nat] :
% 16.19/8.90 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 16.19/8.90 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1405_add_Oassoc,axiom,
% 16.19/8.90 ! [A: int,B: int,C: int] :
% 16.19/8.90 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 16.19/8.90 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1406_add_Oleft__cancel,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real] :
% 16.19/8.90 ( ( ( plus_plus_real @ A @ B )
% 16.19/8.90 = ( plus_plus_real @ A @ C ) )
% 16.19/8.90 = ( B = C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1407_add_Oleft__cancel,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat] :
% 16.19/8.90 ( ( ( plus_plus_rat @ A @ B )
% 16.19/8.90 = ( plus_plus_rat @ A @ C ) )
% 16.19/8.90 = ( B = C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1408_add_Oleft__cancel,axiom,
% 16.19/8.90 ! [A: int,B: int,C: int] :
% 16.19/8.90 ( ( ( plus_plus_int @ A @ B )
% 16.19/8.90 = ( plus_plus_int @ A @ C ) )
% 16.19/8.90 = ( B = C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1409_add_Oright__cancel,axiom,
% 16.19/8.90 ! [B: real,A: real,C: real] :
% 16.19/8.90 ( ( ( plus_plus_real @ B @ A )
% 16.19/8.90 = ( plus_plus_real @ C @ A ) )
% 16.19/8.90 = ( B = C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1410_add_Oright__cancel,axiom,
% 16.19/8.90 ! [B: rat,A: rat,C: rat] :
% 16.19/8.90 ( ( ( plus_plus_rat @ B @ A )
% 16.19/8.90 = ( plus_plus_rat @ C @ A ) )
% 16.19/8.90 = ( B = C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1411_add_Oright__cancel,axiom,
% 16.19/8.90 ! [B: int,A: int,C: int] :
% 16.19/8.90 ( ( ( plus_plus_int @ B @ A )
% 16.19/8.90 = ( plus_plus_int @ C @ A ) )
% 16.19/8.90 = ( B = C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1412_add_Ocommute,axiom,
% 16.19/8.90 ( plus_plus_real
% 16.19/8.90 = ( ^ [A5: real,B5: real] : ( plus_plus_real @ B5 @ A5 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1413_add_Ocommute,axiom,
% 16.19/8.90 ( plus_plus_rat
% 16.19/8.90 = ( ^ [A5: rat,B5: rat] : ( plus_plus_rat @ B5 @ A5 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1414_add_Ocommute,axiom,
% 16.19/8.90 ( plus_plus_nat
% 16.19/8.90 = ( ^ [A5: nat,B5: nat] : ( plus_plus_nat @ B5 @ A5 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1415_add_Ocommute,axiom,
% 16.19/8.90 ( plus_plus_int
% 16.19/8.90 = ( ^ [A5: int,B5: int] : ( plus_plus_int @ B5 @ A5 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1416_add_Oleft__commute,axiom,
% 16.19/8.90 ! [B: real,A: real,C: real] :
% 16.19/8.90 ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
% 16.19/8.90 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1417_add_Oleft__commute,axiom,
% 16.19/8.90 ! [B: rat,A: rat,C: rat] :
% 16.19/8.90 ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
% 16.19/8.90 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1418_add_Oleft__commute,axiom,
% 16.19/8.90 ! [B: nat,A: nat,C: nat] :
% 16.19/8.90 ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 16.19/8.90 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1419_add_Oleft__commute,axiom,
% 16.19/8.90 ! [B: int,A: int,C: int] :
% 16.19/8.90 ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
% 16.19/8.90 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1420_add__left__imp__eq,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real] :
% 16.19/8.90 ( ( ( plus_plus_real @ A @ B )
% 16.19/8.90 = ( plus_plus_real @ A @ C ) )
% 16.19/8.90 => ( B = C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1421_add__left__imp__eq,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat] :
% 16.19/8.90 ( ( ( plus_plus_rat @ A @ B )
% 16.19/8.90 = ( plus_plus_rat @ A @ C ) )
% 16.19/8.90 => ( B = C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1422_add__left__imp__eq,axiom,
% 16.19/8.90 ! [A: nat,B: nat,C: nat] :
% 16.19/8.90 ( ( ( plus_plus_nat @ A @ B )
% 16.19/8.90 = ( plus_plus_nat @ A @ C ) )
% 16.19/8.90 => ( B = C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1423_add__left__imp__eq,axiom,
% 16.19/8.90 ! [A: int,B: int,C: int] :
% 16.19/8.90 ( ( ( plus_plus_int @ A @ B )
% 16.19/8.90 = ( plus_plus_int @ A @ C ) )
% 16.19/8.90 => ( B = C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1424_add__right__imp__eq,axiom,
% 16.19/8.90 ! [B: real,A: real,C: real] :
% 16.19/8.90 ( ( ( plus_plus_real @ B @ A )
% 16.19/8.90 = ( plus_plus_real @ C @ A ) )
% 16.19/8.90 => ( B = C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1425_add__right__imp__eq,axiom,
% 16.19/8.90 ! [B: rat,A: rat,C: rat] :
% 16.19/8.90 ( ( ( plus_plus_rat @ B @ A )
% 16.19/8.90 = ( plus_plus_rat @ C @ A ) )
% 16.19/8.90 => ( B = C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1426_add__right__imp__eq,axiom,
% 16.19/8.90 ! [B: nat,A: nat,C: nat] :
% 16.19/8.90 ( ( ( plus_plus_nat @ B @ A )
% 16.19/8.90 = ( plus_plus_nat @ C @ A ) )
% 16.19/8.90 => ( B = C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1427_add__right__imp__eq,axiom,
% 16.19/8.90 ! [B: int,A: int,C: int] :
% 16.19/8.90 ( ( ( plus_plus_int @ B @ A )
% 16.19/8.90 = ( plus_plus_int @ C @ A ) )
% 16.19/8.90 => ( B = C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1428_one__reorient,axiom,
% 16.19/8.90 ! [X: complex] :
% 16.19/8.90 ( ( one_one_complex = X )
% 16.19/8.90 = ( X = one_one_complex ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1429_one__reorient,axiom,
% 16.19/8.90 ! [X: real] :
% 16.19/8.90 ( ( one_one_real = X )
% 16.19/8.90 = ( X = one_one_real ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1430_one__reorient,axiom,
% 16.19/8.90 ! [X: rat] :
% 16.19/8.90 ( ( one_one_rat = X )
% 16.19/8.90 = ( X = one_one_rat ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1431_one__reorient,axiom,
% 16.19/8.90 ! [X: nat] :
% 16.19/8.90 ( ( one_one_nat = X )
% 16.19/8.90 = ( X = one_one_nat ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1432_one__reorient,axiom,
% 16.19/8.90 ! [X: int] :
% 16.19/8.90 ( ( one_one_int = X )
% 16.19/8.90 = ( X = one_one_int ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1433_diff__right__commute,axiom,
% 16.19/8.90 ! [A: real,C: real,B: real] :
% 16.19/8.90 ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
% 16.19/8.90 = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1434_diff__right__commute,axiom,
% 16.19/8.90 ! [A: rat,C: rat,B: rat] :
% 16.19/8.90 ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
% 16.19/8.90 = ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1435_diff__right__commute,axiom,
% 16.19/8.90 ! [A: nat,C: nat,B: nat] :
% 16.19/8.90 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 16.19/8.90 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1436_diff__right__commute,axiom,
% 16.19/8.90 ! [A: int,C: int,B: int] :
% 16.19/8.90 ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
% 16.19/8.90 = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1437_diff__eq__diff__eq,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real,D: real] :
% 16.19/8.90 ( ( ( minus_minus_real @ A @ B )
% 16.19/8.90 = ( minus_minus_real @ C @ D ) )
% 16.19/8.90 => ( ( A = B )
% 16.19/8.90 = ( C = D ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1438_diff__eq__diff__eq,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat,D: rat] :
% 16.19/8.90 ( ( ( minus_minus_rat @ A @ B )
% 16.19/8.90 = ( minus_minus_rat @ C @ D ) )
% 16.19/8.90 => ( ( A = B )
% 16.19/8.90 = ( C = D ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1439_diff__eq__diff__eq,axiom,
% 16.19/8.90 ! [A: int,B: int,C: int,D: int] :
% 16.19/8.90 ( ( ( minus_minus_int @ A @ B )
% 16.19/8.90 = ( minus_minus_int @ C @ D ) )
% 16.19/8.90 => ( ( A = B )
% 16.19/8.90 = ( C = D ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1440_add__mono__thms__linordered__semiring_I3_J,axiom,
% 16.19/8.90 ! [I2: real,J: real,K: real,L2: real] :
% 16.19/8.90 ( ( ( ord_less_eq_real @ I2 @ J )
% 16.19/8.90 & ( K = L2 ) )
% 16.19/8.90 => ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1441_add__mono__thms__linordered__semiring_I3_J,axiom,
% 16.19/8.90 ! [I2: rat,J: rat,K: rat,L2: rat] :
% 16.19/8.90 ( ( ( ord_less_eq_rat @ I2 @ J )
% 16.19/8.90 & ( K = L2 ) )
% 16.19/8.90 => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1442_add__mono__thms__linordered__semiring_I3_J,axiom,
% 16.19/8.90 ! [I2: nat,J: nat,K: nat,L2: nat] :
% 16.19/8.90 ( ( ( ord_less_eq_nat @ I2 @ J )
% 16.19/8.90 & ( K = L2 ) )
% 16.19/8.90 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1443_add__mono__thms__linordered__semiring_I3_J,axiom,
% 16.19/8.90 ! [I2: int,J: int,K: int,L2: int] :
% 16.19/8.90 ( ( ( ord_less_eq_int @ I2 @ J )
% 16.19/8.90 & ( K = L2 ) )
% 16.19/8.90 => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1444_add__mono__thms__linordered__semiring_I2_J,axiom,
% 16.19/8.90 ! [I2: real,J: real,K: real,L2: real] :
% 16.19/8.90 ( ( ( I2 = J )
% 16.19/8.90 & ( ord_less_eq_real @ K @ L2 ) )
% 16.19/8.90 => ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1445_add__mono__thms__linordered__semiring_I2_J,axiom,
% 16.19/8.90 ! [I2: rat,J: rat,K: rat,L2: rat] :
% 16.19/8.90 ( ( ( I2 = J )
% 16.19/8.90 & ( ord_less_eq_rat @ K @ L2 ) )
% 16.19/8.90 => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1446_add__mono__thms__linordered__semiring_I2_J,axiom,
% 16.19/8.90 ! [I2: nat,J: nat,K: nat,L2: nat] :
% 16.19/8.90 ( ( ( I2 = J )
% 16.19/8.90 & ( ord_less_eq_nat @ K @ L2 ) )
% 16.19/8.90 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1447_add__mono__thms__linordered__semiring_I2_J,axiom,
% 16.19/8.90 ! [I2: int,J: int,K: int,L2: int] :
% 16.19/8.90 ( ( ( I2 = J )
% 16.19/8.90 & ( ord_less_eq_int @ K @ L2 ) )
% 16.19/8.90 => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1448_add__mono__thms__linordered__semiring_I1_J,axiom,
% 16.19/8.90 ! [I2: real,J: real,K: real,L2: real] :
% 16.19/8.90 ( ( ( ord_less_eq_real @ I2 @ J )
% 16.19/8.90 & ( ord_less_eq_real @ K @ L2 ) )
% 16.19/8.90 => ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1449_add__mono__thms__linordered__semiring_I1_J,axiom,
% 16.19/8.90 ! [I2: rat,J: rat,K: rat,L2: rat] :
% 16.19/8.90 ( ( ( ord_less_eq_rat @ I2 @ J )
% 16.19/8.90 & ( ord_less_eq_rat @ K @ L2 ) )
% 16.19/8.90 => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1450_add__mono__thms__linordered__semiring_I1_J,axiom,
% 16.19/8.90 ! [I2: nat,J: nat,K: nat,L2: nat] :
% 16.19/8.90 ( ( ( ord_less_eq_nat @ I2 @ J )
% 16.19/8.90 & ( ord_less_eq_nat @ K @ L2 ) )
% 16.19/8.90 => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1451_add__mono__thms__linordered__semiring_I1_J,axiom,
% 16.19/8.90 ! [I2: int,J: int,K: int,L2: int] :
% 16.19/8.90 ( ( ( ord_less_eq_int @ I2 @ J )
% 16.19/8.90 & ( ord_less_eq_int @ K @ L2 ) )
% 16.19/8.90 => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1452_add__mono,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real,D: real] :
% 16.19/8.90 ( ( ord_less_eq_real @ A @ B )
% 16.19/8.90 => ( ( ord_less_eq_real @ C @ D )
% 16.19/8.90 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1453_add__mono,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat,D: rat] :
% 16.19/8.90 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.90 => ( ( ord_less_eq_rat @ C @ D )
% 16.19/8.90 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1454_add__mono,axiom,
% 16.19/8.90 ! [A: nat,B: nat,C: nat,D: nat] :
% 16.19/8.90 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.90 => ( ( ord_less_eq_nat @ C @ D )
% 16.19/8.90 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1455_add__mono,axiom,
% 16.19/8.90 ! [A: int,B: int,C: int,D: int] :
% 16.19/8.90 ( ( ord_less_eq_int @ A @ B )
% 16.19/8.90 => ( ( ord_less_eq_int @ C @ D )
% 16.19/8.90 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1456_add__left__mono,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real] :
% 16.19/8.90 ( ( ord_less_eq_real @ A @ B )
% 16.19/8.90 => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1457_add__left__mono,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat] :
% 16.19/8.90 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.90 => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1458_add__left__mono,axiom,
% 16.19/8.90 ! [A: nat,B: nat,C: nat] :
% 16.19/8.90 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.90 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1459_add__left__mono,axiom,
% 16.19/8.90 ! [A: int,B: int,C: int] :
% 16.19/8.90 ( ( ord_less_eq_int @ A @ B )
% 16.19/8.90 => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1460_less__eqE,axiom,
% 16.19/8.90 ! [A: nat,B: nat] :
% 16.19/8.90 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.90 => ~ ! [C3: nat] :
% 16.19/8.90 ( B
% 16.19/8.90 != ( plus_plus_nat @ A @ C3 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1461_add__right__mono,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real] :
% 16.19/8.90 ( ( ord_less_eq_real @ A @ B )
% 16.19/8.90 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1462_add__right__mono,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat] :
% 16.19/8.90 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.90 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1463_add__right__mono,axiom,
% 16.19/8.90 ! [A: nat,B: nat,C: nat] :
% 16.19/8.90 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.90 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1464_add__right__mono,axiom,
% 16.19/8.90 ! [A: int,B: int,C: int] :
% 16.19/8.90 ( ( ord_less_eq_int @ A @ B )
% 16.19/8.90 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1465_le__iff__add,axiom,
% 16.19/8.90 ( ord_less_eq_nat
% 16.19/8.90 = ( ^ [A5: nat,B5: nat] :
% 16.19/8.90 ? [C4: nat] :
% 16.19/8.90 ( B5
% 16.19/8.90 = ( plus_plus_nat @ A5 @ C4 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1466_add__le__imp__le__left,axiom,
% 16.19/8.90 ! [C: real,A: real,B: real] :
% 16.19/8.90 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 16.19/8.90 => ( ord_less_eq_real @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1467_add__le__imp__le__left,axiom,
% 16.19/8.90 ! [C: rat,A: rat,B: rat] :
% 16.19/8.90 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 16.19/8.90 => ( ord_less_eq_rat @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1468_add__le__imp__le__left,axiom,
% 16.19/8.90 ! [C: nat,A: nat,B: nat] :
% 16.19/8.90 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 16.19/8.90 => ( ord_less_eq_nat @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1469_add__le__imp__le__left,axiom,
% 16.19/8.90 ! [C: int,A: int,B: int] :
% 16.19/8.90 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 16.19/8.90 => ( ord_less_eq_int @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1470_add__le__imp__le__right,axiom,
% 16.19/8.90 ! [A: real,C: real,B: real] :
% 16.19/8.90 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 16.19/8.90 => ( ord_less_eq_real @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1471_add__le__imp__le__right,axiom,
% 16.19/8.90 ! [A: rat,C: rat,B: rat] :
% 16.19/8.90 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 16.19/8.90 => ( ord_less_eq_rat @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1472_add__le__imp__le__right,axiom,
% 16.19/8.90 ! [A: nat,C: nat,B: nat] :
% 16.19/8.90 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 16.19/8.90 => ( ord_less_eq_nat @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1473_add__le__imp__le__right,axiom,
% 16.19/8.90 ! [A: int,C: int,B: int] :
% 16.19/8.90 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 16.19/8.90 => ( ord_less_eq_int @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1474_add__mono__thms__linordered__field_I5_J,axiom,
% 16.19/8.90 ! [I2: real,J: real,K: real,L2: real] :
% 16.19/8.90 ( ( ( ord_less_real @ I2 @ J )
% 16.19/8.90 & ( ord_less_real @ K @ L2 ) )
% 16.19/8.90 => ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1475_add__mono__thms__linordered__field_I5_J,axiom,
% 16.19/8.90 ! [I2: rat,J: rat,K: rat,L2: rat] :
% 16.19/8.90 ( ( ( ord_less_rat @ I2 @ J )
% 16.19/8.90 & ( ord_less_rat @ K @ L2 ) )
% 16.19/8.90 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1476_add__mono__thms__linordered__field_I5_J,axiom,
% 16.19/8.90 ! [I2: nat,J: nat,K: nat,L2: nat] :
% 16.19/8.90 ( ( ( ord_less_nat @ I2 @ J )
% 16.19/8.90 & ( ord_less_nat @ K @ L2 ) )
% 16.19/8.90 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1477_add__mono__thms__linordered__field_I5_J,axiom,
% 16.19/8.90 ! [I2: int,J: int,K: int,L2: int] :
% 16.19/8.90 ( ( ( ord_less_int @ I2 @ J )
% 16.19/8.90 & ( ord_less_int @ K @ L2 ) )
% 16.19/8.90 => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1478_add__mono__thms__linordered__field_I2_J,axiom,
% 16.19/8.90 ! [I2: real,J: real,K: real,L2: real] :
% 16.19/8.90 ( ( ( I2 = J )
% 16.19/8.90 & ( ord_less_real @ K @ L2 ) )
% 16.19/8.90 => ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1479_add__mono__thms__linordered__field_I2_J,axiom,
% 16.19/8.90 ! [I2: rat,J: rat,K: rat,L2: rat] :
% 16.19/8.90 ( ( ( I2 = J )
% 16.19/8.90 & ( ord_less_rat @ K @ L2 ) )
% 16.19/8.90 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1480_add__mono__thms__linordered__field_I2_J,axiom,
% 16.19/8.90 ! [I2: nat,J: nat,K: nat,L2: nat] :
% 16.19/8.90 ( ( ( I2 = J )
% 16.19/8.90 & ( ord_less_nat @ K @ L2 ) )
% 16.19/8.90 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1481_add__mono__thms__linordered__field_I2_J,axiom,
% 16.19/8.90 ! [I2: int,J: int,K: int,L2: int] :
% 16.19/8.90 ( ( ( I2 = J )
% 16.19/8.90 & ( ord_less_int @ K @ L2 ) )
% 16.19/8.90 => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1482_add__mono__thms__linordered__field_I1_J,axiom,
% 16.19/8.90 ! [I2: real,J: real,K: real,L2: real] :
% 16.19/8.90 ( ( ( ord_less_real @ I2 @ J )
% 16.19/8.90 & ( K = L2 ) )
% 16.19/8.90 => ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1483_add__mono__thms__linordered__field_I1_J,axiom,
% 16.19/8.90 ! [I2: rat,J: rat,K: rat,L2: rat] :
% 16.19/8.90 ( ( ( ord_less_rat @ I2 @ J )
% 16.19/8.90 & ( K = L2 ) )
% 16.19/8.90 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1484_add__mono__thms__linordered__field_I1_J,axiom,
% 16.19/8.90 ! [I2: nat,J: nat,K: nat,L2: nat] :
% 16.19/8.90 ( ( ( ord_less_nat @ I2 @ J )
% 16.19/8.90 & ( K = L2 ) )
% 16.19/8.90 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1485_add__mono__thms__linordered__field_I1_J,axiom,
% 16.19/8.90 ! [I2: int,J: int,K: int,L2: int] :
% 16.19/8.90 ( ( ( ord_less_int @ I2 @ J )
% 16.19/8.90 & ( K = L2 ) )
% 16.19/8.90 => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1486_add__strict__mono,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real,D: real] :
% 16.19/8.90 ( ( ord_less_real @ A @ B )
% 16.19/8.90 => ( ( ord_less_real @ C @ D )
% 16.19/8.90 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1487_add__strict__mono,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat,D: rat] :
% 16.19/8.90 ( ( ord_less_rat @ A @ B )
% 16.19/8.90 => ( ( ord_less_rat @ C @ D )
% 16.19/8.90 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1488_add__strict__mono,axiom,
% 16.19/8.90 ! [A: nat,B: nat,C: nat,D: nat] :
% 16.19/8.90 ( ( ord_less_nat @ A @ B )
% 16.19/8.90 => ( ( ord_less_nat @ C @ D )
% 16.19/8.90 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1489_add__strict__mono,axiom,
% 16.19/8.90 ! [A: int,B: int,C: int,D: int] :
% 16.19/8.90 ( ( ord_less_int @ A @ B )
% 16.19/8.90 => ( ( ord_less_int @ C @ D )
% 16.19/8.90 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1490_add__strict__left__mono,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real] :
% 16.19/8.90 ( ( ord_less_real @ A @ B )
% 16.19/8.90 => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1491_add__strict__left__mono,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat] :
% 16.19/8.90 ( ( ord_less_rat @ A @ B )
% 16.19/8.90 => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1492_add__strict__left__mono,axiom,
% 16.19/8.90 ! [A: nat,B: nat,C: nat] :
% 16.19/8.90 ( ( ord_less_nat @ A @ B )
% 16.19/8.90 => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1493_add__strict__left__mono,axiom,
% 16.19/8.90 ! [A: int,B: int,C: int] :
% 16.19/8.90 ( ( ord_less_int @ A @ B )
% 16.19/8.90 => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1494_add__strict__right__mono,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real] :
% 16.19/8.90 ( ( ord_less_real @ A @ B )
% 16.19/8.90 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1495_add__strict__right__mono,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat] :
% 16.19/8.90 ( ( ord_less_rat @ A @ B )
% 16.19/8.90 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1496_add__strict__right__mono,axiom,
% 16.19/8.90 ! [A: nat,B: nat,C: nat] :
% 16.19/8.90 ( ( ord_less_nat @ A @ B )
% 16.19/8.90 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1497_add__strict__right__mono,axiom,
% 16.19/8.90 ! [A: int,B: int,C: int] :
% 16.19/8.90 ( ( ord_less_int @ A @ B )
% 16.19/8.90 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1498_add__less__imp__less__left,axiom,
% 16.19/8.90 ! [C: real,A: real,B: real] :
% 16.19/8.90 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 16.19/8.90 => ( ord_less_real @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1499_add__less__imp__less__left,axiom,
% 16.19/8.90 ! [C: rat,A: rat,B: rat] :
% 16.19/8.90 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 16.19/8.90 => ( ord_less_rat @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1500_add__less__imp__less__left,axiom,
% 16.19/8.90 ! [C: nat,A: nat,B: nat] :
% 16.19/8.90 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 16.19/8.90 => ( ord_less_nat @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1501_add__less__imp__less__left,axiom,
% 16.19/8.90 ! [C: int,A: int,B: int] :
% 16.19/8.90 ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 16.19/8.90 => ( ord_less_int @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1502_add__less__imp__less__right,axiom,
% 16.19/8.90 ! [A: real,C: real,B: real] :
% 16.19/8.90 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 16.19/8.90 => ( ord_less_real @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1503_add__less__imp__less__right,axiom,
% 16.19/8.90 ! [A: rat,C: rat,B: rat] :
% 16.19/8.90 ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 16.19/8.90 => ( ord_less_rat @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1504_add__less__imp__less__right,axiom,
% 16.19/8.90 ! [A: nat,C: nat,B: nat] :
% 16.19/8.90 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 16.19/8.90 => ( ord_less_nat @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1505_add__less__imp__less__right,axiom,
% 16.19/8.90 ! [A: int,C: int,B: int] :
% 16.19/8.90 ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 16.19/8.90 => ( ord_less_int @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1506_diff__eq__diff__less__eq,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real,D: real] :
% 16.19/8.90 ( ( ( minus_minus_real @ A @ B )
% 16.19/8.90 = ( minus_minus_real @ C @ D ) )
% 16.19/8.90 => ( ( ord_less_eq_real @ A @ B )
% 16.19/8.90 = ( ord_less_eq_real @ C @ D ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1507_diff__eq__diff__less__eq,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat,D: rat] :
% 16.19/8.90 ( ( ( minus_minus_rat @ A @ B )
% 16.19/8.90 = ( minus_minus_rat @ C @ D ) )
% 16.19/8.90 => ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.90 = ( ord_less_eq_rat @ C @ D ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1508_diff__eq__diff__less__eq,axiom,
% 16.19/8.90 ! [A: int,B: int,C: int,D: int] :
% 16.19/8.90 ( ( ( minus_minus_int @ A @ B )
% 16.19/8.90 = ( minus_minus_int @ C @ D ) )
% 16.19/8.90 => ( ( ord_less_eq_int @ A @ B )
% 16.19/8.90 = ( ord_less_eq_int @ C @ D ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1509_diff__right__mono,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real] :
% 16.19/8.90 ( ( ord_less_eq_real @ A @ B )
% 16.19/8.90 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1510_diff__right__mono,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat] :
% 16.19/8.90 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.90 => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1511_diff__right__mono,axiom,
% 16.19/8.90 ! [A: int,B: int,C: int] :
% 16.19/8.90 ( ( ord_less_eq_int @ A @ B )
% 16.19/8.90 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1512_diff__left__mono,axiom,
% 16.19/8.90 ! [B: real,A: real,C: real] :
% 16.19/8.90 ( ( ord_less_eq_real @ B @ A )
% 16.19/8.90 => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1513_diff__left__mono,axiom,
% 16.19/8.90 ! [B: rat,A: rat,C: rat] :
% 16.19/8.90 ( ( ord_less_eq_rat @ B @ A )
% 16.19/8.90 => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1514_diff__left__mono,axiom,
% 16.19/8.90 ! [B: int,A: int,C: int] :
% 16.19/8.90 ( ( ord_less_eq_int @ B @ A )
% 16.19/8.90 => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1515_diff__mono,axiom,
% 16.19/8.90 ! [A: real,B: real,D: real,C: real] :
% 16.19/8.90 ( ( ord_less_eq_real @ A @ B )
% 16.19/8.90 => ( ( ord_less_eq_real @ D @ C )
% 16.19/8.90 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1516_diff__mono,axiom,
% 16.19/8.90 ! [A: rat,B: rat,D: rat,C: rat] :
% 16.19/8.90 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.90 => ( ( ord_less_eq_rat @ D @ C )
% 16.19/8.90 => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1517_diff__mono,axiom,
% 16.19/8.90 ! [A: int,B: int,D: int,C: int] :
% 16.19/8.90 ( ( ord_less_eq_int @ A @ B )
% 16.19/8.90 => ( ( ord_less_eq_int @ D @ C )
% 16.19/8.90 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1518_diff__strict__right__mono,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real] :
% 16.19/8.90 ( ( ord_less_real @ A @ B )
% 16.19/8.90 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1519_diff__strict__right__mono,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat] :
% 16.19/8.90 ( ( ord_less_rat @ A @ B )
% 16.19/8.90 => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1520_diff__strict__right__mono,axiom,
% 16.19/8.90 ! [A: int,B: int,C: int] :
% 16.19/8.90 ( ( ord_less_int @ A @ B )
% 16.19/8.90 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1521_diff__strict__left__mono,axiom,
% 16.19/8.90 ! [B: real,A: real,C: real] :
% 16.19/8.90 ( ( ord_less_real @ B @ A )
% 16.19/8.90 => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1522_diff__strict__left__mono,axiom,
% 16.19/8.90 ! [B: rat,A: rat,C: rat] :
% 16.19/8.90 ( ( ord_less_rat @ B @ A )
% 16.19/8.90 => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1523_diff__strict__left__mono,axiom,
% 16.19/8.90 ! [B: int,A: int,C: int] :
% 16.19/8.90 ( ( ord_less_int @ B @ A )
% 16.19/8.90 => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1524_diff__eq__diff__less,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real,D: real] :
% 16.19/8.90 ( ( ( minus_minus_real @ A @ B )
% 16.19/8.90 = ( minus_minus_real @ C @ D ) )
% 16.19/8.90 => ( ( ord_less_real @ A @ B )
% 16.19/8.90 = ( ord_less_real @ C @ D ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1525_diff__eq__diff__less,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat,D: rat] :
% 16.19/8.90 ( ( ( minus_minus_rat @ A @ B )
% 16.19/8.90 = ( minus_minus_rat @ C @ D ) )
% 16.19/8.90 => ( ( ord_less_rat @ A @ B )
% 16.19/8.90 = ( ord_less_rat @ C @ D ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1526_diff__eq__diff__less,axiom,
% 16.19/8.90 ! [A: int,B: int,C: int,D: int] :
% 16.19/8.90 ( ( ( minus_minus_int @ A @ B )
% 16.19/8.90 = ( minus_minus_int @ C @ D ) )
% 16.19/8.90 => ( ( ord_less_int @ A @ B )
% 16.19/8.90 = ( ord_less_int @ C @ D ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1527_diff__strict__mono,axiom,
% 16.19/8.90 ! [A: real,B: real,D: real,C: real] :
% 16.19/8.90 ( ( ord_less_real @ A @ B )
% 16.19/8.90 => ( ( ord_less_real @ D @ C )
% 16.19/8.90 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1528_diff__strict__mono,axiom,
% 16.19/8.90 ! [A: rat,B: rat,D: rat,C: rat] :
% 16.19/8.90 ( ( ord_less_rat @ A @ B )
% 16.19/8.90 => ( ( ord_less_rat @ D @ C )
% 16.19/8.90 => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1529_diff__strict__mono,axiom,
% 16.19/8.90 ! [A: int,B: int,D: int,C: int] :
% 16.19/8.90 ( ( ord_less_int @ A @ B )
% 16.19/8.90 => ( ( ord_less_int @ D @ C )
% 16.19/8.90 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1530_comm__monoid__mult__class_Omult__1,axiom,
% 16.19/8.90 ! [A: complex] :
% 16.19/8.90 ( ( times_times_complex @ one_one_complex @ A )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1531_comm__monoid__mult__class_Omult__1,axiom,
% 16.19/8.90 ! [A: real] :
% 16.19/8.90 ( ( times_times_real @ one_one_real @ A )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1532_comm__monoid__mult__class_Omult__1,axiom,
% 16.19/8.90 ! [A: rat] :
% 16.19/8.90 ( ( times_times_rat @ one_one_rat @ A )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1533_comm__monoid__mult__class_Omult__1,axiom,
% 16.19/8.90 ! [A: nat] :
% 16.19/8.90 ( ( times_times_nat @ one_one_nat @ A )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1534_comm__monoid__mult__class_Omult__1,axiom,
% 16.19/8.90 ! [A: int] :
% 16.19/8.90 ( ( times_times_int @ one_one_int @ A )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1535_mult_Ocomm__neutral,axiom,
% 16.19/8.90 ! [A: complex] :
% 16.19/8.90 ( ( times_times_complex @ A @ one_one_complex )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1536_mult_Ocomm__neutral,axiom,
% 16.19/8.90 ! [A: real] :
% 16.19/8.90 ( ( times_times_real @ A @ one_one_real )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1537_mult_Ocomm__neutral,axiom,
% 16.19/8.90 ! [A: rat] :
% 16.19/8.90 ( ( times_times_rat @ A @ one_one_rat )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1538_mult_Ocomm__neutral,axiom,
% 16.19/8.90 ! [A: nat] :
% 16.19/8.90 ( ( times_times_nat @ A @ one_one_nat )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1539_mult_Ocomm__neutral,axiom,
% 16.19/8.90 ! [A: int] :
% 16.19/8.90 ( ( times_times_int @ A @ one_one_int )
% 16.19/8.90 = A ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1540_times__divide__times__eq,axiom,
% 16.19/8.90 ! [X: complex,Y: complex,Z: complex,W: complex] :
% 16.19/8.90 ( ( times_times_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
% 16.19/8.90 = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ Y @ W ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1541_times__divide__times__eq,axiom,
% 16.19/8.90 ! [X: real,Y: real,Z: real,W: real] :
% 16.19/8.90 ( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
% 16.19/8.90 = ( divide_divide_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ W ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1542_times__divide__times__eq,axiom,
% 16.19/8.90 ! [X: rat,Y: rat,Z: rat,W: rat] :
% 16.19/8.90 ( ( times_times_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W ) )
% 16.19/8.90 = ( divide_divide_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ W ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1543_divide__divide__times__eq,axiom,
% 16.19/8.90 ! [X: complex,Y: complex,Z: complex,W: complex] :
% 16.19/8.90 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
% 16.19/8.90 = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ W ) @ ( times_times_complex @ Y @ Z ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1544_divide__divide__times__eq,axiom,
% 16.19/8.90 ! [X: real,Y: real,Z: real,W: real] :
% 16.19/8.90 ( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z @ W ) )
% 16.19/8.90 = ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y @ Z ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1545_divide__divide__times__eq,axiom,
% 16.19/8.90 ! [X: rat,Y: rat,Z: rat,W: rat] :
% 16.19/8.90 ( ( divide_divide_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z @ W ) )
% 16.19/8.90 = ( divide_divide_rat @ ( times_times_rat @ X @ W ) @ ( times_times_rat @ Y @ Z ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1546_divide__divide__eq__left_H,axiom,
% 16.19/8.90 ! [A: complex,B: complex,C: complex] :
% 16.19/8.90 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
% 16.19/8.90 = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1547_divide__divide__eq__left_H,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real] :
% 16.19/8.90 ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
% 16.19/8.90 = ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1548_divide__divide__eq__left_H,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat] :
% 16.19/8.90 ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
% 16.19/8.90 = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1549_group__cancel_Osub1,axiom,
% 16.19/8.90 ! [A2: real,K: real,A: real,B: real] :
% 16.19/8.90 ( ( A2
% 16.19/8.90 = ( plus_plus_real @ K @ A ) )
% 16.19/8.90 => ( ( minus_minus_real @ A2 @ B )
% 16.19/8.90 = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1550_group__cancel_Osub1,axiom,
% 16.19/8.90 ! [A2: rat,K: rat,A: rat,B: rat] :
% 16.19/8.90 ( ( A2
% 16.19/8.90 = ( plus_plus_rat @ K @ A ) )
% 16.19/8.90 => ( ( minus_minus_rat @ A2 @ B )
% 16.19/8.90 = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1551_group__cancel_Osub1,axiom,
% 16.19/8.90 ! [A2: int,K: int,A: int,B: int] :
% 16.19/8.90 ( ( A2
% 16.19/8.90 = ( plus_plus_int @ K @ A ) )
% 16.19/8.90 => ( ( minus_minus_int @ A2 @ B )
% 16.19/8.90 = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1552_diff__eq__eq,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real] :
% 16.19/8.90 ( ( ( minus_minus_real @ A @ B )
% 16.19/8.90 = C )
% 16.19/8.90 = ( A
% 16.19/8.90 = ( plus_plus_real @ C @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1553_diff__eq__eq,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat] :
% 16.19/8.90 ( ( ( minus_minus_rat @ A @ B )
% 16.19/8.90 = C )
% 16.19/8.90 = ( A
% 16.19/8.90 = ( plus_plus_rat @ C @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1554_diff__eq__eq,axiom,
% 16.19/8.90 ! [A: int,B: int,C: int] :
% 16.19/8.90 ( ( ( minus_minus_int @ A @ B )
% 16.19/8.90 = C )
% 16.19/8.90 = ( A
% 16.19/8.90 = ( plus_plus_int @ C @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1555_eq__diff__eq,axiom,
% 16.19/8.90 ! [A: real,C: real,B: real] :
% 16.19/8.90 ( ( A
% 16.19/8.90 = ( minus_minus_real @ C @ B ) )
% 16.19/8.90 = ( ( plus_plus_real @ A @ B )
% 16.19/8.90 = C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1556_eq__diff__eq,axiom,
% 16.19/8.90 ! [A: rat,C: rat,B: rat] :
% 16.19/8.90 ( ( A
% 16.19/8.90 = ( minus_minus_rat @ C @ B ) )
% 16.19/8.90 = ( ( plus_plus_rat @ A @ B )
% 16.19/8.90 = C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1557_eq__diff__eq,axiom,
% 16.19/8.90 ! [A: int,C: int,B: int] :
% 16.19/8.90 ( ( A
% 16.19/8.90 = ( minus_minus_int @ C @ B ) )
% 16.19/8.90 = ( ( plus_plus_int @ A @ B )
% 16.19/8.90 = C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1558_add__diff__eq,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real] :
% 16.19/8.90 ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
% 16.19/8.90 = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1559_add__diff__eq,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat] :
% 16.19/8.90 ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 16.19/8.90 = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1560_add__diff__eq,axiom,
% 16.19/8.90 ! [A: int,B: int,C: int] :
% 16.19/8.90 ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
% 16.19/8.90 = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1561_diff__diff__eq2,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real] :
% 16.19/8.90 ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
% 16.19/8.90 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1562_diff__diff__eq2,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat] :
% 16.19/8.90 ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 16.19/8.90 = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1563_diff__diff__eq2,axiom,
% 16.19/8.90 ! [A: int,B: int,C: int] :
% 16.19/8.90 ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
% 16.19/8.90 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1564_diff__add__eq,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real] :
% 16.19/8.90 ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
% 16.19/8.90 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1565_diff__add__eq,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat] :
% 16.19/8.90 ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 16.19/8.90 = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1566_diff__add__eq,axiom,
% 16.19/8.90 ! [A: int,B: int,C: int] :
% 16.19/8.90 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
% 16.19/8.90 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1567_diff__add__eq__diff__diff__swap,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real] :
% 16.19/8.90 ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
% 16.19/8.90 = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1568_diff__add__eq__diff__diff__swap,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat] :
% 16.19/8.90 ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 16.19/8.90 = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1569_diff__add__eq__diff__diff__swap,axiom,
% 16.19/8.90 ! [A: int,B: int,C: int] :
% 16.19/8.90 ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
% 16.19/8.90 = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1570_add__implies__diff,axiom,
% 16.19/8.90 ! [C: real,B: real,A: real] :
% 16.19/8.90 ( ( ( plus_plus_real @ C @ B )
% 16.19/8.90 = A )
% 16.19/8.90 => ( C
% 16.19/8.90 = ( minus_minus_real @ A @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1571_add__implies__diff,axiom,
% 16.19/8.90 ! [C: rat,B: rat,A: rat] :
% 16.19/8.90 ( ( ( plus_plus_rat @ C @ B )
% 16.19/8.90 = A )
% 16.19/8.90 => ( C
% 16.19/8.90 = ( minus_minus_rat @ A @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1572_add__implies__diff,axiom,
% 16.19/8.90 ! [C: nat,B: nat,A: nat] :
% 16.19/8.90 ( ( ( plus_plus_nat @ C @ B )
% 16.19/8.90 = A )
% 16.19/8.90 => ( C
% 16.19/8.90 = ( minus_minus_nat @ A @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1573_add__implies__diff,axiom,
% 16.19/8.90 ! [C: int,B: int,A: int] :
% 16.19/8.90 ( ( ( plus_plus_int @ C @ B )
% 16.19/8.90 = A )
% 16.19/8.90 => ( C
% 16.19/8.90 = ( minus_minus_int @ A @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1574_diff__diff__eq,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real] :
% 16.19/8.90 ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
% 16.19/8.90 = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1575_diff__diff__eq,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat] :
% 16.19/8.90 ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 16.19/8.90 = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1576_diff__diff__eq,axiom,
% 16.19/8.90 ! [A: nat,B: nat,C: nat] :
% 16.19/8.90 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
% 16.19/8.90 = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1577_diff__diff__eq,axiom,
% 16.19/8.90 ! [A: int,B: int,C: int] :
% 16.19/8.90 ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
% 16.19/8.90 = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1578_add__divide__distrib,axiom,
% 16.19/8.90 ! [A: complex,B: complex,C: complex] :
% 16.19/8.90 ( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B ) @ C )
% 16.19/8.90 = ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1579_add__divide__distrib,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real] :
% 16.19/8.90 ( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
% 16.19/8.90 = ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1580_add__divide__distrib,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat] :
% 16.19/8.90 ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 16.19/8.90 = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1581_diff__divide__distrib,axiom,
% 16.19/8.90 ! [A: complex,B: complex,C: complex] :
% 16.19/8.90 ( ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ B ) @ C )
% 16.19/8.90 = ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1582_diff__divide__distrib,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real] :
% 16.19/8.90 ( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
% 16.19/8.90 = ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1583_diff__divide__distrib,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat] :
% 16.19/8.90 ( ( divide_divide_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 16.19/8.90 = ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1584_max__add__distrib__left,axiom,
% 16.19/8.90 ! [X: real,Y: real,Z: real] :
% 16.19/8.90 ( ( plus_plus_real @ ( ord_max_real @ X @ Y ) @ Z )
% 16.19/8.90 = ( ord_max_real @ ( plus_plus_real @ X @ Z ) @ ( plus_plus_real @ Y @ Z ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1585_max__add__distrib__left,axiom,
% 16.19/8.90 ! [X: rat,Y: rat,Z: rat] :
% 16.19/8.90 ( ( plus_plus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
% 16.19/8.90 = ( ord_max_rat @ ( plus_plus_rat @ X @ Z ) @ ( plus_plus_rat @ Y @ Z ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1586_max__add__distrib__left,axiom,
% 16.19/8.90 ! [X: nat,Y: nat,Z: nat] :
% 16.19/8.90 ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z )
% 16.19/8.90 = ( ord_max_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y @ Z ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1587_max__add__distrib__left,axiom,
% 16.19/8.90 ! [X: int,Y: int,Z: int] :
% 16.19/8.90 ( ( plus_plus_int @ ( ord_max_int @ X @ Y ) @ Z )
% 16.19/8.90 = ( ord_max_int @ ( plus_plus_int @ X @ Z ) @ ( plus_plus_int @ Y @ Z ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1588_max__add__distrib__right,axiom,
% 16.19/8.90 ! [X: real,Y: real,Z: real] :
% 16.19/8.90 ( ( plus_plus_real @ X @ ( ord_max_real @ Y @ Z ) )
% 16.19/8.90 = ( ord_max_real @ ( plus_plus_real @ X @ Y ) @ ( plus_plus_real @ X @ Z ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1589_max__add__distrib__right,axiom,
% 16.19/8.90 ! [X: rat,Y: rat,Z: rat] :
% 16.19/8.90 ( ( plus_plus_rat @ X @ ( ord_max_rat @ Y @ Z ) )
% 16.19/8.90 = ( ord_max_rat @ ( plus_plus_rat @ X @ Y ) @ ( plus_plus_rat @ X @ Z ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1590_max__add__distrib__right,axiom,
% 16.19/8.90 ! [X: nat,Y: nat,Z: nat] :
% 16.19/8.90 ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z ) )
% 16.19/8.90 = ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1591_max__add__distrib__right,axiom,
% 16.19/8.90 ! [X: int,Y: int,Z: int] :
% 16.19/8.90 ( ( plus_plus_int @ X @ ( ord_max_int @ Y @ Z ) )
% 16.19/8.90 = ( ord_max_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1592_max__diff__distrib__left,axiom,
% 16.19/8.90 ! [X: real,Y: real,Z: real] :
% 16.19/8.90 ( ( minus_minus_real @ ( ord_max_real @ X @ Y ) @ Z )
% 16.19/8.90 = ( ord_max_real @ ( minus_minus_real @ X @ Z ) @ ( minus_minus_real @ Y @ Z ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1593_max__diff__distrib__left,axiom,
% 16.19/8.90 ! [X: rat,Y: rat,Z: rat] :
% 16.19/8.90 ( ( minus_minus_rat @ ( ord_max_rat @ X @ Y ) @ Z )
% 16.19/8.90 = ( ord_max_rat @ ( minus_minus_rat @ X @ Z ) @ ( minus_minus_rat @ Y @ Z ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1594_max__diff__distrib__left,axiom,
% 16.19/8.90 ! [X: int,Y: int,Z: int] :
% 16.19/8.90 ( ( minus_minus_int @ ( ord_max_int @ X @ Y ) @ Z )
% 16.19/8.90 = ( ord_max_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y @ Z ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1595_add__less__le__mono,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real,D: real] :
% 16.19/8.90 ( ( ord_less_real @ A @ B )
% 16.19/8.90 => ( ( ord_less_eq_real @ C @ D )
% 16.19/8.90 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1596_add__less__le__mono,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat,D: rat] :
% 16.19/8.90 ( ( ord_less_rat @ A @ B )
% 16.19/8.90 => ( ( ord_less_eq_rat @ C @ D )
% 16.19/8.90 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1597_add__less__le__mono,axiom,
% 16.19/8.90 ! [A: nat,B: nat,C: nat,D: nat] :
% 16.19/8.90 ( ( ord_less_nat @ A @ B )
% 16.19/8.90 => ( ( ord_less_eq_nat @ C @ D )
% 16.19/8.90 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1598_add__less__le__mono,axiom,
% 16.19/8.90 ! [A: int,B: int,C: int,D: int] :
% 16.19/8.90 ( ( ord_less_int @ A @ B )
% 16.19/8.90 => ( ( ord_less_eq_int @ C @ D )
% 16.19/8.90 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1599_add__le__less__mono,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real,D: real] :
% 16.19/8.90 ( ( ord_less_eq_real @ A @ B )
% 16.19/8.90 => ( ( ord_less_real @ C @ D )
% 16.19/8.90 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1600_add__le__less__mono,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat,D: rat] :
% 16.19/8.90 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.90 => ( ( ord_less_rat @ C @ D )
% 16.19/8.90 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1601_add__le__less__mono,axiom,
% 16.19/8.90 ! [A: nat,B: nat,C: nat,D: nat] :
% 16.19/8.90 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.90 => ( ( ord_less_nat @ C @ D )
% 16.19/8.90 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1602_add__le__less__mono,axiom,
% 16.19/8.90 ! [A: int,B: int,C: int,D: int] :
% 16.19/8.90 ( ( ord_less_eq_int @ A @ B )
% 16.19/8.90 => ( ( ord_less_int @ C @ D )
% 16.19/8.90 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1603_add__mono__thms__linordered__field_I3_J,axiom,
% 16.19/8.90 ! [I2: real,J: real,K: real,L2: real] :
% 16.19/8.90 ( ( ( ord_less_real @ I2 @ J )
% 16.19/8.90 & ( ord_less_eq_real @ K @ L2 ) )
% 16.19/8.90 => ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1604_add__mono__thms__linordered__field_I3_J,axiom,
% 16.19/8.90 ! [I2: rat,J: rat,K: rat,L2: rat] :
% 16.19/8.90 ( ( ( ord_less_rat @ I2 @ J )
% 16.19/8.90 & ( ord_less_eq_rat @ K @ L2 ) )
% 16.19/8.90 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1605_add__mono__thms__linordered__field_I3_J,axiom,
% 16.19/8.90 ! [I2: nat,J: nat,K: nat,L2: nat] :
% 16.19/8.90 ( ( ( ord_less_nat @ I2 @ J )
% 16.19/8.90 & ( ord_less_eq_nat @ K @ L2 ) )
% 16.19/8.90 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1606_add__mono__thms__linordered__field_I3_J,axiom,
% 16.19/8.90 ! [I2: int,J: int,K: int,L2: int] :
% 16.19/8.90 ( ( ( ord_less_int @ I2 @ J )
% 16.19/8.90 & ( ord_less_eq_int @ K @ L2 ) )
% 16.19/8.90 => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1607_add__mono__thms__linordered__field_I4_J,axiom,
% 16.19/8.90 ! [I2: real,J: real,K: real,L2: real] :
% 16.19/8.90 ( ( ( ord_less_eq_real @ I2 @ J )
% 16.19/8.90 & ( ord_less_real @ K @ L2 ) )
% 16.19/8.90 => ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1608_add__mono__thms__linordered__field_I4_J,axiom,
% 16.19/8.90 ! [I2: rat,J: rat,K: rat,L2: rat] :
% 16.19/8.90 ( ( ( ord_less_eq_rat @ I2 @ J )
% 16.19/8.90 & ( ord_less_rat @ K @ L2 ) )
% 16.19/8.90 => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1609_add__mono__thms__linordered__field_I4_J,axiom,
% 16.19/8.90 ! [I2: nat,J: nat,K: nat,L2: nat] :
% 16.19/8.90 ( ( ( ord_less_eq_nat @ I2 @ J )
% 16.19/8.90 & ( ord_less_nat @ K @ L2 ) )
% 16.19/8.90 => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1610_add__mono__thms__linordered__field_I4_J,axiom,
% 16.19/8.90 ! [I2: int,J: int,K: int,L2: int] :
% 16.19/8.90 ( ( ( ord_less_eq_int @ I2 @ J )
% 16.19/8.90 & ( ord_less_int @ K @ L2 ) )
% 16.19/8.90 => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1611_diff__le__eq,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real] :
% 16.19/8.90 ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
% 16.19/8.90 = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1612_diff__le__eq,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat] :
% 16.19/8.90 ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 16.19/8.90 = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1613_diff__le__eq,axiom,
% 16.19/8.90 ! [A: int,B: int,C: int] :
% 16.19/8.90 ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
% 16.19/8.90 = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1614_le__diff__eq,axiom,
% 16.19/8.90 ! [A: real,C: real,B: real] :
% 16.19/8.90 ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
% 16.19/8.90 = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1615_le__diff__eq,axiom,
% 16.19/8.90 ! [A: rat,C: rat,B: rat] :
% 16.19/8.90 ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 16.19/8.90 = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1616_le__diff__eq,axiom,
% 16.19/8.90 ! [A: int,C: int,B: int] :
% 16.19/8.90 ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
% 16.19/8.90 = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1617_diff__add,axiom,
% 16.19/8.90 ! [A: nat,B: nat] :
% 16.19/8.90 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.90 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
% 16.19/8.90 = B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1618_le__add__diff,axiom,
% 16.19/8.90 ! [A: nat,B: nat,C: nat] :
% 16.19/8.90 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.90 => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1619_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 16.19/8.90 ! [A: nat,B: nat,C: nat] :
% 16.19/8.90 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.90 => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 16.19/8.90 = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1620_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 16.19/8.90 ! [A: nat,B: nat,C: nat] :
% 16.19/8.90 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.90 => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 16.19/8.90 = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1621_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 16.19/8.90 ! [A: nat,B: nat,C: nat] :
% 16.19/8.90 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.90 => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
% 16.19/8.90 = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1622_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 16.19/8.90 ! [A: nat,B: nat,C: nat] :
% 16.19/8.90 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.90 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
% 16.19/8.90 = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1623_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 16.19/8.90 ! [A: nat,B: nat,C: nat] :
% 16.19/8.90 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.90 => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
% 16.19/8.90 = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1624_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 16.19/8.90 ! [A: nat,B: nat,C: nat] :
% 16.19/8.90 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.90 => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 16.19/8.90 = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1625_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 16.19/8.90 ! [A: nat,B: nat] :
% 16.19/8.90 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.90 => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
% 16.19/8.90 = B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1626_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 16.19/8.90 ! [A: nat,B: nat,C: nat] :
% 16.19/8.90 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.90 => ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.90 => ( ( ( minus_minus_nat @ B @ A )
% 16.19/8.90 = C )
% 16.19/8.90 = ( B
% 16.19/8.90 = ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1627_less__diff__eq,axiom,
% 16.19/8.90 ! [A: real,C: real,B: real] :
% 16.19/8.90 ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
% 16.19/8.90 = ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1628_less__diff__eq,axiom,
% 16.19/8.90 ! [A: rat,C: rat,B: rat] :
% 16.19/8.90 ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 16.19/8.90 = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1629_less__diff__eq,axiom,
% 16.19/8.90 ! [A: int,C: int,B: int] :
% 16.19/8.90 ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
% 16.19/8.90 = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1630_diff__less__eq,axiom,
% 16.19/8.90 ! [A: real,B: real,C: real] :
% 16.19/8.90 ( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
% 16.19/8.90 = ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1631_diff__less__eq,axiom,
% 16.19/8.90 ! [A: rat,B: rat,C: rat] :
% 16.19/8.90 ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 16.19/8.90 = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1632_diff__less__eq,axiom,
% 16.19/8.90 ! [A: int,B: int,C: int] :
% 16.19/8.90 ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
% 16.19/8.90 = ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1633_gt__half__sum,axiom,
% 16.19/8.90 ! [A: real,B: real] :
% 16.19/8.90 ( ( ord_less_real @ A @ B )
% 16.19/8.90 => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1634_gt__half__sum,axiom,
% 16.19/8.90 ! [A: rat,B: rat] :
% 16.19/8.90 ( ( ord_less_rat @ A @ B )
% 16.19/8.90 => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1635_less__half__sum,axiom,
% 16.19/8.90 ! [A: real,B: real] :
% 16.19/8.90 ( ( ord_less_real @ A @ B )
% 16.19/8.90 => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1636_less__half__sum,axiom,
% 16.19/8.90 ! [A: rat,B: rat] :
% 16.19/8.90 ( ( ord_less_rat @ A @ B )
% 16.19/8.90 => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1637_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
% 16.19/8.90 ! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
% 16.19/8.90 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList2 @ Vc ) @ X )
% 16.19/8.90 = ( ( X = Mi )
% 16.19/8.90 | ( X = Ma )
% 16.19/8.90 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 16.19/8.90 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/8.90 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1638_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
% 16.19/8.90 ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S3: vEBT_VEBT,X: nat] :
% 16.19/8.90 ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S3 ) @ X )
% 16.19/8.90 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 16.19/8.90 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/8.90 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1639_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
% 16.19/8.90 ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X: nat] :
% 16.19/8.90 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) @ X )
% 16.19/8.90 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 16.19/8.90 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/8.90 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1640_divmod__step__eq,axiom,
% 16.19/8.90 ! [L2: num,R: nat,Q2: nat] :
% 16.19/8.90 ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R )
% 16.19/8.90 => ( ( unique5026877609467782581ep_nat @ L2 @ ( product_Pair_nat_nat @ Q2 @ R ) )
% 16.19/8.90 = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ one_one_nat ) @ ( minus_minus_nat @ R @ ( numeral_numeral_nat @ L2 ) ) ) ) )
% 16.19/8.90 & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R )
% 16.19/8.90 => ( ( unique5026877609467782581ep_nat @ L2 @ ( product_Pair_nat_nat @ Q2 @ R ) )
% 16.19/8.90 = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ R ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1641_divmod__step__eq,axiom,
% 16.19/8.90 ! [L2: num,R: int,Q2: int] :
% 16.19/8.90 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R )
% 16.19/8.90 => ( ( unique5024387138958732305ep_int @ L2 @ ( product_Pair_int_int @ Q2 @ R ) )
% 16.19/8.90 = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ one_one_int ) @ ( minus_minus_int @ R @ ( numeral_numeral_int @ L2 ) ) ) ) )
% 16.19/8.90 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R )
% 16.19/8.90 => ( ( unique5024387138958732305ep_int @ L2 @ ( product_Pair_int_int @ Q2 @ R ) )
% 16.19/8.90 = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ R ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1642_divmod__step__eq,axiom,
% 16.19/8.90 ! [L2: num,R: code_integer,Q2: code_integer] :
% 16.19/8.90 ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R )
% 16.19/8.90 => ( ( unique4921790084139445826nteger @ L2 @ ( produc1086072967326762835nteger @ Q2 @ R ) )
% 16.19/8.90 = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R @ ( numera6620942414471956472nteger @ L2 ) ) ) ) )
% 16.19/8.90 & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R )
% 16.19/8.90 => ( ( unique4921790084139445826nteger @ L2 @ ( produc1086072967326762835nteger @ Q2 @ R ) )
% 16.19/8.90 = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ R ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1643_buildup__nothing__in__leaf,axiom,
% 16.19/8.90 ! [N2: nat,X: nat] :
% 16.19/8.90 ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N2 ) @ X ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1644_vebt__succ_Oelims,axiom,
% 16.19/8.90 ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
% 16.19/8.90 ( ( ( vEBT_vebt_succ @ X @ Xa2 )
% 16.19/8.90 = Y )
% 16.19/8.90 => ( ! [Uu2: $o,B2: $o] :
% 16.19/8.90 ( ( X
% 16.19/8.90 = ( vEBT_Leaf @ Uu2 @ B2 ) )
% 16.19/8.90 => ( ( Xa2 = zero_zero_nat )
% 16.19/8.90 => ~ ( ( B2
% 16.19/8.90 => ( Y
% 16.19/8.90 = ( some_nat @ one_one_nat ) ) )
% 16.19/8.90 & ( ~ B2
% 16.19/8.90 => ( Y = none_nat ) ) ) ) )
% 16.19/8.90 => ( ( ? [Uv2: $o,Uw2: $o] :
% 16.19/8.90 ( X
% 16.19/8.90 = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
% 16.19/8.90 => ( ? [N4: nat] :
% 16.19/8.90 ( Xa2
% 16.19/8.90 = ( suc @ N4 ) )
% 16.19/8.90 => ( Y != none_nat ) ) )
% 16.19/8.90 => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 16.19/8.90 ( X
% 16.19/8.90 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
% 16.19/8.90 => ( Y != none_nat ) )
% 16.19/8.90 => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 16.19/8.90 ( X
% 16.19/8.90 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
% 16.19/8.90 => ( Y != none_nat ) )
% 16.19/8.90 => ( ( ? [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
% 16.19/8.90 ( X
% 16.19/8.90 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) )
% 16.19/8.90 => ( Y != none_nat ) )
% 16.19/8.90 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 16.19/8.90 ( ( X
% 16.19/8.90 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 16.19/8.90 => ~ ( ( ( ord_less_nat @ Xa2 @ Mi2 )
% 16.19/8.90 => ( Y
% 16.19/8.90 = ( some_nat @ Mi2 ) ) )
% 16.19/8.90 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 16.19/8.90 => ( Y
% 16.19/8.90 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.90 @ ( if_option_nat
% 16.19/8.90 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/8.90 != none_nat )
% 16.19/8.90 & ( 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 @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 16.19/8.90 @ ( 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 @ TreeList3 @ ( 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 ) ) ) ) ) )
% 16.19/8.90 @ ( if_option_nat
% 16.19/8.90 @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/8.90 = none_nat )
% 16.19/8.90 @ none_nat
% 16.19/8.90 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 16.19/8.90 @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1645_vebt__pred_Oelims,axiom,
% 16.19/8.90 ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
% 16.19/8.90 ( ( ( vEBT_vebt_pred @ X @ Xa2 )
% 16.19/8.90 = Y )
% 16.19/8.90 => ( ( ? [Uu2: $o,Uv2: $o] :
% 16.19/8.90 ( X
% 16.19/8.90 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 16.19/8.90 => ( ( Xa2 = zero_zero_nat )
% 16.19/8.90 => ( Y != none_nat ) ) )
% 16.19/8.90 => ( ! [A3: $o] :
% 16.19/8.90 ( ? [Uw2: $o] :
% 16.19/8.90 ( X
% 16.19/8.90 = ( vEBT_Leaf @ A3 @ Uw2 ) )
% 16.19/8.90 => ( ( Xa2
% 16.19/8.90 = ( suc @ zero_zero_nat ) )
% 16.19/8.90 => ~ ( ( A3
% 16.19/8.90 => ( Y
% 16.19/8.90 = ( some_nat @ zero_zero_nat ) ) )
% 16.19/8.90 & ( ~ A3
% 16.19/8.90 => ( Y = none_nat ) ) ) ) )
% 16.19/8.90 => ( ! [A3: $o,B2: $o] :
% 16.19/8.90 ( ( X
% 16.19/8.90 = ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/8.90 => ( ? [Va2: nat] :
% 16.19/8.90 ( Xa2
% 16.19/8.90 = ( suc @ ( suc @ Va2 ) ) )
% 16.19/8.90 => ~ ( ( B2
% 16.19/8.90 => ( Y
% 16.19/8.90 = ( some_nat @ one_one_nat ) ) )
% 16.19/8.90 & ( ~ B2
% 16.19/8.90 => ( ( A3
% 16.19/8.90 => ( Y
% 16.19/8.90 = ( some_nat @ zero_zero_nat ) ) )
% 16.19/8.90 & ( ~ A3
% 16.19/8.90 => ( Y = none_nat ) ) ) ) ) ) )
% 16.19/8.90 => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
% 16.19/8.90 ( X
% 16.19/8.90 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
% 16.19/8.90 => ( Y != none_nat ) )
% 16.19/8.90 => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] :
% 16.19/8.90 ( X
% 16.19/8.90 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) )
% 16.19/8.90 => ( Y != none_nat ) )
% 16.19/8.90 => ( ( ? [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
% 16.19/8.90 ( X
% 16.19/8.90 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) )
% 16.19/8.90 => ( Y != none_nat ) )
% 16.19/8.90 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 16.19/8.90 ( ( X
% 16.19/8.90 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 16.19/8.90 => ~ ( ( ( ord_less_nat @ Ma2 @ Xa2 )
% 16.19/8.90 => ( Y
% 16.19/8.90 = ( some_nat @ Ma2 ) ) )
% 16.19/8.90 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 16.19/8.90 => ( Y
% 16.19/8.90 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.90 @ ( if_option_nat
% 16.19/8.90 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/8.90 != none_nat )
% 16.19/8.90 & ( 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 @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 16.19/8.90 @ ( 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 @ TreeList3 @ ( 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 ) ) ) ) ) )
% 16.19/8.90 @ ( if_option_nat
% 16.19/8.90 @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/8.90 = none_nat )
% 16.19/8.90 @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
% 16.19/8.90 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 16.19/8.90 @ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1646_discrete,axiom,
% 16.19/8.90 ( ord_less_nat
% 16.19/8.90 = ( ^ [A5: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A5 @ one_one_nat ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1647_discrete,axiom,
% 16.19/8.90 ( ord_less_int
% 16.19/8.90 = ( ^ [A5: int] : ( ord_less_eq_int @ ( plus_plus_int @ A5 @ one_one_int ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1648_vebt__delete_Oelims,axiom,
% 16.19/8.90 ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
% 16.19/8.90 ( ( ( vEBT_vebt_delete @ X @ Xa2 )
% 16.19/8.90 = Y )
% 16.19/8.90 => ( ! [A3: $o,B2: $o] :
% 16.19/8.90 ( ( X
% 16.19/8.90 = ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/8.90 => ( ( Xa2 = zero_zero_nat )
% 16.19/8.90 => ( Y
% 16.19/8.90 != ( vEBT_Leaf @ $false @ B2 ) ) ) )
% 16.19/8.90 => ( ! [A3: $o] :
% 16.19/8.90 ( ? [B2: $o] :
% 16.19/8.90 ( X
% 16.19/8.90 = ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/8.90 => ( ( Xa2
% 16.19/8.90 = ( suc @ zero_zero_nat ) )
% 16.19/8.90 => ( Y
% 16.19/8.90 != ( vEBT_Leaf @ A3 @ $false ) ) ) )
% 16.19/8.90 => ( ! [A3: $o,B2: $o] :
% 16.19/8.90 ( ( X
% 16.19/8.90 = ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/8.90 => ( ? [N4: nat] :
% 16.19/8.90 ( Xa2
% 16.19/8.90 = ( suc @ ( suc @ N4 ) ) )
% 16.19/8.90 => ( Y
% 16.19/8.90 != ( vEBT_Leaf @ A3 @ B2 ) ) ) )
% 16.19/8.90 => ( ! [Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 16.19/8.90 ( ( X
% 16.19/8.90 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 16.19/8.90 => ( Y
% 16.19/8.90 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) ) )
% 16.19/8.90 => ( ! [Mi2: nat,Ma2: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT] :
% 16.19/8.90 ( ( X
% 16.19/8.90 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) )
% 16.19/8.90 => ( Y
% 16.19/8.90 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) ) )
% 16.19/8.90 => ( ! [Mi2: nat,Ma2: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT] :
% 16.19/8.90 ( ( X
% 16.19/8.90 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) )
% 16.19/8.90 => ( Y
% 16.19/8.90 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) ) )
% 16.19/8.90 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 16.19/8.90 ( ( X
% 16.19/8.90 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 16.19/8.90 => ~ ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
% 16.19/8.90 | ( ord_less_nat @ Ma2 @ Xa2 ) )
% 16.19/8.90 => ( Y
% 16.19/8.90 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) )
% 16.19/8.90 & ( ~ ( ( ord_less_nat @ Xa2 @ Mi2 )
% 16.19/8.90 | ( ord_less_nat @ Ma2 @ Xa2 ) )
% 16.19/8.90 => ( ( ( ( Xa2 = Mi2 )
% 16.19/8.90 & ( Xa2 = Ma2 ) )
% 16.19/8.90 => ( Y
% 16.19/8.90 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) )
% 16.19/8.90 & ( ~ ( ( Xa2 = Mi2 )
% 16.19/8.90 & ( Xa2 = Ma2 ) )
% 16.19/8.90 => ( Y
% 16.19/8.90 = ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.90 @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/8.90 @ ( vEBT_Node
% 16.19/8.90 @ ( some_P7363390416028606310at_nat
% 16.19/8.90 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
% 16.19/8.90 @ ( if_nat
% 16.19/8.90 @ ( ( ( Xa2 = Mi2 )
% 16.19/8.90 => ( ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
% 16.19/8.90 = Ma2 ) )
% 16.19/8.90 & ( ( Xa2 != Mi2 )
% 16.19/8.90 => ( Xa2 = Ma2 ) ) )
% 16.19/8.90 @ ( if_nat
% 16.19/8.90 @ ( ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/8.90 = none_nat )
% 16.19/8.90 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
% 16.19/8.90 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
% 16.19/8.90 @ Ma2 ) ) )
% 16.19/8.90 @ ( suc @ ( suc @ Va2 ) )
% 16.19/8.90 @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/8.90 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/8.90 @ ( vEBT_Node
% 16.19/8.90 @ ( some_P7363390416028606310at_nat
% 16.19/8.90 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
% 16.19/8.90 @ ( if_nat
% 16.19/8.90 @ ( ( ( Xa2 = Mi2 )
% 16.19/8.90 => ( ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
% 16.19/8.90 = Ma2 ) )
% 16.19/8.90 & ( ( Xa2 != Mi2 )
% 16.19/8.90 => ( Xa2 = Ma2 ) ) )
% 16.19/8.90 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 16.19/8.90 @ Ma2 ) ) )
% 16.19/8.90 @ ( suc @ ( suc @ Va2 ) )
% 16.19/8.90 @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/8.90 @ Summary2 ) )
% 16.19/8.90 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1649_low__def,axiom,
% 16.19/8.90 ( vEBT_VEBT_low
% 16.19/8.90 = ( ^ [X2: nat,N: nat] : ( modulo_modulo_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1650_valid__0__not,axiom,
% 16.19/8.90 ! [T: vEBT_VEBT] :
% 16.19/8.90 ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1651_valid__tree__deg__neq__0,axiom,
% 16.19/8.90 ! [T: vEBT_VEBT] :
% 16.19/8.90 ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1652_buildup__nothing__in__min__max,axiom,
% 16.19/8.90 ! [N2: nat,X: nat] :
% 16.19/8.90 ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N2 ) @ X ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1653_deg__not__0,axiom,
% 16.19/8.90 ! [T: vEBT_VEBT,N2: nat] :
% 16.19/8.90 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.19/8.90 => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1654_Leaf__0__not,axiom,
% 16.19/8.90 ! [A: $o,B: $o] :
% 16.19/8.90 ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1655_deg1Leaf,axiom,
% 16.19/8.90 ! [T: vEBT_VEBT] :
% 16.19/8.90 ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 16.19/8.90 = ( ? [A5: $o,B5: $o] :
% 16.19/8.90 ( T
% 16.19/8.90 = ( vEBT_Leaf @ A5 @ B5 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1656_deg__1__Leaf,axiom,
% 16.19/8.90 ! [T: vEBT_VEBT] :
% 16.19/8.90 ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 16.19/8.90 => ? [A3: $o,B2: $o] :
% 16.19/8.90 ( T
% 16.19/8.90 = ( vEBT_Leaf @ A3 @ B2 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1657_deg__1__Leafy,axiom,
% 16.19/8.90 ! [T: vEBT_VEBT,N2: nat] :
% 16.19/8.90 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.19/8.90 => ( ( N2 = one_one_nat )
% 16.19/8.90 => ? [A3: $o,B2: $o] :
% 16.19/8.90 ( T
% 16.19/8.90 = ( vEBT_Leaf @ A3 @ B2 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1658_both__member__options__def,axiom,
% 16.19/8.90 ( vEBT_V8194947554948674370ptions
% 16.19/8.90 = ( ^ [T2: vEBT_VEBT,X2: nat] :
% 16.19/8.90 ( ( vEBT_V5719532721284313246member @ T2 @ X2 )
% 16.19/8.90 | ( vEBT_VEBT_membermima @ T2 @ X2 ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1659_buildup__gives__valid,axiom,
% 16.19/8.90 ! [N2: nat] :
% 16.19/8.90 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.90 => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N2 ) @ N2 ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1660_zdiv__numeral__Bit0,axiom,
% 16.19/8.90 ! [V: num,W: num] :
% 16.19/8.90 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 16.19/8.90 = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1661_mod__mod__trivial,axiom,
% 16.19/8.90 ! [A: nat,B: nat] :
% 16.19/8.90 ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 16.19/8.90 = ( modulo_modulo_nat @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1662_mod__mod__trivial,axiom,
% 16.19/8.90 ! [A: int,B: int] :
% 16.19/8.90 ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 16.19/8.90 = ( modulo_modulo_int @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1663_mod__mod__trivial,axiom,
% 16.19/8.90 ! [A: code_integer,B: code_integer] :
% 16.19/8.90 ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 16.19/8.90 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1664_member__valid__both__member__options,axiom,
% 16.19/8.90 ! [Tree: vEBT_VEBT,N2: nat,X: nat] :
% 16.19/8.90 ( ( vEBT_invar_vebt @ Tree @ N2 )
% 16.19/8.90 => ( ( vEBT_vebt_member @ Tree @ X )
% 16.19/8.90 => ( ( vEBT_V5719532721284313246member @ Tree @ X )
% 16.19/8.90 | ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1665_VEBT_Oinject_I2_J,axiom,
% 16.19/8.90 ! [X21: $o,X222: $o,Y21: $o,Y222: $o] :
% 16.19/8.90 ( ( ( vEBT_Leaf @ X21 @ X222 )
% 16.19/8.90 = ( vEBT_Leaf @ Y21 @ Y222 ) )
% 16.19/8.90 = ( ( X21 = Y21 )
% 16.19/8.90 & ( X222 = Y222 ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1666_le__zero__eq,axiom,
% 16.19/8.90 ! [N2: nat] :
% 16.19/8.90 ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
% 16.19/8.90 = ( N2 = zero_zero_nat ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1667_not__gr__zero,axiom,
% 16.19/8.90 ! [N2: nat] :
% 16.19/8.90 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
% 16.19/8.90 = ( N2 = zero_zero_nat ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1668_mult__cancel__right,axiom,
% 16.19/8.90 ! [A: complex,C: complex,B: complex] :
% 16.19/8.90 ( ( ( times_times_complex @ A @ C )
% 16.19/8.90 = ( times_times_complex @ B @ C ) )
% 16.19/8.90 = ( ( C = zero_zero_complex )
% 16.19/8.90 | ( A = B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1669_mult__cancel__right,axiom,
% 16.19/8.90 ! [A: real,C: real,B: real] :
% 16.19/8.90 ( ( ( times_times_real @ A @ C )
% 16.19/8.90 = ( times_times_real @ B @ C ) )
% 16.19/8.90 = ( ( C = zero_zero_real )
% 16.19/8.90 | ( A = B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1670_mult__cancel__right,axiom,
% 16.19/8.90 ! [A: rat,C: rat,B: rat] :
% 16.19/8.90 ( ( ( times_times_rat @ A @ C )
% 16.19/8.90 = ( times_times_rat @ B @ C ) )
% 16.19/8.90 = ( ( C = zero_zero_rat )
% 16.19/8.90 | ( A = B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1671_mult__cancel__right,axiom,
% 16.19/8.90 ! [A: nat,C: nat,B: nat] :
% 16.19/8.90 ( ( ( times_times_nat @ A @ C )
% 16.19/8.90 = ( times_times_nat @ B @ C ) )
% 16.19/8.90 = ( ( C = zero_zero_nat )
% 16.19/8.90 | ( A = B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1672_mult__cancel__right,axiom,
% 16.19/8.90 ! [A: int,C: int,B: int] :
% 16.19/8.90 ( ( ( times_times_int @ A @ C )
% 16.19/8.90 = ( times_times_int @ B @ C ) )
% 16.19/8.90 = ( ( C = zero_zero_int )
% 16.19/8.90 | ( A = B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1673_mult__cancel__left,axiom,
% 16.19/8.90 ! [C: complex,A: complex,B: complex] :
% 16.19/8.90 ( ( ( times_times_complex @ C @ A )
% 16.19/8.90 = ( times_times_complex @ C @ B ) )
% 16.19/8.90 = ( ( C = zero_zero_complex )
% 16.19/8.90 | ( A = B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1674_mult__cancel__left,axiom,
% 16.19/8.90 ! [C: real,A: real,B: real] :
% 16.19/8.90 ( ( ( times_times_real @ C @ A )
% 16.19/8.90 = ( times_times_real @ C @ B ) )
% 16.19/8.90 = ( ( C = zero_zero_real )
% 16.19/8.90 | ( A = B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1675_mult__cancel__left,axiom,
% 16.19/8.90 ! [C: rat,A: rat,B: rat] :
% 16.19/8.90 ( ( ( times_times_rat @ C @ A )
% 16.19/8.90 = ( times_times_rat @ C @ B ) )
% 16.19/8.90 = ( ( C = zero_zero_rat )
% 16.19/8.90 | ( A = B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1676_mult__cancel__left,axiom,
% 16.19/8.90 ! [C: nat,A: nat,B: nat] :
% 16.19/8.90 ( ( ( times_times_nat @ C @ A )
% 16.19/8.90 = ( times_times_nat @ C @ B ) )
% 16.19/8.90 = ( ( C = zero_zero_nat )
% 16.19/8.90 | ( A = B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1677_mult__cancel__left,axiom,
% 16.19/8.90 ! [C: int,A: int,B: int] :
% 16.19/8.90 ( ( ( times_times_int @ C @ A )
% 16.19/8.90 = ( times_times_int @ C @ B ) )
% 16.19/8.90 = ( ( C = zero_zero_int )
% 16.19/8.90 | ( A = B ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1678_mult__eq__0__iff,axiom,
% 16.19/8.90 ! [A: complex,B: complex] :
% 16.19/8.90 ( ( ( times_times_complex @ A @ B )
% 16.19/8.90 = zero_zero_complex )
% 16.19/8.90 = ( ( A = zero_zero_complex )
% 16.19/8.90 | ( B = zero_zero_complex ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1679_mult__eq__0__iff,axiom,
% 16.19/8.90 ! [A: real,B: real] :
% 16.19/8.90 ( ( ( times_times_real @ A @ B )
% 16.19/8.90 = zero_zero_real )
% 16.19/8.90 = ( ( A = zero_zero_real )
% 16.19/8.90 | ( B = zero_zero_real ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1680_mult__eq__0__iff,axiom,
% 16.19/8.90 ! [A: rat,B: rat] :
% 16.19/8.90 ( ( ( times_times_rat @ A @ B )
% 16.19/8.90 = zero_zero_rat )
% 16.19/8.90 = ( ( A = zero_zero_rat )
% 16.19/8.90 | ( B = zero_zero_rat ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1681_mult__eq__0__iff,axiom,
% 16.19/8.90 ! [A: nat,B: nat] :
% 16.19/8.90 ( ( ( times_times_nat @ A @ B )
% 16.19/8.90 = zero_zero_nat )
% 16.19/8.90 = ( ( A = zero_zero_nat )
% 16.19/8.90 | ( B = zero_zero_nat ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1682_mult__eq__0__iff,axiom,
% 16.19/8.90 ! [A: int,B: int] :
% 16.19/8.90 ( ( ( times_times_int @ A @ B )
% 16.19/8.90 = zero_zero_int )
% 16.19/8.90 = ( ( A = zero_zero_int )
% 16.19/8.90 | ( B = zero_zero_int ) ) ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1683_mult__zero__right,axiom,
% 16.19/8.90 ! [A: complex] :
% 16.19/8.90 ( ( times_times_complex @ A @ zero_zero_complex )
% 16.19/8.90 = zero_zero_complex ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1684_mult__zero__right,axiom,
% 16.19/8.90 ! [A: real] :
% 16.19/8.90 ( ( times_times_real @ A @ zero_zero_real )
% 16.19/8.90 = zero_zero_real ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1685_mult__zero__right,axiom,
% 16.19/8.90 ! [A: rat] :
% 16.19/8.90 ( ( times_times_rat @ A @ zero_zero_rat )
% 16.19/8.90 = zero_zero_rat ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1686_mult__zero__right,axiom,
% 16.19/8.90 ! [A: nat] :
% 16.19/8.90 ( ( times_times_nat @ A @ zero_zero_nat )
% 16.19/8.90 = zero_zero_nat ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1687_mult__zero__right,axiom,
% 16.19/8.90 ! [A: int] :
% 16.19/8.90 ( ( times_times_int @ A @ zero_zero_int )
% 16.19/8.90 = zero_zero_int ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1688_mult__zero__left,axiom,
% 16.19/8.90 ! [A: complex] :
% 16.19/8.90 ( ( times_times_complex @ zero_zero_complex @ A )
% 16.19/8.90 = zero_zero_complex ) ).
% 16.19/8.90
% 16.19/8.90 thf(fact_1689_mult__zero__left,axiom,
% 16.19/8.90 ! [A: real] :
% 16.19/8.90 ( ( times_times_real @ zero_zero_real @ A )
% 16.19/8.90 = zero_zero_real ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1690_mult__zero__left,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( times_times_rat @ zero_zero_rat @ A )
% 16.19/8.91 = zero_zero_rat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1691_mult__zero__left,axiom,
% 16.19/8.91 ! [A: nat] :
% 16.19/8.91 ( ( times_times_nat @ zero_zero_nat @ A )
% 16.19/8.91 = zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1692_mult__zero__left,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( times_times_int @ zero_zero_int @ A )
% 16.19/8.91 = zero_zero_int ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1693_add__0,axiom,
% 16.19/8.91 ! [A: complex] :
% 16.19/8.91 ( ( plus_plus_complex @ zero_zero_complex @ A )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1694_add__0,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( plus_plus_real @ zero_zero_real @ A )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1695_add__0,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1696_add__0,axiom,
% 16.19/8.91 ! [A: nat] :
% 16.19/8.91 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1697_add__0,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( plus_plus_int @ zero_zero_int @ A )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1698_zero__eq__add__iff__both__eq__0,axiom,
% 16.19/8.91 ! [X: nat,Y: nat] :
% 16.19/8.91 ( ( zero_zero_nat
% 16.19/8.91 = ( plus_plus_nat @ X @ Y ) )
% 16.19/8.91 = ( ( X = zero_zero_nat )
% 16.19/8.91 & ( Y = zero_zero_nat ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1699_add__eq__0__iff__both__eq__0,axiom,
% 16.19/8.91 ! [X: nat,Y: nat] :
% 16.19/8.91 ( ( ( plus_plus_nat @ X @ Y )
% 16.19/8.91 = zero_zero_nat )
% 16.19/8.91 = ( ( X = zero_zero_nat )
% 16.19/8.91 & ( Y = zero_zero_nat ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1700_add__cancel__right__right,axiom,
% 16.19/8.91 ! [A: complex,B: complex] :
% 16.19/8.91 ( ( A
% 16.19/8.91 = ( plus_plus_complex @ A @ B ) )
% 16.19/8.91 = ( B = zero_zero_complex ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1701_add__cancel__right__right,axiom,
% 16.19/8.91 ! [A: real,B: real] :
% 16.19/8.91 ( ( A
% 16.19/8.91 = ( plus_plus_real @ A @ B ) )
% 16.19/8.91 = ( B = zero_zero_real ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1702_add__cancel__right__right,axiom,
% 16.19/8.91 ! [A: rat,B: rat] :
% 16.19/8.91 ( ( A
% 16.19/8.91 = ( plus_plus_rat @ A @ B ) )
% 16.19/8.91 = ( B = zero_zero_rat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1703_add__cancel__right__right,axiom,
% 16.19/8.91 ! [A: nat,B: nat] :
% 16.19/8.91 ( ( A
% 16.19/8.91 = ( plus_plus_nat @ A @ B ) )
% 16.19/8.91 = ( B = zero_zero_nat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1704_add__cancel__right__right,axiom,
% 16.19/8.91 ! [A: int,B: int] :
% 16.19/8.91 ( ( A
% 16.19/8.91 = ( plus_plus_int @ A @ B ) )
% 16.19/8.91 = ( B = zero_zero_int ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1705_add__cancel__right__left,axiom,
% 16.19/8.91 ! [A: complex,B: complex] :
% 16.19/8.91 ( ( A
% 16.19/8.91 = ( plus_plus_complex @ B @ A ) )
% 16.19/8.91 = ( B = zero_zero_complex ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1706_add__cancel__right__left,axiom,
% 16.19/8.91 ! [A: real,B: real] :
% 16.19/8.91 ( ( A
% 16.19/8.91 = ( plus_plus_real @ B @ A ) )
% 16.19/8.91 = ( B = zero_zero_real ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1707_add__cancel__right__left,axiom,
% 16.19/8.91 ! [A: rat,B: rat] :
% 16.19/8.91 ( ( A
% 16.19/8.91 = ( plus_plus_rat @ B @ A ) )
% 16.19/8.91 = ( B = zero_zero_rat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1708_add__cancel__right__left,axiom,
% 16.19/8.91 ! [A: nat,B: nat] :
% 16.19/8.91 ( ( A
% 16.19/8.91 = ( plus_plus_nat @ B @ A ) )
% 16.19/8.91 = ( B = zero_zero_nat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1709_add__cancel__right__left,axiom,
% 16.19/8.91 ! [A: int,B: int] :
% 16.19/8.91 ( ( A
% 16.19/8.91 = ( plus_plus_int @ B @ A ) )
% 16.19/8.91 = ( B = zero_zero_int ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1710_add__cancel__left__right,axiom,
% 16.19/8.91 ! [A: complex,B: complex] :
% 16.19/8.91 ( ( ( plus_plus_complex @ A @ B )
% 16.19/8.91 = A )
% 16.19/8.91 = ( B = zero_zero_complex ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1711_add__cancel__left__right,axiom,
% 16.19/8.91 ! [A: real,B: real] :
% 16.19/8.91 ( ( ( plus_plus_real @ A @ B )
% 16.19/8.91 = A )
% 16.19/8.91 = ( B = zero_zero_real ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1712_add__cancel__left__right,axiom,
% 16.19/8.91 ! [A: rat,B: rat] :
% 16.19/8.91 ( ( ( plus_plus_rat @ A @ B )
% 16.19/8.91 = A )
% 16.19/8.91 = ( B = zero_zero_rat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1713_add__cancel__left__right,axiom,
% 16.19/8.91 ! [A: nat,B: nat] :
% 16.19/8.91 ( ( ( plus_plus_nat @ A @ B )
% 16.19/8.91 = A )
% 16.19/8.91 = ( B = zero_zero_nat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1714_add__cancel__left__right,axiom,
% 16.19/8.91 ! [A: int,B: int] :
% 16.19/8.91 ( ( ( plus_plus_int @ A @ B )
% 16.19/8.91 = A )
% 16.19/8.91 = ( B = zero_zero_int ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1715_add__cancel__left__left,axiom,
% 16.19/8.91 ! [B: complex,A: complex] :
% 16.19/8.91 ( ( ( plus_plus_complex @ B @ A )
% 16.19/8.91 = A )
% 16.19/8.91 = ( B = zero_zero_complex ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1716_add__cancel__left__left,axiom,
% 16.19/8.91 ! [B: real,A: real] :
% 16.19/8.91 ( ( ( plus_plus_real @ B @ A )
% 16.19/8.91 = A )
% 16.19/8.91 = ( B = zero_zero_real ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1717_add__cancel__left__left,axiom,
% 16.19/8.91 ! [B: rat,A: rat] :
% 16.19/8.91 ( ( ( plus_plus_rat @ B @ A )
% 16.19/8.91 = A )
% 16.19/8.91 = ( B = zero_zero_rat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1718_add__cancel__left__left,axiom,
% 16.19/8.91 ! [B: nat,A: nat] :
% 16.19/8.91 ( ( ( plus_plus_nat @ B @ A )
% 16.19/8.91 = A )
% 16.19/8.91 = ( B = zero_zero_nat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1719_add__cancel__left__left,axiom,
% 16.19/8.91 ! [B: int,A: int] :
% 16.19/8.91 ( ( ( plus_plus_int @ B @ A )
% 16.19/8.91 = A )
% 16.19/8.91 = ( B = zero_zero_int ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1720_double__zero__sym,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( zero_zero_real
% 16.19/8.91 = ( plus_plus_real @ A @ A ) )
% 16.19/8.91 = ( A = zero_zero_real ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1721_double__zero__sym,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( zero_zero_rat
% 16.19/8.91 = ( plus_plus_rat @ A @ A ) )
% 16.19/8.91 = ( A = zero_zero_rat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1722_double__zero__sym,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( zero_zero_int
% 16.19/8.91 = ( plus_plus_int @ A @ A ) )
% 16.19/8.91 = ( A = zero_zero_int ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1723_add_Oright__neutral,axiom,
% 16.19/8.91 ! [A: complex] :
% 16.19/8.91 ( ( plus_plus_complex @ A @ zero_zero_complex )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1724_add_Oright__neutral,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( plus_plus_real @ A @ zero_zero_real )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1725_add_Oright__neutral,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1726_add_Oright__neutral,axiom,
% 16.19/8.91 ! [A: nat] :
% 16.19/8.91 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1727_add_Oright__neutral,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( plus_plus_int @ A @ zero_zero_int )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1728_diff__self,axiom,
% 16.19/8.91 ! [A: complex] :
% 16.19/8.91 ( ( minus_minus_complex @ A @ A )
% 16.19/8.91 = zero_zero_complex ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1729_diff__self,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( minus_minus_real @ A @ A )
% 16.19/8.91 = zero_zero_real ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1730_diff__self,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( minus_minus_rat @ A @ A )
% 16.19/8.91 = zero_zero_rat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1731_diff__self,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( minus_minus_int @ A @ A )
% 16.19/8.91 = zero_zero_int ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1732_diff__0__right,axiom,
% 16.19/8.91 ! [A: complex] :
% 16.19/8.91 ( ( minus_minus_complex @ A @ zero_zero_complex )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1733_diff__0__right,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( minus_minus_real @ A @ zero_zero_real )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1734_diff__0__right,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( minus_minus_rat @ A @ zero_zero_rat )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1735_diff__0__right,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( minus_minus_int @ A @ zero_zero_int )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1736_zero__diff,axiom,
% 16.19/8.91 ! [A: nat] :
% 16.19/8.91 ( ( minus_minus_nat @ zero_zero_nat @ A )
% 16.19/8.91 = zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1737_diff__zero,axiom,
% 16.19/8.91 ! [A: complex] :
% 16.19/8.91 ( ( minus_minus_complex @ A @ zero_zero_complex )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1738_diff__zero,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( minus_minus_real @ A @ zero_zero_real )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1739_diff__zero,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( minus_minus_rat @ A @ zero_zero_rat )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1740_diff__zero,axiom,
% 16.19/8.91 ! [A: nat] :
% 16.19/8.91 ( ( minus_minus_nat @ A @ zero_zero_nat )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1741_diff__zero,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( minus_minus_int @ A @ zero_zero_int )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1742_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 16.19/8.91 ! [A: complex] :
% 16.19/8.91 ( ( minus_minus_complex @ A @ A )
% 16.19/8.91 = zero_zero_complex ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1743_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( minus_minus_real @ A @ A )
% 16.19/8.91 = zero_zero_real ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1744_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( minus_minus_rat @ A @ A )
% 16.19/8.91 = zero_zero_rat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1745_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 16.19/8.91 ! [A: nat] :
% 16.19/8.91 ( ( minus_minus_nat @ A @ A )
% 16.19/8.91 = zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1746_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( minus_minus_int @ A @ A )
% 16.19/8.91 = zero_zero_int ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1747_bits__div__by__0,axiom,
% 16.19/8.91 ! [A: nat] :
% 16.19/8.91 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 16.19/8.91 = zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1748_bits__div__by__0,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( divide_divide_int @ A @ zero_zero_int )
% 16.19/8.91 = zero_zero_int ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1749_bits__div__0,axiom,
% 16.19/8.91 ! [A: nat] :
% 16.19/8.91 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 16.19/8.91 = zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1750_bits__div__0,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( divide_divide_int @ zero_zero_int @ A )
% 16.19/8.91 = zero_zero_int ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1751_div__by__0,axiom,
% 16.19/8.91 ! [A: complex] :
% 16.19/8.91 ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 16.19/8.91 = zero_zero_complex ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1752_div__by__0,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( divide_divide_real @ A @ zero_zero_real )
% 16.19/8.91 = zero_zero_real ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1753_div__by__0,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( divide_divide_rat @ A @ zero_zero_rat )
% 16.19/8.91 = zero_zero_rat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1754_div__by__0,axiom,
% 16.19/8.91 ! [A: nat] :
% 16.19/8.91 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 16.19/8.91 = zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1755_div__by__0,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( divide_divide_int @ A @ zero_zero_int )
% 16.19/8.91 = zero_zero_int ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1756_div__0,axiom,
% 16.19/8.91 ! [A: complex] :
% 16.19/8.91 ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
% 16.19/8.91 = zero_zero_complex ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1757_div__0,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( divide_divide_real @ zero_zero_real @ A )
% 16.19/8.91 = zero_zero_real ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1758_div__0,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( divide_divide_rat @ zero_zero_rat @ A )
% 16.19/8.91 = zero_zero_rat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1759_div__0,axiom,
% 16.19/8.91 ! [A: nat] :
% 16.19/8.91 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 16.19/8.91 = zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1760_div__0,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( divide_divide_int @ zero_zero_int @ A )
% 16.19/8.91 = zero_zero_int ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1761_division__ring__divide__zero,axiom,
% 16.19/8.91 ! [A: complex] :
% 16.19/8.91 ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 16.19/8.91 = zero_zero_complex ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1762_division__ring__divide__zero,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( divide_divide_real @ A @ zero_zero_real )
% 16.19/8.91 = zero_zero_real ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1763_division__ring__divide__zero,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( divide_divide_rat @ A @ zero_zero_rat )
% 16.19/8.91 = zero_zero_rat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1764_divide__cancel__right,axiom,
% 16.19/8.91 ! [A: complex,C: complex,B: complex] :
% 16.19/8.91 ( ( ( divide1717551699836669952omplex @ A @ C )
% 16.19/8.91 = ( divide1717551699836669952omplex @ B @ C ) )
% 16.19/8.91 = ( ( C = zero_zero_complex )
% 16.19/8.91 | ( A = B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1765_divide__cancel__right,axiom,
% 16.19/8.91 ! [A: real,C: real,B: real] :
% 16.19/8.91 ( ( ( divide_divide_real @ A @ C )
% 16.19/8.91 = ( divide_divide_real @ B @ C ) )
% 16.19/8.91 = ( ( C = zero_zero_real )
% 16.19/8.91 | ( A = B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1766_divide__cancel__right,axiom,
% 16.19/8.91 ! [A: rat,C: rat,B: rat] :
% 16.19/8.91 ( ( ( divide_divide_rat @ A @ C )
% 16.19/8.91 = ( divide_divide_rat @ B @ C ) )
% 16.19/8.91 = ( ( C = zero_zero_rat )
% 16.19/8.91 | ( A = B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1767_divide__cancel__left,axiom,
% 16.19/8.91 ! [C: complex,A: complex,B: complex] :
% 16.19/8.91 ( ( ( divide1717551699836669952omplex @ C @ A )
% 16.19/8.91 = ( divide1717551699836669952omplex @ C @ B ) )
% 16.19/8.91 = ( ( C = zero_zero_complex )
% 16.19/8.91 | ( A = B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1768_divide__cancel__left,axiom,
% 16.19/8.91 ! [C: real,A: real,B: real] :
% 16.19/8.91 ( ( ( divide_divide_real @ C @ A )
% 16.19/8.91 = ( divide_divide_real @ C @ B ) )
% 16.19/8.91 = ( ( C = zero_zero_real )
% 16.19/8.91 | ( A = B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1769_divide__cancel__left,axiom,
% 16.19/8.91 ! [C: rat,A: rat,B: rat] :
% 16.19/8.91 ( ( ( divide_divide_rat @ C @ A )
% 16.19/8.91 = ( divide_divide_rat @ C @ B ) )
% 16.19/8.91 = ( ( C = zero_zero_rat )
% 16.19/8.91 | ( A = B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1770_divide__eq__0__iff,axiom,
% 16.19/8.91 ! [A: complex,B: complex] :
% 16.19/8.91 ( ( ( divide1717551699836669952omplex @ A @ B )
% 16.19/8.91 = zero_zero_complex )
% 16.19/8.91 = ( ( A = zero_zero_complex )
% 16.19/8.91 | ( B = zero_zero_complex ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1771_divide__eq__0__iff,axiom,
% 16.19/8.91 ! [A: real,B: real] :
% 16.19/8.91 ( ( ( divide_divide_real @ A @ B )
% 16.19/8.91 = zero_zero_real )
% 16.19/8.91 = ( ( A = zero_zero_real )
% 16.19/8.91 | ( B = zero_zero_real ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1772_divide__eq__0__iff,axiom,
% 16.19/8.91 ! [A: rat,B: rat] :
% 16.19/8.91 ( ( ( divide_divide_rat @ A @ B )
% 16.19/8.91 = zero_zero_rat )
% 16.19/8.91 = ( ( A = zero_zero_rat )
% 16.19/8.91 | ( B = zero_zero_rat ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1773_bits__mod__0,axiom,
% 16.19/8.91 ! [A: nat] :
% 16.19/8.91 ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 16.19/8.91 = zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1774_bits__mod__0,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( modulo_modulo_int @ zero_zero_int @ A )
% 16.19/8.91 = zero_zero_int ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1775_bits__mod__0,axiom,
% 16.19/8.91 ! [A: code_integer] :
% 16.19/8.91 ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 16.19/8.91 = zero_z3403309356797280102nteger ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1776_bot__nat__0_Onot__eq__extremum,axiom,
% 16.19/8.91 ! [A: nat] :
% 16.19/8.91 ( ( A != zero_zero_nat )
% 16.19/8.91 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1777_neq0__conv,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ( ( N2 != zero_zero_nat )
% 16.19/8.91 = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1778_less__nat__zero__code,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1779_bot__nat__0_Oextremum,axiom,
% 16.19/8.91 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1780_le0,axiom,
% 16.19/8.91 ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1781_mod__add__self2,axiom,
% 16.19/8.91 ! [A: nat,B: nat] :
% 16.19/8.91 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 16.19/8.91 = ( modulo_modulo_nat @ A @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1782_mod__add__self2,axiom,
% 16.19/8.91 ! [A: int,B: int] :
% 16.19/8.91 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
% 16.19/8.91 = ( modulo_modulo_int @ A @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1783_mod__add__self2,axiom,
% 16.19/8.91 ! [A: code_integer,B: code_integer] :
% 16.19/8.91 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
% 16.19/8.91 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1784_mod__add__self1,axiom,
% 16.19/8.91 ! [B: nat,A: nat] :
% 16.19/8.91 ( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 16.19/8.91 = ( modulo_modulo_nat @ A @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1785_mod__add__self1,axiom,
% 16.19/8.91 ! [B: int,A: int] :
% 16.19/8.91 ( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
% 16.19/8.91 = ( modulo_modulo_int @ A @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1786_mod__add__self1,axiom,
% 16.19/8.91 ! [B: code_integer,A: code_integer] :
% 16.19/8.91 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
% 16.19/8.91 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1787_Nat_Oadd__0__right,axiom,
% 16.19/8.91 ! [M: nat] :
% 16.19/8.91 ( ( plus_plus_nat @ M @ zero_zero_nat )
% 16.19/8.91 = M ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1788_add__is__0,axiom,
% 16.19/8.91 ! [M: nat,N2: nat] :
% 16.19/8.91 ( ( ( plus_plus_nat @ M @ N2 )
% 16.19/8.91 = zero_zero_nat )
% 16.19/8.91 = ( ( M = zero_zero_nat )
% 16.19/8.91 & ( N2 = zero_zero_nat ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1789_minus__mod__self2,axiom,
% 16.19/8.91 ! [A: int,B: int] :
% 16.19/8.91 ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ B )
% 16.19/8.91 = ( modulo_modulo_int @ A @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1790_minus__mod__self2,axiom,
% 16.19/8.91 ! [A: code_integer,B: code_integer] :
% 16.19/8.91 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ B )
% 16.19/8.91 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1791_mult__cancel2,axiom,
% 16.19/8.91 ! [M: nat,K: nat,N2: nat] :
% 16.19/8.91 ( ( ( times_times_nat @ M @ K )
% 16.19/8.91 = ( times_times_nat @ N2 @ K ) )
% 16.19/8.91 = ( ( M = N2 )
% 16.19/8.91 | ( K = zero_zero_nat ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1792_mult__cancel1,axiom,
% 16.19/8.91 ! [K: nat,M: nat,N2: nat] :
% 16.19/8.91 ( ( ( times_times_nat @ K @ M )
% 16.19/8.91 = ( times_times_nat @ K @ N2 ) )
% 16.19/8.91 = ( ( M = N2 )
% 16.19/8.91 | ( K = zero_zero_nat ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1793_mult__0__right,axiom,
% 16.19/8.91 ! [M: nat] :
% 16.19/8.91 ( ( times_times_nat @ M @ zero_zero_nat )
% 16.19/8.91 = zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1794_mult__is__0,axiom,
% 16.19/8.91 ! [M: nat,N2: nat] :
% 16.19/8.91 ( ( ( times_times_nat @ M @ N2 )
% 16.19/8.91 = zero_zero_nat )
% 16.19/8.91 = ( ( M = zero_zero_nat )
% 16.19/8.91 | ( N2 = zero_zero_nat ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1795_diff__0__eq__0,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ( ( minus_minus_nat @ zero_zero_nat @ N2 )
% 16.19/8.91 = zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1796_diff__self__eq__0,axiom,
% 16.19/8.91 ! [M: nat] :
% 16.19/8.91 ( ( minus_minus_nat @ M @ M )
% 16.19/8.91 = zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1797_mod__less,axiom,
% 16.19/8.91 ! [M: nat,N2: nat] :
% 16.19/8.91 ( ( ord_less_nat @ M @ N2 )
% 16.19/8.91 => ( ( modulo_modulo_nat @ M @ N2 )
% 16.19/8.91 = M ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1798_max__0R,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ( ( ord_max_nat @ N2 @ zero_zero_nat )
% 16.19/8.91 = N2 ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1799_max__0L,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ( ( ord_max_nat @ zero_zero_nat @ N2 )
% 16.19/8.91 = N2 ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1800_max__nat_Oright__neutral,axiom,
% 16.19/8.91 ! [A: nat] :
% 16.19/8.91 ( ( ord_max_nat @ A @ zero_zero_nat )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1801_max__nat_Oneutr__eq__iff,axiom,
% 16.19/8.91 ! [A: nat,B: nat] :
% 16.19/8.91 ( ( zero_zero_nat
% 16.19/8.91 = ( ord_max_nat @ A @ B ) )
% 16.19/8.91 = ( ( A = zero_zero_nat )
% 16.19/8.91 & ( B = zero_zero_nat ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1802_max__nat_Oleft__neutral,axiom,
% 16.19/8.91 ! [A: nat] :
% 16.19/8.91 ( ( ord_max_nat @ zero_zero_nat @ A )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1803_max__nat_Oeq__neutr__iff,axiom,
% 16.19/8.91 ! [A: nat,B: nat] :
% 16.19/8.91 ( ( ( ord_max_nat @ A @ B )
% 16.19/8.91 = zero_zero_nat )
% 16.19/8.91 = ( ( A = zero_zero_nat )
% 16.19/8.91 & ( B = zero_zero_nat ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1804_zero__le__double__add__iff__zero__le__single__add,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 16.19/8.91 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1805_zero__le__double__add__iff__zero__le__single__add,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 16.19/8.91 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1806_zero__le__double__add__iff__zero__le__single__add,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 16.19/8.91 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1807_double__add__le__zero__iff__single__add__le__zero,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 16.19/8.91 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1808_double__add__le__zero__iff__single__add__le__zero,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 16.19/8.91 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1809_double__add__le__zero__iff__single__add__le__zero,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 16.19/8.91 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1810_le__add__same__cancel2,axiom,
% 16.19/8.91 ! [A: real,B: real] :
% 16.19/8.91 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
% 16.19/8.91 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1811_le__add__same__cancel2,axiom,
% 16.19/8.91 ! [A: rat,B: rat] :
% 16.19/8.91 ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 16.19/8.91 = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1812_le__add__same__cancel2,axiom,
% 16.19/8.91 ! [A: nat,B: nat] :
% 16.19/8.91 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 16.19/8.91 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1813_le__add__same__cancel2,axiom,
% 16.19/8.91 ! [A: int,B: int] :
% 16.19/8.91 ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
% 16.19/8.91 = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1814_le__add__same__cancel1,axiom,
% 16.19/8.91 ! [A: real,B: real] :
% 16.19/8.91 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
% 16.19/8.91 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1815_le__add__same__cancel1,axiom,
% 16.19/8.91 ! [A: rat,B: rat] :
% 16.19/8.91 ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 16.19/8.91 = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1816_le__add__same__cancel1,axiom,
% 16.19/8.91 ! [A: nat,B: nat] :
% 16.19/8.91 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 16.19/8.91 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1817_le__add__same__cancel1,axiom,
% 16.19/8.91 ! [A: int,B: int] :
% 16.19/8.91 ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
% 16.19/8.91 = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1818_add__le__same__cancel2,axiom,
% 16.19/8.91 ! [A: real,B: real] :
% 16.19/8.91 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
% 16.19/8.91 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1819_add__le__same__cancel2,axiom,
% 16.19/8.91 ! [A: rat,B: rat] :
% 16.19/8.91 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 16.19/8.91 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1820_add__le__same__cancel2,axiom,
% 16.19/8.91 ! [A: nat,B: nat] :
% 16.19/8.91 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 16.19/8.91 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1821_add__le__same__cancel2,axiom,
% 16.19/8.91 ! [A: int,B: int] :
% 16.19/8.91 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
% 16.19/8.91 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1822_add__le__same__cancel1,axiom,
% 16.19/8.91 ! [B: real,A: real] :
% 16.19/8.91 ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
% 16.19/8.91 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1823_add__le__same__cancel1,axiom,
% 16.19/8.91 ! [B: rat,A: rat] :
% 16.19/8.91 ( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 16.19/8.91 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1824_add__le__same__cancel1,axiom,
% 16.19/8.91 ! [B: nat,A: nat] :
% 16.19/8.91 ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 16.19/8.91 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1825_add__le__same__cancel1,axiom,
% 16.19/8.91 ! [B: int,A: int] :
% 16.19/8.91 ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
% 16.19/8.91 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1826_add__less__same__cancel1,axiom,
% 16.19/8.91 ! [B: real,A: real] :
% 16.19/8.91 ( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
% 16.19/8.91 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1827_add__less__same__cancel1,axiom,
% 16.19/8.91 ! [B: rat,A: rat] :
% 16.19/8.91 ( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 16.19/8.91 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1828_add__less__same__cancel1,axiom,
% 16.19/8.91 ! [B: nat,A: nat] :
% 16.19/8.91 ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 16.19/8.91 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1829_add__less__same__cancel1,axiom,
% 16.19/8.91 ! [B: int,A: int] :
% 16.19/8.91 ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
% 16.19/8.91 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1830_add__less__same__cancel2,axiom,
% 16.19/8.91 ! [A: real,B: real] :
% 16.19/8.91 ( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
% 16.19/8.91 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1831_add__less__same__cancel2,axiom,
% 16.19/8.91 ! [A: rat,B: rat] :
% 16.19/8.91 ( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 16.19/8.91 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1832_add__less__same__cancel2,axiom,
% 16.19/8.91 ! [A: nat,B: nat] :
% 16.19/8.91 ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 16.19/8.91 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1833_add__less__same__cancel2,axiom,
% 16.19/8.91 ! [A: int,B: int] :
% 16.19/8.91 ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
% 16.19/8.91 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1834_less__add__same__cancel1,axiom,
% 16.19/8.91 ! [A: real,B: real] :
% 16.19/8.91 ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
% 16.19/8.91 = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1835_less__add__same__cancel1,axiom,
% 16.19/8.91 ! [A: rat,B: rat] :
% 16.19/8.91 ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 16.19/8.91 = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1836_less__add__same__cancel1,axiom,
% 16.19/8.91 ! [A: nat,B: nat] :
% 16.19/8.91 ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 16.19/8.91 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1837_less__add__same__cancel1,axiom,
% 16.19/8.91 ! [A: int,B: int] :
% 16.19/8.91 ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
% 16.19/8.91 = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1838_less__add__same__cancel2,axiom,
% 16.19/8.91 ! [A: real,B: real] :
% 16.19/8.91 ( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
% 16.19/8.91 = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1839_less__add__same__cancel2,axiom,
% 16.19/8.91 ! [A: rat,B: rat] :
% 16.19/8.91 ( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 16.19/8.91 = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1840_less__add__same__cancel2,axiom,
% 16.19/8.91 ! [A: nat,B: nat] :
% 16.19/8.91 ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 16.19/8.91 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1841_less__add__same__cancel2,axiom,
% 16.19/8.91 ! [A: int,B: int] :
% 16.19/8.91 ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
% 16.19/8.91 = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1842_double__add__less__zero__iff__single__add__less__zero,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 16.19/8.91 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1843_double__add__less__zero__iff__single__add__less__zero,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 16.19/8.91 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1844_double__add__less__zero__iff__single__add__less__zero,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 16.19/8.91 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1845_zero__less__double__add__iff__zero__less__single__add,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 16.19/8.91 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1846_zero__less__double__add__iff__zero__less__single__add,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 16.19/8.91 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1847_zero__less__double__add__iff__zero__less__single__add,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 16.19/8.91 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1848_diff__ge__0__iff__ge,axiom,
% 16.19/8.91 ! [A: real,B: real] :
% 16.19/8.91 ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 16.19/8.91 = ( ord_less_eq_real @ B @ A ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1849_diff__ge__0__iff__ge,axiom,
% 16.19/8.91 ! [A: rat,B: rat] :
% 16.19/8.91 ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 16.19/8.91 = ( ord_less_eq_rat @ B @ A ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1850_diff__ge__0__iff__ge,axiom,
% 16.19/8.91 ! [A: int,B: int] :
% 16.19/8.91 ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 16.19/8.91 = ( ord_less_eq_int @ B @ A ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1851_diff__gt__0__iff__gt,axiom,
% 16.19/8.91 ! [A: real,B: real] :
% 16.19/8.91 ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 16.19/8.91 = ( ord_less_real @ B @ A ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1852_diff__gt__0__iff__gt,axiom,
% 16.19/8.91 ! [A: rat,B: rat] :
% 16.19/8.91 ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 16.19/8.91 = ( ord_less_rat @ B @ A ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1853_diff__gt__0__iff__gt,axiom,
% 16.19/8.91 ! [A: int,B: int] :
% 16.19/8.91 ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 16.19/8.91 = ( ord_less_int @ B @ A ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1854_sum__squares__eq__zero__iff,axiom,
% 16.19/8.91 ! [X: real,Y: real] :
% 16.19/8.91 ( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
% 16.19/8.91 = zero_zero_real )
% 16.19/8.91 = ( ( X = zero_zero_real )
% 16.19/8.91 & ( Y = zero_zero_real ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1855_sum__squares__eq__zero__iff,axiom,
% 16.19/8.91 ! [X: rat,Y: rat] :
% 16.19/8.91 ( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
% 16.19/8.91 = zero_zero_rat )
% 16.19/8.91 = ( ( X = zero_zero_rat )
% 16.19/8.91 & ( Y = zero_zero_rat ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1856_sum__squares__eq__zero__iff,axiom,
% 16.19/8.91 ! [X: int,Y: int] :
% 16.19/8.91 ( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
% 16.19/8.91 = zero_zero_int )
% 16.19/8.91 = ( ( X = zero_zero_int )
% 16.19/8.91 & ( Y = zero_zero_int ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1857_mult__cancel__left1,axiom,
% 16.19/8.91 ! [C: complex,B: complex] :
% 16.19/8.91 ( ( C
% 16.19/8.91 = ( times_times_complex @ C @ B ) )
% 16.19/8.91 = ( ( C = zero_zero_complex )
% 16.19/8.91 | ( B = one_one_complex ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1858_mult__cancel__left1,axiom,
% 16.19/8.91 ! [C: real,B: real] :
% 16.19/8.91 ( ( C
% 16.19/8.91 = ( times_times_real @ C @ B ) )
% 16.19/8.91 = ( ( C = zero_zero_real )
% 16.19/8.91 | ( B = one_one_real ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1859_mult__cancel__left1,axiom,
% 16.19/8.91 ! [C: rat,B: rat] :
% 16.19/8.91 ( ( C
% 16.19/8.91 = ( times_times_rat @ C @ B ) )
% 16.19/8.91 = ( ( C = zero_zero_rat )
% 16.19/8.91 | ( B = one_one_rat ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1860_mult__cancel__left1,axiom,
% 16.19/8.91 ! [C: int,B: int] :
% 16.19/8.91 ( ( C
% 16.19/8.91 = ( times_times_int @ C @ B ) )
% 16.19/8.91 = ( ( C = zero_zero_int )
% 16.19/8.91 | ( B = one_one_int ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1861_mult__cancel__left2,axiom,
% 16.19/8.91 ! [C: complex,A: complex] :
% 16.19/8.91 ( ( ( times_times_complex @ C @ A )
% 16.19/8.91 = C )
% 16.19/8.91 = ( ( C = zero_zero_complex )
% 16.19/8.91 | ( A = one_one_complex ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1862_mult__cancel__left2,axiom,
% 16.19/8.91 ! [C: real,A: real] :
% 16.19/8.91 ( ( ( times_times_real @ C @ A )
% 16.19/8.91 = C )
% 16.19/8.91 = ( ( C = zero_zero_real )
% 16.19/8.91 | ( A = one_one_real ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1863_mult__cancel__left2,axiom,
% 16.19/8.91 ! [C: rat,A: rat] :
% 16.19/8.91 ( ( ( times_times_rat @ C @ A )
% 16.19/8.91 = C )
% 16.19/8.91 = ( ( C = zero_zero_rat )
% 16.19/8.91 | ( A = one_one_rat ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1864_mult__cancel__left2,axiom,
% 16.19/8.91 ! [C: int,A: int] :
% 16.19/8.91 ( ( ( times_times_int @ C @ A )
% 16.19/8.91 = C )
% 16.19/8.91 = ( ( C = zero_zero_int )
% 16.19/8.91 | ( A = one_one_int ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1865_mult__cancel__right1,axiom,
% 16.19/8.91 ! [C: complex,B: complex] :
% 16.19/8.91 ( ( C
% 16.19/8.91 = ( times_times_complex @ B @ C ) )
% 16.19/8.91 = ( ( C = zero_zero_complex )
% 16.19/8.91 | ( B = one_one_complex ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1866_mult__cancel__right1,axiom,
% 16.19/8.91 ! [C: real,B: real] :
% 16.19/8.91 ( ( C
% 16.19/8.91 = ( times_times_real @ B @ C ) )
% 16.19/8.91 = ( ( C = zero_zero_real )
% 16.19/8.91 | ( B = one_one_real ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1867_mult__cancel__right1,axiom,
% 16.19/8.91 ! [C: rat,B: rat] :
% 16.19/8.91 ( ( C
% 16.19/8.91 = ( times_times_rat @ B @ C ) )
% 16.19/8.91 = ( ( C = zero_zero_rat )
% 16.19/8.91 | ( B = one_one_rat ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1868_mult__cancel__right1,axiom,
% 16.19/8.91 ! [C: int,B: int] :
% 16.19/8.91 ( ( C
% 16.19/8.91 = ( times_times_int @ B @ C ) )
% 16.19/8.91 = ( ( C = zero_zero_int )
% 16.19/8.91 | ( B = one_one_int ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1869_mult__cancel__right2,axiom,
% 16.19/8.91 ! [A: complex,C: complex] :
% 16.19/8.91 ( ( ( times_times_complex @ A @ C )
% 16.19/8.91 = C )
% 16.19/8.91 = ( ( C = zero_zero_complex )
% 16.19/8.91 | ( A = one_one_complex ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1870_mult__cancel__right2,axiom,
% 16.19/8.91 ! [A: real,C: real] :
% 16.19/8.91 ( ( ( times_times_real @ A @ C )
% 16.19/8.91 = C )
% 16.19/8.91 = ( ( C = zero_zero_real )
% 16.19/8.91 | ( A = one_one_real ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1871_mult__cancel__right2,axiom,
% 16.19/8.91 ! [A: rat,C: rat] :
% 16.19/8.91 ( ( ( times_times_rat @ A @ C )
% 16.19/8.91 = C )
% 16.19/8.91 = ( ( C = zero_zero_rat )
% 16.19/8.91 | ( A = one_one_rat ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1872_mult__cancel__right2,axiom,
% 16.19/8.91 ! [A: int,C: int] :
% 16.19/8.91 ( ( ( times_times_int @ A @ C )
% 16.19/8.91 = C )
% 16.19/8.91 = ( ( C = zero_zero_int )
% 16.19/8.91 | ( A = one_one_int ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1873_div__mult__mult1__if,axiom,
% 16.19/8.91 ! [C: nat,A: nat,B: nat] :
% 16.19/8.91 ( ( ( C = zero_zero_nat )
% 16.19/8.91 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 16.19/8.91 = zero_zero_nat ) )
% 16.19/8.91 & ( ( C != zero_zero_nat )
% 16.19/8.91 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 16.19/8.91 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1874_div__mult__mult1__if,axiom,
% 16.19/8.91 ! [C: int,A: int,B: int] :
% 16.19/8.91 ( ( ( C = zero_zero_int )
% 16.19/8.91 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 16.19/8.91 = zero_zero_int ) )
% 16.19/8.91 & ( ( C != zero_zero_int )
% 16.19/8.91 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 16.19/8.91 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1875_div__mult__mult2,axiom,
% 16.19/8.91 ! [C: nat,A: nat,B: nat] :
% 16.19/8.91 ( ( C != zero_zero_nat )
% 16.19/8.91 => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 16.19/8.91 = ( divide_divide_nat @ A @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1876_div__mult__mult2,axiom,
% 16.19/8.91 ! [C: int,A: int,B: int] :
% 16.19/8.91 ( ( C != zero_zero_int )
% 16.19/8.91 => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 16.19/8.91 = ( divide_divide_int @ A @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1877_div__mult__mult1,axiom,
% 16.19/8.91 ! [C: nat,A: nat,B: nat] :
% 16.19/8.91 ( ( C != zero_zero_nat )
% 16.19/8.91 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 16.19/8.91 = ( divide_divide_nat @ A @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1878_div__mult__mult1,axiom,
% 16.19/8.91 ! [C: int,A: int,B: int] :
% 16.19/8.91 ( ( C != zero_zero_int )
% 16.19/8.91 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 16.19/8.91 = ( divide_divide_int @ A @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1879_nonzero__mult__div__cancel__right,axiom,
% 16.19/8.91 ! [B: complex,A: complex] :
% 16.19/8.91 ( ( B != zero_zero_complex )
% 16.19/8.91 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
% 16.19/8.91 = A ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1880_nonzero__mult__div__cancel__right,axiom,
% 16.19/8.91 ! [B: real,A: real] :
% 16.19/8.91 ( ( B != zero_zero_real )
% 16.19/8.91 => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
% 16.19/8.91 = A ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1881_nonzero__mult__div__cancel__right,axiom,
% 16.19/8.91 ! [B: rat,A: rat] :
% 16.19/8.91 ( ( B != zero_zero_rat )
% 16.19/8.91 => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
% 16.19/8.91 = A ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1882_nonzero__mult__div__cancel__right,axiom,
% 16.19/8.91 ! [B: nat,A: nat] :
% 16.19/8.91 ( ( B != zero_zero_nat )
% 16.19/8.91 => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
% 16.19/8.91 = A ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1883_nonzero__mult__div__cancel__right,axiom,
% 16.19/8.91 ! [B: int,A: int] :
% 16.19/8.91 ( ( B != zero_zero_int )
% 16.19/8.91 => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
% 16.19/8.91 = A ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1884_nonzero__mult__div__cancel__left,axiom,
% 16.19/8.91 ! [A: complex,B: complex] :
% 16.19/8.91 ( ( A != zero_zero_complex )
% 16.19/8.91 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
% 16.19/8.91 = B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1885_nonzero__mult__div__cancel__left,axiom,
% 16.19/8.91 ! [A: real,B: real] :
% 16.19/8.91 ( ( A != zero_zero_real )
% 16.19/8.91 => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
% 16.19/8.91 = B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1886_nonzero__mult__div__cancel__left,axiom,
% 16.19/8.91 ! [A: rat,B: rat] :
% 16.19/8.91 ( ( A != zero_zero_rat )
% 16.19/8.91 => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
% 16.19/8.91 = B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1887_nonzero__mult__div__cancel__left,axiom,
% 16.19/8.91 ! [A: nat,B: nat] :
% 16.19/8.91 ( ( A != zero_zero_nat )
% 16.19/8.91 => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
% 16.19/8.91 = B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1888_nonzero__mult__div__cancel__left,axiom,
% 16.19/8.91 ! [A: int,B: int] :
% 16.19/8.91 ( ( A != zero_zero_int )
% 16.19/8.91 => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
% 16.19/8.91 = B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1889_nonzero__mult__divide__mult__cancel__right2,axiom,
% 16.19/8.91 ! [C: complex,A: complex,B: complex] :
% 16.19/8.91 ( ( C != zero_zero_complex )
% 16.19/8.91 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B ) )
% 16.19/8.91 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1890_nonzero__mult__divide__mult__cancel__right2,axiom,
% 16.19/8.91 ! [C: real,A: real,B: real] :
% 16.19/8.91 ( ( C != zero_zero_real )
% 16.19/8.91 => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
% 16.19/8.91 = ( divide_divide_real @ A @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1891_nonzero__mult__divide__mult__cancel__right2,axiom,
% 16.19/8.91 ! [C: rat,A: rat,B: rat] :
% 16.19/8.91 ( ( C != zero_zero_rat )
% 16.19/8.91 => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
% 16.19/8.91 = ( divide_divide_rat @ A @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1892_nonzero__mult__divide__mult__cancel__right,axiom,
% 16.19/8.91 ! [C: complex,A: complex,B: complex] :
% 16.19/8.91 ( ( C != zero_zero_complex )
% 16.19/8.91 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 16.19/8.91 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1893_nonzero__mult__divide__mult__cancel__right,axiom,
% 16.19/8.91 ! [C: real,A: real,B: real] :
% 16.19/8.91 ( ( C != zero_zero_real )
% 16.19/8.91 => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 16.19/8.91 = ( divide_divide_real @ A @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1894_nonzero__mult__divide__mult__cancel__right,axiom,
% 16.19/8.91 ! [C: rat,A: rat,B: rat] :
% 16.19/8.91 ( ( C != zero_zero_rat )
% 16.19/8.91 => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 16.19/8.91 = ( divide_divide_rat @ A @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1895_nonzero__mult__divide__mult__cancel__left2,axiom,
% 16.19/8.91 ! [C: complex,A: complex,B: complex] :
% 16.19/8.91 ( ( C != zero_zero_complex )
% 16.19/8.91 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B @ C ) )
% 16.19/8.91 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1896_nonzero__mult__divide__mult__cancel__left2,axiom,
% 16.19/8.91 ! [C: real,A: real,B: real] :
% 16.19/8.91 ( ( C != zero_zero_real )
% 16.19/8.91 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
% 16.19/8.91 = ( divide_divide_real @ A @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1897_nonzero__mult__divide__mult__cancel__left2,axiom,
% 16.19/8.91 ! [C: rat,A: rat,B: rat] :
% 16.19/8.91 ( ( C != zero_zero_rat )
% 16.19/8.91 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
% 16.19/8.91 = ( divide_divide_rat @ A @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1898_nonzero__mult__divide__mult__cancel__left,axiom,
% 16.19/8.91 ! [C: complex,A: complex,B: complex] :
% 16.19/8.91 ( ( C != zero_zero_complex )
% 16.19/8.91 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 16.19/8.91 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1899_nonzero__mult__divide__mult__cancel__left,axiom,
% 16.19/8.91 ! [C: real,A: real,B: real] :
% 16.19/8.91 ( ( C != zero_zero_real )
% 16.19/8.91 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 16.19/8.91 = ( divide_divide_real @ A @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1900_nonzero__mult__divide__mult__cancel__left,axiom,
% 16.19/8.91 ! [C: rat,A: rat,B: rat] :
% 16.19/8.91 ( ( C != zero_zero_rat )
% 16.19/8.91 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 16.19/8.91 = ( divide_divide_rat @ A @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1901_mult__divide__mult__cancel__left__if,axiom,
% 16.19/8.91 ! [C: complex,A: complex,B: complex] :
% 16.19/8.91 ( ( ( C = zero_zero_complex )
% 16.19/8.91 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 16.19/8.91 = zero_zero_complex ) )
% 16.19/8.91 & ( ( C != zero_zero_complex )
% 16.19/8.91 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 16.19/8.91 = ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1902_mult__divide__mult__cancel__left__if,axiom,
% 16.19/8.91 ! [C: real,A: real,B: real] :
% 16.19/8.91 ( ( ( C = zero_zero_real )
% 16.19/8.91 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 16.19/8.91 = zero_zero_real ) )
% 16.19/8.91 & ( ( C != zero_zero_real )
% 16.19/8.91 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 16.19/8.91 = ( divide_divide_real @ A @ B ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1903_mult__divide__mult__cancel__left__if,axiom,
% 16.19/8.91 ! [C: rat,A: rat,B: rat] :
% 16.19/8.91 ( ( ( C = zero_zero_rat )
% 16.19/8.91 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 16.19/8.91 = zero_zero_rat ) )
% 16.19/8.91 & ( ( C != zero_zero_rat )
% 16.19/8.91 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 16.19/8.91 = ( divide_divide_rat @ A @ B ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1904_diff__add__zero,axiom,
% 16.19/8.91 ! [A: nat,B: nat] :
% 16.19/8.91 ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 16.19/8.91 = zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1905_diff__numeral__special_I9_J,axiom,
% 16.19/8.91 ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
% 16.19/8.91 = zero_zero_complex ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1906_diff__numeral__special_I9_J,axiom,
% 16.19/8.91 ( ( minus_minus_real @ one_one_real @ one_one_real )
% 16.19/8.91 = zero_zero_real ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1907_diff__numeral__special_I9_J,axiom,
% 16.19/8.91 ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
% 16.19/8.91 = zero_zero_rat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1908_diff__numeral__special_I9_J,axiom,
% 16.19/8.91 ( ( minus_minus_int @ one_one_int @ one_one_int )
% 16.19/8.91 = zero_zero_int ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1909_div__self,axiom,
% 16.19/8.91 ! [A: complex] :
% 16.19/8.91 ( ( A != zero_zero_complex )
% 16.19/8.91 => ( ( divide1717551699836669952omplex @ A @ A )
% 16.19/8.91 = one_one_complex ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1910_div__self,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( A != zero_zero_real )
% 16.19/8.91 => ( ( divide_divide_real @ A @ A )
% 16.19/8.91 = one_one_real ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1911_div__self,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( A != zero_zero_rat )
% 16.19/8.91 => ( ( divide_divide_rat @ A @ A )
% 16.19/8.91 = one_one_rat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1912_div__self,axiom,
% 16.19/8.91 ! [A: nat] :
% 16.19/8.91 ( ( A != zero_zero_nat )
% 16.19/8.91 => ( ( divide_divide_nat @ A @ A )
% 16.19/8.91 = one_one_nat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1913_div__self,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( A != zero_zero_int )
% 16.19/8.91 => ( ( divide_divide_int @ A @ A )
% 16.19/8.91 = one_one_int ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1914_zero__eq__1__divide__iff,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( zero_zero_real
% 16.19/8.91 = ( divide_divide_real @ one_one_real @ A ) )
% 16.19/8.91 = ( A = zero_zero_real ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1915_zero__eq__1__divide__iff,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( zero_zero_rat
% 16.19/8.91 = ( divide_divide_rat @ one_one_rat @ A ) )
% 16.19/8.91 = ( A = zero_zero_rat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1916_one__divide__eq__0__iff,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( ( divide_divide_real @ one_one_real @ A )
% 16.19/8.91 = zero_zero_real )
% 16.19/8.91 = ( A = zero_zero_real ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1917_one__divide__eq__0__iff,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( ( divide_divide_rat @ one_one_rat @ A )
% 16.19/8.91 = zero_zero_rat )
% 16.19/8.91 = ( A = zero_zero_rat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1918_eq__divide__eq__1,axiom,
% 16.19/8.91 ! [B: real,A: real] :
% 16.19/8.91 ( ( one_one_real
% 16.19/8.91 = ( divide_divide_real @ B @ A ) )
% 16.19/8.91 = ( ( A != zero_zero_real )
% 16.19/8.91 & ( A = B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1919_eq__divide__eq__1,axiom,
% 16.19/8.91 ! [B: rat,A: rat] :
% 16.19/8.91 ( ( one_one_rat
% 16.19/8.91 = ( divide_divide_rat @ B @ A ) )
% 16.19/8.91 = ( ( A != zero_zero_rat )
% 16.19/8.91 & ( A = B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1920_divide__eq__eq__1,axiom,
% 16.19/8.91 ! [B: real,A: real] :
% 16.19/8.91 ( ( ( divide_divide_real @ B @ A )
% 16.19/8.91 = one_one_real )
% 16.19/8.91 = ( ( A != zero_zero_real )
% 16.19/8.91 & ( A = B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1921_divide__eq__eq__1,axiom,
% 16.19/8.91 ! [B: rat,A: rat] :
% 16.19/8.91 ( ( ( divide_divide_rat @ B @ A )
% 16.19/8.91 = one_one_rat )
% 16.19/8.91 = ( ( A != zero_zero_rat )
% 16.19/8.91 & ( A = B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1922_divide__self__if,axiom,
% 16.19/8.91 ! [A: complex] :
% 16.19/8.91 ( ( ( A = zero_zero_complex )
% 16.19/8.91 => ( ( divide1717551699836669952omplex @ A @ A )
% 16.19/8.91 = zero_zero_complex ) )
% 16.19/8.91 & ( ( A != zero_zero_complex )
% 16.19/8.91 => ( ( divide1717551699836669952omplex @ A @ A )
% 16.19/8.91 = one_one_complex ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1923_divide__self__if,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( ( A = zero_zero_real )
% 16.19/8.91 => ( ( divide_divide_real @ A @ A )
% 16.19/8.91 = zero_zero_real ) )
% 16.19/8.91 & ( ( A != zero_zero_real )
% 16.19/8.91 => ( ( divide_divide_real @ A @ A )
% 16.19/8.91 = one_one_real ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1924_divide__self__if,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( ( A = zero_zero_rat )
% 16.19/8.91 => ( ( divide_divide_rat @ A @ A )
% 16.19/8.91 = zero_zero_rat ) )
% 16.19/8.91 & ( ( A != zero_zero_rat )
% 16.19/8.91 => ( ( divide_divide_rat @ A @ A )
% 16.19/8.91 = one_one_rat ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1925_divide__self,axiom,
% 16.19/8.91 ! [A: complex] :
% 16.19/8.91 ( ( A != zero_zero_complex )
% 16.19/8.91 => ( ( divide1717551699836669952omplex @ A @ A )
% 16.19/8.91 = one_one_complex ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1926_divide__self,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( A != zero_zero_real )
% 16.19/8.91 => ( ( divide_divide_real @ A @ A )
% 16.19/8.91 = one_one_real ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1927_divide__self,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( A != zero_zero_rat )
% 16.19/8.91 => ( ( divide_divide_rat @ A @ A )
% 16.19/8.91 = one_one_rat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1928_one__eq__divide__iff,axiom,
% 16.19/8.91 ! [A: complex,B: complex] :
% 16.19/8.91 ( ( one_one_complex
% 16.19/8.91 = ( divide1717551699836669952omplex @ A @ B ) )
% 16.19/8.91 = ( ( B != zero_zero_complex )
% 16.19/8.91 & ( A = B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1929_one__eq__divide__iff,axiom,
% 16.19/8.91 ! [A: real,B: real] :
% 16.19/8.91 ( ( one_one_real
% 16.19/8.91 = ( divide_divide_real @ A @ B ) )
% 16.19/8.91 = ( ( B != zero_zero_real )
% 16.19/8.91 & ( A = B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1930_one__eq__divide__iff,axiom,
% 16.19/8.91 ! [A: rat,B: rat] :
% 16.19/8.91 ( ( one_one_rat
% 16.19/8.91 = ( divide_divide_rat @ A @ B ) )
% 16.19/8.91 = ( ( B != zero_zero_rat )
% 16.19/8.91 & ( A = B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1931_divide__eq__1__iff,axiom,
% 16.19/8.91 ! [A: complex,B: complex] :
% 16.19/8.91 ( ( ( divide1717551699836669952omplex @ A @ B )
% 16.19/8.91 = one_one_complex )
% 16.19/8.91 = ( ( B != zero_zero_complex )
% 16.19/8.91 & ( A = B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1932_divide__eq__1__iff,axiom,
% 16.19/8.91 ! [A: real,B: real] :
% 16.19/8.91 ( ( ( divide_divide_real @ A @ B )
% 16.19/8.91 = one_one_real )
% 16.19/8.91 = ( ( B != zero_zero_real )
% 16.19/8.91 & ( A = B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1933_divide__eq__1__iff,axiom,
% 16.19/8.91 ! [A: rat,B: rat] :
% 16.19/8.91 ( ( ( divide_divide_rat @ A @ B )
% 16.19/8.91 = one_one_rat )
% 16.19/8.91 = ( ( B != zero_zero_rat )
% 16.19/8.91 & ( A = B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1934_power__0__Suc,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ( ( power_power_rat @ zero_zero_rat @ ( suc @ N2 ) )
% 16.19/8.91 = zero_zero_rat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1935_power__0__Suc,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ( ( power_power_nat @ zero_zero_nat @ ( suc @ N2 ) )
% 16.19/8.91 = zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1936_power__0__Suc,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ( ( power_power_real @ zero_zero_real @ ( suc @ N2 ) )
% 16.19/8.91 = zero_zero_real ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1937_power__0__Suc,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ( ( power_power_int @ zero_zero_int @ ( suc @ N2 ) )
% 16.19/8.91 = zero_zero_int ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1938_power__0__Suc,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ( ( power_power_complex @ zero_zero_complex @ ( suc @ N2 ) )
% 16.19/8.91 = zero_zero_complex ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1939_power__zero__numeral,axiom,
% 16.19/8.91 ! [K: num] :
% 16.19/8.91 ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
% 16.19/8.91 = zero_zero_rat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1940_power__zero__numeral,axiom,
% 16.19/8.91 ! [K: num] :
% 16.19/8.91 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
% 16.19/8.91 = zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1941_power__zero__numeral,axiom,
% 16.19/8.91 ! [K: num] :
% 16.19/8.91 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
% 16.19/8.91 = zero_zero_real ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1942_power__zero__numeral,axiom,
% 16.19/8.91 ! [K: num] :
% 16.19/8.91 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
% 16.19/8.91 = zero_zero_int ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1943_power__zero__numeral,axiom,
% 16.19/8.91 ! [K: num] :
% 16.19/8.91 ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
% 16.19/8.91 = zero_zero_complex ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1944_mod__mult__self1__is__0,axiom,
% 16.19/8.91 ! [B: nat,A: nat] :
% 16.19/8.91 ( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
% 16.19/8.91 = zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1945_mod__mult__self1__is__0,axiom,
% 16.19/8.91 ! [B: int,A: int] :
% 16.19/8.91 ( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
% 16.19/8.91 = zero_zero_int ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1946_mod__mult__self1__is__0,axiom,
% 16.19/8.91 ! [B: code_integer,A: code_integer] :
% 16.19/8.91 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
% 16.19/8.91 = zero_z3403309356797280102nteger ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1947_mod__mult__self2__is__0,axiom,
% 16.19/8.91 ! [A: nat,B: nat] :
% 16.19/8.91 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
% 16.19/8.91 = zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1948_mod__mult__self2__is__0,axiom,
% 16.19/8.91 ! [A: int,B: int] :
% 16.19/8.91 ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
% 16.19/8.91 = zero_zero_int ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1949_mod__mult__self2__is__0,axiom,
% 16.19/8.91 ! [A: code_integer,B: code_integer] :
% 16.19/8.91 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
% 16.19/8.91 = zero_z3403309356797280102nteger ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1950_bits__mod__by__1,axiom,
% 16.19/8.91 ! [A: nat] :
% 16.19/8.91 ( ( modulo_modulo_nat @ A @ one_one_nat )
% 16.19/8.91 = zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1951_bits__mod__by__1,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( modulo_modulo_int @ A @ one_one_int )
% 16.19/8.91 = zero_zero_int ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1952_bits__mod__by__1,axiom,
% 16.19/8.91 ! [A: code_integer] :
% 16.19/8.91 ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 16.19/8.91 = zero_z3403309356797280102nteger ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1953_mod__by__1,axiom,
% 16.19/8.91 ! [A: nat] :
% 16.19/8.91 ( ( modulo_modulo_nat @ A @ one_one_nat )
% 16.19/8.91 = zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1954_mod__by__1,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( modulo_modulo_int @ A @ one_one_int )
% 16.19/8.91 = zero_zero_int ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1955_mod__by__1,axiom,
% 16.19/8.91 ! [A: code_integer] :
% 16.19/8.91 ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 16.19/8.91 = zero_z3403309356797280102nteger ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1956_power__Suc0__right,axiom,
% 16.19/8.91 ! [A: nat] :
% 16.19/8.91 ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1957_power__Suc0__right,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1958_power__Suc0__right,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1959_power__Suc0__right,axiom,
% 16.19/8.91 ! [A: complex] :
% 16.19/8.91 ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1960_bits__mod__div__trivial,axiom,
% 16.19/8.91 ! [A: nat,B: nat] :
% 16.19/8.91 ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 16.19/8.91 = zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1961_bits__mod__div__trivial,axiom,
% 16.19/8.91 ! [A: int,B: int] :
% 16.19/8.91 ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 16.19/8.91 = zero_zero_int ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1962_bits__mod__div__trivial,axiom,
% 16.19/8.91 ! [A: code_integer,B: code_integer] :
% 16.19/8.91 ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 16.19/8.91 = zero_z3403309356797280102nteger ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1963_mod__div__trivial,axiom,
% 16.19/8.91 ! [A: nat,B: nat] :
% 16.19/8.91 ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 16.19/8.91 = zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1964_mod__div__trivial,axiom,
% 16.19/8.91 ! [A: int,B: int] :
% 16.19/8.91 ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 16.19/8.91 = zero_zero_int ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1965_mod__div__trivial,axiom,
% 16.19/8.91 ! [A: code_integer,B: code_integer] :
% 16.19/8.91 ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 16.19/8.91 = zero_z3403309356797280102nteger ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1966_mod__mult__self4,axiom,
% 16.19/8.91 ! [B: nat,C: nat,A: nat] :
% 16.19/8.91 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 16.19/8.91 = ( modulo_modulo_nat @ A @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1967_mod__mult__self4,axiom,
% 16.19/8.91 ! [B: int,C: int,A: int] :
% 16.19/8.91 ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 16.19/8.91 = ( modulo_modulo_int @ A @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1968_mod__mult__self4,axiom,
% 16.19/8.91 ! [B: code_integer,C: code_integer,A: code_integer] :
% 16.19/8.91 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
% 16.19/8.91 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1969_mod__mult__self3,axiom,
% 16.19/8.91 ! [C: nat,B: nat,A: nat] :
% 16.19/8.91 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 16.19/8.91 = ( modulo_modulo_nat @ A @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1970_mod__mult__self3,axiom,
% 16.19/8.91 ! [C: int,B: int,A: int] :
% 16.19/8.91 ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 16.19/8.91 = ( modulo_modulo_int @ A @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1971_mod__mult__self3,axiom,
% 16.19/8.91 ! [C: code_integer,B: code_integer,A: code_integer] :
% 16.19/8.91 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
% 16.19/8.91 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1972_mod__mult__self2,axiom,
% 16.19/8.91 ! [A: nat,B: nat,C: nat] :
% 16.19/8.91 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 16.19/8.91 = ( modulo_modulo_nat @ A @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1973_mod__mult__self2,axiom,
% 16.19/8.91 ! [A: int,B: int,C: int] :
% 16.19/8.91 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 16.19/8.91 = ( modulo_modulo_int @ A @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1974_mod__mult__self2,axiom,
% 16.19/8.91 ! [A: code_integer,B: code_integer,C: code_integer] :
% 16.19/8.91 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
% 16.19/8.91 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1975_mod__mult__self1,axiom,
% 16.19/8.91 ! [A: nat,C: nat,B: nat] :
% 16.19/8.91 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 16.19/8.91 = ( modulo_modulo_nat @ A @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1976_mod__mult__self1,axiom,
% 16.19/8.91 ! [A: int,C: int,B: int] :
% 16.19/8.91 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 16.19/8.91 = ( modulo_modulo_int @ A @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1977_mod__mult__self1,axiom,
% 16.19/8.91 ! [A: code_integer,C: code_integer,B: code_integer] :
% 16.19/8.91 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
% 16.19/8.91 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1978_less__Suc0,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 16.19/8.91 = ( N2 = zero_zero_nat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1979_zero__less__Suc,axiom,
% 16.19/8.91 ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1980_max__0__1_I4_J,axiom,
% 16.19/8.91 ! [X: num] :
% 16.19/8.91 ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ zero_zero_real )
% 16.19/8.91 = ( numeral_numeral_real @ X ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1981_max__0__1_I4_J,axiom,
% 16.19/8.91 ! [X: num] :
% 16.19/8.91 ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ zero_zero_rat )
% 16.19/8.91 = ( numeral_numeral_rat @ X ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1982_max__0__1_I4_J,axiom,
% 16.19/8.91 ! [X: num] :
% 16.19/8.91 ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ zero_zero_nat )
% 16.19/8.91 = ( numeral_numeral_nat @ X ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1983_max__0__1_I4_J,axiom,
% 16.19/8.91 ! [X: num] :
% 16.19/8.91 ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ zero_zero_int )
% 16.19/8.91 = ( numeral_numeral_int @ X ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1984_max__0__1_I3_J,axiom,
% 16.19/8.91 ! [X: num] :
% 16.19/8.91 ( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X ) )
% 16.19/8.91 = ( numeral_numeral_real @ X ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1985_max__0__1_I3_J,axiom,
% 16.19/8.91 ! [X: num] :
% 16.19/8.91 ( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X ) )
% 16.19/8.91 = ( numeral_numeral_rat @ X ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1986_max__0__1_I3_J,axiom,
% 16.19/8.91 ! [X: num] :
% 16.19/8.91 ( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X ) )
% 16.19/8.91 = ( numeral_numeral_nat @ X ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1987_max__0__1_I3_J,axiom,
% 16.19/8.91 ! [X: num] :
% 16.19/8.91 ( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X ) )
% 16.19/8.91 = ( numeral_numeral_int @ X ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1988_one__eq__mult__iff,axiom,
% 16.19/8.91 ! [M: nat,N2: nat] :
% 16.19/8.91 ( ( ( suc @ zero_zero_nat )
% 16.19/8.91 = ( times_times_nat @ M @ N2 ) )
% 16.19/8.91 = ( ( M
% 16.19/8.91 = ( suc @ zero_zero_nat ) )
% 16.19/8.91 & ( N2
% 16.19/8.91 = ( suc @ zero_zero_nat ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1989_mult__eq__1__iff,axiom,
% 16.19/8.91 ! [M: nat,N2: nat] :
% 16.19/8.91 ( ( ( times_times_nat @ M @ N2 )
% 16.19/8.91 = ( suc @ zero_zero_nat ) )
% 16.19/8.91 = ( ( M
% 16.19/8.91 = ( suc @ zero_zero_nat ) )
% 16.19/8.91 & ( N2
% 16.19/8.91 = ( suc @ zero_zero_nat ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1990_add__gr__0,axiom,
% 16.19/8.91 ! [M: nat,N2: nat] :
% 16.19/8.91 ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N2 ) )
% 16.19/8.91 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 16.19/8.91 | ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1991_max__0__1_I1_J,axiom,
% 16.19/8.91 ( ( ord_max_real @ zero_zero_real @ one_one_real )
% 16.19/8.91 = one_one_real ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1992_max__0__1_I1_J,axiom,
% 16.19/8.91 ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
% 16.19/8.91 = one_one_rat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1993_max__0__1_I1_J,axiom,
% 16.19/8.91 ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
% 16.19/8.91 = one_one_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1994_max__0__1_I1_J,axiom,
% 16.19/8.91 ( ( ord_max_int @ zero_zero_int @ one_one_int )
% 16.19/8.91 = one_one_int ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1995_max__0__1_I2_J,axiom,
% 16.19/8.91 ( ( ord_max_real @ one_one_real @ zero_zero_real )
% 16.19/8.91 = one_one_real ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1996_max__0__1_I2_J,axiom,
% 16.19/8.91 ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
% 16.19/8.91 = one_one_rat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1997_max__0__1_I2_J,axiom,
% 16.19/8.91 ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
% 16.19/8.91 = one_one_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1998_max__0__1_I2_J,axiom,
% 16.19/8.91 ( ( ord_max_int @ one_one_int @ zero_zero_int )
% 16.19/8.91 = one_one_int ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_1999_div__by__Suc__0,axiom,
% 16.19/8.91 ! [M: nat] :
% 16.19/8.91 ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
% 16.19/8.91 = M ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2000_mult__less__cancel2,axiom,
% 16.19/8.91 ! [M: nat,K: nat,N2: nat] :
% 16.19/8.91 ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
% 16.19/8.91 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 16.19/8.91 & ( ord_less_nat @ M @ N2 ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2001_nat__0__less__mult__iff,axiom,
% 16.19/8.91 ! [M: nat,N2: nat] :
% 16.19/8.91 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N2 ) )
% 16.19/8.91 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 16.19/8.91 & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2002_nat__mult__less__cancel__disj,axiom,
% 16.19/8.91 ! [K: nat,M: nat,N2: nat] :
% 16.19/8.91 ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 16.19/8.91 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 16.19/8.91 & ( ord_less_nat @ M @ N2 ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2003_power__Suc__0,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 16.19/8.91 = ( suc @ zero_zero_nat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2004_nat__power__eq__Suc__0__iff,axiom,
% 16.19/8.91 ! [X: nat,M: nat] :
% 16.19/8.91 ( ( ( power_power_nat @ X @ M )
% 16.19/8.91 = ( suc @ zero_zero_nat ) )
% 16.19/8.91 = ( ( M = zero_zero_nat )
% 16.19/8.91 | ( X
% 16.19/8.91 = ( suc @ zero_zero_nat ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2005_zero__less__diff,axiom,
% 16.19/8.91 ! [N2: nat,M: nat] :
% 16.19/8.91 ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M ) )
% 16.19/8.91 = ( ord_less_nat @ M @ N2 ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2006_div__less,axiom,
% 16.19/8.91 ! [M: nat,N2: nat] :
% 16.19/8.91 ( ( ord_less_nat @ M @ N2 )
% 16.19/8.91 => ( ( divide_divide_nat @ M @ N2 )
% 16.19/8.91 = zero_zero_nat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2007_nat__zero__less__power__iff,axiom,
% 16.19/8.91 ! [X: nat,N2: nat] :
% 16.19/8.91 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N2 ) )
% 16.19/8.91 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 16.19/8.91 | ( N2 = zero_zero_nat ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2008_diff__is__0__eq_H,axiom,
% 16.19/8.91 ! [M: nat,N2: nat] :
% 16.19/8.91 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/8.91 => ( ( minus_minus_nat @ M @ N2 )
% 16.19/8.91 = zero_zero_nat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2009_diff__is__0__eq,axiom,
% 16.19/8.91 ! [M: nat,N2: nat] :
% 16.19/8.91 ( ( ( minus_minus_nat @ M @ N2 )
% 16.19/8.91 = zero_zero_nat )
% 16.19/8.91 = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2010_less__one,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ( ( ord_less_nat @ N2 @ one_one_nat )
% 16.19/8.91 = ( N2 = zero_zero_nat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2011_nat__mult__div__cancel__disj,axiom,
% 16.19/8.91 ! [K: nat,M: nat,N2: nat] :
% 16.19/8.91 ( ( ( K = zero_zero_nat )
% 16.19/8.91 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 16.19/8.91 = zero_zero_nat ) )
% 16.19/8.91 & ( ( K != zero_zero_nat )
% 16.19/8.91 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 16.19/8.91 = ( divide_divide_nat @ M @ N2 ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2012_mod__by__Suc__0,axiom,
% 16.19/8.91 ! [M: nat] :
% 16.19/8.91 ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
% 16.19/8.91 = zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2013_zero__le__divide__1__iff,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 16.19/8.91 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2014_zero__le__divide__1__iff,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 16.19/8.91 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2015_divide__le__0__1__iff,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 16.19/8.91 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2016_divide__le__0__1__iff,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 16.19/8.91 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2017_eq__divide__eq__numeral1_I1_J,axiom,
% 16.19/8.91 ! [A: complex,B: complex,W: num] :
% 16.19/8.91 ( ( A
% 16.19/8.91 = ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) ) )
% 16.19/8.91 = ( ( ( ( numera6690914467698888265omplex @ W )
% 16.19/8.91 != zero_zero_complex )
% 16.19/8.91 => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) )
% 16.19/8.91 = B ) )
% 16.19/8.91 & ( ( ( numera6690914467698888265omplex @ W )
% 16.19/8.91 = zero_zero_complex )
% 16.19/8.91 => ( A = zero_zero_complex ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2018_eq__divide__eq__numeral1_I1_J,axiom,
% 16.19/8.91 ! [A: real,B: real,W: num] :
% 16.19/8.91 ( ( A
% 16.19/8.91 = ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 16.19/8.91 = ( ( ( ( numeral_numeral_real @ W )
% 16.19/8.91 != zero_zero_real )
% 16.19/8.91 => ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
% 16.19/8.91 = B ) )
% 16.19/8.91 & ( ( ( numeral_numeral_real @ W )
% 16.19/8.91 = zero_zero_real )
% 16.19/8.91 => ( A = zero_zero_real ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2019_eq__divide__eq__numeral1_I1_J,axiom,
% 16.19/8.91 ! [A: rat,B: rat,W: num] :
% 16.19/8.91 ( ( A
% 16.19/8.91 = ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 16.19/8.91 = ( ( ( ( numeral_numeral_rat @ W )
% 16.19/8.91 != zero_zero_rat )
% 16.19/8.91 => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
% 16.19/8.91 = B ) )
% 16.19/8.91 & ( ( ( numeral_numeral_rat @ W )
% 16.19/8.91 = zero_zero_rat )
% 16.19/8.91 => ( A = zero_zero_rat ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2020_divide__eq__eq__numeral1_I1_J,axiom,
% 16.19/8.91 ! [B: complex,W: num,A: complex] :
% 16.19/8.91 ( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) )
% 16.19/8.91 = A )
% 16.19/8.91 = ( ( ( ( numera6690914467698888265omplex @ W )
% 16.19/8.91 != zero_zero_complex )
% 16.19/8.91 => ( B
% 16.19/8.91 = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) ) )
% 16.19/8.91 & ( ( ( numera6690914467698888265omplex @ W )
% 16.19/8.91 = zero_zero_complex )
% 16.19/8.91 => ( A = zero_zero_complex ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2021_divide__eq__eq__numeral1_I1_J,axiom,
% 16.19/8.91 ! [B: real,W: num,A: real] :
% 16.19/8.91 ( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) )
% 16.19/8.91 = A )
% 16.19/8.91 = ( ( ( ( numeral_numeral_real @ W )
% 16.19/8.91 != zero_zero_real )
% 16.19/8.91 => ( B
% 16.19/8.91 = ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
% 16.19/8.91 & ( ( ( numeral_numeral_real @ W )
% 16.19/8.91 = zero_zero_real )
% 16.19/8.91 => ( A = zero_zero_real ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2022_divide__eq__eq__numeral1_I1_J,axiom,
% 16.19/8.91 ! [B: rat,W: num,A: rat] :
% 16.19/8.91 ( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) )
% 16.19/8.91 = A )
% 16.19/8.91 = ( ( ( ( numeral_numeral_rat @ W )
% 16.19/8.91 != zero_zero_rat )
% 16.19/8.91 => ( B
% 16.19/8.91 = ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
% 16.19/8.91 & ( ( ( numeral_numeral_rat @ W )
% 16.19/8.91 = zero_zero_rat )
% 16.19/8.91 => ( A = zero_zero_rat ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2023_divide__less__0__1__iff,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 16.19/8.91 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2024_divide__less__0__1__iff,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 16.19/8.91 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2025_divide__less__eq__1__neg,axiom,
% 16.19/8.91 ! [A: real,B: real] :
% 16.19/8.91 ( ( ord_less_real @ A @ zero_zero_real )
% 16.19/8.91 => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 16.19/8.91 = ( ord_less_real @ A @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2026_divide__less__eq__1__neg,axiom,
% 16.19/8.91 ! [A: rat,B: rat] :
% 16.19/8.91 ( ( ord_less_rat @ A @ zero_zero_rat )
% 16.19/8.91 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 16.19/8.91 = ( ord_less_rat @ A @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2027_divide__less__eq__1__pos,axiom,
% 16.19/8.91 ! [A: real,B: real] :
% 16.19/8.91 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.91 => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 16.19/8.91 = ( ord_less_real @ B @ A ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2028_divide__less__eq__1__pos,axiom,
% 16.19/8.91 ! [A: rat,B: rat] :
% 16.19/8.91 ( ( ord_less_rat @ zero_zero_rat @ A )
% 16.19/8.91 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 16.19/8.91 = ( ord_less_rat @ B @ A ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2029_less__divide__eq__1__neg,axiom,
% 16.19/8.91 ! [A: real,B: real] :
% 16.19/8.91 ( ( ord_less_real @ A @ zero_zero_real )
% 16.19/8.91 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 16.19/8.91 = ( ord_less_real @ B @ A ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2030_less__divide__eq__1__neg,axiom,
% 16.19/8.91 ! [A: rat,B: rat] :
% 16.19/8.91 ( ( ord_less_rat @ A @ zero_zero_rat )
% 16.19/8.91 => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 16.19/8.91 = ( ord_less_rat @ B @ A ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2031_less__divide__eq__1__pos,axiom,
% 16.19/8.91 ! [A: real,B: real] :
% 16.19/8.91 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.91 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 16.19/8.91 = ( ord_less_real @ A @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2032_less__divide__eq__1__pos,axiom,
% 16.19/8.91 ! [A: rat,B: rat] :
% 16.19/8.91 ( ( ord_less_rat @ zero_zero_rat @ A )
% 16.19/8.91 => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 16.19/8.91 = ( ord_less_rat @ A @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2033_zero__less__divide__1__iff,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 16.19/8.91 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2034_zero__less__divide__1__iff,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 16.19/8.91 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2035_div__mult__self4,axiom,
% 16.19/8.91 ! [B: nat,C: nat,A: nat] :
% 16.19/8.91 ( ( B != zero_zero_nat )
% 16.19/8.91 => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 16.19/8.91 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2036_div__mult__self4,axiom,
% 16.19/8.91 ! [B: int,C: int,A: int] :
% 16.19/8.91 ( ( B != zero_zero_int )
% 16.19/8.91 => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 16.19/8.91 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2037_div__mult__self3,axiom,
% 16.19/8.91 ! [B: nat,C: nat,A: nat] :
% 16.19/8.91 ( ( B != zero_zero_nat )
% 16.19/8.91 => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 16.19/8.91 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2038_div__mult__self3,axiom,
% 16.19/8.91 ! [B: int,C: int,A: int] :
% 16.19/8.91 ( ( B != zero_zero_int )
% 16.19/8.91 => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 16.19/8.91 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2039_div__mult__self2,axiom,
% 16.19/8.91 ! [B: nat,A: nat,C: nat] :
% 16.19/8.91 ( ( B != zero_zero_nat )
% 16.19/8.91 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 16.19/8.91 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2040_div__mult__self2,axiom,
% 16.19/8.91 ! [B: int,A: int,C: int] :
% 16.19/8.91 ( ( B != zero_zero_int )
% 16.19/8.91 => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 16.19/8.91 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2041_div__mult__self1,axiom,
% 16.19/8.91 ! [B: nat,A: nat,C: nat] :
% 16.19/8.91 ( ( B != zero_zero_nat )
% 16.19/8.91 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 16.19/8.91 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2042_div__mult__self1,axiom,
% 16.19/8.91 ! [B: int,A: int,C: int] :
% 16.19/8.91 ( ( B != zero_zero_int )
% 16.19/8.91 => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 16.19/8.91 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2043_nonzero__divide__mult__cancel__right,axiom,
% 16.19/8.91 ! [B: complex,A: complex] :
% 16.19/8.91 ( ( B != zero_zero_complex )
% 16.19/8.91 => ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
% 16.19/8.91 = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2044_nonzero__divide__mult__cancel__right,axiom,
% 16.19/8.91 ! [B: real,A: real] :
% 16.19/8.91 ( ( B != zero_zero_real )
% 16.19/8.91 => ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
% 16.19/8.91 = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2045_nonzero__divide__mult__cancel__right,axiom,
% 16.19/8.91 ! [B: rat,A: rat] :
% 16.19/8.91 ( ( B != zero_zero_rat )
% 16.19/8.91 => ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
% 16.19/8.91 = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2046_nonzero__divide__mult__cancel__left,axiom,
% 16.19/8.91 ! [A: complex,B: complex] :
% 16.19/8.91 ( ( A != zero_zero_complex )
% 16.19/8.91 => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
% 16.19/8.91 = ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2047_nonzero__divide__mult__cancel__left,axiom,
% 16.19/8.91 ! [A: real,B: real] :
% 16.19/8.91 ( ( A != zero_zero_real )
% 16.19/8.91 => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
% 16.19/8.91 = ( divide_divide_real @ one_one_real @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2048_nonzero__divide__mult__cancel__left,axiom,
% 16.19/8.91 ! [A: rat,B: rat] :
% 16.19/8.91 ( ( A != zero_zero_rat )
% 16.19/8.91 => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
% 16.19/8.91 = ( divide_divide_rat @ one_one_rat @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2049_power__eq__0__iff,axiom,
% 16.19/8.91 ! [A: rat,N2: nat] :
% 16.19/8.91 ( ( ( power_power_rat @ A @ N2 )
% 16.19/8.91 = zero_zero_rat )
% 16.19/8.91 = ( ( A = zero_zero_rat )
% 16.19/8.91 & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2050_power__eq__0__iff,axiom,
% 16.19/8.91 ! [A: nat,N2: nat] :
% 16.19/8.91 ( ( ( power_power_nat @ A @ N2 )
% 16.19/8.91 = zero_zero_nat )
% 16.19/8.91 = ( ( A = zero_zero_nat )
% 16.19/8.91 & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2051_power__eq__0__iff,axiom,
% 16.19/8.91 ! [A: real,N2: nat] :
% 16.19/8.91 ( ( ( power_power_real @ A @ N2 )
% 16.19/8.91 = zero_zero_real )
% 16.19/8.91 = ( ( A = zero_zero_real )
% 16.19/8.91 & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2052_power__eq__0__iff,axiom,
% 16.19/8.91 ! [A: int,N2: nat] :
% 16.19/8.91 ( ( ( power_power_int @ A @ N2 )
% 16.19/8.91 = zero_zero_int )
% 16.19/8.91 = ( ( A = zero_zero_int )
% 16.19/8.91 & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2053_power__eq__0__iff,axiom,
% 16.19/8.91 ! [A: complex,N2: nat] :
% 16.19/8.91 ( ( ( power_power_complex @ A @ N2 )
% 16.19/8.91 = zero_zero_complex )
% 16.19/8.91 = ( ( A = zero_zero_complex )
% 16.19/8.91 & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2054_Suc__pred,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.91 => ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
% 16.19/8.91 = N2 ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2055_one__le__mult__iff,axiom,
% 16.19/8.91 ! [M: nat,N2: nat] :
% 16.19/8.91 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) )
% 16.19/8.91 = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
% 16.19/8.91 & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2056_mult__le__cancel2,axiom,
% 16.19/8.91 ! [M: nat,K: nat,N2: nat] :
% 16.19/8.91 ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
% 16.19/8.91 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 16.19/8.91 => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2057_nat__mult__le__cancel__disj,axiom,
% 16.19/8.91 ! [K: nat,M: nat,N2: nat] :
% 16.19/8.91 ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 16.19/8.91 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 16.19/8.91 => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2058_div__mult__self1__is__m,axiom,
% 16.19/8.91 ! [N2: nat,M: nat] :
% 16.19/8.91 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.91 => ( ( divide_divide_nat @ ( times_times_nat @ N2 @ M ) @ N2 )
% 16.19/8.91 = M ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2059_div__mult__self__is__m,axiom,
% 16.19/8.91 ! [N2: nat,M: nat] :
% 16.19/8.91 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.91 => ( ( divide_divide_nat @ ( times_times_nat @ M @ N2 ) @ N2 )
% 16.19/8.91 = M ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2060_Suc__mod__mult__self4,axiom,
% 16.19/8.91 ! [N2: nat,K: nat,M: nat] :
% 16.19/8.91 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N2 @ K ) @ M ) ) @ N2 )
% 16.19/8.91 = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2061_Suc__mod__mult__self3,axiom,
% 16.19/8.91 ! [K: nat,N2: nat,M: nat] :
% 16.19/8.91 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N2 ) @ M ) ) @ N2 )
% 16.19/8.91 = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2062_Suc__mod__mult__self2,axiom,
% 16.19/8.91 ! [M: nat,N2: nat,K: nat] :
% 16.19/8.91 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N2 @ K ) ) ) @ N2 )
% 16.19/8.91 = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2063_Suc__mod__mult__self1,axiom,
% 16.19/8.91 ! [M: nat,K: nat,N2: nat] :
% 16.19/8.91 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N2 ) ) ) @ N2 )
% 16.19/8.91 = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2064_divide__le__eq__1__neg,axiom,
% 16.19/8.91 ! [A: real,B: real] :
% 16.19/8.91 ( ( ord_less_real @ A @ zero_zero_real )
% 16.19/8.91 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 16.19/8.91 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2065_divide__le__eq__1__neg,axiom,
% 16.19/8.91 ! [A: rat,B: rat] :
% 16.19/8.91 ( ( ord_less_rat @ A @ zero_zero_rat )
% 16.19/8.91 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 16.19/8.91 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2066_divide__le__eq__1__pos,axiom,
% 16.19/8.91 ! [A: real,B: real] :
% 16.19/8.91 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.91 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 16.19/8.91 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2067_divide__le__eq__1__pos,axiom,
% 16.19/8.91 ! [A: rat,B: rat] :
% 16.19/8.91 ( ( ord_less_rat @ zero_zero_rat @ A )
% 16.19/8.91 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 16.19/8.91 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2068_le__divide__eq__1__neg,axiom,
% 16.19/8.91 ! [A: real,B: real] :
% 16.19/8.91 ( ( ord_less_real @ A @ zero_zero_real )
% 16.19/8.91 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 16.19/8.91 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2069_le__divide__eq__1__neg,axiom,
% 16.19/8.91 ! [A: rat,B: rat] :
% 16.19/8.91 ( ( ord_less_rat @ A @ zero_zero_rat )
% 16.19/8.91 => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 16.19/8.91 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2070_le__divide__eq__1__pos,axiom,
% 16.19/8.91 ! [A: real,B: real] :
% 16.19/8.91 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.91 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 16.19/8.91 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2071_le__divide__eq__1__pos,axiom,
% 16.19/8.91 ! [A: rat,B: rat] :
% 16.19/8.91 ( ( ord_less_rat @ zero_zero_rat @ A )
% 16.19/8.91 => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 16.19/8.91 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2072_power__strict__decreasing__iff,axiom,
% 16.19/8.91 ! [B: real,M: nat,N2: nat] :
% 16.19/8.91 ( ( ord_less_real @ zero_zero_real @ B )
% 16.19/8.91 => ( ( ord_less_real @ B @ one_one_real )
% 16.19/8.91 => ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N2 ) )
% 16.19/8.91 = ( ord_less_nat @ N2 @ M ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2073_power__strict__decreasing__iff,axiom,
% 16.19/8.91 ! [B: rat,M: nat,N2: nat] :
% 16.19/8.91 ( ( ord_less_rat @ zero_zero_rat @ B )
% 16.19/8.91 => ( ( ord_less_rat @ B @ one_one_rat )
% 16.19/8.91 => ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N2 ) )
% 16.19/8.91 = ( ord_less_nat @ N2 @ M ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2074_power__strict__decreasing__iff,axiom,
% 16.19/8.91 ! [B: nat,M: nat,N2: nat] :
% 16.19/8.91 ( ( ord_less_nat @ zero_zero_nat @ B )
% 16.19/8.91 => ( ( ord_less_nat @ B @ one_one_nat )
% 16.19/8.91 => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N2 ) )
% 16.19/8.91 = ( ord_less_nat @ N2 @ M ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2075_power__strict__decreasing__iff,axiom,
% 16.19/8.91 ! [B: int,M: nat,N2: nat] :
% 16.19/8.91 ( ( ord_less_int @ zero_zero_int @ B )
% 16.19/8.91 => ( ( ord_less_int @ B @ one_one_int )
% 16.19/8.91 => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N2 ) )
% 16.19/8.91 = ( ord_less_nat @ N2 @ M ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2076_zero__eq__power2,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.91 = zero_zero_rat )
% 16.19/8.91 = ( A = zero_zero_rat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2077_zero__eq__power2,axiom,
% 16.19/8.91 ! [A: nat] :
% 16.19/8.91 ( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.91 = zero_zero_nat )
% 16.19/8.91 = ( A = zero_zero_nat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2078_zero__eq__power2,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.91 = zero_zero_real )
% 16.19/8.91 = ( A = zero_zero_real ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2079_zero__eq__power2,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.91 = zero_zero_int )
% 16.19/8.91 = ( A = zero_zero_int ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2080_zero__eq__power2,axiom,
% 16.19/8.91 ! [A: complex] :
% 16.19/8.91 ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.91 = zero_zero_complex )
% 16.19/8.91 = ( A = zero_zero_complex ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2081_power__mono__iff,axiom,
% 16.19/8.91 ! [A: real,B: real,N2: nat] :
% 16.19/8.91 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/8.91 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 16.19/8.91 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.91 => ( ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) )
% 16.19/8.91 = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2082_power__mono__iff,axiom,
% 16.19/8.91 ! [A: rat,B: rat,N2: nat] :
% 16.19/8.91 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/8.91 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 16.19/8.91 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.91 => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) )
% 16.19/8.91 = ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2083_power__mono__iff,axiom,
% 16.19/8.91 ! [A: nat,B: nat,N2: nat] :
% 16.19/8.91 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 16.19/8.91 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 16.19/8.91 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.91 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
% 16.19/8.91 = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2084_power__mono__iff,axiom,
% 16.19/8.91 ! [A: int,B: int,N2: nat] :
% 16.19/8.91 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.91 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 16.19/8.91 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.91 => ( ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
% 16.19/8.91 = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2085_one__mod__two__eq__one,axiom,
% 16.19/8.91 ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.91 = one_one_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2086_one__mod__two__eq__one,axiom,
% 16.19/8.91 ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/8.91 = one_one_int ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2087_one__mod__two__eq__one,axiom,
% 16.19/8.91 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 16.19/8.91 = one_one_Code_integer ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2088_bits__one__mod__two__eq__one,axiom,
% 16.19/8.91 ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.91 = one_one_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2089_bits__one__mod__two__eq__one,axiom,
% 16.19/8.91 ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/8.91 = one_one_int ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2090_bits__one__mod__two__eq__one,axiom,
% 16.19/8.91 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 16.19/8.91 = one_one_Code_integer ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2091_mod2__Suc__Suc,axiom,
% 16.19/8.91 ! [M: nat] :
% 16.19/8.91 ( ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.91 = ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2092_Suc__diff__1,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.91 => ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
% 16.19/8.91 = N2 ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2093_Suc__times__numeral__mod__eq,axiom,
% 16.19/8.91 ! [K: num,N2: nat] :
% 16.19/8.91 ( ( ( numeral_numeral_nat @ K )
% 16.19/8.91 != one_one_nat )
% 16.19/8.91 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N2 ) ) @ ( numeral_numeral_nat @ K ) )
% 16.19/8.91 = one_one_nat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2094_one__div__two__eq__zero,axiom,
% 16.19/8.91 ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.91 = zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2095_one__div__two__eq__zero,axiom,
% 16.19/8.91 ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/8.91 = zero_zero_int ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2096_bits__1__div__2,axiom,
% 16.19/8.91 ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.91 = zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2097_bits__1__div__2,axiom,
% 16.19/8.91 ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/8.91 = zero_zero_int ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2098_power2__eq__iff__nonneg,axiom,
% 16.19/8.91 ! [X: real,Y: real] :
% 16.19/8.91 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/8.91 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/8.91 => ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.91 = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.91 = ( X = Y ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2099_power2__eq__iff__nonneg,axiom,
% 16.19/8.91 ! [X: rat,Y: rat] :
% 16.19/8.91 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 16.19/8.91 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 16.19/8.91 => ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.91 = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.91 = ( X = Y ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2100_power2__eq__iff__nonneg,axiom,
% 16.19/8.91 ! [X: nat,Y: nat] :
% 16.19/8.91 ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 16.19/8.91 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 16.19/8.91 => ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.91 = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.91 = ( X = Y ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2101_power2__eq__iff__nonneg,axiom,
% 16.19/8.91 ! [X: int,Y: int] :
% 16.19/8.91 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 16.19/8.91 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 16.19/8.91 => ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.91 = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.91 = ( X = Y ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2102_power2__less__eq__zero__iff,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
% 16.19/8.91 = ( A = zero_zero_real ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2103_power2__less__eq__zero__iff,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
% 16.19/8.91 = ( A = zero_zero_rat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2104_power2__less__eq__zero__iff,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 16.19/8.91 = ( A = zero_zero_int ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2105_zero__less__power2,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.91 = ( A != zero_zero_real ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2106_zero__less__power2,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.91 = ( A != zero_zero_rat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2107_zero__less__power2,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.91 = ( A != zero_zero_int ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2108_power__decreasing__iff,axiom,
% 16.19/8.91 ! [B: real,M: nat,N2: nat] :
% 16.19/8.91 ( ( ord_less_real @ zero_zero_real @ B )
% 16.19/8.91 => ( ( ord_less_real @ B @ one_one_real )
% 16.19/8.91 => ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N2 ) )
% 16.19/8.91 = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2109_power__decreasing__iff,axiom,
% 16.19/8.91 ! [B: rat,M: nat,N2: nat] :
% 16.19/8.91 ( ( ord_less_rat @ zero_zero_rat @ B )
% 16.19/8.91 => ( ( ord_less_rat @ B @ one_one_rat )
% 16.19/8.91 => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N2 ) )
% 16.19/8.91 = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2110_power__decreasing__iff,axiom,
% 16.19/8.91 ! [B: nat,M: nat,N2: nat] :
% 16.19/8.91 ( ( ord_less_nat @ zero_zero_nat @ B )
% 16.19/8.91 => ( ( ord_less_nat @ B @ one_one_nat )
% 16.19/8.91 => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N2 ) )
% 16.19/8.91 = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2111_power__decreasing__iff,axiom,
% 16.19/8.91 ! [B: int,M: nat,N2: nat] :
% 16.19/8.91 ( ( ord_less_int @ zero_zero_int @ B )
% 16.19/8.91 => ( ( ord_less_int @ B @ one_one_int )
% 16.19/8.91 => ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N2 ) )
% 16.19/8.91 = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2112_sum__power2__eq__zero__iff,axiom,
% 16.19/8.91 ! [X: rat,Y: rat] :
% 16.19/8.91 ( ( ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.91 = zero_zero_rat )
% 16.19/8.91 = ( ( X = zero_zero_rat )
% 16.19/8.91 & ( Y = zero_zero_rat ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2113_sum__power2__eq__zero__iff,axiom,
% 16.19/8.91 ! [X: real,Y: real] :
% 16.19/8.91 ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.91 = zero_zero_real )
% 16.19/8.91 = ( ( X = zero_zero_real )
% 16.19/8.91 & ( Y = zero_zero_real ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2114_sum__power2__eq__zero__iff,axiom,
% 16.19/8.91 ! [X: int,Y: int] :
% 16.19/8.91 ( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.91 = zero_zero_int )
% 16.19/8.91 = ( ( X = zero_zero_int )
% 16.19/8.91 & ( Y = zero_zero_int ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2115_not__mod__2__eq__1__eq__0,axiom,
% 16.19/8.91 ! [A: nat] :
% 16.19/8.91 ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.91 != one_one_nat )
% 16.19/8.91 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.91 = zero_zero_nat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2116_not__mod__2__eq__1__eq__0,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/8.91 != one_one_int )
% 16.19/8.91 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/8.91 = zero_zero_int ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2117_not__mod__2__eq__1__eq__0,axiom,
% 16.19/8.91 ! [A: code_integer] :
% 16.19/8.91 ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 16.19/8.91 != one_one_Code_integer )
% 16.19/8.91 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 16.19/8.91 = zero_z3403309356797280102nteger ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2118_not__mod__2__eq__0__eq__1,axiom,
% 16.19/8.91 ! [A: nat] :
% 16.19/8.91 ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.91 != zero_zero_nat )
% 16.19/8.91 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.91 = one_one_nat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2119_not__mod__2__eq__0__eq__1,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/8.91 != zero_zero_int )
% 16.19/8.91 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/8.91 = one_one_int ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2120_not__mod__2__eq__0__eq__1,axiom,
% 16.19/8.91 ! [A: code_integer] :
% 16.19/8.91 ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 16.19/8.91 != zero_z3403309356797280102nteger )
% 16.19/8.91 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 16.19/8.91 = one_one_Code_integer ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2121_not__mod2__eq__Suc__0__eq__0,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ( ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.91 != ( suc @ zero_zero_nat ) )
% 16.19/8.91 = ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.91 = zero_zero_nat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2122_add__self__mod__2,axiom,
% 16.19/8.91 ! [M: nat] :
% 16.19/8.91 ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.91 = zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2123_mod2__gr__0,axiom,
% 16.19/8.91 ! [M: nat] :
% 16.19/8.91 ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.91 = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.91 = one_one_nat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2124_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 16.19/8.91 ! [B: nat,A: nat] :
% 16.19/8.91 ( ( ord_less_nat @ zero_zero_nat @ B )
% 16.19/8.91 => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B ) @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2125_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 16.19/8.91 ! [B: int,A: int] :
% 16.19/8.91 ( ( ord_less_int @ zero_zero_int @ B )
% 16.19/8.91 => ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2126_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 16.19/8.91 ! [B: code_integer,A: code_integer] :
% 16.19/8.91 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 16.19/8.91 => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2127_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 16.19/8.91 ! [A: code_integer,B: code_integer] :
% 16.19/8.91 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 16.19/8.91 => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2128_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 16.19/8.91 ! [A: nat,B: nat] :
% 16.19/8.91 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 16.19/8.91 => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B ) @ A ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2129_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 16.19/8.91 ! [A: int,B: int] :
% 16.19/8.91 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.91 => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ A ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2130_mod__Suc,axiom,
% 16.19/8.91 ! [M: nat,N2: nat] :
% 16.19/8.91 ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N2 ) )
% 16.19/8.91 = N2 )
% 16.19/8.91 => ( ( modulo_modulo_nat @ ( suc @ M ) @ N2 )
% 16.19/8.91 = zero_zero_nat ) )
% 16.19/8.91 & ( ( ( suc @ ( modulo_modulo_nat @ M @ N2 ) )
% 16.19/8.91 != N2 )
% 16.19/8.91 => ( ( modulo_modulo_nat @ ( suc @ M ) @ N2 )
% 16.19/8.91 = ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2131_VEBT_Osize_I4_J,axiom,
% 16.19/8.91 ! [X21: $o,X222: $o] :
% 16.19/8.91 ( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 16.19/8.91 = zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2132_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
% 16.19/8.91 ! [Uu: $o,Uv: $o,Uw: nat] :
% 16.19/8.91 ~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2133_mod__eq__0D,axiom,
% 16.19/8.91 ! [M: nat,D: nat] :
% 16.19/8.91 ( ( ( modulo_modulo_nat @ M @ D )
% 16.19/8.91 = zero_zero_nat )
% 16.19/8.91 => ? [Q3: nat] :
% 16.19/8.91 ( M
% 16.19/8.91 = ( times_times_nat @ D @ Q3 ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2134_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
% 16.19/8.91 ! [A: $o,B: $o,X: nat] :
% 16.19/8.91 ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X )
% 16.19/8.91 = ( ( ( X = zero_zero_nat )
% 16.19/8.91 => A )
% 16.19/8.91 & ( ( X != zero_zero_nat )
% 16.19/8.91 => ( ( ( X = one_one_nat )
% 16.19/8.91 => B )
% 16.19/8.91 & ( X = one_one_nat ) ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2135_mod__eq__self__iff__div__eq__0,axiom,
% 16.19/8.91 ! [A: nat,B: nat] :
% 16.19/8.91 ( ( ( modulo_modulo_nat @ A @ B )
% 16.19/8.91 = A )
% 16.19/8.91 = ( ( divide_divide_nat @ A @ B )
% 16.19/8.91 = zero_zero_nat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2136_mod__eq__self__iff__div__eq__0,axiom,
% 16.19/8.91 ! [A: int,B: int] :
% 16.19/8.91 ( ( ( modulo_modulo_int @ A @ B )
% 16.19/8.91 = A )
% 16.19/8.91 = ( ( divide_divide_int @ A @ B )
% 16.19/8.91 = zero_zero_int ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2137_mod__eq__self__iff__div__eq__0,axiom,
% 16.19/8.91 ! [A: code_integer,B: code_integer] :
% 16.19/8.91 ( ( ( modulo364778990260209775nteger @ A @ B )
% 16.19/8.91 = A )
% 16.19/8.91 = ( ( divide6298287555418463151nteger @ A @ B )
% 16.19/8.91 = zero_z3403309356797280102nteger ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2138_VEBT__internal_Ovalid_H_Ocases,axiom,
% 16.19/8.91 ! [X: produc9072475918466114483BT_nat] :
% 16.19/8.91 ( ! [Uu2: $o,Uv2: $o,D4: nat] :
% 16.19/8.91 ( X
% 16.19/8.91 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D4 ) )
% 16.19/8.91 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
% 16.19/8.91 ( X
% 16.19/8.91 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Deg3 ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2139_mod__less__divisor,axiom,
% 16.19/8.91 ! [N2: nat,M: nat] :
% 16.19/8.91 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.91 => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N2 ) @ N2 ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2140_vebt__delete_Osimps_I1_J,axiom,
% 16.19/8.91 ! [A: $o,B: $o] :
% 16.19/8.91 ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat )
% 16.19/8.91 = ( vEBT_Leaf @ $false @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2141_VEBT__internal_Onaive__member_Ocases,axiom,
% 16.19/8.91 ! [X: produc9072475918466114483BT_nat] :
% 16.19/8.91 ( ! [A3: $o,B2: $o,X4: nat] :
% 16.19/8.91 ( X
% 16.19/8.91 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ X4 ) )
% 16.19/8.91 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
% 16.19/8.91 ( X
% 16.19/8.91 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
% 16.19/8.91 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT,X4: nat] :
% 16.19/8.91 ( X
% 16.19/8.91 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) @ X4 ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2142_vebt__buildup_Osimps_I1_J,axiom,
% 16.19/8.91 ( ( vEBT_vebt_buildup @ zero_zero_nat )
% 16.19/8.91 = ( vEBT_Leaf @ $false @ $false ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2143_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 16.19/8.91 ! [B: code_integer,A: code_integer] :
% 16.19/8.91 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 16.19/8.91 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2144_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 16.19/8.91 ! [B: nat,A: nat] :
% 16.19/8.91 ( ( ord_less_nat @ zero_zero_nat @ B )
% 16.19/8.91 => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2145_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 16.19/8.91 ! [B: int,A: int] :
% 16.19/8.91 ( ( ord_less_int @ zero_zero_int @ B )
% 16.19/8.91 => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2146_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 16.19/8.91 ! [A: code_integer,B: code_integer] :
% 16.19/8.91 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 16.19/8.91 => ( ( ord_le6747313008572928689nteger @ A @ B )
% 16.19/8.91 => ( ( modulo364778990260209775nteger @ A @ B )
% 16.19/8.91 = A ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2147_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 16.19/8.91 ! [A: nat,B: nat] :
% 16.19/8.91 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 16.19/8.91 => ( ( ord_less_nat @ A @ B )
% 16.19/8.91 => ( ( modulo_modulo_nat @ A @ B )
% 16.19/8.91 = A ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2148_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 16.19/8.91 ! [A: int,B: int] :
% 16.19/8.91 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.91 => ( ( ord_less_int @ A @ B )
% 16.19/8.91 => ( ( modulo_modulo_int @ A @ B )
% 16.19/8.91 = A ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2149_cong__exp__iff__simps_I2_J,axiom,
% 16.19/8.91 ! [N2: num,Q2: num] :
% 16.19/8.91 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 16.19/8.91 = zero_zero_nat )
% 16.19/8.91 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q2 ) )
% 16.19/8.91 = zero_zero_nat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2150_cong__exp__iff__simps_I2_J,axiom,
% 16.19/8.91 ! [N2: num,Q2: num] :
% 16.19/8.91 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 16.19/8.91 = zero_zero_int )
% 16.19/8.91 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q2 ) )
% 16.19/8.91 = zero_zero_int ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2151_cong__exp__iff__simps_I2_J,axiom,
% 16.19/8.91 ! [N2: num,Q2: num] :
% 16.19/8.91 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 16.19/8.91 = zero_z3403309356797280102nteger )
% 16.19/8.91 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 16.19/8.91 = zero_z3403309356797280102nteger ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2152_cong__exp__iff__simps_I1_J,axiom,
% 16.19/8.91 ! [N2: num] :
% 16.19/8.91 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ one ) )
% 16.19/8.91 = zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2153_cong__exp__iff__simps_I1_J,axiom,
% 16.19/8.91 ! [N2: num] :
% 16.19/8.91 ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ one ) )
% 16.19/8.91 = zero_zero_int ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2154_cong__exp__iff__simps_I1_J,axiom,
% 16.19/8.91 ! [N2: num] :
% 16.19/8.91 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ one ) )
% 16.19/8.91 = zero_z3403309356797280102nteger ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2155_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
% 16.19/8.91 ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
% 16.19/8.91 ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2156_VEBT__internal_Ooption__shift_Ocases,axiom,
% 16.19/8.91 ! [X: produc8306885398267862888on_nat] :
% 16.19/8.91 ( ! [Uu2: nat > nat > nat,Uv2: option_nat] :
% 16.19/8.91 ( X
% 16.19/8.91 != ( produc8929957630744042906on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
% 16.19/8.91 => ( ! [Uw2: nat > nat > nat,V2: nat] :
% 16.19/8.91 ( X
% 16.19/8.91 != ( produc8929957630744042906on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
% 16.19/8.91 => ~ ! [F2: nat > nat > nat,A3: nat,B2: nat] :
% 16.19/8.91 ( X
% 16.19/8.91 != ( produc8929957630744042906on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A3 ) @ ( some_nat @ B2 ) ) ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2157_VEBT__internal_Ooption__shift_Ocases,axiom,
% 16.19/8.91 ! [X: produc5542196010084753463at_nat] :
% 16.19/8.91 ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
% 16.19/8.91 ( X
% 16.19/8.91 != ( produc2899441246263362727at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
% 16.19/8.91 => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V2: product_prod_nat_nat] :
% 16.19/8.91 ( X
% 16.19/8.91 != ( produc2899441246263362727at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
% 16.19/8.91 => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A3: product_prod_nat_nat,B2: product_prod_nat_nat] :
% 16.19/8.91 ( X
% 16.19/8.91 != ( produc2899441246263362727at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A3 ) @ ( some_P7363390416028606310at_nat @ B2 ) ) ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2158_VEBT__internal_Ooption__shift_Ocases,axiom,
% 16.19/8.91 ! [X: produc1193250871479095198on_num] :
% 16.19/8.91 ( ! [Uu2: num > num > num,Uv2: option_num] :
% 16.19/8.91 ( X
% 16.19/8.91 != ( produc5778274026573060048on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
% 16.19/8.91 => ( ! [Uw2: num > num > num,V2: num] :
% 16.19/8.91 ( X
% 16.19/8.91 != ( produc5778274026573060048on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
% 16.19/8.91 => ~ ! [F2: num > num > num,A3: num,B2: num] :
% 16.19/8.91 ( X
% 16.19/8.91 != ( produc5778274026573060048on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ A3 ) @ ( some_num @ B2 ) ) ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2159_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 16.19/8.91 ! [X: produc2233624965454879586on_nat] :
% 16.19/8.91 ( ! [Uu2: nat > nat > $o,Uv2: option_nat] :
% 16.19/8.91 ( X
% 16.19/8.91 != ( produc4035269172776083154on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
% 16.19/8.91 => ( ! [Uw2: nat > nat > $o,V2: nat] :
% 16.19/8.91 ( X
% 16.19/8.91 != ( produc4035269172776083154on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
% 16.19/8.91 => ~ ! [F2: nat > nat > $o,X4: nat,Y4: nat] :
% 16.19/8.91 ( X
% 16.19/8.91 != ( produc4035269172776083154on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ X4 ) @ ( some_nat @ Y4 ) ) ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2160_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 16.19/8.91 ! [X: produc5491161045314408544at_nat] :
% 16.19/8.91 ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > $o,Uv2: option4927543243414619207at_nat] :
% 16.19/8.91 ( X
% 16.19/8.91 != ( produc3994169339658061776at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
% 16.19/8.91 => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > $o,V2: product_prod_nat_nat] :
% 16.19/8.91 ( X
% 16.19/8.91 != ( produc3994169339658061776at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
% 16.19/8.91 => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,X4: product_prod_nat_nat,Y4: product_prod_nat_nat] :
% 16.19/8.91 ( X
% 16.19/8.91 != ( produc3994169339658061776at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X4 ) @ ( some_P7363390416028606310at_nat @ Y4 ) ) ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2161_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 16.19/8.91 ! [X: produc7036089656553540234on_num] :
% 16.19/8.91 ( ! [Uu2: num > num > $o,Uv2: option_num] :
% 16.19/8.91 ( X
% 16.19/8.91 != ( produc3576312749637752826on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
% 16.19/8.91 => ( ! [Uw2: num > num > $o,V2: num] :
% 16.19/8.91 ( X
% 16.19/8.91 != ( produc3576312749637752826on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
% 16.19/8.91 => ~ ! [F2: num > num > $o,X4: num,Y4: num] :
% 16.19/8.91 ( X
% 16.19/8.91 != ( produc3576312749637752826on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ X4 ) @ ( some_num @ Y4 ) ) ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2162_mod__le__divisor,axiom,
% 16.19/8.91 ! [N2: nat,M: nat] :
% 16.19/8.91 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.91 => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N2 ) @ N2 ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2163_invar__vebt_Ointros_I1_J,axiom,
% 16.19/8.91 ! [A: $o,B: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2164_vebt__delete_Osimps_I2_J,axiom,
% 16.19/8.91 ! [A: $o,B: $o] :
% 16.19/8.91 ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) )
% 16.19/8.91 = ( vEBT_Leaf @ A @ $false ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2165_power__0__left,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ( ( ( N2 = zero_zero_nat )
% 16.19/8.91 => ( ( power_power_rat @ zero_zero_rat @ N2 )
% 16.19/8.91 = one_one_rat ) )
% 16.19/8.91 & ( ( N2 != zero_zero_nat )
% 16.19/8.91 => ( ( power_power_rat @ zero_zero_rat @ N2 )
% 16.19/8.91 = zero_zero_rat ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2166_power__0__left,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ( ( ( N2 = zero_zero_nat )
% 16.19/8.91 => ( ( power_power_nat @ zero_zero_nat @ N2 )
% 16.19/8.91 = one_one_nat ) )
% 16.19/8.91 & ( ( N2 != zero_zero_nat )
% 16.19/8.91 => ( ( power_power_nat @ zero_zero_nat @ N2 )
% 16.19/8.91 = zero_zero_nat ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2167_power__0__left,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ( ( ( N2 = zero_zero_nat )
% 16.19/8.91 => ( ( power_power_real @ zero_zero_real @ N2 )
% 16.19/8.91 = one_one_real ) )
% 16.19/8.91 & ( ( N2 != zero_zero_nat )
% 16.19/8.91 => ( ( power_power_real @ zero_zero_real @ N2 )
% 16.19/8.91 = zero_zero_real ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2168_power__0__left,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ( ( ( N2 = zero_zero_nat )
% 16.19/8.91 => ( ( power_power_int @ zero_zero_int @ N2 )
% 16.19/8.91 = one_one_int ) )
% 16.19/8.91 & ( ( N2 != zero_zero_nat )
% 16.19/8.91 => ( ( power_power_int @ zero_zero_int @ N2 )
% 16.19/8.91 = zero_zero_int ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2169_power__0__left,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ( ( ( N2 = zero_zero_nat )
% 16.19/8.91 => ( ( power_power_complex @ zero_zero_complex @ N2 )
% 16.19/8.91 = one_one_complex ) )
% 16.19/8.91 & ( ( N2 != zero_zero_nat )
% 16.19/8.91 => ( ( power_power_complex @ zero_zero_complex @ N2 )
% 16.19/8.91 = zero_zero_complex ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2170_zero__power,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.91 => ( ( power_power_rat @ zero_zero_rat @ N2 )
% 16.19/8.91 = zero_zero_rat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2171_zero__power,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.91 => ( ( power_power_nat @ zero_zero_nat @ N2 )
% 16.19/8.91 = zero_zero_nat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2172_zero__power,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.91 => ( ( power_power_real @ zero_zero_real @ N2 )
% 16.19/8.91 = zero_zero_real ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2173_zero__power,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.91 => ( ( power_power_int @ zero_zero_int @ N2 )
% 16.19/8.91 = zero_zero_int ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2174_zero__power,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.91 => ( ( power_power_complex @ zero_zero_complex @ N2 )
% 16.19/8.91 = zero_zero_complex ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2175_mod__mult__right__eq,axiom,
% 16.19/8.91 ! [A: nat,B: nat,C: nat] :
% 16.19/8.91 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 16.19/8.91 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2176_mod__mult__right__eq,axiom,
% 16.19/8.91 ! [A: int,B: int,C: int] :
% 16.19/8.91 ( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 16.19/8.91 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2177_mod__mult__right__eq,axiom,
% 16.19/8.91 ! [A: code_integer,B: code_integer,C: code_integer] :
% 16.19/8.91 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 16.19/8.91 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2178_mod__mult__left__eq,axiom,
% 16.19/8.91 ! [A: nat,C: nat,B: nat] :
% 16.19/8.91 ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
% 16.19/8.91 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2179_mod__mult__left__eq,axiom,
% 16.19/8.91 ! [A: int,C: int,B: int] :
% 16.19/8.91 ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 16.19/8.91 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2180_mod__mult__left__eq,axiom,
% 16.19/8.91 ! [A: code_integer,C: code_integer,B: code_integer] :
% 16.19/8.91 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 16.19/8.91 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2181_mult__mod__right,axiom,
% 16.19/8.91 ! [C: nat,A: nat,B: nat] :
% 16.19/8.91 ( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 16.19/8.91 = ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2182_mult__mod__right,axiom,
% 16.19/8.91 ! [C: int,A: int,B: int] :
% 16.19/8.91 ( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 16.19/8.91 = ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2183_mult__mod__right,axiom,
% 16.19/8.91 ! [C: code_integer,A: code_integer,B: code_integer] :
% 16.19/8.91 ( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 16.19/8.91 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2184_mod__mult__mult2,axiom,
% 16.19/8.91 ! [A: nat,C: nat,B: nat] :
% 16.19/8.91 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 16.19/8.91 = ( times_times_nat @ ( modulo_modulo_nat @ A @ B ) @ C ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2185_mod__mult__mult2,axiom,
% 16.19/8.91 ! [A: int,C: int,B: int] :
% 16.19/8.91 ( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 16.19/8.91 = ( times_times_int @ ( modulo_modulo_int @ A @ B ) @ C ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2186_mod__mult__mult2,axiom,
% 16.19/8.91 ! [A: code_integer,C: code_integer,B: code_integer] :
% 16.19/8.91 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 16.19/8.91 = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2187_mod__mult__cong,axiom,
% 16.19/8.91 ! [A: nat,C: nat,A6: nat,B: nat,B6: nat] :
% 16.19/8.91 ( ( ( modulo_modulo_nat @ A @ C )
% 16.19/8.91 = ( modulo_modulo_nat @ A6 @ C ) )
% 16.19/8.91 => ( ( ( modulo_modulo_nat @ B @ C )
% 16.19/8.91 = ( modulo_modulo_nat @ B6 @ C ) )
% 16.19/8.91 => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
% 16.19/8.91 = ( modulo_modulo_nat @ ( times_times_nat @ A6 @ B6 ) @ C ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2188_mod__mult__cong,axiom,
% 16.19/8.91 ! [A: int,C: int,A6: int,B: int,B6: int] :
% 16.19/8.91 ( ( ( modulo_modulo_int @ A @ C )
% 16.19/8.91 = ( modulo_modulo_int @ A6 @ C ) )
% 16.19/8.91 => ( ( ( modulo_modulo_int @ B @ C )
% 16.19/8.91 = ( modulo_modulo_int @ B6 @ C ) )
% 16.19/8.91 => ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C )
% 16.19/8.91 = ( modulo_modulo_int @ ( times_times_int @ A6 @ B6 ) @ C ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2189_mod__mult__cong,axiom,
% 16.19/8.91 ! [A: code_integer,C: code_integer,A6: code_integer,B: code_integer,B6: code_integer] :
% 16.19/8.91 ( ( ( modulo364778990260209775nteger @ A @ C )
% 16.19/8.91 = ( modulo364778990260209775nteger @ A6 @ C ) )
% 16.19/8.91 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 16.19/8.91 = ( modulo364778990260209775nteger @ B6 @ C ) )
% 16.19/8.91 => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 16.19/8.91 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A6 @ B6 ) @ C ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2190_mod__mult__eq,axiom,
% 16.19/8.91 ! [A: nat,C: nat,B: nat] :
% 16.19/8.91 ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 16.19/8.91 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2191_mod__mult__eq,axiom,
% 16.19/8.91 ! [A: int,C: int,B: int] :
% 16.19/8.91 ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 16.19/8.91 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2192_mod__mult__eq,axiom,
% 16.19/8.91 ! [A: code_integer,C: code_integer,B: code_integer] :
% 16.19/8.91 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 16.19/8.91 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2193_mod__add__right__eq,axiom,
% 16.19/8.91 ! [A: nat,B: nat,C: nat] :
% 16.19/8.91 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 16.19/8.91 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2194_mod__add__right__eq,axiom,
% 16.19/8.91 ! [A: int,B: int,C: int] :
% 16.19/8.91 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 16.19/8.91 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2195_mod__add__right__eq,axiom,
% 16.19/8.91 ! [A: code_integer,B: code_integer,C: code_integer] :
% 16.19/8.91 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 16.19/8.91 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2196_mod__add__left__eq,axiom,
% 16.19/8.91 ! [A: nat,C: nat,B: nat] :
% 16.19/8.91 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
% 16.19/8.91 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2197_mod__add__left__eq,axiom,
% 16.19/8.91 ! [A: int,C: int,B: int] :
% 16.19/8.91 ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 16.19/8.91 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2198_mod__add__left__eq,axiom,
% 16.19/8.91 ! [A: code_integer,C: code_integer,B: code_integer] :
% 16.19/8.91 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 16.19/8.91 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2199_mod__add__cong,axiom,
% 16.19/8.91 ! [A: nat,C: nat,A6: nat,B: nat,B6: nat] :
% 16.19/8.91 ( ( ( modulo_modulo_nat @ A @ C )
% 16.19/8.91 = ( modulo_modulo_nat @ A6 @ C ) )
% 16.19/8.91 => ( ( ( modulo_modulo_nat @ B @ C )
% 16.19/8.91 = ( modulo_modulo_nat @ B6 @ C ) )
% 16.19/8.91 => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 16.19/8.91 = ( modulo_modulo_nat @ ( plus_plus_nat @ A6 @ B6 ) @ C ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2200_mod__add__cong,axiom,
% 16.19/8.91 ! [A: int,C: int,A6: int,B: int,B6: int] :
% 16.19/8.91 ( ( ( modulo_modulo_int @ A @ C )
% 16.19/8.91 = ( modulo_modulo_int @ A6 @ C ) )
% 16.19/8.91 => ( ( ( modulo_modulo_int @ B @ C )
% 16.19/8.91 = ( modulo_modulo_int @ B6 @ C ) )
% 16.19/8.91 => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
% 16.19/8.91 = ( modulo_modulo_int @ ( plus_plus_int @ A6 @ B6 ) @ C ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2201_mod__add__cong,axiom,
% 16.19/8.91 ! [A: code_integer,C: code_integer,A6: code_integer,B: code_integer,B6: code_integer] :
% 16.19/8.91 ( ( ( modulo364778990260209775nteger @ A @ C )
% 16.19/8.91 = ( modulo364778990260209775nteger @ A6 @ C ) )
% 16.19/8.91 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 16.19/8.91 = ( modulo364778990260209775nteger @ B6 @ C ) )
% 16.19/8.91 => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 16.19/8.91 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A6 @ B6 ) @ C ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2202_mod__add__eq,axiom,
% 16.19/8.91 ! [A: nat,C: nat,B: nat] :
% 16.19/8.91 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 16.19/8.91 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2203_mod__add__eq,axiom,
% 16.19/8.91 ! [A: int,C: int,B: int] :
% 16.19/8.91 ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 16.19/8.91 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2204_mod__add__eq,axiom,
% 16.19/8.91 ! [A: code_integer,C: code_integer,B: code_integer] :
% 16.19/8.91 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 16.19/8.91 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2205_mod__diff__right__eq,axiom,
% 16.19/8.91 ! [A: int,B: int,C: int] :
% 16.19/8.91 ( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 16.19/8.91 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2206_mod__diff__right__eq,axiom,
% 16.19/8.91 ! [A: code_integer,B: code_integer,C: code_integer] :
% 16.19/8.91 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 16.19/8.91 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2207_mod__diff__left__eq,axiom,
% 16.19/8.91 ! [A: int,C: int,B: int] :
% 16.19/8.91 ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 16.19/8.91 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2208_mod__diff__left__eq,axiom,
% 16.19/8.91 ! [A: code_integer,C: code_integer,B: code_integer] :
% 16.19/8.91 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 16.19/8.91 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2209_mod__diff__cong,axiom,
% 16.19/8.91 ! [A: int,C: int,A6: int,B: int,B6: int] :
% 16.19/8.91 ( ( ( modulo_modulo_int @ A @ C )
% 16.19/8.91 = ( modulo_modulo_int @ A6 @ C ) )
% 16.19/8.91 => ( ( ( modulo_modulo_int @ B @ C )
% 16.19/8.91 = ( modulo_modulo_int @ B6 @ C ) )
% 16.19/8.91 => ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C )
% 16.19/8.91 = ( modulo_modulo_int @ ( minus_minus_int @ A6 @ B6 ) @ C ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2210_mod__diff__cong,axiom,
% 16.19/8.91 ! [A: code_integer,C: code_integer,A6: code_integer,B: code_integer,B6: code_integer] :
% 16.19/8.91 ( ( ( modulo364778990260209775nteger @ A @ C )
% 16.19/8.91 = ( modulo364778990260209775nteger @ A6 @ C ) )
% 16.19/8.91 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 16.19/8.91 = ( modulo364778990260209775nteger @ B6 @ C ) )
% 16.19/8.91 => ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
% 16.19/8.91 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A6 @ B6 ) @ C ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2211_mod__diff__eq,axiom,
% 16.19/8.91 ! [A: int,C: int,B: int] :
% 16.19/8.91 ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 16.19/8.91 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2212_mod__diff__eq,axiom,
% 16.19/8.91 ! [A: code_integer,C: code_integer,B: code_integer] :
% 16.19/8.91 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 16.19/8.91 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2213_power__mod,axiom,
% 16.19/8.91 ! [A: nat,B: nat,N2: nat] :
% 16.19/8.91 ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B ) @ N2 ) @ B )
% 16.19/8.91 = ( modulo_modulo_nat @ ( power_power_nat @ A @ N2 ) @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2214_power__mod,axiom,
% 16.19/8.91 ! [A: int,B: int,N2: nat] :
% 16.19/8.91 ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B ) @ N2 ) @ B )
% 16.19/8.91 = ( modulo_modulo_int @ ( power_power_int @ A @ N2 ) @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2215_power__mod,axiom,
% 16.19/8.91 ! [A: code_integer,B: code_integer,N2: nat] :
% 16.19/8.91 ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B ) @ N2 ) @ B )
% 16.19/8.91 = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ B ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2216_vebt__member_Osimps_I1_J,axiom,
% 16.19/8.91 ! [A: $o,B: $o,X: nat] :
% 16.19/8.91 ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B ) @ X )
% 16.19/8.91 = ( ( ( X = zero_zero_nat )
% 16.19/8.91 => A )
% 16.19/8.91 & ( ( X != zero_zero_nat )
% 16.19/8.91 => ( ( ( X = one_one_nat )
% 16.19/8.91 => B )
% 16.19/8.91 & ( X = one_one_nat ) ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2217_mod__Suc__Suc__eq,axiom,
% 16.19/8.91 ! [M: nat,N2: nat] :
% 16.19/8.91 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) ) @ N2 )
% 16.19/8.91 = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N2 ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2218_mod__Suc__eq,axiom,
% 16.19/8.91 ! [M: nat,N2: nat] :
% 16.19/8.91 ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) @ N2 )
% 16.19/8.91 = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2219_VEBT_Odistinct_I1_J,axiom,
% 16.19/8.91 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] :
% 16.19/8.91 ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 16.19/8.91 != ( vEBT_Leaf @ X21 @ X222 ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2220_VEBT_Oexhaust,axiom,
% 16.19/8.91 ! [Y: vEBT_VEBT] :
% 16.19/8.91 ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
% 16.19/8.91 ( Y
% 16.19/8.91 != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
% 16.19/8.91 => ~ ! [X212: $o,X223: $o] :
% 16.19/8.91 ( Y
% 16.19/8.91 != ( vEBT_Leaf @ X212 @ X223 ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2221_vebt__buildup_Osimps_I2_J,axiom,
% 16.19/8.91 ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
% 16.19/8.91 = ( vEBT_Leaf @ $false @ $false ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2222_mod__less__eq__dividend,axiom,
% 16.19/8.91 ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N2 ) @ M ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2223_vebt__insert_Osimps_I1_J,axiom,
% 16.19/8.91 ! [X: nat,A: $o,B: $o] :
% 16.19/8.91 ( ( ( X = zero_zero_nat )
% 16.19/8.91 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
% 16.19/8.91 = ( vEBT_Leaf @ $true @ B ) ) )
% 16.19/8.91 & ( ( X != zero_zero_nat )
% 16.19/8.91 => ( ( ( X = one_one_nat )
% 16.19/8.91 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
% 16.19/8.91 = ( vEBT_Leaf @ A @ $true ) ) )
% 16.19/8.91 & ( ( X != one_one_nat )
% 16.19/8.91 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
% 16.19/8.91 = ( vEBT_Leaf @ A @ B ) ) ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2224_vebt__pred_Osimps_I1_J,axiom,
% 16.19/8.91 ! [Uu: $o,Uv: $o] :
% 16.19/8.91 ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat )
% 16.19/8.91 = none_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2225_le__numeral__extra_I3_J,axiom,
% 16.19/8.91 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2226_le__numeral__extra_I3_J,axiom,
% 16.19/8.91 ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2227_le__numeral__extra_I3_J,axiom,
% 16.19/8.91 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2228_le__numeral__extra_I3_J,axiom,
% 16.19/8.91 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2229_zero__le,axiom,
% 16.19/8.91 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2230_less__numeral__extra_I3_J,axiom,
% 16.19/8.91 ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2231_less__numeral__extra_I3_J,axiom,
% 16.19/8.91 ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2232_less__numeral__extra_I3_J,axiom,
% 16.19/8.91 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2233_less__numeral__extra_I3_J,axiom,
% 16.19/8.91 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2234_field__lbound__gt__zero,axiom,
% 16.19/8.91 ! [D1: real,D22: real] :
% 16.19/8.91 ( ( ord_less_real @ zero_zero_real @ D1 )
% 16.19/8.91 => ( ( ord_less_real @ zero_zero_real @ D22 )
% 16.19/8.91 => ? [E2: real] :
% 16.19/8.91 ( ( ord_less_real @ zero_zero_real @ E2 )
% 16.19/8.91 & ( ord_less_real @ E2 @ D1 )
% 16.19/8.91 & ( ord_less_real @ E2 @ D22 ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2235_field__lbound__gt__zero,axiom,
% 16.19/8.91 ! [D1: rat,D22: rat] :
% 16.19/8.91 ( ( ord_less_rat @ zero_zero_rat @ D1 )
% 16.19/8.91 => ( ( ord_less_rat @ zero_zero_rat @ D22 )
% 16.19/8.91 => ? [E2: rat] :
% 16.19/8.91 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 16.19/8.91 & ( ord_less_rat @ E2 @ D1 )
% 16.19/8.91 & ( ord_less_rat @ E2 @ D22 ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2236_gr__zeroI,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ( ( N2 != zero_zero_nat )
% 16.19/8.91 => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2237_not__less__zero,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2238_gr__implies__not__zero,axiom,
% 16.19/8.91 ! [M: nat,N2: nat] :
% 16.19/8.91 ( ( ord_less_nat @ M @ N2 )
% 16.19/8.91 => ( N2 != zero_zero_nat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2239_zero__less__iff__neq__zero,axiom,
% 16.19/8.91 ! [N2: nat] :
% 16.19/8.91 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.91 = ( N2 != zero_zero_nat ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2240_zero__neq__numeral,axiom,
% 16.19/8.91 ! [N2: num] :
% 16.19/8.91 ( zero_zero_complex
% 16.19/8.91 != ( numera6690914467698888265omplex @ N2 ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2241_zero__neq__numeral,axiom,
% 16.19/8.91 ! [N2: num] :
% 16.19/8.91 ( zero_zero_real
% 16.19/8.91 != ( numeral_numeral_real @ N2 ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2242_zero__neq__numeral,axiom,
% 16.19/8.91 ! [N2: num] :
% 16.19/8.91 ( zero_zero_rat
% 16.19/8.91 != ( numeral_numeral_rat @ N2 ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2243_zero__neq__numeral,axiom,
% 16.19/8.91 ! [N2: num] :
% 16.19/8.91 ( zero_zero_nat
% 16.19/8.91 != ( numeral_numeral_nat @ N2 ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2244_zero__neq__numeral,axiom,
% 16.19/8.91 ! [N2: num] :
% 16.19/8.91 ( zero_zero_int
% 16.19/8.91 != ( numeral_numeral_int @ N2 ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2245_cong__exp__iff__simps_I9_J,axiom,
% 16.19/8.91 ! [M: num,Q2: num,N2: num] :
% 16.19/8.91 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 16.19/8.91 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 16.19/8.91 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 16.19/8.91 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2246_cong__exp__iff__simps_I9_J,axiom,
% 16.19/8.91 ! [M: num,Q2: num,N2: num] :
% 16.19/8.91 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 16.19/8.91 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 16.19/8.91 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 16.19/8.91 = ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2247_cong__exp__iff__simps_I9_J,axiom,
% 16.19/8.91 ! [M: num,Q2: num,N2: num] :
% 16.19/8.91 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 16.19/8.91 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 16.19/8.91 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 16.19/8.91 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2248_cong__exp__iff__simps_I4_J,axiom,
% 16.19/8.91 ! [M: num,N2: num] :
% 16.19/8.91 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
% 16.19/8.91 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ one ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2249_cong__exp__iff__simps_I4_J,axiom,
% 16.19/8.91 ! [M: num,N2: num] :
% 16.19/8.91 ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
% 16.19/8.91 = ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ one ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2250_cong__exp__iff__simps_I4_J,axiom,
% 16.19/8.91 ! [M: num,N2: num] :
% 16.19/8.91 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
% 16.19/8.91 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2251_mult__right__cancel,axiom,
% 16.19/8.91 ! [C: complex,A: complex,B: complex] :
% 16.19/8.91 ( ( C != zero_zero_complex )
% 16.19/8.91 => ( ( ( times_times_complex @ A @ C )
% 16.19/8.91 = ( times_times_complex @ B @ C ) )
% 16.19/8.91 = ( A = B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2252_mult__right__cancel,axiom,
% 16.19/8.91 ! [C: real,A: real,B: real] :
% 16.19/8.91 ( ( C != zero_zero_real )
% 16.19/8.91 => ( ( ( times_times_real @ A @ C )
% 16.19/8.91 = ( times_times_real @ B @ C ) )
% 16.19/8.91 = ( A = B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2253_mult__right__cancel,axiom,
% 16.19/8.91 ! [C: rat,A: rat,B: rat] :
% 16.19/8.91 ( ( C != zero_zero_rat )
% 16.19/8.91 => ( ( ( times_times_rat @ A @ C )
% 16.19/8.91 = ( times_times_rat @ B @ C ) )
% 16.19/8.91 = ( A = B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2254_mult__right__cancel,axiom,
% 16.19/8.91 ! [C: nat,A: nat,B: nat] :
% 16.19/8.91 ( ( C != zero_zero_nat )
% 16.19/8.91 => ( ( ( times_times_nat @ A @ C )
% 16.19/8.91 = ( times_times_nat @ B @ C ) )
% 16.19/8.91 = ( A = B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2255_mult__right__cancel,axiom,
% 16.19/8.91 ! [C: int,A: int,B: int] :
% 16.19/8.91 ( ( C != zero_zero_int )
% 16.19/8.91 => ( ( ( times_times_int @ A @ C )
% 16.19/8.91 = ( times_times_int @ B @ C ) )
% 16.19/8.91 = ( A = B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2256_mult__left__cancel,axiom,
% 16.19/8.91 ! [C: complex,A: complex,B: complex] :
% 16.19/8.91 ( ( C != zero_zero_complex )
% 16.19/8.91 => ( ( ( times_times_complex @ C @ A )
% 16.19/8.91 = ( times_times_complex @ C @ B ) )
% 16.19/8.91 = ( A = B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2257_mult__left__cancel,axiom,
% 16.19/8.91 ! [C: real,A: real,B: real] :
% 16.19/8.91 ( ( C != zero_zero_real )
% 16.19/8.91 => ( ( ( times_times_real @ C @ A )
% 16.19/8.91 = ( times_times_real @ C @ B ) )
% 16.19/8.91 = ( A = B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2258_mult__left__cancel,axiom,
% 16.19/8.91 ! [C: rat,A: rat,B: rat] :
% 16.19/8.91 ( ( C != zero_zero_rat )
% 16.19/8.91 => ( ( ( times_times_rat @ C @ A )
% 16.19/8.91 = ( times_times_rat @ C @ B ) )
% 16.19/8.91 = ( A = B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2259_mult__left__cancel,axiom,
% 16.19/8.91 ! [C: nat,A: nat,B: nat] :
% 16.19/8.91 ( ( C != zero_zero_nat )
% 16.19/8.91 => ( ( ( times_times_nat @ C @ A )
% 16.19/8.91 = ( times_times_nat @ C @ B ) )
% 16.19/8.91 = ( A = B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2260_mult__left__cancel,axiom,
% 16.19/8.91 ! [C: int,A: int,B: int] :
% 16.19/8.91 ( ( C != zero_zero_int )
% 16.19/8.91 => ( ( ( times_times_int @ C @ A )
% 16.19/8.91 = ( times_times_int @ C @ B ) )
% 16.19/8.91 = ( A = B ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2261_no__zero__divisors,axiom,
% 16.19/8.91 ! [A: complex,B: complex] :
% 16.19/8.91 ( ( A != zero_zero_complex )
% 16.19/8.91 => ( ( B != zero_zero_complex )
% 16.19/8.91 => ( ( times_times_complex @ A @ B )
% 16.19/8.91 != zero_zero_complex ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2262_no__zero__divisors,axiom,
% 16.19/8.91 ! [A: real,B: real] :
% 16.19/8.91 ( ( A != zero_zero_real )
% 16.19/8.91 => ( ( B != zero_zero_real )
% 16.19/8.91 => ( ( times_times_real @ A @ B )
% 16.19/8.91 != zero_zero_real ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2263_no__zero__divisors,axiom,
% 16.19/8.91 ! [A: rat,B: rat] :
% 16.19/8.91 ( ( A != zero_zero_rat )
% 16.19/8.91 => ( ( B != zero_zero_rat )
% 16.19/8.91 => ( ( times_times_rat @ A @ B )
% 16.19/8.91 != zero_zero_rat ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2264_no__zero__divisors,axiom,
% 16.19/8.91 ! [A: nat,B: nat] :
% 16.19/8.91 ( ( A != zero_zero_nat )
% 16.19/8.91 => ( ( B != zero_zero_nat )
% 16.19/8.91 => ( ( times_times_nat @ A @ B )
% 16.19/8.91 != zero_zero_nat ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2265_no__zero__divisors,axiom,
% 16.19/8.91 ! [A: int,B: int] :
% 16.19/8.91 ( ( A != zero_zero_int )
% 16.19/8.91 => ( ( B != zero_zero_int )
% 16.19/8.91 => ( ( times_times_int @ A @ B )
% 16.19/8.91 != zero_zero_int ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2266_divisors__zero,axiom,
% 16.19/8.91 ! [A: complex,B: complex] :
% 16.19/8.91 ( ( ( times_times_complex @ A @ B )
% 16.19/8.91 = zero_zero_complex )
% 16.19/8.91 => ( ( A = zero_zero_complex )
% 16.19/8.91 | ( B = zero_zero_complex ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2267_divisors__zero,axiom,
% 16.19/8.91 ! [A: real,B: real] :
% 16.19/8.91 ( ( ( times_times_real @ A @ B )
% 16.19/8.91 = zero_zero_real )
% 16.19/8.91 => ( ( A = zero_zero_real )
% 16.19/8.91 | ( B = zero_zero_real ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2268_divisors__zero,axiom,
% 16.19/8.91 ! [A: rat,B: rat] :
% 16.19/8.91 ( ( ( times_times_rat @ A @ B )
% 16.19/8.91 = zero_zero_rat )
% 16.19/8.91 => ( ( A = zero_zero_rat )
% 16.19/8.91 | ( B = zero_zero_rat ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2269_divisors__zero,axiom,
% 16.19/8.91 ! [A: nat,B: nat] :
% 16.19/8.91 ( ( ( times_times_nat @ A @ B )
% 16.19/8.91 = zero_zero_nat )
% 16.19/8.91 => ( ( A = zero_zero_nat )
% 16.19/8.91 | ( B = zero_zero_nat ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2270_divisors__zero,axiom,
% 16.19/8.91 ! [A: int,B: int] :
% 16.19/8.91 ( ( ( times_times_int @ A @ B )
% 16.19/8.91 = zero_zero_int )
% 16.19/8.91 => ( ( A = zero_zero_int )
% 16.19/8.91 | ( B = zero_zero_int ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2271_mult__not__zero,axiom,
% 16.19/8.91 ! [A: complex,B: complex] :
% 16.19/8.91 ( ( ( times_times_complex @ A @ B )
% 16.19/8.91 != zero_zero_complex )
% 16.19/8.91 => ( ( A != zero_zero_complex )
% 16.19/8.91 & ( B != zero_zero_complex ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2272_mult__not__zero,axiom,
% 16.19/8.91 ! [A: real,B: real] :
% 16.19/8.91 ( ( ( times_times_real @ A @ B )
% 16.19/8.91 != zero_zero_real )
% 16.19/8.91 => ( ( A != zero_zero_real )
% 16.19/8.91 & ( B != zero_zero_real ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2273_mult__not__zero,axiom,
% 16.19/8.91 ! [A: rat,B: rat] :
% 16.19/8.91 ( ( ( times_times_rat @ A @ B )
% 16.19/8.91 != zero_zero_rat )
% 16.19/8.91 => ( ( A != zero_zero_rat )
% 16.19/8.91 & ( B != zero_zero_rat ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2274_mult__not__zero,axiom,
% 16.19/8.91 ! [A: nat,B: nat] :
% 16.19/8.91 ( ( ( times_times_nat @ A @ B )
% 16.19/8.91 != zero_zero_nat )
% 16.19/8.91 => ( ( A != zero_zero_nat )
% 16.19/8.91 & ( B != zero_zero_nat ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2275_mult__not__zero,axiom,
% 16.19/8.91 ! [A: int,B: int] :
% 16.19/8.91 ( ( ( times_times_int @ A @ B )
% 16.19/8.91 != zero_zero_int )
% 16.19/8.91 => ( ( A != zero_zero_int )
% 16.19/8.91 & ( B != zero_zero_int ) ) ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2276_add_Ogroup__left__neutral,axiom,
% 16.19/8.91 ! [A: complex] :
% 16.19/8.91 ( ( plus_plus_complex @ zero_zero_complex @ A )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2277_add_Ogroup__left__neutral,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( plus_plus_real @ zero_zero_real @ A )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2278_add_Ogroup__left__neutral,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2279_add_Ogroup__left__neutral,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( plus_plus_int @ zero_zero_int @ A )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2280_add_Ocomm__neutral,axiom,
% 16.19/8.91 ! [A: complex] :
% 16.19/8.91 ( ( plus_plus_complex @ A @ zero_zero_complex )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2281_add_Ocomm__neutral,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( plus_plus_real @ A @ zero_zero_real )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2282_add_Ocomm__neutral,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2283_add_Ocomm__neutral,axiom,
% 16.19/8.91 ! [A: nat] :
% 16.19/8.91 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2284_add_Ocomm__neutral,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( plus_plus_int @ A @ zero_zero_int )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2285_comm__monoid__add__class_Oadd__0,axiom,
% 16.19/8.91 ! [A: complex] :
% 16.19/8.91 ( ( plus_plus_complex @ zero_zero_complex @ A )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2286_comm__monoid__add__class_Oadd__0,axiom,
% 16.19/8.91 ! [A: real] :
% 16.19/8.91 ( ( plus_plus_real @ zero_zero_real @ A )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2287_comm__monoid__add__class_Oadd__0,axiom,
% 16.19/8.91 ! [A: rat] :
% 16.19/8.91 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2288_comm__monoid__add__class_Oadd__0,axiom,
% 16.19/8.91 ! [A: nat] :
% 16.19/8.91 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2289_comm__monoid__add__class_Oadd__0,axiom,
% 16.19/8.91 ! [A: int] :
% 16.19/8.91 ( ( plus_plus_int @ zero_zero_int @ A )
% 16.19/8.91 = A ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2290_zero__neq__one,axiom,
% 16.19/8.91 zero_zero_complex != one_one_complex ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2291_zero__neq__one,axiom,
% 16.19/8.91 zero_zero_real != one_one_real ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2292_zero__neq__one,axiom,
% 16.19/8.91 zero_zero_rat != one_one_rat ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2293_zero__neq__one,axiom,
% 16.19/8.91 zero_zero_nat != one_one_nat ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2294_zero__neq__one,axiom,
% 16.19/8.91 zero_zero_int != one_one_int ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2295_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
% 16.19/8.91 ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
% 16.19/8.91 ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz ) ).
% 16.19/8.91
% 16.19/8.91 thf(fact_2296_eq__iff__diff__eq__0,axiom,
% 16.19/8.91 ( ( ^ [Y3: complex,Z2: complex] : ( Y3 = Z2 ) )
% 16.19/8.92 = ( ^ [A5: complex,B5: complex] :
% 16.19/8.92 ( ( minus_minus_complex @ A5 @ B5 )
% 16.19/8.92 = zero_zero_complex ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2297_eq__iff__diff__eq__0,axiom,
% 16.19/8.92 ( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
% 16.19/8.92 = ( ^ [A5: real,B5: real] :
% 16.19/8.92 ( ( minus_minus_real @ A5 @ B5 )
% 16.19/8.92 = zero_zero_real ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2298_eq__iff__diff__eq__0,axiom,
% 16.19/8.92 ( ( ^ [Y3: rat,Z2: rat] : ( Y3 = Z2 ) )
% 16.19/8.92 = ( ^ [A5: rat,B5: rat] :
% 16.19/8.92 ( ( minus_minus_rat @ A5 @ B5 )
% 16.19/8.92 = zero_zero_rat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2299_eq__iff__diff__eq__0,axiom,
% 16.19/8.92 ( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
% 16.19/8.92 = ( ^ [A5: int,B5: int] :
% 16.19/8.92 ( ( minus_minus_int @ A5 @ B5 )
% 16.19/8.92 = zero_zero_int ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2300_power__not__zero,axiom,
% 16.19/8.92 ! [A: rat,N2: nat] :
% 16.19/8.92 ( ( A != zero_zero_rat )
% 16.19/8.92 => ( ( power_power_rat @ A @ N2 )
% 16.19/8.92 != zero_zero_rat ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2301_power__not__zero,axiom,
% 16.19/8.92 ! [A: nat,N2: nat] :
% 16.19/8.92 ( ( A != zero_zero_nat )
% 16.19/8.92 => ( ( power_power_nat @ A @ N2 )
% 16.19/8.92 != zero_zero_nat ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2302_power__not__zero,axiom,
% 16.19/8.92 ! [A: real,N2: nat] :
% 16.19/8.92 ( ( A != zero_zero_real )
% 16.19/8.92 => ( ( power_power_real @ A @ N2 )
% 16.19/8.92 != zero_zero_real ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2303_power__not__zero,axiom,
% 16.19/8.92 ! [A: int,N2: nat] :
% 16.19/8.92 ( ( A != zero_zero_int )
% 16.19/8.92 => ( ( power_power_int @ A @ N2 )
% 16.19/8.92 != zero_zero_int ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2304_power__not__zero,axiom,
% 16.19/8.92 ! [A: complex,N2: nat] :
% 16.19/8.92 ( ( A != zero_zero_complex )
% 16.19/8.92 => ( ( power_power_complex @ A @ N2 )
% 16.19/8.92 != zero_zero_complex ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2305_num_Osize_I4_J,axiom,
% 16.19/8.92 ( ( size_size_num @ one )
% 16.19/8.92 = zero_zero_nat ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2306_VEBT__internal_OminNull_Osimps_I3_J,axiom,
% 16.19/8.92 ! [Uu: $o] :
% 16.19/8.92 ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2307_VEBT__internal_OminNull_Osimps_I2_J,axiom,
% 16.19/8.92 ! [Uv: $o] :
% 16.19/8.92 ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2308_VEBT__internal_OminNull_Osimps_I1_J,axiom,
% 16.19/8.92 vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2309_vebt__buildup_Ocases,axiom,
% 16.19/8.92 ! [X: nat] :
% 16.19/8.92 ( ( X != zero_zero_nat )
% 16.19/8.92 => ( ( X
% 16.19/8.92 != ( suc @ zero_zero_nat ) )
% 16.19/8.92 => ~ ! [Va2: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( suc @ ( suc @ Va2 ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2310_nat_Odistinct_I1_J,axiom,
% 16.19/8.92 ! [X22: nat] :
% 16.19/8.92 ( zero_zero_nat
% 16.19/8.92 != ( suc @ X22 ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2311_old_Onat_Odistinct_I2_J,axiom,
% 16.19/8.92 ! [Nat2: nat] :
% 16.19/8.92 ( ( suc @ Nat2 )
% 16.19/8.92 != zero_zero_nat ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2312_old_Onat_Odistinct_I1_J,axiom,
% 16.19/8.92 ! [Nat2: nat] :
% 16.19/8.92 ( zero_zero_nat
% 16.19/8.92 != ( suc @ Nat2 ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2313_nat_OdiscI,axiom,
% 16.19/8.92 ! [Nat: nat,X22: nat] :
% 16.19/8.92 ( ( Nat
% 16.19/8.92 = ( suc @ X22 ) )
% 16.19/8.92 => ( Nat != zero_zero_nat ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2314_old_Onat_Oexhaust,axiom,
% 16.19/8.92 ! [Y: nat] :
% 16.19/8.92 ( ( Y != zero_zero_nat )
% 16.19/8.92 => ~ ! [Nat3: nat] :
% 16.19/8.92 ( Y
% 16.19/8.92 != ( suc @ Nat3 ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2315_nat__induct,axiom,
% 16.19/8.92 ! [P: nat > $o,N2: nat] :
% 16.19/8.92 ( ( P @ zero_zero_nat )
% 16.19/8.92 => ( ! [N4: nat] :
% 16.19/8.92 ( ( P @ N4 )
% 16.19/8.92 => ( P @ ( suc @ N4 ) ) )
% 16.19/8.92 => ( P @ N2 ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2316_diff__induct,axiom,
% 16.19/8.92 ! [P: nat > nat > $o,M: nat,N2: nat] :
% 16.19/8.92 ( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
% 16.19/8.92 => ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
% 16.19/8.92 => ( ! [X4: nat,Y4: nat] :
% 16.19/8.92 ( ( P @ X4 @ Y4 )
% 16.19/8.92 => ( P @ ( suc @ X4 ) @ ( suc @ Y4 ) ) )
% 16.19/8.92 => ( P @ M @ N2 ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2317_zero__induct,axiom,
% 16.19/8.92 ! [P: nat > $o,K: nat] :
% 16.19/8.92 ( ( P @ K )
% 16.19/8.92 => ( ! [N4: nat] :
% 16.19/8.92 ( ( P @ ( suc @ N4 ) )
% 16.19/8.92 => ( P @ N4 ) )
% 16.19/8.92 => ( P @ zero_zero_nat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2318_Suc__neq__Zero,axiom,
% 16.19/8.92 ! [M: nat] :
% 16.19/8.92 ( ( suc @ M )
% 16.19/8.92 != zero_zero_nat ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2319_Zero__neq__Suc,axiom,
% 16.19/8.92 ! [M: nat] :
% 16.19/8.92 ( zero_zero_nat
% 16.19/8.92 != ( suc @ M ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2320_Zero__not__Suc,axiom,
% 16.19/8.92 ! [M: nat] :
% 16.19/8.92 ( zero_zero_nat
% 16.19/8.92 != ( suc @ M ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2321_not0__implies__Suc,axiom,
% 16.19/8.92 ! [N2: nat] :
% 16.19/8.92 ( ( N2 != zero_zero_nat )
% 16.19/8.92 => ? [M2: nat] :
% 16.19/8.92 ( N2
% 16.19/8.92 = ( suc @ M2 ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2322_bot__nat__0_Oextremum__strict,axiom,
% 16.19/8.92 ! [A: nat] :
% 16.19/8.92 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2323_gr0I,axiom,
% 16.19/8.92 ! [N2: nat] :
% 16.19/8.92 ( ( N2 != zero_zero_nat )
% 16.19/8.92 => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2324_not__gr0,axiom,
% 16.19/8.92 ! [N2: nat] :
% 16.19/8.92 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
% 16.19/8.92 = ( N2 = zero_zero_nat ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2325_not__less0,axiom,
% 16.19/8.92 ! [N2: nat] :
% 16.19/8.92 ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2326_less__zeroE,axiom,
% 16.19/8.92 ! [N2: nat] :
% 16.19/8.92 ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2327_gr__implies__not0,axiom,
% 16.19/8.92 ! [M: nat,N2: nat] :
% 16.19/8.92 ( ( ord_less_nat @ M @ N2 )
% 16.19/8.92 => ( N2 != zero_zero_nat ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2328_infinite__descent0,axiom,
% 16.19/8.92 ! [P: nat > $o,N2: nat] :
% 16.19/8.92 ( ( P @ zero_zero_nat )
% 16.19/8.92 => ( ! [N4: nat] :
% 16.19/8.92 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 16.19/8.92 => ( ~ ( P @ N4 )
% 16.19/8.92 => ? [M5: nat] :
% 16.19/8.92 ( ( ord_less_nat @ M5 @ N4 )
% 16.19/8.92 & ~ ( P @ M5 ) ) ) )
% 16.19/8.92 => ( P @ N2 ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2329_less__eq__nat_Osimps_I1_J,axiom,
% 16.19/8.92 ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2330_bot__nat__0_Oextremum__unique,axiom,
% 16.19/8.92 ! [A: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 16.19/8.92 = ( A = zero_zero_nat ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2331_bot__nat__0_Oextremum__uniqueI,axiom,
% 16.19/8.92 ! [A: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 16.19/8.92 => ( A = zero_zero_nat ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2332_le__0__eq,axiom,
% 16.19/8.92 ! [N2: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
% 16.19/8.92 = ( N2 = zero_zero_nat ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2333_add__eq__self__zero,axiom,
% 16.19/8.92 ! [M: nat,N2: nat] :
% 16.19/8.92 ( ( ( plus_plus_nat @ M @ N2 )
% 16.19/8.92 = M )
% 16.19/8.92 => ( N2 = zero_zero_nat ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2334_plus__nat_Oadd__0,axiom,
% 16.19/8.92 ! [N2: nat] :
% 16.19/8.92 ( ( plus_plus_nat @ zero_zero_nat @ N2 )
% 16.19/8.92 = N2 ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2335_mult__0,axiom,
% 16.19/8.92 ! [N2: nat] :
% 16.19/8.92 ( ( times_times_nat @ zero_zero_nat @ N2 )
% 16.19/8.92 = zero_zero_nat ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2336_nat__mult__eq__cancel__disj,axiom,
% 16.19/8.92 ! [K: nat,M: nat,N2: nat] :
% 16.19/8.92 ( ( ( times_times_nat @ K @ M )
% 16.19/8.92 = ( times_times_nat @ K @ N2 ) )
% 16.19/8.92 = ( ( K = zero_zero_nat )
% 16.19/8.92 | ( M = N2 ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2337_minus__nat_Odiff__0,axiom,
% 16.19/8.92 ! [M: nat] :
% 16.19/8.92 ( ( minus_minus_nat @ M @ zero_zero_nat )
% 16.19/8.92 = M ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2338_diffs0__imp__equal,axiom,
% 16.19/8.92 ! [M: nat,N2: nat] :
% 16.19/8.92 ( ( ( minus_minus_nat @ M @ N2 )
% 16.19/8.92 = zero_zero_nat )
% 16.19/8.92 => ( ( ( minus_minus_nat @ N2 @ M )
% 16.19/8.92 = zero_zero_nat )
% 16.19/8.92 => ( M = N2 ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2339_mod__geq,axiom,
% 16.19/8.92 ! [M: nat,N2: nat] :
% 16.19/8.92 ( ~ ( ord_less_nat @ M @ N2 )
% 16.19/8.92 => ( ( modulo_modulo_nat @ M @ N2 )
% 16.19/8.92 = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2340_nat__mod__eq__iff,axiom,
% 16.19/8.92 ! [X: nat,N2: nat,Y: nat] :
% 16.19/8.92 ( ( ( modulo_modulo_nat @ X @ N2 )
% 16.19/8.92 = ( modulo_modulo_nat @ Y @ N2 ) )
% 16.19/8.92 = ( ? [Q1: nat,Q22: nat] :
% 16.19/8.92 ( ( plus_plus_nat @ X @ ( times_times_nat @ N2 @ Q1 ) )
% 16.19/8.92 = ( plus_plus_nat @ Y @ ( times_times_nat @ N2 @ Q22 ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2341_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 16.19/8.92 ! [C: code_integer,A: code_integer,B: code_integer] :
% 16.19/8.92 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 16.19/8.92 => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 16.19/8.92 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2342_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 16.19/8.92 ! [C: nat,A: nat,B: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 16.19/8.92 => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
% 16.19/8.92 = ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2343_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 16.19/8.92 ! [C: int,A: int,B: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 16.19/8.92 => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
% 16.19/8.92 = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2344_split__mod,axiom,
% 16.19/8.92 ! [P: nat > $o,M: nat,N2: nat] :
% 16.19/8.92 ( ( P @ ( modulo_modulo_nat @ M @ N2 ) )
% 16.19/8.92 = ( ( ( N2 = zero_zero_nat )
% 16.19/8.92 => ( P @ M ) )
% 16.19/8.92 & ( ( N2 != zero_zero_nat )
% 16.19/8.92 => ! [I3: nat,J3: nat] :
% 16.19/8.92 ( ( ord_less_nat @ J3 @ N2 )
% 16.19/8.92 => ( ( M
% 16.19/8.92 = ( plus_plus_nat @ ( times_times_nat @ N2 @ I3 ) @ J3 ) )
% 16.19/8.92 => ( P @ J3 ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2345_power__eq__iff__eq__base,axiom,
% 16.19/8.92 ! [N2: nat,A: real,B: real] :
% 16.19/8.92 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.92 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/8.92 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 16.19/8.92 => ( ( ( power_power_real @ A @ N2 )
% 16.19/8.92 = ( power_power_real @ B @ N2 ) )
% 16.19/8.92 = ( A = B ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2346_power__eq__iff__eq__base,axiom,
% 16.19/8.92 ! [N2: nat,A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/8.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 16.19/8.92 => ( ( ( power_power_rat @ A @ N2 )
% 16.19/8.92 = ( power_power_rat @ B @ N2 ) )
% 16.19/8.92 = ( A = B ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2347_power__eq__iff__eq__base,axiom,
% 16.19/8.92 ! [N2: nat,A: nat,B: nat] :
% 16.19/8.92 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 16.19/8.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 16.19/8.92 => ( ( ( power_power_nat @ A @ N2 )
% 16.19/8.92 = ( power_power_nat @ B @ N2 ) )
% 16.19/8.92 = ( A = B ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2348_power__eq__iff__eq__base,axiom,
% 16.19/8.92 ! [N2: nat,A: int,B: int] :
% 16.19/8.92 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.92 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.92 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 16.19/8.92 => ( ( ( power_power_int @ A @ N2 )
% 16.19/8.92 = ( power_power_int @ B @ N2 ) )
% 16.19/8.92 = ( A = B ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2349_power__eq__imp__eq__base,axiom,
% 16.19/8.92 ! [A: real,N2: nat,B: real] :
% 16.19/8.92 ( ( ( power_power_real @ A @ N2 )
% 16.19/8.92 = ( power_power_real @ B @ N2 ) )
% 16.19/8.92 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/8.92 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 16.19/8.92 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.92 => ( A = B ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2350_power__eq__imp__eq__base,axiom,
% 16.19/8.92 ! [A: rat,N2: nat,B: rat] :
% 16.19/8.92 ( ( ( power_power_rat @ A @ N2 )
% 16.19/8.92 = ( power_power_rat @ B @ N2 ) )
% 16.19/8.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/8.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 16.19/8.92 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.92 => ( A = B ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2351_power__eq__imp__eq__base,axiom,
% 16.19/8.92 ! [A: nat,N2: nat,B: nat] :
% 16.19/8.92 ( ( ( power_power_nat @ A @ N2 )
% 16.19/8.92 = ( power_power_nat @ B @ N2 ) )
% 16.19/8.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 16.19/8.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 16.19/8.92 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.92 => ( A = B ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2352_power__eq__imp__eq__base,axiom,
% 16.19/8.92 ! [A: int,N2: nat,B: int] :
% 16.19/8.92 ( ( ( power_power_int @ A @ N2 )
% 16.19/8.92 = ( power_power_int @ B @ N2 ) )
% 16.19/8.92 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.92 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 16.19/8.92 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.92 => ( A = B ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2353_lambda__zero,axiom,
% 16.19/8.92 ( ( ^ [H: complex] : zero_zero_complex )
% 16.19/8.92 = ( times_times_complex @ zero_zero_complex ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2354_lambda__zero,axiom,
% 16.19/8.92 ( ( ^ [H: real] : zero_zero_real )
% 16.19/8.92 = ( times_times_real @ zero_zero_real ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2355_lambda__zero,axiom,
% 16.19/8.92 ( ( ^ [H: rat] : zero_zero_rat )
% 16.19/8.92 = ( times_times_rat @ zero_zero_rat ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2356_lambda__zero,axiom,
% 16.19/8.92 ( ( ^ [H: nat] : zero_zero_nat )
% 16.19/8.92 = ( times_times_nat @ zero_zero_nat ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2357_lambda__zero,axiom,
% 16.19/8.92 ( ( ^ [H: int] : zero_zero_int )
% 16.19/8.92 = ( times_times_int @ zero_zero_int ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2358_VEBT__internal_Omembermima_Ocases,axiom,
% 16.19/8.92 ! [X: produc9072475918466114483BT_nat] :
% 16.19/8.92 ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
% 16.19/8.92 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
% 16.19/8.92 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X4: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X4 ) )
% 16.19/8.92 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ X4 ) )
% 16.19/8.92 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT,X4: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ X4 ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2359_vebt__delete_Ocases,axiom,
% 16.19/8.92 ! [X: produc9072475918466114483BT_nat] :
% 16.19/8.92 ( ! [A3: $o,B2: $o] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ zero_zero_nat ) )
% 16.19/8.92 => ( ! [A3: $o,B2: $o] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ zero_zero_nat ) ) )
% 16.19/8.92 => ( ! [A3: $o,B2: $o,N4: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ N4 ) ) ) )
% 16.19/8.92 => ( ! [Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu2: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) @ Uu2 ) )
% 16.19/8.92 => ( ! [Mi2: nat,Ma2: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X4: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) @ X4 ) )
% 16.19/8.92 => ( ! [Mi2: nat,Ma2: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X4: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) @ X4 ) )
% 16.19/8.92 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X4 ) ) ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2360_vebt__member_Ocases,axiom,
% 16.19/8.92 ! [X: produc9072475918466114483BT_nat] :
% 16.19/8.92 ( ! [A3: $o,B2: $o,X4: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ X4 ) )
% 16.19/8.92 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X4: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X4 ) )
% 16.19/8.92 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X4: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X4 ) )
% 16.19/8.92 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X4 ) )
% 16.19/8.92 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X4 ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2361_vebt__insert_Ocases,axiom,
% 16.19/8.92 ! [X: produc9072475918466114483BT_nat] :
% 16.19/8.92 ( ! [A3: $o,B2: $o,X4: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ X4 ) )
% 16.19/8.92 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,X4: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) @ X4 ) )
% 16.19/8.92 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT,X4: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) @ X4 ) )
% 16.19/8.92 => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ X4 ) )
% 16.19/8.92 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X4 ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2362_vebt__succ_Ocases,axiom,
% 16.19/8.92 ! [X: produc9072475918466114483BT_nat] :
% 16.19/8.92 ( ! [Uu2: $o,B2: $o] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B2 ) @ zero_zero_nat ) )
% 16.19/8.92 => ( ! [Uv2: $o,Uw2: $o,N4: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N4 ) ) )
% 16.19/8.92 => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Va3 ) )
% 16.19/8.92 => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Ve ) )
% 16.19/8.92 => ( ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi ) )
% 16.19/8.92 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X4 ) ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2363_vebt__pred_Ocases,axiom,
% 16.19/8.92 ! [X: produc9072475918466114483BT_nat] :
% 16.19/8.92 ( ! [Uu2: $o,Uv2: $o] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) )
% 16.19/8.92 => ( ! [A3: $o,Uw2: $o] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) )
% 16.19/8.92 => ( ! [A3: $o,B2: $o,Va2: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ Va2 ) ) ) )
% 16.19/8.92 => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Vb2 ) )
% 16.19/8.92 => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) @ Vf ) )
% 16.19/8.92 => ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj ) )
% 16.19/8.92 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X4 ) ) ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2364_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
% 16.19/8.92 ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
% 16.19/8.92 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va @ Vb ) @ X )
% 16.19/8.92 = ( ( X = Mi )
% 16.19/8.92 | ( X = Ma ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2365_divmod__digit__0_I2_J,axiom,
% 16.19/8.92 ! [B: nat,A: nat] :
% 16.19/8.92 ( ( ord_less_nat @ zero_zero_nat @ B )
% 16.19/8.92 => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 16.19/8.92 => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
% 16.19/8.92 = ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2366_divmod__digit__0_I2_J,axiom,
% 16.19/8.92 ! [B: int,A: int] :
% 16.19/8.92 ( ( ord_less_int @ zero_zero_int @ B )
% 16.19/8.92 => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 16.19/8.92 => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
% 16.19/8.92 = ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2367_divmod__digit__0_I2_J,axiom,
% 16.19/8.92 ! [B: code_integer,A: code_integer] :
% 16.19/8.92 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 16.19/8.92 => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 16.19/8.92 => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
% 16.19/8.92 = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2368_cong__exp__iff__simps_I6_J,axiom,
% 16.19/8.92 ! [Q2: num,N2: num] :
% 16.19/8.92 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 16.19/8.92 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2369_cong__exp__iff__simps_I6_J,axiom,
% 16.19/8.92 ! [Q2: num,N2: num] :
% 16.19/8.92 ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 16.19/8.92 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2370_cong__exp__iff__simps_I6_J,axiom,
% 16.19/8.92 ! [Q2: num,N2: num] :
% 16.19/8.92 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 16.19/8.92 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2371_cong__exp__iff__simps_I8_J,axiom,
% 16.19/8.92 ! [M: num,Q2: num] :
% 16.19/8.92 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 16.19/8.92 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2372_cong__exp__iff__simps_I8_J,axiom,
% 16.19/8.92 ! [M: num,Q2: num] :
% 16.19/8.92 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 16.19/8.92 != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2373_cong__exp__iff__simps_I8_J,axiom,
% 16.19/8.92 ! [M: num,Q2: num] :
% 16.19/8.92 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 16.19/8.92 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2374_mod__eqE,axiom,
% 16.19/8.92 ! [A: int,C: int,B: int] :
% 16.19/8.92 ( ( ( modulo_modulo_int @ A @ C )
% 16.19/8.92 = ( modulo_modulo_int @ B @ C ) )
% 16.19/8.92 => ~ ! [D4: int] :
% 16.19/8.92 ( B
% 16.19/8.92 != ( plus_plus_int @ A @ ( times_times_int @ C @ D4 ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2375_mod__eqE,axiom,
% 16.19/8.92 ! [A: code_integer,C: code_integer,B: code_integer] :
% 16.19/8.92 ( ( ( modulo364778990260209775nteger @ A @ C )
% 16.19/8.92 = ( modulo364778990260209775nteger @ B @ C ) )
% 16.19/8.92 => ~ ! [D4: code_integer] :
% 16.19/8.92 ( B
% 16.19/8.92 != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D4 ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2376_div__add1__eq,axiom,
% 16.19/8.92 ! [A: nat,B: nat,C: nat] :
% 16.19/8.92 ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 16.19/8.92 = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2377_div__add1__eq,axiom,
% 16.19/8.92 ! [A: int,B: int,C: int] :
% 16.19/8.92 ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 16.19/8.92 = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2378_div__add1__eq,axiom,
% 16.19/8.92 ! [A: code_integer,B: code_integer,C: code_integer] :
% 16.19/8.92 ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 16.19/8.92 = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2379_Suc__times__mod__eq,axiom,
% 16.19/8.92 ! [M: nat,N2: nat] :
% 16.19/8.92 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 16.19/8.92 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N2 ) ) @ M )
% 16.19/8.92 = one_one_nat ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2380_mod__induct,axiom,
% 16.19/8.92 ! [P: nat > $o,N2: nat,P4: nat,M: nat] :
% 16.19/8.92 ( ( P @ N2 )
% 16.19/8.92 => ( ( ord_less_nat @ N2 @ P4 )
% 16.19/8.92 => ( ( ord_less_nat @ M @ P4 )
% 16.19/8.92 => ( ! [N4: nat] :
% 16.19/8.92 ( ( ord_less_nat @ N4 @ P4 )
% 16.19/8.92 => ( ( P @ N4 )
% 16.19/8.92 => ( P @ ( modulo_modulo_nat @ ( suc @ N4 ) @ P4 ) ) ) )
% 16.19/8.92 => ( P @ M ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2381_mod__Suc__le__divisor,axiom,
% 16.19/8.92 ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N2 ) ) @ N2 ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2382_nat__mod__eq__lemma,axiom,
% 16.19/8.92 ! [X: nat,N2: nat,Y: nat] :
% 16.19/8.92 ( ( ( modulo_modulo_nat @ X @ N2 )
% 16.19/8.92 = ( modulo_modulo_nat @ Y @ N2 ) )
% 16.19/8.92 => ( ( ord_less_eq_nat @ Y @ X )
% 16.19/8.92 => ? [Q3: nat] :
% 16.19/8.92 ( X
% 16.19/8.92 = ( plus_plus_nat @ Y @ ( times_times_nat @ N2 @ Q3 ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2383_mod__eq__nat2E,axiom,
% 16.19/8.92 ! [M: nat,Q2: nat,N2: nat] :
% 16.19/8.92 ( ( ( modulo_modulo_nat @ M @ Q2 )
% 16.19/8.92 = ( modulo_modulo_nat @ N2 @ Q2 ) )
% 16.19/8.92 => ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/8.92 => ~ ! [S2: nat] :
% 16.19/8.92 ( N2
% 16.19/8.92 != ( plus_plus_nat @ M @ ( times_times_nat @ Q2 @ S2 ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2384_mod__eq__nat1E,axiom,
% 16.19/8.92 ! [M: nat,Q2: nat,N2: nat] :
% 16.19/8.92 ( ( ( modulo_modulo_nat @ M @ Q2 )
% 16.19/8.92 = ( modulo_modulo_nat @ N2 @ Q2 ) )
% 16.19/8.92 => ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/8.92 => ~ ! [S2: nat] :
% 16.19/8.92 ( M
% 16.19/8.92 != ( plus_plus_nat @ N2 @ ( times_times_nat @ Q2 @ S2 ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2385_power__strict__mono,axiom,
% 16.19/8.92 ! [A: real,B: real,N2: nat] :
% 16.19/8.92 ( ( ord_less_real @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/8.92 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.92 => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2386_power__strict__mono,axiom,
% 16.19/8.92 ! [A: rat,B: rat,N2: nat] :
% 16.19/8.92 ( ( ord_less_rat @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/8.92 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.92 => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2387_power__strict__mono,axiom,
% 16.19/8.92 ! [A: nat,B: nat,N2: nat] :
% 16.19/8.92 ( ( ord_less_nat @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 16.19/8.92 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.92 => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2388_power__strict__mono,axiom,
% 16.19/8.92 ! [A: int,B: int,N2: nat] :
% 16.19/8.92 ( ( ord_less_int @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.92 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.92 => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2389_mod__if,axiom,
% 16.19/8.92 ( modulo_modulo_nat
% 16.19/8.92 = ( ^ [M6: nat,N: nat] : ( if_nat @ ( ord_less_nat @ M6 @ N ) @ M6 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M6 @ N ) @ N ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2390_vebt__delete_Osimps_I3_J,axiom,
% 16.19/8.92 ! [A: $o,B: $o,N2: nat] :
% 16.19/8.92 ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N2 ) ) )
% 16.19/8.92 = ( vEBT_Leaf @ A @ B ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2391_le__mod__geq,axiom,
% 16.19/8.92 ! [N2: nat,M: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/8.92 => ( ( modulo_modulo_nat @ M @ N2 )
% 16.19/8.92 = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2392_vebt__maxt_Osimps_I1_J,axiom,
% 16.19/8.92 ! [B: $o,A: $o] :
% 16.19/8.92 ( ( B
% 16.19/8.92 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 16.19/8.92 = ( some_nat @ one_one_nat ) ) )
% 16.19/8.92 & ( ~ B
% 16.19/8.92 => ( ( A
% 16.19/8.92 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 16.19/8.92 = ( some_nat @ zero_zero_nat ) ) )
% 16.19/8.92 & ( ~ A
% 16.19/8.92 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 16.19/8.92 = none_nat ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2393_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 16.19/8.92 => ( ( ord_less_nat @ A @ B )
% 16.19/8.92 => ( ( divide_divide_nat @ A @ B )
% 16.19/8.92 = zero_zero_nat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2394_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.92 => ( ( ord_less_int @ A @ B )
% 16.19/8.92 => ( ( divide_divide_int @ A @ B )
% 16.19/8.92 = zero_zero_int ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2395_div__positive,axiom,
% 16.19/8.92 ! [B: nat,A: nat] :
% 16.19/8.92 ( ( ord_less_nat @ zero_zero_nat @ B )
% 16.19/8.92 => ( ( ord_less_eq_nat @ B @ A )
% 16.19/8.92 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2396_div__positive,axiom,
% 16.19/8.92 ! [B: int,A: int] :
% 16.19/8.92 ( ( ord_less_int @ zero_zero_int @ B )
% 16.19/8.92 => ( ( ord_less_eq_int @ B @ A )
% 16.19/8.92 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2397_vebt__mint_Osimps_I1_J,axiom,
% 16.19/8.92 ! [A: $o,B: $o] :
% 16.19/8.92 ( ( A
% 16.19/8.92 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 16.19/8.92 = ( some_nat @ zero_zero_nat ) ) )
% 16.19/8.92 & ( ~ A
% 16.19/8.92 => ( ( B
% 16.19/8.92 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 16.19/8.92 = ( some_nat @ one_one_nat ) ) )
% 16.19/8.92 & ( ~ B
% 16.19/8.92 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 16.19/8.92 = none_nat ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2398_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 16.19/8.92 ! [C: nat,A: nat,B: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 16.19/8.92 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 16.19/8.92 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2399_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 16.19/8.92 ! [C: int,A: int,B: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 16.19/8.92 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 16.19/8.92 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2400_vebt__pred_Osimps_I2_J,axiom,
% 16.19/8.92 ! [A: $o,Uw: $o] :
% 16.19/8.92 ( ( A
% 16.19/8.92 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
% 16.19/8.92 = ( some_nat @ zero_zero_nat ) ) )
% 16.19/8.92 & ( ~ A
% 16.19/8.92 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw ) @ ( suc @ zero_zero_nat ) )
% 16.19/8.92 = none_nat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2401_vebt__succ_Osimps_I1_J,axiom,
% 16.19/8.92 ! [B: $o,Uu: $o] :
% 16.19/8.92 ( ( B
% 16.19/8.92 => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
% 16.19/8.92 = ( some_nat @ one_one_nat ) ) )
% 16.19/8.92 & ( ~ B
% 16.19/8.92 => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
% 16.19/8.92 = none_nat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2402_zero__le__numeral,axiom,
% 16.19/8.92 ! [N2: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2403_zero__le__numeral,axiom,
% 16.19/8.92 ! [N2: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N2 ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2404_zero__le__numeral,axiom,
% 16.19/8.92 ! [N2: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2405_zero__le__numeral,axiom,
% 16.19/8.92 ! [N2: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2406_not__numeral__le__zero,axiom,
% 16.19/8.92 ! [N2: num] :
% 16.19/8.92 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2407_not__numeral__le__zero,axiom,
% 16.19/8.92 ! [N2: num] :
% 16.19/8.92 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ zero_zero_rat ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2408_not__numeral__le__zero,axiom,
% 16.19/8.92 ! [N2: num] :
% 16.19/8.92 ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2409_not__numeral__le__zero,axiom,
% 16.19/8.92 ! [N2: num] :
% 16.19/8.92 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2410_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 16.19/8.92 ! [A: real,B: real,C: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 16.19/8.92 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2411_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 16.19/8.92 ! [A: rat,B: rat,C: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 16.19/8.92 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2412_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 16.19/8.92 ! [A: nat,B: nat,C: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 16.19/8.92 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2413_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 16.19/8.92 ! [A: int,B: int,C: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 16.19/8.92 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2414_zero__le__mult__iff,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 16.19/8.92 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/8.92 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 16.19/8.92 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 16.19/8.92 & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2415_zero__le__mult__iff,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 16.19/8.92 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/8.92 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 16.19/8.92 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 16.19/8.92 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2416_zero__le__mult__iff,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 16.19/8.92 = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.92 & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 16.19/8.92 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 16.19/8.92 & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2417_mult__nonneg__nonpos2,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/8.92 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 16.19/8.92 => ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2418_mult__nonneg__nonpos2,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/8.92 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 16.19/8.92 => ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2419_mult__nonneg__nonpos2,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 16.19/8.92 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 16.19/8.92 => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2420_mult__nonneg__nonpos2,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.92 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 16.19/8.92 => ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2421_mult__nonpos__nonneg,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 16.19/8.92 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 16.19/8.92 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2422_mult__nonpos__nonneg,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 16.19/8.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 16.19/8.92 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2423_mult__nonpos__nonneg,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 16.19/8.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 16.19/8.92 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2424_mult__nonpos__nonneg,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 16.19/8.92 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 16.19/8.92 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2425_mult__nonneg__nonpos,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/8.92 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 16.19/8.92 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2426_mult__nonneg__nonpos,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/8.92 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 16.19/8.92 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2427_mult__nonneg__nonpos,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 16.19/8.92 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 16.19/8.92 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2428_mult__nonneg__nonpos,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.92 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 16.19/8.92 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2429_mult__nonneg__nonneg,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/8.92 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 16.19/8.92 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2430_mult__nonneg__nonneg,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/8.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 16.19/8.92 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2431_mult__nonneg__nonneg,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 16.19/8.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 16.19/8.92 => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2432_mult__nonneg__nonneg,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.92 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 16.19/8.92 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2433_split__mult__neg__le,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/8.92 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 16.19/8.92 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 16.19/8.92 & ( ord_less_eq_real @ zero_zero_real @ B ) ) )
% 16.19/8.92 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2434_split__mult__neg__le,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/8.92 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 16.19/8.92 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 16.19/8.92 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
% 16.19/8.92 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2435_split__mult__neg__le,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.92 ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 16.19/8.92 & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
% 16.19/8.92 | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 16.19/8.92 & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
% 16.19/8.92 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2436_split__mult__neg__le,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.92 & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 16.19/8.92 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 16.19/8.92 & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
% 16.19/8.92 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2437_mult__le__0__iff,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 16.19/8.92 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/8.92 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 16.19/8.92 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 16.19/8.92 & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2438_mult__le__0__iff,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 16.19/8.92 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/8.92 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 16.19/8.92 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 16.19/8.92 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2439_mult__le__0__iff,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 16.19/8.92 = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.92 & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 16.19/8.92 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 16.19/8.92 & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2440_mult__right__mono,axiom,
% 16.19/8.92 ! [A: real,B: real,C: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 16.19/8.92 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2441_mult__right__mono,axiom,
% 16.19/8.92 ! [A: rat,B: rat,C: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 16.19/8.92 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2442_mult__right__mono,axiom,
% 16.19/8.92 ! [A: nat,B: nat,C: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 16.19/8.92 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2443_mult__right__mono,axiom,
% 16.19/8.92 ! [A: int,B: int,C: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 16.19/8.92 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2444_mult__right__mono__neg,axiom,
% 16.19/8.92 ! [B: real,A: real,C: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ B @ A )
% 16.19/8.92 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 16.19/8.92 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2445_mult__right__mono__neg,axiom,
% 16.19/8.92 ! [B: rat,A: rat,C: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ B @ A )
% 16.19/8.92 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 16.19/8.92 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2446_mult__right__mono__neg,axiom,
% 16.19/8.92 ! [B: int,A: int,C: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ B @ A )
% 16.19/8.92 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 16.19/8.92 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2447_mult__left__mono,axiom,
% 16.19/8.92 ! [A: real,B: real,C: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 16.19/8.92 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2448_mult__left__mono,axiom,
% 16.19/8.92 ! [A: rat,B: rat,C: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 16.19/8.92 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2449_mult__left__mono,axiom,
% 16.19/8.92 ! [A: nat,B: nat,C: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 16.19/8.92 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2450_mult__left__mono,axiom,
% 16.19/8.92 ! [A: int,B: int,C: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 16.19/8.92 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2451_mult__nonpos__nonpos,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 16.19/8.92 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 16.19/8.92 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2452_mult__nonpos__nonpos,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 16.19/8.92 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 16.19/8.92 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2453_mult__nonpos__nonpos,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 16.19/8.92 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 16.19/8.92 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2454_mult__left__mono__neg,axiom,
% 16.19/8.92 ! [B: real,A: real,C: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ B @ A )
% 16.19/8.92 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 16.19/8.92 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2455_mult__left__mono__neg,axiom,
% 16.19/8.92 ! [B: rat,A: rat,C: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ B @ A )
% 16.19/8.92 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 16.19/8.92 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2456_mult__left__mono__neg,axiom,
% 16.19/8.92 ! [B: int,A: int,C: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ B @ A )
% 16.19/8.92 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 16.19/8.92 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2457_split__mult__pos__le,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/8.92 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 16.19/8.92 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 16.19/8.92 & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 16.19/8.92 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2458_split__mult__pos__le,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/8.92 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 16.19/8.92 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 16.19/8.92 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 16.19/8.92 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2459_split__mult__pos__le,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.92 & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 16.19/8.92 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 16.19/8.92 & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 16.19/8.92 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2460_zero__le__square,axiom,
% 16.19/8.92 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2461_zero__le__square,axiom,
% 16.19/8.92 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2462_zero__le__square,axiom,
% 16.19/8.92 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2463_mult__mono_H,axiom,
% 16.19/8.92 ! [A: real,B: real,C: real,D: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_real @ C @ D )
% 16.19/8.92 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/8.92 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 16.19/8.92 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2464_mult__mono_H,axiom,
% 16.19/8.92 ! [A: rat,B: rat,C: rat,D: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_rat @ C @ D )
% 16.19/8.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/8.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 16.19/8.92 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2465_mult__mono_H,axiom,
% 16.19/8.92 ! [A: nat,B: nat,C: nat,D: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_nat @ C @ D )
% 16.19/8.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 16.19/8.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 16.19/8.92 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2466_mult__mono_H,axiom,
% 16.19/8.92 ! [A: int,B: int,C: int,D: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_int @ C @ D )
% 16.19/8.92 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.92 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 16.19/8.92 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2467_mult__mono,axiom,
% 16.19/8.92 ! [A: real,B: real,C: real,D: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_real @ C @ D )
% 16.19/8.92 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 16.19/8.92 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 16.19/8.92 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2468_mult__mono,axiom,
% 16.19/8.92 ! [A: rat,B: rat,C: rat,D: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_rat @ C @ D )
% 16.19/8.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 16.19/8.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 16.19/8.92 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2469_mult__mono,axiom,
% 16.19/8.92 ! [A: nat,B: nat,C: nat,D: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_nat @ C @ D )
% 16.19/8.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 16.19/8.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 16.19/8.92 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2470_mult__mono,axiom,
% 16.19/8.92 ! [A: int,B: int,C: int,D: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_int @ C @ D )
% 16.19/8.92 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 16.19/8.92 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 16.19/8.92 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2471_zero__less__numeral,axiom,
% 16.19/8.92 ! [N2: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2472_zero__less__numeral,axiom,
% 16.19/8.92 ! [N2: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N2 ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2473_zero__less__numeral,axiom,
% 16.19/8.92 ! [N2: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2474_zero__less__numeral,axiom,
% 16.19/8.92 ! [N2: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2475_not__numeral__less__zero,axiom,
% 16.19/8.92 ! [N2: num] :
% 16.19/8.92 ~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2476_not__numeral__less__zero,axiom,
% 16.19/8.92 ! [N2: num] :
% 16.19/8.92 ~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ zero_zero_rat ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2477_not__numeral__less__zero,axiom,
% 16.19/8.92 ! [N2: num] :
% 16.19/8.92 ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2478_not__numeral__less__zero,axiom,
% 16.19/8.92 ! [N2: num] :
% 16.19/8.92 ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2479_add__nonpos__eq__0__iff,axiom,
% 16.19/8.92 ! [X: real,Y: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 16.19/8.92 => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 16.19/8.92 => ( ( ( plus_plus_real @ X @ Y )
% 16.19/8.92 = zero_zero_real )
% 16.19/8.92 = ( ( X = zero_zero_real )
% 16.19/8.92 & ( Y = zero_zero_real ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2480_add__nonpos__eq__0__iff,axiom,
% 16.19/8.92 ! [X: rat,Y: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 16.19/8.92 => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
% 16.19/8.92 => ( ( ( plus_plus_rat @ X @ Y )
% 16.19/8.92 = zero_zero_rat )
% 16.19/8.92 = ( ( X = zero_zero_rat )
% 16.19/8.92 & ( Y = zero_zero_rat ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2481_add__nonpos__eq__0__iff,axiom,
% 16.19/8.92 ! [X: nat,Y: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ X @ zero_zero_nat )
% 16.19/8.92 => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
% 16.19/8.92 => ( ( ( plus_plus_nat @ X @ Y )
% 16.19/8.92 = zero_zero_nat )
% 16.19/8.92 = ( ( X = zero_zero_nat )
% 16.19/8.92 & ( Y = zero_zero_nat ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2482_add__nonpos__eq__0__iff,axiom,
% 16.19/8.92 ! [X: int,Y: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ X @ zero_zero_int )
% 16.19/8.92 => ( ( ord_less_eq_int @ Y @ zero_zero_int )
% 16.19/8.92 => ( ( ( plus_plus_int @ X @ Y )
% 16.19/8.92 = zero_zero_int )
% 16.19/8.92 = ( ( X = zero_zero_int )
% 16.19/8.92 & ( Y = zero_zero_int ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2483_add__nonneg__eq__0__iff,axiom,
% 16.19/8.92 ! [X: real,Y: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/8.92 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/8.92 => ( ( ( plus_plus_real @ X @ Y )
% 16.19/8.92 = zero_zero_real )
% 16.19/8.92 = ( ( X = zero_zero_real )
% 16.19/8.92 & ( Y = zero_zero_real ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2484_add__nonneg__eq__0__iff,axiom,
% 16.19/8.92 ! [X: rat,Y: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 16.19/8.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 16.19/8.92 => ( ( ( plus_plus_rat @ X @ Y )
% 16.19/8.92 = zero_zero_rat )
% 16.19/8.92 = ( ( X = zero_zero_rat )
% 16.19/8.92 & ( Y = zero_zero_rat ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2485_add__nonneg__eq__0__iff,axiom,
% 16.19/8.92 ! [X: nat,Y: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 16.19/8.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 16.19/8.92 => ( ( ( plus_plus_nat @ X @ Y )
% 16.19/8.92 = zero_zero_nat )
% 16.19/8.92 = ( ( X = zero_zero_nat )
% 16.19/8.92 & ( Y = zero_zero_nat ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2486_add__nonneg__eq__0__iff,axiom,
% 16.19/8.92 ! [X: int,Y: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 16.19/8.92 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 16.19/8.92 => ( ( ( plus_plus_int @ X @ Y )
% 16.19/8.92 = zero_zero_int )
% 16.19/8.92 = ( ( X = zero_zero_int )
% 16.19/8.92 & ( Y = zero_zero_int ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2487_add__nonpos__nonpos,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 16.19/8.92 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 16.19/8.92 => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2488_add__nonpos__nonpos,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 16.19/8.92 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 16.19/8.92 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2489_add__nonpos__nonpos,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 16.19/8.92 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 16.19/8.92 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2490_add__nonpos__nonpos,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 16.19/8.92 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 16.19/8.92 => ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2491_add__nonneg__nonneg,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/8.92 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 16.19/8.92 => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2492_add__nonneg__nonneg,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/8.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 16.19/8.92 => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2493_add__nonneg__nonneg,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 16.19/8.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 16.19/8.92 => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2494_add__nonneg__nonneg,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.92 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 16.19/8.92 => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2495_add__increasing2,axiom,
% 16.19/8.92 ! [C: real,B: real,A: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ zero_zero_real @ C )
% 16.19/8.92 => ( ( ord_less_eq_real @ B @ A )
% 16.19/8.92 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2496_add__increasing2,axiom,
% 16.19/8.92 ! [C: rat,B: rat,A: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 16.19/8.92 => ( ( ord_less_eq_rat @ B @ A )
% 16.19/8.92 => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2497_add__increasing2,axiom,
% 16.19/8.92 ! [C: nat,B: nat,A: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 16.19/8.92 => ( ( ord_less_eq_nat @ B @ A )
% 16.19/8.92 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2498_add__increasing2,axiom,
% 16.19/8.92 ! [C: int,B: int,A: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 16.19/8.92 => ( ( ord_less_eq_int @ B @ A )
% 16.19/8.92 => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2499_add__decreasing2,axiom,
% 16.19/8.92 ! [C: real,A: real,B: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ C @ zero_zero_real )
% 16.19/8.92 => ( ( ord_less_eq_real @ A @ B )
% 16.19/8.92 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2500_add__decreasing2,axiom,
% 16.19/8.92 ! [C: rat,A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 16.19/8.92 => ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.92 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2501_add__decreasing2,axiom,
% 16.19/8.92 ! [C: nat,A: nat,B: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 16.19/8.92 => ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.92 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2502_add__decreasing2,axiom,
% 16.19/8.92 ! [C: int,A: int,B: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ C @ zero_zero_int )
% 16.19/8.92 => ( ( ord_less_eq_int @ A @ B )
% 16.19/8.92 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2503_add__increasing,axiom,
% 16.19/8.92 ! [A: real,B: real,C: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/8.92 => ( ( ord_less_eq_real @ B @ C )
% 16.19/8.92 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2504_add__increasing,axiom,
% 16.19/8.92 ! [A: rat,B: rat,C: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/8.92 => ( ( ord_less_eq_rat @ B @ C )
% 16.19/8.92 => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2505_add__increasing,axiom,
% 16.19/8.92 ! [A: nat,B: nat,C: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 16.19/8.92 => ( ( ord_less_eq_nat @ B @ C )
% 16.19/8.92 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2506_add__increasing,axiom,
% 16.19/8.92 ! [A: int,B: int,C: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.92 => ( ( ord_less_eq_int @ B @ C )
% 16.19/8.92 => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2507_add__decreasing,axiom,
% 16.19/8.92 ! [A: real,C: real,B: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 16.19/8.92 => ( ( ord_less_eq_real @ C @ B )
% 16.19/8.92 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2508_add__decreasing,axiom,
% 16.19/8.92 ! [A: rat,C: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 16.19/8.92 => ( ( ord_less_eq_rat @ C @ B )
% 16.19/8.92 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2509_add__decreasing,axiom,
% 16.19/8.92 ! [A: nat,C: nat,B: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 16.19/8.92 => ( ( ord_less_eq_nat @ C @ B )
% 16.19/8.92 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2510_add__decreasing,axiom,
% 16.19/8.92 ! [A: int,C: int,B: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 16.19/8.92 => ( ( ord_less_eq_int @ C @ B )
% 16.19/8.92 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2511_not__one__le__zero,axiom,
% 16.19/8.92 ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2512_not__one__le__zero,axiom,
% 16.19/8.92 ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2513_not__one__le__zero,axiom,
% 16.19/8.92 ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2514_not__one__le__zero,axiom,
% 16.19/8.92 ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2515_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 16.19/8.92 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2516_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 16.19/8.92 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2517_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 16.19/8.92 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2518_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 16.19/8.92 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2519_zero__less__one__class_Ozero__le__one,axiom,
% 16.19/8.92 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2520_zero__less__one__class_Ozero__le__one,axiom,
% 16.19/8.92 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2521_zero__less__one__class_Ozero__le__one,axiom,
% 16.19/8.92 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2522_zero__less__one__class_Ozero__le__one,axiom,
% 16.19/8.92 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2523_mult__neg__neg,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ord_less_real @ A @ zero_zero_real )
% 16.19/8.92 => ( ( ord_less_real @ B @ zero_zero_real )
% 16.19/8.92 => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2524_mult__neg__neg,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_rat @ A @ zero_zero_rat )
% 16.19/8.92 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 16.19/8.92 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2525_mult__neg__neg,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( ord_less_int @ A @ zero_zero_int )
% 16.19/8.92 => ( ( ord_less_int @ B @ zero_zero_int )
% 16.19/8.92 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2526_not__square__less__zero,axiom,
% 16.19/8.92 ! [A: real] :
% 16.19/8.92 ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2527_not__square__less__zero,axiom,
% 16.19/8.92 ! [A: rat] :
% 16.19/8.92 ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2528_not__square__less__zero,axiom,
% 16.19/8.92 ! [A: int] :
% 16.19/8.92 ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2529_mult__less__0__iff,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 16.19/8.92 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.92 & ( ord_less_real @ B @ zero_zero_real ) )
% 16.19/8.92 | ( ( ord_less_real @ A @ zero_zero_real )
% 16.19/8.92 & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2530_mult__less__0__iff,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 16.19/8.92 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 16.19/8.92 & ( ord_less_rat @ B @ zero_zero_rat ) )
% 16.19/8.92 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 16.19/8.92 & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2531_mult__less__0__iff,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 16.19/8.92 = ( ( ( ord_less_int @ zero_zero_int @ A )
% 16.19/8.92 & ( ord_less_int @ B @ zero_zero_int ) )
% 16.19/8.92 | ( ( ord_less_int @ A @ zero_zero_int )
% 16.19/8.92 & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2532_mult__neg__pos,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ord_less_real @ A @ zero_zero_real )
% 16.19/8.92 => ( ( ord_less_real @ zero_zero_real @ B )
% 16.19/8.92 => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2533_mult__neg__pos,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_rat @ A @ zero_zero_rat )
% 16.19/8.92 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 16.19/8.92 => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2534_mult__neg__pos,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.92 ( ( ord_less_nat @ A @ zero_zero_nat )
% 16.19/8.92 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 16.19/8.92 => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2535_mult__neg__pos,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( ord_less_int @ A @ zero_zero_int )
% 16.19/8.92 => ( ( ord_less_int @ zero_zero_int @ B )
% 16.19/8.92 => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2536_mult__pos__neg,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.92 => ( ( ord_less_real @ B @ zero_zero_real )
% 16.19/8.92 => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2537_mult__pos__neg,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_rat @ zero_zero_rat @ A )
% 16.19/8.92 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 16.19/8.92 => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2538_mult__pos__neg,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.92 ( ( ord_less_nat @ zero_zero_nat @ A )
% 16.19/8.92 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 16.19/8.92 => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2539_mult__pos__neg,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( ord_less_int @ zero_zero_int @ A )
% 16.19/8.92 => ( ( ord_less_int @ B @ zero_zero_int )
% 16.19/8.92 => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2540_mult__pos__pos,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.92 => ( ( ord_less_real @ zero_zero_real @ B )
% 16.19/8.92 => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2541_mult__pos__pos,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_rat @ zero_zero_rat @ A )
% 16.19/8.92 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 16.19/8.92 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2542_mult__pos__pos,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.92 ( ( ord_less_nat @ zero_zero_nat @ A )
% 16.19/8.92 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 16.19/8.92 => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2543_mult__pos__pos,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( ord_less_int @ zero_zero_int @ A )
% 16.19/8.92 => ( ( ord_less_int @ zero_zero_int @ B )
% 16.19/8.92 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2544_mult__pos__neg2,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.92 => ( ( ord_less_real @ B @ zero_zero_real )
% 16.19/8.92 => ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2545_mult__pos__neg2,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_rat @ zero_zero_rat @ A )
% 16.19/8.92 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 16.19/8.92 => ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2546_mult__pos__neg2,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.92 ( ( ord_less_nat @ zero_zero_nat @ A )
% 16.19/8.92 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 16.19/8.92 => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2547_mult__pos__neg2,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( ord_less_int @ zero_zero_int @ A )
% 16.19/8.92 => ( ( ord_less_int @ B @ zero_zero_int )
% 16.19/8.92 => ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2548_zero__less__mult__iff,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 16.19/8.92 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.92 & ( ord_less_real @ zero_zero_real @ B ) )
% 16.19/8.92 | ( ( ord_less_real @ A @ zero_zero_real )
% 16.19/8.92 & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2549_zero__less__mult__iff,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 16.19/8.92 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 16.19/8.92 & ( ord_less_rat @ zero_zero_rat @ B ) )
% 16.19/8.92 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 16.19/8.92 & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2550_zero__less__mult__iff,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 16.19/8.92 = ( ( ( ord_less_int @ zero_zero_int @ A )
% 16.19/8.92 & ( ord_less_int @ zero_zero_int @ B ) )
% 16.19/8.92 | ( ( ord_less_int @ A @ zero_zero_int )
% 16.19/8.92 & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2551_zero__less__mult__pos,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 16.19/8.92 => ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.92 => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2552_zero__less__mult__pos,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 16.19/8.92 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 16.19/8.92 => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2553_zero__less__mult__pos,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.92 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
% 16.19/8.92 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 16.19/8.92 => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2554_zero__less__mult__pos,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 16.19/8.92 => ( ( ord_less_int @ zero_zero_int @ A )
% 16.19/8.92 => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2555_zero__less__mult__pos2,axiom,
% 16.19/8.92 ! [B: real,A: real] :
% 16.19/8.92 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
% 16.19/8.92 => ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.92 => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2556_zero__less__mult__pos2,axiom,
% 16.19/8.92 ! [B: rat,A: rat] :
% 16.19/8.92 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
% 16.19/8.92 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 16.19/8.92 => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2557_zero__less__mult__pos2,axiom,
% 16.19/8.92 ! [B: nat,A: nat] :
% 16.19/8.92 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
% 16.19/8.92 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 16.19/8.92 => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2558_zero__less__mult__pos2,axiom,
% 16.19/8.92 ! [B: int,A: int] :
% 16.19/8.92 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
% 16.19/8.92 => ( ( ord_less_int @ zero_zero_int @ A )
% 16.19/8.92 => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2559_mult__less__cancel__left__neg,axiom,
% 16.19/8.92 ! [C: real,A: real,B: real] :
% 16.19/8.92 ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.92 => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 16.19/8.92 = ( ord_less_real @ B @ A ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2560_mult__less__cancel__left__neg,axiom,
% 16.19/8.92 ! [C: rat,A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.92 => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 16.19/8.92 = ( ord_less_rat @ B @ A ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2561_mult__less__cancel__left__neg,axiom,
% 16.19/8.92 ! [C: int,A: int,B: int] :
% 16.19/8.92 ( ( ord_less_int @ C @ zero_zero_int )
% 16.19/8.92 => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 16.19/8.92 = ( ord_less_int @ B @ A ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2562_mult__less__cancel__left__pos,axiom,
% 16.19/8.92 ! [C: real,A: real,B: real] :
% 16.19/8.92 ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.92 => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 16.19/8.92 = ( ord_less_real @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2563_mult__less__cancel__left__pos,axiom,
% 16.19/8.92 ! [C: rat,A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.92 => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 16.19/8.92 = ( ord_less_rat @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2564_mult__less__cancel__left__pos,axiom,
% 16.19/8.92 ! [C: int,A: int,B: int] :
% 16.19/8.92 ( ( ord_less_int @ zero_zero_int @ C )
% 16.19/8.92 => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 16.19/8.92 = ( ord_less_int @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2565_mult__strict__left__mono__neg,axiom,
% 16.19/8.92 ! [B: real,A: real,C: real] :
% 16.19/8.92 ( ( ord_less_real @ B @ A )
% 16.19/8.92 => ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.92 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2566_mult__strict__left__mono__neg,axiom,
% 16.19/8.92 ! [B: rat,A: rat,C: rat] :
% 16.19/8.92 ( ( ord_less_rat @ B @ A )
% 16.19/8.92 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.92 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2567_mult__strict__left__mono__neg,axiom,
% 16.19/8.92 ! [B: int,A: int,C: int] :
% 16.19/8.92 ( ( ord_less_int @ B @ A )
% 16.19/8.92 => ( ( ord_less_int @ C @ zero_zero_int )
% 16.19/8.92 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2568_mult__strict__left__mono,axiom,
% 16.19/8.92 ! [A: real,B: real,C: real] :
% 16.19/8.92 ( ( ord_less_real @ A @ B )
% 16.19/8.92 => ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.92 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2569_mult__strict__left__mono,axiom,
% 16.19/8.92 ! [A: rat,B: rat,C: rat] :
% 16.19/8.92 ( ( ord_less_rat @ A @ B )
% 16.19/8.92 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.92 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2570_mult__strict__left__mono,axiom,
% 16.19/8.92 ! [A: nat,B: nat,C: nat] :
% 16.19/8.92 ( ( ord_less_nat @ A @ B )
% 16.19/8.92 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 16.19/8.92 => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2571_mult__strict__left__mono,axiom,
% 16.19/8.92 ! [A: int,B: int,C: int] :
% 16.19/8.92 ( ( ord_less_int @ A @ B )
% 16.19/8.92 => ( ( ord_less_int @ zero_zero_int @ C )
% 16.19/8.92 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2572_mult__less__cancel__left__disj,axiom,
% 16.19/8.92 ! [C: real,A: real,B: real] :
% 16.19/8.92 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 16.19/8.92 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.92 & ( ord_less_real @ A @ B ) )
% 16.19/8.92 | ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.92 & ( ord_less_real @ B @ A ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2573_mult__less__cancel__left__disj,axiom,
% 16.19/8.92 ! [C: rat,A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 16.19/8.92 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.92 & ( ord_less_rat @ A @ B ) )
% 16.19/8.92 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.92 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2574_mult__less__cancel__left__disj,axiom,
% 16.19/8.92 ! [C: int,A: int,B: int] :
% 16.19/8.92 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 16.19/8.92 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 16.19/8.92 & ( ord_less_int @ A @ B ) )
% 16.19/8.92 | ( ( ord_less_int @ C @ zero_zero_int )
% 16.19/8.92 & ( ord_less_int @ B @ A ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2575_mult__strict__right__mono__neg,axiom,
% 16.19/8.92 ! [B: real,A: real,C: real] :
% 16.19/8.92 ( ( ord_less_real @ B @ A )
% 16.19/8.92 => ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.92 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2576_mult__strict__right__mono__neg,axiom,
% 16.19/8.92 ! [B: rat,A: rat,C: rat] :
% 16.19/8.92 ( ( ord_less_rat @ B @ A )
% 16.19/8.92 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.92 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2577_mult__strict__right__mono__neg,axiom,
% 16.19/8.92 ! [B: int,A: int,C: int] :
% 16.19/8.92 ( ( ord_less_int @ B @ A )
% 16.19/8.92 => ( ( ord_less_int @ C @ zero_zero_int )
% 16.19/8.92 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2578_mult__strict__right__mono,axiom,
% 16.19/8.92 ! [A: real,B: real,C: real] :
% 16.19/8.92 ( ( ord_less_real @ A @ B )
% 16.19/8.92 => ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.92 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2579_mult__strict__right__mono,axiom,
% 16.19/8.92 ! [A: rat,B: rat,C: rat] :
% 16.19/8.92 ( ( ord_less_rat @ A @ B )
% 16.19/8.92 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.92 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2580_mult__strict__right__mono,axiom,
% 16.19/8.92 ! [A: nat,B: nat,C: nat] :
% 16.19/8.92 ( ( ord_less_nat @ A @ B )
% 16.19/8.92 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 16.19/8.92 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2581_mult__strict__right__mono,axiom,
% 16.19/8.92 ! [A: int,B: int,C: int] :
% 16.19/8.92 ( ( ord_less_int @ A @ B )
% 16.19/8.92 => ( ( ord_less_int @ zero_zero_int @ C )
% 16.19/8.92 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2582_mult__less__cancel__right__disj,axiom,
% 16.19/8.92 ! [A: real,C: real,B: real] :
% 16.19/8.92 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 16.19/8.92 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.92 & ( ord_less_real @ A @ B ) )
% 16.19/8.92 | ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.92 & ( ord_less_real @ B @ A ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2583_mult__less__cancel__right__disj,axiom,
% 16.19/8.92 ! [A: rat,C: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 16.19/8.92 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.92 & ( ord_less_rat @ A @ B ) )
% 16.19/8.92 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.92 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2584_mult__less__cancel__right__disj,axiom,
% 16.19/8.92 ! [A: int,C: int,B: int] :
% 16.19/8.92 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 16.19/8.92 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 16.19/8.92 & ( ord_less_int @ A @ B ) )
% 16.19/8.92 | ( ( ord_less_int @ C @ zero_zero_int )
% 16.19/8.92 & ( ord_less_int @ B @ A ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2585_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 16.19/8.92 ! [A: real,B: real,C: real] :
% 16.19/8.92 ( ( ord_less_real @ A @ B )
% 16.19/8.92 => ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.92 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2586_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 16.19/8.92 ! [A: rat,B: rat,C: rat] :
% 16.19/8.92 ( ( ord_less_rat @ A @ B )
% 16.19/8.92 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.92 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2587_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 16.19/8.92 ! [A: nat,B: nat,C: nat] :
% 16.19/8.92 ( ( ord_less_nat @ A @ B )
% 16.19/8.92 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 16.19/8.92 => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2588_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 16.19/8.92 ! [A: int,B: int,C: int] :
% 16.19/8.92 ( ( ord_less_int @ A @ B )
% 16.19/8.92 => ( ( ord_less_int @ zero_zero_int @ C )
% 16.19/8.92 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2589_add__less__zeroD,axiom,
% 16.19/8.92 ! [X: real,Y: real] :
% 16.19/8.92 ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
% 16.19/8.92 => ( ( ord_less_real @ X @ zero_zero_real )
% 16.19/8.92 | ( ord_less_real @ Y @ zero_zero_real ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2590_add__less__zeroD,axiom,
% 16.19/8.92 ! [X: rat,Y: rat] :
% 16.19/8.92 ( ( ord_less_rat @ ( plus_plus_rat @ X @ Y ) @ zero_zero_rat )
% 16.19/8.92 => ( ( ord_less_rat @ X @ zero_zero_rat )
% 16.19/8.92 | ( ord_less_rat @ Y @ zero_zero_rat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2591_add__less__zeroD,axiom,
% 16.19/8.92 ! [X: int,Y: int] :
% 16.19/8.92 ( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
% 16.19/8.92 => ( ( ord_less_int @ X @ zero_zero_int )
% 16.19/8.92 | ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2592_add__neg__neg,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ord_less_real @ A @ zero_zero_real )
% 16.19/8.92 => ( ( ord_less_real @ B @ zero_zero_real )
% 16.19/8.92 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2593_add__neg__neg,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_rat @ A @ zero_zero_rat )
% 16.19/8.92 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 16.19/8.92 => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2594_add__neg__neg,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.92 ( ( ord_less_nat @ A @ zero_zero_nat )
% 16.19/8.92 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 16.19/8.92 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2595_add__neg__neg,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( ord_less_int @ A @ zero_zero_int )
% 16.19/8.92 => ( ( ord_less_int @ B @ zero_zero_int )
% 16.19/8.92 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2596_add__pos__pos,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.92 => ( ( ord_less_real @ zero_zero_real @ B )
% 16.19/8.92 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2597_add__pos__pos,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_rat @ zero_zero_rat @ A )
% 16.19/8.92 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 16.19/8.92 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2598_add__pos__pos,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.92 ( ( ord_less_nat @ zero_zero_nat @ A )
% 16.19/8.92 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 16.19/8.92 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2599_add__pos__pos,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( ord_less_int @ zero_zero_int @ A )
% 16.19/8.92 => ( ( ord_less_int @ zero_zero_int @ B )
% 16.19/8.92 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2600_canonically__ordered__monoid__add__class_OlessE,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.92 ( ( ord_less_nat @ A @ B )
% 16.19/8.92 => ~ ! [C3: nat] :
% 16.19/8.92 ( ( B
% 16.19/8.92 = ( plus_plus_nat @ A @ C3 ) )
% 16.19/8.92 => ( C3 = zero_zero_nat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2601_pos__add__strict,axiom,
% 16.19/8.92 ! [A: real,B: real,C: real] :
% 16.19/8.92 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.92 => ( ( ord_less_real @ B @ C )
% 16.19/8.92 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2602_pos__add__strict,axiom,
% 16.19/8.92 ! [A: rat,B: rat,C: rat] :
% 16.19/8.92 ( ( ord_less_rat @ zero_zero_rat @ A )
% 16.19/8.92 => ( ( ord_less_rat @ B @ C )
% 16.19/8.92 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2603_pos__add__strict,axiom,
% 16.19/8.92 ! [A: nat,B: nat,C: nat] :
% 16.19/8.92 ( ( ord_less_nat @ zero_zero_nat @ A )
% 16.19/8.92 => ( ( ord_less_nat @ B @ C )
% 16.19/8.92 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2604_pos__add__strict,axiom,
% 16.19/8.92 ! [A: int,B: int,C: int] :
% 16.19/8.92 ( ( ord_less_int @ zero_zero_int @ A )
% 16.19/8.92 => ( ( ord_less_int @ B @ C )
% 16.19/8.92 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2605_less__numeral__extra_I1_J,axiom,
% 16.19/8.92 ord_less_real @ zero_zero_real @ one_one_real ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2606_less__numeral__extra_I1_J,axiom,
% 16.19/8.92 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2607_less__numeral__extra_I1_J,axiom,
% 16.19/8.92 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2608_less__numeral__extra_I1_J,axiom,
% 16.19/8.92 ord_less_int @ zero_zero_int @ one_one_int ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2609_zero__less__one,axiom,
% 16.19/8.92 ord_less_real @ zero_zero_real @ one_one_real ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2610_zero__less__one,axiom,
% 16.19/8.92 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2611_zero__less__one,axiom,
% 16.19/8.92 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2612_zero__less__one,axiom,
% 16.19/8.92 ord_less_int @ zero_zero_int @ one_one_int ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2613_not__one__less__zero,axiom,
% 16.19/8.92 ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2614_not__one__less__zero,axiom,
% 16.19/8.92 ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2615_not__one__less__zero,axiom,
% 16.19/8.92 ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2616_not__one__less__zero,axiom,
% 16.19/8.92 ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2617_le__iff__diff__le__0,axiom,
% 16.19/8.92 ( ord_less_eq_real
% 16.19/8.92 = ( ^ [A5: real,B5: real] : ( ord_less_eq_real @ ( minus_minus_real @ A5 @ B5 ) @ zero_zero_real ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2618_le__iff__diff__le__0,axiom,
% 16.19/8.92 ( ord_less_eq_rat
% 16.19/8.92 = ( ^ [A5: rat,B5: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A5 @ B5 ) @ zero_zero_rat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2619_le__iff__diff__le__0,axiom,
% 16.19/8.92 ( ord_less_eq_int
% 16.19/8.92 = ( ^ [A5: int,B5: int] : ( ord_less_eq_int @ ( minus_minus_int @ A5 @ B5 ) @ zero_zero_int ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2620_divide__right__mono__neg,axiom,
% 16.19/8.92 ! [A: real,B: real,C: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 16.19/8.92 => ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2621_divide__right__mono__neg,axiom,
% 16.19/8.92 ! [A: rat,B: rat,C: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 16.19/8.92 => ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2622_divide__nonpos__nonpos,axiom,
% 16.19/8.92 ! [X: real,Y: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 16.19/8.92 => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 16.19/8.92 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2623_divide__nonpos__nonpos,axiom,
% 16.19/8.92 ! [X: rat,Y: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 16.19/8.92 => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
% 16.19/8.92 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2624_divide__nonpos__nonneg,axiom,
% 16.19/8.92 ! [X: real,Y: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 16.19/8.92 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/8.92 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2625_divide__nonpos__nonneg,axiom,
% 16.19/8.92 ! [X: rat,Y: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 16.19/8.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 16.19/8.92 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2626_divide__nonneg__nonpos,axiom,
% 16.19/8.92 ! [X: real,Y: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/8.92 => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 16.19/8.92 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2627_divide__nonneg__nonpos,axiom,
% 16.19/8.92 ! [X: rat,Y: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 16.19/8.92 => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
% 16.19/8.92 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2628_divide__nonneg__nonneg,axiom,
% 16.19/8.92 ! [X: real,Y: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/8.92 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/8.92 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2629_divide__nonneg__nonneg,axiom,
% 16.19/8.92 ! [X: rat,Y: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 16.19/8.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 16.19/8.92 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2630_zero__le__divide__iff,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 16.19/8.92 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/8.92 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 16.19/8.92 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 16.19/8.92 & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2631_zero__le__divide__iff,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 16.19/8.92 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/8.92 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 16.19/8.92 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 16.19/8.92 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2632_divide__right__mono,axiom,
% 16.19/8.92 ! [A: real,B: real,C: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 16.19/8.92 => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2633_divide__right__mono,axiom,
% 16.19/8.92 ! [A: rat,B: rat,C: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 16.19/8.92 => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2634_divide__le__0__iff,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 16.19/8.92 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/8.92 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 16.19/8.92 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 16.19/8.92 & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2635_divide__le__0__iff,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 16.19/8.92 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/8.92 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 16.19/8.92 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 16.19/8.92 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2636_less__iff__diff__less__0,axiom,
% 16.19/8.92 ( ord_less_real
% 16.19/8.92 = ( ^ [A5: real,B5: real] : ( ord_less_real @ ( minus_minus_real @ A5 @ B5 ) @ zero_zero_real ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2637_less__iff__diff__less__0,axiom,
% 16.19/8.92 ( ord_less_rat
% 16.19/8.92 = ( ^ [A5: rat,B5: rat] : ( ord_less_rat @ ( minus_minus_rat @ A5 @ B5 ) @ zero_zero_rat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2638_less__iff__diff__less__0,axiom,
% 16.19/8.92 ( ord_less_int
% 16.19/8.92 = ( ^ [A5: int,B5: int] : ( ord_less_int @ ( minus_minus_int @ A5 @ B5 ) @ zero_zero_int ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2639_power__mono,axiom,
% 16.19/8.92 ! [A: real,B: real,N2: nat] :
% 16.19/8.92 ( ( ord_less_eq_real @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/8.92 => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2640_power__mono,axiom,
% 16.19/8.92 ! [A: rat,B: rat,N2: nat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/8.92 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2641_power__mono,axiom,
% 16.19/8.92 ! [A: nat,B: nat,N2: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 16.19/8.92 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2642_power__mono,axiom,
% 16.19/8.92 ! [A: int,B: int,N2: nat] :
% 16.19/8.92 ( ( ord_less_eq_int @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.92 => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2643_zero__le__power,axiom,
% 16.19/8.92 ! [A: real,N2: nat] :
% 16.19/8.92 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/8.92 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2644_zero__le__power,axiom,
% 16.19/8.92 ! [A: rat,N2: nat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/8.92 => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2645_zero__le__power,axiom,
% 16.19/8.92 ! [A: nat,N2: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 16.19/8.92 => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2646_zero__le__power,axiom,
% 16.19/8.92 ! [A: int,N2: nat] :
% 16.19/8.92 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.92 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2647_divide__neg__neg,axiom,
% 16.19/8.92 ! [X: real,Y: real] :
% 16.19/8.92 ( ( ord_less_real @ X @ zero_zero_real )
% 16.19/8.92 => ( ( ord_less_real @ Y @ zero_zero_real )
% 16.19/8.92 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2648_divide__neg__neg,axiom,
% 16.19/8.92 ! [X: rat,Y: rat] :
% 16.19/8.92 ( ( ord_less_rat @ X @ zero_zero_rat )
% 16.19/8.92 => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 16.19/8.92 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2649_divide__neg__pos,axiom,
% 16.19/8.92 ! [X: real,Y: real] :
% 16.19/8.92 ( ( ord_less_real @ X @ zero_zero_real )
% 16.19/8.92 => ( ( ord_less_real @ zero_zero_real @ Y )
% 16.19/8.92 => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2650_divide__neg__pos,axiom,
% 16.19/8.92 ! [X: rat,Y: rat] :
% 16.19/8.92 ( ( ord_less_rat @ X @ zero_zero_rat )
% 16.19/8.92 => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 16.19/8.92 => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2651_divide__pos__neg,axiom,
% 16.19/8.92 ! [X: real,Y: real] :
% 16.19/8.92 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/8.92 => ( ( ord_less_real @ Y @ zero_zero_real )
% 16.19/8.92 => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2652_divide__pos__neg,axiom,
% 16.19/8.92 ! [X: rat,Y: rat] :
% 16.19/8.92 ( ( ord_less_rat @ zero_zero_rat @ X )
% 16.19/8.92 => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 16.19/8.92 => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2653_divide__pos__pos,axiom,
% 16.19/8.92 ! [X: real,Y: real] :
% 16.19/8.92 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/8.92 => ( ( ord_less_real @ zero_zero_real @ Y )
% 16.19/8.92 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2654_divide__pos__pos,axiom,
% 16.19/8.92 ! [X: rat,Y: rat] :
% 16.19/8.92 ( ( ord_less_rat @ zero_zero_rat @ X )
% 16.19/8.92 => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 16.19/8.92 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2655_divide__less__0__iff,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 16.19/8.92 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.92 & ( ord_less_real @ B @ zero_zero_real ) )
% 16.19/8.92 | ( ( ord_less_real @ A @ zero_zero_real )
% 16.19/8.92 & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2656_divide__less__0__iff,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 16.19/8.92 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 16.19/8.92 & ( ord_less_rat @ B @ zero_zero_rat ) )
% 16.19/8.92 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 16.19/8.92 & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2657_divide__less__cancel,axiom,
% 16.19/8.92 ! [A: real,C: real,B: real] :
% 16.19/8.92 ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 16.19/8.92 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.92 => ( ord_less_real @ A @ B ) )
% 16.19/8.92 & ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.92 => ( ord_less_real @ B @ A ) )
% 16.19/8.92 & ( C != zero_zero_real ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2658_divide__less__cancel,axiom,
% 16.19/8.92 ! [A: rat,C: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 16.19/8.92 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.92 => ( ord_less_rat @ A @ B ) )
% 16.19/8.92 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.92 => ( ord_less_rat @ B @ A ) )
% 16.19/8.92 & ( C != zero_zero_rat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2659_zero__less__divide__iff,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 16.19/8.92 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.92 & ( ord_less_real @ zero_zero_real @ B ) )
% 16.19/8.92 | ( ( ord_less_real @ A @ zero_zero_real )
% 16.19/8.92 & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2660_zero__less__divide__iff,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 16.19/8.92 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 16.19/8.92 & ( ord_less_rat @ zero_zero_rat @ B ) )
% 16.19/8.92 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 16.19/8.92 & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2661_divide__strict__right__mono,axiom,
% 16.19/8.92 ! [A: real,B: real,C: real] :
% 16.19/8.92 ( ( ord_less_real @ A @ B )
% 16.19/8.92 => ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.92 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2662_divide__strict__right__mono,axiom,
% 16.19/8.92 ! [A: rat,B: rat,C: rat] :
% 16.19/8.92 ( ( ord_less_rat @ A @ B )
% 16.19/8.92 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.92 => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2663_divide__strict__right__mono__neg,axiom,
% 16.19/8.92 ! [B: real,A: real,C: real] :
% 16.19/8.92 ( ( ord_less_real @ B @ A )
% 16.19/8.92 => ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.92 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2664_divide__strict__right__mono__neg,axiom,
% 16.19/8.92 ! [B: rat,A: rat,C: rat] :
% 16.19/8.92 ( ( ord_less_rat @ B @ A )
% 16.19/8.92 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.92 => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2665_zero__less__power,axiom,
% 16.19/8.92 ! [A: real,N2: nat] :
% 16.19/8.92 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.92 => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2666_zero__less__power,axiom,
% 16.19/8.92 ! [A: rat,N2: nat] :
% 16.19/8.92 ( ( ord_less_rat @ zero_zero_rat @ A )
% 16.19/8.92 => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2667_zero__less__power,axiom,
% 16.19/8.92 ! [A: nat,N2: nat] :
% 16.19/8.92 ( ( ord_less_nat @ zero_zero_nat @ A )
% 16.19/8.92 => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2668_zero__less__power,axiom,
% 16.19/8.92 ! [A: int,N2: nat] :
% 16.19/8.92 ( ( ord_less_int @ zero_zero_int @ A )
% 16.19/8.92 => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2669_nonzero__eq__divide__eq,axiom,
% 16.19/8.92 ! [C: complex,A: complex,B: complex] :
% 16.19/8.92 ( ( C != zero_zero_complex )
% 16.19/8.92 => ( ( A
% 16.19/8.92 = ( divide1717551699836669952omplex @ B @ C ) )
% 16.19/8.92 = ( ( times_times_complex @ A @ C )
% 16.19/8.92 = B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2670_nonzero__eq__divide__eq,axiom,
% 16.19/8.92 ! [C: real,A: real,B: real] :
% 16.19/8.92 ( ( C != zero_zero_real )
% 16.19/8.92 => ( ( A
% 16.19/8.92 = ( divide_divide_real @ B @ C ) )
% 16.19/8.92 = ( ( times_times_real @ A @ C )
% 16.19/8.92 = B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2671_nonzero__eq__divide__eq,axiom,
% 16.19/8.92 ! [C: rat,A: rat,B: rat] :
% 16.19/8.92 ( ( C != zero_zero_rat )
% 16.19/8.92 => ( ( A
% 16.19/8.92 = ( divide_divide_rat @ B @ C ) )
% 16.19/8.92 = ( ( times_times_rat @ A @ C )
% 16.19/8.92 = B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2672_nonzero__divide__eq__eq,axiom,
% 16.19/8.92 ! [C: complex,B: complex,A: complex] :
% 16.19/8.92 ( ( C != zero_zero_complex )
% 16.19/8.92 => ( ( ( divide1717551699836669952omplex @ B @ C )
% 16.19/8.92 = A )
% 16.19/8.92 = ( B
% 16.19/8.92 = ( times_times_complex @ A @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2673_nonzero__divide__eq__eq,axiom,
% 16.19/8.92 ! [C: real,B: real,A: real] :
% 16.19/8.92 ( ( C != zero_zero_real )
% 16.19/8.92 => ( ( ( divide_divide_real @ B @ C )
% 16.19/8.92 = A )
% 16.19/8.92 = ( B
% 16.19/8.92 = ( times_times_real @ A @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2674_nonzero__divide__eq__eq,axiom,
% 16.19/8.92 ! [C: rat,B: rat,A: rat] :
% 16.19/8.92 ( ( C != zero_zero_rat )
% 16.19/8.92 => ( ( ( divide_divide_rat @ B @ C )
% 16.19/8.92 = A )
% 16.19/8.92 = ( B
% 16.19/8.92 = ( times_times_rat @ A @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2675_eq__divide__imp,axiom,
% 16.19/8.92 ! [C: complex,A: complex,B: complex] :
% 16.19/8.92 ( ( C != zero_zero_complex )
% 16.19/8.92 => ( ( ( times_times_complex @ A @ C )
% 16.19/8.92 = B )
% 16.19/8.92 => ( A
% 16.19/8.92 = ( divide1717551699836669952omplex @ B @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2676_eq__divide__imp,axiom,
% 16.19/8.92 ! [C: real,A: real,B: real] :
% 16.19/8.92 ( ( C != zero_zero_real )
% 16.19/8.92 => ( ( ( times_times_real @ A @ C )
% 16.19/8.92 = B )
% 16.19/8.92 => ( A
% 16.19/8.92 = ( divide_divide_real @ B @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2677_eq__divide__imp,axiom,
% 16.19/8.92 ! [C: rat,A: rat,B: rat] :
% 16.19/8.92 ( ( C != zero_zero_rat )
% 16.19/8.92 => ( ( ( times_times_rat @ A @ C )
% 16.19/8.92 = B )
% 16.19/8.92 => ( A
% 16.19/8.92 = ( divide_divide_rat @ B @ C ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2678_divide__eq__imp,axiom,
% 16.19/8.92 ! [C: complex,B: complex,A: complex] :
% 16.19/8.92 ( ( C != zero_zero_complex )
% 16.19/8.92 => ( ( B
% 16.19/8.92 = ( times_times_complex @ A @ C ) )
% 16.19/8.92 => ( ( divide1717551699836669952omplex @ B @ C )
% 16.19/8.92 = A ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2679_divide__eq__imp,axiom,
% 16.19/8.92 ! [C: real,B: real,A: real] :
% 16.19/8.92 ( ( C != zero_zero_real )
% 16.19/8.92 => ( ( B
% 16.19/8.92 = ( times_times_real @ A @ C ) )
% 16.19/8.92 => ( ( divide_divide_real @ B @ C )
% 16.19/8.92 = A ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2680_divide__eq__imp,axiom,
% 16.19/8.92 ! [C: rat,B: rat,A: rat] :
% 16.19/8.92 ( ( C != zero_zero_rat )
% 16.19/8.92 => ( ( B
% 16.19/8.92 = ( times_times_rat @ A @ C ) )
% 16.19/8.92 => ( ( divide_divide_rat @ B @ C )
% 16.19/8.92 = A ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2681_eq__divide__eq,axiom,
% 16.19/8.92 ! [A: complex,B: complex,C: complex] :
% 16.19/8.92 ( ( A
% 16.19/8.92 = ( divide1717551699836669952omplex @ B @ C ) )
% 16.19/8.92 = ( ( ( C != zero_zero_complex )
% 16.19/8.92 => ( ( times_times_complex @ A @ C )
% 16.19/8.92 = B ) )
% 16.19/8.92 & ( ( C = zero_zero_complex )
% 16.19/8.92 => ( A = zero_zero_complex ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2682_eq__divide__eq,axiom,
% 16.19/8.92 ! [A: real,B: real,C: real] :
% 16.19/8.92 ( ( A
% 16.19/8.92 = ( divide_divide_real @ B @ C ) )
% 16.19/8.92 = ( ( ( C != zero_zero_real )
% 16.19/8.92 => ( ( times_times_real @ A @ C )
% 16.19/8.92 = B ) )
% 16.19/8.92 & ( ( C = zero_zero_real )
% 16.19/8.92 => ( A = zero_zero_real ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2683_eq__divide__eq,axiom,
% 16.19/8.92 ! [A: rat,B: rat,C: rat] :
% 16.19/8.92 ( ( A
% 16.19/8.92 = ( divide_divide_rat @ B @ C ) )
% 16.19/8.92 = ( ( ( C != zero_zero_rat )
% 16.19/8.92 => ( ( times_times_rat @ A @ C )
% 16.19/8.92 = B ) )
% 16.19/8.92 & ( ( C = zero_zero_rat )
% 16.19/8.92 => ( A = zero_zero_rat ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2684_divide__eq__eq,axiom,
% 16.19/8.92 ! [B: complex,C: complex,A: complex] :
% 16.19/8.92 ( ( ( divide1717551699836669952omplex @ B @ C )
% 16.19/8.92 = A )
% 16.19/8.92 = ( ( ( C != zero_zero_complex )
% 16.19/8.92 => ( B
% 16.19/8.92 = ( times_times_complex @ A @ C ) ) )
% 16.19/8.92 & ( ( C = zero_zero_complex )
% 16.19/8.92 => ( A = zero_zero_complex ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2685_divide__eq__eq,axiom,
% 16.19/8.92 ! [B: real,C: real,A: real] :
% 16.19/8.92 ( ( ( divide_divide_real @ B @ C )
% 16.19/8.92 = A )
% 16.19/8.92 = ( ( ( C != zero_zero_real )
% 16.19/8.92 => ( B
% 16.19/8.92 = ( times_times_real @ A @ C ) ) )
% 16.19/8.92 & ( ( C = zero_zero_real )
% 16.19/8.92 => ( A = zero_zero_real ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2686_divide__eq__eq,axiom,
% 16.19/8.92 ! [B: rat,C: rat,A: rat] :
% 16.19/8.92 ( ( ( divide_divide_rat @ B @ C )
% 16.19/8.92 = A )
% 16.19/8.92 = ( ( ( C != zero_zero_rat )
% 16.19/8.92 => ( B
% 16.19/8.92 = ( times_times_rat @ A @ C ) ) )
% 16.19/8.92 & ( ( C = zero_zero_rat )
% 16.19/8.92 => ( A = zero_zero_rat ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2687_frac__eq__eq,axiom,
% 16.19/8.92 ! [Y: complex,Z: complex,X: complex,W: complex] :
% 16.19/8.92 ( ( Y != zero_zero_complex )
% 16.19/8.92 => ( ( Z != zero_zero_complex )
% 16.19/8.92 => ( ( ( divide1717551699836669952omplex @ X @ Y )
% 16.19/8.92 = ( divide1717551699836669952omplex @ W @ Z ) )
% 16.19/8.92 = ( ( times_times_complex @ X @ Z )
% 16.19/8.92 = ( times_times_complex @ W @ Y ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2688_frac__eq__eq,axiom,
% 16.19/8.92 ! [Y: real,Z: real,X: real,W: real] :
% 16.19/8.92 ( ( Y != zero_zero_real )
% 16.19/8.92 => ( ( Z != zero_zero_real )
% 16.19/8.92 => ( ( ( divide_divide_real @ X @ Y )
% 16.19/8.92 = ( divide_divide_real @ W @ Z ) )
% 16.19/8.92 = ( ( times_times_real @ X @ Z )
% 16.19/8.92 = ( times_times_real @ W @ Y ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2689_frac__eq__eq,axiom,
% 16.19/8.92 ! [Y: rat,Z: rat,X: rat,W: rat] :
% 16.19/8.92 ( ( Y != zero_zero_rat )
% 16.19/8.92 => ( ( Z != zero_zero_rat )
% 16.19/8.92 => ( ( ( divide_divide_rat @ X @ Y )
% 16.19/8.92 = ( divide_divide_rat @ W @ Z ) )
% 16.19/8.92 = ( ( times_times_rat @ X @ Z )
% 16.19/8.92 = ( times_times_rat @ W @ Y ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2690_right__inverse__eq,axiom,
% 16.19/8.92 ! [B: complex,A: complex] :
% 16.19/8.92 ( ( B != zero_zero_complex )
% 16.19/8.92 => ( ( ( divide1717551699836669952omplex @ A @ B )
% 16.19/8.92 = one_one_complex )
% 16.19/8.92 = ( A = B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2691_right__inverse__eq,axiom,
% 16.19/8.92 ! [B: real,A: real] :
% 16.19/8.92 ( ( B != zero_zero_real )
% 16.19/8.92 => ( ( ( divide_divide_real @ A @ B )
% 16.19/8.92 = one_one_real )
% 16.19/8.92 = ( A = B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2692_right__inverse__eq,axiom,
% 16.19/8.92 ! [B: rat,A: rat] :
% 16.19/8.92 ( ( B != zero_zero_rat )
% 16.19/8.92 => ( ( ( divide_divide_rat @ A @ B )
% 16.19/8.92 = one_one_rat )
% 16.19/8.92 = ( A = B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2693_power__0,axiom,
% 16.19/8.92 ! [A: rat] :
% 16.19/8.92 ( ( power_power_rat @ A @ zero_zero_nat )
% 16.19/8.92 = one_one_rat ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2694_power__0,axiom,
% 16.19/8.92 ! [A: nat] :
% 16.19/8.92 ( ( power_power_nat @ A @ zero_zero_nat )
% 16.19/8.92 = one_one_nat ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2695_power__0,axiom,
% 16.19/8.92 ! [A: real] :
% 16.19/8.92 ( ( power_power_real @ A @ zero_zero_nat )
% 16.19/8.92 = one_one_real ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2696_power__0,axiom,
% 16.19/8.92 ! [A: int] :
% 16.19/8.92 ( ( power_power_int @ A @ zero_zero_nat )
% 16.19/8.92 = one_one_int ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2697_power__0,axiom,
% 16.19/8.92 ! [A: complex] :
% 16.19/8.92 ( ( power_power_complex @ A @ zero_zero_nat )
% 16.19/8.92 = one_one_complex ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2698_divmod__digit__0_I1_J,axiom,
% 16.19/8.92 ! [B: nat,A: nat] :
% 16.19/8.92 ( ( ord_less_nat @ zero_zero_nat @ B )
% 16.19/8.92 => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 16.19/8.92 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 16.19/8.92 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2699_divmod__digit__0_I1_J,axiom,
% 16.19/8.92 ! [B: int,A: int] :
% 16.19/8.92 ( ( ord_less_int @ zero_zero_int @ B )
% 16.19/8.92 => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 16.19/8.92 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 16.19/8.92 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2700_divmod__digit__0_I1_J,axiom,
% 16.19/8.92 ! [B: code_integer,A: code_integer] :
% 16.19/8.92 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 16.19/8.92 => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 16.19/8.92 => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 16.19/8.92 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2701_Ex__less__Suc2,axiom,
% 16.19/8.92 ! [N2: nat,P: nat > $o] :
% 16.19/8.92 ( ( ? [I3: nat] :
% 16.19/8.92 ( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
% 16.19/8.92 & ( P @ I3 ) ) )
% 16.19/8.92 = ( ( P @ zero_zero_nat )
% 16.19/8.92 | ? [I3: nat] :
% 16.19/8.92 ( ( ord_less_nat @ I3 @ N2 )
% 16.19/8.92 & ( P @ ( suc @ I3 ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2702_gr0__conv__Suc,axiom,
% 16.19/8.92 ! [N2: nat] :
% 16.19/8.92 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.92 = ( ? [M6: nat] :
% 16.19/8.92 ( N2
% 16.19/8.92 = ( suc @ M6 ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2703_All__less__Suc2,axiom,
% 16.19/8.92 ! [N2: nat,P: nat > $o] :
% 16.19/8.92 ( ( ! [I3: nat] :
% 16.19/8.92 ( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
% 16.19/8.92 => ( P @ I3 ) ) )
% 16.19/8.92 = ( ( P @ zero_zero_nat )
% 16.19/8.92 & ! [I3: nat] :
% 16.19/8.92 ( ( ord_less_nat @ I3 @ N2 )
% 16.19/8.92 => ( P @ ( suc @ I3 ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2704_gr0__implies__Suc,axiom,
% 16.19/8.92 ! [N2: nat] :
% 16.19/8.92 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.92 => ? [M2: nat] :
% 16.19/8.92 ( N2
% 16.19/8.92 = ( suc @ M2 ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2705_less__Suc__eq__0__disj,axiom,
% 16.19/8.92 ! [M: nat,N2: nat] :
% 16.19/8.92 ( ( ord_less_nat @ M @ ( suc @ N2 ) )
% 16.19/8.92 = ( ( M = zero_zero_nat )
% 16.19/8.92 | ? [J3: nat] :
% 16.19/8.92 ( ( M
% 16.19/8.92 = ( suc @ J3 ) )
% 16.19/8.92 & ( ord_less_nat @ J3 @ N2 ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2706_one__is__add,axiom,
% 16.19/8.92 ! [M: nat,N2: nat] :
% 16.19/8.92 ( ( ( suc @ zero_zero_nat )
% 16.19/8.92 = ( plus_plus_nat @ M @ N2 ) )
% 16.19/8.92 = ( ( ( M
% 16.19/8.92 = ( suc @ zero_zero_nat ) )
% 16.19/8.92 & ( N2 = zero_zero_nat ) )
% 16.19/8.92 | ( ( M = zero_zero_nat )
% 16.19/8.92 & ( N2
% 16.19/8.92 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2707_add__is__1,axiom,
% 16.19/8.92 ! [M: nat,N2: nat] :
% 16.19/8.92 ( ( ( plus_plus_nat @ M @ N2 )
% 16.19/8.92 = ( suc @ zero_zero_nat ) )
% 16.19/8.92 = ( ( ( M
% 16.19/8.92 = ( suc @ zero_zero_nat ) )
% 16.19/8.92 & ( N2 = zero_zero_nat ) )
% 16.19/8.92 | ( ( M = zero_zero_nat )
% 16.19/8.92 & ( N2
% 16.19/8.92 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2708_option_Osize_I4_J,axiom,
% 16.19/8.92 ! [X22: nat] :
% 16.19/8.92 ( ( size_size_option_nat @ ( some_nat @ X22 ) )
% 16.19/8.92 = ( suc @ zero_zero_nat ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2709_option_Osize_I4_J,axiom,
% 16.19/8.92 ! [X22: product_prod_nat_nat] :
% 16.19/8.92 ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
% 16.19/8.92 = ( suc @ zero_zero_nat ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2710_option_Osize_I4_J,axiom,
% 16.19/8.92 ! [X22: num] :
% 16.19/8.92 ( ( size_size_option_num @ ( some_num @ X22 ) )
% 16.19/8.92 = ( suc @ zero_zero_nat ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2711_ex__least__nat__le,axiom,
% 16.19/8.92 ! [P: nat > $o,N2: nat] :
% 16.19/8.92 ( ( P @ N2 )
% 16.19/8.92 => ( ~ ( P @ zero_zero_nat )
% 16.19/8.92 => ? [K2: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ K2 @ N2 )
% 16.19/8.92 & ! [I: nat] :
% 16.19/8.92 ( ( ord_less_nat @ I @ K2 )
% 16.19/8.92 => ~ ( P @ I ) )
% 16.19/8.92 & ( P @ K2 ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2712_less__imp__add__positive,axiom,
% 16.19/8.92 ! [I2: nat,J: nat] :
% 16.19/8.92 ( ( ord_less_nat @ I2 @ J )
% 16.19/8.92 => ? [K2: nat] :
% 16.19/8.92 ( ( ord_less_nat @ zero_zero_nat @ K2 )
% 16.19/8.92 & ( ( plus_plus_nat @ I2 @ K2 )
% 16.19/8.92 = J ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2713_option_Osize_I3_J,axiom,
% 16.19/8.92 ( ( size_size_option_nat @ none_nat )
% 16.19/8.92 = ( suc @ zero_zero_nat ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2714_option_Osize_I3_J,axiom,
% 16.19/8.92 ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
% 16.19/8.92 = ( suc @ zero_zero_nat ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2715_option_Osize_I3_J,axiom,
% 16.19/8.92 ( ( size_size_option_num @ none_num )
% 16.19/8.92 = ( suc @ zero_zero_nat ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2716_mult__less__mono1,axiom,
% 16.19/8.92 ! [I2: nat,J: nat,K: nat] :
% 16.19/8.92 ( ( ord_less_nat @ I2 @ J )
% 16.19/8.92 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 16.19/8.92 => ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2717_mult__less__mono2,axiom,
% 16.19/8.92 ! [I2: nat,J: nat,K: nat] :
% 16.19/8.92 ( ( ord_less_nat @ I2 @ J )
% 16.19/8.92 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 16.19/8.92 => ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2718_nat__mult__eq__cancel1,axiom,
% 16.19/8.92 ! [K: nat,M: nat,N2: nat] :
% 16.19/8.92 ( ( ord_less_nat @ zero_zero_nat @ K )
% 16.19/8.92 => ( ( ( times_times_nat @ K @ M )
% 16.19/8.92 = ( times_times_nat @ K @ N2 ) )
% 16.19/8.92 = ( M = N2 ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2719_nat__mult__less__cancel1,axiom,
% 16.19/8.92 ! [K: nat,M: nat,N2: nat] :
% 16.19/8.92 ( ( ord_less_nat @ zero_zero_nat @ K )
% 16.19/8.92 => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 16.19/8.92 = ( ord_less_nat @ M @ N2 ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2720_diff__less,axiom,
% 16.19/8.92 ! [N2: nat,M: nat] :
% 16.19/8.92 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.92 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 16.19/8.92 => ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2721_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 16.19/8.92 ! [M: nat,N2: nat] :
% 16.19/8.92 ( ( ( divide_divide_nat @ M @ N2 )
% 16.19/8.92 = zero_zero_nat )
% 16.19/8.92 = ( ( ord_less_nat @ M @ N2 )
% 16.19/8.92 | ( N2 = zero_zero_nat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2722_One__nat__def,axiom,
% 16.19/8.92 ( one_one_nat
% 16.19/8.92 = ( suc @ zero_zero_nat ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2723_nat__power__less__imp__less,axiom,
% 16.19/8.92 ! [I2: nat,M: nat,N2: nat] :
% 16.19/8.92 ( ( ord_less_nat @ zero_zero_nat @ I2 )
% 16.19/8.92 => ( ( ord_less_nat @ ( power_power_nat @ I2 @ M ) @ ( power_power_nat @ I2 @ N2 ) )
% 16.19/8.92 => ( ord_less_nat @ M @ N2 ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2724_diff__add__0,axiom,
% 16.19/8.92 ! [N2: nat,M: nat] :
% 16.19/8.92 ( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) )
% 16.19/8.92 = zero_zero_nat ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2725_bits__stable__imp__add__self,axiom,
% 16.19/8.92 ! [A: nat] :
% 16.19/8.92 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.92 = A )
% 16.19/8.92 => ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.92 = zero_zero_nat ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2726_bits__stable__imp__add__self,axiom,
% 16.19/8.92 ! [A: int] :
% 16.19/8.92 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/8.92 = A )
% 16.19/8.92 => ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 16.19/8.92 = zero_zero_int ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2727_bits__stable__imp__add__self,axiom,
% 16.19/8.92 ! [A: code_integer] :
% 16.19/8.92 ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 16.19/8.92 = A )
% 16.19/8.92 => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 16.19/8.92 = zero_z3403309356797280102nteger ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2728_mult__eq__self__implies__10,axiom,
% 16.19/8.92 ! [M: nat,N2: nat] :
% 16.19/8.92 ( ( M
% 16.19/8.92 = ( times_times_nat @ M @ N2 ) )
% 16.19/8.92 => ( ( N2 = one_one_nat )
% 16.19/8.92 | ( M = zero_zero_nat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2729_vebt__insert_Osimps_I2_J,axiom,
% 16.19/8.92 ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X: nat] :
% 16.19/8.92 ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S3 ) @ X )
% 16.19/8.92 = ( vEBT_Node @ Info @ zero_zero_nat @ Ts2 @ S3 ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2730_vebt__pred_Osimps_I3_J,axiom,
% 16.19/8.92 ! [B: $o,A: $o,Va: nat] :
% 16.19/8.92 ( ( B
% 16.19/8.92 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
% 16.19/8.92 = ( some_nat @ one_one_nat ) ) )
% 16.19/8.92 & ( ~ B
% 16.19/8.92 => ( ( A
% 16.19/8.92 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
% 16.19/8.92 = ( some_nat @ zero_zero_nat ) ) )
% 16.19/8.92 & ( ~ A
% 16.19/8.92 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
% 16.19/8.92 = none_nat ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2731_mod__double__modulus,axiom,
% 16.19/8.92 ! [M: code_integer,X: code_integer] :
% 16.19/8.92 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
% 16.19/8.92 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 16.19/8.92 => ( ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 16.19/8.92 = ( modulo364778990260209775nteger @ X @ M ) )
% 16.19/8.92 | ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 16.19/8.92 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X @ M ) @ M ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2732_mod__double__modulus,axiom,
% 16.19/8.92 ! [M: nat,X: nat] :
% 16.19/8.92 ( ( ord_less_nat @ zero_zero_nat @ M )
% 16.19/8.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 16.19/8.92 => ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 16.19/8.92 = ( modulo_modulo_nat @ X @ M ) )
% 16.19/8.92 | ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 16.19/8.92 = ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M ) @ M ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2733_mod__double__modulus,axiom,
% 16.19/8.92 ! [M: int,X: int] :
% 16.19/8.92 ( ( ord_less_int @ zero_zero_int @ M )
% 16.19/8.92 => ( ( ord_less_eq_int @ zero_zero_int @ X )
% 16.19/8.92 => ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 16.19/8.92 = ( modulo_modulo_int @ X @ M ) )
% 16.19/8.92 | ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 16.19/8.92 = ( plus_plus_int @ ( modulo_modulo_int @ X @ M ) @ M ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2734_divmod__digit__1_I2_J,axiom,
% 16.19/8.92 ! [A: code_integer,B: code_integer] :
% 16.19/8.92 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 16.19/8.92 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 16.19/8.92 => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 16.19/8.92 => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 16.19/8.92 = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2735_divmod__digit__1_I2_J,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 16.19/8.92 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 16.19/8.92 => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 16.19/8.92 => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 16.19/8.92 = ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2736_divmod__digit__1_I2_J,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.92 => ( ( ord_less_int @ zero_zero_int @ B )
% 16.19/8.92 => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 16.19/8.92 => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 16.19/8.92 = ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2737_div__mult1__eq,axiom,
% 16.19/8.92 ! [A: nat,B: nat,C: nat] :
% 16.19/8.92 ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
% 16.19/8.92 = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2738_div__mult1__eq,axiom,
% 16.19/8.92 ! [A: int,B: int,C: int] :
% 16.19/8.92 ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C )
% 16.19/8.92 = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2739_div__mult1__eq,axiom,
% 16.19/8.92 ! [A: code_integer,B: code_integer,C: code_integer] :
% 16.19/8.92 ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 16.19/8.92 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2740_mult__div__mod__eq,axiom,
% 16.19/8.92 ! [B: nat,A: nat] :
% 16.19/8.92 ( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
% 16.19/8.92 = A ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2741_mult__div__mod__eq,axiom,
% 16.19/8.92 ! [B: int,A: int] :
% 16.19/8.92 ( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
% 16.19/8.92 = A ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2742_mult__div__mod__eq,axiom,
% 16.19/8.92 ! [B: code_integer,A: code_integer] :
% 16.19/8.92 ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
% 16.19/8.92 = A ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2743_mod__mult__div__eq,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.92 ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
% 16.19/8.92 = A ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2744_mod__mult__div__eq,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
% 16.19/8.92 = A ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2745_mod__mult__div__eq,axiom,
% 16.19/8.92 ! [A: code_integer,B: code_integer] :
% 16.19/8.92 ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
% 16.19/8.92 = A ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2746_mod__div__mult__eq,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.92 ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
% 16.19/8.92 = A ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2747_mod__div__mult__eq,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
% 16.19/8.92 = A ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2748_mod__div__mult__eq,axiom,
% 16.19/8.92 ! [A: code_integer,B: code_integer] :
% 16.19/8.92 ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
% 16.19/8.92 = A ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2749_div__mult__mod__eq,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.92 ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
% 16.19/8.92 = A ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2750_div__mult__mod__eq,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
% 16.19/8.92 = A ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2751_div__mult__mod__eq,axiom,
% 16.19/8.92 ! [A: code_integer,B: code_integer] :
% 16.19/8.92 ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
% 16.19/8.92 = A ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2752_mod__div__decomp,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.92 ( A
% 16.19/8.92 = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2753_mod__div__decomp,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( A
% 16.19/8.92 = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2754_mod__div__decomp,axiom,
% 16.19/8.92 ! [A: code_integer,B: code_integer] :
% 16.19/8.92 ( A
% 16.19/8.92 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2755_cancel__div__mod__rules_I1_J,axiom,
% 16.19/8.92 ! [A: nat,B: nat,C: nat] :
% 16.19/8.92 ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
% 16.19/8.92 = ( plus_plus_nat @ A @ C ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2756_cancel__div__mod__rules_I1_J,axiom,
% 16.19/8.92 ! [A: int,B: int,C: int] :
% 16.19/8.92 ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
% 16.19/8.92 = ( plus_plus_int @ A @ C ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2757_cancel__div__mod__rules_I1_J,axiom,
% 16.19/8.92 ! [A: code_integer,B: code_integer,C: code_integer] :
% 16.19/8.92 ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
% 16.19/8.92 = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2758_cancel__div__mod__rules_I2_J,axiom,
% 16.19/8.92 ! [B: nat,A: nat,C: nat] :
% 16.19/8.92 ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
% 16.19/8.92 = ( plus_plus_nat @ A @ C ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2759_cancel__div__mod__rules_I2_J,axiom,
% 16.19/8.92 ! [B: int,A: int,C: int] :
% 16.19/8.92 ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
% 16.19/8.92 = ( plus_plus_int @ A @ C ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2760_cancel__div__mod__rules_I2_J,axiom,
% 16.19/8.92 ! [B: code_integer,A: code_integer,C: code_integer] :
% 16.19/8.92 ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
% 16.19/8.92 = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2761_minus__mult__div__eq__mod,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.92 ( ( minus_minus_nat @ A @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
% 16.19/8.92 = ( modulo_modulo_nat @ A @ B ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2762_minus__mult__div__eq__mod,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( minus_minus_int @ A @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
% 16.19/8.92 = ( modulo_modulo_int @ A @ B ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2763_minus__mult__div__eq__mod,axiom,
% 16.19/8.92 ! [A: code_integer,B: code_integer] :
% 16.19/8.92 ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
% 16.19/8.92 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2764_minus__mod__eq__mult__div,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.92 ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
% 16.19/8.92 = ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2765_minus__mod__eq__mult__div,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
% 16.19/8.92 = ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2766_minus__mod__eq__mult__div,axiom,
% 16.19/8.92 ! [A: code_integer,B: code_integer] :
% 16.19/8.92 ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
% 16.19/8.92 = ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2767_minus__mod__eq__div__mult,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.92 ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
% 16.19/8.92 = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2768_minus__mod__eq__div__mult,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
% 16.19/8.92 = ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2769_minus__mod__eq__div__mult,axiom,
% 16.19/8.92 ! [A: code_integer,B: code_integer] :
% 16.19/8.92 ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
% 16.19/8.92 = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2770_minus__div__mult__eq__mod,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.92 ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
% 16.19/8.92 = ( modulo_modulo_nat @ A @ B ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2771_minus__div__mult__eq__mod,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
% 16.19/8.92 = ( modulo_modulo_int @ A @ B ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2772_minus__div__mult__eq__mod,axiom,
% 16.19/8.92 ! [A: code_integer,B: code_integer] :
% 16.19/8.92 ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
% 16.19/8.92 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2773_mod__mult2__eq,axiom,
% 16.19/8.92 ! [M: nat,N2: nat,Q2: nat] :
% 16.19/8.92 ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N2 @ Q2 ) )
% 16.19/8.92 = ( plus_plus_nat @ ( times_times_nat @ N2 @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N2 ) @ Q2 ) ) @ ( modulo_modulo_nat @ M @ N2 ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2774_modulo__nat__def,axiom,
% 16.19/8.92 ( modulo_modulo_nat
% 16.19/8.92 = ( ^ [M6: nat,N: nat] : ( minus_minus_nat @ M6 @ ( times_times_nat @ ( divide_divide_nat @ M6 @ N ) @ N ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2775_VEBT__internal_OminNull_Ocases,axiom,
% 16.19/8.92 ! [X: vEBT_VEBT] :
% 16.19/8.92 ( ( X
% 16.19/8.92 != ( vEBT_Leaf @ $false @ $false ) )
% 16.19/8.92 => ( ! [Uv2: $o] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( vEBT_Leaf @ $true @ Uv2 ) )
% 16.19/8.92 => ( ! [Uu2: $o] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( vEBT_Leaf @ Uu2 @ $true ) )
% 16.19/8.92 => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 16.19/8.92 => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2776_div__geq,axiom,
% 16.19/8.92 ! [N2: nat,M: nat] :
% 16.19/8.92 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.92 => ( ~ ( ord_less_nat @ M @ N2 )
% 16.19/8.92 => ( ( divide_divide_nat @ M @ N2 )
% 16.19/8.92 = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2777_VEBT__internal_OminNull_Oelims_I3_J,axiom,
% 16.19/8.92 ! [X: vEBT_VEBT] :
% 16.19/8.92 ( ~ ( vEBT_VEBT_minNull @ X )
% 16.19/8.92 => ( ! [Uv2: $o] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( vEBT_Leaf @ $true @ Uv2 ) )
% 16.19/8.92 => ( ! [Uu2: $o] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( vEBT_Leaf @ Uu2 @ $true ) )
% 16.19/8.92 => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2778_vebt__succ_Osimps_I2_J,axiom,
% 16.19/8.92 ! [Uv: $o,Uw: $o,N2: nat] :
% 16.19/8.92 ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) )
% 16.19/8.92 = none_nat ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2779_VEBT__internal_OminNull_Oelims_I2_J,axiom,
% 16.19/8.92 ! [X: vEBT_VEBT] :
% 16.19/8.92 ( ( vEBT_VEBT_minNull @ X )
% 16.19/8.92 => ( ( X
% 16.19/8.92 != ( vEBT_Leaf @ $false @ $false ) )
% 16.19/8.92 => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 16.19/8.92 ( X
% 16.19/8.92 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2780_mult__le__cancel__left,axiom,
% 16.19/8.92 ! [C: real,A: real,B: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 16.19/8.92 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.92 => ( ord_less_eq_real @ A @ B ) )
% 16.19/8.92 & ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.92 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2781_mult__le__cancel__left,axiom,
% 16.19/8.92 ! [C: rat,A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 16.19/8.92 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.92 => ( ord_less_eq_rat @ A @ B ) )
% 16.19/8.92 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.92 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2782_mult__le__cancel__left,axiom,
% 16.19/8.92 ! [C: int,A: int,B: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 16.19/8.92 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 16.19/8.92 => ( ord_less_eq_int @ A @ B ) )
% 16.19/8.92 & ( ( ord_less_int @ C @ zero_zero_int )
% 16.19/8.92 => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2783_mult__le__cancel__right,axiom,
% 16.19/8.92 ! [A: real,C: real,B: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 16.19/8.92 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.92 => ( ord_less_eq_real @ A @ B ) )
% 16.19/8.92 & ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.92 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2784_mult__le__cancel__right,axiom,
% 16.19/8.92 ! [A: rat,C: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 16.19/8.92 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.92 => ( ord_less_eq_rat @ A @ B ) )
% 16.19/8.92 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.92 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2785_mult__le__cancel__right,axiom,
% 16.19/8.92 ! [A: int,C: int,B: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 16.19/8.92 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 16.19/8.92 => ( ord_less_eq_int @ A @ B ) )
% 16.19/8.92 & ( ( ord_less_int @ C @ zero_zero_int )
% 16.19/8.92 => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2786_mult__left__less__imp__less,axiom,
% 16.19/8.92 ! [C: real,A: real,B: real] :
% 16.19/8.92 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 16.19/8.92 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 16.19/8.92 => ( ord_less_real @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2787_mult__left__less__imp__less,axiom,
% 16.19/8.92 ! [C: rat,A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 16.19/8.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 16.19/8.92 => ( ord_less_rat @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2788_mult__left__less__imp__less,axiom,
% 16.19/8.92 ! [C: nat,A: nat,B: nat] :
% 16.19/8.92 ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 16.19/8.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 16.19/8.92 => ( ord_less_nat @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2789_mult__left__less__imp__less,axiom,
% 16.19/8.92 ! [C: int,A: int,B: int] :
% 16.19/8.92 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 16.19/8.92 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 16.19/8.92 => ( ord_less_int @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2790_mult__strict__mono,axiom,
% 16.19/8.92 ! [A: real,B: real,C: real,D: real] :
% 16.19/8.92 ( ( ord_less_real @ A @ B )
% 16.19/8.92 => ( ( ord_less_real @ C @ D )
% 16.19/8.92 => ( ( ord_less_real @ zero_zero_real @ B )
% 16.19/8.92 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 16.19/8.92 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2791_mult__strict__mono,axiom,
% 16.19/8.92 ! [A: rat,B: rat,C: rat,D: rat] :
% 16.19/8.92 ( ( ord_less_rat @ A @ B )
% 16.19/8.92 => ( ( ord_less_rat @ C @ D )
% 16.19/8.92 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 16.19/8.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 16.19/8.92 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2792_mult__strict__mono,axiom,
% 16.19/8.92 ! [A: nat,B: nat,C: nat,D: nat] :
% 16.19/8.92 ( ( ord_less_nat @ A @ B )
% 16.19/8.92 => ( ( ord_less_nat @ C @ D )
% 16.19/8.92 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 16.19/8.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 16.19/8.92 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2793_mult__strict__mono,axiom,
% 16.19/8.92 ! [A: int,B: int,C: int,D: int] :
% 16.19/8.92 ( ( ord_less_int @ A @ B )
% 16.19/8.92 => ( ( ord_less_int @ C @ D )
% 16.19/8.92 => ( ( ord_less_int @ zero_zero_int @ B )
% 16.19/8.92 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 16.19/8.92 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2794_mult__less__cancel__left,axiom,
% 16.19/8.92 ! [C: real,A: real,B: real] :
% 16.19/8.92 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 16.19/8.92 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 16.19/8.92 => ( ord_less_real @ A @ B ) )
% 16.19/8.92 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 16.19/8.92 => ( ord_less_real @ B @ A ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2795_mult__less__cancel__left,axiom,
% 16.19/8.92 ! [C: rat,A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 16.19/8.92 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 16.19/8.92 => ( ord_less_rat @ A @ B ) )
% 16.19/8.92 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 16.19/8.92 => ( ord_less_rat @ B @ A ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2796_mult__less__cancel__left,axiom,
% 16.19/8.92 ! [C: int,A: int,B: int] :
% 16.19/8.92 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 16.19/8.92 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 16.19/8.92 => ( ord_less_int @ A @ B ) )
% 16.19/8.92 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 16.19/8.92 => ( ord_less_int @ B @ A ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2797_mult__right__less__imp__less,axiom,
% 16.19/8.92 ! [A: real,C: real,B: real] :
% 16.19/8.92 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 16.19/8.92 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 16.19/8.92 => ( ord_less_real @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2798_mult__right__less__imp__less,axiom,
% 16.19/8.92 ! [A: rat,C: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 16.19/8.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 16.19/8.92 => ( ord_less_rat @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2799_mult__right__less__imp__less,axiom,
% 16.19/8.92 ! [A: nat,C: nat,B: nat] :
% 16.19/8.92 ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 16.19/8.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 16.19/8.92 => ( ord_less_nat @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2800_mult__right__less__imp__less,axiom,
% 16.19/8.92 ! [A: int,C: int,B: int] :
% 16.19/8.92 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 16.19/8.92 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 16.19/8.92 => ( ord_less_int @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2801_mult__strict__mono_H,axiom,
% 16.19/8.92 ! [A: real,B: real,C: real,D: real] :
% 16.19/8.92 ( ( ord_less_real @ A @ B )
% 16.19/8.92 => ( ( ord_less_real @ C @ D )
% 16.19/8.92 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/8.92 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 16.19/8.92 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2802_mult__strict__mono_H,axiom,
% 16.19/8.92 ! [A: rat,B: rat,C: rat,D: rat] :
% 16.19/8.92 ( ( ord_less_rat @ A @ B )
% 16.19/8.92 => ( ( ord_less_rat @ C @ D )
% 16.19/8.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/8.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 16.19/8.92 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2803_mult__strict__mono_H,axiom,
% 16.19/8.92 ! [A: nat,B: nat,C: nat,D: nat] :
% 16.19/8.92 ( ( ord_less_nat @ A @ B )
% 16.19/8.92 => ( ( ord_less_nat @ C @ D )
% 16.19/8.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 16.19/8.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 16.19/8.92 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2804_mult__strict__mono_H,axiom,
% 16.19/8.92 ! [A: int,B: int,C: int,D: int] :
% 16.19/8.92 ( ( ord_less_int @ A @ B )
% 16.19/8.92 => ( ( ord_less_int @ C @ D )
% 16.19/8.92 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.92 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 16.19/8.92 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2805_mult__less__cancel__right,axiom,
% 16.19/8.92 ! [A: real,C: real,B: real] :
% 16.19/8.92 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 16.19/8.92 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 16.19/8.92 => ( ord_less_real @ A @ B ) )
% 16.19/8.92 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 16.19/8.92 => ( ord_less_real @ B @ A ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2806_mult__less__cancel__right,axiom,
% 16.19/8.92 ! [A: rat,C: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 16.19/8.92 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 16.19/8.92 => ( ord_less_rat @ A @ B ) )
% 16.19/8.92 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 16.19/8.92 => ( ord_less_rat @ B @ A ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2807_mult__less__cancel__right,axiom,
% 16.19/8.92 ! [A: int,C: int,B: int] :
% 16.19/8.92 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 16.19/8.92 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 16.19/8.92 => ( ord_less_int @ A @ B ) )
% 16.19/8.92 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 16.19/8.92 => ( ord_less_int @ B @ A ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2808_mult__le__cancel__left__neg,axiom,
% 16.19/8.92 ! [C: real,A: real,B: real] :
% 16.19/8.92 ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.92 => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 16.19/8.92 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2809_mult__le__cancel__left__neg,axiom,
% 16.19/8.92 ! [C: rat,A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.92 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 16.19/8.92 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2810_mult__le__cancel__left__neg,axiom,
% 16.19/8.92 ! [C: int,A: int,B: int] :
% 16.19/8.92 ( ( ord_less_int @ C @ zero_zero_int )
% 16.19/8.92 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 16.19/8.92 = ( ord_less_eq_int @ B @ A ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2811_mult__le__cancel__left__pos,axiom,
% 16.19/8.92 ! [C: real,A: real,B: real] :
% 16.19/8.92 ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.92 => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 16.19/8.92 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2812_mult__le__cancel__left__pos,axiom,
% 16.19/8.92 ! [C: rat,A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.92 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 16.19/8.92 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2813_mult__le__cancel__left__pos,axiom,
% 16.19/8.92 ! [C: int,A: int,B: int] :
% 16.19/8.92 ( ( ord_less_int @ zero_zero_int @ C )
% 16.19/8.92 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 16.19/8.92 = ( ord_less_eq_int @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2814_mult__left__le__imp__le,axiom,
% 16.19/8.92 ! [C: real,A: real,B: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 16.19/8.92 => ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.92 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2815_mult__left__le__imp__le,axiom,
% 16.19/8.92 ! [C: rat,A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 16.19/8.92 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.92 => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2816_mult__left__le__imp__le,axiom,
% 16.19/8.92 ! [C: nat,A: nat,B: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 16.19/8.92 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 16.19/8.92 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2817_mult__left__le__imp__le,axiom,
% 16.19/8.92 ! [C: int,A: int,B: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 16.19/8.92 => ( ( ord_less_int @ zero_zero_int @ C )
% 16.19/8.92 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2818_mult__right__le__imp__le,axiom,
% 16.19/8.92 ! [A: real,C: real,B: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 16.19/8.92 => ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.92 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2819_mult__right__le__imp__le,axiom,
% 16.19/8.92 ! [A: rat,C: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 16.19/8.92 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.92 => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2820_mult__right__le__imp__le,axiom,
% 16.19/8.92 ! [A: nat,C: nat,B: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 16.19/8.92 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 16.19/8.92 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2821_mult__right__le__imp__le,axiom,
% 16.19/8.92 ! [A: int,C: int,B: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 16.19/8.92 => ( ( ord_less_int @ zero_zero_int @ C )
% 16.19/8.92 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2822_mult__le__less__imp__less,axiom,
% 16.19/8.92 ! [A: real,B: real,C: real,D: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ A @ B )
% 16.19/8.92 => ( ( ord_less_real @ C @ D )
% 16.19/8.92 => ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.92 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 16.19/8.92 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2823_mult__le__less__imp__less,axiom,
% 16.19/8.92 ! [A: rat,B: rat,C: rat,D: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.92 => ( ( ord_less_rat @ C @ D )
% 16.19/8.92 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 16.19/8.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 16.19/8.92 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2824_mult__le__less__imp__less,axiom,
% 16.19/8.92 ! [A: nat,B: nat,C: nat,D: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.92 => ( ( ord_less_nat @ C @ D )
% 16.19/8.92 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 16.19/8.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 16.19/8.92 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2825_mult__le__less__imp__less,axiom,
% 16.19/8.92 ! [A: int,B: int,C: int,D: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ A @ B )
% 16.19/8.92 => ( ( ord_less_int @ C @ D )
% 16.19/8.92 => ( ( ord_less_int @ zero_zero_int @ A )
% 16.19/8.92 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 16.19/8.92 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2826_mult__less__le__imp__less,axiom,
% 16.19/8.92 ! [A: real,B: real,C: real,D: real] :
% 16.19/8.92 ( ( ord_less_real @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_real @ C @ D )
% 16.19/8.92 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/8.92 => ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.92 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2827_mult__less__le__imp__less,axiom,
% 16.19/8.92 ! [A: rat,B: rat,C: rat,D: rat] :
% 16.19/8.92 ( ( ord_less_rat @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_rat @ C @ D )
% 16.19/8.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/8.92 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.92 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2828_mult__less__le__imp__less,axiom,
% 16.19/8.92 ! [A: nat,B: nat,C: nat,D: nat] :
% 16.19/8.92 ( ( ord_less_nat @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_nat @ C @ D )
% 16.19/8.92 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 16.19/8.92 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 16.19/8.92 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2829_mult__less__le__imp__less,axiom,
% 16.19/8.92 ! [A: int,B: int,C: int,D: int] :
% 16.19/8.92 ( ( ord_less_int @ A @ B )
% 16.19/8.92 => ( ( ord_less_eq_int @ C @ D )
% 16.19/8.92 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.92 => ( ( ord_less_int @ zero_zero_int @ C )
% 16.19/8.92 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2830_field__le__epsilon,axiom,
% 16.19/8.92 ! [X: real,Y: real] :
% 16.19/8.92 ( ! [E2: real] :
% 16.19/8.92 ( ( ord_less_real @ zero_zero_real @ E2 )
% 16.19/8.92 => ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E2 ) ) )
% 16.19/8.92 => ( ord_less_eq_real @ X @ Y ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2831_field__le__epsilon,axiom,
% 16.19/8.92 ! [X: rat,Y: rat] :
% 16.19/8.92 ( ! [E2: rat] :
% 16.19/8.92 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 16.19/8.92 => ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y @ E2 ) ) )
% 16.19/8.92 => ( ord_less_eq_rat @ X @ Y ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2832_add__neg__nonpos,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ord_less_real @ A @ zero_zero_real )
% 16.19/8.92 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 16.19/8.92 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2833_add__neg__nonpos,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_rat @ A @ zero_zero_rat )
% 16.19/8.92 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 16.19/8.92 => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2834_add__neg__nonpos,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.92 ( ( ord_less_nat @ A @ zero_zero_nat )
% 16.19/8.92 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 16.19/8.92 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2835_add__neg__nonpos,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( ord_less_int @ A @ zero_zero_int )
% 16.19/8.92 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 16.19/8.92 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2836_add__nonneg__pos,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/8.92 => ( ( ord_less_real @ zero_zero_real @ B )
% 16.19/8.92 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2837_add__nonneg__pos,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/8.92 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 16.19/8.92 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2838_add__nonneg__pos,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 16.19/8.92 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 16.19/8.92 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2839_add__nonneg__pos,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.92 => ( ( ord_less_int @ zero_zero_int @ B )
% 16.19/8.92 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2840_add__nonpos__neg,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 16.19/8.92 => ( ( ord_less_real @ B @ zero_zero_real )
% 16.19/8.92 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2841_add__nonpos__neg,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 16.19/8.92 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 16.19/8.92 => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2842_add__nonpos__neg,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.92 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 16.19/8.92 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 16.19/8.92 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2843_add__nonpos__neg,axiom,
% 16.19/8.92 ! [A: int,B: int] :
% 16.19/8.92 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 16.19/8.92 => ( ( ord_less_int @ B @ zero_zero_int )
% 16.19/8.92 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2844_add__pos__nonneg,axiom,
% 16.19/8.92 ! [A: real,B: real] :
% 16.19/8.92 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.92 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 16.19/8.92 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2845_add__pos__nonneg,axiom,
% 16.19/8.92 ! [A: rat,B: rat] :
% 16.19/8.92 ( ( ord_less_rat @ zero_zero_rat @ A )
% 16.19/8.92 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 16.19/8.92 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 16.19/8.92
% 16.19/8.92 thf(fact_2846_add__pos__nonneg,axiom,
% 16.19/8.92 ! [A: nat,B: nat] :
% 16.19/8.93 ( ( ord_less_nat @ zero_zero_nat @ A )
% 16.19/8.93 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 16.19/8.93 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2847_add__pos__nonneg,axiom,
% 16.19/8.93 ! [A: int,B: int] :
% 16.19/8.93 ( ( ord_less_int @ zero_zero_int @ A )
% 16.19/8.93 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 16.19/8.93 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2848_add__strict__increasing,axiom,
% 16.19/8.93 ! [A: real,B: real,C: real] :
% 16.19/8.93 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.93 => ( ( ord_less_eq_real @ B @ C )
% 16.19/8.93 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2849_add__strict__increasing,axiom,
% 16.19/8.93 ! [A: rat,B: rat,C: rat] :
% 16.19/8.93 ( ( ord_less_rat @ zero_zero_rat @ A )
% 16.19/8.93 => ( ( ord_less_eq_rat @ B @ C )
% 16.19/8.93 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2850_add__strict__increasing,axiom,
% 16.19/8.93 ! [A: nat,B: nat,C: nat] :
% 16.19/8.93 ( ( ord_less_nat @ zero_zero_nat @ A )
% 16.19/8.93 => ( ( ord_less_eq_nat @ B @ C )
% 16.19/8.93 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2851_add__strict__increasing,axiom,
% 16.19/8.93 ! [A: int,B: int,C: int] :
% 16.19/8.93 ( ( ord_less_int @ zero_zero_int @ A )
% 16.19/8.93 => ( ( ord_less_eq_int @ B @ C )
% 16.19/8.93 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2852_add__strict__increasing2,axiom,
% 16.19/8.93 ! [A: real,B: real,C: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/8.93 => ( ( ord_less_real @ B @ C )
% 16.19/8.93 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2853_add__strict__increasing2,axiom,
% 16.19/8.93 ! [A: rat,B: rat,C: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/8.93 => ( ( ord_less_rat @ B @ C )
% 16.19/8.93 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2854_add__strict__increasing2,axiom,
% 16.19/8.93 ! [A: nat,B: nat,C: nat] :
% 16.19/8.93 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 16.19/8.93 => ( ( ord_less_nat @ B @ C )
% 16.19/8.93 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2855_add__strict__increasing2,axiom,
% 16.19/8.93 ! [A: int,B: int,C: int] :
% 16.19/8.93 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.93 => ( ( ord_less_int @ B @ C )
% 16.19/8.93 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2856_frac__le,axiom,
% 16.19/8.93 ! [Y: real,X: real,W: real,Z: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/8.93 => ( ( ord_less_eq_real @ X @ Y )
% 16.19/8.93 => ( ( ord_less_real @ zero_zero_real @ W )
% 16.19/8.93 => ( ( ord_less_eq_real @ W @ Z )
% 16.19/8.93 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2857_frac__le,axiom,
% 16.19/8.93 ! [Y: rat,X: rat,W: rat,Z: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 16.19/8.93 => ( ( ord_less_eq_rat @ X @ Y )
% 16.19/8.93 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 16.19/8.93 => ( ( ord_less_eq_rat @ W @ Z )
% 16.19/8.93 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2858_frac__less,axiom,
% 16.19/8.93 ! [X: real,Y: real,W: real,Z: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/8.93 => ( ( ord_less_real @ X @ Y )
% 16.19/8.93 => ( ( ord_less_real @ zero_zero_real @ W )
% 16.19/8.93 => ( ( ord_less_eq_real @ W @ Z )
% 16.19/8.93 => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2859_frac__less,axiom,
% 16.19/8.93 ! [X: rat,Y: rat,W: rat,Z: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 16.19/8.93 => ( ( ord_less_rat @ X @ Y )
% 16.19/8.93 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 16.19/8.93 => ( ( ord_less_eq_rat @ W @ Z )
% 16.19/8.93 => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2860_frac__less2,axiom,
% 16.19/8.93 ! [X: real,Y: real,W: real,Z: real] :
% 16.19/8.93 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/8.93 => ( ( ord_less_eq_real @ X @ Y )
% 16.19/8.93 => ( ( ord_less_real @ zero_zero_real @ W )
% 16.19/8.93 => ( ( ord_less_real @ W @ Z )
% 16.19/8.93 => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2861_frac__less2,axiom,
% 16.19/8.93 ! [X: rat,Y: rat,W: rat,Z: rat] :
% 16.19/8.93 ( ( ord_less_rat @ zero_zero_rat @ X )
% 16.19/8.93 => ( ( ord_less_eq_rat @ X @ Y )
% 16.19/8.93 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 16.19/8.93 => ( ( ord_less_rat @ W @ Z )
% 16.19/8.93 => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2862_divide__le__cancel,axiom,
% 16.19/8.93 ! [A: real,C: real,B: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 16.19/8.93 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ord_less_eq_real @ A @ B ) )
% 16.19/8.93 & ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2863_divide__le__cancel,axiom,
% 16.19/8.93 ! [A: rat,C: rat,B: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 16.19/8.93 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ord_less_eq_rat @ A @ B ) )
% 16.19/8.93 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2864_divide__nonneg__neg,axiom,
% 16.19/8.93 ! [X: real,Y: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/8.93 => ( ( ord_less_real @ Y @ zero_zero_real )
% 16.19/8.93 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2865_divide__nonneg__neg,axiom,
% 16.19/8.93 ! [X: rat,Y: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 16.19/8.93 => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 16.19/8.93 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2866_divide__nonneg__pos,axiom,
% 16.19/8.93 ! [X: real,Y: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/8.93 => ( ( ord_less_real @ zero_zero_real @ Y )
% 16.19/8.93 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2867_divide__nonneg__pos,axiom,
% 16.19/8.93 ! [X: rat,Y: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 16.19/8.93 => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 16.19/8.93 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2868_divide__nonpos__neg,axiom,
% 16.19/8.93 ! [X: real,Y: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 16.19/8.93 => ( ( ord_less_real @ Y @ zero_zero_real )
% 16.19/8.93 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2869_divide__nonpos__neg,axiom,
% 16.19/8.93 ! [X: rat,Y: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 16.19/8.93 => ( ( ord_less_rat @ Y @ zero_zero_rat )
% 16.19/8.93 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2870_divide__nonpos__pos,axiom,
% 16.19/8.93 ! [X: real,Y: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 16.19/8.93 => ( ( ord_less_real @ zero_zero_real @ Y )
% 16.19/8.93 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2871_divide__nonpos__pos,axiom,
% 16.19/8.93 ! [X: rat,Y: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 16.19/8.93 => ( ( ord_less_rat @ zero_zero_rat @ Y )
% 16.19/8.93 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2872_sum__squares__le__zero__iff,axiom,
% 16.19/8.93 ! [X: real,Y: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
% 16.19/8.93 = ( ( X = zero_zero_real )
% 16.19/8.93 & ( Y = zero_zero_real ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2873_sum__squares__le__zero__iff,axiom,
% 16.19/8.93 ! [X: rat,Y: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
% 16.19/8.93 = ( ( X = zero_zero_rat )
% 16.19/8.93 & ( Y = zero_zero_rat ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2874_sum__squares__le__zero__iff,axiom,
% 16.19/8.93 ! [X: int,Y: int] :
% 16.19/8.93 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
% 16.19/8.93 = ( ( X = zero_zero_int )
% 16.19/8.93 & ( Y = zero_zero_int ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2875_sum__squares__ge__zero,axiom,
% 16.19/8.93 ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2876_sum__squares__ge__zero,axiom,
% 16.19/8.93 ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2877_sum__squares__ge__zero,axiom,
% 16.19/8.93 ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2878_mult__left__le,axiom,
% 16.19/8.93 ! [C: real,A: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ C @ one_one_real )
% 16.19/8.93 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/8.93 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2879_mult__left__le,axiom,
% 16.19/8.93 ! [C: rat,A: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ C @ one_one_rat )
% 16.19/8.93 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/8.93 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2880_mult__left__le,axiom,
% 16.19/8.93 ! [C: nat,A: nat] :
% 16.19/8.93 ( ( ord_less_eq_nat @ C @ one_one_nat )
% 16.19/8.93 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 16.19/8.93 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2881_mult__left__le,axiom,
% 16.19/8.93 ! [C: int,A: int] :
% 16.19/8.93 ( ( ord_less_eq_int @ C @ one_one_int )
% 16.19/8.93 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.93 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2882_mult__le__one,axiom,
% 16.19/8.93 ! [A: real,B: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ A @ one_one_real )
% 16.19/8.93 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 16.19/8.93 => ( ( ord_less_eq_real @ B @ one_one_real )
% 16.19/8.93 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2883_mult__le__one,axiom,
% 16.19/8.93 ! [A: rat,B: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ A @ one_one_rat )
% 16.19/8.93 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 16.19/8.93 => ( ( ord_less_eq_rat @ B @ one_one_rat )
% 16.19/8.93 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2884_mult__le__one,axiom,
% 16.19/8.93 ! [A: nat,B: nat] :
% 16.19/8.93 ( ( ord_less_eq_nat @ A @ one_one_nat )
% 16.19/8.93 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 16.19/8.93 => ( ( ord_less_eq_nat @ B @ one_one_nat )
% 16.19/8.93 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2885_mult__le__one,axiom,
% 16.19/8.93 ! [A: int,B: int] :
% 16.19/8.93 ( ( ord_less_eq_int @ A @ one_one_int )
% 16.19/8.93 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 16.19/8.93 => ( ( ord_less_eq_int @ B @ one_one_int )
% 16.19/8.93 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2886_mult__right__le__one__le,axiom,
% 16.19/8.93 ! [X: real,Y: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/8.93 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/8.93 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 16.19/8.93 => ( ord_less_eq_real @ ( times_times_real @ X @ Y ) @ X ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2887_mult__right__le__one__le,axiom,
% 16.19/8.93 ! [X: rat,Y: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 16.19/8.93 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 16.19/8.93 => ( ( ord_less_eq_rat @ Y @ one_one_rat )
% 16.19/8.93 => ( ord_less_eq_rat @ ( times_times_rat @ X @ Y ) @ X ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2888_mult__right__le__one__le,axiom,
% 16.19/8.93 ! [X: int,Y: int] :
% 16.19/8.93 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 16.19/8.93 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 16.19/8.93 => ( ( ord_less_eq_int @ Y @ one_one_int )
% 16.19/8.93 => ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2889_mult__left__le__one__le,axiom,
% 16.19/8.93 ! [X: real,Y: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/8.93 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/8.93 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 16.19/8.93 => ( ord_less_eq_real @ ( times_times_real @ Y @ X ) @ X ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2890_mult__left__le__one__le,axiom,
% 16.19/8.93 ! [X: rat,Y: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 16.19/8.93 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 16.19/8.93 => ( ( ord_less_eq_rat @ Y @ one_one_rat )
% 16.19/8.93 => ( ord_less_eq_rat @ ( times_times_rat @ Y @ X ) @ X ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2891_mult__left__le__one__le,axiom,
% 16.19/8.93 ! [X: int,Y: int] :
% 16.19/8.93 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 16.19/8.93 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 16.19/8.93 => ( ( ord_less_eq_int @ Y @ one_one_int )
% 16.19/8.93 => ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2892_power__less__imp__less__base,axiom,
% 16.19/8.93 ! [A: real,N2: nat,B: real] :
% 16.19/8.93 ( ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) )
% 16.19/8.93 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 16.19/8.93 => ( ord_less_real @ A @ B ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2893_power__less__imp__less__base,axiom,
% 16.19/8.93 ! [A: rat,N2: nat,B: rat] :
% 16.19/8.93 ( ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) )
% 16.19/8.93 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 16.19/8.93 => ( ord_less_rat @ A @ B ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2894_power__less__imp__less__base,axiom,
% 16.19/8.93 ! [A: nat,N2: nat,B: nat] :
% 16.19/8.93 ( ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
% 16.19/8.93 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 16.19/8.93 => ( ord_less_nat @ A @ B ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2895_power__less__imp__less__base,axiom,
% 16.19/8.93 ! [A: int,N2: nat,B: int] :
% 16.19/8.93 ( ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
% 16.19/8.93 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 16.19/8.93 => ( ord_less_int @ A @ B ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2896_sum__squares__gt__zero__iff,axiom,
% 16.19/8.93 ! [X: real,Y: real] :
% 16.19/8.93 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) )
% 16.19/8.93 = ( ( X != zero_zero_real )
% 16.19/8.93 | ( Y != zero_zero_real ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2897_sum__squares__gt__zero__iff,axiom,
% 16.19/8.93 ! [X: rat,Y: rat] :
% 16.19/8.93 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) )
% 16.19/8.93 = ( ( X != zero_zero_rat )
% 16.19/8.93 | ( Y != zero_zero_rat ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2898_sum__squares__gt__zero__iff,axiom,
% 16.19/8.93 ! [X: int,Y: int] :
% 16.19/8.93 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
% 16.19/8.93 = ( ( X != zero_zero_int )
% 16.19/8.93 | ( Y != zero_zero_int ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2899_not__sum__squares__lt__zero,axiom,
% 16.19/8.93 ! [X: real,Y: real] :
% 16.19/8.93 ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2900_not__sum__squares__lt__zero,axiom,
% 16.19/8.93 ! [X: rat,Y: rat] :
% 16.19/8.93 ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2901_not__sum__squares__lt__zero,axiom,
% 16.19/8.93 ! [X: int,Y: int] :
% 16.19/8.93 ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2902_zero__less__two,axiom,
% 16.19/8.93 ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2903_zero__less__two,axiom,
% 16.19/8.93 ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2904_zero__less__two,axiom,
% 16.19/8.93 ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2905_zero__less__two,axiom,
% 16.19/8.93 ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2906_divide__strict__left__mono__neg,axiom,
% 16.19/8.93 ! [A: real,B: real,C: real] :
% 16.19/8.93 ( ( ord_less_real @ A @ B )
% 16.19/8.93 => ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 16.19/8.93 => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2907_divide__strict__left__mono__neg,axiom,
% 16.19/8.93 ! [A: rat,B: rat,C: rat] :
% 16.19/8.93 ( ( ord_less_rat @ A @ B )
% 16.19/8.93 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 16.19/8.93 => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2908_divide__strict__left__mono,axiom,
% 16.19/8.93 ! [B: real,A: real,C: real] :
% 16.19/8.93 ( ( ord_less_real @ B @ A )
% 16.19/8.93 => ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 16.19/8.93 => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2909_divide__strict__left__mono,axiom,
% 16.19/8.93 ! [B: rat,A: rat,C: rat] :
% 16.19/8.93 ( ( ord_less_rat @ B @ A )
% 16.19/8.93 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 16.19/8.93 => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2910_mult__imp__less__div__pos,axiom,
% 16.19/8.93 ! [Y: real,Z: real,X: real] :
% 16.19/8.93 ( ( ord_less_real @ zero_zero_real @ Y )
% 16.19/8.93 => ( ( ord_less_real @ ( times_times_real @ Z @ Y ) @ X )
% 16.19/8.93 => ( ord_less_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2911_mult__imp__less__div__pos,axiom,
% 16.19/8.93 ! [Y: rat,Z: rat,X: rat] :
% 16.19/8.93 ( ( ord_less_rat @ zero_zero_rat @ Y )
% 16.19/8.93 => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y ) @ X )
% 16.19/8.93 => ( ord_less_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2912_mult__imp__div__pos__less,axiom,
% 16.19/8.93 ! [Y: real,X: real,Z: real] :
% 16.19/8.93 ( ( ord_less_real @ zero_zero_real @ Y )
% 16.19/8.93 => ( ( ord_less_real @ X @ ( times_times_real @ Z @ Y ) )
% 16.19/8.93 => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2913_mult__imp__div__pos__less,axiom,
% 16.19/8.93 ! [Y: rat,X: rat,Z: rat] :
% 16.19/8.93 ( ( ord_less_rat @ zero_zero_rat @ Y )
% 16.19/8.93 => ( ( ord_less_rat @ X @ ( times_times_rat @ Z @ Y ) )
% 16.19/8.93 => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2914_pos__less__divide__eq,axiom,
% 16.19/8.93 ! [C: real,A: real,B: real] :
% 16.19/8.93 ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 16.19/8.93 = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2915_pos__less__divide__eq,axiom,
% 16.19/8.93 ! [C: rat,A: rat,B: rat] :
% 16.19/8.93 ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 16.19/8.93 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2916_pos__divide__less__eq,axiom,
% 16.19/8.93 ! [C: real,B: real,A: real] :
% 16.19/8.93 ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 16.19/8.93 = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2917_pos__divide__less__eq,axiom,
% 16.19/8.93 ! [C: rat,B: rat,A: rat] :
% 16.19/8.93 ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 16.19/8.93 = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2918_neg__less__divide__eq,axiom,
% 16.19/8.93 ! [C: real,A: real,B: real] :
% 16.19/8.93 ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 16.19/8.93 = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2919_neg__less__divide__eq,axiom,
% 16.19/8.93 ! [C: rat,A: rat,B: rat] :
% 16.19/8.93 ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 16.19/8.93 = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2920_neg__divide__less__eq,axiom,
% 16.19/8.93 ! [C: real,B: real,A: real] :
% 16.19/8.93 ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 16.19/8.93 = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2921_neg__divide__less__eq,axiom,
% 16.19/8.93 ! [C: rat,B: rat,A: rat] :
% 16.19/8.93 ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 16.19/8.93 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2922_less__divide__eq,axiom,
% 16.19/8.93 ! [A: real,B: real,C: real] :
% 16.19/8.93 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 16.19/8.93 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 16.19/8.93 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 16.19/8.93 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2923_less__divide__eq,axiom,
% 16.19/8.93 ! [A: rat,B: rat,C: rat] :
% 16.19/8.93 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 16.19/8.93 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 16.19/8.93 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 16.19/8.93 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2924_divide__less__eq,axiom,
% 16.19/8.93 ! [B: real,C: real,A: real] :
% 16.19/8.93 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 16.19/8.93 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 16.19/8.93 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 16.19/8.93 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2925_divide__less__eq,axiom,
% 16.19/8.93 ! [B: rat,C: rat,A: rat] :
% 16.19/8.93 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 16.19/8.93 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 16.19/8.93 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 16.19/8.93 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2926_power__le__one,axiom,
% 16.19/8.93 ! [A: real,N2: nat] :
% 16.19/8.93 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/8.93 => ( ( ord_less_eq_real @ A @ one_one_real )
% 16.19/8.93 => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ one_one_real ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2927_power__le__one,axiom,
% 16.19/8.93 ! [A: rat,N2: nat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/8.93 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 16.19/8.93 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ one_one_rat ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2928_power__le__one,axiom,
% 16.19/8.93 ! [A: nat,N2: nat] :
% 16.19/8.93 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 16.19/8.93 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 16.19/8.93 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ one_one_nat ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2929_power__le__one,axiom,
% 16.19/8.93 ! [A: int,N2: nat] :
% 16.19/8.93 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.93 => ( ( ord_less_eq_int @ A @ one_one_int )
% 16.19/8.93 => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ one_one_int ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2930_eq__divide__eq__numeral_I1_J,axiom,
% 16.19/8.93 ! [W: num,B: complex,C: complex] :
% 16.19/8.93 ( ( ( numera6690914467698888265omplex @ W )
% 16.19/8.93 = ( divide1717551699836669952omplex @ B @ C ) )
% 16.19/8.93 = ( ( ( C != zero_zero_complex )
% 16.19/8.93 => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C )
% 16.19/8.93 = B ) )
% 16.19/8.93 & ( ( C = zero_zero_complex )
% 16.19/8.93 => ( ( numera6690914467698888265omplex @ W )
% 16.19/8.93 = zero_zero_complex ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2931_eq__divide__eq__numeral_I1_J,axiom,
% 16.19/8.93 ! [W: num,B: real,C: real] :
% 16.19/8.93 ( ( ( numeral_numeral_real @ W )
% 16.19/8.93 = ( divide_divide_real @ B @ C ) )
% 16.19/8.93 = ( ( ( C != zero_zero_real )
% 16.19/8.93 => ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
% 16.19/8.93 = B ) )
% 16.19/8.93 & ( ( C = zero_zero_real )
% 16.19/8.93 => ( ( numeral_numeral_real @ W )
% 16.19/8.93 = zero_zero_real ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2932_eq__divide__eq__numeral_I1_J,axiom,
% 16.19/8.93 ! [W: num,B: rat,C: rat] :
% 16.19/8.93 ( ( ( numeral_numeral_rat @ W )
% 16.19/8.93 = ( divide_divide_rat @ B @ C ) )
% 16.19/8.93 = ( ( ( C != zero_zero_rat )
% 16.19/8.93 => ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
% 16.19/8.93 = B ) )
% 16.19/8.93 & ( ( C = zero_zero_rat )
% 16.19/8.93 => ( ( numeral_numeral_rat @ W )
% 16.19/8.93 = zero_zero_rat ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2933_divide__eq__eq__numeral_I1_J,axiom,
% 16.19/8.93 ! [B: complex,C: complex,W: num] :
% 16.19/8.93 ( ( ( divide1717551699836669952omplex @ B @ C )
% 16.19/8.93 = ( numera6690914467698888265omplex @ W ) )
% 16.19/8.93 = ( ( ( C != zero_zero_complex )
% 16.19/8.93 => ( B
% 16.19/8.93 = ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C ) ) )
% 16.19/8.93 & ( ( C = zero_zero_complex )
% 16.19/8.93 => ( ( numera6690914467698888265omplex @ W )
% 16.19/8.93 = zero_zero_complex ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2934_divide__eq__eq__numeral_I1_J,axiom,
% 16.19/8.93 ! [B: real,C: real,W: num] :
% 16.19/8.93 ( ( ( divide_divide_real @ B @ C )
% 16.19/8.93 = ( numeral_numeral_real @ W ) )
% 16.19/8.93 = ( ( ( C != zero_zero_real )
% 16.19/8.93 => ( B
% 16.19/8.93 = ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 16.19/8.93 & ( ( C = zero_zero_real )
% 16.19/8.93 => ( ( numeral_numeral_real @ W )
% 16.19/8.93 = zero_zero_real ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2935_divide__eq__eq__numeral_I1_J,axiom,
% 16.19/8.93 ! [B: rat,C: rat,W: num] :
% 16.19/8.93 ( ( ( divide_divide_rat @ B @ C )
% 16.19/8.93 = ( numeral_numeral_rat @ W ) )
% 16.19/8.93 = ( ( ( C != zero_zero_rat )
% 16.19/8.93 => ( B
% 16.19/8.93 = ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 16.19/8.93 & ( ( C = zero_zero_rat )
% 16.19/8.93 => ( ( numeral_numeral_rat @ W )
% 16.19/8.93 = zero_zero_rat ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2936_divide__less__eq__1,axiom,
% 16.19/8.93 ! [B: real,A: real] :
% 16.19/8.93 ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 16.19/8.93 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.93 & ( ord_less_real @ B @ A ) )
% 16.19/8.93 | ( ( ord_less_real @ A @ zero_zero_real )
% 16.19/8.93 & ( ord_less_real @ A @ B ) )
% 16.19/8.93 | ( A = zero_zero_real ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2937_divide__less__eq__1,axiom,
% 16.19/8.93 ! [B: rat,A: rat] :
% 16.19/8.93 ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 16.19/8.93 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 16.19/8.93 & ( ord_less_rat @ B @ A ) )
% 16.19/8.93 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 16.19/8.93 & ( ord_less_rat @ A @ B ) )
% 16.19/8.93 | ( A = zero_zero_rat ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2938_less__divide__eq__1,axiom,
% 16.19/8.93 ! [B: real,A: real] :
% 16.19/8.93 ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 16.19/8.93 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.93 & ( ord_less_real @ A @ B ) )
% 16.19/8.93 | ( ( ord_less_real @ A @ zero_zero_real )
% 16.19/8.93 & ( ord_less_real @ B @ A ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2939_less__divide__eq__1,axiom,
% 16.19/8.93 ! [B: rat,A: rat] :
% 16.19/8.93 ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 16.19/8.93 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 16.19/8.93 & ( ord_less_rat @ A @ B ) )
% 16.19/8.93 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 16.19/8.93 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2940_divide__add__eq__iff,axiom,
% 16.19/8.93 ! [Z: complex,X: complex,Y: complex] :
% 16.19/8.93 ( ( Z != zero_zero_complex )
% 16.19/8.93 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y )
% 16.19/8.93 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2941_divide__add__eq__iff,axiom,
% 16.19/8.93 ! [Z: real,X: real,Y: real] :
% 16.19/8.93 ( ( Z != zero_zero_real )
% 16.19/8.93 => ( ( plus_plus_real @ ( divide_divide_real @ X @ Z ) @ Y )
% 16.19/8.93 = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2942_divide__add__eq__iff,axiom,
% 16.19/8.93 ! [Z: rat,X: rat,Y: rat] :
% 16.19/8.93 ( ( Z != zero_zero_rat )
% 16.19/8.93 => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
% 16.19/8.93 = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2943_add__divide__eq__iff,axiom,
% 16.19/8.93 ! [Z: complex,X: complex,Y: complex] :
% 16.19/8.93 ( ( Z != zero_zero_complex )
% 16.19/8.93 => ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z ) )
% 16.19/8.93 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ Y ) @ Z ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2944_add__divide__eq__iff,axiom,
% 16.19/8.93 ! [Z: real,X: real,Y: real] :
% 16.19/8.93 ( ( Z != zero_zero_real )
% 16.19/8.93 => ( ( plus_plus_real @ X @ ( divide_divide_real @ Y @ Z ) )
% 16.19/8.93 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2945_add__divide__eq__iff,axiom,
% 16.19/8.93 ! [Z: rat,X: rat,Y: rat] :
% 16.19/8.93 ( ( Z != zero_zero_rat )
% 16.19/8.93 => ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
% 16.19/8.93 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2946_add__num__frac,axiom,
% 16.19/8.93 ! [Y: complex,Z: complex,X: complex] :
% 16.19/8.93 ( ( Y != zero_zero_complex )
% 16.19/8.93 => ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X @ Y ) )
% 16.19/8.93 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2947_add__num__frac,axiom,
% 16.19/8.93 ! [Y: real,Z: real,X: real] :
% 16.19/8.93 ( ( Y != zero_zero_real )
% 16.19/8.93 => ( ( plus_plus_real @ Z @ ( divide_divide_real @ X @ Y ) )
% 16.19/8.93 = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2948_add__num__frac,axiom,
% 16.19/8.93 ! [Y: rat,Z: rat,X: rat] :
% 16.19/8.93 ( ( Y != zero_zero_rat )
% 16.19/8.93 => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X @ Y ) )
% 16.19/8.93 = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2949_add__frac__num,axiom,
% 16.19/8.93 ! [Y: complex,X: complex,Z: complex] :
% 16.19/8.93 ( ( Y != zero_zero_complex )
% 16.19/8.93 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ Z )
% 16.19/8.93 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2950_add__frac__num,axiom,
% 16.19/8.93 ! [Y: real,X: real,Z: real] :
% 16.19/8.93 ( ( Y != zero_zero_real )
% 16.19/8.93 => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ Z )
% 16.19/8.93 = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2951_add__frac__num,axiom,
% 16.19/8.93 ! [Y: rat,X: rat,Z: rat] :
% 16.19/8.93 ( ( Y != zero_zero_rat )
% 16.19/8.93 => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ Z )
% 16.19/8.93 = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2952_add__frac__eq,axiom,
% 16.19/8.93 ! [Y: complex,Z: complex,X: complex,W: complex] :
% 16.19/8.93 ( ( Y != zero_zero_complex )
% 16.19/8.93 => ( ( Z != zero_zero_complex )
% 16.19/8.93 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
% 16.19/8.93 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2953_add__frac__eq,axiom,
% 16.19/8.93 ! [Y: real,Z: real,X: real,W: real] :
% 16.19/8.93 ( ( Y != zero_zero_real )
% 16.19/8.93 => ( ( Z != zero_zero_real )
% 16.19/8.93 => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
% 16.19/8.93 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2954_add__frac__eq,axiom,
% 16.19/8.93 ! [Y: rat,Z: rat,X: rat,W: rat] :
% 16.19/8.93 ( ( Y != zero_zero_rat )
% 16.19/8.93 => ( ( Z != zero_zero_rat )
% 16.19/8.93 => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
% 16.19/8.93 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2955_add__divide__eq__if__simps_I1_J,axiom,
% 16.19/8.93 ! [Z: complex,A: complex,B: complex] :
% 16.19/8.93 ( ( ( Z = zero_zero_complex )
% 16.19/8.93 => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 16.19/8.93 = A ) )
% 16.19/8.93 & ( ( Z != zero_zero_complex )
% 16.19/8.93 => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 16.19/8.93 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2956_add__divide__eq__if__simps_I1_J,axiom,
% 16.19/8.93 ! [Z: real,A: real,B: real] :
% 16.19/8.93 ( ( ( Z = zero_zero_real )
% 16.19/8.93 => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 16.19/8.93 = A ) )
% 16.19/8.93 & ( ( Z != zero_zero_real )
% 16.19/8.93 => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 16.19/8.93 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2957_add__divide__eq__if__simps_I1_J,axiom,
% 16.19/8.93 ! [Z: rat,A: rat,B: rat] :
% 16.19/8.93 ( ( ( Z = zero_zero_rat )
% 16.19/8.93 => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 16.19/8.93 = A ) )
% 16.19/8.93 & ( ( Z != zero_zero_rat )
% 16.19/8.93 => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 16.19/8.93 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2958_add__divide__eq__if__simps_I2_J,axiom,
% 16.19/8.93 ! [Z: complex,A: complex,B: complex] :
% 16.19/8.93 ( ( ( Z = zero_zero_complex )
% 16.19/8.93 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 16.19/8.93 = B ) )
% 16.19/8.93 & ( ( Z != zero_zero_complex )
% 16.19/8.93 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 16.19/8.93 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2959_add__divide__eq__if__simps_I2_J,axiom,
% 16.19/8.93 ! [Z: real,A: real,B: real] :
% 16.19/8.93 ( ( ( Z = zero_zero_real )
% 16.19/8.93 => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 16.19/8.93 = B ) )
% 16.19/8.93 & ( ( Z != zero_zero_real )
% 16.19/8.93 => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 16.19/8.93 = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2960_add__divide__eq__if__simps_I2_J,axiom,
% 16.19/8.93 ! [Z: rat,A: rat,B: rat] :
% 16.19/8.93 ( ( ( Z = zero_zero_rat )
% 16.19/8.93 => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 16.19/8.93 = B ) )
% 16.19/8.93 & ( ( Z != zero_zero_rat )
% 16.19/8.93 => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 16.19/8.93 = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2961_power__le__imp__le__base,axiom,
% 16.19/8.93 ! [A: real,N2: nat,B: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ ( power_power_real @ B @ ( suc @ N2 ) ) )
% 16.19/8.93 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 16.19/8.93 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2962_power__le__imp__le__base,axiom,
% 16.19/8.93 ! [A: rat,N2: nat,B: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ ( power_power_rat @ B @ ( suc @ N2 ) ) )
% 16.19/8.93 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 16.19/8.93 => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2963_power__le__imp__le__base,axiom,
% 16.19/8.93 ! [A: nat,N2: nat,B: nat] :
% 16.19/8.93 ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ ( power_power_nat @ B @ ( suc @ N2 ) ) )
% 16.19/8.93 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 16.19/8.93 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2964_power__le__imp__le__base,axiom,
% 16.19/8.93 ! [A: int,N2: nat,B: int] :
% 16.19/8.93 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ ( power_power_int @ B @ ( suc @ N2 ) ) )
% 16.19/8.93 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 16.19/8.93 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2965_power__inject__base,axiom,
% 16.19/8.93 ! [A: real,N2: nat,B: real] :
% 16.19/8.93 ( ( ( power_power_real @ A @ ( suc @ N2 ) )
% 16.19/8.93 = ( power_power_real @ B @ ( suc @ N2 ) ) )
% 16.19/8.93 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/8.93 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 16.19/8.93 => ( A = B ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2966_power__inject__base,axiom,
% 16.19/8.93 ! [A: rat,N2: nat,B: rat] :
% 16.19/8.93 ( ( ( power_power_rat @ A @ ( suc @ N2 ) )
% 16.19/8.93 = ( power_power_rat @ B @ ( suc @ N2 ) ) )
% 16.19/8.93 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/8.93 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 16.19/8.93 => ( A = B ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2967_power__inject__base,axiom,
% 16.19/8.93 ! [A: nat,N2: nat,B: nat] :
% 16.19/8.93 ( ( ( power_power_nat @ A @ ( suc @ N2 ) )
% 16.19/8.93 = ( power_power_nat @ B @ ( suc @ N2 ) ) )
% 16.19/8.93 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 16.19/8.93 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 16.19/8.93 => ( A = B ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2968_power__inject__base,axiom,
% 16.19/8.93 ! [A: int,N2: nat,B: int] :
% 16.19/8.93 ( ( ( power_power_int @ A @ ( suc @ N2 ) )
% 16.19/8.93 = ( power_power_int @ B @ ( suc @ N2 ) ) )
% 16.19/8.93 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.93 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 16.19/8.93 => ( A = B ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2969_div__add__self2,axiom,
% 16.19/8.93 ! [B: nat,A: nat] :
% 16.19/8.93 ( ( B != zero_zero_nat )
% 16.19/8.93 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 16.19/8.93 = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2970_div__add__self2,axiom,
% 16.19/8.93 ! [B: int,A: int] :
% 16.19/8.93 ( ( B != zero_zero_int )
% 16.19/8.93 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
% 16.19/8.93 = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2971_div__add__self1,axiom,
% 16.19/8.93 ! [B: nat,A: nat] :
% 16.19/8.93 ( ( B != zero_zero_nat )
% 16.19/8.93 => ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 16.19/8.93 = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2972_div__add__self1,axiom,
% 16.19/8.93 ! [B: int,A: int] :
% 16.19/8.93 ( ( B != zero_zero_int )
% 16.19/8.93 => ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
% 16.19/8.93 = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2973_divide__diff__eq__iff,axiom,
% 16.19/8.93 ! [Z: complex,X: complex,Y: complex] :
% 16.19/8.93 ( ( Z != zero_zero_complex )
% 16.19/8.93 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y )
% 16.19/8.93 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2974_divide__diff__eq__iff,axiom,
% 16.19/8.93 ! [Z: real,X: real,Y: real] :
% 16.19/8.93 ( ( Z != zero_zero_real )
% 16.19/8.93 => ( ( minus_minus_real @ ( divide_divide_real @ X @ Z ) @ Y )
% 16.19/8.93 = ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2975_divide__diff__eq__iff,axiom,
% 16.19/8.93 ! [Z: rat,X: rat,Y: rat] :
% 16.19/8.93 ( ( Z != zero_zero_rat )
% 16.19/8.93 => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z ) @ Y )
% 16.19/8.93 = ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2976_diff__divide__eq__iff,axiom,
% 16.19/8.93 ! [Z: complex,X: complex,Y: complex] :
% 16.19/8.93 ( ( Z != zero_zero_complex )
% 16.19/8.93 => ( ( minus_minus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z ) )
% 16.19/8.93 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ Y ) @ Z ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2977_diff__divide__eq__iff,axiom,
% 16.19/8.93 ! [Z: real,X: real,Y: real] :
% 16.19/8.93 ( ( Z != zero_zero_real )
% 16.19/8.93 => ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z ) )
% 16.19/8.93 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ Y ) @ Z ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2978_diff__divide__eq__iff,axiom,
% 16.19/8.93 ! [Z: rat,X: rat,Y: rat] :
% 16.19/8.93 ( ( Z != zero_zero_rat )
% 16.19/8.93 => ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y @ Z ) )
% 16.19/8.93 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ Y ) @ Z ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2979_diff__frac__eq,axiom,
% 16.19/8.93 ! [Y: complex,Z: complex,X: complex,W: complex] :
% 16.19/8.93 ( ( Y != zero_zero_complex )
% 16.19/8.93 => ( ( Z != zero_zero_complex )
% 16.19/8.93 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
% 16.19/8.93 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2980_diff__frac__eq,axiom,
% 16.19/8.93 ! [Y: real,Z: real,X: real,W: real] :
% 16.19/8.93 ( ( Y != zero_zero_real )
% 16.19/8.93 => ( ( Z != zero_zero_real )
% 16.19/8.93 => ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
% 16.19/8.93 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2981_diff__frac__eq,axiom,
% 16.19/8.93 ! [Y: rat,Z: rat,X: rat,W: rat] :
% 16.19/8.93 ( ( Y != zero_zero_rat )
% 16.19/8.93 => ( ( Z != zero_zero_rat )
% 16.19/8.93 => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
% 16.19/8.93 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2982_add__divide__eq__if__simps_I4_J,axiom,
% 16.19/8.93 ! [Z: complex,A: complex,B: complex] :
% 16.19/8.93 ( ( ( Z = zero_zero_complex )
% 16.19/8.93 => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 16.19/8.93 = A ) )
% 16.19/8.93 & ( ( Z != zero_zero_complex )
% 16.19/8.93 => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 16.19/8.93 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2983_add__divide__eq__if__simps_I4_J,axiom,
% 16.19/8.93 ! [Z: real,A: real,B: real] :
% 16.19/8.93 ( ( ( Z = zero_zero_real )
% 16.19/8.93 => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 16.19/8.93 = A ) )
% 16.19/8.93 & ( ( Z != zero_zero_real )
% 16.19/8.93 => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 16.19/8.93 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2984_add__divide__eq__if__simps_I4_J,axiom,
% 16.19/8.93 ! [Z: rat,A: rat,B: rat] :
% 16.19/8.93 ( ( ( Z = zero_zero_rat )
% 16.19/8.93 => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 16.19/8.93 = A ) )
% 16.19/8.93 & ( ( Z != zero_zero_rat )
% 16.19/8.93 => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 16.19/8.93 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2985_numeral__1__eq__Suc__0,axiom,
% 16.19/8.93 ( ( numeral_numeral_nat @ one )
% 16.19/8.93 = ( suc @ zero_zero_nat ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2986_num_Osize_I5_J,axiom,
% 16.19/8.93 ! [X22: num] :
% 16.19/8.93 ( ( size_size_num @ ( bit0 @ X22 ) )
% 16.19/8.93 = ( plus_plus_nat @ ( size_size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2987_ex__least__nat__less,axiom,
% 16.19/8.93 ! [P: nat > $o,N2: nat] :
% 16.19/8.93 ( ( P @ N2 )
% 16.19/8.93 => ( ~ ( P @ zero_zero_nat )
% 16.19/8.93 => ? [K2: nat] :
% 16.19/8.93 ( ( ord_less_nat @ K2 @ N2 )
% 16.19/8.93 & ! [I: nat] :
% 16.19/8.93 ( ( ord_less_eq_nat @ I @ K2 )
% 16.19/8.93 => ~ ( P @ I ) )
% 16.19/8.93 & ( P @ ( suc @ K2 ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2988_one__less__mult,axiom,
% 16.19/8.93 ! [N2: nat,M: nat] :
% 16.19/8.93 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 16.19/8.93 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 16.19/8.93 => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2989_n__less__m__mult__n,axiom,
% 16.19/8.93 ! [N2: nat,M: nat] :
% 16.19/8.93 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.93 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 16.19/8.93 => ( ord_less_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2990_n__less__n__mult__m,axiom,
% 16.19/8.93 ! [N2: nat,M: nat] :
% 16.19/8.93 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.93 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 16.19/8.93 => ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2991_diff__Suc__less,axiom,
% 16.19/8.93 ! [N2: nat,I2: nat] :
% 16.19/8.93 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.93 => ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) @ N2 ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2992_power__gt__expt,axiom,
% 16.19/8.93 ! [N2: nat,K: nat] :
% 16.19/8.93 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 16.19/8.93 => ( ord_less_nat @ K @ ( power_power_nat @ N2 @ K ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2993_nat__mult__le__cancel1,axiom,
% 16.19/8.93 ! [K: nat,M: nat,N2: nat] :
% 16.19/8.93 ( ( ord_less_nat @ zero_zero_nat @ K )
% 16.19/8.93 => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 16.19/8.93 = ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2994_nat__induct__non__zero,axiom,
% 16.19/8.93 ! [N2: nat,P: nat > $o] :
% 16.19/8.93 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.93 => ( ( P @ one_one_nat )
% 16.19/8.93 => ( ! [N4: nat] :
% 16.19/8.93 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 16.19/8.93 => ( ( P @ N4 )
% 16.19/8.93 => ( P @ ( suc @ N4 ) ) ) )
% 16.19/8.93 => ( P @ N2 ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2995_length__pos__if__in__set,axiom,
% 16.19/8.93 ! [X: nat,Xs: list_nat] :
% 16.19/8.93 ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 16.19/8.93 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2996_length__pos__if__in__set,axiom,
% 16.19/8.93 ! [X: real,Xs: list_real] :
% 16.19/8.93 ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 16.19/8.93 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2997_length__pos__if__in__set,axiom,
% 16.19/8.93 ! [X: complex,Xs: list_complex] :
% 16.19/8.93 ( ( member_complex @ X @ ( set_complex2 @ Xs ) )
% 16.19/8.93 => ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2998_length__pos__if__in__set,axiom,
% 16.19/8.93 ! [X: set_nat,Xs: list_set_nat] :
% 16.19/8.93 ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 16.19/8.93 => ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_2999_length__pos__if__in__set,axiom,
% 16.19/8.93 ! [X: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 16.19/8.93 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 16.19/8.93 => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3000_length__pos__if__in__set,axiom,
% 16.19/8.93 ! [X: $o,Xs: list_o] :
% 16.19/8.93 ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 16.19/8.93 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3001_length__pos__if__in__set,axiom,
% 16.19/8.93 ! [X: int,Xs: list_int] :
% 16.19/8.93 ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 16.19/8.93 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3002_nat__one__le__power,axiom,
% 16.19/8.93 ! [I2: nat,N2: nat] :
% 16.19/8.93 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I2 )
% 16.19/8.93 => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I2 @ N2 ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3003_nat__diff__split,axiom,
% 16.19/8.93 ! [P: nat > $o,A: nat,B: nat] :
% 16.19/8.93 ( ( P @ ( minus_minus_nat @ A @ B ) )
% 16.19/8.93 = ( ( ( ord_less_nat @ A @ B )
% 16.19/8.93 => ( P @ zero_zero_nat ) )
% 16.19/8.93 & ! [D2: nat] :
% 16.19/8.93 ( ( A
% 16.19/8.93 = ( plus_plus_nat @ B @ D2 ) )
% 16.19/8.93 => ( P @ D2 ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3004_nat__diff__split__asm,axiom,
% 16.19/8.93 ! [P: nat > $o,A: nat,B: nat] :
% 16.19/8.93 ( ( P @ ( minus_minus_nat @ A @ B ) )
% 16.19/8.93 = ( ~ ( ( ( ord_less_nat @ A @ B )
% 16.19/8.93 & ~ ( P @ zero_zero_nat ) )
% 16.19/8.93 | ? [D2: nat] :
% 16.19/8.93 ( ( A
% 16.19/8.93 = ( plus_plus_nat @ B @ D2 ) )
% 16.19/8.93 & ~ ( P @ D2 ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3005_div__le__mono2,axiom,
% 16.19/8.93 ! [M: nat,N2: nat,K: nat] :
% 16.19/8.93 ( ( ord_less_nat @ zero_zero_nat @ M )
% 16.19/8.93 => ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/8.93 => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N2 ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3006_div__greater__zero__iff,axiom,
% 16.19/8.93 ! [M: nat,N2: nat] :
% 16.19/8.93 ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N2 ) )
% 16.19/8.93 = ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/8.93 & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3007_nat__mult__div__cancel1,axiom,
% 16.19/8.93 ! [K: nat,M: nat,N2: nat] :
% 16.19/8.93 ( ( ord_less_nat @ zero_zero_nat @ K )
% 16.19/8.93 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 16.19/8.93 = ( divide_divide_nat @ M @ N2 ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3008_div__less__iff__less__mult,axiom,
% 16.19/8.93 ! [Q2: nat,M: nat,N2: nat] :
% 16.19/8.93 ( ( ord_less_nat @ zero_zero_nat @ Q2 )
% 16.19/8.93 => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N2 )
% 16.19/8.93 = ( ord_less_nat @ M @ ( times_times_nat @ N2 @ Q2 ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3009_div__less__dividend,axiom,
% 16.19/8.93 ! [N2: nat,M: nat] :
% 16.19/8.93 ( ( ord_less_nat @ one_one_nat @ N2 )
% 16.19/8.93 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 16.19/8.93 => ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3010_div__eq__dividend__iff,axiom,
% 16.19/8.93 ! [M: nat,N2: nat] :
% 16.19/8.93 ( ( ord_less_nat @ zero_zero_nat @ M )
% 16.19/8.93 => ( ( ( divide_divide_nat @ M @ N2 )
% 16.19/8.93 = M )
% 16.19/8.93 = ( N2 = one_one_nat ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3011_divmod__digit__1_I1_J,axiom,
% 16.19/8.93 ! [A: code_integer,B: code_integer] :
% 16.19/8.93 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 16.19/8.93 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 16.19/8.93 => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 16.19/8.93 => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
% 16.19/8.93 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3012_divmod__digit__1_I1_J,axiom,
% 16.19/8.93 ! [A: nat,B: nat] :
% 16.19/8.93 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 16.19/8.93 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 16.19/8.93 => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 16.19/8.93 => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_nat )
% 16.19/8.93 = ( divide_divide_nat @ A @ B ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3013_divmod__digit__1_I1_J,axiom,
% 16.19/8.93 ! [A: int,B: int] :
% 16.19/8.93 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.93 => ( ( ord_less_int @ zero_zero_int @ B )
% 16.19/8.93 => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 16.19/8.93 => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_int )
% 16.19/8.93 = ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3014_vebt__insert_Osimps_I3_J,axiom,
% 16.19/8.93 ! [Info: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X: nat] :
% 16.19/8.93 ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) @ X )
% 16.19/8.93 = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3015_vebt__member_Osimps_I3_J,axiom,
% 16.19/8.93 ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
% 16.19/8.93 ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3016_vebt__mint_Ocases,axiom,
% 16.19/8.93 ! [X: vEBT_VEBT] :
% 16.19/8.93 ( ! [A3: $o,B2: $o] :
% 16.19/8.93 ( X
% 16.19/8.93 != ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/8.93 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 16.19/8.93 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3017_VEBT__internal_OminNull_Oelims_I1_J,axiom,
% 16.19/8.93 ! [X: vEBT_VEBT,Y: $o] :
% 16.19/8.93 ( ( ( vEBT_VEBT_minNull @ X )
% 16.19/8.93 = Y )
% 16.19/8.93 => ( ( ( X
% 16.19/8.93 = ( vEBT_Leaf @ $false @ $false ) )
% 16.19/8.93 => ~ Y )
% 16.19/8.93 => ( ( ? [Uv2: $o] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Leaf @ $true @ Uv2 ) )
% 16.19/8.93 => Y )
% 16.19/8.93 => ( ( ? [Uu2: $o] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Leaf @ Uu2 @ $true ) )
% 16.19/8.93 => Y )
% 16.19/8.93 => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 16.19/8.93 => ~ Y )
% 16.19/8.93 => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 16.19/8.93 => Y ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3018_vebt__maxt_Oelims,axiom,
% 16.19/8.93 ! [X: vEBT_VEBT,Y: option_nat] :
% 16.19/8.93 ( ( ( vEBT_vebt_maxt @ X )
% 16.19/8.93 = Y )
% 16.19/8.93 => ( ! [A3: $o,B2: $o] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/8.93 => ~ ( ( B2
% 16.19/8.93 => ( Y
% 16.19/8.93 = ( some_nat @ one_one_nat ) ) )
% 16.19/8.93 & ( ~ B2
% 16.19/8.93 => ( ( A3
% 16.19/8.93 => ( Y
% 16.19/8.93 = ( some_nat @ zero_zero_nat ) ) )
% 16.19/8.93 & ( ~ A3
% 16.19/8.93 => ( Y = none_nat ) ) ) ) ) )
% 16.19/8.93 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 16.19/8.93 => ( Y != none_nat ) )
% 16.19/8.93 => ~ ! [Mi2: nat,Ma2: nat] :
% 16.19/8.93 ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 16.19/8.93 => ( Y
% 16.19/8.93 != ( some_nat @ Ma2 ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3019_vebt__mint_Oelims,axiom,
% 16.19/8.93 ! [X: vEBT_VEBT,Y: option_nat] :
% 16.19/8.93 ( ( ( vEBT_vebt_mint @ X )
% 16.19/8.93 = Y )
% 16.19/8.93 => ( ! [A3: $o,B2: $o] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/8.93 => ~ ( ( A3
% 16.19/8.93 => ( Y
% 16.19/8.93 = ( some_nat @ zero_zero_nat ) ) )
% 16.19/8.93 & ( ~ A3
% 16.19/8.93 => ( ( B2
% 16.19/8.93 => ( Y
% 16.19/8.93 = ( some_nat @ one_one_nat ) ) )
% 16.19/8.93 & ( ~ B2
% 16.19/8.93 => ( Y = none_nat ) ) ) ) ) )
% 16.19/8.93 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 16.19/8.93 => ( Y != none_nat ) )
% 16.19/8.93 => ~ ! [Mi2: nat] :
% 16.19/8.93 ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 16.19/8.93 => ( Y
% 16.19/8.93 != ( some_nat @ Mi2 ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3020_mult__le__cancel__left1,axiom,
% 16.19/8.93 ! [C: real,B: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
% 16.19/8.93 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ord_less_eq_real @ one_one_real @ B ) )
% 16.19/8.93 & ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3021_mult__le__cancel__left1,axiom,
% 16.19/8.93 ! [C: rat,B: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
% 16.19/8.93 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 16.19/8.93 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3022_mult__le__cancel__left1,axiom,
% 16.19/8.93 ! [C: int,B: int] :
% 16.19/8.93 ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
% 16.19/8.93 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 16.19/8.93 => ( ord_less_eq_int @ one_one_int @ B ) )
% 16.19/8.93 & ( ( ord_less_int @ C @ zero_zero_int )
% 16.19/8.93 => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3023_mult__le__cancel__left2,axiom,
% 16.19/8.93 ! [C: real,A: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
% 16.19/8.93 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ord_less_eq_real @ A @ one_one_real ) )
% 16.19/8.93 & ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3024_mult__le__cancel__left2,axiom,
% 16.19/8.93 ! [C: rat,A: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
% 16.19/8.93 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 16.19/8.93 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3025_mult__le__cancel__left2,axiom,
% 16.19/8.93 ! [C: int,A: int] :
% 16.19/8.93 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
% 16.19/8.93 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 16.19/8.93 => ( ord_less_eq_int @ A @ one_one_int ) )
% 16.19/8.93 & ( ( ord_less_int @ C @ zero_zero_int )
% 16.19/8.93 => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3026_mult__le__cancel__right1,axiom,
% 16.19/8.93 ! [C: real,B: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
% 16.19/8.93 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ord_less_eq_real @ one_one_real @ B ) )
% 16.19/8.93 & ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3027_mult__le__cancel__right1,axiom,
% 16.19/8.93 ! [C: rat,B: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
% 16.19/8.93 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 16.19/8.93 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3028_mult__le__cancel__right1,axiom,
% 16.19/8.93 ! [C: int,B: int] :
% 16.19/8.93 ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
% 16.19/8.93 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 16.19/8.93 => ( ord_less_eq_int @ one_one_int @ B ) )
% 16.19/8.93 & ( ( ord_less_int @ C @ zero_zero_int )
% 16.19/8.93 => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3029_mult__le__cancel__right2,axiom,
% 16.19/8.93 ! [A: real,C: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
% 16.19/8.93 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ord_less_eq_real @ A @ one_one_real ) )
% 16.19/8.93 & ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3030_mult__le__cancel__right2,axiom,
% 16.19/8.93 ! [A: rat,C: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
% 16.19/8.93 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 16.19/8.93 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3031_mult__le__cancel__right2,axiom,
% 16.19/8.93 ! [A: int,C: int] :
% 16.19/8.93 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
% 16.19/8.93 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 16.19/8.93 => ( ord_less_eq_int @ A @ one_one_int ) )
% 16.19/8.93 & ( ( ord_less_int @ C @ zero_zero_int )
% 16.19/8.93 => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3032_mult__less__cancel__left1,axiom,
% 16.19/8.93 ! [C: real,B: real] :
% 16.19/8.93 ( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
% 16.19/8.93 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ord_less_real @ one_one_real @ B ) )
% 16.19/8.93 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3033_mult__less__cancel__left1,axiom,
% 16.19/8.93 ! [C: rat,B: rat] :
% 16.19/8.93 ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
% 16.19/8.93 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ord_less_rat @ one_one_rat @ B ) )
% 16.19/8.93 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3034_mult__less__cancel__left1,axiom,
% 16.19/8.93 ! [C: int,B: int] :
% 16.19/8.93 ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
% 16.19/8.93 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 16.19/8.93 => ( ord_less_int @ one_one_int @ B ) )
% 16.19/8.93 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 16.19/8.93 => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3035_mult__less__cancel__left2,axiom,
% 16.19/8.93 ! [C: real,A: real] :
% 16.19/8.93 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
% 16.19/8.93 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ord_less_real @ A @ one_one_real ) )
% 16.19/8.93 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3036_mult__less__cancel__left2,axiom,
% 16.19/8.93 ! [C: rat,A: rat] :
% 16.19/8.93 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
% 16.19/8.93 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ord_less_rat @ A @ one_one_rat ) )
% 16.19/8.93 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3037_mult__less__cancel__left2,axiom,
% 16.19/8.93 ! [C: int,A: int] :
% 16.19/8.93 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
% 16.19/8.93 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 16.19/8.93 => ( ord_less_int @ A @ one_one_int ) )
% 16.19/8.93 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 16.19/8.93 => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3038_mult__less__cancel__right1,axiom,
% 16.19/8.93 ! [C: real,B: real] :
% 16.19/8.93 ( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
% 16.19/8.93 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ord_less_real @ one_one_real @ B ) )
% 16.19/8.93 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3039_mult__less__cancel__right1,axiom,
% 16.19/8.93 ! [C: rat,B: rat] :
% 16.19/8.93 ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
% 16.19/8.93 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ord_less_rat @ one_one_rat @ B ) )
% 16.19/8.93 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3040_mult__less__cancel__right1,axiom,
% 16.19/8.93 ! [C: int,B: int] :
% 16.19/8.93 ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
% 16.19/8.93 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 16.19/8.93 => ( ord_less_int @ one_one_int @ B ) )
% 16.19/8.93 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 16.19/8.93 => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3041_mult__less__cancel__right2,axiom,
% 16.19/8.93 ! [A: real,C: real] :
% 16.19/8.93 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
% 16.19/8.93 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ord_less_real @ A @ one_one_real ) )
% 16.19/8.93 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3042_mult__less__cancel__right2,axiom,
% 16.19/8.93 ! [A: rat,C: rat] :
% 16.19/8.93 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
% 16.19/8.93 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ord_less_rat @ A @ one_one_rat ) )
% 16.19/8.93 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3043_mult__less__cancel__right2,axiom,
% 16.19/8.93 ! [A: int,C: int] :
% 16.19/8.93 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
% 16.19/8.93 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 16.19/8.93 => ( ord_less_int @ A @ one_one_int ) )
% 16.19/8.93 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 16.19/8.93 => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3044_field__le__mult__one__interval,axiom,
% 16.19/8.93 ! [X: real,Y: real] :
% 16.19/8.93 ( ! [Z5: real] :
% 16.19/8.93 ( ( ord_less_real @ zero_zero_real @ Z5 )
% 16.19/8.93 => ( ( ord_less_real @ Z5 @ one_one_real )
% 16.19/8.93 => ( ord_less_eq_real @ ( times_times_real @ Z5 @ X ) @ Y ) ) )
% 16.19/8.93 => ( ord_less_eq_real @ X @ Y ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3045_field__le__mult__one__interval,axiom,
% 16.19/8.93 ! [X: rat,Y: rat] :
% 16.19/8.93 ( ! [Z5: rat] :
% 16.19/8.93 ( ( ord_less_rat @ zero_zero_rat @ Z5 )
% 16.19/8.93 => ( ( ord_less_rat @ Z5 @ one_one_rat )
% 16.19/8.93 => ( ord_less_eq_rat @ ( times_times_rat @ Z5 @ X ) @ Y ) ) )
% 16.19/8.93 => ( ord_less_eq_rat @ X @ Y ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3046_divide__left__mono__neg,axiom,
% 16.19/8.93 ! [A: real,B: real,C: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ A @ B )
% 16.19/8.93 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 16.19/8.93 => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3047_divide__left__mono__neg,axiom,
% 16.19/8.93 ! [A: rat,B: rat,C: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.93 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 16.19/8.93 => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3048_mult__imp__le__div__pos,axiom,
% 16.19/8.93 ! [Y: real,Z: real,X: real] :
% 16.19/8.93 ( ( ord_less_real @ zero_zero_real @ Y )
% 16.19/8.93 => ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y ) @ X )
% 16.19/8.93 => ( ord_less_eq_real @ Z @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3049_mult__imp__le__div__pos,axiom,
% 16.19/8.93 ! [Y: rat,Z: rat,X: rat] :
% 16.19/8.93 ( ( ord_less_rat @ zero_zero_rat @ Y )
% 16.19/8.93 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y ) @ X )
% 16.19/8.93 => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3050_mult__imp__div__pos__le,axiom,
% 16.19/8.93 ! [Y: real,X: real,Z: real] :
% 16.19/8.93 ( ( ord_less_real @ zero_zero_real @ Y )
% 16.19/8.93 => ( ( ord_less_eq_real @ X @ ( times_times_real @ Z @ Y ) )
% 16.19/8.93 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ Z ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3051_mult__imp__div__pos__le,axiom,
% 16.19/8.93 ! [Y: rat,X: rat,Z: rat] :
% 16.19/8.93 ( ( ord_less_rat @ zero_zero_rat @ Y )
% 16.19/8.93 => ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z @ Y ) )
% 16.19/8.93 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ Z ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3052_pos__le__divide__eq,axiom,
% 16.19/8.93 ! [C: real,A: real,B: real] :
% 16.19/8.93 ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 16.19/8.93 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3053_pos__le__divide__eq,axiom,
% 16.19/8.93 ! [C: rat,A: rat,B: rat] :
% 16.19/8.93 ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 16.19/8.93 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3054_pos__divide__le__eq,axiom,
% 16.19/8.93 ! [C: real,B: real,A: real] :
% 16.19/8.93 ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 16.19/8.93 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3055_pos__divide__le__eq,axiom,
% 16.19/8.93 ! [C: rat,B: rat,A: rat] :
% 16.19/8.93 ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 16.19/8.93 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3056_neg__le__divide__eq,axiom,
% 16.19/8.93 ! [C: real,A: real,B: real] :
% 16.19/8.93 ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 16.19/8.93 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3057_neg__le__divide__eq,axiom,
% 16.19/8.93 ! [C: rat,A: rat,B: rat] :
% 16.19/8.93 ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 16.19/8.93 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3058_neg__divide__le__eq,axiom,
% 16.19/8.93 ! [C: real,B: real,A: real] :
% 16.19/8.93 ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 16.19/8.93 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3059_neg__divide__le__eq,axiom,
% 16.19/8.93 ! [C: rat,B: rat,A: rat] :
% 16.19/8.93 ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 16.19/8.93 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3060_divide__left__mono,axiom,
% 16.19/8.93 ! [B: real,A: real,C: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ B @ A )
% 16.19/8.93 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 16.19/8.93 => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3061_divide__left__mono,axiom,
% 16.19/8.93 ! [B: rat,A: rat,C: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ B @ A )
% 16.19/8.93 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 16.19/8.93 => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3062_le__divide__eq,axiom,
% 16.19/8.93 ! [A: real,B: real,C: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 16.19/8.93 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 16.19/8.93 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 16.19/8.93 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3063_le__divide__eq,axiom,
% 16.19/8.93 ! [A: rat,B: rat,C: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 16.19/8.93 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 16.19/8.93 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 16.19/8.93 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3064_divide__le__eq,axiom,
% 16.19/8.93 ! [B: real,C: real,A: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 16.19/8.93 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 16.19/8.93 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 16.19/8.93 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3065_divide__le__eq,axiom,
% 16.19/8.93 ! [B: rat,C: rat,A: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 16.19/8.93 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 16.19/8.93 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 16.19/8.93 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3066_divide__le__eq__1,axiom,
% 16.19/8.93 ! [B: real,A: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 16.19/8.93 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.93 & ( ord_less_eq_real @ B @ A ) )
% 16.19/8.93 | ( ( ord_less_real @ A @ zero_zero_real )
% 16.19/8.93 & ( ord_less_eq_real @ A @ B ) )
% 16.19/8.93 | ( A = zero_zero_real ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3067_divide__le__eq__1,axiom,
% 16.19/8.93 ! [B: rat,A: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 16.19/8.93 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 16.19/8.93 & ( ord_less_eq_rat @ B @ A ) )
% 16.19/8.93 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 16.19/8.93 & ( ord_less_eq_rat @ A @ B ) )
% 16.19/8.93 | ( A = zero_zero_rat ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3068_le__divide__eq__1,axiom,
% 16.19/8.93 ! [B: real,A: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 16.19/8.93 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.93 & ( ord_less_eq_real @ A @ B ) )
% 16.19/8.93 | ( ( ord_less_real @ A @ zero_zero_real )
% 16.19/8.93 & ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3069_le__divide__eq__1,axiom,
% 16.19/8.93 ! [B: rat,A: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 16.19/8.93 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 16.19/8.93 & ( ord_less_eq_rat @ A @ B ) )
% 16.19/8.93 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 16.19/8.93 & ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3070_convex__bound__le,axiom,
% 16.19/8.93 ! [X: real,A: real,Y: real,U: real,V: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ X @ A )
% 16.19/8.93 => ( ( ord_less_eq_real @ Y @ A )
% 16.19/8.93 => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 16.19/8.93 => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 16.19/8.93 => ( ( ( plus_plus_real @ U @ V )
% 16.19/8.93 = one_one_real )
% 16.19/8.93 => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3071_convex__bound__le,axiom,
% 16.19/8.93 ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ X @ A )
% 16.19/8.93 => ( ( ord_less_eq_rat @ Y @ A )
% 16.19/8.93 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 16.19/8.93 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 16.19/8.93 => ( ( ( plus_plus_rat @ U @ V )
% 16.19/8.93 = one_one_rat )
% 16.19/8.93 => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3072_convex__bound__le,axiom,
% 16.19/8.93 ! [X: int,A: int,Y: int,U: int,V: int] :
% 16.19/8.93 ( ( ord_less_eq_int @ X @ A )
% 16.19/8.93 => ( ( ord_less_eq_int @ Y @ A )
% 16.19/8.93 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 16.19/8.93 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 16.19/8.93 => ( ( ( plus_plus_int @ U @ V )
% 16.19/8.93 = one_one_int )
% 16.19/8.93 => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3073_less__divide__eq__numeral_I1_J,axiom,
% 16.19/8.93 ! [W: num,B: real,C: real] :
% 16.19/8.93 ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
% 16.19/8.93 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 16.19/8.93 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 16.19/8.93 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3074_less__divide__eq__numeral_I1_J,axiom,
% 16.19/8.93 ! [W: num,B: rat,C: rat] :
% 16.19/8.93 ( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
% 16.19/8.93 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 16.19/8.93 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 16.19/8.93 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3075_divide__less__eq__numeral_I1_J,axiom,
% 16.19/8.93 ! [B: real,C: real,W: num] :
% 16.19/8.93 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
% 16.19/8.93 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 16.19/8.93 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 16.19/8.93 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3076_divide__less__eq__numeral_I1_J,axiom,
% 16.19/8.93 ! [B: rat,C: rat,W: num] :
% 16.19/8.93 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
% 16.19/8.93 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 16.19/8.93 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 16.19/8.93 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3077_frac__le__eq,axiom,
% 16.19/8.93 ! [Y: real,Z: real,X: real,W: real] :
% 16.19/8.93 ( ( Y != zero_zero_real )
% 16.19/8.93 => ( ( Z != zero_zero_real )
% 16.19/8.93 => ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
% 16.19/8.93 = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3078_frac__le__eq,axiom,
% 16.19/8.93 ! [Y: rat,Z: rat,X: rat,W: rat] :
% 16.19/8.93 ( ( Y != zero_zero_rat )
% 16.19/8.93 => ( ( Z != zero_zero_rat )
% 16.19/8.93 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
% 16.19/8.93 = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3079_power__Suc__less,axiom,
% 16.19/8.93 ! [A: real,N2: nat] :
% 16.19/8.93 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.93 => ( ( ord_less_real @ A @ one_one_real )
% 16.19/8.93 => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) @ ( power_power_real @ A @ N2 ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3080_power__Suc__less,axiom,
% 16.19/8.93 ! [A: rat,N2: nat] :
% 16.19/8.93 ( ( ord_less_rat @ zero_zero_rat @ A )
% 16.19/8.93 => ( ( ord_less_rat @ A @ one_one_rat )
% 16.19/8.93 => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) @ ( power_power_rat @ A @ N2 ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3081_power__Suc__less,axiom,
% 16.19/8.93 ! [A: nat,N2: nat] :
% 16.19/8.93 ( ( ord_less_nat @ zero_zero_nat @ A )
% 16.19/8.93 => ( ( ord_less_nat @ A @ one_one_nat )
% 16.19/8.93 => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3082_power__Suc__less,axiom,
% 16.19/8.93 ! [A: int,N2: nat] :
% 16.19/8.93 ( ( ord_less_int @ zero_zero_int @ A )
% 16.19/8.93 => ( ( ord_less_int @ A @ one_one_int )
% 16.19/8.93 => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) @ ( power_power_int @ A @ N2 ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3083_frac__less__eq,axiom,
% 16.19/8.93 ! [Y: real,Z: real,X: real,W: real] :
% 16.19/8.93 ( ( Y != zero_zero_real )
% 16.19/8.93 => ( ( Z != zero_zero_real )
% 16.19/8.93 => ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W @ Z ) )
% 16.19/8.93 = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3084_frac__less__eq,axiom,
% 16.19/8.93 ! [Y: rat,Z: rat,X: rat,W: rat] :
% 16.19/8.93 ( ( Y != zero_zero_rat )
% 16.19/8.93 => ( ( Z != zero_zero_rat )
% 16.19/8.93 => ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W @ Z ) )
% 16.19/8.93 = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3085_power__Suc__le__self,axiom,
% 16.19/8.93 ! [A: real,N2: nat] :
% 16.19/8.93 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/8.93 => ( ( ord_less_eq_real @ A @ one_one_real )
% 16.19/8.93 => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ A ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3086_power__Suc__le__self,axiom,
% 16.19/8.93 ! [A: rat,N2: nat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/8.93 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 16.19/8.93 => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ A ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3087_power__Suc__le__self,axiom,
% 16.19/8.93 ! [A: nat,N2: nat] :
% 16.19/8.93 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 16.19/8.93 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 16.19/8.93 => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ A ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3088_power__Suc__le__self,axiom,
% 16.19/8.93 ! [A: int,N2: nat] :
% 16.19/8.93 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.93 => ( ( ord_less_eq_int @ A @ one_one_int )
% 16.19/8.93 => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ A ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3089_power__Suc__less__one,axiom,
% 16.19/8.93 ! [A: real,N2: nat] :
% 16.19/8.93 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.93 => ( ( ord_less_real @ A @ one_one_real )
% 16.19/8.93 => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ one_one_real ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3090_power__Suc__less__one,axiom,
% 16.19/8.93 ! [A: rat,N2: nat] :
% 16.19/8.93 ( ( ord_less_rat @ zero_zero_rat @ A )
% 16.19/8.93 => ( ( ord_less_rat @ A @ one_one_rat )
% 16.19/8.93 => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ one_one_rat ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3091_power__Suc__less__one,axiom,
% 16.19/8.93 ! [A: nat,N2: nat] :
% 16.19/8.93 ( ( ord_less_nat @ zero_zero_nat @ A )
% 16.19/8.93 => ( ( ord_less_nat @ A @ one_one_nat )
% 16.19/8.93 => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ one_one_nat ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3092_power__Suc__less__one,axiom,
% 16.19/8.93 ! [A: int,N2: nat] :
% 16.19/8.93 ( ( ord_less_int @ zero_zero_int @ A )
% 16.19/8.93 => ( ( ord_less_int @ A @ one_one_int )
% 16.19/8.93 => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ one_one_int ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3093_power__strict__decreasing,axiom,
% 16.19/8.93 ! [N2: nat,N3: nat,A: real] :
% 16.19/8.93 ( ( ord_less_nat @ N2 @ N3 )
% 16.19/8.93 => ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.93 => ( ( ord_less_real @ A @ one_one_real )
% 16.19/8.93 => ( ord_less_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3094_power__strict__decreasing,axiom,
% 16.19/8.93 ! [N2: nat,N3: nat,A: rat] :
% 16.19/8.93 ( ( ord_less_nat @ N2 @ N3 )
% 16.19/8.93 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 16.19/8.93 => ( ( ord_less_rat @ A @ one_one_rat )
% 16.19/8.93 => ( ord_less_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3095_power__strict__decreasing,axiom,
% 16.19/8.93 ! [N2: nat,N3: nat,A: nat] :
% 16.19/8.93 ( ( ord_less_nat @ N2 @ N3 )
% 16.19/8.93 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 16.19/8.93 => ( ( ord_less_nat @ A @ one_one_nat )
% 16.19/8.93 => ( ord_less_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3096_power__strict__decreasing,axiom,
% 16.19/8.93 ! [N2: nat,N3: nat,A: int] :
% 16.19/8.93 ( ( ord_less_nat @ N2 @ N3 )
% 16.19/8.93 => ( ( ord_less_int @ zero_zero_int @ A )
% 16.19/8.93 => ( ( ord_less_int @ A @ one_one_int )
% 16.19/8.93 => ( ord_less_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3097_zero__power2,axiom,
% 16.19/8.93 ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.93 = zero_zero_rat ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3098_zero__power2,axiom,
% 16.19/8.93 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.93 = zero_zero_nat ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3099_zero__power2,axiom,
% 16.19/8.93 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.93 = zero_zero_real ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3100_zero__power2,axiom,
% 16.19/8.93 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.93 = zero_zero_int ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3101_zero__power2,axiom,
% 16.19/8.93 ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.93 = zero_zero_complex ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3102_power__decreasing,axiom,
% 16.19/8.93 ! [N2: nat,N3: nat,A: real] :
% 16.19/8.93 ( ( ord_less_eq_nat @ N2 @ N3 )
% 16.19/8.93 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/8.93 => ( ( ord_less_eq_real @ A @ one_one_real )
% 16.19/8.93 => ( ord_less_eq_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3103_power__decreasing,axiom,
% 16.19/8.93 ! [N2: nat,N3: nat,A: rat] :
% 16.19/8.93 ( ( ord_less_eq_nat @ N2 @ N3 )
% 16.19/8.93 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/8.93 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 16.19/8.93 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3104_power__decreasing,axiom,
% 16.19/8.93 ! [N2: nat,N3: nat,A: nat] :
% 16.19/8.93 ( ( ord_less_eq_nat @ N2 @ N3 )
% 16.19/8.93 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 16.19/8.93 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 16.19/8.93 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3105_power__decreasing,axiom,
% 16.19/8.93 ! [N2: nat,N3: nat,A: int] :
% 16.19/8.93 ( ( ord_less_eq_nat @ N2 @ N3 )
% 16.19/8.93 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.93 => ( ( ord_less_eq_int @ A @ one_one_int )
% 16.19/8.93 => ( ord_less_eq_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3106_self__le__power,axiom,
% 16.19/8.93 ! [A: real,N2: nat] :
% 16.19/8.93 ( ( ord_less_eq_real @ one_one_real @ A )
% 16.19/8.93 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.93 => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3107_self__le__power,axiom,
% 16.19/8.93 ! [A: rat,N2: nat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ one_one_rat @ A )
% 16.19/8.93 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.93 => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3108_self__le__power,axiom,
% 16.19/8.93 ! [A: nat,N2: nat] :
% 16.19/8.93 ( ( ord_less_eq_nat @ one_one_nat @ A )
% 16.19/8.93 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.93 => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3109_self__le__power,axiom,
% 16.19/8.93 ! [A: int,N2: nat] :
% 16.19/8.93 ( ( ord_less_eq_int @ one_one_int @ A )
% 16.19/8.93 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.93 => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3110_numeral__2__eq__2,axiom,
% 16.19/8.93 ( ( numeral_numeral_nat @ ( bit0 @ one ) )
% 16.19/8.93 = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3111_one__less__power,axiom,
% 16.19/8.93 ! [A: real,N2: nat] :
% 16.19/8.93 ( ( ord_less_real @ one_one_real @ A )
% 16.19/8.93 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.93 => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3112_one__less__power,axiom,
% 16.19/8.93 ! [A: rat,N2: nat] :
% 16.19/8.93 ( ( ord_less_rat @ one_one_rat @ A )
% 16.19/8.93 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.93 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3113_one__less__power,axiom,
% 16.19/8.93 ! [A: nat,N2: nat] :
% 16.19/8.93 ( ( ord_less_nat @ one_one_nat @ A )
% 16.19/8.93 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.93 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3114_one__less__power,axiom,
% 16.19/8.93 ! [A: int,N2: nat] :
% 16.19/8.93 ( ( ord_less_int @ one_one_int @ A )
% 16.19/8.93 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.93 => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3115_pos2,axiom,
% 16.19/8.93 ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3116_power__diff,axiom,
% 16.19/8.93 ! [A: complex,N2: nat,M: nat] :
% 16.19/8.93 ( ( A != zero_zero_complex )
% 16.19/8.93 => ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/8.93 => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N2 ) )
% 16.19/8.93 = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N2 ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3117_power__diff,axiom,
% 16.19/8.93 ! [A: real,N2: nat,M: nat] :
% 16.19/8.93 ( ( A != zero_zero_real )
% 16.19/8.93 => ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/8.93 => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N2 ) )
% 16.19/8.93 = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3118_power__diff,axiom,
% 16.19/8.93 ! [A: rat,N2: nat,M: nat] :
% 16.19/8.93 ( ( A != zero_zero_rat )
% 16.19/8.93 => ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/8.93 => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N2 ) )
% 16.19/8.93 = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3119_power__diff,axiom,
% 16.19/8.93 ! [A: nat,N2: nat,M: nat] :
% 16.19/8.93 ( ( A != zero_zero_nat )
% 16.19/8.93 => ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/8.93 => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N2 ) )
% 16.19/8.93 = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3120_power__diff,axiom,
% 16.19/8.93 ! [A: int,N2: nat,M: nat] :
% 16.19/8.93 ( ( A != zero_zero_int )
% 16.19/8.93 => ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/8.93 => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N2 ) )
% 16.19/8.93 = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3121_div__if,axiom,
% 16.19/8.93 ( divide_divide_nat
% 16.19/8.93 = ( ^ [M6: nat,N: nat] :
% 16.19/8.93 ( if_nat
% 16.19/8.93 @ ( ( ord_less_nat @ M6 @ N )
% 16.19/8.93 | ( N = zero_zero_nat ) )
% 16.19/8.93 @ zero_zero_nat
% 16.19/8.93 @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M6 @ N ) @ N ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3122_Suc__pred_H,axiom,
% 16.19/8.93 ! [N2: nat] :
% 16.19/8.93 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.93 => ( N2
% 16.19/8.93 = ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3123_Suc__diff__eq__diff__pred,axiom,
% 16.19/8.93 ! [N2: nat,M: nat] :
% 16.19/8.93 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.93 => ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
% 16.19/8.93 = ( minus_minus_nat @ M @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3124_less__eq__div__iff__mult__less__eq,axiom,
% 16.19/8.93 ! [Q2: nat,M: nat,N2: nat] :
% 16.19/8.93 ( ( ord_less_nat @ zero_zero_nat @ Q2 )
% 16.19/8.93 => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N2 @ Q2 ) )
% 16.19/8.93 = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q2 ) @ N2 ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3125_dividend__less__times__div,axiom,
% 16.19/8.93 ! [N2: nat,M: nat] :
% 16.19/8.93 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.93 => ( ord_less_nat @ M @ ( plus_plus_nat @ N2 @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3126_dividend__less__div__times,axiom,
% 16.19/8.93 ! [N2: nat,M: nat] :
% 16.19/8.93 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.93 => ( ord_less_nat @ M @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3127_split__div,axiom,
% 16.19/8.93 ! [P: nat > $o,M: nat,N2: nat] :
% 16.19/8.93 ( ( P @ ( divide_divide_nat @ M @ N2 ) )
% 16.19/8.93 = ( ( ( N2 = zero_zero_nat )
% 16.19/8.93 => ( P @ zero_zero_nat ) )
% 16.19/8.93 & ( ( N2 != zero_zero_nat )
% 16.19/8.93 => ! [I3: nat,J3: nat] :
% 16.19/8.93 ( ( ord_less_nat @ J3 @ N2 )
% 16.19/8.93 => ( ( M
% 16.19/8.93 = ( plus_plus_nat @ ( times_times_nat @ N2 @ I3 ) @ J3 ) )
% 16.19/8.93 => ( P @ I3 ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3128_add__eq__if,axiom,
% 16.19/8.93 ( plus_plus_nat
% 16.19/8.93 = ( ^ [M6: nat,N: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3129_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
% 16.19/8.93 ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 16.19/8.93 ( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
% 16.19/8.93 = Y )
% 16.19/8.93 => ( ! [A3: $o,B2: $o] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/8.93 => ( Y
% 16.19/8.93 = ( ~ ( ( ( Xa2 = zero_zero_nat )
% 16.19/8.93 => A3 )
% 16.19/8.93 & ( ( Xa2 != zero_zero_nat )
% 16.19/8.93 => ( ( ( Xa2 = one_one_nat )
% 16.19/8.93 => B2 )
% 16.19/8.93 & ( Xa2 = one_one_nat ) ) ) ) ) ) )
% 16.19/8.93 => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 16.19/8.93 => Y )
% 16.19/8.93 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 16.19/8.93 ( ? [S2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
% 16.19/8.93 => ( Y
% 16.19/8.93 = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.93 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 16.19/8.93 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3130_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
% 16.19/8.93 ! [X: vEBT_VEBT,Xa2: nat] :
% 16.19/8.93 ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
% 16.19/8.93 => ( ! [A3: $o,B2: $o] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/8.93 => ~ ( ( ( Xa2 = zero_zero_nat )
% 16.19/8.93 => A3 )
% 16.19/8.93 & ( ( Xa2 != zero_zero_nat )
% 16.19/8.93 => ( ( ( Xa2 = one_one_nat )
% 16.19/8.93 => B2 )
% 16.19/8.93 & ( Xa2 = one_one_nat ) ) ) ) )
% 16.19/8.93 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 16.19/8.93 ( ? [S2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
% 16.19/8.93 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.93 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 16.19/8.93 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3131_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
% 16.19/8.93 ! [X: vEBT_VEBT,Xa2: nat] :
% 16.19/8.93 ( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
% 16.19/8.93 => ( ! [A3: $o,B2: $o] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/8.93 => ( ( ( Xa2 = zero_zero_nat )
% 16.19/8.93 => A3 )
% 16.19/8.93 & ( ( Xa2 != zero_zero_nat )
% 16.19/8.93 => ( ( ( Xa2 = one_one_nat )
% 16.19/8.93 => B2 )
% 16.19/8.93 & ( Xa2 = one_one_nat ) ) ) ) )
% 16.19/8.93 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 16.19/8.93 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 16.19/8.93 ( ? [S2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
% 16.19/8.93 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.93 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 16.19/8.93 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3132_mult__eq__if,axiom,
% 16.19/8.93 ( times_times_nat
% 16.19/8.93 = ( ^ [M6: nat,N: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N @ ( times_times_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3133_vebt__member_Osimps_I4_J,axiom,
% 16.19/8.93 ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
% 16.19/8.93 ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3134_vebt__delete_Osimps_I5_J,axiom,
% 16.19/8.93 ! [Mi: nat,Ma: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT,X: nat] :
% 16.19/8.93 ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) @ X )
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3135_vebt__pred_Osimps_I5_J,axiom,
% 16.19/8.93 ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
% 16.19/8.93 ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve2 ) @ Vf2 )
% 16.19/8.93 = none_nat ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3136_vebt__succ_Osimps_I4_J,axiom,
% 16.19/8.93 ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve2: nat] :
% 16.19/8.93 ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve2 )
% 16.19/8.93 = none_nat ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3137_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
% 16.19/8.93 ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 16.19/8.93 ( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
% 16.19/8.93 = Y )
% 16.19/8.93 => ( ( ? [Uu2: $o,Uv2: $o] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 16.19/8.93 => Y )
% 16.19/8.93 => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 16.19/8.93 => Y )
% 16.19/8.93 => ( ! [Mi2: nat,Ma2: nat] :
% 16.19/8.93 ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 16.19/8.93 => ( Y
% 16.19/8.93 = ( ~ ( ( Xa2 = Mi2 )
% 16.19/8.93 | ( Xa2 = Ma2 ) ) ) ) )
% 16.19/8.93 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 16.19/8.93 ( ? [Vc2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 16.19/8.93 => ( Y
% 16.19/8.93 = ( ~ ( ( Xa2 = Mi2 )
% 16.19/8.93 | ( Xa2 = Ma2 )
% 16.19/8.93 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.93 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 16.19/8.93 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) )
% 16.19/8.93 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 16.19/8.93 ( ? [Vd2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 16.19/8.93 => ( Y
% 16.19/8.93 = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.93 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 16.19/8.93 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3138_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
% 16.19/8.93 ! [X: vEBT_VEBT,Xa2: nat] :
% 16.19/8.93 ( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
% 16.19/8.93 => ( ! [Uu2: $o,Uv2: $o] :
% 16.19/8.93 ( X
% 16.19/8.93 != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 16.19/8.93 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 16.19/8.93 => ( ! [Mi2: nat,Ma2: nat] :
% 16.19/8.93 ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 16.19/8.93 => ( ( Xa2 = Mi2 )
% 16.19/8.93 | ( Xa2 = Ma2 ) ) )
% 16.19/8.93 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 16.19/8.93 ( ? [Vc2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 16.19/8.93 => ( ( Xa2 = Mi2 )
% 16.19/8.93 | ( Xa2 = Ma2 )
% 16.19/8.93 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.93 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 16.19/8.93 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 16.19/8.93 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 16.19/8.93 ( ? [Vd2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 16.19/8.93 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.93 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 16.19/8.93 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3139_le__divide__eq__numeral_I1_J,axiom,
% 16.19/8.93 ! [W: num,B: real,C: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
% 16.19/8.93 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 16.19/8.93 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 16.19/8.93 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3140_le__divide__eq__numeral_I1_J,axiom,
% 16.19/8.93 ! [W: num,B: rat,C: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
% 16.19/8.93 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 16.19/8.93 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 16.19/8.93 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3141_divide__le__eq__numeral_I1_J,axiom,
% 16.19/8.93 ! [B: real,C: real,W: num] :
% 16.19/8.93 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
% 16.19/8.93 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 16.19/8.93 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.93 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 16.19/8.93 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.93 => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3142_divide__le__eq__numeral_I1_J,axiom,
% 16.19/8.93 ! [B: rat,C: rat,W: num] :
% 16.19/8.93 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
% 16.19/8.93 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 16.19/8.93 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.93 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 16.19/8.93 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.93 => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3143_convex__bound__lt,axiom,
% 16.19/8.93 ! [X: real,A: real,Y: real,U: real,V: real] :
% 16.19/8.93 ( ( ord_less_real @ X @ A )
% 16.19/8.93 => ( ( ord_less_real @ Y @ A )
% 16.19/8.93 => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 16.19/8.93 => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 16.19/8.93 => ( ( ( plus_plus_real @ U @ V )
% 16.19/8.93 = one_one_real )
% 16.19/8.93 => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3144_convex__bound__lt,axiom,
% 16.19/8.93 ! [X: rat,A: rat,Y: rat,U: rat,V: rat] :
% 16.19/8.93 ( ( ord_less_rat @ X @ A )
% 16.19/8.93 => ( ( ord_less_rat @ Y @ A )
% 16.19/8.93 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 16.19/8.93 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 16.19/8.93 => ( ( ( plus_plus_rat @ U @ V )
% 16.19/8.93 = one_one_rat )
% 16.19/8.93 => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3145_convex__bound__lt,axiom,
% 16.19/8.93 ! [X: int,A: int,Y: int,U: int,V: int] :
% 16.19/8.93 ( ( ord_less_int @ X @ A )
% 16.19/8.93 => ( ( ord_less_int @ Y @ A )
% 16.19/8.93 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 16.19/8.93 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 16.19/8.93 => ( ( ( plus_plus_int @ U @ V )
% 16.19/8.93 = one_one_int )
% 16.19/8.93 => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3146_half__gt__zero__iff,axiom,
% 16.19/8.93 ! [A: real] :
% 16.19/8.93 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/8.93 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3147_half__gt__zero__iff,axiom,
% 16.19/8.93 ! [A: rat] :
% 16.19/8.93 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 16.19/8.93 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3148_half__gt__zero,axiom,
% 16.19/8.93 ! [A: real] :
% 16.19/8.93 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.93 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3149_half__gt__zero,axiom,
% 16.19/8.93 ! [A: rat] :
% 16.19/8.93 ( ( ord_less_rat @ zero_zero_rat @ A )
% 16.19/8.93 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3150_scaling__mono,axiom,
% 16.19/8.93 ! [U: real,V: real,R: real,S3: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ U @ V )
% 16.19/8.93 => ( ( ord_less_eq_real @ zero_zero_real @ R )
% 16.19/8.93 => ( ( ord_less_eq_real @ R @ S3 )
% 16.19/8.93 => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R @ ( minus_minus_real @ V @ U ) ) @ S3 ) ) @ V ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3151_scaling__mono,axiom,
% 16.19/8.93 ! [U: rat,V: rat,R: rat,S3: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ U @ V )
% 16.19/8.93 => ( ( ord_less_eq_rat @ zero_zero_rat @ R )
% 16.19/8.93 => ( ( ord_less_eq_rat @ R @ S3 )
% 16.19/8.93 => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R @ ( minus_minus_rat @ V @ U ) ) @ S3 ) ) @ V ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3152_power2__le__imp__le,axiom,
% 16.19/8.93 ! [X: real,Y: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.93 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/8.93 => ( ord_less_eq_real @ X @ Y ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3153_power2__le__imp__le,axiom,
% 16.19/8.93 ! [X: rat,Y: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.93 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 16.19/8.93 => ( ord_less_eq_rat @ X @ Y ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3154_power2__le__imp__le,axiom,
% 16.19/8.93 ! [X: nat,Y: nat] :
% 16.19/8.93 ( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.93 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 16.19/8.93 => ( ord_less_eq_nat @ X @ Y ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3155_power2__le__imp__le,axiom,
% 16.19/8.93 ! [X: int,Y: int] :
% 16.19/8.93 ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.93 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 16.19/8.93 => ( ord_less_eq_int @ X @ Y ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3156_power2__eq__imp__eq,axiom,
% 16.19/8.93 ! [X: real,Y: real] :
% 16.19/8.93 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.93 = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.93 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/8.93 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/8.93 => ( X = Y ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3157_power2__eq__imp__eq,axiom,
% 16.19/8.93 ! [X: rat,Y: rat] :
% 16.19/8.93 ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.93 = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.93 => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 16.19/8.93 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 16.19/8.93 => ( X = Y ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3158_power2__eq__imp__eq,axiom,
% 16.19/8.93 ! [X: nat,Y: nat] :
% 16.19/8.93 ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.93 = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.93 => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 16.19/8.93 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 16.19/8.93 => ( X = Y ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3159_power2__eq__imp__eq,axiom,
% 16.19/8.93 ! [X: int,Y: int] :
% 16.19/8.93 ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.93 = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.93 => ( ( ord_less_eq_int @ zero_zero_int @ X )
% 16.19/8.93 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 16.19/8.93 => ( X = Y ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3160_zero__le__power2,axiom,
% 16.19/8.93 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3161_zero__le__power2,axiom,
% 16.19/8.93 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3162_zero__le__power2,axiom,
% 16.19/8.93 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3163_power2__less__0,axiom,
% 16.19/8.93 ! [A: real] :
% 16.19/8.93 ~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3164_power2__less__0,axiom,
% 16.19/8.93 ! [A: rat] :
% 16.19/8.93 ~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3165_power2__less__0,axiom,
% 16.19/8.93 ! [A: int] :
% 16.19/8.93 ~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3166_exp__add__not__zero__imp__right,axiom,
% 16.19/8.93 ! [M: nat,N2: nat] :
% 16.19/8.93 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
% 16.19/8.93 != zero_zero_nat )
% 16.19/8.93 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.93 != zero_zero_nat ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3167_exp__add__not__zero__imp__right,axiom,
% 16.19/8.93 ! [M: nat,N2: nat] :
% 16.19/8.93 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
% 16.19/8.93 != zero_zero_int )
% 16.19/8.93 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.93 != zero_zero_int ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3168_exp__add__not__zero__imp__left,axiom,
% 16.19/8.93 ! [M: nat,N2: nat] :
% 16.19/8.93 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
% 16.19/8.93 != zero_zero_nat )
% 16.19/8.93 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 16.19/8.93 != zero_zero_nat ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3169_exp__add__not__zero__imp__left,axiom,
% 16.19/8.93 ! [M: nat,N2: nat] :
% 16.19/8.93 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
% 16.19/8.93 != zero_zero_int )
% 16.19/8.93 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 16.19/8.93 != zero_zero_int ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3170_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 16.19/8.93 ! [N2: nat,M: nat] :
% 16.19/8.93 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.93 != zero_zero_nat )
% 16.19/8.93 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) )
% 16.19/8.93 != zero_zero_nat ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3171_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 16.19/8.93 ! [N2: nat,M: nat] :
% 16.19/8.93 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.93 != zero_zero_int )
% 16.19/8.93 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) )
% 16.19/8.93 != zero_zero_int ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3172_power__diff__power__eq,axiom,
% 16.19/8.93 ! [A: nat,N2: nat,M: nat] :
% 16.19/8.93 ( ( A != zero_zero_nat )
% 16.19/8.93 => ( ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/8.93 => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
% 16.19/8.93 = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N2 ) ) ) )
% 16.19/8.93 & ( ~ ( ord_less_eq_nat @ N2 @ M )
% 16.19/8.93 => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
% 16.19/8.93 = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3173_power__diff__power__eq,axiom,
% 16.19/8.93 ! [A: int,N2: nat,M: nat] :
% 16.19/8.93 ( ( A != zero_zero_int )
% 16.19/8.93 => ( ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/8.93 => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
% 16.19/8.93 = ( power_power_int @ A @ ( minus_minus_nat @ M @ N2 ) ) ) )
% 16.19/8.93 & ( ~ ( ord_less_eq_nat @ N2 @ M )
% 16.19/8.93 => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
% 16.19/8.93 = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3174_less__2__cases__iff,axiom,
% 16.19/8.93 ! [N2: nat] :
% 16.19/8.93 ( ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.93 = ( ( N2 = zero_zero_nat )
% 16.19/8.93 | ( N2
% 16.19/8.93 = ( suc @ zero_zero_nat ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3175_less__2__cases,axiom,
% 16.19/8.93 ! [N2: nat] :
% 16.19/8.93 ( ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.93 => ( ( N2 = zero_zero_nat )
% 16.19/8.93 | ( N2
% 16.19/8.93 = ( suc @ zero_zero_nat ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3176_nat__induct2,axiom,
% 16.19/8.93 ! [P: nat > $o,N2: nat] :
% 16.19/8.93 ( ( P @ zero_zero_nat )
% 16.19/8.93 => ( ( P @ one_one_nat )
% 16.19/8.93 => ( ! [N4: nat] :
% 16.19/8.93 ( ( P @ N4 )
% 16.19/8.93 => ( P @ ( plus_plus_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/8.93 => ( P @ N2 ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3177_power__eq__if,axiom,
% 16.19/8.93 ( power_power_complex
% 16.19/8.93 = ( ^ [P5: complex,M6: nat] : ( if_complex @ ( M6 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P5 @ ( power_power_complex @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3178_power__eq__if,axiom,
% 16.19/8.93 ( power_power_real
% 16.19/8.93 = ( ^ [P5: real,M6: nat] : ( if_real @ ( M6 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P5 @ ( power_power_real @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3179_power__eq__if,axiom,
% 16.19/8.93 ( power_power_rat
% 16.19/8.93 = ( ^ [P5: rat,M6: nat] : ( if_rat @ ( M6 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P5 @ ( power_power_rat @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3180_power__eq__if,axiom,
% 16.19/8.93 ( power_power_nat
% 16.19/8.93 = ( ^ [P5: nat,M6: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P5 @ ( power_power_nat @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3181_power__eq__if,axiom,
% 16.19/8.93 ( power_power_int
% 16.19/8.93 = ( ^ [P5: int,M6: nat] : ( if_int @ ( M6 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P5 @ ( power_power_int @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3182_power__minus__mult,axiom,
% 16.19/8.93 ! [N2: nat,A: complex] :
% 16.19/8.93 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.93 => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
% 16.19/8.93 = ( power_power_complex @ A @ N2 ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3183_power__minus__mult,axiom,
% 16.19/8.93 ! [N2: nat,A: real] :
% 16.19/8.93 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.93 => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
% 16.19/8.93 = ( power_power_real @ A @ N2 ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3184_power__minus__mult,axiom,
% 16.19/8.93 ! [N2: nat,A: rat] :
% 16.19/8.93 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.93 => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
% 16.19/8.93 = ( power_power_rat @ A @ N2 ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3185_power__minus__mult,axiom,
% 16.19/8.93 ! [N2: nat,A: nat] :
% 16.19/8.93 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.93 => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
% 16.19/8.93 = ( power_power_nat @ A @ N2 ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3186_power__minus__mult,axiom,
% 16.19/8.93 ! [N2: nat,A: int] :
% 16.19/8.93 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.93 => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
% 16.19/8.93 = ( power_power_int @ A @ N2 ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3187_split__div_H,axiom,
% 16.19/8.93 ! [P: nat > $o,M: nat,N2: nat] :
% 16.19/8.93 ( ( P @ ( divide_divide_nat @ M @ N2 ) )
% 16.19/8.93 = ( ( ( N2 = zero_zero_nat )
% 16.19/8.93 & ( P @ zero_zero_nat ) )
% 16.19/8.93 | ? [Q4: nat] :
% 16.19/8.93 ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q4 ) @ M )
% 16.19/8.93 & ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q4 ) ) )
% 16.19/8.93 & ( P @ Q4 ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3188_le__div__geq,axiom,
% 16.19/8.93 ! [N2: nat,M: nat] :
% 16.19/8.93 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.93 => ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/8.93 => ( ( divide_divide_nat @ M @ N2 )
% 16.19/8.93 = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3189_div__exp__mod__exp__eq,axiom,
% 16.19/8.93 ! [A: nat,N2: nat,M: nat] :
% 16.19/8.93 ( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 16.19/8.93 = ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3190_div__exp__mod__exp__eq,axiom,
% 16.19/8.93 ! [A: int,N2: nat,M: nat] :
% 16.19/8.93 ( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 16.19/8.93 = ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3191_div__exp__mod__exp__eq,axiom,
% 16.19/8.93 ! [A: code_integer,N2: nat,M: nat] :
% 16.19/8.93 ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 16.19/8.93 = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3192_vebt__delete_Osimps_I6_J,axiom,
% 16.19/8.93 ! [Mi: nat,Ma: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT,X: nat] :
% 16.19/8.93 ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) @ X )
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3193_vebt__pred_Osimps_I6_J,axiom,
% 16.19/8.93 ! [V: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
% 16.19/8.93 ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 )
% 16.19/8.93 = none_nat ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3194_vebt__succ_Osimps_I5_J,axiom,
% 16.19/8.93 ! [V: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
% 16.19/8.93 ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 )
% 16.19/8.93 = none_nat ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3195_power2__less__imp__less,axiom,
% 16.19/8.93 ! [X: real,Y: real] :
% 16.19/8.93 ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.93 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/8.93 => ( ord_less_real @ X @ Y ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3196_power2__less__imp__less,axiom,
% 16.19/8.93 ! [X: rat,Y: rat] :
% 16.19/8.93 ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.93 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 16.19/8.93 => ( ord_less_rat @ X @ Y ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3197_power2__less__imp__less,axiom,
% 16.19/8.93 ! [X: nat,Y: nat] :
% 16.19/8.93 ( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.93 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
% 16.19/8.93 => ( ord_less_nat @ X @ Y ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3198_power2__less__imp__less,axiom,
% 16.19/8.93 ! [X: int,Y: int] :
% 16.19/8.93 ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.93 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 16.19/8.93 => ( ord_less_int @ X @ Y ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3199_sum__power2__le__zero__iff,axiom,
% 16.19/8.93 ! [X: real,Y: real] :
% 16.19/8.93 ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
% 16.19/8.93 = ( ( X = zero_zero_real )
% 16.19/8.93 & ( Y = zero_zero_real ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3200_sum__power2__le__zero__iff,axiom,
% 16.19/8.93 ! [X: rat,Y: rat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
% 16.19/8.93 = ( ( X = zero_zero_rat )
% 16.19/8.93 & ( Y = zero_zero_rat ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3201_sum__power2__le__zero__iff,axiom,
% 16.19/8.93 ! [X: int,Y: int] :
% 16.19/8.93 ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
% 16.19/8.93 = ( ( X = zero_zero_int )
% 16.19/8.93 & ( Y = zero_zero_int ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3202_sum__power2__ge__zero,axiom,
% 16.19/8.93 ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3203_sum__power2__ge__zero,axiom,
% 16.19/8.93 ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3204_sum__power2__ge__zero,axiom,
% 16.19/8.93 ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3205_sum__power2__gt__zero__iff,axiom,
% 16.19/8.93 ! [X: real,Y: real] :
% 16.19/8.93 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/8.93 = ( ( X != zero_zero_real )
% 16.19/8.93 | ( Y != zero_zero_real ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3206_sum__power2__gt__zero__iff,axiom,
% 16.19/8.93 ! [X: rat,Y: rat] :
% 16.19/8.93 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/8.93 = ( ( X != zero_zero_rat )
% 16.19/8.93 | ( Y != zero_zero_rat ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3207_sum__power2__gt__zero__iff,axiom,
% 16.19/8.93 ! [X: int,Y: int] :
% 16.19/8.93 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/8.93 = ( ( X != zero_zero_int )
% 16.19/8.93 | ( Y != zero_zero_int ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3208_not__sum__power2__lt__zero,axiom,
% 16.19/8.93 ! [X: real,Y: real] :
% 16.19/8.93 ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3209_not__sum__power2__lt__zero,axiom,
% 16.19/8.93 ! [X: rat,Y: rat] :
% 16.19/8.93 ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3210_not__sum__power2__lt__zero,axiom,
% 16.19/8.93 ! [X: int,Y: int] :
% 16.19/8.93 ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3211_zero__le__even__power_H,axiom,
% 16.19/8.93 ! [A: real,N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3212_zero__le__even__power_H,axiom,
% 16.19/8.93 ! [A: rat,N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3213_zero__le__even__power_H,axiom,
% 16.19/8.93 ! [A: int,N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3214_nat__bit__induct,axiom,
% 16.19/8.93 ! [P: nat > $o,N2: nat] :
% 16.19/8.93 ( ( P @ zero_zero_nat )
% 16.19/8.93 => ( ! [N4: nat] :
% 16.19/8.93 ( ( P @ N4 )
% 16.19/8.93 => ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 16.19/8.93 => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
% 16.19/8.93 => ( ! [N4: nat] :
% 16.19/8.93 ( ( P @ N4 )
% 16.19/8.93 => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
% 16.19/8.93 => ( P @ N2 ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3215_Suc__n__div__2__gt__zero,axiom,
% 16.19/8.93 ! [N2: nat] :
% 16.19/8.93 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.93 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3216_div__2__gt__zero,axiom,
% 16.19/8.93 ! [N2: nat] :
% 16.19/8.93 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 16.19/8.93 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3217_mult__exp__mod__exp__eq,axiom,
% 16.19/8.93 ! [M: nat,N2: nat,A: nat] :
% 16.19/8.93 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/8.93 => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.93 = ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3218_mult__exp__mod__exp__eq,axiom,
% 16.19/8.93 ! [M: nat,N2: nat,A: int] :
% 16.19/8.93 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/8.93 => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.93 = ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3219_mult__exp__mod__exp__eq,axiom,
% 16.19/8.93 ! [M: nat,N2: nat,A: code_integer] :
% 16.19/8.93 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/8.93 => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.93 = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3220_odd__0__le__power__imp__0__le,axiom,
% 16.19/8.93 ! [A: real,N2: nat] :
% 16.19/8.93 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 16.19/8.93 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3221_odd__0__le__power__imp__0__le,axiom,
% 16.19/8.93 ! [A: rat,N2: nat] :
% 16.19/8.93 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 16.19/8.93 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3222_odd__0__le__power__imp__0__le,axiom,
% 16.19/8.93 ! [A: int,N2: nat] :
% 16.19/8.93 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 16.19/8.93 => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3223_odd__power__less__zero,axiom,
% 16.19/8.93 ! [A: real,N2: nat] :
% 16.19/8.93 ( ( ord_less_real @ A @ zero_zero_real )
% 16.19/8.93 => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_real ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3224_odd__power__less__zero,axiom,
% 16.19/8.93 ! [A: rat,N2: nat] :
% 16.19/8.93 ( ( ord_less_rat @ A @ zero_zero_rat )
% 16.19/8.93 => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_rat ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3225_odd__power__less__zero,axiom,
% 16.19/8.93 ! [A: int,N2: nat] :
% 16.19/8.93 ( ( ord_less_int @ A @ zero_zero_int )
% 16.19/8.93 => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_int ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3226_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
% 16.19/8.93 ! [X: nat,N2: nat,M: nat] :
% 16.19/8.93 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
% 16.19/8.93 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.93 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 16.19/8.93 => ( ord_less_nat @ ( vEBT_VEBT_high @ X @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3227_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
% 16.19/8.93 ! [X: nat,N2: nat,M: nat] :
% 16.19/8.93 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
% 16.19/8.93 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.93 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 16.19/8.93 => ( ord_less_nat @ ( vEBT_VEBT_low @ X @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3228_vebt__member_Oelims_I2_J,axiom,
% 16.19/8.93 ! [X: vEBT_VEBT,Xa2: nat] :
% 16.19/8.93 ( ( vEBT_vebt_member @ X @ Xa2 )
% 16.19/8.93 => ( ! [A3: $o,B2: $o] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/8.93 => ~ ( ( ( Xa2 = zero_zero_nat )
% 16.19/8.93 => A3 )
% 16.19/8.93 & ( ( Xa2 != zero_zero_nat )
% 16.19/8.93 => ( ( ( Xa2 = one_one_nat )
% 16.19/8.93 => B2 )
% 16.19/8.93 & ( Xa2 = one_one_nat ) ) ) ) )
% 16.19/8.93 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
% 16.19/8.93 ( ? [Summary2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 16.19/8.93 => ~ ( ( Xa2 != Mi2 )
% 16.19/8.93 => ( ( Xa2 != Ma2 )
% 16.19/8.93 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 16.19/8.93 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 16.19/8.93 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 16.19/8.93 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 16.19/8.93 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.93 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 16.19/8.93 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3229_vebt__member_Oelims_I3_J,axiom,
% 16.19/8.93 ! [X: vEBT_VEBT,Xa2: nat] :
% 16.19/8.93 ( ~ ( vEBT_vebt_member @ X @ Xa2 )
% 16.19/8.93 => ( ! [A3: $o,B2: $o] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/8.93 => ( ( ( Xa2 = zero_zero_nat )
% 16.19/8.93 => A3 )
% 16.19/8.93 & ( ( Xa2 != zero_zero_nat )
% 16.19/8.93 => ( ( ( Xa2 = one_one_nat )
% 16.19/8.93 => B2 )
% 16.19/8.93 & ( Xa2 = one_one_nat ) ) ) ) )
% 16.19/8.93 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 16.19/8.93 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 16.19/8.93 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 16.19/8.93 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
% 16.19/8.93 ( ? [Summary2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 16.19/8.93 => ( ( Xa2 != Mi2 )
% 16.19/8.93 => ( ( Xa2 != Ma2 )
% 16.19/8.93 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 16.19/8.93 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 16.19/8.93 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 16.19/8.93 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 16.19/8.93 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.93 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 16.19/8.93 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3230_vebt__member_Oelims_I1_J,axiom,
% 16.19/8.93 ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 16.19/8.93 ( ( ( vEBT_vebt_member @ X @ Xa2 )
% 16.19/8.93 = Y )
% 16.19/8.93 => ( ! [A3: $o,B2: $o] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/8.93 => ( Y
% 16.19/8.93 = ( ~ ( ( ( Xa2 = zero_zero_nat )
% 16.19/8.93 => A3 )
% 16.19/8.93 & ( ( Xa2 != zero_zero_nat )
% 16.19/8.93 => ( ( ( Xa2 = one_one_nat )
% 16.19/8.93 => B2 )
% 16.19/8.93 & ( Xa2 = one_one_nat ) ) ) ) ) ) )
% 16.19/8.93 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 16.19/8.93 => Y )
% 16.19/8.93 => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 16.19/8.93 => Y )
% 16.19/8.93 => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 16.19/8.93 => Y )
% 16.19/8.93 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
% 16.19/8.93 ( ? [Summary2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 16.19/8.93 => ( Y
% 16.19/8.93 = ( ~ ( ( Xa2 != Mi2 )
% 16.19/8.93 => ( ( Xa2 != Ma2 )
% 16.19/8.93 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 16.19/8.93 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 16.19/8.93 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 16.19/8.93 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 16.19/8.93 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.93 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 16.19/8.93 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3231_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
% 16.19/8.93 ! [X: vEBT_VEBT,Xa2: nat] :
% 16.19/8.93 ( ( vEBT_VEBT_membermima @ X @ Xa2 )
% 16.19/8.93 => ( ! [Mi2: nat,Ma2: nat] :
% 16.19/8.93 ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 16.19/8.93 => ~ ( ( Xa2 = Mi2 )
% 16.19/8.93 | ( Xa2 = Ma2 ) ) )
% 16.19/8.93 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 16.19/8.93 ( ? [Vc2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 16.19/8.93 => ~ ( ( Xa2 = Mi2 )
% 16.19/8.93 | ( Xa2 = Ma2 )
% 16.19/8.93 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.93 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 16.19/8.93 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 16.19/8.93 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 16.19/8.93 ( ? [Vd2: vEBT_VEBT] :
% 16.19/8.93 ( X
% 16.19/8.93 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 16.19/8.93 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.93 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 16.19/8.93 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3232_arith__geo__mean,axiom,
% 16.19/8.93 ! [U: real,X: real,Y: real] :
% 16.19/8.93 ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.93 = ( times_times_real @ X @ Y ) )
% 16.19/8.93 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/8.93 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/8.93 => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3233_arith__geo__mean,axiom,
% 16.19/8.93 ! [U: rat,X: rat,Y: rat] :
% 16.19/8.93 ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.93 = ( times_times_rat @ X @ Y ) )
% 16.19/8.93 => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 16.19/8.93 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 16.19/8.93 => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3234_invar__vebt_Osimps,axiom,
% 16.19/8.93 ( vEBT_invar_vebt
% 16.19/8.93 = ( ^ [A1: vEBT_VEBT,A22: nat] :
% 16.19/8.93 ( ( ? [A5: $o,B5: $o] :
% 16.19/8.93 ( A1
% 16.19/8.93 = ( vEBT_Leaf @ A5 @ B5 ) )
% 16.19/8.93 & ( A22
% 16.19/8.93 = ( suc @ zero_zero_nat ) ) )
% 16.19/8.93 | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT] :
% 16.19/8.93 ( ( A1
% 16.19/8.93 = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
% 16.19/8.93 & ! [X2: vEBT_VEBT] :
% 16.19/8.93 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 16.19/8.93 => ( vEBT_invar_vebt @ X2 @ N ) )
% 16.19/8.93 & ( vEBT_invar_vebt @ Summary3 @ N )
% 16.19/8.93 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 16.19/8.93 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 16.19/8.93 & ( A22
% 16.19/8.93 = ( plus_plus_nat @ N @ N ) )
% 16.19/8.93 & ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X3 )
% 16.19/8.93 & ! [X2: vEBT_VEBT] :
% 16.19/8.93 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 16.19/8.93 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 16.19/8.93 | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT] :
% 16.19/8.93 ( ( A1
% 16.19/8.93 = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
% 16.19/8.93 & ! [X2: vEBT_VEBT] :
% 16.19/8.93 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 16.19/8.93 => ( vEBT_invar_vebt @ X2 @ N ) )
% 16.19/8.93 & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N ) )
% 16.19/8.93 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 16.19/8.93 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
% 16.19/8.93 & ( A22
% 16.19/8.93 = ( plus_plus_nat @ N @ ( suc @ N ) ) )
% 16.19/8.93 & ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X3 )
% 16.19/8.93 & ! [X2: vEBT_VEBT] :
% 16.19/8.93 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 16.19/8.93 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 16.19/8.93 | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 16.19/8.93 ( ( A1
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
% 16.19/8.93 & ! [X2: vEBT_VEBT] :
% 16.19/8.93 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 16.19/8.93 => ( vEBT_invar_vebt @ X2 @ N ) )
% 16.19/8.93 & ( vEBT_invar_vebt @ Summary3 @ N )
% 16.19/8.93 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 16.19/8.93 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 16.19/8.93 & ( A22
% 16.19/8.93 = ( plus_plus_nat @ N @ N ) )
% 16.19/8.93 & ! [I3: nat] :
% 16.19/8.93 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 16.19/8.93 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X3 ) )
% 16.19/8.93 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
% 16.19/8.93 & ( ( Mi3 = Ma3 )
% 16.19/8.93 => ! [X2: vEBT_VEBT] :
% 16.19/8.93 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 16.19/8.93 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 16.19/8.93 & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 16.19/8.93 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
% 16.19/8.93 & ( ( Mi3 != Ma3 )
% 16.19/8.93 => ! [I3: nat] :
% 16.19/8.93 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 16.19/8.93 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N )
% 16.19/8.93 = I3 )
% 16.19/8.93 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) )
% 16.19/8.93 & ! [X2: nat] :
% 16.19/8.93 ( ( ( ( vEBT_VEBT_high @ X2 @ N )
% 16.19/8.93 = I3 )
% 16.19/8.93 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X2 @ N ) ) )
% 16.19/8.93 => ( ( ord_less_nat @ Mi3 @ X2 )
% 16.19/8.93 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) )
% 16.19/8.93 | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 16.19/8.93 ( ( A1
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
% 16.19/8.93 & ! [X2: vEBT_VEBT] :
% 16.19/8.93 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 16.19/8.93 => ( vEBT_invar_vebt @ X2 @ N ) )
% 16.19/8.93 & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N ) )
% 16.19/8.93 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 16.19/8.93 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
% 16.19/8.93 & ( A22
% 16.19/8.93 = ( plus_plus_nat @ N @ ( suc @ N ) ) )
% 16.19/8.93 & ! [I3: nat] :
% 16.19/8.93 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
% 16.19/8.93 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X3 ) )
% 16.19/8.93 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
% 16.19/8.93 & ( ( Mi3 = Ma3 )
% 16.19/8.93 => ! [X2: vEBT_VEBT] :
% 16.19/8.93 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 16.19/8.93 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 16.19/8.93 & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 16.19/8.93 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
% 16.19/8.93 & ( ( Mi3 != Ma3 )
% 16.19/8.93 => ! [I3: nat] :
% 16.19/8.93 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
% 16.19/8.93 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N )
% 16.19/8.93 = I3 )
% 16.19/8.93 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) )
% 16.19/8.93 & ! [X2: nat] :
% 16.19/8.93 ( ( ( ( vEBT_VEBT_high @ X2 @ N )
% 16.19/8.93 = I3 )
% 16.19/8.93 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X2 @ N ) ) )
% 16.19/8.93 => ( ( ord_less_nat @ Mi3 @ X2 )
% 16.19/8.93 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3235_invar__vebt_Ocases,axiom,
% 16.19/8.93 ! [A12: vEBT_VEBT,A23: nat] :
% 16.19/8.93 ( ( vEBT_invar_vebt @ A12 @ A23 )
% 16.19/8.93 => ( ( ? [A3: $o,B2: $o] :
% 16.19/8.93 ( A12
% 16.19/8.93 = ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/8.93 => ( A23
% 16.19/8.93 != ( suc @ zero_zero_nat ) ) )
% 16.19/8.93 => ( ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat] :
% 16.19/8.93 ( ( A12
% 16.19/8.93 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 16.19/8.93 => ( ( A23 = Deg2 )
% 16.19/8.93 => ( ! [X5: vEBT_VEBT] :
% 16.19/8.93 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 16.19/8.93 => ( vEBT_invar_vebt @ X5 @ N4 ) )
% 16.19/8.93 => ( ( vEBT_invar_vebt @ Summary2 @ M2 )
% 16.19/8.93 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 16.19/8.93 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 16.19/8.93 => ( ( M2 = N4 )
% 16.19/8.93 => ( ( Deg2
% 16.19/8.93 = ( plus_plus_nat @ N4 @ M2 ) )
% 16.19/8.93 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
% 16.19/8.93 => ~ ! [X5: vEBT_VEBT] :
% 16.19/8.93 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 16.19/8.93 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
% 16.19/8.93 => ( ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat] :
% 16.19/8.93 ( ( A12
% 16.19/8.93 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 16.19/8.93 => ( ( A23 = Deg2 )
% 16.19/8.93 => ( ! [X5: vEBT_VEBT] :
% 16.19/8.93 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 16.19/8.93 => ( vEBT_invar_vebt @ X5 @ N4 ) )
% 16.19/8.93 => ( ( vEBT_invar_vebt @ Summary2 @ M2 )
% 16.19/8.93 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 16.19/8.93 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 16.19/8.93 => ( ( M2
% 16.19/8.93 = ( suc @ N4 ) )
% 16.19/8.93 => ( ( Deg2
% 16.19/8.93 = ( plus_plus_nat @ N4 @ M2 ) )
% 16.19/8.93 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
% 16.19/8.93 => ~ ! [X5: vEBT_VEBT] :
% 16.19/8.93 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 16.19/8.93 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
% 16.19/8.93 => ( ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 16.19/8.93 ( ( A12
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
% 16.19/8.93 => ( ( A23 = Deg2 )
% 16.19/8.93 => ( ! [X5: vEBT_VEBT] :
% 16.19/8.93 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 16.19/8.93 => ( vEBT_invar_vebt @ X5 @ N4 ) )
% 16.19/8.93 => ( ( vEBT_invar_vebt @ Summary2 @ M2 )
% 16.19/8.93 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 16.19/8.93 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 16.19/8.93 => ( ( M2 = N4 )
% 16.19/8.93 => ( ( Deg2
% 16.19/8.93 = ( plus_plus_nat @ N4 @ M2 ) )
% 16.19/8.93 => ( ! [I: nat] :
% 16.19/8.93 ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 16.19/8.93 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ X3 ) )
% 16.19/8.93 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
% 16.19/8.93 => ( ( ( Mi2 = Ma2 )
% 16.19/8.93 => ! [X5: vEBT_VEBT] :
% 16.19/8.93 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 16.19/8.93 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
% 16.19/8.93 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 16.19/8.93 => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 16.19/8.93 => ~ ( ( Mi2 != Ma2 )
% 16.19/8.93 => ! [I: nat] :
% 16.19/8.93 ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 16.19/8.93 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N4 )
% 16.19/8.93 = I )
% 16.19/8.93 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ ( vEBT_VEBT_low @ Ma2 @ N4 ) ) )
% 16.19/8.93 & ! [X5: nat] :
% 16.19/8.93 ( ( ( ( vEBT_VEBT_high @ X5 @ N4 )
% 16.19/8.93 = I )
% 16.19/8.93 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ ( vEBT_VEBT_low @ X5 @ N4 ) ) )
% 16.19/8.93 => ( ( ord_less_nat @ Mi2 @ X5 )
% 16.19/8.93 & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 16.19/8.93 => ~ ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M2: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 16.19/8.93 ( ( A12
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
% 16.19/8.93 => ( ( A23 = Deg2 )
% 16.19/8.93 => ( ! [X5: vEBT_VEBT] :
% 16.19/8.93 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 16.19/8.93 => ( vEBT_invar_vebt @ X5 @ N4 ) )
% 16.19/8.93 => ( ( vEBT_invar_vebt @ Summary2 @ M2 )
% 16.19/8.93 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 16.19/8.93 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 16.19/8.93 => ( ( M2
% 16.19/8.93 = ( suc @ N4 ) )
% 16.19/8.93 => ( ( Deg2
% 16.19/8.93 = ( plus_plus_nat @ N4 @ M2 ) )
% 16.19/8.93 => ( ! [I: nat] :
% 16.19/8.93 ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 16.19/8.93 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ X3 ) )
% 16.19/8.93 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
% 16.19/8.93 => ( ( ( Mi2 = Ma2 )
% 16.19/8.93 => ! [X5: vEBT_VEBT] :
% 16.19/8.93 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 16.19/8.93 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
% 16.19/8.93 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 16.19/8.93 => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 16.19/8.93 => ~ ( ( Mi2 != Ma2 )
% 16.19/8.93 => ! [I: nat] :
% 16.19/8.93 ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 16.19/8.93 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N4 )
% 16.19/8.93 = I )
% 16.19/8.93 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ ( vEBT_VEBT_low @ Ma2 @ N4 ) ) )
% 16.19/8.93 & ! [X5: nat] :
% 16.19/8.93 ( ( ( ( vEBT_VEBT_high @ X5 @ N4 )
% 16.19/8.93 = I )
% 16.19/8.93 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ ( vEBT_VEBT_low @ X5 @ N4 ) ) )
% 16.19/8.93 => ( ( ord_less_nat @ Mi2 @ X5 )
% 16.19/8.93 & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3236_vebt__insert_Oelims,axiom,
% 16.19/8.93 ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
% 16.19/8.93 ( ( ( vEBT_vebt_insert @ X @ Xa2 )
% 16.19/8.93 = Y )
% 16.19/8.93 => ( ! [A3: $o,B2: $o] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/8.93 => ~ ( ( ( Xa2 = zero_zero_nat )
% 16.19/8.93 => ( Y
% 16.19/8.93 = ( vEBT_Leaf @ $true @ B2 ) ) )
% 16.19/8.93 & ( ( Xa2 != zero_zero_nat )
% 16.19/8.93 => ( ( ( Xa2 = one_one_nat )
% 16.19/8.93 => ( Y
% 16.19/8.93 = ( vEBT_Leaf @ A3 @ $true ) ) )
% 16.19/8.93 & ( ( Xa2 != one_one_nat )
% 16.19/8.93 => ( Y
% 16.19/8.93 = ( vEBT_Leaf @ A3 @ B2 ) ) ) ) ) ) )
% 16.19/8.93 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) )
% 16.19/8.93 => ( Y
% 16.19/8.93 != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) ) )
% 16.19/8.93 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) )
% 16.19/8.93 => ( Y
% 16.19/8.93 != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) ) )
% 16.19/8.93 => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
% 16.19/8.93 => ( Y
% 16.19/8.93 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) ) )
% 16.19/8.93 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 16.19/8.93 => ( Y
% 16.19/8.93 != ( if_VEBT_VEBT
% 16.19/8.93 @ ( ( 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 @ TreeList3 ) )
% 16.19/8.93 & ~ ( ( Xa2 = Mi2 )
% 16.19/8.93 | ( Xa2 = Ma2 ) ) )
% 16.19/8.93 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 ) )
% 16.19/8.93 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3237_verit__le__mono__div,axiom,
% 16.19/8.93 ! [A2: nat,B3: nat,N2: nat] :
% 16.19/8.93 ( ( ord_less_nat @ A2 @ B3 )
% 16.19/8.93 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.93 => ( ord_less_eq_nat
% 16.19/8.93 @ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N2 )
% 16.19/8.93 @ ( if_nat
% 16.19/8.93 @ ( ( modulo_modulo_nat @ B3 @ N2 )
% 16.19/8.93 = zero_zero_nat )
% 16.19/8.93 @ one_one_nat
% 16.19/8.93 @ zero_zero_nat ) )
% 16.19/8.93 @ ( divide_divide_nat @ B3 @ N2 ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3238_inrange,axiom,
% 16.19/8.93 ! [T: vEBT_VEBT,N2: nat] :
% 16.19/8.93 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.19/8.93 => ( 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 ) ) @ N2 ) @ one_one_nat ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3239_set__bit__0,axiom,
% 16.19/8.93 ! [A: int] :
% 16.19/8.93 ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
% 16.19/8.93 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3240_set__bit__0,axiom,
% 16.19/8.93 ! [A: nat] :
% 16.19/8.93 ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
% 16.19/8.93 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3241_vebt__succ_Opelims,axiom,
% 16.19/8.93 ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
% 16.19/8.93 ( ( ( vEBT_vebt_succ @ X @ Xa2 )
% 16.19/8.93 = Y )
% 16.19/8.93 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 16.19/8.93 => ( ! [Uu2: $o,B2: $o] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Leaf @ Uu2 @ B2 ) )
% 16.19/8.93 => ( ( Xa2 = zero_zero_nat )
% 16.19/8.93 => ( ( ( B2
% 16.19/8.93 => ( Y
% 16.19/8.93 = ( some_nat @ one_one_nat ) ) )
% 16.19/8.93 & ( ~ B2
% 16.19/8.93 => ( Y = none_nat ) ) )
% 16.19/8.93 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B2 ) @ zero_zero_nat ) ) ) ) )
% 16.19/8.93 => ( ! [Uv2: $o,Uw2: $o] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
% 16.19/8.93 => ! [N4: nat] :
% 16.19/8.93 ( ( Xa2
% 16.19/8.93 = ( suc @ N4 ) )
% 16.19/8.93 => ( ( Y = none_nat )
% 16.19/8.93 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N4 ) ) ) ) ) )
% 16.19/8.93 => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
% 16.19/8.93 => ( ( Y = none_nat )
% 16.19/8.93 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
% 16.19/8.93 => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
% 16.19/8.93 => ( ( Y = none_nat )
% 16.19/8.93 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa2 ) ) ) )
% 16.19/8.93 => ( ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) )
% 16.19/8.93 => ( ( Y = none_nat )
% 16.19/8.93 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Xa2 ) ) ) )
% 16.19/8.93 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 16.19/8.93 => ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
% 16.19/8.93 => ( Y
% 16.19/8.93 = ( some_nat @ Mi2 ) ) )
% 16.19/8.93 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 16.19/8.93 => ( Y
% 16.19/8.93 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.93 @ ( if_option_nat
% 16.19/8.93 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/8.93 != none_nat )
% 16.19/8.93 & ( 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 @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 16.19/8.93 @ ( 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 @ TreeList3 @ ( 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 ) ) ) ) ) )
% 16.19/8.93 @ ( if_option_nat
% 16.19/8.93 @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/8.93 = none_nat )
% 16.19/8.93 @ none_nat
% 16.19/8.93 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 16.19/8.93 @ none_nat ) ) ) )
% 16.19/8.93 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3242_vebt__pred_Opelims,axiom,
% 16.19/8.93 ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
% 16.19/8.93 ( ( ( vEBT_vebt_pred @ X @ Xa2 )
% 16.19/8.93 = Y )
% 16.19/8.93 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 16.19/8.93 => ( ! [Uu2: $o,Uv2: $o] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 16.19/8.93 => ( ( Xa2 = zero_zero_nat )
% 16.19/8.93 => ( ( Y = none_nat )
% 16.19/8.93 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
% 16.19/8.93 => ( ! [A3: $o,Uw2: $o] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Leaf @ A3 @ Uw2 ) )
% 16.19/8.93 => ( ( Xa2
% 16.19/8.93 = ( suc @ zero_zero_nat ) )
% 16.19/8.93 => ( ( ( A3
% 16.19/8.93 => ( Y
% 16.19/8.93 = ( some_nat @ zero_zero_nat ) ) )
% 16.19/8.93 & ( ~ A3
% 16.19/8.93 => ( Y = none_nat ) ) )
% 16.19/8.93 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
% 16.19/8.93 => ( ! [A3: $o,B2: $o] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/8.93 => ! [Va2: nat] :
% 16.19/8.93 ( ( Xa2
% 16.19/8.93 = ( suc @ ( suc @ Va2 ) ) )
% 16.19/8.93 => ( ( ( B2
% 16.19/8.93 => ( Y
% 16.19/8.93 = ( some_nat @ one_one_nat ) ) )
% 16.19/8.93 & ( ~ B2
% 16.19/8.93 => ( ( A3
% 16.19/8.93 => ( Y
% 16.19/8.93 = ( some_nat @ zero_zero_nat ) ) )
% 16.19/8.93 & ( ~ A3
% 16.19/8.93 => ( Y = none_nat ) ) ) ) )
% 16.19/8.93 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) )
% 16.19/8.93 => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
% 16.19/8.93 => ( ( Y = none_nat )
% 16.19/8.93 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Xa2 ) ) ) )
% 16.19/8.93 => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve: vEBT_VEBT] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) )
% 16.19/8.93 => ( ( Y = none_nat )
% 16.19/8.93 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve ) @ Xa2 ) ) ) )
% 16.19/8.93 => ( ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) )
% 16.19/8.93 => ( ( Y = none_nat )
% 16.19/8.93 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Xa2 ) ) ) )
% 16.19/8.93 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 16.19/8.93 => ( ( ( ( ord_less_nat @ Ma2 @ Xa2 )
% 16.19/8.93 => ( Y
% 16.19/8.93 = ( some_nat @ Ma2 ) ) )
% 16.19/8.93 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 16.19/8.93 => ( Y
% 16.19/8.93 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.93 @ ( if_option_nat
% 16.19/8.93 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/8.93 != none_nat )
% 16.19/8.93 & ( 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 @ TreeList3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 16.19/8.93 @ ( 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 @ TreeList3 @ ( 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 ) ) ) ) ) )
% 16.19/8.93 @ ( if_option_nat
% 16.19/8.93 @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/8.93 = none_nat )
% 16.19/8.93 @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
% 16.19/8.93 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 16.19/8.93 @ none_nat ) ) ) )
% 16.19/8.93 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3243_double__eq__0__iff,axiom,
% 16.19/8.93 ! [A: real] :
% 16.19/8.93 ( ( ( plus_plus_real @ A @ A )
% 16.19/8.93 = zero_zero_real )
% 16.19/8.93 = ( A = zero_zero_real ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3244_double__eq__0__iff,axiom,
% 16.19/8.93 ! [A: rat] :
% 16.19/8.93 ( ( ( plus_plus_rat @ A @ A )
% 16.19/8.93 = zero_zero_rat )
% 16.19/8.93 = ( A = zero_zero_rat ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3245_double__eq__0__iff,axiom,
% 16.19/8.93 ! [A: int] :
% 16.19/8.93 ( ( ( plus_plus_int @ A @ A )
% 16.19/8.93 = zero_zero_int )
% 16.19/8.93 = ( A = zero_zero_int ) ) ).
% 16.19/8.93
% 16.19/8.93 thf(fact_3246_vebt__delete_Opelims,axiom,
% 16.19/8.93 ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
% 16.19/8.93 ( ( ( vEBT_vebt_delete @ X @ Xa2 )
% 16.19/8.93 = Y )
% 16.19/8.93 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 16.19/8.93 => ( ! [A3: $o,B2: $o] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/8.93 => ( ( Xa2 = zero_zero_nat )
% 16.19/8.93 => ( ( Y
% 16.19/8.93 = ( vEBT_Leaf @ $false @ B2 ) )
% 16.19/8.93 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ zero_zero_nat ) ) ) ) )
% 16.19/8.93 => ( ! [A3: $o,B2: $o] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/8.93 => ( ( Xa2
% 16.19/8.93 = ( suc @ zero_zero_nat ) )
% 16.19/8.93 => ( ( Y
% 16.19/8.93 = ( vEBT_Leaf @ A3 @ $false ) )
% 16.19/8.93 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
% 16.19/8.93 => ( ! [A3: $o,B2: $o] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/8.93 => ! [N4: nat] :
% 16.19/8.93 ( ( Xa2
% 16.19/8.93 = ( suc @ ( suc @ N4 ) ) )
% 16.19/8.93 => ( ( Y
% 16.19/8.93 = ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/8.93 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( suc @ N4 ) ) ) ) ) ) )
% 16.19/8.93 => ( ! [Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 16.19/8.93 => ( ( Y
% 16.19/8.93 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 16.19/8.93 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) )
% 16.19/8.93 => ( ! [Mi2: nat,Ma2: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) )
% 16.19/8.93 => ( ( Y
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) )
% 16.19/8.93 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) @ Xa2 ) ) ) )
% 16.19/8.93 => ( ! [Mi2: nat,Ma2: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) )
% 16.19/8.93 => ( ( Y
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) )
% 16.19/8.93 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) @ Xa2 ) ) ) )
% 16.19/8.93 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 16.19/8.93 ( ( X
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 16.19/8.93 => ( ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
% 16.19/8.93 | ( ord_less_nat @ Ma2 @ Xa2 ) )
% 16.19/8.93 => ( Y
% 16.19/8.93 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) )
% 16.19/8.93 & ( ~ ( ( ord_less_nat @ Xa2 @ Mi2 )
% 16.19/8.93 | ( ord_less_nat @ Ma2 @ Xa2 ) )
% 16.19/8.93 => ( ( ( ( Xa2 = Mi2 )
% 16.19/8.93 & ( Xa2 = Ma2 ) )
% 16.19/8.93 => ( Y
% 16.19/8.93 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) )
% 16.19/8.93 & ( ~ ( ( Xa2 = Mi2 )
% 16.19/8.93 & ( Xa2 = Ma2 ) )
% 16.19/8.93 => ( Y
% 16.19/8.93 = ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.93 @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/8.93 @ ( vEBT_Node
% 16.19/8.93 @ ( some_P7363390416028606310at_nat
% 16.19/8.93 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
% 16.19/8.93 @ ( if_nat
% 16.19/8.93 @ ( ( ( Xa2 = Mi2 )
% 16.19/8.93 => ( ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
% 16.19/8.93 = Ma2 ) )
% 16.19/8.93 & ( ( Xa2 != Mi2 )
% 16.19/8.93 => ( Xa2 = Ma2 ) ) )
% 16.19/8.93 @ ( if_nat
% 16.19/8.93 @ ( ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/8.93 = none_nat )
% 16.19/8.93 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
% 16.19/8.93 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
% 16.19/8.93 @ Ma2 ) ) )
% 16.19/8.93 @ ( suc @ ( suc @ Va2 ) )
% 16.19/8.93 @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/8.94 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/8.94 @ ( vEBT_Node
% 16.19/8.94 @ ( some_P7363390416028606310at_nat
% 16.19/8.94 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
% 16.19/8.94 @ ( if_nat
% 16.19/8.94 @ ( ( ( Xa2 = Mi2 )
% 16.19/8.94 => ( ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
% 16.19/8.94 = Ma2 ) )
% 16.19/8.94 & ( ( Xa2 != Mi2 )
% 16.19/8.94 => ( Xa2 = Ma2 ) ) )
% 16.19/8.94 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 16.19/8.94 @ Ma2 ) ) )
% 16.19/8.94 @ ( suc @ ( suc @ Va2 ) )
% 16.19/8.94 @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/8.94 @ Summary2 ) )
% 16.19/8.94 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) ) ) ) ) )
% 16.19/8.94 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3247_unset__bit__0,axiom,
% 16.19/8.94 ! [A: int] :
% 16.19/8.94 ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
% 16.19/8.94 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3248_unset__bit__0,axiom,
% 16.19/8.94 ! [A: nat] :
% 16.19/8.94 ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
% 16.19/8.94 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3249_div__mod__decomp,axiom,
% 16.19/8.94 ! [A2: nat,N2: nat] :
% 16.19/8.94 ( A2
% 16.19/8.94 = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N2 ) @ N2 ) @ ( modulo_modulo_nat @ A2 @ N2 ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3250_unset__bit__Suc,axiom,
% 16.19/8.94 ! [N2: nat,A: code_integer] :
% 16.19/8.94 ( ( bit_se8260200283734997820nteger @ ( suc @ N2 ) @ A )
% 16.19/8.94 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3251_unset__bit__Suc,axiom,
% 16.19/8.94 ! [N2: nat,A: int] :
% 16.19/8.94 ( ( bit_se4203085406695923979it_int @ ( suc @ N2 ) @ A )
% 16.19/8.94 = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3252_unset__bit__Suc,axiom,
% 16.19/8.94 ! [N2: nat,A: nat] :
% 16.19/8.94 ( ( bit_se4205575877204974255it_nat @ ( suc @ N2 ) @ A )
% 16.19/8.94 = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3253_verit__eq__simplify_I8_J,axiom,
% 16.19/8.94 ! [X22: num,Y22: num] :
% 16.19/8.94 ( ( ( bit0 @ X22 )
% 16.19/8.94 = ( bit0 @ Y22 ) )
% 16.19/8.94 = ( X22 = Y22 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3254_div__neg__neg__trivial,axiom,
% 16.19/8.94 ! [K: int,L2: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 16.19/8.94 => ( ( ord_less_int @ L2 @ K )
% 16.19/8.94 => ( ( divide_divide_int @ K @ L2 )
% 16.19/8.94 = zero_zero_int ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3255_div__pos__pos__trivial,axiom,
% 16.19/8.94 ! [K: int,L2: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 16.19/8.94 => ( ( ord_less_int @ K @ L2 )
% 16.19/8.94 => ( ( divide_divide_int @ K @ L2 )
% 16.19/8.94 = zero_zero_int ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3256_i0__less,axiom,
% 16.19/8.94 ! [N2: extended_enat] :
% 16.19/8.94 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 )
% 16.19/8.94 = ( N2 != zero_z5237406670263579293d_enat ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3257_idiff__0__right,axiom,
% 16.19/8.94 ! [N2: extended_enat] :
% 16.19/8.94 ( ( minus_3235023915231533773d_enat @ N2 @ zero_z5237406670263579293d_enat )
% 16.19/8.94 = N2 ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3258_idiff__0,axiom,
% 16.19/8.94 ! [N2: extended_enat] :
% 16.19/8.94 ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N2 )
% 16.19/8.94 = zero_z5237406670263579293d_enat ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3259_not__real__square__gt__zero,axiom,
% 16.19/8.94 ! [X: real] :
% 16.19/8.94 ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
% 16.19/8.94 = ( X = zero_zero_real ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3260_zmod__numeral__Bit0,axiom,
% 16.19/8.94 ! [V: num,W: num] :
% 16.19/8.94 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 16.19/8.94 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3261_half__negative__int__iff,axiom,
% 16.19/8.94 ! [K: int] :
% 16.19/8.94 ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 16.19/8.94 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3262_half__nonnegative__int__iff,axiom,
% 16.19/8.94 ! [K: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 16.19/8.94 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3263_split__zdiv,axiom,
% 16.19/8.94 ! [P: int > $o,N2: int,K: int] :
% 16.19/8.94 ( ( P @ ( divide_divide_int @ N2 @ K ) )
% 16.19/8.94 = ( ( ( K = zero_zero_int )
% 16.19/8.94 => ( P @ zero_zero_int ) )
% 16.19/8.94 & ( ( ord_less_int @ zero_zero_int @ K )
% 16.19/8.94 => ! [I3: int,J3: int] :
% 16.19/8.94 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 16.19/8.94 & ( ord_less_int @ J3 @ K )
% 16.19/8.94 & ( N2
% 16.19/8.94 = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
% 16.19/8.94 => ( P @ I3 ) ) )
% 16.19/8.94 & ( ( ord_less_int @ K @ zero_zero_int )
% 16.19/8.94 => ! [I3: int,J3: int] :
% 16.19/8.94 ( ( ( ord_less_int @ K @ J3 )
% 16.19/8.94 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 16.19/8.94 & ( N2
% 16.19/8.94 = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
% 16.19/8.94 => ( P @ I3 ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3264_zdiv__mono1,axiom,
% 16.19/8.94 ! [A: int,A6: int,B: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ A @ A6 )
% 16.19/8.94 => ( ( ord_less_int @ zero_zero_int @ B )
% 16.19/8.94 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A6 @ B ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3265_zdiv__mono2,axiom,
% 16.19/8.94 ! [A: int,B6: int,B: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.94 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 16.19/8.94 => ( ( ord_less_eq_int @ B6 @ B )
% 16.19/8.94 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B6 ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3266_div__pos__geq,axiom,
% 16.19/8.94 ! [L2: int,K: int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ L2 )
% 16.19/8.94 => ( ( ord_less_eq_int @ L2 @ K )
% 16.19/8.94 => ( ( divide_divide_int @ K @ L2 )
% 16.19/8.94 = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L2 ) @ L2 ) @ one_one_int ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3267_zdiv__eq__0__iff,axiom,
% 16.19/8.94 ! [I2: int,K: int] :
% 16.19/8.94 ( ( ( divide_divide_int @ I2 @ K )
% 16.19/8.94 = zero_zero_int )
% 16.19/8.94 = ( ( K = zero_zero_int )
% 16.19/8.94 | ( ( ord_less_eq_int @ zero_zero_int @ I2 )
% 16.19/8.94 & ( ord_less_int @ I2 @ K ) )
% 16.19/8.94 | ( ( ord_less_eq_int @ I2 @ zero_zero_int )
% 16.19/8.94 & ( ord_less_int @ K @ I2 ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3268_int__div__neg__eq,axiom,
% 16.19/8.94 ! [A: int,B: int,Q2: int,R: int] :
% 16.19/8.94 ( ( A
% 16.19/8.94 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 16.19/8.94 => ( ( ord_less_eq_int @ R @ zero_zero_int )
% 16.19/8.94 => ( ( ord_less_int @ B @ R )
% 16.19/8.94 => ( ( divide_divide_int @ A @ B )
% 16.19/8.94 = Q2 ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3269_int__div__pos__eq,axiom,
% 16.19/8.94 ! [A: int,B: int,Q2: int,R: int] :
% 16.19/8.94 ( ( A
% 16.19/8.94 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 16.19/8.94 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 16.19/8.94 => ( ( ord_less_int @ R @ B )
% 16.19/8.94 => ( ( divide_divide_int @ A @ B )
% 16.19/8.94 = Q2 ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3270_zdiv__mono1__neg,axiom,
% 16.19/8.94 ! [A: int,A6: int,B: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ A @ A6 )
% 16.19/8.94 => ( ( ord_less_int @ B @ zero_zero_int )
% 16.19/8.94 => ( ord_less_eq_int @ ( divide_divide_int @ A6 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3271_zdiv__mono2__neg,axiom,
% 16.19/8.94 ! [A: int,B6: int,B: int] :
% 16.19/8.94 ( ( ord_less_int @ A @ zero_zero_int )
% 16.19/8.94 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 16.19/8.94 => ( ( ord_less_eq_int @ B6 @ B )
% 16.19/8.94 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B6 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3272_div__int__pos__iff,axiom,
% 16.19/8.94 ! [K: int,L2: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L2 ) )
% 16.19/8.94 = ( ( K = zero_zero_int )
% 16.19/8.94 | ( L2 = zero_zero_int )
% 16.19/8.94 | ( ( ord_less_eq_int @ zero_zero_int @ K )
% 16.19/8.94 & ( ord_less_eq_int @ zero_zero_int @ L2 ) )
% 16.19/8.94 | ( ( ord_less_int @ K @ zero_zero_int )
% 16.19/8.94 & ( ord_less_int @ L2 @ zero_zero_int ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3273_split__neg__lemma,axiom,
% 16.19/8.94 ! [K: int,P: int > int > $o,N2: int] :
% 16.19/8.94 ( ( ord_less_int @ K @ zero_zero_int )
% 16.19/8.94 => ( ( P @ ( divide_divide_int @ N2 @ K ) @ ( modulo_modulo_int @ N2 @ K ) )
% 16.19/8.94 = ( ! [I3: int,J3: int] :
% 16.19/8.94 ( ( ( ord_less_int @ K @ J3 )
% 16.19/8.94 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 16.19/8.94 & ( N2
% 16.19/8.94 = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
% 16.19/8.94 => ( P @ I3 @ J3 ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3274_split__pos__lemma,axiom,
% 16.19/8.94 ! [K: int,P: int > int > $o,N2: int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ K )
% 16.19/8.94 => ( ( P @ ( divide_divide_int @ N2 @ K ) @ ( modulo_modulo_int @ N2 @ K ) )
% 16.19/8.94 = ( ! [I3: int,J3: int] :
% 16.19/8.94 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 16.19/8.94 & ( ord_less_int @ J3 @ K )
% 16.19/8.94 & ( N2
% 16.19/8.94 = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
% 16.19/8.94 => ( P @ I3 @ J3 ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3275_div__positive__int,axiom,
% 16.19/8.94 ! [L2: int,K: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ L2 @ K )
% 16.19/8.94 => ( ( ord_less_int @ zero_zero_int @ L2 )
% 16.19/8.94 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L2 ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3276_verit__le__mono__div__int,axiom,
% 16.19/8.94 ! [A2: int,B3: int,N2: int] :
% 16.19/8.94 ( ( ord_less_int @ A2 @ B3 )
% 16.19/8.94 => ( ( ord_less_int @ zero_zero_int @ N2 )
% 16.19/8.94 => ( ord_less_eq_int
% 16.19/8.94 @ ( plus_plus_int @ ( divide_divide_int @ A2 @ N2 )
% 16.19/8.94 @ ( if_int
% 16.19/8.94 @ ( ( modulo_modulo_int @ B3 @ N2 )
% 16.19/8.94 = zero_zero_int )
% 16.19/8.94 @ one_one_int
% 16.19/8.94 @ zero_zero_int ) )
% 16.19/8.94 @ ( divide_divide_int @ B3 @ N2 ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3277_div__nonneg__neg__le0,axiom,
% 16.19/8.94 ! [A: int,B: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.94 => ( ( ord_less_int @ B @ zero_zero_int )
% 16.19/8.94 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3278_div__nonpos__pos__le0,axiom,
% 16.19/8.94 ! [A: int,B: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 16.19/8.94 => ( ( ord_less_int @ zero_zero_int @ B )
% 16.19/8.94 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3279_pos__imp__zdiv__pos__iff,axiom,
% 16.19/8.94 ! [K: int,I2: int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ K )
% 16.19/8.94 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I2 @ K ) )
% 16.19/8.94 = ( ord_less_eq_int @ K @ I2 ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3280_neg__imp__zdiv__nonneg__iff,axiom,
% 16.19/8.94 ! [B: int,A: int] :
% 16.19/8.94 ( ( ord_less_int @ B @ zero_zero_int )
% 16.19/8.94 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 16.19/8.94 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3281_pos__imp__zdiv__nonneg__iff,axiom,
% 16.19/8.94 ! [B: int,A: int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ B )
% 16.19/8.94 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 16.19/8.94 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3282_nonneg1__imp__zdiv__pos__iff,axiom,
% 16.19/8.94 ! [A: int,B: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.94 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 16.19/8.94 = ( ( ord_less_eq_int @ B @ A )
% 16.19/8.94 & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3283_zmod__zmult2__eq,axiom,
% 16.19/8.94 ! [C: int,A: int,B: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 16.19/8.94 => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
% 16.19/8.94 = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3284_pos__imp__zdiv__neg__iff,axiom,
% 16.19/8.94 ! [B: int,A: int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ B )
% 16.19/8.94 => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 16.19/8.94 = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3285_neg__imp__zdiv__neg__iff,axiom,
% 16.19/8.94 ! [B: int,A: int] :
% 16.19/8.94 ( ( ord_less_int @ B @ zero_zero_int )
% 16.19/8.94 => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 16.19/8.94 = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3286_div__neg__pos__less0,axiom,
% 16.19/8.94 ! [A: int,B: int] :
% 16.19/8.94 ( ( ord_less_int @ A @ zero_zero_int )
% 16.19/8.94 => ( ( ord_less_int @ zero_zero_int @ B )
% 16.19/8.94 => ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3287_zdiv__mono__strict,axiom,
% 16.19/8.94 ! [A2: int,B3: int,N2: int] :
% 16.19/8.94 ( ( ord_less_int @ A2 @ B3 )
% 16.19/8.94 => ( ( ord_less_int @ zero_zero_int @ N2 )
% 16.19/8.94 => ( ( ( modulo_modulo_int @ A2 @ N2 )
% 16.19/8.94 = zero_zero_int )
% 16.19/8.94 => ( ( ( modulo_modulo_int @ B3 @ N2 )
% 16.19/8.94 = zero_zero_int )
% 16.19/8.94 => ( ord_less_int @ ( divide_divide_int @ A2 @ N2 ) @ ( divide_divide_int @ B3 @ N2 ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3288_int__div__less__self,axiom,
% 16.19/8.94 ! [X: int,K: int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ X )
% 16.19/8.94 => ( ( ord_less_int @ one_one_int @ K )
% 16.19/8.94 => ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3289_zdiv__zmult2__eq,axiom,
% 16.19/8.94 ! [C: int,A: int,B: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 16.19/8.94 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 16.19/8.94 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3290_div__mod__decomp__int,axiom,
% 16.19/8.94 ! [A2: int,N2: int] :
% 16.19/8.94 ( A2
% 16.19/8.94 = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ N2 ) @ N2 ) @ ( modulo_modulo_int @ A2 @ N2 ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3291_neg__zmod__mult__2,axiom,
% 16.19/8.94 ! [A: int,B: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 16.19/8.94 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 16.19/8.94 = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3292_pos__zmod__mult__2,axiom,
% 16.19/8.94 ! [A: int,B: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.94 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 16.19/8.94 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ A ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3293_not__iless0,axiom,
% 16.19/8.94 ! [N2: extended_enat] :
% 16.19/8.94 ~ ( ord_le72135733267957522d_enat @ N2 @ zero_z5237406670263579293d_enat ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3294_enat__0__less__mult__iff,axiom,
% 16.19/8.94 ! [M: extended_enat,N2: extended_enat] :
% 16.19/8.94 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N2 ) )
% 16.19/8.94 = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
% 16.19/8.94 & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3295_all__nat__less,axiom,
% 16.19/8.94 ! [N2: nat,P: nat > $o] :
% 16.19/8.94 ( ( ! [M6: nat] :
% 16.19/8.94 ( ( ord_less_eq_nat @ M6 @ N2 )
% 16.19/8.94 => ( P @ M6 ) ) )
% 16.19/8.94 = ( ! [X2: nat] :
% 16.19/8.94 ( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 16.19/8.94 => ( P @ X2 ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3296_ex__nat__less,axiom,
% 16.19/8.94 ! [N2: nat,P: nat > $o] :
% 16.19/8.94 ( ( ? [M6: nat] :
% 16.19/8.94 ( ( ord_less_eq_nat @ M6 @ N2 )
% 16.19/8.94 & ( P @ M6 ) ) )
% 16.19/8.94 = ( ? [X2: nat] :
% 16.19/8.94 ( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 16.19/8.94 & ( P @ X2 ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3297_not__exp__less__eq__0__int,axiom,
% 16.19/8.94 ! [N2: nat] :
% 16.19/8.94 ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_int ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3298_pos__zdiv__mult__2,axiom,
% 16.19/8.94 ! [A: int,B: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/8.94 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 16.19/8.94 = ( divide_divide_int @ B @ A ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3299_neg__zdiv__mult__2,axiom,
% 16.19/8.94 ! [A: int,B: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 16.19/8.94 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 16.19/8.94 = ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3300_verit__la__disequality,axiom,
% 16.19/8.94 ! [A: rat,B: rat] :
% 16.19/8.94 ( ( A = B )
% 16.19/8.94 | ~ ( ord_less_eq_rat @ A @ B )
% 16.19/8.94 | ~ ( ord_less_eq_rat @ B @ A ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3301_verit__la__disequality,axiom,
% 16.19/8.94 ! [A: num,B: num] :
% 16.19/8.94 ( ( A = B )
% 16.19/8.94 | ~ ( ord_less_eq_num @ A @ B )
% 16.19/8.94 | ~ ( ord_less_eq_num @ B @ A ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3302_verit__la__disequality,axiom,
% 16.19/8.94 ! [A: nat,B: nat] :
% 16.19/8.94 ( ( A = B )
% 16.19/8.94 | ~ ( ord_less_eq_nat @ A @ B )
% 16.19/8.94 | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3303_verit__la__disequality,axiom,
% 16.19/8.94 ! [A: int,B: int] :
% 16.19/8.94 ( ( A = B )
% 16.19/8.94 | ~ ( ord_less_eq_int @ A @ B )
% 16.19/8.94 | ~ ( ord_less_eq_int @ B @ A ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3304_verit__comp__simplify1_I2_J,axiom,
% 16.19/8.94 ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3305_verit__comp__simplify1_I2_J,axiom,
% 16.19/8.94 ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3306_verit__comp__simplify1_I2_J,axiom,
% 16.19/8.94 ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3307_verit__comp__simplify1_I2_J,axiom,
% 16.19/8.94 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3308_verit__comp__simplify1_I2_J,axiom,
% 16.19/8.94 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3309_verit__comp__simplify1_I1_J,axiom,
% 16.19/8.94 ! [A: real] :
% 16.19/8.94 ~ ( ord_less_real @ A @ A ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3310_verit__comp__simplify1_I1_J,axiom,
% 16.19/8.94 ! [A: rat] :
% 16.19/8.94 ~ ( ord_less_rat @ A @ A ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3311_verit__comp__simplify1_I1_J,axiom,
% 16.19/8.94 ! [A: num] :
% 16.19/8.94 ~ ( ord_less_num @ A @ A ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3312_verit__comp__simplify1_I1_J,axiom,
% 16.19/8.94 ! [A: nat] :
% 16.19/8.94 ~ ( ord_less_nat @ A @ A ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3313_verit__comp__simplify1_I1_J,axiom,
% 16.19/8.94 ! [A: int] :
% 16.19/8.94 ~ ( ord_less_int @ A @ A ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3314_realpow__pos__nth2,axiom,
% 16.19/8.94 ! [A: real,N2: nat] :
% 16.19/8.94 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.94 => ? [R3: real] :
% 16.19/8.94 ( ( ord_less_real @ zero_zero_real @ R3 )
% 16.19/8.94 & ( ( power_power_real @ R3 @ ( suc @ N2 ) )
% 16.19/8.94 = A ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3315_real__arch__pow__inv,axiom,
% 16.19/8.94 ! [Y: real,X: real] :
% 16.19/8.94 ( ( ord_less_real @ zero_zero_real @ Y )
% 16.19/8.94 => ( ( ord_less_real @ X @ one_one_real )
% 16.19/8.94 => ? [N4: nat] : ( ord_less_real @ ( power_power_real @ X @ N4 ) @ Y ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3316_int__power__div__base,axiom,
% 16.19/8.94 ! [M: nat,K: int] :
% 16.19/8.94 ( ( ord_less_nat @ zero_zero_nat @ M )
% 16.19/8.94 => ( ( ord_less_int @ zero_zero_int @ K )
% 16.19/8.94 => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
% 16.19/8.94 = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3317_realpow__pos__nth__unique,axiom,
% 16.19/8.94 ! [N2: nat,A: real] :
% 16.19/8.94 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.94 => ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.94 => ? [X4: real] :
% 16.19/8.94 ( ( ord_less_real @ zero_zero_real @ X4 )
% 16.19/8.94 & ( ( power_power_real @ X4 @ N2 )
% 16.19/8.94 = A )
% 16.19/8.94 & ! [Y5: real] :
% 16.19/8.94 ( ( ( ord_less_real @ zero_zero_real @ Y5 )
% 16.19/8.94 & ( ( power_power_real @ Y5 @ N2 )
% 16.19/8.94 = A ) )
% 16.19/8.94 => ( Y5 = X4 ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3318_realpow__pos__nth,axiom,
% 16.19/8.94 ! [N2: nat,A: real] :
% 16.19/8.94 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.94 => ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/8.94 => ? [R3: real] :
% 16.19/8.94 ( ( ord_less_real @ zero_zero_real @ R3 )
% 16.19/8.94 & ( ( power_power_real @ R3 @ N2 )
% 16.19/8.94 = A ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3319_verit__comp__simplify1_I3_J,axiom,
% 16.19/8.94 ! [B6: real,A6: real] :
% 16.19/8.94 ( ( ~ ( ord_less_eq_real @ B6 @ A6 ) )
% 16.19/8.94 = ( ord_less_real @ A6 @ B6 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3320_verit__comp__simplify1_I3_J,axiom,
% 16.19/8.94 ! [B6: rat,A6: rat] :
% 16.19/8.94 ( ( ~ ( ord_less_eq_rat @ B6 @ A6 ) )
% 16.19/8.94 = ( ord_less_rat @ A6 @ B6 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3321_verit__comp__simplify1_I3_J,axiom,
% 16.19/8.94 ! [B6: num,A6: num] :
% 16.19/8.94 ( ( ~ ( ord_less_eq_num @ B6 @ A6 ) )
% 16.19/8.94 = ( ord_less_num @ A6 @ B6 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3322_verit__comp__simplify1_I3_J,axiom,
% 16.19/8.94 ! [B6: nat,A6: nat] :
% 16.19/8.94 ( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
% 16.19/8.94 = ( ord_less_nat @ A6 @ B6 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3323_verit__comp__simplify1_I3_J,axiom,
% 16.19/8.94 ! [B6: int,A6: int] :
% 16.19/8.94 ( ( ~ ( ord_less_eq_int @ B6 @ A6 ) )
% 16.19/8.94 = ( ord_less_int @ A6 @ B6 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3324_verit__sum__simplify,axiom,
% 16.19/8.94 ! [A: complex] :
% 16.19/8.94 ( ( plus_plus_complex @ A @ zero_zero_complex )
% 16.19/8.94 = A ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3325_verit__sum__simplify,axiom,
% 16.19/8.94 ! [A: real] :
% 16.19/8.94 ( ( plus_plus_real @ A @ zero_zero_real )
% 16.19/8.94 = A ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3326_verit__sum__simplify,axiom,
% 16.19/8.94 ! [A: rat] :
% 16.19/8.94 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 16.19/8.94 = A ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3327_verit__sum__simplify,axiom,
% 16.19/8.94 ! [A: nat] :
% 16.19/8.94 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 16.19/8.94 = A ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3328_verit__sum__simplify,axiom,
% 16.19/8.94 ! [A: int] :
% 16.19/8.94 ( ( plus_plus_int @ A @ zero_zero_int )
% 16.19/8.94 = A ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3329_verit__eq__simplify_I10_J,axiom,
% 16.19/8.94 ! [X22: num] :
% 16.19/8.94 ( one
% 16.19/8.94 != ( bit0 @ X22 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3330_max__def__raw,axiom,
% 16.19/8.94 ( ord_max_set_nat
% 16.19/8.94 = ( ^ [A5: set_nat,B5: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A5 @ B5 ) @ B5 @ A5 ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3331_max__def__raw,axiom,
% 16.19/8.94 ( ord_max_rat
% 16.19/8.94 = ( ^ [A5: rat,B5: rat] : ( if_rat @ ( ord_less_eq_rat @ A5 @ B5 ) @ B5 @ A5 ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3332_max__def__raw,axiom,
% 16.19/8.94 ( ord_max_num
% 16.19/8.94 = ( ^ [A5: num,B5: num] : ( if_num @ ( ord_less_eq_num @ A5 @ B5 ) @ B5 @ A5 ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3333_max__def__raw,axiom,
% 16.19/8.94 ( ord_max_nat
% 16.19/8.94 = ( ^ [A5: nat,B5: nat] : ( if_nat @ ( ord_less_eq_nat @ A5 @ B5 ) @ B5 @ A5 ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3334_max__def__raw,axiom,
% 16.19/8.94 ( ord_max_int
% 16.19/8.94 = ( ^ [A5: int,B5: int] : ( if_int @ ( ord_less_eq_int @ A5 @ B5 ) @ B5 @ A5 ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3335_div__less__mono,axiom,
% 16.19/8.94 ! [A2: nat,B3: nat,N2: nat] :
% 16.19/8.94 ( ( ord_less_nat @ A2 @ B3 )
% 16.19/8.94 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.94 => ( ( ( modulo_modulo_nat @ A2 @ N2 )
% 16.19/8.94 = zero_zero_nat )
% 16.19/8.94 => ( ( ( modulo_modulo_nat @ B3 @ N2 )
% 16.19/8.94 = zero_zero_nat )
% 16.19/8.94 => ( ord_less_nat @ ( divide_divide_nat @ A2 @ N2 ) @ ( divide_divide_nat @ B3 @ N2 ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3336_set__bit__Suc,axiom,
% 16.19/8.94 ! [N2: nat,A: code_integer] :
% 16.19/8.94 ( ( bit_se2793503036327961859nteger @ ( suc @ N2 ) @ A )
% 16.19/8.94 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3337_set__bit__Suc,axiom,
% 16.19/8.94 ! [N2: nat,A: int] :
% 16.19/8.94 ( ( bit_se7879613467334960850it_int @ ( suc @ N2 ) @ A )
% 16.19/8.94 = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3338_set__bit__Suc,axiom,
% 16.19/8.94 ! [N2: nat,A: nat] :
% 16.19/8.94 ( ( bit_se7882103937844011126it_nat @ ( suc @ N2 ) @ A )
% 16.19/8.94 = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3339_vebt__insert_Opelims,axiom,
% 16.19/8.94 ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
% 16.19/8.94 ( ( ( vEBT_vebt_insert @ X @ Xa2 )
% 16.19/8.94 = Y )
% 16.19/8.94 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 16.19/8.94 => ( ! [A3: $o,B2: $o] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/8.94 => ( ( ( ( Xa2 = zero_zero_nat )
% 16.19/8.94 => ( Y
% 16.19/8.94 = ( vEBT_Leaf @ $true @ B2 ) ) )
% 16.19/8.94 & ( ( Xa2 != zero_zero_nat )
% 16.19/8.94 => ( ( ( Xa2 = one_one_nat )
% 16.19/8.94 => ( Y
% 16.19/8.94 = ( vEBT_Leaf @ A3 @ $true ) ) )
% 16.19/8.94 & ( ( Xa2 != one_one_nat )
% 16.19/8.94 => ( Y
% 16.19/8.94 = ( vEBT_Leaf @ A3 @ B2 ) ) ) ) ) )
% 16.19/8.94 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa2 ) ) ) )
% 16.19/8.94 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) )
% 16.19/8.94 => ( ( Y
% 16.19/8.94 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) )
% 16.19/8.94 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts @ S2 ) @ Xa2 ) ) ) )
% 16.19/8.94 => ( ! [Info2: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) )
% 16.19/8.94 => ( ( Y
% 16.19/8.94 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) )
% 16.19/8.94 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts @ S2 ) @ Xa2 ) ) ) )
% 16.19/8.94 => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
% 16.19/8.94 => ( ( Y
% 16.19/8.94 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
% 16.19/8.94 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) )
% 16.19/8.94 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 16.19/8.94 => ( ( Y
% 16.19/8.94 = ( if_VEBT_VEBT
% 16.19/8.94 @ ( ( 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 @ TreeList3 ) )
% 16.19/8.94 & ~ ( ( Xa2 = Mi2 )
% 16.19/8.94 | ( Xa2 = Ma2 ) ) )
% 16.19/8.94 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 @ TreeList3 @ ( 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 ) )
% 16.19/8.94 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) ) )
% 16.19/8.94 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3340_vebt__member_Opelims_I1_J,axiom,
% 16.19/8.94 ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 16.19/8.94 ( ( ( vEBT_vebt_member @ X @ Xa2 )
% 16.19/8.94 = Y )
% 16.19/8.94 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 16.19/8.94 => ( ! [A3: $o,B2: $o] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/8.94 => ( ( Y
% 16.19/8.94 = ( ( ( Xa2 = zero_zero_nat )
% 16.19/8.94 => A3 )
% 16.19/8.94 & ( ( Xa2 != zero_zero_nat )
% 16.19/8.94 => ( ( ( Xa2 = one_one_nat )
% 16.19/8.94 => B2 )
% 16.19/8.94 & ( Xa2 = one_one_nat ) ) ) ) )
% 16.19/8.94 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa2 ) ) ) )
% 16.19/8.94 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 16.19/8.94 => ( ~ Y
% 16.19/8.94 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
% 16.19/8.94 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 16.19/8.94 => ( ~ Y
% 16.19/8.94 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
% 16.19/8.94 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 16.19/8.94 => ( ~ Y
% 16.19/8.94 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
% 16.19/8.94 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 16.19/8.94 => ( ( Y
% 16.19/8.94 = ( ( Xa2 != Mi2 )
% 16.19/8.94 => ( ( Xa2 != Ma2 )
% 16.19/8.94 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 16.19/8.94 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 16.19/8.94 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 16.19/8.94 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 16.19/8.94 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.94 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 16.19/8.94 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) )
% 16.19/8.94 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3341_vebt__member_Opelims_I3_J,axiom,
% 16.19/8.94 ! [X: vEBT_VEBT,Xa2: nat] :
% 16.19/8.94 ( ~ ( vEBT_vebt_member @ X @ Xa2 )
% 16.19/8.94 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 16.19/8.94 => ( ! [A3: $o,B2: $o] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/8.94 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa2 ) )
% 16.19/8.94 => ( ( ( Xa2 = zero_zero_nat )
% 16.19/8.94 => A3 )
% 16.19/8.94 & ( ( Xa2 != zero_zero_nat )
% 16.19/8.94 => ( ( ( Xa2 = one_one_nat )
% 16.19/8.94 => B2 )
% 16.19/8.94 & ( Xa2 = one_one_nat ) ) ) ) ) )
% 16.19/8.94 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 16.19/8.94 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) )
% 16.19/8.94 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 16.19/8.94 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) )
% 16.19/8.94 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 16.19/8.94 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
% 16.19/8.94 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 16.19/8.94 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) )
% 16.19/8.94 => ( ( Xa2 != Mi2 )
% 16.19/8.94 => ( ( Xa2 != Ma2 )
% 16.19/8.94 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 16.19/8.94 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 16.19/8.94 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 16.19/8.94 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 16.19/8.94 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.94 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 16.19/8.94 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3342_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
% 16.19/8.94 ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 16.19/8.94 ( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
% 16.19/8.94 = Y )
% 16.19/8.94 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 16.19/8.94 => ( ! [A3: $o,B2: $o] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/8.94 => ( ( Y
% 16.19/8.94 = ( ( ( Xa2 = zero_zero_nat )
% 16.19/8.94 => A3 )
% 16.19/8.94 & ( ( Xa2 != zero_zero_nat )
% 16.19/8.94 => ( ( ( Xa2 = one_one_nat )
% 16.19/8.94 => B2 )
% 16.19/8.94 & ( Xa2 = one_one_nat ) ) ) ) )
% 16.19/8.94 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa2 ) ) ) )
% 16.19/8.94 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 16.19/8.94 => ( ~ Y
% 16.19/8.94 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
% 16.19/8.94 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
% 16.19/8.94 => ( ( Y
% 16.19/8.94 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.94 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 16.19/8.94 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 16.19/8.94 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) @ Xa2 ) ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3343_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
% 16.19/8.94 ! [X: vEBT_VEBT,Xa2: nat] :
% 16.19/8.94 ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
% 16.19/8.94 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 16.19/8.94 => ( ! [A3: $o,B2: $o] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/8.94 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa2 ) )
% 16.19/8.94 => ~ ( ( ( Xa2 = zero_zero_nat )
% 16.19/8.94 => A3 )
% 16.19/8.94 & ( ( Xa2 != zero_zero_nat )
% 16.19/8.94 => ( ( ( Xa2 = one_one_nat )
% 16.19/8.94 => B2 )
% 16.19/8.94 & ( Xa2 = one_one_nat ) ) ) ) ) )
% 16.19/8.94 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
% 16.19/8.94 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) @ Xa2 ) )
% 16.19/8.94 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.94 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 16.19/8.94 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3344_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
% 16.19/8.94 ! [X: vEBT_VEBT,Xa2: nat] :
% 16.19/8.94 ( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
% 16.19/8.94 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 16.19/8.94 => ( ! [A3: $o,B2: $o] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/8.94 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa2 ) )
% 16.19/8.94 => ( ( ( Xa2 = zero_zero_nat )
% 16.19/8.94 => A3 )
% 16.19/8.94 & ( ( Xa2 != zero_zero_nat )
% 16.19/8.94 => ( ( ( Xa2 = one_one_nat )
% 16.19/8.94 => B2 )
% 16.19/8.94 & ( Xa2 = one_one_nat ) ) ) ) ) )
% 16.19/8.94 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 16.19/8.94 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) )
% 16.19/8.94 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
% 16.19/8.94 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) @ Xa2 ) )
% 16.19/8.94 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.94 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 16.19/8.94 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3345_vebt__member_Opelims_I2_J,axiom,
% 16.19/8.94 ! [X: vEBT_VEBT,Xa2: nat] :
% 16.19/8.94 ( ( vEBT_vebt_member @ X @ Xa2 )
% 16.19/8.94 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 16.19/8.94 => ( ! [A3: $o,B2: $o] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/8.94 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B2 ) @ Xa2 ) )
% 16.19/8.94 => ~ ( ( ( Xa2 = zero_zero_nat )
% 16.19/8.94 => A3 )
% 16.19/8.94 & ( ( Xa2 != zero_zero_nat )
% 16.19/8.94 => ( ( ( Xa2 = one_one_nat )
% 16.19/8.94 => B2 )
% 16.19/8.94 & ( Xa2 = one_one_nat ) ) ) ) ) )
% 16.19/8.94 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 16.19/8.94 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) )
% 16.19/8.94 => ~ ( ( Xa2 != Mi2 )
% 16.19/8.94 => ( ( Xa2 != Ma2 )
% 16.19/8.94 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 16.19/8.94 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 16.19/8.94 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 16.19/8.94 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 16.19/8.94 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.94 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 16.19/8.94 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3346_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
% 16.19/8.94 ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 16.19/8.94 ( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
% 16.19/8.94 = Y )
% 16.19/8.94 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 16.19/8.94 => ( ! [Uu2: $o,Uv2: $o] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 16.19/8.94 => ( ~ Y
% 16.19/8.94 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
% 16.19/8.94 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 16.19/8.94 => ( ~ Y
% 16.19/8.94 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) ) )
% 16.19/8.94 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 16.19/8.94 => ( ( Y
% 16.19/8.94 = ( ( Xa2 = Mi2 )
% 16.19/8.94 | ( Xa2 = Ma2 ) ) )
% 16.19/8.94 => ~ ( 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 ) ) ) )
% 16.19/8.94 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 16.19/8.94 => ( ( Y
% 16.19/8.94 = ( ( Xa2 = Mi2 )
% 16.19/8.94 | ( Xa2 = Ma2 )
% 16.19/8.94 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.94 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 16.19/8.94 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 16.19/8.94 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) ) ) )
% 16.19/8.94 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 16.19/8.94 => ( ( Y
% 16.19/8.94 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.94 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 16.19/8.94 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 16.19/8.94 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3347_set__bit__nonnegative__int__iff,axiom,
% 16.19/8.94 ! [N2: nat,K: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N2 @ K ) )
% 16.19/8.94 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3348_unset__bit__nonnegative__int__iff,axiom,
% 16.19/8.94 ! [N2: nat,K: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N2 @ K ) )
% 16.19/8.94 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3349_unset__bit__negative__int__iff,axiom,
% 16.19/8.94 ! [N2: nat,K: int] :
% 16.19/8.94 ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N2 @ K ) @ zero_zero_int )
% 16.19/8.94 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3350_set__bit__negative__int__iff,axiom,
% 16.19/8.94 ! [N2: nat,K: int] :
% 16.19/8.94 ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N2 @ K ) @ zero_zero_int )
% 16.19/8.94 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3351_zle__add1__eq__le,axiom,
% 16.19/8.94 ! [W: int,Z: int] :
% 16.19/8.94 ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 16.19/8.94 = ( ord_less_eq_int @ W @ Z ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3352_zle__diff1__eq,axiom,
% 16.19/8.94 ! [W: int,Z: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
% 16.19/8.94 = ( ord_less_int @ W @ Z ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3353_mod__neg__neg__trivial,axiom,
% 16.19/8.94 ! [K: int,L2: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 16.19/8.94 => ( ( ord_less_int @ L2 @ K )
% 16.19/8.94 => ( ( modulo_modulo_int @ K @ L2 )
% 16.19/8.94 = K ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3354_mod__pos__pos__trivial,axiom,
% 16.19/8.94 ! [K: int,L2: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 16.19/8.94 => ( ( ord_less_int @ K @ L2 )
% 16.19/8.94 => ( ( modulo_modulo_int @ K @ L2 )
% 16.19/8.94 = K ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3355_add1__zle__eq,axiom,
% 16.19/8.94 ! [W: int,Z: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
% 16.19/8.94 = ( ord_less_int @ W @ Z ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3356_le__imp__0__less,axiom,
% 16.19/8.94 ! [Z: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 16.19/8.94 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3357_split__zmod,axiom,
% 16.19/8.94 ! [P: int > $o,N2: int,K: int] :
% 16.19/8.94 ( ( P @ ( modulo_modulo_int @ N2 @ K ) )
% 16.19/8.94 = ( ( ( K = zero_zero_int )
% 16.19/8.94 => ( P @ N2 ) )
% 16.19/8.94 & ( ( ord_less_int @ zero_zero_int @ K )
% 16.19/8.94 => ! [I3: int,J3: int] :
% 16.19/8.94 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 16.19/8.94 & ( ord_less_int @ J3 @ K )
% 16.19/8.94 & ( N2
% 16.19/8.94 = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
% 16.19/8.94 => ( P @ J3 ) ) )
% 16.19/8.94 & ( ( ord_less_int @ K @ zero_zero_int )
% 16.19/8.94 => ! [I3: int,J3: int] :
% 16.19/8.94 ( ( ( ord_less_int @ K @ J3 )
% 16.19/8.94 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 16.19/8.94 & ( N2
% 16.19/8.94 = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
% 16.19/8.94 => ( P @ J3 ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3358_mod__pos__geq,axiom,
% 16.19/8.94 ! [L2: int,K: int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ L2 )
% 16.19/8.94 => ( ( ord_less_eq_int @ L2 @ K )
% 16.19/8.94 => ( ( modulo_modulo_int @ K @ L2 )
% 16.19/8.94 = ( modulo_modulo_int @ ( minus_minus_int @ K @ L2 ) @ L2 ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3359_q__pos__lemma,axiom,
% 16.19/8.94 ! [B6: int,Q5: int,R4: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q5 ) @ R4 ) )
% 16.19/8.94 => ( ( ord_less_int @ R4 @ B6 )
% 16.19/8.94 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 16.19/8.94 => ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3360_int__less__induct,axiom,
% 16.19/8.94 ! [I2: int,K: int,P: int > $o] :
% 16.19/8.94 ( ( ord_less_int @ I2 @ K )
% 16.19/8.94 => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
% 16.19/8.94 => ( ! [I4: int] :
% 16.19/8.94 ( ( ord_less_int @ I4 @ K )
% 16.19/8.94 => ( ( P @ I4 )
% 16.19/8.94 => ( P @ ( minus_minus_int @ I4 @ one_one_int ) ) ) )
% 16.19/8.94 => ( P @ I2 ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3361_neg__mod__conj,axiom,
% 16.19/8.94 ! [B: int,A: int] :
% 16.19/8.94 ( ( ord_less_int @ B @ zero_zero_int )
% 16.19/8.94 => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
% 16.19/8.94 & ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3362_pos__mod__conj,axiom,
% 16.19/8.94 ! [B: int,A: int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ B )
% 16.19/8.94 => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
% 16.19/8.94 & ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3363_int__mod__neg__eq,axiom,
% 16.19/8.94 ! [A: int,B: int,Q2: int,R: int] :
% 16.19/8.94 ( ( A
% 16.19/8.94 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 16.19/8.94 => ( ( ord_less_eq_int @ R @ zero_zero_int )
% 16.19/8.94 => ( ( ord_less_int @ B @ R )
% 16.19/8.94 => ( ( modulo_modulo_int @ A @ B )
% 16.19/8.94 = R ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3364_int__mod__pos__eq,axiom,
% 16.19/8.94 ! [A: int,B: int,Q2: int,R: int] :
% 16.19/8.94 ( ( A
% 16.19/8.94 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 16.19/8.94 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 16.19/8.94 => ( ( ord_less_int @ R @ B )
% 16.19/8.94 => ( ( modulo_modulo_int @ A @ B )
% 16.19/8.94 = R ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3365_zless__imp__add1__zle,axiom,
% 16.19/8.94 ! [W: int,Z: int] :
% 16.19/8.94 ( ( ord_less_int @ W @ Z )
% 16.19/8.94 => ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3366_zdiv__mono2__lemma,axiom,
% 16.19/8.94 ! [B: int,Q2: int,R: int,B6: int,Q5: int,R4: int] :
% 16.19/8.94 ( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R )
% 16.19/8.94 = ( plus_plus_int @ ( times_times_int @ B6 @ Q5 ) @ R4 ) )
% 16.19/8.94 => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q5 ) @ R4 ) )
% 16.19/8.94 => ( ( ord_less_int @ R4 @ B6 )
% 16.19/8.94 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 16.19/8.94 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 16.19/8.94 => ( ( ord_less_eq_int @ B6 @ B )
% 16.19/8.94 => ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3367_zmod__trivial__iff,axiom,
% 16.19/8.94 ! [I2: int,K: int] :
% 16.19/8.94 ( ( ( modulo_modulo_int @ I2 @ K )
% 16.19/8.94 = I2 )
% 16.19/8.94 = ( ( K = zero_zero_int )
% 16.19/8.94 | ( ( ord_less_eq_int @ zero_zero_int @ I2 )
% 16.19/8.94 & ( ord_less_int @ I2 @ K ) )
% 16.19/8.94 | ( ( ord_less_eq_int @ I2 @ zero_zero_int )
% 16.19/8.94 & ( ord_less_int @ K @ I2 ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3368_zdiv__mono2__neg__lemma,axiom,
% 16.19/8.94 ! [B: int,Q2: int,R: int,B6: int,Q5: int,R4: int] :
% 16.19/8.94 ( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R )
% 16.19/8.94 = ( plus_plus_int @ ( times_times_int @ B6 @ Q5 ) @ R4 ) )
% 16.19/8.94 => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q5 ) @ R4 ) @ zero_zero_int )
% 16.19/8.94 => ( ( ord_less_int @ R @ B )
% 16.19/8.94 => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 16.19/8.94 => ( ( ord_less_int @ zero_zero_int @ B6 )
% 16.19/8.94 => ( ( ord_less_eq_int @ B6 @ B )
% 16.19/8.94 => ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3369_int__one__le__iff__zero__less,axiom,
% 16.19/8.94 ! [Z: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ one_one_int @ Z )
% 16.19/8.94 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3370_unique__quotient__lemma,axiom,
% 16.19/8.94 ! [B: int,Q5: int,R4: int,Q2: int,R: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 16.19/8.94 => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 16.19/8.94 => ( ( ord_less_int @ R4 @ B )
% 16.19/8.94 => ( ( ord_less_int @ R @ B )
% 16.19/8.94 => ( ord_less_eq_int @ Q5 @ Q2 ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3371_neg__mod__sign,axiom,
% 16.19/8.94 ! [L2: int,K: int] :
% 16.19/8.94 ( ( ord_less_int @ L2 @ zero_zero_int )
% 16.19/8.94 => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L2 ) @ zero_zero_int ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3372_Euclidean__Division_Opos__mod__sign,axiom,
% 16.19/8.94 ! [L2: int,K: int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ L2 )
% 16.19/8.94 => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L2 ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3373_unique__quotient__lemma__neg,axiom,
% 16.19/8.94 ! [B: int,Q5: int,R4: int,Q2: int,R: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 16.19/8.94 => ( ( ord_less_eq_int @ R @ zero_zero_int )
% 16.19/8.94 => ( ( ord_less_int @ B @ R )
% 16.19/8.94 => ( ( ord_less_int @ B @ R4 )
% 16.19/8.94 => ( ord_less_eq_int @ Q2 @ Q5 ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3374_mod__pos__neg__trivial,axiom,
% 16.19/8.94 ! [K: int,L2: int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ K )
% 16.19/8.94 => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L2 ) @ zero_zero_int )
% 16.19/8.94 => ( ( modulo_modulo_int @ K @ L2 )
% 16.19/8.94 = ( plus_plus_int @ K @ L2 ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3375_minus__int__code_I1_J,axiom,
% 16.19/8.94 ! [K: int] :
% 16.19/8.94 ( ( minus_minus_int @ K @ zero_zero_int )
% 16.19/8.94 = K ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3376_Euclidean__Division_Opos__mod__bound,axiom,
% 16.19/8.94 ! [L2: int,K: int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ L2 )
% 16.19/8.94 => ( ord_less_int @ ( modulo_modulo_int @ K @ L2 ) @ L2 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3377_neg__mod__bound,axiom,
% 16.19/8.94 ! [L2: int,K: int] :
% 16.19/8.94 ( ( ord_less_int @ L2 @ zero_zero_int )
% 16.19/8.94 => ( ord_less_int @ L2 @ ( modulo_modulo_int @ K @ L2 ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3378_less__int__code_I1_J,axiom,
% 16.19/8.94 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3379_zmult__zless__mono2,axiom,
% 16.19/8.94 ! [I2: int,J: int,K: int] :
% 16.19/8.94 ( ( ord_less_int @ I2 @ J )
% 16.19/8.94 => ( ( ord_less_int @ zero_zero_int @ K )
% 16.19/8.94 => ( ord_less_int @ ( times_times_int @ K @ I2 ) @ ( times_times_int @ K @ J ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3380_pos__zmult__eq__1__iff,axiom,
% 16.19/8.94 ! [M: int,N2: int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ M )
% 16.19/8.94 => ( ( ( times_times_int @ M @ N2 )
% 16.19/8.94 = one_one_int )
% 16.19/8.94 = ( ( M = one_one_int )
% 16.19/8.94 & ( N2 = one_one_int ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3381_odd__less__0__iff,axiom,
% 16.19/8.94 ! [Z: int] :
% 16.19/8.94 ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
% 16.19/8.94 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3382_zless__add1__eq,axiom,
% 16.19/8.94 ! [W: int,Z: int] :
% 16.19/8.94 ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 16.19/8.94 = ( ( ord_less_int @ W @ Z )
% 16.19/8.94 | ( W = Z ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3383_int__gr__induct,axiom,
% 16.19/8.94 ! [K: int,I2: int,P: int > $o] :
% 16.19/8.94 ( ( ord_less_int @ K @ I2 )
% 16.19/8.94 => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
% 16.19/8.94 => ( ! [I4: int] :
% 16.19/8.94 ( ( ord_less_int @ K @ I4 )
% 16.19/8.94 => ( ( P @ I4 )
% 16.19/8.94 => ( P @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
% 16.19/8.94 => ( P @ I2 ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3384_zero__one__enat__neq_I1_J,axiom,
% 16.19/8.94 zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3385_odd__nonzero,axiom,
% 16.19/8.94 ! [Z: int] :
% 16.19/8.94 ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
% 16.19/8.94 != zero_zero_int ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3386_set__bit__greater__eq,axiom,
% 16.19/8.94 ! [K: int,N2: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N2 @ K ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3387_unset__bit__less__eq,axiom,
% 16.19/8.94 ! [N2: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N2 @ K ) @ K ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3388_int__distrib_I4_J,axiom,
% 16.19/8.94 ! [W: int,Z1: int,Z22: int] :
% 16.19/8.94 ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
% 16.19/8.94 = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3389_int__distrib_I3_J,axiom,
% 16.19/8.94 ! [Z1: int,Z22: int,W: int] :
% 16.19/8.94 ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
% 16.19/8.94 = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3390_int__le__induct,axiom,
% 16.19/8.94 ! [I2: int,K: int,P: int > $o] :
% 16.19/8.94 ( ( ord_less_eq_int @ I2 @ K )
% 16.19/8.94 => ( ( P @ K )
% 16.19/8.94 => ( ! [I4: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ I4 @ K )
% 16.19/8.94 => ( ( P @ I4 )
% 16.19/8.94 => ( P @ ( minus_minus_int @ I4 @ one_one_int ) ) ) )
% 16.19/8.94 => ( P @ I2 ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3391_int__ge__induct,axiom,
% 16.19/8.94 ! [K: int,I2: int,P: int > $o] :
% 16.19/8.94 ( ( ord_less_eq_int @ K @ I2 )
% 16.19/8.94 => ( ( P @ K )
% 16.19/8.94 => ( ! [I4: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ K @ I4 )
% 16.19/8.94 => ( ( P @ I4 )
% 16.19/8.94 => ( P @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
% 16.19/8.94 => ( P @ I2 ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3392_int__induct,axiom,
% 16.19/8.94 ! [P: int > $o,K: int,I2: int] :
% 16.19/8.94 ( ( P @ K )
% 16.19/8.94 => ( ! [I4: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ K @ I4 )
% 16.19/8.94 => ( ( P @ I4 )
% 16.19/8.94 => ( P @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
% 16.19/8.94 => ( ! [I4: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ I4 @ K )
% 16.19/8.94 => ( ( P @ I4 )
% 16.19/8.94 => ( P @ ( minus_minus_int @ I4 @ one_one_int ) ) ) )
% 16.19/8.94 => ( P @ I2 ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3393_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
% 16.19/8.94 ! [X: vEBT_VEBT,Xa2: nat] :
% 16.19/8.94 ( ( vEBT_VEBT_membermima @ X @ Xa2 )
% 16.19/8.94 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 16.19/8.94 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 16.19/8.94 => ( ( 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 ) )
% 16.19/8.94 => ~ ( ( Xa2 = Mi2 )
% 16.19/8.94 | ( Xa2 = Ma2 ) ) ) )
% 16.19/8.94 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 16.19/8.94 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
% 16.19/8.94 => ~ ( ( Xa2 = Mi2 )
% 16.19/8.94 | ( Xa2 = Ma2 )
% 16.19/8.94 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.94 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 16.19/8.94 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 16.19/8.94 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 16.19/8.94 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) )
% 16.19/8.94 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.94 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 16.19/8.94 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3394_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
% 16.19/8.94 ! [X: vEBT_VEBT,Xa2: nat] :
% 16.19/8.94 ( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
% 16.19/8.94 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 16.19/8.94 => ( ! [Uu2: $o,Uv2: $o] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 16.19/8.94 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) )
% 16.19/8.94 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 16.19/8.94 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) )
% 16.19/8.94 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 16.19/8.94 => ( ( 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 ) )
% 16.19/8.94 => ( ( Xa2 = Mi2 )
% 16.19/8.94 | ( Xa2 = Ma2 ) ) ) )
% 16.19/8.94 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 16.19/8.94 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
% 16.19/8.94 => ( ( Xa2 = Mi2 )
% 16.19/8.94 | ( Xa2 = Ma2 )
% 16.19/8.94 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.94 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 16.19/8.94 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 16.19/8.94 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 16.19/8.94 ( ( X
% 16.19/8.94 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 16.19/8.94 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa2 ) )
% 16.19/8.94 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 16.19/8.94 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( 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 ) ) ) ) ) )
% 16.19/8.94 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3395_atLeastatMost__empty,axiom,
% 16.19/8.94 ! [B: rat,A: rat] :
% 16.19/8.94 ( ( ord_less_rat @ B @ A )
% 16.19/8.94 => ( ( set_or633870826150836451st_rat @ A @ B )
% 16.19/8.94 = bot_bot_set_rat ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3396_atLeastatMost__empty,axiom,
% 16.19/8.94 ! [B: num,A: num] :
% 16.19/8.94 ( ( ord_less_num @ B @ A )
% 16.19/8.94 => ( ( set_or7049704709247886629st_num @ A @ B )
% 16.19/8.94 = bot_bot_set_num ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3397_atLeastatMost__empty,axiom,
% 16.19/8.94 ! [B: nat,A: nat] :
% 16.19/8.94 ( ( ord_less_nat @ B @ A )
% 16.19/8.94 => ( ( set_or1269000886237332187st_nat @ A @ B )
% 16.19/8.94 = bot_bot_set_nat ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3398_atLeastatMost__empty,axiom,
% 16.19/8.94 ! [B: int,A: int] :
% 16.19/8.94 ( ( ord_less_int @ B @ A )
% 16.19/8.94 => ( ( set_or1266510415728281911st_int @ A @ B )
% 16.19/8.94 = bot_bot_set_int ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3399_atLeastatMost__empty,axiom,
% 16.19/8.94 ! [B: real,A: real] :
% 16.19/8.94 ( ( ord_less_real @ B @ A )
% 16.19/8.94 => ( ( set_or1222579329274155063t_real @ A @ B )
% 16.19/8.94 = bot_bot_set_real ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3400_atLeastatMost__subset__iff,axiom,
% 16.19/8.94 ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
% 16.19/8.94 ( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
% 16.19/8.94 = ( ~ ( ord_less_eq_set_nat @ A @ B )
% 16.19/8.94 | ( ( ord_less_eq_set_nat @ C @ A )
% 16.19/8.94 & ( ord_less_eq_set_nat @ B @ D ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3401_atLeastatMost__subset__iff,axiom,
% 16.19/8.94 ! [A: rat,B: rat,C: rat,D: rat] :
% 16.19/8.94 ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 16.19/8.94 = ( ~ ( ord_less_eq_rat @ A @ B )
% 16.19/8.94 | ( ( ord_less_eq_rat @ C @ A )
% 16.19/8.94 & ( ord_less_eq_rat @ B @ D ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3402_atLeastatMost__subset__iff,axiom,
% 16.19/8.94 ! [A: num,B: num,C: num,D: num] :
% 16.19/8.94 ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 16.19/8.94 = ( ~ ( ord_less_eq_num @ A @ B )
% 16.19/8.94 | ( ( ord_less_eq_num @ C @ A )
% 16.19/8.94 & ( ord_less_eq_num @ B @ D ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3403_atLeastatMost__subset__iff,axiom,
% 16.19/8.94 ! [A: nat,B: nat,C: nat,D: nat] :
% 16.19/8.94 ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 16.19/8.94 = ( ~ ( ord_less_eq_nat @ A @ B )
% 16.19/8.94 | ( ( ord_less_eq_nat @ C @ A )
% 16.19/8.94 & ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3404_atLeastatMost__subset__iff,axiom,
% 16.19/8.94 ! [A: int,B: int,C: int,D: int] :
% 16.19/8.94 ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 16.19/8.94 = ( ~ ( ord_less_eq_int @ A @ B )
% 16.19/8.94 | ( ( ord_less_eq_int @ C @ A )
% 16.19/8.94 & ( ord_less_eq_int @ B @ D ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3405_atLeastatMost__subset__iff,axiom,
% 16.19/8.94 ! [A: real,B: real,C: real,D: real] :
% 16.19/8.94 ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 16.19/8.94 = ( ~ ( ord_less_eq_real @ A @ B )
% 16.19/8.94 | ( ( ord_less_eq_real @ C @ A )
% 16.19/8.94 & ( ord_less_eq_real @ B @ D ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3406_atLeastatMost__empty__iff2,axiom,
% 16.19/8.94 ! [A: set_nat,B: set_nat] :
% 16.19/8.94 ( ( bot_bot_set_set_nat
% 16.19/8.94 = ( set_or4548717258645045905et_nat @ A @ B ) )
% 16.19/8.94 = ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3407_atLeastatMost__empty__iff2,axiom,
% 16.19/8.94 ! [A: rat,B: rat] :
% 16.19/8.94 ( ( bot_bot_set_rat
% 16.19/8.94 = ( set_or633870826150836451st_rat @ A @ B ) )
% 16.19/8.94 = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3408_atLeastatMost__empty__iff2,axiom,
% 16.19/8.94 ! [A: num,B: num] :
% 16.19/8.94 ( ( bot_bot_set_num
% 16.19/8.94 = ( set_or7049704709247886629st_num @ A @ B ) )
% 16.19/8.94 = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3409_atLeastatMost__empty__iff2,axiom,
% 16.19/8.94 ! [A: nat,B: nat] :
% 16.19/8.94 ( ( bot_bot_set_nat
% 16.19/8.94 = ( set_or1269000886237332187st_nat @ A @ B ) )
% 16.19/8.94 = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3410_atLeastatMost__empty__iff2,axiom,
% 16.19/8.94 ! [A: int,B: int] :
% 16.19/8.94 ( ( bot_bot_set_int
% 16.19/8.94 = ( set_or1266510415728281911st_int @ A @ B ) )
% 16.19/8.94 = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3411_atLeastatMost__empty__iff2,axiom,
% 16.19/8.94 ! [A: real,B: real] :
% 16.19/8.94 ( ( bot_bot_set_real
% 16.19/8.94 = ( set_or1222579329274155063t_real @ A @ B ) )
% 16.19/8.94 = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3412_atLeastatMost__empty__iff,axiom,
% 16.19/8.94 ! [A: set_nat,B: set_nat] :
% 16.19/8.94 ( ( ( set_or4548717258645045905et_nat @ A @ B )
% 16.19/8.94 = bot_bot_set_set_nat )
% 16.19/8.94 = ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3413_atLeastatMost__empty__iff,axiom,
% 16.19/8.94 ! [A: rat,B: rat] :
% 16.19/8.94 ( ( ( set_or633870826150836451st_rat @ A @ B )
% 16.19/8.94 = bot_bot_set_rat )
% 16.19/8.94 = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3414_atLeastatMost__empty__iff,axiom,
% 16.19/8.94 ! [A: num,B: num] :
% 16.19/8.94 ( ( ( set_or7049704709247886629st_num @ A @ B )
% 16.19/8.94 = bot_bot_set_num )
% 16.19/8.94 = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3415_atLeastatMost__empty__iff,axiom,
% 16.19/8.94 ! [A: nat,B: nat] :
% 16.19/8.94 ( ( ( set_or1269000886237332187st_nat @ A @ B )
% 16.19/8.94 = bot_bot_set_nat )
% 16.19/8.94 = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3416_atLeastatMost__empty__iff,axiom,
% 16.19/8.94 ! [A: int,B: int] :
% 16.19/8.94 ( ( ( set_or1266510415728281911st_int @ A @ B )
% 16.19/8.94 = bot_bot_set_int )
% 16.19/8.94 = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3417_atLeastatMost__empty__iff,axiom,
% 16.19/8.94 ! [A: real,B: real] :
% 16.19/8.94 ( ( ( set_or1222579329274155063t_real @ A @ B )
% 16.19/8.94 = bot_bot_set_real )
% 16.19/8.94 = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3418_decr__mult__lemma,axiom,
% 16.19/8.94 ! [D: int,P: int > $o,K: int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ D )
% 16.19/8.94 => ( ! [X4: int] :
% 16.19/8.94 ( ( P @ X4 )
% 16.19/8.94 => ( P @ ( minus_minus_int @ X4 @ D ) ) )
% 16.19/8.94 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 16.19/8.94 => ! [X5: int] :
% 16.19/8.94 ( ( P @ X5 )
% 16.19/8.94 => ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3419_incr__mult__lemma,axiom,
% 16.19/8.94 ! [D: int,P: int > $o,K: int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ D )
% 16.19/8.94 => ( ! [X4: int] :
% 16.19/8.94 ( ( P @ X4 )
% 16.19/8.94 => ( P @ ( plus_plus_int @ X4 @ D ) ) )
% 16.19/8.94 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 16.19/8.94 => ! [X5: int] :
% 16.19/8.94 ( ( P @ X5 )
% 16.19/8.94 => ( P @ ( plus_plus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3420_Icc__eq__Icc,axiom,
% 16.19/8.94 ! [L2: set_nat,H2: set_nat,L3: set_nat,H3: set_nat] :
% 16.19/8.94 ( ( ( set_or4548717258645045905et_nat @ L2 @ H2 )
% 16.19/8.94 = ( set_or4548717258645045905et_nat @ L3 @ H3 ) )
% 16.19/8.94 = ( ( ( L2 = L3 )
% 16.19/8.94 & ( H2 = H3 ) )
% 16.19/8.94 | ( ~ ( ord_less_eq_set_nat @ L2 @ H2 )
% 16.19/8.94 & ~ ( ord_less_eq_set_nat @ L3 @ H3 ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3421_Icc__eq__Icc,axiom,
% 16.19/8.94 ! [L2: rat,H2: rat,L3: rat,H3: rat] :
% 16.19/8.94 ( ( ( set_or633870826150836451st_rat @ L2 @ H2 )
% 16.19/8.94 = ( set_or633870826150836451st_rat @ L3 @ H3 ) )
% 16.19/8.94 = ( ( ( L2 = L3 )
% 16.19/8.94 & ( H2 = H3 ) )
% 16.19/8.94 | ( ~ ( ord_less_eq_rat @ L2 @ H2 )
% 16.19/8.94 & ~ ( ord_less_eq_rat @ L3 @ H3 ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3422_Icc__eq__Icc,axiom,
% 16.19/8.94 ! [L2: num,H2: num,L3: num,H3: num] :
% 16.19/8.94 ( ( ( set_or7049704709247886629st_num @ L2 @ H2 )
% 16.19/8.94 = ( set_or7049704709247886629st_num @ L3 @ H3 ) )
% 16.19/8.94 = ( ( ( L2 = L3 )
% 16.19/8.94 & ( H2 = H3 ) )
% 16.19/8.94 | ( ~ ( ord_less_eq_num @ L2 @ H2 )
% 16.19/8.94 & ~ ( ord_less_eq_num @ L3 @ H3 ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3423_Icc__eq__Icc,axiom,
% 16.19/8.94 ! [L2: nat,H2: nat,L3: nat,H3: nat] :
% 16.19/8.94 ( ( ( set_or1269000886237332187st_nat @ L2 @ H2 )
% 16.19/8.94 = ( set_or1269000886237332187st_nat @ L3 @ H3 ) )
% 16.19/8.94 = ( ( ( L2 = L3 )
% 16.19/8.94 & ( H2 = H3 ) )
% 16.19/8.94 | ( ~ ( ord_less_eq_nat @ L2 @ H2 )
% 16.19/8.94 & ~ ( ord_less_eq_nat @ L3 @ H3 ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3424_Icc__eq__Icc,axiom,
% 16.19/8.94 ! [L2: int,H2: int,L3: int,H3: int] :
% 16.19/8.94 ( ( ( set_or1266510415728281911st_int @ L2 @ H2 )
% 16.19/8.94 = ( set_or1266510415728281911st_int @ L3 @ H3 ) )
% 16.19/8.94 = ( ( ( L2 = L3 )
% 16.19/8.94 & ( H2 = H3 ) )
% 16.19/8.94 | ( ~ ( ord_less_eq_int @ L2 @ H2 )
% 16.19/8.94 & ~ ( ord_less_eq_int @ L3 @ H3 ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3425_Icc__eq__Icc,axiom,
% 16.19/8.94 ! [L2: real,H2: real,L3: real,H3: real] :
% 16.19/8.94 ( ( ( set_or1222579329274155063t_real @ L2 @ H2 )
% 16.19/8.94 = ( set_or1222579329274155063t_real @ L3 @ H3 ) )
% 16.19/8.94 = ( ( ( L2 = L3 )
% 16.19/8.94 & ( H2 = H3 ) )
% 16.19/8.94 | ( ~ ( ord_less_eq_real @ L2 @ H2 )
% 16.19/8.94 & ~ ( ord_less_eq_real @ L3 @ H3 ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3426_atLeastAtMost__iff,axiom,
% 16.19/8.94 ! [I2: set_nat,L2: set_nat,U: set_nat] :
% 16.19/8.94 ( ( member_set_nat @ I2 @ ( set_or4548717258645045905et_nat @ L2 @ U ) )
% 16.19/8.94 = ( ( ord_less_eq_set_nat @ L2 @ I2 )
% 16.19/8.94 & ( ord_less_eq_set_nat @ I2 @ U ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3427_atLeastAtMost__iff,axiom,
% 16.19/8.94 ! [I2: rat,L2: rat,U: rat] :
% 16.19/8.94 ( ( member_rat @ I2 @ ( set_or633870826150836451st_rat @ L2 @ U ) )
% 16.19/8.94 = ( ( ord_less_eq_rat @ L2 @ I2 )
% 16.19/8.94 & ( ord_less_eq_rat @ I2 @ U ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3428_atLeastAtMost__iff,axiom,
% 16.19/8.94 ! [I2: num,L2: num,U: num] :
% 16.19/8.94 ( ( member_num @ I2 @ ( set_or7049704709247886629st_num @ L2 @ U ) )
% 16.19/8.94 = ( ( ord_less_eq_num @ L2 @ I2 )
% 16.19/8.94 & ( ord_less_eq_num @ I2 @ U ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3429_atLeastAtMost__iff,axiom,
% 16.19/8.94 ! [I2: nat,L2: nat,U: nat] :
% 16.19/8.94 ( ( member_nat @ I2 @ ( set_or1269000886237332187st_nat @ L2 @ U ) )
% 16.19/8.94 = ( ( ord_less_eq_nat @ L2 @ I2 )
% 16.19/8.94 & ( ord_less_eq_nat @ I2 @ U ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3430_atLeastAtMost__iff,axiom,
% 16.19/8.94 ! [I2: int,L2: int,U: int] :
% 16.19/8.94 ( ( member_int @ I2 @ ( set_or1266510415728281911st_int @ L2 @ U ) )
% 16.19/8.94 = ( ( ord_less_eq_int @ L2 @ I2 )
% 16.19/8.94 & ( ord_less_eq_int @ I2 @ U ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3431_atLeastAtMost__iff,axiom,
% 16.19/8.94 ! [I2: real,L2: real,U: real] :
% 16.19/8.94 ( ( member_real @ I2 @ ( set_or1222579329274155063t_real @ L2 @ U ) )
% 16.19/8.94 = ( ( ord_less_eq_real @ L2 @ I2 )
% 16.19/8.94 & ( ord_less_eq_real @ I2 @ U ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3432_aset_I2_J,axiom,
% 16.19/8.94 ! [D3: int,A2: set_int,P: int > $o,Q: int > $o] :
% 16.19/8.94 ( ! [X4: int] :
% 16.19/8.94 ( ! [Xa: int] :
% 16.19/8.94 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 => ! [Xb: int] :
% 16.19/8.94 ( ( member_int @ Xb @ A2 )
% 16.19/8.94 => ( X4
% 16.19/8.94 != ( minus_minus_int @ Xb @ Xa ) ) ) )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 => ( P @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 16.19/8.94 => ( ! [X4: int] :
% 16.19/8.94 ( ! [Xa: int] :
% 16.19/8.94 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 => ! [Xb: int] :
% 16.19/8.94 ( ( member_int @ Xb @ A2 )
% 16.19/8.94 => ( X4
% 16.19/8.94 != ( minus_minus_int @ Xb @ Xa ) ) ) )
% 16.19/8.94 => ( ( Q @ X4 )
% 16.19/8.94 => ( Q @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 16.19/8.94 => ! [X5: int] :
% 16.19/8.94 ( ! [Xa3: int] :
% 16.19/8.94 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 => ! [Xb3: int] :
% 16.19/8.94 ( ( member_int @ Xb3 @ A2 )
% 16.19/8.94 => ( X5
% 16.19/8.94 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 16.19/8.94 => ( ( ( P @ X5 )
% 16.19/8.94 | ( Q @ X5 ) )
% 16.19/8.94 => ( ( P @ ( plus_plus_int @ X5 @ D3 ) )
% 16.19/8.94 | ( Q @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3433_aset_I1_J,axiom,
% 16.19/8.94 ! [D3: int,A2: set_int,P: int > $o,Q: int > $o] :
% 16.19/8.94 ( ! [X4: int] :
% 16.19/8.94 ( ! [Xa: int] :
% 16.19/8.94 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 => ! [Xb: int] :
% 16.19/8.94 ( ( member_int @ Xb @ A2 )
% 16.19/8.94 => ( X4
% 16.19/8.94 != ( minus_minus_int @ Xb @ Xa ) ) ) )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 => ( P @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 16.19/8.94 => ( ! [X4: int] :
% 16.19/8.94 ( ! [Xa: int] :
% 16.19/8.94 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 => ! [Xb: int] :
% 16.19/8.94 ( ( member_int @ Xb @ A2 )
% 16.19/8.94 => ( X4
% 16.19/8.94 != ( minus_minus_int @ Xb @ Xa ) ) ) )
% 16.19/8.94 => ( ( Q @ X4 )
% 16.19/8.94 => ( Q @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 16.19/8.94 => ! [X5: int] :
% 16.19/8.94 ( ! [Xa3: int] :
% 16.19/8.94 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 => ! [Xb3: int] :
% 16.19/8.94 ( ( member_int @ Xb3 @ A2 )
% 16.19/8.94 => ( X5
% 16.19/8.94 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 16.19/8.94 => ( ( ( P @ X5 )
% 16.19/8.94 & ( Q @ X5 ) )
% 16.19/8.94 => ( ( P @ ( plus_plus_int @ X5 @ D3 ) )
% 16.19/8.94 & ( Q @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3434_bset_I2_J,axiom,
% 16.19/8.94 ! [D3: int,B3: set_int,P: int > $o,Q: int > $o] :
% 16.19/8.94 ( ! [X4: int] :
% 16.19/8.94 ( ! [Xa: int] :
% 16.19/8.94 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 => ! [Xb: int] :
% 16.19/8.94 ( ( member_int @ Xb @ B3 )
% 16.19/8.94 => ( X4
% 16.19/8.94 != ( plus_plus_int @ Xb @ Xa ) ) ) )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 => ( P @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 16.19/8.94 => ( ! [X4: int] :
% 16.19/8.94 ( ! [Xa: int] :
% 16.19/8.94 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 => ! [Xb: int] :
% 16.19/8.94 ( ( member_int @ Xb @ B3 )
% 16.19/8.94 => ( X4
% 16.19/8.94 != ( plus_plus_int @ Xb @ Xa ) ) ) )
% 16.19/8.94 => ( ( Q @ X4 )
% 16.19/8.94 => ( Q @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 16.19/8.94 => ! [X5: int] :
% 16.19/8.94 ( ! [Xa3: int] :
% 16.19/8.94 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 => ! [Xb3: int] :
% 16.19/8.94 ( ( member_int @ Xb3 @ B3 )
% 16.19/8.94 => ( X5
% 16.19/8.94 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 16.19/8.94 => ( ( ( P @ X5 )
% 16.19/8.94 | ( Q @ X5 ) )
% 16.19/8.94 => ( ( P @ ( minus_minus_int @ X5 @ D3 ) )
% 16.19/8.94 | ( Q @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3435_bset_I1_J,axiom,
% 16.19/8.94 ! [D3: int,B3: set_int,P: int > $o,Q: int > $o] :
% 16.19/8.94 ( ! [X4: int] :
% 16.19/8.94 ( ! [Xa: int] :
% 16.19/8.94 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 => ! [Xb: int] :
% 16.19/8.94 ( ( member_int @ Xb @ B3 )
% 16.19/8.94 => ( X4
% 16.19/8.94 != ( plus_plus_int @ Xb @ Xa ) ) ) )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 => ( P @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 16.19/8.94 => ( ! [X4: int] :
% 16.19/8.94 ( ! [Xa: int] :
% 16.19/8.94 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 => ! [Xb: int] :
% 16.19/8.94 ( ( member_int @ Xb @ B3 )
% 16.19/8.94 => ( X4
% 16.19/8.94 != ( plus_plus_int @ Xb @ Xa ) ) ) )
% 16.19/8.94 => ( ( Q @ X4 )
% 16.19/8.94 => ( Q @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 16.19/8.94 => ! [X5: int] :
% 16.19/8.94 ( ! [Xa3: int] :
% 16.19/8.94 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 => ! [Xb3: int] :
% 16.19/8.94 ( ( member_int @ Xb3 @ B3 )
% 16.19/8.94 => ( X5
% 16.19/8.94 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 16.19/8.94 => ( ( ( P @ X5 )
% 16.19/8.94 & ( Q @ X5 ) )
% 16.19/8.94 => ( ( P @ ( minus_minus_int @ X5 @ D3 ) )
% 16.19/8.94 & ( Q @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3436_pinf_I1_J,axiom,
% 16.19/8.94 ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
% 16.19/8.94 ( ? [Z4: real] :
% 16.19/8.94 ! [X4: real] :
% 16.19/8.94 ( ( ord_less_real @ Z4 @ X4 )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 = ( P6 @ X4 ) ) )
% 16.19/8.94 => ( ? [Z4: real] :
% 16.19/8.94 ! [X4: real] :
% 16.19/8.94 ( ( ord_less_real @ Z4 @ X4 )
% 16.19/8.94 => ( ( Q @ X4 )
% 16.19/8.94 = ( Q6 @ X4 ) ) )
% 16.19/8.94 => ? [Z5: real] :
% 16.19/8.94 ! [X5: real] :
% 16.19/8.94 ( ( ord_less_real @ Z5 @ X5 )
% 16.19/8.94 => ( ( ( P @ X5 )
% 16.19/8.94 & ( Q @ X5 ) )
% 16.19/8.94 = ( ( P6 @ X5 )
% 16.19/8.94 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3437_pinf_I1_J,axiom,
% 16.19/8.94 ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 16.19/8.94 ( ? [Z4: rat] :
% 16.19/8.94 ! [X4: rat] :
% 16.19/8.94 ( ( ord_less_rat @ Z4 @ X4 )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 = ( P6 @ X4 ) ) )
% 16.19/8.94 => ( ? [Z4: rat] :
% 16.19/8.94 ! [X4: rat] :
% 16.19/8.94 ( ( ord_less_rat @ Z4 @ X4 )
% 16.19/8.94 => ( ( Q @ X4 )
% 16.19/8.94 = ( Q6 @ X4 ) ) )
% 16.19/8.94 => ? [Z5: rat] :
% 16.19/8.94 ! [X5: rat] :
% 16.19/8.94 ( ( ord_less_rat @ Z5 @ X5 )
% 16.19/8.94 => ( ( ( P @ X5 )
% 16.19/8.94 & ( Q @ X5 ) )
% 16.19/8.94 = ( ( P6 @ X5 )
% 16.19/8.94 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3438_pinf_I1_J,axiom,
% 16.19/8.94 ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
% 16.19/8.94 ( ? [Z4: num] :
% 16.19/8.94 ! [X4: num] :
% 16.19/8.94 ( ( ord_less_num @ Z4 @ X4 )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 = ( P6 @ X4 ) ) )
% 16.19/8.94 => ( ? [Z4: num] :
% 16.19/8.94 ! [X4: num] :
% 16.19/8.94 ( ( ord_less_num @ Z4 @ X4 )
% 16.19/8.94 => ( ( Q @ X4 )
% 16.19/8.94 = ( Q6 @ X4 ) ) )
% 16.19/8.94 => ? [Z5: num] :
% 16.19/8.94 ! [X5: num] :
% 16.19/8.94 ( ( ord_less_num @ Z5 @ X5 )
% 16.19/8.94 => ( ( ( P @ X5 )
% 16.19/8.94 & ( Q @ X5 ) )
% 16.19/8.94 = ( ( P6 @ X5 )
% 16.19/8.94 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3439_pinf_I1_J,axiom,
% 16.19/8.94 ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 16.19/8.94 ( ? [Z4: nat] :
% 16.19/8.94 ! [X4: nat] :
% 16.19/8.94 ( ( ord_less_nat @ Z4 @ X4 )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 = ( P6 @ X4 ) ) )
% 16.19/8.94 => ( ? [Z4: nat] :
% 16.19/8.94 ! [X4: nat] :
% 16.19/8.94 ( ( ord_less_nat @ Z4 @ X4 )
% 16.19/8.94 => ( ( Q @ X4 )
% 16.19/8.94 = ( Q6 @ X4 ) ) )
% 16.19/8.94 => ? [Z5: nat] :
% 16.19/8.94 ! [X5: nat] :
% 16.19/8.94 ( ( ord_less_nat @ Z5 @ X5 )
% 16.19/8.94 => ( ( ( P @ X5 )
% 16.19/8.94 & ( Q @ X5 ) )
% 16.19/8.94 = ( ( P6 @ X5 )
% 16.19/8.94 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3440_pinf_I1_J,axiom,
% 16.19/8.94 ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
% 16.19/8.94 ( ? [Z4: int] :
% 16.19/8.94 ! [X4: int] :
% 16.19/8.94 ( ( ord_less_int @ Z4 @ X4 )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 = ( P6 @ X4 ) ) )
% 16.19/8.94 => ( ? [Z4: int] :
% 16.19/8.94 ! [X4: int] :
% 16.19/8.94 ( ( ord_less_int @ Z4 @ X4 )
% 16.19/8.94 => ( ( Q @ X4 )
% 16.19/8.94 = ( Q6 @ X4 ) ) )
% 16.19/8.94 => ? [Z5: int] :
% 16.19/8.94 ! [X5: int] :
% 16.19/8.94 ( ( ord_less_int @ Z5 @ X5 )
% 16.19/8.94 => ( ( ( P @ X5 )
% 16.19/8.94 & ( Q @ X5 ) )
% 16.19/8.94 = ( ( P6 @ X5 )
% 16.19/8.94 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3441_pinf_I2_J,axiom,
% 16.19/8.94 ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
% 16.19/8.94 ( ? [Z4: real] :
% 16.19/8.94 ! [X4: real] :
% 16.19/8.94 ( ( ord_less_real @ Z4 @ X4 )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 = ( P6 @ X4 ) ) )
% 16.19/8.94 => ( ? [Z4: real] :
% 16.19/8.94 ! [X4: real] :
% 16.19/8.94 ( ( ord_less_real @ Z4 @ X4 )
% 16.19/8.94 => ( ( Q @ X4 )
% 16.19/8.94 = ( Q6 @ X4 ) ) )
% 16.19/8.94 => ? [Z5: real] :
% 16.19/8.94 ! [X5: real] :
% 16.19/8.94 ( ( ord_less_real @ Z5 @ X5 )
% 16.19/8.94 => ( ( ( P @ X5 )
% 16.19/8.94 | ( Q @ X5 ) )
% 16.19/8.94 = ( ( P6 @ X5 )
% 16.19/8.94 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3442_pinf_I2_J,axiom,
% 16.19/8.94 ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 16.19/8.94 ( ? [Z4: rat] :
% 16.19/8.94 ! [X4: rat] :
% 16.19/8.94 ( ( ord_less_rat @ Z4 @ X4 )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 = ( P6 @ X4 ) ) )
% 16.19/8.94 => ( ? [Z4: rat] :
% 16.19/8.94 ! [X4: rat] :
% 16.19/8.94 ( ( ord_less_rat @ Z4 @ X4 )
% 16.19/8.94 => ( ( Q @ X4 )
% 16.19/8.94 = ( Q6 @ X4 ) ) )
% 16.19/8.94 => ? [Z5: rat] :
% 16.19/8.94 ! [X5: rat] :
% 16.19/8.94 ( ( ord_less_rat @ Z5 @ X5 )
% 16.19/8.94 => ( ( ( P @ X5 )
% 16.19/8.94 | ( Q @ X5 ) )
% 16.19/8.94 = ( ( P6 @ X5 )
% 16.19/8.94 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3443_pinf_I2_J,axiom,
% 16.19/8.94 ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
% 16.19/8.94 ( ? [Z4: num] :
% 16.19/8.94 ! [X4: num] :
% 16.19/8.94 ( ( ord_less_num @ Z4 @ X4 )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 = ( P6 @ X4 ) ) )
% 16.19/8.94 => ( ? [Z4: num] :
% 16.19/8.94 ! [X4: num] :
% 16.19/8.94 ( ( ord_less_num @ Z4 @ X4 )
% 16.19/8.94 => ( ( Q @ X4 )
% 16.19/8.94 = ( Q6 @ X4 ) ) )
% 16.19/8.94 => ? [Z5: num] :
% 16.19/8.94 ! [X5: num] :
% 16.19/8.94 ( ( ord_less_num @ Z5 @ X5 )
% 16.19/8.94 => ( ( ( P @ X5 )
% 16.19/8.94 | ( Q @ X5 ) )
% 16.19/8.94 = ( ( P6 @ X5 )
% 16.19/8.94 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3444_pinf_I2_J,axiom,
% 16.19/8.94 ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 16.19/8.94 ( ? [Z4: nat] :
% 16.19/8.94 ! [X4: nat] :
% 16.19/8.94 ( ( ord_less_nat @ Z4 @ X4 )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 = ( P6 @ X4 ) ) )
% 16.19/8.94 => ( ? [Z4: nat] :
% 16.19/8.94 ! [X4: nat] :
% 16.19/8.94 ( ( ord_less_nat @ Z4 @ X4 )
% 16.19/8.94 => ( ( Q @ X4 )
% 16.19/8.94 = ( Q6 @ X4 ) ) )
% 16.19/8.94 => ? [Z5: nat] :
% 16.19/8.94 ! [X5: nat] :
% 16.19/8.94 ( ( ord_less_nat @ Z5 @ X5 )
% 16.19/8.94 => ( ( ( P @ X5 )
% 16.19/8.94 | ( Q @ X5 ) )
% 16.19/8.94 = ( ( P6 @ X5 )
% 16.19/8.94 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3445_pinf_I2_J,axiom,
% 16.19/8.94 ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
% 16.19/8.94 ( ? [Z4: int] :
% 16.19/8.94 ! [X4: int] :
% 16.19/8.94 ( ( ord_less_int @ Z4 @ X4 )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 = ( P6 @ X4 ) ) )
% 16.19/8.94 => ( ? [Z4: int] :
% 16.19/8.94 ! [X4: int] :
% 16.19/8.94 ( ( ord_less_int @ Z4 @ X4 )
% 16.19/8.94 => ( ( Q @ X4 )
% 16.19/8.94 = ( Q6 @ X4 ) ) )
% 16.19/8.94 => ? [Z5: int] :
% 16.19/8.94 ! [X5: int] :
% 16.19/8.94 ( ( ord_less_int @ Z5 @ X5 )
% 16.19/8.94 => ( ( ( P @ X5 )
% 16.19/8.94 | ( Q @ X5 ) )
% 16.19/8.94 = ( ( P6 @ X5 )
% 16.19/8.94 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3446_pinf_I3_J,axiom,
% 16.19/8.94 ! [T: real] :
% 16.19/8.94 ? [Z5: real] :
% 16.19/8.94 ! [X5: real] :
% 16.19/8.94 ( ( ord_less_real @ Z5 @ X5 )
% 16.19/8.94 => ( X5 != T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3447_pinf_I3_J,axiom,
% 16.19/8.94 ! [T: rat] :
% 16.19/8.94 ? [Z5: rat] :
% 16.19/8.94 ! [X5: rat] :
% 16.19/8.94 ( ( ord_less_rat @ Z5 @ X5 )
% 16.19/8.94 => ( X5 != T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3448_pinf_I3_J,axiom,
% 16.19/8.94 ! [T: num] :
% 16.19/8.94 ? [Z5: num] :
% 16.19/8.94 ! [X5: num] :
% 16.19/8.94 ( ( ord_less_num @ Z5 @ X5 )
% 16.19/8.94 => ( X5 != T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3449_pinf_I3_J,axiom,
% 16.19/8.94 ! [T: nat] :
% 16.19/8.94 ? [Z5: nat] :
% 16.19/8.94 ! [X5: nat] :
% 16.19/8.94 ( ( ord_less_nat @ Z5 @ X5 )
% 16.19/8.94 => ( X5 != T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3450_pinf_I3_J,axiom,
% 16.19/8.94 ! [T: int] :
% 16.19/8.94 ? [Z5: int] :
% 16.19/8.94 ! [X5: int] :
% 16.19/8.94 ( ( ord_less_int @ Z5 @ X5 )
% 16.19/8.94 => ( X5 != T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3451_pinf_I4_J,axiom,
% 16.19/8.94 ! [T: real] :
% 16.19/8.94 ? [Z5: real] :
% 16.19/8.94 ! [X5: real] :
% 16.19/8.94 ( ( ord_less_real @ Z5 @ X5 )
% 16.19/8.94 => ( X5 != T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3452_pinf_I4_J,axiom,
% 16.19/8.94 ! [T: rat] :
% 16.19/8.94 ? [Z5: rat] :
% 16.19/8.94 ! [X5: rat] :
% 16.19/8.94 ( ( ord_less_rat @ Z5 @ X5 )
% 16.19/8.94 => ( X5 != T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3453_pinf_I4_J,axiom,
% 16.19/8.94 ! [T: num] :
% 16.19/8.94 ? [Z5: num] :
% 16.19/8.94 ! [X5: num] :
% 16.19/8.94 ( ( ord_less_num @ Z5 @ X5 )
% 16.19/8.94 => ( X5 != T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3454_pinf_I4_J,axiom,
% 16.19/8.94 ! [T: nat] :
% 16.19/8.94 ? [Z5: nat] :
% 16.19/8.94 ! [X5: nat] :
% 16.19/8.94 ( ( ord_less_nat @ Z5 @ X5 )
% 16.19/8.94 => ( X5 != T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3455_pinf_I4_J,axiom,
% 16.19/8.94 ! [T: int] :
% 16.19/8.94 ? [Z5: int] :
% 16.19/8.94 ! [X5: int] :
% 16.19/8.94 ( ( ord_less_int @ Z5 @ X5 )
% 16.19/8.94 => ( X5 != T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3456_pinf_I5_J,axiom,
% 16.19/8.94 ! [T: real] :
% 16.19/8.94 ? [Z5: real] :
% 16.19/8.94 ! [X5: real] :
% 16.19/8.94 ( ( ord_less_real @ Z5 @ X5 )
% 16.19/8.94 => ~ ( ord_less_real @ X5 @ T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3457_pinf_I5_J,axiom,
% 16.19/8.94 ! [T: rat] :
% 16.19/8.94 ? [Z5: rat] :
% 16.19/8.94 ! [X5: rat] :
% 16.19/8.94 ( ( ord_less_rat @ Z5 @ X5 )
% 16.19/8.94 => ~ ( ord_less_rat @ X5 @ T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3458_pinf_I5_J,axiom,
% 16.19/8.94 ! [T: num] :
% 16.19/8.94 ? [Z5: num] :
% 16.19/8.94 ! [X5: num] :
% 16.19/8.94 ( ( ord_less_num @ Z5 @ X5 )
% 16.19/8.94 => ~ ( ord_less_num @ X5 @ T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3459_pinf_I5_J,axiom,
% 16.19/8.94 ! [T: nat] :
% 16.19/8.94 ? [Z5: nat] :
% 16.19/8.94 ! [X5: nat] :
% 16.19/8.94 ( ( ord_less_nat @ Z5 @ X5 )
% 16.19/8.94 => ~ ( ord_less_nat @ X5 @ T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3460_pinf_I5_J,axiom,
% 16.19/8.94 ! [T: int] :
% 16.19/8.94 ? [Z5: int] :
% 16.19/8.94 ! [X5: int] :
% 16.19/8.94 ( ( ord_less_int @ Z5 @ X5 )
% 16.19/8.94 => ~ ( ord_less_int @ X5 @ T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3461_pinf_I7_J,axiom,
% 16.19/8.94 ! [T: real] :
% 16.19/8.94 ? [Z5: real] :
% 16.19/8.94 ! [X5: real] :
% 16.19/8.94 ( ( ord_less_real @ Z5 @ X5 )
% 16.19/8.94 => ( ord_less_real @ T @ X5 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3462_pinf_I7_J,axiom,
% 16.19/8.94 ! [T: rat] :
% 16.19/8.94 ? [Z5: rat] :
% 16.19/8.94 ! [X5: rat] :
% 16.19/8.94 ( ( ord_less_rat @ Z5 @ X5 )
% 16.19/8.94 => ( ord_less_rat @ T @ X5 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3463_pinf_I7_J,axiom,
% 16.19/8.94 ! [T: num] :
% 16.19/8.94 ? [Z5: num] :
% 16.19/8.94 ! [X5: num] :
% 16.19/8.94 ( ( ord_less_num @ Z5 @ X5 )
% 16.19/8.94 => ( ord_less_num @ T @ X5 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3464_pinf_I7_J,axiom,
% 16.19/8.94 ! [T: nat] :
% 16.19/8.94 ? [Z5: nat] :
% 16.19/8.94 ! [X5: nat] :
% 16.19/8.94 ( ( ord_less_nat @ Z5 @ X5 )
% 16.19/8.94 => ( ord_less_nat @ T @ X5 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3465_pinf_I7_J,axiom,
% 16.19/8.94 ! [T: int] :
% 16.19/8.94 ? [Z5: int] :
% 16.19/8.94 ! [X5: int] :
% 16.19/8.94 ( ( ord_less_int @ Z5 @ X5 )
% 16.19/8.94 => ( ord_less_int @ T @ X5 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3466_minf_I1_J,axiom,
% 16.19/8.94 ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
% 16.19/8.94 ( ? [Z4: real] :
% 16.19/8.94 ! [X4: real] :
% 16.19/8.94 ( ( ord_less_real @ X4 @ Z4 )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 = ( P6 @ X4 ) ) )
% 16.19/8.94 => ( ? [Z4: real] :
% 16.19/8.94 ! [X4: real] :
% 16.19/8.94 ( ( ord_less_real @ X4 @ Z4 )
% 16.19/8.94 => ( ( Q @ X4 )
% 16.19/8.94 = ( Q6 @ X4 ) ) )
% 16.19/8.94 => ? [Z5: real] :
% 16.19/8.94 ! [X5: real] :
% 16.19/8.94 ( ( ord_less_real @ X5 @ Z5 )
% 16.19/8.94 => ( ( ( P @ X5 )
% 16.19/8.94 & ( Q @ X5 ) )
% 16.19/8.94 = ( ( P6 @ X5 )
% 16.19/8.94 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3467_minf_I1_J,axiom,
% 16.19/8.94 ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 16.19/8.94 ( ? [Z4: rat] :
% 16.19/8.94 ! [X4: rat] :
% 16.19/8.94 ( ( ord_less_rat @ X4 @ Z4 )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 = ( P6 @ X4 ) ) )
% 16.19/8.94 => ( ? [Z4: rat] :
% 16.19/8.94 ! [X4: rat] :
% 16.19/8.94 ( ( ord_less_rat @ X4 @ Z4 )
% 16.19/8.94 => ( ( Q @ X4 )
% 16.19/8.94 = ( Q6 @ X4 ) ) )
% 16.19/8.94 => ? [Z5: rat] :
% 16.19/8.94 ! [X5: rat] :
% 16.19/8.94 ( ( ord_less_rat @ X5 @ Z5 )
% 16.19/8.94 => ( ( ( P @ X5 )
% 16.19/8.94 & ( Q @ X5 ) )
% 16.19/8.94 = ( ( P6 @ X5 )
% 16.19/8.94 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3468_minf_I1_J,axiom,
% 16.19/8.94 ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
% 16.19/8.94 ( ? [Z4: num] :
% 16.19/8.94 ! [X4: num] :
% 16.19/8.94 ( ( ord_less_num @ X4 @ Z4 )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 = ( P6 @ X4 ) ) )
% 16.19/8.94 => ( ? [Z4: num] :
% 16.19/8.94 ! [X4: num] :
% 16.19/8.94 ( ( ord_less_num @ X4 @ Z4 )
% 16.19/8.94 => ( ( Q @ X4 )
% 16.19/8.94 = ( Q6 @ X4 ) ) )
% 16.19/8.94 => ? [Z5: num] :
% 16.19/8.94 ! [X5: num] :
% 16.19/8.94 ( ( ord_less_num @ X5 @ Z5 )
% 16.19/8.94 => ( ( ( P @ X5 )
% 16.19/8.94 & ( Q @ X5 ) )
% 16.19/8.94 = ( ( P6 @ X5 )
% 16.19/8.94 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3469_minf_I1_J,axiom,
% 16.19/8.94 ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 16.19/8.94 ( ? [Z4: nat] :
% 16.19/8.94 ! [X4: nat] :
% 16.19/8.94 ( ( ord_less_nat @ X4 @ Z4 )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 = ( P6 @ X4 ) ) )
% 16.19/8.94 => ( ? [Z4: nat] :
% 16.19/8.94 ! [X4: nat] :
% 16.19/8.94 ( ( ord_less_nat @ X4 @ Z4 )
% 16.19/8.94 => ( ( Q @ X4 )
% 16.19/8.94 = ( Q6 @ X4 ) ) )
% 16.19/8.94 => ? [Z5: nat] :
% 16.19/8.94 ! [X5: nat] :
% 16.19/8.94 ( ( ord_less_nat @ X5 @ Z5 )
% 16.19/8.94 => ( ( ( P @ X5 )
% 16.19/8.94 & ( Q @ X5 ) )
% 16.19/8.94 = ( ( P6 @ X5 )
% 16.19/8.94 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3470_minf_I1_J,axiom,
% 16.19/8.94 ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
% 16.19/8.94 ( ? [Z4: int] :
% 16.19/8.94 ! [X4: int] :
% 16.19/8.94 ( ( ord_less_int @ X4 @ Z4 )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 = ( P6 @ X4 ) ) )
% 16.19/8.94 => ( ? [Z4: int] :
% 16.19/8.94 ! [X4: int] :
% 16.19/8.94 ( ( ord_less_int @ X4 @ Z4 )
% 16.19/8.94 => ( ( Q @ X4 )
% 16.19/8.94 = ( Q6 @ X4 ) ) )
% 16.19/8.94 => ? [Z5: int] :
% 16.19/8.94 ! [X5: int] :
% 16.19/8.94 ( ( ord_less_int @ X5 @ Z5 )
% 16.19/8.94 => ( ( ( P @ X5 )
% 16.19/8.94 & ( Q @ X5 ) )
% 16.19/8.94 = ( ( P6 @ X5 )
% 16.19/8.94 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3471_minf_I2_J,axiom,
% 16.19/8.94 ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
% 16.19/8.94 ( ? [Z4: real] :
% 16.19/8.94 ! [X4: real] :
% 16.19/8.94 ( ( ord_less_real @ X4 @ Z4 )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 = ( P6 @ X4 ) ) )
% 16.19/8.94 => ( ? [Z4: real] :
% 16.19/8.94 ! [X4: real] :
% 16.19/8.94 ( ( ord_less_real @ X4 @ Z4 )
% 16.19/8.94 => ( ( Q @ X4 )
% 16.19/8.94 = ( Q6 @ X4 ) ) )
% 16.19/8.94 => ? [Z5: real] :
% 16.19/8.94 ! [X5: real] :
% 16.19/8.94 ( ( ord_less_real @ X5 @ Z5 )
% 16.19/8.94 => ( ( ( P @ X5 )
% 16.19/8.94 | ( Q @ X5 ) )
% 16.19/8.94 = ( ( P6 @ X5 )
% 16.19/8.94 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3472_minf_I2_J,axiom,
% 16.19/8.94 ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 16.19/8.94 ( ? [Z4: rat] :
% 16.19/8.94 ! [X4: rat] :
% 16.19/8.94 ( ( ord_less_rat @ X4 @ Z4 )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 = ( P6 @ X4 ) ) )
% 16.19/8.94 => ( ? [Z4: rat] :
% 16.19/8.94 ! [X4: rat] :
% 16.19/8.94 ( ( ord_less_rat @ X4 @ Z4 )
% 16.19/8.94 => ( ( Q @ X4 )
% 16.19/8.94 = ( Q6 @ X4 ) ) )
% 16.19/8.94 => ? [Z5: rat] :
% 16.19/8.94 ! [X5: rat] :
% 16.19/8.94 ( ( ord_less_rat @ X5 @ Z5 )
% 16.19/8.94 => ( ( ( P @ X5 )
% 16.19/8.94 | ( Q @ X5 ) )
% 16.19/8.94 = ( ( P6 @ X5 )
% 16.19/8.94 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3473_minf_I2_J,axiom,
% 16.19/8.94 ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
% 16.19/8.94 ( ? [Z4: num] :
% 16.19/8.94 ! [X4: num] :
% 16.19/8.94 ( ( ord_less_num @ X4 @ Z4 )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 = ( P6 @ X4 ) ) )
% 16.19/8.94 => ( ? [Z4: num] :
% 16.19/8.94 ! [X4: num] :
% 16.19/8.94 ( ( ord_less_num @ X4 @ Z4 )
% 16.19/8.94 => ( ( Q @ X4 )
% 16.19/8.94 = ( Q6 @ X4 ) ) )
% 16.19/8.94 => ? [Z5: num] :
% 16.19/8.94 ! [X5: num] :
% 16.19/8.94 ( ( ord_less_num @ X5 @ Z5 )
% 16.19/8.94 => ( ( ( P @ X5 )
% 16.19/8.94 | ( Q @ X5 ) )
% 16.19/8.94 = ( ( P6 @ X5 )
% 16.19/8.94 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3474_minf_I2_J,axiom,
% 16.19/8.94 ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 16.19/8.94 ( ? [Z4: nat] :
% 16.19/8.94 ! [X4: nat] :
% 16.19/8.94 ( ( ord_less_nat @ X4 @ Z4 )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 = ( P6 @ X4 ) ) )
% 16.19/8.94 => ( ? [Z4: nat] :
% 16.19/8.94 ! [X4: nat] :
% 16.19/8.94 ( ( ord_less_nat @ X4 @ Z4 )
% 16.19/8.94 => ( ( Q @ X4 )
% 16.19/8.94 = ( Q6 @ X4 ) ) )
% 16.19/8.94 => ? [Z5: nat] :
% 16.19/8.94 ! [X5: nat] :
% 16.19/8.94 ( ( ord_less_nat @ X5 @ Z5 )
% 16.19/8.94 => ( ( ( P @ X5 )
% 16.19/8.94 | ( Q @ X5 ) )
% 16.19/8.94 = ( ( P6 @ X5 )
% 16.19/8.94 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3475_minf_I2_J,axiom,
% 16.19/8.94 ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
% 16.19/8.94 ( ? [Z4: int] :
% 16.19/8.94 ! [X4: int] :
% 16.19/8.94 ( ( ord_less_int @ X4 @ Z4 )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 = ( P6 @ X4 ) ) )
% 16.19/8.94 => ( ? [Z4: int] :
% 16.19/8.94 ! [X4: int] :
% 16.19/8.94 ( ( ord_less_int @ X4 @ Z4 )
% 16.19/8.94 => ( ( Q @ X4 )
% 16.19/8.94 = ( Q6 @ X4 ) ) )
% 16.19/8.94 => ? [Z5: int] :
% 16.19/8.94 ! [X5: int] :
% 16.19/8.94 ( ( ord_less_int @ X5 @ Z5 )
% 16.19/8.94 => ( ( ( P @ X5 )
% 16.19/8.94 | ( Q @ X5 ) )
% 16.19/8.94 = ( ( P6 @ X5 )
% 16.19/8.94 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3476_minf_I3_J,axiom,
% 16.19/8.94 ! [T: real] :
% 16.19/8.94 ? [Z5: real] :
% 16.19/8.94 ! [X5: real] :
% 16.19/8.94 ( ( ord_less_real @ X5 @ Z5 )
% 16.19/8.94 => ( X5 != T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3477_minf_I3_J,axiom,
% 16.19/8.94 ! [T: rat] :
% 16.19/8.94 ? [Z5: rat] :
% 16.19/8.94 ! [X5: rat] :
% 16.19/8.94 ( ( ord_less_rat @ X5 @ Z5 )
% 16.19/8.94 => ( X5 != T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3478_minf_I3_J,axiom,
% 16.19/8.94 ! [T: num] :
% 16.19/8.94 ? [Z5: num] :
% 16.19/8.94 ! [X5: num] :
% 16.19/8.94 ( ( ord_less_num @ X5 @ Z5 )
% 16.19/8.94 => ( X5 != T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3479_minf_I3_J,axiom,
% 16.19/8.94 ! [T: nat] :
% 16.19/8.94 ? [Z5: nat] :
% 16.19/8.94 ! [X5: nat] :
% 16.19/8.94 ( ( ord_less_nat @ X5 @ Z5 )
% 16.19/8.94 => ( X5 != T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3480_minf_I3_J,axiom,
% 16.19/8.94 ! [T: int] :
% 16.19/8.94 ? [Z5: int] :
% 16.19/8.94 ! [X5: int] :
% 16.19/8.94 ( ( ord_less_int @ X5 @ Z5 )
% 16.19/8.94 => ( X5 != T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3481_minf_I4_J,axiom,
% 16.19/8.94 ! [T: real] :
% 16.19/8.94 ? [Z5: real] :
% 16.19/8.94 ! [X5: real] :
% 16.19/8.94 ( ( ord_less_real @ X5 @ Z5 )
% 16.19/8.94 => ( X5 != T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3482_minf_I4_J,axiom,
% 16.19/8.94 ! [T: rat] :
% 16.19/8.94 ? [Z5: rat] :
% 16.19/8.94 ! [X5: rat] :
% 16.19/8.94 ( ( ord_less_rat @ X5 @ Z5 )
% 16.19/8.94 => ( X5 != T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3483_minf_I4_J,axiom,
% 16.19/8.94 ! [T: num] :
% 16.19/8.94 ? [Z5: num] :
% 16.19/8.94 ! [X5: num] :
% 16.19/8.94 ( ( ord_less_num @ X5 @ Z5 )
% 16.19/8.94 => ( X5 != T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3484_minf_I4_J,axiom,
% 16.19/8.94 ! [T: nat] :
% 16.19/8.94 ? [Z5: nat] :
% 16.19/8.94 ! [X5: nat] :
% 16.19/8.94 ( ( ord_less_nat @ X5 @ Z5 )
% 16.19/8.94 => ( X5 != T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3485_minf_I4_J,axiom,
% 16.19/8.94 ! [T: int] :
% 16.19/8.94 ? [Z5: int] :
% 16.19/8.94 ! [X5: int] :
% 16.19/8.94 ( ( ord_less_int @ X5 @ Z5 )
% 16.19/8.94 => ( X5 != T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3486_minf_I5_J,axiom,
% 16.19/8.94 ! [T: real] :
% 16.19/8.94 ? [Z5: real] :
% 16.19/8.94 ! [X5: real] :
% 16.19/8.94 ( ( ord_less_real @ X5 @ Z5 )
% 16.19/8.94 => ( ord_less_real @ X5 @ T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3487_minf_I5_J,axiom,
% 16.19/8.94 ! [T: rat] :
% 16.19/8.94 ? [Z5: rat] :
% 16.19/8.94 ! [X5: rat] :
% 16.19/8.94 ( ( ord_less_rat @ X5 @ Z5 )
% 16.19/8.94 => ( ord_less_rat @ X5 @ T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3488_minf_I5_J,axiom,
% 16.19/8.94 ! [T: num] :
% 16.19/8.94 ? [Z5: num] :
% 16.19/8.94 ! [X5: num] :
% 16.19/8.94 ( ( ord_less_num @ X5 @ Z5 )
% 16.19/8.94 => ( ord_less_num @ X5 @ T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3489_minf_I5_J,axiom,
% 16.19/8.94 ! [T: nat] :
% 16.19/8.94 ? [Z5: nat] :
% 16.19/8.94 ! [X5: nat] :
% 16.19/8.94 ( ( ord_less_nat @ X5 @ Z5 )
% 16.19/8.94 => ( ord_less_nat @ X5 @ T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3490_minf_I5_J,axiom,
% 16.19/8.94 ! [T: int] :
% 16.19/8.94 ? [Z5: int] :
% 16.19/8.94 ! [X5: int] :
% 16.19/8.94 ( ( ord_less_int @ X5 @ Z5 )
% 16.19/8.94 => ( ord_less_int @ X5 @ T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3491_minf_I7_J,axiom,
% 16.19/8.94 ! [T: real] :
% 16.19/8.94 ? [Z5: real] :
% 16.19/8.94 ! [X5: real] :
% 16.19/8.94 ( ( ord_less_real @ X5 @ Z5 )
% 16.19/8.94 => ~ ( ord_less_real @ T @ X5 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3492_minf_I7_J,axiom,
% 16.19/8.94 ! [T: rat] :
% 16.19/8.94 ? [Z5: rat] :
% 16.19/8.94 ! [X5: rat] :
% 16.19/8.94 ( ( ord_less_rat @ X5 @ Z5 )
% 16.19/8.94 => ~ ( ord_less_rat @ T @ X5 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3493_minf_I7_J,axiom,
% 16.19/8.94 ! [T: num] :
% 16.19/8.94 ? [Z5: num] :
% 16.19/8.94 ! [X5: num] :
% 16.19/8.94 ( ( ord_less_num @ X5 @ Z5 )
% 16.19/8.94 => ~ ( ord_less_num @ T @ X5 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3494_minf_I7_J,axiom,
% 16.19/8.94 ! [T: nat] :
% 16.19/8.94 ? [Z5: nat] :
% 16.19/8.94 ! [X5: nat] :
% 16.19/8.94 ( ( ord_less_nat @ X5 @ Z5 )
% 16.19/8.94 => ~ ( ord_less_nat @ T @ X5 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3495_minf_I7_J,axiom,
% 16.19/8.94 ! [T: int] :
% 16.19/8.94 ? [Z5: int] :
% 16.19/8.94 ! [X5: int] :
% 16.19/8.94 ( ( ord_less_int @ X5 @ Z5 )
% 16.19/8.94 => ~ ( ord_less_int @ T @ X5 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3496_bounded__Max__nat,axiom,
% 16.19/8.94 ! [P: nat > $o,X: nat,M7: nat] :
% 16.19/8.94 ( ( P @ X )
% 16.19/8.94 => ( ! [X4: nat] :
% 16.19/8.94 ( ( P @ X4 )
% 16.19/8.94 => ( ord_less_eq_nat @ X4 @ M7 ) )
% 16.19/8.94 => ~ ! [M2: nat] :
% 16.19/8.94 ( ( P @ M2 )
% 16.19/8.94 => ~ ! [X5: nat] :
% 16.19/8.94 ( ( P @ X5 )
% 16.19/8.94 => ( ord_less_eq_nat @ X5 @ M2 ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3497_fold__atLeastAtMost__nat_Ocases,axiom,
% 16.19/8.94 ! [X: produc3368934014287244435at_num] :
% 16.19/8.94 ~ ! [F2: nat > num > num,A3: nat,B2: nat,Acc: num] :
% 16.19/8.94 ( X
% 16.19/8.94 != ( produc851828971589881931at_num @ F2 @ ( produc1195630363706982562at_num @ A3 @ ( product_Pair_nat_num @ B2 @ Acc ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3498_fold__atLeastAtMost__nat_Ocases,axiom,
% 16.19/8.94 ! [X: produc4471711990508489141at_nat] :
% 16.19/8.94 ~ ! [F2: nat > nat > nat,A3: nat,B2: nat,Acc: nat] :
% 16.19/8.94 ( X
% 16.19/8.94 != ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A3 @ ( product_Pair_nat_nat @ B2 @ Acc ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3499_periodic__finite__ex,axiom,
% 16.19/8.94 ! [D: int,P: int > $o] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ D )
% 16.19/8.94 => ( ! [X4: int,K2: int] :
% 16.19/8.94 ( ( P @ X4 )
% 16.19/8.94 = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
% 16.19/8.94 => ( ( ? [X3: int] : ( P @ X3 ) )
% 16.19/8.94 = ( ? [X2: int] :
% 16.19/8.94 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
% 16.19/8.94 & ( P @ X2 ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3500_bset_I3_J,axiom,
% 16.19/8.94 ! [D3: int,T: int,B3: set_int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ D3 )
% 16.19/8.94 => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B3 )
% 16.19/8.94 => ! [X5: int] :
% 16.19/8.94 ( ! [Xa3: int] :
% 16.19/8.94 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 => ! [Xb3: int] :
% 16.19/8.94 ( ( member_int @ Xb3 @ B3 )
% 16.19/8.94 => ( X5
% 16.19/8.94 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 16.19/8.94 => ( ( X5 = T )
% 16.19/8.94 => ( ( minus_minus_int @ X5 @ D3 )
% 16.19/8.94 = T ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3501_bset_I4_J,axiom,
% 16.19/8.94 ! [D3: int,T: int,B3: set_int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ D3 )
% 16.19/8.94 => ( ( member_int @ T @ B3 )
% 16.19/8.94 => ! [X5: int] :
% 16.19/8.94 ( ! [Xa3: int] :
% 16.19/8.94 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 => ! [Xb3: int] :
% 16.19/8.94 ( ( member_int @ Xb3 @ B3 )
% 16.19/8.94 => ( X5
% 16.19/8.94 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 16.19/8.94 => ( ( X5 != T )
% 16.19/8.94 => ( ( minus_minus_int @ X5 @ D3 )
% 16.19/8.94 != T ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3502_bset_I5_J,axiom,
% 16.19/8.94 ! [D3: int,B3: set_int,T: int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ D3 )
% 16.19/8.94 => ! [X5: int] :
% 16.19/8.94 ( ! [Xa3: int] :
% 16.19/8.94 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 => ! [Xb3: int] :
% 16.19/8.94 ( ( member_int @ Xb3 @ B3 )
% 16.19/8.94 => ( X5
% 16.19/8.94 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 16.19/8.94 => ( ( ord_less_int @ X5 @ T )
% 16.19/8.94 => ( ord_less_int @ ( minus_minus_int @ X5 @ D3 ) @ T ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3503_bset_I7_J,axiom,
% 16.19/8.94 ! [D3: int,T: int,B3: set_int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ D3 )
% 16.19/8.94 => ( ( member_int @ T @ B3 )
% 16.19/8.94 => ! [X5: int] :
% 16.19/8.94 ( ! [Xa3: int] :
% 16.19/8.94 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 => ! [Xb3: int] :
% 16.19/8.94 ( ( member_int @ Xb3 @ B3 )
% 16.19/8.94 => ( X5
% 16.19/8.94 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 16.19/8.94 => ( ( ord_less_int @ T @ X5 )
% 16.19/8.94 => ( ord_less_int @ T @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3504_aset_I3_J,axiom,
% 16.19/8.94 ! [D3: int,T: int,A2: set_int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ D3 )
% 16.19/8.94 => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
% 16.19/8.94 => ! [X5: int] :
% 16.19/8.94 ( ! [Xa3: int] :
% 16.19/8.94 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 => ! [Xb3: int] :
% 16.19/8.94 ( ( member_int @ Xb3 @ A2 )
% 16.19/8.94 => ( X5
% 16.19/8.94 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 16.19/8.94 => ( ( X5 = T )
% 16.19/8.94 => ( ( plus_plus_int @ X5 @ D3 )
% 16.19/8.94 = T ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3505_aset_I4_J,axiom,
% 16.19/8.94 ! [D3: int,T: int,A2: set_int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ D3 )
% 16.19/8.94 => ( ( member_int @ T @ A2 )
% 16.19/8.94 => ! [X5: int] :
% 16.19/8.94 ( ! [Xa3: int] :
% 16.19/8.94 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 => ! [Xb3: int] :
% 16.19/8.94 ( ( member_int @ Xb3 @ A2 )
% 16.19/8.94 => ( X5
% 16.19/8.94 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 16.19/8.94 => ( ( X5 != T )
% 16.19/8.94 => ( ( plus_plus_int @ X5 @ D3 )
% 16.19/8.94 != T ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3506_aset_I5_J,axiom,
% 16.19/8.94 ! [D3: int,T: int,A2: set_int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ D3 )
% 16.19/8.94 => ( ( member_int @ T @ A2 )
% 16.19/8.94 => ! [X5: int] :
% 16.19/8.94 ( ! [Xa3: int] :
% 16.19/8.94 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 => ! [Xb3: int] :
% 16.19/8.94 ( ( member_int @ Xb3 @ A2 )
% 16.19/8.94 => ( X5
% 16.19/8.94 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 16.19/8.94 => ( ( ord_less_int @ X5 @ T )
% 16.19/8.94 => ( ord_less_int @ ( plus_plus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3507_aset_I7_J,axiom,
% 16.19/8.94 ! [D3: int,A2: set_int,T: int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ D3 )
% 16.19/8.94 => ! [X5: int] :
% 16.19/8.94 ( ! [Xa3: int] :
% 16.19/8.94 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 => ! [Xb3: int] :
% 16.19/8.94 ( ( member_int @ Xb3 @ A2 )
% 16.19/8.94 => ( X5
% 16.19/8.94 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 16.19/8.94 => ( ( ord_less_int @ T @ X5 )
% 16.19/8.94 => ( ord_less_int @ T @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3508_bset_I6_J,axiom,
% 16.19/8.94 ! [D3: int,B3: set_int,T: int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ D3 )
% 16.19/8.94 => ! [X5: int] :
% 16.19/8.94 ( ! [Xa3: int] :
% 16.19/8.94 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 => ! [Xb3: int] :
% 16.19/8.94 ( ( member_int @ Xb3 @ B3 )
% 16.19/8.94 => ( X5
% 16.19/8.94 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 16.19/8.94 => ( ( ord_less_eq_int @ X5 @ T )
% 16.19/8.94 => ( ord_less_eq_int @ ( minus_minus_int @ X5 @ D3 ) @ T ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3509_bset_I8_J,axiom,
% 16.19/8.94 ! [D3: int,T: int,B3: set_int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ D3 )
% 16.19/8.94 => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B3 )
% 16.19/8.94 => ! [X5: int] :
% 16.19/8.94 ( ! [Xa3: int] :
% 16.19/8.94 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 => ! [Xb3: int] :
% 16.19/8.94 ( ( member_int @ Xb3 @ B3 )
% 16.19/8.94 => ( X5
% 16.19/8.94 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 16.19/8.94 => ( ( ord_less_eq_int @ T @ X5 )
% 16.19/8.94 => ( ord_less_eq_int @ T @ ( minus_minus_int @ X5 @ D3 ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3510_aset_I6_J,axiom,
% 16.19/8.94 ! [D3: int,T: int,A2: set_int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ D3 )
% 16.19/8.94 => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
% 16.19/8.94 => ! [X5: int] :
% 16.19/8.94 ( ! [Xa3: int] :
% 16.19/8.94 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 => ! [Xb3: int] :
% 16.19/8.94 ( ( member_int @ Xb3 @ A2 )
% 16.19/8.94 => ( X5
% 16.19/8.94 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 16.19/8.94 => ( ( ord_less_eq_int @ X5 @ T )
% 16.19/8.94 => ( ord_less_eq_int @ ( plus_plus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3511_aset_I8_J,axiom,
% 16.19/8.94 ! [D3: int,A2: set_int,T: int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ D3 )
% 16.19/8.94 => ! [X5: int] :
% 16.19/8.94 ( ! [Xa3: int] :
% 16.19/8.94 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 => ! [Xb3: int] :
% 16.19/8.94 ( ( member_int @ Xb3 @ A2 )
% 16.19/8.94 => ( X5
% 16.19/8.94 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 16.19/8.94 => ( ( ord_less_eq_int @ T @ X5 )
% 16.19/8.94 => ( ord_less_eq_int @ T @ ( plus_plus_int @ X5 @ D3 ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3512_cppi,axiom,
% 16.19/8.94 ! [D3: int,P: int > $o,P6: int > $o,A2: set_int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ D3 )
% 16.19/8.94 => ( ? [Z4: int] :
% 16.19/8.94 ! [X4: int] :
% 16.19/8.94 ( ( ord_less_int @ Z4 @ X4 )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 = ( P6 @ X4 ) ) )
% 16.19/8.94 => ( ! [X4: int] :
% 16.19/8.94 ( ! [Xa: int] :
% 16.19/8.94 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 => ! [Xb: int] :
% 16.19/8.94 ( ( member_int @ Xb @ A2 )
% 16.19/8.94 => ( X4
% 16.19/8.94 != ( minus_minus_int @ Xb @ Xa ) ) ) )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 => ( P @ ( plus_plus_int @ X4 @ D3 ) ) ) )
% 16.19/8.94 => ( ! [X4: int,K2: int] :
% 16.19/8.94 ( ( P6 @ X4 )
% 16.19/8.94 = ( P6 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D3 ) ) ) )
% 16.19/8.94 => ( ( ? [X3: int] : ( P @ X3 ) )
% 16.19/8.94 = ( ? [X2: int] :
% 16.19/8.94 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 & ( P6 @ X2 ) )
% 16.19/8.94 | ? [X2: int] :
% 16.19/8.94 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 & ? [Y2: int] :
% 16.19/8.94 ( ( member_int @ Y2 @ A2 )
% 16.19/8.94 & ( P @ ( minus_minus_int @ Y2 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3513_cpmi,axiom,
% 16.19/8.94 ! [D3: int,P: int > $o,P6: int > $o,B3: set_int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ D3 )
% 16.19/8.94 => ( ? [Z4: int] :
% 16.19/8.94 ! [X4: int] :
% 16.19/8.94 ( ( ord_less_int @ X4 @ Z4 )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 = ( P6 @ X4 ) ) )
% 16.19/8.94 => ( ! [X4: int] :
% 16.19/8.94 ( ! [Xa: int] :
% 16.19/8.94 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 => ! [Xb: int] :
% 16.19/8.94 ( ( member_int @ Xb @ B3 )
% 16.19/8.94 => ( X4
% 16.19/8.94 != ( plus_plus_int @ Xb @ Xa ) ) ) )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 => ( P @ ( minus_minus_int @ X4 @ D3 ) ) ) )
% 16.19/8.94 => ( ! [X4: int,K2: int] :
% 16.19/8.94 ( ( P6 @ X4 )
% 16.19/8.94 = ( P6 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D3 ) ) ) )
% 16.19/8.94 => ( ( ? [X3: int] : ( P @ X3 ) )
% 16.19/8.94 = ( ? [X2: int] :
% 16.19/8.94 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 & ( P6 @ X2 ) )
% 16.19/8.94 | ? [X2: int] :
% 16.19/8.94 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.94 & ? [Y2: int] :
% 16.19/8.94 ( ( member_int @ Y2 @ B3 )
% 16.19/8.94 & ( P @ ( plus_plus_int @ Y2 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3514_pinf_I6_J,axiom,
% 16.19/8.94 ! [T: real] :
% 16.19/8.94 ? [Z5: real] :
% 16.19/8.94 ! [X5: real] :
% 16.19/8.94 ( ( ord_less_real @ Z5 @ X5 )
% 16.19/8.94 => ~ ( ord_less_eq_real @ X5 @ T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3515_pinf_I6_J,axiom,
% 16.19/8.94 ! [T: rat] :
% 16.19/8.94 ? [Z5: rat] :
% 16.19/8.94 ! [X5: rat] :
% 16.19/8.94 ( ( ord_less_rat @ Z5 @ X5 )
% 16.19/8.94 => ~ ( ord_less_eq_rat @ X5 @ T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3516_pinf_I6_J,axiom,
% 16.19/8.94 ! [T: num] :
% 16.19/8.94 ? [Z5: num] :
% 16.19/8.94 ! [X5: num] :
% 16.19/8.94 ( ( ord_less_num @ Z5 @ X5 )
% 16.19/8.94 => ~ ( ord_less_eq_num @ X5 @ T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3517_pinf_I6_J,axiom,
% 16.19/8.94 ! [T: nat] :
% 16.19/8.94 ? [Z5: nat] :
% 16.19/8.94 ! [X5: nat] :
% 16.19/8.94 ( ( ord_less_nat @ Z5 @ X5 )
% 16.19/8.94 => ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3518_pinf_I6_J,axiom,
% 16.19/8.94 ! [T: int] :
% 16.19/8.94 ? [Z5: int] :
% 16.19/8.94 ! [X5: int] :
% 16.19/8.94 ( ( ord_less_int @ Z5 @ X5 )
% 16.19/8.94 => ~ ( ord_less_eq_int @ X5 @ T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3519_pinf_I8_J,axiom,
% 16.19/8.94 ! [T: real] :
% 16.19/8.94 ? [Z5: real] :
% 16.19/8.94 ! [X5: real] :
% 16.19/8.94 ( ( ord_less_real @ Z5 @ X5 )
% 16.19/8.94 => ( ord_less_eq_real @ T @ X5 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3520_pinf_I8_J,axiom,
% 16.19/8.94 ! [T: rat] :
% 16.19/8.94 ? [Z5: rat] :
% 16.19/8.94 ! [X5: rat] :
% 16.19/8.94 ( ( ord_less_rat @ Z5 @ X5 )
% 16.19/8.94 => ( ord_less_eq_rat @ T @ X5 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3521_pinf_I8_J,axiom,
% 16.19/8.94 ! [T: num] :
% 16.19/8.94 ? [Z5: num] :
% 16.19/8.94 ! [X5: num] :
% 16.19/8.94 ( ( ord_less_num @ Z5 @ X5 )
% 16.19/8.94 => ( ord_less_eq_num @ T @ X5 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3522_pinf_I8_J,axiom,
% 16.19/8.94 ! [T: nat] :
% 16.19/8.94 ? [Z5: nat] :
% 16.19/8.94 ! [X5: nat] :
% 16.19/8.94 ( ( ord_less_nat @ Z5 @ X5 )
% 16.19/8.94 => ( ord_less_eq_nat @ T @ X5 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3523_pinf_I8_J,axiom,
% 16.19/8.94 ! [T: int] :
% 16.19/8.94 ? [Z5: int] :
% 16.19/8.94 ! [X5: int] :
% 16.19/8.94 ( ( ord_less_int @ Z5 @ X5 )
% 16.19/8.94 => ( ord_less_eq_int @ T @ X5 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3524_minf_I6_J,axiom,
% 16.19/8.94 ! [T: real] :
% 16.19/8.94 ? [Z5: real] :
% 16.19/8.94 ! [X5: real] :
% 16.19/8.94 ( ( ord_less_real @ X5 @ Z5 )
% 16.19/8.94 => ( ord_less_eq_real @ X5 @ T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3525_minf_I6_J,axiom,
% 16.19/8.94 ! [T: rat] :
% 16.19/8.94 ? [Z5: rat] :
% 16.19/8.94 ! [X5: rat] :
% 16.19/8.94 ( ( ord_less_rat @ X5 @ Z5 )
% 16.19/8.94 => ( ord_less_eq_rat @ X5 @ T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3526_minf_I6_J,axiom,
% 16.19/8.94 ! [T: num] :
% 16.19/8.94 ? [Z5: num] :
% 16.19/8.94 ! [X5: num] :
% 16.19/8.94 ( ( ord_less_num @ X5 @ Z5 )
% 16.19/8.94 => ( ord_less_eq_num @ X5 @ T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3527_minf_I6_J,axiom,
% 16.19/8.94 ! [T: nat] :
% 16.19/8.94 ? [Z5: nat] :
% 16.19/8.94 ! [X5: nat] :
% 16.19/8.94 ( ( ord_less_nat @ X5 @ Z5 )
% 16.19/8.94 => ( ord_less_eq_nat @ X5 @ T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3528_minf_I6_J,axiom,
% 16.19/8.94 ! [T: int] :
% 16.19/8.94 ? [Z5: int] :
% 16.19/8.94 ! [X5: int] :
% 16.19/8.94 ( ( ord_less_int @ X5 @ Z5 )
% 16.19/8.94 => ( ord_less_eq_int @ X5 @ T ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3529_minf_I8_J,axiom,
% 16.19/8.94 ! [T: real] :
% 16.19/8.94 ? [Z5: real] :
% 16.19/8.94 ! [X5: real] :
% 16.19/8.94 ( ( ord_less_real @ X5 @ Z5 )
% 16.19/8.94 => ~ ( ord_less_eq_real @ T @ X5 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3530_minf_I8_J,axiom,
% 16.19/8.94 ! [T: rat] :
% 16.19/8.94 ? [Z5: rat] :
% 16.19/8.94 ! [X5: rat] :
% 16.19/8.94 ( ( ord_less_rat @ X5 @ Z5 )
% 16.19/8.94 => ~ ( ord_less_eq_rat @ T @ X5 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3531_minf_I8_J,axiom,
% 16.19/8.94 ! [T: num] :
% 16.19/8.94 ? [Z5: num] :
% 16.19/8.94 ! [X5: num] :
% 16.19/8.94 ( ( ord_less_num @ X5 @ Z5 )
% 16.19/8.94 => ~ ( ord_less_eq_num @ T @ X5 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3532_minf_I8_J,axiom,
% 16.19/8.94 ! [T: nat] :
% 16.19/8.94 ? [Z5: nat] :
% 16.19/8.94 ! [X5: nat] :
% 16.19/8.94 ( ( ord_less_nat @ X5 @ Z5 )
% 16.19/8.94 => ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3533_minf_I8_J,axiom,
% 16.19/8.94 ! [T: int] :
% 16.19/8.94 ? [Z5: int] :
% 16.19/8.94 ! [X5: int] :
% 16.19/8.94 ( ( ord_less_int @ X5 @ Z5 )
% 16.19/8.94 => ~ ( ord_less_eq_int @ T @ X5 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3534_inf__period_I2_J,axiom,
% 16.19/8.94 ! [P: real > $o,D3: real,Q: real > $o] :
% 16.19/8.94 ( ! [X4: real,K2: real] :
% 16.19/8.94 ( ( P @ X4 )
% 16.19/8.94 = ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K2 @ D3 ) ) ) )
% 16.19/8.94 => ( ! [X4: real,K2: real] :
% 16.19/8.94 ( ( Q @ X4 )
% 16.19/8.94 = ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K2 @ D3 ) ) ) )
% 16.19/8.94 => ! [X5: real,K4: real] :
% 16.19/8.94 ( ( ( P @ X5 )
% 16.19/8.94 | ( Q @ X5 ) )
% 16.19/8.94 = ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) )
% 16.19/8.94 | ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3535_inf__period_I2_J,axiom,
% 16.19/8.94 ! [P: rat > $o,D3: rat,Q: rat > $o] :
% 16.19/8.94 ( ! [X4: rat,K2: rat] :
% 16.19/8.94 ( ( P @ X4 )
% 16.19/8.94 = ( P @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K2 @ D3 ) ) ) )
% 16.19/8.94 => ( ! [X4: rat,K2: rat] :
% 16.19/8.94 ( ( Q @ X4 )
% 16.19/8.94 = ( Q @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K2 @ D3 ) ) ) )
% 16.19/8.94 => ! [X5: rat,K4: rat] :
% 16.19/8.94 ( ( ( P @ X5 )
% 16.19/8.94 | ( Q @ X5 ) )
% 16.19/8.94 = ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) )
% 16.19/8.94 | ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3536_inf__period_I2_J,axiom,
% 16.19/8.94 ! [P: int > $o,D3: int,Q: int > $o] :
% 16.19/8.94 ( ! [X4: int,K2: int] :
% 16.19/8.94 ( ( P @ X4 )
% 16.19/8.94 = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D3 ) ) ) )
% 16.19/8.94 => ( ! [X4: int,K2: int] :
% 16.19/8.94 ( ( Q @ X4 )
% 16.19/8.94 = ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D3 ) ) ) )
% 16.19/8.94 => ! [X5: int,K4: int] :
% 16.19/8.94 ( ( ( P @ X5 )
% 16.19/8.94 | ( Q @ X5 ) )
% 16.19/8.94 = ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) )
% 16.19/8.94 | ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3537_inf__period_I1_J,axiom,
% 16.19/8.94 ! [P: real > $o,D3: real,Q: real > $o] :
% 16.19/8.94 ( ! [X4: real,K2: real] :
% 16.19/8.94 ( ( P @ X4 )
% 16.19/8.94 = ( P @ ( minus_minus_real @ X4 @ ( times_times_real @ K2 @ D3 ) ) ) )
% 16.19/8.94 => ( ! [X4: real,K2: real] :
% 16.19/8.94 ( ( Q @ X4 )
% 16.19/8.94 = ( Q @ ( minus_minus_real @ X4 @ ( times_times_real @ K2 @ D3 ) ) ) )
% 16.19/8.94 => ! [X5: real,K4: real] :
% 16.19/8.94 ( ( ( P @ X5 )
% 16.19/8.94 & ( Q @ X5 ) )
% 16.19/8.94 = ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) )
% 16.19/8.94 & ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3538_inf__period_I1_J,axiom,
% 16.19/8.94 ! [P: rat > $o,D3: rat,Q: rat > $o] :
% 16.19/8.94 ( ! [X4: rat,K2: rat] :
% 16.19/8.94 ( ( P @ X4 )
% 16.19/8.94 = ( P @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K2 @ D3 ) ) ) )
% 16.19/8.94 => ( ! [X4: rat,K2: rat] :
% 16.19/8.94 ( ( Q @ X4 )
% 16.19/8.94 = ( Q @ ( minus_minus_rat @ X4 @ ( times_times_rat @ K2 @ D3 ) ) ) )
% 16.19/8.94 => ! [X5: rat,K4: rat] :
% 16.19/8.94 ( ( ( P @ X5 )
% 16.19/8.94 & ( Q @ X5 ) )
% 16.19/8.94 = ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) )
% 16.19/8.94 & ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3539_inf__period_I1_J,axiom,
% 16.19/8.94 ! [P: int > $o,D3: int,Q: int > $o] :
% 16.19/8.94 ( ! [X4: int,K2: int] :
% 16.19/8.94 ( ( P @ X4 )
% 16.19/8.94 = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D3 ) ) ) )
% 16.19/8.94 => ( ! [X4: int,K2: int] :
% 16.19/8.94 ( ( Q @ X4 )
% 16.19/8.94 = ( Q @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D3 ) ) ) )
% 16.19/8.94 => ! [X5: int,K4: int] :
% 16.19/8.94 ( ( ( P @ X5 )
% 16.19/8.94 & ( Q @ X5 ) )
% 16.19/8.94 = ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) )
% 16.19/8.94 & ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3540_atLeastatMost__psubset__iff,axiom,
% 16.19/8.94 ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
% 16.19/8.94 ( ( ord_less_set_set_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
% 16.19/8.94 = ( ( ~ ( ord_less_eq_set_nat @ A @ B )
% 16.19/8.94 | ( ( ord_less_eq_set_nat @ C @ A )
% 16.19/8.94 & ( ord_less_eq_set_nat @ B @ D )
% 16.19/8.94 & ( ( ord_less_set_nat @ C @ A )
% 16.19/8.94 | ( ord_less_set_nat @ B @ D ) ) ) )
% 16.19/8.94 & ( ord_less_eq_set_nat @ C @ D ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3541_atLeastatMost__psubset__iff,axiom,
% 16.19/8.94 ! [A: rat,B: rat,C: rat,D: rat] :
% 16.19/8.94 ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 16.19/8.94 = ( ( ~ ( ord_less_eq_rat @ A @ B )
% 16.19/8.94 | ( ( ord_less_eq_rat @ C @ A )
% 16.19/8.94 & ( ord_less_eq_rat @ B @ D )
% 16.19/8.94 & ( ( ord_less_rat @ C @ A )
% 16.19/8.94 | ( ord_less_rat @ B @ D ) ) ) )
% 16.19/8.94 & ( ord_less_eq_rat @ C @ D ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3542_atLeastatMost__psubset__iff,axiom,
% 16.19/8.94 ! [A: num,B: num,C: num,D: num] :
% 16.19/8.94 ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 16.19/8.94 = ( ( ~ ( ord_less_eq_num @ A @ B )
% 16.19/8.94 | ( ( ord_less_eq_num @ C @ A )
% 16.19/8.94 & ( ord_less_eq_num @ B @ D )
% 16.19/8.94 & ( ( ord_less_num @ C @ A )
% 16.19/8.94 | ( ord_less_num @ B @ D ) ) ) )
% 16.19/8.94 & ( ord_less_eq_num @ C @ D ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3543_atLeastatMost__psubset__iff,axiom,
% 16.19/8.94 ! [A: nat,B: nat,C: nat,D: nat] :
% 16.19/8.94 ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 16.19/8.94 = ( ( ~ ( ord_less_eq_nat @ A @ B )
% 16.19/8.94 | ( ( ord_less_eq_nat @ C @ A )
% 16.19/8.94 & ( ord_less_eq_nat @ B @ D )
% 16.19/8.94 & ( ( ord_less_nat @ C @ A )
% 16.19/8.94 | ( ord_less_nat @ B @ D ) ) ) )
% 16.19/8.94 & ( ord_less_eq_nat @ C @ D ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3544_atLeastatMost__psubset__iff,axiom,
% 16.19/8.94 ! [A: int,B: int,C: int,D: int] :
% 16.19/8.94 ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 16.19/8.94 = ( ( ~ ( ord_less_eq_int @ A @ B )
% 16.19/8.94 | ( ( ord_less_eq_int @ C @ A )
% 16.19/8.94 & ( ord_less_eq_int @ B @ D )
% 16.19/8.94 & ( ( ord_less_int @ C @ A )
% 16.19/8.94 | ( ord_less_int @ B @ D ) ) ) )
% 16.19/8.94 & ( ord_less_eq_int @ C @ D ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3545_atLeastatMost__psubset__iff,axiom,
% 16.19/8.94 ! [A: real,B: real,C: real,D: real] :
% 16.19/8.94 ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 16.19/8.94 = ( ( ~ ( ord_less_eq_real @ A @ B )
% 16.19/8.94 | ( ( ord_less_eq_real @ C @ A )
% 16.19/8.94 & ( ord_less_eq_real @ B @ D )
% 16.19/8.94 & ( ( ord_less_real @ C @ A )
% 16.19/8.94 | ( ord_less_real @ B @ D ) ) ) )
% 16.19/8.94 & ( ord_less_eq_real @ C @ D ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3546_plusinfinity,axiom,
% 16.19/8.94 ! [D: int,P6: int > $o,P: int > $o] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ D )
% 16.19/8.94 => ( ! [X4: int,K2: int] :
% 16.19/8.94 ( ( P6 @ X4 )
% 16.19/8.94 = ( P6 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
% 16.19/8.94 => ( ? [Z4: int] :
% 16.19/8.94 ! [X4: int] :
% 16.19/8.94 ( ( ord_less_int @ Z4 @ X4 )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 = ( P6 @ X4 ) ) )
% 16.19/8.94 => ( ? [X_12: int] : ( P6 @ X_12 )
% 16.19/8.94 => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3547_minusinfinity,axiom,
% 16.19/8.94 ! [D: int,P1: int > $o,P: int > $o] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ D )
% 16.19/8.94 => ( ! [X4: int,K2: int] :
% 16.19/8.94 ( ( P1 @ X4 )
% 16.19/8.94 = ( P1 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
% 16.19/8.94 => ( ? [Z4: int] :
% 16.19/8.94 ! [X4: int] :
% 16.19/8.94 ( ( ord_less_int @ X4 @ Z4 )
% 16.19/8.94 => ( ( P @ X4 )
% 16.19/8.94 = ( P1 @ X4 ) ) )
% 16.19/8.94 => ( ? [X_12: int] : ( P1 @ X_12 )
% 16.19/8.94 => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3548_flip__bit__Suc,axiom,
% 16.19/8.94 ! [N2: nat,A: code_integer] :
% 16.19/8.94 ( ( bit_se1345352211410354436nteger @ ( suc @ N2 ) @ A )
% 16.19/8.94 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3549_flip__bit__Suc,axiom,
% 16.19/8.94 ! [N2: nat,A: int] :
% 16.19/8.94 ( ( bit_se2159334234014336723it_int @ ( suc @ N2 ) @ A )
% 16.19/8.94 = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3550_flip__bit__Suc,axiom,
% 16.19/8.94 ! [N2: nat,A: nat] :
% 16.19/8.94 ( ( bit_se2161824704523386999it_nat @ ( suc @ N2 ) @ A )
% 16.19/8.94 = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3551_Bolzano,axiom,
% 16.19/8.94 ! [A: real,B: real,P: real > real > $o] :
% 16.19/8.94 ( ( ord_less_eq_real @ A @ B )
% 16.19/8.94 => ( ! [A3: real,B2: real,C3: real] :
% 16.19/8.94 ( ( P @ A3 @ B2 )
% 16.19/8.94 => ( ( P @ B2 @ C3 )
% 16.19/8.94 => ( ( ord_less_eq_real @ A3 @ B2 )
% 16.19/8.94 => ( ( ord_less_eq_real @ B2 @ C3 )
% 16.19/8.94 => ( P @ A3 @ C3 ) ) ) ) )
% 16.19/8.94 => ( ! [X4: real] :
% 16.19/8.94 ( ( ord_less_eq_real @ A @ X4 )
% 16.19/8.94 => ( ( ord_less_eq_real @ X4 @ B )
% 16.19/8.94 => ? [D5: real] :
% 16.19/8.94 ( ( ord_less_real @ zero_zero_real @ D5 )
% 16.19/8.94 & ! [A3: real,B2: real] :
% 16.19/8.94 ( ( ( ord_less_eq_real @ A3 @ X4 )
% 16.19/8.94 & ( ord_less_eq_real @ X4 @ B2 )
% 16.19/8.94 & ( ord_less_real @ ( minus_minus_real @ B2 @ A3 ) @ D5 ) )
% 16.19/8.94 => ( P @ A3 @ B2 ) ) ) ) )
% 16.19/8.94 => ( P @ A @ B ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3552_mult__le__cancel__iff2,axiom,
% 16.19/8.94 ! [Z: real,X: real,Y: real] :
% 16.19/8.94 ( ( ord_less_real @ zero_zero_real @ Z )
% 16.19/8.94 => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ ( times_times_real @ Z @ Y ) )
% 16.19/8.94 = ( ord_less_eq_real @ X @ Y ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3553_mult__le__cancel__iff2,axiom,
% 16.19/8.94 ! [Z: rat,X: rat,Y: rat] :
% 16.19/8.94 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 16.19/8.94 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ ( times_times_rat @ Z @ Y ) )
% 16.19/8.94 = ( ord_less_eq_rat @ X @ Y ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3554_mult__le__cancel__iff2,axiom,
% 16.19/8.94 ! [Z: int,X: int,Y: int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ Z )
% 16.19/8.94 => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y ) )
% 16.19/8.94 = ( ord_less_eq_int @ X @ Y ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3555_mult__le__cancel__iff1,axiom,
% 16.19/8.94 ! [Z: real,X: real,Y: real] :
% 16.19/8.94 ( ( ord_less_real @ zero_zero_real @ Z )
% 16.19/8.94 => ( ( ord_less_eq_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
% 16.19/8.94 = ( ord_less_eq_real @ X @ Y ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3556_mult__le__cancel__iff1,axiom,
% 16.19/8.94 ! [Z: rat,X: rat,Y: rat] :
% 16.19/8.94 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 16.19/8.94 => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
% 16.19/8.94 = ( ord_less_eq_rat @ X @ Y ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3557_mult__le__cancel__iff1,axiom,
% 16.19/8.94 ! [Z: int,X: int,Y: int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ Z )
% 16.19/8.94 => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
% 16.19/8.94 = ( ord_less_eq_int @ X @ Y ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3558_divides__aux__eq,axiom,
% 16.19/8.94 ! [Q2: nat,R: nat] :
% 16.19/8.94 ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q2 @ R ) )
% 16.19/8.94 = ( R = zero_zero_nat ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3559_divides__aux__eq,axiom,
% 16.19/8.94 ! [Q2: int,R: int] :
% 16.19/8.94 ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q2 @ R ) )
% 16.19/8.94 = ( R = zero_zero_int ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3560_neg__eucl__rel__int__mult__2,axiom,
% 16.19/8.94 ! [B: int,A: int,Q2: int,R: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ B @ zero_zero_int )
% 16.19/8.94 => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q2 @ R ) )
% 16.19/8.94 => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q2 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R ) @ one_one_int ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3561_flip__bit__nonnegative__int__iff,axiom,
% 16.19/8.94 ! [N2: nat,K: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N2 @ K ) )
% 16.19/8.94 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3562_flip__bit__negative__int__iff,axiom,
% 16.19/8.94 ! [N2: nat,K: int] :
% 16.19/8.94 ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N2 @ K ) @ zero_zero_int )
% 16.19/8.94 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3563_div__int__unique,axiom,
% 16.19/8.94 ! [K: int,L2: int,Q2: int,R: int] :
% 16.19/8.94 ( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R ) )
% 16.19/8.94 => ( ( divide_divide_int @ K @ L2 )
% 16.19/8.94 = Q2 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3564_eucl__rel__int,axiom,
% 16.19/8.94 ! [K: int,L2: int] : ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L2 ) @ ( modulo_modulo_int @ K @ L2 ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3565_eucl__rel__int__iff,axiom,
% 16.19/8.94 ! [K: int,L2: int,Q2: int,R: int] :
% 16.19/8.94 ( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R ) )
% 16.19/8.94 = ( ( K
% 16.19/8.94 = ( plus_plus_int @ ( times_times_int @ L2 @ Q2 ) @ R ) )
% 16.19/8.94 & ( ( ord_less_int @ zero_zero_int @ L2 )
% 16.19/8.94 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 16.19/8.94 & ( ord_less_int @ R @ L2 ) ) )
% 16.19/8.94 & ( ~ ( ord_less_int @ zero_zero_int @ L2 )
% 16.19/8.94 => ( ( ( ord_less_int @ L2 @ zero_zero_int )
% 16.19/8.94 => ( ( ord_less_int @ L2 @ R )
% 16.19/8.94 & ( ord_less_eq_int @ R @ zero_zero_int ) ) )
% 16.19/8.94 & ( ~ ( ord_less_int @ L2 @ zero_zero_int )
% 16.19/8.94 => ( Q2 = zero_zero_int ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3566_mult__less__iff1,axiom,
% 16.19/8.94 ! [Z: real,X: real,Y: real] :
% 16.19/8.94 ( ( ord_less_real @ zero_zero_real @ Z )
% 16.19/8.94 => ( ( ord_less_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y @ Z ) )
% 16.19/8.94 = ( ord_less_real @ X @ Y ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3567_mult__less__iff1,axiom,
% 16.19/8.94 ! [Z: rat,X: rat,Y: rat] :
% 16.19/8.94 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 16.19/8.94 => ( ( ord_less_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y @ Z ) )
% 16.19/8.94 = ( ord_less_rat @ X @ Y ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3568_mult__less__iff1,axiom,
% 16.19/8.94 ! [Z: int,X: int,Y: int] :
% 16.19/8.94 ( ( ord_less_int @ zero_zero_int @ Z )
% 16.19/8.94 => ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
% 16.19/8.94 = ( ord_less_int @ X @ Y ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3569_pos__eucl__rel__int__mult__2,axiom,
% 16.19/8.94 ! [B: int,A: int,Q2: int,R: int] :
% 16.19/8.94 ( ( ord_less_eq_int @ zero_zero_int @ B )
% 16.19/8.94 => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R ) )
% 16.19/8.94 => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q2 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3570_product__nth,axiom,
% 16.19/8.94 ! [N2: nat,Xs: list_num,Ys: list_num] :
% 16.19/8.94 ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_num @ Xs ) @ ( size_size_list_num @ Ys ) ) )
% 16.19/8.94 => ( ( nth_Pr6456567536196504476um_num @ ( product_num_num @ Xs @ Ys ) @ N2 )
% 16.19/8.94 = ( product_Pair_num_num @ ( nth_num @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_num @ Ys ) ) ) @ ( nth_num @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_num @ Ys ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3571_product__nth,axiom,
% 16.19/8.94 ! [N2: nat,Xs: list_nat,Ys: list_num] :
% 16.19/8.94 ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_num @ Ys ) ) )
% 16.19/8.94 => ( ( nth_Pr8326237132889035090at_num @ ( product_nat_num @ Xs @ Ys ) @ N2 )
% 16.19/8.94 = ( product_Pair_nat_num @ ( nth_nat @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_num @ Ys ) ) ) @ ( nth_num @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_num @ Ys ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3572_product__nth,axiom,
% 16.19/8.94 ! [N2: nat,Xs: list_nat,Ys: list_nat] :
% 16.19/8.94 ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
% 16.19/8.94 => ( ( nth_Pr7617993195940197384at_nat @ ( product_nat_nat @ Xs @ Ys ) @ N2 )
% 16.19/8.94 = ( product_Pair_nat_nat @ ( nth_nat @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3573_product__nth,axiom,
% 16.19/8.94 ! [N2: nat,Xs: list_Code_integer,Ys: list_o] :
% 16.19/8.94 ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s3445333598471063425nteger @ Xs ) @ ( size_size_list_o @ Ys ) ) )
% 16.19/8.94 => ( ( nth_Pr8522763379788166057eger_o @ ( produc3607205314601156340eger_o @ Xs @ Ys ) @ N2 )
% 16.19/8.94 = ( produc6677183202524767010eger_o @ ( nth_Code_integer @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3574_product__nth,axiom,
% 16.19/8.94 ! [N2: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 16.19/8.94 ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
% 16.19/8.94 => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys ) @ N2 )
% 16.19/8.94 = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3575_product__nth,axiom,
% 16.19/8.94 ! [N2: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
% 16.19/8.94 ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys ) ) )
% 16.19/8.94 => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys ) @ N2 )
% 16.19/8.94 = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3576_product__nth,axiom,
% 16.19/8.94 ! [N2: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
% 16.19/8.94 ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys ) ) )
% 16.19/8.94 => ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys ) @ N2 )
% 16.19/8.94 = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3577_product__nth,axiom,
% 16.19/8.94 ! [N2: nat,Xs: list_o,Ys: list_VEBT_VEBT] :
% 16.19/8.94 ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
% 16.19/8.94 => ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs @ Ys ) @ N2 )
% 16.19/8.94 = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs @ ( divide_divide_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3578_product__nth,axiom,
% 16.19/8.94 ! [N2: nat,Xs: list_o,Ys: list_o] :
% 16.19/8.94 ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) ) )
% 16.19/8.94 => ( ( nth_Product_prod_o_o @ ( product_o_o @ Xs @ Ys ) @ N2 )
% 16.19/8.94 = ( product_Pair_o_o @ ( nth_o @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3579_product__nth,axiom,
% 16.19/8.94 ! [N2: nat,Xs: list_o,Ys: list_int] :
% 16.19/8.94 ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys ) ) )
% 16.19/8.94 => ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs @ Ys ) @ N2 )
% 16.19/8.94 = ( product_Pair_o_int @ ( nth_o @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3580_obtain__set__pred,axiom,
% 16.19/8.94 ! [Z: nat,X: nat,A2: set_nat] :
% 16.19/8.94 ( ( ord_less_nat @ Z @ X )
% 16.19/8.94 => ( ( vEBT_VEBT_min_in_set @ A2 @ Z )
% 16.19/8.94 => ( ( finite_finite_nat @ A2 )
% 16.19/8.94 => ? [X_1: nat] : ( vEBT_is_pred_in_set @ A2 @ X @ X_1 ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3581_obtain__set__succ,axiom,
% 16.19/8.94 ! [X: nat,Z: nat,A2: set_nat,B3: set_nat] :
% 16.19/8.94 ( ( ord_less_nat @ X @ Z )
% 16.19/8.94 => ( ( vEBT_VEBT_max_in_set @ A2 @ Z )
% 16.19/8.94 => ( ( finite_finite_nat @ B3 )
% 16.19/8.94 => ( ( A2 = B3 )
% 16.19/8.94 => ? [X_1: nat] : ( vEBT_is_succ_in_set @ A2 @ X @ X_1 ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3582_set__vebt__finite,axiom,
% 16.19/8.94 ! [T: vEBT_VEBT,N2: nat] :
% 16.19/8.94 ( ( vEBT_invar_vebt @ T @ N2 )
% 16.19/8.94 => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3583_succ__none__empty,axiom,
% 16.19/8.94 ! [Xs: set_nat,A: nat] :
% 16.19/8.94 ( ~ ? [X_1: nat] : ( vEBT_is_succ_in_set @ Xs @ A @ X_1 )
% 16.19/8.94 => ( ( finite_finite_nat @ Xs )
% 16.19/8.94 => ~ ? [X5: nat] :
% 16.19/8.94 ( ( member_nat @ X5 @ Xs )
% 16.19/8.94 & ( ord_less_nat @ A @ X5 ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3584_pred__none__empty,axiom,
% 16.19/8.94 ! [Xs: set_nat,A: nat] :
% 16.19/8.94 ( ~ ? [X_1: nat] : ( vEBT_is_pred_in_set @ Xs @ A @ X_1 )
% 16.19/8.94 => ( ( finite_finite_nat @ Xs )
% 16.19/8.94 => ~ ? [X5: nat] :
% 16.19/8.94 ( ( member_nat @ X5 @ Xs )
% 16.19/8.94 & ( ord_less_nat @ X5 @ A ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3585_old_Oprod_Oinject,axiom,
% 16.19/8.94 ! [A: code_integer,B: $o,A6: code_integer,B6: $o] :
% 16.19/8.94 ( ( ( produc6677183202524767010eger_o @ A @ B )
% 16.19/8.94 = ( produc6677183202524767010eger_o @ A6 @ B6 ) )
% 16.19/8.94 = ( ( A = A6 )
% 16.19/8.94 & ( B = B6 ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3586_old_Oprod_Oinject,axiom,
% 16.19/8.94 ! [A: num,B: num,A6: num,B6: num] :
% 16.19/8.94 ( ( ( product_Pair_num_num @ A @ B )
% 16.19/8.94 = ( product_Pair_num_num @ A6 @ B6 ) )
% 16.19/8.94 = ( ( A = A6 )
% 16.19/8.94 & ( B = B6 ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3587_old_Oprod_Oinject,axiom,
% 16.19/8.94 ! [A: nat,B: num,A6: nat,B6: num] :
% 16.19/8.94 ( ( ( product_Pair_nat_num @ A @ B )
% 16.19/8.94 = ( product_Pair_nat_num @ A6 @ B6 ) )
% 16.19/8.94 = ( ( A = A6 )
% 16.19/8.94 & ( B = B6 ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3588_old_Oprod_Oinject,axiom,
% 16.19/8.94 ! [A: nat,B: nat,A6: nat,B6: nat] :
% 16.19/8.94 ( ( ( product_Pair_nat_nat @ A @ B )
% 16.19/8.94 = ( product_Pair_nat_nat @ A6 @ B6 ) )
% 16.19/8.94 = ( ( A = A6 )
% 16.19/8.94 & ( B = B6 ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3589_old_Oprod_Oinject,axiom,
% 16.19/8.94 ! [A: int,B: int,A6: int,B6: int] :
% 16.19/8.94 ( ( ( product_Pair_int_int @ A @ B )
% 16.19/8.94 = ( product_Pair_int_int @ A6 @ B6 ) )
% 16.19/8.94 = ( ( A = A6 )
% 16.19/8.94 & ( B = B6 ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3590_prod_Oinject,axiom,
% 16.19/8.94 ! [X1: code_integer,X22: $o,Y1: code_integer,Y22: $o] :
% 16.19/8.94 ( ( ( produc6677183202524767010eger_o @ X1 @ X22 )
% 16.19/8.94 = ( produc6677183202524767010eger_o @ Y1 @ Y22 ) )
% 16.19/8.94 = ( ( X1 = Y1 )
% 16.19/8.94 & ( X22 = Y22 ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3591_prod_Oinject,axiom,
% 16.19/8.94 ! [X1: num,X22: num,Y1: num,Y22: num] :
% 16.19/8.94 ( ( ( product_Pair_num_num @ X1 @ X22 )
% 16.19/8.94 = ( product_Pair_num_num @ Y1 @ Y22 ) )
% 16.19/8.94 = ( ( X1 = Y1 )
% 16.19/8.94 & ( X22 = Y22 ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3592_prod_Oinject,axiom,
% 16.19/8.94 ! [X1: nat,X22: num,Y1: nat,Y22: num] :
% 16.19/8.94 ( ( ( product_Pair_nat_num @ X1 @ X22 )
% 16.19/8.94 = ( product_Pair_nat_num @ Y1 @ Y22 ) )
% 16.19/8.94 = ( ( X1 = Y1 )
% 16.19/8.94 & ( X22 = Y22 ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3593_prod_Oinject,axiom,
% 16.19/8.94 ! [X1: nat,X22: nat,Y1: nat,Y22: nat] :
% 16.19/8.94 ( ( ( product_Pair_nat_nat @ X1 @ X22 )
% 16.19/8.94 = ( product_Pair_nat_nat @ Y1 @ Y22 ) )
% 16.19/8.94 = ( ( X1 = Y1 )
% 16.19/8.94 & ( X22 = Y22 ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3594_prod_Oinject,axiom,
% 16.19/8.94 ! [X1: int,X22: int,Y1: int,Y22: int] :
% 16.19/8.94 ( ( ( product_Pair_int_int @ X1 @ X22 )
% 16.19/8.94 = ( product_Pair_int_int @ Y1 @ Y22 ) )
% 16.19/8.94 = ( ( X1 = Y1 )
% 16.19/8.94 & ( X22 = Y22 ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3595_List_Ofinite__set,axiom,
% 16.19/8.94 ! [Xs: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3596_List_Ofinite__set,axiom,
% 16.19/8.94 ! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3597_List_Ofinite__set,axiom,
% 16.19/8.94 ! [Xs: list_int] : ( finite_finite_int @ ( set_int2 @ Xs ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3598_List_Ofinite__set,axiom,
% 16.19/8.94 ! [Xs: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3599_infinite__Icc__iff,axiom,
% 16.19/8.94 ! [A: rat,B: rat] :
% 16.19/8.94 ( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) )
% 16.19/8.94 = ( ord_less_rat @ A @ B ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3600_infinite__Icc__iff,axiom,
% 16.19/8.94 ! [A: real,B: real] :
% 16.19/8.94 ( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) )
% 16.19/8.94 = ( ord_less_real @ A @ B ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3601_length__product,axiom,
% 16.19/8.94 ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 16.19/8.94 ( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys ) )
% 16.19/8.94 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3602_length__product,axiom,
% 16.19/8.94 ! [Xs: list_VEBT_VEBT,Ys: list_o] :
% 16.19/8.94 ( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys ) )
% 16.19/8.94 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3603_length__product,axiom,
% 16.19/8.94 ! [Xs: list_VEBT_VEBT,Ys: list_int] :
% 16.19/8.94 ( ( size_s3661962791536183091BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys ) )
% 16.19/8.94 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3604_length__product,axiom,
% 16.19/8.94 ! [Xs: list_o,Ys: list_VEBT_VEBT] :
% 16.19/8.94 ( ( size_s4313452262239582901T_VEBT @ ( product_o_VEBT_VEBT @ Xs @ Ys ) )
% 16.19/8.94 = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3605_length__product,axiom,
% 16.19/8.94 ! [Xs: list_o,Ys: list_o] :
% 16.19/8.94 ( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs @ Ys ) )
% 16.19/8.94 = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3606_length__product,axiom,
% 16.19/8.94 ! [Xs: list_o,Ys: list_int] :
% 16.19/8.94 ( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs @ Ys ) )
% 16.19/8.94 = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3607_length__product,axiom,
% 16.19/8.94 ! [Xs: list_int,Ys: list_VEBT_VEBT] :
% 16.19/8.94 ( ( size_s6639371672096860321T_VEBT @ ( produc662631939642741121T_VEBT @ Xs @ Ys ) )
% 16.19/8.94 = ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3608_length__product,axiom,
% 16.19/8.94 ! [Xs: list_int,Ys: list_o] :
% 16.19/8.94 ( ( size_s4246224855604898693_int_o @ ( product_int_o @ Xs @ Ys ) )
% 16.19/8.94 = ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3609_length__product,axiom,
% 16.19/8.94 ! [Xs: list_int,Ys: list_int] :
% 16.19/8.94 ( ( size_s5157815400016825771nt_int @ ( product_int_int @ Xs @ Ys ) )
% 16.19/8.94 = ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3610_finite__nat__set__iff__bounded,axiom,
% 16.19/8.94 ( finite_finite_nat
% 16.19/8.94 = ( ^ [N6: set_nat] :
% 16.19/8.94 ? [M6: nat] :
% 16.19/8.94 ! [X2: nat] :
% 16.19/8.94 ( ( member_nat @ X2 @ N6 )
% 16.19/8.94 => ( ord_less_nat @ X2 @ M6 ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3611_bounded__nat__set__is__finite,axiom,
% 16.19/8.94 ! [N3: set_nat,N2: nat] :
% 16.19/8.94 ( ! [X4: nat] :
% 16.19/8.94 ( ( member_nat @ X4 @ N3 )
% 16.19/8.94 => ( ord_less_nat @ X4 @ N2 ) )
% 16.19/8.94 => ( finite_finite_nat @ N3 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3612_finite__nat__set__iff__bounded__le,axiom,
% 16.19/8.94 ( finite_finite_nat
% 16.19/8.94 = ( ^ [N6: set_nat] :
% 16.19/8.94 ? [M6: nat] :
% 16.19/8.94 ! [X2: nat] :
% 16.19/8.94 ( ( member_nat @ X2 @ N6 )
% 16.19/8.94 => ( ord_less_eq_nat @ X2 @ M6 ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3613_finite__list,axiom,
% 16.19/8.94 ! [A2: set_VEBT_VEBT] :
% 16.19/8.94 ( ( finite5795047828879050333T_VEBT @ A2 )
% 16.19/8.94 => ? [Xs3: list_VEBT_VEBT] :
% 16.19/8.94 ( ( set_VEBT_VEBT2 @ Xs3 )
% 16.19/8.94 = A2 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3614_finite__list,axiom,
% 16.19/8.94 ! [A2: set_nat] :
% 16.19/8.94 ( ( finite_finite_nat @ A2 )
% 16.19/8.94 => ? [Xs3: list_nat] :
% 16.19/8.94 ( ( set_nat2 @ Xs3 )
% 16.19/8.94 = A2 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3615_finite__list,axiom,
% 16.19/8.94 ! [A2: set_int] :
% 16.19/8.94 ( ( finite_finite_int @ A2 )
% 16.19/8.94 => ? [Xs3: list_int] :
% 16.19/8.94 ( ( set_int2 @ Xs3 )
% 16.19/8.94 = A2 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3616_finite__list,axiom,
% 16.19/8.94 ! [A2: set_complex] :
% 16.19/8.94 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.94 => ? [Xs3: list_complex] :
% 16.19/8.94 ( ( set_complex2 @ Xs3 )
% 16.19/8.94 = A2 ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3617_finite__M__bounded__by__nat,axiom,
% 16.19/8.94 ! [P: nat > $o,I2: nat] :
% 16.19/8.94 ( finite_finite_nat
% 16.19/8.94 @ ( collect_nat
% 16.19/8.94 @ ^ [K3: nat] :
% 16.19/8.94 ( ( P @ K3 )
% 16.19/8.94 & ( ord_less_nat @ K3 @ I2 ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3618_finite__less__ub,axiom,
% 16.19/8.94 ! [F: nat > nat,U: nat] :
% 16.19/8.94 ( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F @ N4 ) )
% 16.19/8.94 => ( finite_finite_nat
% 16.19/8.94 @ ( collect_nat
% 16.19/8.94 @ ^ [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ U ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3619_finite__lists__length__eq,axiom,
% 16.19/8.94 ! [A2: set_complex,N2: nat] :
% 16.19/8.94 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.94 => ( finite8712137658972009173omplex
% 16.19/8.94 @ ( collect_list_complex
% 16.19/8.94 @ ^ [Xs2: list_complex] :
% 16.19/8.94 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
% 16.19/8.94 & ( ( size_s3451745648224563538omplex @ Xs2 )
% 16.19/8.94 = N2 ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3620_finite__lists__length__eq,axiom,
% 16.19/8.94 ! [A2: set_VEBT_VEBT,N2: nat] :
% 16.19/8.94 ( ( finite5795047828879050333T_VEBT @ A2 )
% 16.19/8.94 => ( finite3004134309566078307T_VEBT
% 16.19/8.94 @ ( collec5608196760682091941T_VEBT
% 16.19/8.94 @ ^ [Xs2: list_VEBT_VEBT] :
% 16.19/8.94 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
% 16.19/8.94 & ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 16.19/8.94 = N2 ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3621_finite__lists__length__eq,axiom,
% 16.19/8.94 ! [A2: set_o,N2: nat] :
% 16.19/8.94 ( ( finite_finite_o @ A2 )
% 16.19/8.94 => ( finite_finite_list_o
% 16.19/8.94 @ ( collect_list_o
% 16.19/8.94 @ ^ [Xs2: list_o] :
% 16.19/8.94 ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A2 )
% 16.19/8.94 & ( ( size_size_list_o @ Xs2 )
% 16.19/8.94 = N2 ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3622_finite__lists__length__eq,axiom,
% 16.19/8.94 ! [A2: set_int,N2: nat] :
% 16.19/8.94 ( ( finite_finite_int @ A2 )
% 16.19/8.94 => ( finite3922522038869484883st_int
% 16.19/8.94 @ ( collect_list_int
% 16.19/8.94 @ ^ [Xs2: list_int] :
% 16.19/8.94 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
% 16.19/8.94 & ( ( size_size_list_int @ Xs2 )
% 16.19/8.94 = N2 ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3623_finite__lists__length__eq,axiom,
% 16.19/8.94 ! [A2: set_nat,N2: nat] :
% 16.19/8.94 ( ( finite_finite_nat @ A2 )
% 16.19/8.94 => ( finite8100373058378681591st_nat
% 16.19/8.94 @ ( collect_list_nat
% 16.19/8.94 @ ^ [Xs2: list_nat] :
% 16.19/8.94 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
% 16.19/8.94 & ( ( size_size_list_nat @ Xs2 )
% 16.19/8.94 = N2 ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3624_infinite__Icc,axiom,
% 16.19/8.94 ! [A: rat,B: rat] :
% 16.19/8.94 ( ( ord_less_rat @ A @ B )
% 16.19/8.94 => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3625_infinite__Icc,axiom,
% 16.19/8.94 ! [A: real,B: real] :
% 16.19/8.94 ( ( ord_less_real @ A @ B )
% 16.19/8.94 => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3626_finite__lists__length__le,axiom,
% 16.19/8.94 ! [A2: set_complex,N2: nat] :
% 16.19/8.94 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.94 => ( finite8712137658972009173omplex
% 16.19/8.94 @ ( collect_list_complex
% 16.19/8.94 @ ^ [Xs2: list_complex] :
% 16.19/8.94 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
% 16.19/8.94 & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs2 ) @ N2 ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3627_finite__lists__length__le,axiom,
% 16.19/8.94 ! [A2: set_VEBT_VEBT,N2: nat] :
% 16.19/8.94 ( ( finite5795047828879050333T_VEBT @ A2 )
% 16.19/8.94 => ( finite3004134309566078307T_VEBT
% 16.19/8.94 @ ( collec5608196760682091941T_VEBT
% 16.19/8.94 @ ^ [Xs2: list_VEBT_VEBT] :
% 16.19/8.94 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
% 16.19/8.94 & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ N2 ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3628_finite__lists__length__le,axiom,
% 16.19/8.94 ! [A2: set_o,N2: nat] :
% 16.19/8.94 ( ( finite_finite_o @ A2 )
% 16.19/8.94 => ( finite_finite_list_o
% 16.19/8.94 @ ( collect_list_o
% 16.19/8.94 @ ^ [Xs2: list_o] :
% 16.19/8.94 ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A2 )
% 16.19/8.94 & ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ N2 ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3629_finite__lists__length__le,axiom,
% 16.19/8.94 ! [A2: set_int,N2: nat] :
% 16.19/8.94 ( ( finite_finite_int @ A2 )
% 16.19/8.94 => ( finite3922522038869484883st_int
% 16.19/8.94 @ ( collect_list_int
% 16.19/8.94 @ ^ [Xs2: list_int] :
% 16.19/8.94 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
% 16.19/8.94 & ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ N2 ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3630_finite__lists__length__le,axiom,
% 16.19/8.94 ! [A2: set_nat,N2: nat] :
% 16.19/8.94 ( ( finite_finite_nat @ A2 )
% 16.19/8.94 => ( finite8100373058378681591st_nat
% 16.19/8.94 @ ( collect_list_nat
% 16.19/8.94 @ ^ [Xs2: list_nat] :
% 16.19/8.94 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
% 16.19/8.94 & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N2 ) ) ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3631_Pair__inject,axiom,
% 16.19/8.94 ! [A: code_integer,B: $o,A6: code_integer,B6: $o] :
% 16.19/8.94 ( ( ( produc6677183202524767010eger_o @ A @ B )
% 16.19/8.94 = ( produc6677183202524767010eger_o @ A6 @ B6 ) )
% 16.19/8.94 => ~ ( ( A = A6 )
% 16.19/8.94 => ( B = ~ B6 ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3632_Pair__inject,axiom,
% 16.19/8.94 ! [A: num,B: num,A6: num,B6: num] :
% 16.19/8.94 ( ( ( product_Pair_num_num @ A @ B )
% 16.19/8.94 = ( product_Pair_num_num @ A6 @ B6 ) )
% 16.19/8.94 => ~ ( ( A = A6 )
% 16.19/8.94 => ( B != B6 ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3633_Pair__inject,axiom,
% 16.19/8.94 ! [A: nat,B: num,A6: nat,B6: num] :
% 16.19/8.94 ( ( ( product_Pair_nat_num @ A @ B )
% 16.19/8.94 = ( product_Pair_nat_num @ A6 @ B6 ) )
% 16.19/8.94 => ~ ( ( A = A6 )
% 16.19/8.94 => ( B != B6 ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3634_Pair__inject,axiom,
% 16.19/8.94 ! [A: nat,B: nat,A6: nat,B6: nat] :
% 16.19/8.94 ( ( ( product_Pair_nat_nat @ A @ B )
% 16.19/8.94 = ( product_Pair_nat_nat @ A6 @ B6 ) )
% 16.19/8.94 => ~ ( ( A = A6 )
% 16.19/8.94 => ( B != B6 ) ) ) ).
% 16.19/8.94
% 16.19/8.94 thf(fact_3635_Pair__inject,axiom,
% 16.19/8.94 ! [A: int,B: int,A6: int,B6: int] :
% 16.19/8.94 ( ( ( product_Pair_int_int @ A @ B )
% 16.19/8.95 = ( product_Pair_int_int @ A6 @ B6 ) )
% 16.19/8.95 => ~ ( ( A = A6 )
% 16.19/8.95 => ( B != B6 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3636_prod__cases,axiom,
% 16.19/8.95 ! [P: produc6271795597528267376eger_o > $o,P4: produc6271795597528267376eger_o] :
% 16.19/8.95 ( ! [A3: code_integer,B2: $o] : ( P @ ( produc6677183202524767010eger_o @ A3 @ B2 ) )
% 16.19/8.95 => ( P @ P4 ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3637_prod__cases,axiom,
% 16.19/8.95 ! [P: product_prod_num_num > $o,P4: product_prod_num_num] :
% 16.19/8.95 ( ! [A3: num,B2: num] : ( P @ ( product_Pair_num_num @ A3 @ B2 ) )
% 16.19/8.95 => ( P @ P4 ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3638_prod__cases,axiom,
% 16.19/8.95 ! [P: product_prod_nat_num > $o,P4: product_prod_nat_num] :
% 16.19/8.95 ( ! [A3: nat,B2: num] : ( P @ ( product_Pair_nat_num @ A3 @ B2 ) )
% 16.19/8.95 => ( P @ P4 ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3639_prod__cases,axiom,
% 16.19/8.95 ! [P: product_prod_nat_nat > $o,P4: product_prod_nat_nat] :
% 16.19/8.95 ( ! [A3: nat,B2: nat] : ( P @ ( product_Pair_nat_nat @ A3 @ B2 ) )
% 16.19/8.95 => ( P @ P4 ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3640_prod__cases,axiom,
% 16.19/8.95 ! [P: product_prod_int_int > $o,P4: product_prod_int_int] :
% 16.19/8.95 ( ! [A3: int,B2: int] : ( P @ ( product_Pair_int_int @ A3 @ B2 ) )
% 16.19/8.95 => ( P @ P4 ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3641_surj__pair,axiom,
% 16.19/8.95 ! [P4: produc6271795597528267376eger_o] :
% 16.19/8.95 ? [X4: code_integer,Y4: $o] :
% 16.19/8.95 ( P4
% 16.19/8.95 = ( produc6677183202524767010eger_o @ X4 @ Y4 ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3642_surj__pair,axiom,
% 16.19/8.95 ! [P4: product_prod_num_num] :
% 16.19/8.95 ? [X4: num,Y4: num] :
% 16.19/8.95 ( P4
% 16.19/8.95 = ( product_Pair_num_num @ X4 @ Y4 ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3643_surj__pair,axiom,
% 16.19/8.95 ! [P4: product_prod_nat_num] :
% 16.19/8.95 ? [X4: nat,Y4: num] :
% 16.19/8.95 ( P4
% 16.19/8.95 = ( product_Pair_nat_num @ X4 @ Y4 ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3644_surj__pair,axiom,
% 16.19/8.95 ! [P4: product_prod_nat_nat] :
% 16.19/8.95 ? [X4: nat,Y4: nat] :
% 16.19/8.95 ( P4
% 16.19/8.95 = ( product_Pair_nat_nat @ X4 @ Y4 ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3645_surj__pair,axiom,
% 16.19/8.95 ! [P4: product_prod_int_int] :
% 16.19/8.95 ? [X4: int,Y4: int] :
% 16.19/8.95 ( P4
% 16.19/8.95 = ( product_Pair_int_int @ X4 @ Y4 ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3646_old_Oprod_Oexhaust,axiom,
% 16.19/8.95 ! [Y: produc6271795597528267376eger_o] :
% 16.19/8.95 ~ ! [A3: code_integer,B2: $o] :
% 16.19/8.95 ( Y
% 16.19/8.95 != ( produc6677183202524767010eger_o @ A3 @ B2 ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3647_old_Oprod_Oexhaust,axiom,
% 16.19/8.95 ! [Y: product_prod_num_num] :
% 16.19/8.95 ~ ! [A3: num,B2: num] :
% 16.19/8.95 ( Y
% 16.19/8.95 != ( product_Pair_num_num @ A3 @ B2 ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3648_old_Oprod_Oexhaust,axiom,
% 16.19/8.95 ! [Y: product_prod_nat_num] :
% 16.19/8.95 ~ ! [A3: nat,B2: num] :
% 16.19/8.95 ( Y
% 16.19/8.95 != ( product_Pair_nat_num @ A3 @ B2 ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3649_old_Oprod_Oexhaust,axiom,
% 16.19/8.95 ! [Y: product_prod_nat_nat] :
% 16.19/8.95 ~ ! [A3: nat,B2: nat] :
% 16.19/8.95 ( Y
% 16.19/8.95 != ( product_Pair_nat_nat @ A3 @ B2 ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3650_old_Oprod_Oexhaust,axiom,
% 16.19/8.95 ! [Y: product_prod_int_int] :
% 16.19/8.95 ~ ! [A3: int,B2: int] :
% 16.19/8.95 ( Y
% 16.19/8.95 != ( product_Pair_int_int @ A3 @ B2 ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3651_finite__Collect__le__nat,axiom,
% 16.19/8.95 ! [K: nat] :
% 16.19/8.95 ( finite_finite_nat
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [N: nat] : ( ord_less_eq_nat @ N @ K ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3652_finite__Collect__less__nat,axiom,
% 16.19/8.95 ! [K: nat] :
% 16.19/8.95 ( finite_finite_nat
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [N: nat] : ( ord_less_nat @ N @ K ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3653_finite__Collect__subsets,axiom,
% 16.19/8.95 ! [A2: set_int] :
% 16.19/8.95 ( ( finite_finite_int @ A2 )
% 16.19/8.95 => ( finite6197958912794628473et_int
% 16.19/8.95 @ ( collect_set_int
% 16.19/8.95 @ ^ [B4: set_int] : ( ord_less_eq_set_int @ B4 @ A2 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3654_finite__Collect__subsets,axiom,
% 16.19/8.95 ! [A2: set_complex] :
% 16.19/8.95 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.95 => ( finite6551019134538273531omplex
% 16.19/8.95 @ ( collect_set_complex
% 16.19/8.95 @ ^ [B4: set_complex] : ( ord_le211207098394363844omplex @ B4 @ A2 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3655_finite__Collect__subsets,axiom,
% 16.19/8.95 ! [A2: set_nat] :
% 16.19/8.95 ( ( finite_finite_nat @ A2 )
% 16.19/8.95 => ( finite1152437895449049373et_nat
% 16.19/8.95 @ ( collect_set_nat
% 16.19/8.95 @ ^ [B4: set_nat] : ( ord_less_eq_set_nat @ B4 @ A2 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3656_finite__roots__unity,axiom,
% 16.19/8.95 ! [N2: nat] :
% 16.19/8.95 ( ( ord_less_eq_nat @ one_one_nat @ N2 )
% 16.19/8.95 => ( finite_finite_real
% 16.19/8.95 @ ( collect_real
% 16.19/8.95 @ ^ [Z3: real] :
% 16.19/8.95 ( ( power_power_real @ Z3 @ N2 )
% 16.19/8.95 = one_one_real ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3657_finite__roots__unity,axiom,
% 16.19/8.95 ! [N2: nat] :
% 16.19/8.95 ( ( ord_less_eq_nat @ one_one_nat @ N2 )
% 16.19/8.95 => ( finite3207457112153483333omplex
% 16.19/8.95 @ ( collect_complex
% 16.19/8.95 @ ^ [Z3: complex] :
% 16.19/8.95 ( ( power_power_complex @ Z3 @ N2 )
% 16.19/8.95 = one_one_complex ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3658_finite__Diff,axiom,
% 16.19/8.95 ! [A2: set_int,B3: set_int] :
% 16.19/8.95 ( ( finite_finite_int @ A2 )
% 16.19/8.95 => ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B3 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3659_finite__Diff,axiom,
% 16.19/8.95 ! [A2: set_complex,B3: set_complex] :
% 16.19/8.95 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.95 => ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3660_finite__Diff,axiom,
% 16.19/8.95 ! [A2: set_nat,B3: set_nat] :
% 16.19/8.95 ( ( finite_finite_nat @ A2 )
% 16.19/8.95 => ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3661_finite__Diff2,axiom,
% 16.19/8.95 ! [B3: set_int,A2: set_int] :
% 16.19/8.95 ( ( finite_finite_int @ B3 )
% 16.19/8.95 => ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B3 ) )
% 16.19/8.95 = ( finite_finite_int @ A2 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3662_finite__Diff2,axiom,
% 16.19/8.95 ! [B3: set_complex,A2: set_complex] :
% 16.19/8.95 ( ( finite3207457112153483333omplex @ B3 )
% 16.19/8.95 => ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 16.19/8.95 = ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3663_finite__Diff2,axiom,
% 16.19/8.95 ! [B3: set_nat,A2: set_nat] :
% 16.19/8.95 ( ( finite_finite_nat @ B3 )
% 16.19/8.95 => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 16.19/8.95 = ( finite_finite_nat @ A2 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3664_finite__Collect__conjI,axiom,
% 16.19/8.95 ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
% 16.19/8.95 ( ( ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
% 16.19/8.95 | ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ Q ) ) )
% 16.19/8.95 => ( finite2998713641127702882nt_int
% 16.19/8.95 @ ( collec213857154873943460nt_int
% 16.19/8.95 @ ^ [X2: product_prod_int_int] :
% 16.19/8.95 ( ( P @ X2 )
% 16.19/8.95 & ( Q @ X2 ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3665_finite__Collect__conjI,axiom,
% 16.19/8.95 ! [P: set_nat > $o,Q: set_nat > $o] :
% 16.19/8.95 ( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
% 16.19/8.95 | ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) )
% 16.19/8.95 => ( finite1152437895449049373et_nat
% 16.19/8.95 @ ( collect_set_nat
% 16.19/8.95 @ ^ [X2: set_nat] :
% 16.19/8.95 ( ( P @ X2 )
% 16.19/8.95 & ( Q @ X2 ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3666_finite__Collect__conjI,axiom,
% 16.19/8.95 ! [P: nat > $o,Q: nat > $o] :
% 16.19/8.95 ( ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 16.19/8.95 | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
% 16.19/8.95 => ( finite_finite_nat
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [X2: nat] :
% 16.19/8.95 ( ( P @ X2 )
% 16.19/8.95 & ( Q @ X2 ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3667_finite__Collect__conjI,axiom,
% 16.19/8.95 ! [P: int > $o,Q: int > $o] :
% 16.19/8.95 ( ( ( finite_finite_int @ ( collect_int @ P ) )
% 16.19/8.95 | ( finite_finite_int @ ( collect_int @ Q ) ) )
% 16.19/8.95 => ( finite_finite_int
% 16.19/8.95 @ ( collect_int
% 16.19/8.95 @ ^ [X2: int] :
% 16.19/8.95 ( ( P @ X2 )
% 16.19/8.95 & ( Q @ X2 ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3668_finite__Collect__conjI,axiom,
% 16.19/8.95 ! [P: complex > $o,Q: complex > $o] :
% 16.19/8.95 ( ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
% 16.19/8.95 | ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) )
% 16.19/8.95 => ( finite3207457112153483333omplex
% 16.19/8.95 @ ( collect_complex
% 16.19/8.95 @ ^ [X2: complex] :
% 16.19/8.95 ( ( P @ X2 )
% 16.19/8.95 & ( Q @ X2 ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3669_finite__Collect__disjI,axiom,
% 16.19/8.95 ! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
% 16.19/8.95 ( ( finite2998713641127702882nt_int
% 16.19/8.95 @ ( collec213857154873943460nt_int
% 16.19/8.95 @ ^ [X2: product_prod_int_int] :
% 16.19/8.95 ( ( P @ X2 )
% 16.19/8.95 | ( Q @ X2 ) ) ) )
% 16.19/8.95 = ( ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
% 16.19/8.95 & ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ Q ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3670_finite__Collect__disjI,axiom,
% 16.19/8.95 ! [P: set_nat > $o,Q: set_nat > $o] :
% 16.19/8.95 ( ( finite1152437895449049373et_nat
% 16.19/8.95 @ ( collect_set_nat
% 16.19/8.95 @ ^ [X2: set_nat] :
% 16.19/8.95 ( ( P @ X2 )
% 16.19/8.95 | ( Q @ X2 ) ) ) )
% 16.19/8.95 = ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
% 16.19/8.95 & ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3671_finite__Collect__disjI,axiom,
% 16.19/8.95 ! [P: nat > $o,Q: nat > $o] :
% 16.19/8.95 ( ( finite_finite_nat
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [X2: nat] :
% 16.19/8.95 ( ( P @ X2 )
% 16.19/8.95 | ( Q @ X2 ) ) ) )
% 16.19/8.95 = ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 16.19/8.95 & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3672_finite__Collect__disjI,axiom,
% 16.19/8.95 ! [P: int > $o,Q: int > $o] :
% 16.19/8.95 ( ( finite_finite_int
% 16.19/8.95 @ ( collect_int
% 16.19/8.95 @ ^ [X2: int] :
% 16.19/8.95 ( ( P @ X2 )
% 16.19/8.95 | ( Q @ X2 ) ) ) )
% 16.19/8.95 = ( ( finite_finite_int @ ( collect_int @ P ) )
% 16.19/8.95 & ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3673_finite__Collect__disjI,axiom,
% 16.19/8.95 ! [P: complex > $o,Q: complex > $o] :
% 16.19/8.95 ( ( finite3207457112153483333omplex
% 16.19/8.95 @ ( collect_complex
% 16.19/8.95 @ ^ [X2: complex] :
% 16.19/8.95 ( ( P @ X2 )
% 16.19/8.95 | ( Q @ X2 ) ) ) )
% 16.19/8.95 = ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
% 16.19/8.95 & ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3674_finite__interval__int1,axiom,
% 16.19/8.95 ! [A: int,B: int] :
% 16.19/8.95 ( finite_finite_int
% 16.19/8.95 @ ( collect_int
% 16.19/8.95 @ ^ [I3: int] :
% 16.19/8.95 ( ( ord_less_eq_int @ A @ I3 )
% 16.19/8.95 & ( ord_less_eq_int @ I3 @ B ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3675_finite__interval__int4,axiom,
% 16.19/8.95 ! [A: int,B: int] :
% 16.19/8.95 ( finite_finite_int
% 16.19/8.95 @ ( collect_int
% 16.19/8.95 @ ^ [I3: int] :
% 16.19/8.95 ( ( ord_less_int @ A @ I3 )
% 16.19/8.95 & ( ord_less_int @ I3 @ B ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3676_finite__interval__int3,axiom,
% 16.19/8.95 ! [A: int,B: int] :
% 16.19/8.95 ( finite_finite_int
% 16.19/8.95 @ ( collect_int
% 16.19/8.95 @ ^ [I3: int] :
% 16.19/8.95 ( ( ord_less_int @ A @ I3 )
% 16.19/8.95 & ( ord_less_eq_int @ I3 @ B ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3677_finite__interval__int2,axiom,
% 16.19/8.95 ! [A: int,B: int] :
% 16.19/8.95 ( finite_finite_int
% 16.19/8.95 @ ( collect_int
% 16.19/8.95 @ ^ [I3: int] :
% 16.19/8.95 ( ( ord_less_eq_int @ A @ I3 )
% 16.19/8.95 & ( ord_less_int @ I3 @ B ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3678_finite__maxlen,axiom,
% 16.19/8.95 ! [M7: set_list_VEBT_VEBT] :
% 16.19/8.95 ( ( finite3004134309566078307T_VEBT @ M7 )
% 16.19/8.95 => ? [N4: nat] :
% 16.19/8.95 ! [X5: list_VEBT_VEBT] :
% 16.19/8.95 ( ( member2936631157270082147T_VEBT @ X5 @ M7 )
% 16.19/8.95 => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X5 ) @ N4 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3679_finite__maxlen,axiom,
% 16.19/8.95 ! [M7: set_list_o] :
% 16.19/8.95 ( ( finite_finite_list_o @ M7 )
% 16.19/8.95 => ? [N4: nat] :
% 16.19/8.95 ! [X5: list_o] :
% 16.19/8.95 ( ( member_list_o @ X5 @ M7 )
% 16.19/8.95 => ( ord_less_nat @ ( size_size_list_o @ X5 ) @ N4 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3680_finite__maxlen,axiom,
% 16.19/8.95 ! [M7: set_list_int] :
% 16.19/8.95 ( ( finite3922522038869484883st_int @ M7 )
% 16.19/8.95 => ? [N4: nat] :
% 16.19/8.95 ! [X5: list_int] :
% 16.19/8.95 ( ( member_list_int @ X5 @ M7 )
% 16.19/8.95 => ( ord_less_nat @ ( size_size_list_int @ X5 ) @ N4 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3681_pigeonhole__infinite__rel,axiom,
% 16.19/8.95 ! [A2: set_real,B3: set_nat,R2: real > nat > $o] :
% 16.19/8.95 ( ~ ( finite_finite_real @ A2 )
% 16.19/8.95 => ( ( finite_finite_nat @ B3 )
% 16.19/8.95 => ( ! [X4: real] :
% 16.19/8.95 ( ( member_real @ X4 @ A2 )
% 16.19/8.95 => ? [Xa: nat] :
% 16.19/8.95 ( ( member_nat @ Xa @ B3 )
% 16.19/8.95 & ( R2 @ X4 @ Xa ) ) )
% 16.19/8.95 => ? [X4: nat] :
% 16.19/8.95 ( ( member_nat @ X4 @ B3 )
% 16.19/8.95 & ~ ( finite_finite_real
% 16.19/8.95 @ ( collect_real
% 16.19/8.95 @ ^ [A5: real] :
% 16.19/8.95 ( ( member_real @ A5 @ A2 )
% 16.19/8.95 & ( R2 @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3682_pigeonhole__infinite__rel,axiom,
% 16.19/8.95 ! [A2: set_real,B3: set_int,R2: real > int > $o] :
% 16.19/8.95 ( ~ ( finite_finite_real @ A2 )
% 16.19/8.95 => ( ( finite_finite_int @ B3 )
% 16.19/8.95 => ( ! [X4: real] :
% 16.19/8.95 ( ( member_real @ X4 @ A2 )
% 16.19/8.95 => ? [Xa: int] :
% 16.19/8.95 ( ( member_int @ Xa @ B3 )
% 16.19/8.95 & ( R2 @ X4 @ Xa ) ) )
% 16.19/8.95 => ? [X4: int] :
% 16.19/8.95 ( ( member_int @ X4 @ B3 )
% 16.19/8.95 & ~ ( finite_finite_real
% 16.19/8.95 @ ( collect_real
% 16.19/8.95 @ ^ [A5: real] :
% 16.19/8.95 ( ( member_real @ A5 @ A2 )
% 16.19/8.95 & ( R2 @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3683_pigeonhole__infinite__rel,axiom,
% 16.19/8.95 ! [A2: set_real,B3: set_complex,R2: real > complex > $o] :
% 16.19/8.95 ( ~ ( finite_finite_real @ A2 )
% 16.19/8.95 => ( ( finite3207457112153483333omplex @ B3 )
% 16.19/8.95 => ( ! [X4: real] :
% 16.19/8.95 ( ( member_real @ X4 @ A2 )
% 16.19/8.95 => ? [Xa: complex] :
% 16.19/8.95 ( ( member_complex @ Xa @ B3 )
% 16.19/8.95 & ( R2 @ X4 @ Xa ) ) )
% 16.19/8.95 => ? [X4: complex] :
% 16.19/8.95 ( ( member_complex @ X4 @ B3 )
% 16.19/8.95 & ~ ( finite_finite_real
% 16.19/8.95 @ ( collect_real
% 16.19/8.95 @ ^ [A5: real] :
% 16.19/8.95 ( ( member_real @ A5 @ A2 )
% 16.19/8.95 & ( R2 @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3684_pigeonhole__infinite__rel,axiom,
% 16.19/8.95 ! [A2: set_nat,B3: set_nat,R2: nat > nat > $o] :
% 16.19/8.95 ( ~ ( finite_finite_nat @ A2 )
% 16.19/8.95 => ( ( finite_finite_nat @ B3 )
% 16.19/8.95 => ( ! [X4: nat] :
% 16.19/8.95 ( ( member_nat @ X4 @ A2 )
% 16.19/8.95 => ? [Xa: nat] :
% 16.19/8.95 ( ( member_nat @ Xa @ B3 )
% 16.19/8.95 & ( R2 @ X4 @ Xa ) ) )
% 16.19/8.95 => ? [X4: nat] :
% 16.19/8.95 ( ( member_nat @ X4 @ B3 )
% 16.19/8.95 & ~ ( finite_finite_nat
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [A5: nat] :
% 16.19/8.95 ( ( member_nat @ A5 @ A2 )
% 16.19/8.95 & ( R2 @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3685_pigeonhole__infinite__rel,axiom,
% 16.19/8.95 ! [A2: set_nat,B3: set_int,R2: nat > int > $o] :
% 16.19/8.95 ( ~ ( finite_finite_nat @ A2 )
% 16.19/8.95 => ( ( finite_finite_int @ B3 )
% 16.19/8.95 => ( ! [X4: nat] :
% 16.19/8.95 ( ( member_nat @ X4 @ A2 )
% 16.19/8.95 => ? [Xa: int] :
% 16.19/8.95 ( ( member_int @ Xa @ B3 )
% 16.19/8.95 & ( R2 @ X4 @ Xa ) ) )
% 16.19/8.95 => ? [X4: int] :
% 16.19/8.95 ( ( member_int @ X4 @ B3 )
% 16.19/8.95 & ~ ( finite_finite_nat
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [A5: nat] :
% 16.19/8.95 ( ( member_nat @ A5 @ A2 )
% 16.19/8.95 & ( R2 @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3686_pigeonhole__infinite__rel,axiom,
% 16.19/8.95 ! [A2: set_nat,B3: set_complex,R2: nat > complex > $o] :
% 16.19/8.95 ( ~ ( finite_finite_nat @ A2 )
% 16.19/8.95 => ( ( finite3207457112153483333omplex @ B3 )
% 16.19/8.95 => ( ! [X4: nat] :
% 16.19/8.95 ( ( member_nat @ X4 @ A2 )
% 16.19/8.95 => ? [Xa: complex] :
% 16.19/8.95 ( ( member_complex @ Xa @ B3 )
% 16.19/8.95 & ( R2 @ X4 @ Xa ) ) )
% 16.19/8.95 => ? [X4: complex] :
% 16.19/8.95 ( ( member_complex @ X4 @ B3 )
% 16.19/8.95 & ~ ( finite_finite_nat
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [A5: nat] :
% 16.19/8.95 ( ( member_nat @ A5 @ A2 )
% 16.19/8.95 & ( R2 @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3687_pigeonhole__infinite__rel,axiom,
% 16.19/8.95 ! [A2: set_int,B3: set_nat,R2: int > nat > $o] :
% 16.19/8.95 ( ~ ( finite_finite_int @ A2 )
% 16.19/8.95 => ( ( finite_finite_nat @ B3 )
% 16.19/8.95 => ( ! [X4: int] :
% 16.19/8.95 ( ( member_int @ X4 @ A2 )
% 16.19/8.95 => ? [Xa: nat] :
% 16.19/8.95 ( ( member_nat @ Xa @ B3 )
% 16.19/8.95 & ( R2 @ X4 @ Xa ) ) )
% 16.19/8.95 => ? [X4: nat] :
% 16.19/8.95 ( ( member_nat @ X4 @ B3 )
% 16.19/8.95 & ~ ( finite_finite_int
% 16.19/8.95 @ ( collect_int
% 16.19/8.95 @ ^ [A5: int] :
% 16.19/8.95 ( ( member_int @ A5 @ A2 )
% 16.19/8.95 & ( R2 @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3688_pigeonhole__infinite__rel,axiom,
% 16.19/8.95 ! [A2: set_int,B3: set_int,R2: int > int > $o] :
% 16.19/8.95 ( ~ ( finite_finite_int @ A2 )
% 16.19/8.95 => ( ( finite_finite_int @ B3 )
% 16.19/8.95 => ( ! [X4: int] :
% 16.19/8.95 ( ( member_int @ X4 @ A2 )
% 16.19/8.95 => ? [Xa: int] :
% 16.19/8.95 ( ( member_int @ Xa @ B3 )
% 16.19/8.95 & ( R2 @ X4 @ Xa ) ) )
% 16.19/8.95 => ? [X4: int] :
% 16.19/8.95 ( ( member_int @ X4 @ B3 )
% 16.19/8.95 & ~ ( finite_finite_int
% 16.19/8.95 @ ( collect_int
% 16.19/8.95 @ ^ [A5: int] :
% 16.19/8.95 ( ( member_int @ A5 @ A2 )
% 16.19/8.95 & ( R2 @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3689_pigeonhole__infinite__rel,axiom,
% 16.19/8.95 ! [A2: set_int,B3: set_complex,R2: int > complex > $o] :
% 16.19/8.95 ( ~ ( finite_finite_int @ A2 )
% 16.19/8.95 => ( ( finite3207457112153483333omplex @ B3 )
% 16.19/8.95 => ( ! [X4: int] :
% 16.19/8.95 ( ( member_int @ X4 @ A2 )
% 16.19/8.95 => ? [Xa: complex] :
% 16.19/8.95 ( ( member_complex @ Xa @ B3 )
% 16.19/8.95 & ( R2 @ X4 @ Xa ) ) )
% 16.19/8.95 => ? [X4: complex] :
% 16.19/8.95 ( ( member_complex @ X4 @ B3 )
% 16.19/8.95 & ~ ( finite_finite_int
% 16.19/8.95 @ ( collect_int
% 16.19/8.95 @ ^ [A5: int] :
% 16.19/8.95 ( ( member_int @ A5 @ A2 )
% 16.19/8.95 & ( R2 @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3690_pigeonhole__infinite__rel,axiom,
% 16.19/8.95 ! [A2: set_complex,B3: set_nat,R2: complex > nat > $o] :
% 16.19/8.95 ( ~ ( finite3207457112153483333omplex @ A2 )
% 16.19/8.95 => ( ( finite_finite_nat @ B3 )
% 16.19/8.95 => ( ! [X4: complex] :
% 16.19/8.95 ( ( member_complex @ X4 @ A2 )
% 16.19/8.95 => ? [Xa: nat] :
% 16.19/8.95 ( ( member_nat @ Xa @ B3 )
% 16.19/8.95 & ( R2 @ X4 @ Xa ) ) )
% 16.19/8.95 => ? [X4: nat] :
% 16.19/8.95 ( ( member_nat @ X4 @ B3 )
% 16.19/8.95 & ~ ( finite3207457112153483333omplex
% 16.19/8.95 @ ( collect_complex
% 16.19/8.95 @ ^ [A5: complex] :
% 16.19/8.95 ( ( member_complex @ A5 @ A2 )
% 16.19/8.95 & ( R2 @ A5 @ X4 ) ) ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3691_not__finite__existsD,axiom,
% 16.19/8.95 ! [P: product_prod_int_int > $o] :
% 16.19/8.95 ( ~ ( finite2998713641127702882nt_int @ ( collec213857154873943460nt_int @ P ) )
% 16.19/8.95 => ? [X_1: product_prod_int_int] : ( P @ X_1 ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3692_not__finite__existsD,axiom,
% 16.19/8.95 ! [P: set_nat > $o] :
% 16.19/8.95 ( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
% 16.19/8.95 => ? [X_1: set_nat] : ( P @ X_1 ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3693_not__finite__existsD,axiom,
% 16.19/8.95 ! [P: nat > $o] :
% 16.19/8.95 ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
% 16.19/8.95 => ? [X_1: nat] : ( P @ X_1 ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3694_not__finite__existsD,axiom,
% 16.19/8.95 ! [P: int > $o] :
% 16.19/8.95 ( ~ ( finite_finite_int @ ( collect_int @ P ) )
% 16.19/8.95 => ? [X_1: int] : ( P @ X_1 ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3695_not__finite__existsD,axiom,
% 16.19/8.95 ! [P: complex > $o] :
% 16.19/8.95 ( ~ ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
% 16.19/8.95 => ? [X_1: complex] : ( P @ X_1 ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3696_finite__has__maximal2,axiom,
% 16.19/8.95 ! [A2: set_real,A: real] :
% 16.19/8.95 ( ( finite_finite_real @ A2 )
% 16.19/8.95 => ( ( member_real @ A @ A2 )
% 16.19/8.95 => ? [X4: real] :
% 16.19/8.95 ( ( member_real @ X4 @ A2 )
% 16.19/8.95 & ( ord_less_eq_real @ A @ X4 )
% 16.19/8.95 & ! [Xa: real] :
% 16.19/8.95 ( ( member_real @ Xa @ A2 )
% 16.19/8.95 => ( ( ord_less_eq_real @ X4 @ Xa )
% 16.19/8.95 => ( X4 = Xa ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3697_finite__has__maximal2,axiom,
% 16.19/8.95 ! [A2: set_set_nat,A: set_nat] :
% 16.19/8.95 ( ( finite1152437895449049373et_nat @ A2 )
% 16.19/8.95 => ( ( member_set_nat @ A @ A2 )
% 16.19/8.95 => ? [X4: set_nat] :
% 16.19/8.95 ( ( member_set_nat @ X4 @ A2 )
% 16.19/8.95 & ( ord_less_eq_set_nat @ A @ X4 )
% 16.19/8.95 & ! [Xa: set_nat] :
% 16.19/8.95 ( ( member_set_nat @ Xa @ A2 )
% 16.19/8.95 => ( ( ord_less_eq_set_nat @ X4 @ Xa )
% 16.19/8.95 => ( X4 = Xa ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3698_finite__has__maximal2,axiom,
% 16.19/8.95 ! [A2: set_rat,A: rat] :
% 16.19/8.95 ( ( finite_finite_rat @ A2 )
% 16.19/8.95 => ( ( member_rat @ A @ A2 )
% 16.19/8.95 => ? [X4: rat] :
% 16.19/8.95 ( ( member_rat @ X4 @ A2 )
% 16.19/8.95 & ( ord_less_eq_rat @ A @ X4 )
% 16.19/8.95 & ! [Xa: rat] :
% 16.19/8.95 ( ( member_rat @ Xa @ A2 )
% 16.19/8.95 => ( ( ord_less_eq_rat @ X4 @ Xa )
% 16.19/8.95 => ( X4 = Xa ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3699_finite__has__maximal2,axiom,
% 16.19/8.95 ! [A2: set_num,A: num] :
% 16.19/8.95 ( ( finite_finite_num @ A2 )
% 16.19/8.95 => ( ( member_num @ A @ A2 )
% 16.19/8.95 => ? [X4: num] :
% 16.19/8.95 ( ( member_num @ X4 @ A2 )
% 16.19/8.95 & ( ord_less_eq_num @ A @ X4 )
% 16.19/8.95 & ! [Xa: num] :
% 16.19/8.95 ( ( member_num @ Xa @ A2 )
% 16.19/8.95 => ( ( ord_less_eq_num @ X4 @ Xa )
% 16.19/8.95 => ( X4 = Xa ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3700_finite__has__maximal2,axiom,
% 16.19/8.95 ! [A2: set_nat,A: nat] :
% 16.19/8.95 ( ( finite_finite_nat @ A2 )
% 16.19/8.95 => ( ( member_nat @ A @ A2 )
% 16.19/8.95 => ? [X4: nat] :
% 16.19/8.95 ( ( member_nat @ X4 @ A2 )
% 16.19/8.95 & ( ord_less_eq_nat @ A @ X4 )
% 16.19/8.95 & ! [Xa: nat] :
% 16.19/8.95 ( ( member_nat @ Xa @ A2 )
% 16.19/8.95 => ( ( ord_less_eq_nat @ X4 @ Xa )
% 16.19/8.95 => ( X4 = Xa ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3701_finite__has__maximal2,axiom,
% 16.19/8.95 ! [A2: set_int,A: int] :
% 16.19/8.95 ( ( finite_finite_int @ A2 )
% 16.19/8.95 => ( ( member_int @ A @ A2 )
% 16.19/8.95 => ? [X4: int] :
% 16.19/8.95 ( ( member_int @ X4 @ A2 )
% 16.19/8.95 & ( ord_less_eq_int @ A @ X4 )
% 16.19/8.95 & ! [Xa: int] :
% 16.19/8.95 ( ( member_int @ Xa @ A2 )
% 16.19/8.95 => ( ( ord_less_eq_int @ X4 @ Xa )
% 16.19/8.95 => ( X4 = Xa ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3702_finite__has__minimal2,axiom,
% 16.19/8.95 ! [A2: set_real,A: real] :
% 16.19/8.95 ( ( finite_finite_real @ A2 )
% 16.19/8.95 => ( ( member_real @ A @ A2 )
% 16.19/8.95 => ? [X4: real] :
% 16.19/8.95 ( ( member_real @ X4 @ A2 )
% 16.19/8.95 & ( ord_less_eq_real @ X4 @ A )
% 16.19/8.95 & ! [Xa: real] :
% 16.19/8.95 ( ( member_real @ Xa @ A2 )
% 16.19/8.95 => ( ( ord_less_eq_real @ Xa @ X4 )
% 16.19/8.95 => ( X4 = Xa ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3703_finite__has__minimal2,axiom,
% 16.19/8.95 ! [A2: set_set_nat,A: set_nat] :
% 16.19/8.95 ( ( finite1152437895449049373et_nat @ A2 )
% 16.19/8.95 => ( ( member_set_nat @ A @ A2 )
% 16.19/8.95 => ? [X4: set_nat] :
% 16.19/8.95 ( ( member_set_nat @ X4 @ A2 )
% 16.19/8.95 & ( ord_less_eq_set_nat @ X4 @ A )
% 16.19/8.95 & ! [Xa: set_nat] :
% 16.19/8.95 ( ( member_set_nat @ Xa @ A2 )
% 16.19/8.95 => ( ( ord_less_eq_set_nat @ Xa @ X4 )
% 16.19/8.95 => ( X4 = Xa ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3704_finite__has__minimal2,axiom,
% 16.19/8.95 ! [A2: set_rat,A: rat] :
% 16.19/8.95 ( ( finite_finite_rat @ A2 )
% 16.19/8.95 => ( ( member_rat @ A @ A2 )
% 16.19/8.95 => ? [X4: rat] :
% 16.19/8.95 ( ( member_rat @ X4 @ A2 )
% 16.19/8.95 & ( ord_less_eq_rat @ X4 @ A )
% 16.19/8.95 & ! [Xa: rat] :
% 16.19/8.95 ( ( member_rat @ Xa @ A2 )
% 16.19/8.95 => ( ( ord_less_eq_rat @ Xa @ X4 )
% 16.19/8.95 => ( X4 = Xa ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3705_finite__has__minimal2,axiom,
% 16.19/8.95 ! [A2: set_num,A: num] :
% 16.19/8.95 ( ( finite_finite_num @ A2 )
% 16.19/8.95 => ( ( member_num @ A @ A2 )
% 16.19/8.95 => ? [X4: num] :
% 16.19/8.95 ( ( member_num @ X4 @ A2 )
% 16.19/8.95 & ( ord_less_eq_num @ X4 @ A )
% 16.19/8.95 & ! [Xa: num] :
% 16.19/8.95 ( ( member_num @ Xa @ A2 )
% 16.19/8.95 => ( ( ord_less_eq_num @ Xa @ X4 )
% 16.19/8.95 => ( X4 = Xa ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3706_finite__has__minimal2,axiom,
% 16.19/8.95 ! [A2: set_nat,A: nat] :
% 16.19/8.95 ( ( finite_finite_nat @ A2 )
% 16.19/8.95 => ( ( member_nat @ A @ A2 )
% 16.19/8.95 => ? [X4: nat] :
% 16.19/8.95 ( ( member_nat @ X4 @ A2 )
% 16.19/8.95 & ( ord_less_eq_nat @ X4 @ A )
% 16.19/8.95 & ! [Xa: nat] :
% 16.19/8.95 ( ( member_nat @ Xa @ A2 )
% 16.19/8.95 => ( ( ord_less_eq_nat @ Xa @ X4 )
% 16.19/8.95 => ( X4 = Xa ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3707_finite__has__minimal2,axiom,
% 16.19/8.95 ! [A2: set_int,A: int] :
% 16.19/8.95 ( ( finite_finite_int @ A2 )
% 16.19/8.95 => ( ( member_int @ A @ A2 )
% 16.19/8.95 => ? [X4: int] :
% 16.19/8.95 ( ( member_int @ X4 @ A2 )
% 16.19/8.95 & ( ord_less_eq_int @ X4 @ A )
% 16.19/8.95 & ! [Xa: int] :
% 16.19/8.95 ( ( member_int @ Xa @ A2 )
% 16.19/8.95 => ( ( ord_less_eq_int @ Xa @ X4 )
% 16.19/8.95 => ( X4 = Xa ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3708_finite__subset,axiom,
% 16.19/8.95 ! [A2: set_int,B3: set_int] :
% 16.19/8.95 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 16.19/8.95 => ( ( finite_finite_int @ B3 )
% 16.19/8.95 => ( finite_finite_int @ A2 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3709_finite__subset,axiom,
% 16.19/8.95 ! [A2: set_complex,B3: set_complex] :
% 16.19/8.95 ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 16.19/8.95 => ( ( finite3207457112153483333omplex @ B3 )
% 16.19/8.95 => ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3710_finite__subset,axiom,
% 16.19/8.95 ! [A2: set_nat,B3: set_nat] :
% 16.19/8.95 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 16.19/8.95 => ( ( finite_finite_nat @ B3 )
% 16.19/8.95 => ( finite_finite_nat @ A2 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3711_infinite__super,axiom,
% 16.19/8.95 ! [S: set_int,T3: set_int] :
% 16.19/8.95 ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/8.95 => ( ~ ( finite_finite_int @ S )
% 16.19/8.95 => ~ ( finite_finite_int @ T3 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3712_infinite__super,axiom,
% 16.19/8.95 ! [S: set_complex,T3: set_complex] :
% 16.19/8.95 ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/8.95 => ( ~ ( finite3207457112153483333omplex @ S )
% 16.19/8.95 => ~ ( finite3207457112153483333omplex @ T3 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3713_infinite__super,axiom,
% 16.19/8.95 ! [S: set_nat,T3: set_nat] :
% 16.19/8.95 ( ( ord_less_eq_set_nat @ S @ T3 )
% 16.19/8.95 => ( ~ ( finite_finite_nat @ S )
% 16.19/8.95 => ~ ( finite_finite_nat @ T3 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3714_rev__finite__subset,axiom,
% 16.19/8.95 ! [B3: set_int,A2: set_int] :
% 16.19/8.95 ( ( finite_finite_int @ B3 )
% 16.19/8.95 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 16.19/8.95 => ( finite_finite_int @ A2 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3715_rev__finite__subset,axiom,
% 16.19/8.95 ! [B3: set_complex,A2: set_complex] :
% 16.19/8.95 ( ( finite3207457112153483333omplex @ B3 )
% 16.19/8.95 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 16.19/8.95 => ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3716_rev__finite__subset,axiom,
% 16.19/8.95 ! [B3: set_nat,A2: set_nat] :
% 16.19/8.95 ( ( finite_finite_nat @ B3 )
% 16.19/8.95 => ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 16.19/8.95 => ( finite_finite_nat @ A2 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3717_Diff__infinite__finite,axiom,
% 16.19/8.95 ! [T3: set_int,S: set_int] :
% 16.19/8.95 ( ( finite_finite_int @ T3 )
% 16.19/8.95 => ( ~ ( finite_finite_int @ S )
% 16.19/8.95 => ~ ( finite_finite_int @ ( minus_minus_set_int @ S @ T3 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3718_Diff__infinite__finite,axiom,
% 16.19/8.95 ! [T3: set_complex,S: set_complex] :
% 16.19/8.95 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/8.95 => ( ~ ( finite3207457112153483333omplex @ S )
% 16.19/8.95 => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S @ T3 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3719_Diff__infinite__finite,axiom,
% 16.19/8.95 ! [T3: set_nat,S: set_nat] :
% 16.19/8.95 ( ( finite_finite_nat @ T3 )
% 16.19/8.95 => ( ~ ( finite_finite_nat @ S )
% 16.19/8.95 => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T3 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3720_finite__psubset__induct,axiom,
% 16.19/8.95 ! [A2: set_nat,P: set_nat > $o] :
% 16.19/8.95 ( ( finite_finite_nat @ A2 )
% 16.19/8.95 => ( ! [A7: set_nat] :
% 16.19/8.95 ( ( finite_finite_nat @ A7 )
% 16.19/8.95 => ( ! [B7: set_nat] :
% 16.19/8.95 ( ( ord_less_set_nat @ B7 @ A7 )
% 16.19/8.95 => ( P @ B7 ) )
% 16.19/8.95 => ( P @ A7 ) ) )
% 16.19/8.95 => ( P @ A2 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3721_finite__psubset__induct,axiom,
% 16.19/8.95 ! [A2: set_int,P: set_int > $o] :
% 16.19/8.95 ( ( finite_finite_int @ A2 )
% 16.19/8.95 => ( ! [A7: set_int] :
% 16.19/8.95 ( ( finite_finite_int @ A7 )
% 16.19/8.95 => ( ! [B7: set_int] :
% 16.19/8.95 ( ( ord_less_set_int @ B7 @ A7 )
% 16.19/8.95 => ( P @ B7 ) )
% 16.19/8.95 => ( P @ A7 ) ) )
% 16.19/8.95 => ( P @ A2 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3722_finite__psubset__induct,axiom,
% 16.19/8.95 ! [A2: set_complex,P: set_complex > $o] :
% 16.19/8.95 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.95 => ( ! [A7: set_complex] :
% 16.19/8.95 ( ( finite3207457112153483333omplex @ A7 )
% 16.19/8.95 => ( ! [B7: set_complex] :
% 16.19/8.95 ( ( ord_less_set_complex @ B7 @ A7 )
% 16.19/8.95 => ( P @ B7 ) )
% 16.19/8.95 => ( P @ A7 ) ) )
% 16.19/8.95 => ( P @ A2 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3723_finite__has__minimal,axiom,
% 16.19/8.95 ! [A2: set_real] :
% 16.19/8.95 ( ( finite_finite_real @ A2 )
% 16.19/8.95 => ( ( A2 != bot_bot_set_real )
% 16.19/8.95 => ? [X4: real] :
% 16.19/8.95 ( ( member_real @ X4 @ A2 )
% 16.19/8.95 & ! [Xa: real] :
% 16.19/8.95 ( ( member_real @ Xa @ A2 )
% 16.19/8.95 => ( ( ord_less_eq_real @ Xa @ X4 )
% 16.19/8.95 => ( X4 = Xa ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3724_finite__has__minimal,axiom,
% 16.19/8.95 ! [A2: set_set_nat] :
% 16.19/8.95 ( ( finite1152437895449049373et_nat @ A2 )
% 16.19/8.95 => ( ( A2 != bot_bot_set_set_nat )
% 16.19/8.95 => ? [X4: set_nat] :
% 16.19/8.95 ( ( member_set_nat @ X4 @ A2 )
% 16.19/8.95 & ! [Xa: set_nat] :
% 16.19/8.95 ( ( member_set_nat @ Xa @ A2 )
% 16.19/8.95 => ( ( ord_less_eq_set_nat @ Xa @ X4 )
% 16.19/8.95 => ( X4 = Xa ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3725_finite__has__minimal,axiom,
% 16.19/8.95 ! [A2: set_rat] :
% 16.19/8.95 ( ( finite_finite_rat @ A2 )
% 16.19/8.95 => ( ( A2 != bot_bot_set_rat )
% 16.19/8.95 => ? [X4: rat] :
% 16.19/8.95 ( ( member_rat @ X4 @ A2 )
% 16.19/8.95 & ! [Xa: rat] :
% 16.19/8.95 ( ( member_rat @ Xa @ A2 )
% 16.19/8.95 => ( ( ord_less_eq_rat @ Xa @ X4 )
% 16.19/8.95 => ( X4 = Xa ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3726_finite__has__minimal,axiom,
% 16.19/8.95 ! [A2: set_num] :
% 16.19/8.95 ( ( finite_finite_num @ A2 )
% 16.19/8.95 => ( ( A2 != bot_bot_set_num )
% 16.19/8.95 => ? [X4: num] :
% 16.19/8.95 ( ( member_num @ X4 @ A2 )
% 16.19/8.95 & ! [Xa: num] :
% 16.19/8.95 ( ( member_num @ Xa @ A2 )
% 16.19/8.95 => ( ( ord_less_eq_num @ Xa @ X4 )
% 16.19/8.95 => ( X4 = Xa ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3727_finite__has__minimal,axiom,
% 16.19/8.95 ! [A2: set_nat] :
% 16.19/8.95 ( ( finite_finite_nat @ A2 )
% 16.19/8.95 => ( ( A2 != bot_bot_set_nat )
% 16.19/8.95 => ? [X4: nat] :
% 16.19/8.95 ( ( member_nat @ X4 @ A2 )
% 16.19/8.95 & ! [Xa: nat] :
% 16.19/8.95 ( ( member_nat @ Xa @ A2 )
% 16.19/8.95 => ( ( ord_less_eq_nat @ Xa @ X4 )
% 16.19/8.95 => ( X4 = Xa ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3728_finite__has__minimal,axiom,
% 16.19/8.95 ! [A2: set_int] :
% 16.19/8.95 ( ( finite_finite_int @ A2 )
% 16.19/8.95 => ( ( A2 != bot_bot_set_int )
% 16.19/8.95 => ? [X4: int] :
% 16.19/8.95 ( ( member_int @ X4 @ A2 )
% 16.19/8.95 & ! [Xa: int] :
% 16.19/8.95 ( ( member_int @ Xa @ A2 )
% 16.19/8.95 => ( ( ord_less_eq_int @ Xa @ X4 )
% 16.19/8.95 => ( X4 = Xa ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3729_finite__has__maximal,axiom,
% 16.19/8.95 ! [A2: set_real] :
% 16.19/8.95 ( ( finite_finite_real @ A2 )
% 16.19/8.95 => ( ( A2 != bot_bot_set_real )
% 16.19/8.95 => ? [X4: real] :
% 16.19/8.95 ( ( member_real @ X4 @ A2 )
% 16.19/8.95 & ! [Xa: real] :
% 16.19/8.95 ( ( member_real @ Xa @ A2 )
% 16.19/8.95 => ( ( ord_less_eq_real @ X4 @ Xa )
% 16.19/8.95 => ( X4 = Xa ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3730_finite__has__maximal,axiom,
% 16.19/8.95 ! [A2: set_set_nat] :
% 16.19/8.95 ( ( finite1152437895449049373et_nat @ A2 )
% 16.19/8.95 => ( ( A2 != bot_bot_set_set_nat )
% 16.19/8.95 => ? [X4: set_nat] :
% 16.19/8.95 ( ( member_set_nat @ X4 @ A2 )
% 16.19/8.95 & ! [Xa: set_nat] :
% 16.19/8.95 ( ( member_set_nat @ Xa @ A2 )
% 16.19/8.95 => ( ( ord_less_eq_set_nat @ X4 @ Xa )
% 16.19/8.95 => ( X4 = Xa ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3731_finite__has__maximal,axiom,
% 16.19/8.95 ! [A2: set_rat] :
% 16.19/8.95 ( ( finite_finite_rat @ A2 )
% 16.19/8.95 => ( ( A2 != bot_bot_set_rat )
% 16.19/8.95 => ? [X4: rat] :
% 16.19/8.95 ( ( member_rat @ X4 @ A2 )
% 16.19/8.95 & ! [Xa: rat] :
% 16.19/8.95 ( ( member_rat @ Xa @ A2 )
% 16.19/8.95 => ( ( ord_less_eq_rat @ X4 @ Xa )
% 16.19/8.95 => ( X4 = Xa ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3732_finite__has__maximal,axiom,
% 16.19/8.95 ! [A2: set_num] :
% 16.19/8.95 ( ( finite_finite_num @ A2 )
% 16.19/8.95 => ( ( A2 != bot_bot_set_num )
% 16.19/8.95 => ? [X4: num] :
% 16.19/8.95 ( ( member_num @ X4 @ A2 )
% 16.19/8.95 & ! [Xa: num] :
% 16.19/8.95 ( ( member_num @ Xa @ A2 )
% 16.19/8.95 => ( ( ord_less_eq_num @ X4 @ Xa )
% 16.19/8.95 => ( X4 = Xa ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3733_finite__has__maximal,axiom,
% 16.19/8.95 ! [A2: set_nat] :
% 16.19/8.95 ( ( finite_finite_nat @ A2 )
% 16.19/8.95 => ( ( A2 != bot_bot_set_nat )
% 16.19/8.95 => ? [X4: nat] :
% 16.19/8.95 ( ( member_nat @ X4 @ A2 )
% 16.19/8.95 & ! [Xa: nat] :
% 16.19/8.95 ( ( member_nat @ Xa @ A2 )
% 16.19/8.95 => ( ( ord_less_eq_nat @ X4 @ Xa )
% 16.19/8.95 => ( X4 = Xa ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3734_finite__has__maximal,axiom,
% 16.19/8.95 ! [A2: set_int] :
% 16.19/8.95 ( ( finite_finite_int @ A2 )
% 16.19/8.95 => ( ( A2 != bot_bot_set_int )
% 16.19/8.95 => ? [X4: int] :
% 16.19/8.95 ( ( member_int @ X4 @ A2 )
% 16.19/8.95 & ! [Xa: int] :
% 16.19/8.95 ( ( member_int @ Xa @ A2 )
% 16.19/8.95 => ( ( ord_less_eq_int @ X4 @ Xa )
% 16.19/8.95 => ( X4 = Xa ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3735_arcosh__1,axiom,
% 16.19/8.95 ( ( arcosh_real @ one_one_real )
% 16.19/8.95 = zero_zero_real ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3736_finite__nth__roots,axiom,
% 16.19/8.95 ! [N2: nat,C: complex] :
% 16.19/8.95 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.95 => ( finite3207457112153483333omplex
% 16.19/8.95 @ ( collect_complex
% 16.19/8.95 @ ^ [Z3: complex] :
% 16.19/8.95 ( ( power_power_complex @ Z3 @ N2 )
% 16.19/8.95 = C ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3737_prod_Ofinite__Collect__op,axiom,
% 16.19/8.95 ! [I6: set_real,X: real > complex,Y: real > complex] :
% 16.19/8.95 ( ( finite_finite_real
% 16.19/8.95 @ ( collect_real
% 16.19/8.95 @ ^ [I3: real] :
% 16.19/8.95 ( ( member_real @ I3 @ I6 )
% 16.19/8.95 & ( ( X @ I3 )
% 16.19/8.95 != one_one_complex ) ) ) )
% 16.19/8.95 => ( ( finite_finite_real
% 16.19/8.95 @ ( collect_real
% 16.19/8.95 @ ^ [I3: real] :
% 16.19/8.95 ( ( member_real @ I3 @ I6 )
% 16.19/8.95 & ( ( Y @ I3 )
% 16.19/8.95 != one_one_complex ) ) ) )
% 16.19/8.95 => ( finite_finite_real
% 16.19/8.95 @ ( collect_real
% 16.19/8.95 @ ^ [I3: real] :
% 16.19/8.95 ( ( member_real @ I3 @ I6 )
% 16.19/8.95 & ( ( times_times_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
% 16.19/8.95 != one_one_complex ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3738_prod_Ofinite__Collect__op,axiom,
% 16.19/8.95 ! [I6: set_nat,X: nat > complex,Y: nat > complex] :
% 16.19/8.95 ( ( finite_finite_nat
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [I3: nat] :
% 16.19/8.95 ( ( member_nat @ I3 @ I6 )
% 16.19/8.95 & ( ( X @ I3 )
% 16.19/8.95 != one_one_complex ) ) ) )
% 16.19/8.95 => ( ( finite_finite_nat
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [I3: nat] :
% 16.19/8.95 ( ( member_nat @ I3 @ I6 )
% 16.19/8.95 & ( ( Y @ I3 )
% 16.19/8.95 != one_one_complex ) ) ) )
% 16.19/8.95 => ( finite_finite_nat
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [I3: nat] :
% 16.19/8.95 ( ( member_nat @ I3 @ I6 )
% 16.19/8.95 & ( ( times_times_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
% 16.19/8.95 != one_one_complex ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3739_prod_Ofinite__Collect__op,axiom,
% 16.19/8.95 ! [I6: set_int,X: int > complex,Y: int > complex] :
% 16.19/8.95 ( ( finite_finite_int
% 16.19/8.95 @ ( collect_int
% 16.19/8.95 @ ^ [I3: int] :
% 16.19/8.95 ( ( member_int @ I3 @ I6 )
% 16.19/8.95 & ( ( X @ I3 )
% 16.19/8.95 != one_one_complex ) ) ) )
% 16.19/8.95 => ( ( finite_finite_int
% 16.19/8.95 @ ( collect_int
% 16.19/8.95 @ ^ [I3: int] :
% 16.19/8.95 ( ( member_int @ I3 @ I6 )
% 16.19/8.95 & ( ( Y @ I3 )
% 16.19/8.95 != one_one_complex ) ) ) )
% 16.19/8.95 => ( finite_finite_int
% 16.19/8.95 @ ( collect_int
% 16.19/8.95 @ ^ [I3: int] :
% 16.19/8.95 ( ( member_int @ I3 @ I6 )
% 16.19/8.95 & ( ( times_times_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
% 16.19/8.95 != one_one_complex ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3740_prod_Ofinite__Collect__op,axiom,
% 16.19/8.95 ! [I6: set_complex,X: complex > complex,Y: complex > complex] :
% 16.19/8.95 ( ( finite3207457112153483333omplex
% 16.19/8.95 @ ( collect_complex
% 16.19/8.95 @ ^ [I3: complex] :
% 16.19/8.95 ( ( member_complex @ I3 @ I6 )
% 16.19/8.95 & ( ( X @ I3 )
% 16.19/8.95 != one_one_complex ) ) ) )
% 16.19/8.95 => ( ( finite3207457112153483333omplex
% 16.19/8.95 @ ( collect_complex
% 16.19/8.95 @ ^ [I3: complex] :
% 16.19/8.95 ( ( member_complex @ I3 @ I6 )
% 16.19/8.95 & ( ( Y @ I3 )
% 16.19/8.95 != one_one_complex ) ) ) )
% 16.19/8.95 => ( finite3207457112153483333omplex
% 16.19/8.95 @ ( collect_complex
% 16.19/8.95 @ ^ [I3: complex] :
% 16.19/8.95 ( ( member_complex @ I3 @ I6 )
% 16.19/8.95 & ( ( times_times_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
% 16.19/8.95 != one_one_complex ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3741_prod_Ofinite__Collect__op,axiom,
% 16.19/8.95 ! [I6: set_real,X: real > real,Y: real > real] :
% 16.19/8.95 ( ( finite_finite_real
% 16.19/8.95 @ ( collect_real
% 16.19/8.95 @ ^ [I3: real] :
% 16.19/8.95 ( ( member_real @ I3 @ I6 )
% 16.19/8.95 & ( ( X @ I3 )
% 16.19/8.95 != one_one_real ) ) ) )
% 16.19/8.95 => ( ( finite_finite_real
% 16.19/8.95 @ ( collect_real
% 16.19/8.95 @ ^ [I3: real] :
% 16.19/8.95 ( ( member_real @ I3 @ I6 )
% 16.19/8.95 & ( ( Y @ I3 )
% 16.19/8.95 != one_one_real ) ) ) )
% 16.19/8.95 => ( finite_finite_real
% 16.19/8.95 @ ( collect_real
% 16.19/8.95 @ ^ [I3: real] :
% 16.19/8.95 ( ( member_real @ I3 @ I6 )
% 16.19/8.95 & ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
% 16.19/8.95 != one_one_real ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3742_prod_Ofinite__Collect__op,axiom,
% 16.19/8.95 ! [I6: set_nat,X: nat > real,Y: nat > real] :
% 16.19/8.95 ( ( finite_finite_nat
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [I3: nat] :
% 16.19/8.95 ( ( member_nat @ I3 @ I6 )
% 16.19/8.95 & ( ( X @ I3 )
% 16.19/8.95 != one_one_real ) ) ) )
% 16.19/8.95 => ( ( finite_finite_nat
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [I3: nat] :
% 16.19/8.95 ( ( member_nat @ I3 @ I6 )
% 16.19/8.95 & ( ( Y @ I3 )
% 16.19/8.95 != one_one_real ) ) ) )
% 16.19/8.95 => ( finite_finite_nat
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [I3: nat] :
% 16.19/8.95 ( ( member_nat @ I3 @ I6 )
% 16.19/8.95 & ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
% 16.19/8.95 != one_one_real ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3743_prod_Ofinite__Collect__op,axiom,
% 16.19/8.95 ! [I6: set_int,X: int > real,Y: int > real] :
% 16.19/8.95 ( ( finite_finite_int
% 16.19/8.95 @ ( collect_int
% 16.19/8.95 @ ^ [I3: int] :
% 16.19/8.95 ( ( member_int @ I3 @ I6 )
% 16.19/8.95 & ( ( X @ I3 )
% 16.19/8.95 != one_one_real ) ) ) )
% 16.19/8.95 => ( ( finite_finite_int
% 16.19/8.95 @ ( collect_int
% 16.19/8.95 @ ^ [I3: int] :
% 16.19/8.95 ( ( member_int @ I3 @ I6 )
% 16.19/8.95 & ( ( Y @ I3 )
% 16.19/8.95 != one_one_real ) ) ) )
% 16.19/8.95 => ( finite_finite_int
% 16.19/8.95 @ ( collect_int
% 16.19/8.95 @ ^ [I3: int] :
% 16.19/8.95 ( ( member_int @ I3 @ I6 )
% 16.19/8.95 & ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
% 16.19/8.95 != one_one_real ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3744_prod_Ofinite__Collect__op,axiom,
% 16.19/8.95 ! [I6: set_complex,X: complex > real,Y: complex > real] :
% 16.19/8.95 ( ( finite3207457112153483333omplex
% 16.19/8.95 @ ( collect_complex
% 16.19/8.95 @ ^ [I3: complex] :
% 16.19/8.95 ( ( member_complex @ I3 @ I6 )
% 16.19/8.95 & ( ( X @ I3 )
% 16.19/8.95 != one_one_real ) ) ) )
% 16.19/8.95 => ( ( finite3207457112153483333omplex
% 16.19/8.95 @ ( collect_complex
% 16.19/8.95 @ ^ [I3: complex] :
% 16.19/8.95 ( ( member_complex @ I3 @ I6 )
% 16.19/8.95 & ( ( Y @ I3 )
% 16.19/8.95 != one_one_real ) ) ) )
% 16.19/8.95 => ( finite3207457112153483333omplex
% 16.19/8.95 @ ( collect_complex
% 16.19/8.95 @ ^ [I3: complex] :
% 16.19/8.95 ( ( member_complex @ I3 @ I6 )
% 16.19/8.95 & ( ( times_times_real @ ( X @ I3 ) @ ( Y @ I3 ) )
% 16.19/8.95 != one_one_real ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3745_prod_Ofinite__Collect__op,axiom,
% 16.19/8.95 ! [I6: set_real,X: real > rat,Y: real > rat] :
% 16.19/8.95 ( ( finite_finite_real
% 16.19/8.95 @ ( collect_real
% 16.19/8.95 @ ^ [I3: real] :
% 16.19/8.95 ( ( member_real @ I3 @ I6 )
% 16.19/8.95 & ( ( X @ I3 )
% 16.19/8.95 != one_one_rat ) ) ) )
% 16.19/8.95 => ( ( finite_finite_real
% 16.19/8.95 @ ( collect_real
% 16.19/8.95 @ ^ [I3: real] :
% 16.19/8.95 ( ( member_real @ I3 @ I6 )
% 16.19/8.95 & ( ( Y @ I3 )
% 16.19/8.95 != one_one_rat ) ) ) )
% 16.19/8.95 => ( finite_finite_real
% 16.19/8.95 @ ( collect_real
% 16.19/8.95 @ ^ [I3: real] :
% 16.19/8.95 ( ( member_real @ I3 @ I6 )
% 16.19/8.95 & ( ( times_times_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
% 16.19/8.95 != one_one_rat ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3746_prod_Ofinite__Collect__op,axiom,
% 16.19/8.95 ! [I6: set_nat,X: nat > rat,Y: nat > rat] :
% 16.19/8.95 ( ( finite_finite_nat
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [I3: nat] :
% 16.19/8.95 ( ( member_nat @ I3 @ I6 )
% 16.19/8.95 & ( ( X @ I3 )
% 16.19/8.95 != one_one_rat ) ) ) )
% 16.19/8.95 => ( ( finite_finite_nat
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [I3: nat] :
% 16.19/8.95 ( ( member_nat @ I3 @ I6 )
% 16.19/8.95 & ( ( Y @ I3 )
% 16.19/8.95 != one_one_rat ) ) ) )
% 16.19/8.95 => ( finite_finite_nat
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [I3: nat] :
% 16.19/8.95 ( ( member_nat @ I3 @ I6 )
% 16.19/8.95 & ( ( times_times_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
% 16.19/8.95 != one_one_rat ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3747_sum_Ofinite__Collect__op,axiom,
% 16.19/8.95 ! [I6: set_real,X: real > complex,Y: real > complex] :
% 16.19/8.95 ( ( finite_finite_real
% 16.19/8.95 @ ( collect_real
% 16.19/8.95 @ ^ [I3: real] :
% 16.19/8.95 ( ( member_real @ I3 @ I6 )
% 16.19/8.95 & ( ( X @ I3 )
% 16.19/8.95 != zero_zero_complex ) ) ) )
% 16.19/8.95 => ( ( finite_finite_real
% 16.19/8.95 @ ( collect_real
% 16.19/8.95 @ ^ [I3: real] :
% 16.19/8.95 ( ( member_real @ I3 @ I6 )
% 16.19/8.95 & ( ( Y @ I3 )
% 16.19/8.95 != zero_zero_complex ) ) ) )
% 16.19/8.95 => ( finite_finite_real
% 16.19/8.95 @ ( collect_real
% 16.19/8.95 @ ^ [I3: real] :
% 16.19/8.95 ( ( member_real @ I3 @ I6 )
% 16.19/8.95 & ( ( plus_plus_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
% 16.19/8.95 != zero_zero_complex ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3748_sum_Ofinite__Collect__op,axiom,
% 16.19/8.95 ! [I6: set_nat,X: nat > complex,Y: nat > complex] :
% 16.19/8.95 ( ( finite_finite_nat
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [I3: nat] :
% 16.19/8.95 ( ( member_nat @ I3 @ I6 )
% 16.19/8.95 & ( ( X @ I3 )
% 16.19/8.95 != zero_zero_complex ) ) ) )
% 16.19/8.95 => ( ( finite_finite_nat
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [I3: nat] :
% 16.19/8.95 ( ( member_nat @ I3 @ I6 )
% 16.19/8.95 & ( ( Y @ I3 )
% 16.19/8.95 != zero_zero_complex ) ) ) )
% 16.19/8.95 => ( finite_finite_nat
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [I3: nat] :
% 16.19/8.95 ( ( member_nat @ I3 @ I6 )
% 16.19/8.95 & ( ( plus_plus_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
% 16.19/8.95 != zero_zero_complex ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3749_sum_Ofinite__Collect__op,axiom,
% 16.19/8.95 ! [I6: set_int,X: int > complex,Y: int > complex] :
% 16.19/8.95 ( ( finite_finite_int
% 16.19/8.95 @ ( collect_int
% 16.19/8.95 @ ^ [I3: int] :
% 16.19/8.95 ( ( member_int @ I3 @ I6 )
% 16.19/8.95 & ( ( X @ I3 )
% 16.19/8.95 != zero_zero_complex ) ) ) )
% 16.19/8.95 => ( ( finite_finite_int
% 16.19/8.95 @ ( collect_int
% 16.19/8.95 @ ^ [I3: int] :
% 16.19/8.95 ( ( member_int @ I3 @ I6 )
% 16.19/8.95 & ( ( Y @ I3 )
% 16.19/8.95 != zero_zero_complex ) ) ) )
% 16.19/8.95 => ( finite_finite_int
% 16.19/8.95 @ ( collect_int
% 16.19/8.95 @ ^ [I3: int] :
% 16.19/8.95 ( ( member_int @ I3 @ I6 )
% 16.19/8.95 & ( ( plus_plus_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
% 16.19/8.95 != zero_zero_complex ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3750_sum_Ofinite__Collect__op,axiom,
% 16.19/8.95 ! [I6: set_complex,X: complex > complex,Y: complex > complex] :
% 16.19/8.95 ( ( finite3207457112153483333omplex
% 16.19/8.95 @ ( collect_complex
% 16.19/8.95 @ ^ [I3: complex] :
% 16.19/8.95 ( ( member_complex @ I3 @ I6 )
% 16.19/8.95 & ( ( X @ I3 )
% 16.19/8.95 != zero_zero_complex ) ) ) )
% 16.19/8.95 => ( ( finite3207457112153483333omplex
% 16.19/8.95 @ ( collect_complex
% 16.19/8.95 @ ^ [I3: complex] :
% 16.19/8.95 ( ( member_complex @ I3 @ I6 )
% 16.19/8.95 & ( ( Y @ I3 )
% 16.19/8.95 != zero_zero_complex ) ) ) )
% 16.19/8.95 => ( finite3207457112153483333omplex
% 16.19/8.95 @ ( collect_complex
% 16.19/8.95 @ ^ [I3: complex] :
% 16.19/8.95 ( ( member_complex @ I3 @ I6 )
% 16.19/8.95 & ( ( plus_plus_complex @ ( X @ I3 ) @ ( Y @ I3 ) )
% 16.19/8.95 != zero_zero_complex ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3751_sum_Ofinite__Collect__op,axiom,
% 16.19/8.95 ! [I6: set_real,X: real > real,Y: real > real] :
% 16.19/8.95 ( ( finite_finite_real
% 16.19/8.95 @ ( collect_real
% 16.19/8.95 @ ^ [I3: real] :
% 16.19/8.95 ( ( member_real @ I3 @ I6 )
% 16.19/8.95 & ( ( X @ I3 )
% 16.19/8.95 != zero_zero_real ) ) ) )
% 16.19/8.95 => ( ( finite_finite_real
% 16.19/8.95 @ ( collect_real
% 16.19/8.95 @ ^ [I3: real] :
% 16.19/8.95 ( ( member_real @ I3 @ I6 )
% 16.19/8.95 & ( ( Y @ I3 )
% 16.19/8.95 != zero_zero_real ) ) ) )
% 16.19/8.95 => ( finite_finite_real
% 16.19/8.95 @ ( collect_real
% 16.19/8.95 @ ^ [I3: real] :
% 16.19/8.95 ( ( member_real @ I3 @ I6 )
% 16.19/8.95 & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
% 16.19/8.95 != zero_zero_real ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3752_sum_Ofinite__Collect__op,axiom,
% 16.19/8.95 ! [I6: set_nat,X: nat > real,Y: nat > real] :
% 16.19/8.95 ( ( finite_finite_nat
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [I3: nat] :
% 16.19/8.95 ( ( member_nat @ I3 @ I6 )
% 16.19/8.95 & ( ( X @ I3 )
% 16.19/8.95 != zero_zero_real ) ) ) )
% 16.19/8.95 => ( ( finite_finite_nat
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [I3: nat] :
% 16.19/8.95 ( ( member_nat @ I3 @ I6 )
% 16.19/8.95 & ( ( Y @ I3 )
% 16.19/8.95 != zero_zero_real ) ) ) )
% 16.19/8.95 => ( finite_finite_nat
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [I3: nat] :
% 16.19/8.95 ( ( member_nat @ I3 @ I6 )
% 16.19/8.95 & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
% 16.19/8.95 != zero_zero_real ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3753_sum_Ofinite__Collect__op,axiom,
% 16.19/8.95 ! [I6: set_int,X: int > real,Y: int > real] :
% 16.19/8.95 ( ( finite_finite_int
% 16.19/8.95 @ ( collect_int
% 16.19/8.95 @ ^ [I3: int] :
% 16.19/8.95 ( ( member_int @ I3 @ I6 )
% 16.19/8.95 & ( ( X @ I3 )
% 16.19/8.95 != zero_zero_real ) ) ) )
% 16.19/8.95 => ( ( finite_finite_int
% 16.19/8.95 @ ( collect_int
% 16.19/8.95 @ ^ [I3: int] :
% 16.19/8.95 ( ( member_int @ I3 @ I6 )
% 16.19/8.95 & ( ( Y @ I3 )
% 16.19/8.95 != zero_zero_real ) ) ) )
% 16.19/8.95 => ( finite_finite_int
% 16.19/8.95 @ ( collect_int
% 16.19/8.95 @ ^ [I3: int] :
% 16.19/8.95 ( ( member_int @ I3 @ I6 )
% 16.19/8.95 & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
% 16.19/8.95 != zero_zero_real ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3754_sum_Ofinite__Collect__op,axiom,
% 16.19/8.95 ! [I6: set_complex,X: complex > real,Y: complex > real] :
% 16.19/8.95 ( ( finite3207457112153483333omplex
% 16.19/8.95 @ ( collect_complex
% 16.19/8.95 @ ^ [I3: complex] :
% 16.19/8.95 ( ( member_complex @ I3 @ I6 )
% 16.19/8.95 & ( ( X @ I3 )
% 16.19/8.95 != zero_zero_real ) ) ) )
% 16.19/8.95 => ( ( finite3207457112153483333omplex
% 16.19/8.95 @ ( collect_complex
% 16.19/8.95 @ ^ [I3: complex] :
% 16.19/8.95 ( ( member_complex @ I3 @ I6 )
% 16.19/8.95 & ( ( Y @ I3 )
% 16.19/8.95 != zero_zero_real ) ) ) )
% 16.19/8.95 => ( finite3207457112153483333omplex
% 16.19/8.95 @ ( collect_complex
% 16.19/8.95 @ ^ [I3: complex] :
% 16.19/8.95 ( ( member_complex @ I3 @ I6 )
% 16.19/8.95 & ( ( plus_plus_real @ ( X @ I3 ) @ ( Y @ I3 ) )
% 16.19/8.95 != zero_zero_real ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3755_sum_Ofinite__Collect__op,axiom,
% 16.19/8.95 ! [I6: set_real,X: real > rat,Y: real > rat] :
% 16.19/8.95 ( ( finite_finite_real
% 16.19/8.95 @ ( collect_real
% 16.19/8.95 @ ^ [I3: real] :
% 16.19/8.95 ( ( member_real @ I3 @ I6 )
% 16.19/8.95 & ( ( X @ I3 )
% 16.19/8.95 != zero_zero_rat ) ) ) )
% 16.19/8.95 => ( ( finite_finite_real
% 16.19/8.95 @ ( collect_real
% 16.19/8.95 @ ^ [I3: real] :
% 16.19/8.95 ( ( member_real @ I3 @ I6 )
% 16.19/8.95 & ( ( Y @ I3 )
% 16.19/8.95 != zero_zero_rat ) ) ) )
% 16.19/8.95 => ( finite_finite_real
% 16.19/8.95 @ ( collect_real
% 16.19/8.95 @ ^ [I3: real] :
% 16.19/8.95 ( ( member_real @ I3 @ I6 )
% 16.19/8.95 & ( ( plus_plus_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
% 16.19/8.95 != zero_zero_rat ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3756_sum_Ofinite__Collect__op,axiom,
% 16.19/8.95 ! [I6: set_nat,X: nat > rat,Y: nat > rat] :
% 16.19/8.95 ( ( finite_finite_nat
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [I3: nat] :
% 16.19/8.95 ( ( member_nat @ I3 @ I6 )
% 16.19/8.95 & ( ( X @ I3 )
% 16.19/8.95 != zero_zero_rat ) ) ) )
% 16.19/8.95 => ( ( finite_finite_nat
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [I3: nat] :
% 16.19/8.95 ( ( member_nat @ I3 @ I6 )
% 16.19/8.95 & ( ( Y @ I3 )
% 16.19/8.95 != zero_zero_rat ) ) ) )
% 16.19/8.95 => ( finite_finite_nat
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [I3: nat] :
% 16.19/8.95 ( ( member_nat @ I3 @ I6 )
% 16.19/8.95 & ( ( plus_plus_rat @ ( X @ I3 ) @ ( Y @ I3 ) )
% 16.19/8.95 != zero_zero_rat ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3757_gcd__nat__induct,axiom,
% 16.19/8.95 ! [P: nat > nat > $o,M: nat,N2: nat] :
% 16.19/8.95 ( ! [M2: nat] : ( P @ M2 @ zero_zero_nat )
% 16.19/8.95 => ( ! [M2: nat,N4: nat] :
% 16.19/8.95 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 16.19/8.95 => ( ( P @ N4 @ ( modulo_modulo_nat @ M2 @ N4 ) )
% 16.19/8.95 => ( P @ M2 @ N4 ) ) )
% 16.19/8.95 => ( P @ M @ N2 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3758_Euclid__induct,axiom,
% 16.19/8.95 ! [P: nat > nat > $o,A: nat,B: nat] :
% 16.19/8.95 ( ! [A3: nat,B2: nat] :
% 16.19/8.95 ( ( P @ A3 @ B2 )
% 16.19/8.95 = ( P @ B2 @ A3 ) )
% 16.19/8.95 => ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
% 16.19/8.95 => ( ! [A3: nat,B2: nat] :
% 16.19/8.95 ( ( P @ A3 @ B2 )
% 16.19/8.95 => ( P @ A3 @ ( plus_plus_nat @ A3 @ B2 ) ) )
% 16.19/8.95 => ( P @ A @ B ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3759_concat__bit__Suc,axiom,
% 16.19/8.95 ! [N2: nat,K: int,L2: int] :
% 16.19/8.95 ( ( bit_concat_bit @ ( suc @ N2 ) @ K @ L2 )
% 16.19/8.95 = ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N2 @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L2 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3760_dbl__simps_I3_J,axiom,
% 16.19/8.95 ( ( neg_nu7009210354673126013omplex @ one_one_complex )
% 16.19/8.95 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3761_dbl__simps_I3_J,axiom,
% 16.19/8.95 ( ( neg_numeral_dbl_real @ one_one_real )
% 16.19/8.95 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3762_dbl__simps_I3_J,axiom,
% 16.19/8.95 ( ( neg_numeral_dbl_rat @ one_one_rat )
% 16.19/8.95 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3763_dbl__simps_I3_J,axiom,
% 16.19/8.95 ( ( neg_numeral_dbl_int @ one_one_int )
% 16.19/8.95 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3764_even__succ__mod__exp,axiom,
% 16.19/8.95 ! [A: nat,N2: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 16.19/8.95 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.95 => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.95 = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3765_even__succ__mod__exp,axiom,
% 16.19/8.95 ! [A: int,N2: nat] :
% 16.19/8.95 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 16.19/8.95 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.95 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.95 = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3766_even__succ__mod__exp,axiom,
% 16.19/8.95 ! [A: code_integer,N2: nat] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 16.19/8.95 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.95 => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.95 = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3767_even__succ__div__exp,axiom,
% 16.19/8.95 ! [A: code_integer,N2: nat] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 16.19/8.95 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.95 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.95 = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3768_even__succ__div__exp,axiom,
% 16.19/8.95 ! [A: nat,N2: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 16.19/8.95 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.95 => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.95 = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3769_even__succ__div__exp,axiom,
% 16.19/8.95 ! [A: int,N2: nat] :
% 16.19/8.95 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 16.19/8.95 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.95 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.95 = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3770_order__refl,axiom,
% 16.19/8.95 ! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3771_order__refl,axiom,
% 16.19/8.95 ! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3772_order__refl,axiom,
% 16.19/8.95 ! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3773_order__refl,axiom,
% 16.19/8.95 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3774_order__refl,axiom,
% 16.19/8.95 ! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3775_dual__order_Orefl,axiom,
% 16.19/8.95 ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3776_dual__order_Orefl,axiom,
% 16.19/8.95 ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3777_dual__order_Orefl,axiom,
% 16.19/8.95 ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3778_dual__order_Orefl,axiom,
% 16.19/8.95 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3779_dual__order_Orefl,axiom,
% 16.19/8.95 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3780_nat__dvd__1__iff__1,axiom,
% 16.19/8.95 ! [M: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ M @ one_one_nat )
% 16.19/8.95 = ( M = one_one_nat ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3781_dvd__add__triv__right__iff,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
% 16.19/8.95 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3782_dvd__add__triv__right__iff,axiom,
% 16.19/8.95 ! [A: real,B: real] :
% 16.19/8.95 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ A ) )
% 16.19/8.95 = ( dvd_dvd_real @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3783_dvd__add__triv__right__iff,axiom,
% 16.19/8.95 ! [A: rat,B: rat] :
% 16.19/8.95 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 16.19/8.95 = ( dvd_dvd_rat @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3784_dvd__add__triv__right__iff,axiom,
% 16.19/8.95 ! [A: nat,B: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 16.19/8.95 = ( dvd_dvd_nat @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3785_dvd__add__triv__right__iff,axiom,
% 16.19/8.95 ! [A: int,B: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
% 16.19/8.95 = ( dvd_dvd_int @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3786_dvd__add__triv__left__iff,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 16.19/8.95 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3787_dvd__add__triv__left__iff,axiom,
% 16.19/8.95 ! [A: real,B: real] :
% 16.19/8.95 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B ) )
% 16.19/8.95 = ( dvd_dvd_real @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3788_dvd__add__triv__left__iff,axiom,
% 16.19/8.95 ! [A: rat,B: rat] :
% 16.19/8.95 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 16.19/8.95 = ( dvd_dvd_rat @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3789_dvd__add__triv__left__iff,axiom,
% 16.19/8.95 ! [A: nat,B: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 16.19/8.95 = ( dvd_dvd_nat @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3790_dvd__add__triv__left__iff,axiom,
% 16.19/8.95 ! [A: int,B: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
% 16.19/8.95 = ( dvd_dvd_int @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3791_dvd__1__left,axiom,
% 16.19/8.95 ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3792_dvd__1__iff__1,axiom,
% 16.19/8.95 ! [M: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
% 16.19/8.95 = ( M
% 16.19/8.95 = ( suc @ zero_zero_nat ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3793_div__dvd__div,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer,C: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ A @ B )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ A @ C )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B @ A ) @ ( divide6298287555418463151nteger @ C @ A ) )
% 16.19/8.95 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3794_div__dvd__div,axiom,
% 16.19/8.95 ! [A: nat,B: nat,C: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ A @ B )
% 16.19/8.95 => ( ( dvd_dvd_nat @ A @ C )
% 16.19/8.95 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C @ A ) )
% 16.19/8.95 = ( dvd_dvd_nat @ B @ C ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3795_div__dvd__div,axiom,
% 16.19/8.95 ! [A: int,B: int,C: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ A @ B )
% 16.19/8.95 => ( ( dvd_dvd_int @ A @ C )
% 16.19/8.95 => ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C @ A ) )
% 16.19/8.95 = ( dvd_dvd_int @ B @ C ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3796_nat__mult__dvd__cancel__disj,axiom,
% 16.19/8.95 ! [K: nat,M: nat,N2: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 16.19/8.95 = ( ( K = zero_zero_nat )
% 16.19/8.95 | ( dvd_dvd_nat @ M @ N2 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3797_concat__bit__0,axiom,
% 16.19/8.95 ! [K: int,L2: int] :
% 16.19/8.95 ( ( bit_concat_bit @ zero_zero_nat @ K @ L2 )
% 16.19/8.95 = L2 ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3798_dbl__simps_I2_J,axiom,
% 16.19/8.95 ( ( neg_nu7009210354673126013omplex @ zero_zero_complex )
% 16.19/8.95 = zero_zero_complex ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3799_dbl__simps_I2_J,axiom,
% 16.19/8.95 ( ( neg_numeral_dbl_real @ zero_zero_real )
% 16.19/8.95 = zero_zero_real ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3800_dbl__simps_I2_J,axiom,
% 16.19/8.95 ( ( neg_numeral_dbl_rat @ zero_zero_rat )
% 16.19/8.95 = zero_zero_rat ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3801_dbl__simps_I2_J,axiom,
% 16.19/8.95 ( ( neg_numeral_dbl_int @ zero_zero_int )
% 16.19/8.95 = zero_zero_int ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3802_dvd__times__right__cancel__iff,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer,C: code_integer] :
% 16.19/8.95 ( ( A != zero_z3403309356797280102nteger )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
% 16.19/8.95 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3803_dvd__times__right__cancel__iff,axiom,
% 16.19/8.95 ! [A: nat,B: nat,C: nat] :
% 16.19/8.95 ( ( A != zero_zero_nat )
% 16.19/8.95 => ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
% 16.19/8.95 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3804_dvd__times__right__cancel__iff,axiom,
% 16.19/8.95 ! [A: int,B: int,C: int] :
% 16.19/8.95 ( ( A != zero_zero_int )
% 16.19/8.95 => ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
% 16.19/8.95 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3805_dvd__times__left__cancel__iff,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer,C: code_integer] :
% 16.19/8.95 ( ( A != zero_z3403309356797280102nteger )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) )
% 16.19/8.95 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3806_dvd__times__left__cancel__iff,axiom,
% 16.19/8.95 ! [A: nat,B: nat,C: nat] :
% 16.19/8.95 ( ( A != zero_zero_nat )
% 16.19/8.95 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
% 16.19/8.95 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3807_dvd__times__left__cancel__iff,axiom,
% 16.19/8.95 ! [A: int,B: int,C: int] :
% 16.19/8.95 ( ( A != zero_zero_int )
% 16.19/8.95 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
% 16.19/8.95 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3808_dvd__mult__cancel__right,axiom,
% 16.19/8.95 ! [A: code_integer,C: code_integer,B: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 16.19/8.95 = ( ( C = zero_z3403309356797280102nteger )
% 16.19/8.95 | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3809_dvd__mult__cancel__right,axiom,
% 16.19/8.95 ! [A: complex,C: complex,B: complex] :
% 16.19/8.95 ( ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 16.19/8.95 = ( ( C = zero_zero_complex )
% 16.19/8.95 | ( dvd_dvd_complex @ A @ B ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3810_dvd__mult__cancel__right,axiom,
% 16.19/8.95 ! [A: real,C: real,B: real] :
% 16.19/8.95 ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 16.19/8.95 = ( ( C = zero_zero_real )
% 16.19/8.95 | ( dvd_dvd_real @ A @ B ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3811_dvd__mult__cancel__right,axiom,
% 16.19/8.95 ! [A: rat,C: rat,B: rat] :
% 16.19/8.95 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 16.19/8.95 = ( ( C = zero_zero_rat )
% 16.19/8.95 | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3812_dvd__mult__cancel__right,axiom,
% 16.19/8.95 ! [A: int,C: int,B: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 16.19/8.95 = ( ( C = zero_zero_int )
% 16.19/8.95 | ( dvd_dvd_int @ A @ B ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3813_dvd__mult__cancel__left,axiom,
% 16.19/8.95 ! [C: code_integer,A: code_integer,B: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 16.19/8.95 = ( ( C = zero_z3403309356797280102nteger )
% 16.19/8.95 | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3814_dvd__mult__cancel__left,axiom,
% 16.19/8.95 ! [C: complex,A: complex,B: complex] :
% 16.19/8.95 ( ( dvd_dvd_complex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 16.19/8.95 = ( ( C = zero_zero_complex )
% 16.19/8.95 | ( dvd_dvd_complex @ A @ B ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3815_dvd__mult__cancel__left,axiom,
% 16.19/8.95 ! [C: real,A: real,B: real] :
% 16.19/8.95 ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 16.19/8.95 = ( ( C = zero_zero_real )
% 16.19/8.95 | ( dvd_dvd_real @ A @ B ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3816_dvd__mult__cancel__left,axiom,
% 16.19/8.95 ! [C: rat,A: rat,B: rat] :
% 16.19/8.95 ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 16.19/8.95 = ( ( C = zero_zero_rat )
% 16.19/8.95 | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3817_dvd__mult__cancel__left,axiom,
% 16.19/8.95 ! [C: int,A: int,B: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 16.19/8.95 = ( ( C = zero_zero_int )
% 16.19/8.95 | ( dvd_dvd_int @ A @ B ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3818_dvd__add__times__triv__right__iff,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer,C: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C @ A ) ) )
% 16.19/8.95 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3819_dvd__add__times__triv__right__iff,axiom,
% 16.19/8.95 ! [A: real,B: real,C: real] :
% 16.19/8.95 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ ( times_times_real @ C @ A ) ) )
% 16.19/8.95 = ( dvd_dvd_real @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3820_dvd__add__times__triv__right__iff,axiom,
% 16.19/8.95 ! [A: rat,B: rat,C: rat] :
% 16.19/8.95 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A ) ) )
% 16.19/8.95 = ( dvd_dvd_rat @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3821_dvd__add__times__triv__right__iff,axiom,
% 16.19/8.95 ! [A: nat,B: nat,C: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
% 16.19/8.95 = ( dvd_dvd_nat @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3822_dvd__add__times__triv__right__iff,axiom,
% 16.19/8.95 ! [A: int,B: int,C: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
% 16.19/8.95 = ( dvd_dvd_int @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3823_dvd__add__times__triv__left__iff,axiom,
% 16.19/8.95 ! [A: code_integer,C: code_integer,B: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B ) )
% 16.19/8.95 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3824_dvd__add__times__triv__left__iff,axiom,
% 16.19/8.95 ! [A: real,C: real,B: real] :
% 16.19/8.95 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B ) )
% 16.19/8.95 = ( dvd_dvd_real @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3825_dvd__add__times__triv__left__iff,axiom,
% 16.19/8.95 ! [A: rat,C: rat,B: rat] :
% 16.19/8.95 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B ) )
% 16.19/8.95 = ( dvd_dvd_rat @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3826_dvd__add__times__triv__left__iff,axiom,
% 16.19/8.95 ! [A: nat,C: nat,B: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
% 16.19/8.95 = ( dvd_dvd_nat @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3827_dvd__add__times__triv__left__iff,axiom,
% 16.19/8.95 ! [A: int,C: int,B: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
% 16.19/8.95 = ( dvd_dvd_int @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3828_unit__prod,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 16.19/8.95 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3829_unit__prod,axiom,
% 16.19/8.95 ! [A: nat,B: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 16.19/8.95 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 16.19/8.95 => ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3830_unit__prod,axiom,
% 16.19/8.95 ! [A: int,B: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ A @ one_one_int )
% 16.19/8.95 => ( ( dvd_dvd_int @ B @ one_one_int )
% 16.19/8.95 => ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3831_dvd__mult__div__cancel,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ A @ B )
% 16.19/8.95 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ A ) )
% 16.19/8.95 = B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3832_dvd__mult__div__cancel,axiom,
% 16.19/8.95 ! [A: nat,B: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ A @ B )
% 16.19/8.95 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
% 16.19/8.95 = B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3833_dvd__mult__div__cancel,axiom,
% 16.19/8.95 ! [A: int,B: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ A @ B )
% 16.19/8.95 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
% 16.19/8.95 = B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3834_dvd__div__mult__self,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ A @ B )
% 16.19/8.95 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
% 16.19/8.95 = B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3835_dvd__div__mult__self,axiom,
% 16.19/8.95 ! [A: nat,B: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ A @ B )
% 16.19/8.95 => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
% 16.19/8.95 = B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3836_dvd__div__mult__self,axiom,
% 16.19/8.95 ! [A: int,B: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ A @ B )
% 16.19/8.95 => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
% 16.19/8.95 = B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3837_div__add,axiom,
% 16.19/8.95 ! [C: code_integer,A: code_integer,B: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ C @ A )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ C @ B )
% 16.19/8.95 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 16.19/8.95 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3838_div__add,axiom,
% 16.19/8.95 ! [C: nat,A: nat,B: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ C @ A )
% 16.19/8.95 => ( ( dvd_dvd_nat @ C @ B )
% 16.19/8.95 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 16.19/8.95 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3839_div__add,axiom,
% 16.19/8.95 ! [C: int,A: int,B: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ C @ A )
% 16.19/8.95 => ( ( dvd_dvd_int @ C @ B )
% 16.19/8.95 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 16.19/8.95 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3840_unit__div__1__div__1,axiom,
% 16.19/8.95 ! [A: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 16.19/8.95 => ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 16.19/8.95 = A ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3841_unit__div__1__div__1,axiom,
% 16.19/8.95 ! [A: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 16.19/8.95 => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
% 16.19/8.95 = A ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3842_unit__div__1__div__1,axiom,
% 16.19/8.95 ! [A: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ A @ one_one_int )
% 16.19/8.95 => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
% 16.19/8.95 = A ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3843_unit__div__1__unit,axiom,
% 16.19/8.95 ! [A: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 16.19/8.95 => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3844_unit__div__1__unit,axiom,
% 16.19/8.95 ! [A: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 16.19/8.95 => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3845_unit__div__1__unit,axiom,
% 16.19/8.95 ! [A: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ A @ one_one_int )
% 16.19/8.95 => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3846_unit__div,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 16.19/8.95 => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3847_unit__div,axiom,
% 16.19/8.95 ! [A: nat,B: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 16.19/8.95 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 16.19/8.95 => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3848_unit__div,axiom,
% 16.19/8.95 ! [A: int,B: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ A @ one_one_int )
% 16.19/8.95 => ( ( dvd_dvd_int @ B @ one_one_int )
% 16.19/8.95 => ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3849_div__diff,axiom,
% 16.19/8.95 ! [C: code_integer,A: code_integer,B: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ C @ A )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ C @ B )
% 16.19/8.95 => ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
% 16.19/8.95 = ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3850_div__diff,axiom,
% 16.19/8.95 ! [C: int,A: int,B: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ C @ A )
% 16.19/8.95 => ( ( dvd_dvd_int @ C @ B )
% 16.19/8.95 => ( ( divide_divide_int @ ( minus_minus_int @ A @ B ) @ C )
% 16.19/8.95 = ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3851_concat__bit__nonnegative__iff,axiom,
% 16.19/8.95 ! [N2: nat,K: int,L2: int] :
% 16.19/8.95 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N2 @ K @ L2 ) )
% 16.19/8.95 = ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3852_concat__bit__negative__iff,axiom,
% 16.19/8.95 ! [N2: nat,K: int,L2: int] :
% 16.19/8.95 ( ( ord_less_int @ ( bit_concat_bit @ N2 @ K @ L2 ) @ zero_zero_int )
% 16.19/8.95 = ( ord_less_int @ L2 @ zero_zero_int ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3853_dbl__simps_I5_J,axiom,
% 16.19/8.95 ! [K: num] :
% 16.19/8.95 ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
% 16.19/8.95 = ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3854_dbl__simps_I5_J,axiom,
% 16.19/8.95 ! [K: num] :
% 16.19/8.95 ( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
% 16.19/8.95 = ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3855_dbl__simps_I5_J,axiom,
% 16.19/8.95 ! [K: num] :
% 16.19/8.95 ( ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) )
% 16.19/8.95 = ( numeral_numeral_rat @ ( bit0 @ K ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3856_dbl__simps_I5_J,axiom,
% 16.19/8.95 ! [K: num] :
% 16.19/8.95 ( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
% 16.19/8.95 = ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3857_even__Suc__Suc__iff,axiom,
% 16.19/8.95 ! [N2: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N2 ) ) )
% 16.19/8.95 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3858_even__Suc,axiom,
% 16.19/8.95 ! [N2: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) )
% 16.19/8.95 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3859_unit__div__mult__self,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 16.19/8.95 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
% 16.19/8.95 = B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3860_unit__div__mult__self,axiom,
% 16.19/8.95 ! [A: nat,B: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 16.19/8.95 => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
% 16.19/8.95 = B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3861_unit__div__mult__self,axiom,
% 16.19/8.95 ! [A: int,B: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ A @ one_one_int )
% 16.19/8.95 => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
% 16.19/8.95 = B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3862_unit__mult__div__div,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 16.19/8.95 => ( ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 16.19/8.95 = ( divide6298287555418463151nteger @ B @ A ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3863_unit__mult__div__div,axiom,
% 16.19/8.95 ! [A: nat,B: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 16.19/8.95 => ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
% 16.19/8.95 = ( divide_divide_nat @ B @ A ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3864_unit__mult__div__div,axiom,
% 16.19/8.95 ! [A: int,B: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ A @ one_one_int )
% 16.19/8.95 => ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
% 16.19/8.95 = ( divide_divide_int @ B @ A ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3865_pow__divides__pow__iff,axiom,
% 16.19/8.95 ! [N2: nat,A: nat,B: nat] :
% 16.19/8.95 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.95 => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
% 16.19/8.95 = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3866_pow__divides__pow__iff,axiom,
% 16.19/8.95 ! [N2: nat,A: int,B: int] :
% 16.19/8.95 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.95 => ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
% 16.19/8.95 = ( dvd_dvd_int @ A @ B ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3867_even__mult__iff,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B ) )
% 16.19/8.95 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 16.19/8.95 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3868_even__mult__iff,axiom,
% 16.19/8.95 ! [A: nat,B: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) )
% 16.19/8.95 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 16.19/8.95 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3869_even__mult__iff,axiom,
% 16.19/8.95 ! [A: int,B: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) )
% 16.19/8.95 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 16.19/8.95 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3870_even__add,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 16.19/8.95 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 16.19/8.95 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3871_even__add,axiom,
% 16.19/8.95 ! [A: nat,B: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
% 16.19/8.95 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 16.19/8.95 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3872_even__add,axiom,
% 16.19/8.95 ! [A: int,B: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
% 16.19/8.95 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 16.19/8.95 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3873_odd__add,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer] :
% 16.19/8.95 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) )
% 16.19/8.95 = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 16.19/8.95 != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3874_odd__add,axiom,
% 16.19/8.95 ! [A: nat,B: nat] :
% 16.19/8.95 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
% 16.19/8.95 = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 16.19/8.95 != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3875_odd__add,axiom,
% 16.19/8.95 ! [A: int,B: int] :
% 16.19/8.95 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
% 16.19/8.95 = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 16.19/8.95 != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3876_even__mod__2__iff,axiom,
% 16.19/8.95 ! [A: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.95 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3877_even__mod__2__iff,axiom,
% 16.19/8.95 ! [A: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 16.19/8.95 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3878_even__mod__2__iff,axiom,
% 16.19/8.95 ! [A: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 16.19/8.95 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3879_even__Suc__div__two,axiom,
% 16.19/8.95 ! [N2: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.95 => ( ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.95 = ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3880_odd__Suc__div__two,axiom,
% 16.19/8.95 ! [N2: nat] :
% 16.19/8.95 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.95 => ( ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.95 = ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3881_zero__le__power__eq__numeral,axiom,
% 16.19/8.95 ! [A: real,W: num] :
% 16.19/8.95 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 16.19/8.95 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 16.19/8.95 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 16.19/8.95 & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3882_zero__le__power__eq__numeral,axiom,
% 16.19/8.95 ! [A: rat,W: num] :
% 16.19/8.95 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 16.19/8.95 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 16.19/8.95 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 16.19/8.95 & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3883_zero__le__power__eq__numeral,axiom,
% 16.19/8.95 ! [A: int,W: num] :
% 16.19/8.95 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 16.19/8.95 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 16.19/8.95 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 16.19/8.95 & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3884_power__less__zero__eq__numeral,axiom,
% 16.19/8.95 ! [A: real,W: num] :
% 16.19/8.95 ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 16.19/8.95 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 16.19/8.95 & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3885_power__less__zero__eq__numeral,axiom,
% 16.19/8.95 ! [A: rat,W: num] :
% 16.19/8.95 ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 16.19/8.95 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 16.19/8.95 & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3886_power__less__zero__eq__numeral,axiom,
% 16.19/8.95 ! [A: int,W: num] :
% 16.19/8.95 ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 16.19/8.95 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 16.19/8.95 & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3887_power__less__zero__eq,axiom,
% 16.19/8.95 ! [A: real,N2: nat] :
% 16.19/8.95 ( ( ord_less_real @ ( power_power_real @ A @ N2 ) @ zero_zero_real )
% 16.19/8.95 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.95 & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3888_power__less__zero__eq,axiom,
% 16.19/8.95 ! [A: rat,N2: nat] :
% 16.19/8.95 ( ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ zero_zero_rat )
% 16.19/8.95 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.95 & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3889_power__less__zero__eq,axiom,
% 16.19/8.95 ! [A: int,N2: nat] :
% 16.19/8.95 ( ( ord_less_int @ ( power_power_int @ A @ N2 ) @ zero_zero_int )
% 16.19/8.95 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.95 & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3890_even__plus__one__iff,axiom,
% 16.19/8.95 ! [A: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
% 16.19/8.95 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3891_even__plus__one__iff,axiom,
% 16.19/8.95 ! [A: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
% 16.19/8.95 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3892_even__plus__one__iff,axiom,
% 16.19/8.95 ! [A: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
% 16.19/8.95 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3893_even__diff,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B ) )
% 16.19/8.95 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3894_even__diff,axiom,
% 16.19/8.95 ! [A: int,B: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
% 16.19/8.95 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3895_odd__Suc__minus__one,axiom,
% 16.19/8.95 ! [N2: nat] :
% 16.19/8.95 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.95 => ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
% 16.19/8.95 = N2 ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3896_even__diff__nat,axiom,
% 16.19/8.95 ! [M: nat,N2: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) )
% 16.19/8.95 = ( ( ord_less_nat @ M @ N2 )
% 16.19/8.95 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3897_zero__less__power__eq__numeral,axiom,
% 16.19/8.95 ! [A: real,W: num] :
% 16.19/8.95 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 16.19/8.95 = ( ( ( numeral_numeral_nat @ W )
% 16.19/8.95 = zero_zero_nat )
% 16.19/8.95 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 16.19/8.95 & ( A != zero_zero_real ) )
% 16.19/8.95 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 16.19/8.95 & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3898_zero__less__power__eq__numeral,axiom,
% 16.19/8.95 ! [A: rat,W: num] :
% 16.19/8.95 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 16.19/8.95 = ( ( ( numeral_numeral_nat @ W )
% 16.19/8.95 = zero_zero_nat )
% 16.19/8.95 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 16.19/8.95 & ( A != zero_zero_rat ) )
% 16.19/8.95 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 16.19/8.95 & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3899_zero__less__power__eq__numeral,axiom,
% 16.19/8.95 ! [A: int,W: num] :
% 16.19/8.95 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 16.19/8.95 = ( ( ( numeral_numeral_nat @ W )
% 16.19/8.95 = zero_zero_nat )
% 16.19/8.95 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 16.19/8.95 & ( A != zero_zero_int ) )
% 16.19/8.95 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 16.19/8.95 & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3900_even__succ__div__2,axiom,
% 16.19/8.95 ! [A: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 16.19/8.95 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 16.19/8.95 = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3901_even__succ__div__2,axiom,
% 16.19/8.95 ! [A: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 16.19/8.95 => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.95 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3902_even__succ__div__2,axiom,
% 16.19/8.95 ! [A: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 16.19/8.95 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/8.95 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3903_even__succ__div__two,axiom,
% 16.19/8.95 ! [A: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 16.19/8.95 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 16.19/8.95 = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3904_even__succ__div__two,axiom,
% 16.19/8.95 ! [A: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 16.19/8.95 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.95 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3905_even__succ__div__two,axiom,
% 16.19/8.95 ! [A: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 16.19/8.95 => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/8.95 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3906_odd__succ__div__two,axiom,
% 16.19/8.95 ! [A: code_integer] :
% 16.19/8.95 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 16.19/8.95 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 16.19/8.95 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3907_odd__succ__div__two,axiom,
% 16.19/8.95 ! [A: nat] :
% 16.19/8.95 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 16.19/8.95 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.95 = ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3908_odd__succ__div__two,axiom,
% 16.19/8.95 ! [A: int] :
% 16.19/8.95 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 16.19/8.95 => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/8.95 = ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3909_even__power,axiom,
% 16.19/8.95 ! [A: code_integer,N2: nat] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N2 ) )
% 16.19/8.95 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 16.19/8.95 & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3910_even__power,axiom,
% 16.19/8.95 ! [A: nat,N2: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N2 ) )
% 16.19/8.95 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 16.19/8.95 & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3911_even__power,axiom,
% 16.19/8.95 ! [A: int,N2: nat] :
% 16.19/8.95 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N2 ) )
% 16.19/8.95 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 16.19/8.95 & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3912_odd__two__times__div__two__nat,axiom,
% 16.19/8.95 ! [N2: nat] :
% 16.19/8.95 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.95 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.95 = ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3913_odd__two__times__div__two__succ,axiom,
% 16.19/8.95 ! [A: code_integer] :
% 16.19/8.95 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 16.19/8.95 => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
% 16.19/8.95 = A ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3914_odd__two__times__div__two__succ,axiom,
% 16.19/8.95 ! [A: nat] :
% 16.19/8.95 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 16.19/8.95 => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
% 16.19/8.95 = A ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3915_odd__two__times__div__two__succ,axiom,
% 16.19/8.95 ! [A: int] :
% 16.19/8.95 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 16.19/8.95 => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
% 16.19/8.95 = A ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3916_power__le__zero__eq__numeral,axiom,
% 16.19/8.95 ! [A: real,W: num] :
% 16.19/8.95 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 16.19/8.95 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 16.19/8.95 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 16.19/8.95 & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 16.19/8.95 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 16.19/8.95 & ( A = zero_zero_real ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3917_power__le__zero__eq__numeral,axiom,
% 16.19/8.95 ! [A: rat,W: num] :
% 16.19/8.95 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 16.19/8.95 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 16.19/8.95 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 16.19/8.95 & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 16.19/8.95 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 16.19/8.95 & ( A = zero_zero_rat ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3918_power__le__zero__eq__numeral,axiom,
% 16.19/8.95 ! [A: int,W: num] :
% 16.19/8.95 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 16.19/8.95 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 16.19/8.95 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 16.19/8.95 & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 16.19/8.95 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 16.19/8.95 & ( A = zero_zero_int ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3919_semiring__parity__class_Oeven__mask__iff,axiom,
% 16.19/8.95 ! [N2: nat] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) @ one_one_Code_integer ) )
% 16.19/8.95 = ( N2 = zero_zero_nat ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3920_semiring__parity__class_Oeven__mask__iff,axiom,
% 16.19/8.95 ! [N2: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) )
% 16.19/8.95 = ( N2 = zero_zero_nat ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3921_semiring__parity__class_Oeven__mask__iff,axiom,
% 16.19/8.95 ! [N2: nat] :
% 16.19/8.95 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) )
% 16.19/8.95 = ( N2 = zero_zero_nat ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3922_dvd__triv__right,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ A ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3923_dvd__triv__right,axiom,
% 16.19/8.95 ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3924_dvd__triv__right,axiom,
% 16.19/8.95 ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3925_dvd__triv__right,axiom,
% 16.19/8.95 ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3926_dvd__triv__right,axiom,
% 16.19/8.95 ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3927_dvd__mult__right,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer,C: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 16.19/8.95 => ( dvd_dvd_Code_integer @ B @ C ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3928_dvd__mult__right,axiom,
% 16.19/8.95 ! [A: real,B: real,C: real] :
% 16.19/8.95 ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
% 16.19/8.95 => ( dvd_dvd_real @ B @ C ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3929_dvd__mult__right,axiom,
% 16.19/8.95 ! [A: rat,B: rat,C: rat] :
% 16.19/8.95 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
% 16.19/8.95 => ( dvd_dvd_rat @ B @ C ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3930_dvd__mult__right,axiom,
% 16.19/8.95 ! [A: nat,B: nat,C: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 16.19/8.95 => ( dvd_dvd_nat @ B @ C ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3931_dvd__mult__right,axiom,
% 16.19/8.95 ! [A: int,B: int,C: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 16.19/8.95 => ( dvd_dvd_int @ B @ C ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3932_mult__dvd__mono,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ A @ B )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ C @ D )
% 16.19/8.95 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3933_mult__dvd__mono,axiom,
% 16.19/8.95 ! [A: real,B: real,C: real,D: real] :
% 16.19/8.95 ( ( dvd_dvd_real @ A @ B )
% 16.19/8.95 => ( ( dvd_dvd_real @ C @ D )
% 16.19/8.95 => ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3934_mult__dvd__mono,axiom,
% 16.19/8.95 ! [A: rat,B: rat,C: rat,D: rat] :
% 16.19/8.95 ( ( dvd_dvd_rat @ A @ B )
% 16.19/8.95 => ( ( dvd_dvd_rat @ C @ D )
% 16.19/8.95 => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3935_mult__dvd__mono,axiom,
% 16.19/8.95 ! [A: nat,B: nat,C: nat,D: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ A @ B )
% 16.19/8.95 => ( ( dvd_dvd_nat @ C @ D )
% 16.19/8.95 => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3936_mult__dvd__mono,axiom,
% 16.19/8.95 ! [A: int,B: int,C: int,D: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ A @ B )
% 16.19/8.95 => ( ( dvd_dvd_int @ C @ D )
% 16.19/8.95 => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3937_dvd__triv__left,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3938_dvd__triv__left,axiom,
% 16.19/8.95 ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3939_dvd__triv__left,axiom,
% 16.19/8.95 ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3940_dvd__triv__left,axiom,
% 16.19/8.95 ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3941_dvd__triv__left,axiom,
% 16.19/8.95 ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3942_dvd__mult__left,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer,C: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 16.19/8.95 => ( dvd_dvd_Code_integer @ A @ C ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3943_dvd__mult__left,axiom,
% 16.19/8.95 ! [A: real,B: real,C: real] :
% 16.19/8.95 ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
% 16.19/8.95 => ( dvd_dvd_real @ A @ C ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3944_dvd__mult__left,axiom,
% 16.19/8.95 ! [A: rat,B: rat,C: rat] :
% 16.19/8.95 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
% 16.19/8.95 => ( dvd_dvd_rat @ A @ C ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3945_dvd__mult__left,axiom,
% 16.19/8.95 ! [A: nat,B: nat,C: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 16.19/8.95 => ( dvd_dvd_nat @ A @ C ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3946_dvd__mult__left,axiom,
% 16.19/8.95 ! [A: int,B: int,C: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 16.19/8.95 => ( dvd_dvd_int @ A @ C ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3947_dvd__mult2,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer,C: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ A @ B )
% 16.19/8.95 => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3948_dvd__mult2,axiom,
% 16.19/8.95 ! [A: real,B: real,C: real] :
% 16.19/8.95 ( ( dvd_dvd_real @ A @ B )
% 16.19/8.95 => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3949_dvd__mult2,axiom,
% 16.19/8.95 ! [A: rat,B: rat,C: rat] :
% 16.19/8.95 ( ( dvd_dvd_rat @ A @ B )
% 16.19/8.95 => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3950_dvd__mult2,axiom,
% 16.19/8.95 ! [A: nat,B: nat,C: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ A @ B )
% 16.19/8.95 => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3951_dvd__mult2,axiom,
% 16.19/8.95 ! [A: int,B: int,C: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ A @ B )
% 16.19/8.95 => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3952_dvd__mult,axiom,
% 16.19/8.95 ! [A: code_integer,C: code_integer,B: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ A @ C )
% 16.19/8.95 => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3953_dvd__mult,axiom,
% 16.19/8.95 ! [A: real,C: real,B: real] :
% 16.19/8.95 ( ( dvd_dvd_real @ A @ C )
% 16.19/8.95 => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3954_dvd__mult,axiom,
% 16.19/8.95 ! [A: rat,C: rat,B: rat] :
% 16.19/8.95 ( ( dvd_dvd_rat @ A @ C )
% 16.19/8.95 => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3955_dvd__mult,axiom,
% 16.19/8.95 ! [A: nat,C: nat,B: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ A @ C )
% 16.19/8.95 => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3956_dvd__mult,axiom,
% 16.19/8.95 ! [A: int,C: int,B: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ A @ C )
% 16.19/8.95 => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3957_dvd__def,axiom,
% 16.19/8.95 ( dvd_dvd_Code_integer
% 16.19/8.95 = ( ^ [B5: code_integer,A5: code_integer] :
% 16.19/8.95 ? [K3: code_integer] :
% 16.19/8.95 ( A5
% 16.19/8.95 = ( times_3573771949741848930nteger @ B5 @ K3 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3958_dvd__def,axiom,
% 16.19/8.95 ( dvd_dvd_real
% 16.19/8.95 = ( ^ [B5: real,A5: real] :
% 16.19/8.95 ? [K3: real] :
% 16.19/8.95 ( A5
% 16.19/8.95 = ( times_times_real @ B5 @ K3 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3959_dvd__def,axiom,
% 16.19/8.95 ( dvd_dvd_rat
% 16.19/8.95 = ( ^ [B5: rat,A5: rat] :
% 16.19/8.95 ? [K3: rat] :
% 16.19/8.95 ( A5
% 16.19/8.95 = ( times_times_rat @ B5 @ K3 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3960_dvd__def,axiom,
% 16.19/8.95 ( dvd_dvd_nat
% 16.19/8.95 = ( ^ [B5: nat,A5: nat] :
% 16.19/8.95 ? [K3: nat] :
% 16.19/8.95 ( A5
% 16.19/8.95 = ( times_times_nat @ B5 @ K3 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3961_dvd__def,axiom,
% 16.19/8.95 ( dvd_dvd_int
% 16.19/8.95 = ( ^ [B5: int,A5: int] :
% 16.19/8.95 ? [K3: int] :
% 16.19/8.95 ( A5
% 16.19/8.95 = ( times_times_int @ B5 @ K3 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3962_dvdI,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer,K: code_integer] :
% 16.19/8.95 ( ( A
% 16.19/8.95 = ( times_3573771949741848930nteger @ B @ K ) )
% 16.19/8.95 => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3963_dvdI,axiom,
% 16.19/8.95 ! [A: real,B: real,K: real] :
% 16.19/8.95 ( ( A
% 16.19/8.95 = ( times_times_real @ B @ K ) )
% 16.19/8.95 => ( dvd_dvd_real @ B @ A ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3964_dvdI,axiom,
% 16.19/8.95 ! [A: rat,B: rat,K: rat] :
% 16.19/8.95 ( ( A
% 16.19/8.95 = ( times_times_rat @ B @ K ) )
% 16.19/8.95 => ( dvd_dvd_rat @ B @ A ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3965_dvdI,axiom,
% 16.19/8.95 ! [A: nat,B: nat,K: nat] :
% 16.19/8.95 ( ( A
% 16.19/8.95 = ( times_times_nat @ B @ K ) )
% 16.19/8.95 => ( dvd_dvd_nat @ B @ A ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3966_dvdI,axiom,
% 16.19/8.95 ! [A: int,B: int,K: int] :
% 16.19/8.95 ( ( A
% 16.19/8.95 = ( times_times_int @ B @ K ) )
% 16.19/8.95 => ( dvd_dvd_int @ B @ A ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3967_dvdE,axiom,
% 16.19/8.95 ! [B: code_integer,A: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ B @ A )
% 16.19/8.95 => ~ ! [K2: code_integer] :
% 16.19/8.95 ( A
% 16.19/8.95 != ( times_3573771949741848930nteger @ B @ K2 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3968_dvdE,axiom,
% 16.19/8.95 ! [B: real,A: real] :
% 16.19/8.95 ( ( dvd_dvd_real @ B @ A )
% 16.19/8.95 => ~ ! [K2: real] :
% 16.19/8.95 ( A
% 16.19/8.95 != ( times_times_real @ B @ K2 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3969_dvdE,axiom,
% 16.19/8.95 ! [B: rat,A: rat] :
% 16.19/8.95 ( ( dvd_dvd_rat @ B @ A )
% 16.19/8.95 => ~ ! [K2: rat] :
% 16.19/8.95 ( A
% 16.19/8.95 != ( times_times_rat @ B @ K2 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3970_dvdE,axiom,
% 16.19/8.95 ! [B: nat,A: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ B @ A )
% 16.19/8.95 => ~ ! [K2: nat] :
% 16.19/8.95 ( A
% 16.19/8.95 != ( times_times_nat @ B @ K2 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3971_dvdE,axiom,
% 16.19/8.95 ! [B: int,A: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ B @ A )
% 16.19/8.95 => ~ ! [K2: int] :
% 16.19/8.95 ( A
% 16.19/8.95 != ( times_times_int @ B @ K2 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3972_division__decomp,axiom,
% 16.19/8.95 ! [A: nat,B: nat,C: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
% 16.19/8.95 => ? [B8: nat,C5: nat] :
% 16.19/8.95 ( ( A
% 16.19/8.95 = ( times_times_nat @ B8 @ C5 ) )
% 16.19/8.95 & ( dvd_dvd_nat @ B8 @ B )
% 16.19/8.95 & ( dvd_dvd_nat @ C5 @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3973_division__decomp,axiom,
% 16.19/8.95 ! [A: int,B: int,C: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
% 16.19/8.95 => ? [B8: int,C5: int] :
% 16.19/8.95 ( ( A
% 16.19/8.95 = ( times_times_int @ B8 @ C5 ) )
% 16.19/8.95 & ( dvd_dvd_int @ B8 @ B )
% 16.19/8.95 & ( dvd_dvd_int @ C5 @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3974_dvd__productE,axiom,
% 16.19/8.95 ! [P4: nat,A: nat,B: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ P4 @ ( times_times_nat @ A @ B ) )
% 16.19/8.95 => ~ ! [X4: nat,Y4: nat] :
% 16.19/8.95 ( ( P4
% 16.19/8.95 = ( times_times_nat @ X4 @ Y4 ) )
% 16.19/8.95 => ( ( dvd_dvd_nat @ X4 @ A )
% 16.19/8.95 => ~ ( dvd_dvd_nat @ Y4 @ B ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3975_dvd__productE,axiom,
% 16.19/8.95 ! [P4: int,A: int,B: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ P4 @ ( times_times_int @ A @ B ) )
% 16.19/8.95 => ~ ! [X4: int,Y4: int] :
% 16.19/8.95 ( ( P4
% 16.19/8.95 = ( times_times_int @ X4 @ Y4 ) )
% 16.19/8.95 => ( ( dvd_dvd_int @ X4 @ A )
% 16.19/8.95 => ~ ( dvd_dvd_int @ Y4 @ B ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3976_dvd__add__right__iff,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer,C: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ A @ B )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 16.19/8.95 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3977_dvd__add__right__iff,axiom,
% 16.19/8.95 ! [A: real,B: real,C: real] :
% 16.19/8.95 ( ( dvd_dvd_real @ A @ B )
% 16.19/8.95 => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
% 16.19/8.95 = ( dvd_dvd_real @ A @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3978_dvd__add__right__iff,axiom,
% 16.19/8.95 ! [A: rat,B: rat,C: rat] :
% 16.19/8.95 ( ( dvd_dvd_rat @ A @ B )
% 16.19/8.95 => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 16.19/8.95 = ( dvd_dvd_rat @ A @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3979_dvd__add__right__iff,axiom,
% 16.19/8.95 ! [A: nat,B: nat,C: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ A @ B )
% 16.19/8.95 => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 16.19/8.95 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3980_dvd__add__right__iff,axiom,
% 16.19/8.95 ! [A: int,B: int,C: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ A @ B )
% 16.19/8.95 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
% 16.19/8.95 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3981_dvd__add__left__iff,axiom,
% 16.19/8.95 ! [A: code_integer,C: code_integer,B: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ A @ C )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 16.19/8.95 = ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3982_dvd__add__left__iff,axiom,
% 16.19/8.95 ! [A: real,C: real,B: real] :
% 16.19/8.95 ( ( dvd_dvd_real @ A @ C )
% 16.19/8.95 => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
% 16.19/8.95 = ( dvd_dvd_real @ A @ B ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3983_dvd__add__left__iff,axiom,
% 16.19/8.95 ! [A: rat,C: rat,B: rat] :
% 16.19/8.95 ( ( dvd_dvd_rat @ A @ C )
% 16.19/8.95 => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 16.19/8.95 = ( dvd_dvd_rat @ A @ B ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3984_dvd__add__left__iff,axiom,
% 16.19/8.95 ! [A: nat,C: nat,B: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ A @ C )
% 16.19/8.95 => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 16.19/8.95 = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3985_dvd__add__left__iff,axiom,
% 16.19/8.95 ! [A: int,C: int,B: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ A @ C )
% 16.19/8.95 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
% 16.19/8.95 = ( dvd_dvd_int @ A @ B ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3986_dvd__add,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer,C: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ A @ B )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ A @ C )
% 16.19/8.95 => ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3987_dvd__add,axiom,
% 16.19/8.95 ! [A: real,B: real,C: real] :
% 16.19/8.95 ( ( dvd_dvd_real @ A @ B )
% 16.19/8.95 => ( ( dvd_dvd_real @ A @ C )
% 16.19/8.95 => ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3988_dvd__add,axiom,
% 16.19/8.95 ! [A: rat,B: rat,C: rat] :
% 16.19/8.95 ( ( dvd_dvd_rat @ A @ B )
% 16.19/8.95 => ( ( dvd_dvd_rat @ A @ C )
% 16.19/8.95 => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3989_dvd__add,axiom,
% 16.19/8.95 ! [A: nat,B: nat,C: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ A @ B )
% 16.19/8.95 => ( ( dvd_dvd_nat @ A @ C )
% 16.19/8.95 => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3990_dvd__add,axiom,
% 16.19/8.95 ! [A: int,B: int,C: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ A @ B )
% 16.19/8.95 => ( ( dvd_dvd_int @ A @ C )
% 16.19/8.95 => ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3991_one__dvd,axiom,
% 16.19/8.95 ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3992_one__dvd,axiom,
% 16.19/8.95 ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3993_one__dvd,axiom,
% 16.19/8.95 ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3994_one__dvd,axiom,
% 16.19/8.95 ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3995_one__dvd,axiom,
% 16.19/8.95 ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3996_one__dvd,axiom,
% 16.19/8.95 ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3997_unit__imp__dvd,axiom,
% 16.19/8.95 ! [B: code_integer,A: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 16.19/8.95 => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3998_unit__imp__dvd,axiom,
% 16.19/8.95 ! [B: nat,A: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 16.19/8.95 => ( dvd_dvd_nat @ B @ A ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_3999_unit__imp__dvd,axiom,
% 16.19/8.95 ! [B: int,A: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ B @ one_one_int )
% 16.19/8.95 => ( dvd_dvd_int @ B @ A ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4000_dvd__unit__imp__unit,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ A @ B )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 16.19/8.95 => ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4001_dvd__unit__imp__unit,axiom,
% 16.19/8.95 ! [A: nat,B: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ A @ B )
% 16.19/8.95 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 16.19/8.95 => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4002_dvd__unit__imp__unit,axiom,
% 16.19/8.95 ! [A: int,B: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ A @ B )
% 16.19/8.95 => ( ( dvd_dvd_int @ B @ one_one_int )
% 16.19/8.95 => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4003_dvd__diff__commute,axiom,
% 16.19/8.95 ! [A: code_integer,C: code_integer,B: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ C @ B ) )
% 16.19/8.95 = ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ B @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4004_dvd__diff__commute,axiom,
% 16.19/8.95 ! [A: int,C: int,B: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ A @ ( minus_minus_int @ C @ B ) )
% 16.19/8.95 = ( dvd_dvd_int @ A @ ( minus_minus_int @ B @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4005_dvd__diff,axiom,
% 16.19/8.95 ! [X: code_integer,Y: code_integer,Z: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ X @ Y )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ X @ Z )
% 16.19/8.95 => ( dvd_dvd_Code_integer @ X @ ( minus_8373710615458151222nteger @ Y @ Z ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4006_dvd__diff,axiom,
% 16.19/8.95 ! [X: real,Y: real,Z: real] :
% 16.19/8.95 ( ( dvd_dvd_real @ X @ Y )
% 16.19/8.95 => ( ( dvd_dvd_real @ X @ Z )
% 16.19/8.95 => ( dvd_dvd_real @ X @ ( minus_minus_real @ Y @ Z ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4007_dvd__diff,axiom,
% 16.19/8.95 ! [X: rat,Y: rat,Z: rat] :
% 16.19/8.95 ( ( dvd_dvd_rat @ X @ Y )
% 16.19/8.95 => ( ( dvd_dvd_rat @ X @ Z )
% 16.19/8.95 => ( dvd_dvd_rat @ X @ ( minus_minus_rat @ Y @ Z ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4008_dvd__diff,axiom,
% 16.19/8.95 ! [X: int,Y: int,Z: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ X @ Y )
% 16.19/8.95 => ( ( dvd_dvd_int @ X @ Z )
% 16.19/8.95 => ( dvd_dvd_int @ X @ ( minus_minus_int @ Y @ Z ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4009_div__div__div__same,axiom,
% 16.19/8.95 ! [D: code_integer,B: code_integer,A: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ D @ B )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ B @ A )
% 16.19/8.95 => ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ D ) @ ( divide6298287555418463151nteger @ B @ D ) )
% 16.19/8.95 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4010_div__div__div__same,axiom,
% 16.19/8.95 ! [D: nat,B: nat,A: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ D @ B )
% 16.19/8.95 => ( ( dvd_dvd_nat @ B @ A )
% 16.19/8.95 => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B @ D ) )
% 16.19/8.95 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4011_div__div__div__same,axiom,
% 16.19/8.95 ! [D: int,B: int,A: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ D @ B )
% 16.19/8.95 => ( ( dvd_dvd_int @ B @ A )
% 16.19/8.95 => ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B @ D ) )
% 16.19/8.95 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4012_dvd__div__eq__cancel,axiom,
% 16.19/8.95 ! [A: code_integer,C: code_integer,B: code_integer] :
% 16.19/8.95 ( ( ( divide6298287555418463151nteger @ A @ C )
% 16.19/8.95 = ( divide6298287555418463151nteger @ B @ C ) )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ C @ A )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ C @ B )
% 16.19/8.95 => ( A = B ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4013_dvd__div__eq__cancel,axiom,
% 16.19/8.95 ! [A: complex,C: complex,B: complex] :
% 16.19/8.95 ( ( ( divide1717551699836669952omplex @ A @ C )
% 16.19/8.95 = ( divide1717551699836669952omplex @ B @ C ) )
% 16.19/8.95 => ( ( dvd_dvd_complex @ C @ A )
% 16.19/8.95 => ( ( dvd_dvd_complex @ C @ B )
% 16.19/8.95 => ( A = B ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4014_dvd__div__eq__cancel,axiom,
% 16.19/8.95 ! [A: real,C: real,B: real] :
% 16.19/8.95 ( ( ( divide_divide_real @ A @ C )
% 16.19/8.95 = ( divide_divide_real @ B @ C ) )
% 16.19/8.95 => ( ( dvd_dvd_real @ C @ A )
% 16.19/8.95 => ( ( dvd_dvd_real @ C @ B )
% 16.19/8.95 => ( A = B ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4015_dvd__div__eq__cancel,axiom,
% 16.19/8.95 ! [A: rat,C: rat,B: rat] :
% 16.19/8.95 ( ( ( divide_divide_rat @ A @ C )
% 16.19/8.95 = ( divide_divide_rat @ B @ C ) )
% 16.19/8.95 => ( ( dvd_dvd_rat @ C @ A )
% 16.19/8.95 => ( ( dvd_dvd_rat @ C @ B )
% 16.19/8.95 => ( A = B ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4016_dvd__div__eq__cancel,axiom,
% 16.19/8.95 ! [A: nat,C: nat,B: nat] :
% 16.19/8.95 ( ( ( divide_divide_nat @ A @ C )
% 16.19/8.95 = ( divide_divide_nat @ B @ C ) )
% 16.19/8.95 => ( ( dvd_dvd_nat @ C @ A )
% 16.19/8.95 => ( ( dvd_dvd_nat @ C @ B )
% 16.19/8.95 => ( A = B ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4017_dvd__div__eq__cancel,axiom,
% 16.19/8.95 ! [A: int,C: int,B: int] :
% 16.19/8.95 ( ( ( divide_divide_int @ A @ C )
% 16.19/8.95 = ( divide_divide_int @ B @ C ) )
% 16.19/8.95 => ( ( dvd_dvd_int @ C @ A )
% 16.19/8.95 => ( ( dvd_dvd_int @ C @ B )
% 16.19/8.95 => ( A = B ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4018_dvd__div__eq__iff,axiom,
% 16.19/8.95 ! [C: code_integer,A: code_integer,B: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ C @ A )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ C @ B )
% 16.19/8.95 => ( ( ( divide6298287555418463151nteger @ A @ C )
% 16.19/8.95 = ( divide6298287555418463151nteger @ B @ C ) )
% 16.19/8.95 = ( A = B ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4019_dvd__div__eq__iff,axiom,
% 16.19/8.95 ! [C: complex,A: complex,B: complex] :
% 16.19/8.95 ( ( dvd_dvd_complex @ C @ A )
% 16.19/8.95 => ( ( dvd_dvd_complex @ C @ B )
% 16.19/8.95 => ( ( ( divide1717551699836669952omplex @ A @ C )
% 16.19/8.95 = ( divide1717551699836669952omplex @ B @ C ) )
% 16.19/8.95 = ( A = B ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4020_dvd__div__eq__iff,axiom,
% 16.19/8.95 ! [C: real,A: real,B: real] :
% 16.19/8.95 ( ( dvd_dvd_real @ C @ A )
% 16.19/8.95 => ( ( dvd_dvd_real @ C @ B )
% 16.19/8.95 => ( ( ( divide_divide_real @ A @ C )
% 16.19/8.95 = ( divide_divide_real @ B @ C ) )
% 16.19/8.95 = ( A = B ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4021_dvd__div__eq__iff,axiom,
% 16.19/8.95 ! [C: rat,A: rat,B: rat] :
% 16.19/8.95 ( ( dvd_dvd_rat @ C @ A )
% 16.19/8.95 => ( ( dvd_dvd_rat @ C @ B )
% 16.19/8.95 => ( ( ( divide_divide_rat @ A @ C )
% 16.19/8.95 = ( divide_divide_rat @ B @ C ) )
% 16.19/8.95 = ( A = B ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4022_dvd__div__eq__iff,axiom,
% 16.19/8.95 ! [C: nat,A: nat,B: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ C @ A )
% 16.19/8.95 => ( ( dvd_dvd_nat @ C @ B )
% 16.19/8.95 => ( ( ( divide_divide_nat @ A @ C )
% 16.19/8.95 = ( divide_divide_nat @ B @ C ) )
% 16.19/8.95 = ( A = B ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4023_dvd__div__eq__iff,axiom,
% 16.19/8.95 ! [C: int,A: int,B: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ C @ A )
% 16.19/8.95 => ( ( dvd_dvd_int @ C @ B )
% 16.19/8.95 => ( ( ( divide_divide_int @ A @ C )
% 16.19/8.95 = ( divide_divide_int @ B @ C ) )
% 16.19/8.95 = ( A = B ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4024_dvd__power__same,axiom,
% 16.19/8.95 ! [X: code_integer,Y: code_integer,N2: nat] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ X @ Y )
% 16.19/8.95 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N2 ) @ ( power_8256067586552552935nteger @ Y @ N2 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4025_dvd__power__same,axiom,
% 16.19/8.95 ! [X: nat,Y: nat,N2: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ X @ Y )
% 16.19/8.95 => ( dvd_dvd_nat @ ( power_power_nat @ X @ N2 ) @ ( power_power_nat @ Y @ N2 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4026_dvd__power__same,axiom,
% 16.19/8.95 ! [X: real,Y: real,N2: nat] :
% 16.19/8.95 ( ( dvd_dvd_real @ X @ Y )
% 16.19/8.95 => ( dvd_dvd_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ N2 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4027_dvd__power__same,axiom,
% 16.19/8.95 ! [X: int,Y: int,N2: nat] :
% 16.19/8.95 ( ( dvd_dvd_int @ X @ Y )
% 16.19/8.95 => ( dvd_dvd_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ N2 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4028_dvd__power__same,axiom,
% 16.19/8.95 ! [X: complex,Y: complex,N2: nat] :
% 16.19/8.95 ( ( dvd_dvd_complex @ X @ Y )
% 16.19/8.95 => ( dvd_dvd_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ Y @ N2 ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4029_mod__mod__cancel,axiom,
% 16.19/8.95 ! [C: nat,B: nat,A: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ C @ B )
% 16.19/8.95 => ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ C )
% 16.19/8.95 = ( modulo_modulo_nat @ A @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4030_mod__mod__cancel,axiom,
% 16.19/8.95 ! [C: int,B: int,A: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ C @ B )
% 16.19/8.95 => ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ C )
% 16.19/8.95 = ( modulo_modulo_int @ A @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4031_mod__mod__cancel,axiom,
% 16.19/8.95 ! [C: code_integer,B: code_integer,A: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ C @ B )
% 16.19/8.95 => ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C )
% 16.19/8.95 = ( modulo364778990260209775nteger @ A @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4032_dvd__mod,axiom,
% 16.19/8.95 ! [K: nat,M: nat,N2: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ K @ M )
% 16.19/8.95 => ( ( dvd_dvd_nat @ K @ N2 )
% 16.19/8.95 => ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4033_dvd__mod,axiom,
% 16.19/8.95 ! [K: int,M: int,N2: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ K @ M )
% 16.19/8.95 => ( ( dvd_dvd_int @ K @ N2 )
% 16.19/8.95 => ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N2 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4034_dvd__mod,axiom,
% 16.19/8.95 ! [K: code_integer,M: code_integer,N2: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ K @ M )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ K @ N2 )
% 16.19/8.95 => ( dvd_dvd_Code_integer @ K @ ( modulo364778990260209775nteger @ M @ N2 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4035_dvd__diff__nat,axiom,
% 16.19/8.95 ! [K: nat,M: nat,N2: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ K @ M )
% 16.19/8.95 => ( ( dvd_dvd_nat @ K @ N2 )
% 16.19/8.95 => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4036_zdvd__zdiffD,axiom,
% 16.19/8.95 ! [K: int,M: int,N2: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N2 ) )
% 16.19/8.95 => ( ( dvd_dvd_int @ K @ N2 )
% 16.19/8.95 => ( dvd_dvd_int @ K @ M ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4037_subset__divisors__dvd,axiom,
% 16.19/8.95 ! [A: complex,B: complex] :
% 16.19/8.95 ( ( ord_le211207098394363844omplex
% 16.19/8.95 @ ( collect_complex
% 16.19/8.95 @ ^ [C4: complex] : ( dvd_dvd_complex @ C4 @ A ) )
% 16.19/8.95 @ ( collect_complex
% 16.19/8.95 @ ^ [C4: complex] : ( dvd_dvd_complex @ C4 @ B ) ) )
% 16.19/8.95 = ( dvd_dvd_complex @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4038_subset__divisors__dvd,axiom,
% 16.19/8.95 ! [A: int,B: int] :
% 16.19/8.95 ( ( ord_less_eq_set_int
% 16.19/8.95 @ ( collect_int
% 16.19/8.95 @ ^ [C4: int] : ( dvd_dvd_int @ C4 @ A ) )
% 16.19/8.95 @ ( collect_int
% 16.19/8.95 @ ^ [C4: int] : ( dvd_dvd_int @ C4 @ B ) ) )
% 16.19/8.95 = ( dvd_dvd_int @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4039_subset__divisors__dvd,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer] :
% 16.19/8.95 ( ( ord_le7084787975880047091nteger
% 16.19/8.95 @ ( collect_Code_integer
% 16.19/8.95 @ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ A ) )
% 16.19/8.95 @ ( collect_Code_integer
% 16.19/8.95 @ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ B ) ) )
% 16.19/8.95 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4040_subset__divisors__dvd,axiom,
% 16.19/8.95 ! [A: nat,B: nat] :
% 16.19/8.95 ( ( ord_less_eq_set_nat
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ A ) )
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ B ) ) )
% 16.19/8.95 = ( dvd_dvd_nat @ A @ B ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4041_concat__bit__assoc,axiom,
% 16.19/8.95 ! [N2: nat,K: int,M: nat,L2: int,R: int] :
% 16.19/8.95 ( ( bit_concat_bit @ N2 @ K @ ( bit_concat_bit @ M @ L2 @ R ) )
% 16.19/8.95 = ( bit_concat_bit @ ( plus_plus_nat @ M @ N2 ) @ ( bit_concat_bit @ N2 @ K @ L2 ) @ R ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4042_strict__subset__divisors__dvd,axiom,
% 16.19/8.95 ! [A: complex,B: complex] :
% 16.19/8.95 ( ( ord_less_set_complex
% 16.19/8.95 @ ( collect_complex
% 16.19/8.95 @ ^ [C4: complex] : ( dvd_dvd_complex @ C4 @ A ) )
% 16.19/8.95 @ ( collect_complex
% 16.19/8.95 @ ^ [C4: complex] : ( dvd_dvd_complex @ C4 @ B ) ) )
% 16.19/8.95 = ( ( dvd_dvd_complex @ A @ B )
% 16.19/8.95 & ~ ( dvd_dvd_complex @ B @ A ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4043_strict__subset__divisors__dvd,axiom,
% 16.19/8.95 ! [A: nat,B: nat] :
% 16.19/8.95 ( ( ord_less_set_nat
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ A ) )
% 16.19/8.95 @ ( collect_nat
% 16.19/8.95 @ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ B ) ) )
% 16.19/8.95 = ( ( dvd_dvd_nat @ A @ B )
% 16.19/8.95 & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4044_strict__subset__divisors__dvd,axiom,
% 16.19/8.95 ! [A: int,B: int] :
% 16.19/8.95 ( ( ord_less_set_int
% 16.19/8.95 @ ( collect_int
% 16.19/8.95 @ ^ [C4: int] : ( dvd_dvd_int @ C4 @ A ) )
% 16.19/8.95 @ ( collect_int
% 16.19/8.95 @ ^ [C4: int] : ( dvd_dvd_int @ C4 @ B ) ) )
% 16.19/8.95 = ( ( dvd_dvd_int @ A @ B )
% 16.19/8.95 & ~ ( dvd_dvd_int @ B @ A ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4045_strict__subset__divisors__dvd,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer] :
% 16.19/8.95 ( ( ord_le1307284697595431911nteger
% 16.19/8.95 @ ( collect_Code_integer
% 16.19/8.95 @ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ A ) )
% 16.19/8.95 @ ( collect_Code_integer
% 16.19/8.95 @ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ B ) ) )
% 16.19/8.95 = ( ( dvd_dvd_Code_integer @ A @ B )
% 16.19/8.95 & ~ ( dvd_dvd_Code_integer @ B @ A ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4046_not__is__unit__0,axiom,
% 16.19/8.95 ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4047_not__is__unit__0,axiom,
% 16.19/8.95 ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4048_not__is__unit__0,axiom,
% 16.19/8.95 ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4049_minf_I10_J,axiom,
% 16.19/8.95 ! [D: code_integer,S3: code_integer] :
% 16.19/8.95 ? [Z5: code_integer] :
% 16.19/8.95 ! [X5: code_integer] :
% 16.19/8.95 ( ( ord_le6747313008572928689nteger @ X5 @ Z5 )
% 16.19/8.95 => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S3 ) ) )
% 16.19/8.95 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S3 ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4050_minf_I10_J,axiom,
% 16.19/8.95 ! [D: real,S3: real] :
% 16.19/8.95 ? [Z5: real] :
% 16.19/8.95 ! [X5: real] :
% 16.19/8.95 ( ( ord_less_real @ X5 @ Z5 )
% 16.19/8.95 => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S3 ) ) )
% 16.19/8.95 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S3 ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4051_minf_I10_J,axiom,
% 16.19/8.95 ! [D: rat,S3: rat] :
% 16.19/8.95 ? [Z5: rat] :
% 16.19/8.95 ! [X5: rat] :
% 16.19/8.95 ( ( ord_less_rat @ X5 @ Z5 )
% 16.19/8.95 => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S3 ) ) )
% 16.19/8.95 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S3 ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4052_minf_I10_J,axiom,
% 16.19/8.95 ! [D: nat,S3: nat] :
% 16.19/8.95 ? [Z5: nat] :
% 16.19/8.95 ! [X5: nat] :
% 16.19/8.95 ( ( ord_less_nat @ X5 @ Z5 )
% 16.19/8.95 => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S3 ) ) )
% 16.19/8.95 = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S3 ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4053_minf_I10_J,axiom,
% 16.19/8.95 ! [D: int,S3: int] :
% 16.19/8.95 ? [Z5: int] :
% 16.19/8.95 ! [X5: int] :
% 16.19/8.95 ( ( ord_less_int @ X5 @ Z5 )
% 16.19/8.95 => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S3 ) ) )
% 16.19/8.95 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S3 ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4054_minf_I9_J,axiom,
% 16.19/8.95 ! [D: code_integer,S3: code_integer] :
% 16.19/8.95 ? [Z5: code_integer] :
% 16.19/8.95 ! [X5: code_integer] :
% 16.19/8.95 ( ( ord_le6747313008572928689nteger @ X5 @ Z5 )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S3 ) )
% 16.19/8.95 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S3 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4055_minf_I9_J,axiom,
% 16.19/8.95 ! [D: real,S3: real] :
% 16.19/8.95 ? [Z5: real] :
% 16.19/8.95 ! [X5: real] :
% 16.19/8.95 ( ( ord_less_real @ X5 @ Z5 )
% 16.19/8.95 => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S3 ) )
% 16.19/8.95 = ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S3 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4056_minf_I9_J,axiom,
% 16.19/8.95 ! [D: rat,S3: rat] :
% 16.19/8.95 ? [Z5: rat] :
% 16.19/8.95 ! [X5: rat] :
% 16.19/8.95 ( ( ord_less_rat @ X5 @ Z5 )
% 16.19/8.95 => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S3 ) )
% 16.19/8.95 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S3 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4057_minf_I9_J,axiom,
% 16.19/8.95 ! [D: nat,S3: nat] :
% 16.19/8.95 ? [Z5: nat] :
% 16.19/8.95 ! [X5: nat] :
% 16.19/8.95 ( ( ord_less_nat @ X5 @ Z5 )
% 16.19/8.95 => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S3 ) )
% 16.19/8.95 = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S3 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4058_minf_I9_J,axiom,
% 16.19/8.95 ! [D: int,S3: int] :
% 16.19/8.95 ? [Z5: int] :
% 16.19/8.95 ! [X5: int] :
% 16.19/8.95 ( ( ord_less_int @ X5 @ Z5 )
% 16.19/8.95 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S3 ) )
% 16.19/8.95 = ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S3 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4059_pinf_I10_J,axiom,
% 16.19/8.95 ! [D: code_integer,S3: code_integer] :
% 16.19/8.95 ? [Z5: code_integer] :
% 16.19/8.95 ! [X5: code_integer] :
% 16.19/8.95 ( ( ord_le6747313008572928689nteger @ Z5 @ X5 )
% 16.19/8.95 => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S3 ) ) )
% 16.19/8.95 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S3 ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4060_pinf_I10_J,axiom,
% 16.19/8.95 ! [D: real,S3: real] :
% 16.19/8.95 ? [Z5: real] :
% 16.19/8.95 ! [X5: real] :
% 16.19/8.95 ( ( ord_less_real @ Z5 @ X5 )
% 16.19/8.95 => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S3 ) ) )
% 16.19/8.95 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S3 ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4061_pinf_I10_J,axiom,
% 16.19/8.95 ! [D: rat,S3: rat] :
% 16.19/8.95 ? [Z5: rat] :
% 16.19/8.95 ! [X5: rat] :
% 16.19/8.95 ( ( ord_less_rat @ Z5 @ X5 )
% 16.19/8.95 => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S3 ) ) )
% 16.19/8.95 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S3 ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4062_pinf_I10_J,axiom,
% 16.19/8.95 ! [D: nat,S3: nat] :
% 16.19/8.95 ? [Z5: nat] :
% 16.19/8.95 ! [X5: nat] :
% 16.19/8.95 ( ( ord_less_nat @ Z5 @ X5 )
% 16.19/8.95 => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S3 ) ) )
% 16.19/8.95 = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S3 ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4063_pinf_I10_J,axiom,
% 16.19/8.95 ! [D: int,S3: int] :
% 16.19/8.95 ? [Z5: int] :
% 16.19/8.95 ! [X5: int] :
% 16.19/8.95 ( ( ord_less_int @ Z5 @ X5 )
% 16.19/8.95 => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S3 ) ) )
% 16.19/8.95 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S3 ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4064_pinf_I9_J,axiom,
% 16.19/8.95 ! [D: code_integer,S3: code_integer] :
% 16.19/8.95 ? [Z5: code_integer] :
% 16.19/8.95 ! [X5: code_integer] :
% 16.19/8.95 ( ( ord_le6747313008572928689nteger @ Z5 @ X5 )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S3 ) )
% 16.19/8.95 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S3 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4065_pinf_I9_J,axiom,
% 16.19/8.95 ! [D: real,S3: real] :
% 16.19/8.95 ? [Z5: real] :
% 16.19/8.95 ! [X5: real] :
% 16.19/8.95 ( ( ord_less_real @ Z5 @ X5 )
% 16.19/8.95 => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S3 ) )
% 16.19/8.95 = ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S3 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4066_pinf_I9_J,axiom,
% 16.19/8.95 ! [D: rat,S3: rat] :
% 16.19/8.95 ? [Z5: rat] :
% 16.19/8.95 ! [X5: rat] :
% 16.19/8.95 ( ( ord_less_rat @ Z5 @ X5 )
% 16.19/8.95 => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S3 ) )
% 16.19/8.95 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S3 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4067_pinf_I9_J,axiom,
% 16.19/8.95 ! [D: nat,S3: nat] :
% 16.19/8.95 ? [Z5: nat] :
% 16.19/8.95 ! [X5: nat] :
% 16.19/8.95 ( ( ord_less_nat @ Z5 @ X5 )
% 16.19/8.95 => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S3 ) )
% 16.19/8.95 = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S3 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4068_pinf_I9_J,axiom,
% 16.19/8.95 ! [D: int,S3: int] :
% 16.19/8.95 ? [Z5: int] :
% 16.19/8.95 ! [X5: int] :
% 16.19/8.95 ( ( ord_less_int @ Z5 @ X5 )
% 16.19/8.95 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S3 ) )
% 16.19/8.95 = ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S3 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4069_dvd__div__eq__0__iff,axiom,
% 16.19/8.95 ! [B: code_integer,A: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ B @ A )
% 16.19/8.95 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 16.19/8.95 = zero_z3403309356797280102nteger )
% 16.19/8.95 = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4070_dvd__div__eq__0__iff,axiom,
% 16.19/8.95 ! [B: complex,A: complex] :
% 16.19/8.95 ( ( dvd_dvd_complex @ B @ A )
% 16.19/8.95 => ( ( ( divide1717551699836669952omplex @ A @ B )
% 16.19/8.95 = zero_zero_complex )
% 16.19/8.95 = ( A = zero_zero_complex ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4071_dvd__div__eq__0__iff,axiom,
% 16.19/8.95 ! [B: real,A: real] :
% 16.19/8.95 ( ( dvd_dvd_real @ B @ A )
% 16.19/8.95 => ( ( ( divide_divide_real @ A @ B )
% 16.19/8.95 = zero_zero_real )
% 16.19/8.95 = ( A = zero_zero_real ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4072_dvd__div__eq__0__iff,axiom,
% 16.19/8.95 ! [B: rat,A: rat] :
% 16.19/8.95 ( ( dvd_dvd_rat @ B @ A )
% 16.19/8.95 => ( ( ( divide_divide_rat @ A @ B )
% 16.19/8.95 = zero_zero_rat )
% 16.19/8.95 = ( A = zero_zero_rat ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4073_dvd__div__eq__0__iff,axiom,
% 16.19/8.95 ! [B: nat,A: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ B @ A )
% 16.19/8.95 => ( ( ( divide_divide_nat @ A @ B )
% 16.19/8.95 = zero_zero_nat )
% 16.19/8.95 = ( A = zero_zero_nat ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4074_dvd__div__eq__0__iff,axiom,
% 16.19/8.95 ! [B: int,A: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ B @ A )
% 16.19/8.95 => ( ( ( divide_divide_int @ A @ B )
% 16.19/8.95 = zero_zero_int )
% 16.19/8.95 = ( A = zero_zero_int ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4075_is__unit__mult__iff,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer )
% 16.19/8.95 = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 16.19/8.95 & ( dvd_dvd_Code_integer @ B @ one_one_Code_integer ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4076_is__unit__mult__iff,axiom,
% 16.19/8.95 ! [A: nat,B: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
% 16.19/8.95 = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 16.19/8.95 & ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4077_is__unit__mult__iff,axiom,
% 16.19/8.95 ! [A: int,B: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
% 16.19/8.95 = ( ( dvd_dvd_int @ A @ one_one_int )
% 16.19/8.95 & ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4078_dvd__mult__unit__iff,axiom,
% 16.19/8.95 ! [B: code_integer,A: code_integer,C: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) )
% 16.19/8.95 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4079_dvd__mult__unit__iff,axiom,
% 16.19/8.95 ! [B: nat,A: nat,C: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 16.19/8.95 => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
% 16.19/8.95 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4080_dvd__mult__unit__iff,axiom,
% 16.19/8.95 ! [B: int,A: int,C: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ B @ one_one_int )
% 16.19/8.95 => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
% 16.19/8.95 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4081_mult__unit__dvd__iff,axiom,
% 16.19/8.95 ! [B: code_integer,A: code_integer,C: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 16.19/8.95 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4082_mult__unit__dvd__iff,axiom,
% 16.19/8.95 ! [B: nat,A: nat,C: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 16.19/8.95 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 16.19/8.95 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4083_mult__unit__dvd__iff,axiom,
% 16.19/8.95 ! [B: int,A: int,C: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ B @ one_one_int )
% 16.19/8.95 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 16.19/8.95 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4084_dvd__mult__unit__iff_H,axiom,
% 16.19/8.95 ! [B: code_integer,A: code_integer,C: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 16.19/8.95 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4085_dvd__mult__unit__iff_H,axiom,
% 16.19/8.95 ! [B: nat,A: nat,C: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 16.19/8.95 => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
% 16.19/8.95 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4086_dvd__mult__unit__iff_H,axiom,
% 16.19/8.95 ! [B: int,A: int,C: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ B @ one_one_int )
% 16.19/8.95 => ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
% 16.19/8.95 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4087_mult__unit__dvd__iff_H,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer,C: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 16.19/8.95 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4088_mult__unit__dvd__iff_H,axiom,
% 16.19/8.95 ! [A: nat,B: nat,C: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 16.19/8.95 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 16.19/8.95 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4089_mult__unit__dvd__iff_H,axiom,
% 16.19/8.95 ! [A: int,B: int,C: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ A @ one_one_int )
% 16.19/8.95 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 16.19/8.95 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4090_unit__mult__left__cancel,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer,C: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 16.19/8.95 => ( ( ( times_3573771949741848930nteger @ A @ B )
% 16.19/8.95 = ( times_3573771949741848930nteger @ A @ C ) )
% 16.19/8.95 = ( B = C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4091_unit__mult__left__cancel,axiom,
% 16.19/8.95 ! [A: nat,B: nat,C: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 16.19/8.95 => ( ( ( times_times_nat @ A @ B )
% 16.19/8.95 = ( times_times_nat @ A @ C ) )
% 16.19/8.95 = ( B = C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4092_unit__mult__left__cancel,axiom,
% 16.19/8.95 ! [A: int,B: int,C: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ A @ one_one_int )
% 16.19/8.95 => ( ( ( times_times_int @ A @ B )
% 16.19/8.95 = ( times_times_int @ A @ C ) )
% 16.19/8.95 = ( B = C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4093_unit__mult__right__cancel,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer,C: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 16.19/8.95 => ( ( ( times_3573771949741848930nteger @ B @ A )
% 16.19/8.95 = ( times_3573771949741848930nteger @ C @ A ) )
% 16.19/8.95 = ( B = C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4094_unit__mult__right__cancel,axiom,
% 16.19/8.95 ! [A: nat,B: nat,C: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 16.19/8.95 => ( ( ( times_times_nat @ B @ A )
% 16.19/8.95 = ( times_times_nat @ C @ A ) )
% 16.19/8.95 = ( B = C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4095_unit__mult__right__cancel,axiom,
% 16.19/8.95 ! [A: int,B: int,C: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ A @ one_one_int )
% 16.19/8.95 => ( ( ( times_times_int @ B @ A )
% 16.19/8.95 = ( times_times_int @ C @ A ) )
% 16.19/8.95 = ( B = C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4096_div__mult__div__if__dvd,axiom,
% 16.19/8.95 ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ B @ A )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ D @ C )
% 16.19/8.95 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ ( divide6298287555418463151nteger @ C @ D ) )
% 16.19/8.95 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4097_div__mult__div__if__dvd,axiom,
% 16.19/8.95 ! [B: nat,A: nat,D: nat,C: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ B @ A )
% 16.19/8.95 => ( ( dvd_dvd_nat @ D @ C )
% 16.19/8.95 => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C @ D ) )
% 16.19/8.95 = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4098_div__mult__div__if__dvd,axiom,
% 16.19/8.95 ! [B: int,A: int,D: int,C: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ B @ A )
% 16.19/8.95 => ( ( dvd_dvd_int @ D @ C )
% 16.19/8.95 => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C @ D ) )
% 16.19/8.95 = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4099_dvd__mult__imp__div,axiom,
% 16.19/8.95 ! [A: code_integer,C: code_integer,B: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B )
% 16.19/8.95 => ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4100_dvd__mult__imp__div,axiom,
% 16.19/8.95 ! [A: nat,C: nat,B: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B )
% 16.19/8.95 => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4101_dvd__mult__imp__div,axiom,
% 16.19/8.95 ! [A: int,C: int,B: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B )
% 16.19/8.95 => ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4102_dvd__div__mult2__eq,axiom,
% 16.19/8.95 ! [B: code_integer,C: code_integer,A: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ C ) @ A )
% 16.19/8.95 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 16.19/8.95 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4103_dvd__div__mult2__eq,axiom,
% 16.19/8.95 ! [B: nat,C: nat,A: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A )
% 16.19/8.95 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 16.19/8.95 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4104_dvd__div__mult2__eq,axiom,
% 16.19/8.95 ! [B: int,C: int,A: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ ( times_times_int @ B @ C ) @ A )
% 16.19/8.95 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 16.19/8.95 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4105_div__div__eq__right,axiom,
% 16.19/8.95 ! [C: code_integer,B: code_integer,A: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ C @ B )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ B @ A )
% 16.19/8.95 => ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 16.19/8.95 = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4106_div__div__eq__right,axiom,
% 16.19/8.95 ! [C: nat,B: nat,A: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ C @ B )
% 16.19/8.95 => ( ( dvd_dvd_nat @ B @ A )
% 16.19/8.95 => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 16.19/8.95 = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4107_div__div__eq__right,axiom,
% 16.19/8.95 ! [C: int,B: int,A: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ C @ B )
% 16.19/8.95 => ( ( dvd_dvd_int @ B @ A )
% 16.19/8.95 => ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C ) )
% 16.19/8.95 = ( times_times_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4108_div__mult__swap,axiom,
% 16.19/8.95 ! [C: code_integer,B: code_integer,A: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ C @ B )
% 16.19/8.95 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 16.19/8.95 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4109_div__mult__swap,axiom,
% 16.19/8.95 ! [C: nat,B: nat,A: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ C @ B )
% 16.19/8.95 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 16.19/8.95 = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4110_div__mult__swap,axiom,
% 16.19/8.95 ! [C: int,B: int,A: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ C @ B )
% 16.19/8.95 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
% 16.19/8.95 = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4111_dvd__div__mult,axiom,
% 16.19/8.95 ! [C: code_integer,B: code_integer,A: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ C @ B )
% 16.19/8.95 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ C ) @ A )
% 16.19/8.95 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B @ A ) @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4112_dvd__div__mult,axiom,
% 16.19/8.95 ! [C: nat,B: nat,A: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ C @ B )
% 16.19/8.95 => ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A )
% 16.19/8.95 = ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4113_dvd__div__mult,axiom,
% 16.19/8.95 ! [C: int,B: int,A: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ C @ B )
% 16.19/8.95 => ( ( times_times_int @ ( divide_divide_int @ B @ C ) @ A )
% 16.19/8.95 = ( divide_divide_int @ ( times_times_int @ B @ A ) @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4114_div__plus__div__distrib__dvd__right,axiom,
% 16.19/8.95 ! [C: code_integer,B: code_integer,A: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ C @ B )
% 16.19/8.95 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 16.19/8.95 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4115_div__plus__div__distrib__dvd__right,axiom,
% 16.19/8.95 ! [C: nat,B: nat,A: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ C @ B )
% 16.19/8.95 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 16.19/8.95 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4116_div__plus__div__distrib__dvd__right,axiom,
% 16.19/8.95 ! [C: int,B: int,A: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ C @ B )
% 16.19/8.95 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 16.19/8.95 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4117_div__plus__div__distrib__dvd__left,axiom,
% 16.19/8.95 ! [C: code_integer,A: code_integer,B: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ C @ A )
% 16.19/8.95 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 16.19/8.95 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4118_div__plus__div__distrib__dvd__left,axiom,
% 16.19/8.95 ! [C: nat,A: nat,B: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ C @ A )
% 16.19/8.95 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 16.19/8.95 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4119_div__plus__div__distrib__dvd__left,axiom,
% 16.19/8.95 ! [C: int,A: int,B: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ C @ A )
% 16.19/8.95 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 16.19/8.95 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4120_dvd__div__unit__iff,axiom,
% 16.19/8.95 ! [B: code_integer,A: code_integer,C: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B ) )
% 16.19/8.95 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4121_dvd__div__unit__iff,axiom,
% 16.19/8.95 ! [B: nat,A: nat,C: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 16.19/8.95 => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B ) )
% 16.19/8.95 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4122_dvd__div__unit__iff,axiom,
% 16.19/8.95 ! [B: int,A: int,C: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ B @ one_one_int )
% 16.19/8.95 => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B ) )
% 16.19/8.95 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4123_div__unit__dvd__iff,axiom,
% 16.19/8.95 ! [B: code_integer,A: code_integer,C: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 16.19/8.95 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 16.19/8.95 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4124_div__unit__dvd__iff,axiom,
% 16.19/8.95 ! [B: nat,A: nat,C: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 16.19/8.95 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 16.19/8.95 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4125_div__unit__dvd__iff,axiom,
% 16.19/8.95 ! [B: int,A: int,C: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ B @ one_one_int )
% 16.19/8.95 => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 16.19/8.95 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4126_unit__div__cancel,axiom,
% 16.19/8.95 ! [A: code_integer,B: code_integer,C: code_integer] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 16.19/8.95 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 16.19/8.95 = ( divide6298287555418463151nteger @ C @ A ) )
% 16.19/8.95 = ( B = C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4127_unit__div__cancel,axiom,
% 16.19/8.95 ! [A: nat,B: nat,C: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 16.19/8.95 => ( ( ( divide_divide_nat @ B @ A )
% 16.19/8.95 = ( divide_divide_nat @ C @ A ) )
% 16.19/8.95 = ( B = C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4128_unit__div__cancel,axiom,
% 16.19/8.95 ! [A: int,B: int,C: int] :
% 16.19/8.95 ( ( dvd_dvd_int @ A @ one_one_int )
% 16.19/8.95 => ( ( ( divide_divide_int @ B @ A )
% 16.19/8.95 = ( divide_divide_int @ C @ A ) )
% 16.19/8.95 = ( B = C ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4129_div__power,axiom,
% 16.19/8.95 ! [B: code_integer,A: code_integer,N2: nat] :
% 16.19/8.95 ( ( dvd_dvd_Code_integer @ B @ A )
% 16.19/8.95 => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B ) @ N2 )
% 16.19/8.95 = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( power_8256067586552552935nteger @ B @ N2 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4130_div__power,axiom,
% 16.19/8.95 ! [B: nat,A: nat,N2: nat] :
% 16.19/8.95 ( ( dvd_dvd_nat @ B @ A )
% 16.19/8.95 => ( ( power_power_nat @ ( divide_divide_nat @ A @ B ) @ N2 )
% 16.19/8.95 = ( divide_divide_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ).
% 16.19/8.95
% 16.19/8.95 thf(fact_4131_div__power,axiom,
% 16.19/8.95 ! [B: int,A: int,N2: nat] :
% 16.19/8.95 ( ( dvd_dvd_int @ B @ A )
% 16.19/8.95 => ( ( power_power_int @ ( divide_divide_int @ A @ B ) @ N2 )
% 16.19/8.95 = ( divide_divide_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4132_dvd__power__le,axiom,
% 16.19/8.96 ! [X: code_integer,Y: code_integer,N2: nat,M: nat] :
% 16.19/8.96 ( ( dvd_dvd_Code_integer @ X @ Y )
% 16.19/8.96 => ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/8.96 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N2 ) @ ( power_8256067586552552935nteger @ Y @ M ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4133_dvd__power__le,axiom,
% 16.19/8.96 ! [X: nat,Y: nat,N2: nat,M: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ X @ Y )
% 16.19/8.96 => ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/8.96 => ( dvd_dvd_nat @ ( power_power_nat @ X @ N2 ) @ ( power_power_nat @ Y @ M ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4134_dvd__power__le,axiom,
% 16.19/8.96 ! [X: real,Y: real,N2: nat,M: nat] :
% 16.19/8.96 ( ( dvd_dvd_real @ X @ Y )
% 16.19/8.96 => ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/8.96 => ( dvd_dvd_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ M ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4135_dvd__power__le,axiom,
% 16.19/8.96 ! [X: int,Y: int,N2: nat,M: nat] :
% 16.19/8.96 ( ( dvd_dvd_int @ X @ Y )
% 16.19/8.96 => ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/8.96 => ( dvd_dvd_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ M ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4136_dvd__power__le,axiom,
% 16.19/8.96 ! [X: complex,Y: complex,N2: nat,M: nat] :
% 16.19/8.96 ( ( dvd_dvd_complex @ X @ Y )
% 16.19/8.96 => ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/8.96 => ( dvd_dvd_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ Y @ M ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4137_power__le__dvd,axiom,
% 16.19/8.96 ! [A: code_integer,N2: nat,B: code_integer,M: nat] :
% 16.19/8.96 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) @ B )
% 16.19/8.96 => ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/8.96 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4138_power__le__dvd,axiom,
% 16.19/8.96 ! [A: nat,N2: nat,B: nat,M: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ B )
% 16.19/8.96 => ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/8.96 => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4139_power__le__dvd,axiom,
% 16.19/8.96 ! [A: real,N2: nat,B: real,M: nat] :
% 16.19/8.96 ( ( dvd_dvd_real @ ( power_power_real @ A @ N2 ) @ B )
% 16.19/8.96 => ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/8.96 => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4140_power__le__dvd,axiom,
% 16.19/8.96 ! [A: int,N2: nat,B: int,M: nat] :
% 16.19/8.96 ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ B )
% 16.19/8.96 => ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/8.96 => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4141_power__le__dvd,axiom,
% 16.19/8.96 ! [A: complex,N2: nat,B: complex,M: nat] :
% 16.19/8.96 ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N2 ) @ B )
% 16.19/8.96 => ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/8.96 => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4142_le__imp__power__dvd,axiom,
% 16.19/8.96 ! [M: nat,N2: nat,A: code_integer] :
% 16.19/8.96 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/8.96 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4143_le__imp__power__dvd,axiom,
% 16.19/8.96 ! [M: nat,N2: nat,A: nat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/8.96 => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4144_le__imp__power__dvd,axiom,
% 16.19/8.96 ! [M: nat,N2: nat,A: real] :
% 16.19/8.96 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/8.96 => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4145_le__imp__power__dvd,axiom,
% 16.19/8.96 ! [M: nat,N2: nat,A: int] :
% 16.19/8.96 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/8.96 => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4146_le__imp__power__dvd,axiom,
% 16.19/8.96 ! [M: nat,N2: nat,A: complex] :
% 16.19/8.96 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/8.96 => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4147_dvd__minus__mod,axiom,
% 16.19/8.96 ! [B: nat,A: nat] : ( dvd_dvd_nat @ B @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4148_dvd__minus__mod,axiom,
% 16.19/8.96 ! [B: int,A: int] : ( dvd_dvd_int @ B @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4149_dvd__minus__mod,axiom,
% 16.19/8.96 ! [B: code_integer,A: code_integer] : ( dvd_dvd_Code_integer @ B @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4150_mod__eq__dvd__iff,axiom,
% 16.19/8.96 ! [A: int,C: int,B: int] :
% 16.19/8.96 ( ( ( modulo_modulo_int @ A @ C )
% 16.19/8.96 = ( modulo_modulo_int @ B @ C ) )
% 16.19/8.96 = ( dvd_dvd_int @ C @ ( minus_minus_int @ A @ B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4151_mod__eq__dvd__iff,axiom,
% 16.19/8.96 ! [A: code_integer,C: code_integer,B: code_integer] :
% 16.19/8.96 ( ( ( modulo364778990260209775nteger @ A @ C )
% 16.19/8.96 = ( modulo364778990260209775nteger @ B @ C ) )
% 16.19/8.96 = ( dvd_dvd_Code_integer @ C @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4152_nat__dvd__not__less,axiom,
% 16.19/8.96 ! [M: nat,N2: nat] :
% 16.19/8.96 ( ( ord_less_nat @ zero_zero_nat @ M )
% 16.19/8.96 => ( ( ord_less_nat @ M @ N2 )
% 16.19/8.96 => ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4153_dvd__pos__nat,axiom,
% 16.19/8.96 ! [N2: nat,M: nat] :
% 16.19/8.96 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.96 => ( ( dvd_dvd_nat @ M @ N2 )
% 16.19/8.96 => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4154_dvd__minus__self,axiom,
% 16.19/8.96 ! [M: nat,N2: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ M ) )
% 16.19/8.96 = ( ( ord_less_nat @ N2 @ M )
% 16.19/8.96 | ( dvd_dvd_nat @ M @ N2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4155_bezout__lemma__nat,axiom,
% 16.19/8.96 ! [D: nat,A: nat,B: nat,X: nat,Y: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ D @ A )
% 16.19/8.96 => ( ( dvd_dvd_nat @ D @ B )
% 16.19/8.96 => ( ( ( ( times_times_nat @ A @ X )
% 16.19/8.96 = ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D ) )
% 16.19/8.96 | ( ( times_times_nat @ B @ X )
% 16.19/8.96 = ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D ) ) )
% 16.19/8.96 => ? [X4: nat,Y4: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ D @ A )
% 16.19/8.96 & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
% 16.19/8.96 & ( ( ( times_times_nat @ A @ X4 )
% 16.19/8.96 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y4 ) @ D ) )
% 16.19/8.96 | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X4 )
% 16.19/8.96 = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D ) ) ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4156_bezout__add__nat,axiom,
% 16.19/8.96 ! [A: nat,B: nat] :
% 16.19/8.96 ? [D4: nat,X4: nat,Y4: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ D4 @ A )
% 16.19/8.96 & ( dvd_dvd_nat @ D4 @ B )
% 16.19/8.96 & ( ( ( times_times_nat @ A @ X4 )
% 16.19/8.96 = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D4 ) )
% 16.19/8.96 | ( ( times_times_nat @ B @ X4 )
% 16.19/8.96 = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D4 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4157_less__eq__dvd__minus,axiom,
% 16.19/8.96 ! [M: nat,N2: nat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/8.96 => ( ( dvd_dvd_nat @ M @ N2 )
% 16.19/8.96 = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4158_dvd__diffD1,axiom,
% 16.19/8.96 ! [K: nat,M: nat,N2: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
% 16.19/8.96 => ( ( dvd_dvd_nat @ K @ M )
% 16.19/8.96 => ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/8.96 => ( dvd_dvd_nat @ K @ N2 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4159_dvd__diffD,axiom,
% 16.19/8.96 ! [K: nat,M: nat,N2: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
% 16.19/8.96 => ( ( dvd_dvd_nat @ K @ N2 )
% 16.19/8.96 => ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/8.96 => ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4160_zdvd__not__zless,axiom,
% 16.19/8.96 ! [M: int,N2: int] :
% 16.19/8.96 ( ( ord_less_int @ zero_zero_int @ M )
% 16.19/8.96 => ( ( ord_less_int @ M @ N2 )
% 16.19/8.96 => ~ ( dvd_dvd_int @ N2 @ M ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4161_bezout1__nat,axiom,
% 16.19/8.96 ! [A: nat,B: nat] :
% 16.19/8.96 ? [D4: nat,X4: nat,Y4: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ D4 @ A )
% 16.19/8.96 & ( dvd_dvd_nat @ D4 @ B )
% 16.19/8.96 & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X4 ) @ ( times_times_nat @ B @ Y4 ) )
% 16.19/8.96 = D4 )
% 16.19/8.96 | ( ( minus_minus_nat @ ( times_times_nat @ B @ X4 ) @ ( times_times_nat @ A @ Y4 ) )
% 16.19/8.96 = D4 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4162_dbl__def,axiom,
% 16.19/8.96 ( neg_numeral_dbl_real
% 16.19/8.96 = ( ^ [X2: real] : ( plus_plus_real @ X2 @ X2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4163_dbl__def,axiom,
% 16.19/8.96 ( neg_numeral_dbl_rat
% 16.19/8.96 = ( ^ [X2: rat] : ( plus_plus_rat @ X2 @ X2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4164_dbl__def,axiom,
% 16.19/8.96 ( neg_numeral_dbl_int
% 16.19/8.96 = ( ^ [X2: int] : ( plus_plus_int @ X2 @ X2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4165_finite__divisors__int,axiom,
% 16.19/8.96 ! [I2: int] :
% 16.19/8.96 ( ( I2 != zero_zero_int )
% 16.19/8.96 => ( finite_finite_int
% 16.19/8.96 @ ( collect_int
% 16.19/8.96 @ ^ [D2: int] : ( dvd_dvd_int @ D2 @ I2 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4166_div2__even__ext__nat,axiom,
% 16.19/8.96 ! [X: nat,Y: nat] :
% 16.19/8.96 ( ( ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.96 = ( divide_divide_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.96 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X )
% 16.19/8.96 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y ) )
% 16.19/8.96 => ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4167_unity__coeff__ex,axiom,
% 16.19/8.96 ! [P: code_integer > $o,L2: code_integer] :
% 16.19/8.96 ( ( ? [X2: code_integer] : ( P @ ( times_3573771949741848930nteger @ L2 @ X2 ) ) )
% 16.19/8.96 = ( ? [X2: code_integer] :
% 16.19/8.96 ( ( dvd_dvd_Code_integer @ L2 @ ( plus_p5714425477246183910nteger @ X2 @ zero_z3403309356797280102nteger ) )
% 16.19/8.96 & ( P @ X2 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4168_unity__coeff__ex,axiom,
% 16.19/8.96 ! [P: complex > $o,L2: complex] :
% 16.19/8.96 ( ( ? [X2: complex] : ( P @ ( times_times_complex @ L2 @ X2 ) ) )
% 16.19/8.96 = ( ? [X2: complex] :
% 16.19/8.96 ( ( dvd_dvd_complex @ L2 @ ( plus_plus_complex @ X2 @ zero_zero_complex ) )
% 16.19/8.96 & ( P @ X2 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4169_unity__coeff__ex,axiom,
% 16.19/8.96 ! [P: real > $o,L2: real] :
% 16.19/8.96 ( ( ? [X2: real] : ( P @ ( times_times_real @ L2 @ X2 ) ) )
% 16.19/8.96 = ( ? [X2: real] :
% 16.19/8.96 ( ( dvd_dvd_real @ L2 @ ( plus_plus_real @ X2 @ zero_zero_real ) )
% 16.19/8.96 & ( P @ X2 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4170_unity__coeff__ex,axiom,
% 16.19/8.96 ! [P: rat > $o,L2: rat] :
% 16.19/8.96 ( ( ? [X2: rat] : ( P @ ( times_times_rat @ L2 @ X2 ) ) )
% 16.19/8.96 = ( ? [X2: rat] :
% 16.19/8.96 ( ( dvd_dvd_rat @ L2 @ ( plus_plus_rat @ X2 @ zero_zero_rat ) )
% 16.19/8.96 & ( P @ X2 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4171_unity__coeff__ex,axiom,
% 16.19/8.96 ! [P: nat > $o,L2: nat] :
% 16.19/8.96 ( ( ? [X2: nat] : ( P @ ( times_times_nat @ L2 @ X2 ) ) )
% 16.19/8.96 = ( ? [X2: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ L2 @ ( plus_plus_nat @ X2 @ zero_zero_nat ) )
% 16.19/8.96 & ( P @ X2 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4172_unity__coeff__ex,axiom,
% 16.19/8.96 ! [P: int > $o,L2: int] :
% 16.19/8.96 ( ( ? [X2: int] : ( P @ ( times_times_int @ L2 @ X2 ) ) )
% 16.19/8.96 = ( ? [X2: int] :
% 16.19/8.96 ( ( dvd_dvd_int @ L2 @ ( plus_plus_int @ X2 @ zero_zero_int ) )
% 16.19/8.96 & ( P @ X2 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4173_unit__dvdE,axiom,
% 16.19/8.96 ! [A: code_integer,B: code_integer] :
% 16.19/8.96 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 16.19/8.96 => ~ ( ( A != zero_z3403309356797280102nteger )
% 16.19/8.96 => ! [C3: code_integer] :
% 16.19/8.96 ( B
% 16.19/8.96 != ( times_3573771949741848930nteger @ A @ C3 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4174_unit__dvdE,axiom,
% 16.19/8.96 ! [A: nat,B: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 16.19/8.96 => ~ ( ( A != zero_zero_nat )
% 16.19/8.96 => ! [C3: nat] :
% 16.19/8.96 ( B
% 16.19/8.96 != ( times_times_nat @ A @ C3 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4175_unit__dvdE,axiom,
% 16.19/8.96 ! [A: int,B: int] :
% 16.19/8.96 ( ( dvd_dvd_int @ A @ one_one_int )
% 16.19/8.96 => ~ ( ( A != zero_zero_int )
% 16.19/8.96 => ! [C3: int] :
% 16.19/8.96 ( B
% 16.19/8.96 != ( times_times_int @ A @ C3 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4176_dvd__div__div__eq__mult,axiom,
% 16.19/8.96 ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 16.19/8.96 ( ( A != zero_z3403309356797280102nteger )
% 16.19/8.96 => ( ( C != zero_z3403309356797280102nteger )
% 16.19/8.96 => ( ( dvd_dvd_Code_integer @ A @ B )
% 16.19/8.96 => ( ( dvd_dvd_Code_integer @ C @ D )
% 16.19/8.96 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 16.19/8.96 = ( divide6298287555418463151nteger @ D @ C ) )
% 16.19/8.96 = ( ( times_3573771949741848930nteger @ B @ C )
% 16.19/8.96 = ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4177_dvd__div__div__eq__mult,axiom,
% 16.19/8.96 ! [A: nat,C: nat,B: nat,D: nat] :
% 16.19/8.96 ( ( A != zero_zero_nat )
% 16.19/8.96 => ( ( C != zero_zero_nat )
% 16.19/8.96 => ( ( dvd_dvd_nat @ A @ B )
% 16.19/8.96 => ( ( dvd_dvd_nat @ C @ D )
% 16.19/8.96 => ( ( ( divide_divide_nat @ B @ A )
% 16.19/8.96 = ( divide_divide_nat @ D @ C ) )
% 16.19/8.96 = ( ( times_times_nat @ B @ C )
% 16.19/8.96 = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4178_dvd__div__div__eq__mult,axiom,
% 16.19/8.96 ! [A: int,C: int,B: int,D: int] :
% 16.19/8.96 ( ( A != zero_zero_int )
% 16.19/8.96 => ( ( C != zero_zero_int )
% 16.19/8.96 => ( ( dvd_dvd_int @ A @ B )
% 16.19/8.96 => ( ( dvd_dvd_int @ C @ D )
% 16.19/8.96 => ( ( ( divide_divide_int @ B @ A )
% 16.19/8.96 = ( divide_divide_int @ D @ C ) )
% 16.19/8.96 = ( ( times_times_int @ B @ C )
% 16.19/8.96 = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4179_dvd__div__iff__mult,axiom,
% 16.19/8.96 ! [C: code_integer,B: code_integer,A: code_integer] :
% 16.19/8.96 ( ( C != zero_z3403309356797280102nteger )
% 16.19/8.96 => ( ( dvd_dvd_Code_integer @ C @ B )
% 16.19/8.96 => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 16.19/8.96 = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4180_dvd__div__iff__mult,axiom,
% 16.19/8.96 ! [C: nat,B: nat,A: nat] :
% 16.19/8.96 ( ( C != zero_zero_nat )
% 16.19/8.96 => ( ( dvd_dvd_nat @ C @ B )
% 16.19/8.96 => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 16.19/8.96 = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4181_dvd__div__iff__mult,axiom,
% 16.19/8.96 ! [C: int,B: int,A: int] :
% 16.19/8.96 ( ( C != zero_zero_int )
% 16.19/8.96 => ( ( dvd_dvd_int @ C @ B )
% 16.19/8.96 => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
% 16.19/8.96 = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4182_div__dvd__iff__mult,axiom,
% 16.19/8.96 ! [B: code_integer,A: code_integer,C: code_integer] :
% 16.19/8.96 ( ( B != zero_z3403309356797280102nteger )
% 16.19/8.96 => ( ( dvd_dvd_Code_integer @ B @ A )
% 16.19/8.96 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 16.19/8.96 = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4183_div__dvd__iff__mult,axiom,
% 16.19/8.96 ! [B: nat,A: nat,C: nat] :
% 16.19/8.96 ( ( B != zero_zero_nat )
% 16.19/8.96 => ( ( dvd_dvd_nat @ B @ A )
% 16.19/8.96 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 16.19/8.96 = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4184_div__dvd__iff__mult,axiom,
% 16.19/8.96 ! [B: int,A: int,C: int] :
% 16.19/8.96 ( ( B != zero_zero_int )
% 16.19/8.96 => ( ( dvd_dvd_int @ B @ A )
% 16.19/8.96 => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 16.19/8.96 = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4185_dvd__div__eq__mult,axiom,
% 16.19/8.96 ! [A: code_integer,B: code_integer,C: code_integer] :
% 16.19/8.96 ( ( A != zero_z3403309356797280102nteger )
% 16.19/8.96 => ( ( dvd_dvd_Code_integer @ A @ B )
% 16.19/8.96 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 16.19/8.96 = C )
% 16.19/8.96 = ( B
% 16.19/8.96 = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4186_dvd__div__eq__mult,axiom,
% 16.19/8.96 ! [A: nat,B: nat,C: nat] :
% 16.19/8.96 ( ( A != zero_zero_nat )
% 16.19/8.96 => ( ( dvd_dvd_nat @ A @ B )
% 16.19/8.96 => ( ( ( divide_divide_nat @ B @ A )
% 16.19/8.96 = C )
% 16.19/8.96 = ( B
% 16.19/8.96 = ( times_times_nat @ C @ A ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4187_dvd__div__eq__mult,axiom,
% 16.19/8.96 ! [A: int,B: int,C: int] :
% 16.19/8.96 ( ( A != zero_zero_int )
% 16.19/8.96 => ( ( dvd_dvd_int @ A @ B )
% 16.19/8.96 => ( ( ( divide_divide_int @ B @ A )
% 16.19/8.96 = C )
% 16.19/8.96 = ( B
% 16.19/8.96 = ( times_times_int @ C @ A ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4188_even__numeral,axiom,
% 16.19/8.96 ! [N2: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4189_even__numeral,axiom,
% 16.19/8.96 ! [N2: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4190_even__numeral,axiom,
% 16.19/8.96 ! [N2: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4191_unit__div__eq__0__iff,axiom,
% 16.19/8.96 ! [B: code_integer,A: code_integer] :
% 16.19/8.96 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 16.19/8.96 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 16.19/8.96 = zero_z3403309356797280102nteger )
% 16.19/8.96 = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4192_unit__div__eq__0__iff,axiom,
% 16.19/8.96 ! [B: nat,A: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 16.19/8.96 => ( ( ( divide_divide_nat @ A @ B )
% 16.19/8.96 = zero_zero_nat )
% 16.19/8.96 = ( A = zero_zero_nat ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4193_unit__div__eq__0__iff,axiom,
% 16.19/8.96 ! [B: int,A: int] :
% 16.19/8.96 ( ( dvd_dvd_int @ B @ one_one_int )
% 16.19/8.96 => ( ( ( divide_divide_int @ A @ B )
% 16.19/8.96 = zero_zero_int )
% 16.19/8.96 = ( A = zero_zero_int ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4194_inf__period_I3_J,axiom,
% 16.19/8.96 ! [D: code_integer,D3: code_integer,T: code_integer] :
% 16.19/8.96 ( ( dvd_dvd_Code_integer @ D @ D3 )
% 16.19/8.96 => ! [X5: code_integer,K4: code_integer] :
% 16.19/8.96 ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ T ) )
% 16.19/8.96 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K4 @ D3 ) ) @ T ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4195_inf__period_I3_J,axiom,
% 16.19/8.96 ! [D: real,D3: real,T: real] :
% 16.19/8.96 ( ( dvd_dvd_real @ D @ D3 )
% 16.19/8.96 => ! [X5: real,K4: real] :
% 16.19/8.96 ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) )
% 16.19/8.96 = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) @ T ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4196_inf__period_I3_J,axiom,
% 16.19/8.96 ! [D: rat,D3: rat,T: rat] :
% 16.19/8.96 ( ( dvd_dvd_rat @ D @ D3 )
% 16.19/8.96 => ! [X5: rat,K4: rat] :
% 16.19/8.96 ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) )
% 16.19/8.96 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) @ T ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4197_inf__period_I3_J,axiom,
% 16.19/8.96 ! [D: int,D3: int,T: int] :
% 16.19/8.96 ( ( dvd_dvd_int @ D @ D3 )
% 16.19/8.96 => ! [X5: int,K4: int] :
% 16.19/8.96 ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 16.19/8.96 = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) @ T ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4198_inf__period_I4_J,axiom,
% 16.19/8.96 ! [D: code_integer,D3: code_integer,T: code_integer] :
% 16.19/8.96 ( ( dvd_dvd_Code_integer @ D @ D3 )
% 16.19/8.96 => ! [X5: code_integer,K4: code_integer] :
% 16.19/8.96 ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ T ) ) )
% 16.19/8.96 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K4 @ D3 ) ) @ T ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4199_inf__period_I4_J,axiom,
% 16.19/8.96 ! [D: real,D3: real,T: real] :
% 16.19/8.96 ( ( dvd_dvd_real @ D @ D3 )
% 16.19/8.96 => ! [X5: real,K4: real] :
% 16.19/8.96 ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) ) )
% 16.19/8.96 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) @ T ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4200_inf__period_I4_J,axiom,
% 16.19/8.96 ! [D: rat,D3: rat,T: rat] :
% 16.19/8.96 ( ( dvd_dvd_rat @ D @ D3 )
% 16.19/8.96 => ! [X5: rat,K4: rat] :
% 16.19/8.96 ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) ) )
% 16.19/8.96 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) @ T ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4201_inf__period_I4_J,axiom,
% 16.19/8.96 ! [D: int,D3: int,T: int] :
% 16.19/8.96 ( ( dvd_dvd_int @ D @ D3 )
% 16.19/8.96 => ! [X5: int,K4: int] :
% 16.19/8.96 ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) ) )
% 16.19/8.96 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) @ T ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4202_is__unit__div__mult2__eq,axiom,
% 16.19/8.96 ! [B: code_integer,C: code_integer,A: code_integer] :
% 16.19/8.96 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 16.19/8.96 => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 16.19/8.96 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 16.19/8.96 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4203_is__unit__div__mult2__eq,axiom,
% 16.19/8.96 ! [B: nat,C: nat,A: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 16.19/8.96 => ( ( dvd_dvd_nat @ C @ one_one_nat )
% 16.19/8.96 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 16.19/8.96 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4204_is__unit__div__mult2__eq,axiom,
% 16.19/8.96 ! [B: int,C: int,A: int] :
% 16.19/8.96 ( ( dvd_dvd_int @ B @ one_one_int )
% 16.19/8.96 => ( ( dvd_dvd_int @ C @ one_one_int )
% 16.19/8.96 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 16.19/8.96 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4205_unit__div__mult__swap,axiom,
% 16.19/8.96 ! [C: code_integer,A: code_integer,B: code_integer] :
% 16.19/8.96 ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 16.19/8.96 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 16.19/8.96 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4206_unit__div__mult__swap,axiom,
% 16.19/8.96 ! [C: nat,A: nat,B: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ C @ one_one_nat )
% 16.19/8.96 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 16.19/8.96 = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4207_unit__div__mult__swap,axiom,
% 16.19/8.96 ! [C: int,A: int,B: int] :
% 16.19/8.96 ( ( dvd_dvd_int @ C @ one_one_int )
% 16.19/8.96 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
% 16.19/8.96 = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4208_unit__div__commute,axiom,
% 16.19/8.96 ! [B: code_integer,A: code_integer,C: code_integer] :
% 16.19/8.96 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 16.19/8.96 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 16.19/8.96 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4209_unit__div__commute,axiom,
% 16.19/8.96 ! [B: nat,A: nat,C: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 16.19/8.96 => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 16.19/8.96 = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4210_unit__div__commute,axiom,
% 16.19/8.96 ! [B: int,A: int,C: int] :
% 16.19/8.96 ( ( dvd_dvd_int @ B @ one_one_int )
% 16.19/8.96 => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ C )
% 16.19/8.96 = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4211_div__mult__unit2,axiom,
% 16.19/8.96 ! [C: code_integer,B: code_integer,A: code_integer] :
% 16.19/8.96 ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 16.19/8.96 => ( ( dvd_dvd_Code_integer @ B @ A )
% 16.19/8.96 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 16.19/8.96 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4212_div__mult__unit2,axiom,
% 16.19/8.96 ! [C: nat,B: nat,A: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ C @ one_one_nat )
% 16.19/8.96 => ( ( dvd_dvd_nat @ B @ A )
% 16.19/8.96 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 16.19/8.96 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4213_div__mult__unit2,axiom,
% 16.19/8.96 ! [C: int,B: int,A: int] :
% 16.19/8.96 ( ( dvd_dvd_int @ C @ one_one_int )
% 16.19/8.96 => ( ( dvd_dvd_int @ B @ A )
% 16.19/8.96 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 16.19/8.96 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4214_unit__eq__div2,axiom,
% 16.19/8.96 ! [B: code_integer,A: code_integer,C: code_integer] :
% 16.19/8.96 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 16.19/8.96 => ( ( A
% 16.19/8.96 = ( divide6298287555418463151nteger @ C @ B ) )
% 16.19/8.96 = ( ( times_3573771949741848930nteger @ A @ B )
% 16.19/8.96 = C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4215_unit__eq__div2,axiom,
% 16.19/8.96 ! [B: nat,A: nat,C: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 16.19/8.96 => ( ( A
% 16.19/8.96 = ( divide_divide_nat @ C @ B ) )
% 16.19/8.96 = ( ( times_times_nat @ A @ B )
% 16.19/8.96 = C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4216_unit__eq__div2,axiom,
% 16.19/8.96 ! [B: int,A: int,C: int] :
% 16.19/8.96 ( ( dvd_dvd_int @ B @ one_one_int )
% 16.19/8.96 => ( ( A
% 16.19/8.96 = ( divide_divide_int @ C @ B ) )
% 16.19/8.96 = ( ( times_times_int @ A @ B )
% 16.19/8.96 = C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4217_unit__eq__div1,axiom,
% 16.19/8.96 ! [B: code_integer,A: code_integer,C: code_integer] :
% 16.19/8.96 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 16.19/8.96 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 16.19/8.96 = C )
% 16.19/8.96 = ( A
% 16.19/8.96 = ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4218_unit__eq__div1,axiom,
% 16.19/8.96 ! [B: nat,A: nat,C: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 16.19/8.96 => ( ( ( divide_divide_nat @ A @ B )
% 16.19/8.96 = C )
% 16.19/8.96 = ( A
% 16.19/8.96 = ( times_times_nat @ C @ B ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4219_unit__eq__div1,axiom,
% 16.19/8.96 ! [B: int,A: int,C: int] :
% 16.19/8.96 ( ( dvd_dvd_int @ B @ one_one_int )
% 16.19/8.96 => ( ( ( divide_divide_int @ A @ B )
% 16.19/8.96 = C )
% 16.19/8.96 = ( A
% 16.19/8.96 = ( times_times_int @ C @ B ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4220_is__unit__power__iff,axiom,
% 16.19/8.96 ! [A: code_integer,N2: nat] :
% 16.19/8.96 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) @ one_one_Code_integer )
% 16.19/8.96 = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 16.19/8.96 | ( N2 = zero_zero_nat ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4221_is__unit__power__iff,axiom,
% 16.19/8.96 ! [A: nat,N2: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ one_one_nat )
% 16.19/8.96 = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 16.19/8.96 | ( N2 = zero_zero_nat ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4222_is__unit__power__iff,axiom,
% 16.19/8.96 ! [A: int,N2: nat] :
% 16.19/8.96 ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ one_one_int )
% 16.19/8.96 = ( ( dvd_dvd_int @ A @ one_one_int )
% 16.19/8.96 | ( N2 = zero_zero_nat ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4223_unit__imp__mod__eq__0,axiom,
% 16.19/8.96 ! [B: nat,A: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 16.19/8.96 => ( ( modulo_modulo_nat @ A @ B )
% 16.19/8.96 = zero_zero_nat ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4224_unit__imp__mod__eq__0,axiom,
% 16.19/8.96 ! [B: int,A: int] :
% 16.19/8.96 ( ( dvd_dvd_int @ B @ one_one_int )
% 16.19/8.96 => ( ( modulo_modulo_int @ A @ B )
% 16.19/8.96 = zero_zero_int ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4225_unit__imp__mod__eq__0,axiom,
% 16.19/8.96 ! [B: code_integer,A: code_integer] :
% 16.19/8.96 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 16.19/8.96 => ( ( modulo364778990260209775nteger @ A @ B )
% 16.19/8.96 = zero_z3403309356797280102nteger ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4226_dvd__imp__le,axiom,
% 16.19/8.96 ! [K: nat,N2: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ K @ N2 )
% 16.19/8.96 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.96 => ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4227_dvd__mult__cancel,axiom,
% 16.19/8.96 ! [K: nat,M: nat,N2: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 16.19/8.96 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 16.19/8.96 => ( dvd_dvd_nat @ M @ N2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4228_nat__mult__dvd__cancel1,axiom,
% 16.19/8.96 ! [K: nat,M: nat,N2: nat] :
% 16.19/8.96 ( ( ord_less_nat @ zero_zero_nat @ K )
% 16.19/8.96 => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 16.19/8.96 = ( dvd_dvd_nat @ M @ N2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4229_bezout__add__strong__nat,axiom,
% 16.19/8.96 ! [A: nat,B: nat] :
% 16.19/8.96 ( ( A != zero_zero_nat )
% 16.19/8.96 => ? [D4: nat,X4: nat,Y4: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ D4 @ A )
% 16.19/8.96 & ( dvd_dvd_nat @ D4 @ B )
% 16.19/8.96 & ( ( times_times_nat @ A @ X4 )
% 16.19/8.96 = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D4 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4230_zdvd__imp__le,axiom,
% 16.19/8.96 ! [Z: int,N2: int] :
% 16.19/8.96 ( ( dvd_dvd_int @ Z @ N2 )
% 16.19/8.96 => ( ( ord_less_int @ zero_zero_int @ N2 )
% 16.19/8.96 => ( ord_less_eq_int @ Z @ N2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4231_nle__le,axiom,
% 16.19/8.96 ! [A: rat,B: rat] :
% 16.19/8.96 ( ( ~ ( ord_less_eq_rat @ A @ B ) )
% 16.19/8.96 = ( ( ord_less_eq_rat @ B @ A )
% 16.19/8.96 & ( B != A ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4232_nle__le,axiom,
% 16.19/8.96 ! [A: num,B: num] :
% 16.19/8.96 ( ( ~ ( ord_less_eq_num @ A @ B ) )
% 16.19/8.96 = ( ( ord_less_eq_num @ B @ A )
% 16.19/8.96 & ( B != A ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4233_nle__le,axiom,
% 16.19/8.96 ! [A: nat,B: nat] :
% 16.19/8.96 ( ( ~ ( ord_less_eq_nat @ A @ B ) )
% 16.19/8.96 = ( ( ord_less_eq_nat @ B @ A )
% 16.19/8.96 & ( B != A ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4234_nle__le,axiom,
% 16.19/8.96 ! [A: int,B: int] :
% 16.19/8.96 ( ( ~ ( ord_less_eq_int @ A @ B ) )
% 16.19/8.96 = ( ( ord_less_eq_int @ B @ A )
% 16.19/8.96 & ( B != A ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4235_le__cases3,axiom,
% 16.19/8.96 ! [X: rat,Y: rat,Z: rat] :
% 16.19/8.96 ( ( ( ord_less_eq_rat @ X @ Y )
% 16.19/8.96 => ~ ( ord_less_eq_rat @ Y @ Z ) )
% 16.19/8.96 => ( ( ( ord_less_eq_rat @ Y @ X )
% 16.19/8.96 => ~ ( ord_less_eq_rat @ X @ Z ) )
% 16.19/8.96 => ( ( ( ord_less_eq_rat @ X @ Z )
% 16.19/8.96 => ~ ( ord_less_eq_rat @ Z @ Y ) )
% 16.19/8.96 => ( ( ( ord_less_eq_rat @ Z @ Y )
% 16.19/8.96 => ~ ( ord_less_eq_rat @ Y @ X ) )
% 16.19/8.96 => ( ( ( ord_less_eq_rat @ Y @ Z )
% 16.19/8.96 => ~ ( ord_less_eq_rat @ Z @ X ) )
% 16.19/8.96 => ~ ( ( ord_less_eq_rat @ Z @ X )
% 16.19/8.96 => ~ ( ord_less_eq_rat @ X @ Y ) ) ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4236_le__cases3,axiom,
% 16.19/8.96 ! [X: num,Y: num,Z: num] :
% 16.19/8.96 ( ( ( ord_less_eq_num @ X @ Y )
% 16.19/8.96 => ~ ( ord_less_eq_num @ Y @ Z ) )
% 16.19/8.96 => ( ( ( ord_less_eq_num @ Y @ X )
% 16.19/8.96 => ~ ( ord_less_eq_num @ X @ Z ) )
% 16.19/8.96 => ( ( ( ord_less_eq_num @ X @ Z )
% 16.19/8.96 => ~ ( ord_less_eq_num @ Z @ Y ) )
% 16.19/8.96 => ( ( ( ord_less_eq_num @ Z @ Y )
% 16.19/8.96 => ~ ( ord_less_eq_num @ Y @ X ) )
% 16.19/8.96 => ( ( ( ord_less_eq_num @ Y @ Z )
% 16.19/8.96 => ~ ( ord_less_eq_num @ Z @ X ) )
% 16.19/8.96 => ~ ( ( ord_less_eq_num @ Z @ X )
% 16.19/8.96 => ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4237_le__cases3,axiom,
% 16.19/8.96 ! [X: nat,Y: nat,Z: nat] :
% 16.19/8.96 ( ( ( ord_less_eq_nat @ X @ Y )
% 16.19/8.96 => ~ ( ord_less_eq_nat @ Y @ Z ) )
% 16.19/8.96 => ( ( ( ord_less_eq_nat @ Y @ X )
% 16.19/8.96 => ~ ( ord_less_eq_nat @ X @ Z ) )
% 16.19/8.96 => ( ( ( ord_less_eq_nat @ X @ Z )
% 16.19/8.96 => ~ ( ord_less_eq_nat @ Z @ Y ) )
% 16.19/8.96 => ( ( ( ord_less_eq_nat @ Z @ Y )
% 16.19/8.96 => ~ ( ord_less_eq_nat @ Y @ X ) )
% 16.19/8.96 => ( ( ( ord_less_eq_nat @ Y @ Z )
% 16.19/8.96 => ~ ( ord_less_eq_nat @ Z @ X ) )
% 16.19/8.96 => ~ ( ( ord_less_eq_nat @ Z @ X )
% 16.19/8.96 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4238_le__cases3,axiom,
% 16.19/8.96 ! [X: int,Y: int,Z: int] :
% 16.19/8.96 ( ( ( ord_less_eq_int @ X @ Y )
% 16.19/8.96 => ~ ( ord_less_eq_int @ Y @ Z ) )
% 16.19/8.96 => ( ( ( ord_less_eq_int @ Y @ X )
% 16.19/8.96 => ~ ( ord_less_eq_int @ X @ Z ) )
% 16.19/8.96 => ( ( ( ord_less_eq_int @ X @ Z )
% 16.19/8.96 => ~ ( ord_less_eq_int @ Z @ Y ) )
% 16.19/8.96 => ( ( ( ord_less_eq_int @ Z @ Y )
% 16.19/8.96 => ~ ( ord_less_eq_int @ Y @ X ) )
% 16.19/8.96 => ( ( ( ord_less_eq_int @ Y @ Z )
% 16.19/8.96 => ~ ( ord_less_eq_int @ Z @ X ) )
% 16.19/8.96 => ~ ( ( ord_less_eq_int @ Z @ X )
% 16.19/8.96 => ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4239_order__class_Oorder__eq__iff,axiom,
% 16.19/8.96 ( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
% 16.19/8.96 = ( ^ [X2: set_nat,Y2: set_nat] :
% 16.19/8.96 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 16.19/8.96 & ( ord_less_eq_set_nat @ Y2 @ X2 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4240_order__class_Oorder__eq__iff,axiom,
% 16.19/8.96 ( ( ^ [Y3: rat,Z2: rat] : ( Y3 = Z2 ) )
% 16.19/8.96 = ( ^ [X2: rat,Y2: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ X2 @ Y2 )
% 16.19/8.96 & ( ord_less_eq_rat @ Y2 @ X2 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4241_order__class_Oorder__eq__iff,axiom,
% 16.19/8.96 ( ( ^ [Y3: num,Z2: num] : ( Y3 = Z2 ) )
% 16.19/8.96 = ( ^ [X2: num,Y2: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ X2 @ Y2 )
% 16.19/8.96 & ( ord_less_eq_num @ Y2 @ X2 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4242_order__class_Oorder__eq__iff,axiom,
% 16.19/8.96 ( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
% 16.19/8.96 = ( ^ [X2: nat,Y2: nat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 16.19/8.96 & ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4243_order__class_Oorder__eq__iff,axiom,
% 16.19/8.96 ( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
% 16.19/8.96 = ( ^ [X2: int,Y2: int] :
% 16.19/8.96 ( ( ord_less_eq_int @ X2 @ Y2 )
% 16.19/8.96 & ( ord_less_eq_int @ Y2 @ X2 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4244_ord__eq__le__trans,axiom,
% 16.19/8.96 ! [A: set_nat,B: set_nat,C: set_nat] :
% 16.19/8.96 ( ( A = B )
% 16.19/8.96 => ( ( ord_less_eq_set_nat @ B @ C )
% 16.19/8.96 => ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4245_ord__eq__le__trans,axiom,
% 16.19/8.96 ! [A: rat,B: rat,C: rat] :
% 16.19/8.96 ( ( A = B )
% 16.19/8.96 => ( ( ord_less_eq_rat @ B @ C )
% 16.19/8.96 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4246_ord__eq__le__trans,axiom,
% 16.19/8.96 ! [A: num,B: num,C: num] :
% 16.19/8.96 ( ( A = B )
% 16.19/8.96 => ( ( ord_less_eq_num @ B @ C )
% 16.19/8.96 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4247_ord__eq__le__trans,axiom,
% 16.19/8.96 ! [A: nat,B: nat,C: nat] :
% 16.19/8.96 ( ( A = B )
% 16.19/8.96 => ( ( ord_less_eq_nat @ B @ C )
% 16.19/8.96 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4248_ord__eq__le__trans,axiom,
% 16.19/8.96 ! [A: int,B: int,C: int] :
% 16.19/8.96 ( ( A = B )
% 16.19/8.96 => ( ( ord_less_eq_int @ B @ C )
% 16.19/8.96 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4249_ord__le__eq__trans,axiom,
% 16.19/8.96 ! [A: set_nat,B: set_nat,C: set_nat] :
% 16.19/8.96 ( ( ord_less_eq_set_nat @ A @ B )
% 16.19/8.96 => ( ( B = C )
% 16.19/8.96 => ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4250_ord__le__eq__trans,axiom,
% 16.19/8.96 ! [A: rat,B: rat,C: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.96 => ( ( B = C )
% 16.19/8.96 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4251_ord__le__eq__trans,axiom,
% 16.19/8.96 ! [A: num,B: num,C: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ A @ B )
% 16.19/8.96 => ( ( B = C )
% 16.19/8.96 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4252_ord__le__eq__trans,axiom,
% 16.19/8.96 ! [A: nat,B: nat,C: nat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.96 => ( ( B = C )
% 16.19/8.96 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4253_ord__le__eq__trans,axiom,
% 16.19/8.96 ! [A: int,B: int,C: int] :
% 16.19/8.96 ( ( ord_less_eq_int @ A @ B )
% 16.19/8.96 => ( ( B = C )
% 16.19/8.96 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4254_order__antisym,axiom,
% 16.19/8.96 ! [X: set_nat,Y: set_nat] :
% 16.19/8.96 ( ( ord_less_eq_set_nat @ X @ Y )
% 16.19/8.96 => ( ( ord_less_eq_set_nat @ Y @ X )
% 16.19/8.96 => ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4255_order__antisym,axiom,
% 16.19/8.96 ! [X: rat,Y: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ X @ Y )
% 16.19/8.96 => ( ( ord_less_eq_rat @ Y @ X )
% 16.19/8.96 => ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4256_order__antisym,axiom,
% 16.19/8.96 ! [X: num,Y: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ X @ Y )
% 16.19/8.96 => ( ( ord_less_eq_num @ Y @ X )
% 16.19/8.96 => ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4257_order__antisym,axiom,
% 16.19/8.96 ! [X: nat,Y: nat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ X @ Y )
% 16.19/8.96 => ( ( ord_less_eq_nat @ Y @ X )
% 16.19/8.96 => ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4258_order__antisym,axiom,
% 16.19/8.96 ! [X: int,Y: int] :
% 16.19/8.96 ( ( ord_less_eq_int @ X @ Y )
% 16.19/8.96 => ( ( ord_less_eq_int @ Y @ X )
% 16.19/8.96 => ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4259_order_Otrans,axiom,
% 16.19/8.96 ! [A: set_nat,B: set_nat,C: set_nat] :
% 16.19/8.96 ( ( ord_less_eq_set_nat @ A @ B )
% 16.19/8.96 => ( ( ord_less_eq_set_nat @ B @ C )
% 16.19/8.96 => ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4260_order_Otrans,axiom,
% 16.19/8.96 ! [A: rat,B: rat,C: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.96 => ( ( ord_less_eq_rat @ B @ C )
% 16.19/8.96 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4261_order_Otrans,axiom,
% 16.19/8.96 ! [A: num,B: num,C: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ A @ B )
% 16.19/8.96 => ( ( ord_less_eq_num @ B @ C )
% 16.19/8.96 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4262_order_Otrans,axiom,
% 16.19/8.96 ! [A: nat,B: nat,C: nat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.96 => ( ( ord_less_eq_nat @ B @ C )
% 16.19/8.96 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4263_order_Otrans,axiom,
% 16.19/8.96 ! [A: int,B: int,C: int] :
% 16.19/8.96 ( ( ord_less_eq_int @ A @ B )
% 16.19/8.96 => ( ( ord_less_eq_int @ B @ C )
% 16.19/8.96 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4264_order__trans,axiom,
% 16.19/8.96 ! [X: set_nat,Y: set_nat,Z: set_nat] :
% 16.19/8.96 ( ( ord_less_eq_set_nat @ X @ Y )
% 16.19/8.96 => ( ( ord_less_eq_set_nat @ Y @ Z )
% 16.19/8.96 => ( ord_less_eq_set_nat @ X @ Z ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4265_order__trans,axiom,
% 16.19/8.96 ! [X: rat,Y: rat,Z: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ X @ Y )
% 16.19/8.96 => ( ( ord_less_eq_rat @ Y @ Z )
% 16.19/8.96 => ( ord_less_eq_rat @ X @ Z ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4266_order__trans,axiom,
% 16.19/8.96 ! [X: num,Y: num,Z: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ X @ Y )
% 16.19/8.96 => ( ( ord_less_eq_num @ Y @ Z )
% 16.19/8.96 => ( ord_less_eq_num @ X @ Z ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4267_order__trans,axiom,
% 16.19/8.96 ! [X: nat,Y: nat,Z: nat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ X @ Y )
% 16.19/8.96 => ( ( ord_less_eq_nat @ Y @ Z )
% 16.19/8.96 => ( ord_less_eq_nat @ X @ Z ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4268_order__trans,axiom,
% 16.19/8.96 ! [X: int,Y: int,Z: int] :
% 16.19/8.96 ( ( ord_less_eq_int @ X @ Y )
% 16.19/8.96 => ( ( ord_less_eq_int @ Y @ Z )
% 16.19/8.96 => ( ord_less_eq_int @ X @ Z ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4269_linorder__wlog,axiom,
% 16.19/8.96 ! [P: rat > rat > $o,A: rat,B: rat] :
% 16.19/8.96 ( ! [A3: rat,B2: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ A3 @ B2 )
% 16.19/8.96 => ( P @ A3 @ B2 ) )
% 16.19/8.96 => ( ! [A3: rat,B2: rat] :
% 16.19/8.96 ( ( P @ B2 @ A3 )
% 16.19/8.96 => ( P @ A3 @ B2 ) )
% 16.19/8.96 => ( P @ A @ B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4270_linorder__wlog,axiom,
% 16.19/8.96 ! [P: num > num > $o,A: num,B: num] :
% 16.19/8.96 ( ! [A3: num,B2: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ A3 @ B2 )
% 16.19/8.96 => ( P @ A3 @ B2 ) )
% 16.19/8.96 => ( ! [A3: num,B2: num] :
% 16.19/8.96 ( ( P @ B2 @ A3 )
% 16.19/8.96 => ( P @ A3 @ B2 ) )
% 16.19/8.96 => ( P @ A @ B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4271_linorder__wlog,axiom,
% 16.19/8.96 ! [P: nat > nat > $o,A: nat,B: nat] :
% 16.19/8.96 ( ! [A3: nat,B2: nat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ A3 @ B2 )
% 16.19/8.96 => ( P @ A3 @ B2 ) )
% 16.19/8.96 => ( ! [A3: nat,B2: nat] :
% 16.19/8.96 ( ( P @ B2 @ A3 )
% 16.19/8.96 => ( P @ A3 @ B2 ) )
% 16.19/8.96 => ( P @ A @ B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4272_linorder__wlog,axiom,
% 16.19/8.96 ! [P: int > int > $o,A: int,B: int] :
% 16.19/8.96 ( ! [A3: int,B2: int] :
% 16.19/8.96 ( ( ord_less_eq_int @ A3 @ B2 )
% 16.19/8.96 => ( P @ A3 @ B2 ) )
% 16.19/8.96 => ( ! [A3: int,B2: int] :
% 16.19/8.96 ( ( P @ B2 @ A3 )
% 16.19/8.96 => ( P @ A3 @ B2 ) )
% 16.19/8.96 => ( P @ A @ B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4273_dual__order_Oeq__iff,axiom,
% 16.19/8.96 ( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
% 16.19/8.96 = ( ^ [A5: set_nat,B5: set_nat] :
% 16.19/8.96 ( ( ord_less_eq_set_nat @ B5 @ A5 )
% 16.19/8.96 & ( ord_less_eq_set_nat @ A5 @ B5 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4274_dual__order_Oeq__iff,axiom,
% 16.19/8.96 ( ( ^ [Y3: rat,Z2: rat] : ( Y3 = Z2 ) )
% 16.19/8.96 = ( ^ [A5: rat,B5: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ B5 @ A5 )
% 16.19/8.96 & ( ord_less_eq_rat @ A5 @ B5 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4275_dual__order_Oeq__iff,axiom,
% 16.19/8.96 ( ( ^ [Y3: num,Z2: num] : ( Y3 = Z2 ) )
% 16.19/8.96 = ( ^ [A5: num,B5: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ B5 @ A5 )
% 16.19/8.96 & ( ord_less_eq_num @ A5 @ B5 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4276_dual__order_Oeq__iff,axiom,
% 16.19/8.96 ( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
% 16.19/8.96 = ( ^ [A5: nat,B5: nat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ B5 @ A5 )
% 16.19/8.96 & ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4277_dual__order_Oeq__iff,axiom,
% 16.19/8.96 ( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
% 16.19/8.96 = ( ^ [A5: int,B5: int] :
% 16.19/8.96 ( ( ord_less_eq_int @ B5 @ A5 )
% 16.19/8.96 & ( ord_less_eq_int @ A5 @ B5 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4278_dual__order_Oantisym,axiom,
% 16.19/8.96 ! [B: set_nat,A: set_nat] :
% 16.19/8.96 ( ( ord_less_eq_set_nat @ B @ A )
% 16.19/8.96 => ( ( ord_less_eq_set_nat @ A @ B )
% 16.19/8.96 => ( A = B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4279_dual__order_Oantisym,axiom,
% 16.19/8.96 ! [B: rat,A: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ B @ A )
% 16.19/8.96 => ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.96 => ( A = B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4280_dual__order_Oantisym,axiom,
% 16.19/8.96 ! [B: num,A: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ B @ A )
% 16.19/8.96 => ( ( ord_less_eq_num @ A @ B )
% 16.19/8.96 => ( A = B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4281_dual__order_Oantisym,axiom,
% 16.19/8.96 ! [B: nat,A: nat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ B @ A )
% 16.19/8.96 => ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.96 => ( A = B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4282_dual__order_Oantisym,axiom,
% 16.19/8.96 ! [B: int,A: int] :
% 16.19/8.96 ( ( ord_less_eq_int @ B @ A )
% 16.19/8.96 => ( ( ord_less_eq_int @ A @ B )
% 16.19/8.96 => ( A = B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4283_dual__order_Otrans,axiom,
% 16.19/8.96 ! [B: set_nat,A: set_nat,C: set_nat] :
% 16.19/8.96 ( ( ord_less_eq_set_nat @ B @ A )
% 16.19/8.96 => ( ( ord_less_eq_set_nat @ C @ B )
% 16.19/8.96 => ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4284_dual__order_Otrans,axiom,
% 16.19/8.96 ! [B: rat,A: rat,C: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ B @ A )
% 16.19/8.96 => ( ( ord_less_eq_rat @ C @ B )
% 16.19/8.96 => ( ord_less_eq_rat @ C @ A ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4285_dual__order_Otrans,axiom,
% 16.19/8.96 ! [B: num,A: num,C: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ B @ A )
% 16.19/8.96 => ( ( ord_less_eq_num @ C @ B )
% 16.19/8.96 => ( ord_less_eq_num @ C @ A ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4286_dual__order_Otrans,axiom,
% 16.19/8.96 ! [B: nat,A: nat,C: nat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ B @ A )
% 16.19/8.96 => ( ( ord_less_eq_nat @ C @ B )
% 16.19/8.96 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4287_dual__order_Otrans,axiom,
% 16.19/8.96 ! [B: int,A: int,C: int] :
% 16.19/8.96 ( ( ord_less_eq_int @ B @ A )
% 16.19/8.96 => ( ( ord_less_eq_int @ C @ B )
% 16.19/8.96 => ( ord_less_eq_int @ C @ A ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4288_antisym,axiom,
% 16.19/8.96 ! [A: set_nat,B: set_nat] :
% 16.19/8.96 ( ( ord_less_eq_set_nat @ A @ B )
% 16.19/8.96 => ( ( ord_less_eq_set_nat @ B @ A )
% 16.19/8.96 => ( A = B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4289_antisym,axiom,
% 16.19/8.96 ! [A: rat,B: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.96 => ( ( ord_less_eq_rat @ B @ A )
% 16.19/8.96 => ( A = B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4290_antisym,axiom,
% 16.19/8.96 ! [A: num,B: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ A @ B )
% 16.19/8.96 => ( ( ord_less_eq_num @ B @ A )
% 16.19/8.96 => ( A = B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4291_antisym,axiom,
% 16.19/8.96 ! [A: nat,B: nat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.96 => ( ( ord_less_eq_nat @ B @ A )
% 16.19/8.96 => ( A = B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4292_antisym,axiom,
% 16.19/8.96 ! [A: int,B: int] :
% 16.19/8.96 ( ( ord_less_eq_int @ A @ B )
% 16.19/8.96 => ( ( ord_less_eq_int @ B @ A )
% 16.19/8.96 => ( A = B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4293_Orderings_Oorder__eq__iff,axiom,
% 16.19/8.96 ( ( ^ [Y3: set_nat,Z2: set_nat] : ( Y3 = Z2 ) )
% 16.19/8.96 = ( ^ [A5: set_nat,B5: set_nat] :
% 16.19/8.96 ( ( ord_less_eq_set_nat @ A5 @ B5 )
% 16.19/8.96 & ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4294_Orderings_Oorder__eq__iff,axiom,
% 16.19/8.96 ( ( ^ [Y3: rat,Z2: rat] : ( Y3 = Z2 ) )
% 16.19/8.96 = ( ^ [A5: rat,B5: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ A5 @ B5 )
% 16.19/8.96 & ( ord_less_eq_rat @ B5 @ A5 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4295_Orderings_Oorder__eq__iff,axiom,
% 16.19/8.96 ( ( ^ [Y3: num,Z2: num] : ( Y3 = Z2 ) )
% 16.19/8.96 = ( ^ [A5: num,B5: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ A5 @ B5 )
% 16.19/8.96 & ( ord_less_eq_num @ B5 @ A5 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4296_Orderings_Oorder__eq__iff,axiom,
% 16.19/8.96 ( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
% 16.19/8.96 = ( ^ [A5: nat,B5: nat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ A5 @ B5 )
% 16.19/8.96 & ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4297_Orderings_Oorder__eq__iff,axiom,
% 16.19/8.96 ( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
% 16.19/8.96 = ( ^ [A5: int,B5: int] :
% 16.19/8.96 ( ( ord_less_eq_int @ A5 @ B5 )
% 16.19/8.96 & ( ord_less_eq_int @ B5 @ A5 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4298_order__subst1,axiom,
% 16.19/8.96 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_eq_rat @ B @ C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4299_order__subst1,axiom,
% 16.19/8.96 ! [A: rat,F: num > rat,B: num,C: num] :
% 16.19/8.96 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_eq_num @ B @ C )
% 16.19/8.96 => ( ! [X4: num,Y4: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4300_order__subst1,axiom,
% 16.19/8.96 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_eq_nat @ B @ C )
% 16.19/8.96 => ( ! [X4: nat,Y4: nat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4301_order__subst1,axiom,
% 16.19/8.96 ! [A: rat,F: int > rat,B: int,C: int] :
% 16.19/8.96 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_eq_int @ B @ C )
% 16.19/8.96 => ( ! [X4: int,Y4: int] :
% 16.19/8.96 ( ( ord_less_eq_int @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4302_order__subst1,axiom,
% 16.19/8.96 ! [A: num,F: rat > num,B: rat,C: rat] :
% 16.19/8.96 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_eq_rat @ B @ C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4303_order__subst1,axiom,
% 16.19/8.96 ! [A: num,F: num > num,B: num,C: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_eq_num @ B @ C )
% 16.19/8.96 => ( ! [X4: num,Y4: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4304_order__subst1,axiom,
% 16.19/8.96 ! [A: num,F: nat > num,B: nat,C: nat] :
% 16.19/8.96 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_eq_nat @ B @ C )
% 16.19/8.96 => ( ! [X4: nat,Y4: nat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4305_order__subst1,axiom,
% 16.19/8.96 ! [A: num,F: int > num,B: int,C: int] :
% 16.19/8.96 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_eq_int @ B @ C )
% 16.19/8.96 => ( ! [X4: int,Y4: int] :
% 16.19/8.96 ( ( ord_less_eq_int @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4306_order__subst1,axiom,
% 16.19/8.96 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_eq_rat @ B @ C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4307_order__subst1,axiom,
% 16.19/8.96 ! [A: nat,F: num > nat,B: num,C: num] :
% 16.19/8.96 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_eq_num @ B @ C )
% 16.19/8.96 => ( ! [X4: num,Y4: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4308_order__subst2,axiom,
% 16.19/8.96 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.96 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4309_order__subst2,axiom,
% 16.19/8.96 ! [A: rat,B: rat,F: rat > num,C: num] :
% 16.19/8.96 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.96 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4310_order__subst2,axiom,
% 16.19/8.96 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.96 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4311_order__subst2,axiom,
% 16.19/8.96 ! [A: rat,B: rat,F: rat > int,C: int] :
% 16.19/8.96 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.96 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4312_order__subst2,axiom,
% 16.19/8.96 ! [A: num,B: num,F: num > rat,C: rat] :
% 16.19/8.96 ( ( ord_less_eq_num @ A @ B )
% 16.19/8.96 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 16.19/8.96 => ( ! [X4: num,Y4: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4313_order__subst2,axiom,
% 16.19/8.96 ! [A: num,B: num,F: num > num,C: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ A @ B )
% 16.19/8.96 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 16.19/8.96 => ( ! [X4: num,Y4: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4314_order__subst2,axiom,
% 16.19/8.96 ! [A: num,B: num,F: num > nat,C: nat] :
% 16.19/8.96 ( ( ord_less_eq_num @ A @ B )
% 16.19/8.96 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 16.19/8.96 => ( ! [X4: num,Y4: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4315_order__subst2,axiom,
% 16.19/8.96 ! [A: num,B: num,F: num > int,C: int] :
% 16.19/8.96 ( ( ord_less_eq_num @ A @ B )
% 16.19/8.96 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 16.19/8.96 => ( ! [X4: num,Y4: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4316_order__subst2,axiom,
% 16.19/8.96 ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.96 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 16.19/8.96 => ( ! [X4: nat,Y4: nat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4317_order__subst2,axiom,
% 16.19/8.96 ! [A: nat,B: nat,F: nat > num,C: num] :
% 16.19/8.96 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.96 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 16.19/8.96 => ( ! [X4: nat,Y4: nat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4318_order__eq__refl,axiom,
% 16.19/8.96 ! [X: set_nat,Y: set_nat] :
% 16.19/8.96 ( ( X = Y )
% 16.19/8.96 => ( ord_less_eq_set_nat @ X @ Y ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4319_order__eq__refl,axiom,
% 16.19/8.96 ! [X: rat,Y: rat] :
% 16.19/8.96 ( ( X = Y )
% 16.19/8.96 => ( ord_less_eq_rat @ X @ Y ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4320_order__eq__refl,axiom,
% 16.19/8.96 ! [X: num,Y: num] :
% 16.19/8.96 ( ( X = Y )
% 16.19/8.96 => ( ord_less_eq_num @ X @ Y ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4321_order__eq__refl,axiom,
% 16.19/8.96 ! [X: nat,Y: nat] :
% 16.19/8.96 ( ( X = Y )
% 16.19/8.96 => ( ord_less_eq_nat @ X @ Y ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4322_order__eq__refl,axiom,
% 16.19/8.96 ! [X: int,Y: int] :
% 16.19/8.96 ( ( X = Y )
% 16.19/8.96 => ( ord_less_eq_int @ X @ Y ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4323_linorder__linear,axiom,
% 16.19/8.96 ! [X: rat,Y: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ X @ Y )
% 16.19/8.96 | ( ord_less_eq_rat @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4324_linorder__linear,axiom,
% 16.19/8.96 ! [X: num,Y: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ X @ Y )
% 16.19/8.96 | ( ord_less_eq_num @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4325_linorder__linear,axiom,
% 16.19/8.96 ! [X: nat,Y: nat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ X @ Y )
% 16.19/8.96 | ( ord_less_eq_nat @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4326_linorder__linear,axiom,
% 16.19/8.96 ! [X: int,Y: int] :
% 16.19/8.96 ( ( ord_less_eq_int @ X @ Y )
% 16.19/8.96 | ( ord_less_eq_int @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4327_ord__eq__le__subst,axiom,
% 16.19/8.96 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 16.19/8.96 ( ( A
% 16.19/8.96 = ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_eq_rat @ B @ C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4328_ord__eq__le__subst,axiom,
% 16.19/8.96 ! [A: num,F: rat > num,B: rat,C: rat] :
% 16.19/8.96 ( ( A
% 16.19/8.96 = ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_eq_rat @ B @ C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4329_ord__eq__le__subst,axiom,
% 16.19/8.96 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 16.19/8.96 ( ( A
% 16.19/8.96 = ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_eq_rat @ B @ C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4330_ord__eq__le__subst,axiom,
% 16.19/8.96 ! [A: int,F: rat > int,B: rat,C: rat] :
% 16.19/8.96 ( ( A
% 16.19/8.96 = ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_eq_rat @ B @ C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4331_ord__eq__le__subst,axiom,
% 16.19/8.96 ! [A: rat,F: num > rat,B: num,C: num] :
% 16.19/8.96 ( ( A
% 16.19/8.96 = ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_eq_num @ B @ C )
% 16.19/8.96 => ( ! [X4: num,Y4: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4332_ord__eq__le__subst,axiom,
% 16.19/8.96 ! [A: num,F: num > num,B: num,C: num] :
% 16.19/8.96 ( ( A
% 16.19/8.96 = ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_eq_num @ B @ C )
% 16.19/8.96 => ( ! [X4: num,Y4: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4333_ord__eq__le__subst,axiom,
% 16.19/8.96 ! [A: nat,F: num > nat,B: num,C: num] :
% 16.19/8.96 ( ( A
% 16.19/8.96 = ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_eq_num @ B @ C )
% 16.19/8.96 => ( ! [X4: num,Y4: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4334_ord__eq__le__subst,axiom,
% 16.19/8.96 ! [A: int,F: num > int,B: num,C: num] :
% 16.19/8.96 ( ( A
% 16.19/8.96 = ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_eq_num @ B @ C )
% 16.19/8.96 => ( ! [X4: num,Y4: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4335_ord__eq__le__subst,axiom,
% 16.19/8.96 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 16.19/8.96 ( ( A
% 16.19/8.96 = ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_eq_nat @ B @ C )
% 16.19/8.96 => ( ! [X4: nat,Y4: nat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4336_ord__eq__le__subst,axiom,
% 16.19/8.96 ! [A: num,F: nat > num,B: nat,C: nat] :
% 16.19/8.96 ( ( A
% 16.19/8.96 = ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_eq_nat @ B @ C )
% 16.19/8.96 => ( ! [X4: nat,Y4: nat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4337_ord__le__eq__subst,axiom,
% 16.19/8.96 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.96 => ( ( ( F @ B )
% 16.19/8.96 = C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4338_ord__le__eq__subst,axiom,
% 16.19/8.96 ! [A: rat,B: rat,F: rat > num,C: num] :
% 16.19/8.96 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.96 => ( ( ( F @ B )
% 16.19/8.96 = C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4339_ord__le__eq__subst,axiom,
% 16.19/8.96 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.96 => ( ( ( F @ B )
% 16.19/8.96 = C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4340_ord__le__eq__subst,axiom,
% 16.19/8.96 ! [A: rat,B: rat,F: rat > int,C: int] :
% 16.19/8.96 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.96 => ( ( ( F @ B )
% 16.19/8.96 = C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4341_ord__le__eq__subst,axiom,
% 16.19/8.96 ! [A: num,B: num,F: num > rat,C: rat] :
% 16.19/8.96 ( ( ord_less_eq_num @ A @ B )
% 16.19/8.96 => ( ( ( F @ B )
% 16.19/8.96 = C )
% 16.19/8.96 => ( ! [X4: num,Y4: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4342_ord__le__eq__subst,axiom,
% 16.19/8.96 ! [A: num,B: num,F: num > num,C: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ A @ B )
% 16.19/8.96 => ( ( ( F @ B )
% 16.19/8.96 = C )
% 16.19/8.96 => ( ! [X4: num,Y4: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4343_ord__le__eq__subst,axiom,
% 16.19/8.96 ! [A: num,B: num,F: num > nat,C: nat] :
% 16.19/8.96 ( ( ord_less_eq_num @ A @ B )
% 16.19/8.96 => ( ( ( F @ B )
% 16.19/8.96 = C )
% 16.19/8.96 => ( ! [X4: num,Y4: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4344_ord__le__eq__subst,axiom,
% 16.19/8.96 ! [A: num,B: num,F: num > int,C: int] :
% 16.19/8.96 ( ( ord_less_eq_num @ A @ B )
% 16.19/8.96 => ( ( ( F @ B )
% 16.19/8.96 = C )
% 16.19/8.96 => ( ! [X4: num,Y4: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4345_ord__le__eq__subst,axiom,
% 16.19/8.96 ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.96 => ( ( ( F @ B )
% 16.19/8.96 = C )
% 16.19/8.96 => ( ! [X4: nat,Y4: nat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4346_ord__le__eq__subst,axiom,
% 16.19/8.96 ! [A: nat,B: nat,F: nat > num,C: num] :
% 16.19/8.96 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.96 => ( ( ( F @ B )
% 16.19/8.96 = C )
% 16.19/8.96 => ( ! [X4: nat,Y4: nat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4347_linorder__le__cases,axiom,
% 16.19/8.96 ! [X: rat,Y: rat] :
% 16.19/8.96 ( ~ ( ord_less_eq_rat @ X @ Y )
% 16.19/8.96 => ( ord_less_eq_rat @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4348_linorder__le__cases,axiom,
% 16.19/8.96 ! [X: num,Y: num] :
% 16.19/8.96 ( ~ ( ord_less_eq_num @ X @ Y )
% 16.19/8.96 => ( ord_less_eq_num @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4349_linorder__le__cases,axiom,
% 16.19/8.96 ! [X: nat,Y: nat] :
% 16.19/8.96 ( ~ ( ord_less_eq_nat @ X @ Y )
% 16.19/8.96 => ( ord_less_eq_nat @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4350_linorder__le__cases,axiom,
% 16.19/8.96 ! [X: int,Y: int] :
% 16.19/8.96 ( ~ ( ord_less_eq_int @ X @ Y )
% 16.19/8.96 => ( ord_less_eq_int @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4351_order__antisym__conv,axiom,
% 16.19/8.96 ! [Y: set_nat,X: set_nat] :
% 16.19/8.96 ( ( ord_less_eq_set_nat @ Y @ X )
% 16.19/8.96 => ( ( ord_less_eq_set_nat @ X @ Y )
% 16.19/8.96 = ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4352_order__antisym__conv,axiom,
% 16.19/8.96 ! [Y: rat,X: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ Y @ X )
% 16.19/8.96 => ( ( ord_less_eq_rat @ X @ Y )
% 16.19/8.96 = ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4353_order__antisym__conv,axiom,
% 16.19/8.96 ! [Y: num,X: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ Y @ X )
% 16.19/8.96 => ( ( ord_less_eq_num @ X @ Y )
% 16.19/8.96 = ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4354_order__antisym__conv,axiom,
% 16.19/8.96 ! [Y: nat,X: nat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ Y @ X )
% 16.19/8.96 => ( ( ord_less_eq_nat @ X @ Y )
% 16.19/8.96 = ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4355_order__antisym__conv,axiom,
% 16.19/8.96 ! [Y: int,X: int] :
% 16.19/8.96 ( ( ord_less_eq_int @ Y @ X )
% 16.19/8.96 => ( ( ord_less_eq_int @ X @ Y )
% 16.19/8.96 = ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4356_mod__greater__zero__iff__not__dvd,axiom,
% 16.19/8.96 ! [M: nat,N2: nat] :
% 16.19/8.96 ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N2 ) )
% 16.19/8.96 = ( ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4357_lt__ex,axiom,
% 16.19/8.96 ! [X: real] :
% 16.19/8.96 ? [Y4: real] : ( ord_less_real @ Y4 @ X ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4358_lt__ex,axiom,
% 16.19/8.96 ! [X: rat] :
% 16.19/8.96 ? [Y4: rat] : ( ord_less_rat @ Y4 @ X ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4359_lt__ex,axiom,
% 16.19/8.96 ! [X: int] :
% 16.19/8.96 ? [Y4: int] : ( ord_less_int @ Y4 @ X ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4360_gt__ex,axiom,
% 16.19/8.96 ! [X: real] :
% 16.19/8.96 ? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4361_gt__ex,axiom,
% 16.19/8.96 ! [X: rat] :
% 16.19/8.96 ? [X_1: rat] : ( ord_less_rat @ X @ X_1 ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4362_gt__ex,axiom,
% 16.19/8.96 ! [X: nat] :
% 16.19/8.96 ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4363_gt__ex,axiom,
% 16.19/8.96 ! [X: int] :
% 16.19/8.96 ? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4364_dense,axiom,
% 16.19/8.96 ! [X: real,Y: real] :
% 16.19/8.96 ( ( ord_less_real @ X @ Y )
% 16.19/8.96 => ? [Z5: real] :
% 16.19/8.96 ( ( ord_less_real @ X @ Z5 )
% 16.19/8.96 & ( ord_less_real @ Z5 @ Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4365_dense,axiom,
% 16.19/8.96 ! [X: rat,Y: rat] :
% 16.19/8.96 ( ( ord_less_rat @ X @ Y )
% 16.19/8.96 => ? [Z5: rat] :
% 16.19/8.96 ( ( ord_less_rat @ X @ Z5 )
% 16.19/8.96 & ( ord_less_rat @ Z5 @ Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4366_less__imp__neq,axiom,
% 16.19/8.96 ! [X: real,Y: real] :
% 16.19/8.96 ( ( ord_less_real @ X @ Y )
% 16.19/8.96 => ( X != Y ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4367_less__imp__neq,axiom,
% 16.19/8.96 ! [X: rat,Y: rat] :
% 16.19/8.96 ( ( ord_less_rat @ X @ Y )
% 16.19/8.96 => ( X != Y ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4368_less__imp__neq,axiom,
% 16.19/8.96 ! [X: num,Y: num] :
% 16.19/8.96 ( ( ord_less_num @ X @ Y )
% 16.19/8.96 => ( X != Y ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4369_less__imp__neq,axiom,
% 16.19/8.96 ! [X: nat,Y: nat] :
% 16.19/8.96 ( ( ord_less_nat @ X @ Y )
% 16.19/8.96 => ( X != Y ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4370_less__imp__neq,axiom,
% 16.19/8.96 ! [X: int,Y: int] :
% 16.19/8.96 ( ( ord_less_int @ X @ Y )
% 16.19/8.96 => ( X != Y ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4371_order_Oasym,axiom,
% 16.19/8.96 ! [A: real,B: real] :
% 16.19/8.96 ( ( ord_less_real @ A @ B )
% 16.19/8.96 => ~ ( ord_less_real @ B @ A ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4372_order_Oasym,axiom,
% 16.19/8.96 ! [A: rat,B: rat] :
% 16.19/8.96 ( ( ord_less_rat @ A @ B )
% 16.19/8.96 => ~ ( ord_less_rat @ B @ A ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4373_order_Oasym,axiom,
% 16.19/8.96 ! [A: num,B: num] :
% 16.19/8.96 ( ( ord_less_num @ A @ B )
% 16.19/8.96 => ~ ( ord_less_num @ B @ A ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4374_order_Oasym,axiom,
% 16.19/8.96 ! [A: nat,B: nat] :
% 16.19/8.96 ( ( ord_less_nat @ A @ B )
% 16.19/8.96 => ~ ( ord_less_nat @ B @ A ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4375_order_Oasym,axiom,
% 16.19/8.96 ! [A: int,B: int] :
% 16.19/8.96 ( ( ord_less_int @ A @ B )
% 16.19/8.96 => ~ ( ord_less_int @ B @ A ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4376_ord__eq__less__trans,axiom,
% 16.19/8.96 ! [A: real,B: real,C: real] :
% 16.19/8.96 ( ( A = B )
% 16.19/8.96 => ( ( ord_less_real @ B @ C )
% 16.19/8.96 => ( ord_less_real @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4377_ord__eq__less__trans,axiom,
% 16.19/8.96 ! [A: rat,B: rat,C: rat] :
% 16.19/8.96 ( ( A = B )
% 16.19/8.96 => ( ( ord_less_rat @ B @ C )
% 16.19/8.96 => ( ord_less_rat @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4378_ord__eq__less__trans,axiom,
% 16.19/8.96 ! [A: num,B: num,C: num] :
% 16.19/8.96 ( ( A = B )
% 16.19/8.96 => ( ( ord_less_num @ B @ C )
% 16.19/8.96 => ( ord_less_num @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4379_ord__eq__less__trans,axiom,
% 16.19/8.96 ! [A: nat,B: nat,C: nat] :
% 16.19/8.96 ( ( A = B )
% 16.19/8.96 => ( ( ord_less_nat @ B @ C )
% 16.19/8.96 => ( ord_less_nat @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4380_ord__eq__less__trans,axiom,
% 16.19/8.96 ! [A: int,B: int,C: int] :
% 16.19/8.96 ( ( A = B )
% 16.19/8.96 => ( ( ord_less_int @ B @ C )
% 16.19/8.96 => ( ord_less_int @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4381_ord__less__eq__trans,axiom,
% 16.19/8.96 ! [A: real,B: real,C: real] :
% 16.19/8.96 ( ( ord_less_real @ A @ B )
% 16.19/8.96 => ( ( B = C )
% 16.19/8.96 => ( ord_less_real @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4382_ord__less__eq__trans,axiom,
% 16.19/8.96 ! [A: rat,B: rat,C: rat] :
% 16.19/8.96 ( ( ord_less_rat @ A @ B )
% 16.19/8.96 => ( ( B = C )
% 16.19/8.96 => ( ord_less_rat @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4383_ord__less__eq__trans,axiom,
% 16.19/8.96 ! [A: num,B: num,C: num] :
% 16.19/8.96 ( ( ord_less_num @ A @ B )
% 16.19/8.96 => ( ( B = C )
% 16.19/8.96 => ( ord_less_num @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4384_ord__less__eq__trans,axiom,
% 16.19/8.96 ! [A: nat,B: nat,C: nat] :
% 16.19/8.96 ( ( ord_less_nat @ A @ B )
% 16.19/8.96 => ( ( B = C )
% 16.19/8.96 => ( ord_less_nat @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4385_ord__less__eq__trans,axiom,
% 16.19/8.96 ! [A: int,B: int,C: int] :
% 16.19/8.96 ( ( ord_less_int @ A @ B )
% 16.19/8.96 => ( ( B = C )
% 16.19/8.96 => ( ord_less_int @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4386_less__induct,axiom,
% 16.19/8.96 ! [P: nat > $o,A: nat] :
% 16.19/8.96 ( ! [X4: nat] :
% 16.19/8.96 ( ! [Y5: nat] :
% 16.19/8.96 ( ( ord_less_nat @ Y5 @ X4 )
% 16.19/8.96 => ( P @ Y5 ) )
% 16.19/8.96 => ( P @ X4 ) )
% 16.19/8.96 => ( P @ A ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4387_antisym__conv3,axiom,
% 16.19/8.96 ! [Y: real,X: real] :
% 16.19/8.96 ( ~ ( ord_less_real @ Y @ X )
% 16.19/8.96 => ( ( ~ ( ord_less_real @ X @ Y ) )
% 16.19/8.96 = ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4388_antisym__conv3,axiom,
% 16.19/8.96 ! [Y: rat,X: rat] :
% 16.19/8.96 ( ~ ( ord_less_rat @ Y @ X )
% 16.19/8.96 => ( ( ~ ( ord_less_rat @ X @ Y ) )
% 16.19/8.96 = ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4389_antisym__conv3,axiom,
% 16.19/8.96 ! [Y: num,X: num] :
% 16.19/8.96 ( ~ ( ord_less_num @ Y @ X )
% 16.19/8.96 => ( ( ~ ( ord_less_num @ X @ Y ) )
% 16.19/8.96 = ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4390_antisym__conv3,axiom,
% 16.19/8.96 ! [Y: nat,X: nat] :
% 16.19/8.96 ( ~ ( ord_less_nat @ Y @ X )
% 16.19/8.96 => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 16.19/8.96 = ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4391_antisym__conv3,axiom,
% 16.19/8.96 ! [Y: int,X: int] :
% 16.19/8.96 ( ~ ( ord_less_int @ Y @ X )
% 16.19/8.96 => ( ( ~ ( ord_less_int @ X @ Y ) )
% 16.19/8.96 = ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4392_linorder__cases,axiom,
% 16.19/8.96 ! [X: real,Y: real] :
% 16.19/8.96 ( ~ ( ord_less_real @ X @ Y )
% 16.19/8.96 => ( ( X != Y )
% 16.19/8.96 => ( ord_less_real @ Y @ X ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4393_linorder__cases,axiom,
% 16.19/8.96 ! [X: rat,Y: rat] :
% 16.19/8.96 ( ~ ( ord_less_rat @ X @ Y )
% 16.19/8.96 => ( ( X != Y )
% 16.19/8.96 => ( ord_less_rat @ Y @ X ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4394_linorder__cases,axiom,
% 16.19/8.96 ! [X: num,Y: num] :
% 16.19/8.96 ( ~ ( ord_less_num @ X @ Y )
% 16.19/8.96 => ( ( X != Y )
% 16.19/8.96 => ( ord_less_num @ Y @ X ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4395_linorder__cases,axiom,
% 16.19/8.96 ! [X: nat,Y: nat] :
% 16.19/8.96 ( ~ ( ord_less_nat @ X @ Y )
% 16.19/8.96 => ( ( X != Y )
% 16.19/8.96 => ( ord_less_nat @ Y @ X ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4396_linorder__cases,axiom,
% 16.19/8.96 ! [X: int,Y: int] :
% 16.19/8.96 ( ~ ( ord_less_int @ X @ Y )
% 16.19/8.96 => ( ( X != Y )
% 16.19/8.96 => ( ord_less_int @ Y @ X ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4397_dual__order_Oasym,axiom,
% 16.19/8.96 ! [B: real,A: real] :
% 16.19/8.96 ( ( ord_less_real @ B @ A )
% 16.19/8.96 => ~ ( ord_less_real @ A @ B ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4398_dual__order_Oasym,axiom,
% 16.19/8.96 ! [B: rat,A: rat] :
% 16.19/8.96 ( ( ord_less_rat @ B @ A )
% 16.19/8.96 => ~ ( ord_less_rat @ A @ B ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4399_dual__order_Oasym,axiom,
% 16.19/8.96 ! [B: num,A: num] :
% 16.19/8.96 ( ( ord_less_num @ B @ A )
% 16.19/8.96 => ~ ( ord_less_num @ A @ B ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4400_dual__order_Oasym,axiom,
% 16.19/8.96 ! [B: nat,A: nat] :
% 16.19/8.96 ( ( ord_less_nat @ B @ A )
% 16.19/8.96 => ~ ( ord_less_nat @ A @ B ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4401_dual__order_Oasym,axiom,
% 16.19/8.96 ! [B: int,A: int] :
% 16.19/8.96 ( ( ord_less_int @ B @ A )
% 16.19/8.96 => ~ ( ord_less_int @ A @ B ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4402_dual__order_Oirrefl,axiom,
% 16.19/8.96 ! [A: real] :
% 16.19/8.96 ~ ( ord_less_real @ A @ A ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4403_dual__order_Oirrefl,axiom,
% 16.19/8.96 ! [A: rat] :
% 16.19/8.96 ~ ( ord_less_rat @ A @ A ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4404_dual__order_Oirrefl,axiom,
% 16.19/8.96 ! [A: num] :
% 16.19/8.96 ~ ( ord_less_num @ A @ A ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4405_dual__order_Oirrefl,axiom,
% 16.19/8.96 ! [A: nat] :
% 16.19/8.96 ~ ( ord_less_nat @ A @ A ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4406_dual__order_Oirrefl,axiom,
% 16.19/8.96 ! [A: int] :
% 16.19/8.96 ~ ( ord_less_int @ A @ A ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4407_exists__least__iff,axiom,
% 16.19/8.96 ( ( ^ [P2: nat > $o] :
% 16.19/8.96 ? [X7: nat] : ( P2 @ X7 ) )
% 16.19/8.96 = ( ^ [P3: nat > $o] :
% 16.19/8.96 ? [N: nat] :
% 16.19/8.96 ( ( P3 @ N )
% 16.19/8.96 & ! [M6: nat] :
% 16.19/8.96 ( ( ord_less_nat @ M6 @ N )
% 16.19/8.96 => ~ ( P3 @ M6 ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4408_linorder__less__wlog,axiom,
% 16.19/8.96 ! [P: real > real > $o,A: real,B: real] :
% 16.19/8.96 ( ! [A3: real,B2: real] :
% 16.19/8.96 ( ( ord_less_real @ A3 @ B2 )
% 16.19/8.96 => ( P @ A3 @ B2 ) )
% 16.19/8.96 => ( ! [A3: real] : ( P @ A3 @ A3 )
% 16.19/8.96 => ( ! [A3: real,B2: real] :
% 16.19/8.96 ( ( P @ B2 @ A3 )
% 16.19/8.96 => ( P @ A3 @ B2 ) )
% 16.19/8.96 => ( P @ A @ B ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4409_linorder__less__wlog,axiom,
% 16.19/8.96 ! [P: rat > rat > $o,A: rat,B: rat] :
% 16.19/8.96 ( ! [A3: rat,B2: rat] :
% 16.19/8.96 ( ( ord_less_rat @ A3 @ B2 )
% 16.19/8.96 => ( P @ A3 @ B2 ) )
% 16.19/8.96 => ( ! [A3: rat] : ( P @ A3 @ A3 )
% 16.19/8.96 => ( ! [A3: rat,B2: rat] :
% 16.19/8.96 ( ( P @ B2 @ A3 )
% 16.19/8.96 => ( P @ A3 @ B2 ) )
% 16.19/8.96 => ( P @ A @ B ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4410_linorder__less__wlog,axiom,
% 16.19/8.96 ! [P: num > num > $o,A: num,B: num] :
% 16.19/8.96 ( ! [A3: num,B2: num] :
% 16.19/8.96 ( ( ord_less_num @ A3 @ B2 )
% 16.19/8.96 => ( P @ A3 @ B2 ) )
% 16.19/8.96 => ( ! [A3: num] : ( P @ A3 @ A3 )
% 16.19/8.96 => ( ! [A3: num,B2: num] :
% 16.19/8.96 ( ( P @ B2 @ A3 )
% 16.19/8.96 => ( P @ A3 @ B2 ) )
% 16.19/8.96 => ( P @ A @ B ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4411_linorder__less__wlog,axiom,
% 16.19/8.96 ! [P: nat > nat > $o,A: nat,B: nat] :
% 16.19/8.96 ( ! [A3: nat,B2: nat] :
% 16.19/8.96 ( ( ord_less_nat @ A3 @ B2 )
% 16.19/8.96 => ( P @ A3 @ B2 ) )
% 16.19/8.96 => ( ! [A3: nat] : ( P @ A3 @ A3 )
% 16.19/8.96 => ( ! [A3: nat,B2: nat] :
% 16.19/8.96 ( ( P @ B2 @ A3 )
% 16.19/8.96 => ( P @ A3 @ B2 ) )
% 16.19/8.96 => ( P @ A @ B ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4412_linorder__less__wlog,axiom,
% 16.19/8.96 ! [P: int > int > $o,A: int,B: int] :
% 16.19/8.96 ( ! [A3: int,B2: int] :
% 16.19/8.96 ( ( ord_less_int @ A3 @ B2 )
% 16.19/8.96 => ( P @ A3 @ B2 ) )
% 16.19/8.96 => ( ! [A3: int] : ( P @ A3 @ A3 )
% 16.19/8.96 => ( ! [A3: int,B2: int] :
% 16.19/8.96 ( ( P @ B2 @ A3 )
% 16.19/8.96 => ( P @ A3 @ B2 ) )
% 16.19/8.96 => ( P @ A @ B ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4413_order_Ostrict__trans,axiom,
% 16.19/8.96 ! [A: real,B: real,C: real] :
% 16.19/8.96 ( ( ord_less_real @ A @ B )
% 16.19/8.96 => ( ( ord_less_real @ B @ C )
% 16.19/8.96 => ( ord_less_real @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4414_order_Ostrict__trans,axiom,
% 16.19/8.96 ! [A: rat,B: rat,C: rat] :
% 16.19/8.96 ( ( ord_less_rat @ A @ B )
% 16.19/8.96 => ( ( ord_less_rat @ B @ C )
% 16.19/8.96 => ( ord_less_rat @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4415_order_Ostrict__trans,axiom,
% 16.19/8.96 ! [A: num,B: num,C: num] :
% 16.19/8.96 ( ( ord_less_num @ A @ B )
% 16.19/8.96 => ( ( ord_less_num @ B @ C )
% 16.19/8.96 => ( ord_less_num @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4416_order_Ostrict__trans,axiom,
% 16.19/8.96 ! [A: nat,B: nat,C: nat] :
% 16.19/8.96 ( ( ord_less_nat @ A @ B )
% 16.19/8.96 => ( ( ord_less_nat @ B @ C )
% 16.19/8.96 => ( ord_less_nat @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4417_order_Ostrict__trans,axiom,
% 16.19/8.96 ! [A: int,B: int,C: int] :
% 16.19/8.96 ( ( ord_less_int @ A @ B )
% 16.19/8.96 => ( ( ord_less_int @ B @ C )
% 16.19/8.96 => ( ord_less_int @ A @ C ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4418_not__less__iff__gr__or__eq,axiom,
% 16.19/8.96 ! [X: real,Y: real] :
% 16.19/8.96 ( ( ~ ( ord_less_real @ X @ Y ) )
% 16.19/8.96 = ( ( ord_less_real @ Y @ X )
% 16.19/8.96 | ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4419_not__less__iff__gr__or__eq,axiom,
% 16.19/8.96 ! [X: rat,Y: rat] :
% 16.19/8.96 ( ( ~ ( ord_less_rat @ X @ Y ) )
% 16.19/8.96 = ( ( ord_less_rat @ Y @ X )
% 16.19/8.96 | ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4420_not__less__iff__gr__or__eq,axiom,
% 16.19/8.96 ! [X: num,Y: num] :
% 16.19/8.96 ( ( ~ ( ord_less_num @ X @ Y ) )
% 16.19/8.96 = ( ( ord_less_num @ Y @ X )
% 16.19/8.96 | ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4421_not__less__iff__gr__or__eq,axiom,
% 16.19/8.96 ! [X: nat,Y: nat] :
% 16.19/8.96 ( ( ~ ( ord_less_nat @ X @ Y ) )
% 16.19/8.96 = ( ( ord_less_nat @ Y @ X )
% 16.19/8.96 | ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4422_not__less__iff__gr__or__eq,axiom,
% 16.19/8.96 ! [X: int,Y: int] :
% 16.19/8.96 ( ( ~ ( ord_less_int @ X @ Y ) )
% 16.19/8.96 = ( ( ord_less_int @ Y @ X )
% 16.19/8.96 | ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4423_dual__order_Ostrict__trans,axiom,
% 16.19/8.96 ! [B: real,A: real,C: real] :
% 16.19/8.96 ( ( ord_less_real @ B @ A )
% 16.19/8.96 => ( ( ord_less_real @ C @ B )
% 16.19/8.96 => ( ord_less_real @ C @ A ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4424_dual__order_Ostrict__trans,axiom,
% 16.19/8.96 ! [B: rat,A: rat,C: rat] :
% 16.19/8.96 ( ( ord_less_rat @ B @ A )
% 16.19/8.96 => ( ( ord_less_rat @ C @ B )
% 16.19/8.96 => ( ord_less_rat @ C @ A ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4425_dual__order_Ostrict__trans,axiom,
% 16.19/8.96 ! [B: num,A: num,C: num] :
% 16.19/8.96 ( ( ord_less_num @ B @ A )
% 16.19/8.96 => ( ( ord_less_num @ C @ B )
% 16.19/8.96 => ( ord_less_num @ C @ A ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4426_dual__order_Ostrict__trans,axiom,
% 16.19/8.96 ! [B: nat,A: nat,C: nat] :
% 16.19/8.96 ( ( ord_less_nat @ B @ A )
% 16.19/8.96 => ( ( ord_less_nat @ C @ B )
% 16.19/8.96 => ( ord_less_nat @ C @ A ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4427_dual__order_Ostrict__trans,axiom,
% 16.19/8.96 ! [B: int,A: int,C: int] :
% 16.19/8.96 ( ( ord_less_int @ B @ A )
% 16.19/8.96 => ( ( ord_less_int @ C @ B )
% 16.19/8.96 => ( ord_less_int @ C @ A ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4428_order_Ostrict__implies__not__eq,axiom,
% 16.19/8.96 ! [A: real,B: real] :
% 16.19/8.96 ( ( ord_less_real @ A @ B )
% 16.19/8.96 => ( A != B ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4429_order_Ostrict__implies__not__eq,axiom,
% 16.19/8.96 ! [A: rat,B: rat] :
% 16.19/8.96 ( ( ord_less_rat @ A @ B )
% 16.19/8.96 => ( A != B ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4430_order_Ostrict__implies__not__eq,axiom,
% 16.19/8.96 ! [A: num,B: num] :
% 16.19/8.96 ( ( ord_less_num @ A @ B )
% 16.19/8.96 => ( A != B ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4431_order_Ostrict__implies__not__eq,axiom,
% 16.19/8.96 ! [A: nat,B: nat] :
% 16.19/8.96 ( ( ord_less_nat @ A @ B )
% 16.19/8.96 => ( A != B ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4432_order_Ostrict__implies__not__eq,axiom,
% 16.19/8.96 ! [A: int,B: int] :
% 16.19/8.96 ( ( ord_less_int @ A @ B )
% 16.19/8.96 => ( A != B ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4433_dual__order_Ostrict__implies__not__eq,axiom,
% 16.19/8.96 ! [B: real,A: real] :
% 16.19/8.96 ( ( ord_less_real @ B @ A )
% 16.19/8.96 => ( A != B ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4434_dual__order_Ostrict__implies__not__eq,axiom,
% 16.19/8.96 ! [B: rat,A: rat] :
% 16.19/8.96 ( ( ord_less_rat @ B @ A )
% 16.19/8.96 => ( A != B ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4435_dual__order_Ostrict__implies__not__eq,axiom,
% 16.19/8.96 ! [B: num,A: num] :
% 16.19/8.96 ( ( ord_less_num @ B @ A )
% 16.19/8.96 => ( A != B ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4436_dual__order_Ostrict__implies__not__eq,axiom,
% 16.19/8.96 ! [B: nat,A: nat] :
% 16.19/8.96 ( ( ord_less_nat @ B @ A )
% 16.19/8.96 => ( A != B ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4437_dual__order_Ostrict__implies__not__eq,axiom,
% 16.19/8.96 ! [B: int,A: int] :
% 16.19/8.96 ( ( ord_less_int @ B @ A )
% 16.19/8.96 => ( A != B ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4438_linorder__neqE,axiom,
% 16.19/8.96 ! [X: real,Y: real] :
% 16.19/8.96 ( ( X != Y )
% 16.19/8.96 => ( ~ ( ord_less_real @ X @ Y )
% 16.19/8.96 => ( ord_less_real @ Y @ X ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4439_linorder__neqE,axiom,
% 16.19/8.96 ! [X: rat,Y: rat] :
% 16.19/8.96 ( ( X != Y )
% 16.19/8.96 => ( ~ ( ord_less_rat @ X @ Y )
% 16.19/8.96 => ( ord_less_rat @ Y @ X ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4440_linorder__neqE,axiom,
% 16.19/8.96 ! [X: num,Y: num] :
% 16.19/8.96 ( ( X != Y )
% 16.19/8.96 => ( ~ ( ord_less_num @ X @ Y )
% 16.19/8.96 => ( ord_less_num @ Y @ X ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4441_linorder__neqE,axiom,
% 16.19/8.96 ! [X: nat,Y: nat] :
% 16.19/8.96 ( ( X != Y )
% 16.19/8.96 => ( ~ ( ord_less_nat @ X @ Y )
% 16.19/8.96 => ( ord_less_nat @ Y @ X ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4442_linorder__neqE,axiom,
% 16.19/8.96 ! [X: int,Y: int] :
% 16.19/8.96 ( ( X != Y )
% 16.19/8.96 => ( ~ ( ord_less_int @ X @ Y )
% 16.19/8.96 => ( ord_less_int @ Y @ X ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4443_order__less__asym,axiom,
% 16.19/8.96 ! [X: real,Y: real] :
% 16.19/8.96 ( ( ord_less_real @ X @ Y )
% 16.19/8.96 => ~ ( ord_less_real @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4444_order__less__asym,axiom,
% 16.19/8.96 ! [X: rat,Y: rat] :
% 16.19/8.96 ( ( ord_less_rat @ X @ Y )
% 16.19/8.96 => ~ ( ord_less_rat @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4445_order__less__asym,axiom,
% 16.19/8.96 ! [X: num,Y: num] :
% 16.19/8.96 ( ( ord_less_num @ X @ Y )
% 16.19/8.96 => ~ ( ord_less_num @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4446_order__less__asym,axiom,
% 16.19/8.96 ! [X: nat,Y: nat] :
% 16.19/8.96 ( ( ord_less_nat @ X @ Y )
% 16.19/8.96 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4447_order__less__asym,axiom,
% 16.19/8.96 ! [X: int,Y: int] :
% 16.19/8.96 ( ( ord_less_int @ X @ Y )
% 16.19/8.96 => ~ ( ord_less_int @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4448_linorder__neq__iff,axiom,
% 16.19/8.96 ! [X: real,Y: real] :
% 16.19/8.96 ( ( X != Y )
% 16.19/8.96 = ( ( ord_less_real @ X @ Y )
% 16.19/8.96 | ( ord_less_real @ Y @ X ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4449_linorder__neq__iff,axiom,
% 16.19/8.96 ! [X: rat,Y: rat] :
% 16.19/8.96 ( ( X != Y )
% 16.19/8.96 = ( ( ord_less_rat @ X @ Y )
% 16.19/8.96 | ( ord_less_rat @ Y @ X ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4450_linorder__neq__iff,axiom,
% 16.19/8.96 ! [X: num,Y: num] :
% 16.19/8.96 ( ( X != Y )
% 16.19/8.96 = ( ( ord_less_num @ X @ Y )
% 16.19/8.96 | ( ord_less_num @ Y @ X ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4451_linorder__neq__iff,axiom,
% 16.19/8.96 ! [X: nat,Y: nat] :
% 16.19/8.96 ( ( X != Y )
% 16.19/8.96 = ( ( ord_less_nat @ X @ Y )
% 16.19/8.96 | ( ord_less_nat @ Y @ X ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4452_linorder__neq__iff,axiom,
% 16.19/8.96 ! [X: int,Y: int] :
% 16.19/8.96 ( ( X != Y )
% 16.19/8.96 = ( ( ord_less_int @ X @ Y )
% 16.19/8.96 | ( ord_less_int @ Y @ X ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4453_order__less__asym_H,axiom,
% 16.19/8.96 ! [A: real,B: real] :
% 16.19/8.96 ( ( ord_less_real @ A @ B )
% 16.19/8.96 => ~ ( ord_less_real @ B @ A ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4454_order__less__asym_H,axiom,
% 16.19/8.96 ! [A: rat,B: rat] :
% 16.19/8.96 ( ( ord_less_rat @ A @ B )
% 16.19/8.96 => ~ ( ord_less_rat @ B @ A ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4455_order__less__asym_H,axiom,
% 16.19/8.96 ! [A: num,B: num] :
% 16.19/8.96 ( ( ord_less_num @ A @ B )
% 16.19/8.96 => ~ ( ord_less_num @ B @ A ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4456_order__less__asym_H,axiom,
% 16.19/8.96 ! [A: nat,B: nat] :
% 16.19/8.96 ( ( ord_less_nat @ A @ B )
% 16.19/8.96 => ~ ( ord_less_nat @ B @ A ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4457_order__less__asym_H,axiom,
% 16.19/8.96 ! [A: int,B: int] :
% 16.19/8.96 ( ( ord_less_int @ A @ B )
% 16.19/8.96 => ~ ( ord_less_int @ B @ A ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4458_order__less__trans,axiom,
% 16.19/8.96 ! [X: real,Y: real,Z: real] :
% 16.19/8.96 ( ( ord_less_real @ X @ Y )
% 16.19/8.96 => ( ( ord_less_real @ Y @ Z )
% 16.19/8.96 => ( ord_less_real @ X @ Z ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4459_order__less__trans,axiom,
% 16.19/8.96 ! [X: rat,Y: rat,Z: rat] :
% 16.19/8.96 ( ( ord_less_rat @ X @ Y )
% 16.19/8.96 => ( ( ord_less_rat @ Y @ Z )
% 16.19/8.96 => ( ord_less_rat @ X @ Z ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4460_order__less__trans,axiom,
% 16.19/8.96 ! [X: num,Y: num,Z: num] :
% 16.19/8.96 ( ( ord_less_num @ X @ Y )
% 16.19/8.96 => ( ( ord_less_num @ Y @ Z )
% 16.19/8.96 => ( ord_less_num @ X @ Z ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4461_order__less__trans,axiom,
% 16.19/8.96 ! [X: nat,Y: nat,Z: nat] :
% 16.19/8.96 ( ( ord_less_nat @ X @ Y )
% 16.19/8.96 => ( ( ord_less_nat @ Y @ Z )
% 16.19/8.96 => ( ord_less_nat @ X @ Z ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4462_order__less__trans,axiom,
% 16.19/8.96 ! [X: int,Y: int,Z: int] :
% 16.19/8.96 ( ( ord_less_int @ X @ Y )
% 16.19/8.96 => ( ( ord_less_int @ Y @ Z )
% 16.19/8.96 => ( ord_less_int @ X @ Z ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4463_ord__eq__less__subst,axiom,
% 16.19/8.96 ! [A: real,F: real > real,B: real,C: real] :
% 16.19/8.96 ( ( A
% 16.19/8.96 = ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_real @ B @ C )
% 16.19/8.96 => ( ! [X4: real,Y4: real] :
% 16.19/8.96 ( ( ord_less_real @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4464_ord__eq__less__subst,axiom,
% 16.19/8.96 ! [A: rat,F: real > rat,B: real,C: real] :
% 16.19/8.96 ( ( A
% 16.19/8.96 = ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_real @ B @ C )
% 16.19/8.96 => ( ! [X4: real,Y4: real] :
% 16.19/8.96 ( ( ord_less_real @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4465_ord__eq__less__subst,axiom,
% 16.19/8.96 ! [A: num,F: real > num,B: real,C: real] :
% 16.19/8.96 ( ( A
% 16.19/8.96 = ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_real @ B @ C )
% 16.19/8.96 => ( ! [X4: real,Y4: real] :
% 16.19/8.96 ( ( ord_less_real @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4466_ord__eq__less__subst,axiom,
% 16.19/8.96 ! [A: nat,F: real > nat,B: real,C: real] :
% 16.19/8.96 ( ( A
% 16.19/8.96 = ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_real @ B @ C )
% 16.19/8.96 => ( ! [X4: real,Y4: real] :
% 16.19/8.96 ( ( ord_less_real @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4467_ord__eq__less__subst,axiom,
% 16.19/8.96 ! [A: int,F: real > int,B: real,C: real] :
% 16.19/8.96 ( ( A
% 16.19/8.96 = ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_real @ B @ C )
% 16.19/8.96 => ( ! [X4: real,Y4: real] :
% 16.19/8.96 ( ( ord_less_real @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4468_ord__eq__less__subst,axiom,
% 16.19/8.96 ! [A: real,F: rat > real,B: rat,C: rat] :
% 16.19/8.96 ( ( A
% 16.19/8.96 = ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_rat @ B @ C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4469_ord__eq__less__subst,axiom,
% 16.19/8.96 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 16.19/8.96 ( ( A
% 16.19/8.96 = ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_rat @ B @ C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4470_ord__eq__less__subst,axiom,
% 16.19/8.96 ! [A: num,F: rat > num,B: rat,C: rat] :
% 16.19/8.96 ( ( A
% 16.19/8.96 = ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_rat @ B @ C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4471_ord__eq__less__subst,axiom,
% 16.19/8.96 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 16.19/8.96 ( ( A
% 16.19/8.96 = ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_rat @ B @ C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4472_ord__eq__less__subst,axiom,
% 16.19/8.96 ! [A: int,F: rat > int,B: rat,C: rat] :
% 16.19/8.96 ( ( A
% 16.19/8.96 = ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_rat @ B @ C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4473_ord__less__eq__subst,axiom,
% 16.19/8.96 ! [A: real,B: real,F: real > real,C: real] :
% 16.19/8.96 ( ( ord_less_real @ A @ B )
% 16.19/8.96 => ( ( ( F @ B )
% 16.19/8.96 = C )
% 16.19/8.96 => ( ! [X4: real,Y4: real] :
% 16.19/8.96 ( ( ord_less_real @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4474_ord__less__eq__subst,axiom,
% 16.19/8.96 ! [A: real,B: real,F: real > rat,C: rat] :
% 16.19/8.96 ( ( ord_less_real @ A @ B )
% 16.19/8.96 => ( ( ( F @ B )
% 16.19/8.96 = C )
% 16.19/8.96 => ( ! [X4: real,Y4: real] :
% 16.19/8.96 ( ( ord_less_real @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4475_ord__less__eq__subst,axiom,
% 16.19/8.96 ! [A: real,B: real,F: real > num,C: num] :
% 16.19/8.96 ( ( ord_less_real @ A @ B )
% 16.19/8.96 => ( ( ( F @ B )
% 16.19/8.96 = C )
% 16.19/8.96 => ( ! [X4: real,Y4: real] :
% 16.19/8.96 ( ( ord_less_real @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4476_ord__less__eq__subst,axiom,
% 16.19/8.96 ! [A: real,B: real,F: real > nat,C: nat] :
% 16.19/8.96 ( ( ord_less_real @ A @ B )
% 16.19/8.96 => ( ( ( F @ B )
% 16.19/8.96 = C )
% 16.19/8.96 => ( ! [X4: real,Y4: real] :
% 16.19/8.96 ( ( ord_less_real @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4477_ord__less__eq__subst,axiom,
% 16.19/8.96 ! [A: real,B: real,F: real > int,C: int] :
% 16.19/8.96 ( ( ord_less_real @ A @ B )
% 16.19/8.96 => ( ( ( F @ B )
% 16.19/8.96 = C )
% 16.19/8.96 => ( ! [X4: real,Y4: real] :
% 16.19/8.96 ( ( ord_less_real @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4478_ord__less__eq__subst,axiom,
% 16.19/8.96 ! [A: rat,B: rat,F: rat > real,C: real] :
% 16.19/8.96 ( ( ord_less_rat @ A @ B )
% 16.19/8.96 => ( ( ( F @ B )
% 16.19/8.96 = C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4479_ord__less__eq__subst,axiom,
% 16.19/8.96 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 16.19/8.96 ( ( ord_less_rat @ A @ B )
% 16.19/8.96 => ( ( ( F @ B )
% 16.19/8.96 = C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4480_ord__less__eq__subst,axiom,
% 16.19/8.96 ! [A: rat,B: rat,F: rat > num,C: num] :
% 16.19/8.96 ( ( ord_less_rat @ A @ B )
% 16.19/8.96 => ( ( ( F @ B )
% 16.19/8.96 = C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4481_ord__less__eq__subst,axiom,
% 16.19/8.96 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 16.19/8.96 ( ( ord_less_rat @ A @ B )
% 16.19/8.96 => ( ( ( F @ B )
% 16.19/8.96 = C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4482_ord__less__eq__subst,axiom,
% 16.19/8.96 ! [A: rat,B: rat,F: rat > int,C: int] :
% 16.19/8.96 ( ( ord_less_rat @ A @ B )
% 16.19/8.96 => ( ( ( F @ B )
% 16.19/8.96 = C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4483_order__less__irrefl,axiom,
% 16.19/8.96 ! [X: real] :
% 16.19/8.96 ~ ( ord_less_real @ X @ X ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4484_order__less__irrefl,axiom,
% 16.19/8.96 ! [X: rat] :
% 16.19/8.96 ~ ( ord_less_rat @ X @ X ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4485_order__less__irrefl,axiom,
% 16.19/8.96 ! [X: num] :
% 16.19/8.96 ~ ( ord_less_num @ X @ X ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4486_order__less__irrefl,axiom,
% 16.19/8.96 ! [X: nat] :
% 16.19/8.96 ~ ( ord_less_nat @ X @ X ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4487_order__less__irrefl,axiom,
% 16.19/8.96 ! [X: int] :
% 16.19/8.96 ~ ( ord_less_int @ X @ X ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4488_order__less__subst1,axiom,
% 16.19/8.96 ! [A: real,F: real > real,B: real,C: real] :
% 16.19/8.96 ( ( ord_less_real @ A @ ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_real @ B @ C )
% 16.19/8.96 => ( ! [X4: real,Y4: real] :
% 16.19/8.96 ( ( ord_less_real @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4489_order__less__subst1,axiom,
% 16.19/8.96 ! [A: real,F: rat > real,B: rat,C: rat] :
% 16.19/8.96 ( ( ord_less_real @ A @ ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_rat @ B @ C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4490_order__less__subst1,axiom,
% 16.19/8.96 ! [A: real,F: num > real,B: num,C: num] :
% 16.19/8.96 ( ( ord_less_real @ A @ ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_num @ B @ C )
% 16.19/8.96 => ( ! [X4: num,Y4: num] :
% 16.19/8.96 ( ( ord_less_num @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4491_order__less__subst1,axiom,
% 16.19/8.96 ! [A: real,F: nat > real,B: nat,C: nat] :
% 16.19/8.96 ( ( ord_less_real @ A @ ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_nat @ B @ C )
% 16.19/8.96 => ( ! [X4: nat,Y4: nat] :
% 16.19/8.96 ( ( ord_less_nat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4492_order__less__subst1,axiom,
% 16.19/8.96 ! [A: real,F: int > real,B: int,C: int] :
% 16.19/8.96 ( ( ord_less_real @ A @ ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_int @ B @ C )
% 16.19/8.96 => ( ! [X4: int,Y4: int] :
% 16.19/8.96 ( ( ord_less_int @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4493_order__less__subst1,axiom,
% 16.19/8.96 ! [A: rat,F: real > rat,B: real,C: real] :
% 16.19/8.96 ( ( ord_less_rat @ A @ ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_real @ B @ C )
% 16.19/8.96 => ( ! [X4: real,Y4: real] :
% 16.19/8.96 ( ( ord_less_real @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4494_order__less__subst1,axiom,
% 16.19/8.96 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 16.19/8.96 ( ( ord_less_rat @ A @ ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_rat @ B @ C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4495_order__less__subst1,axiom,
% 16.19/8.96 ! [A: rat,F: num > rat,B: num,C: num] :
% 16.19/8.96 ( ( ord_less_rat @ A @ ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_num @ B @ C )
% 16.19/8.96 => ( ! [X4: num,Y4: num] :
% 16.19/8.96 ( ( ord_less_num @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4496_order__less__subst1,axiom,
% 16.19/8.96 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 16.19/8.96 ( ( ord_less_rat @ A @ ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_nat @ B @ C )
% 16.19/8.96 => ( ! [X4: nat,Y4: nat] :
% 16.19/8.96 ( ( ord_less_nat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4497_order__less__subst1,axiom,
% 16.19/8.96 ! [A: rat,F: int > rat,B: int,C: int] :
% 16.19/8.96 ( ( ord_less_rat @ A @ ( F @ B ) )
% 16.19/8.96 => ( ( ord_less_int @ B @ C )
% 16.19/8.96 => ( ! [X4: int,Y4: int] :
% 16.19/8.96 ( ( ord_less_int @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4498_order__less__subst2,axiom,
% 16.19/8.96 ! [A: real,B: real,F: real > real,C: real] :
% 16.19/8.96 ( ( ord_less_real @ A @ B )
% 16.19/8.96 => ( ( ord_less_real @ ( F @ B ) @ C )
% 16.19/8.96 => ( ! [X4: real,Y4: real] :
% 16.19/8.96 ( ( ord_less_real @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4499_order__less__subst2,axiom,
% 16.19/8.96 ! [A: real,B: real,F: real > rat,C: rat] :
% 16.19/8.96 ( ( ord_less_real @ A @ B )
% 16.19/8.96 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 16.19/8.96 => ( ! [X4: real,Y4: real] :
% 16.19/8.96 ( ( ord_less_real @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4500_order__less__subst2,axiom,
% 16.19/8.96 ! [A: real,B: real,F: real > num,C: num] :
% 16.19/8.96 ( ( ord_less_real @ A @ B )
% 16.19/8.96 => ( ( ord_less_num @ ( F @ B ) @ C )
% 16.19/8.96 => ( ! [X4: real,Y4: real] :
% 16.19/8.96 ( ( ord_less_real @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4501_order__less__subst2,axiom,
% 16.19/8.96 ! [A: real,B: real,F: real > nat,C: nat] :
% 16.19/8.96 ( ( ord_less_real @ A @ B )
% 16.19/8.96 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 16.19/8.96 => ( ! [X4: real,Y4: real] :
% 16.19/8.96 ( ( ord_less_real @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4502_order__less__subst2,axiom,
% 16.19/8.96 ! [A: real,B: real,F: real > int,C: int] :
% 16.19/8.96 ( ( ord_less_real @ A @ B )
% 16.19/8.96 => ( ( ord_less_int @ ( F @ B ) @ C )
% 16.19/8.96 => ( ! [X4: real,Y4: real] :
% 16.19/8.96 ( ( ord_less_real @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4503_order__less__subst2,axiom,
% 16.19/8.96 ! [A: rat,B: rat,F: rat > real,C: real] :
% 16.19/8.96 ( ( ord_less_rat @ A @ B )
% 16.19/8.96 => ( ( ord_less_real @ ( F @ B ) @ C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4504_order__less__subst2,axiom,
% 16.19/8.96 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 16.19/8.96 ( ( ord_less_rat @ A @ B )
% 16.19/8.96 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4505_order__less__subst2,axiom,
% 16.19/8.96 ! [A: rat,B: rat,F: rat > num,C: num] :
% 16.19/8.96 ( ( ord_less_rat @ A @ B )
% 16.19/8.96 => ( ( ord_less_num @ ( F @ B ) @ C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4506_order__less__subst2,axiom,
% 16.19/8.96 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 16.19/8.96 ( ( ord_less_rat @ A @ B )
% 16.19/8.96 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4507_order__less__subst2,axiom,
% 16.19/8.96 ! [A: rat,B: rat,F: rat > int,C: int] :
% 16.19/8.96 ( ( ord_less_rat @ A @ B )
% 16.19/8.96 => ( ( ord_less_int @ ( F @ B ) @ C )
% 16.19/8.96 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.96 ( ( ord_less_rat @ X4 @ Y4 )
% 16.19/8.96 => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.96 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4508_order__less__not__sym,axiom,
% 16.19/8.96 ! [X: real,Y: real] :
% 16.19/8.96 ( ( ord_less_real @ X @ Y )
% 16.19/8.96 => ~ ( ord_less_real @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4509_order__less__not__sym,axiom,
% 16.19/8.96 ! [X: rat,Y: rat] :
% 16.19/8.96 ( ( ord_less_rat @ X @ Y )
% 16.19/8.96 => ~ ( ord_less_rat @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4510_order__less__not__sym,axiom,
% 16.19/8.96 ! [X: num,Y: num] :
% 16.19/8.96 ( ( ord_less_num @ X @ Y )
% 16.19/8.96 => ~ ( ord_less_num @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4511_order__less__not__sym,axiom,
% 16.19/8.96 ! [X: nat,Y: nat] :
% 16.19/8.96 ( ( ord_less_nat @ X @ Y )
% 16.19/8.96 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4512_order__less__not__sym,axiom,
% 16.19/8.96 ! [X: int,Y: int] :
% 16.19/8.96 ( ( ord_less_int @ X @ Y )
% 16.19/8.96 => ~ ( ord_less_int @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4513_order__less__imp__triv,axiom,
% 16.19/8.96 ! [X: real,Y: real,P: $o] :
% 16.19/8.96 ( ( ord_less_real @ X @ Y )
% 16.19/8.96 => ( ( ord_less_real @ Y @ X )
% 16.19/8.96 => P ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4514_order__less__imp__triv,axiom,
% 16.19/8.96 ! [X: rat,Y: rat,P: $o] :
% 16.19/8.96 ( ( ord_less_rat @ X @ Y )
% 16.19/8.96 => ( ( ord_less_rat @ Y @ X )
% 16.19/8.96 => P ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4515_order__less__imp__triv,axiom,
% 16.19/8.96 ! [X: num,Y: num,P: $o] :
% 16.19/8.96 ( ( ord_less_num @ X @ Y )
% 16.19/8.96 => ( ( ord_less_num @ Y @ X )
% 16.19/8.96 => P ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4516_order__less__imp__triv,axiom,
% 16.19/8.96 ! [X: nat,Y: nat,P: $o] :
% 16.19/8.96 ( ( ord_less_nat @ X @ Y )
% 16.19/8.96 => ( ( ord_less_nat @ Y @ X )
% 16.19/8.96 => P ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4517_order__less__imp__triv,axiom,
% 16.19/8.96 ! [X: int,Y: int,P: $o] :
% 16.19/8.96 ( ( ord_less_int @ X @ Y )
% 16.19/8.96 => ( ( ord_less_int @ Y @ X )
% 16.19/8.96 => P ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4518_linorder__less__linear,axiom,
% 16.19/8.96 ! [X: real,Y: real] :
% 16.19/8.96 ( ( ord_less_real @ X @ Y )
% 16.19/8.96 | ( X = Y )
% 16.19/8.96 | ( ord_less_real @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4519_linorder__less__linear,axiom,
% 16.19/8.96 ! [X: rat,Y: rat] :
% 16.19/8.96 ( ( ord_less_rat @ X @ Y )
% 16.19/8.96 | ( X = Y )
% 16.19/8.96 | ( ord_less_rat @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4520_linorder__less__linear,axiom,
% 16.19/8.96 ! [X: num,Y: num] :
% 16.19/8.96 ( ( ord_less_num @ X @ Y )
% 16.19/8.96 | ( X = Y )
% 16.19/8.96 | ( ord_less_num @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4521_linorder__less__linear,axiom,
% 16.19/8.96 ! [X: nat,Y: nat] :
% 16.19/8.96 ( ( ord_less_nat @ X @ Y )
% 16.19/8.96 | ( X = Y )
% 16.19/8.96 | ( ord_less_nat @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4522_linorder__less__linear,axiom,
% 16.19/8.96 ! [X: int,Y: int] :
% 16.19/8.96 ( ( ord_less_int @ X @ Y )
% 16.19/8.96 | ( X = Y )
% 16.19/8.96 | ( ord_less_int @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4523_order__less__imp__not__eq,axiom,
% 16.19/8.96 ! [X: real,Y: real] :
% 16.19/8.96 ( ( ord_less_real @ X @ Y )
% 16.19/8.96 => ( X != Y ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4524_order__less__imp__not__eq,axiom,
% 16.19/8.96 ! [X: rat,Y: rat] :
% 16.19/8.96 ( ( ord_less_rat @ X @ Y )
% 16.19/8.96 => ( X != Y ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4525_order__less__imp__not__eq,axiom,
% 16.19/8.96 ! [X: num,Y: num] :
% 16.19/8.96 ( ( ord_less_num @ X @ Y )
% 16.19/8.96 => ( X != Y ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4526_order__less__imp__not__eq,axiom,
% 16.19/8.96 ! [X: nat,Y: nat] :
% 16.19/8.96 ( ( ord_less_nat @ X @ Y )
% 16.19/8.96 => ( X != Y ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4527_order__less__imp__not__eq,axiom,
% 16.19/8.96 ! [X: int,Y: int] :
% 16.19/8.96 ( ( ord_less_int @ X @ Y )
% 16.19/8.96 => ( X != Y ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4528_order__less__imp__not__eq2,axiom,
% 16.19/8.96 ! [X: real,Y: real] :
% 16.19/8.96 ( ( ord_less_real @ X @ Y )
% 16.19/8.96 => ( Y != X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4529_order__less__imp__not__eq2,axiom,
% 16.19/8.96 ! [X: rat,Y: rat] :
% 16.19/8.96 ( ( ord_less_rat @ X @ Y )
% 16.19/8.96 => ( Y != X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4530_order__less__imp__not__eq2,axiom,
% 16.19/8.96 ! [X: num,Y: num] :
% 16.19/8.96 ( ( ord_less_num @ X @ Y )
% 16.19/8.96 => ( Y != X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4531_order__less__imp__not__eq2,axiom,
% 16.19/8.96 ! [X: nat,Y: nat] :
% 16.19/8.96 ( ( ord_less_nat @ X @ Y )
% 16.19/8.96 => ( Y != X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4532_order__less__imp__not__eq2,axiom,
% 16.19/8.96 ! [X: int,Y: int] :
% 16.19/8.96 ( ( ord_less_int @ X @ Y )
% 16.19/8.96 => ( Y != X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4533_order__less__imp__not__less,axiom,
% 16.19/8.96 ! [X: real,Y: real] :
% 16.19/8.96 ( ( ord_less_real @ X @ Y )
% 16.19/8.96 => ~ ( ord_less_real @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4534_order__less__imp__not__less,axiom,
% 16.19/8.96 ! [X: rat,Y: rat] :
% 16.19/8.96 ( ( ord_less_rat @ X @ Y )
% 16.19/8.96 => ~ ( ord_less_rat @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4535_order__less__imp__not__less,axiom,
% 16.19/8.96 ! [X: num,Y: num] :
% 16.19/8.96 ( ( ord_less_num @ X @ Y )
% 16.19/8.96 => ~ ( ord_less_num @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4536_order__less__imp__not__less,axiom,
% 16.19/8.96 ! [X: nat,Y: nat] :
% 16.19/8.96 ( ( ord_less_nat @ X @ Y )
% 16.19/8.96 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4537_order__less__imp__not__less,axiom,
% 16.19/8.96 ! [X: int,Y: int] :
% 16.19/8.96 ( ( ord_less_int @ X @ Y )
% 16.19/8.96 => ~ ( ord_less_int @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4538_mod__eq__dvd__iff__nat,axiom,
% 16.19/8.96 ! [N2: nat,M: nat,Q2: nat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/8.96 => ( ( ( modulo_modulo_nat @ M @ Q2 )
% 16.19/8.96 = ( modulo_modulo_nat @ N2 @ Q2 ) )
% 16.19/8.96 = ( dvd_dvd_nat @ Q2 @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4539_prod__decode__aux_Ocases,axiom,
% 16.19/8.96 ! [X: product_prod_nat_nat] :
% 16.19/8.96 ~ ! [K2: nat,M2: nat] :
% 16.19/8.96 ( X
% 16.19/8.96 != ( product_Pair_nat_nat @ K2 @ M2 ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4540_finite__divisors__nat,axiom,
% 16.19/8.96 ! [M: nat] :
% 16.19/8.96 ( ( ord_less_nat @ zero_zero_nat @ M )
% 16.19/8.96 => ( finite_finite_nat
% 16.19/8.96 @ ( collect_nat
% 16.19/8.96 @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4541_even__zero,axiom,
% 16.19/8.96 dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4542_even__zero,axiom,
% 16.19/8.96 dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4543_even__zero,axiom,
% 16.19/8.96 dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4544_evenE,axiom,
% 16.19/8.96 ! [A: code_integer] :
% 16.19/8.96 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 => ~ ! [B2: code_integer] :
% 16.19/8.96 ( A
% 16.19/8.96 != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4545_evenE,axiom,
% 16.19/8.96 ! [A: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 => ~ ! [B2: nat] :
% 16.19/8.96 ( A
% 16.19/8.96 != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4546_evenE,axiom,
% 16.19/8.96 ! [A: int] :
% 16.19/8.96 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 => ~ ! [B2: int] :
% 16.19/8.96 ( A
% 16.19/8.96 != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4547_is__unit__div__mult__cancel__right,axiom,
% 16.19/8.96 ! [A: code_integer,B: code_integer] :
% 16.19/8.96 ( ( A != zero_z3403309356797280102nteger )
% 16.19/8.96 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 16.19/8.96 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ A ) )
% 16.19/8.96 = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4548_is__unit__div__mult__cancel__right,axiom,
% 16.19/8.96 ! [A: nat,B: nat] :
% 16.19/8.96 ( ( A != zero_zero_nat )
% 16.19/8.96 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 16.19/8.96 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
% 16.19/8.96 = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4549_is__unit__div__mult__cancel__right,axiom,
% 16.19/8.96 ! [A: int,B: int] :
% 16.19/8.96 ( ( A != zero_zero_int )
% 16.19/8.96 => ( ( dvd_dvd_int @ B @ one_one_int )
% 16.19/8.96 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
% 16.19/8.96 = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4550_is__unit__div__mult__cancel__left,axiom,
% 16.19/8.96 ! [A: code_integer,B: code_integer] :
% 16.19/8.96 ( ( A != zero_z3403309356797280102nteger )
% 16.19/8.96 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 16.19/8.96 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B ) )
% 16.19/8.96 = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4551_is__unit__div__mult__cancel__left,axiom,
% 16.19/8.96 ! [A: nat,B: nat] :
% 16.19/8.96 ( ( A != zero_zero_nat )
% 16.19/8.96 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 16.19/8.96 => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
% 16.19/8.96 = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4552_is__unit__div__mult__cancel__left,axiom,
% 16.19/8.96 ! [A: int,B: int] :
% 16.19/8.96 ( ( A != zero_zero_int )
% 16.19/8.96 => ( ( dvd_dvd_int @ B @ one_one_int )
% 16.19/8.96 => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
% 16.19/8.96 = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4553_is__unitE,axiom,
% 16.19/8.96 ! [A: code_integer,C: code_integer] :
% 16.19/8.96 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 16.19/8.96 => ~ ( ( A != zero_z3403309356797280102nteger )
% 16.19/8.96 => ! [B2: code_integer] :
% 16.19/8.96 ( ( B2 != zero_z3403309356797280102nteger )
% 16.19/8.96 => ( ( dvd_dvd_Code_integer @ B2 @ one_one_Code_integer )
% 16.19/8.96 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
% 16.19/8.96 = B2 )
% 16.19/8.96 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B2 )
% 16.19/8.96 = A )
% 16.19/8.96 => ( ( ( times_3573771949741848930nteger @ A @ B2 )
% 16.19/8.96 = one_one_Code_integer )
% 16.19/8.96 => ( ( divide6298287555418463151nteger @ C @ A )
% 16.19/8.96 != ( times_3573771949741848930nteger @ C @ B2 ) ) ) ) ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4554_is__unitE,axiom,
% 16.19/8.96 ! [A: nat,C: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 16.19/8.96 => ~ ( ( A != zero_zero_nat )
% 16.19/8.96 => ! [B2: nat] :
% 16.19/8.96 ( ( B2 != zero_zero_nat )
% 16.19/8.96 => ( ( dvd_dvd_nat @ B2 @ one_one_nat )
% 16.19/8.96 => ( ( ( divide_divide_nat @ one_one_nat @ A )
% 16.19/8.96 = B2 )
% 16.19/8.96 => ( ( ( divide_divide_nat @ one_one_nat @ B2 )
% 16.19/8.96 = A )
% 16.19/8.96 => ( ( ( times_times_nat @ A @ B2 )
% 16.19/8.96 = one_one_nat )
% 16.19/8.96 => ( ( divide_divide_nat @ C @ A )
% 16.19/8.96 != ( times_times_nat @ C @ B2 ) ) ) ) ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4555_is__unitE,axiom,
% 16.19/8.96 ! [A: int,C: int] :
% 16.19/8.96 ( ( dvd_dvd_int @ A @ one_one_int )
% 16.19/8.96 => ~ ( ( A != zero_zero_int )
% 16.19/8.96 => ! [B2: int] :
% 16.19/8.96 ( ( B2 != zero_zero_int )
% 16.19/8.96 => ( ( dvd_dvd_int @ B2 @ one_one_int )
% 16.19/8.96 => ( ( ( divide_divide_int @ one_one_int @ A )
% 16.19/8.96 = B2 )
% 16.19/8.96 => ( ( ( divide_divide_int @ one_one_int @ B2 )
% 16.19/8.96 = A )
% 16.19/8.96 => ( ( ( times_times_int @ A @ B2 )
% 16.19/8.96 = one_one_int )
% 16.19/8.96 => ( ( divide_divide_int @ C @ A )
% 16.19/8.96 != ( times_times_int @ C @ B2 ) ) ) ) ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4556_odd__even__add,axiom,
% 16.19/8.96 ! [A: code_integer,B: code_integer] :
% 16.19/8.96 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B )
% 16.19/8.96 => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4557_odd__even__add,axiom,
% 16.19/8.96 ! [A: nat,B: nat] :
% 16.19/8.96 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 16.19/8.96 => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4558_odd__even__add,axiom,
% 16.19/8.96 ! [A: int,B: int] :
% 16.19/8.96 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
% 16.19/8.96 => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4559_odd__one,axiom,
% 16.19/8.96 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4560_odd__one,axiom,
% 16.19/8.96 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4561_odd__one,axiom,
% 16.19/8.96 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4562_bit__eq__rec,axiom,
% 16.19/8.96 ( ( ^ [Y3: code_integer,Z2: code_integer] : ( Y3 = Z2 ) )
% 16.19/8.96 = ( ^ [A5: code_integer,B5: code_integer] :
% 16.19/8.96 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 )
% 16.19/8.96 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B5 ) )
% 16.19/8.96 & ( ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 16.19/8.96 = ( divide6298287555418463151nteger @ B5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4563_bit__eq__rec,axiom,
% 16.19/8.96 ( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
% 16.19/8.96 = ( ^ [A5: nat,B5: nat] :
% 16.19/8.96 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 )
% 16.19/8.96 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B5 ) )
% 16.19/8.96 & ( ( divide_divide_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.96 = ( divide_divide_nat @ B5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4564_bit__eq__rec,axiom,
% 16.19/8.96 ( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
% 16.19/8.96 = ( ^ [A5: int,B5: int] :
% 16.19/8.96 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 )
% 16.19/8.96 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B5 ) )
% 16.19/8.96 & ( ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/8.96 = ( divide_divide_int @ B5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4565_dvd__power__iff,axiom,
% 16.19/8.96 ! [X: code_integer,M: nat,N2: nat] :
% 16.19/8.96 ( ( X != zero_z3403309356797280102nteger )
% 16.19/8.96 => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ N2 ) )
% 16.19/8.96 = ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
% 16.19/8.96 | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4566_dvd__power__iff,axiom,
% 16.19/8.96 ! [X: nat,M: nat,N2: nat] :
% 16.19/8.96 ( ( X != zero_zero_nat )
% 16.19/8.96 => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N2 ) )
% 16.19/8.96 = ( ( dvd_dvd_nat @ X @ one_one_nat )
% 16.19/8.96 | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4567_dvd__power__iff,axiom,
% 16.19/8.96 ! [X: int,M: nat,N2: nat] :
% 16.19/8.96 ( ( X != zero_zero_int )
% 16.19/8.96 => ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N2 ) )
% 16.19/8.96 = ( ( dvd_dvd_int @ X @ one_one_int )
% 16.19/8.96 | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4568_dvd__power,axiom,
% 16.19/8.96 ! [N2: nat,X: code_integer] :
% 16.19/8.96 ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.96 | ( X = one_one_Code_integer ) )
% 16.19/8.96 => ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4569_dvd__power,axiom,
% 16.19/8.96 ! [N2: nat,X: rat] :
% 16.19/8.96 ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.96 | ( X = one_one_rat ) )
% 16.19/8.96 => ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4570_dvd__power,axiom,
% 16.19/8.96 ! [N2: nat,X: nat] :
% 16.19/8.96 ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.96 | ( X = one_one_nat ) )
% 16.19/8.96 => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4571_dvd__power,axiom,
% 16.19/8.96 ! [N2: nat,X: real] :
% 16.19/8.96 ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.96 | ( X = one_one_real ) )
% 16.19/8.96 => ( dvd_dvd_real @ X @ ( power_power_real @ X @ N2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4572_dvd__power,axiom,
% 16.19/8.96 ! [N2: nat,X: int] :
% 16.19/8.96 ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.96 | ( X = one_one_int ) )
% 16.19/8.96 => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4573_dvd__power,axiom,
% 16.19/8.96 ! [N2: nat,X: complex] :
% 16.19/8.96 ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.96 | ( X = one_one_complex ) )
% 16.19/8.96 => ( dvd_dvd_complex @ X @ ( power_power_complex @ X @ N2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4574_even__even__mod__4__iff,axiom,
% 16.19/8.96 ! [N2: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.96 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4575_dvd__mult__cancel1,axiom,
% 16.19/8.96 ! [M: nat,N2: nat] :
% 16.19/8.96 ( ( ord_less_nat @ zero_zero_nat @ M )
% 16.19/8.96 => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N2 ) @ M )
% 16.19/8.96 = ( N2 = one_one_nat ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4576_dvd__mult__cancel2,axiom,
% 16.19/8.96 ! [M: nat,N2: nat] :
% 16.19/8.96 ( ( ord_less_nat @ zero_zero_nat @ M )
% 16.19/8.96 => ( ( dvd_dvd_nat @ ( times_times_nat @ N2 @ M ) @ M )
% 16.19/8.96 = ( N2 = one_one_nat ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4577_dvd__minus__add,axiom,
% 16.19/8.96 ! [Q2: nat,N2: nat,R: nat,M: nat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ Q2 @ N2 )
% 16.19/8.96 => ( ( ord_less_eq_nat @ Q2 @ ( times_times_nat @ R @ M ) )
% 16.19/8.96 => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ Q2 ) )
% 16.19/8.96 = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N2 @ ( minus_minus_nat @ ( times_times_nat @ R @ M ) @ Q2 ) ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4578_power__dvd__imp__le,axiom,
% 16.19/8.96 ! [I2: nat,M: nat,N2: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ ( power_power_nat @ I2 @ M ) @ ( power_power_nat @ I2 @ N2 ) )
% 16.19/8.96 => ( ( ord_less_nat @ one_one_nat @ I2 )
% 16.19/8.96 => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4579_mod__nat__eqI,axiom,
% 16.19/8.96 ! [R: nat,N2: nat,M: nat] :
% 16.19/8.96 ( ( ord_less_nat @ R @ N2 )
% 16.19/8.96 => ( ( ord_less_eq_nat @ R @ M )
% 16.19/8.96 => ( ( dvd_dvd_nat @ N2 @ ( minus_minus_nat @ M @ R ) )
% 16.19/8.96 => ( ( modulo_modulo_nat @ M @ N2 )
% 16.19/8.96 = R ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4580_mod__int__pos__iff,axiom,
% 16.19/8.96 ! [K: int,L2: int] :
% 16.19/8.96 ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L2 ) )
% 16.19/8.96 = ( ( dvd_dvd_int @ L2 @ K )
% 16.19/8.96 | ( ( L2 = zero_zero_int )
% 16.19/8.96 & ( ord_less_eq_int @ zero_zero_int @ K ) )
% 16.19/8.96 | ( ord_less_int @ zero_zero_int @ L2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4581_bset_I9_J,axiom,
% 16.19/8.96 ! [D: int,D3: int,B3: set_int,T: int] :
% 16.19/8.96 ( ( dvd_dvd_int @ D @ D3 )
% 16.19/8.96 => ! [X5: int] :
% 16.19/8.96 ( ! [Xa3: int] :
% 16.19/8.96 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.96 => ! [Xb3: int] :
% 16.19/8.96 ( ( member_int @ Xb3 @ B3 )
% 16.19/8.96 => ( X5
% 16.19/8.96 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 16.19/8.96 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 16.19/8.96 => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4582_bset_I10_J,axiom,
% 16.19/8.96 ! [D: int,D3: int,B3: set_int,T: int] :
% 16.19/8.96 ( ( dvd_dvd_int @ D @ D3 )
% 16.19/8.96 => ! [X5: int] :
% 16.19/8.96 ( ! [Xa3: int] :
% 16.19/8.96 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.96 => ! [Xb3: int] :
% 16.19/8.96 ( ( member_int @ Xb3 @ B3 )
% 16.19/8.96 => ( X5
% 16.19/8.96 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 16.19/8.96 => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 16.19/8.96 => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4583_aset_I9_J,axiom,
% 16.19/8.96 ! [D: int,D3: int,A2: set_int,T: int] :
% 16.19/8.96 ( ( dvd_dvd_int @ D @ D3 )
% 16.19/8.96 => ! [X5: int] :
% 16.19/8.96 ( ! [Xa3: int] :
% 16.19/8.96 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.96 => ! [Xb3: int] :
% 16.19/8.96 ( ( member_int @ Xb3 @ A2 )
% 16.19/8.96 => ( X5
% 16.19/8.96 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 16.19/8.96 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 16.19/8.96 => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4584_aset_I10_J,axiom,
% 16.19/8.96 ! [D: int,D3: int,A2: set_int,T: int] :
% 16.19/8.96 ( ( dvd_dvd_int @ D @ D3 )
% 16.19/8.96 => ! [X5: int] :
% 16.19/8.96 ( ! [Xa3: int] :
% 16.19/8.96 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
% 16.19/8.96 => ! [Xb3: int] :
% 16.19/8.96 ( ( member_int @ Xb3 @ A2 )
% 16.19/8.96 => ( X5
% 16.19/8.96 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 16.19/8.96 => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 16.19/8.96 => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D3 ) @ T ) ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4585_even__two__times__div__two,axiom,
% 16.19/8.96 ! [A: code_integer] :
% 16.19/8.96 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 16.19/8.96 = A ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4586_even__two__times__div__two,axiom,
% 16.19/8.96 ! [A: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.96 = A ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4587_even__two__times__div__two,axiom,
% 16.19/8.96 ! [A: int] :
% 16.19/8.96 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 16.19/8.96 = A ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4588_even__iff__mod__2__eq__zero,axiom,
% 16.19/8.96 ! [A: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.96 = zero_zero_nat ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4589_even__iff__mod__2__eq__zero,axiom,
% 16.19/8.96 ! [A: int] :
% 16.19/8.96 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/8.96 = zero_zero_int ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4590_even__iff__mod__2__eq__zero,axiom,
% 16.19/8.96 ! [A: code_integer] :
% 16.19/8.96 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 16.19/8.96 = zero_z3403309356797280102nteger ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4591_odd__iff__mod__2__eq__one,axiom,
% 16.19/8.96 ! [A: nat] :
% 16.19/8.96 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 16.19/8.96 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.96 = one_one_nat ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4592_odd__iff__mod__2__eq__one,axiom,
% 16.19/8.96 ! [A: int] :
% 16.19/8.96 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 16.19/8.96 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/8.96 = one_one_int ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4593_odd__iff__mod__2__eq__one,axiom,
% 16.19/8.96 ! [A: code_integer] :
% 16.19/8.96 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 16.19/8.96 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 16.19/8.96 = one_one_Code_integer ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4594_power__mono__odd,axiom,
% 16.19/8.96 ! [N2: nat,A: real,B: real] :
% 16.19/8.96 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.96 => ( ( ord_less_eq_real @ A @ B )
% 16.19/8.96 => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4595_power__mono__odd,axiom,
% 16.19/8.96 ! [N2: nat,A: rat,B: rat] :
% 16.19/8.96 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.96 => ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.96 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4596_power__mono__odd,axiom,
% 16.19/8.96 ! [N2: nat,A: int,B: int] :
% 16.19/8.96 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.96 => ( ( ord_less_eq_int @ A @ B )
% 16.19/8.96 => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4597_odd__pos,axiom,
% 16.19/8.96 ! [N2: nat] :
% 16.19/8.96 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.96 => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4598_dvd__power__iff__le,axiom,
% 16.19/8.96 ! [K: nat,M: nat,N2: nat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 16.19/8.96 => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N2 ) )
% 16.19/8.96 = ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4599_even__unset__bit__iff,axiom,
% 16.19/8.96 ! [M: nat,A: code_integer] :
% 16.19/8.96 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A ) )
% 16.19/8.96 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 | ( M = zero_zero_nat ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4600_even__unset__bit__iff,axiom,
% 16.19/8.96 ! [M: nat,A: int] :
% 16.19/8.96 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 16.19/8.96 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 | ( M = zero_zero_nat ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4601_even__unset__bit__iff,axiom,
% 16.19/8.96 ! [M: nat,A: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 16.19/8.96 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 | ( M = zero_zero_nat ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4602_even__set__bit__iff,axiom,
% 16.19/8.96 ! [M: nat,A: code_integer] :
% 16.19/8.96 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A ) )
% 16.19/8.96 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 & ( M != zero_zero_nat ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4603_even__set__bit__iff,axiom,
% 16.19/8.96 ! [M: nat,A: int] :
% 16.19/8.96 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 16.19/8.96 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 & ( M != zero_zero_nat ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4604_even__set__bit__iff,axiom,
% 16.19/8.96 ! [M: nat,A: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 16.19/8.96 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 & ( M != zero_zero_nat ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4605_even__flip__bit__iff,axiom,
% 16.19/8.96 ! [M: nat,A: code_integer] :
% 16.19/8.96 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A ) )
% 16.19/8.96 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 != ( M = zero_zero_nat ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4606_even__flip__bit__iff,axiom,
% 16.19/8.96 ! [M: nat,A: int] :
% 16.19/8.96 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 16.19/8.96 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 != ( M = zero_zero_nat ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4607_even__flip__bit__iff,axiom,
% 16.19/8.96 ! [M: nat,A: nat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 16.19/8.96 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 != ( M = zero_zero_nat ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4608_even__diff__iff,axiom,
% 16.19/8.96 ! [K: int,L2: int] :
% 16.19/8.96 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L2 ) )
% 16.19/8.96 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4609_oddE,axiom,
% 16.19/8.96 ! [A: code_integer] :
% 16.19/8.96 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 => ~ ! [B2: code_integer] :
% 16.19/8.96 ( A
% 16.19/8.96 != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) @ one_one_Code_integer ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4610_oddE,axiom,
% 16.19/8.96 ! [A: nat] :
% 16.19/8.96 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 => ~ ! [B2: nat] :
% 16.19/8.96 ( A
% 16.19/8.96 != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) @ one_one_nat ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4611_oddE,axiom,
% 16.19/8.96 ! [A: int] :
% 16.19/8.96 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 => ~ ! [B2: int] :
% 16.19/8.96 ( A
% 16.19/8.96 != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) @ one_one_int ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4612_parity__cases,axiom,
% 16.19/8.96 ! [A: nat] :
% 16.19/8.96 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.96 != zero_zero_nat ) )
% 16.19/8.96 => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.96 != one_one_nat ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4613_parity__cases,axiom,
% 16.19/8.96 ! [A: int] :
% 16.19/8.96 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/8.96 != zero_zero_int ) )
% 16.19/8.96 => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/8.96 != one_one_int ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4614_parity__cases,axiom,
% 16.19/8.96 ! [A: code_integer] :
% 16.19/8.96 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 16.19/8.96 != zero_z3403309356797280102nteger ) )
% 16.19/8.96 => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 16.19/8.96 != one_one_Code_integer ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4615_mod2__eq__if,axiom,
% 16.19/8.96 ! [A: nat] :
% 16.19/8.96 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.96 = zero_zero_nat ) )
% 16.19/8.96 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.96 = one_one_nat ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4616_mod2__eq__if,axiom,
% 16.19/8.96 ! [A: int] :
% 16.19/8.96 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/8.96 = zero_zero_int ) )
% 16.19/8.96 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/8.96 = one_one_int ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4617_mod2__eq__if,axiom,
% 16.19/8.96 ! [A: code_integer] :
% 16.19/8.96 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 16.19/8.96 = zero_z3403309356797280102nteger ) )
% 16.19/8.96 & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 16.19/8.96 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 16.19/8.96 = one_one_Code_integer ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4618_zero__le__even__power,axiom,
% 16.19/8.96 ! [N2: nat,A: real] :
% 16.19/8.96 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.96 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4619_zero__le__even__power,axiom,
% 16.19/8.96 ! [N2: nat,A: rat] :
% 16.19/8.96 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.96 => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4620_zero__le__even__power,axiom,
% 16.19/8.96 ! [N2: nat,A: int] :
% 16.19/8.96 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.96 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4621_zero__le__odd__power,axiom,
% 16.19/8.96 ! [N2: nat,A: real] :
% 16.19/8.96 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.96 => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
% 16.19/8.96 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4622_zero__le__odd__power,axiom,
% 16.19/8.96 ! [N2: nat,A: rat] :
% 16.19/8.96 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.96 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
% 16.19/8.96 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4623_zero__le__odd__power,axiom,
% 16.19/8.96 ! [N2: nat,A: int] :
% 16.19/8.96 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.96 => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
% 16.19/8.96 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4624_zero__le__power__eq,axiom,
% 16.19/8.96 ! [A: real,N2: nat] :
% 16.19/8.96 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
% 16.19/8.96 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.96 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.96 & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4625_zero__le__power__eq,axiom,
% 16.19/8.96 ! [A: rat,N2: nat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
% 16.19/8.96 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.96 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.96 & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4626_zero__le__power__eq,axiom,
% 16.19/8.96 ! [A: int,N2: nat] :
% 16.19/8.96 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
% 16.19/8.96 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.96 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.96 & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4627_leD,axiom,
% 16.19/8.96 ! [Y: real,X: real] :
% 16.19/8.96 ( ( ord_less_eq_real @ Y @ X )
% 16.19/8.96 => ~ ( ord_less_real @ X @ Y ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4628_leD,axiom,
% 16.19/8.96 ! [Y: set_nat,X: set_nat] :
% 16.19/8.96 ( ( ord_less_eq_set_nat @ Y @ X )
% 16.19/8.96 => ~ ( ord_less_set_nat @ X @ Y ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4629_leD,axiom,
% 16.19/8.96 ! [Y: rat,X: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ Y @ X )
% 16.19/8.96 => ~ ( ord_less_rat @ X @ Y ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4630_leD,axiom,
% 16.19/8.96 ! [Y: num,X: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ Y @ X )
% 16.19/8.96 => ~ ( ord_less_num @ X @ Y ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4631_leD,axiom,
% 16.19/8.96 ! [Y: nat,X: nat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ Y @ X )
% 16.19/8.96 => ~ ( ord_less_nat @ X @ Y ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4632_leD,axiom,
% 16.19/8.96 ! [Y: int,X: int] :
% 16.19/8.96 ( ( ord_less_eq_int @ Y @ X )
% 16.19/8.96 => ~ ( ord_less_int @ X @ Y ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4633_leI,axiom,
% 16.19/8.96 ! [X: real,Y: real] :
% 16.19/8.96 ( ~ ( ord_less_real @ X @ Y )
% 16.19/8.96 => ( ord_less_eq_real @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4634_leI,axiom,
% 16.19/8.96 ! [X: rat,Y: rat] :
% 16.19/8.96 ( ~ ( ord_less_rat @ X @ Y )
% 16.19/8.96 => ( ord_less_eq_rat @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4635_leI,axiom,
% 16.19/8.96 ! [X: num,Y: num] :
% 16.19/8.96 ( ~ ( ord_less_num @ X @ Y )
% 16.19/8.96 => ( ord_less_eq_num @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4636_leI,axiom,
% 16.19/8.96 ! [X: nat,Y: nat] :
% 16.19/8.96 ( ~ ( ord_less_nat @ X @ Y )
% 16.19/8.96 => ( ord_less_eq_nat @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4637_leI,axiom,
% 16.19/8.96 ! [X: int,Y: int] :
% 16.19/8.96 ( ~ ( ord_less_int @ X @ Y )
% 16.19/8.96 => ( ord_less_eq_int @ Y @ X ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4638_nless__le,axiom,
% 16.19/8.96 ! [A: real,B: real] :
% 16.19/8.96 ( ( ~ ( ord_less_real @ A @ B ) )
% 16.19/8.96 = ( ~ ( ord_less_eq_real @ A @ B )
% 16.19/8.96 | ( A = B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4639_nless__le,axiom,
% 16.19/8.96 ! [A: set_nat,B: set_nat] :
% 16.19/8.96 ( ( ~ ( ord_less_set_nat @ A @ B ) )
% 16.19/8.96 = ( ~ ( ord_less_eq_set_nat @ A @ B )
% 16.19/8.96 | ( A = B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4640_nless__le,axiom,
% 16.19/8.96 ! [A: rat,B: rat] :
% 16.19/8.96 ( ( ~ ( ord_less_rat @ A @ B ) )
% 16.19/8.96 = ( ~ ( ord_less_eq_rat @ A @ B )
% 16.19/8.96 | ( A = B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4641_nless__le,axiom,
% 16.19/8.96 ! [A: num,B: num] :
% 16.19/8.96 ( ( ~ ( ord_less_num @ A @ B ) )
% 16.19/8.96 = ( ~ ( ord_less_eq_num @ A @ B )
% 16.19/8.96 | ( A = B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4642_nless__le,axiom,
% 16.19/8.96 ! [A: nat,B: nat] :
% 16.19/8.96 ( ( ~ ( ord_less_nat @ A @ B ) )
% 16.19/8.96 = ( ~ ( ord_less_eq_nat @ A @ B )
% 16.19/8.96 | ( A = B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4643_nless__le,axiom,
% 16.19/8.96 ! [A: int,B: int] :
% 16.19/8.96 ( ( ~ ( ord_less_int @ A @ B ) )
% 16.19/8.96 = ( ~ ( ord_less_eq_int @ A @ B )
% 16.19/8.96 | ( A = B ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4644_antisym__conv1,axiom,
% 16.19/8.96 ! [X: real,Y: real] :
% 16.19/8.96 ( ~ ( ord_less_real @ X @ Y )
% 16.19/8.96 => ( ( ord_less_eq_real @ X @ Y )
% 16.19/8.96 = ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4645_antisym__conv1,axiom,
% 16.19/8.96 ! [X: set_nat,Y: set_nat] :
% 16.19/8.96 ( ~ ( ord_less_set_nat @ X @ Y )
% 16.19/8.96 => ( ( ord_less_eq_set_nat @ X @ Y )
% 16.19/8.96 = ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4646_antisym__conv1,axiom,
% 16.19/8.96 ! [X: rat,Y: rat] :
% 16.19/8.96 ( ~ ( ord_less_rat @ X @ Y )
% 16.19/8.96 => ( ( ord_less_eq_rat @ X @ Y )
% 16.19/8.96 = ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4647_antisym__conv1,axiom,
% 16.19/8.96 ! [X: num,Y: num] :
% 16.19/8.96 ( ~ ( ord_less_num @ X @ Y )
% 16.19/8.96 => ( ( ord_less_eq_num @ X @ Y )
% 16.19/8.96 = ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4648_antisym__conv1,axiom,
% 16.19/8.96 ! [X: nat,Y: nat] :
% 16.19/8.96 ( ~ ( ord_less_nat @ X @ Y )
% 16.19/8.96 => ( ( ord_less_eq_nat @ X @ Y )
% 16.19/8.96 = ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4649_antisym__conv1,axiom,
% 16.19/8.96 ! [X: int,Y: int] :
% 16.19/8.96 ( ~ ( ord_less_int @ X @ Y )
% 16.19/8.96 => ( ( ord_less_eq_int @ X @ Y )
% 16.19/8.96 = ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4650_antisym__conv2,axiom,
% 16.19/8.96 ! [X: real,Y: real] :
% 16.19/8.96 ( ( ord_less_eq_real @ X @ Y )
% 16.19/8.96 => ( ( ~ ( ord_less_real @ X @ Y ) )
% 16.19/8.96 = ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4651_antisym__conv2,axiom,
% 16.19/8.96 ! [X: set_nat,Y: set_nat] :
% 16.19/8.96 ( ( ord_less_eq_set_nat @ X @ Y )
% 16.19/8.96 => ( ( ~ ( ord_less_set_nat @ X @ Y ) )
% 16.19/8.96 = ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4652_antisym__conv2,axiom,
% 16.19/8.96 ! [X: rat,Y: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ X @ Y )
% 16.19/8.96 => ( ( ~ ( ord_less_rat @ X @ Y ) )
% 16.19/8.96 = ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4653_antisym__conv2,axiom,
% 16.19/8.96 ! [X: num,Y: num] :
% 16.19/8.96 ( ( ord_less_eq_num @ X @ Y )
% 16.19/8.96 => ( ( ~ ( ord_less_num @ X @ Y ) )
% 16.19/8.96 = ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4654_antisym__conv2,axiom,
% 16.19/8.96 ! [X: nat,Y: nat] :
% 16.19/8.96 ( ( ord_less_eq_nat @ X @ Y )
% 16.19/8.96 => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 16.19/8.96 = ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4655_antisym__conv2,axiom,
% 16.19/8.96 ! [X: int,Y: int] :
% 16.19/8.96 ( ( ord_less_eq_int @ X @ Y )
% 16.19/8.96 => ( ( ~ ( ord_less_int @ X @ Y ) )
% 16.19/8.96 = ( X = Y ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4656_dense__ge,axiom,
% 16.19/8.96 ! [Z: real,Y: real] :
% 16.19/8.96 ( ! [X4: real] :
% 16.19/8.96 ( ( ord_less_real @ Z @ X4 )
% 16.19/8.96 => ( ord_less_eq_real @ Y @ X4 ) )
% 16.19/8.96 => ( ord_less_eq_real @ Y @ Z ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4657_dense__ge,axiom,
% 16.19/8.96 ! [Z: rat,Y: rat] :
% 16.19/8.96 ( ! [X4: rat] :
% 16.19/8.96 ( ( ord_less_rat @ Z @ X4 )
% 16.19/8.96 => ( ord_less_eq_rat @ Y @ X4 ) )
% 16.19/8.96 => ( ord_less_eq_rat @ Y @ Z ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4658_dense__le,axiom,
% 16.19/8.96 ! [Y: real,Z: real] :
% 16.19/8.96 ( ! [X4: real] :
% 16.19/8.96 ( ( ord_less_real @ X4 @ Y )
% 16.19/8.96 => ( ord_less_eq_real @ X4 @ Z ) )
% 16.19/8.96 => ( ord_less_eq_real @ Y @ Z ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4659_dense__le,axiom,
% 16.19/8.96 ! [Y: rat,Z: rat] :
% 16.19/8.96 ( ! [X4: rat] :
% 16.19/8.96 ( ( ord_less_rat @ X4 @ Y )
% 16.19/8.96 => ( ord_less_eq_rat @ X4 @ Z ) )
% 16.19/8.96 => ( ord_less_eq_rat @ Y @ Z ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4660_less__le__not__le,axiom,
% 16.19/8.96 ( ord_less_real
% 16.19/8.96 = ( ^ [X2: real,Y2: real] :
% 16.19/8.96 ( ( ord_less_eq_real @ X2 @ Y2 )
% 16.19/8.96 & ~ ( ord_less_eq_real @ Y2 @ X2 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4661_less__le__not__le,axiom,
% 16.19/8.96 ( ord_less_set_nat
% 16.19/8.96 = ( ^ [X2: set_nat,Y2: set_nat] :
% 16.19/8.96 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 16.19/8.96 & ~ ( ord_less_eq_set_nat @ Y2 @ X2 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4662_less__le__not__le,axiom,
% 16.19/8.96 ( ord_less_rat
% 16.19/8.96 = ( ^ [X2: rat,Y2: rat] :
% 16.19/8.96 ( ( ord_less_eq_rat @ X2 @ Y2 )
% 16.19/8.96 & ~ ( ord_less_eq_rat @ Y2 @ X2 ) ) ) ) ).
% 16.19/8.96
% 16.19/8.96 thf(fact_4663_less__le__not__le,axiom,
% 16.19/8.96 ( ord_less_num
% 16.19/8.96 = ( ^ [X2: num,Y2: num] :
% 16.19/8.97 ( ( ord_less_eq_num @ X2 @ Y2 )
% 16.19/8.97 & ~ ( ord_less_eq_num @ Y2 @ X2 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4664_less__le__not__le,axiom,
% 16.19/8.97 ( ord_less_nat
% 16.19/8.97 = ( ^ [X2: nat,Y2: nat] :
% 16.19/8.97 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 16.19/8.97 & ~ ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4665_less__le__not__le,axiom,
% 16.19/8.97 ( ord_less_int
% 16.19/8.97 = ( ^ [X2: int,Y2: int] :
% 16.19/8.97 ( ( ord_less_eq_int @ X2 @ Y2 )
% 16.19/8.97 & ~ ( ord_less_eq_int @ Y2 @ X2 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4666_not__le__imp__less,axiom,
% 16.19/8.97 ! [Y: real,X: real] :
% 16.19/8.97 ( ~ ( ord_less_eq_real @ Y @ X )
% 16.19/8.97 => ( ord_less_real @ X @ Y ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4667_not__le__imp__less,axiom,
% 16.19/8.97 ! [Y: rat,X: rat] :
% 16.19/8.97 ( ~ ( ord_less_eq_rat @ Y @ X )
% 16.19/8.97 => ( ord_less_rat @ X @ Y ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4668_not__le__imp__less,axiom,
% 16.19/8.97 ! [Y: num,X: num] :
% 16.19/8.97 ( ~ ( ord_less_eq_num @ Y @ X )
% 16.19/8.97 => ( ord_less_num @ X @ Y ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4669_not__le__imp__less,axiom,
% 16.19/8.97 ! [Y: nat,X: nat] :
% 16.19/8.97 ( ~ ( ord_less_eq_nat @ Y @ X )
% 16.19/8.97 => ( ord_less_nat @ X @ Y ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4670_not__le__imp__less,axiom,
% 16.19/8.97 ! [Y: int,X: int] :
% 16.19/8.97 ( ~ ( ord_less_eq_int @ Y @ X )
% 16.19/8.97 => ( ord_less_int @ X @ Y ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4671_order_Oorder__iff__strict,axiom,
% 16.19/8.97 ( ord_less_eq_real
% 16.19/8.97 = ( ^ [A5: real,B5: real] :
% 16.19/8.97 ( ( ord_less_real @ A5 @ B5 )
% 16.19/8.97 | ( A5 = B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4672_order_Oorder__iff__strict,axiom,
% 16.19/8.97 ( ord_less_eq_set_nat
% 16.19/8.97 = ( ^ [A5: set_nat,B5: set_nat] :
% 16.19/8.97 ( ( ord_less_set_nat @ A5 @ B5 )
% 16.19/8.97 | ( A5 = B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4673_order_Oorder__iff__strict,axiom,
% 16.19/8.97 ( ord_less_eq_rat
% 16.19/8.97 = ( ^ [A5: rat,B5: rat] :
% 16.19/8.97 ( ( ord_less_rat @ A5 @ B5 )
% 16.19/8.97 | ( A5 = B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4674_order_Oorder__iff__strict,axiom,
% 16.19/8.97 ( ord_less_eq_num
% 16.19/8.97 = ( ^ [A5: num,B5: num] :
% 16.19/8.97 ( ( ord_less_num @ A5 @ B5 )
% 16.19/8.97 | ( A5 = B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4675_order_Oorder__iff__strict,axiom,
% 16.19/8.97 ( ord_less_eq_nat
% 16.19/8.97 = ( ^ [A5: nat,B5: nat] :
% 16.19/8.97 ( ( ord_less_nat @ A5 @ B5 )
% 16.19/8.97 | ( A5 = B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4676_order_Oorder__iff__strict,axiom,
% 16.19/8.97 ( ord_less_eq_int
% 16.19/8.97 = ( ^ [A5: int,B5: int] :
% 16.19/8.97 ( ( ord_less_int @ A5 @ B5 )
% 16.19/8.97 | ( A5 = B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4677_order_Ostrict__iff__order,axiom,
% 16.19/8.97 ( ord_less_real
% 16.19/8.97 = ( ^ [A5: real,B5: real] :
% 16.19/8.97 ( ( ord_less_eq_real @ A5 @ B5 )
% 16.19/8.97 & ( A5 != B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4678_order_Ostrict__iff__order,axiom,
% 16.19/8.97 ( ord_less_set_nat
% 16.19/8.97 = ( ^ [A5: set_nat,B5: set_nat] :
% 16.19/8.97 ( ( ord_less_eq_set_nat @ A5 @ B5 )
% 16.19/8.97 & ( A5 != B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4679_order_Ostrict__iff__order,axiom,
% 16.19/8.97 ( ord_less_rat
% 16.19/8.97 = ( ^ [A5: rat,B5: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ A5 @ B5 )
% 16.19/8.97 & ( A5 != B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4680_order_Ostrict__iff__order,axiom,
% 16.19/8.97 ( ord_less_num
% 16.19/8.97 = ( ^ [A5: num,B5: num] :
% 16.19/8.97 ( ( ord_less_eq_num @ A5 @ B5 )
% 16.19/8.97 & ( A5 != B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4681_order_Ostrict__iff__order,axiom,
% 16.19/8.97 ( ord_less_nat
% 16.19/8.97 = ( ^ [A5: nat,B5: nat] :
% 16.19/8.97 ( ( ord_less_eq_nat @ A5 @ B5 )
% 16.19/8.97 & ( A5 != B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4682_order_Ostrict__iff__order,axiom,
% 16.19/8.97 ( ord_less_int
% 16.19/8.97 = ( ^ [A5: int,B5: int] :
% 16.19/8.97 ( ( ord_less_eq_int @ A5 @ B5 )
% 16.19/8.97 & ( A5 != B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4683_order_Ostrict__trans1,axiom,
% 16.19/8.97 ! [A: real,B: real,C: real] :
% 16.19/8.97 ( ( ord_less_eq_real @ A @ B )
% 16.19/8.97 => ( ( ord_less_real @ B @ C )
% 16.19/8.97 => ( ord_less_real @ A @ C ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4684_order_Ostrict__trans1,axiom,
% 16.19/8.97 ! [A: set_nat,B: set_nat,C: set_nat] :
% 16.19/8.97 ( ( ord_less_eq_set_nat @ A @ B )
% 16.19/8.97 => ( ( ord_less_set_nat @ B @ C )
% 16.19/8.97 => ( ord_less_set_nat @ A @ C ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4685_order_Ostrict__trans1,axiom,
% 16.19/8.97 ! [A: rat,B: rat,C: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.97 => ( ( ord_less_rat @ B @ C )
% 16.19/8.97 => ( ord_less_rat @ A @ C ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4686_order_Ostrict__trans1,axiom,
% 16.19/8.97 ! [A: num,B: num,C: num] :
% 16.19/8.97 ( ( ord_less_eq_num @ A @ B )
% 16.19/8.97 => ( ( ord_less_num @ B @ C )
% 16.19/8.97 => ( ord_less_num @ A @ C ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4687_order_Ostrict__trans1,axiom,
% 16.19/8.97 ! [A: nat,B: nat,C: nat] :
% 16.19/8.97 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.97 => ( ( ord_less_nat @ B @ C )
% 16.19/8.97 => ( ord_less_nat @ A @ C ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4688_order_Ostrict__trans1,axiom,
% 16.19/8.97 ! [A: int,B: int,C: int] :
% 16.19/8.97 ( ( ord_less_eq_int @ A @ B )
% 16.19/8.97 => ( ( ord_less_int @ B @ C )
% 16.19/8.97 => ( ord_less_int @ A @ C ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4689_order_Ostrict__trans2,axiom,
% 16.19/8.97 ! [A: real,B: real,C: real] :
% 16.19/8.97 ( ( ord_less_real @ A @ B )
% 16.19/8.97 => ( ( ord_less_eq_real @ B @ C )
% 16.19/8.97 => ( ord_less_real @ A @ C ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4690_order_Ostrict__trans2,axiom,
% 16.19/8.97 ! [A: set_nat,B: set_nat,C: set_nat] :
% 16.19/8.97 ( ( ord_less_set_nat @ A @ B )
% 16.19/8.97 => ( ( ord_less_eq_set_nat @ B @ C )
% 16.19/8.97 => ( ord_less_set_nat @ A @ C ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4691_order_Ostrict__trans2,axiom,
% 16.19/8.97 ! [A: rat,B: rat,C: rat] :
% 16.19/8.97 ( ( ord_less_rat @ A @ B )
% 16.19/8.97 => ( ( ord_less_eq_rat @ B @ C )
% 16.19/8.97 => ( ord_less_rat @ A @ C ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4692_order_Ostrict__trans2,axiom,
% 16.19/8.97 ! [A: num,B: num,C: num] :
% 16.19/8.97 ( ( ord_less_num @ A @ B )
% 16.19/8.97 => ( ( ord_less_eq_num @ B @ C )
% 16.19/8.97 => ( ord_less_num @ A @ C ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4693_order_Ostrict__trans2,axiom,
% 16.19/8.97 ! [A: nat,B: nat,C: nat] :
% 16.19/8.97 ( ( ord_less_nat @ A @ B )
% 16.19/8.97 => ( ( ord_less_eq_nat @ B @ C )
% 16.19/8.97 => ( ord_less_nat @ A @ C ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4694_order_Ostrict__trans2,axiom,
% 16.19/8.97 ! [A: int,B: int,C: int] :
% 16.19/8.97 ( ( ord_less_int @ A @ B )
% 16.19/8.97 => ( ( ord_less_eq_int @ B @ C )
% 16.19/8.97 => ( ord_less_int @ A @ C ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4695_order_Ostrict__iff__not,axiom,
% 16.19/8.97 ( ord_less_real
% 16.19/8.97 = ( ^ [A5: real,B5: real] :
% 16.19/8.97 ( ( ord_less_eq_real @ A5 @ B5 )
% 16.19/8.97 & ~ ( ord_less_eq_real @ B5 @ A5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4696_order_Ostrict__iff__not,axiom,
% 16.19/8.97 ( ord_less_set_nat
% 16.19/8.97 = ( ^ [A5: set_nat,B5: set_nat] :
% 16.19/8.97 ( ( ord_less_eq_set_nat @ A5 @ B5 )
% 16.19/8.97 & ~ ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4697_order_Ostrict__iff__not,axiom,
% 16.19/8.97 ( ord_less_rat
% 16.19/8.97 = ( ^ [A5: rat,B5: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ A5 @ B5 )
% 16.19/8.97 & ~ ( ord_less_eq_rat @ B5 @ A5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4698_order_Ostrict__iff__not,axiom,
% 16.19/8.97 ( ord_less_num
% 16.19/8.97 = ( ^ [A5: num,B5: num] :
% 16.19/8.97 ( ( ord_less_eq_num @ A5 @ B5 )
% 16.19/8.97 & ~ ( ord_less_eq_num @ B5 @ A5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4699_order_Ostrict__iff__not,axiom,
% 16.19/8.97 ( ord_less_nat
% 16.19/8.97 = ( ^ [A5: nat,B5: nat] :
% 16.19/8.97 ( ( ord_less_eq_nat @ A5 @ B5 )
% 16.19/8.97 & ~ ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4700_order_Ostrict__iff__not,axiom,
% 16.19/8.97 ( ord_less_int
% 16.19/8.97 = ( ^ [A5: int,B5: int] :
% 16.19/8.97 ( ( ord_less_eq_int @ A5 @ B5 )
% 16.19/8.97 & ~ ( ord_less_eq_int @ B5 @ A5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4701_dense__ge__bounded,axiom,
% 16.19/8.97 ! [Z: real,X: real,Y: real] :
% 16.19/8.97 ( ( ord_less_real @ Z @ X )
% 16.19/8.97 => ( ! [W2: real] :
% 16.19/8.97 ( ( ord_less_real @ Z @ W2 )
% 16.19/8.97 => ( ( ord_less_real @ W2 @ X )
% 16.19/8.97 => ( ord_less_eq_real @ Y @ W2 ) ) )
% 16.19/8.97 => ( ord_less_eq_real @ Y @ Z ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4702_dense__ge__bounded,axiom,
% 16.19/8.97 ! [Z: rat,X: rat,Y: rat] :
% 16.19/8.97 ( ( ord_less_rat @ Z @ X )
% 16.19/8.97 => ( ! [W2: rat] :
% 16.19/8.97 ( ( ord_less_rat @ Z @ W2 )
% 16.19/8.97 => ( ( ord_less_rat @ W2 @ X )
% 16.19/8.97 => ( ord_less_eq_rat @ Y @ W2 ) ) )
% 16.19/8.97 => ( ord_less_eq_rat @ Y @ Z ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4703_dense__le__bounded,axiom,
% 16.19/8.97 ! [X: real,Y: real,Z: real] :
% 16.19/8.97 ( ( ord_less_real @ X @ Y )
% 16.19/8.97 => ( ! [W2: real] :
% 16.19/8.97 ( ( ord_less_real @ X @ W2 )
% 16.19/8.97 => ( ( ord_less_real @ W2 @ Y )
% 16.19/8.97 => ( ord_less_eq_real @ W2 @ Z ) ) )
% 16.19/8.97 => ( ord_less_eq_real @ Y @ Z ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4704_dense__le__bounded,axiom,
% 16.19/8.97 ! [X: rat,Y: rat,Z: rat] :
% 16.19/8.97 ( ( ord_less_rat @ X @ Y )
% 16.19/8.97 => ( ! [W2: rat] :
% 16.19/8.97 ( ( ord_less_rat @ X @ W2 )
% 16.19/8.97 => ( ( ord_less_rat @ W2 @ Y )
% 16.19/8.97 => ( ord_less_eq_rat @ W2 @ Z ) ) )
% 16.19/8.97 => ( ord_less_eq_rat @ Y @ Z ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4705_dual__order_Oorder__iff__strict,axiom,
% 16.19/8.97 ( ord_less_eq_real
% 16.19/8.97 = ( ^ [B5: real,A5: real] :
% 16.19/8.97 ( ( ord_less_real @ B5 @ A5 )
% 16.19/8.97 | ( A5 = B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4706_dual__order_Oorder__iff__strict,axiom,
% 16.19/8.97 ( ord_less_eq_set_nat
% 16.19/8.97 = ( ^ [B5: set_nat,A5: set_nat] :
% 16.19/8.97 ( ( ord_less_set_nat @ B5 @ A5 )
% 16.19/8.97 | ( A5 = B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4707_dual__order_Oorder__iff__strict,axiom,
% 16.19/8.97 ( ord_less_eq_rat
% 16.19/8.97 = ( ^ [B5: rat,A5: rat] :
% 16.19/8.97 ( ( ord_less_rat @ B5 @ A5 )
% 16.19/8.97 | ( A5 = B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4708_dual__order_Oorder__iff__strict,axiom,
% 16.19/8.97 ( ord_less_eq_num
% 16.19/8.97 = ( ^ [B5: num,A5: num] :
% 16.19/8.97 ( ( ord_less_num @ B5 @ A5 )
% 16.19/8.97 | ( A5 = B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4709_dual__order_Oorder__iff__strict,axiom,
% 16.19/8.97 ( ord_less_eq_nat
% 16.19/8.97 = ( ^ [B5: nat,A5: nat] :
% 16.19/8.97 ( ( ord_less_nat @ B5 @ A5 )
% 16.19/8.97 | ( A5 = B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4710_dual__order_Oorder__iff__strict,axiom,
% 16.19/8.97 ( ord_less_eq_int
% 16.19/8.97 = ( ^ [B5: int,A5: int] :
% 16.19/8.97 ( ( ord_less_int @ B5 @ A5 )
% 16.19/8.97 | ( A5 = B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4711_dual__order_Ostrict__iff__order,axiom,
% 16.19/8.97 ( ord_less_real
% 16.19/8.97 = ( ^ [B5: real,A5: real] :
% 16.19/8.97 ( ( ord_less_eq_real @ B5 @ A5 )
% 16.19/8.97 & ( A5 != B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4712_dual__order_Ostrict__iff__order,axiom,
% 16.19/8.97 ( ord_less_set_nat
% 16.19/8.97 = ( ^ [B5: set_nat,A5: set_nat] :
% 16.19/8.97 ( ( ord_less_eq_set_nat @ B5 @ A5 )
% 16.19/8.97 & ( A5 != B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4713_dual__order_Ostrict__iff__order,axiom,
% 16.19/8.97 ( ord_less_rat
% 16.19/8.97 = ( ^ [B5: rat,A5: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ B5 @ A5 )
% 16.19/8.97 & ( A5 != B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4714_dual__order_Ostrict__iff__order,axiom,
% 16.19/8.97 ( ord_less_num
% 16.19/8.97 = ( ^ [B5: num,A5: num] :
% 16.19/8.97 ( ( ord_less_eq_num @ B5 @ A5 )
% 16.19/8.97 & ( A5 != B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4715_dual__order_Ostrict__iff__order,axiom,
% 16.19/8.97 ( ord_less_nat
% 16.19/8.97 = ( ^ [B5: nat,A5: nat] :
% 16.19/8.97 ( ( ord_less_eq_nat @ B5 @ A5 )
% 16.19/8.97 & ( A5 != B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4716_dual__order_Ostrict__iff__order,axiom,
% 16.19/8.97 ( ord_less_int
% 16.19/8.97 = ( ^ [B5: int,A5: int] :
% 16.19/8.97 ( ( ord_less_eq_int @ B5 @ A5 )
% 16.19/8.97 & ( A5 != B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4717_dual__order_Ostrict__trans1,axiom,
% 16.19/8.97 ! [B: real,A: real,C: real] :
% 16.19/8.97 ( ( ord_less_eq_real @ B @ A )
% 16.19/8.97 => ( ( ord_less_real @ C @ B )
% 16.19/8.97 => ( ord_less_real @ C @ A ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4718_dual__order_Ostrict__trans1,axiom,
% 16.19/8.97 ! [B: set_nat,A: set_nat,C: set_nat] :
% 16.19/8.97 ( ( ord_less_eq_set_nat @ B @ A )
% 16.19/8.97 => ( ( ord_less_set_nat @ C @ B )
% 16.19/8.97 => ( ord_less_set_nat @ C @ A ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4719_dual__order_Ostrict__trans1,axiom,
% 16.19/8.97 ! [B: rat,A: rat,C: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ B @ A )
% 16.19/8.97 => ( ( ord_less_rat @ C @ B )
% 16.19/8.97 => ( ord_less_rat @ C @ A ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4720_dual__order_Ostrict__trans1,axiom,
% 16.19/8.97 ! [B: num,A: num,C: num] :
% 16.19/8.97 ( ( ord_less_eq_num @ B @ A )
% 16.19/8.97 => ( ( ord_less_num @ C @ B )
% 16.19/8.97 => ( ord_less_num @ C @ A ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4721_dual__order_Ostrict__trans1,axiom,
% 16.19/8.97 ! [B: nat,A: nat,C: nat] :
% 16.19/8.97 ( ( ord_less_eq_nat @ B @ A )
% 16.19/8.97 => ( ( ord_less_nat @ C @ B )
% 16.19/8.97 => ( ord_less_nat @ C @ A ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4722_dual__order_Ostrict__trans1,axiom,
% 16.19/8.97 ! [B: int,A: int,C: int] :
% 16.19/8.97 ( ( ord_less_eq_int @ B @ A )
% 16.19/8.97 => ( ( ord_less_int @ C @ B )
% 16.19/8.97 => ( ord_less_int @ C @ A ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4723_dual__order_Ostrict__trans2,axiom,
% 16.19/8.97 ! [B: real,A: real,C: real] :
% 16.19/8.97 ( ( ord_less_real @ B @ A )
% 16.19/8.97 => ( ( ord_less_eq_real @ C @ B )
% 16.19/8.97 => ( ord_less_real @ C @ A ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4724_dual__order_Ostrict__trans2,axiom,
% 16.19/8.97 ! [B: set_nat,A: set_nat,C: set_nat] :
% 16.19/8.97 ( ( ord_less_set_nat @ B @ A )
% 16.19/8.97 => ( ( ord_less_eq_set_nat @ C @ B )
% 16.19/8.97 => ( ord_less_set_nat @ C @ A ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4725_dual__order_Ostrict__trans2,axiom,
% 16.19/8.97 ! [B: rat,A: rat,C: rat] :
% 16.19/8.97 ( ( ord_less_rat @ B @ A )
% 16.19/8.97 => ( ( ord_less_eq_rat @ C @ B )
% 16.19/8.97 => ( ord_less_rat @ C @ A ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4726_dual__order_Ostrict__trans2,axiom,
% 16.19/8.97 ! [B: num,A: num,C: num] :
% 16.19/8.97 ( ( ord_less_num @ B @ A )
% 16.19/8.97 => ( ( ord_less_eq_num @ C @ B )
% 16.19/8.97 => ( ord_less_num @ C @ A ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4727_dual__order_Ostrict__trans2,axiom,
% 16.19/8.97 ! [B: nat,A: nat,C: nat] :
% 16.19/8.97 ( ( ord_less_nat @ B @ A )
% 16.19/8.97 => ( ( ord_less_eq_nat @ C @ B )
% 16.19/8.97 => ( ord_less_nat @ C @ A ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4728_dual__order_Ostrict__trans2,axiom,
% 16.19/8.97 ! [B: int,A: int,C: int] :
% 16.19/8.97 ( ( ord_less_int @ B @ A )
% 16.19/8.97 => ( ( ord_less_eq_int @ C @ B )
% 16.19/8.97 => ( ord_less_int @ C @ A ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4729_dual__order_Ostrict__iff__not,axiom,
% 16.19/8.97 ( ord_less_real
% 16.19/8.97 = ( ^ [B5: real,A5: real] :
% 16.19/8.97 ( ( ord_less_eq_real @ B5 @ A5 )
% 16.19/8.97 & ~ ( ord_less_eq_real @ A5 @ B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4730_dual__order_Ostrict__iff__not,axiom,
% 16.19/8.97 ( ord_less_set_nat
% 16.19/8.97 = ( ^ [B5: set_nat,A5: set_nat] :
% 16.19/8.97 ( ( ord_less_eq_set_nat @ B5 @ A5 )
% 16.19/8.97 & ~ ( ord_less_eq_set_nat @ A5 @ B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4731_dual__order_Ostrict__iff__not,axiom,
% 16.19/8.97 ( ord_less_rat
% 16.19/8.97 = ( ^ [B5: rat,A5: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ B5 @ A5 )
% 16.19/8.97 & ~ ( ord_less_eq_rat @ A5 @ B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4732_dual__order_Ostrict__iff__not,axiom,
% 16.19/8.97 ( ord_less_num
% 16.19/8.97 = ( ^ [B5: num,A5: num] :
% 16.19/8.97 ( ( ord_less_eq_num @ B5 @ A5 )
% 16.19/8.97 & ~ ( ord_less_eq_num @ A5 @ B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4733_dual__order_Ostrict__iff__not,axiom,
% 16.19/8.97 ( ord_less_nat
% 16.19/8.97 = ( ^ [B5: nat,A5: nat] :
% 16.19/8.97 ( ( ord_less_eq_nat @ B5 @ A5 )
% 16.19/8.97 & ~ ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4734_dual__order_Ostrict__iff__not,axiom,
% 16.19/8.97 ( ord_less_int
% 16.19/8.97 = ( ^ [B5: int,A5: int] :
% 16.19/8.97 ( ( ord_less_eq_int @ B5 @ A5 )
% 16.19/8.97 & ~ ( ord_less_eq_int @ A5 @ B5 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4735_order_Ostrict__implies__order,axiom,
% 16.19/8.97 ! [A: real,B: real] :
% 16.19/8.97 ( ( ord_less_real @ A @ B )
% 16.19/8.97 => ( ord_less_eq_real @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4736_order_Ostrict__implies__order,axiom,
% 16.19/8.97 ! [A: set_nat,B: set_nat] :
% 16.19/8.97 ( ( ord_less_set_nat @ A @ B )
% 16.19/8.97 => ( ord_less_eq_set_nat @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4737_order_Ostrict__implies__order,axiom,
% 16.19/8.97 ! [A: rat,B: rat] :
% 16.19/8.97 ( ( ord_less_rat @ A @ B )
% 16.19/8.97 => ( ord_less_eq_rat @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4738_order_Ostrict__implies__order,axiom,
% 16.19/8.97 ! [A: num,B: num] :
% 16.19/8.97 ( ( ord_less_num @ A @ B )
% 16.19/8.97 => ( ord_less_eq_num @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4739_order_Ostrict__implies__order,axiom,
% 16.19/8.97 ! [A: nat,B: nat] :
% 16.19/8.97 ( ( ord_less_nat @ A @ B )
% 16.19/8.97 => ( ord_less_eq_nat @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4740_order_Ostrict__implies__order,axiom,
% 16.19/8.97 ! [A: int,B: int] :
% 16.19/8.97 ( ( ord_less_int @ A @ B )
% 16.19/8.97 => ( ord_less_eq_int @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4741_dual__order_Ostrict__implies__order,axiom,
% 16.19/8.97 ! [B: real,A: real] :
% 16.19/8.97 ( ( ord_less_real @ B @ A )
% 16.19/8.97 => ( ord_less_eq_real @ B @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4742_dual__order_Ostrict__implies__order,axiom,
% 16.19/8.97 ! [B: set_nat,A: set_nat] :
% 16.19/8.97 ( ( ord_less_set_nat @ B @ A )
% 16.19/8.97 => ( ord_less_eq_set_nat @ B @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4743_dual__order_Ostrict__implies__order,axiom,
% 16.19/8.97 ! [B: rat,A: rat] :
% 16.19/8.97 ( ( ord_less_rat @ B @ A )
% 16.19/8.97 => ( ord_less_eq_rat @ B @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4744_dual__order_Ostrict__implies__order,axiom,
% 16.19/8.97 ! [B: num,A: num] :
% 16.19/8.97 ( ( ord_less_num @ B @ A )
% 16.19/8.97 => ( ord_less_eq_num @ B @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4745_dual__order_Ostrict__implies__order,axiom,
% 16.19/8.97 ! [B: nat,A: nat] :
% 16.19/8.97 ( ( ord_less_nat @ B @ A )
% 16.19/8.97 => ( ord_less_eq_nat @ B @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4746_dual__order_Ostrict__implies__order,axiom,
% 16.19/8.97 ! [B: int,A: int] :
% 16.19/8.97 ( ( ord_less_int @ B @ A )
% 16.19/8.97 => ( ord_less_eq_int @ B @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4747_order__le__less,axiom,
% 16.19/8.97 ( ord_less_eq_real
% 16.19/8.97 = ( ^ [X2: real,Y2: real] :
% 16.19/8.97 ( ( ord_less_real @ X2 @ Y2 )
% 16.19/8.97 | ( X2 = Y2 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4748_order__le__less,axiom,
% 16.19/8.97 ( ord_less_eq_set_nat
% 16.19/8.97 = ( ^ [X2: set_nat,Y2: set_nat] :
% 16.19/8.97 ( ( ord_less_set_nat @ X2 @ Y2 )
% 16.19/8.97 | ( X2 = Y2 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4749_order__le__less,axiom,
% 16.19/8.97 ( ord_less_eq_rat
% 16.19/8.97 = ( ^ [X2: rat,Y2: rat] :
% 16.19/8.97 ( ( ord_less_rat @ X2 @ Y2 )
% 16.19/8.97 | ( X2 = Y2 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4750_order__le__less,axiom,
% 16.19/8.97 ( ord_less_eq_num
% 16.19/8.97 = ( ^ [X2: num,Y2: num] :
% 16.19/8.97 ( ( ord_less_num @ X2 @ Y2 )
% 16.19/8.97 | ( X2 = Y2 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4751_order__le__less,axiom,
% 16.19/8.97 ( ord_less_eq_nat
% 16.19/8.97 = ( ^ [X2: nat,Y2: nat] :
% 16.19/8.97 ( ( ord_less_nat @ X2 @ Y2 )
% 16.19/8.97 | ( X2 = Y2 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4752_order__le__less,axiom,
% 16.19/8.97 ( ord_less_eq_int
% 16.19/8.97 = ( ^ [X2: int,Y2: int] :
% 16.19/8.97 ( ( ord_less_int @ X2 @ Y2 )
% 16.19/8.97 | ( X2 = Y2 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4753_order__less__le,axiom,
% 16.19/8.97 ( ord_less_real
% 16.19/8.97 = ( ^ [X2: real,Y2: real] :
% 16.19/8.97 ( ( ord_less_eq_real @ X2 @ Y2 )
% 16.19/8.97 & ( X2 != Y2 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4754_order__less__le,axiom,
% 16.19/8.97 ( ord_less_set_nat
% 16.19/8.97 = ( ^ [X2: set_nat,Y2: set_nat] :
% 16.19/8.97 ( ( ord_less_eq_set_nat @ X2 @ Y2 )
% 16.19/8.97 & ( X2 != Y2 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4755_order__less__le,axiom,
% 16.19/8.97 ( ord_less_rat
% 16.19/8.97 = ( ^ [X2: rat,Y2: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ X2 @ Y2 )
% 16.19/8.97 & ( X2 != Y2 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4756_order__less__le,axiom,
% 16.19/8.97 ( ord_less_num
% 16.19/8.97 = ( ^ [X2: num,Y2: num] :
% 16.19/8.97 ( ( ord_less_eq_num @ X2 @ Y2 )
% 16.19/8.97 & ( X2 != Y2 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4757_order__less__le,axiom,
% 16.19/8.97 ( ord_less_nat
% 16.19/8.97 = ( ^ [X2: nat,Y2: nat] :
% 16.19/8.97 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 16.19/8.97 & ( X2 != Y2 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4758_order__less__le,axiom,
% 16.19/8.97 ( ord_less_int
% 16.19/8.97 = ( ^ [X2: int,Y2: int] :
% 16.19/8.97 ( ( ord_less_eq_int @ X2 @ Y2 )
% 16.19/8.97 & ( X2 != Y2 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4759_linorder__not__le,axiom,
% 16.19/8.97 ! [X: real,Y: real] :
% 16.19/8.97 ( ( ~ ( ord_less_eq_real @ X @ Y ) )
% 16.19/8.97 = ( ord_less_real @ Y @ X ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4760_linorder__not__le,axiom,
% 16.19/8.97 ! [X: rat,Y: rat] :
% 16.19/8.97 ( ( ~ ( ord_less_eq_rat @ X @ Y ) )
% 16.19/8.97 = ( ord_less_rat @ Y @ X ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4761_linorder__not__le,axiom,
% 16.19/8.97 ! [X: num,Y: num] :
% 16.19/8.97 ( ( ~ ( ord_less_eq_num @ X @ Y ) )
% 16.19/8.97 = ( ord_less_num @ Y @ X ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4762_linorder__not__le,axiom,
% 16.19/8.97 ! [X: nat,Y: nat] :
% 16.19/8.97 ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
% 16.19/8.97 = ( ord_less_nat @ Y @ X ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4763_linorder__not__le,axiom,
% 16.19/8.97 ! [X: int,Y: int] :
% 16.19/8.97 ( ( ~ ( ord_less_eq_int @ X @ Y ) )
% 16.19/8.97 = ( ord_less_int @ Y @ X ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4764_linorder__not__less,axiom,
% 16.19/8.97 ! [X: real,Y: real] :
% 16.19/8.97 ( ( ~ ( ord_less_real @ X @ Y ) )
% 16.19/8.97 = ( ord_less_eq_real @ Y @ X ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4765_linorder__not__less,axiom,
% 16.19/8.97 ! [X: rat,Y: rat] :
% 16.19/8.97 ( ( ~ ( ord_less_rat @ X @ Y ) )
% 16.19/8.97 = ( ord_less_eq_rat @ Y @ X ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4766_linorder__not__less,axiom,
% 16.19/8.97 ! [X: num,Y: num] :
% 16.19/8.97 ( ( ~ ( ord_less_num @ X @ Y ) )
% 16.19/8.97 = ( ord_less_eq_num @ Y @ X ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4767_linorder__not__less,axiom,
% 16.19/8.97 ! [X: nat,Y: nat] :
% 16.19/8.97 ( ( ~ ( ord_less_nat @ X @ Y ) )
% 16.19/8.97 = ( ord_less_eq_nat @ Y @ X ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4768_linorder__not__less,axiom,
% 16.19/8.97 ! [X: int,Y: int] :
% 16.19/8.97 ( ( ~ ( ord_less_int @ X @ Y ) )
% 16.19/8.97 = ( ord_less_eq_int @ Y @ X ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4769_order__less__imp__le,axiom,
% 16.19/8.97 ! [X: real,Y: real] :
% 16.19/8.97 ( ( ord_less_real @ X @ Y )
% 16.19/8.97 => ( ord_less_eq_real @ X @ Y ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4770_order__less__imp__le,axiom,
% 16.19/8.97 ! [X: set_nat,Y: set_nat] :
% 16.19/8.97 ( ( ord_less_set_nat @ X @ Y )
% 16.19/8.97 => ( ord_less_eq_set_nat @ X @ Y ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4771_order__less__imp__le,axiom,
% 16.19/8.97 ! [X: rat,Y: rat] :
% 16.19/8.97 ( ( ord_less_rat @ X @ Y )
% 16.19/8.97 => ( ord_less_eq_rat @ X @ Y ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4772_order__less__imp__le,axiom,
% 16.19/8.97 ! [X: num,Y: num] :
% 16.19/8.97 ( ( ord_less_num @ X @ Y )
% 16.19/8.97 => ( ord_less_eq_num @ X @ Y ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4773_order__less__imp__le,axiom,
% 16.19/8.97 ! [X: nat,Y: nat] :
% 16.19/8.97 ( ( ord_less_nat @ X @ Y )
% 16.19/8.97 => ( ord_less_eq_nat @ X @ Y ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4774_order__less__imp__le,axiom,
% 16.19/8.97 ! [X: int,Y: int] :
% 16.19/8.97 ( ( ord_less_int @ X @ Y )
% 16.19/8.97 => ( ord_less_eq_int @ X @ Y ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4775_order__le__neq__trans,axiom,
% 16.19/8.97 ! [A: real,B: real] :
% 16.19/8.97 ( ( ord_less_eq_real @ A @ B )
% 16.19/8.97 => ( ( A != B )
% 16.19/8.97 => ( ord_less_real @ A @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4776_order__le__neq__trans,axiom,
% 16.19/8.97 ! [A: set_nat,B: set_nat] :
% 16.19/8.97 ( ( ord_less_eq_set_nat @ A @ B )
% 16.19/8.97 => ( ( A != B )
% 16.19/8.97 => ( ord_less_set_nat @ A @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4777_order__le__neq__trans,axiom,
% 16.19/8.97 ! [A: rat,B: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.97 => ( ( A != B )
% 16.19/8.97 => ( ord_less_rat @ A @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4778_order__le__neq__trans,axiom,
% 16.19/8.97 ! [A: num,B: num] :
% 16.19/8.97 ( ( ord_less_eq_num @ A @ B )
% 16.19/8.97 => ( ( A != B )
% 16.19/8.97 => ( ord_less_num @ A @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4779_order__le__neq__trans,axiom,
% 16.19/8.97 ! [A: nat,B: nat] :
% 16.19/8.97 ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.97 => ( ( A != B )
% 16.19/8.97 => ( ord_less_nat @ A @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4780_order__le__neq__trans,axiom,
% 16.19/8.97 ! [A: int,B: int] :
% 16.19/8.97 ( ( ord_less_eq_int @ A @ B )
% 16.19/8.97 => ( ( A != B )
% 16.19/8.97 => ( ord_less_int @ A @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4781_order__neq__le__trans,axiom,
% 16.19/8.97 ! [A: real,B: real] :
% 16.19/8.97 ( ( A != B )
% 16.19/8.97 => ( ( ord_less_eq_real @ A @ B )
% 16.19/8.97 => ( ord_less_real @ A @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4782_order__neq__le__trans,axiom,
% 16.19/8.97 ! [A: set_nat,B: set_nat] :
% 16.19/8.97 ( ( A != B )
% 16.19/8.97 => ( ( ord_less_eq_set_nat @ A @ B )
% 16.19/8.97 => ( ord_less_set_nat @ A @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4783_order__neq__le__trans,axiom,
% 16.19/8.97 ! [A: rat,B: rat] :
% 16.19/8.97 ( ( A != B )
% 16.19/8.97 => ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.97 => ( ord_less_rat @ A @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4784_order__neq__le__trans,axiom,
% 16.19/8.97 ! [A: num,B: num] :
% 16.19/8.97 ( ( A != B )
% 16.19/8.97 => ( ( ord_less_eq_num @ A @ B )
% 16.19/8.97 => ( ord_less_num @ A @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4785_order__neq__le__trans,axiom,
% 16.19/8.97 ! [A: nat,B: nat] :
% 16.19/8.97 ( ( A != B )
% 16.19/8.97 => ( ( ord_less_eq_nat @ A @ B )
% 16.19/8.97 => ( ord_less_nat @ A @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4786_order__neq__le__trans,axiom,
% 16.19/8.97 ! [A: int,B: int] :
% 16.19/8.97 ( ( A != B )
% 16.19/8.97 => ( ( ord_less_eq_int @ A @ B )
% 16.19/8.97 => ( ord_less_int @ A @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4787_order__le__less__trans,axiom,
% 16.19/8.97 ! [X: real,Y: real,Z: real] :
% 16.19/8.97 ( ( ord_less_eq_real @ X @ Y )
% 16.19/8.97 => ( ( ord_less_real @ Y @ Z )
% 16.19/8.97 => ( ord_less_real @ X @ Z ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4788_order__le__less__trans,axiom,
% 16.19/8.97 ! [X: set_nat,Y: set_nat,Z: set_nat] :
% 16.19/8.97 ( ( ord_less_eq_set_nat @ X @ Y )
% 16.19/8.97 => ( ( ord_less_set_nat @ Y @ Z )
% 16.19/8.97 => ( ord_less_set_nat @ X @ Z ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4789_order__le__less__trans,axiom,
% 16.19/8.97 ! [X: rat,Y: rat,Z: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ X @ Y )
% 16.19/8.97 => ( ( ord_less_rat @ Y @ Z )
% 16.19/8.97 => ( ord_less_rat @ X @ Z ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4790_order__le__less__trans,axiom,
% 16.19/8.97 ! [X: num,Y: num,Z: num] :
% 16.19/8.97 ( ( ord_less_eq_num @ X @ Y )
% 16.19/8.97 => ( ( ord_less_num @ Y @ Z )
% 16.19/8.97 => ( ord_less_num @ X @ Z ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4791_order__le__less__trans,axiom,
% 16.19/8.97 ! [X: nat,Y: nat,Z: nat] :
% 16.19/8.97 ( ( ord_less_eq_nat @ X @ Y )
% 16.19/8.97 => ( ( ord_less_nat @ Y @ Z )
% 16.19/8.97 => ( ord_less_nat @ X @ Z ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4792_order__le__less__trans,axiom,
% 16.19/8.97 ! [X: int,Y: int,Z: int] :
% 16.19/8.97 ( ( ord_less_eq_int @ X @ Y )
% 16.19/8.97 => ( ( ord_less_int @ Y @ Z )
% 16.19/8.97 => ( ord_less_int @ X @ Z ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4793_order__less__le__trans,axiom,
% 16.19/8.97 ! [X: real,Y: real,Z: real] :
% 16.19/8.97 ( ( ord_less_real @ X @ Y )
% 16.19/8.97 => ( ( ord_less_eq_real @ Y @ Z )
% 16.19/8.97 => ( ord_less_real @ X @ Z ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4794_order__less__le__trans,axiom,
% 16.19/8.97 ! [X: set_nat,Y: set_nat,Z: set_nat] :
% 16.19/8.97 ( ( ord_less_set_nat @ X @ Y )
% 16.19/8.97 => ( ( ord_less_eq_set_nat @ Y @ Z )
% 16.19/8.97 => ( ord_less_set_nat @ X @ Z ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4795_order__less__le__trans,axiom,
% 16.19/8.97 ! [X: rat,Y: rat,Z: rat] :
% 16.19/8.97 ( ( ord_less_rat @ X @ Y )
% 16.19/8.97 => ( ( ord_less_eq_rat @ Y @ Z )
% 16.19/8.97 => ( ord_less_rat @ X @ Z ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4796_order__less__le__trans,axiom,
% 16.19/8.97 ! [X: num,Y: num,Z: num] :
% 16.19/8.97 ( ( ord_less_num @ X @ Y )
% 16.19/8.97 => ( ( ord_less_eq_num @ Y @ Z )
% 16.19/8.97 => ( ord_less_num @ X @ Z ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4797_order__less__le__trans,axiom,
% 16.19/8.97 ! [X: nat,Y: nat,Z: nat] :
% 16.19/8.97 ( ( ord_less_nat @ X @ Y )
% 16.19/8.97 => ( ( ord_less_eq_nat @ Y @ Z )
% 16.19/8.97 => ( ord_less_nat @ X @ Z ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4798_order__less__le__trans,axiom,
% 16.19/8.97 ! [X: int,Y: int,Z: int] :
% 16.19/8.97 ( ( ord_less_int @ X @ Y )
% 16.19/8.97 => ( ( ord_less_eq_int @ Y @ Z )
% 16.19/8.97 => ( ord_less_int @ X @ Z ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4799_order__le__less__subst1,axiom,
% 16.19/8.97 ! [A: real,F: real > real,B: real,C: real] :
% 16.19/8.97 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 16.19/8.97 => ( ( ord_less_real @ B @ C )
% 16.19/8.97 => ( ! [X4: real,Y4: real] :
% 16.19/8.97 ( ( ord_less_real @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4800_order__le__less__subst1,axiom,
% 16.19/8.97 ! [A: real,F: rat > real,B: rat,C: rat] :
% 16.19/8.97 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 16.19/8.97 => ( ( ord_less_rat @ B @ C )
% 16.19/8.97 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.97 ( ( ord_less_rat @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4801_order__le__less__subst1,axiom,
% 16.19/8.97 ! [A: real,F: num > real,B: num,C: num] :
% 16.19/8.97 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 16.19/8.97 => ( ( ord_less_num @ B @ C )
% 16.19/8.97 => ( ! [X4: num,Y4: num] :
% 16.19/8.97 ( ( ord_less_num @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4802_order__le__less__subst1,axiom,
% 16.19/8.97 ! [A: real,F: nat > real,B: nat,C: nat] :
% 16.19/8.97 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 16.19/8.97 => ( ( ord_less_nat @ B @ C )
% 16.19/8.97 => ( ! [X4: nat,Y4: nat] :
% 16.19/8.97 ( ( ord_less_nat @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4803_order__le__less__subst1,axiom,
% 16.19/8.97 ! [A: real,F: int > real,B: int,C: int] :
% 16.19/8.97 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 16.19/8.97 => ( ( ord_less_int @ B @ C )
% 16.19/8.97 => ( ! [X4: int,Y4: int] :
% 16.19/8.97 ( ( ord_less_int @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4804_order__le__less__subst1,axiom,
% 16.19/8.97 ! [A: rat,F: real > rat,B: real,C: real] :
% 16.19/8.97 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 16.19/8.97 => ( ( ord_less_real @ B @ C )
% 16.19/8.97 => ( ! [X4: real,Y4: real] :
% 16.19/8.97 ( ( ord_less_real @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4805_order__le__less__subst1,axiom,
% 16.19/8.97 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 16.19/8.97 => ( ( ord_less_rat @ B @ C )
% 16.19/8.97 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.97 ( ( ord_less_rat @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4806_order__le__less__subst1,axiom,
% 16.19/8.97 ! [A: rat,F: num > rat,B: num,C: num] :
% 16.19/8.97 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 16.19/8.97 => ( ( ord_less_num @ B @ C )
% 16.19/8.97 => ( ! [X4: num,Y4: num] :
% 16.19/8.97 ( ( ord_less_num @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4807_order__le__less__subst1,axiom,
% 16.19/8.97 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 16.19/8.97 => ( ( ord_less_nat @ B @ C )
% 16.19/8.97 => ( ! [X4: nat,Y4: nat] :
% 16.19/8.97 ( ( ord_less_nat @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4808_order__le__less__subst1,axiom,
% 16.19/8.97 ! [A: rat,F: int > rat,B: int,C: int] :
% 16.19/8.97 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 16.19/8.97 => ( ( ord_less_int @ B @ C )
% 16.19/8.97 => ( ! [X4: int,Y4: int] :
% 16.19/8.97 ( ( ord_less_int @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4809_order__le__less__subst2,axiom,
% 16.19/8.97 ! [A: rat,B: rat,F: rat > real,C: real] :
% 16.19/8.97 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.97 => ( ( ord_less_real @ ( F @ B ) @ C )
% 16.19/8.97 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4810_order__le__less__subst2,axiom,
% 16.19/8.97 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.97 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 16.19/8.97 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4811_order__le__less__subst2,axiom,
% 16.19/8.97 ! [A: rat,B: rat,F: rat > num,C: num] :
% 16.19/8.97 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.97 => ( ( ord_less_num @ ( F @ B ) @ C )
% 16.19/8.97 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4812_order__le__less__subst2,axiom,
% 16.19/8.97 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.97 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 16.19/8.97 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4813_order__le__less__subst2,axiom,
% 16.19/8.97 ! [A: rat,B: rat,F: rat > int,C: int] :
% 16.19/8.97 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.97 => ( ( ord_less_int @ ( F @ B ) @ C )
% 16.19/8.97 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4814_order__le__less__subst2,axiom,
% 16.19/8.97 ! [A: num,B: num,F: num > real,C: real] :
% 16.19/8.97 ( ( ord_less_eq_num @ A @ B )
% 16.19/8.97 => ( ( ord_less_real @ ( F @ B ) @ C )
% 16.19/8.97 => ( ! [X4: num,Y4: num] :
% 16.19/8.97 ( ( ord_less_eq_num @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4815_order__le__less__subst2,axiom,
% 16.19/8.97 ! [A: num,B: num,F: num > rat,C: rat] :
% 16.19/8.97 ( ( ord_less_eq_num @ A @ B )
% 16.19/8.97 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 16.19/8.97 => ( ! [X4: num,Y4: num] :
% 16.19/8.97 ( ( ord_less_eq_num @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4816_order__le__less__subst2,axiom,
% 16.19/8.97 ! [A: num,B: num,F: num > num,C: num] :
% 16.19/8.97 ( ( ord_less_eq_num @ A @ B )
% 16.19/8.97 => ( ( ord_less_num @ ( F @ B ) @ C )
% 16.19/8.97 => ( ! [X4: num,Y4: num] :
% 16.19/8.97 ( ( ord_less_eq_num @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4817_order__le__less__subst2,axiom,
% 16.19/8.97 ! [A: num,B: num,F: num > nat,C: nat] :
% 16.19/8.97 ( ( ord_less_eq_num @ A @ B )
% 16.19/8.97 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 16.19/8.97 => ( ! [X4: num,Y4: num] :
% 16.19/8.97 ( ( ord_less_eq_num @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4818_order__le__less__subst2,axiom,
% 16.19/8.97 ! [A: num,B: num,F: num > int,C: int] :
% 16.19/8.97 ( ( ord_less_eq_num @ A @ B )
% 16.19/8.97 => ( ( ord_less_int @ ( F @ B ) @ C )
% 16.19/8.97 => ( ! [X4: num,Y4: num] :
% 16.19/8.97 ( ( ord_less_eq_num @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4819_order__less__le__subst1,axiom,
% 16.19/8.97 ! [A: real,F: rat > real,B: rat,C: rat] :
% 16.19/8.97 ( ( ord_less_real @ A @ ( F @ B ) )
% 16.19/8.97 => ( ( ord_less_eq_rat @ B @ C )
% 16.19/8.97 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4820_order__less__le__subst1,axiom,
% 16.19/8.97 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 16.19/8.97 ( ( ord_less_rat @ A @ ( F @ B ) )
% 16.19/8.97 => ( ( ord_less_eq_rat @ B @ C )
% 16.19/8.97 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4821_order__less__le__subst1,axiom,
% 16.19/8.97 ! [A: num,F: rat > num,B: rat,C: rat] :
% 16.19/8.97 ( ( ord_less_num @ A @ ( F @ B ) )
% 16.19/8.97 => ( ( ord_less_eq_rat @ B @ C )
% 16.19/8.97 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4822_order__less__le__subst1,axiom,
% 16.19/8.97 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 16.19/8.97 ( ( ord_less_nat @ A @ ( F @ B ) )
% 16.19/8.97 => ( ( ord_less_eq_rat @ B @ C )
% 16.19/8.97 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4823_order__less__le__subst1,axiom,
% 16.19/8.97 ! [A: int,F: rat > int,B: rat,C: rat] :
% 16.19/8.97 ( ( ord_less_int @ A @ ( F @ B ) )
% 16.19/8.97 => ( ( ord_less_eq_rat @ B @ C )
% 16.19/8.97 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4824_order__less__le__subst1,axiom,
% 16.19/8.97 ! [A: real,F: num > real,B: num,C: num] :
% 16.19/8.97 ( ( ord_less_real @ A @ ( F @ B ) )
% 16.19/8.97 => ( ( ord_less_eq_num @ B @ C )
% 16.19/8.97 => ( ! [X4: num,Y4: num] :
% 16.19/8.97 ( ( ord_less_eq_num @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4825_order__less__le__subst1,axiom,
% 16.19/8.97 ! [A: rat,F: num > rat,B: num,C: num] :
% 16.19/8.97 ( ( ord_less_rat @ A @ ( F @ B ) )
% 16.19/8.97 => ( ( ord_less_eq_num @ B @ C )
% 16.19/8.97 => ( ! [X4: num,Y4: num] :
% 16.19/8.97 ( ( ord_less_eq_num @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4826_order__less__le__subst1,axiom,
% 16.19/8.97 ! [A: num,F: num > num,B: num,C: num] :
% 16.19/8.97 ( ( ord_less_num @ A @ ( F @ B ) )
% 16.19/8.97 => ( ( ord_less_eq_num @ B @ C )
% 16.19/8.97 => ( ! [X4: num,Y4: num] :
% 16.19/8.97 ( ( ord_less_eq_num @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4827_order__less__le__subst1,axiom,
% 16.19/8.97 ! [A: nat,F: num > nat,B: num,C: num] :
% 16.19/8.97 ( ( ord_less_nat @ A @ ( F @ B ) )
% 16.19/8.97 => ( ( ord_less_eq_num @ B @ C )
% 16.19/8.97 => ( ! [X4: num,Y4: num] :
% 16.19/8.97 ( ( ord_less_eq_num @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4828_order__less__le__subst1,axiom,
% 16.19/8.97 ! [A: int,F: num > int,B: num,C: num] :
% 16.19/8.97 ( ( ord_less_int @ A @ ( F @ B ) )
% 16.19/8.97 => ( ( ord_less_eq_num @ B @ C )
% 16.19/8.97 => ( ! [X4: num,Y4: num] :
% 16.19/8.97 ( ( ord_less_eq_num @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4829_order__less__le__subst2,axiom,
% 16.19/8.97 ! [A: real,B: real,F: real > real,C: real] :
% 16.19/8.97 ( ( ord_less_real @ A @ B )
% 16.19/8.97 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 16.19/8.97 => ( ! [X4: real,Y4: real] :
% 16.19/8.97 ( ( ord_less_real @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4830_order__less__le__subst2,axiom,
% 16.19/8.97 ! [A: rat,B: rat,F: rat > real,C: real] :
% 16.19/8.97 ( ( ord_less_rat @ A @ B )
% 16.19/8.97 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 16.19/8.97 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.97 ( ( ord_less_rat @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4831_order__less__le__subst2,axiom,
% 16.19/8.97 ! [A: num,B: num,F: num > real,C: real] :
% 16.19/8.97 ( ( ord_less_num @ A @ B )
% 16.19/8.97 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 16.19/8.97 => ( ! [X4: num,Y4: num] :
% 16.19/8.97 ( ( ord_less_num @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4832_order__less__le__subst2,axiom,
% 16.19/8.97 ! [A: nat,B: nat,F: nat > real,C: real] :
% 16.19/8.97 ( ( ord_less_nat @ A @ B )
% 16.19/8.97 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 16.19/8.97 => ( ! [X4: nat,Y4: nat] :
% 16.19/8.97 ( ( ord_less_nat @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4833_order__less__le__subst2,axiom,
% 16.19/8.97 ! [A: int,B: int,F: int > real,C: real] :
% 16.19/8.97 ( ( ord_less_int @ A @ B )
% 16.19/8.97 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 16.19/8.97 => ( ! [X4: int,Y4: int] :
% 16.19/8.97 ( ( ord_less_int @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4834_order__less__le__subst2,axiom,
% 16.19/8.97 ! [A: real,B: real,F: real > rat,C: rat] :
% 16.19/8.97 ( ( ord_less_real @ A @ B )
% 16.19/8.97 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 16.19/8.97 => ( ! [X4: real,Y4: real] :
% 16.19/8.97 ( ( ord_less_real @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4835_order__less__le__subst2,axiom,
% 16.19/8.97 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 16.19/8.97 ( ( ord_less_rat @ A @ B )
% 16.19/8.97 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 16.19/8.97 => ( ! [X4: rat,Y4: rat] :
% 16.19/8.97 ( ( ord_less_rat @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4836_order__less__le__subst2,axiom,
% 16.19/8.97 ! [A: num,B: num,F: num > rat,C: rat] :
% 16.19/8.97 ( ( ord_less_num @ A @ B )
% 16.19/8.97 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 16.19/8.97 => ( ! [X4: num,Y4: num] :
% 16.19/8.97 ( ( ord_less_num @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4837_order__less__le__subst2,axiom,
% 16.19/8.97 ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 16.19/8.97 ( ( ord_less_nat @ A @ B )
% 16.19/8.97 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 16.19/8.97 => ( ! [X4: nat,Y4: nat] :
% 16.19/8.97 ( ( ord_less_nat @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4838_order__less__le__subst2,axiom,
% 16.19/8.97 ! [A: int,B: int,F: int > rat,C: rat] :
% 16.19/8.97 ( ( ord_less_int @ A @ B )
% 16.19/8.97 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 16.19/8.97 => ( ! [X4: int,Y4: int] :
% 16.19/8.97 ( ( ord_less_int @ X4 @ Y4 )
% 16.19/8.97 => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
% 16.19/8.97 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4839_linorder__le__less__linear,axiom,
% 16.19/8.97 ! [X: real,Y: real] :
% 16.19/8.97 ( ( ord_less_eq_real @ X @ Y )
% 16.19/8.97 | ( ord_less_real @ Y @ X ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4840_linorder__le__less__linear,axiom,
% 16.19/8.97 ! [X: rat,Y: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ X @ Y )
% 16.19/8.97 | ( ord_less_rat @ Y @ X ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4841_linorder__le__less__linear,axiom,
% 16.19/8.97 ! [X: num,Y: num] :
% 16.19/8.97 ( ( ord_less_eq_num @ X @ Y )
% 16.19/8.97 | ( ord_less_num @ Y @ X ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4842_linorder__le__less__linear,axiom,
% 16.19/8.97 ! [X: nat,Y: nat] :
% 16.19/8.97 ( ( ord_less_eq_nat @ X @ Y )
% 16.19/8.97 | ( ord_less_nat @ Y @ X ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4843_linorder__le__less__linear,axiom,
% 16.19/8.97 ! [X: int,Y: int] :
% 16.19/8.97 ( ( ord_less_eq_int @ X @ Y )
% 16.19/8.97 | ( ord_less_int @ Y @ X ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4844_order__le__imp__less__or__eq,axiom,
% 16.19/8.97 ! [X: real,Y: real] :
% 16.19/8.97 ( ( ord_less_eq_real @ X @ Y )
% 16.19/8.97 => ( ( ord_less_real @ X @ Y )
% 16.19/8.97 | ( X = Y ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4845_order__le__imp__less__or__eq,axiom,
% 16.19/8.97 ! [X: set_nat,Y: set_nat] :
% 16.19/8.97 ( ( ord_less_eq_set_nat @ X @ Y )
% 16.19/8.97 => ( ( ord_less_set_nat @ X @ Y )
% 16.19/8.97 | ( X = Y ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4846_order__le__imp__less__or__eq,axiom,
% 16.19/8.97 ! [X: rat,Y: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ X @ Y )
% 16.19/8.97 => ( ( ord_less_rat @ X @ Y )
% 16.19/8.97 | ( X = Y ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4847_order__le__imp__less__or__eq,axiom,
% 16.19/8.97 ! [X: num,Y: num] :
% 16.19/8.97 ( ( ord_less_eq_num @ X @ Y )
% 16.19/8.97 => ( ( ord_less_num @ X @ Y )
% 16.19/8.97 | ( X = Y ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4848_order__le__imp__less__or__eq,axiom,
% 16.19/8.97 ! [X: nat,Y: nat] :
% 16.19/8.97 ( ( ord_less_eq_nat @ X @ Y )
% 16.19/8.97 => ( ( ord_less_nat @ X @ Y )
% 16.19/8.97 | ( X = Y ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4849_order__le__imp__less__or__eq,axiom,
% 16.19/8.97 ! [X: int,Y: int] :
% 16.19/8.97 ( ( ord_less_eq_int @ X @ Y )
% 16.19/8.97 => ( ( ord_less_int @ X @ Y )
% 16.19/8.97 | ( X = Y ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4850_bot_Oextremum,axiom,
% 16.19/8.97 ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4851_bot_Oextremum,axiom,
% 16.19/8.97 ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4852_bot_Oextremum,axiom,
% 16.19/8.97 ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4853_bot_Oextremum,axiom,
% 16.19/8.97 ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4854_bot_Oextremum__unique,axiom,
% 16.19/8.97 ! [A: set_int] :
% 16.19/8.97 ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 16.19/8.97 = ( A = bot_bot_set_int ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4855_bot_Oextremum__unique,axiom,
% 16.19/8.97 ! [A: set_real] :
% 16.19/8.97 ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 16.19/8.97 = ( A = bot_bot_set_real ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4856_bot_Oextremum__unique,axiom,
% 16.19/8.97 ! [A: set_nat] :
% 16.19/8.97 ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 16.19/8.97 = ( A = bot_bot_set_nat ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4857_bot_Oextremum__unique,axiom,
% 16.19/8.97 ! [A: nat] :
% 16.19/8.97 ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 16.19/8.97 = ( A = bot_bot_nat ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4858_bot_Oextremum__uniqueI,axiom,
% 16.19/8.97 ! [A: set_int] :
% 16.19/8.97 ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 16.19/8.97 => ( A = bot_bot_set_int ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4859_bot_Oextremum__uniqueI,axiom,
% 16.19/8.97 ! [A: set_real] :
% 16.19/8.97 ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 16.19/8.97 => ( A = bot_bot_set_real ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4860_bot_Oextremum__uniqueI,axiom,
% 16.19/8.97 ! [A: set_nat] :
% 16.19/8.97 ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 16.19/8.97 => ( A = bot_bot_set_nat ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4861_bot_Oextremum__uniqueI,axiom,
% 16.19/8.97 ! [A: nat] :
% 16.19/8.97 ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 16.19/8.97 => ( A = bot_bot_nat ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4862_bot_Oextremum__strict,axiom,
% 16.19/8.97 ! [A: set_nat] :
% 16.19/8.97 ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4863_bot_Oextremum__strict,axiom,
% 16.19/8.97 ! [A: set_int] :
% 16.19/8.97 ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4864_bot_Oextremum__strict,axiom,
% 16.19/8.97 ! [A: set_real] :
% 16.19/8.97 ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4865_bot_Oextremum__strict,axiom,
% 16.19/8.97 ! [A: nat] :
% 16.19/8.97 ~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4866_bot_Onot__eq__extremum,axiom,
% 16.19/8.97 ! [A: set_nat] :
% 16.19/8.97 ( ( A != bot_bot_set_nat )
% 16.19/8.97 = ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4867_bot_Onot__eq__extremum,axiom,
% 16.19/8.97 ! [A: set_int] :
% 16.19/8.97 ( ( A != bot_bot_set_int )
% 16.19/8.97 = ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4868_bot_Onot__eq__extremum,axiom,
% 16.19/8.97 ! [A: set_real] :
% 16.19/8.97 ( ( A != bot_bot_set_real )
% 16.19/8.97 = ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4869_bot_Onot__eq__extremum,axiom,
% 16.19/8.97 ! [A: nat] :
% 16.19/8.97 ( ( A != bot_bot_nat )
% 16.19/8.97 = ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4870_list__decode_Ocases,axiom,
% 16.19/8.97 ! [X: nat] :
% 16.19/8.97 ( ( X != zero_zero_nat )
% 16.19/8.97 => ~ ! [N4: nat] :
% 16.19/8.97 ( X
% 16.19/8.97 != ( suc @ N4 ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4871_zero__less__power__eq,axiom,
% 16.19/8.97 ! [A: real,N2: nat] :
% 16.19/8.97 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
% 16.19/8.97 = ( ( N2 = zero_zero_nat )
% 16.19/8.97 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.97 & ( A != zero_zero_real ) )
% 16.19/8.97 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.97 & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4872_zero__less__power__eq,axiom,
% 16.19/8.97 ! [A: rat,N2: nat] :
% 16.19/8.97 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
% 16.19/8.97 = ( ( N2 = zero_zero_nat )
% 16.19/8.97 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.97 & ( A != zero_zero_rat ) )
% 16.19/8.97 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.97 & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4873_zero__less__power__eq,axiom,
% 16.19/8.97 ! [A: int,N2: nat] :
% 16.19/8.97 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
% 16.19/8.97 = ( ( N2 = zero_zero_nat )
% 16.19/8.97 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.97 & ( A != zero_zero_int ) )
% 16.19/8.97 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.97 & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4874_max__def,axiom,
% 16.19/8.97 ( ord_max_set_nat
% 16.19/8.97 = ( ^ [A5: set_nat,B5: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A5 @ B5 ) @ B5 @ A5 ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4875_max__def,axiom,
% 16.19/8.97 ( ord_max_rat
% 16.19/8.97 = ( ^ [A5: rat,B5: rat] : ( if_rat @ ( ord_less_eq_rat @ A5 @ B5 ) @ B5 @ A5 ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4876_max__def,axiom,
% 16.19/8.97 ( ord_max_num
% 16.19/8.97 = ( ^ [A5: num,B5: num] : ( if_num @ ( ord_less_eq_num @ A5 @ B5 ) @ B5 @ A5 ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4877_max__def,axiom,
% 16.19/8.97 ( ord_max_nat
% 16.19/8.97 = ( ^ [A5: nat,B5: nat] : ( if_nat @ ( ord_less_eq_nat @ A5 @ B5 ) @ B5 @ A5 ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4878_max__def,axiom,
% 16.19/8.97 ( ord_max_int
% 16.19/8.97 = ( ^ [A5: int,B5: int] : ( if_int @ ( ord_less_eq_int @ A5 @ B5 ) @ B5 @ A5 ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4879_max__absorb1,axiom,
% 16.19/8.97 ! [Y: set_nat,X: set_nat] :
% 16.19/8.97 ( ( ord_less_eq_set_nat @ Y @ X )
% 16.19/8.97 => ( ( ord_max_set_nat @ X @ Y )
% 16.19/8.97 = X ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4880_max__absorb1,axiom,
% 16.19/8.97 ! [Y: rat,X: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ Y @ X )
% 16.19/8.97 => ( ( ord_max_rat @ X @ Y )
% 16.19/8.97 = X ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4881_max__absorb1,axiom,
% 16.19/8.97 ! [Y: num,X: num] :
% 16.19/8.97 ( ( ord_less_eq_num @ Y @ X )
% 16.19/8.97 => ( ( ord_max_num @ X @ Y )
% 16.19/8.97 = X ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4882_max__absorb1,axiom,
% 16.19/8.97 ! [Y: nat,X: nat] :
% 16.19/8.97 ( ( ord_less_eq_nat @ Y @ X )
% 16.19/8.97 => ( ( ord_max_nat @ X @ Y )
% 16.19/8.97 = X ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4883_max__absorb1,axiom,
% 16.19/8.97 ! [Y: int,X: int] :
% 16.19/8.97 ( ( ord_less_eq_int @ Y @ X )
% 16.19/8.97 => ( ( ord_max_int @ X @ Y )
% 16.19/8.97 = X ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4884_max__absorb2,axiom,
% 16.19/8.97 ! [X: set_nat,Y: set_nat] :
% 16.19/8.97 ( ( ord_less_eq_set_nat @ X @ Y )
% 16.19/8.97 => ( ( ord_max_set_nat @ X @ Y )
% 16.19/8.97 = Y ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4885_max__absorb2,axiom,
% 16.19/8.97 ! [X: rat,Y: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ X @ Y )
% 16.19/8.97 => ( ( ord_max_rat @ X @ Y )
% 16.19/8.97 = Y ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4886_max__absorb2,axiom,
% 16.19/8.97 ! [X: num,Y: num] :
% 16.19/8.97 ( ( ord_less_eq_num @ X @ Y )
% 16.19/8.97 => ( ( ord_max_num @ X @ Y )
% 16.19/8.97 = Y ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4887_max__absorb2,axiom,
% 16.19/8.97 ! [X: nat,Y: nat] :
% 16.19/8.97 ( ( ord_less_eq_nat @ X @ Y )
% 16.19/8.97 => ( ( ord_max_nat @ X @ Y )
% 16.19/8.97 = Y ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4888_max__absorb2,axiom,
% 16.19/8.97 ! [X: int,Y: int] :
% 16.19/8.97 ( ( ord_less_eq_int @ X @ Y )
% 16.19/8.97 => ( ( ord_max_int @ X @ Y )
% 16.19/8.97 = Y ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4889_even__mask__div__iff_H,axiom,
% 16.19/8.97 ! [M: nat,N2: nat] :
% 16.19/8.97 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) )
% 16.19/8.97 = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4890_even__mask__div__iff_H,axiom,
% 16.19/8.97 ! [M: nat,N2: nat] :
% 16.19/8.97 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 16.19/8.97 = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4891_even__mask__div__iff_H,axiom,
% 16.19/8.97 ! [M: nat,N2: nat] :
% 16.19/8.97 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
% 16.19/8.97 = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4892_power__le__zero__eq,axiom,
% 16.19/8.97 ! [A: real,N2: nat] :
% 16.19/8.97 ( ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ zero_zero_real )
% 16.19/8.97 = ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.97 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.97 & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 16.19/8.97 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.97 & ( A = zero_zero_real ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4893_power__le__zero__eq,axiom,
% 16.19/8.97 ! [A: rat,N2: nat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ zero_zero_rat )
% 16.19/8.97 = ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.97 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.97 & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 16.19/8.97 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.97 & ( A = zero_zero_rat ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4894_power__le__zero__eq,axiom,
% 16.19/8.97 ! [A: int,N2: nat] :
% 16.19/8.97 ( ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ zero_zero_int )
% 16.19/8.97 = ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/8.97 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.97 & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 16.19/8.97 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.97 & ( A = zero_zero_int ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4895_even__mod__4__div__2,axiom,
% 16.19/8.97 ! [N2: nat] :
% 16.19/8.97 ( ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 16.19/8.97 = ( suc @ zero_zero_nat ) )
% 16.19/8.97 => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4896_even__mask__div__iff,axiom,
% 16.19/8.97 ! [M: nat,N2: nat] :
% 16.19/8.97 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) )
% 16.19/8.97 = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.97 = zero_z3403309356797280102nteger )
% 16.19/8.97 | ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4897_even__mask__div__iff,axiom,
% 16.19/8.97 ! [M: nat,N2: nat] :
% 16.19/8.97 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 16.19/8.97 = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.97 = zero_zero_nat )
% 16.19/8.97 | ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4898_even__mask__div__iff,axiom,
% 16.19/8.97 ! [M: nat,N2: nat] :
% 16.19/8.97 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
% 16.19/8.97 = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.97 = zero_zero_int )
% 16.19/8.97 | ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4899_even__mult__exp__div__exp__iff,axiom,
% 16.19/8.97 ! [A: code_integer,M: nat,N2: nat] :
% 16.19/8.97 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) )
% 16.19/8.97 = ( ( ord_less_nat @ N2 @ M )
% 16.19/8.97 | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.97 = zero_z3403309356797280102nteger )
% 16.19/8.97 | ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/8.97 & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4900_even__mult__exp__div__exp__iff,axiom,
% 16.19/8.97 ! [A: nat,M: nat,N2: nat] :
% 16.19/8.97 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 16.19/8.97 = ( ( ord_less_nat @ N2 @ M )
% 16.19/8.97 | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.97 = zero_zero_nat )
% 16.19/8.97 | ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/8.97 & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4901_even__mult__exp__div__exp__iff,axiom,
% 16.19/8.97 ! [A: int,M: nat,N2: nat] :
% 16.19/8.97 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
% 16.19/8.97 = ( ( ord_less_nat @ N2 @ M )
% 16.19/8.97 | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.97 = zero_zero_int )
% 16.19/8.97 | ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/8.97 & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4902_ex__min__if__finite,axiom,
% 16.19/8.97 ! [S: set_real] :
% 16.19/8.97 ( ( finite_finite_real @ S )
% 16.19/8.97 => ( ( S != bot_bot_set_real )
% 16.19/8.97 => ? [X4: real] :
% 16.19/8.97 ( ( member_real @ X4 @ S )
% 16.19/8.97 & ~ ? [Xa: real] :
% 16.19/8.97 ( ( member_real @ Xa @ S )
% 16.19/8.97 & ( ord_less_real @ Xa @ X4 ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4903_ex__min__if__finite,axiom,
% 16.19/8.97 ! [S: set_rat] :
% 16.19/8.97 ( ( finite_finite_rat @ S )
% 16.19/8.97 => ( ( S != bot_bot_set_rat )
% 16.19/8.97 => ? [X4: rat] :
% 16.19/8.97 ( ( member_rat @ X4 @ S )
% 16.19/8.97 & ~ ? [Xa: rat] :
% 16.19/8.97 ( ( member_rat @ Xa @ S )
% 16.19/8.97 & ( ord_less_rat @ Xa @ X4 ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4904_ex__min__if__finite,axiom,
% 16.19/8.97 ! [S: set_num] :
% 16.19/8.97 ( ( finite_finite_num @ S )
% 16.19/8.97 => ( ( S != bot_bot_set_num )
% 16.19/8.97 => ? [X4: num] :
% 16.19/8.97 ( ( member_num @ X4 @ S )
% 16.19/8.97 & ~ ? [Xa: num] :
% 16.19/8.97 ( ( member_num @ Xa @ S )
% 16.19/8.97 & ( ord_less_num @ Xa @ X4 ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4905_ex__min__if__finite,axiom,
% 16.19/8.97 ! [S: set_nat] :
% 16.19/8.97 ( ( finite_finite_nat @ S )
% 16.19/8.97 => ( ( S != bot_bot_set_nat )
% 16.19/8.97 => ? [X4: nat] :
% 16.19/8.97 ( ( member_nat @ X4 @ S )
% 16.19/8.97 & ~ ? [Xa: nat] :
% 16.19/8.97 ( ( member_nat @ Xa @ S )
% 16.19/8.97 & ( ord_less_nat @ Xa @ X4 ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4906_ex__min__if__finite,axiom,
% 16.19/8.97 ! [S: set_int] :
% 16.19/8.97 ( ( finite_finite_int @ S )
% 16.19/8.97 => ( ( S != bot_bot_set_int )
% 16.19/8.97 => ? [X4: int] :
% 16.19/8.97 ( ( member_int @ X4 @ S )
% 16.19/8.97 & ~ ? [Xa: int] :
% 16.19/8.97 ( ( member_int @ Xa @ S )
% 16.19/8.97 & ( ord_less_int @ Xa @ X4 ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4907_infinite__growing,axiom,
% 16.19/8.97 ! [X8: set_real] :
% 16.19/8.97 ( ( X8 != bot_bot_set_real )
% 16.19/8.97 => ( ! [X4: real] :
% 16.19/8.97 ( ( member_real @ X4 @ X8 )
% 16.19/8.97 => ? [Xa: real] :
% 16.19/8.97 ( ( member_real @ Xa @ X8 )
% 16.19/8.97 & ( ord_less_real @ X4 @ Xa ) ) )
% 16.19/8.97 => ~ ( finite_finite_real @ X8 ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4908_infinite__growing,axiom,
% 16.19/8.97 ! [X8: set_rat] :
% 16.19/8.97 ( ( X8 != bot_bot_set_rat )
% 16.19/8.97 => ( ! [X4: rat] :
% 16.19/8.97 ( ( member_rat @ X4 @ X8 )
% 16.19/8.97 => ? [Xa: rat] :
% 16.19/8.97 ( ( member_rat @ Xa @ X8 )
% 16.19/8.97 & ( ord_less_rat @ X4 @ Xa ) ) )
% 16.19/8.97 => ~ ( finite_finite_rat @ X8 ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4909_infinite__growing,axiom,
% 16.19/8.97 ! [X8: set_num] :
% 16.19/8.97 ( ( X8 != bot_bot_set_num )
% 16.19/8.97 => ( ! [X4: num] :
% 16.19/8.97 ( ( member_num @ X4 @ X8 )
% 16.19/8.97 => ? [Xa: num] :
% 16.19/8.97 ( ( member_num @ Xa @ X8 )
% 16.19/8.97 & ( ord_less_num @ X4 @ Xa ) ) )
% 16.19/8.97 => ~ ( finite_finite_num @ X8 ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4910_infinite__growing,axiom,
% 16.19/8.97 ! [X8: set_nat] :
% 16.19/8.97 ( ( X8 != bot_bot_set_nat )
% 16.19/8.97 => ( ! [X4: nat] :
% 16.19/8.97 ( ( member_nat @ X4 @ X8 )
% 16.19/8.97 => ? [Xa: nat] :
% 16.19/8.97 ( ( member_nat @ Xa @ X8 )
% 16.19/8.97 & ( ord_less_nat @ X4 @ Xa ) ) )
% 16.19/8.97 => ~ ( finite_finite_nat @ X8 ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4911_infinite__growing,axiom,
% 16.19/8.97 ! [X8: set_int] :
% 16.19/8.97 ( ( X8 != bot_bot_set_int )
% 16.19/8.97 => ( ! [X4: int] :
% 16.19/8.97 ( ( member_int @ X4 @ X8 )
% 16.19/8.97 => ? [Xa: int] :
% 16.19/8.97 ( ( member_int @ Xa @ X8 )
% 16.19/8.97 & ( ord_less_int @ X4 @ Xa ) ) )
% 16.19/8.97 => ~ ( finite_finite_int @ X8 ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4912_triangle__def,axiom,
% 16.19/8.97 ( nat_triangle
% 16.19/8.97 = ( ^ [N: nat] : ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4913_vebt__buildup_Oelims,axiom,
% 16.19/8.97 ! [X: nat,Y: vEBT_VEBT] :
% 16.19/8.97 ( ( ( vEBT_vebt_buildup @ X )
% 16.19/8.97 = Y )
% 16.19/8.97 => ( ( ( X = zero_zero_nat )
% 16.19/8.97 => ( Y
% 16.19/8.97 != ( vEBT_Leaf @ $false @ $false ) ) )
% 16.19/8.97 => ( ( ( X
% 16.19/8.97 = ( suc @ zero_zero_nat ) )
% 16.19/8.97 => ( Y
% 16.19/8.97 != ( vEBT_Leaf @ $false @ $false ) ) )
% 16.19/8.97 => ~ ! [Va2: nat] :
% 16.19/8.97 ( ( X
% 16.19/8.97 = ( suc @ ( suc @ Va2 ) ) )
% 16.19/8.97 => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 16.19/8.97 => ( Y
% 16.19/8.97 = ( 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 ) ) ) ) ) ) )
% 16.19/8.97 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 16.19/8.97 => ( Y
% 16.19/8.97 = ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4914_flip__bit__0,axiom,
% 16.19/8.97 ! [A: code_integer] :
% 16.19/8.97 ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
% 16.19/8.97 = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4915_flip__bit__0,axiom,
% 16.19/8.97 ! [A: int] :
% 16.19/8.97 ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
% 16.19/8.97 = ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4916_flip__bit__0,axiom,
% 16.19/8.97 ! [A: nat] :
% 16.19/8.97 ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
% 16.19/8.97 = ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4917_option_Osize__gen_I2_J,axiom,
% 16.19/8.97 ! [X: nat > nat,X22: nat] :
% 16.19/8.97 ( ( size_option_nat @ X @ ( some_nat @ X22 ) )
% 16.19/8.97 = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4918_option_Osize__gen_I2_J,axiom,
% 16.19/8.97 ! [X: product_prod_nat_nat > nat,X22: product_prod_nat_nat] :
% 16.19/8.97 ( ( size_o8335143837870341156at_nat @ X @ ( some_P7363390416028606310at_nat @ X22 ) )
% 16.19/8.97 = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4919_option_Osize__gen_I2_J,axiom,
% 16.19/8.97 ! [X: num > nat,X22: num] :
% 16.19/8.97 ( ( size_option_num @ X @ ( some_num @ X22 ) )
% 16.19/8.97 = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4920_signed__take__bit__Suc,axiom,
% 16.19/8.97 ! [N2: nat,A: code_integer] :
% 16.19/8.97 ( ( bit_ri6519982836138164636nteger @ ( suc @ N2 ) @ A )
% 16.19/8.97 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4921_signed__take__bit__Suc,axiom,
% 16.19/8.97 ! [N2: nat,A: int] :
% 16.19/8.97 ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ A )
% 16.19/8.97 = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4922_set__decode__Suc,axiom,
% 16.19/8.97 ! [N2: nat,X: nat] :
% 16.19/8.97 ( ( member_nat @ ( suc @ N2 ) @ ( nat_set_decode @ X ) )
% 16.19/8.97 = ( member_nat @ N2 @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4923_diff__shunt__var,axiom,
% 16.19/8.97 ! [X: set_int,Y: set_int] :
% 16.19/8.97 ( ( ( minus_minus_set_int @ X @ Y )
% 16.19/8.97 = bot_bot_set_int )
% 16.19/8.97 = ( ord_less_eq_set_int @ X @ Y ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4924_diff__shunt__var,axiom,
% 16.19/8.97 ! [X: set_real,Y: set_real] :
% 16.19/8.97 ( ( ( minus_minus_set_real @ X @ Y )
% 16.19/8.97 = bot_bot_set_real )
% 16.19/8.97 = ( ord_less_eq_set_real @ X @ Y ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4925_diff__shunt__var,axiom,
% 16.19/8.97 ! [X: set_nat,Y: set_nat] :
% 16.19/8.97 ( ( ( minus_minus_set_nat @ X @ Y )
% 16.19/8.97 = bot_bot_set_nat )
% 16.19/8.97 = ( ord_less_eq_set_nat @ X @ Y ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4926_intind,axiom,
% 16.19/8.97 ! [I2: nat,N2: nat,P: int > $o,X: int] :
% 16.19/8.97 ( ( ord_less_nat @ I2 @ N2 )
% 16.19/8.97 => ( ( P @ X )
% 16.19/8.97 => ( P @ ( nth_int @ ( replicate_int @ N2 @ X ) @ I2 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4927_intind,axiom,
% 16.19/8.97 ! [I2: nat,N2: nat,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
% 16.19/8.97 ( ( ord_less_nat @ I2 @ N2 )
% 16.19/8.97 => ( ( P @ X )
% 16.19/8.97 => ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X ) @ I2 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4928_of__bool__less__eq__iff,axiom,
% 16.19/8.97 ! [P: $o,Q: $o] :
% 16.19/8.97 ( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 16.19/8.97 = ( P
% 16.19/8.97 => Q ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4929_of__bool__less__eq__iff,axiom,
% 16.19/8.97 ! [P: $o,Q: $o] :
% 16.19/8.97 ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 16.19/8.97 = ( P
% 16.19/8.97 => Q ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4930_of__bool__less__eq__iff,axiom,
% 16.19/8.97 ! [P: $o,Q: $o] :
% 16.19/8.97 ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 16.19/8.97 = ( P
% 16.19/8.97 => Q ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4931_of__bool__less__eq__iff,axiom,
% 16.19/8.97 ! [P: $o,Q: $o] :
% 16.19/8.97 ( ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 16.19/8.97 = ( P
% 16.19/8.97 => Q ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4932_of__bool__less__iff,axiom,
% 16.19/8.97 ! [P: $o,Q: $o] :
% 16.19/8.97 ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
% 16.19/8.97 = ( ~ P
% 16.19/8.97 & Q ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4933_of__bool__less__iff,axiom,
% 16.19/8.97 ! [P: $o,Q: $o] :
% 16.19/8.97 ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 16.19/8.97 = ( ~ P
% 16.19/8.97 & Q ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4934_of__bool__less__iff,axiom,
% 16.19/8.97 ! [P: $o,Q: $o] :
% 16.19/8.97 ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 16.19/8.97 = ( ~ P
% 16.19/8.97 & Q ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4935_of__bool__less__iff,axiom,
% 16.19/8.97 ! [P: $o,Q: $o] :
% 16.19/8.97 ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 16.19/8.97 = ( ~ P
% 16.19/8.97 & Q ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4936_of__bool__less__iff,axiom,
% 16.19/8.97 ! [P: $o,Q: $o] :
% 16.19/8.97 ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 16.19/8.97 = ( ~ P
% 16.19/8.97 & Q ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4937_of__bool__eq__1__iff,axiom,
% 16.19/8.97 ! [P: $o] :
% 16.19/8.97 ( ( ( zero_n1201886186963655149omplex @ P )
% 16.19/8.97 = one_one_complex )
% 16.19/8.97 = P ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4938_of__bool__eq__1__iff,axiom,
% 16.19/8.97 ! [P: $o] :
% 16.19/8.97 ( ( ( zero_n3304061248610475627l_real @ P )
% 16.19/8.97 = one_one_real )
% 16.19/8.97 = P ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4939_of__bool__eq__1__iff,axiom,
% 16.19/8.97 ! [P: $o] :
% 16.19/8.97 ( ( ( zero_n2052037380579107095ol_rat @ P )
% 16.19/8.97 = one_one_rat )
% 16.19/8.97 = P ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4940_of__bool__eq__1__iff,axiom,
% 16.19/8.97 ! [P: $o] :
% 16.19/8.97 ( ( ( zero_n2687167440665602831ol_nat @ P )
% 16.19/8.97 = one_one_nat )
% 16.19/8.97 = P ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4941_of__bool__eq__1__iff,axiom,
% 16.19/8.97 ! [P: $o] :
% 16.19/8.97 ( ( ( zero_n2684676970156552555ol_int @ P )
% 16.19/8.97 = one_one_int )
% 16.19/8.97 = P ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4942_of__bool__eq__1__iff,axiom,
% 16.19/8.97 ! [P: $o] :
% 16.19/8.97 ( ( ( zero_n356916108424825756nteger @ P )
% 16.19/8.97 = one_one_Code_integer )
% 16.19/8.97 = P ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4943_of__bool__eq_I2_J,axiom,
% 16.19/8.97 ( ( zero_n1201886186963655149omplex @ $true )
% 16.19/8.97 = one_one_complex ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4944_of__bool__eq_I2_J,axiom,
% 16.19/8.97 ( ( zero_n3304061248610475627l_real @ $true )
% 16.19/8.97 = one_one_real ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4945_of__bool__eq_I2_J,axiom,
% 16.19/8.97 ( ( zero_n2052037380579107095ol_rat @ $true )
% 16.19/8.97 = one_one_rat ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4946_of__bool__eq_I2_J,axiom,
% 16.19/8.97 ( ( zero_n2687167440665602831ol_nat @ $true )
% 16.19/8.97 = one_one_nat ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4947_of__bool__eq_I2_J,axiom,
% 16.19/8.97 ( ( zero_n2684676970156552555ol_int @ $true )
% 16.19/8.97 = one_one_int ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4948_of__bool__eq_I2_J,axiom,
% 16.19/8.97 ( ( zero_n356916108424825756nteger @ $true )
% 16.19/8.97 = one_one_Code_integer ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4949_signed__take__bit__of__0,axiom,
% 16.19/8.97 ! [N2: nat] :
% 16.19/8.97 ( ( bit_ri631733984087533419it_int @ N2 @ zero_zero_int )
% 16.19/8.97 = zero_zero_int ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4950_replicate__eq__replicate,axiom,
% 16.19/8.97 ! [M: nat,X: vEBT_VEBT,N2: nat,Y: vEBT_VEBT] :
% 16.19/8.97 ( ( ( replicate_VEBT_VEBT @ M @ X )
% 16.19/8.97 = ( replicate_VEBT_VEBT @ N2 @ Y ) )
% 16.19/8.97 = ( ( M = N2 )
% 16.19/8.97 & ( ( M != zero_zero_nat )
% 16.19/8.97 => ( X = Y ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4951_length__replicate,axiom,
% 16.19/8.97 ! [N2: nat,X: vEBT_VEBT] :
% 16.19/8.97 ( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X ) )
% 16.19/8.97 = N2 ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4952_length__replicate,axiom,
% 16.19/8.97 ! [N2: nat,X: $o] :
% 16.19/8.97 ( ( size_size_list_o @ ( replicate_o @ N2 @ X ) )
% 16.19/8.97 = N2 ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4953_length__replicate,axiom,
% 16.19/8.97 ! [N2: nat,X: int] :
% 16.19/8.97 ( ( size_size_list_int @ ( replicate_int @ N2 @ X ) )
% 16.19/8.97 = N2 ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4954_zero__less__of__bool__iff,axiom,
% 16.19/8.97 ! [P: $o] :
% 16.19/8.97 ( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) )
% 16.19/8.97 = P ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4955_zero__less__of__bool__iff,axiom,
% 16.19/8.97 ! [P: $o] :
% 16.19/8.97 ( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 16.19/8.97 = P ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4956_zero__less__of__bool__iff,axiom,
% 16.19/8.97 ! [P: $o] :
% 16.19/8.97 ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 16.19/8.97 = P ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4957_zero__less__of__bool__iff,axiom,
% 16.19/8.97 ! [P: $o] :
% 16.19/8.97 ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 16.19/8.97 = P ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4958_zero__less__of__bool__iff,axiom,
% 16.19/8.97 ! [P: $o] :
% 16.19/8.97 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
% 16.19/8.97 = P ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4959_of__bool__less__one__iff,axiom,
% 16.19/8.97 ! [P: $o] :
% 16.19/8.97 ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real )
% 16.19/8.97 = ~ P ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4960_of__bool__less__one__iff,axiom,
% 16.19/8.97 ! [P: $o] :
% 16.19/8.97 ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
% 16.19/8.97 = ~ P ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4961_of__bool__less__one__iff,axiom,
% 16.19/8.97 ! [P: $o] :
% 16.19/8.97 ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
% 16.19/8.97 = ~ P ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4962_of__bool__less__one__iff,axiom,
% 16.19/8.97 ! [P: $o] :
% 16.19/8.97 ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
% 16.19/8.97 = ~ P ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4963_of__bool__less__one__iff,axiom,
% 16.19/8.97 ! [P: $o] :
% 16.19/8.97 ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer )
% 16.19/8.97 = ~ P ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4964_of__bool__not__iff,axiom,
% 16.19/8.97 ! [P: $o] :
% 16.19/8.97 ( ( zero_n1201886186963655149omplex @ ~ P )
% 16.19/8.97 = ( minus_minus_complex @ one_one_complex @ ( zero_n1201886186963655149omplex @ P ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4965_of__bool__not__iff,axiom,
% 16.19/8.97 ! [P: $o] :
% 16.19/8.97 ( ( zero_n3304061248610475627l_real @ ~ P )
% 16.19/8.97 = ( minus_minus_real @ one_one_real @ ( zero_n3304061248610475627l_real @ P ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4966_of__bool__not__iff,axiom,
% 16.19/8.97 ! [P: $o] :
% 16.19/8.97 ( ( zero_n2052037380579107095ol_rat @ ~ P )
% 16.19/8.97 = ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4967_of__bool__not__iff,axiom,
% 16.19/8.97 ! [P: $o] :
% 16.19/8.97 ( ( zero_n2684676970156552555ol_int @ ~ P )
% 16.19/8.97 = ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4968_of__bool__not__iff,axiom,
% 16.19/8.97 ! [P: $o] :
% 16.19/8.97 ( ( zero_n356916108424825756nteger @ ~ P )
% 16.19/8.97 = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( zero_n356916108424825756nteger @ P ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4969_Suc__0__mod__eq,axiom,
% 16.19/8.97 ! [N2: nat] :
% 16.19/8.97 ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 16.19/8.97 = ( zero_n2687167440665602831ol_nat
% 16.19/8.97 @ ( N2
% 16.19/8.97 != ( suc @ zero_zero_nat ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4970_signed__take__bit__Suc__1,axiom,
% 16.19/8.97 ! [N2: nat] :
% 16.19/8.97 ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ one_one_int )
% 16.19/8.97 = one_one_int ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4971_signed__take__bit__numeral__of__1,axiom,
% 16.19/8.97 ! [K: num] :
% 16.19/8.97 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K ) @ one_one_int )
% 16.19/8.97 = one_one_int ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4972_Ball__set__replicate,axiom,
% 16.19/8.97 ! [N2: nat,A: int,P: int > $o] :
% 16.19/8.97 ( ( ! [X2: int] :
% 16.19/8.97 ( ( member_int @ X2 @ ( set_int2 @ ( replicate_int @ N2 @ A ) ) )
% 16.19/8.97 => ( P @ X2 ) ) )
% 16.19/8.97 = ( ( P @ A )
% 16.19/8.97 | ( N2 = zero_zero_nat ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4973_Ball__set__replicate,axiom,
% 16.19/8.97 ! [N2: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
% 16.19/8.97 ( ( ! [X2: vEBT_VEBT] :
% 16.19/8.97 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ A ) ) )
% 16.19/8.97 => ( P @ X2 ) ) )
% 16.19/8.97 = ( ( P @ A )
% 16.19/8.97 | ( N2 = zero_zero_nat ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4974_Bex__set__replicate,axiom,
% 16.19/8.97 ! [N2: nat,A: int,P: int > $o] :
% 16.19/8.97 ( ( ? [X2: int] :
% 16.19/8.97 ( ( member_int @ X2 @ ( set_int2 @ ( replicate_int @ N2 @ A ) ) )
% 16.19/8.97 & ( P @ X2 ) ) )
% 16.19/8.97 = ( ( P @ A )
% 16.19/8.97 & ( N2 != zero_zero_nat ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4975_Bex__set__replicate,axiom,
% 16.19/8.97 ! [N2: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
% 16.19/8.97 ( ( ? [X2: vEBT_VEBT] :
% 16.19/8.97 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ A ) ) )
% 16.19/8.97 & ( P @ X2 ) ) )
% 16.19/8.97 = ( ( P @ A )
% 16.19/8.97 & ( N2 != zero_zero_nat ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4976_in__set__replicate,axiom,
% 16.19/8.97 ! [X: nat,N2: nat,Y: nat] :
% 16.19/8.97 ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N2 @ Y ) ) )
% 16.19/8.97 = ( ( X = Y )
% 16.19/8.97 & ( N2 != zero_zero_nat ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4977_in__set__replicate,axiom,
% 16.19/8.97 ! [X: real,N2: nat,Y: real] :
% 16.19/8.97 ( ( member_real @ X @ ( set_real2 @ ( replicate_real @ N2 @ Y ) ) )
% 16.19/8.97 = ( ( X = Y )
% 16.19/8.97 & ( N2 != zero_zero_nat ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4978_in__set__replicate,axiom,
% 16.19/8.97 ! [X: complex,N2: nat,Y: complex] :
% 16.19/8.97 ( ( member_complex @ X @ ( set_complex2 @ ( replicate_complex @ N2 @ Y ) ) )
% 16.19/8.97 = ( ( X = Y )
% 16.19/8.97 & ( N2 != zero_zero_nat ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4979_in__set__replicate,axiom,
% 16.19/8.97 ! [X: set_nat,N2: nat,Y: set_nat] :
% 16.19/8.97 ( ( member_set_nat @ X @ ( set_set_nat2 @ ( replicate_set_nat @ N2 @ Y ) ) )
% 16.19/8.97 = ( ( X = Y )
% 16.19/8.97 & ( N2 != zero_zero_nat ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4980_in__set__replicate,axiom,
% 16.19/8.97 ! [X: int,N2: nat,Y: int] :
% 16.19/8.97 ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N2 @ Y ) ) )
% 16.19/8.97 = ( ( X = Y )
% 16.19/8.97 & ( N2 != zero_zero_nat ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4981_in__set__replicate,axiom,
% 16.19/8.97 ! [X: vEBT_VEBT,N2: nat,Y: vEBT_VEBT] :
% 16.19/8.97 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ Y ) ) )
% 16.19/8.97 = ( ( X = Y )
% 16.19/8.97 & ( N2 != zero_zero_nat ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4982_nth__replicate,axiom,
% 16.19/8.97 ! [I2: nat,N2: nat,X: int] :
% 16.19/8.97 ( ( ord_less_nat @ I2 @ N2 )
% 16.19/8.97 => ( ( nth_int @ ( replicate_int @ N2 @ X ) @ I2 )
% 16.19/8.97 = X ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4983_nth__replicate,axiom,
% 16.19/8.97 ! [I2: nat,N2: nat,X: vEBT_VEBT] :
% 16.19/8.97 ( ( ord_less_nat @ I2 @ N2 )
% 16.19/8.97 => ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X ) @ I2 )
% 16.19/8.97 = X ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4984_triangle__Suc,axiom,
% 16.19/8.97 ! [N2: nat] :
% 16.19/8.97 ( ( nat_triangle @ ( suc @ N2 ) )
% 16.19/8.97 = ( plus_plus_nat @ ( nat_triangle @ N2 ) @ ( suc @ N2 ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4985_signed__take__bit__Suc__bit0,axiom,
% 16.19/8.97 ! [N2: nat,K: num] :
% 16.19/8.97 ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 16.19/8.97 = ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4986_odd__of__bool__self,axiom,
% 16.19/8.97 ! [P4: $o] :
% 16.19/8.97 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P4 ) ) )
% 16.19/8.97 = P4 ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4987_odd__of__bool__self,axiom,
% 16.19/8.97 ! [P4: $o] :
% 16.19/8.97 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P4 ) ) )
% 16.19/8.97 = P4 ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4988_odd__of__bool__self,axiom,
% 16.19/8.97 ! [P4: $o] :
% 16.19/8.97 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P4 ) ) )
% 16.19/8.97 = P4 ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4989_of__bool__half__eq__0,axiom,
% 16.19/8.97 ! [B: $o] :
% 16.19/8.97 ( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.97 = zero_zero_nat ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4990_of__bool__half__eq__0,axiom,
% 16.19/8.97 ! [B: $o] :
% 16.19/8.97 ( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/8.97 = zero_zero_int ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4991_of__bool__half__eq__0,axiom,
% 16.19/8.97 ! [B: $o] :
% 16.19/8.97 ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 16.19/8.97 = zero_z3403309356797280102nteger ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4992_set__decode__0,axiom,
% 16.19/8.97 ! [X: nat] :
% 16.19/8.97 ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X ) )
% 16.19/8.97 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4993_bits__1__div__exp,axiom,
% 16.19/8.97 ! [N2: nat] :
% 16.19/8.97 ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.97 = ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4994_bits__1__div__exp,axiom,
% 16.19/8.97 ! [N2: nat] :
% 16.19/8.97 ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.97 = ( zero_n2684676970156552555ol_int @ ( N2 = zero_zero_nat ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4995_bits__1__div__exp,axiom,
% 16.19/8.97 ! [N2: nat] :
% 16.19/8.97 ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.97 = ( zero_n356916108424825756nteger @ ( N2 = zero_zero_nat ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4996_one__div__2__pow__eq,axiom,
% 16.19/8.97 ! [N2: nat] :
% 16.19/8.97 ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.97 = ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4997_one__div__2__pow__eq,axiom,
% 16.19/8.97 ! [N2: nat] :
% 16.19/8.97 ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.97 = ( zero_n2684676970156552555ol_int @ ( N2 = zero_zero_nat ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4998_one__div__2__pow__eq,axiom,
% 16.19/8.97 ! [N2: nat] :
% 16.19/8.97 ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.97 = ( zero_n356916108424825756nteger @ ( N2 = zero_zero_nat ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_4999_one__mod__2__pow__eq,axiom,
% 16.19/8.97 ! [N2: nat] :
% 16.19/8.97 ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.97 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5000_one__mod__2__pow__eq,axiom,
% 16.19/8.97 ! [N2: nat] :
% 16.19/8.97 ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.97 = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5001_one__mod__2__pow__eq,axiom,
% 16.19/8.97 ! [N2: nat] :
% 16.19/8.97 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.97 = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5002_dvd__antisym,axiom,
% 16.19/8.97 ! [M: nat,N2: nat] :
% 16.19/8.97 ( ( dvd_dvd_nat @ M @ N2 )
% 16.19/8.97 => ( ( dvd_dvd_nat @ N2 @ M )
% 16.19/8.97 => ( M = N2 ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5003_of__bool__conj,axiom,
% 16.19/8.97 ! [P: $o,Q: $o] :
% 16.19/8.97 ( ( zero_n3304061248610475627l_real
% 16.19/8.97 @ ( P
% 16.19/8.97 & Q ) )
% 16.19/8.97 = ( times_times_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5004_of__bool__conj,axiom,
% 16.19/8.97 ! [P: $o,Q: $o] :
% 16.19/8.97 ( ( zero_n2052037380579107095ol_rat
% 16.19/8.97 @ ( P
% 16.19/8.97 & Q ) )
% 16.19/8.97 = ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5005_of__bool__conj,axiom,
% 16.19/8.97 ! [P: $o,Q: $o] :
% 16.19/8.97 ( ( zero_n2687167440665602831ol_nat
% 16.19/8.97 @ ( P
% 16.19/8.97 & Q ) )
% 16.19/8.97 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5006_of__bool__conj,axiom,
% 16.19/8.97 ! [P: $o,Q: $o] :
% 16.19/8.97 ( ( zero_n2684676970156552555ol_int
% 16.19/8.97 @ ( P
% 16.19/8.97 & Q ) )
% 16.19/8.97 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5007_of__bool__conj,axiom,
% 16.19/8.97 ! [P: $o,Q: $o] :
% 16.19/8.97 ( ( zero_n356916108424825756nteger
% 16.19/8.97 @ ( P
% 16.19/8.97 & Q ) )
% 16.19/8.97 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5008_bot__nat__def,axiom,
% 16.19/8.97 bot_bot_nat = zero_zero_nat ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5009_signed__take__bit__mult,axiom,
% 16.19/8.97 ! [N2: nat,K: int,L2: int] :
% 16.19/8.97 ( ( bit_ri631733984087533419it_int @ N2 @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( bit_ri631733984087533419it_int @ N2 @ L2 ) ) )
% 16.19/8.97 = ( bit_ri631733984087533419it_int @ N2 @ ( times_times_int @ K @ L2 ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5010_signed__take__bit__add,axiom,
% 16.19/8.97 ! [N2: nat,K: int,L2: int] :
% 16.19/8.97 ( ( bit_ri631733984087533419it_int @ N2 @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( bit_ri631733984087533419it_int @ N2 @ L2 ) ) )
% 16.19/8.97 = ( bit_ri631733984087533419it_int @ N2 @ ( plus_plus_int @ K @ L2 ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5011_signed__take__bit__diff,axiom,
% 16.19/8.97 ! [N2: nat,K: int,L2: int] :
% 16.19/8.97 ( ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( bit_ri631733984087533419it_int @ N2 @ L2 ) ) )
% 16.19/8.97 = ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ K @ L2 ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5012_zero__less__eq__of__bool,axiom,
% 16.19/8.97 ! [P: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5013_zero__less__eq__of__bool,axiom,
% 16.19/8.97 ! [P: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5014_zero__less__eq__of__bool,axiom,
% 16.19/8.97 ! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5015_zero__less__eq__of__bool,axiom,
% 16.19/8.97 ! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5016_zero__less__eq__of__bool,axiom,
% 16.19/8.97 ! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5017_of__bool__less__eq__one,axiom,
% 16.19/8.97 ! [P: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5018_of__bool__less__eq__one,axiom,
% 16.19/8.97 ! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5019_of__bool__less__eq__one,axiom,
% 16.19/8.97 ! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5020_of__bool__less__eq__one,axiom,
% 16.19/8.97 ! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5021_of__bool__less__eq__one,axiom,
% 16.19/8.97 ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5022_split__of__bool__asm,axiom,
% 16.19/8.97 ! [P: complex > $o,P4: $o] :
% 16.19/8.97 ( ( P @ ( zero_n1201886186963655149omplex @ P4 ) )
% 16.19/8.97 = ( ~ ( ( P4
% 16.19/8.97 & ~ ( P @ one_one_complex ) )
% 16.19/8.97 | ( ~ P4
% 16.19/8.97 & ~ ( P @ zero_zero_complex ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5023_split__of__bool__asm,axiom,
% 16.19/8.97 ! [P: real > $o,P4: $o] :
% 16.19/8.97 ( ( P @ ( zero_n3304061248610475627l_real @ P4 ) )
% 16.19/8.97 = ( ~ ( ( P4
% 16.19/8.97 & ~ ( P @ one_one_real ) )
% 16.19/8.97 | ( ~ P4
% 16.19/8.97 & ~ ( P @ zero_zero_real ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5024_split__of__bool__asm,axiom,
% 16.19/8.97 ! [P: rat > $o,P4: $o] :
% 16.19/8.97 ( ( P @ ( zero_n2052037380579107095ol_rat @ P4 ) )
% 16.19/8.97 = ( ~ ( ( P4
% 16.19/8.97 & ~ ( P @ one_one_rat ) )
% 16.19/8.97 | ( ~ P4
% 16.19/8.97 & ~ ( P @ zero_zero_rat ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5025_split__of__bool__asm,axiom,
% 16.19/8.97 ! [P: nat > $o,P4: $o] :
% 16.19/8.97 ( ( P @ ( zero_n2687167440665602831ol_nat @ P4 ) )
% 16.19/8.97 = ( ~ ( ( P4
% 16.19/8.97 & ~ ( P @ one_one_nat ) )
% 16.19/8.97 | ( ~ P4
% 16.19/8.97 & ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5026_split__of__bool__asm,axiom,
% 16.19/8.97 ! [P: int > $o,P4: $o] :
% 16.19/8.97 ( ( P @ ( zero_n2684676970156552555ol_int @ P4 ) )
% 16.19/8.97 = ( ~ ( ( P4
% 16.19/8.97 & ~ ( P @ one_one_int ) )
% 16.19/8.97 | ( ~ P4
% 16.19/8.97 & ~ ( P @ zero_zero_int ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5027_split__of__bool__asm,axiom,
% 16.19/8.97 ! [P: code_integer > $o,P4: $o] :
% 16.19/8.97 ( ( P @ ( zero_n356916108424825756nteger @ P4 ) )
% 16.19/8.97 = ( ~ ( ( P4
% 16.19/8.97 & ~ ( P @ one_one_Code_integer ) )
% 16.19/8.97 | ( ~ P4
% 16.19/8.97 & ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5028_split__of__bool,axiom,
% 16.19/8.97 ! [P: complex > $o,P4: $o] :
% 16.19/8.97 ( ( P @ ( zero_n1201886186963655149omplex @ P4 ) )
% 16.19/8.97 = ( ( P4
% 16.19/8.97 => ( P @ one_one_complex ) )
% 16.19/8.97 & ( ~ P4
% 16.19/8.97 => ( P @ zero_zero_complex ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5029_split__of__bool,axiom,
% 16.19/8.97 ! [P: real > $o,P4: $o] :
% 16.19/8.97 ( ( P @ ( zero_n3304061248610475627l_real @ P4 ) )
% 16.19/8.97 = ( ( P4
% 16.19/8.97 => ( P @ one_one_real ) )
% 16.19/8.97 & ( ~ P4
% 16.19/8.97 => ( P @ zero_zero_real ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5030_split__of__bool,axiom,
% 16.19/8.97 ! [P: rat > $o,P4: $o] :
% 16.19/8.97 ( ( P @ ( zero_n2052037380579107095ol_rat @ P4 ) )
% 16.19/8.97 = ( ( P4
% 16.19/8.97 => ( P @ one_one_rat ) )
% 16.19/8.97 & ( ~ P4
% 16.19/8.97 => ( P @ zero_zero_rat ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5031_split__of__bool,axiom,
% 16.19/8.97 ! [P: nat > $o,P4: $o] :
% 16.19/8.97 ( ( P @ ( zero_n2687167440665602831ol_nat @ P4 ) )
% 16.19/8.97 = ( ( P4
% 16.19/8.97 => ( P @ one_one_nat ) )
% 16.19/8.97 & ( ~ P4
% 16.19/8.97 => ( P @ zero_zero_nat ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5032_split__of__bool,axiom,
% 16.19/8.97 ! [P: int > $o,P4: $o] :
% 16.19/8.97 ( ( P @ ( zero_n2684676970156552555ol_int @ P4 ) )
% 16.19/8.97 = ( ( P4
% 16.19/8.97 => ( P @ one_one_int ) )
% 16.19/8.97 & ( ~ P4
% 16.19/8.97 => ( P @ zero_zero_int ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5033_split__of__bool,axiom,
% 16.19/8.97 ! [P: code_integer > $o,P4: $o] :
% 16.19/8.97 ( ( P @ ( zero_n356916108424825756nteger @ P4 ) )
% 16.19/8.97 = ( ( P4
% 16.19/8.97 => ( P @ one_one_Code_integer ) )
% 16.19/8.97 & ( ~ P4
% 16.19/8.97 => ( P @ zero_z3403309356797280102nteger ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5034_of__bool__def,axiom,
% 16.19/8.97 ( zero_n1201886186963655149omplex
% 16.19/8.97 = ( ^ [P5: $o] : ( if_complex @ P5 @ one_one_complex @ zero_zero_complex ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5035_of__bool__def,axiom,
% 16.19/8.97 ( zero_n3304061248610475627l_real
% 16.19/8.97 = ( ^ [P5: $o] : ( if_real @ P5 @ one_one_real @ zero_zero_real ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5036_of__bool__def,axiom,
% 16.19/8.97 ( zero_n2052037380579107095ol_rat
% 16.19/8.97 = ( ^ [P5: $o] : ( if_rat @ P5 @ one_one_rat @ zero_zero_rat ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5037_of__bool__def,axiom,
% 16.19/8.97 ( zero_n2687167440665602831ol_nat
% 16.19/8.97 = ( ^ [P5: $o] : ( if_nat @ P5 @ one_one_nat @ zero_zero_nat ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5038_of__bool__def,axiom,
% 16.19/8.97 ( zero_n2684676970156552555ol_int
% 16.19/8.97 = ( ^ [P5: $o] : ( if_int @ P5 @ one_one_int @ zero_zero_int ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5039_of__bool__def,axiom,
% 16.19/8.97 ( zero_n356916108424825756nteger
% 16.19/8.97 = ( ^ [P5: $o] : ( if_Code_integer @ P5 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5040_replicate__eqI,axiom,
% 16.19/8.97 ! [Xs: list_nat,N2: nat,X: nat] :
% 16.19/8.97 ( ( ( size_size_list_nat @ Xs )
% 16.19/8.97 = N2 )
% 16.19/8.97 => ( ! [Y4: nat] :
% 16.19/8.97 ( ( member_nat @ Y4 @ ( set_nat2 @ Xs ) )
% 16.19/8.97 => ( Y4 = X ) )
% 16.19/8.97 => ( Xs
% 16.19/8.97 = ( replicate_nat @ N2 @ X ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5041_replicate__eqI,axiom,
% 16.19/8.97 ! [Xs: list_real,N2: nat,X: real] :
% 16.19/8.97 ( ( ( size_size_list_real @ Xs )
% 16.19/8.97 = N2 )
% 16.19/8.97 => ( ! [Y4: real] :
% 16.19/8.97 ( ( member_real @ Y4 @ ( set_real2 @ Xs ) )
% 16.19/8.97 => ( Y4 = X ) )
% 16.19/8.97 => ( Xs
% 16.19/8.97 = ( replicate_real @ N2 @ X ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5042_replicate__eqI,axiom,
% 16.19/8.97 ! [Xs: list_complex,N2: nat,X: complex] :
% 16.19/8.97 ( ( ( size_s3451745648224563538omplex @ Xs )
% 16.19/8.97 = N2 )
% 16.19/8.97 => ( ! [Y4: complex] :
% 16.19/8.97 ( ( member_complex @ Y4 @ ( set_complex2 @ Xs ) )
% 16.19/8.97 => ( Y4 = X ) )
% 16.19/8.97 => ( Xs
% 16.19/8.97 = ( replicate_complex @ N2 @ X ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5043_replicate__eqI,axiom,
% 16.19/8.97 ! [Xs: list_set_nat,N2: nat,X: set_nat] :
% 16.19/8.97 ( ( ( size_s3254054031482475050et_nat @ Xs )
% 16.19/8.97 = N2 )
% 16.19/8.97 => ( ! [Y4: set_nat] :
% 16.19/8.97 ( ( member_set_nat @ Y4 @ ( set_set_nat2 @ Xs ) )
% 16.19/8.97 => ( Y4 = X ) )
% 16.19/8.97 => ( Xs
% 16.19/8.97 = ( replicate_set_nat @ N2 @ X ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5044_replicate__eqI,axiom,
% 16.19/8.97 ! [Xs: list_VEBT_VEBT,N2: nat,X: vEBT_VEBT] :
% 16.19/8.97 ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 16.19/8.97 = N2 )
% 16.19/8.97 => ( ! [Y4: vEBT_VEBT] :
% 16.19/8.97 ( ( member_VEBT_VEBT @ Y4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 16.19/8.97 => ( Y4 = X ) )
% 16.19/8.97 => ( Xs
% 16.19/8.97 = ( replicate_VEBT_VEBT @ N2 @ X ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5045_replicate__eqI,axiom,
% 16.19/8.97 ! [Xs: list_o,N2: nat,X: $o] :
% 16.19/8.97 ( ( ( size_size_list_o @ Xs )
% 16.19/8.97 = N2 )
% 16.19/8.97 => ( ! [Y4: $o] :
% 16.19/8.97 ( ( member_o @ Y4 @ ( set_o2 @ Xs ) )
% 16.19/8.97 => ( Y4 = X ) )
% 16.19/8.97 => ( Xs
% 16.19/8.97 = ( replicate_o @ N2 @ X ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5046_replicate__eqI,axiom,
% 16.19/8.97 ! [Xs: list_int,N2: nat,X: int] :
% 16.19/8.97 ( ( ( size_size_list_int @ Xs )
% 16.19/8.97 = N2 )
% 16.19/8.97 => ( ! [Y4: int] :
% 16.19/8.97 ( ( member_int @ Y4 @ ( set_int2 @ Xs ) )
% 16.19/8.97 => ( Y4 = X ) )
% 16.19/8.97 => ( Xs
% 16.19/8.97 = ( replicate_int @ N2 @ X ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5047_replicate__length__same,axiom,
% 16.19/8.97 ! [Xs: list_VEBT_VEBT,X: vEBT_VEBT] :
% 16.19/8.97 ( ! [X4: vEBT_VEBT] :
% 16.19/8.97 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 16.19/8.97 => ( X4 = X ) )
% 16.19/8.97 => ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs ) @ X )
% 16.19/8.97 = Xs ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5048_replicate__length__same,axiom,
% 16.19/8.97 ! [Xs: list_o,X: $o] :
% 16.19/8.97 ( ! [X4: $o] :
% 16.19/8.97 ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
% 16.19/8.97 => ( X4 = X ) )
% 16.19/8.97 => ( ( replicate_o @ ( size_size_list_o @ Xs ) @ X )
% 16.19/8.97 = Xs ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5049_replicate__length__same,axiom,
% 16.19/8.97 ! [Xs: list_int,X: int] :
% 16.19/8.97 ( ! [X4: int] :
% 16.19/8.97 ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
% 16.19/8.97 => ( X4 = X ) )
% 16.19/8.97 => ( ( replicate_int @ ( size_size_list_int @ Xs ) @ X )
% 16.19/8.97 = Xs ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5050_subset__decode__imp__le,axiom,
% 16.19/8.97 ! [M: nat,N2: nat] :
% 16.19/8.97 ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N2 ) )
% 16.19/8.97 => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5051_of__bool__odd__eq__mod__2,axiom,
% 16.19/8.97 ! [A: nat] :
% 16.19/8.97 ( ( zero_n2687167440665602831ol_nat
% 16.19/8.97 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 16.19/8.97 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5052_of__bool__odd__eq__mod__2,axiom,
% 16.19/8.97 ! [A: int] :
% 16.19/8.97 ( ( zero_n2684676970156552555ol_int
% 16.19/8.97 @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 16.19/8.97 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5053_of__bool__odd__eq__mod__2,axiom,
% 16.19/8.97 ! [A: code_integer] :
% 16.19/8.97 ( ( zero_n356916108424825756nteger
% 16.19/8.97 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 16.19/8.97 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5054_signed__take__bit__int__less__exp,axiom,
% 16.19/8.97 ! [N2: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5055_even__signed__take__bit__iff,axiom,
% 16.19/8.97 ! [M: nat,A: code_integer] :
% 16.19/8.97 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ M @ A ) )
% 16.19/8.97 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5056_even__signed__take__bit__iff,axiom,
% 16.19/8.97 ! [M: nat,A: int] :
% 16.19/8.97 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ M @ A ) )
% 16.19/8.97 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5057_bits__induct,axiom,
% 16.19/8.97 ! [P: nat > $o,A: nat] :
% 16.19/8.97 ( ! [A3: nat] :
% 16.19/8.97 ( ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.97 = A3 )
% 16.19/8.97 => ( P @ A3 ) )
% 16.19/8.97 => ( ! [A3: nat,B2: $o] :
% 16.19/8.97 ( ( P @ A3 )
% 16.19/8.97 => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.97 = A3 )
% 16.19/8.97 => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
% 16.19/8.97 => ( P @ A ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5058_bits__induct,axiom,
% 16.19/8.97 ! [P: int > $o,A: int] :
% 16.19/8.97 ( ! [A3: int] :
% 16.19/8.97 ( ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/8.97 = A3 )
% 16.19/8.97 => ( P @ A3 ) )
% 16.19/8.97 => ( ! [A3: int,B2: $o] :
% 16.19/8.97 ( ( P @ A3 )
% 16.19/8.97 => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/8.97 = A3 )
% 16.19/8.97 => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B2 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
% 16.19/8.97 => ( P @ A ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5059_bits__induct,axiom,
% 16.19/8.97 ! [P: code_integer > $o,A: code_integer] :
% 16.19/8.97 ( ! [A3: code_integer] :
% 16.19/8.97 ( ( ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 16.19/8.97 = A3 )
% 16.19/8.97 => ( P @ A3 ) )
% 16.19/8.97 => ( ! [A3: code_integer,B2: $o] :
% 16.19/8.97 ( ( P @ A3 )
% 16.19/8.97 => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B2 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 16.19/8.97 = A3 )
% 16.19/8.97 => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B2 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
% 16.19/8.97 => ( P @ A ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5060_signed__take__bit__int__greater__eq__self__iff,axiom,
% 16.19/8.97 ! [K: int,N2: nat] :
% 16.19/8.97 ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N2 @ K ) )
% 16.19/8.97 = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5061_signed__take__bit__int__less__self__iff,axiom,
% 16.19/8.97 ! [N2: nat,K: int] :
% 16.19/8.97 ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ K )
% 16.19/8.97 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5062_exp__mod__exp,axiom,
% 16.19/8.97 ! [M: nat,N2: nat] :
% 16.19/8.97 ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.97 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5063_exp__mod__exp,axiom,
% 16.19/8.97 ! [M: nat,N2: nat] :
% 16.19/8.97 ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.97 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5064_exp__mod__exp,axiom,
% 16.19/8.97 ! [M: nat,N2: nat] :
% 16.19/8.97 ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.97 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5065_signed__take__bit__int__less__eq,axiom,
% 16.19/8.97 ! [N2: nat,K: int] :
% 16.19/8.97 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K )
% 16.19/8.97 => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5066_option_Osize__gen_I1_J,axiom,
% 16.19/8.97 ! [X: nat > nat] :
% 16.19/8.97 ( ( size_option_nat @ X @ none_nat )
% 16.19/8.97 = ( suc @ zero_zero_nat ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5067_option_Osize__gen_I1_J,axiom,
% 16.19/8.97 ! [X: product_prod_nat_nat > nat] :
% 16.19/8.97 ( ( size_o8335143837870341156at_nat @ X @ none_P5556105721700978146at_nat )
% 16.19/8.97 = ( suc @ zero_zero_nat ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5068_option_Osize__gen_I1_J,axiom,
% 16.19/8.97 ! [X: num > nat] :
% 16.19/8.97 ( ( size_option_num @ X @ none_num )
% 16.19/8.97 = ( suc @ zero_zero_nat ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5069_set__decode__def,axiom,
% 16.19/8.97 ( nat_set_decode
% 16.19/8.97 = ( ^ [X2: nat] :
% 16.19/8.97 ( collect_nat
% 16.19/8.97 @ ^ [N: nat] :
% 16.19/8.97 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5070_exp__div__exp__eq,axiom,
% 16.19/8.97 ! [M: nat,N2: nat] :
% 16.19/8.97 ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.97 = ( times_times_nat
% 16.19/8.97 @ ( zero_n2687167440665602831ol_nat
% 16.19/8.97 @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 16.19/8.97 != zero_zero_nat )
% 16.19/8.97 & ( ord_less_eq_nat @ N2 @ M ) ) )
% 16.19/8.97 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5071_exp__div__exp__eq,axiom,
% 16.19/8.97 ! [M: nat,N2: nat] :
% 16.19/8.97 ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.97 = ( times_times_int
% 16.19/8.97 @ ( zero_n2684676970156552555ol_int
% 16.19/8.97 @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 16.19/8.97 != zero_zero_int )
% 16.19/8.97 & ( ord_less_eq_nat @ N2 @ M ) ) )
% 16.19/8.97 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5072_exp__div__exp__eq,axiom,
% 16.19/8.97 ! [M: nat,N2: nat] :
% 16.19/8.97 ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.97 = ( times_3573771949741848930nteger
% 16.19/8.97 @ ( zero_n356916108424825756nteger
% 16.19/8.97 @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
% 16.19/8.97 != zero_z3403309356797280102nteger )
% 16.19/8.97 & ( ord_less_eq_nat @ N2 @ M ) ) )
% 16.19/8.97 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5073_vebt__buildup_Osimps_I3_J,axiom,
% 16.19/8.97 ! [Va: nat] :
% 16.19/8.97 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 16.19/8.97 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
% 16.19/8.97 = ( 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 ) ) ) ) ) ) )
% 16.19/8.97 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 16.19/8.97 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
% 16.19/8.97 = ( 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 ) ) ) ) ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5074_signed__take__bit__rec,axiom,
% 16.19/8.97 ( bit_ri6519982836138164636nteger
% 16.19/8.97 = ( ^ [N: nat,A5: code_integer] : ( if_Code_integer @ ( N = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5075_signed__take__bit__rec,axiom,
% 16.19/8.97 ( bit_ri631733984087533419it_int
% 16.19/8.97 = ( ^ [N: nat,A5: int] : ( if_int @ ( N = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5076_vebt__buildup_Opelims,axiom,
% 16.19/8.97 ! [X: nat,Y: vEBT_VEBT] :
% 16.19/8.97 ( ( ( vEBT_vebt_buildup @ X )
% 16.19/8.97 = Y )
% 16.19/8.97 => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X )
% 16.19/8.97 => ( ( ( X = zero_zero_nat )
% 16.19/8.97 => ( ( Y
% 16.19/8.97 = ( vEBT_Leaf @ $false @ $false ) )
% 16.19/8.97 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
% 16.19/8.97 => ( ( ( X
% 16.19/8.97 = ( suc @ zero_zero_nat ) )
% 16.19/8.97 => ( ( Y
% 16.19/8.97 = ( vEBT_Leaf @ $false @ $false ) )
% 16.19/8.97 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
% 16.19/8.97 => ~ ! [Va2: nat] :
% 16.19/8.97 ( ( X
% 16.19/8.97 = ( suc @ ( suc @ Va2 ) ) )
% 16.19/8.97 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 16.19/8.97 => ( Y
% 16.19/8.97 = ( 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 ) ) ) ) ) ) )
% 16.19/8.97 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 16.19/8.97 => ( Y
% 16.19/8.97 = ( 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 ) ) ) ) ) ) ) ) )
% 16.19/8.97 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5077_add__scale__eq__noteq,axiom,
% 16.19/8.97 ! [R: complex,A: complex,B: complex,C: complex,D: complex] :
% 16.19/8.97 ( ( R != zero_zero_complex )
% 16.19/8.97 => ( ( ( A = B )
% 16.19/8.97 & ( C != D ) )
% 16.19/8.97 => ( ( plus_plus_complex @ A @ ( times_times_complex @ R @ C ) )
% 16.19/8.97 != ( plus_plus_complex @ B @ ( times_times_complex @ R @ D ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5078_add__scale__eq__noteq,axiom,
% 16.19/8.97 ! [R: real,A: real,B: real,C: real,D: real] :
% 16.19/8.97 ( ( R != zero_zero_real )
% 16.19/8.97 => ( ( ( A = B )
% 16.19/8.97 & ( C != D ) )
% 16.19/8.97 => ( ( plus_plus_real @ A @ ( times_times_real @ R @ C ) )
% 16.19/8.97 != ( plus_plus_real @ B @ ( times_times_real @ R @ D ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5079_add__scale__eq__noteq,axiom,
% 16.19/8.97 ! [R: rat,A: rat,B: rat,C: rat,D: rat] :
% 16.19/8.97 ( ( R != zero_zero_rat )
% 16.19/8.97 => ( ( ( A = B )
% 16.19/8.97 & ( C != D ) )
% 16.19/8.97 => ( ( plus_plus_rat @ A @ ( times_times_rat @ R @ C ) )
% 16.19/8.97 != ( plus_plus_rat @ B @ ( times_times_rat @ R @ D ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5080_add__scale__eq__noteq,axiom,
% 16.19/8.97 ! [R: nat,A: nat,B: nat,C: nat,D: nat] :
% 16.19/8.97 ( ( R != zero_zero_nat )
% 16.19/8.97 => ( ( ( A = B )
% 16.19/8.97 & ( C != D ) )
% 16.19/8.97 => ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C ) )
% 16.19/8.97 != ( plus_plus_nat @ B @ ( times_times_nat @ R @ D ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5081_add__scale__eq__noteq,axiom,
% 16.19/8.97 ! [R: int,A: int,B: int,C: int,D: int] :
% 16.19/8.97 ( ( R != zero_zero_int )
% 16.19/8.97 => ( ( ( A = B )
% 16.19/8.97 & ( C != D ) )
% 16.19/8.97 => ( ( plus_plus_int @ A @ ( times_times_int @ R @ C ) )
% 16.19/8.97 != ( plus_plus_int @ B @ ( times_times_int @ R @ D ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5082_artanh__def,axiom,
% 16.19/8.97 ( artanh_real
% 16.19/8.97 = ( ^ [X2: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5083_Sum__Icc__int,axiom,
% 16.19/8.97 ! [M: int,N2: int] :
% 16.19/8.97 ( ( ord_less_eq_int @ M @ N2 )
% 16.19/8.97 => ( ( groups4538972089207619220nt_int
% 16.19/8.97 @ ^ [X2: int] : X2
% 16.19/8.97 @ ( set_or1266510415728281911st_int @ M @ N2 ) )
% 16.19/8.97 = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N2 @ ( plus_plus_int @ N2 @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5084_divmod__step__def,axiom,
% 16.19/8.97 ( unique5026877609467782581ep_nat
% 16.19/8.97 = ( ^ [L: num] :
% 16.19/8.97 ( produc2626176000494625587at_nat
% 16.19/8.97 @ ^ [Q4: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5085_divmod__step__def,axiom,
% 16.19/8.97 ( unique5024387138958732305ep_int
% 16.19/8.97 = ( ^ [L: num] :
% 16.19/8.97 ( produc4245557441103728435nt_int
% 16.19/8.97 @ ^ [Q4: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5086_divmod__step__def,axiom,
% 16.19/8.97 ( unique4921790084139445826nteger
% 16.19/8.97 = ( ^ [L: num] :
% 16.19/8.97 ( produc6916734918728496179nteger
% 16.19/8.97 @ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5087_even__set__encode__iff,axiom,
% 16.19/8.97 ! [A2: set_nat] :
% 16.19/8.97 ( ( finite_finite_nat @ A2 )
% 16.19/8.97 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A2 ) )
% 16.19/8.97 = ( ~ ( member_nat @ zero_zero_nat @ A2 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5088_Compl__anti__mono,axiom,
% 16.19/8.97 ! [A2: set_nat,B3: set_nat] :
% 16.19/8.97 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 16.19/8.97 => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B3 ) @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5089_Compl__subset__Compl__iff,axiom,
% 16.19/8.97 ! [A2: set_nat,B3: set_nat] :
% 16.19/8.97 ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( uminus5710092332889474511et_nat @ B3 ) )
% 16.19/8.97 = ( ord_less_eq_set_nat @ B3 @ A2 ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5090_neg__le__iff__le,axiom,
% 16.19/8.97 ! [B: real,A: real] :
% 16.19/8.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 16.19/8.97 = ( ord_less_eq_real @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5091_neg__le__iff__le,axiom,
% 16.19/8.97 ! [B: code_integer,A: code_integer] :
% 16.19/8.97 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 16.19/8.97 = ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5092_neg__le__iff__le,axiom,
% 16.19/8.97 ! [B: rat,A: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 16.19/8.97 = ( ord_less_eq_rat @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5093_neg__le__iff__le,axiom,
% 16.19/8.97 ! [B: int,A: int] :
% 16.19/8.97 ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 16.19/8.97 = ( ord_less_eq_int @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5094_compl__le__compl__iff,axiom,
% 16.19/8.97 ! [X: set_nat,Y: set_nat] :
% 16.19/8.97 ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y ) )
% 16.19/8.97 = ( ord_less_eq_set_nat @ Y @ X ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5095_neg__less__iff__less,axiom,
% 16.19/8.97 ! [B: real,A: real] :
% 16.19/8.97 ( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 16.19/8.97 = ( ord_less_real @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5096_neg__less__iff__less,axiom,
% 16.19/8.97 ! [B: int,A: int] :
% 16.19/8.97 ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 16.19/8.97 = ( ord_less_int @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5097_neg__less__iff__less,axiom,
% 16.19/8.97 ! [B: rat,A: rat] :
% 16.19/8.97 ( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 16.19/8.97 = ( ord_less_rat @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5098_neg__less__iff__less,axiom,
% 16.19/8.97 ! [B: code_integer,A: code_integer] :
% 16.19/8.97 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 16.19/8.97 = ( ord_le6747313008572928689nteger @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5099_neg__numeral__eq__iff,axiom,
% 16.19/8.97 ! [M: num,N2: num] :
% 16.19/8.97 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 16.19/8.97 = ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 16.19/8.97 = ( M = N2 ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5100_neg__numeral__eq__iff,axiom,
% 16.19/8.97 ! [M: num,N2: num] :
% 16.19/8.97 ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 16.19/8.97 = ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/8.97 = ( M = N2 ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5101_neg__numeral__eq__iff,axiom,
% 16.19/8.97 ! [M: num,N2: num] :
% 16.19/8.97 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 16.19/8.97 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
% 16.19/8.97 = ( M = N2 ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5102_neg__numeral__eq__iff,axiom,
% 16.19/8.97 ! [M: num,N2: num] :
% 16.19/8.97 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 16.19/8.97 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 16.19/8.97 = ( M = N2 ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5103_neg__numeral__eq__iff,axiom,
% 16.19/8.97 ! [M: num,N2: num] :
% 16.19/8.97 ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 16.19/8.97 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
% 16.19/8.97 = ( M = N2 ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5104_mult__minus__right,axiom,
% 16.19/8.97 ! [A: real,B: real] :
% 16.19/8.97 ( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
% 16.19/8.97 = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5105_mult__minus__right,axiom,
% 16.19/8.97 ! [A: int,B: int] :
% 16.19/8.97 ( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
% 16.19/8.97 = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5106_mult__minus__right,axiom,
% 16.19/8.97 ! [A: complex,B: complex] :
% 16.19/8.97 ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 16.19/8.97 = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5107_mult__minus__right,axiom,
% 16.19/8.97 ! [A: rat,B: rat] :
% 16.19/8.97 ( ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) )
% 16.19/8.97 = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5108_mult__minus__right,axiom,
% 16.19/8.97 ! [A: code_integer,B: code_integer] :
% 16.19/8.97 ( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 16.19/8.97 = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5109_minus__mult__minus,axiom,
% 16.19/8.97 ! [A: real,B: real] :
% 16.19/8.97 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 16.19/8.97 = ( times_times_real @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5110_minus__mult__minus,axiom,
% 16.19/8.97 ! [A: int,B: int] :
% 16.19/8.97 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 16.19/8.97 = ( times_times_int @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5111_minus__mult__minus,axiom,
% 16.19/8.97 ! [A: complex,B: complex] :
% 16.19/8.97 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 16.19/8.97 = ( times_times_complex @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5112_minus__mult__minus,axiom,
% 16.19/8.97 ! [A: rat,B: rat] :
% 16.19/8.97 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 16.19/8.97 = ( times_times_rat @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5113_minus__mult__minus,axiom,
% 16.19/8.97 ! [A: code_integer,B: code_integer] :
% 16.19/8.97 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 16.19/8.97 = ( times_3573771949741848930nteger @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5114_mult__minus__left,axiom,
% 16.19/8.97 ! [A: real,B: real] :
% 16.19/8.97 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
% 16.19/8.97 = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5115_mult__minus__left,axiom,
% 16.19/8.97 ! [A: int,B: int] :
% 16.19/8.97 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
% 16.19/8.97 = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5116_mult__minus__left,axiom,
% 16.19/8.97 ! [A: complex,B: complex] :
% 16.19/8.97 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 16.19/8.97 = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5117_mult__minus__left,axiom,
% 16.19/8.97 ! [A: rat,B: rat] :
% 16.19/8.97 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
% 16.19/8.97 = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5118_mult__minus__left,axiom,
% 16.19/8.97 ! [A: code_integer,B: code_integer] :
% 16.19/8.97 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 16.19/8.97 = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5119_minus__add__distrib,axiom,
% 16.19/8.97 ! [A: real,B: real] :
% 16.19/8.97 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 16.19/8.97 = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5120_minus__add__distrib,axiom,
% 16.19/8.97 ! [A: int,B: int] :
% 16.19/8.97 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 16.19/8.97 = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5121_minus__add__distrib,axiom,
% 16.19/8.97 ! [A: complex,B: complex] :
% 16.19/8.97 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 16.19/8.97 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5122_minus__add__distrib,axiom,
% 16.19/8.97 ! [A: rat,B: rat] :
% 16.19/8.97 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 16.19/8.97 = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5123_minus__add__distrib,axiom,
% 16.19/8.97 ! [A: code_integer,B: code_integer] :
% 16.19/8.97 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 16.19/8.97 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5124_minus__add__cancel,axiom,
% 16.19/8.97 ! [A: real,B: real] :
% 16.19/8.97 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
% 16.19/8.97 = B ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5125_minus__add__cancel,axiom,
% 16.19/8.97 ! [A: int,B: int] :
% 16.19/8.97 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
% 16.19/8.97 = B ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5126_minus__add__cancel,axiom,
% 16.19/8.97 ! [A: complex,B: complex] :
% 16.19/8.97 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
% 16.19/8.97 = B ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5127_minus__add__cancel,axiom,
% 16.19/8.97 ! [A: rat,B: rat] :
% 16.19/8.97 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
% 16.19/8.97 = B ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5128_minus__add__cancel,axiom,
% 16.19/8.97 ! [A: code_integer,B: code_integer] :
% 16.19/8.97 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 16.19/8.97 = B ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5129_add__minus__cancel,axiom,
% 16.19/8.97 ! [A: real,B: real] :
% 16.19/8.97 ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
% 16.19/8.97 = B ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5130_add__minus__cancel,axiom,
% 16.19/8.97 ! [A: int,B: int] :
% 16.19/8.97 ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
% 16.19/8.97 = B ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5131_add__minus__cancel,axiom,
% 16.19/8.97 ! [A: complex,B: complex] :
% 16.19/8.97 ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
% 16.19/8.97 = B ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5132_add__minus__cancel,axiom,
% 16.19/8.97 ! [A: rat,B: rat] :
% 16.19/8.97 ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
% 16.19/8.97 = B ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5133_add__minus__cancel,axiom,
% 16.19/8.97 ! [A: code_integer,B: code_integer] :
% 16.19/8.97 ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
% 16.19/8.97 = B ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5134_minus__diff__eq,axiom,
% 16.19/8.97 ! [A: real,B: real] :
% 16.19/8.97 ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
% 16.19/8.97 = ( minus_minus_real @ B @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5135_minus__diff__eq,axiom,
% 16.19/8.97 ! [A: int,B: int] :
% 16.19/8.97 ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
% 16.19/8.97 = ( minus_minus_int @ B @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5136_minus__diff__eq,axiom,
% 16.19/8.97 ! [A: complex,B: complex] :
% 16.19/8.97 ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
% 16.19/8.97 = ( minus_minus_complex @ B @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5137_minus__diff__eq,axiom,
% 16.19/8.97 ! [A: rat,B: rat] :
% 16.19/8.97 ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
% 16.19/8.97 = ( minus_minus_rat @ B @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5138_minus__diff__eq,axiom,
% 16.19/8.97 ! [A: code_integer,B: code_integer] :
% 16.19/8.97 ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
% 16.19/8.97 = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5139_div__minus__minus,axiom,
% 16.19/8.97 ! [A: int,B: int] :
% 16.19/8.97 ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 16.19/8.97 = ( divide_divide_int @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5140_div__minus__minus,axiom,
% 16.19/8.97 ! [A: code_integer,B: code_integer] :
% 16.19/8.97 ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 16.19/8.97 = ( divide6298287555418463151nteger @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5141_ln__less__cancel__iff,axiom,
% 16.19/8.97 ! [X: real,Y: real] :
% 16.19/8.97 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/8.97 => ( ( ord_less_real @ zero_zero_real @ Y )
% 16.19/8.97 => ( ( ord_less_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
% 16.19/8.97 = ( ord_less_real @ X @ Y ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5142_ln__inj__iff,axiom,
% 16.19/8.97 ! [X: real,Y: real] :
% 16.19/8.97 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/8.97 => ( ( ord_less_real @ zero_zero_real @ Y )
% 16.19/8.97 => ( ( ( ln_ln_real @ X )
% 16.19/8.97 = ( ln_ln_real @ Y ) )
% 16.19/8.97 = ( X = Y ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5143_mod__minus__minus,axiom,
% 16.19/8.97 ! [A: int,B: int] :
% 16.19/8.97 ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 16.19/8.97 = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5144_mod__minus__minus,axiom,
% 16.19/8.97 ! [A: code_integer,B: code_integer] :
% 16.19/8.97 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 16.19/8.97 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5145_real__add__minus__iff,axiom,
% 16.19/8.97 ! [X: real,A: real] :
% 16.19/8.97 ( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
% 16.19/8.97 = zero_zero_real )
% 16.19/8.97 = ( X = A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5146_sum_Oneutral__const,axiom,
% 16.19/8.97 ! [A2: set_int] :
% 16.19/8.97 ( ( groups4538972089207619220nt_int
% 16.19/8.97 @ ^ [Uu3: int] : zero_zero_int
% 16.19/8.97 @ A2 )
% 16.19/8.97 = zero_zero_int ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5147_sum_Oneutral__const,axiom,
% 16.19/8.97 ! [A2: set_complex] :
% 16.19/8.97 ( ( groups7754918857620584856omplex
% 16.19/8.97 @ ^ [Uu3: complex] : zero_zero_complex
% 16.19/8.97 @ A2 )
% 16.19/8.97 = zero_zero_complex ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5148_sum_Oneutral__const,axiom,
% 16.19/8.97 ! [A2: set_nat] :
% 16.19/8.97 ( ( groups3542108847815614940at_nat
% 16.19/8.97 @ ^ [Uu3: nat] : zero_zero_nat
% 16.19/8.97 @ A2 )
% 16.19/8.97 = zero_zero_nat ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5149_sum_Oneutral__const,axiom,
% 16.19/8.97 ! [A2: set_nat] :
% 16.19/8.97 ( ( groups6591440286371151544t_real
% 16.19/8.97 @ ^ [Uu3: nat] : zero_zero_real
% 16.19/8.97 @ A2 )
% 16.19/8.97 = zero_zero_real ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5150_case__prod__conv,axiom,
% 16.19/8.97 ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,A: nat,B: nat] :
% 16.19/8.97 ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
% 16.19/8.97 = ( F @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5151_case__prod__conv,axiom,
% 16.19/8.97 ! [F: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat] :
% 16.19/8.97 ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
% 16.19/8.97 = ( F @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5152_case__prod__conv,axiom,
% 16.19/8.97 ! [F: int > int > product_prod_int_int,A: int,B: int] :
% 16.19/8.97 ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
% 16.19/8.97 = ( F @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5153_case__prod__conv,axiom,
% 16.19/8.97 ! [F: int > int > $o,A: int,B: int] :
% 16.19/8.97 ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
% 16.19/8.97 = ( F @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5154_case__prod__conv,axiom,
% 16.19/8.97 ! [F: int > int > int,A: int,B: int] :
% 16.19/8.97 ( ( produc8211389475949308722nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
% 16.19/8.97 = ( F @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5155_neg__0__le__iff__le,axiom,
% 16.19/8.97 ! [A: real] :
% 16.19/8.97 ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 16.19/8.97 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5156_neg__0__le__iff__le,axiom,
% 16.19/8.97 ! [A: code_integer] :
% 16.19/8.97 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 16.19/8.97 = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5157_neg__0__le__iff__le,axiom,
% 16.19/8.97 ! [A: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 16.19/8.97 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5158_neg__0__le__iff__le,axiom,
% 16.19/8.97 ! [A: int] :
% 16.19/8.97 ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 16.19/8.97 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5159_neg__le__0__iff__le,axiom,
% 16.19/8.97 ! [A: real] :
% 16.19/8.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 16.19/8.97 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5160_neg__le__0__iff__le,axiom,
% 16.19/8.97 ! [A: code_integer] :
% 16.19/8.97 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 16.19/8.97 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5161_neg__le__0__iff__le,axiom,
% 16.19/8.97 ! [A: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 16.19/8.97 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5162_neg__le__0__iff__le,axiom,
% 16.19/8.97 ! [A: int] :
% 16.19/8.97 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 16.19/8.97 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5163_less__eq__neg__nonpos,axiom,
% 16.19/8.97 ! [A: real] :
% 16.19/8.97 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 16.19/8.97 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5164_less__eq__neg__nonpos,axiom,
% 16.19/8.97 ! [A: code_integer] :
% 16.19/8.97 ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 16.19/8.97 = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5165_less__eq__neg__nonpos,axiom,
% 16.19/8.97 ! [A: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
% 16.19/8.97 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5166_less__eq__neg__nonpos,axiom,
% 16.19/8.97 ! [A: int] :
% 16.19/8.97 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
% 16.19/8.97 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5167_neg__less__eq__nonneg,axiom,
% 16.19/8.97 ! [A: real] :
% 16.19/8.97 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 16.19/8.97 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5168_neg__less__eq__nonneg,axiom,
% 16.19/8.97 ! [A: code_integer] :
% 16.19/8.97 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 16.19/8.97 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5169_neg__less__eq__nonneg,axiom,
% 16.19/8.97 ! [A: rat] :
% 16.19/8.97 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
% 16.19/8.97 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5170_neg__less__eq__nonneg,axiom,
% 16.19/8.97 ! [A: int] :
% 16.19/8.97 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
% 16.19/8.97 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5171_less__neg__neg,axiom,
% 16.19/8.97 ! [A: real] :
% 16.19/8.97 ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
% 16.19/8.97 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5172_less__neg__neg,axiom,
% 16.19/8.97 ! [A: int] :
% 16.19/8.97 ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
% 16.19/8.97 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5173_less__neg__neg,axiom,
% 16.19/8.97 ! [A: rat] :
% 16.19/8.97 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
% 16.19/8.97 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5174_less__neg__neg,axiom,
% 16.19/8.97 ! [A: code_integer] :
% 16.19/8.97 ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 16.19/8.97 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5175_neg__less__pos,axiom,
% 16.19/8.97 ! [A: real] :
% 16.19/8.97 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
% 16.19/8.97 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5176_neg__less__pos,axiom,
% 16.19/8.97 ! [A: int] :
% 16.19/8.97 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
% 16.19/8.97 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5177_neg__less__pos,axiom,
% 16.19/8.97 ! [A: rat] :
% 16.19/8.97 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
% 16.19/8.97 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5178_neg__less__pos,axiom,
% 16.19/8.97 ! [A: code_integer] :
% 16.19/8.97 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 16.19/8.97 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5179_neg__0__less__iff__less,axiom,
% 16.19/8.97 ! [A: real] :
% 16.19/8.97 ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 16.19/8.97 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5180_neg__0__less__iff__less,axiom,
% 16.19/8.97 ! [A: int] :
% 16.19/8.97 ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 16.19/8.97 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5181_neg__0__less__iff__less,axiom,
% 16.19/8.97 ! [A: rat] :
% 16.19/8.97 ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 16.19/8.97 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5182_neg__0__less__iff__less,axiom,
% 16.19/8.97 ! [A: code_integer] :
% 16.19/8.97 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 16.19/8.97 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5183_neg__less__0__iff__less,axiom,
% 16.19/8.97 ! [A: real] :
% 16.19/8.97 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 16.19/8.97 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5184_neg__less__0__iff__less,axiom,
% 16.19/8.97 ! [A: int] :
% 16.19/8.97 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 16.19/8.97 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5185_neg__less__0__iff__less,axiom,
% 16.19/8.97 ! [A: rat] :
% 16.19/8.97 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 16.19/8.97 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5186_neg__less__0__iff__less,axiom,
% 16.19/8.97 ! [A: code_integer] :
% 16.19/8.97 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 16.19/8.97 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5187_add_Oright__inverse,axiom,
% 16.19/8.97 ! [A: real] :
% 16.19/8.97 ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
% 16.19/8.97 = zero_zero_real ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5188_add_Oright__inverse,axiom,
% 16.19/8.97 ! [A: int] :
% 16.19/8.97 ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
% 16.19/8.97 = zero_zero_int ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5189_add_Oright__inverse,axiom,
% 16.19/8.97 ! [A: complex] :
% 16.19/8.97 ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
% 16.19/8.97 = zero_zero_complex ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5190_add_Oright__inverse,axiom,
% 16.19/8.97 ! [A: rat] :
% 16.19/8.97 ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
% 16.19/8.97 = zero_zero_rat ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5191_add_Oright__inverse,axiom,
% 16.19/8.97 ! [A: code_integer] :
% 16.19/8.97 ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 16.19/8.97 = zero_z3403309356797280102nteger ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5192_ab__left__minus,axiom,
% 16.19/8.97 ! [A: real] :
% 16.19/8.97 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 16.19/8.97 = zero_zero_real ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5193_ab__left__minus,axiom,
% 16.19/8.97 ! [A: int] :
% 16.19/8.97 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 16.19/8.97 = zero_zero_int ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5194_ab__left__minus,axiom,
% 16.19/8.97 ! [A: complex] :
% 16.19/8.97 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 16.19/8.97 = zero_zero_complex ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5195_ab__left__minus,axiom,
% 16.19/8.97 ! [A: rat] :
% 16.19/8.97 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 16.19/8.97 = zero_zero_rat ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5196_ab__left__minus,axiom,
% 16.19/8.97 ! [A: code_integer] :
% 16.19/8.97 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 16.19/8.97 = zero_z3403309356797280102nteger ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5197_diff__0,axiom,
% 16.19/8.97 ! [A: real] :
% 16.19/8.97 ( ( minus_minus_real @ zero_zero_real @ A )
% 16.19/8.97 = ( uminus_uminus_real @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5198_diff__0,axiom,
% 16.19/8.97 ! [A: int] :
% 16.19/8.97 ( ( minus_minus_int @ zero_zero_int @ A )
% 16.19/8.97 = ( uminus_uminus_int @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5199_diff__0,axiom,
% 16.19/8.97 ! [A: complex] :
% 16.19/8.97 ( ( minus_minus_complex @ zero_zero_complex @ A )
% 16.19/8.97 = ( uminus1482373934393186551omplex @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5200_diff__0,axiom,
% 16.19/8.97 ! [A: rat] :
% 16.19/8.97 ( ( minus_minus_rat @ zero_zero_rat @ A )
% 16.19/8.97 = ( uminus_uminus_rat @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5201_diff__0,axiom,
% 16.19/8.97 ! [A: code_integer] :
% 16.19/8.97 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
% 16.19/8.97 = ( uminus1351360451143612070nteger @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5202_verit__minus__simplify_I3_J,axiom,
% 16.19/8.97 ! [B: real] :
% 16.19/8.97 ( ( minus_minus_real @ zero_zero_real @ B )
% 16.19/8.97 = ( uminus_uminus_real @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5203_verit__minus__simplify_I3_J,axiom,
% 16.19/8.97 ! [B: int] :
% 16.19/8.97 ( ( minus_minus_int @ zero_zero_int @ B )
% 16.19/8.97 = ( uminus_uminus_int @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5204_verit__minus__simplify_I3_J,axiom,
% 16.19/8.97 ! [B: complex] :
% 16.19/8.97 ( ( minus_minus_complex @ zero_zero_complex @ B )
% 16.19/8.97 = ( uminus1482373934393186551omplex @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5205_verit__minus__simplify_I3_J,axiom,
% 16.19/8.97 ! [B: rat] :
% 16.19/8.97 ( ( minus_minus_rat @ zero_zero_rat @ B )
% 16.19/8.97 = ( uminus_uminus_rat @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5206_verit__minus__simplify_I3_J,axiom,
% 16.19/8.97 ! [B: code_integer] :
% 16.19/8.97 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
% 16.19/8.97 = ( uminus1351360451143612070nteger @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5207_add__neg__numeral__simps_I3_J,axiom,
% 16.19/8.97 ! [M: num,N2: num] :
% 16.19/8.97 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 16.19/8.97 = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5208_add__neg__numeral__simps_I3_J,axiom,
% 16.19/8.97 ! [M: num,N2: num] :
% 16.19/8.97 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/8.97 = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5209_add__neg__numeral__simps_I3_J,axiom,
% 16.19/8.97 ! [M: num,N2: num] :
% 16.19/8.97 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
% 16.19/8.97 = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N2 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5210_add__neg__numeral__simps_I3_J,axiom,
% 16.19/8.97 ! [M: num,N2: num] :
% 16.19/8.97 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 16.19/8.97 = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5211_add__neg__numeral__simps_I3_J,axiom,
% 16.19/8.97 ! [M: num,N2: num] :
% 16.19/8.97 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
% 16.19/8.97 = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5212_mult__minus1,axiom,
% 16.19/8.97 ! [Z: real] :
% 16.19/8.97 ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
% 16.19/8.97 = ( uminus_uminus_real @ Z ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5213_mult__minus1,axiom,
% 16.19/8.97 ! [Z: int] :
% 16.19/8.97 ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
% 16.19/8.97 = ( uminus_uminus_int @ Z ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5214_mult__minus1,axiom,
% 16.19/8.97 ! [Z: complex] :
% 16.19/8.97 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
% 16.19/8.97 = ( uminus1482373934393186551omplex @ Z ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5215_mult__minus1,axiom,
% 16.19/8.97 ! [Z: rat] :
% 16.19/8.97 ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
% 16.19/8.97 = ( uminus_uminus_rat @ Z ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5216_mult__minus1,axiom,
% 16.19/8.97 ! [Z: code_integer] :
% 16.19/8.97 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
% 16.19/8.97 = ( uminus1351360451143612070nteger @ Z ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5217_mult__minus1__right,axiom,
% 16.19/8.97 ! [Z: real] :
% 16.19/8.97 ( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
% 16.19/8.97 = ( uminus_uminus_real @ Z ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5218_mult__minus1__right,axiom,
% 16.19/8.97 ! [Z: int] :
% 16.19/8.97 ( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
% 16.19/8.97 = ( uminus_uminus_int @ Z ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5219_mult__minus1__right,axiom,
% 16.19/8.97 ! [Z: complex] :
% 16.19/8.97 ( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 16.19/8.97 = ( uminus1482373934393186551omplex @ Z ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5220_mult__minus1__right,axiom,
% 16.19/8.97 ! [Z: rat] :
% 16.19/8.97 ( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
% 16.19/8.97 = ( uminus_uminus_rat @ Z ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5221_mult__minus1__right,axiom,
% 16.19/8.97 ! [Z: code_integer] :
% 16.19/8.97 ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 16.19/8.97 = ( uminus1351360451143612070nteger @ Z ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5222_uminus__add__conv__diff,axiom,
% 16.19/8.97 ! [A: real,B: real] :
% 16.19/8.97 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
% 16.19/8.97 = ( minus_minus_real @ B @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5223_uminus__add__conv__diff,axiom,
% 16.19/8.97 ! [A: int,B: int] :
% 16.19/8.97 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
% 16.19/8.97 = ( minus_minus_int @ B @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5224_uminus__add__conv__diff,axiom,
% 16.19/8.97 ! [A: complex,B: complex] :
% 16.19/8.97 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 16.19/8.97 = ( minus_minus_complex @ B @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5225_uminus__add__conv__diff,axiom,
% 16.19/8.97 ! [A: rat,B: rat] :
% 16.19/8.97 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
% 16.19/8.97 = ( minus_minus_rat @ B @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5226_uminus__add__conv__diff,axiom,
% 16.19/8.97 ! [A: code_integer,B: code_integer] :
% 16.19/8.97 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 16.19/8.97 = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5227_diff__minus__eq__add,axiom,
% 16.19/8.97 ! [A: real,B: real] :
% 16.19/8.97 ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
% 16.19/8.97 = ( plus_plus_real @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5228_diff__minus__eq__add,axiom,
% 16.19/8.97 ! [A: int,B: int] :
% 16.19/8.97 ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
% 16.19/8.97 = ( plus_plus_int @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5229_diff__minus__eq__add,axiom,
% 16.19/8.97 ! [A: complex,B: complex] :
% 16.19/8.97 ( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 16.19/8.97 = ( plus_plus_complex @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5230_diff__minus__eq__add,axiom,
% 16.19/8.97 ! [A: rat,B: rat] :
% 16.19/8.97 ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
% 16.19/8.97 = ( plus_plus_rat @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5231_diff__minus__eq__add,axiom,
% 16.19/8.97 ! [A: code_integer,B: code_integer] :
% 16.19/8.97 ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 16.19/8.97 = ( plus_p5714425477246183910nteger @ A @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5232_div__minus1__right,axiom,
% 16.19/8.97 ! [A: int] :
% 16.19/8.97 ( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 16.19/8.97 = ( uminus_uminus_int @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5233_div__minus1__right,axiom,
% 16.19/8.97 ! [A: code_integer] :
% 16.19/8.97 ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 16.19/8.97 = ( uminus1351360451143612070nteger @ A ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5234_divide__minus1,axiom,
% 16.19/8.97 ! [X: real] :
% 16.19/8.97 ( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
% 16.19/8.97 = ( uminus_uminus_real @ X ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5235_divide__minus1,axiom,
% 16.19/8.97 ! [X: complex] :
% 16.19/8.97 ( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 16.19/8.97 = ( uminus1482373934393186551omplex @ X ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5236_divide__minus1,axiom,
% 16.19/8.97 ! [X: rat] :
% 16.19/8.97 ( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
% 16.19/8.97 = ( uminus_uminus_rat @ X ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5237_minus__mod__self1,axiom,
% 16.19/8.97 ! [B: int,A: int] :
% 16.19/8.97 ( ( modulo_modulo_int @ ( minus_minus_int @ B @ A ) @ B )
% 16.19/8.97 = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5238_minus__mod__self1,axiom,
% 16.19/8.97 ! [B: code_integer,A: code_integer] :
% 16.19/8.97 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ B )
% 16.19/8.97 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5239_ln__le__cancel__iff,axiom,
% 16.19/8.97 ! [X: real,Y: real] :
% 16.19/8.97 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/8.97 => ( ( ord_less_real @ zero_zero_real @ Y )
% 16.19/8.97 => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) )
% 16.19/8.97 = ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5240_ln__eq__zero__iff,axiom,
% 16.19/8.97 ! [X: real] :
% 16.19/8.97 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/8.97 => ( ( ( ln_ln_real @ X )
% 16.19/8.97 = zero_zero_real )
% 16.19/8.97 = ( X = one_one_real ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5241_ln__gt__zero__iff,axiom,
% 16.19/8.97 ! [X: real] :
% 16.19/8.97 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/8.97 => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 16.19/8.97 = ( ord_less_real @ one_one_real @ X ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5242_ln__less__zero__iff,axiom,
% 16.19/8.97 ! [X: real] :
% 16.19/8.97 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/8.97 => ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
% 16.19/8.97 = ( ord_less_real @ X @ one_one_real ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5243_ln__one,axiom,
% 16.19/8.97 ( ( ln_ln_real @ one_one_real )
% 16.19/8.97 = zero_zero_real ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5244_signed__take__bit__of__minus__1,axiom,
% 16.19/8.97 ! [N2: nat] :
% 16.19/8.97 ( ( bit_ri6519982836138164636nteger @ N2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 16.19/8.97 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5245_signed__take__bit__of__minus__1,axiom,
% 16.19/8.97 ! [N2: nat] :
% 16.19/8.97 ( ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ one_one_int ) )
% 16.19/8.97 = ( uminus_uminus_int @ one_one_int ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5246_sum_Odelta,axiom,
% 16.19/8.97 ! [S: set_real,A: real,B: real > complex] :
% 16.19/8.97 ( ( finite_finite_real @ S )
% 16.19/8.97 => ( ( ( member_real @ A @ S )
% 16.19/8.97 => ( ( groups5754745047067104278omplex
% 16.19/8.97 @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 16.19/8.97 @ S )
% 16.19/8.97 = ( B @ A ) ) )
% 16.19/8.97 & ( ~ ( member_real @ A @ S )
% 16.19/8.97 => ( ( groups5754745047067104278omplex
% 16.19/8.97 @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 16.19/8.97 @ S )
% 16.19/8.97 = zero_zero_complex ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5247_sum_Odelta,axiom,
% 16.19/8.97 ! [S: set_nat,A: nat,B: nat > complex] :
% 16.19/8.97 ( ( finite_finite_nat @ S )
% 16.19/8.97 => ( ( ( member_nat @ A @ S )
% 16.19/8.97 => ( ( groups2073611262835488442omplex
% 16.19/8.97 @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 16.19/8.97 @ S )
% 16.19/8.97 = ( B @ A ) ) )
% 16.19/8.97 & ( ~ ( member_nat @ A @ S )
% 16.19/8.97 => ( ( groups2073611262835488442omplex
% 16.19/8.97 @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 16.19/8.97 @ S )
% 16.19/8.97 = zero_zero_complex ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5248_sum_Odelta,axiom,
% 16.19/8.97 ! [S: set_int,A: int,B: int > complex] :
% 16.19/8.97 ( ( finite_finite_int @ S )
% 16.19/8.97 => ( ( ( member_int @ A @ S )
% 16.19/8.97 => ( ( groups3049146728041665814omplex
% 16.19/8.97 @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 16.19/8.97 @ S )
% 16.19/8.97 = ( B @ A ) ) )
% 16.19/8.97 & ( ~ ( member_int @ A @ S )
% 16.19/8.97 => ( ( groups3049146728041665814omplex
% 16.19/8.97 @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 16.19/8.97 @ S )
% 16.19/8.97 = zero_zero_complex ) ) ) ) ).
% 16.19/8.97
% 16.19/8.97 thf(fact_5249_sum_Odelta,axiom,
% 16.19/8.97 ! [S: set_real,A: real,B: real > real] :
% 16.19/8.97 ( ( finite_finite_real @ S )
% 16.19/8.97 => ( ( ( member_real @ A @ S )
% 16.19/8.97 => ( ( groups8097168146408367636l_real
% 16.19/8.97 @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 16.19/8.97 @ S )
% 16.19/8.97 = ( B @ A ) ) )
% 16.19/8.97 & ( ~ ( member_real @ A @ S )
% 16.19/8.97 => ( ( groups8097168146408367636l_real
% 16.19/8.97 @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 16.19/8.97 @ S )
% 16.19/8.97 = zero_zero_real ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5250_sum_Odelta,axiom,
% 16.19/8.98 ! [S: set_int,A: int,B: int > real] :
% 16.19/8.98 ( ( finite_finite_int @ S )
% 16.19/8.98 => ( ( ( member_int @ A @ S )
% 16.19/8.98 => ( ( groups8778361861064173332t_real
% 16.19/8.98 @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 16.19/8.98 @ S )
% 16.19/8.98 = ( B @ A ) ) )
% 16.19/8.98 & ( ~ ( member_int @ A @ S )
% 16.19/8.98 => ( ( groups8778361861064173332t_real
% 16.19/8.98 @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 16.19/8.98 @ S )
% 16.19/8.98 = zero_zero_real ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5251_sum_Odelta,axiom,
% 16.19/8.98 ! [S: set_complex,A: complex,B: complex > real] :
% 16.19/8.98 ( ( finite3207457112153483333omplex @ S )
% 16.19/8.98 => ( ( ( member_complex @ A @ S )
% 16.19/8.98 => ( ( groups5808333547571424918x_real
% 16.19/8.98 @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 16.19/8.98 @ S )
% 16.19/8.98 = ( B @ A ) ) )
% 16.19/8.98 & ( ~ ( member_complex @ A @ S )
% 16.19/8.98 => ( ( groups5808333547571424918x_real
% 16.19/8.98 @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 16.19/8.98 @ S )
% 16.19/8.98 = zero_zero_real ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5252_sum_Odelta,axiom,
% 16.19/8.98 ! [S: set_real,A: real,B: real > rat] :
% 16.19/8.98 ( ( finite_finite_real @ S )
% 16.19/8.98 => ( ( ( member_real @ A @ S )
% 16.19/8.98 => ( ( groups1300246762558778688al_rat
% 16.19/8.98 @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 16.19/8.98 @ S )
% 16.19/8.98 = ( B @ A ) ) )
% 16.19/8.98 & ( ~ ( member_real @ A @ S )
% 16.19/8.98 => ( ( groups1300246762558778688al_rat
% 16.19/8.98 @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 16.19/8.98 @ S )
% 16.19/8.98 = zero_zero_rat ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5253_sum_Odelta,axiom,
% 16.19/8.98 ! [S: set_nat,A: nat,B: nat > rat] :
% 16.19/8.98 ( ( finite_finite_nat @ S )
% 16.19/8.98 => ( ( ( member_nat @ A @ S )
% 16.19/8.98 => ( ( groups2906978787729119204at_rat
% 16.19/8.98 @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 16.19/8.98 @ S )
% 16.19/8.98 = ( B @ A ) ) )
% 16.19/8.98 & ( ~ ( member_nat @ A @ S )
% 16.19/8.98 => ( ( groups2906978787729119204at_rat
% 16.19/8.98 @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 16.19/8.98 @ S )
% 16.19/8.98 = zero_zero_rat ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5254_sum_Odelta,axiom,
% 16.19/8.98 ! [S: set_int,A: int,B: int > rat] :
% 16.19/8.98 ( ( finite_finite_int @ S )
% 16.19/8.98 => ( ( ( member_int @ A @ S )
% 16.19/8.98 => ( ( groups3906332499630173760nt_rat
% 16.19/8.98 @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 16.19/8.98 @ S )
% 16.19/8.98 = ( B @ A ) ) )
% 16.19/8.98 & ( ~ ( member_int @ A @ S )
% 16.19/8.98 => ( ( groups3906332499630173760nt_rat
% 16.19/8.98 @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 16.19/8.98 @ S )
% 16.19/8.98 = zero_zero_rat ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5255_sum_Odelta,axiom,
% 16.19/8.98 ! [S: set_complex,A: complex,B: complex > rat] :
% 16.19/8.98 ( ( finite3207457112153483333omplex @ S )
% 16.19/8.98 => ( ( ( member_complex @ A @ S )
% 16.19/8.98 => ( ( groups5058264527183730370ex_rat
% 16.19/8.98 @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 16.19/8.98 @ S )
% 16.19/8.98 = ( B @ A ) ) )
% 16.19/8.98 & ( ~ ( member_complex @ A @ S )
% 16.19/8.98 => ( ( groups5058264527183730370ex_rat
% 16.19/8.98 @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 16.19/8.98 @ S )
% 16.19/8.98 = zero_zero_rat ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5256_sum_Odelta_H,axiom,
% 16.19/8.98 ! [S: set_real,A: real,B: real > complex] :
% 16.19/8.98 ( ( finite_finite_real @ S )
% 16.19/8.98 => ( ( ( member_real @ A @ S )
% 16.19/8.98 => ( ( groups5754745047067104278omplex
% 16.19/8.98 @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 16.19/8.98 @ S )
% 16.19/8.98 = ( B @ A ) ) )
% 16.19/8.98 & ( ~ ( member_real @ A @ S )
% 16.19/8.98 => ( ( groups5754745047067104278omplex
% 16.19/8.98 @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 16.19/8.98 @ S )
% 16.19/8.98 = zero_zero_complex ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5257_sum_Odelta_H,axiom,
% 16.19/8.98 ! [S: set_nat,A: nat,B: nat > complex] :
% 16.19/8.98 ( ( finite_finite_nat @ S )
% 16.19/8.98 => ( ( ( member_nat @ A @ S )
% 16.19/8.98 => ( ( groups2073611262835488442omplex
% 16.19/8.98 @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 16.19/8.98 @ S )
% 16.19/8.98 = ( B @ A ) ) )
% 16.19/8.98 & ( ~ ( member_nat @ A @ S )
% 16.19/8.98 => ( ( groups2073611262835488442omplex
% 16.19/8.98 @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 16.19/8.98 @ S )
% 16.19/8.98 = zero_zero_complex ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5258_sum_Odelta_H,axiom,
% 16.19/8.98 ! [S: set_int,A: int,B: int > complex] :
% 16.19/8.98 ( ( finite_finite_int @ S )
% 16.19/8.98 => ( ( ( member_int @ A @ S )
% 16.19/8.98 => ( ( groups3049146728041665814omplex
% 16.19/8.98 @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 16.19/8.98 @ S )
% 16.19/8.98 = ( B @ A ) ) )
% 16.19/8.98 & ( ~ ( member_int @ A @ S )
% 16.19/8.98 => ( ( groups3049146728041665814omplex
% 16.19/8.98 @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 16.19/8.98 @ S )
% 16.19/8.98 = zero_zero_complex ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5259_sum_Odelta_H,axiom,
% 16.19/8.98 ! [S: set_real,A: real,B: real > real] :
% 16.19/8.98 ( ( finite_finite_real @ S )
% 16.19/8.98 => ( ( ( member_real @ A @ S )
% 16.19/8.98 => ( ( groups8097168146408367636l_real
% 16.19/8.98 @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 16.19/8.98 @ S )
% 16.19/8.98 = ( B @ A ) ) )
% 16.19/8.98 & ( ~ ( member_real @ A @ S )
% 16.19/8.98 => ( ( groups8097168146408367636l_real
% 16.19/8.98 @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 16.19/8.98 @ S )
% 16.19/8.98 = zero_zero_real ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5260_sum_Odelta_H,axiom,
% 16.19/8.98 ! [S: set_int,A: int,B: int > real] :
% 16.19/8.98 ( ( finite_finite_int @ S )
% 16.19/8.98 => ( ( ( member_int @ A @ S )
% 16.19/8.98 => ( ( groups8778361861064173332t_real
% 16.19/8.98 @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 16.19/8.98 @ S )
% 16.19/8.98 = ( B @ A ) ) )
% 16.19/8.98 & ( ~ ( member_int @ A @ S )
% 16.19/8.98 => ( ( groups8778361861064173332t_real
% 16.19/8.98 @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 16.19/8.98 @ S )
% 16.19/8.98 = zero_zero_real ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5261_sum_Odelta_H,axiom,
% 16.19/8.98 ! [S: set_complex,A: complex,B: complex > real] :
% 16.19/8.98 ( ( finite3207457112153483333omplex @ S )
% 16.19/8.98 => ( ( ( member_complex @ A @ S )
% 16.19/8.98 => ( ( groups5808333547571424918x_real
% 16.19/8.98 @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 16.19/8.98 @ S )
% 16.19/8.98 = ( B @ A ) ) )
% 16.19/8.98 & ( ~ ( member_complex @ A @ S )
% 16.19/8.98 => ( ( groups5808333547571424918x_real
% 16.19/8.98 @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 16.19/8.98 @ S )
% 16.19/8.98 = zero_zero_real ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5262_sum_Odelta_H,axiom,
% 16.19/8.98 ! [S: set_real,A: real,B: real > rat] :
% 16.19/8.98 ( ( finite_finite_real @ S )
% 16.19/8.98 => ( ( ( member_real @ A @ S )
% 16.19/8.98 => ( ( groups1300246762558778688al_rat
% 16.19/8.98 @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 16.19/8.98 @ S )
% 16.19/8.98 = ( B @ A ) ) )
% 16.19/8.98 & ( ~ ( member_real @ A @ S )
% 16.19/8.98 => ( ( groups1300246762558778688al_rat
% 16.19/8.98 @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 16.19/8.98 @ S )
% 16.19/8.98 = zero_zero_rat ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5263_sum_Odelta_H,axiom,
% 16.19/8.98 ! [S: set_nat,A: nat,B: nat > rat] :
% 16.19/8.98 ( ( finite_finite_nat @ S )
% 16.19/8.98 => ( ( ( member_nat @ A @ S )
% 16.19/8.98 => ( ( groups2906978787729119204at_rat
% 16.19/8.98 @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 16.19/8.98 @ S )
% 16.19/8.98 = ( B @ A ) ) )
% 16.19/8.98 & ( ~ ( member_nat @ A @ S )
% 16.19/8.98 => ( ( groups2906978787729119204at_rat
% 16.19/8.98 @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 16.19/8.98 @ S )
% 16.19/8.98 = zero_zero_rat ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5264_sum_Odelta_H,axiom,
% 16.19/8.98 ! [S: set_int,A: int,B: int > rat] :
% 16.19/8.98 ( ( finite_finite_int @ S )
% 16.19/8.98 => ( ( ( member_int @ A @ S )
% 16.19/8.98 => ( ( groups3906332499630173760nt_rat
% 16.19/8.98 @ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 16.19/8.98 @ S )
% 16.19/8.98 = ( B @ A ) ) )
% 16.19/8.98 & ( ~ ( member_int @ A @ S )
% 16.19/8.98 => ( ( groups3906332499630173760nt_rat
% 16.19/8.98 @ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 16.19/8.98 @ S )
% 16.19/8.98 = zero_zero_rat ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5265_sum_Odelta_H,axiom,
% 16.19/8.98 ! [S: set_complex,A: complex,B: complex > rat] :
% 16.19/8.98 ( ( finite3207457112153483333omplex @ S )
% 16.19/8.98 => ( ( ( member_complex @ A @ S )
% 16.19/8.98 => ( ( groups5058264527183730370ex_rat
% 16.19/8.98 @ ^ [K3: complex] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 16.19/8.98 @ S )
% 16.19/8.98 = ( B @ A ) ) )
% 16.19/8.98 & ( ~ ( member_complex @ A @ S )
% 16.19/8.98 => ( ( groups5058264527183730370ex_rat
% 16.19/8.98 @ ^ [K3: complex] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 16.19/8.98 @ S )
% 16.19/8.98 = zero_zero_rat ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5266_dbl__simps_I1_J,axiom,
% 16.19/8.98 ! [K: num] :
% 16.19/8.98 ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 16.19/8.98 = ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5267_dbl__simps_I1_J,axiom,
% 16.19/8.98 ! [K: num] :
% 16.19/8.98 ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 16.19/8.98 = ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5268_dbl__simps_I1_J,axiom,
% 16.19/8.98 ! [K: num] :
% 16.19/8.98 ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 16.19/8.98 = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5269_dbl__simps_I1_J,axiom,
% 16.19/8.98 ! [K: num] :
% 16.19/8.98 ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 16.19/8.98 = ( uminus_uminus_rat @ ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5270_dbl__simps_I1_J,axiom,
% 16.19/8.98 ! [K: num] :
% 16.19/8.98 ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 16.19/8.98 = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5271_add__neg__numeral__special_I8_J,axiom,
% 16.19/8.98 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 16.19/8.98 = zero_zero_real ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5272_add__neg__numeral__special_I8_J,axiom,
% 16.19/8.98 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 16.19/8.98 = zero_zero_int ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5273_add__neg__numeral__special_I8_J,axiom,
% 16.19/8.98 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 16.19/8.98 = zero_zero_complex ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5274_add__neg__numeral__special_I8_J,axiom,
% 16.19/8.98 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 16.19/8.98 = zero_zero_rat ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5275_add__neg__numeral__special_I8_J,axiom,
% 16.19/8.98 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 16.19/8.98 = zero_z3403309356797280102nteger ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5276_add__neg__numeral__special_I7_J,axiom,
% 16.19/8.98 ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 16.19/8.98 = zero_zero_real ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5277_add__neg__numeral__special_I7_J,axiom,
% 16.19/8.98 ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 16.19/8.98 = zero_zero_int ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5278_add__neg__numeral__special_I7_J,axiom,
% 16.19/8.98 ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 16.19/8.98 = zero_zero_complex ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5279_add__neg__numeral__special_I7_J,axiom,
% 16.19/8.98 ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 16.19/8.98 = zero_zero_rat ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5280_add__neg__numeral__special_I7_J,axiom,
% 16.19/8.98 ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 16.19/8.98 = zero_z3403309356797280102nteger ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5281_diff__numeral__special_I12_J,axiom,
% 16.19/8.98 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 16.19/8.98 = zero_zero_real ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5282_diff__numeral__special_I12_J,axiom,
% 16.19/8.98 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 16.19/8.98 = zero_zero_int ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5283_diff__numeral__special_I12_J,axiom,
% 16.19/8.98 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 16.19/8.98 = zero_zero_complex ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5284_diff__numeral__special_I12_J,axiom,
% 16.19/8.98 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 16.19/8.98 = zero_zero_rat ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5285_diff__numeral__special_I12_J,axiom,
% 16.19/8.98 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 16.19/8.98 = zero_z3403309356797280102nteger ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5286_numeral__eq__neg__one__iff,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) )
% 16.19/8.98 = ( uminus_uminus_real @ one_one_real ) )
% 16.19/8.98 = ( N2 = one ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5287_numeral__eq__neg__one__iff,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) )
% 16.19/8.98 = ( uminus_uminus_int @ one_one_int ) )
% 16.19/8.98 = ( N2 = one ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5288_numeral__eq__neg__one__iff,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) )
% 16.19/8.98 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 16.19/8.98 = ( N2 = one ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5289_numeral__eq__neg__one__iff,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) )
% 16.19/8.98 = ( uminus_uminus_rat @ one_one_rat ) )
% 16.19/8.98 = ( N2 = one ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5290_numeral__eq__neg__one__iff,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) )
% 16.19/8.98 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 16.19/8.98 = ( N2 = one ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5291_neg__one__eq__numeral__iff,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( ( ( uminus_uminus_real @ one_one_real )
% 16.19/8.98 = ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 16.19/8.98 = ( N2 = one ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5292_neg__one__eq__numeral__iff,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( ( ( uminus_uminus_int @ one_one_int )
% 16.19/8.98 = ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/8.98 = ( N2 = one ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5293_neg__one__eq__numeral__iff,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( ( ( uminus1482373934393186551omplex @ one_one_complex )
% 16.19/8.98 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
% 16.19/8.98 = ( N2 = one ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5294_neg__one__eq__numeral__iff,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( ( ( uminus_uminus_rat @ one_one_rat )
% 16.19/8.98 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 16.19/8.98 = ( N2 = one ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5295_neg__one__eq__numeral__iff,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
% 16.19/8.98 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
% 16.19/8.98 = ( N2 = one ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5296_left__minus__one__mult__self,axiom,
% 16.19/8.98 ! [N2: nat,A: real] :
% 16.19/8.98 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ A ) )
% 16.19/8.98 = A ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5297_left__minus__one__mult__self,axiom,
% 16.19/8.98 ! [N2: nat,A: int] :
% 16.19/8.98 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ A ) )
% 16.19/8.98 = A ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5298_left__minus__one__mult__self,axiom,
% 16.19/8.98 ! [N2: nat,A: complex] :
% 16.19/8.98 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ A ) )
% 16.19/8.98 = A ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5299_left__minus__one__mult__self,axiom,
% 16.19/8.98 ! [N2: nat,A: rat] :
% 16.19/8.98 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ A ) )
% 16.19/8.98 = A ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5300_left__minus__one__mult__self,axiom,
% 16.19/8.98 ! [N2: nat,A: code_integer] :
% 16.19/8.98 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ A ) )
% 16.19/8.98 = A ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5301_minus__one__mult__self,axiom,
% 16.19/8.98 ! [N2: nat] :
% 16.19/8.98 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) )
% 16.19/8.98 = one_one_real ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5302_minus__one__mult__self,axiom,
% 16.19/8.98 ! [N2: nat] :
% 16.19/8.98 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) )
% 16.19/8.98 = one_one_int ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5303_minus__one__mult__self,axiom,
% 16.19/8.98 ! [N2: nat] :
% 16.19/8.98 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) )
% 16.19/8.98 = one_one_complex ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5304_minus__one__mult__self,axiom,
% 16.19/8.98 ! [N2: nat] :
% 16.19/8.98 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) )
% 16.19/8.98 = one_one_rat ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5305_minus__one__mult__self,axiom,
% 16.19/8.98 ! [N2: nat] :
% 16.19/8.98 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) )
% 16.19/8.98 = one_one_Code_integer ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5306_mod__minus1__right,axiom,
% 16.19/8.98 ! [A: int] :
% 16.19/8.98 ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 16.19/8.98 = zero_zero_int ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5307_mod__minus1__right,axiom,
% 16.19/8.98 ! [A: code_integer] :
% 16.19/8.98 ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 16.19/8.98 = zero_z3403309356797280102nteger ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5308_max__number__of_I2_J,axiom,
% 16.19/8.98 ! [U: num,V: num] :
% 16.19/8.98 ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 16.19/8.98 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 16.19/8.98 = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 16.19/8.98 & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 16.19/8.98 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 16.19/8.98 = ( numeral_numeral_real @ U ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5309_max__number__of_I2_J,axiom,
% 16.19/8.98 ! [U: num,V: num] :
% 16.19/8.98 ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 16.19/8.98 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 16.19/8.98 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 16.19/8.98 & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 16.19/8.98 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 16.19/8.98 = ( numera6620942414471956472nteger @ U ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5310_max__number__of_I2_J,axiom,
% 16.19/8.98 ! [U: num,V: num] :
% 16.19/8.98 ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 16.19/8.98 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 16.19/8.98 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 16.19/8.98 & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 16.19/8.98 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 16.19/8.98 = ( numeral_numeral_rat @ U ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5311_max__number__of_I2_J,axiom,
% 16.19/8.98 ! [U: num,V: num] :
% 16.19/8.98 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 16.19/8.98 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 16.19/8.98 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 16.19/8.98 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 16.19/8.98 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 16.19/8.98 = ( numeral_numeral_int @ U ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5312_max__number__of_I3_J,axiom,
% 16.19/8.98 ! [U: num,V: num] :
% 16.19/8.98 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 16.19/8.98 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 16.19/8.98 = ( numeral_numeral_real @ V ) ) )
% 16.19/8.98 & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 16.19/8.98 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 16.19/8.98 = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5313_max__number__of_I3_J,axiom,
% 16.19/8.98 ! [U: num,V: num] :
% 16.19/8.98 ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 16.19/8.98 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 16.19/8.98 = ( numera6620942414471956472nteger @ V ) ) )
% 16.19/8.98 & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 16.19/8.98 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 16.19/8.98 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5314_max__number__of_I3_J,axiom,
% 16.19/8.98 ! [U: num,V: num] :
% 16.19/8.98 ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 16.19/8.98 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 16.19/8.98 = ( numeral_numeral_rat @ V ) ) )
% 16.19/8.98 & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 16.19/8.98 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 16.19/8.98 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5315_max__number__of_I3_J,axiom,
% 16.19/8.98 ! [U: num,V: num] :
% 16.19/8.98 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 16.19/8.98 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 16.19/8.98 = ( numeral_numeral_int @ V ) ) )
% 16.19/8.98 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 16.19/8.98 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 16.19/8.98 = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5316_max__number__of_I4_J,axiom,
% 16.19/8.98 ! [U: num,V: num] :
% 16.19/8.98 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 16.19/8.98 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 16.19/8.98 = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 16.19/8.98 & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 16.19/8.98 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 16.19/8.98 = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5317_max__number__of_I4_J,axiom,
% 16.19/8.98 ! [U: num,V: num] :
% 16.19/8.98 ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 16.19/8.98 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 16.19/8.98 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 16.19/8.98 & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 16.19/8.98 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 16.19/8.98 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5318_max__number__of_I4_J,axiom,
% 16.19/8.98 ! [U: num,V: num] :
% 16.19/8.98 ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 16.19/8.98 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 16.19/8.98 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 16.19/8.98 & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 16.19/8.98 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 16.19/8.98 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5319_max__number__of_I4_J,axiom,
% 16.19/8.98 ! [U: num,V: num] :
% 16.19/8.98 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 16.19/8.98 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 16.19/8.98 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 16.19/8.98 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 16.19/8.98 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 16.19/8.98 = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5320_ln__ge__zero__iff,axiom,
% 16.19/8.98 ! [X: real] :
% 16.19/8.98 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/8.98 => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 16.19/8.98 = ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5321_ln__le__zero__iff,axiom,
% 16.19/8.98 ! [X: real] :
% 16.19/8.98 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/8.98 => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
% 16.19/8.98 = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5322_semiring__norm_I168_J,axiom,
% 16.19/8.98 ! [V: num,W: num,Y: real] :
% 16.19/8.98 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
% 16.19/8.98 = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5323_semiring__norm_I168_J,axiom,
% 16.19/8.98 ! [V: num,W: num,Y: int] :
% 16.19/8.98 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
% 16.19/8.98 = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5324_semiring__norm_I168_J,axiom,
% 16.19/8.98 ! [V: num,W: num,Y: complex] :
% 16.19/8.98 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
% 16.19/8.98 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5325_semiring__norm_I168_J,axiom,
% 16.19/8.98 ! [V: num,W: num,Y: rat] :
% 16.19/8.98 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
% 16.19/8.98 = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5326_semiring__norm_I168_J,axiom,
% 16.19/8.98 ! [V: num,W: num,Y: code_integer] :
% 16.19/8.98 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
% 16.19/8.98 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5327_diff__numeral__simps_I3_J,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) )
% 16.19/8.98 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5328_diff__numeral__simps_I3_J,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
% 16.19/8.98 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5329_diff__numeral__simps_I3_J,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N2 ) )
% 16.19/8.98 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5330_diff__numeral__simps_I3_J,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) )
% 16.19/8.98 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5331_diff__numeral__simps_I3_J,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) )
% 16.19/8.98 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N2 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5332_diff__numeral__simps_I2_J,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 16.19/8.98 = ( numeral_numeral_real @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5333_diff__numeral__simps_I2_J,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/8.98 = ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5334_diff__numeral__simps_I2_J,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
% 16.19/8.98 = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5335_diff__numeral__simps_I2_J,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 16.19/8.98 = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5336_diff__numeral__simps_I2_J,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
% 16.19/8.98 = ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5337_semiring__norm_I170_J,axiom,
% 16.19/8.98 ! [V: num,W: num,Y: real] :
% 16.19/8.98 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y ) )
% 16.19/8.98 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5338_semiring__norm_I170_J,axiom,
% 16.19/8.98 ! [V: num,W: num,Y: int] :
% 16.19/8.98 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y ) )
% 16.19/8.98 = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5339_semiring__norm_I170_J,axiom,
% 16.19/8.98 ! [V: num,W: num,Y: complex] :
% 16.19/8.98 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Y ) )
% 16.19/8.98 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5340_semiring__norm_I170_J,axiom,
% 16.19/8.98 ! [V: num,W: num,Y: rat] :
% 16.19/8.98 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y ) )
% 16.19/8.98 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5341_semiring__norm_I170_J,axiom,
% 16.19/8.98 ! [V: num,W: num,Y: code_integer] :
% 16.19/8.98 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y ) )
% 16.19/8.98 = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5342_semiring__norm_I171_J,axiom,
% 16.19/8.98 ! [V: num,W: num,Y: real] :
% 16.19/8.98 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
% 16.19/8.98 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5343_semiring__norm_I171_J,axiom,
% 16.19/8.98 ! [V: num,W: num,Y: int] :
% 16.19/8.98 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
% 16.19/8.98 = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5344_semiring__norm_I171_J,axiom,
% 16.19/8.98 ! [V: num,W: num,Y: complex] :
% 16.19/8.98 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
% 16.19/8.98 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5345_semiring__norm_I171_J,axiom,
% 16.19/8.98 ! [V: num,W: num,Y: rat] :
% 16.19/8.98 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
% 16.19/8.98 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5346_semiring__norm_I171_J,axiom,
% 16.19/8.98 ! [V: num,W: num,Y: code_integer] :
% 16.19/8.98 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
% 16.19/8.98 = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5347_semiring__norm_I172_J,axiom,
% 16.19/8.98 ! [V: num,W: num,Y: real] :
% 16.19/8.98 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
% 16.19/8.98 = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5348_semiring__norm_I172_J,axiom,
% 16.19/8.98 ! [V: num,W: num,Y: int] :
% 16.19/8.98 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
% 16.19/8.98 = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5349_semiring__norm_I172_J,axiom,
% 16.19/8.98 ! [V: num,W: num,Y: complex] :
% 16.19/8.98 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
% 16.19/8.98 = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5350_semiring__norm_I172_J,axiom,
% 16.19/8.98 ! [V: num,W: num,Y: rat] :
% 16.19/8.98 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
% 16.19/8.98 = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5351_semiring__norm_I172_J,axiom,
% 16.19/8.98 ! [V: num,W: num,Y: code_integer] :
% 16.19/8.98 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
% 16.19/8.98 = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5352_mult__neg__numeral__simps_I1_J,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 16.19/8.98 = ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5353_mult__neg__numeral__simps_I1_J,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/8.98 = ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5354_mult__neg__numeral__simps_I1_J,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
% 16.19/8.98 = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5355_mult__neg__numeral__simps_I1_J,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 16.19/8.98 = ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5356_mult__neg__numeral__simps_I1_J,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
% 16.19/8.98 = ( numera6620942414471956472nteger @ ( times_times_num @ M @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5357_mult__neg__numeral__simps_I2_J,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) )
% 16.19/8.98 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5358_mult__neg__numeral__simps_I2_J,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
% 16.19/8.98 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5359_mult__neg__numeral__simps_I2_J,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N2 ) )
% 16.19/8.98 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5360_mult__neg__numeral__simps_I2_J,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) )
% 16.19/8.98 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5361_mult__neg__numeral__simps_I2_J,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) )
% 16.19/8.98 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5362_mult__neg__numeral__simps_I3_J,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 16.19/8.98 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5363_mult__neg__numeral__simps_I3_J,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/8.98 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5364_mult__neg__numeral__simps_I3_J,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
% 16.19/8.98 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5365_mult__neg__numeral__simps_I3_J,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 16.19/8.98 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5366_mult__neg__numeral__simps_I3_J,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
% 16.19/8.98 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N2 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5367_neg__numeral__le__iff,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 16.19/8.98 = ( ord_less_eq_num @ N2 @ M ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5368_neg__numeral__le__iff,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
% 16.19/8.98 = ( ord_less_eq_num @ N2 @ M ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5369_neg__numeral__le__iff,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 16.19/8.98 = ( ord_less_eq_num @ N2 @ M ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5370_neg__numeral__le__iff,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/8.98 = ( ord_less_eq_num @ N2 @ M ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5371_neg__numeral__less__iff,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 16.19/8.98 = ( ord_less_num @ N2 @ M ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5372_neg__numeral__less__iff,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/8.98 = ( ord_less_num @ N2 @ M ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5373_neg__numeral__less__iff,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 16.19/8.98 = ( ord_less_num @ N2 @ M ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5374_neg__numeral__less__iff,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
% 16.19/8.98 = ( ord_less_num @ N2 @ M ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5375_not__neg__one__le__neg__numeral__iff,axiom,
% 16.19/8.98 ! [M: num] :
% 16.19/8.98 ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
% 16.19/8.98 = ( M != one ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5376_not__neg__one__le__neg__numeral__iff,axiom,
% 16.19/8.98 ! [M: num] :
% 16.19/8.98 ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
% 16.19/8.98 = ( M != one ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5377_not__neg__one__le__neg__numeral__iff,axiom,
% 16.19/8.98 ! [M: num] :
% 16.19/8.98 ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
% 16.19/8.98 = ( M != one ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5378_not__neg__one__le__neg__numeral__iff,axiom,
% 16.19/8.98 ! [M: num] :
% 16.19/8.98 ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
% 16.19/8.98 = ( M != one ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5379_le__divide__eq__numeral1_I2_J,axiom,
% 16.19/8.98 ! [A: real,B: real,W: num] :
% 16.19/8.98 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 16.19/8.98 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5380_le__divide__eq__numeral1_I2_J,axiom,
% 16.19/8.98 ! [A: rat,B: rat,W: num] :
% 16.19/8.98 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 16.19/8.98 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5381_divide__le__eq__numeral1_I2_J,axiom,
% 16.19/8.98 ! [B: real,W: num,A: real] :
% 16.19/8.98 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 16.19/8.98 = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5382_divide__le__eq__numeral1_I2_J,axiom,
% 16.19/8.98 ! [B: rat,W: num,A: rat] :
% 16.19/8.98 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 16.19/8.98 = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5383_eq__divide__eq__numeral1_I2_J,axiom,
% 16.19/8.98 ! [A: real,B: real,W: num] :
% 16.19/8.98 ( ( A
% 16.19/8.98 = ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 16.19/8.98 = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 16.19/8.98 != zero_zero_real )
% 16.19/8.98 => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 16.19/8.98 = B ) )
% 16.19/8.98 & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 16.19/8.98 = zero_zero_real )
% 16.19/8.98 => ( A = zero_zero_real ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5384_eq__divide__eq__numeral1_I2_J,axiom,
% 16.19/8.98 ! [A: complex,B: complex,W: num] :
% 16.19/8.98 ( ( A
% 16.19/8.98 = ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) )
% 16.19/8.98 = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 16.19/8.98 != zero_zero_complex )
% 16.19/8.98 => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 16.19/8.98 = B ) )
% 16.19/8.98 & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 16.19/8.98 = zero_zero_complex )
% 16.19/8.98 => ( A = zero_zero_complex ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5385_eq__divide__eq__numeral1_I2_J,axiom,
% 16.19/8.98 ! [A: rat,B: rat,W: num] :
% 16.19/8.98 ( ( A
% 16.19/8.98 = ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 16.19/8.98 = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 16.19/8.98 != zero_zero_rat )
% 16.19/8.98 => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 16.19/8.98 = B ) )
% 16.19/8.98 & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 16.19/8.98 = zero_zero_rat )
% 16.19/8.98 => ( A = zero_zero_rat ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5386_divide__eq__eq__numeral1_I2_J,axiom,
% 16.19/8.98 ! [B: real,W: num,A: real] :
% 16.19/8.98 ( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 16.19/8.98 = A )
% 16.19/8.98 = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 16.19/8.98 != zero_zero_real )
% 16.19/8.98 => ( B
% 16.19/8.98 = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
% 16.19/8.98 & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 16.19/8.98 = zero_zero_real )
% 16.19/8.98 => ( A = zero_zero_real ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5387_divide__eq__eq__numeral1_I2_J,axiom,
% 16.19/8.98 ! [B: complex,W: num,A: complex] :
% 16.19/8.98 ( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 16.19/8.98 = A )
% 16.19/8.98 = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 16.19/8.98 != zero_zero_complex )
% 16.19/8.98 => ( B
% 16.19/8.98 = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) )
% 16.19/8.98 & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 16.19/8.98 = zero_zero_complex )
% 16.19/8.98 => ( A = zero_zero_complex ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5388_divide__eq__eq__numeral1_I2_J,axiom,
% 16.19/8.98 ! [B: rat,W: num,A: rat] :
% 16.19/8.98 ( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 16.19/8.98 = A )
% 16.19/8.98 = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 16.19/8.98 != zero_zero_rat )
% 16.19/8.98 => ( B
% 16.19/8.98 = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
% 16.19/8.98 & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 16.19/8.98 = zero_zero_rat )
% 16.19/8.98 => ( A = zero_zero_rat ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5389_neg__numeral__less__neg__one__iff,axiom,
% 16.19/8.98 ! [M: num] :
% 16.19/8.98 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 16.19/8.98 = ( M != one ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5390_neg__numeral__less__neg__one__iff,axiom,
% 16.19/8.98 ! [M: num] :
% 16.19/8.98 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 16.19/8.98 = ( M != one ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5391_neg__numeral__less__neg__one__iff,axiom,
% 16.19/8.98 ! [M: num] :
% 16.19/8.98 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 16.19/8.98 = ( M != one ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5392_neg__numeral__less__neg__one__iff,axiom,
% 16.19/8.98 ! [M: num] :
% 16.19/8.98 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 16.19/8.98 = ( M != one ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5393_less__divide__eq__numeral1_I2_J,axiom,
% 16.19/8.98 ! [A: real,B: real,W: num] :
% 16.19/8.98 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 16.19/8.98 = ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5394_less__divide__eq__numeral1_I2_J,axiom,
% 16.19/8.98 ! [A: rat,B: rat,W: num] :
% 16.19/8.98 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 16.19/8.98 = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5395_divide__less__eq__numeral1_I2_J,axiom,
% 16.19/8.98 ! [B: real,W: num,A: real] :
% 16.19/8.98 ( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 16.19/8.98 = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5396_divide__less__eq__numeral1_I2_J,axiom,
% 16.19/8.98 ! [B: rat,W: num,A: rat] :
% 16.19/8.98 ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 16.19/8.98 = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5397_power2__minus,axiom,
% 16.19/8.98 ! [A: real] :
% 16.19/8.98 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.98 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5398_power2__minus,axiom,
% 16.19/8.98 ! [A: int] :
% 16.19/8.98 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.98 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5399_power2__minus,axiom,
% 16.19/8.98 ! [A: complex] :
% 16.19/8.98 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.98 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5400_power2__minus,axiom,
% 16.19/8.98 ! [A: rat] :
% 16.19/8.98 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.98 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5401_power2__minus,axiom,
% 16.19/8.98 ! [A: code_integer] :
% 16.19/8.98 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.98 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5402_add__neg__numeral__special_I9_J,axiom,
% 16.19/8.98 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 16.19/8.98 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5403_add__neg__numeral__special_I9_J,axiom,
% 16.19/8.98 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 16.19/8.98 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5404_add__neg__numeral__special_I9_J,axiom,
% 16.19/8.98 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 16.19/8.98 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5405_add__neg__numeral__special_I9_J,axiom,
% 16.19/8.98 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 16.19/8.98 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5406_add__neg__numeral__special_I9_J,axiom,
% 16.19/8.98 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 16.19/8.98 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5407_diff__numeral__special_I10_J,axiom,
% 16.19/8.98 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 16.19/8.98 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5408_diff__numeral__special_I10_J,axiom,
% 16.19/8.98 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 16.19/8.98 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5409_diff__numeral__special_I10_J,axiom,
% 16.19/8.98 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 16.19/8.98 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5410_diff__numeral__special_I10_J,axiom,
% 16.19/8.98 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 16.19/8.98 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5411_diff__numeral__special_I10_J,axiom,
% 16.19/8.98 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 16.19/8.98 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5412_diff__numeral__special_I11_J,axiom,
% 16.19/8.98 ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 16.19/8.98 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5413_diff__numeral__special_I11_J,axiom,
% 16.19/8.98 ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 16.19/8.98 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5414_diff__numeral__special_I11_J,axiom,
% 16.19/8.98 ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 16.19/8.98 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5415_diff__numeral__special_I11_J,axiom,
% 16.19/8.98 ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 16.19/8.98 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5416_diff__numeral__special_I11_J,axiom,
% 16.19/8.98 ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 16.19/8.98 = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5417_minus__1__div__2__eq,axiom,
% 16.19/8.98 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/8.98 = ( uminus_uminus_int @ one_one_int ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5418_minus__1__div__2__eq,axiom,
% 16.19/8.98 ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 16.19/8.98 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5419_bits__minus__1__mod__2__eq,axiom,
% 16.19/8.98 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/8.98 = one_one_int ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5420_bits__minus__1__mod__2__eq,axiom,
% 16.19/8.98 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 16.19/8.98 = one_one_Code_integer ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5421_minus__1__mod__2__eq,axiom,
% 16.19/8.98 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/8.98 = one_one_int ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5422_minus__1__mod__2__eq,axiom,
% 16.19/8.98 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 16.19/8.98 = one_one_Code_integer ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5423_Power_Oring__1__class_Opower__minus__even,axiom,
% 16.19/8.98 ! [A: real,N2: nat] :
% 16.19/8.98 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.98 = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5424_Power_Oring__1__class_Opower__minus__even,axiom,
% 16.19/8.98 ! [A: int,N2: nat] :
% 16.19/8.98 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.98 = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5425_Power_Oring__1__class_Opower__minus__even,axiom,
% 16.19/8.98 ! [A: complex,N2: nat] :
% 16.19/8.98 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.98 = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5426_Power_Oring__1__class_Opower__minus__even,axiom,
% 16.19/8.98 ! [A: rat,N2: nat] :
% 16.19/8.98 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.98 = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5427_Power_Oring__1__class_Opower__minus__even,axiom,
% 16.19/8.98 ! [A: code_integer,N2: nat] :
% 16.19/8.98 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.98 = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5428_Parity_Oring__1__class_Opower__minus__even,axiom,
% 16.19/8.98 ! [N2: nat,A: real] :
% 16.19/8.98 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.98 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 )
% 16.19/8.98 = ( power_power_real @ A @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5429_Parity_Oring__1__class_Opower__minus__even,axiom,
% 16.19/8.98 ! [N2: nat,A: int] :
% 16.19/8.98 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.98 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
% 16.19/8.98 = ( power_power_int @ A @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5430_Parity_Oring__1__class_Opower__minus__even,axiom,
% 16.19/8.98 ! [N2: nat,A: complex] :
% 16.19/8.98 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.98 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 )
% 16.19/8.98 = ( power_power_complex @ A @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5431_Parity_Oring__1__class_Opower__minus__even,axiom,
% 16.19/8.98 ! [N2: nat,A: rat] :
% 16.19/8.98 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.98 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 )
% 16.19/8.98 = ( power_power_rat @ A @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5432_Parity_Oring__1__class_Opower__minus__even,axiom,
% 16.19/8.98 ! [N2: nat,A: code_integer] :
% 16.19/8.98 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.98 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 )
% 16.19/8.98 = ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5433_power__minus__odd,axiom,
% 16.19/8.98 ! [N2: nat,A: real] :
% 16.19/8.98 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.98 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 )
% 16.19/8.98 = ( uminus_uminus_real @ ( power_power_real @ A @ N2 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5434_power__minus__odd,axiom,
% 16.19/8.98 ! [N2: nat,A: int] :
% 16.19/8.98 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.98 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
% 16.19/8.98 = ( uminus_uminus_int @ ( power_power_int @ A @ N2 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5435_power__minus__odd,axiom,
% 16.19/8.98 ! [N2: nat,A: complex] :
% 16.19/8.98 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.98 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 )
% 16.19/8.98 = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N2 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5436_power__minus__odd,axiom,
% 16.19/8.98 ! [N2: nat,A: rat] :
% 16.19/8.98 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.98 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 )
% 16.19/8.98 = ( uminus_uminus_rat @ ( power_power_rat @ A @ N2 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5437_power__minus__odd,axiom,
% 16.19/8.98 ! [N2: nat,A: code_integer] :
% 16.19/8.98 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.98 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 )
% 16.19/8.98 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5438_diff__numeral__special_I4_J,axiom,
% 16.19/8.98 ! [M: num] :
% 16.19/8.98 ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real )
% 16.19/8.98 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5439_diff__numeral__special_I4_J,axiom,
% 16.19/8.98 ! [M: num] :
% 16.19/8.98 ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
% 16.19/8.98 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5440_diff__numeral__special_I4_J,axiom,
% 16.19/8.98 ! [M: num] :
% 16.19/8.98 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ one_one_complex )
% 16.19/8.98 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5441_diff__numeral__special_I4_J,axiom,
% 16.19/8.98 ! [M: num] :
% 16.19/8.98 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
% 16.19/8.98 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5442_diff__numeral__special_I4_J,axiom,
% 16.19/8.98 ! [M: num] :
% 16.19/8.98 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
% 16.19/8.98 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5443_diff__numeral__special_I3_J,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 16.19/8.98 = ( numeral_numeral_real @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5444_diff__numeral__special_I3_J,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/8.98 = ( numeral_numeral_int @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5445_diff__numeral__special_I3_J,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
% 16.19/8.98 = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5446_diff__numeral__special_I3_J,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 16.19/8.98 = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5447_diff__numeral__special_I3_J,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
% 16.19/8.98 = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5448_signed__take__bit__Suc__minus__bit0,axiom,
% 16.19/8.98 ! [N2: nat,K: num] :
% 16.19/8.98 ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 16.19/8.98 = ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5449_dbl__simps_I4_J,axiom,
% 16.19/8.98 ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ one_one_real ) )
% 16.19/8.98 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5450_dbl__simps_I4_J,axiom,
% 16.19/8.98 ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
% 16.19/8.98 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5451_dbl__simps_I4_J,axiom,
% 16.19/8.98 ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 16.19/8.98 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5452_dbl__simps_I4_J,axiom,
% 16.19/8.98 ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 16.19/8.98 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5453_dbl__simps_I4_J,axiom,
% 16.19/8.98 ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 16.19/8.98 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5454_power__minus1__even,axiom,
% 16.19/8.98 ! [N2: nat] :
% 16.19/8.98 ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.98 = one_one_real ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5455_power__minus1__even,axiom,
% 16.19/8.98 ! [N2: nat] :
% 16.19/8.98 ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.98 = one_one_int ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5456_power__minus1__even,axiom,
% 16.19/8.98 ! [N2: nat] :
% 16.19/8.98 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.98 = one_one_complex ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5457_power__minus1__even,axiom,
% 16.19/8.98 ! [N2: nat] :
% 16.19/8.98 ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.98 = one_one_rat ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5458_power__minus1__even,axiom,
% 16.19/8.98 ! [N2: nat] :
% 16.19/8.98 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/8.98 = one_one_Code_integer ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5459_neg__one__even__power,axiom,
% 16.19/8.98 ! [N2: nat] :
% 16.19/8.98 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.98 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 )
% 16.19/8.98 = one_one_real ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5460_neg__one__even__power,axiom,
% 16.19/8.98 ! [N2: nat] :
% 16.19/8.98 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.98 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
% 16.19/8.98 = one_one_int ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5461_neg__one__even__power,axiom,
% 16.19/8.98 ! [N2: nat] :
% 16.19/8.98 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.98 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 )
% 16.19/8.98 = one_one_complex ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5462_neg__one__even__power,axiom,
% 16.19/8.98 ! [N2: nat] :
% 16.19/8.98 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.98 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
% 16.19/8.98 = one_one_rat ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5463_neg__one__even__power,axiom,
% 16.19/8.98 ! [N2: nat] :
% 16.19/8.98 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.98 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
% 16.19/8.98 = one_one_Code_integer ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5464_neg__one__odd__power,axiom,
% 16.19/8.98 ! [N2: nat] :
% 16.19/8.98 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.98 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 )
% 16.19/8.98 = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5465_neg__one__odd__power,axiom,
% 16.19/8.98 ! [N2: nat] :
% 16.19/8.98 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.98 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
% 16.19/8.98 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5466_neg__one__odd__power,axiom,
% 16.19/8.98 ! [N2: nat] :
% 16.19/8.98 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.98 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 )
% 16.19/8.98 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5467_neg__one__odd__power,axiom,
% 16.19/8.98 ! [N2: nat] :
% 16.19/8.98 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.98 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
% 16.19/8.98 = ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5468_neg__one__odd__power,axiom,
% 16.19/8.98 ! [N2: nat] :
% 16.19/8.98 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/8.98 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
% 16.19/8.98 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5469_signed__take__bit__0,axiom,
% 16.19/8.98 ! [A: code_integer] :
% 16.19/8.98 ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
% 16.19/8.98 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5470_signed__take__bit__0,axiom,
% 16.19/8.98 ! [A: int] :
% 16.19/8.98 ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
% 16.19/8.98 = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5471_signed__take__bit__minus,axiom,
% 16.19/8.98 ! [N2: nat,K: int] :
% 16.19/8.98 ( ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) ) )
% 16.19/8.98 = ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ K ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5472_sum_Oswap,axiom,
% 16.19/8.98 ! [G: int > int > int,B3: set_int,A2: set_int] :
% 16.19/8.98 ( ( groups4538972089207619220nt_int
% 16.19/8.98 @ ^ [I3: int] : ( groups4538972089207619220nt_int @ ( G @ I3 ) @ B3 )
% 16.19/8.98 @ A2 )
% 16.19/8.98 = ( groups4538972089207619220nt_int
% 16.19/8.98 @ ^ [J3: int] :
% 16.19/8.98 ( groups4538972089207619220nt_int
% 16.19/8.98 @ ^ [I3: int] : ( G @ I3 @ J3 )
% 16.19/8.98 @ A2 )
% 16.19/8.98 @ B3 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5473_sum_Oswap,axiom,
% 16.19/8.98 ! [G: complex > complex > complex,B3: set_complex,A2: set_complex] :
% 16.19/8.98 ( ( groups7754918857620584856omplex
% 16.19/8.98 @ ^ [I3: complex] : ( groups7754918857620584856omplex @ ( G @ I3 ) @ B3 )
% 16.19/8.98 @ A2 )
% 16.19/8.98 = ( groups7754918857620584856omplex
% 16.19/8.98 @ ^ [J3: complex] :
% 16.19/8.98 ( groups7754918857620584856omplex
% 16.19/8.98 @ ^ [I3: complex] : ( G @ I3 @ J3 )
% 16.19/8.98 @ A2 )
% 16.19/8.98 @ B3 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5474_sum_Oswap,axiom,
% 16.19/8.98 ! [G: nat > nat > nat,B3: set_nat,A2: set_nat] :
% 16.19/8.98 ( ( groups3542108847815614940at_nat
% 16.19/8.98 @ ^ [I3: nat] : ( groups3542108847815614940at_nat @ ( G @ I3 ) @ B3 )
% 16.19/8.98 @ A2 )
% 16.19/8.98 = ( groups3542108847815614940at_nat
% 16.19/8.98 @ ^ [J3: nat] :
% 16.19/8.98 ( groups3542108847815614940at_nat
% 16.19/8.98 @ ^ [I3: nat] : ( G @ I3 @ J3 )
% 16.19/8.98 @ A2 )
% 16.19/8.98 @ B3 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5475_sum_Oswap,axiom,
% 16.19/8.98 ! [G: nat > nat > real,B3: set_nat,A2: set_nat] :
% 16.19/8.98 ( ( groups6591440286371151544t_real
% 16.19/8.98 @ ^ [I3: nat] : ( groups6591440286371151544t_real @ ( G @ I3 ) @ B3 )
% 16.19/8.98 @ A2 )
% 16.19/8.98 = ( groups6591440286371151544t_real
% 16.19/8.98 @ ^ [J3: nat] :
% 16.19/8.98 ( groups6591440286371151544t_real
% 16.19/8.98 @ ^ [I3: nat] : ( G @ I3 @ J3 )
% 16.19/8.98 @ A2 )
% 16.19/8.98 @ B3 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5476_sum__negf,axiom,
% 16.19/8.98 ! [F: int > int,A2: set_int] :
% 16.19/8.98 ( ( groups4538972089207619220nt_int
% 16.19/8.98 @ ^ [X2: int] : ( uminus_uminus_int @ ( F @ X2 ) )
% 16.19/8.98 @ A2 )
% 16.19/8.98 = ( uminus_uminus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5477_sum__negf,axiom,
% 16.19/8.98 ! [F: complex > complex,A2: set_complex] :
% 16.19/8.98 ( ( groups7754918857620584856omplex
% 16.19/8.98 @ ^ [X2: complex] : ( uminus1482373934393186551omplex @ ( F @ X2 ) )
% 16.19/8.98 @ A2 )
% 16.19/8.98 = ( uminus1482373934393186551omplex @ ( groups7754918857620584856omplex @ F @ A2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5478_sum__negf,axiom,
% 16.19/8.98 ! [F: nat > real,A2: set_nat] :
% 16.19/8.98 ( ( groups6591440286371151544t_real
% 16.19/8.98 @ ^ [X2: nat] : ( uminus_uminus_real @ ( F @ X2 ) )
% 16.19/8.98 @ A2 )
% 16.19/8.98 = ( uminus_uminus_real @ ( groups6591440286371151544t_real @ F @ A2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5479_prod_Ocase__distrib,axiom,
% 16.19/8.98 ! [H2: $o > $o,F: int > int > $o,Prod: product_prod_int_int] :
% 16.19/8.98 ( ( H2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
% 16.19/8.98 = ( produc4947309494688390418_int_o
% 16.19/8.98 @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
% 16.19/8.98 @ Prod ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5480_prod_Ocase__distrib,axiom,
% 16.19/8.98 ! [H2: $o > int,F: int > int > $o,Prod: product_prod_int_int] :
% 16.19/8.98 ( ( H2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
% 16.19/8.98 = ( produc8211389475949308722nt_int
% 16.19/8.98 @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
% 16.19/8.98 @ Prod ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5481_prod_Ocase__distrib,axiom,
% 16.19/8.98 ! [H2: int > $o,F: int > int > int,Prod: product_prod_int_int] :
% 16.19/8.98 ( ( H2 @ ( produc8211389475949308722nt_int @ F @ Prod ) )
% 16.19/8.98 = ( produc4947309494688390418_int_o
% 16.19/8.98 @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
% 16.19/8.98 @ Prod ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5482_prod_Ocase__distrib,axiom,
% 16.19/8.98 ! [H2: int > int,F: int > int > int,Prod: product_prod_int_int] :
% 16.19/8.98 ( ( H2 @ ( produc8211389475949308722nt_int @ F @ Prod ) )
% 16.19/8.98 = ( produc8211389475949308722nt_int
% 16.19/8.98 @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
% 16.19/8.98 @ Prod ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5483_prod_Ocase__distrib,axiom,
% 16.19/8.98 ! [H2: product_prod_int_int > $o,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
% 16.19/8.98 ( ( H2 @ ( produc4245557441103728435nt_int @ F @ Prod ) )
% 16.19/8.98 = ( produc4947309494688390418_int_o
% 16.19/8.98 @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
% 16.19/8.98 @ Prod ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5484_prod_Ocase__distrib,axiom,
% 16.19/8.98 ! [H2: product_prod_int_int > int,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
% 16.19/8.98 ( ( H2 @ ( produc4245557441103728435nt_int @ F @ Prod ) )
% 16.19/8.98 = ( produc8211389475949308722nt_int
% 16.19/8.98 @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
% 16.19/8.98 @ Prod ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5485_prod_Ocase__distrib,axiom,
% 16.19/8.98 ! [H2: $o > product_prod_int_int,F: int > int > $o,Prod: product_prod_int_int] :
% 16.19/8.98 ( ( H2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
% 16.19/8.98 = ( produc4245557441103728435nt_int
% 16.19/8.98 @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
% 16.19/8.98 @ Prod ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5486_prod_Ocase__distrib,axiom,
% 16.19/8.98 ! [H2: int > product_prod_int_int,F: int > int > int,Prod: product_prod_int_int] :
% 16.19/8.98 ( ( H2 @ ( produc8211389475949308722nt_int @ F @ Prod ) )
% 16.19/8.98 = ( produc4245557441103728435nt_int
% 16.19/8.98 @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
% 16.19/8.98 @ Prod ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5487_prod_Ocase__distrib,axiom,
% 16.19/8.98 ! [H2: product_prod_int_int > product_prod_int_int,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
% 16.19/8.98 ( ( H2 @ ( produc4245557441103728435nt_int @ F @ Prod ) )
% 16.19/8.98 = ( produc4245557441103728435nt_int
% 16.19/8.98 @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
% 16.19/8.98 @ Prod ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5488_prod_Ocase__distrib,axiom,
% 16.19/8.98 ! [H2: ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o,F: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
% 16.19/8.98 ( ( H2 @ ( produc8739625826339149834_nat_o @ F @ Prod ) )
% 16.19/8.98 = ( produc8739625826339149834_nat_o
% 16.19/8.98 @ ^ [X15: nat,X24: nat] : ( H2 @ ( F @ X15 @ X24 ) )
% 16.19/8.98 @ Prod ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5489_le__minus__iff,axiom,
% 16.19/8.98 ! [A: real,B: real] :
% 16.19/8.98 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
% 16.19/8.98 = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5490_le__minus__iff,axiom,
% 16.19/8.98 ! [A: code_integer,B: code_integer] :
% 16.19/8.98 ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 16.19/8.98 = ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5491_le__minus__iff,axiom,
% 16.19/8.98 ! [A: rat,B: rat] :
% 16.19/8.98 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
% 16.19/8.98 = ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5492_le__minus__iff,axiom,
% 16.19/8.98 ! [A: int,B: int] :
% 16.19/8.98 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
% 16.19/8.98 = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5493_minus__le__iff,axiom,
% 16.19/8.98 ! [A: real,B: real] :
% 16.19/8.98 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 16.19/8.98 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5494_minus__le__iff,axiom,
% 16.19/8.98 ! [A: code_integer,B: code_integer] :
% 16.19/8.98 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 16.19/8.98 = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5495_minus__le__iff,axiom,
% 16.19/8.98 ! [A: rat,B: rat] :
% 16.19/8.98 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 16.19/8.98 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5496_minus__le__iff,axiom,
% 16.19/8.98 ! [A: int,B: int] :
% 16.19/8.98 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 16.19/8.98 = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5497_le__imp__neg__le,axiom,
% 16.19/8.98 ! [A: real,B: real] :
% 16.19/8.98 ( ( ord_less_eq_real @ A @ B )
% 16.19/8.98 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5498_le__imp__neg__le,axiom,
% 16.19/8.98 ! [A: code_integer,B: code_integer] :
% 16.19/8.98 ( ( ord_le3102999989581377725nteger @ A @ B )
% 16.19/8.98 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5499_le__imp__neg__le,axiom,
% 16.19/8.98 ! [A: rat,B: rat] :
% 16.19/8.98 ( ( ord_less_eq_rat @ A @ B )
% 16.19/8.98 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5500_le__imp__neg__le,axiom,
% 16.19/8.98 ! [A: int,B: int] :
% 16.19/8.98 ( ( ord_less_eq_int @ A @ B )
% 16.19/8.98 => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5501_compl__le__swap2,axiom,
% 16.19/8.98 ! [Y: set_nat,X: set_nat] :
% 16.19/8.98 ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ X )
% 16.19/8.98 => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5502_compl__le__swap1,axiom,
% 16.19/8.98 ! [Y: set_nat,X: set_nat] :
% 16.19/8.98 ( ( ord_less_eq_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) )
% 16.19/8.98 => ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5503_compl__mono,axiom,
% 16.19/8.98 ! [X: set_nat,Y: set_nat] :
% 16.19/8.98 ( ( ord_less_eq_set_nat @ X @ Y )
% 16.19/8.98 => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ ( uminus5710092332889474511et_nat @ X ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5504_less__minus__iff,axiom,
% 16.19/8.98 ! [A: real,B: real] :
% 16.19/8.98 ( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
% 16.19/8.98 = ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5505_less__minus__iff,axiom,
% 16.19/8.98 ! [A: int,B: int] :
% 16.19/8.98 ( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
% 16.19/8.98 = ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5506_less__minus__iff,axiom,
% 16.19/8.98 ! [A: rat,B: rat] :
% 16.19/8.98 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
% 16.19/8.98 = ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5507_less__minus__iff,axiom,
% 16.19/8.98 ! [A: code_integer,B: code_integer] :
% 16.19/8.98 ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 16.19/8.98 = ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5508_minus__less__iff,axiom,
% 16.19/8.98 ! [A: real,B: real] :
% 16.19/8.98 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
% 16.19/8.98 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5509_minus__less__iff,axiom,
% 16.19/8.98 ! [A: int,B: int] :
% 16.19/8.98 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
% 16.19/8.98 = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5510_minus__less__iff,axiom,
% 16.19/8.98 ! [A: rat,B: rat] :
% 16.19/8.98 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
% 16.19/8.98 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5511_minus__less__iff,axiom,
% 16.19/8.98 ! [A: code_integer,B: code_integer] :
% 16.19/8.98 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 16.19/8.98 = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5512_verit__negate__coefficient_I2_J,axiom,
% 16.19/8.98 ! [A: real,B: real] :
% 16.19/8.98 ( ( ord_less_real @ A @ B )
% 16.19/8.98 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5513_verit__negate__coefficient_I2_J,axiom,
% 16.19/8.98 ! [A: int,B: int] :
% 16.19/8.98 ( ( ord_less_int @ A @ B )
% 16.19/8.98 => ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5514_verit__negate__coefficient_I2_J,axiom,
% 16.19/8.98 ! [A: rat,B: rat] :
% 16.19/8.98 ( ( ord_less_rat @ A @ B )
% 16.19/8.98 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5515_verit__negate__coefficient_I2_J,axiom,
% 16.19/8.98 ! [A: code_integer,B: code_integer] :
% 16.19/8.98 ( ( ord_le6747313008572928689nteger @ A @ B )
% 16.19/8.98 => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5516_numeral__neq__neg__numeral,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( numeral_numeral_real @ M )
% 16.19/8.98 != ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5517_numeral__neq__neg__numeral,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( numeral_numeral_int @ M )
% 16.19/8.98 != ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5518_numeral__neq__neg__numeral,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( numera6690914467698888265omplex @ M )
% 16.19/8.98 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5519_numeral__neq__neg__numeral,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( numeral_numeral_rat @ M )
% 16.19/8.98 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5520_numeral__neq__neg__numeral,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( numera6620942414471956472nteger @ M )
% 16.19/8.98 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5521_neg__numeral__neq__numeral,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 16.19/8.98 != ( numeral_numeral_real @ N2 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5522_neg__numeral__neq__numeral,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 16.19/8.98 != ( numeral_numeral_int @ N2 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5523_neg__numeral__neq__numeral,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 16.19/8.98 != ( numera6690914467698888265omplex @ N2 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5524_neg__numeral__neq__numeral,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 16.19/8.98 != ( numeral_numeral_rat @ N2 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5525_neg__numeral__neq__numeral,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 16.19/8.98 != ( numera6620942414471956472nteger @ N2 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5526_minus__mult__commute,axiom,
% 16.19/8.98 ! [A: real,B: real] :
% 16.19/8.98 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
% 16.19/8.98 = ( times_times_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5527_minus__mult__commute,axiom,
% 16.19/8.98 ! [A: int,B: int] :
% 16.19/8.98 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
% 16.19/8.98 = ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5528_minus__mult__commute,axiom,
% 16.19/8.98 ! [A: complex,B: complex] :
% 16.19/8.98 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 16.19/8.98 = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5529_minus__mult__commute,axiom,
% 16.19/8.98 ! [A: rat,B: rat] :
% 16.19/8.98 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
% 16.19/8.98 = ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5530_minus__mult__commute,axiom,
% 16.19/8.98 ! [A: code_integer,B: code_integer] :
% 16.19/8.98 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 16.19/8.98 = ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5531_square__eq__iff,axiom,
% 16.19/8.98 ! [A: real,B: real] :
% 16.19/8.98 ( ( ( times_times_real @ A @ A )
% 16.19/8.98 = ( times_times_real @ B @ B ) )
% 16.19/8.98 = ( ( A = B )
% 16.19/8.98 | ( A
% 16.19/8.98 = ( uminus_uminus_real @ B ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5532_square__eq__iff,axiom,
% 16.19/8.98 ! [A: int,B: int] :
% 16.19/8.98 ( ( ( times_times_int @ A @ A )
% 16.19/8.98 = ( times_times_int @ B @ B ) )
% 16.19/8.98 = ( ( A = B )
% 16.19/8.98 | ( A
% 16.19/8.98 = ( uminus_uminus_int @ B ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5533_square__eq__iff,axiom,
% 16.19/8.98 ! [A: complex,B: complex] :
% 16.19/8.98 ( ( ( times_times_complex @ A @ A )
% 16.19/8.98 = ( times_times_complex @ B @ B ) )
% 16.19/8.98 = ( ( A = B )
% 16.19/8.98 | ( A
% 16.19/8.98 = ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5534_square__eq__iff,axiom,
% 16.19/8.98 ! [A: rat,B: rat] :
% 16.19/8.98 ( ( ( times_times_rat @ A @ A )
% 16.19/8.98 = ( times_times_rat @ B @ B ) )
% 16.19/8.98 = ( ( A = B )
% 16.19/8.98 | ( A
% 16.19/8.98 = ( uminus_uminus_rat @ B ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5535_square__eq__iff,axiom,
% 16.19/8.98 ! [A: code_integer,B: code_integer] :
% 16.19/8.98 ( ( ( times_3573771949741848930nteger @ A @ A )
% 16.19/8.98 = ( times_3573771949741848930nteger @ B @ B ) )
% 16.19/8.98 = ( ( A = B )
% 16.19/8.98 | ( A
% 16.19/8.98 = ( uminus1351360451143612070nteger @ B ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5536_is__num__normalize_I8_J,axiom,
% 16.19/8.98 ! [A: real,B: real] :
% 16.19/8.98 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 16.19/8.98 = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5537_is__num__normalize_I8_J,axiom,
% 16.19/8.98 ! [A: int,B: int] :
% 16.19/8.98 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 16.19/8.98 = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5538_is__num__normalize_I8_J,axiom,
% 16.19/8.98 ! [A: complex,B: complex] :
% 16.19/8.98 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 16.19/8.98 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5539_is__num__normalize_I8_J,axiom,
% 16.19/8.98 ! [A: rat,B: rat] :
% 16.19/8.98 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 16.19/8.98 = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5540_is__num__normalize_I8_J,axiom,
% 16.19/8.98 ! [A: code_integer,B: code_integer] :
% 16.19/8.98 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 16.19/8.98 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5541_add_Oinverse__distrib__swap,axiom,
% 16.19/8.98 ! [A: real,B: real] :
% 16.19/8.98 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 16.19/8.98 = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5542_add_Oinverse__distrib__swap,axiom,
% 16.19/8.98 ! [A: int,B: int] :
% 16.19/8.98 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 16.19/8.98 = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5543_add_Oinverse__distrib__swap,axiom,
% 16.19/8.98 ! [A: complex,B: complex] :
% 16.19/8.98 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 16.19/8.98 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5544_add_Oinverse__distrib__swap,axiom,
% 16.19/8.98 ! [A: rat,B: rat] :
% 16.19/8.98 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 16.19/8.98 = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5545_add_Oinverse__distrib__swap,axiom,
% 16.19/8.98 ! [A: code_integer,B: code_integer] :
% 16.19/8.98 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 16.19/8.98 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5546_group__cancel_Oneg1,axiom,
% 16.19/8.98 ! [A2: real,K: real,A: real] :
% 16.19/8.98 ( ( A2
% 16.19/8.98 = ( plus_plus_real @ K @ A ) )
% 16.19/8.98 => ( ( uminus_uminus_real @ A2 )
% 16.19/8.98 = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5547_group__cancel_Oneg1,axiom,
% 16.19/8.98 ! [A2: int,K: int,A: int] :
% 16.19/8.98 ( ( A2
% 16.19/8.98 = ( plus_plus_int @ K @ A ) )
% 16.19/8.98 => ( ( uminus_uminus_int @ A2 )
% 16.19/8.98 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5548_group__cancel_Oneg1,axiom,
% 16.19/8.98 ! [A2: complex,K: complex,A: complex] :
% 16.19/8.98 ( ( A2
% 16.19/8.98 = ( plus_plus_complex @ K @ A ) )
% 16.19/8.98 => ( ( uminus1482373934393186551omplex @ A2 )
% 16.19/8.98 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5549_group__cancel_Oneg1,axiom,
% 16.19/8.98 ! [A2: rat,K: rat,A: rat] :
% 16.19/8.98 ( ( A2
% 16.19/8.98 = ( plus_plus_rat @ K @ A ) )
% 16.19/8.98 => ( ( uminus_uminus_rat @ A2 )
% 16.19/8.98 = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5550_group__cancel_Oneg1,axiom,
% 16.19/8.98 ! [A2: code_integer,K: code_integer,A: code_integer] :
% 16.19/8.98 ( ( A2
% 16.19/8.98 = ( plus_p5714425477246183910nteger @ K @ A ) )
% 16.19/8.98 => ( ( uminus1351360451143612070nteger @ A2 )
% 16.19/8.98 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5551_one__neq__neg__one,axiom,
% 16.19/8.98 ( one_one_real
% 16.19/8.98 != ( uminus_uminus_real @ one_one_real ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5552_one__neq__neg__one,axiom,
% 16.19/8.98 ( one_one_int
% 16.19/8.98 != ( uminus_uminus_int @ one_one_int ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5553_one__neq__neg__one,axiom,
% 16.19/8.98 ( one_one_complex
% 16.19/8.98 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5554_one__neq__neg__one,axiom,
% 16.19/8.98 ( one_one_rat
% 16.19/8.98 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5555_one__neq__neg__one,axiom,
% 16.19/8.98 ( one_one_Code_integer
% 16.19/8.98 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5556_minus__diff__minus,axiom,
% 16.19/8.98 ! [A: real,B: real] :
% 16.19/8.98 ( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 16.19/8.98 = ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5557_minus__diff__minus,axiom,
% 16.19/8.98 ! [A: int,B: int] :
% 16.19/8.98 ( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 16.19/8.98 = ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5558_minus__diff__minus,axiom,
% 16.19/8.98 ! [A: complex,B: complex] :
% 16.19/8.98 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 16.19/8.98 = ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5559_minus__diff__minus,axiom,
% 16.19/8.98 ! [A: rat,B: rat] :
% 16.19/8.98 ( ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 16.19/8.98 = ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5560_minus__diff__minus,axiom,
% 16.19/8.98 ! [A: code_integer,B: code_integer] :
% 16.19/8.98 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 16.19/8.98 = ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5561_minus__diff__commute,axiom,
% 16.19/8.98 ! [B: real,A: real] :
% 16.19/8.98 ( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
% 16.19/8.98 = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5562_minus__diff__commute,axiom,
% 16.19/8.98 ! [B: int,A: int] :
% 16.19/8.98 ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
% 16.19/8.98 = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5563_minus__diff__commute,axiom,
% 16.19/8.98 ! [B: complex,A: complex] :
% 16.19/8.98 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B ) @ A )
% 16.19/8.98 = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5564_minus__diff__commute,axiom,
% 16.19/8.98 ! [B: rat,A: rat] :
% 16.19/8.98 ( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
% 16.19/8.98 = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5565_minus__diff__commute,axiom,
% 16.19/8.98 ! [B: code_integer,A: code_integer] :
% 16.19/8.98 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A )
% 16.19/8.98 = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5566_div__minus__right,axiom,
% 16.19/8.98 ! [A: int,B: int] :
% 16.19/8.98 ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 16.19/8.98 = ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5567_div__minus__right,axiom,
% 16.19/8.98 ! [A: code_integer,B: code_integer] :
% 16.19/8.98 ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 16.19/8.98 = ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5568_minus__divide__left,axiom,
% 16.19/8.98 ! [A: real,B: real] :
% 16.19/8.98 ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 16.19/8.98 = ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5569_minus__divide__left,axiom,
% 16.19/8.98 ! [A: complex,B: complex] :
% 16.19/8.98 ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 16.19/8.98 = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5570_minus__divide__left,axiom,
% 16.19/8.98 ! [A: rat,B: rat] :
% 16.19/8.98 ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 16.19/8.98 = ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5571_minus__divide__divide,axiom,
% 16.19/8.98 ! [A: real,B: real] :
% 16.19/8.98 ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 16.19/8.98 = ( divide_divide_real @ A @ B ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5572_minus__divide__divide,axiom,
% 16.19/8.98 ! [A: complex,B: complex] :
% 16.19/8.98 ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 16.19/8.98 = ( divide1717551699836669952omplex @ A @ B ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5573_minus__divide__divide,axiom,
% 16.19/8.98 ! [A: rat,B: rat] :
% 16.19/8.98 ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 16.19/8.98 = ( divide_divide_rat @ A @ B ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5574_minus__divide__right,axiom,
% 16.19/8.98 ! [A: real,B: real] :
% 16.19/8.98 ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 16.19/8.98 = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5575_minus__divide__right,axiom,
% 16.19/8.98 ! [A: complex,B: complex] :
% 16.19/8.98 ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 16.19/8.98 = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5576_minus__divide__right,axiom,
% 16.19/8.98 ! [A: rat,B: rat] :
% 16.19/8.98 ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 16.19/8.98 = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5577_old_Oprod_Ocase,axiom,
% 16.19/8.98 ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X1: nat,X22: nat] :
% 16.19/8.98 ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
% 16.19/8.98 = ( F @ X1 @ X22 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5578_old_Oprod_Ocase,axiom,
% 16.19/8.98 ! [F: nat > nat > product_prod_nat_nat > $o,X1: nat,X22: nat] :
% 16.19/8.98 ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
% 16.19/8.98 = ( F @ X1 @ X22 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5579_old_Oprod_Ocase,axiom,
% 16.19/8.98 ! [F: int > int > product_prod_int_int,X1: int,X22: int] :
% 16.19/8.98 ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
% 16.19/8.98 = ( F @ X1 @ X22 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5580_old_Oprod_Ocase,axiom,
% 16.19/8.98 ! [F: int > int > $o,X1: int,X22: int] :
% 16.19/8.98 ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
% 16.19/8.98 = ( F @ X1 @ X22 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5581_old_Oprod_Ocase,axiom,
% 16.19/8.98 ! [F: int > int > int,X1: int,X22: int] :
% 16.19/8.98 ( ( produc8211389475949308722nt_int @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
% 16.19/8.98 = ( F @ X1 @ X22 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5582_mod__minus__right,axiom,
% 16.19/8.98 ! [A: int,B: int] :
% 16.19/8.98 ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 16.19/8.98 = ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5583_mod__minus__right,axiom,
% 16.19/8.98 ! [A: code_integer,B: code_integer] :
% 16.19/8.98 ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 16.19/8.98 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5584_mod__minus__cong,axiom,
% 16.19/8.98 ! [A: int,B: int,A6: int] :
% 16.19/8.98 ( ( ( modulo_modulo_int @ A @ B )
% 16.19/8.98 = ( modulo_modulo_int @ A6 @ B ) )
% 16.19/8.98 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 16.19/8.98 = ( modulo_modulo_int @ ( uminus_uminus_int @ A6 ) @ B ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5585_mod__minus__cong,axiom,
% 16.19/8.98 ! [A: code_integer,B: code_integer,A6: code_integer] :
% 16.19/8.98 ( ( ( modulo364778990260209775nteger @ A @ B )
% 16.19/8.98 = ( modulo364778990260209775nteger @ A6 @ B ) )
% 16.19/8.98 => ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 16.19/8.98 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A6 ) @ B ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5586_mod__minus__eq,axiom,
% 16.19/8.98 ! [A: int,B: int] :
% 16.19/8.98 ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) @ B )
% 16.19/8.98 = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5587_mod__minus__eq,axiom,
% 16.19/8.98 ! [A: code_integer,B: code_integer] :
% 16.19/8.98 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) @ B )
% 16.19/8.98 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5588_sum__mono,axiom,
% 16.19/8.98 ! [K5: set_nat,F: nat > rat,G: nat > rat] :
% 16.19/8.98 ( ! [I4: nat] :
% 16.19/8.98 ( ( member_nat @ I4 @ K5 )
% 16.19/8.98 => ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 16.19/8.98 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K5 ) @ ( groups2906978787729119204at_rat @ G @ K5 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5589_sum__mono,axiom,
% 16.19/8.98 ! [K5: set_real,F: real > rat,G: real > rat] :
% 16.19/8.98 ( ! [I4: real] :
% 16.19/8.98 ( ( member_real @ I4 @ K5 )
% 16.19/8.98 => ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 16.19/8.98 => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K5 ) @ ( groups1300246762558778688al_rat @ G @ K5 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5590_sum__mono,axiom,
% 16.19/8.98 ! [K5: set_int,F: int > rat,G: int > rat] :
% 16.19/8.98 ( ! [I4: int] :
% 16.19/8.98 ( ( member_int @ I4 @ K5 )
% 16.19/8.98 => ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 16.19/8.98 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K5 ) @ ( groups3906332499630173760nt_rat @ G @ K5 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5591_sum__mono,axiom,
% 16.19/8.98 ! [K5: set_complex,F: complex > rat,G: complex > rat] :
% 16.19/8.98 ( ! [I4: complex] :
% 16.19/8.98 ( ( member_complex @ I4 @ K5 )
% 16.19/8.98 => ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 16.19/8.98 => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ K5 ) @ ( groups5058264527183730370ex_rat @ G @ K5 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5592_sum__mono,axiom,
% 16.19/8.98 ! [K5: set_real,F: real > nat,G: real > nat] :
% 16.19/8.98 ( ! [I4: real] :
% 16.19/8.98 ( ( member_real @ I4 @ K5 )
% 16.19/8.98 => ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 16.19/8.98 => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K5 ) @ ( groups1935376822645274424al_nat @ G @ K5 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5593_sum__mono,axiom,
% 16.19/8.98 ! [K5: set_int,F: int > nat,G: int > nat] :
% 16.19/8.98 ( ! [I4: int] :
% 16.19/8.98 ( ( member_int @ I4 @ K5 )
% 16.19/8.98 => ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 16.19/8.98 => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K5 ) @ ( groups4541462559716669496nt_nat @ G @ K5 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5594_sum__mono,axiom,
% 16.19/8.98 ! [K5: set_complex,F: complex > nat,G: complex > nat] :
% 16.19/8.98 ( ! [I4: complex] :
% 16.19/8.98 ( ( member_complex @ I4 @ K5 )
% 16.19/8.98 => ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 16.19/8.98 => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ K5 ) @ ( groups5693394587270226106ex_nat @ G @ K5 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5595_sum__mono,axiom,
% 16.19/8.98 ! [K5: set_nat,F: nat > int,G: nat > int] :
% 16.19/8.98 ( ! [I4: nat] :
% 16.19/8.98 ( ( member_nat @ I4 @ K5 )
% 16.19/8.98 => ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 16.19/8.98 => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K5 ) @ ( groups3539618377306564664at_int @ G @ K5 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5596_sum__mono,axiom,
% 16.19/8.98 ! [K5: set_real,F: real > int,G: real > int] :
% 16.19/8.98 ( ! [I4: real] :
% 16.19/8.98 ( ( member_real @ I4 @ K5 )
% 16.19/8.98 => ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 16.19/8.98 => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K5 ) @ ( groups1932886352136224148al_int @ G @ K5 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5597_sum__mono,axiom,
% 16.19/8.98 ! [K5: set_complex,F: complex > int,G: complex > int] :
% 16.19/8.98 ( ! [I4: complex] :
% 16.19/8.98 ( ( member_complex @ I4 @ K5 )
% 16.19/8.98 => ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 16.19/8.98 => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ K5 ) @ ( groups5690904116761175830ex_int @ G @ K5 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5598_sum__product,axiom,
% 16.19/8.98 ! [F: int > int,A2: set_int,G: int > int,B3: set_int] :
% 16.19/8.98 ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ B3 ) )
% 16.19/8.98 = ( groups4538972089207619220nt_int
% 16.19/8.98 @ ^ [I3: int] :
% 16.19/8.98 ( groups4538972089207619220nt_int
% 16.19/8.98 @ ^ [J3: int] : ( times_times_int @ ( F @ I3 ) @ ( G @ J3 ) )
% 16.19/8.98 @ B3 )
% 16.19/8.98 @ A2 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5599_sum__product,axiom,
% 16.19/8.98 ! [F: complex > complex,A2: set_complex,G: complex > complex,B3: set_complex] :
% 16.19/8.98 ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ B3 ) )
% 16.19/8.98 = ( groups7754918857620584856omplex
% 16.19/8.98 @ ^ [I3: complex] :
% 16.19/8.98 ( groups7754918857620584856omplex
% 16.19/8.98 @ ^ [J3: complex] : ( times_times_complex @ ( F @ I3 ) @ ( G @ J3 ) )
% 16.19/8.98 @ B3 )
% 16.19/8.98 @ A2 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5600_sum__product,axiom,
% 16.19/8.98 ! [F: nat > nat,A2: set_nat,G: nat > nat,B3: set_nat] :
% 16.19/8.98 ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ B3 ) )
% 16.19/8.98 = ( groups3542108847815614940at_nat
% 16.19/8.98 @ ^ [I3: nat] :
% 16.19/8.98 ( groups3542108847815614940at_nat
% 16.19/8.98 @ ^ [J3: nat] : ( times_times_nat @ ( F @ I3 ) @ ( G @ J3 ) )
% 16.19/8.98 @ B3 )
% 16.19/8.98 @ A2 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5601_sum__product,axiom,
% 16.19/8.98 ! [F: nat > real,A2: set_nat,G: nat > real,B3: set_nat] :
% 16.19/8.98 ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ B3 ) )
% 16.19/8.98 = ( groups6591440286371151544t_real
% 16.19/8.98 @ ^ [I3: nat] :
% 16.19/8.98 ( groups6591440286371151544t_real
% 16.19/8.98 @ ^ [J3: nat] : ( times_times_real @ ( F @ I3 ) @ ( G @ J3 ) )
% 16.19/8.98 @ B3 )
% 16.19/8.98 @ A2 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5602_sum__distrib__right,axiom,
% 16.19/8.98 ! [F: int > int,A2: set_int,R: int] :
% 16.19/8.98 ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ R )
% 16.19/8.98 = ( groups4538972089207619220nt_int
% 16.19/8.98 @ ^ [N: int] : ( times_times_int @ ( F @ N ) @ R )
% 16.19/8.98 @ A2 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5603_sum__distrib__right,axiom,
% 16.19/8.98 ! [F: complex > complex,A2: set_complex,R: complex] :
% 16.19/8.98 ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R )
% 16.19/8.98 = ( groups7754918857620584856omplex
% 16.19/8.98 @ ^ [N: complex] : ( times_times_complex @ ( F @ N ) @ R )
% 16.19/8.98 @ A2 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5604_sum__distrib__right,axiom,
% 16.19/8.98 ! [F: nat > nat,A2: set_nat,R: nat] :
% 16.19/8.98 ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ R )
% 16.19/8.98 = ( groups3542108847815614940at_nat
% 16.19/8.98 @ ^ [N: nat] : ( times_times_nat @ ( F @ N ) @ R )
% 16.19/8.98 @ A2 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5605_sum__distrib__right,axiom,
% 16.19/8.98 ! [F: nat > real,A2: set_nat,R: real] :
% 16.19/8.98 ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R )
% 16.19/8.98 = ( groups6591440286371151544t_real
% 16.19/8.98 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ R )
% 16.19/8.98 @ A2 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5606_sum__distrib__left,axiom,
% 16.19/8.98 ! [R: int,F: int > int,A2: set_int] :
% 16.19/8.98 ( ( times_times_int @ R @ ( groups4538972089207619220nt_int @ F @ A2 ) )
% 16.19/8.98 = ( groups4538972089207619220nt_int
% 16.19/8.98 @ ^ [N: int] : ( times_times_int @ R @ ( F @ N ) )
% 16.19/8.98 @ A2 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5607_sum__distrib__left,axiom,
% 16.19/8.98 ! [R: complex,F: complex > complex,A2: set_complex] :
% 16.19/8.98 ( ( times_times_complex @ R @ ( groups7754918857620584856omplex @ F @ A2 ) )
% 16.19/8.98 = ( groups7754918857620584856omplex
% 16.19/8.98 @ ^ [N: complex] : ( times_times_complex @ R @ ( F @ N ) )
% 16.19/8.98 @ A2 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5608_sum__distrib__left,axiom,
% 16.19/8.98 ! [R: nat,F: nat > nat,A2: set_nat] :
% 16.19/8.98 ( ( times_times_nat @ R @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 16.19/8.98 = ( groups3542108847815614940at_nat
% 16.19/8.98 @ ^ [N: nat] : ( times_times_nat @ R @ ( F @ N ) )
% 16.19/8.98 @ A2 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5609_sum__distrib__left,axiom,
% 16.19/8.98 ! [R: real,F: nat > real,A2: set_nat] :
% 16.19/8.98 ( ( times_times_real @ R @ ( groups6591440286371151544t_real @ F @ A2 ) )
% 16.19/8.98 = ( groups6591440286371151544t_real
% 16.19/8.98 @ ^ [N: nat] : ( times_times_real @ R @ ( F @ N ) )
% 16.19/8.98 @ A2 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5610_sum_Odistrib,axiom,
% 16.19/8.98 ! [G: int > int,H2: int > int,A2: set_int] :
% 16.19/8.98 ( ( groups4538972089207619220nt_int
% 16.19/8.98 @ ^ [X2: int] : ( plus_plus_int @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 16.19/8.98 @ A2 )
% 16.19/8.98 = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A2 ) @ ( groups4538972089207619220nt_int @ H2 @ A2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5611_sum_Odistrib,axiom,
% 16.19/8.98 ! [G: complex > complex,H2: complex > complex,A2: set_complex] :
% 16.19/8.98 ( ( groups7754918857620584856omplex
% 16.19/8.98 @ ^ [X2: complex] : ( plus_plus_complex @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 16.19/8.98 @ A2 )
% 16.19/8.98 = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A2 ) @ ( groups7754918857620584856omplex @ H2 @ A2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5612_sum_Odistrib,axiom,
% 16.19/8.98 ! [G: nat > nat,H2: nat > nat,A2: set_nat] :
% 16.19/8.98 ( ( groups3542108847815614940at_nat
% 16.19/8.98 @ ^ [X2: nat] : ( plus_plus_nat @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 16.19/8.98 @ A2 )
% 16.19/8.98 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A2 ) @ ( groups3542108847815614940at_nat @ H2 @ A2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5613_sum_Odistrib,axiom,
% 16.19/8.98 ! [G: nat > real,H2: nat > real,A2: set_nat] :
% 16.19/8.98 ( ( groups6591440286371151544t_real
% 16.19/8.98 @ ^ [X2: nat] : ( plus_plus_real @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 16.19/8.98 @ A2 )
% 16.19/8.98 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A2 ) @ ( groups6591440286371151544t_real @ H2 @ A2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5614_sum__subtractf,axiom,
% 16.19/8.98 ! [F: int > int,G: int > int,A2: set_int] :
% 16.19/8.98 ( ( groups4538972089207619220nt_int
% 16.19/8.98 @ ^ [X2: int] : ( minus_minus_int @ ( F @ X2 ) @ ( G @ X2 ) )
% 16.19/8.98 @ A2 )
% 16.19/8.98 = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5615_sum__subtractf,axiom,
% 16.19/8.98 ! [F: complex > complex,G: complex > complex,A2: set_complex] :
% 16.19/8.98 ( ( groups7754918857620584856omplex
% 16.19/8.98 @ ^ [X2: complex] : ( minus_minus_complex @ ( F @ X2 ) @ ( G @ X2 ) )
% 16.19/8.98 @ A2 )
% 16.19/8.98 = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ A2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5616_sum__subtractf,axiom,
% 16.19/8.98 ! [F: nat > real,G: nat > real,A2: set_nat] :
% 16.19/8.98 ( ( groups6591440286371151544t_real
% 16.19/8.98 @ ^ [X2: nat] : ( minus_minus_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 16.19/8.98 @ A2 )
% 16.19/8.98 = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ A2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5617_sum__divide__distrib,axiom,
% 16.19/8.98 ! [F: complex > complex,A2: set_complex,R: complex] :
% 16.19/8.98 ( ( divide1717551699836669952omplex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R )
% 16.19/8.98 = ( groups7754918857620584856omplex
% 16.19/8.98 @ ^ [N: complex] : ( divide1717551699836669952omplex @ ( F @ N ) @ R )
% 16.19/8.98 @ A2 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5618_sum__divide__distrib,axiom,
% 16.19/8.98 ! [F: nat > real,A2: set_nat,R: real] :
% 16.19/8.98 ( ( divide_divide_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R )
% 16.19/8.98 = ( groups6591440286371151544t_real
% 16.19/8.98 @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ R )
% 16.19/8.98 @ A2 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5619_sum_Oswap__restrict,axiom,
% 16.19/8.98 ! [A2: set_real,B3: set_int,G: real > int > int,R2: real > int > $o] :
% 16.19/8.98 ( ( finite_finite_real @ A2 )
% 16.19/8.98 => ( ( finite_finite_int @ B3 )
% 16.19/8.98 => ( ( groups1932886352136224148al_int
% 16.19/8.98 @ ^ [X2: real] :
% 16.19/8.98 ( groups4538972089207619220nt_int @ ( G @ X2 )
% 16.19/8.98 @ ( collect_int
% 16.19/8.98 @ ^ [Y2: int] :
% 16.19/8.98 ( ( member_int @ Y2 @ B3 )
% 16.19/8.98 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/8.98 @ A2 )
% 16.19/8.98 = ( groups4538972089207619220nt_int
% 16.19/8.98 @ ^ [Y2: int] :
% 16.19/8.98 ( groups1932886352136224148al_int
% 16.19/8.98 @ ^ [X2: real] : ( G @ X2 @ Y2 )
% 16.19/8.98 @ ( collect_real
% 16.19/8.98 @ ^ [X2: real] :
% 16.19/8.98 ( ( member_real @ X2 @ A2 )
% 16.19/8.98 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/8.98 @ B3 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5620_sum_Oswap__restrict,axiom,
% 16.19/8.98 ! [A2: set_nat,B3: set_int,G: nat > int > int,R2: nat > int > $o] :
% 16.19/8.98 ( ( finite_finite_nat @ A2 )
% 16.19/8.98 => ( ( finite_finite_int @ B3 )
% 16.19/8.98 => ( ( groups3539618377306564664at_int
% 16.19/8.98 @ ^ [X2: nat] :
% 16.19/8.98 ( groups4538972089207619220nt_int @ ( G @ X2 )
% 16.19/8.98 @ ( collect_int
% 16.19/8.98 @ ^ [Y2: int] :
% 16.19/8.98 ( ( member_int @ Y2 @ B3 )
% 16.19/8.98 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/8.98 @ A2 )
% 16.19/8.98 = ( groups4538972089207619220nt_int
% 16.19/8.98 @ ^ [Y2: int] :
% 16.19/8.98 ( groups3539618377306564664at_int
% 16.19/8.98 @ ^ [X2: nat] : ( G @ X2 @ Y2 )
% 16.19/8.98 @ ( collect_nat
% 16.19/8.98 @ ^ [X2: nat] :
% 16.19/8.98 ( ( member_nat @ X2 @ A2 )
% 16.19/8.98 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/8.98 @ B3 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5621_sum_Oswap__restrict,axiom,
% 16.19/8.98 ! [A2: set_complex,B3: set_int,G: complex > int > int,R2: complex > int > $o] :
% 16.19/8.98 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.98 => ( ( finite_finite_int @ B3 )
% 16.19/8.98 => ( ( groups5690904116761175830ex_int
% 16.19/8.98 @ ^ [X2: complex] :
% 16.19/8.98 ( groups4538972089207619220nt_int @ ( G @ X2 )
% 16.19/8.98 @ ( collect_int
% 16.19/8.98 @ ^ [Y2: int] :
% 16.19/8.98 ( ( member_int @ Y2 @ B3 )
% 16.19/8.98 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/8.98 @ A2 )
% 16.19/8.98 = ( groups4538972089207619220nt_int
% 16.19/8.98 @ ^ [Y2: int] :
% 16.19/8.98 ( groups5690904116761175830ex_int
% 16.19/8.98 @ ^ [X2: complex] : ( G @ X2 @ Y2 )
% 16.19/8.98 @ ( collect_complex
% 16.19/8.98 @ ^ [X2: complex] :
% 16.19/8.98 ( ( member_complex @ X2 @ A2 )
% 16.19/8.98 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/8.98 @ B3 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5622_sum_Oswap__restrict,axiom,
% 16.19/8.98 ! [A2: set_real,B3: set_complex,G: real > complex > complex,R2: real > complex > $o] :
% 16.19/8.98 ( ( finite_finite_real @ A2 )
% 16.19/8.98 => ( ( finite3207457112153483333omplex @ B3 )
% 16.19/8.98 => ( ( groups5754745047067104278omplex
% 16.19/8.98 @ ^ [X2: real] :
% 16.19/8.98 ( groups7754918857620584856omplex @ ( G @ X2 )
% 16.19/8.98 @ ( collect_complex
% 16.19/8.98 @ ^ [Y2: complex] :
% 16.19/8.98 ( ( member_complex @ Y2 @ B3 )
% 16.19/8.98 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/8.98 @ A2 )
% 16.19/8.98 = ( groups7754918857620584856omplex
% 16.19/8.98 @ ^ [Y2: complex] :
% 16.19/8.98 ( groups5754745047067104278omplex
% 16.19/8.98 @ ^ [X2: real] : ( G @ X2 @ Y2 )
% 16.19/8.98 @ ( collect_real
% 16.19/8.98 @ ^ [X2: real] :
% 16.19/8.98 ( ( member_real @ X2 @ A2 )
% 16.19/8.98 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/8.98 @ B3 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5623_sum_Oswap__restrict,axiom,
% 16.19/8.98 ! [A2: set_nat,B3: set_complex,G: nat > complex > complex,R2: nat > complex > $o] :
% 16.19/8.98 ( ( finite_finite_nat @ A2 )
% 16.19/8.98 => ( ( finite3207457112153483333omplex @ B3 )
% 16.19/8.98 => ( ( groups2073611262835488442omplex
% 16.19/8.98 @ ^ [X2: nat] :
% 16.19/8.98 ( groups7754918857620584856omplex @ ( G @ X2 )
% 16.19/8.98 @ ( collect_complex
% 16.19/8.98 @ ^ [Y2: complex] :
% 16.19/8.98 ( ( member_complex @ Y2 @ B3 )
% 16.19/8.98 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/8.98 @ A2 )
% 16.19/8.98 = ( groups7754918857620584856omplex
% 16.19/8.98 @ ^ [Y2: complex] :
% 16.19/8.98 ( groups2073611262835488442omplex
% 16.19/8.98 @ ^ [X2: nat] : ( G @ X2 @ Y2 )
% 16.19/8.98 @ ( collect_nat
% 16.19/8.98 @ ^ [X2: nat] :
% 16.19/8.98 ( ( member_nat @ X2 @ A2 )
% 16.19/8.98 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/8.98 @ B3 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5624_sum_Oswap__restrict,axiom,
% 16.19/8.98 ! [A2: set_int,B3: set_complex,G: int > complex > complex,R2: int > complex > $o] :
% 16.19/8.98 ( ( finite_finite_int @ A2 )
% 16.19/8.98 => ( ( finite3207457112153483333omplex @ B3 )
% 16.19/8.98 => ( ( groups3049146728041665814omplex
% 16.19/8.98 @ ^ [X2: int] :
% 16.19/8.98 ( groups7754918857620584856omplex @ ( G @ X2 )
% 16.19/8.98 @ ( collect_complex
% 16.19/8.98 @ ^ [Y2: complex] :
% 16.19/8.98 ( ( member_complex @ Y2 @ B3 )
% 16.19/8.98 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/8.98 @ A2 )
% 16.19/8.98 = ( groups7754918857620584856omplex
% 16.19/8.98 @ ^ [Y2: complex] :
% 16.19/8.98 ( groups3049146728041665814omplex
% 16.19/8.98 @ ^ [X2: int] : ( G @ X2 @ Y2 )
% 16.19/8.98 @ ( collect_int
% 16.19/8.98 @ ^ [X2: int] :
% 16.19/8.98 ( ( member_int @ X2 @ A2 )
% 16.19/8.98 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/8.98 @ B3 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5625_sum_Oswap__restrict,axiom,
% 16.19/8.98 ! [A2: set_real,B3: set_nat,G: real > nat > nat,R2: real > nat > $o] :
% 16.19/8.98 ( ( finite_finite_real @ A2 )
% 16.19/8.98 => ( ( finite_finite_nat @ B3 )
% 16.19/8.98 => ( ( groups1935376822645274424al_nat
% 16.19/8.98 @ ^ [X2: real] :
% 16.19/8.98 ( groups3542108847815614940at_nat @ ( G @ X2 )
% 16.19/8.98 @ ( collect_nat
% 16.19/8.98 @ ^ [Y2: nat] :
% 16.19/8.98 ( ( member_nat @ Y2 @ B3 )
% 16.19/8.98 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/8.98 @ A2 )
% 16.19/8.98 = ( groups3542108847815614940at_nat
% 16.19/8.98 @ ^ [Y2: nat] :
% 16.19/8.98 ( groups1935376822645274424al_nat
% 16.19/8.98 @ ^ [X2: real] : ( G @ X2 @ Y2 )
% 16.19/8.98 @ ( collect_real
% 16.19/8.98 @ ^ [X2: real] :
% 16.19/8.98 ( ( member_real @ X2 @ A2 )
% 16.19/8.98 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/8.98 @ B3 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5626_sum_Oswap__restrict,axiom,
% 16.19/8.98 ! [A2: set_int,B3: set_nat,G: int > nat > nat,R2: int > nat > $o] :
% 16.19/8.98 ( ( finite_finite_int @ A2 )
% 16.19/8.98 => ( ( finite_finite_nat @ B3 )
% 16.19/8.98 => ( ( groups4541462559716669496nt_nat
% 16.19/8.98 @ ^ [X2: int] :
% 16.19/8.98 ( groups3542108847815614940at_nat @ ( G @ X2 )
% 16.19/8.98 @ ( collect_nat
% 16.19/8.98 @ ^ [Y2: nat] :
% 16.19/8.98 ( ( member_nat @ Y2 @ B3 )
% 16.19/8.98 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/8.98 @ A2 )
% 16.19/8.98 = ( groups3542108847815614940at_nat
% 16.19/8.98 @ ^ [Y2: nat] :
% 16.19/8.98 ( groups4541462559716669496nt_nat
% 16.19/8.98 @ ^ [X2: int] : ( G @ X2 @ Y2 )
% 16.19/8.98 @ ( collect_int
% 16.19/8.98 @ ^ [X2: int] :
% 16.19/8.98 ( ( member_int @ X2 @ A2 )
% 16.19/8.98 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/8.98 @ B3 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5627_sum_Oswap__restrict,axiom,
% 16.19/8.98 ! [A2: set_complex,B3: set_nat,G: complex > nat > nat,R2: complex > nat > $o] :
% 16.19/8.98 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.98 => ( ( finite_finite_nat @ B3 )
% 16.19/8.98 => ( ( groups5693394587270226106ex_nat
% 16.19/8.98 @ ^ [X2: complex] :
% 16.19/8.98 ( groups3542108847815614940at_nat @ ( G @ X2 )
% 16.19/8.98 @ ( collect_nat
% 16.19/8.98 @ ^ [Y2: nat] :
% 16.19/8.98 ( ( member_nat @ Y2 @ B3 )
% 16.19/8.98 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/8.98 @ A2 )
% 16.19/8.98 = ( groups3542108847815614940at_nat
% 16.19/8.98 @ ^ [Y2: nat] :
% 16.19/8.98 ( groups5693394587270226106ex_nat
% 16.19/8.98 @ ^ [X2: complex] : ( G @ X2 @ Y2 )
% 16.19/8.98 @ ( collect_complex
% 16.19/8.98 @ ^ [X2: complex] :
% 16.19/8.98 ( ( member_complex @ X2 @ A2 )
% 16.19/8.98 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/8.98 @ B3 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5628_sum_Oswap__restrict,axiom,
% 16.19/8.98 ! [A2: set_real,B3: set_nat,G: real > nat > real,R2: real > nat > $o] :
% 16.19/8.98 ( ( finite_finite_real @ A2 )
% 16.19/8.98 => ( ( finite_finite_nat @ B3 )
% 16.19/8.98 => ( ( groups8097168146408367636l_real
% 16.19/8.98 @ ^ [X2: real] :
% 16.19/8.98 ( groups6591440286371151544t_real @ ( G @ X2 )
% 16.19/8.98 @ ( collect_nat
% 16.19/8.98 @ ^ [Y2: nat] :
% 16.19/8.98 ( ( member_nat @ Y2 @ B3 )
% 16.19/8.98 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/8.98 @ A2 )
% 16.19/8.98 = ( groups6591440286371151544t_real
% 16.19/8.98 @ ^ [Y2: nat] :
% 16.19/8.98 ( groups8097168146408367636l_real
% 16.19/8.98 @ ^ [X2: real] : ( G @ X2 @ Y2 )
% 16.19/8.98 @ ( collect_real
% 16.19/8.98 @ ^ [X2: real] :
% 16.19/8.98 ( ( member_real @ X2 @ A2 )
% 16.19/8.98 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/8.98 @ B3 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5629_mod__sum__eq,axiom,
% 16.19/8.98 ! [F: int > int,A: int,A2: set_int] :
% 16.19/8.98 ( ( modulo_modulo_int
% 16.19/8.98 @ ( groups4538972089207619220nt_int
% 16.19/8.98 @ ^ [I3: int] : ( modulo_modulo_int @ ( F @ I3 ) @ A )
% 16.19/8.98 @ A2 )
% 16.19/8.98 @ A )
% 16.19/8.98 = ( modulo_modulo_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ A ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5630_mod__sum__eq,axiom,
% 16.19/8.98 ! [F: nat > nat,A: nat,A2: set_nat] :
% 16.19/8.98 ( ( modulo_modulo_nat
% 16.19/8.98 @ ( groups3542108847815614940at_nat
% 16.19/8.98 @ ^ [I3: nat] : ( modulo_modulo_nat @ ( F @ I3 ) @ A )
% 16.19/8.98 @ A2 )
% 16.19/8.98 @ A )
% 16.19/8.98 = ( modulo_modulo_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ A ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5631_cond__case__prod__eta,axiom,
% 16.19/8.98 ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
% 16.19/8.98 ( ! [X4: nat,Y4: nat] :
% 16.19/8.98 ( ( F @ X4 @ Y4 )
% 16.19/8.98 = ( G @ ( product_Pair_nat_nat @ X4 @ Y4 ) ) )
% 16.19/8.98 => ( ( produc27273713700761075at_nat @ F )
% 16.19/8.98 = G ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5632_cond__case__prod__eta,axiom,
% 16.19/8.98 ! [F: nat > nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 16.19/8.98 ( ! [X4: nat,Y4: nat] :
% 16.19/8.98 ( ( F @ X4 @ Y4 )
% 16.19/8.98 = ( G @ ( product_Pair_nat_nat @ X4 @ Y4 ) ) )
% 16.19/8.98 => ( ( produc8739625826339149834_nat_o @ F )
% 16.19/8.98 = G ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5633_cond__case__prod__eta,axiom,
% 16.19/8.98 ! [F: int > int > product_prod_int_int,G: product_prod_int_int > product_prod_int_int] :
% 16.19/8.98 ( ! [X4: int,Y4: int] :
% 16.19/8.98 ( ( F @ X4 @ Y4 )
% 16.19/8.98 = ( G @ ( product_Pair_int_int @ X4 @ Y4 ) ) )
% 16.19/8.98 => ( ( produc4245557441103728435nt_int @ F )
% 16.19/8.98 = G ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5634_cond__case__prod__eta,axiom,
% 16.19/8.98 ! [F: int > int > $o,G: product_prod_int_int > $o] :
% 16.19/8.98 ( ! [X4: int,Y4: int] :
% 16.19/8.98 ( ( F @ X4 @ Y4 )
% 16.19/8.98 = ( G @ ( product_Pair_int_int @ X4 @ Y4 ) ) )
% 16.19/8.98 => ( ( produc4947309494688390418_int_o @ F )
% 16.19/8.98 = G ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5635_cond__case__prod__eta,axiom,
% 16.19/8.98 ! [F: int > int > int,G: product_prod_int_int > int] :
% 16.19/8.98 ( ! [X4: int,Y4: int] :
% 16.19/8.98 ( ( F @ X4 @ Y4 )
% 16.19/8.98 = ( G @ ( product_Pair_int_int @ X4 @ Y4 ) ) )
% 16.19/8.98 => ( ( produc8211389475949308722nt_int @ F )
% 16.19/8.98 = G ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5636_case__prod__eta,axiom,
% 16.19/8.98 ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
% 16.19/8.98 ( ( produc27273713700761075at_nat
% 16.19/8.98 @ ^ [X2: nat,Y2: nat] : ( F @ ( product_Pair_nat_nat @ X2 @ Y2 ) ) )
% 16.19/8.98 = F ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5637_case__prod__eta,axiom,
% 16.19/8.98 ! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 16.19/8.98 ( ( produc8739625826339149834_nat_o
% 16.19/8.98 @ ^ [X2: nat,Y2: nat] : ( F @ ( product_Pair_nat_nat @ X2 @ Y2 ) ) )
% 16.19/8.98 = F ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5638_case__prod__eta,axiom,
% 16.19/8.98 ! [F: product_prod_int_int > product_prod_int_int] :
% 16.19/8.98 ( ( produc4245557441103728435nt_int
% 16.19/8.98 @ ^ [X2: int,Y2: int] : ( F @ ( product_Pair_int_int @ X2 @ Y2 ) ) )
% 16.19/8.98 = F ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5639_case__prod__eta,axiom,
% 16.19/8.98 ! [F: product_prod_int_int > $o] :
% 16.19/8.98 ( ( produc4947309494688390418_int_o
% 16.19/8.98 @ ^ [X2: int,Y2: int] : ( F @ ( product_Pair_int_int @ X2 @ Y2 ) ) )
% 16.19/8.98 = F ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5640_case__prod__eta,axiom,
% 16.19/8.98 ! [F: product_prod_int_int > int] :
% 16.19/8.98 ( ( produc8211389475949308722nt_int
% 16.19/8.98 @ ^ [X2: int,Y2: int] : ( F @ ( product_Pair_int_int @ X2 @ Y2 ) ) )
% 16.19/8.98 = F ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5641_case__prodE2,axiom,
% 16.19/8.98 ! [Q: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,P: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Z: product_prod_nat_nat] :
% 16.19/8.98 ( ( Q @ ( produc27273713700761075at_nat @ P @ Z ) )
% 16.19/8.98 => ~ ! [X4: nat,Y4: nat] :
% 16.19/8.98 ( ( Z
% 16.19/8.98 = ( product_Pair_nat_nat @ X4 @ Y4 ) )
% 16.19/8.98 => ~ ( Q @ ( P @ X4 @ Y4 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5642_case__prodE2,axiom,
% 16.19/8.98 ! [Q: ( product_prod_nat_nat > $o ) > $o,P: nat > nat > product_prod_nat_nat > $o,Z: product_prod_nat_nat] :
% 16.19/8.98 ( ( Q @ ( produc8739625826339149834_nat_o @ P @ Z ) )
% 16.19/8.98 => ~ ! [X4: nat,Y4: nat] :
% 16.19/8.98 ( ( Z
% 16.19/8.98 = ( product_Pair_nat_nat @ X4 @ Y4 ) )
% 16.19/8.98 => ~ ( Q @ ( P @ X4 @ Y4 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5643_case__prodE2,axiom,
% 16.19/8.98 ! [Q: product_prod_int_int > $o,P: int > int > product_prod_int_int,Z: product_prod_int_int] :
% 16.19/8.98 ( ( Q @ ( produc4245557441103728435nt_int @ P @ Z ) )
% 16.19/8.98 => ~ ! [X4: int,Y4: int] :
% 16.19/8.98 ( ( Z
% 16.19/8.98 = ( product_Pair_int_int @ X4 @ Y4 ) )
% 16.19/8.98 => ~ ( Q @ ( P @ X4 @ Y4 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5644_case__prodE2,axiom,
% 16.19/8.98 ! [Q: $o > $o,P: int > int > $o,Z: product_prod_int_int] :
% 16.19/8.98 ( ( Q @ ( produc4947309494688390418_int_o @ P @ Z ) )
% 16.19/8.98 => ~ ! [X4: int,Y4: int] :
% 16.19/8.98 ( ( Z
% 16.19/8.98 = ( product_Pair_int_int @ X4 @ Y4 ) )
% 16.19/8.98 => ~ ( Q @ ( P @ X4 @ Y4 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5645_case__prodE2,axiom,
% 16.19/8.98 ! [Q: int > $o,P: int > int > int,Z: product_prod_int_int] :
% 16.19/8.98 ( ( Q @ ( produc8211389475949308722nt_int @ P @ Z ) )
% 16.19/8.98 => ~ ! [X4: int,Y4: int] :
% 16.19/8.98 ( ( Z
% 16.19/8.98 = ( product_Pair_int_int @ X4 @ Y4 ) )
% 16.19/8.98 => ~ ( Q @ ( P @ X4 @ Y4 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5646_sum__nonpos,axiom,
% 16.19/8.98 ! [A2: set_real,F: real > real] :
% 16.19/8.98 ( ! [X4: real] :
% 16.19/8.98 ( ( member_real @ X4 @ A2 )
% 16.19/8.98 => ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
% 16.19/8.98 => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ zero_zero_real ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5647_sum__nonpos,axiom,
% 16.19/8.98 ! [A2: set_int,F: int > real] :
% 16.19/8.98 ( ! [X4: int] :
% 16.19/8.98 ( ( member_int @ X4 @ A2 )
% 16.19/8.98 => ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
% 16.19/8.98 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ zero_zero_real ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5648_sum__nonpos,axiom,
% 16.19/8.98 ! [A2: set_complex,F: complex > real] :
% 16.19/8.98 ( ! [X4: complex] :
% 16.19/8.98 ( ( member_complex @ X4 @ A2 )
% 16.19/8.98 => ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
% 16.19/8.98 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ zero_zero_real ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5649_sum__nonpos,axiom,
% 16.19/8.98 ! [A2: set_nat,F: nat > rat] :
% 16.19/8.98 ( ! [X4: nat] :
% 16.19/8.98 ( ( member_nat @ X4 @ A2 )
% 16.19/8.98 => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
% 16.19/8.98 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5650_sum__nonpos,axiom,
% 16.19/8.98 ! [A2: set_real,F: real > rat] :
% 16.19/8.98 ( ! [X4: real] :
% 16.19/8.98 ( ( member_real @ X4 @ A2 )
% 16.19/8.98 => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
% 16.19/8.98 => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5651_sum__nonpos,axiom,
% 16.19/8.98 ! [A2: set_int,F: int > rat] :
% 16.19/8.98 ( ! [X4: int] :
% 16.19/8.98 ( ( member_int @ X4 @ A2 )
% 16.19/8.98 => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
% 16.19/8.98 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5652_sum__nonpos,axiom,
% 16.19/8.98 ! [A2: set_complex,F: complex > rat] :
% 16.19/8.98 ( ! [X4: complex] :
% 16.19/8.98 ( ( member_complex @ X4 @ A2 )
% 16.19/8.98 => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
% 16.19/8.98 => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5653_sum__nonpos,axiom,
% 16.19/8.98 ! [A2: set_real,F: real > nat] :
% 16.19/8.98 ( ! [X4: real] :
% 16.19/8.98 ( ( member_real @ X4 @ A2 )
% 16.19/8.98 => ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
% 16.19/8.98 => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5654_sum__nonpos,axiom,
% 16.19/8.98 ! [A2: set_int,F: int > nat] :
% 16.19/8.98 ( ! [X4: int] :
% 16.19/8.98 ( ( member_int @ X4 @ A2 )
% 16.19/8.98 => ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
% 16.19/8.98 => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5655_sum__nonpos,axiom,
% 16.19/8.98 ! [A2: set_complex,F: complex > nat] :
% 16.19/8.98 ( ! [X4: complex] :
% 16.19/8.98 ( ( member_complex @ X4 @ A2 )
% 16.19/8.98 => ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
% 16.19/8.98 => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5656_sum__nonneg,axiom,
% 16.19/8.98 ! [A2: set_real,F: real > real] :
% 16.19/8.98 ( ! [X4: real] :
% 16.19/8.98 ( ( member_real @ X4 @ A2 )
% 16.19/8.98 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 16.19/8.98 => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5657_sum__nonneg,axiom,
% 16.19/8.98 ! [A2: set_int,F: int > real] :
% 16.19/8.98 ( ! [X4: int] :
% 16.19/8.98 ( ( member_int @ X4 @ A2 )
% 16.19/8.98 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 16.19/8.98 => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5658_sum__nonneg,axiom,
% 16.19/8.98 ! [A2: set_complex,F: complex > real] :
% 16.19/8.98 ( ! [X4: complex] :
% 16.19/8.98 ( ( member_complex @ X4 @ A2 )
% 16.19/8.98 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 16.19/8.98 => ( ord_less_eq_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ A2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5659_sum__nonneg,axiom,
% 16.19/8.98 ! [A2: set_nat,F: nat > rat] :
% 16.19/8.98 ( ! [X4: nat] :
% 16.19/8.98 ( ( member_nat @ X4 @ A2 )
% 16.19/8.98 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 16.19/8.98 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5660_sum__nonneg,axiom,
% 16.19/8.98 ! [A2: set_real,F: real > rat] :
% 16.19/8.98 ( ! [X4: real] :
% 16.19/8.98 ( ( member_real @ X4 @ A2 )
% 16.19/8.98 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 16.19/8.98 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5661_sum__nonneg,axiom,
% 16.19/8.98 ! [A2: set_int,F: int > rat] :
% 16.19/8.98 ( ! [X4: int] :
% 16.19/8.98 ( ( member_int @ X4 @ A2 )
% 16.19/8.98 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 16.19/8.98 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5662_sum__nonneg,axiom,
% 16.19/8.98 ! [A2: set_complex,F: complex > rat] :
% 16.19/8.98 ( ! [X4: complex] :
% 16.19/8.98 ( ( member_complex @ X4 @ A2 )
% 16.19/8.98 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 16.19/8.98 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5663_sum__nonneg,axiom,
% 16.19/8.98 ! [A2: set_real,F: real > nat] :
% 16.19/8.98 ( ! [X4: real] :
% 16.19/8.98 ( ( member_real @ X4 @ A2 )
% 16.19/8.98 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 16.19/8.98 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5664_sum__nonneg,axiom,
% 16.19/8.98 ! [A2: set_int,F: int > nat] :
% 16.19/8.98 ( ! [X4: int] :
% 16.19/8.98 ( ( member_int @ X4 @ A2 )
% 16.19/8.98 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 16.19/8.98 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5665_sum__nonneg,axiom,
% 16.19/8.98 ! [A2: set_complex,F: complex > nat] :
% 16.19/8.98 ( ! [X4: complex] :
% 16.19/8.98 ( ( member_complex @ X4 @ A2 )
% 16.19/8.98 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 16.19/8.98 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5666_ln__add__one__self__le__self2,axiom,
% 16.19/8.98 ! [X: real] :
% 16.19/8.98 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 16.19/8.98 => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5667_sum__mono__inv,axiom,
% 16.19/8.98 ! [F: real > rat,I6: set_real,G: real > rat,I2: real] :
% 16.19/8.98 ( ( ( groups1300246762558778688al_rat @ F @ I6 )
% 16.19/8.98 = ( groups1300246762558778688al_rat @ G @ I6 ) )
% 16.19/8.98 => ( ! [I4: real] :
% 16.19/8.98 ( ( member_real @ I4 @ I6 )
% 16.19/8.98 => ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 16.19/8.98 => ( ( member_real @ I2 @ I6 )
% 16.19/8.98 => ( ( finite_finite_real @ I6 )
% 16.19/8.98 => ( ( F @ I2 )
% 16.19/8.98 = ( G @ I2 ) ) ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5668_sum__mono__inv,axiom,
% 16.19/8.98 ! [F: nat > rat,I6: set_nat,G: nat > rat,I2: nat] :
% 16.19/8.98 ( ( ( groups2906978787729119204at_rat @ F @ I6 )
% 16.19/8.98 = ( groups2906978787729119204at_rat @ G @ I6 ) )
% 16.19/8.98 => ( ! [I4: nat] :
% 16.19/8.98 ( ( member_nat @ I4 @ I6 )
% 16.19/8.98 => ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 16.19/8.98 => ( ( member_nat @ I2 @ I6 )
% 16.19/8.98 => ( ( finite_finite_nat @ I6 )
% 16.19/8.98 => ( ( F @ I2 )
% 16.19/8.98 = ( G @ I2 ) ) ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5669_sum__mono__inv,axiom,
% 16.19/8.98 ! [F: int > rat,I6: set_int,G: int > rat,I2: int] :
% 16.19/8.98 ( ( ( groups3906332499630173760nt_rat @ F @ I6 )
% 16.19/8.98 = ( groups3906332499630173760nt_rat @ G @ I6 ) )
% 16.19/8.98 => ( ! [I4: int] :
% 16.19/8.98 ( ( member_int @ I4 @ I6 )
% 16.19/8.98 => ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 16.19/8.98 => ( ( member_int @ I2 @ I6 )
% 16.19/8.98 => ( ( finite_finite_int @ I6 )
% 16.19/8.98 => ( ( F @ I2 )
% 16.19/8.98 = ( G @ I2 ) ) ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5670_sum__mono__inv,axiom,
% 16.19/8.98 ! [F: complex > rat,I6: set_complex,G: complex > rat,I2: complex] :
% 16.19/8.98 ( ( ( groups5058264527183730370ex_rat @ F @ I6 )
% 16.19/8.98 = ( groups5058264527183730370ex_rat @ G @ I6 ) )
% 16.19/8.98 => ( ! [I4: complex] :
% 16.19/8.98 ( ( member_complex @ I4 @ I6 )
% 16.19/8.98 => ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 16.19/8.98 => ( ( member_complex @ I2 @ I6 )
% 16.19/8.98 => ( ( finite3207457112153483333omplex @ I6 )
% 16.19/8.98 => ( ( F @ I2 )
% 16.19/8.98 = ( G @ I2 ) ) ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5671_sum__mono__inv,axiom,
% 16.19/8.98 ! [F: real > nat,I6: set_real,G: real > nat,I2: real] :
% 16.19/8.98 ( ( ( groups1935376822645274424al_nat @ F @ I6 )
% 16.19/8.98 = ( groups1935376822645274424al_nat @ G @ I6 ) )
% 16.19/8.98 => ( ! [I4: real] :
% 16.19/8.98 ( ( member_real @ I4 @ I6 )
% 16.19/8.98 => ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 16.19/8.98 => ( ( member_real @ I2 @ I6 )
% 16.19/8.98 => ( ( finite_finite_real @ I6 )
% 16.19/8.98 => ( ( F @ I2 )
% 16.19/8.98 = ( G @ I2 ) ) ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5672_sum__mono__inv,axiom,
% 16.19/8.98 ! [F: int > nat,I6: set_int,G: int > nat,I2: int] :
% 16.19/8.98 ( ( ( groups4541462559716669496nt_nat @ F @ I6 )
% 16.19/8.98 = ( groups4541462559716669496nt_nat @ G @ I6 ) )
% 16.19/8.98 => ( ! [I4: int] :
% 16.19/8.98 ( ( member_int @ I4 @ I6 )
% 16.19/8.98 => ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 16.19/8.98 => ( ( member_int @ I2 @ I6 )
% 16.19/8.98 => ( ( finite_finite_int @ I6 )
% 16.19/8.98 => ( ( F @ I2 )
% 16.19/8.98 = ( G @ I2 ) ) ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5673_sum__mono__inv,axiom,
% 16.19/8.98 ! [F: complex > nat,I6: set_complex,G: complex > nat,I2: complex] :
% 16.19/8.98 ( ( ( groups5693394587270226106ex_nat @ F @ I6 )
% 16.19/8.98 = ( groups5693394587270226106ex_nat @ G @ I6 ) )
% 16.19/8.98 => ( ! [I4: complex] :
% 16.19/8.98 ( ( member_complex @ I4 @ I6 )
% 16.19/8.98 => ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 16.19/8.98 => ( ( member_complex @ I2 @ I6 )
% 16.19/8.98 => ( ( finite3207457112153483333omplex @ I6 )
% 16.19/8.98 => ( ( F @ I2 )
% 16.19/8.98 = ( G @ I2 ) ) ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5674_sum__mono__inv,axiom,
% 16.19/8.98 ! [F: real > int,I6: set_real,G: real > int,I2: real] :
% 16.19/8.98 ( ( ( groups1932886352136224148al_int @ F @ I6 )
% 16.19/8.98 = ( groups1932886352136224148al_int @ G @ I6 ) )
% 16.19/8.98 => ( ! [I4: real] :
% 16.19/8.98 ( ( member_real @ I4 @ I6 )
% 16.19/8.98 => ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 16.19/8.98 => ( ( member_real @ I2 @ I6 )
% 16.19/8.98 => ( ( finite_finite_real @ I6 )
% 16.19/8.98 => ( ( F @ I2 )
% 16.19/8.98 = ( G @ I2 ) ) ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5675_sum__mono__inv,axiom,
% 16.19/8.98 ! [F: nat > int,I6: set_nat,G: nat > int,I2: nat] :
% 16.19/8.98 ( ( ( groups3539618377306564664at_int @ F @ I6 )
% 16.19/8.98 = ( groups3539618377306564664at_int @ G @ I6 ) )
% 16.19/8.98 => ( ! [I4: nat] :
% 16.19/8.98 ( ( member_nat @ I4 @ I6 )
% 16.19/8.98 => ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 16.19/8.98 => ( ( member_nat @ I2 @ I6 )
% 16.19/8.98 => ( ( finite_finite_nat @ I6 )
% 16.19/8.98 => ( ( F @ I2 )
% 16.19/8.98 = ( G @ I2 ) ) ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5676_sum__mono__inv,axiom,
% 16.19/8.98 ! [F: complex > int,I6: set_complex,G: complex > int,I2: complex] :
% 16.19/8.98 ( ( ( groups5690904116761175830ex_int @ F @ I6 )
% 16.19/8.98 = ( groups5690904116761175830ex_int @ G @ I6 ) )
% 16.19/8.98 => ( ! [I4: complex] :
% 16.19/8.98 ( ( member_complex @ I4 @ I6 )
% 16.19/8.98 => ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
% 16.19/8.98 => ( ( member_complex @ I2 @ I6 )
% 16.19/8.98 => ( ( finite3207457112153483333omplex @ I6 )
% 16.19/8.98 => ( ( F @ I2 )
% 16.19/8.98 = ( G @ I2 ) ) ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5677_ln__less__self,axiom,
% 16.19/8.98 ! [X: real] :
% 16.19/8.98 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/8.98 => ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5678_neg__numeral__le__numeral,axiom,
% 16.19/8.98 ! [M: num,N2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5679_neg__numeral__le__numeral,axiom,
% 16.19/8.98 ! [M: num,N2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5680_neg__numeral__le__numeral,axiom,
% 16.19/8.98 ! [M: num,N2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5681_neg__numeral__le__numeral,axiom,
% 16.19/8.98 ! [M: num,N2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5682_not__numeral__le__neg__numeral,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5683_not__numeral__le__neg__numeral,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5684_not__numeral__le__neg__numeral,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5685_not__numeral__le__neg__numeral,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5686_zero__neq__neg__numeral,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( zero_zero_real
% 16.19/8.98 != ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5687_zero__neq__neg__numeral,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( zero_zero_int
% 16.19/8.98 != ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5688_zero__neq__neg__numeral,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( zero_zero_complex
% 16.19/8.98 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5689_zero__neq__neg__numeral,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( zero_zero_rat
% 16.19/8.98 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5690_zero__neq__neg__numeral,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( zero_z3403309356797280102nteger
% 16.19/8.98 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5691_neg__numeral__less__numeral,axiom,
% 16.19/8.98 ! [M: num,N2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5692_neg__numeral__less__numeral,axiom,
% 16.19/8.98 ! [M: num,N2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5693_neg__numeral__less__numeral,axiom,
% 16.19/8.98 ! [M: num,N2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5694_neg__numeral__less__numeral,axiom,
% 16.19/8.98 ! [M: num,N2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5695_not__numeral__less__neg__numeral,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5696_not__numeral__less__neg__numeral,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5697_not__numeral__less__neg__numeral,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5698_not__numeral__less__neg__numeral,axiom,
% 16.19/8.98 ! [M: num,N2: num] :
% 16.19/8.98 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5699_neg__eq__iff__add__eq__0,axiom,
% 16.19/8.98 ! [A: real,B: real] :
% 16.19/8.98 ( ( ( uminus_uminus_real @ A )
% 16.19/8.98 = B )
% 16.19/8.98 = ( ( plus_plus_real @ A @ B )
% 16.19/8.98 = zero_zero_real ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5700_neg__eq__iff__add__eq__0,axiom,
% 16.19/8.98 ! [A: int,B: int] :
% 16.19/8.98 ( ( ( uminus_uminus_int @ A )
% 16.19/8.98 = B )
% 16.19/8.98 = ( ( plus_plus_int @ A @ B )
% 16.19/8.98 = zero_zero_int ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5701_neg__eq__iff__add__eq__0,axiom,
% 16.19/8.98 ! [A: complex,B: complex] :
% 16.19/8.98 ( ( ( uminus1482373934393186551omplex @ A )
% 16.19/8.98 = B )
% 16.19/8.98 = ( ( plus_plus_complex @ A @ B )
% 16.19/8.98 = zero_zero_complex ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5702_neg__eq__iff__add__eq__0,axiom,
% 16.19/8.98 ! [A: rat,B: rat] :
% 16.19/8.98 ( ( ( uminus_uminus_rat @ A )
% 16.19/8.98 = B )
% 16.19/8.98 = ( ( plus_plus_rat @ A @ B )
% 16.19/8.98 = zero_zero_rat ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5703_neg__eq__iff__add__eq__0,axiom,
% 16.19/8.98 ! [A: code_integer,B: code_integer] :
% 16.19/8.98 ( ( ( uminus1351360451143612070nteger @ A )
% 16.19/8.98 = B )
% 16.19/8.98 = ( ( plus_p5714425477246183910nteger @ A @ B )
% 16.19/8.98 = zero_z3403309356797280102nteger ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5704_eq__neg__iff__add__eq__0,axiom,
% 16.19/8.98 ! [A: real,B: real] :
% 16.19/8.98 ( ( A
% 16.19/8.98 = ( uminus_uminus_real @ B ) )
% 16.19/8.98 = ( ( plus_plus_real @ A @ B )
% 16.19/8.98 = zero_zero_real ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5705_eq__neg__iff__add__eq__0,axiom,
% 16.19/8.98 ! [A: int,B: int] :
% 16.19/8.98 ( ( A
% 16.19/8.98 = ( uminus_uminus_int @ B ) )
% 16.19/8.98 = ( ( plus_plus_int @ A @ B )
% 16.19/8.98 = zero_zero_int ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5706_eq__neg__iff__add__eq__0,axiom,
% 16.19/8.98 ! [A: complex,B: complex] :
% 16.19/8.98 ( ( A
% 16.19/8.98 = ( uminus1482373934393186551omplex @ B ) )
% 16.19/8.98 = ( ( plus_plus_complex @ A @ B )
% 16.19/8.98 = zero_zero_complex ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5707_eq__neg__iff__add__eq__0,axiom,
% 16.19/8.98 ! [A: rat,B: rat] :
% 16.19/8.98 ( ( A
% 16.19/8.98 = ( uminus_uminus_rat @ B ) )
% 16.19/8.98 = ( ( plus_plus_rat @ A @ B )
% 16.19/8.98 = zero_zero_rat ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5708_eq__neg__iff__add__eq__0,axiom,
% 16.19/8.98 ! [A: code_integer,B: code_integer] :
% 16.19/8.98 ( ( A
% 16.19/8.98 = ( uminus1351360451143612070nteger @ B ) )
% 16.19/8.98 = ( ( plus_p5714425477246183910nteger @ A @ B )
% 16.19/8.98 = zero_z3403309356797280102nteger ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5709_add_Oinverse__unique,axiom,
% 16.19/8.98 ! [A: real,B: real] :
% 16.19/8.98 ( ( ( plus_plus_real @ A @ B )
% 16.19/8.98 = zero_zero_real )
% 16.19/8.98 => ( ( uminus_uminus_real @ A )
% 16.19/8.98 = B ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5710_add_Oinverse__unique,axiom,
% 16.19/8.98 ! [A: int,B: int] :
% 16.19/8.98 ( ( ( plus_plus_int @ A @ B )
% 16.19/8.98 = zero_zero_int )
% 16.19/8.98 => ( ( uminus_uminus_int @ A )
% 16.19/8.98 = B ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5711_add_Oinverse__unique,axiom,
% 16.19/8.98 ! [A: complex,B: complex] :
% 16.19/8.98 ( ( ( plus_plus_complex @ A @ B )
% 16.19/8.98 = zero_zero_complex )
% 16.19/8.98 => ( ( uminus1482373934393186551omplex @ A )
% 16.19/8.98 = B ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5712_add_Oinverse__unique,axiom,
% 16.19/8.98 ! [A: rat,B: rat] :
% 16.19/8.98 ( ( ( plus_plus_rat @ A @ B )
% 16.19/8.98 = zero_zero_rat )
% 16.19/8.98 => ( ( uminus_uminus_rat @ A )
% 16.19/8.98 = B ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5713_add_Oinverse__unique,axiom,
% 16.19/8.98 ! [A: code_integer,B: code_integer] :
% 16.19/8.98 ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 16.19/8.98 = zero_z3403309356797280102nteger )
% 16.19/8.98 => ( ( uminus1351360451143612070nteger @ A )
% 16.19/8.98 = B ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5714_ab__group__add__class_Oab__left__minus,axiom,
% 16.19/8.98 ! [A: real] :
% 16.19/8.98 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 16.19/8.98 = zero_zero_real ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5715_ab__group__add__class_Oab__left__minus,axiom,
% 16.19/8.98 ! [A: int] :
% 16.19/8.98 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 16.19/8.98 = zero_zero_int ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5716_ab__group__add__class_Oab__left__minus,axiom,
% 16.19/8.98 ! [A: complex] :
% 16.19/8.98 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 16.19/8.98 = zero_zero_complex ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5717_ab__group__add__class_Oab__left__minus,axiom,
% 16.19/8.98 ! [A: rat] :
% 16.19/8.98 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 16.19/8.98 = zero_zero_rat ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5718_ab__group__add__class_Oab__left__minus,axiom,
% 16.19/8.98 ! [A: code_integer] :
% 16.19/8.98 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 16.19/8.98 = zero_z3403309356797280102nteger ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5719_add__eq__0__iff,axiom,
% 16.19/8.98 ! [A: real,B: real] :
% 16.19/8.98 ( ( ( plus_plus_real @ A @ B )
% 16.19/8.98 = zero_zero_real )
% 16.19/8.98 = ( B
% 16.19/8.98 = ( uminus_uminus_real @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5720_add__eq__0__iff,axiom,
% 16.19/8.98 ! [A: int,B: int] :
% 16.19/8.98 ( ( ( plus_plus_int @ A @ B )
% 16.19/8.98 = zero_zero_int )
% 16.19/8.98 = ( B
% 16.19/8.98 = ( uminus_uminus_int @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5721_add__eq__0__iff,axiom,
% 16.19/8.98 ! [A: complex,B: complex] :
% 16.19/8.98 ( ( ( plus_plus_complex @ A @ B )
% 16.19/8.98 = zero_zero_complex )
% 16.19/8.98 = ( B
% 16.19/8.98 = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5722_add__eq__0__iff,axiom,
% 16.19/8.98 ! [A: rat,B: rat] :
% 16.19/8.98 ( ( ( plus_plus_rat @ A @ B )
% 16.19/8.98 = zero_zero_rat )
% 16.19/8.98 = ( B
% 16.19/8.98 = ( uminus_uminus_rat @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5723_add__eq__0__iff,axiom,
% 16.19/8.98 ! [A: code_integer,B: code_integer] :
% 16.19/8.98 ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 16.19/8.98 = zero_z3403309356797280102nteger )
% 16.19/8.98 = ( B
% 16.19/8.98 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5724_le__minus__one__simps_I2_J,axiom,
% 16.19/8.98 ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5725_le__minus__one__simps_I2_J,axiom,
% 16.19/8.98 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5726_le__minus__one__simps_I2_J,axiom,
% 16.19/8.98 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5727_le__minus__one__simps_I2_J,axiom,
% 16.19/8.98 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5728_le__minus__one__simps_I4_J,axiom,
% 16.19/8.98 ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5729_le__minus__one__simps_I4_J,axiom,
% 16.19/8.98 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5730_le__minus__one__simps_I4_J,axiom,
% 16.19/8.98 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5731_le__minus__one__simps_I4_J,axiom,
% 16.19/8.98 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5732_zero__neq__neg__one,axiom,
% 16.19/8.98 ( zero_zero_real
% 16.19/8.98 != ( uminus_uminus_real @ one_one_real ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5733_zero__neq__neg__one,axiom,
% 16.19/8.98 ( zero_zero_int
% 16.19/8.98 != ( uminus_uminus_int @ one_one_int ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5734_zero__neq__neg__one,axiom,
% 16.19/8.98 ( zero_zero_complex
% 16.19/8.98 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5735_zero__neq__neg__one,axiom,
% 16.19/8.98 ( zero_zero_rat
% 16.19/8.98 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5736_zero__neq__neg__one,axiom,
% 16.19/8.98 ( zero_z3403309356797280102nteger
% 16.19/8.98 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5737_numeral__times__minus__swap,axiom,
% 16.19/8.98 ! [W: num,X: real] :
% 16.19/8.98 ( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X ) )
% 16.19/8.98 = ( times_times_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5738_numeral__times__minus__swap,axiom,
% 16.19/8.98 ! [W: num,X: int] :
% 16.19/8.98 ( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X ) )
% 16.19/8.98 = ( times_times_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5739_numeral__times__minus__swap,axiom,
% 16.19/8.98 ! [W: num,X: complex] :
% 16.19/8.98 ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ ( uminus1482373934393186551omplex @ X ) )
% 16.19/8.98 = ( times_times_complex @ X @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5740_numeral__times__minus__swap,axiom,
% 16.19/8.98 ! [W: num,X: rat] :
% 16.19/8.98 ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X ) )
% 16.19/8.98 = ( times_times_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5741_numeral__times__minus__swap,axiom,
% 16.19/8.98 ! [W: num,X: code_integer] :
% 16.19/8.98 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X ) )
% 16.19/8.98 = ( times_3573771949741848930nteger @ X @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5742_less__minus__one__simps_I4_J,axiom,
% 16.19/8.98 ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5743_less__minus__one__simps_I4_J,axiom,
% 16.19/8.98 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5744_less__minus__one__simps_I4_J,axiom,
% 16.19/8.98 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5745_less__minus__one__simps_I4_J,axiom,
% 16.19/8.98 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5746_less__minus__one__simps_I2_J,axiom,
% 16.19/8.98 ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5747_less__minus__one__simps_I2_J,axiom,
% 16.19/8.98 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5748_less__minus__one__simps_I2_J,axiom,
% 16.19/8.98 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5749_less__minus__one__simps_I2_J,axiom,
% 16.19/8.98 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5750_nonzero__minus__divide__right,axiom,
% 16.19/8.98 ! [B: real,A: real] :
% 16.19/8.98 ( ( B != zero_zero_real )
% 16.19/8.98 => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 16.19/8.98 = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5751_nonzero__minus__divide__right,axiom,
% 16.19/8.98 ! [B: complex,A: complex] :
% 16.19/8.98 ( ( B != zero_zero_complex )
% 16.19/8.98 => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 16.19/8.98 = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5752_nonzero__minus__divide__right,axiom,
% 16.19/8.98 ! [B: rat,A: rat] :
% 16.19/8.98 ( ( B != zero_zero_rat )
% 16.19/8.98 => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 16.19/8.98 = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5753_nonzero__minus__divide__divide,axiom,
% 16.19/8.98 ! [B: real,A: real] :
% 16.19/8.98 ( ( B != zero_zero_real )
% 16.19/8.98 => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 16.19/8.98 = ( divide_divide_real @ A @ B ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5754_nonzero__minus__divide__divide,axiom,
% 16.19/8.98 ! [B: complex,A: complex] :
% 16.19/8.98 ( ( B != zero_zero_complex )
% 16.19/8.98 => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 16.19/8.98 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5755_nonzero__minus__divide__divide,axiom,
% 16.19/8.98 ! [B: rat,A: rat] :
% 16.19/8.98 ( ( B != zero_zero_rat )
% 16.19/8.98 => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 16.19/8.98 = ( divide_divide_rat @ A @ B ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5756_numeral__neq__neg__one,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( ( numeral_numeral_real @ N2 )
% 16.19/8.98 != ( uminus_uminus_real @ one_one_real ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5757_numeral__neq__neg__one,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( ( numeral_numeral_int @ N2 )
% 16.19/8.98 != ( uminus_uminus_int @ one_one_int ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5758_numeral__neq__neg__one,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( ( numera6690914467698888265omplex @ N2 )
% 16.19/8.98 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5759_numeral__neq__neg__one,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( ( numeral_numeral_rat @ N2 )
% 16.19/8.98 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5760_numeral__neq__neg__one,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( ( numera6620942414471956472nteger @ N2 )
% 16.19/8.98 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5761_one__neq__neg__numeral,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( one_one_real
% 16.19/8.98 != ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5762_one__neq__neg__numeral,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( one_one_int
% 16.19/8.98 != ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5763_one__neq__neg__numeral,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( one_one_complex
% 16.19/8.98 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5764_one__neq__neg__numeral,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( one_one_rat
% 16.19/8.98 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5765_one__neq__neg__numeral,axiom,
% 16.19/8.98 ! [N2: num] :
% 16.19/8.98 ( one_one_Code_integer
% 16.19/8.98 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5766_square__eq__1__iff,axiom,
% 16.19/8.98 ! [X: real] :
% 16.19/8.98 ( ( ( times_times_real @ X @ X )
% 16.19/8.98 = one_one_real )
% 16.19/8.98 = ( ( X = one_one_real )
% 16.19/8.98 | ( X
% 16.19/8.98 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5767_square__eq__1__iff,axiom,
% 16.19/8.98 ! [X: int] :
% 16.19/8.98 ( ( ( times_times_int @ X @ X )
% 16.19/8.98 = one_one_int )
% 16.19/8.98 = ( ( X = one_one_int )
% 16.19/8.98 | ( X
% 16.19/8.98 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5768_square__eq__1__iff,axiom,
% 16.19/8.98 ! [X: complex] :
% 16.19/8.98 ( ( ( times_times_complex @ X @ X )
% 16.19/8.98 = one_one_complex )
% 16.19/8.98 = ( ( X = one_one_complex )
% 16.19/8.98 | ( X
% 16.19/8.98 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5769_square__eq__1__iff,axiom,
% 16.19/8.98 ! [X: rat] :
% 16.19/8.98 ( ( ( times_times_rat @ X @ X )
% 16.19/8.98 = one_one_rat )
% 16.19/8.98 = ( ( X = one_one_rat )
% 16.19/8.98 | ( X
% 16.19/8.98 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5770_square__eq__1__iff,axiom,
% 16.19/8.98 ! [X: code_integer] :
% 16.19/8.98 ( ( ( times_3573771949741848930nteger @ X @ X )
% 16.19/8.98 = one_one_Code_integer )
% 16.19/8.98 = ( ( X = one_one_Code_integer )
% 16.19/8.98 | ( X
% 16.19/8.98 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5771_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 16.19/8.98 ( minus_minus_real
% 16.19/8.98 = ( ^ [A5: real,B5: real] : ( plus_plus_real @ A5 @ ( uminus_uminus_real @ B5 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5772_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 16.19/8.98 ( minus_minus_int
% 16.19/8.98 = ( ^ [A5: int,B5: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B5 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5773_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 16.19/8.98 ( minus_minus_complex
% 16.19/8.98 = ( ^ [A5: complex,B5: complex] : ( plus_plus_complex @ A5 @ ( uminus1482373934393186551omplex @ B5 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5774_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 16.19/8.98 ( minus_minus_rat
% 16.19/8.98 = ( ^ [A5: rat,B5: rat] : ( plus_plus_rat @ A5 @ ( uminus_uminus_rat @ B5 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5775_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 16.19/8.98 ( minus_8373710615458151222nteger
% 16.19/8.98 = ( ^ [A5: code_integer,B5: code_integer] : ( plus_p5714425477246183910nteger @ A5 @ ( uminus1351360451143612070nteger @ B5 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5776_diff__conv__add__uminus,axiom,
% 16.19/8.98 ( minus_minus_real
% 16.19/8.98 = ( ^ [A5: real,B5: real] : ( plus_plus_real @ A5 @ ( uminus_uminus_real @ B5 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5777_diff__conv__add__uminus,axiom,
% 16.19/8.98 ( minus_minus_int
% 16.19/8.98 = ( ^ [A5: int,B5: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B5 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5778_diff__conv__add__uminus,axiom,
% 16.19/8.98 ( minus_minus_complex
% 16.19/8.98 = ( ^ [A5: complex,B5: complex] : ( plus_plus_complex @ A5 @ ( uminus1482373934393186551omplex @ B5 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5779_diff__conv__add__uminus,axiom,
% 16.19/8.98 ( minus_minus_rat
% 16.19/8.98 = ( ^ [A5: rat,B5: rat] : ( plus_plus_rat @ A5 @ ( uminus_uminus_rat @ B5 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5780_diff__conv__add__uminus,axiom,
% 16.19/8.98 ( minus_8373710615458151222nteger
% 16.19/8.98 = ( ^ [A5: code_integer,B5: code_integer] : ( plus_p5714425477246183910nteger @ A5 @ ( uminus1351360451143612070nteger @ B5 ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5781_group__cancel_Osub2,axiom,
% 16.19/8.98 ! [B3: real,K: real,B: real,A: real] :
% 16.19/8.98 ( ( B3
% 16.19/8.98 = ( plus_plus_real @ K @ B ) )
% 16.19/8.98 => ( ( minus_minus_real @ A @ B3 )
% 16.19/8.98 = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5782_group__cancel_Osub2,axiom,
% 16.19/8.98 ! [B3: int,K: int,B: int,A: int] :
% 16.19/8.98 ( ( B3
% 16.19/8.98 = ( plus_plus_int @ K @ B ) )
% 16.19/8.98 => ( ( minus_minus_int @ A @ B3 )
% 16.19/8.98 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5783_group__cancel_Osub2,axiom,
% 16.19/8.98 ! [B3: complex,K: complex,B: complex,A: complex] :
% 16.19/8.98 ( ( B3
% 16.19/8.98 = ( plus_plus_complex @ K @ B ) )
% 16.19/8.98 => ( ( minus_minus_complex @ A @ B3 )
% 16.19/8.98 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5784_group__cancel_Osub2,axiom,
% 16.19/8.98 ! [B3: rat,K: rat,B: rat,A: rat] :
% 16.19/8.98 ( ( B3
% 16.19/8.98 = ( plus_plus_rat @ K @ B ) )
% 16.19/8.98 => ( ( minus_minus_rat @ A @ B3 )
% 16.19/8.98 = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5785_group__cancel_Osub2,axiom,
% 16.19/8.98 ! [B3: code_integer,K: code_integer,B: code_integer,A: code_integer] :
% 16.19/8.98 ( ( B3
% 16.19/8.98 = ( plus_p5714425477246183910nteger @ K @ B ) )
% 16.19/8.98 => ( ( minus_8373710615458151222nteger @ A @ B3 )
% 16.19/8.98 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5786_dvd__div__neg,axiom,
% 16.19/8.98 ! [B: real,A: real] :
% 16.19/8.98 ( ( dvd_dvd_real @ B @ A )
% 16.19/8.98 => ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) )
% 16.19/8.98 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5787_dvd__div__neg,axiom,
% 16.19/8.98 ! [B: int,A: int] :
% 16.19/8.98 ( ( dvd_dvd_int @ B @ A )
% 16.19/8.98 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 16.19/8.98 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5788_dvd__div__neg,axiom,
% 16.19/8.98 ! [B: complex,A: complex] :
% 16.19/8.98 ( ( dvd_dvd_complex @ B @ A )
% 16.19/8.98 => ( ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 16.19/8.98 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5789_dvd__div__neg,axiom,
% 16.19/8.98 ! [B: rat,A: rat] :
% 16.19/8.98 ( ( dvd_dvd_rat @ B @ A )
% 16.19/8.98 => ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) )
% 16.19/8.98 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5790_dvd__div__neg,axiom,
% 16.19/8.98 ! [B: code_integer,A: code_integer] :
% 16.19/8.98 ( ( dvd_dvd_Code_integer @ B @ A )
% 16.19/8.98 => ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 16.19/8.98 = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5791_dvd__neg__div,axiom,
% 16.19/8.98 ! [B: real,A: real] :
% 16.19/8.98 ( ( dvd_dvd_real @ B @ A )
% 16.19/8.98 => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B )
% 16.19/8.98 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5792_dvd__neg__div,axiom,
% 16.19/8.98 ! [B: int,A: int] :
% 16.19/8.98 ( ( dvd_dvd_int @ B @ A )
% 16.19/8.98 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 16.19/8.98 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5793_dvd__neg__div,axiom,
% 16.19/8.98 ! [B: complex,A: complex] :
% 16.19/8.98 ( ( dvd_dvd_complex @ B @ A )
% 16.19/8.98 => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 16.19/8.98 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5794_dvd__neg__div,axiom,
% 16.19/8.98 ! [B: rat,A: rat] :
% 16.19/8.98 ( ( dvd_dvd_rat @ B @ A )
% 16.19/8.98 => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B )
% 16.19/8.98 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5795_dvd__neg__div,axiom,
% 16.19/8.98 ! [B: code_integer,A: code_integer] :
% 16.19/8.98 ( ( dvd_dvd_Code_integer @ B @ A )
% 16.19/8.98 => ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 16.19/8.98 = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5796_subset__Compl__self__eq,axiom,
% 16.19/8.98 ! [A2: set_int] :
% 16.19/8.98 ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ A2 ) )
% 16.19/8.98 = ( A2 = bot_bot_set_int ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5797_subset__Compl__self__eq,axiom,
% 16.19/8.98 ! [A2: set_real] :
% 16.19/8.98 ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ A2 ) )
% 16.19/8.98 = ( A2 = bot_bot_set_real ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5798_subset__Compl__self__eq,axiom,
% 16.19/8.98 ! [A2: set_nat] :
% 16.19/8.98 ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
% 16.19/8.98 = ( A2 = bot_bot_set_nat ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5799_real__minus__mult__self__le,axiom,
% 16.19/8.98 ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5800_zmult__eq__1__iff,axiom,
% 16.19/8.98 ! [M: int,N2: int] :
% 16.19/8.98 ( ( ( times_times_int @ M @ N2 )
% 16.19/8.98 = one_one_int )
% 16.19/8.98 = ( ( ( M = one_one_int )
% 16.19/8.98 & ( N2 = one_one_int ) )
% 16.19/8.98 | ( ( M
% 16.19/8.98 = ( uminus_uminus_int @ one_one_int ) )
% 16.19/8.98 & ( N2
% 16.19/8.98 = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5801_pos__zmult__eq__1__iff__lemma,axiom,
% 16.19/8.98 ! [M: int,N2: int] :
% 16.19/8.98 ( ( ( times_times_int @ M @ N2 )
% 16.19/8.98 = one_one_int )
% 16.19/8.98 => ( ( M = one_one_int )
% 16.19/8.98 | ( M
% 16.19/8.98 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5802_minus__int__code_I2_J,axiom,
% 16.19/8.98 ! [L2: int] :
% 16.19/8.98 ( ( minus_minus_int @ zero_zero_int @ L2 )
% 16.19/8.98 = ( uminus_uminus_int @ L2 ) ) ).
% 16.19/8.98
% 16.19/8.98 thf(fact_5803_sum_Ointer__filter,axiom,
% 16.19/8.98 ! [A2: set_real,G: real > complex,P: real > $o] :
% 16.19/8.98 ( ( finite_finite_real @ A2 )
% 16.19/8.98 => ( ( groups5754745047067104278omplex @ G
% 16.19/8.98 @ ( collect_real
% 16.19/8.99 @ ^ [X2: real] :
% 16.19/8.99 ( ( member_real @ X2 @ A2 )
% 16.19/8.99 & ( P @ X2 ) ) ) )
% 16.19/8.99 = ( groups5754745047067104278omplex
% 16.19/8.99 @ ^ [X2: real] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_complex )
% 16.19/8.99 @ A2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5804_sum_Ointer__filter,axiom,
% 16.19/8.99 ! [A2: set_nat,G: nat > complex,P: nat > $o] :
% 16.19/8.99 ( ( finite_finite_nat @ A2 )
% 16.19/8.99 => ( ( groups2073611262835488442omplex @ G
% 16.19/8.99 @ ( collect_nat
% 16.19/8.99 @ ^ [X2: nat] :
% 16.19/8.99 ( ( member_nat @ X2 @ A2 )
% 16.19/8.99 & ( P @ X2 ) ) ) )
% 16.19/8.99 = ( groups2073611262835488442omplex
% 16.19/8.99 @ ^ [X2: nat] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_complex )
% 16.19/8.99 @ A2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5805_sum_Ointer__filter,axiom,
% 16.19/8.99 ! [A2: set_int,G: int > complex,P: int > $o] :
% 16.19/8.99 ( ( finite_finite_int @ A2 )
% 16.19/8.99 => ( ( groups3049146728041665814omplex @ G
% 16.19/8.99 @ ( collect_int
% 16.19/8.99 @ ^ [X2: int] :
% 16.19/8.99 ( ( member_int @ X2 @ A2 )
% 16.19/8.99 & ( P @ X2 ) ) ) )
% 16.19/8.99 = ( groups3049146728041665814omplex
% 16.19/8.99 @ ^ [X2: int] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_complex )
% 16.19/8.99 @ A2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5806_sum_Ointer__filter,axiom,
% 16.19/8.99 ! [A2: set_real,G: real > real,P: real > $o] :
% 16.19/8.99 ( ( finite_finite_real @ A2 )
% 16.19/8.99 => ( ( groups8097168146408367636l_real @ G
% 16.19/8.99 @ ( collect_real
% 16.19/8.99 @ ^ [X2: real] :
% 16.19/8.99 ( ( member_real @ X2 @ A2 )
% 16.19/8.99 & ( P @ X2 ) ) ) )
% 16.19/8.99 = ( groups8097168146408367636l_real
% 16.19/8.99 @ ^ [X2: real] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_real )
% 16.19/8.99 @ A2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5807_sum_Ointer__filter,axiom,
% 16.19/8.99 ! [A2: set_int,G: int > real,P: int > $o] :
% 16.19/8.99 ( ( finite_finite_int @ A2 )
% 16.19/8.99 => ( ( groups8778361861064173332t_real @ G
% 16.19/8.99 @ ( collect_int
% 16.19/8.99 @ ^ [X2: int] :
% 16.19/8.99 ( ( member_int @ X2 @ A2 )
% 16.19/8.99 & ( P @ X2 ) ) ) )
% 16.19/8.99 = ( groups8778361861064173332t_real
% 16.19/8.99 @ ^ [X2: int] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_real )
% 16.19/8.99 @ A2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5808_sum_Ointer__filter,axiom,
% 16.19/8.99 ! [A2: set_complex,G: complex > real,P: complex > $o] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.99 => ( ( groups5808333547571424918x_real @ G
% 16.19/8.99 @ ( collect_complex
% 16.19/8.99 @ ^ [X2: complex] :
% 16.19/8.99 ( ( member_complex @ X2 @ A2 )
% 16.19/8.99 & ( P @ X2 ) ) ) )
% 16.19/8.99 = ( groups5808333547571424918x_real
% 16.19/8.99 @ ^ [X2: complex] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_real )
% 16.19/8.99 @ A2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5809_sum_Ointer__filter,axiom,
% 16.19/8.99 ! [A2: set_real,G: real > rat,P: real > $o] :
% 16.19/8.99 ( ( finite_finite_real @ A2 )
% 16.19/8.99 => ( ( groups1300246762558778688al_rat @ G
% 16.19/8.99 @ ( collect_real
% 16.19/8.99 @ ^ [X2: real] :
% 16.19/8.99 ( ( member_real @ X2 @ A2 )
% 16.19/8.99 & ( P @ X2 ) ) ) )
% 16.19/8.99 = ( groups1300246762558778688al_rat
% 16.19/8.99 @ ^ [X2: real] : ( if_rat @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_rat )
% 16.19/8.99 @ A2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5810_sum_Ointer__filter,axiom,
% 16.19/8.99 ! [A2: set_nat,G: nat > rat,P: nat > $o] :
% 16.19/8.99 ( ( finite_finite_nat @ A2 )
% 16.19/8.99 => ( ( groups2906978787729119204at_rat @ G
% 16.19/8.99 @ ( collect_nat
% 16.19/8.99 @ ^ [X2: nat] :
% 16.19/8.99 ( ( member_nat @ X2 @ A2 )
% 16.19/8.99 & ( P @ X2 ) ) ) )
% 16.19/8.99 = ( groups2906978787729119204at_rat
% 16.19/8.99 @ ^ [X2: nat] : ( if_rat @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_rat )
% 16.19/8.99 @ A2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5811_sum_Ointer__filter,axiom,
% 16.19/8.99 ! [A2: set_int,G: int > rat,P: int > $o] :
% 16.19/8.99 ( ( finite_finite_int @ A2 )
% 16.19/8.99 => ( ( groups3906332499630173760nt_rat @ G
% 16.19/8.99 @ ( collect_int
% 16.19/8.99 @ ^ [X2: int] :
% 16.19/8.99 ( ( member_int @ X2 @ A2 )
% 16.19/8.99 & ( P @ X2 ) ) ) )
% 16.19/8.99 = ( groups3906332499630173760nt_rat
% 16.19/8.99 @ ^ [X2: int] : ( if_rat @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_rat )
% 16.19/8.99 @ A2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5812_sum_Ointer__filter,axiom,
% 16.19/8.99 ! [A2: set_complex,G: complex > rat,P: complex > $o] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.99 => ( ( groups5058264527183730370ex_rat @ G
% 16.19/8.99 @ ( collect_complex
% 16.19/8.99 @ ^ [X2: complex] :
% 16.19/8.99 ( ( member_complex @ X2 @ A2 )
% 16.19/8.99 & ( P @ X2 ) ) ) )
% 16.19/8.99 = ( groups5058264527183730370ex_rat
% 16.19/8.99 @ ^ [X2: complex] : ( if_rat @ ( P @ X2 ) @ ( G @ X2 ) @ zero_zero_rat )
% 16.19/8.99 @ A2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5813_minus__real__def,axiom,
% 16.19/8.99 ( minus_minus_real
% 16.19/8.99 = ( ^ [X2: real,Y2: real] : ( plus_plus_real @ X2 @ ( uminus_uminus_real @ Y2 ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5814_ln__one__minus__pos__upper__bound,axiom,
% 16.19/8.99 ! [X: real] :
% 16.19/8.99 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/8.99 => ( ( ord_less_real @ X @ one_one_real )
% 16.19/8.99 => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) @ ( uminus_uminus_real @ X ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5815_sum__nonneg__eq__0__iff,axiom,
% 16.19/8.99 ! [A2: set_real,F: real > real] :
% 16.19/8.99 ( ( finite_finite_real @ A2 )
% 16.19/8.99 => ( ! [X4: real] :
% 16.19/8.99 ( ( member_real @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 16.19/8.99 => ( ( ( groups8097168146408367636l_real @ F @ A2 )
% 16.19/8.99 = zero_zero_real )
% 16.19/8.99 = ( ! [X2: real] :
% 16.19/8.99 ( ( member_real @ X2 @ A2 )
% 16.19/8.99 => ( ( F @ X2 )
% 16.19/8.99 = zero_zero_real ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5816_sum__nonneg__eq__0__iff,axiom,
% 16.19/8.99 ! [A2: set_int,F: int > real] :
% 16.19/8.99 ( ( finite_finite_int @ A2 )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 16.19/8.99 => ( ( ( groups8778361861064173332t_real @ F @ A2 )
% 16.19/8.99 = zero_zero_real )
% 16.19/8.99 = ( ! [X2: int] :
% 16.19/8.99 ( ( member_int @ X2 @ A2 )
% 16.19/8.99 => ( ( F @ X2 )
% 16.19/8.99 = zero_zero_real ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5817_sum__nonneg__eq__0__iff,axiom,
% 16.19/8.99 ! [A2: set_complex,F: complex > real] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 16.19/8.99 => ( ( ( groups5808333547571424918x_real @ F @ A2 )
% 16.19/8.99 = zero_zero_real )
% 16.19/8.99 = ( ! [X2: complex] :
% 16.19/8.99 ( ( member_complex @ X2 @ A2 )
% 16.19/8.99 => ( ( F @ X2 )
% 16.19/8.99 = zero_zero_real ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5818_sum__nonneg__eq__0__iff,axiom,
% 16.19/8.99 ! [A2: set_real,F: real > rat] :
% 16.19/8.99 ( ( finite_finite_real @ A2 )
% 16.19/8.99 => ( ! [X4: real] :
% 16.19/8.99 ( ( member_real @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 16.19/8.99 => ( ( ( groups1300246762558778688al_rat @ F @ A2 )
% 16.19/8.99 = zero_zero_rat )
% 16.19/8.99 = ( ! [X2: real] :
% 16.19/8.99 ( ( member_real @ X2 @ A2 )
% 16.19/8.99 => ( ( F @ X2 )
% 16.19/8.99 = zero_zero_rat ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5819_sum__nonneg__eq__0__iff,axiom,
% 16.19/8.99 ! [A2: set_nat,F: nat > rat] :
% 16.19/8.99 ( ( finite_finite_nat @ A2 )
% 16.19/8.99 => ( ! [X4: nat] :
% 16.19/8.99 ( ( member_nat @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 16.19/8.99 => ( ( ( groups2906978787729119204at_rat @ F @ A2 )
% 16.19/8.99 = zero_zero_rat )
% 16.19/8.99 = ( ! [X2: nat] :
% 16.19/8.99 ( ( member_nat @ X2 @ A2 )
% 16.19/8.99 => ( ( F @ X2 )
% 16.19/8.99 = zero_zero_rat ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5820_sum__nonneg__eq__0__iff,axiom,
% 16.19/8.99 ! [A2: set_int,F: int > rat] :
% 16.19/8.99 ( ( finite_finite_int @ A2 )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 16.19/8.99 => ( ( ( groups3906332499630173760nt_rat @ F @ A2 )
% 16.19/8.99 = zero_zero_rat )
% 16.19/8.99 = ( ! [X2: int] :
% 16.19/8.99 ( ( member_int @ X2 @ A2 )
% 16.19/8.99 => ( ( F @ X2 )
% 16.19/8.99 = zero_zero_rat ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5821_sum__nonneg__eq__0__iff,axiom,
% 16.19/8.99 ! [A2: set_complex,F: complex > rat] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 16.19/8.99 => ( ( ( groups5058264527183730370ex_rat @ F @ A2 )
% 16.19/8.99 = zero_zero_rat )
% 16.19/8.99 = ( ! [X2: complex] :
% 16.19/8.99 ( ( member_complex @ X2 @ A2 )
% 16.19/8.99 => ( ( F @ X2 )
% 16.19/8.99 = zero_zero_rat ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5822_sum__nonneg__eq__0__iff,axiom,
% 16.19/8.99 ! [A2: set_real,F: real > nat] :
% 16.19/8.99 ( ( finite_finite_real @ A2 )
% 16.19/8.99 => ( ! [X4: real] :
% 16.19/8.99 ( ( member_real @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 16.19/8.99 => ( ( ( groups1935376822645274424al_nat @ F @ A2 )
% 16.19/8.99 = zero_zero_nat )
% 16.19/8.99 = ( ! [X2: real] :
% 16.19/8.99 ( ( member_real @ X2 @ A2 )
% 16.19/8.99 => ( ( F @ X2 )
% 16.19/8.99 = zero_zero_nat ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5823_sum__nonneg__eq__0__iff,axiom,
% 16.19/8.99 ! [A2: set_int,F: int > nat] :
% 16.19/8.99 ( ( finite_finite_int @ A2 )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 16.19/8.99 => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
% 16.19/8.99 = zero_zero_nat )
% 16.19/8.99 = ( ! [X2: int] :
% 16.19/8.99 ( ( member_int @ X2 @ A2 )
% 16.19/8.99 => ( ( F @ X2 )
% 16.19/8.99 = zero_zero_nat ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5824_sum__nonneg__eq__0__iff,axiom,
% 16.19/8.99 ! [A2: set_complex,F: complex > nat] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 16.19/8.99 => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
% 16.19/8.99 = zero_zero_nat )
% 16.19/8.99 = ( ! [X2: complex] :
% 16.19/8.99 ( ( member_complex @ X2 @ A2 )
% 16.19/8.99 => ( ( F @ X2 )
% 16.19/8.99 = zero_zero_nat ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5825_sum__le__included,axiom,
% 16.19/8.99 ! [S3: set_int,T: set_int,G: int > real,I2: int > int,F: int > real] :
% 16.19/8.99 ( ( finite_finite_int @ S3 )
% 16.19/8.99 => ( ( finite_finite_int @ T )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ T )
% 16.19/8.99 => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ S3 )
% 16.19/8.99 => ? [Xa: int] :
% 16.19/8.99 ( ( member_int @ Xa @ T )
% 16.19/8.99 & ( ( I2 @ Xa )
% 16.19/8.99 = X4 )
% 16.19/8.99 & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 16.19/8.99 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S3 ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5826_sum__le__included,axiom,
% 16.19/8.99 ! [S3: set_int,T: set_complex,G: complex > real,I2: complex > int,F: int > real] :
% 16.19/8.99 ( ( finite_finite_int @ S3 )
% 16.19/8.99 => ( ( finite3207457112153483333omplex @ T )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ T )
% 16.19/8.99 => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ S3 )
% 16.19/8.99 => ? [Xa: complex] :
% 16.19/8.99 ( ( member_complex @ Xa @ T )
% 16.19/8.99 & ( ( I2 @ Xa )
% 16.19/8.99 = X4 )
% 16.19/8.99 & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 16.19/8.99 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S3 ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5827_sum__le__included,axiom,
% 16.19/8.99 ! [S3: set_complex,T: set_int,G: int > real,I2: int > complex,F: complex > real] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ S3 )
% 16.19/8.99 => ( ( finite_finite_int @ T )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ T )
% 16.19/8.99 => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ S3 )
% 16.19/8.99 => ? [Xa: int] :
% 16.19/8.99 ( ( member_int @ Xa @ T )
% 16.19/8.99 & ( ( I2 @ Xa )
% 16.19/8.99 = X4 )
% 16.19/8.99 & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 16.19/8.99 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S3 ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5828_sum__le__included,axiom,
% 16.19/8.99 ! [S3: set_complex,T: set_complex,G: complex > real,I2: complex > complex,F: complex > real] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ S3 )
% 16.19/8.99 => ( ( finite3207457112153483333omplex @ T )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ T )
% 16.19/8.99 => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ S3 )
% 16.19/8.99 => ? [Xa: complex] :
% 16.19/8.99 ( ( member_complex @ Xa @ T )
% 16.19/8.99 & ( ( I2 @ Xa )
% 16.19/8.99 = X4 )
% 16.19/8.99 & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 16.19/8.99 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S3 ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5829_sum__le__included,axiom,
% 16.19/8.99 ! [S3: set_nat,T: set_nat,G: nat > rat,I2: nat > nat,F: nat > rat] :
% 16.19/8.99 ( ( finite_finite_nat @ S3 )
% 16.19/8.99 => ( ( finite_finite_nat @ T )
% 16.19/8.99 => ( ! [X4: nat] :
% 16.19/8.99 ( ( member_nat @ X4 @ T )
% 16.19/8.99 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ! [X4: nat] :
% 16.19/8.99 ( ( member_nat @ X4 @ S3 )
% 16.19/8.99 => ? [Xa: nat] :
% 16.19/8.99 ( ( member_nat @ Xa @ T )
% 16.19/8.99 & ( ( I2 @ Xa )
% 16.19/8.99 = X4 )
% 16.19/8.99 & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 16.19/8.99 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S3 ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5830_sum__le__included,axiom,
% 16.19/8.99 ! [S3: set_nat,T: set_int,G: int > rat,I2: int > nat,F: nat > rat] :
% 16.19/8.99 ( ( finite_finite_nat @ S3 )
% 16.19/8.99 => ( ( finite_finite_int @ T )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ T )
% 16.19/8.99 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ! [X4: nat] :
% 16.19/8.99 ( ( member_nat @ X4 @ S3 )
% 16.19/8.99 => ? [Xa: int] :
% 16.19/8.99 ( ( member_int @ Xa @ T )
% 16.19/8.99 & ( ( I2 @ Xa )
% 16.19/8.99 = X4 )
% 16.19/8.99 & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 16.19/8.99 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S3 ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5831_sum__le__included,axiom,
% 16.19/8.99 ! [S3: set_nat,T: set_complex,G: complex > rat,I2: complex > nat,F: nat > rat] :
% 16.19/8.99 ( ( finite_finite_nat @ S3 )
% 16.19/8.99 => ( ( finite3207457112153483333omplex @ T )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ T )
% 16.19/8.99 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ! [X4: nat] :
% 16.19/8.99 ( ( member_nat @ X4 @ S3 )
% 16.19/8.99 => ? [Xa: complex] :
% 16.19/8.99 ( ( member_complex @ Xa @ T )
% 16.19/8.99 & ( ( I2 @ Xa )
% 16.19/8.99 = X4 )
% 16.19/8.99 & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 16.19/8.99 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S3 ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5832_sum__le__included,axiom,
% 16.19/8.99 ! [S3: set_int,T: set_nat,G: nat > rat,I2: nat > int,F: int > rat] :
% 16.19/8.99 ( ( finite_finite_int @ S3 )
% 16.19/8.99 => ( ( finite_finite_nat @ T )
% 16.19/8.99 => ( ! [X4: nat] :
% 16.19/8.99 ( ( member_nat @ X4 @ T )
% 16.19/8.99 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ S3 )
% 16.19/8.99 => ? [Xa: nat] :
% 16.19/8.99 ( ( member_nat @ Xa @ T )
% 16.19/8.99 & ( ( I2 @ Xa )
% 16.19/8.99 = X4 )
% 16.19/8.99 & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 16.19/8.99 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S3 ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5833_sum__le__included,axiom,
% 16.19/8.99 ! [S3: set_int,T: set_int,G: int > rat,I2: int > int,F: int > rat] :
% 16.19/8.99 ( ( finite_finite_int @ S3 )
% 16.19/8.99 => ( ( finite_finite_int @ T )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ T )
% 16.19/8.99 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ S3 )
% 16.19/8.99 => ? [Xa: int] :
% 16.19/8.99 ( ( member_int @ Xa @ T )
% 16.19/8.99 & ( ( I2 @ Xa )
% 16.19/8.99 = X4 )
% 16.19/8.99 & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 16.19/8.99 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S3 ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5834_sum__le__included,axiom,
% 16.19/8.99 ! [S3: set_int,T: set_complex,G: complex > rat,I2: complex > int,F: int > rat] :
% 16.19/8.99 ( ( finite_finite_int @ S3 )
% 16.19/8.99 => ( ( finite3207457112153483333omplex @ T )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ T )
% 16.19/8.99 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ S3 )
% 16.19/8.99 => ? [Xa: complex] :
% 16.19/8.99 ( ( member_complex @ Xa @ T )
% 16.19/8.99 & ( ( I2 @ Xa )
% 16.19/8.99 = X4 )
% 16.19/8.99 & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 16.19/8.99 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S3 ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5835_sum__strict__mono__ex1,axiom,
% 16.19/8.99 ! [A2: set_int,F: int > real,G: int > real] :
% 16.19/8.99 ( ( finite_finite_int @ A2 )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ? [X5: int] :
% 16.19/8.99 ( ( member_int @ X5 @ A2 )
% 16.19/8.99 & ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 16.19/8.99 => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5836_sum__strict__mono__ex1,axiom,
% 16.19/8.99 ! [A2: set_complex,F: complex > real,G: complex > real] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ? [X5: complex] :
% 16.19/8.99 ( ( member_complex @ X5 @ A2 )
% 16.19/8.99 & ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 16.19/8.99 => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5837_sum__strict__mono__ex1,axiom,
% 16.19/8.99 ! [A2: set_nat,F: nat > rat,G: nat > rat] :
% 16.19/8.99 ( ( finite_finite_nat @ A2 )
% 16.19/8.99 => ( ! [X4: nat] :
% 16.19/8.99 ( ( member_nat @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ? [X5: nat] :
% 16.19/8.99 ( ( member_nat @ X5 @ A2 )
% 16.19/8.99 & ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 16.19/8.99 => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5838_sum__strict__mono__ex1,axiom,
% 16.19/8.99 ! [A2: set_int,F: int > rat,G: int > rat] :
% 16.19/8.99 ( ( finite_finite_int @ A2 )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ? [X5: int] :
% 16.19/8.99 ( ( member_int @ X5 @ A2 )
% 16.19/8.99 & ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 16.19/8.99 => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5839_sum__strict__mono__ex1,axiom,
% 16.19/8.99 ! [A2: set_complex,F: complex > rat,G: complex > rat] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ? [X5: complex] :
% 16.19/8.99 ( ( member_complex @ X5 @ A2 )
% 16.19/8.99 & ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 16.19/8.99 => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5840_sum__strict__mono__ex1,axiom,
% 16.19/8.99 ! [A2: set_int,F: int > nat,G: int > nat] :
% 16.19/8.99 ( ( finite_finite_int @ A2 )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ? [X5: int] :
% 16.19/8.99 ( ( member_int @ X5 @ A2 )
% 16.19/8.99 & ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 16.19/8.99 => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5841_sum__strict__mono__ex1,axiom,
% 16.19/8.99 ! [A2: set_complex,F: complex > nat,G: complex > nat] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ? [X5: complex] :
% 16.19/8.99 ( ( member_complex @ X5 @ A2 )
% 16.19/8.99 & ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 16.19/8.99 => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5842_sum__strict__mono__ex1,axiom,
% 16.19/8.99 ! [A2: set_nat,F: nat > int,G: nat > int] :
% 16.19/8.99 ( ( finite_finite_nat @ A2 )
% 16.19/8.99 => ( ! [X4: nat] :
% 16.19/8.99 ( ( member_nat @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ? [X5: nat] :
% 16.19/8.99 ( ( member_nat @ X5 @ A2 )
% 16.19/8.99 & ( ord_less_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 16.19/8.99 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ G @ A2 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5843_sum__strict__mono__ex1,axiom,
% 16.19/8.99 ! [A2: set_complex,F: complex > int,G: complex > int] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ? [X5: complex] :
% 16.19/8.99 ( ( member_complex @ X5 @ A2 )
% 16.19/8.99 & ( ord_less_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 16.19/8.99 => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ G @ A2 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5844_sum__strict__mono__ex1,axiom,
% 16.19/8.99 ! [A2: set_int,F: int > int,G: int > int] :
% 16.19/8.99 ( ( finite_finite_int @ A2 )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ? [X5: int] :
% 16.19/8.99 ( ( member_int @ X5 @ A2 )
% 16.19/8.99 & ( ord_less_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 16.19/8.99 => ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5845_sum_Orelated,axiom,
% 16.19/8.99 ! [R2: complex > complex > $o,S: set_nat,H2: nat > complex,G: nat > complex] :
% 16.19/8.99 ( ( R2 @ zero_zero_complex @ zero_zero_complex )
% 16.19/8.99 => ( ! [X16: complex,Y15: complex,X23: complex,Y23: complex] :
% 16.19/8.99 ( ( ( R2 @ X16 @ X23 )
% 16.19/8.99 & ( R2 @ Y15 @ Y23 ) )
% 16.19/8.99 => ( R2 @ ( plus_plus_complex @ X16 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
% 16.19/8.99 => ( ( finite_finite_nat @ S )
% 16.19/8.99 => ( ! [X4: nat] :
% 16.19/8.99 ( ( member_nat @ X4 @ S )
% 16.19/8.99 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( R2 @ ( groups2073611262835488442omplex @ H2 @ S ) @ ( groups2073611262835488442omplex @ G @ S ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5846_sum_Orelated,axiom,
% 16.19/8.99 ! [R2: complex > complex > $o,S: set_int,H2: int > complex,G: int > complex] :
% 16.19/8.99 ( ( R2 @ zero_zero_complex @ zero_zero_complex )
% 16.19/8.99 => ( ! [X16: complex,Y15: complex,X23: complex,Y23: complex] :
% 16.19/8.99 ( ( ( R2 @ X16 @ X23 )
% 16.19/8.99 & ( R2 @ Y15 @ Y23 ) )
% 16.19/8.99 => ( R2 @ ( plus_plus_complex @ X16 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
% 16.19/8.99 => ( ( finite_finite_int @ S )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ S )
% 16.19/8.99 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( R2 @ ( groups3049146728041665814omplex @ H2 @ S ) @ ( groups3049146728041665814omplex @ G @ S ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5847_sum_Orelated,axiom,
% 16.19/8.99 ! [R2: real > real > $o,S: set_int,H2: int > real,G: int > real] :
% 16.19/8.99 ( ( R2 @ zero_zero_real @ zero_zero_real )
% 16.19/8.99 => ( ! [X16: real,Y15: real,X23: real,Y23: real] :
% 16.19/8.99 ( ( ( R2 @ X16 @ X23 )
% 16.19/8.99 & ( R2 @ Y15 @ Y23 ) )
% 16.19/8.99 => ( R2 @ ( plus_plus_real @ X16 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
% 16.19/8.99 => ( ( finite_finite_int @ S )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ S )
% 16.19/8.99 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( R2 @ ( groups8778361861064173332t_real @ H2 @ S ) @ ( groups8778361861064173332t_real @ G @ S ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5848_sum_Orelated,axiom,
% 16.19/8.99 ! [R2: real > real > $o,S: set_complex,H2: complex > real,G: complex > real] :
% 16.19/8.99 ( ( R2 @ zero_zero_real @ zero_zero_real )
% 16.19/8.99 => ( ! [X16: real,Y15: real,X23: real,Y23: real] :
% 16.19/8.99 ( ( ( R2 @ X16 @ X23 )
% 16.19/8.99 & ( R2 @ Y15 @ Y23 ) )
% 16.19/8.99 => ( R2 @ ( plus_plus_real @ X16 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
% 16.19/8.99 => ( ( finite3207457112153483333omplex @ S )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ S )
% 16.19/8.99 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( R2 @ ( groups5808333547571424918x_real @ H2 @ S ) @ ( groups5808333547571424918x_real @ G @ S ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5849_sum_Orelated,axiom,
% 16.19/8.99 ! [R2: rat > rat > $o,S: set_nat,H2: nat > rat,G: nat > rat] :
% 16.19/8.99 ( ( R2 @ zero_zero_rat @ zero_zero_rat )
% 16.19/8.99 => ( ! [X16: rat,Y15: rat,X23: rat,Y23: rat] :
% 16.19/8.99 ( ( ( R2 @ X16 @ X23 )
% 16.19/8.99 & ( R2 @ Y15 @ Y23 ) )
% 16.19/8.99 => ( R2 @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
% 16.19/8.99 => ( ( finite_finite_nat @ S )
% 16.19/8.99 => ( ! [X4: nat] :
% 16.19/8.99 ( ( member_nat @ X4 @ S )
% 16.19/8.99 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( R2 @ ( groups2906978787729119204at_rat @ H2 @ S ) @ ( groups2906978787729119204at_rat @ G @ S ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5850_sum_Orelated,axiom,
% 16.19/8.99 ! [R2: rat > rat > $o,S: set_int,H2: int > rat,G: int > rat] :
% 16.19/8.99 ( ( R2 @ zero_zero_rat @ zero_zero_rat )
% 16.19/8.99 => ( ! [X16: rat,Y15: rat,X23: rat,Y23: rat] :
% 16.19/8.99 ( ( ( R2 @ X16 @ X23 )
% 16.19/8.99 & ( R2 @ Y15 @ Y23 ) )
% 16.19/8.99 => ( R2 @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
% 16.19/8.99 => ( ( finite_finite_int @ S )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ S )
% 16.19/8.99 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( R2 @ ( groups3906332499630173760nt_rat @ H2 @ S ) @ ( groups3906332499630173760nt_rat @ G @ S ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5851_sum_Orelated,axiom,
% 16.19/8.99 ! [R2: rat > rat > $o,S: set_complex,H2: complex > rat,G: complex > rat] :
% 16.19/8.99 ( ( R2 @ zero_zero_rat @ zero_zero_rat )
% 16.19/8.99 => ( ! [X16: rat,Y15: rat,X23: rat,Y23: rat] :
% 16.19/8.99 ( ( ( R2 @ X16 @ X23 )
% 16.19/8.99 & ( R2 @ Y15 @ Y23 ) )
% 16.19/8.99 => ( R2 @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
% 16.19/8.99 => ( ( finite3207457112153483333omplex @ S )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ S )
% 16.19/8.99 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( R2 @ ( groups5058264527183730370ex_rat @ H2 @ S ) @ ( groups5058264527183730370ex_rat @ G @ S ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5852_sum_Orelated,axiom,
% 16.19/8.99 ! [R2: nat > nat > $o,S: set_int,H2: int > nat,G: int > nat] :
% 16.19/8.99 ( ( R2 @ zero_zero_nat @ zero_zero_nat )
% 16.19/8.99 => ( ! [X16: nat,Y15: nat,X23: nat,Y23: nat] :
% 16.19/8.99 ( ( ( R2 @ X16 @ X23 )
% 16.19/8.99 & ( R2 @ Y15 @ Y23 ) )
% 16.19/8.99 => ( R2 @ ( plus_plus_nat @ X16 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
% 16.19/8.99 => ( ( finite_finite_int @ S )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ S )
% 16.19/8.99 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( R2 @ ( groups4541462559716669496nt_nat @ H2 @ S ) @ ( groups4541462559716669496nt_nat @ G @ S ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5853_sum_Orelated,axiom,
% 16.19/8.99 ! [R2: nat > nat > $o,S: set_complex,H2: complex > nat,G: complex > nat] :
% 16.19/8.99 ( ( R2 @ zero_zero_nat @ zero_zero_nat )
% 16.19/8.99 => ( ! [X16: nat,Y15: nat,X23: nat,Y23: nat] :
% 16.19/8.99 ( ( ( R2 @ X16 @ X23 )
% 16.19/8.99 & ( R2 @ Y15 @ Y23 ) )
% 16.19/8.99 => ( R2 @ ( plus_plus_nat @ X16 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
% 16.19/8.99 => ( ( finite3207457112153483333omplex @ S )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ S )
% 16.19/8.99 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( R2 @ ( groups5693394587270226106ex_nat @ H2 @ S ) @ ( groups5693394587270226106ex_nat @ G @ S ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5854_sum_Orelated,axiom,
% 16.19/8.99 ! [R2: int > int > $o,S: set_nat,H2: nat > int,G: nat > int] :
% 16.19/8.99 ( ( R2 @ zero_zero_int @ zero_zero_int )
% 16.19/8.99 => ( ! [X16: int,Y15: int,X23: int,Y23: int] :
% 16.19/8.99 ( ( ( R2 @ X16 @ X23 )
% 16.19/8.99 & ( R2 @ Y15 @ Y23 ) )
% 16.19/8.99 => ( R2 @ ( plus_plus_int @ X16 @ Y15 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
% 16.19/8.99 => ( ( finite_finite_nat @ S )
% 16.19/8.99 => ( ! [X4: nat] :
% 16.19/8.99 ( ( member_nat @ X4 @ S )
% 16.19/8.99 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( R2 @ ( groups3539618377306564664at_int @ H2 @ S ) @ ( groups3539618377306564664at_int @ G @ S ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5855_sum__strict__mono,axiom,
% 16.19/8.99 ! [A2: set_complex,F: complex > real,G: complex > real] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.99 => ( ( A2 != bot_bot_set_complex )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5856_sum__strict__mono,axiom,
% 16.19/8.99 ! [A2: set_int,F: int > real,G: int > real] :
% 16.19/8.99 ( ( finite_finite_int @ A2 )
% 16.19/8.99 => ( ( A2 != bot_bot_set_int )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5857_sum__strict__mono,axiom,
% 16.19/8.99 ! [A2: set_real,F: real > real,G: real > real] :
% 16.19/8.99 ( ( finite_finite_real @ A2 )
% 16.19/8.99 => ( ( A2 != bot_bot_set_real )
% 16.19/8.99 => ( ! [X4: real] :
% 16.19/8.99 ( ( member_real @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5858_sum__strict__mono,axiom,
% 16.19/8.99 ! [A2: set_complex,F: complex > rat,G: complex > rat] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.99 => ( ( A2 != bot_bot_set_complex )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5859_sum__strict__mono,axiom,
% 16.19/8.99 ! [A2: set_nat,F: nat > rat,G: nat > rat] :
% 16.19/8.99 ( ( finite_finite_nat @ A2 )
% 16.19/8.99 => ( ( A2 != bot_bot_set_nat )
% 16.19/8.99 => ( ! [X4: nat] :
% 16.19/8.99 ( ( member_nat @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5860_sum__strict__mono,axiom,
% 16.19/8.99 ! [A2: set_int,F: int > rat,G: int > rat] :
% 16.19/8.99 ( ( finite_finite_int @ A2 )
% 16.19/8.99 => ( ( A2 != bot_bot_set_int )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5861_sum__strict__mono,axiom,
% 16.19/8.99 ! [A2: set_real,F: real > rat,G: real > rat] :
% 16.19/8.99 ( ( finite_finite_real @ A2 )
% 16.19/8.99 => ( ( A2 != bot_bot_set_real )
% 16.19/8.99 => ( ! [X4: real] :
% 16.19/8.99 ( ( member_real @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5862_sum__strict__mono,axiom,
% 16.19/8.99 ! [A2: set_complex,F: complex > nat,G: complex > nat] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.99 => ( ( A2 != bot_bot_set_complex )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5863_sum__strict__mono,axiom,
% 16.19/8.99 ! [A2: set_int,F: int > nat,G: int > nat] :
% 16.19/8.99 ( ( finite_finite_int @ A2 )
% 16.19/8.99 => ( ( A2 != bot_bot_set_int )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5864_sum__strict__mono,axiom,
% 16.19/8.99 ! [A2: set_real,F: real > nat,G: real > nat] :
% 16.19/8.99 ( ( finite_finite_real @ A2 )
% 16.19/8.99 => ( ( A2 != bot_bot_set_real )
% 16.19/8.99 => ( ! [X4: real] :
% 16.19/8.99 ( ( member_real @ X4 @ A2 )
% 16.19/8.99 => ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 16.19/8.99 => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5865_ln__bound,axiom,
% 16.19/8.99 ! [X: real] :
% 16.19/8.99 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/8.99 => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ X ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5866_ln__gt__zero,axiom,
% 16.19/8.99 ! [X: real] :
% 16.19/8.99 ( ( ord_less_real @ one_one_real @ X )
% 16.19/8.99 => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5867_ln__less__zero,axiom,
% 16.19/8.99 ! [X: real] :
% 16.19/8.99 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/8.99 => ( ( ord_less_real @ X @ one_one_real )
% 16.19/8.99 => ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5868_ln__gt__zero__imp__gt__one,axiom,
% 16.19/8.99 ! [X: real] :
% 16.19/8.99 ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 16.19/8.99 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/8.99 => ( ord_less_real @ one_one_real @ X ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5869_ln__ge__zero,axiom,
% 16.19/8.99 ! [X: real] :
% 16.19/8.99 ( ( ord_less_eq_real @ one_one_real @ X )
% 16.19/8.99 => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5870_sum_Oreindex__bij__witness__not__neutral,axiom,
% 16.19/8.99 ! [S4: set_real,T4: set_real,S: set_real,I2: real > real,J: real > real,T3: set_real,G: real > complex,H2: real > complex] :
% 16.19/8.99 ( ( finite_finite_real @ S4 )
% 16.19/8.99 => ( ( finite_finite_real @ T4 )
% 16.19/8.99 => ( ! [A3: real] :
% 16.19/8.99 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 16.19/8.99 => ( ( I2 @ ( J @ A3 ) )
% 16.19/8.99 = A3 ) )
% 16.19/8.99 => ( ! [A3: real] :
% 16.19/8.99 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 16.19/8.99 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 16.19/8.99 => ( ! [B2: real] :
% 16.19/8.99 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 16.19/8.99 => ( ( J @ ( I2 @ B2 ) )
% 16.19/8.99 = B2 ) )
% 16.19/8.99 => ( ! [B2: real] :
% 16.19/8.99 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 16.19/8.99 => ( member_real @ ( I2 @ B2 ) @ ( minus_minus_set_real @ S @ S4 ) ) )
% 16.19/8.99 => ( ! [A3: real] :
% 16.19/8.99 ( ( member_real @ A3 @ S4 )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_complex ) )
% 16.19/8.99 => ( ! [B2: real] :
% 16.19/8.99 ( ( member_real @ B2 @ T4 )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_complex ) )
% 16.19/8.99 => ( ! [A3: real] :
% 16.19/8.99 ( ( member_real @ A3 @ S )
% 16.19/8.99 => ( ( H2 @ ( J @ A3 ) )
% 16.19/8.99 = ( G @ A3 ) ) )
% 16.19/8.99 => ( ( groups5754745047067104278omplex @ G @ S )
% 16.19/8.99 = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5871_sum_Oreindex__bij__witness__not__neutral,axiom,
% 16.19/8.99 ! [S4: set_real,T4: set_int,S: set_real,I2: int > real,J: real > int,T3: set_int,G: real > complex,H2: int > complex] :
% 16.19/8.99 ( ( finite_finite_real @ S4 )
% 16.19/8.99 => ( ( finite_finite_int @ T4 )
% 16.19/8.99 => ( ! [A3: real] :
% 16.19/8.99 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 16.19/8.99 => ( ( I2 @ ( J @ A3 ) )
% 16.19/8.99 = A3 ) )
% 16.19/8.99 => ( ! [A3: real] :
% 16.19/8.99 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 16.19/8.99 => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 16.19/8.99 => ( ! [B2: int] :
% 16.19/8.99 ( ( member_int @ B2 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 16.19/8.99 => ( ( J @ ( I2 @ B2 ) )
% 16.19/8.99 = B2 ) )
% 16.19/8.99 => ( ! [B2: int] :
% 16.19/8.99 ( ( member_int @ B2 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 16.19/8.99 => ( member_real @ ( I2 @ B2 ) @ ( minus_minus_set_real @ S @ S4 ) ) )
% 16.19/8.99 => ( ! [A3: real] :
% 16.19/8.99 ( ( member_real @ A3 @ S4 )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_complex ) )
% 16.19/8.99 => ( ! [B2: int] :
% 16.19/8.99 ( ( member_int @ B2 @ T4 )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_complex ) )
% 16.19/8.99 => ( ! [A3: real] :
% 16.19/8.99 ( ( member_real @ A3 @ S )
% 16.19/8.99 => ( ( H2 @ ( J @ A3 ) )
% 16.19/8.99 = ( G @ A3 ) ) )
% 16.19/8.99 => ( ( groups5754745047067104278omplex @ G @ S )
% 16.19/8.99 = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5872_sum_Oreindex__bij__witness__not__neutral,axiom,
% 16.19/8.99 ! [S4: set_int,T4: set_real,S: set_int,I2: real > int,J: int > real,T3: set_real,G: int > complex,H2: real > complex] :
% 16.19/8.99 ( ( finite_finite_int @ S4 )
% 16.19/8.99 => ( ( finite_finite_real @ T4 )
% 16.19/8.99 => ( ! [A3: int] :
% 16.19/8.99 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 16.19/8.99 => ( ( I2 @ ( J @ A3 ) )
% 16.19/8.99 = A3 ) )
% 16.19/8.99 => ( ! [A3: int] :
% 16.19/8.99 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 16.19/8.99 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 16.19/8.99 => ( ! [B2: real] :
% 16.19/8.99 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 16.19/8.99 => ( ( J @ ( I2 @ B2 ) )
% 16.19/8.99 = B2 ) )
% 16.19/8.99 => ( ! [B2: real] :
% 16.19/8.99 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 16.19/8.99 => ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
% 16.19/8.99 => ( ! [A3: int] :
% 16.19/8.99 ( ( member_int @ A3 @ S4 )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_complex ) )
% 16.19/8.99 => ( ! [B2: real] :
% 16.19/8.99 ( ( member_real @ B2 @ T4 )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_complex ) )
% 16.19/8.99 => ( ! [A3: int] :
% 16.19/8.99 ( ( member_int @ A3 @ S )
% 16.19/8.99 => ( ( H2 @ ( J @ A3 ) )
% 16.19/8.99 = ( G @ A3 ) ) )
% 16.19/8.99 => ( ( groups3049146728041665814omplex @ G @ S )
% 16.19/8.99 = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5873_sum_Oreindex__bij__witness__not__neutral,axiom,
% 16.19/8.99 ! [S4: set_int,T4: set_int,S: set_int,I2: int > int,J: int > int,T3: set_int,G: int > complex,H2: int > complex] :
% 16.19/8.99 ( ( finite_finite_int @ S4 )
% 16.19/8.99 => ( ( finite_finite_int @ T4 )
% 16.19/8.99 => ( ! [A3: int] :
% 16.19/8.99 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 16.19/8.99 => ( ( I2 @ ( J @ A3 ) )
% 16.19/8.99 = A3 ) )
% 16.19/8.99 => ( ! [A3: int] :
% 16.19/8.99 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 16.19/8.99 => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 16.19/8.99 => ( ! [B2: int] :
% 16.19/8.99 ( ( member_int @ B2 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 16.19/8.99 => ( ( J @ ( I2 @ B2 ) )
% 16.19/8.99 = B2 ) )
% 16.19/8.99 => ( ! [B2: int] :
% 16.19/8.99 ( ( member_int @ B2 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 16.19/8.99 => ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
% 16.19/8.99 => ( ! [A3: int] :
% 16.19/8.99 ( ( member_int @ A3 @ S4 )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_complex ) )
% 16.19/8.99 => ( ! [B2: int] :
% 16.19/8.99 ( ( member_int @ B2 @ T4 )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_complex ) )
% 16.19/8.99 => ( ! [A3: int] :
% 16.19/8.99 ( ( member_int @ A3 @ S )
% 16.19/8.99 => ( ( H2 @ ( J @ A3 ) )
% 16.19/8.99 = ( G @ A3 ) ) )
% 16.19/8.99 => ( ( groups3049146728041665814omplex @ G @ S )
% 16.19/8.99 = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5874_sum_Oreindex__bij__witness__not__neutral,axiom,
% 16.19/8.99 ! [S4: set_real,T4: set_real,S: set_real,I2: real > real,J: real > real,T3: set_real,G: real > real,H2: real > real] :
% 16.19/8.99 ( ( finite_finite_real @ S4 )
% 16.19/8.99 => ( ( finite_finite_real @ T4 )
% 16.19/8.99 => ( ! [A3: real] :
% 16.19/8.99 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 16.19/8.99 => ( ( I2 @ ( J @ A3 ) )
% 16.19/8.99 = A3 ) )
% 16.19/8.99 => ( ! [A3: real] :
% 16.19/8.99 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 16.19/8.99 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 16.19/8.99 => ( ! [B2: real] :
% 16.19/8.99 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 16.19/8.99 => ( ( J @ ( I2 @ B2 ) )
% 16.19/8.99 = B2 ) )
% 16.19/8.99 => ( ! [B2: real] :
% 16.19/8.99 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 16.19/8.99 => ( member_real @ ( I2 @ B2 ) @ ( minus_minus_set_real @ S @ S4 ) ) )
% 16.19/8.99 => ( ! [A3: real] :
% 16.19/8.99 ( ( member_real @ A3 @ S4 )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ! [B2: real] :
% 16.19/8.99 ( ( member_real @ B2 @ T4 )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ! [A3: real] :
% 16.19/8.99 ( ( member_real @ A3 @ S )
% 16.19/8.99 => ( ( H2 @ ( J @ A3 ) )
% 16.19/8.99 = ( G @ A3 ) ) )
% 16.19/8.99 => ( ( groups8097168146408367636l_real @ G @ S )
% 16.19/8.99 = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5875_sum_Oreindex__bij__witness__not__neutral,axiom,
% 16.19/8.99 ! [S4: set_real,T4: set_int,S: set_real,I2: int > real,J: real > int,T3: set_int,G: real > real,H2: int > real] :
% 16.19/8.99 ( ( finite_finite_real @ S4 )
% 16.19/8.99 => ( ( finite_finite_int @ T4 )
% 16.19/8.99 => ( ! [A3: real] :
% 16.19/8.99 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 16.19/8.99 => ( ( I2 @ ( J @ A3 ) )
% 16.19/8.99 = A3 ) )
% 16.19/8.99 => ( ! [A3: real] :
% 16.19/8.99 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 16.19/8.99 => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 16.19/8.99 => ( ! [B2: int] :
% 16.19/8.99 ( ( member_int @ B2 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 16.19/8.99 => ( ( J @ ( I2 @ B2 ) )
% 16.19/8.99 = B2 ) )
% 16.19/8.99 => ( ! [B2: int] :
% 16.19/8.99 ( ( member_int @ B2 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 16.19/8.99 => ( member_real @ ( I2 @ B2 ) @ ( minus_minus_set_real @ S @ S4 ) ) )
% 16.19/8.99 => ( ! [A3: real] :
% 16.19/8.99 ( ( member_real @ A3 @ S4 )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ! [B2: int] :
% 16.19/8.99 ( ( member_int @ B2 @ T4 )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ! [A3: real] :
% 16.19/8.99 ( ( member_real @ A3 @ S )
% 16.19/8.99 => ( ( H2 @ ( J @ A3 ) )
% 16.19/8.99 = ( G @ A3 ) ) )
% 16.19/8.99 => ( ( groups8097168146408367636l_real @ G @ S )
% 16.19/8.99 = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5876_sum_Oreindex__bij__witness__not__neutral,axiom,
% 16.19/8.99 ! [S4: set_real,T4: set_complex,S: set_real,I2: complex > real,J: real > complex,T3: set_complex,G: real > real,H2: complex > real] :
% 16.19/8.99 ( ( finite_finite_real @ S4 )
% 16.19/8.99 => ( ( finite3207457112153483333omplex @ T4 )
% 16.19/8.99 => ( ! [A3: real] :
% 16.19/8.99 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 16.19/8.99 => ( ( I2 @ ( J @ A3 ) )
% 16.19/8.99 = A3 ) )
% 16.19/8.99 => ( ! [A3: real] :
% 16.19/8.99 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 16.19/8.99 => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 16.19/8.99 => ( ! [B2: complex] :
% 16.19/8.99 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 16.19/8.99 => ( ( J @ ( I2 @ B2 ) )
% 16.19/8.99 = B2 ) )
% 16.19/8.99 => ( ! [B2: complex] :
% 16.19/8.99 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 16.19/8.99 => ( member_real @ ( I2 @ B2 ) @ ( minus_minus_set_real @ S @ S4 ) ) )
% 16.19/8.99 => ( ! [A3: real] :
% 16.19/8.99 ( ( member_real @ A3 @ S4 )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ! [B2: complex] :
% 16.19/8.99 ( ( member_complex @ B2 @ T4 )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ! [A3: real] :
% 16.19/8.99 ( ( member_real @ A3 @ S )
% 16.19/8.99 => ( ( H2 @ ( J @ A3 ) )
% 16.19/8.99 = ( G @ A3 ) ) )
% 16.19/8.99 => ( ( groups8097168146408367636l_real @ G @ S )
% 16.19/8.99 = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5877_sum_Oreindex__bij__witness__not__neutral,axiom,
% 16.19/8.99 ! [S4: set_int,T4: set_real,S: set_int,I2: real > int,J: int > real,T3: set_real,G: int > real,H2: real > real] :
% 16.19/8.99 ( ( finite_finite_int @ S4 )
% 16.19/8.99 => ( ( finite_finite_real @ T4 )
% 16.19/8.99 => ( ! [A3: int] :
% 16.19/8.99 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 16.19/8.99 => ( ( I2 @ ( J @ A3 ) )
% 16.19/8.99 = A3 ) )
% 16.19/8.99 => ( ! [A3: int] :
% 16.19/8.99 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 16.19/8.99 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 16.19/8.99 => ( ! [B2: real] :
% 16.19/8.99 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 16.19/8.99 => ( ( J @ ( I2 @ B2 ) )
% 16.19/8.99 = B2 ) )
% 16.19/8.99 => ( ! [B2: real] :
% 16.19/8.99 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 16.19/8.99 => ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
% 16.19/8.99 => ( ! [A3: int] :
% 16.19/8.99 ( ( member_int @ A3 @ S4 )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ! [B2: real] :
% 16.19/8.99 ( ( member_real @ B2 @ T4 )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ! [A3: int] :
% 16.19/8.99 ( ( member_int @ A3 @ S )
% 16.19/8.99 => ( ( H2 @ ( J @ A3 ) )
% 16.19/8.99 = ( G @ A3 ) ) )
% 16.19/8.99 => ( ( groups8778361861064173332t_real @ G @ S )
% 16.19/8.99 = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5878_sum_Oreindex__bij__witness__not__neutral,axiom,
% 16.19/8.99 ! [S4: set_int,T4: set_int,S: set_int,I2: int > int,J: int > int,T3: set_int,G: int > real,H2: int > real] :
% 16.19/8.99 ( ( finite_finite_int @ S4 )
% 16.19/8.99 => ( ( finite_finite_int @ T4 )
% 16.19/8.99 => ( ! [A3: int] :
% 16.19/8.99 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 16.19/8.99 => ( ( I2 @ ( J @ A3 ) )
% 16.19/8.99 = A3 ) )
% 16.19/8.99 => ( ! [A3: int] :
% 16.19/8.99 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 16.19/8.99 => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 16.19/8.99 => ( ! [B2: int] :
% 16.19/8.99 ( ( member_int @ B2 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 16.19/8.99 => ( ( J @ ( I2 @ B2 ) )
% 16.19/8.99 = B2 ) )
% 16.19/8.99 => ( ! [B2: int] :
% 16.19/8.99 ( ( member_int @ B2 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 16.19/8.99 => ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
% 16.19/8.99 => ( ! [A3: int] :
% 16.19/8.99 ( ( member_int @ A3 @ S4 )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ! [B2: int] :
% 16.19/8.99 ( ( member_int @ B2 @ T4 )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ! [A3: int] :
% 16.19/8.99 ( ( member_int @ A3 @ S )
% 16.19/8.99 => ( ( H2 @ ( J @ A3 ) )
% 16.19/8.99 = ( G @ A3 ) ) )
% 16.19/8.99 => ( ( groups8778361861064173332t_real @ G @ S )
% 16.19/8.99 = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5879_sum_Oreindex__bij__witness__not__neutral,axiom,
% 16.19/8.99 ! [S4: set_int,T4: set_complex,S: set_int,I2: complex > int,J: int > complex,T3: set_complex,G: int > real,H2: complex > real] :
% 16.19/8.99 ( ( finite_finite_int @ S4 )
% 16.19/8.99 => ( ( finite3207457112153483333omplex @ T4 )
% 16.19/8.99 => ( ! [A3: int] :
% 16.19/8.99 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 16.19/8.99 => ( ( I2 @ ( J @ A3 ) )
% 16.19/8.99 = A3 ) )
% 16.19/8.99 => ( ! [A3: int] :
% 16.19/8.99 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 16.19/8.99 => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 16.19/8.99 => ( ! [B2: complex] :
% 16.19/8.99 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 16.19/8.99 => ( ( J @ ( I2 @ B2 ) )
% 16.19/8.99 = B2 ) )
% 16.19/8.99 => ( ! [B2: complex] :
% 16.19/8.99 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 16.19/8.99 => ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
% 16.19/8.99 => ( ! [A3: int] :
% 16.19/8.99 ( ( member_int @ A3 @ S4 )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ! [B2: complex] :
% 16.19/8.99 ( ( member_complex @ B2 @ T4 )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ! [A3: int] :
% 16.19/8.99 ( ( member_int @ A3 @ S )
% 16.19/8.99 => ( ( H2 @ ( J @ A3 ) )
% 16.19/8.99 = ( G @ A3 ) ) )
% 16.19/8.99 => ( ( groups8778361861064173332t_real @ G @ S )
% 16.19/8.99 = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5880_not__zero__le__neg__numeral,axiom,
% 16.19/8.99 ! [N2: num] :
% 16.19/8.99 ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5881_not__zero__le__neg__numeral,axiom,
% 16.19/8.99 ! [N2: num] :
% 16.19/8.99 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5882_not__zero__le__neg__numeral,axiom,
% 16.19/8.99 ! [N2: num] :
% 16.19/8.99 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5883_not__zero__le__neg__numeral,axiom,
% 16.19/8.99 ! [N2: num] :
% 16.19/8.99 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5884_neg__numeral__le__zero,axiom,
% 16.19/8.99 ! [N2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) @ zero_zero_real ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5885_neg__numeral__le__zero,axiom,
% 16.19/8.99 ! [N2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) @ zero_z3403309356797280102nteger ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5886_neg__numeral__le__zero,axiom,
% 16.19/8.99 ! [N2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) @ zero_zero_rat ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5887_neg__numeral__le__zero,axiom,
% 16.19/8.99 ! [N2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ zero_zero_int ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5888_neg__numeral__less__zero,axiom,
% 16.19/8.99 ! [N2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) @ zero_zero_real ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5889_neg__numeral__less__zero,axiom,
% 16.19/8.99 ! [N2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ zero_zero_int ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5890_neg__numeral__less__zero,axiom,
% 16.19/8.99 ! [N2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) @ zero_zero_rat ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5891_neg__numeral__less__zero,axiom,
% 16.19/8.99 ! [N2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) @ zero_z3403309356797280102nteger ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5892_not__zero__less__neg__numeral,axiom,
% 16.19/8.99 ! [N2: num] :
% 16.19/8.99 ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5893_not__zero__less__neg__numeral,axiom,
% 16.19/8.99 ! [N2: num] :
% 16.19/8.99 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5894_not__zero__less__neg__numeral,axiom,
% 16.19/8.99 ! [N2: num] :
% 16.19/8.99 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5895_not__zero__less__neg__numeral,axiom,
% 16.19/8.99 ! [N2: num] :
% 16.19/8.99 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5896_le__minus__one__simps_I3_J,axiom,
% 16.19/8.99 ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5897_le__minus__one__simps_I3_J,axiom,
% 16.19/8.99 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5898_le__minus__one__simps_I3_J,axiom,
% 16.19/8.99 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5899_le__minus__one__simps_I3_J,axiom,
% 16.19/8.99 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5900_le__minus__one__simps_I1_J,axiom,
% 16.19/8.99 ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5901_le__minus__one__simps_I1_J,axiom,
% 16.19/8.99 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5902_le__minus__one__simps_I1_J,axiom,
% 16.19/8.99 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5903_le__minus__one__simps_I1_J,axiom,
% 16.19/8.99 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5904_less__minus__one__simps_I3_J,axiom,
% 16.19/8.99 ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5905_less__minus__one__simps_I3_J,axiom,
% 16.19/8.99 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5906_less__minus__one__simps_I3_J,axiom,
% 16.19/8.99 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5907_less__minus__one__simps_I3_J,axiom,
% 16.19/8.99 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5908_less__minus__one__simps_I1_J,axiom,
% 16.19/8.99 ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5909_less__minus__one__simps_I1_J,axiom,
% 16.19/8.99 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5910_less__minus__one__simps_I1_J,axiom,
% 16.19/8.99 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5911_less__minus__one__simps_I1_J,axiom,
% 16.19/8.99 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5912_not__one__le__neg__numeral,axiom,
% 16.19/8.99 ! [M: num] :
% 16.19/8.99 ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5913_not__one__le__neg__numeral,axiom,
% 16.19/8.99 ! [M: num] :
% 16.19/8.99 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5914_not__one__le__neg__numeral,axiom,
% 16.19/8.99 ! [M: num] :
% 16.19/8.99 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5915_not__one__le__neg__numeral,axiom,
% 16.19/8.99 ! [M: num] :
% 16.19/8.99 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5916_not__numeral__le__neg__one,axiom,
% 16.19/8.99 ! [M: num] :
% 16.19/8.99 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5917_not__numeral__le__neg__one,axiom,
% 16.19/8.99 ! [M: num] :
% 16.19/8.99 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5918_not__numeral__le__neg__one,axiom,
% 16.19/8.99 ! [M: num] :
% 16.19/8.99 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5919_not__numeral__le__neg__one,axiom,
% 16.19/8.99 ! [M: num] :
% 16.19/8.99 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5920_neg__numeral__le__neg__one,axiom,
% 16.19/8.99 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5921_neg__numeral__le__neg__one,axiom,
% 16.19/8.99 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5922_neg__numeral__le__neg__one,axiom,
% 16.19/8.99 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5923_neg__numeral__le__neg__one,axiom,
% 16.19/8.99 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5924_neg__one__le__numeral,axiom,
% 16.19/8.99 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5925_neg__one__le__numeral,axiom,
% 16.19/8.99 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5926_neg__one__le__numeral,axiom,
% 16.19/8.99 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5927_neg__one__le__numeral,axiom,
% 16.19/8.99 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5928_neg__numeral__le__one,axiom,
% 16.19/8.99 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5929_neg__numeral__le__one,axiom,
% 16.19/8.99 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5930_neg__numeral__le__one,axiom,
% 16.19/8.99 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5931_neg__numeral__le__one,axiom,
% 16.19/8.99 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5932_neg__numeral__less__one,axiom,
% 16.19/8.99 ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5933_neg__numeral__less__one,axiom,
% 16.19/8.99 ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5934_neg__numeral__less__one,axiom,
% 16.19/8.99 ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5935_neg__numeral__less__one,axiom,
% 16.19/8.99 ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5936_neg__one__less__numeral,axiom,
% 16.19/8.99 ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5937_neg__one__less__numeral,axiom,
% 16.19/8.99 ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5938_neg__one__less__numeral,axiom,
% 16.19/8.99 ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5939_neg__one__less__numeral,axiom,
% 16.19/8.99 ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5940_not__numeral__less__neg__one,axiom,
% 16.19/8.99 ! [M: num] :
% 16.19/8.99 ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5941_not__numeral__less__neg__one,axiom,
% 16.19/8.99 ! [M: num] :
% 16.19/8.99 ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5942_not__numeral__less__neg__one,axiom,
% 16.19/8.99 ! [M: num] :
% 16.19/8.99 ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5943_not__numeral__less__neg__one,axiom,
% 16.19/8.99 ! [M: num] :
% 16.19/8.99 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5944_not__one__less__neg__numeral,axiom,
% 16.19/8.99 ! [M: num] :
% 16.19/8.99 ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5945_not__one__less__neg__numeral,axiom,
% 16.19/8.99 ! [M: num] :
% 16.19/8.99 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5946_not__one__less__neg__numeral,axiom,
% 16.19/8.99 ! [M: num] :
% 16.19/8.99 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5947_not__one__less__neg__numeral,axiom,
% 16.19/8.99 ! [M: num] :
% 16.19/8.99 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5948_not__neg__one__less__neg__numeral,axiom,
% 16.19/8.99 ! [M: num] :
% 16.19/8.99 ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5949_not__neg__one__less__neg__numeral,axiom,
% 16.19/8.99 ! [M: num] :
% 16.19/8.99 ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5950_not__neg__one__less__neg__numeral,axiom,
% 16.19/8.99 ! [M: num] :
% 16.19/8.99 ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5951_not__neg__one__less__neg__numeral,axiom,
% 16.19/8.99 ! [M: num] :
% 16.19/8.99 ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5952_nonzero__neg__divide__eq__eq2,axiom,
% 16.19/8.99 ! [B: real,C: real,A: real] :
% 16.19/8.99 ( ( B != zero_zero_real )
% 16.19/8.99 => ( ( C
% 16.19/8.99 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
% 16.19/8.99 = ( ( times_times_real @ C @ B )
% 16.19/8.99 = ( uminus_uminus_real @ A ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5953_nonzero__neg__divide__eq__eq2,axiom,
% 16.19/8.99 ! [B: complex,C: complex,A: complex] :
% 16.19/8.99 ( ( B != zero_zero_complex )
% 16.19/8.99 => ( ( C
% 16.19/8.99 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 16.19/8.99 = ( ( times_times_complex @ C @ B )
% 16.19/8.99 = ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5954_nonzero__neg__divide__eq__eq2,axiom,
% 16.19/8.99 ! [B: rat,C: rat,A: rat] :
% 16.19/8.99 ( ( B != zero_zero_rat )
% 16.19/8.99 => ( ( C
% 16.19/8.99 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
% 16.19/8.99 = ( ( times_times_rat @ C @ B )
% 16.19/8.99 = ( uminus_uminus_rat @ A ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5955_nonzero__neg__divide__eq__eq,axiom,
% 16.19/8.99 ! [B: real,A: real,C: real] :
% 16.19/8.99 ( ( B != zero_zero_real )
% 16.19/8.99 => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 16.19/8.99 = C )
% 16.19/8.99 = ( ( uminus_uminus_real @ A )
% 16.19/8.99 = ( times_times_real @ C @ B ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5956_nonzero__neg__divide__eq__eq,axiom,
% 16.19/8.99 ! [B: complex,A: complex,C: complex] :
% 16.19/8.99 ( ( B != zero_zero_complex )
% 16.19/8.99 => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 16.19/8.99 = C )
% 16.19/8.99 = ( ( uminus1482373934393186551omplex @ A )
% 16.19/8.99 = ( times_times_complex @ C @ B ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5957_nonzero__neg__divide__eq__eq,axiom,
% 16.19/8.99 ! [B: rat,A: rat,C: rat] :
% 16.19/8.99 ( ( B != zero_zero_rat )
% 16.19/8.99 => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 16.19/8.99 = C )
% 16.19/8.99 = ( ( uminus_uminus_rat @ A )
% 16.19/8.99 = ( times_times_rat @ C @ B ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5958_minus__divide__eq__eq,axiom,
% 16.19/8.99 ! [B: real,C: real,A: real] :
% 16.19/8.99 ( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) )
% 16.19/8.99 = A )
% 16.19/8.99 = ( ( ( C != zero_zero_real )
% 16.19/8.99 => ( ( uminus_uminus_real @ B )
% 16.19/8.99 = ( times_times_real @ A @ C ) ) )
% 16.19/8.99 & ( ( C = zero_zero_real )
% 16.19/8.99 => ( A = zero_zero_real ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5959_minus__divide__eq__eq,axiom,
% 16.19/8.99 ! [B: complex,C: complex,A: complex] :
% 16.19/8.99 ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) )
% 16.19/8.99 = A )
% 16.19/8.99 = ( ( ( C != zero_zero_complex )
% 16.19/8.99 => ( ( uminus1482373934393186551omplex @ B )
% 16.19/8.99 = ( times_times_complex @ A @ C ) ) )
% 16.19/8.99 & ( ( C = zero_zero_complex )
% 16.19/8.99 => ( A = zero_zero_complex ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5960_minus__divide__eq__eq,axiom,
% 16.19/8.99 ! [B: rat,C: rat,A: rat] :
% 16.19/8.99 ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) )
% 16.19/8.99 = A )
% 16.19/8.99 = ( ( ( C != zero_zero_rat )
% 16.19/8.99 => ( ( uminus_uminus_rat @ B )
% 16.19/8.99 = ( times_times_rat @ A @ C ) ) )
% 16.19/8.99 & ( ( C = zero_zero_rat )
% 16.19/8.99 => ( A = zero_zero_rat ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5961_eq__minus__divide__eq,axiom,
% 16.19/8.99 ! [A: real,B: real,C: real] :
% 16.19/8.99 ( ( A
% 16.19/8.99 = ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 16.19/8.99 = ( ( ( C != zero_zero_real )
% 16.19/8.99 => ( ( times_times_real @ A @ C )
% 16.19/8.99 = ( uminus_uminus_real @ B ) ) )
% 16.19/8.99 & ( ( C = zero_zero_real )
% 16.19/8.99 => ( A = zero_zero_real ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5962_eq__minus__divide__eq,axiom,
% 16.19/8.99 ! [A: complex,B: complex,C: complex] :
% 16.19/8.99 ( ( A
% 16.19/8.99 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) ) )
% 16.19/8.99 = ( ( ( C != zero_zero_complex )
% 16.19/8.99 => ( ( times_times_complex @ A @ C )
% 16.19/8.99 = ( uminus1482373934393186551omplex @ B ) ) )
% 16.19/8.99 & ( ( C = zero_zero_complex )
% 16.19/8.99 => ( A = zero_zero_complex ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5963_eq__minus__divide__eq,axiom,
% 16.19/8.99 ! [A: rat,B: rat,C: rat] :
% 16.19/8.99 ( ( A
% 16.19/8.99 = ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 16.19/8.99 = ( ( ( C != zero_zero_rat )
% 16.19/8.99 => ( ( times_times_rat @ A @ C )
% 16.19/8.99 = ( uminus_uminus_rat @ B ) ) )
% 16.19/8.99 & ( ( C = zero_zero_rat )
% 16.19/8.99 => ( A = zero_zero_rat ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5964_mult__1s__ring__1_I1_J,axiom,
% 16.19/8.99 ! [B: real] :
% 16.19/8.99 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B )
% 16.19/8.99 = ( uminus_uminus_real @ B ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5965_mult__1s__ring__1_I1_J,axiom,
% 16.19/8.99 ! [B: int] :
% 16.19/8.99 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
% 16.19/8.99 = ( uminus_uminus_int @ B ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5966_mult__1s__ring__1_I1_J,axiom,
% 16.19/8.99 ! [B: complex] :
% 16.19/8.99 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B )
% 16.19/8.99 = ( uminus1482373934393186551omplex @ B ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5967_mult__1s__ring__1_I1_J,axiom,
% 16.19/8.99 ! [B: rat] :
% 16.19/8.99 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B )
% 16.19/8.99 = ( uminus_uminus_rat @ B ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5968_mult__1s__ring__1_I1_J,axiom,
% 16.19/8.99 ! [B: code_integer] :
% 16.19/8.99 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B )
% 16.19/8.99 = ( uminus1351360451143612070nteger @ B ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5969_mult__1s__ring__1_I2_J,axiom,
% 16.19/8.99 ! [B: real] :
% 16.19/8.99 ( ( times_times_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
% 16.19/8.99 = ( uminus_uminus_real @ B ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5970_mult__1s__ring__1_I2_J,axiom,
% 16.19/8.99 ! [B: int] :
% 16.19/8.99 ( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
% 16.19/8.99 = ( uminus_uminus_int @ B ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5971_mult__1s__ring__1_I2_J,axiom,
% 16.19/8.99 ! [B: complex] :
% 16.19/8.99 ( ( times_times_complex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
% 16.19/8.99 = ( uminus1482373934393186551omplex @ B ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5972_mult__1s__ring__1_I2_J,axiom,
% 16.19/8.99 ! [B: rat] :
% 16.19/8.99 ( ( times_times_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
% 16.19/8.99 = ( uminus_uminus_rat @ B ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5973_mult__1s__ring__1_I2_J,axiom,
% 16.19/8.99 ! [B: code_integer] :
% 16.19/8.99 ( ( times_3573771949741848930nteger @ B @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
% 16.19/8.99 = ( uminus1351360451143612070nteger @ B ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5974_divide__eq__minus__1__iff,axiom,
% 16.19/8.99 ! [A: real,B: real] :
% 16.19/8.99 ( ( ( divide_divide_real @ A @ B )
% 16.19/8.99 = ( uminus_uminus_real @ one_one_real ) )
% 16.19/8.99 = ( ( B != zero_zero_real )
% 16.19/8.99 & ( A
% 16.19/8.99 = ( uminus_uminus_real @ B ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5975_divide__eq__minus__1__iff,axiom,
% 16.19/8.99 ! [A: complex,B: complex] :
% 16.19/8.99 ( ( ( divide1717551699836669952omplex @ A @ B )
% 16.19/8.99 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 16.19/8.99 = ( ( B != zero_zero_complex )
% 16.19/8.99 & ( A
% 16.19/8.99 = ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5976_divide__eq__minus__1__iff,axiom,
% 16.19/8.99 ! [A: rat,B: rat] :
% 16.19/8.99 ( ( ( divide_divide_rat @ A @ B )
% 16.19/8.99 = ( uminus_uminus_rat @ one_one_rat ) )
% 16.19/8.99 = ( ( B != zero_zero_rat )
% 16.19/8.99 & ( A
% 16.19/8.99 = ( uminus_uminus_rat @ B ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5977_uminus__numeral__One,axiom,
% 16.19/8.99 ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
% 16.19/8.99 = ( uminus_uminus_real @ one_one_real ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5978_uminus__numeral__One,axiom,
% 16.19/8.99 ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
% 16.19/8.99 = ( uminus_uminus_int @ one_one_int ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5979_uminus__numeral__One,axiom,
% 16.19/8.99 ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
% 16.19/8.99 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5980_uminus__numeral__One,axiom,
% 16.19/8.99 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
% 16.19/8.99 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5981_uminus__numeral__One,axiom,
% 16.19/8.99 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
% 16.19/8.99 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5982_power__minus,axiom,
% 16.19/8.99 ! [A: real,N2: nat] :
% 16.19/8.99 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 )
% 16.19/8.99 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( power_power_real @ A @ N2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5983_power__minus,axiom,
% 16.19/8.99 ! [A: int,N2: nat] :
% 16.19/8.99 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
% 16.19/8.99 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ A @ N2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5984_power__minus,axiom,
% 16.19/8.99 ! [A: complex,N2: nat] :
% 16.19/8.99 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 )
% 16.19/8.99 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( power_power_complex @ A @ N2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5985_power__minus,axiom,
% 16.19/8.99 ! [A: rat,N2: nat] :
% 16.19/8.99 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 )
% 16.19/8.99 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( power_power_rat @ A @ N2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5986_power__minus,axiom,
% 16.19/8.99 ! [A: code_integer,N2: nat] :
% 16.19/8.99 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 )
% 16.19/8.99 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5987_power__minus__Bit0,axiom,
% 16.19/8.99 ! [X: real,K: num] :
% 16.19/8.99 ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 16.19/8.99 = ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5988_power__minus__Bit0,axiom,
% 16.19/8.99 ! [X: int,K: num] :
% 16.19/8.99 ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 16.19/8.99 = ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5989_power__minus__Bit0,axiom,
% 16.19/8.99 ! [X: complex,K: num] :
% 16.19/8.99 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 16.19/8.99 = ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5990_power__minus__Bit0,axiom,
% 16.19/8.99 ! [X: rat,K: num] :
% 16.19/8.99 ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 16.19/8.99 = ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5991_power__minus__Bit0,axiom,
% 16.19/8.99 ! [X: code_integer,K: num] :
% 16.19/8.99 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 16.19/8.99 = ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5992_sum__nonneg__0,axiom,
% 16.19/8.99 ! [S3: set_real,F: real > real,I2: real] :
% 16.19/8.99 ( ( finite_finite_real @ S3 )
% 16.19/8.99 => ( ! [I4: real] :
% 16.19/8.99 ( ( member_real @ I4 @ S3 )
% 16.19/8.99 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ( ( groups8097168146408367636l_real @ F @ S3 )
% 16.19/8.99 = zero_zero_real )
% 16.19/8.99 => ( ( member_real @ I2 @ S3 )
% 16.19/8.99 => ( ( F @ I2 )
% 16.19/8.99 = zero_zero_real ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5993_sum__nonneg__0,axiom,
% 16.19/8.99 ! [S3: set_int,F: int > real,I2: int] :
% 16.19/8.99 ( ( finite_finite_int @ S3 )
% 16.19/8.99 => ( ! [I4: int] :
% 16.19/8.99 ( ( member_int @ I4 @ S3 )
% 16.19/8.99 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ( ( groups8778361861064173332t_real @ F @ S3 )
% 16.19/8.99 = zero_zero_real )
% 16.19/8.99 => ( ( member_int @ I2 @ S3 )
% 16.19/8.99 => ( ( F @ I2 )
% 16.19/8.99 = zero_zero_real ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5994_sum__nonneg__0,axiom,
% 16.19/8.99 ! [S3: set_complex,F: complex > real,I2: complex] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ S3 )
% 16.19/8.99 => ( ! [I4: complex] :
% 16.19/8.99 ( ( member_complex @ I4 @ S3 )
% 16.19/8.99 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ( ( groups5808333547571424918x_real @ F @ S3 )
% 16.19/8.99 = zero_zero_real )
% 16.19/8.99 => ( ( member_complex @ I2 @ S3 )
% 16.19/8.99 => ( ( F @ I2 )
% 16.19/8.99 = zero_zero_real ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5995_sum__nonneg__0,axiom,
% 16.19/8.99 ! [S3: set_real,F: real > rat,I2: real] :
% 16.19/8.99 ( ( finite_finite_real @ S3 )
% 16.19/8.99 => ( ! [I4: real] :
% 16.19/8.99 ( ( member_real @ I4 @ S3 )
% 16.19/8.99 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ( ( groups1300246762558778688al_rat @ F @ S3 )
% 16.19/8.99 = zero_zero_rat )
% 16.19/8.99 => ( ( member_real @ I2 @ S3 )
% 16.19/8.99 => ( ( F @ I2 )
% 16.19/8.99 = zero_zero_rat ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5996_sum__nonneg__0,axiom,
% 16.19/8.99 ! [S3: set_nat,F: nat > rat,I2: nat] :
% 16.19/8.99 ( ( finite_finite_nat @ S3 )
% 16.19/8.99 => ( ! [I4: nat] :
% 16.19/8.99 ( ( member_nat @ I4 @ S3 )
% 16.19/8.99 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ( ( groups2906978787729119204at_rat @ F @ S3 )
% 16.19/8.99 = zero_zero_rat )
% 16.19/8.99 => ( ( member_nat @ I2 @ S3 )
% 16.19/8.99 => ( ( F @ I2 )
% 16.19/8.99 = zero_zero_rat ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5997_sum__nonneg__0,axiom,
% 16.19/8.99 ! [S3: set_int,F: int > rat,I2: int] :
% 16.19/8.99 ( ( finite_finite_int @ S3 )
% 16.19/8.99 => ( ! [I4: int] :
% 16.19/8.99 ( ( member_int @ I4 @ S3 )
% 16.19/8.99 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ( ( groups3906332499630173760nt_rat @ F @ S3 )
% 16.19/8.99 = zero_zero_rat )
% 16.19/8.99 => ( ( member_int @ I2 @ S3 )
% 16.19/8.99 => ( ( F @ I2 )
% 16.19/8.99 = zero_zero_rat ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5998_sum__nonneg__0,axiom,
% 16.19/8.99 ! [S3: set_complex,F: complex > rat,I2: complex] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ S3 )
% 16.19/8.99 => ( ! [I4: complex] :
% 16.19/8.99 ( ( member_complex @ I4 @ S3 )
% 16.19/8.99 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ( ( groups5058264527183730370ex_rat @ F @ S3 )
% 16.19/8.99 = zero_zero_rat )
% 16.19/8.99 => ( ( member_complex @ I2 @ S3 )
% 16.19/8.99 => ( ( F @ I2 )
% 16.19/8.99 = zero_zero_rat ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_5999_sum__nonneg__0,axiom,
% 16.19/8.99 ! [S3: set_real,F: real > nat,I2: real] :
% 16.19/8.99 ( ( finite_finite_real @ S3 )
% 16.19/8.99 => ( ! [I4: real] :
% 16.19/8.99 ( ( member_real @ I4 @ S3 )
% 16.19/8.99 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ( ( groups1935376822645274424al_nat @ F @ S3 )
% 16.19/8.99 = zero_zero_nat )
% 16.19/8.99 => ( ( member_real @ I2 @ S3 )
% 16.19/8.99 => ( ( F @ I2 )
% 16.19/8.99 = zero_zero_nat ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6000_sum__nonneg__0,axiom,
% 16.19/8.99 ! [S3: set_int,F: int > nat,I2: int] :
% 16.19/8.99 ( ( finite_finite_int @ S3 )
% 16.19/8.99 => ( ! [I4: int] :
% 16.19/8.99 ( ( member_int @ I4 @ S3 )
% 16.19/8.99 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ( ( groups4541462559716669496nt_nat @ F @ S3 )
% 16.19/8.99 = zero_zero_nat )
% 16.19/8.99 => ( ( member_int @ I2 @ S3 )
% 16.19/8.99 => ( ( F @ I2 )
% 16.19/8.99 = zero_zero_nat ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6001_sum__nonneg__0,axiom,
% 16.19/8.99 ! [S3: set_complex,F: complex > nat,I2: complex] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ S3 )
% 16.19/8.99 => ( ! [I4: complex] :
% 16.19/8.99 ( ( member_complex @ I4 @ S3 )
% 16.19/8.99 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ( ( groups5693394587270226106ex_nat @ F @ S3 )
% 16.19/8.99 = zero_zero_nat )
% 16.19/8.99 => ( ( member_complex @ I2 @ S3 )
% 16.19/8.99 => ( ( F @ I2 )
% 16.19/8.99 = zero_zero_nat ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6002_sum__nonneg__leq__bound,axiom,
% 16.19/8.99 ! [S3: set_real,F: real > real,B3: real,I2: real] :
% 16.19/8.99 ( ( finite_finite_real @ S3 )
% 16.19/8.99 => ( ! [I4: real] :
% 16.19/8.99 ( ( member_real @ I4 @ S3 )
% 16.19/8.99 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ( ( groups8097168146408367636l_real @ F @ S3 )
% 16.19/8.99 = B3 )
% 16.19/8.99 => ( ( member_real @ I2 @ S3 )
% 16.19/8.99 => ( ord_less_eq_real @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6003_sum__nonneg__leq__bound,axiom,
% 16.19/8.99 ! [S3: set_int,F: int > real,B3: real,I2: int] :
% 16.19/8.99 ( ( finite_finite_int @ S3 )
% 16.19/8.99 => ( ! [I4: int] :
% 16.19/8.99 ( ( member_int @ I4 @ S3 )
% 16.19/8.99 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ( ( groups8778361861064173332t_real @ F @ S3 )
% 16.19/8.99 = B3 )
% 16.19/8.99 => ( ( member_int @ I2 @ S3 )
% 16.19/8.99 => ( ord_less_eq_real @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6004_sum__nonneg__leq__bound,axiom,
% 16.19/8.99 ! [S3: set_complex,F: complex > real,B3: real,I2: complex] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ S3 )
% 16.19/8.99 => ( ! [I4: complex] :
% 16.19/8.99 ( ( member_complex @ I4 @ S3 )
% 16.19/8.99 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ( ( groups5808333547571424918x_real @ F @ S3 )
% 16.19/8.99 = B3 )
% 16.19/8.99 => ( ( member_complex @ I2 @ S3 )
% 16.19/8.99 => ( ord_less_eq_real @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6005_sum__nonneg__leq__bound,axiom,
% 16.19/8.99 ! [S3: set_real,F: real > rat,B3: rat,I2: real] :
% 16.19/8.99 ( ( finite_finite_real @ S3 )
% 16.19/8.99 => ( ! [I4: real] :
% 16.19/8.99 ( ( member_real @ I4 @ S3 )
% 16.19/8.99 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ( ( groups1300246762558778688al_rat @ F @ S3 )
% 16.19/8.99 = B3 )
% 16.19/8.99 => ( ( member_real @ I2 @ S3 )
% 16.19/8.99 => ( ord_less_eq_rat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6006_sum__nonneg__leq__bound,axiom,
% 16.19/8.99 ! [S3: set_nat,F: nat > rat,B3: rat,I2: nat] :
% 16.19/8.99 ( ( finite_finite_nat @ S3 )
% 16.19/8.99 => ( ! [I4: nat] :
% 16.19/8.99 ( ( member_nat @ I4 @ S3 )
% 16.19/8.99 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ( ( groups2906978787729119204at_rat @ F @ S3 )
% 16.19/8.99 = B3 )
% 16.19/8.99 => ( ( member_nat @ I2 @ S3 )
% 16.19/8.99 => ( ord_less_eq_rat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6007_sum__nonneg__leq__bound,axiom,
% 16.19/8.99 ! [S3: set_int,F: int > rat,B3: rat,I2: int] :
% 16.19/8.99 ( ( finite_finite_int @ S3 )
% 16.19/8.99 => ( ! [I4: int] :
% 16.19/8.99 ( ( member_int @ I4 @ S3 )
% 16.19/8.99 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ( ( groups3906332499630173760nt_rat @ F @ S3 )
% 16.19/8.99 = B3 )
% 16.19/8.99 => ( ( member_int @ I2 @ S3 )
% 16.19/8.99 => ( ord_less_eq_rat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6008_sum__nonneg__leq__bound,axiom,
% 16.19/8.99 ! [S3: set_complex,F: complex > rat,B3: rat,I2: complex] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ S3 )
% 16.19/8.99 => ( ! [I4: complex] :
% 16.19/8.99 ( ( member_complex @ I4 @ S3 )
% 16.19/8.99 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ( ( groups5058264527183730370ex_rat @ F @ S3 )
% 16.19/8.99 = B3 )
% 16.19/8.99 => ( ( member_complex @ I2 @ S3 )
% 16.19/8.99 => ( ord_less_eq_rat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6009_sum__nonneg__leq__bound,axiom,
% 16.19/8.99 ! [S3: set_real,F: real > nat,B3: nat,I2: real] :
% 16.19/8.99 ( ( finite_finite_real @ S3 )
% 16.19/8.99 => ( ! [I4: real] :
% 16.19/8.99 ( ( member_real @ I4 @ S3 )
% 16.19/8.99 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ( ( groups1935376822645274424al_nat @ F @ S3 )
% 16.19/8.99 = B3 )
% 16.19/8.99 => ( ( member_real @ I2 @ S3 )
% 16.19/8.99 => ( ord_less_eq_nat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6010_sum__nonneg__leq__bound,axiom,
% 16.19/8.99 ! [S3: set_int,F: int > nat,B3: nat,I2: int] :
% 16.19/8.99 ( ( finite_finite_int @ S3 )
% 16.19/8.99 => ( ! [I4: int] :
% 16.19/8.99 ( ( member_int @ I4 @ S3 )
% 16.19/8.99 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ( ( groups4541462559716669496nt_nat @ F @ S3 )
% 16.19/8.99 = B3 )
% 16.19/8.99 => ( ( member_int @ I2 @ S3 )
% 16.19/8.99 => ( ord_less_eq_nat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6011_sum__nonneg__leq__bound,axiom,
% 16.19/8.99 ! [S3: set_complex,F: complex > nat,B3: nat,I2: complex] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ S3 )
% 16.19/8.99 => ( ! [I4: complex] :
% 16.19/8.99 ( ( member_complex @ I4 @ S3 )
% 16.19/8.99 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ( ( groups5693394587270226106ex_nat @ F @ S3 )
% 16.19/8.99 = B3 )
% 16.19/8.99 => ( ( member_complex @ I2 @ S3 )
% 16.19/8.99 => ( ord_less_eq_nat @ ( F @ I2 ) @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6012_sum_Osetdiff__irrelevant,axiom,
% 16.19/8.99 ! [A2: set_int,G: int > complex] :
% 16.19/8.99 ( ( finite_finite_int @ A2 )
% 16.19/8.99 => ( ( groups3049146728041665814omplex @ G
% 16.19/8.99 @ ( minus_minus_set_int @ A2
% 16.19/8.99 @ ( collect_int
% 16.19/8.99 @ ^ [X2: int] :
% 16.19/8.99 ( ( G @ X2 )
% 16.19/8.99 = zero_zero_complex ) ) ) )
% 16.19/8.99 = ( groups3049146728041665814omplex @ G @ A2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6013_sum_Osetdiff__irrelevant,axiom,
% 16.19/8.99 ! [A2: set_int,G: int > real] :
% 16.19/8.99 ( ( finite_finite_int @ A2 )
% 16.19/8.99 => ( ( groups8778361861064173332t_real @ G
% 16.19/8.99 @ ( minus_minus_set_int @ A2
% 16.19/8.99 @ ( collect_int
% 16.19/8.99 @ ^ [X2: int] :
% 16.19/8.99 ( ( G @ X2 )
% 16.19/8.99 = zero_zero_real ) ) ) )
% 16.19/8.99 = ( groups8778361861064173332t_real @ G @ A2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6014_sum_Osetdiff__irrelevant,axiom,
% 16.19/8.99 ! [A2: set_complex,G: complex > real] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.99 => ( ( groups5808333547571424918x_real @ G
% 16.19/8.99 @ ( minus_811609699411566653omplex @ A2
% 16.19/8.99 @ ( collect_complex
% 16.19/8.99 @ ^ [X2: complex] :
% 16.19/8.99 ( ( G @ X2 )
% 16.19/8.99 = zero_zero_real ) ) ) )
% 16.19/8.99 = ( groups5808333547571424918x_real @ G @ A2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6015_sum_Osetdiff__irrelevant,axiom,
% 16.19/8.99 ! [A2: set_int,G: int > rat] :
% 16.19/8.99 ( ( finite_finite_int @ A2 )
% 16.19/8.99 => ( ( groups3906332499630173760nt_rat @ G
% 16.19/8.99 @ ( minus_minus_set_int @ A2
% 16.19/8.99 @ ( collect_int
% 16.19/8.99 @ ^ [X2: int] :
% 16.19/8.99 ( ( G @ X2 )
% 16.19/8.99 = zero_zero_rat ) ) ) )
% 16.19/8.99 = ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6016_sum_Osetdiff__irrelevant,axiom,
% 16.19/8.99 ! [A2: set_complex,G: complex > rat] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.99 => ( ( groups5058264527183730370ex_rat @ G
% 16.19/8.99 @ ( minus_811609699411566653omplex @ A2
% 16.19/8.99 @ ( collect_complex
% 16.19/8.99 @ ^ [X2: complex] :
% 16.19/8.99 ( ( G @ X2 )
% 16.19/8.99 = zero_zero_rat ) ) ) )
% 16.19/8.99 = ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6017_sum_Osetdiff__irrelevant,axiom,
% 16.19/8.99 ! [A2: set_int,G: int > nat] :
% 16.19/8.99 ( ( finite_finite_int @ A2 )
% 16.19/8.99 => ( ( groups4541462559716669496nt_nat @ G
% 16.19/8.99 @ ( minus_minus_set_int @ A2
% 16.19/8.99 @ ( collect_int
% 16.19/8.99 @ ^ [X2: int] :
% 16.19/8.99 ( ( G @ X2 )
% 16.19/8.99 = zero_zero_nat ) ) ) )
% 16.19/8.99 = ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6018_sum_Osetdiff__irrelevant,axiom,
% 16.19/8.99 ! [A2: set_complex,G: complex > nat] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.99 => ( ( groups5693394587270226106ex_nat @ G
% 16.19/8.99 @ ( minus_811609699411566653omplex @ A2
% 16.19/8.99 @ ( collect_complex
% 16.19/8.99 @ ^ [X2: complex] :
% 16.19/8.99 ( ( G @ X2 )
% 16.19/8.99 = zero_zero_nat ) ) ) )
% 16.19/8.99 = ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6019_sum_Osetdiff__irrelevant,axiom,
% 16.19/8.99 ! [A2: set_complex,G: complex > int] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.99 => ( ( groups5690904116761175830ex_int @ G
% 16.19/8.99 @ ( minus_811609699411566653omplex @ A2
% 16.19/8.99 @ ( collect_complex
% 16.19/8.99 @ ^ [X2: complex] :
% 16.19/8.99 ( ( G @ X2 )
% 16.19/8.99 = zero_zero_int ) ) ) )
% 16.19/8.99 = ( groups5690904116761175830ex_int @ G @ A2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6020_sum_Osetdiff__irrelevant,axiom,
% 16.19/8.99 ! [A2: set_nat,G: nat > complex] :
% 16.19/8.99 ( ( finite_finite_nat @ A2 )
% 16.19/8.99 => ( ( groups2073611262835488442omplex @ G
% 16.19/8.99 @ ( minus_minus_set_nat @ A2
% 16.19/8.99 @ ( collect_nat
% 16.19/8.99 @ ^ [X2: nat] :
% 16.19/8.99 ( ( G @ X2 )
% 16.19/8.99 = zero_zero_complex ) ) ) )
% 16.19/8.99 = ( groups2073611262835488442omplex @ G @ A2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6021_sum_Osetdiff__irrelevant,axiom,
% 16.19/8.99 ! [A2: set_nat,G: nat > rat] :
% 16.19/8.99 ( ( finite_finite_nat @ A2 )
% 16.19/8.99 => ( ( groups2906978787729119204at_rat @ G
% 16.19/8.99 @ ( minus_minus_set_nat @ A2
% 16.19/8.99 @ ( collect_nat
% 16.19/8.99 @ ^ [X2: nat] :
% 16.19/8.99 ( ( G @ X2 )
% 16.19/8.99 = zero_zero_rat ) ) ) )
% 16.19/8.99 = ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6022_real__0__less__add__iff,axiom,
% 16.19/8.99 ! [X: real,Y: real] :
% 16.19/8.99 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
% 16.19/8.99 = ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6023_real__add__less__0__iff,axiom,
% 16.19/8.99 ! [X: real,Y: real] :
% 16.19/8.99 ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
% 16.19/8.99 = ( ord_less_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6024_real__add__le__0__iff,axiom,
% 16.19/8.99 ! [X: real,Y: real] :
% 16.19/8.99 ( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
% 16.19/8.99 = ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6025_real__0__le__add__iff,axiom,
% 16.19/8.99 ! [X: real,Y: real] :
% 16.19/8.99 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y ) )
% 16.19/8.99 = ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6026_zmod__zminus1__eq__if,axiom,
% 16.19/8.99 ! [A: int,B: int] :
% 16.19/8.99 ( ( ( ( modulo_modulo_int @ A @ B )
% 16.19/8.99 = zero_zero_int )
% 16.19/8.99 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 16.19/8.99 = zero_zero_int ) )
% 16.19/8.99 & ( ( ( modulo_modulo_int @ A @ B )
% 16.19/8.99 != zero_zero_int )
% 16.19/8.99 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 16.19/8.99 = ( minus_minus_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6027_zmod__zminus2__eq__if,axiom,
% 16.19/8.99 ! [A: int,B: int] :
% 16.19/8.99 ( ( ( ( modulo_modulo_int @ A @ B )
% 16.19/8.99 = zero_zero_int )
% 16.19/8.99 => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 16.19/8.99 = zero_zero_int ) )
% 16.19/8.99 & ( ( ( modulo_modulo_int @ A @ B )
% 16.19/8.99 != zero_zero_int )
% 16.19/8.99 => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 16.19/8.99 = ( minus_minus_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6028_sum__pos2,axiom,
% 16.19/8.99 ! [I6: set_real,I2: real,F: real > real] :
% 16.19/8.99 ( ( finite_finite_real @ I6 )
% 16.19/8.99 => ( ( member_real @ I2 @ I6 )
% 16.19/8.99 => ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
% 16.19/8.99 => ( ! [I4: real] :
% 16.19/8.99 ( ( member_real @ I4 @ I6 )
% 16.19/8.99 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I6 ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6029_sum__pos2,axiom,
% 16.19/8.99 ! [I6: set_int,I2: int,F: int > real] :
% 16.19/8.99 ( ( finite_finite_int @ I6 )
% 16.19/8.99 => ( ( member_int @ I2 @ I6 )
% 16.19/8.99 => ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
% 16.19/8.99 => ( ! [I4: int] :
% 16.19/8.99 ( ( member_int @ I4 @ I6 )
% 16.19/8.99 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I6 ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6030_sum__pos2,axiom,
% 16.19/8.99 ! [I6: set_complex,I2: complex,F: complex > real] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ I6 )
% 16.19/8.99 => ( ( member_complex @ I2 @ I6 )
% 16.19/8.99 => ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
% 16.19/8.99 => ( ! [I4: complex] :
% 16.19/8.99 ( ( member_complex @ I4 @ I6 )
% 16.19/8.99 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I6 ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6031_sum__pos2,axiom,
% 16.19/8.99 ! [I6: set_real,I2: real,F: real > rat] :
% 16.19/8.99 ( ( finite_finite_real @ I6 )
% 16.19/8.99 => ( ( member_real @ I2 @ I6 )
% 16.19/8.99 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
% 16.19/8.99 => ( ! [I4: real] :
% 16.19/8.99 ( ( member_real @ I4 @ I6 )
% 16.19/8.99 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I6 ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6032_sum__pos2,axiom,
% 16.19/8.99 ! [I6: set_nat,I2: nat,F: nat > rat] :
% 16.19/8.99 ( ( finite_finite_nat @ I6 )
% 16.19/8.99 => ( ( member_nat @ I2 @ I6 )
% 16.19/8.99 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
% 16.19/8.99 => ( ! [I4: nat] :
% 16.19/8.99 ( ( member_nat @ I4 @ I6 )
% 16.19/8.99 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I6 ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6033_sum__pos2,axiom,
% 16.19/8.99 ! [I6: set_int,I2: int,F: int > rat] :
% 16.19/8.99 ( ( finite_finite_int @ I6 )
% 16.19/8.99 => ( ( member_int @ I2 @ I6 )
% 16.19/8.99 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
% 16.19/8.99 => ( ! [I4: int] :
% 16.19/8.99 ( ( member_int @ I4 @ I6 )
% 16.19/8.99 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I6 ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6034_sum__pos2,axiom,
% 16.19/8.99 ! [I6: set_complex,I2: complex,F: complex > rat] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ I6 )
% 16.19/8.99 => ( ( member_complex @ I2 @ I6 )
% 16.19/8.99 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) )
% 16.19/8.99 => ( ! [I4: complex] :
% 16.19/8.99 ( ( member_complex @ I4 @ I6 )
% 16.19/8.99 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I6 ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6035_sum__pos2,axiom,
% 16.19/8.99 ! [I6: set_real,I2: real,F: real > nat] :
% 16.19/8.99 ( ( finite_finite_real @ I6 )
% 16.19/8.99 => ( ( member_real @ I2 @ I6 )
% 16.19/8.99 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
% 16.19/8.99 => ( ! [I4: real] :
% 16.19/8.99 ( ( member_real @ I4 @ I6 )
% 16.19/8.99 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I6 ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6036_sum__pos2,axiom,
% 16.19/8.99 ! [I6: set_int,I2: int,F: int > nat] :
% 16.19/8.99 ( ( finite_finite_int @ I6 )
% 16.19/8.99 => ( ( member_int @ I2 @ I6 )
% 16.19/8.99 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
% 16.19/8.99 => ( ! [I4: int] :
% 16.19/8.99 ( ( member_int @ I4 @ I6 )
% 16.19/8.99 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I6 ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6037_sum__pos2,axiom,
% 16.19/8.99 ! [I6: set_complex,I2: complex,F: complex > nat] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ I6 )
% 16.19/8.99 => ( ( member_complex @ I2 @ I6 )
% 16.19/8.99 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
% 16.19/8.99 => ( ! [I4: complex] :
% 16.19/8.99 ( ( member_complex @ I4 @ I6 )
% 16.19/8.99 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I6 ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6038_sum__pos,axiom,
% 16.19/8.99 ! [I6: set_complex,F: complex > real] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ I6 )
% 16.19/8.99 => ( ( I6 != bot_bot_set_complex )
% 16.19/8.99 => ( ! [I4: complex] :
% 16.19/8.99 ( ( member_complex @ I4 @ I6 )
% 16.19/8.99 => ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I6 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6039_sum__pos,axiom,
% 16.19/8.99 ! [I6: set_int,F: int > real] :
% 16.19/8.99 ( ( finite_finite_int @ I6 )
% 16.19/8.99 => ( ( I6 != bot_bot_set_int )
% 16.19/8.99 => ( ! [I4: int] :
% 16.19/8.99 ( ( member_int @ I4 @ I6 )
% 16.19/8.99 => ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I6 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6040_sum__pos,axiom,
% 16.19/8.99 ! [I6: set_real,F: real > real] :
% 16.19/8.99 ( ( finite_finite_real @ I6 )
% 16.19/8.99 => ( ( I6 != bot_bot_set_real )
% 16.19/8.99 => ( ! [I4: real] :
% 16.19/8.99 ( ( member_real @ I4 @ I6 )
% 16.19/8.99 => ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I6 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6041_sum__pos,axiom,
% 16.19/8.99 ! [I6: set_complex,F: complex > rat] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ I6 )
% 16.19/8.99 => ( ( I6 != bot_bot_set_complex )
% 16.19/8.99 => ( ! [I4: complex] :
% 16.19/8.99 ( ( member_complex @ I4 @ I6 )
% 16.19/8.99 => ( ord_less_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I6 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6042_sum__pos,axiom,
% 16.19/8.99 ! [I6: set_nat,F: nat > rat] :
% 16.19/8.99 ( ( finite_finite_nat @ I6 )
% 16.19/8.99 => ( ( I6 != bot_bot_set_nat )
% 16.19/8.99 => ( ! [I4: nat] :
% 16.19/8.99 ( ( member_nat @ I4 @ I6 )
% 16.19/8.99 => ( ord_less_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I6 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6043_sum__pos,axiom,
% 16.19/8.99 ! [I6: set_int,F: int > rat] :
% 16.19/8.99 ( ( finite_finite_int @ I6 )
% 16.19/8.99 => ( ( I6 != bot_bot_set_int )
% 16.19/8.99 => ( ! [I4: int] :
% 16.19/8.99 ( ( member_int @ I4 @ I6 )
% 16.19/8.99 => ( ord_less_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I6 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6044_sum__pos,axiom,
% 16.19/8.99 ! [I6: set_real,F: real > rat] :
% 16.19/8.99 ( ( finite_finite_real @ I6 )
% 16.19/8.99 => ( ( I6 != bot_bot_set_real )
% 16.19/8.99 => ( ! [I4: real] :
% 16.19/8.99 ( ( member_real @ I4 @ I6 )
% 16.19/8.99 => ( ord_less_rat @ zero_zero_rat @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I6 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6045_sum__pos,axiom,
% 16.19/8.99 ! [I6: set_complex,F: complex > nat] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ I6 )
% 16.19/8.99 => ( ( I6 != bot_bot_set_complex )
% 16.19/8.99 => ( ! [I4: complex] :
% 16.19/8.99 ( ( member_complex @ I4 @ I6 )
% 16.19/8.99 => ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I6 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6046_sum__pos,axiom,
% 16.19/8.99 ! [I6: set_int,F: int > nat] :
% 16.19/8.99 ( ( finite_finite_int @ I6 )
% 16.19/8.99 => ( ( I6 != bot_bot_set_int )
% 16.19/8.99 => ( ! [I4: int] :
% 16.19/8.99 ( ( member_int @ I4 @ I6 )
% 16.19/8.99 => ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I6 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6047_sum__pos,axiom,
% 16.19/8.99 ! [I6: set_real,F: real > nat] :
% 16.19/8.99 ( ( finite_finite_real @ I6 )
% 16.19/8.99 => ( ( I6 != bot_bot_set_real )
% 16.19/8.99 => ( ! [I4: real] :
% 16.19/8.99 ( ( member_real @ I4 @ I6 )
% 16.19/8.99 => ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) )
% 16.19/8.99 => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I6 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6048_ln__ge__zero__imp__ge__one,axiom,
% 16.19/8.99 ! [X: real] :
% 16.19/8.99 ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 16.19/8.99 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/8.99 => ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6049_sum_Omono__neutral__cong__right,axiom,
% 16.19/8.99 ! [T3: set_real,S: set_real,G: real > complex,H2: real > complex] :
% 16.19/8.99 ( ( finite_finite_real @ T3 )
% 16.19/8.99 => ( ( ord_less_eq_set_real @ S @ T3 )
% 16.19/8.99 => ( ! [X4: real] :
% 16.19/8.99 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_complex ) )
% 16.19/8.99 => ( ! [X4: real] :
% 16.19/8.99 ( ( member_real @ X4 @ S )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = ( H2 @ X4 ) ) )
% 16.19/8.99 => ( ( groups5754745047067104278omplex @ G @ T3 )
% 16.19/8.99 = ( groups5754745047067104278omplex @ H2 @ S ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6050_sum_Omono__neutral__cong__right,axiom,
% 16.19/8.99 ! [T3: set_int,S: set_int,G: int > complex,H2: int > complex] :
% 16.19/8.99 ( ( finite_finite_int @ T3 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_complex ) )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ S )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = ( H2 @ X4 ) ) )
% 16.19/8.99 => ( ( groups3049146728041665814omplex @ G @ T3 )
% 16.19/8.99 = ( groups3049146728041665814omplex @ H2 @ S ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6051_sum_Omono__neutral__cong__right,axiom,
% 16.19/8.99 ! [T3: set_real,S: set_real,G: real > real,H2: real > real] :
% 16.19/8.99 ( ( finite_finite_real @ T3 )
% 16.19/8.99 => ( ( ord_less_eq_set_real @ S @ T3 )
% 16.19/8.99 => ( ! [X4: real] :
% 16.19/8.99 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ! [X4: real] :
% 16.19/8.99 ( ( member_real @ X4 @ S )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = ( H2 @ X4 ) ) )
% 16.19/8.99 => ( ( groups8097168146408367636l_real @ G @ T3 )
% 16.19/8.99 = ( groups8097168146408367636l_real @ H2 @ S ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6052_sum_Omono__neutral__cong__right,axiom,
% 16.19/8.99 ! [T3: set_int,S: set_int,G: int > real,H2: int > real] :
% 16.19/8.99 ( ( finite_finite_int @ T3 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ S )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = ( H2 @ X4 ) ) )
% 16.19/8.99 => ( ( groups8778361861064173332t_real @ G @ T3 )
% 16.19/8.99 = ( groups8778361861064173332t_real @ H2 @ S ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6053_sum_Omono__neutral__cong__right,axiom,
% 16.19/8.99 ! [T3: set_complex,S: set_complex,G: complex > real,H2: complex > real] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/8.99 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ S )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = ( H2 @ X4 ) ) )
% 16.19/8.99 => ( ( groups5808333547571424918x_real @ G @ T3 )
% 16.19/8.99 = ( groups5808333547571424918x_real @ H2 @ S ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6054_sum_Omono__neutral__cong__right,axiom,
% 16.19/8.99 ! [T3: set_real,S: set_real,G: real > rat,H2: real > rat] :
% 16.19/8.99 ( ( finite_finite_real @ T3 )
% 16.19/8.99 => ( ( ord_less_eq_set_real @ S @ T3 )
% 16.19/8.99 => ( ! [X4: real] :
% 16.19/8.99 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_rat ) )
% 16.19/8.99 => ( ! [X4: real] :
% 16.19/8.99 ( ( member_real @ X4 @ S )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = ( H2 @ X4 ) ) )
% 16.19/8.99 => ( ( groups1300246762558778688al_rat @ G @ T3 )
% 16.19/8.99 = ( groups1300246762558778688al_rat @ H2 @ S ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6055_sum_Omono__neutral__cong__right,axiom,
% 16.19/8.99 ! [T3: set_int,S: set_int,G: int > rat,H2: int > rat] :
% 16.19/8.99 ( ( finite_finite_int @ T3 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_rat ) )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ S )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = ( H2 @ X4 ) ) )
% 16.19/8.99 => ( ( groups3906332499630173760nt_rat @ G @ T3 )
% 16.19/8.99 = ( groups3906332499630173760nt_rat @ H2 @ S ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6056_sum_Omono__neutral__cong__right,axiom,
% 16.19/8.99 ! [T3: set_complex,S: set_complex,G: complex > rat,H2: complex > rat] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/8.99 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_rat ) )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ S )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = ( H2 @ X4 ) ) )
% 16.19/8.99 => ( ( groups5058264527183730370ex_rat @ G @ T3 )
% 16.19/8.99 = ( groups5058264527183730370ex_rat @ H2 @ S ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6057_sum_Omono__neutral__cong__right,axiom,
% 16.19/8.99 ! [T3: set_real,S: set_real,G: real > nat,H2: real > nat] :
% 16.19/8.99 ( ( finite_finite_real @ T3 )
% 16.19/8.99 => ( ( ord_less_eq_set_real @ S @ T3 )
% 16.19/8.99 => ( ! [X4: real] :
% 16.19/8.99 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_nat ) )
% 16.19/8.99 => ( ! [X4: real] :
% 16.19/8.99 ( ( member_real @ X4 @ S )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = ( H2 @ X4 ) ) )
% 16.19/8.99 => ( ( groups1935376822645274424al_nat @ G @ T3 )
% 16.19/8.99 = ( groups1935376822645274424al_nat @ H2 @ S ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6058_sum_Omono__neutral__cong__right,axiom,
% 16.19/8.99 ! [T3: set_int,S: set_int,G: int > nat,H2: int > nat] :
% 16.19/8.99 ( ( finite_finite_int @ T3 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_nat ) )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ S )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = ( H2 @ X4 ) ) )
% 16.19/8.99 => ( ( groups4541462559716669496nt_nat @ G @ T3 )
% 16.19/8.99 = ( groups4541462559716669496nt_nat @ H2 @ S ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6059_sum_Omono__neutral__cong__left,axiom,
% 16.19/8.99 ! [T3: set_real,S: set_real,H2: real > complex,G: real > complex] :
% 16.19/8.99 ( ( finite_finite_real @ T3 )
% 16.19/8.99 => ( ( ord_less_eq_set_real @ S @ T3 )
% 16.19/8.99 => ( ! [X4: real] :
% 16.19/8.99 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 16.19/8.99 => ( ( H2 @ X4 )
% 16.19/8.99 = zero_zero_complex ) )
% 16.19/8.99 => ( ! [X4: real] :
% 16.19/8.99 ( ( member_real @ X4 @ S )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = ( H2 @ X4 ) ) )
% 16.19/8.99 => ( ( groups5754745047067104278omplex @ G @ S )
% 16.19/8.99 = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6060_sum_Omono__neutral__cong__left,axiom,
% 16.19/8.99 ! [T3: set_int,S: set_int,H2: int > complex,G: int > complex] :
% 16.19/8.99 ( ( finite_finite_int @ T3 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/8.99 => ( ( H2 @ X4 )
% 16.19/8.99 = zero_zero_complex ) )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ S )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = ( H2 @ X4 ) ) )
% 16.19/8.99 => ( ( groups3049146728041665814omplex @ G @ S )
% 16.19/8.99 = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6061_sum_Omono__neutral__cong__left,axiom,
% 16.19/8.99 ! [T3: set_real,S: set_real,H2: real > real,G: real > real] :
% 16.19/8.99 ( ( finite_finite_real @ T3 )
% 16.19/8.99 => ( ( ord_less_eq_set_real @ S @ T3 )
% 16.19/8.99 => ( ! [X4: real] :
% 16.19/8.99 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 16.19/8.99 => ( ( H2 @ X4 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ! [X4: real] :
% 16.19/8.99 ( ( member_real @ X4 @ S )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = ( H2 @ X4 ) ) )
% 16.19/8.99 => ( ( groups8097168146408367636l_real @ G @ S )
% 16.19/8.99 = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6062_sum_Omono__neutral__cong__left,axiom,
% 16.19/8.99 ! [T3: set_int,S: set_int,H2: int > real,G: int > real] :
% 16.19/8.99 ( ( finite_finite_int @ T3 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/8.99 => ( ( H2 @ X4 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ S )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = ( H2 @ X4 ) ) )
% 16.19/8.99 => ( ( groups8778361861064173332t_real @ G @ S )
% 16.19/8.99 = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6063_sum_Omono__neutral__cong__left,axiom,
% 16.19/8.99 ! [T3: set_complex,S: set_complex,H2: complex > real,G: complex > real] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/8.99 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 16.19/8.99 => ( ( H2 @ X4 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ S )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = ( H2 @ X4 ) ) )
% 16.19/8.99 => ( ( groups5808333547571424918x_real @ G @ S )
% 16.19/8.99 = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6064_sum_Omono__neutral__cong__left,axiom,
% 16.19/8.99 ! [T3: set_real,S: set_real,H2: real > rat,G: real > rat] :
% 16.19/8.99 ( ( finite_finite_real @ T3 )
% 16.19/8.99 => ( ( ord_less_eq_set_real @ S @ T3 )
% 16.19/8.99 => ( ! [X4: real] :
% 16.19/8.99 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 16.19/8.99 => ( ( H2 @ X4 )
% 16.19/8.99 = zero_zero_rat ) )
% 16.19/8.99 => ( ! [X4: real] :
% 16.19/8.99 ( ( member_real @ X4 @ S )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = ( H2 @ X4 ) ) )
% 16.19/8.99 => ( ( groups1300246762558778688al_rat @ G @ S )
% 16.19/8.99 = ( groups1300246762558778688al_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6065_sum_Omono__neutral__cong__left,axiom,
% 16.19/8.99 ! [T3: set_int,S: set_int,H2: int > rat,G: int > rat] :
% 16.19/8.99 ( ( finite_finite_int @ T3 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/8.99 => ( ( H2 @ X4 )
% 16.19/8.99 = zero_zero_rat ) )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ S )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = ( H2 @ X4 ) ) )
% 16.19/8.99 => ( ( groups3906332499630173760nt_rat @ G @ S )
% 16.19/8.99 = ( groups3906332499630173760nt_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6066_sum_Omono__neutral__cong__left,axiom,
% 16.19/8.99 ! [T3: set_complex,S: set_complex,H2: complex > rat,G: complex > rat] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/8.99 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 16.19/8.99 => ( ( H2 @ X4 )
% 16.19/8.99 = zero_zero_rat ) )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ S )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = ( H2 @ X4 ) ) )
% 16.19/8.99 => ( ( groups5058264527183730370ex_rat @ G @ S )
% 16.19/8.99 = ( groups5058264527183730370ex_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6067_sum_Omono__neutral__cong__left,axiom,
% 16.19/8.99 ! [T3: set_real,S: set_real,H2: real > nat,G: real > nat] :
% 16.19/8.99 ( ( finite_finite_real @ T3 )
% 16.19/8.99 => ( ( ord_less_eq_set_real @ S @ T3 )
% 16.19/8.99 => ( ! [X4: real] :
% 16.19/8.99 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 16.19/8.99 => ( ( H2 @ X4 )
% 16.19/8.99 = zero_zero_nat ) )
% 16.19/8.99 => ( ! [X4: real] :
% 16.19/8.99 ( ( member_real @ X4 @ S )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = ( H2 @ X4 ) ) )
% 16.19/8.99 => ( ( groups1935376822645274424al_nat @ G @ S )
% 16.19/8.99 = ( groups1935376822645274424al_nat @ H2 @ T3 ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6068_sum_Omono__neutral__cong__left,axiom,
% 16.19/8.99 ! [T3: set_int,S: set_int,H2: int > nat,G: int > nat] :
% 16.19/8.99 ( ( finite_finite_int @ T3 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/8.99 => ( ( H2 @ X4 )
% 16.19/8.99 = zero_zero_nat ) )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ S )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = ( H2 @ X4 ) ) )
% 16.19/8.99 => ( ( groups4541462559716669496nt_nat @ G @ S )
% 16.19/8.99 = ( groups4541462559716669496nt_nat @ H2 @ T3 ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6069_sum_Omono__neutral__right,axiom,
% 16.19/8.99 ! [T3: set_int,S: set_int,G: int > complex] :
% 16.19/8.99 ( ( finite_finite_int @ T3 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_complex ) )
% 16.19/8.99 => ( ( groups3049146728041665814omplex @ G @ T3 )
% 16.19/8.99 = ( groups3049146728041665814omplex @ G @ S ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6070_sum_Omono__neutral__right,axiom,
% 16.19/8.99 ! [T3: set_int,S: set_int,G: int > real] :
% 16.19/8.99 ( ( finite_finite_int @ T3 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ( groups8778361861064173332t_real @ G @ T3 )
% 16.19/8.99 = ( groups8778361861064173332t_real @ G @ S ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6071_sum_Omono__neutral__right,axiom,
% 16.19/8.99 ! [T3: set_complex,S: set_complex,G: complex > real] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/8.99 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ( groups5808333547571424918x_real @ G @ T3 )
% 16.19/8.99 = ( groups5808333547571424918x_real @ G @ S ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6072_sum_Omono__neutral__right,axiom,
% 16.19/8.99 ! [T3: set_int,S: set_int,G: int > rat] :
% 16.19/8.99 ( ( finite_finite_int @ T3 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_rat ) )
% 16.19/8.99 => ( ( groups3906332499630173760nt_rat @ G @ T3 )
% 16.19/8.99 = ( groups3906332499630173760nt_rat @ G @ S ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6073_sum_Omono__neutral__right,axiom,
% 16.19/8.99 ! [T3: set_complex,S: set_complex,G: complex > rat] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/8.99 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_rat ) )
% 16.19/8.99 => ( ( groups5058264527183730370ex_rat @ G @ T3 )
% 16.19/8.99 = ( groups5058264527183730370ex_rat @ G @ S ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6074_sum_Omono__neutral__right,axiom,
% 16.19/8.99 ! [T3: set_int,S: set_int,G: int > nat] :
% 16.19/8.99 ( ( finite_finite_int @ T3 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_nat ) )
% 16.19/8.99 => ( ( groups4541462559716669496nt_nat @ G @ T3 )
% 16.19/8.99 = ( groups4541462559716669496nt_nat @ G @ S ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6075_sum_Omono__neutral__right,axiom,
% 16.19/8.99 ! [T3: set_complex,S: set_complex,G: complex > nat] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/8.99 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_nat ) )
% 16.19/8.99 => ( ( groups5693394587270226106ex_nat @ G @ T3 )
% 16.19/8.99 = ( groups5693394587270226106ex_nat @ G @ S ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6076_sum_Omono__neutral__right,axiom,
% 16.19/8.99 ! [T3: set_complex,S: set_complex,G: complex > int] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/8.99 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_int ) )
% 16.19/8.99 => ( ( groups5690904116761175830ex_int @ G @ T3 )
% 16.19/8.99 = ( groups5690904116761175830ex_int @ G @ S ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6077_sum_Omono__neutral__right,axiom,
% 16.19/8.99 ! [T3: set_nat,S: set_nat,G: nat > complex] :
% 16.19/8.99 ( ( finite_finite_nat @ T3 )
% 16.19/8.99 => ( ( ord_less_eq_set_nat @ S @ T3 )
% 16.19/8.99 => ( ! [X4: nat] :
% 16.19/8.99 ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_complex ) )
% 16.19/8.99 => ( ( groups2073611262835488442omplex @ G @ T3 )
% 16.19/8.99 = ( groups2073611262835488442omplex @ G @ S ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6078_sum_Omono__neutral__right,axiom,
% 16.19/8.99 ! [T3: set_nat,S: set_nat,G: nat > rat] :
% 16.19/8.99 ( ( finite_finite_nat @ T3 )
% 16.19/8.99 => ( ( ord_less_eq_set_nat @ S @ T3 )
% 16.19/8.99 => ( ! [X4: nat] :
% 16.19/8.99 ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_rat ) )
% 16.19/8.99 => ( ( groups2906978787729119204at_rat @ G @ T3 )
% 16.19/8.99 = ( groups2906978787729119204at_rat @ G @ S ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6079_sum_Omono__neutral__left,axiom,
% 16.19/8.99 ! [T3: set_int,S: set_int,G: int > complex] :
% 16.19/8.99 ( ( finite_finite_int @ T3 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_complex ) )
% 16.19/8.99 => ( ( groups3049146728041665814omplex @ G @ S )
% 16.19/8.99 = ( groups3049146728041665814omplex @ G @ T3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6080_sum_Omono__neutral__left,axiom,
% 16.19/8.99 ! [T3: set_int,S: set_int,G: int > real] :
% 16.19/8.99 ( ( finite_finite_int @ T3 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ( groups8778361861064173332t_real @ G @ S )
% 16.19/8.99 = ( groups8778361861064173332t_real @ G @ T3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6081_sum_Omono__neutral__left,axiom,
% 16.19/8.99 ! [T3: set_complex,S: set_complex,G: complex > real] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/8.99 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ( groups5808333547571424918x_real @ G @ S )
% 16.19/8.99 = ( groups5808333547571424918x_real @ G @ T3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6082_sum_Omono__neutral__left,axiom,
% 16.19/8.99 ! [T3: set_int,S: set_int,G: int > rat] :
% 16.19/8.99 ( ( finite_finite_int @ T3 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_rat ) )
% 16.19/8.99 => ( ( groups3906332499630173760nt_rat @ G @ S )
% 16.19/8.99 = ( groups3906332499630173760nt_rat @ G @ T3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6083_sum_Omono__neutral__left,axiom,
% 16.19/8.99 ! [T3: set_complex,S: set_complex,G: complex > rat] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/8.99 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_rat ) )
% 16.19/8.99 => ( ( groups5058264527183730370ex_rat @ G @ S )
% 16.19/8.99 = ( groups5058264527183730370ex_rat @ G @ T3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6084_sum_Omono__neutral__left,axiom,
% 16.19/8.99 ! [T3: set_int,S: set_int,G: int > nat] :
% 16.19/8.99 ( ( finite_finite_int @ T3 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/8.99 => ( ! [X4: int] :
% 16.19/8.99 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_nat ) )
% 16.19/8.99 => ( ( groups4541462559716669496nt_nat @ G @ S )
% 16.19/8.99 = ( groups4541462559716669496nt_nat @ G @ T3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6085_sum_Omono__neutral__left,axiom,
% 16.19/8.99 ! [T3: set_complex,S: set_complex,G: complex > nat] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/8.99 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_nat ) )
% 16.19/8.99 => ( ( groups5693394587270226106ex_nat @ G @ S )
% 16.19/8.99 = ( groups5693394587270226106ex_nat @ G @ T3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6086_sum_Omono__neutral__left,axiom,
% 16.19/8.99 ! [T3: set_complex,S: set_complex,G: complex > int] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/8.99 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/8.99 => ( ! [X4: complex] :
% 16.19/8.99 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_int ) )
% 16.19/8.99 => ( ( groups5690904116761175830ex_int @ G @ S )
% 16.19/8.99 = ( groups5690904116761175830ex_int @ G @ T3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6087_sum_Omono__neutral__left,axiom,
% 16.19/8.99 ! [T3: set_nat,S: set_nat,G: nat > complex] :
% 16.19/8.99 ( ( finite_finite_nat @ T3 )
% 16.19/8.99 => ( ( ord_less_eq_set_nat @ S @ T3 )
% 16.19/8.99 => ( ! [X4: nat] :
% 16.19/8.99 ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_complex ) )
% 16.19/8.99 => ( ( groups2073611262835488442omplex @ G @ S )
% 16.19/8.99 = ( groups2073611262835488442omplex @ G @ T3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6088_sum_Omono__neutral__left,axiom,
% 16.19/8.99 ! [T3: set_nat,S: set_nat,G: nat > rat] :
% 16.19/8.99 ( ( finite_finite_nat @ T3 )
% 16.19/8.99 => ( ( ord_less_eq_set_nat @ S @ T3 )
% 16.19/8.99 => ( ! [X4: nat] :
% 16.19/8.99 ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S ) )
% 16.19/8.99 => ( ( G @ X4 )
% 16.19/8.99 = zero_zero_rat ) )
% 16.19/8.99 => ( ( groups2906978787729119204at_rat @ G @ S )
% 16.19/8.99 = ( groups2906978787729119204at_rat @ G @ T3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6089_sum_Osame__carrierI,axiom,
% 16.19/8.99 ! [C2: set_real,A2: set_real,B3: set_real,G: real > complex,H2: real > complex] :
% 16.19/8.99 ( ( finite_finite_real @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 16.19/8.99 => ( ! [A3: real] :
% 16.19/8.99 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_complex ) )
% 16.19/8.99 => ( ! [B2: real] :
% 16.19/8.99 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_complex ) )
% 16.19/8.99 => ( ( ( groups5754745047067104278omplex @ G @ C2 )
% 16.19/8.99 = ( groups5754745047067104278omplex @ H2 @ C2 ) )
% 16.19/8.99 => ( ( groups5754745047067104278omplex @ G @ A2 )
% 16.19/8.99 = ( groups5754745047067104278omplex @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6090_sum_Osame__carrierI,axiom,
% 16.19/8.99 ! [C2: set_int,A2: set_int,B3: set_int,G: int > complex,H2: int > complex] :
% 16.19/8.99 ( ( finite_finite_int @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ A2 @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ B3 @ C2 )
% 16.19/8.99 => ( ! [A3: int] :
% 16.19/8.99 ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_complex ) )
% 16.19/8.99 => ( ! [B2: int] :
% 16.19/8.99 ( ( member_int @ B2 @ ( minus_minus_set_int @ C2 @ B3 ) )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_complex ) )
% 16.19/8.99 => ( ( ( groups3049146728041665814omplex @ G @ C2 )
% 16.19/8.99 = ( groups3049146728041665814omplex @ H2 @ C2 ) )
% 16.19/8.99 => ( ( groups3049146728041665814omplex @ G @ A2 )
% 16.19/8.99 = ( groups3049146728041665814omplex @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6091_sum_Osame__carrierI,axiom,
% 16.19/8.99 ! [C2: set_real,A2: set_real,B3: set_real,G: real > real,H2: real > real] :
% 16.19/8.99 ( ( finite_finite_real @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 16.19/8.99 => ( ! [A3: real] :
% 16.19/8.99 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ! [B2: real] :
% 16.19/8.99 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ( ( groups8097168146408367636l_real @ G @ C2 )
% 16.19/8.99 = ( groups8097168146408367636l_real @ H2 @ C2 ) )
% 16.19/8.99 => ( ( groups8097168146408367636l_real @ G @ A2 )
% 16.19/8.99 = ( groups8097168146408367636l_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6092_sum_Osame__carrierI,axiom,
% 16.19/8.99 ! [C2: set_int,A2: set_int,B3: set_int,G: int > real,H2: int > real] :
% 16.19/8.99 ( ( finite_finite_int @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ A2 @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ B3 @ C2 )
% 16.19/8.99 => ( ! [A3: int] :
% 16.19/8.99 ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ! [B2: int] :
% 16.19/8.99 ( ( member_int @ B2 @ ( minus_minus_set_int @ C2 @ B3 ) )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ( ( groups8778361861064173332t_real @ G @ C2 )
% 16.19/8.99 = ( groups8778361861064173332t_real @ H2 @ C2 ) )
% 16.19/8.99 => ( ( groups8778361861064173332t_real @ G @ A2 )
% 16.19/8.99 = ( groups8778361861064173332t_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6093_sum_Osame__carrierI,axiom,
% 16.19/8.99 ! [C2: set_complex,A2: set_complex,B3: set_complex,G: complex > real,H2: complex > real] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ C2 )
% 16.19/8.99 => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 16.19/8.99 => ( ( ord_le211207098394363844omplex @ B3 @ C2 )
% 16.19/8.99 => ( ! [A3: complex] :
% 16.19/8.99 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ! [B2: complex] :
% 16.19/8.99 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B3 ) )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ( ( groups5808333547571424918x_real @ G @ C2 )
% 16.19/8.99 = ( groups5808333547571424918x_real @ H2 @ C2 ) )
% 16.19/8.99 => ( ( groups5808333547571424918x_real @ G @ A2 )
% 16.19/8.99 = ( groups5808333547571424918x_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6094_sum_Osame__carrierI,axiom,
% 16.19/8.99 ! [C2: set_real,A2: set_real,B3: set_real,G: real > rat,H2: real > rat] :
% 16.19/8.99 ( ( finite_finite_real @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 16.19/8.99 => ( ! [A3: real] :
% 16.19/8.99 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_rat ) )
% 16.19/8.99 => ( ! [B2: real] :
% 16.19/8.99 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_rat ) )
% 16.19/8.99 => ( ( ( groups1300246762558778688al_rat @ G @ C2 )
% 16.19/8.99 = ( groups1300246762558778688al_rat @ H2 @ C2 ) )
% 16.19/8.99 => ( ( groups1300246762558778688al_rat @ G @ A2 )
% 16.19/8.99 = ( groups1300246762558778688al_rat @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6095_sum_Osame__carrierI,axiom,
% 16.19/8.99 ! [C2: set_int,A2: set_int,B3: set_int,G: int > rat,H2: int > rat] :
% 16.19/8.99 ( ( finite_finite_int @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ A2 @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ B3 @ C2 )
% 16.19/8.99 => ( ! [A3: int] :
% 16.19/8.99 ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_rat ) )
% 16.19/8.99 => ( ! [B2: int] :
% 16.19/8.99 ( ( member_int @ B2 @ ( minus_minus_set_int @ C2 @ B3 ) )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_rat ) )
% 16.19/8.99 => ( ( ( groups3906332499630173760nt_rat @ G @ C2 )
% 16.19/8.99 = ( groups3906332499630173760nt_rat @ H2 @ C2 ) )
% 16.19/8.99 => ( ( groups3906332499630173760nt_rat @ G @ A2 )
% 16.19/8.99 = ( groups3906332499630173760nt_rat @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6096_sum_Osame__carrierI,axiom,
% 16.19/8.99 ! [C2: set_complex,A2: set_complex,B3: set_complex,G: complex > rat,H2: complex > rat] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ C2 )
% 16.19/8.99 => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 16.19/8.99 => ( ( ord_le211207098394363844omplex @ B3 @ C2 )
% 16.19/8.99 => ( ! [A3: complex] :
% 16.19/8.99 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_rat ) )
% 16.19/8.99 => ( ! [B2: complex] :
% 16.19/8.99 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B3 ) )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_rat ) )
% 16.19/8.99 => ( ( ( groups5058264527183730370ex_rat @ G @ C2 )
% 16.19/8.99 = ( groups5058264527183730370ex_rat @ H2 @ C2 ) )
% 16.19/8.99 => ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 16.19/8.99 = ( groups5058264527183730370ex_rat @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6097_sum_Osame__carrierI,axiom,
% 16.19/8.99 ! [C2: set_real,A2: set_real,B3: set_real,G: real > nat,H2: real > nat] :
% 16.19/8.99 ( ( finite_finite_real @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 16.19/8.99 => ( ! [A3: real] :
% 16.19/8.99 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_nat ) )
% 16.19/8.99 => ( ! [B2: real] :
% 16.19/8.99 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_nat ) )
% 16.19/8.99 => ( ( ( groups1935376822645274424al_nat @ G @ C2 )
% 16.19/8.99 = ( groups1935376822645274424al_nat @ H2 @ C2 ) )
% 16.19/8.99 => ( ( groups1935376822645274424al_nat @ G @ A2 )
% 16.19/8.99 = ( groups1935376822645274424al_nat @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6098_sum_Osame__carrierI,axiom,
% 16.19/8.99 ! [C2: set_int,A2: set_int,B3: set_int,G: int > nat,H2: int > nat] :
% 16.19/8.99 ( ( finite_finite_int @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ A2 @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ B3 @ C2 )
% 16.19/8.99 => ( ! [A3: int] :
% 16.19/8.99 ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_nat ) )
% 16.19/8.99 => ( ! [B2: int] :
% 16.19/8.99 ( ( member_int @ B2 @ ( minus_minus_set_int @ C2 @ B3 ) )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_nat ) )
% 16.19/8.99 => ( ( ( groups4541462559716669496nt_nat @ G @ C2 )
% 16.19/8.99 = ( groups4541462559716669496nt_nat @ H2 @ C2 ) )
% 16.19/8.99 => ( ( groups4541462559716669496nt_nat @ G @ A2 )
% 16.19/8.99 = ( groups4541462559716669496nt_nat @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6099_sum_Osame__carrier,axiom,
% 16.19/8.99 ! [C2: set_real,A2: set_real,B3: set_real,G: real > complex,H2: real > complex] :
% 16.19/8.99 ( ( finite_finite_real @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 16.19/8.99 => ( ! [A3: real] :
% 16.19/8.99 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_complex ) )
% 16.19/8.99 => ( ! [B2: real] :
% 16.19/8.99 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_complex ) )
% 16.19/8.99 => ( ( ( groups5754745047067104278omplex @ G @ A2 )
% 16.19/8.99 = ( groups5754745047067104278omplex @ H2 @ B3 ) )
% 16.19/8.99 = ( ( groups5754745047067104278omplex @ G @ C2 )
% 16.19/8.99 = ( groups5754745047067104278omplex @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6100_sum_Osame__carrier,axiom,
% 16.19/8.99 ! [C2: set_int,A2: set_int,B3: set_int,G: int > complex,H2: int > complex] :
% 16.19/8.99 ( ( finite_finite_int @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ A2 @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ B3 @ C2 )
% 16.19/8.99 => ( ! [A3: int] :
% 16.19/8.99 ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_complex ) )
% 16.19/8.99 => ( ! [B2: int] :
% 16.19/8.99 ( ( member_int @ B2 @ ( minus_minus_set_int @ C2 @ B3 ) )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_complex ) )
% 16.19/8.99 => ( ( ( groups3049146728041665814omplex @ G @ A2 )
% 16.19/8.99 = ( groups3049146728041665814omplex @ H2 @ B3 ) )
% 16.19/8.99 = ( ( groups3049146728041665814omplex @ G @ C2 )
% 16.19/8.99 = ( groups3049146728041665814omplex @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6101_sum_Osame__carrier,axiom,
% 16.19/8.99 ! [C2: set_real,A2: set_real,B3: set_real,G: real > real,H2: real > real] :
% 16.19/8.99 ( ( finite_finite_real @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 16.19/8.99 => ( ! [A3: real] :
% 16.19/8.99 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ! [B2: real] :
% 16.19/8.99 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ( ( groups8097168146408367636l_real @ G @ A2 )
% 16.19/8.99 = ( groups8097168146408367636l_real @ H2 @ B3 ) )
% 16.19/8.99 = ( ( groups8097168146408367636l_real @ G @ C2 )
% 16.19/8.99 = ( groups8097168146408367636l_real @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6102_sum_Osame__carrier,axiom,
% 16.19/8.99 ! [C2: set_int,A2: set_int,B3: set_int,G: int > real,H2: int > real] :
% 16.19/8.99 ( ( finite_finite_int @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ A2 @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ B3 @ C2 )
% 16.19/8.99 => ( ! [A3: int] :
% 16.19/8.99 ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ! [B2: int] :
% 16.19/8.99 ( ( member_int @ B2 @ ( minus_minus_set_int @ C2 @ B3 ) )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ( ( groups8778361861064173332t_real @ G @ A2 )
% 16.19/8.99 = ( groups8778361861064173332t_real @ H2 @ B3 ) )
% 16.19/8.99 = ( ( groups8778361861064173332t_real @ G @ C2 )
% 16.19/8.99 = ( groups8778361861064173332t_real @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6103_sum_Osame__carrier,axiom,
% 16.19/8.99 ! [C2: set_complex,A2: set_complex,B3: set_complex,G: complex > real,H2: complex > real] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ C2 )
% 16.19/8.99 => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 16.19/8.99 => ( ( ord_le211207098394363844omplex @ B3 @ C2 )
% 16.19/8.99 => ( ! [A3: complex] :
% 16.19/8.99 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ! [B2: complex] :
% 16.19/8.99 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B3 ) )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_real ) )
% 16.19/8.99 => ( ( ( groups5808333547571424918x_real @ G @ A2 )
% 16.19/8.99 = ( groups5808333547571424918x_real @ H2 @ B3 ) )
% 16.19/8.99 = ( ( groups5808333547571424918x_real @ G @ C2 )
% 16.19/8.99 = ( groups5808333547571424918x_real @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6104_sum_Osame__carrier,axiom,
% 16.19/8.99 ! [C2: set_real,A2: set_real,B3: set_real,G: real > rat,H2: real > rat] :
% 16.19/8.99 ( ( finite_finite_real @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 16.19/8.99 => ( ! [A3: real] :
% 16.19/8.99 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_rat ) )
% 16.19/8.99 => ( ! [B2: real] :
% 16.19/8.99 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_rat ) )
% 16.19/8.99 => ( ( ( groups1300246762558778688al_rat @ G @ A2 )
% 16.19/8.99 = ( groups1300246762558778688al_rat @ H2 @ B3 ) )
% 16.19/8.99 = ( ( groups1300246762558778688al_rat @ G @ C2 )
% 16.19/8.99 = ( groups1300246762558778688al_rat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6105_sum_Osame__carrier,axiom,
% 16.19/8.99 ! [C2: set_int,A2: set_int,B3: set_int,G: int > rat,H2: int > rat] :
% 16.19/8.99 ( ( finite_finite_int @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ A2 @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ B3 @ C2 )
% 16.19/8.99 => ( ! [A3: int] :
% 16.19/8.99 ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_rat ) )
% 16.19/8.99 => ( ! [B2: int] :
% 16.19/8.99 ( ( member_int @ B2 @ ( minus_minus_set_int @ C2 @ B3 ) )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_rat ) )
% 16.19/8.99 => ( ( ( groups3906332499630173760nt_rat @ G @ A2 )
% 16.19/8.99 = ( groups3906332499630173760nt_rat @ H2 @ B3 ) )
% 16.19/8.99 = ( ( groups3906332499630173760nt_rat @ G @ C2 )
% 16.19/8.99 = ( groups3906332499630173760nt_rat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6106_sum_Osame__carrier,axiom,
% 16.19/8.99 ! [C2: set_complex,A2: set_complex,B3: set_complex,G: complex > rat,H2: complex > rat] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ C2 )
% 16.19/8.99 => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 16.19/8.99 => ( ( ord_le211207098394363844omplex @ B3 @ C2 )
% 16.19/8.99 => ( ! [A3: complex] :
% 16.19/8.99 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_rat ) )
% 16.19/8.99 => ( ! [B2: complex] :
% 16.19/8.99 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B3 ) )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_rat ) )
% 16.19/8.99 => ( ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 16.19/8.99 = ( groups5058264527183730370ex_rat @ H2 @ B3 ) )
% 16.19/8.99 = ( ( groups5058264527183730370ex_rat @ G @ C2 )
% 16.19/8.99 = ( groups5058264527183730370ex_rat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6107_sum_Osame__carrier,axiom,
% 16.19/8.99 ! [C2: set_real,A2: set_real,B3: set_real,G: real > nat,H2: real > nat] :
% 16.19/8.99 ( ( finite_finite_real @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 16.19/8.99 => ( ! [A3: real] :
% 16.19/8.99 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_nat ) )
% 16.19/8.99 => ( ! [B2: real] :
% 16.19/8.99 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_nat ) )
% 16.19/8.99 => ( ( ( groups1935376822645274424al_nat @ G @ A2 )
% 16.19/8.99 = ( groups1935376822645274424al_nat @ H2 @ B3 ) )
% 16.19/8.99 = ( ( groups1935376822645274424al_nat @ G @ C2 )
% 16.19/8.99 = ( groups1935376822645274424al_nat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6108_sum_Osame__carrier,axiom,
% 16.19/8.99 ! [C2: set_int,A2: set_int,B3: set_int,G: int > nat,H2: int > nat] :
% 16.19/8.99 ( ( finite_finite_int @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ A2 @ C2 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ B3 @ C2 )
% 16.19/8.99 => ( ! [A3: int] :
% 16.19/8.99 ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 16.19/8.99 => ( ( G @ A3 )
% 16.19/8.99 = zero_zero_nat ) )
% 16.19/8.99 => ( ! [B2: int] :
% 16.19/8.99 ( ( member_int @ B2 @ ( minus_minus_set_int @ C2 @ B3 ) )
% 16.19/8.99 => ( ( H2 @ B2 )
% 16.19/8.99 = zero_zero_nat ) )
% 16.19/8.99 => ( ( ( groups4541462559716669496nt_nat @ G @ A2 )
% 16.19/8.99 = ( groups4541462559716669496nt_nat @ H2 @ B3 ) )
% 16.19/8.99 = ( ( groups4541462559716669496nt_nat @ G @ C2 )
% 16.19/8.99 = ( groups4541462559716669496nt_nat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6109_sum_Osubset__diff,axiom,
% 16.19/8.99 ! [B3: set_int,A2: set_int,G: int > real] :
% 16.19/8.99 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 16.19/8.99 => ( ( finite_finite_int @ A2 )
% 16.19/8.99 => ( ( groups8778361861064173332t_real @ G @ A2 )
% 16.19/8.99 = ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups8778361861064173332t_real @ G @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6110_sum_Osubset__diff,axiom,
% 16.19/8.99 ! [B3: set_complex,A2: set_complex,G: complex > real] :
% 16.19/8.99 ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 16.19/8.99 => ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.99 => ( ( groups5808333547571424918x_real @ G @ A2 )
% 16.19/8.99 = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups5808333547571424918x_real @ G @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6111_sum_Osubset__diff,axiom,
% 16.19/8.99 ! [B3: set_int,A2: set_int,G: int > rat] :
% 16.19/8.99 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 16.19/8.99 => ( ( finite_finite_int @ A2 )
% 16.19/8.99 => ( ( groups3906332499630173760nt_rat @ G @ A2 )
% 16.19/8.99 = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups3906332499630173760nt_rat @ G @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6112_sum_Osubset__diff,axiom,
% 16.19/8.99 ! [B3: set_complex,A2: set_complex,G: complex > rat] :
% 16.19/8.99 ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 16.19/8.99 => ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.99 => ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 16.19/8.99 = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups5058264527183730370ex_rat @ G @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6113_sum_Osubset__diff,axiom,
% 16.19/8.99 ! [B3: set_int,A2: set_int,G: int > nat] :
% 16.19/8.99 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 16.19/8.99 => ( ( finite_finite_int @ A2 )
% 16.19/8.99 => ( ( groups4541462559716669496nt_nat @ G @ A2 )
% 16.19/8.99 = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups4541462559716669496nt_nat @ G @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6114_sum_Osubset__diff,axiom,
% 16.19/8.99 ! [B3: set_complex,A2: set_complex,G: complex > nat] :
% 16.19/8.99 ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 16.19/8.99 => ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.99 => ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 16.19/8.99 = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups5693394587270226106ex_nat @ G @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6115_sum_Osubset__diff,axiom,
% 16.19/8.99 ! [B3: set_complex,A2: set_complex,G: complex > int] :
% 16.19/8.99 ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 16.19/8.99 => ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.99 => ( ( groups5690904116761175830ex_int @ G @ A2 )
% 16.19/8.99 = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups5690904116761175830ex_int @ G @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6116_sum_Osubset__diff,axiom,
% 16.19/8.99 ! [B3: set_nat,A2: set_nat,G: nat > rat] :
% 16.19/8.99 ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 16.19/8.99 => ( ( finite_finite_nat @ A2 )
% 16.19/8.99 => ( ( groups2906978787729119204at_rat @ G @ A2 )
% 16.19/8.99 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A2 @ B3 ) ) @ ( groups2906978787729119204at_rat @ G @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6117_sum_Osubset__diff,axiom,
% 16.19/8.99 ! [B3: set_nat,A2: set_nat,G: nat > int] :
% 16.19/8.99 ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 16.19/8.99 => ( ( finite_finite_nat @ A2 )
% 16.19/8.99 => ( ( groups3539618377306564664at_int @ G @ A2 )
% 16.19/8.99 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A2 @ B3 ) ) @ ( groups3539618377306564664at_int @ G @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6118_sum_Osubset__diff,axiom,
% 16.19/8.99 ! [B3: set_int,A2: set_int,G: int > int] :
% 16.19/8.99 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 16.19/8.99 => ( ( finite_finite_int @ A2 )
% 16.19/8.99 => ( ( groups4538972089207619220nt_int @ G @ A2 )
% 16.19/8.99 = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups4538972089207619220nt_int @ G @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6119_sum__diff,axiom,
% 16.19/8.99 ! [A2: set_int,B3: set_int,F: int > real] :
% 16.19/8.99 ( ( finite_finite_int @ A2 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ B3 @ A2 )
% 16.19/8.99 => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ B3 ) )
% 16.19/8.99 = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6120_sum__diff,axiom,
% 16.19/8.99 ! [A2: set_complex,B3: set_complex,F: complex > real] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.99 => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 16.19/8.99 => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 16.19/8.99 = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6121_sum__diff,axiom,
% 16.19/8.99 ! [A2: set_int,B3: set_int,F: int > rat] :
% 16.19/8.99 ( ( finite_finite_int @ A2 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ B3 @ A2 )
% 16.19/8.99 => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ B3 ) )
% 16.19/8.99 = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6122_sum__diff,axiom,
% 16.19/8.99 ! [A2: set_complex,B3: set_complex,F: complex > rat] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.99 => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 16.19/8.99 => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 16.19/8.99 = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6123_sum__diff,axiom,
% 16.19/8.99 ! [A2: set_complex,B3: set_complex,F: complex > int] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.99 => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 16.19/8.99 => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 16.19/8.99 = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6124_sum__diff,axiom,
% 16.19/8.99 ! [A2: set_nat,B3: set_nat,F: nat > rat] :
% 16.19/8.99 ( ( finite_finite_nat @ A2 )
% 16.19/8.99 => ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 16.19/8.99 => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 16.19/8.99 = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6125_sum__diff,axiom,
% 16.19/8.99 ! [A2: set_nat,B3: set_nat,F: nat > int] :
% 16.19/8.99 ( ( finite_finite_nat @ A2 )
% 16.19/8.99 => ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 16.19/8.99 => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 16.19/8.99 = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6126_sum__diff,axiom,
% 16.19/8.99 ! [A2: set_int,B3: set_int,F: int > int] :
% 16.19/8.99 ( ( finite_finite_int @ A2 )
% 16.19/8.99 => ( ( ord_less_eq_set_int @ B3 @ A2 )
% 16.19/8.99 => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ B3 ) )
% 16.19/8.99 = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ F @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6127_sum__diff,axiom,
% 16.19/8.99 ! [A2: set_complex,B3: set_complex,F: complex > complex] :
% 16.19/8.99 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/8.99 => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 16.19/8.99 => ( ( groups7754918857620584856omplex @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 16.19/8.99 = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ F @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6128_sum__diff,axiom,
% 16.19/8.99 ! [A2: set_nat,B3: set_nat,F: nat > real] :
% 16.19/8.99 ( ( finite_finite_nat @ A2 )
% 16.19/8.99 => ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 16.19/8.99 => ( ( groups6591440286371151544t_real @ F @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 16.19/8.99 = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ F @ B3 ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6129_ln__add__one__self__le__self,axiom,
% 16.19/8.99 ! [X: real] :
% 16.19/8.99 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/8.99 => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6130_ln__mult,axiom,
% 16.19/8.99 ! [X: real,Y: real] :
% 16.19/8.99 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/8.99 => ( ( ord_less_real @ zero_zero_real @ Y )
% 16.19/8.99 => ( ( ln_ln_real @ ( times_times_real @ X @ Y ) )
% 16.19/8.99 = ( plus_plus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6131_ln__eq__minus__one,axiom,
% 16.19/8.99 ! [X: real] :
% 16.19/8.99 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/8.99 => ( ( ( ln_ln_real @ X )
% 16.19/8.99 = ( minus_minus_real @ X @ one_one_real ) )
% 16.19/8.99 => ( X = one_one_real ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6132_ln__div,axiom,
% 16.19/8.99 ! [X: real,Y: real] :
% 16.19/8.99 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/8.99 => ( ( ord_less_real @ zero_zero_real @ Y )
% 16.19/8.99 => ( ( ln_ln_real @ ( divide_divide_real @ X @ Y ) )
% 16.19/8.99 = ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6133_pos__minus__divide__less__eq,axiom,
% 16.19/8.99 ! [C: real,B: real,A: real] :
% 16.19/8.99 ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.99 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 16.19/8.99 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6134_pos__minus__divide__less__eq,axiom,
% 16.19/8.99 ! [C: rat,B: rat,A: rat] :
% 16.19/8.99 ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.99 => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 16.19/8.99 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6135_pos__less__minus__divide__eq,axiom,
% 16.19/8.99 ! [C: real,A: real,B: real] :
% 16.19/8.99 ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.99 => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 16.19/8.99 = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6136_pos__less__minus__divide__eq,axiom,
% 16.19/8.99 ! [C: rat,A: rat,B: rat] :
% 16.19/8.99 ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.99 => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 16.19/8.99 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6137_neg__minus__divide__less__eq,axiom,
% 16.19/8.99 ! [C: real,B: real,A: real] :
% 16.19/8.99 ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.99 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 16.19/8.99 = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6138_neg__minus__divide__less__eq,axiom,
% 16.19/8.99 ! [C: rat,B: rat,A: rat] :
% 16.19/8.99 ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.99 => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 16.19/8.99 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6139_neg__less__minus__divide__eq,axiom,
% 16.19/8.99 ! [C: real,A: real,B: real] :
% 16.19/8.99 ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.99 => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 16.19/8.99 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6140_neg__less__minus__divide__eq,axiom,
% 16.19/8.99 ! [C: rat,A: rat,B: rat] :
% 16.19/8.99 ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.99 => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 16.19/8.99 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6141_minus__divide__less__eq,axiom,
% 16.19/8.99 ! [B: real,C: real,A: real] :
% 16.19/8.99 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 16.19/8.99 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.99 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 16.19/8.99 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.99 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.99 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 16.19/8.99 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.99 => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6142_minus__divide__less__eq,axiom,
% 16.19/8.99 ! [B: rat,C: rat,A: rat] :
% 16.19/8.99 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 16.19/8.99 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.99 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 16.19/8.99 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.99 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.99 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 16.19/8.99 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.99 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6143_less__minus__divide__eq,axiom,
% 16.19/8.99 ! [A: real,B: real,C: real] :
% 16.19/8.99 ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 16.19/8.99 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.99 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 16.19/8.99 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 16.19/8.99 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.99 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 16.19/8.99 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 16.19/8.99 => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6144_less__minus__divide__eq,axiom,
% 16.19/8.99 ! [A: rat,B: rat,C: rat] :
% 16.19/8.99 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 16.19/8.99 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.99 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 16.19/8.99 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/8.99 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.99 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 16.19/8.99 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/8.99 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6145_eq__divide__eq__numeral_I2_J,axiom,
% 16.19/8.99 ! [W: num,B: real,C: real] :
% 16.19/8.99 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 16.19/8.99 = ( divide_divide_real @ B @ C ) )
% 16.19/8.99 = ( ( ( C != zero_zero_real )
% 16.19/8.99 => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C )
% 16.19/8.99 = B ) )
% 16.19/8.99 & ( ( C = zero_zero_real )
% 16.19/8.99 => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 16.19/8.99 = zero_zero_real ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6146_eq__divide__eq__numeral_I2_J,axiom,
% 16.19/8.99 ! [W: num,B: complex,C: complex] :
% 16.19/8.99 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 16.19/8.99 = ( divide1717551699836669952omplex @ B @ C ) )
% 16.19/8.99 = ( ( ( C != zero_zero_complex )
% 16.19/8.99 => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C )
% 16.19/8.99 = B ) )
% 16.19/8.99 & ( ( C = zero_zero_complex )
% 16.19/8.99 => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 16.19/8.99 = zero_zero_complex ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6147_eq__divide__eq__numeral_I2_J,axiom,
% 16.19/8.99 ! [W: num,B: rat,C: rat] :
% 16.19/8.99 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 16.19/8.99 = ( divide_divide_rat @ B @ C ) )
% 16.19/8.99 = ( ( ( C != zero_zero_rat )
% 16.19/8.99 => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C )
% 16.19/8.99 = B ) )
% 16.19/8.99 & ( ( C = zero_zero_rat )
% 16.19/8.99 => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 16.19/8.99 = zero_zero_rat ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6148_divide__eq__eq__numeral_I2_J,axiom,
% 16.19/8.99 ! [B: real,C: real,W: num] :
% 16.19/8.99 ( ( ( divide_divide_real @ B @ C )
% 16.19/8.99 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 16.19/8.99 = ( ( ( C != zero_zero_real )
% 16.19/8.99 => ( B
% 16.19/8.99 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 16.19/8.99 & ( ( C = zero_zero_real )
% 16.19/8.99 => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 16.19/8.99 = zero_zero_real ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6149_divide__eq__eq__numeral_I2_J,axiom,
% 16.19/8.99 ! [B: complex,C: complex,W: num] :
% 16.19/8.99 ( ( ( divide1717551699836669952omplex @ B @ C )
% 16.19/8.99 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 16.19/8.99 = ( ( ( C != zero_zero_complex )
% 16.19/8.99 => ( B
% 16.19/8.99 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C ) ) )
% 16.19/8.99 & ( ( C = zero_zero_complex )
% 16.19/8.99 => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 16.19/8.99 = zero_zero_complex ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6150_divide__eq__eq__numeral_I2_J,axiom,
% 16.19/8.99 ! [B: rat,C: rat,W: num] :
% 16.19/8.99 ( ( ( divide_divide_rat @ B @ C )
% 16.19/8.99 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 16.19/8.99 = ( ( ( C != zero_zero_rat )
% 16.19/8.99 => ( B
% 16.19/8.99 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 16.19/8.99 & ( ( C = zero_zero_rat )
% 16.19/8.99 => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 16.19/8.99 = zero_zero_rat ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6151_minus__divide__add__eq__iff,axiom,
% 16.19/8.99 ! [Z: real,X: real,Y: real] :
% 16.19/8.99 ( ( Z != zero_zero_real )
% 16.19/8.99 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
% 16.19/8.99 = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6152_minus__divide__add__eq__iff,axiom,
% 16.19/8.99 ! [Z: complex,X: complex,Y: complex] :
% 16.19/8.99 ( ( Z != zero_zero_complex )
% 16.19/8.99 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y )
% 16.19/8.99 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6153_minus__divide__add__eq__iff,axiom,
% 16.19/8.99 ! [Z: rat,X: rat,Y: rat] :
% 16.19/8.99 ( ( Z != zero_zero_rat )
% 16.19/8.99 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
% 16.19/8.99 = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6154_add__divide__eq__if__simps_I3_J,axiom,
% 16.19/8.99 ! [Z: real,A: real,B: real] :
% 16.19/8.99 ( ( ( Z = zero_zero_real )
% 16.19/8.99 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 16.19/8.99 = B ) )
% 16.19/8.99 & ( ( Z != zero_zero_real )
% 16.19/8.99 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 16.19/8.99 = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6155_add__divide__eq__if__simps_I3_J,axiom,
% 16.19/8.99 ! [Z: complex,A: complex,B: complex] :
% 16.19/8.99 ( ( ( Z = zero_zero_complex )
% 16.19/8.99 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 16.19/8.99 = B ) )
% 16.19/8.99 & ( ( Z != zero_zero_complex )
% 16.19/8.99 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 16.19/8.99 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6156_add__divide__eq__if__simps_I3_J,axiom,
% 16.19/8.99 ! [Z: rat,A: rat,B: rat] :
% 16.19/8.99 ( ( ( Z = zero_zero_rat )
% 16.19/8.99 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 16.19/8.99 = B ) )
% 16.19/8.99 & ( ( Z != zero_zero_rat )
% 16.19/8.99 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 16.19/8.99 = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6157_minus__divide__diff__eq__iff,axiom,
% 16.19/8.99 ! [Z: real,X: real,Y: real] :
% 16.19/8.99 ( ( Z != zero_zero_real )
% 16.19/8.99 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y )
% 16.19/8.99 = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6158_minus__divide__diff__eq__iff,axiom,
% 16.19/8.99 ! [Z: complex,X: complex,Y: complex] :
% 16.19/8.99 ( ( Z != zero_zero_complex )
% 16.19/8.99 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y )
% 16.19/8.99 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6159_minus__divide__diff__eq__iff,axiom,
% 16.19/8.99 ! [Z: rat,X: rat,Y: rat] :
% 16.19/8.99 ( ( Z != zero_zero_rat )
% 16.19/8.99 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y )
% 16.19/8.99 = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6160_add__divide__eq__if__simps_I5_J,axiom,
% 16.19/8.99 ! [Z: real,A: real,B: real] :
% 16.19/8.99 ( ( ( Z = zero_zero_real )
% 16.19/8.99 => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 16.19/8.99 = ( uminus_uminus_real @ B ) ) )
% 16.19/8.99 & ( ( Z != zero_zero_real )
% 16.19/8.99 => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 16.19/8.99 = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6161_add__divide__eq__if__simps_I5_J,axiom,
% 16.19/8.99 ! [Z: complex,A: complex,B: complex] :
% 16.19/8.99 ( ( ( Z = zero_zero_complex )
% 16.19/8.99 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 16.19/8.99 = ( uminus1482373934393186551omplex @ B ) ) )
% 16.19/8.99 & ( ( Z != zero_zero_complex )
% 16.19/8.99 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 16.19/8.99 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6162_add__divide__eq__if__simps_I5_J,axiom,
% 16.19/8.99 ! [Z: rat,A: rat,B: rat] :
% 16.19/8.99 ( ( ( Z = zero_zero_rat )
% 16.19/8.99 => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 16.19/8.99 = ( uminus_uminus_rat @ B ) ) )
% 16.19/8.99 & ( ( Z != zero_zero_rat )
% 16.19/8.99 => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 16.19/8.99 = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6163_add__divide__eq__if__simps_I6_J,axiom,
% 16.19/8.99 ! [Z: real,A: real,B: real] :
% 16.19/8.99 ( ( ( Z = zero_zero_real )
% 16.19/8.99 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 16.19/8.99 = ( uminus_uminus_real @ B ) ) )
% 16.19/8.99 & ( ( Z != zero_zero_real )
% 16.19/8.99 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 16.19/8.99 = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6164_add__divide__eq__if__simps_I6_J,axiom,
% 16.19/8.99 ! [Z: complex,A: complex,B: complex] :
% 16.19/8.99 ( ( ( Z = zero_zero_complex )
% 16.19/8.99 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 16.19/8.99 = ( uminus1482373934393186551omplex @ B ) ) )
% 16.19/8.99 & ( ( Z != zero_zero_complex )
% 16.19/8.99 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 16.19/8.99 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6165_add__divide__eq__if__simps_I6_J,axiom,
% 16.19/8.99 ! [Z: rat,A: rat,B: rat] :
% 16.19/8.99 ( ( ( Z = zero_zero_rat )
% 16.19/8.99 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 16.19/8.99 = ( uminus_uminus_rat @ B ) ) )
% 16.19/8.99 & ( ( Z != zero_zero_rat )
% 16.19/8.99 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 16.19/8.99 = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6166_even__minus,axiom,
% 16.19/8.99 ! [A: int] :
% 16.19/8.99 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A ) )
% 16.19/8.99 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6167_even__minus,axiom,
% 16.19/8.99 ! [A: code_integer] :
% 16.19/8.99 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( uminus1351360451143612070nteger @ A ) )
% 16.19/8.99 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 16.19/8.99
% 16.19/8.99 thf(fact_6168_power2__eq__iff,axiom,
% 16.19/8.99 ! [X: real,Y: real] :
% 16.19/8.99 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/8.99 = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/8.99 = ( ( X = Y )
% 16.19/8.99 | ( X
% 16.19/8.99 = ( uminus_uminus_real @ Y ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6169_power2__eq__iff,axiom,
% 16.19/9.00 ! [X: int,Y: int] :
% 16.19/9.00 ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.00 = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.00 = ( ( X = Y )
% 16.19/9.00 | ( X
% 16.19/9.00 = ( uminus_uminus_int @ Y ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6170_power2__eq__iff,axiom,
% 16.19/9.00 ! [X: complex,Y: complex] :
% 16.19/9.00 ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.00 = ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.00 = ( ( X = Y )
% 16.19/9.00 | ( X
% 16.19/9.00 = ( uminus1482373934393186551omplex @ Y ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6171_power2__eq__iff,axiom,
% 16.19/9.00 ! [X: rat,Y: rat] :
% 16.19/9.00 ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.00 = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.00 = ( ( X = Y )
% 16.19/9.00 | ( X
% 16.19/9.00 = ( uminus_uminus_rat @ Y ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6172_power2__eq__iff,axiom,
% 16.19/9.00 ! [X: code_integer,Y: code_integer] :
% 16.19/9.00 ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.00 = ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.00 = ( ( X = Y )
% 16.19/9.00 | ( X
% 16.19/9.00 = ( uminus1351360451143612070nteger @ Y ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6173_verit__less__mono__div__int2,axiom,
% 16.19/9.00 ! [A2: int,B3: int,N2: int] :
% 16.19/9.00 ( ( ord_less_eq_int @ A2 @ B3 )
% 16.19/9.00 => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N2 ) )
% 16.19/9.00 => ( ord_less_eq_int @ ( divide_divide_int @ B3 @ N2 ) @ ( divide_divide_int @ A2 @ N2 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6174_div__eq__minus1,axiom,
% 16.19/9.00 ! [B: int] :
% 16.19/9.00 ( ( ord_less_int @ zero_zero_int @ B )
% 16.19/9.00 => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 16.19/9.00 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6175_sum__mono2,axiom,
% 16.19/9.00 ! [B3: set_real,A2: set_real,F: real > real] :
% 16.19/9.00 ( ( finite_finite_real @ B3 )
% 16.19/9.00 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 16.19/9.00 => ( ! [B2: real] :
% 16.19/9.00 ( ( member_real @ B2 @ ( minus_minus_set_real @ B3 @ A2 ) )
% 16.19/9.00 => ( ord_less_eq_real @ zero_zero_real @ ( F @ B2 ) ) )
% 16.19/9.00 => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B3 ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6176_sum__mono2,axiom,
% 16.19/9.00 ! [B3: set_int,A2: set_int,F: int > real] :
% 16.19/9.00 ( ( finite_finite_int @ B3 )
% 16.19/9.00 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 16.19/9.00 => ( ! [B2: int] :
% 16.19/9.00 ( ( member_int @ B2 @ ( minus_minus_set_int @ B3 @ A2 ) )
% 16.19/9.00 => ( ord_less_eq_real @ zero_zero_real @ ( F @ B2 ) ) )
% 16.19/9.00 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B3 ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6177_sum__mono2,axiom,
% 16.19/9.00 ! [B3: set_complex,A2: set_complex,F: complex > real] :
% 16.19/9.00 ( ( finite3207457112153483333omplex @ B3 )
% 16.19/9.00 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 16.19/9.00 => ( ! [B2: complex] :
% 16.19/9.00 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
% 16.19/9.00 => ( ord_less_eq_real @ zero_zero_real @ ( F @ B2 ) ) )
% 16.19/9.00 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B3 ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6178_sum__mono2,axiom,
% 16.19/9.00 ! [B3: set_real,A2: set_real,F: real > rat] :
% 16.19/9.00 ( ( finite_finite_real @ B3 )
% 16.19/9.00 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 16.19/9.00 => ( ! [B2: real] :
% 16.19/9.00 ( ( member_real @ B2 @ ( minus_minus_set_real @ B3 @ A2 ) )
% 16.19/9.00 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B2 ) ) )
% 16.19/9.00 => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B3 ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6179_sum__mono2,axiom,
% 16.19/9.00 ! [B3: set_int,A2: set_int,F: int > rat] :
% 16.19/9.00 ( ( finite_finite_int @ B3 )
% 16.19/9.00 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 16.19/9.00 => ( ! [B2: int] :
% 16.19/9.00 ( ( member_int @ B2 @ ( minus_minus_set_int @ B3 @ A2 ) )
% 16.19/9.00 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B2 ) ) )
% 16.19/9.00 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B3 ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6180_sum__mono2,axiom,
% 16.19/9.00 ! [B3: set_complex,A2: set_complex,F: complex > rat] :
% 16.19/9.00 ( ( finite3207457112153483333omplex @ B3 )
% 16.19/9.00 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 16.19/9.00 => ( ! [B2: complex] :
% 16.19/9.00 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
% 16.19/9.00 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B2 ) ) )
% 16.19/9.00 => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B3 ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6181_sum__mono2,axiom,
% 16.19/9.00 ! [B3: set_real,A2: set_real,F: real > nat] :
% 16.19/9.00 ( ( finite_finite_real @ B3 )
% 16.19/9.00 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 16.19/9.00 => ( ! [B2: real] :
% 16.19/9.00 ( ( member_real @ B2 @ ( minus_minus_set_real @ B3 @ A2 ) )
% 16.19/9.00 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B2 ) ) )
% 16.19/9.00 => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B3 ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6182_sum__mono2,axiom,
% 16.19/9.00 ! [B3: set_int,A2: set_int,F: int > nat] :
% 16.19/9.00 ( ( finite_finite_int @ B3 )
% 16.19/9.00 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 16.19/9.00 => ( ! [B2: int] :
% 16.19/9.00 ( ( member_int @ B2 @ ( minus_minus_set_int @ B3 @ A2 ) )
% 16.19/9.00 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B2 ) ) )
% 16.19/9.00 => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B3 ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6183_sum__mono2,axiom,
% 16.19/9.00 ! [B3: set_complex,A2: set_complex,F: complex > nat] :
% 16.19/9.00 ( ( finite3207457112153483333omplex @ B3 )
% 16.19/9.00 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 16.19/9.00 => ( ! [B2: complex] :
% 16.19/9.00 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
% 16.19/9.00 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B2 ) ) )
% 16.19/9.00 => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B3 ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6184_sum__mono2,axiom,
% 16.19/9.00 ! [B3: set_real,A2: set_real,F: real > int] :
% 16.19/9.00 ( ( finite_finite_real @ B3 )
% 16.19/9.00 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 16.19/9.00 => ( ! [B2: real] :
% 16.19/9.00 ( ( member_real @ B2 @ ( minus_minus_set_real @ B3 @ A2 ) )
% 16.19/9.00 => ( ord_less_eq_int @ zero_zero_int @ ( F @ B2 ) ) )
% 16.19/9.00 => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B3 ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6185_ln__le__minus__one,axiom,
% 16.19/9.00 ! [X: real] :
% 16.19/9.00 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.00 => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6186_ln__diff__le,axiom,
% 16.19/9.00 ! [X: real,Y: real] :
% 16.19/9.00 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.00 => ( ( ord_less_real @ zero_zero_real @ Y )
% 16.19/9.00 => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) @ ( divide_divide_real @ ( minus_minus_real @ X @ Y ) @ Y ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6187_le__minus__divide__eq,axiom,
% 16.19/9.00 ! [A: real,B: real,C: real] :
% 16.19/9.00 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 16.19/9.00 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/9.00 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 16.19/9.00 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 16.19/9.00 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/9.00 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 16.19/9.00 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 16.19/9.00 => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6188_le__minus__divide__eq,axiom,
% 16.19/9.00 ! [A: rat,B: rat,C: rat] :
% 16.19/9.00 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 16.19/9.00 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/9.00 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 16.19/9.00 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/9.00 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/9.00 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 16.19/9.00 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/9.00 => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6189_minus__divide__le__eq,axiom,
% 16.19/9.00 ! [B: real,C: real,A: real] :
% 16.19/9.00 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 16.19/9.00 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/9.00 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 16.19/9.00 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 16.19/9.00 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/9.00 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 16.19/9.00 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 16.19/9.00 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6190_minus__divide__le__eq,axiom,
% 16.19/9.00 ! [B: rat,C: rat,A: rat] :
% 16.19/9.00 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 16.19/9.00 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/9.00 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 16.19/9.00 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/9.00 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/9.00 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 16.19/9.00 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/9.00 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6191_neg__le__minus__divide__eq,axiom,
% 16.19/9.00 ! [C: real,A: real,B: real] :
% 16.19/9.00 ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/9.00 => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 16.19/9.00 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6192_neg__le__minus__divide__eq,axiom,
% 16.19/9.00 ! [C: rat,A: rat,B: rat] :
% 16.19/9.00 ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/9.00 => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 16.19/9.00 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6193_neg__minus__divide__le__eq,axiom,
% 16.19/9.00 ! [C: real,B: real,A: real] :
% 16.19/9.00 ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/9.00 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 16.19/9.00 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6194_neg__minus__divide__le__eq,axiom,
% 16.19/9.00 ! [C: rat,B: rat,A: rat] :
% 16.19/9.00 ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/9.00 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 16.19/9.00 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6195_pos__le__minus__divide__eq,axiom,
% 16.19/9.00 ! [C: real,A: real,B: real] :
% 16.19/9.00 ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/9.00 => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 16.19/9.00 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6196_pos__le__minus__divide__eq,axiom,
% 16.19/9.00 ! [C: rat,A: rat,B: rat] :
% 16.19/9.00 ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/9.00 => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 16.19/9.00 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6197_pos__minus__divide__le__eq,axiom,
% 16.19/9.00 ! [C: real,B: real,A: real] :
% 16.19/9.00 ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/9.00 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 16.19/9.00 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6198_pos__minus__divide__le__eq,axiom,
% 16.19/9.00 ! [C: rat,B: rat,A: rat] :
% 16.19/9.00 ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/9.00 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 16.19/9.00 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6199_less__divide__eq__numeral_I2_J,axiom,
% 16.19/9.00 ! [W: num,B: real,C: real] :
% 16.19/9.00 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
% 16.19/9.00 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/9.00 => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 16.19/9.00 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 16.19/9.00 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/9.00 => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 16.19/9.00 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 16.19/9.00 => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6200_less__divide__eq__numeral_I2_J,axiom,
% 16.19/9.00 ! [W: num,B: rat,C: rat] :
% 16.19/9.00 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
% 16.19/9.00 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/9.00 => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 16.19/9.00 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/9.00 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/9.00 => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 16.19/9.00 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/9.00 => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6201_divide__less__eq__numeral_I2_J,axiom,
% 16.19/9.00 ! [B: real,C: real,W: num] :
% 16.19/9.00 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 16.19/9.00 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/9.00 => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 16.19/9.00 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 16.19/9.00 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/9.00 => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 16.19/9.00 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 16.19/9.00 => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6202_divide__less__eq__numeral_I2_J,axiom,
% 16.19/9.00 ! [B: rat,C: rat,W: num] :
% 16.19/9.00 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 16.19/9.00 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/9.00 => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 16.19/9.00 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/9.00 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/9.00 => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 16.19/9.00 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/9.00 => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6203_power2__eq__1__iff,axiom,
% 16.19/9.00 ! [A: real] :
% 16.19/9.00 ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.00 = one_one_real )
% 16.19/9.00 = ( ( A = one_one_real )
% 16.19/9.00 | ( A
% 16.19/9.00 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6204_power2__eq__1__iff,axiom,
% 16.19/9.00 ! [A: int] :
% 16.19/9.00 ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.00 = one_one_int )
% 16.19/9.00 = ( ( A = one_one_int )
% 16.19/9.00 | ( A
% 16.19/9.00 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6205_power2__eq__1__iff,axiom,
% 16.19/9.00 ! [A: complex] :
% 16.19/9.00 ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.00 = one_one_complex )
% 16.19/9.00 = ( ( A = one_one_complex )
% 16.19/9.00 | ( A
% 16.19/9.00 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6206_power2__eq__1__iff,axiom,
% 16.19/9.00 ! [A: rat] :
% 16.19/9.00 ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.00 = one_one_rat )
% 16.19/9.00 = ( ( A = one_one_rat )
% 16.19/9.00 | ( A
% 16.19/9.00 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6207_power2__eq__1__iff,axiom,
% 16.19/9.00 ! [A: code_integer] :
% 16.19/9.00 ( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.00 = one_one_Code_integer )
% 16.19/9.00 = ( ( A = one_one_Code_integer )
% 16.19/9.00 | ( A
% 16.19/9.00 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6208_uminus__power__if,axiom,
% 16.19/9.00 ! [N2: nat,A: real] :
% 16.19/9.00 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.00 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 )
% 16.19/9.00 = ( power_power_real @ A @ N2 ) ) )
% 16.19/9.00 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.00 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 )
% 16.19/9.00 = ( uminus_uminus_real @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6209_uminus__power__if,axiom,
% 16.19/9.00 ! [N2: nat,A: int] :
% 16.19/9.00 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.00 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
% 16.19/9.00 = ( power_power_int @ A @ N2 ) ) )
% 16.19/9.00 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.00 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
% 16.19/9.00 = ( uminus_uminus_int @ ( power_power_int @ A @ N2 ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6210_uminus__power__if,axiom,
% 16.19/9.00 ! [N2: nat,A: complex] :
% 16.19/9.00 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.00 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 )
% 16.19/9.00 = ( power_power_complex @ A @ N2 ) ) )
% 16.19/9.00 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.00 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 )
% 16.19/9.00 = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N2 ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6211_uminus__power__if,axiom,
% 16.19/9.00 ! [N2: nat,A: rat] :
% 16.19/9.00 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.00 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 )
% 16.19/9.00 = ( power_power_rat @ A @ N2 ) ) )
% 16.19/9.00 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.00 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 )
% 16.19/9.00 = ( uminus_uminus_rat @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6212_uminus__power__if,axiom,
% 16.19/9.00 ! [N2: nat,A: code_integer] :
% 16.19/9.00 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.00 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 )
% 16.19/9.00 = ( power_8256067586552552935nteger @ A @ N2 ) ) )
% 16.19/9.00 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.00 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 )
% 16.19/9.00 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6213_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 16.19/9.00 ! [K: nat,N2: nat] :
% 16.19/9.00 ( ( ord_less_eq_nat @ K @ N2 )
% 16.19/9.00 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N2 @ K ) )
% 16.19/9.00 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6214_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 16.19/9.00 ! [K: nat,N2: nat] :
% 16.19/9.00 ( ( ord_less_eq_nat @ K @ N2 )
% 16.19/9.00 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N2 @ K ) )
% 16.19/9.00 = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6215_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 16.19/9.00 ! [K: nat,N2: nat] :
% 16.19/9.00 ( ( ord_less_eq_nat @ K @ N2 )
% 16.19/9.00 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N2 @ K ) )
% 16.19/9.00 = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6216_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 16.19/9.00 ! [K: nat,N2: nat] :
% 16.19/9.00 ( ( ord_less_eq_nat @ K @ N2 )
% 16.19/9.00 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N2 @ K ) )
% 16.19/9.00 = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6217_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 16.19/9.00 ! [K: nat,N2: nat] :
% 16.19/9.00 ( ( ord_less_eq_nat @ K @ N2 )
% 16.19/9.00 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N2 @ K ) )
% 16.19/9.00 = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6218_realpow__square__minus__le,axiom,
% 16.19/9.00 ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6219_ln__one__minus__pos__lower__bound,axiom,
% 16.19/9.00 ! [X: real] :
% 16.19/9.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.00 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.00 => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6220_signed__take__bit__int__less__eq__self__iff,axiom,
% 16.19/9.00 ! [N2: nat,K: int] :
% 16.19/9.00 ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ K )
% 16.19/9.00 = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6221_signed__take__bit__int__greater__eq__minus__exp,axiom,
% 16.19/9.00 ! [N2: nat,K: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( bit_ri631733984087533419it_int @ N2 @ K ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6222_signed__take__bit__int__greater__self__iff,axiom,
% 16.19/9.00 ! [K: int,N2: nat] :
% 16.19/9.00 ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N2 @ K ) )
% 16.19/9.00 = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6223_minus__mod__int__eq,axiom,
% 16.19/9.00 ! [L2: int,K: int] :
% 16.19/9.00 ( ( ord_less_eq_int @ zero_zero_int @ L2 )
% 16.19/9.00 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L2 )
% 16.19/9.00 = ( minus_minus_int @ ( minus_minus_int @ L2 @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L2 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6224_zmod__minus1,axiom,
% 16.19/9.00 ! [B: int] :
% 16.19/9.00 ( ( ord_less_int @ zero_zero_int @ B )
% 16.19/9.00 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 16.19/9.00 = ( minus_minus_int @ B @ one_one_int ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6225_sum__strict__mono2,axiom,
% 16.19/9.00 ! [B3: set_real,A2: set_real,B: real,F: real > real] :
% 16.19/9.00 ( ( finite_finite_real @ B3 )
% 16.19/9.00 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 16.19/9.00 => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A2 ) )
% 16.19/9.00 => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
% 16.19/9.00 => ( ! [X4: real] :
% 16.19/9.00 ( ( member_real @ X4 @ B3 )
% 16.19/9.00 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 16.19/9.00 => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B3 ) ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6226_sum__strict__mono2,axiom,
% 16.19/9.00 ! [B3: set_int,A2: set_int,B: int,F: int > real] :
% 16.19/9.00 ( ( finite_finite_int @ B3 )
% 16.19/9.00 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 16.19/9.00 => ( ( member_int @ B @ ( minus_minus_set_int @ B3 @ A2 ) )
% 16.19/9.00 => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
% 16.19/9.00 => ( ! [X4: int] :
% 16.19/9.00 ( ( member_int @ X4 @ B3 )
% 16.19/9.00 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 16.19/9.00 => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B3 ) ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6227_sum__strict__mono2,axiom,
% 16.19/9.00 ! [B3: set_complex,A2: set_complex,B: complex,F: complex > real] :
% 16.19/9.00 ( ( finite3207457112153483333omplex @ B3 )
% 16.19/9.00 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 16.19/9.00 => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
% 16.19/9.00 => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
% 16.19/9.00 => ( ! [X4: complex] :
% 16.19/9.00 ( ( member_complex @ X4 @ B3 )
% 16.19/9.00 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 16.19/9.00 => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B3 ) ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6228_sum__strict__mono2,axiom,
% 16.19/9.00 ! [B3: set_real,A2: set_real,B: real,F: real > rat] :
% 16.19/9.00 ( ( finite_finite_real @ B3 )
% 16.19/9.00 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 16.19/9.00 => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A2 ) )
% 16.19/9.00 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 16.19/9.00 => ( ! [X4: real] :
% 16.19/9.00 ( ( member_real @ X4 @ B3 )
% 16.19/9.00 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 16.19/9.00 => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B3 ) ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6229_sum__strict__mono2,axiom,
% 16.19/9.00 ! [B3: set_int,A2: set_int,B: int,F: int > rat] :
% 16.19/9.00 ( ( finite_finite_int @ B3 )
% 16.19/9.00 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 16.19/9.00 => ( ( member_int @ B @ ( minus_minus_set_int @ B3 @ A2 ) )
% 16.19/9.00 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 16.19/9.00 => ( ! [X4: int] :
% 16.19/9.00 ( ( member_int @ X4 @ B3 )
% 16.19/9.00 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 16.19/9.00 => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B3 ) ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6230_sum__strict__mono2,axiom,
% 16.19/9.00 ! [B3: set_complex,A2: set_complex,B: complex,F: complex > rat] :
% 16.19/9.00 ( ( finite3207457112153483333omplex @ B3 )
% 16.19/9.00 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 16.19/9.00 => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
% 16.19/9.00 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 16.19/9.00 => ( ! [X4: complex] :
% 16.19/9.00 ( ( member_complex @ X4 @ B3 )
% 16.19/9.00 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 16.19/9.00 => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B3 ) ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6231_sum__strict__mono2,axiom,
% 16.19/9.00 ! [B3: set_real,A2: set_real,B: real,F: real > nat] :
% 16.19/9.00 ( ( finite_finite_real @ B3 )
% 16.19/9.00 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 16.19/9.00 => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A2 ) )
% 16.19/9.00 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
% 16.19/9.00 => ( ! [X4: real] :
% 16.19/9.00 ( ( member_real @ X4 @ B3 )
% 16.19/9.00 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 16.19/9.00 => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B3 ) ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6232_sum__strict__mono2,axiom,
% 16.19/9.00 ! [B3: set_int,A2: set_int,B: int,F: int > nat] :
% 16.19/9.00 ( ( finite_finite_int @ B3 )
% 16.19/9.00 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 16.19/9.00 => ( ( member_int @ B @ ( minus_minus_set_int @ B3 @ A2 ) )
% 16.19/9.00 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
% 16.19/9.00 => ( ! [X4: int] :
% 16.19/9.00 ( ( member_int @ X4 @ B3 )
% 16.19/9.00 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 16.19/9.00 => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B3 ) ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6233_sum__strict__mono2,axiom,
% 16.19/9.00 ! [B3: set_complex,A2: set_complex,B: complex,F: complex > nat] :
% 16.19/9.00 ( ( finite3207457112153483333omplex @ B3 )
% 16.19/9.00 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 16.19/9.00 => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B3 @ A2 ) )
% 16.19/9.00 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
% 16.19/9.00 => ( ! [X4: complex] :
% 16.19/9.00 ( ( member_complex @ X4 @ B3 )
% 16.19/9.00 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 16.19/9.00 => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B3 ) ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6234_sum__strict__mono2,axiom,
% 16.19/9.00 ! [B3: set_real,A2: set_real,B: real,F: real > int] :
% 16.19/9.00 ( ( finite_finite_real @ B3 )
% 16.19/9.00 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 16.19/9.00 => ( ( member_real @ B @ ( minus_minus_set_real @ B3 @ A2 ) )
% 16.19/9.00 => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
% 16.19/9.00 => ( ! [X4: real] :
% 16.19/9.00 ( ( member_real @ X4 @ B3 )
% 16.19/9.00 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 16.19/9.00 => ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B3 ) ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6235_zdiv__zminus1__eq__if,axiom,
% 16.19/9.00 ! [B: int,A: int] :
% 16.19/9.00 ( ( B != zero_zero_int )
% 16.19/9.00 => ( ( ( ( modulo_modulo_int @ A @ B )
% 16.19/9.00 = zero_zero_int )
% 16.19/9.00 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 16.19/9.00 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
% 16.19/9.00 & ( ( ( modulo_modulo_int @ A @ B )
% 16.19/9.00 != zero_zero_int )
% 16.19/9.00 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 16.19/9.00 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6236_zdiv__zminus2__eq__if,axiom,
% 16.19/9.00 ! [B: int,A: int] :
% 16.19/9.00 ( ( B != zero_zero_int )
% 16.19/9.00 => ( ( ( ( modulo_modulo_int @ A @ B )
% 16.19/9.00 = zero_zero_int )
% 16.19/9.00 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 16.19/9.00 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
% 16.19/9.00 & ( ( ( modulo_modulo_int @ A @ B )
% 16.19/9.00 != zero_zero_int )
% 16.19/9.00 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 16.19/9.00 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6237_zminus1__lemma,axiom,
% 16.19/9.00 ! [A: int,B: int,Q2: int,R: int] :
% 16.19/9.00 ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R ) )
% 16.19/9.00 => ( ( B != zero_zero_int )
% 16.19/9.00 => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R = zero_zero_int ) @ ( uminus_uminus_int @ Q2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q2 ) @ one_one_int ) ) @ ( if_int @ ( R = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6238_le__divide__eq__numeral_I2_J,axiom,
% 16.19/9.00 ! [W: num,B: real,C: real] :
% 16.19/9.00 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
% 16.19/9.00 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/9.00 => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 16.19/9.00 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 16.19/9.00 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/9.00 => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 16.19/9.00 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 16.19/9.00 => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6239_le__divide__eq__numeral_I2_J,axiom,
% 16.19/9.00 ! [W: num,B: rat,C: rat] :
% 16.19/9.00 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
% 16.19/9.00 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/9.00 => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 16.19/9.00 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/9.00 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/9.00 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 16.19/9.00 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/9.00 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6240_divide__le__eq__numeral_I2_J,axiom,
% 16.19/9.00 ! [B: real,C: real,W: num] :
% 16.19/9.00 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 16.19/9.00 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/9.00 => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 16.19/9.00 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 16.19/9.00 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 16.19/9.00 => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 16.19/9.00 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 16.19/9.00 => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6241_divide__le__eq__numeral_I2_J,axiom,
% 16.19/9.00 ! [B: rat,C: rat,W: num] :
% 16.19/9.00 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 16.19/9.00 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/9.00 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 16.19/9.00 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 16.19/9.00 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/9.00 => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 16.19/9.00 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 16.19/9.00 => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6242_square__le__1,axiom,
% 16.19/9.00 ! [X: real] :
% 16.19/9.00 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 16.19/9.00 => ( ( ord_less_eq_real @ X @ one_one_real )
% 16.19/9.00 => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6243_square__le__1,axiom,
% 16.19/9.00 ! [X: code_integer] :
% 16.19/9.00 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
% 16.19/9.00 => ( ( ord_le3102999989581377725nteger @ X @ one_one_Code_integer )
% 16.19/9.00 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6244_square__le__1,axiom,
% 16.19/9.00 ! [X: rat] :
% 16.19/9.00 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X )
% 16.19/9.00 => ( ( ord_less_eq_rat @ X @ one_one_rat )
% 16.19/9.00 => ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6245_square__le__1,axiom,
% 16.19/9.00 ! [X: int] :
% 16.19/9.00 ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X )
% 16.19/9.00 => ( ( ord_less_eq_int @ X @ one_one_int )
% 16.19/9.00 => ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6246_minus__power__mult__self,axiom,
% 16.19/9.00 ! [A: real,N2: nat] :
% 16.19/9.00 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 ) )
% 16.19/9.00 = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6247_minus__power__mult__self,axiom,
% 16.19/9.00 ! [A: int,N2: nat] :
% 16.19/9.00 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 ) )
% 16.19/9.00 = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6248_minus__power__mult__self,axiom,
% 16.19/9.00 ! [A: complex,N2: nat] :
% 16.19/9.00 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 ) )
% 16.19/9.00 = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6249_minus__power__mult__self,axiom,
% 16.19/9.00 ! [A: rat,N2: nat] :
% 16.19/9.00 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 ) )
% 16.19/9.00 = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6250_minus__power__mult__self,axiom,
% 16.19/9.00 ! [A: code_integer,N2: nat] :
% 16.19/9.00 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 ) )
% 16.19/9.00 = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6251_minus__one__power__iff,axiom,
% 16.19/9.00 ! [N2: nat] :
% 16.19/9.00 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.00 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 )
% 16.19/9.00 = one_one_real ) )
% 16.19/9.00 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.00 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 )
% 16.19/9.00 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6252_minus__one__power__iff,axiom,
% 16.19/9.00 ! [N2: nat] :
% 16.19/9.00 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.00 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
% 16.19/9.00 = one_one_int ) )
% 16.19/9.00 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.00 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
% 16.19/9.00 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6253_minus__one__power__iff,axiom,
% 16.19/9.00 ! [N2: nat] :
% 16.19/9.00 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.00 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 )
% 16.19/9.00 = one_one_complex ) )
% 16.19/9.00 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.00 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 )
% 16.19/9.00 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6254_minus__one__power__iff,axiom,
% 16.19/9.00 ! [N2: nat] :
% 16.19/9.00 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.00 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
% 16.19/9.00 = one_one_rat ) )
% 16.19/9.00 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.00 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
% 16.19/9.00 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6255_minus__one__power__iff,axiom,
% 16.19/9.00 ! [N2: nat] :
% 16.19/9.00 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.00 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
% 16.19/9.00 = one_one_Code_integer ) )
% 16.19/9.00 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.00 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
% 16.19/9.00 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6256_signed__take__bit__int__eq__self,axiom,
% 16.19/9.00 ! [N2: nat,K: int] :
% 16.19/9.00 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K )
% 16.19/9.00 => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/9.00 => ( ( bit_ri631733984087533419it_int @ N2 @ K )
% 16.19/9.00 = K ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6257_signed__take__bit__int__eq__self__iff,axiom,
% 16.19/9.00 ! [N2: nat,K: int] :
% 16.19/9.00 ( ( ( bit_ri631733984087533419it_int @ N2 @ K )
% 16.19/9.00 = K )
% 16.19/9.00 = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K )
% 16.19/9.00 & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6258_minus__1__div__exp__eq__int,axiom,
% 16.19/9.00 ! [N2: nat] :
% 16.19/9.00 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/9.00 = ( uminus_uminus_int @ one_one_int ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6259_div__pos__neg__trivial,axiom,
% 16.19/9.00 ! [K: int,L2: int] :
% 16.19/9.00 ( ( ord_less_int @ zero_zero_int @ K )
% 16.19/9.00 => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L2 ) @ zero_zero_int )
% 16.19/9.00 => ( ( divide_divide_int @ K @ L2 )
% 16.19/9.00 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6260_add__0__iff,axiom,
% 16.19/9.00 ! [B: complex,A: complex] :
% 16.19/9.00 ( ( B
% 16.19/9.00 = ( plus_plus_complex @ B @ A ) )
% 16.19/9.00 = ( A = zero_zero_complex ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6261_add__0__iff,axiom,
% 16.19/9.00 ! [B: real,A: real] :
% 16.19/9.00 ( ( B
% 16.19/9.00 = ( plus_plus_real @ B @ A ) )
% 16.19/9.00 = ( A = zero_zero_real ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6262_add__0__iff,axiom,
% 16.19/9.00 ! [B: rat,A: rat] :
% 16.19/9.00 ( ( B
% 16.19/9.00 = ( plus_plus_rat @ B @ A ) )
% 16.19/9.00 = ( A = zero_zero_rat ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6263_add__0__iff,axiom,
% 16.19/9.00 ! [B: nat,A: nat] :
% 16.19/9.00 ( ( B
% 16.19/9.00 = ( plus_plus_nat @ B @ A ) )
% 16.19/9.00 = ( A = zero_zero_nat ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6264_add__0__iff,axiom,
% 16.19/9.00 ! [B: int,A: int] :
% 16.19/9.00 ( ( B
% 16.19/9.00 = ( plus_plus_int @ B @ A ) )
% 16.19/9.00 = ( A = zero_zero_int ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6265_crossproduct__noteq,axiom,
% 16.19/9.00 ! [A: real,B: real,C: real,D: real] :
% 16.19/9.00 ( ( ( A != B )
% 16.19/9.00 & ( C != D ) )
% 16.19/9.00 = ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) )
% 16.19/9.00 != ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6266_crossproduct__noteq,axiom,
% 16.19/9.00 ! [A: rat,B: rat,C: rat,D: rat] :
% 16.19/9.00 ( ( ( A != B )
% 16.19/9.00 & ( C != D ) )
% 16.19/9.00 = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) )
% 16.19/9.00 != ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6267_crossproduct__noteq,axiom,
% 16.19/9.00 ! [A: nat,B: nat,C: nat,D: nat] :
% 16.19/9.00 ( ( ( A != B )
% 16.19/9.00 & ( C != D ) )
% 16.19/9.00 = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
% 16.19/9.00 != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6268_crossproduct__noteq,axiom,
% 16.19/9.00 ! [A: int,B: int,C: int,D: int] :
% 16.19/9.00 ( ( ( A != B )
% 16.19/9.00 & ( C != D ) )
% 16.19/9.00 = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
% 16.19/9.00 != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6269_crossproduct__eq,axiom,
% 16.19/9.00 ! [W: real,Y: real,X: real,Z: real] :
% 16.19/9.00 ( ( ( plus_plus_real @ ( times_times_real @ W @ Y ) @ ( times_times_real @ X @ Z ) )
% 16.19/9.00 = ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X @ Y ) ) )
% 16.19/9.00 = ( ( W = X )
% 16.19/9.00 | ( Y = Z ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6270_crossproduct__eq,axiom,
% 16.19/9.00 ! [W: rat,Y: rat,X: rat,Z: rat] :
% 16.19/9.00 ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y ) @ ( times_times_rat @ X @ Z ) )
% 16.19/9.00 = ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X @ Y ) ) )
% 16.19/9.00 = ( ( W = X )
% 16.19/9.00 | ( Y = Z ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6271_crossproduct__eq,axiom,
% 16.19/9.00 ! [W: nat,Y: nat,X: nat,Z: nat] :
% 16.19/9.00 ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z ) )
% 16.19/9.00 = ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y ) ) )
% 16.19/9.00 = ( ( W = X )
% 16.19/9.00 | ( Y = Z ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6272_crossproduct__eq,axiom,
% 16.19/9.00 ! [W: int,Y: int,X: int,Z: int] :
% 16.19/9.00 ( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z ) )
% 16.19/9.00 = ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X @ Y ) ) )
% 16.19/9.00 = ( ( W = X )
% 16.19/9.00 | ( Y = Z ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6273_power__minus1__odd,axiom,
% 16.19/9.00 ! [N2: nat] :
% 16.19/9.00 ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 16.19/9.00 = ( uminus_uminus_real @ one_one_real ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6274_power__minus1__odd,axiom,
% 16.19/9.00 ! [N2: nat] :
% 16.19/9.00 ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 16.19/9.00 = ( uminus_uminus_int @ one_one_int ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6275_power__minus1__odd,axiom,
% 16.19/9.00 ! [N2: nat] :
% 16.19/9.00 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 16.19/9.00 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6276_power__minus1__odd,axiom,
% 16.19/9.00 ! [N2: nat] :
% 16.19/9.00 ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 16.19/9.00 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6277_power__minus1__odd,axiom,
% 16.19/9.00 ! [N2: nat] :
% 16.19/9.00 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 16.19/9.00 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6278_int__bit__induct,axiom,
% 16.19/9.00 ! [P: int > $o,K: int] :
% 16.19/9.00 ( ( P @ zero_zero_int )
% 16.19/9.00 => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
% 16.19/9.00 => ( ! [K2: int] :
% 16.19/9.00 ( ( P @ K2 )
% 16.19/9.00 => ( ( K2 != zero_zero_int )
% 16.19/9.00 => ( P @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
% 16.19/9.00 => ( ! [K2: int] :
% 16.19/9.00 ( ( P @ K2 )
% 16.19/9.00 => ( ( K2
% 16.19/9.00 != ( uminus_uminus_int @ one_one_int ) )
% 16.19/9.00 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
% 16.19/9.00 => ( P @ K ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6279_divmod__step__nat__def,axiom,
% 16.19/9.00 ( unique5026877609467782581ep_nat
% 16.19/9.00 = ( ^ [L: num] :
% 16.19/9.00 ( produc2626176000494625587at_nat
% 16.19/9.00 @ ^ [Q4: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6280_ln__one__plus__pos__lower__bound,axiom,
% 16.19/9.00 ! [X: real] :
% 16.19/9.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.00 => ( ( ord_less_eq_real @ X @ one_one_real )
% 16.19/9.00 => ( ord_less_eq_real @ ( minus_minus_real @ X @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6281_signed__take__bit__int__greater__eq,axiom,
% 16.19/9.00 ! [K: int,N2: nat] :
% 16.19/9.00 ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
% 16.19/9.00 => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) ) @ ( bit_ri631733984087533419it_int @ N2 @ K ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6282_divmod__step__int__def,axiom,
% 16.19/9.00 ( unique5024387138958732305ep_int
% 16.19/9.00 = ( ^ [L: num] :
% 16.19/9.00 ( produc4245557441103728435nt_int
% 16.19/9.00 @ ^ [Q4: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6283_ln__2__less__1,axiom,
% 16.19/9.00 ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6284_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
% 16.19/9.00 ! [X: real] :
% 16.19/9.00 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 16.19/9.00 => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 16.19/9.00 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6285_tanh__ln__real,axiom,
% 16.19/9.00 ! [X: real] :
% 16.19/9.00 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.00 => ( ( tanh_real @ ( ln_ln_real @ X ) )
% 16.19/9.00 = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6286_divmod__algorithm__code_I5_J,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
% 16.19/9.00 = ( produc4245557441103728435nt_int
% 16.19/9.00 @ ^ [Q4: int,R5: int] : ( product_Pair_int_int @ Q4 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) )
% 16.19/9.00 @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6287_divmod__algorithm__code_I5_J,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
% 16.19/9.00 = ( produc2626176000494625587at_nat
% 16.19/9.00 @ ^ [Q4: nat,R5: nat] : ( product_Pair_nat_nat @ Q4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) )
% 16.19/9.00 @ ( unique5055182867167087721od_nat @ M @ N2 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6288_divmod__algorithm__code_I5_J,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
% 16.19/9.00 = ( produc6916734918728496179nteger
% 16.19/9.00 @ ^ [Q4: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q4 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) )
% 16.19/9.00 @ ( unique3479559517661332726nteger @ M @ N2 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6289_divmod__nat__if,axiom,
% 16.19/9.00 ( divmod_nat
% 16.19/9.00 = ( ^ [M6: nat,N: nat] :
% 16.19/9.00 ( if_Pro6206227464963214023at_nat
% 16.19/9.00 @ ( ( N = zero_zero_nat )
% 16.19/9.00 | ( ord_less_nat @ M6 @ N ) )
% 16.19/9.00 @ ( product_Pair_nat_nat @ zero_zero_nat @ M6 )
% 16.19/9.00 @ ( produc2626176000494625587at_nat
% 16.19/9.00 @ ^ [Q4: nat] : ( product_Pair_nat_nat @ ( suc @ Q4 ) )
% 16.19/9.00 @ ( divmod_nat @ ( minus_minus_nat @ M6 @ N ) @ N ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6290_signed__take__bit__Suc__minus__bit1,axiom,
% 16.19/9.00 ! [N2: nat,K: num] :
% 16.19/9.00 ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 16.19/9.00 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6291_abs__ln__one__plus__x__minus__x__bound,axiom,
% 16.19/9.00 ! [X: real] :
% 16.19/9.00 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.00 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6292_semiring__norm_I90_J,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( ( bit1 @ M )
% 16.19/9.00 = ( bit1 @ N2 ) )
% 16.19/9.00 = ( M = N2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6293_case__prodI,axiom,
% 16.19/9.00 ! [F: code_integer > $o > $o,A: code_integer,B: $o] :
% 16.19/9.00 ( ( F @ A @ B )
% 16.19/9.00 => ( produc7828578312038201481er_o_o @ F @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6294_case__prodI,axiom,
% 16.19/9.00 ! [F: num > num > $o,A: num,B: num] :
% 16.19/9.00 ( ( F @ A @ B )
% 16.19/9.00 => ( produc5703948589228662326_num_o @ F @ ( product_Pair_num_num @ A @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6295_case__prodI,axiom,
% 16.19/9.00 ! [F: nat > num > $o,A: nat,B: num] :
% 16.19/9.00 ( ( F @ A @ B )
% 16.19/9.00 => ( produc4927758841916487424_num_o @ F @ ( product_Pair_nat_num @ A @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6296_case__prodI,axiom,
% 16.19/9.00 ! [F: nat > nat > $o,A: nat,B: nat] :
% 16.19/9.00 ( ( F @ A @ B )
% 16.19/9.00 => ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6297_case__prodI,axiom,
% 16.19/9.00 ! [F: int > int > $o,A: int,B: int] :
% 16.19/9.00 ( ( F @ A @ B )
% 16.19/9.00 => ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6298_case__prodI2,axiom,
% 16.19/9.00 ! [P4: produc6271795597528267376eger_o,C: code_integer > $o > $o] :
% 16.19/9.00 ( ! [A3: code_integer,B2: $o] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( produc6677183202524767010eger_o @ A3 @ B2 ) )
% 16.19/9.00 => ( C @ A3 @ B2 ) )
% 16.19/9.00 => ( produc7828578312038201481er_o_o @ C @ P4 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6299_case__prodI2,axiom,
% 16.19/9.00 ! [P4: product_prod_num_num,C: num > num > $o] :
% 16.19/9.00 ( ! [A3: num,B2: num] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( product_Pair_num_num @ A3 @ B2 ) )
% 16.19/9.00 => ( C @ A3 @ B2 ) )
% 16.19/9.00 => ( produc5703948589228662326_num_o @ C @ P4 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6300_case__prodI2,axiom,
% 16.19/9.00 ! [P4: product_prod_nat_num,C: nat > num > $o] :
% 16.19/9.00 ( ! [A3: nat,B2: num] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( product_Pair_nat_num @ A3 @ B2 ) )
% 16.19/9.00 => ( C @ A3 @ B2 ) )
% 16.19/9.00 => ( produc4927758841916487424_num_o @ C @ P4 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6301_case__prodI2,axiom,
% 16.19/9.00 ! [P4: product_prod_nat_nat,C: nat > nat > $o] :
% 16.19/9.00 ( ! [A3: nat,B2: nat] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( product_Pair_nat_nat @ A3 @ B2 ) )
% 16.19/9.00 => ( C @ A3 @ B2 ) )
% 16.19/9.00 => ( produc6081775807080527818_nat_o @ C @ P4 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6302_case__prodI2,axiom,
% 16.19/9.00 ! [P4: product_prod_int_int,C: int > int > $o] :
% 16.19/9.00 ( ! [A3: int,B2: int] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( product_Pair_int_int @ A3 @ B2 ) )
% 16.19/9.00 => ( C @ A3 @ B2 ) )
% 16.19/9.00 => ( produc4947309494688390418_int_o @ C @ P4 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6303_mem__case__prodI,axiom,
% 16.19/9.00 ! [Z: nat,C: code_integer > $o > set_nat,A: code_integer,B: $o] :
% 16.19/9.00 ( ( member_nat @ Z @ ( C @ A @ B ) )
% 16.19/9.00 => ( member_nat @ Z @ ( produc5431169771168744661et_nat @ C @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6304_mem__case__prodI,axiom,
% 16.19/9.00 ! [Z: real,C: code_integer > $o > set_real,A: code_integer,B: $o] :
% 16.19/9.00 ( ( member_real @ Z @ ( C @ A @ B ) )
% 16.19/9.00 => ( member_real @ Z @ ( produc242741666403216561t_real @ C @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6305_mem__case__prodI,axiom,
% 16.19/9.00 ! [Z: int,C: code_integer > $o > set_int,A: code_integer,B: $o] :
% 16.19/9.00 ( ( member_int @ Z @ ( C @ A @ B ) )
% 16.19/9.00 => ( member_int @ Z @ ( produc1253318751659547953et_int @ C @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6306_mem__case__prodI,axiom,
% 16.19/9.00 ! [Z: complex,C: code_integer > $o > set_complex,A: code_integer,B: $o] :
% 16.19/9.00 ( ( member_complex @ Z @ ( C @ A @ B ) )
% 16.19/9.00 => ( member_complex @ Z @ ( produc1043322548047392435omplex @ C @ ( produc6677183202524767010eger_o @ A @ B ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6307_mem__case__prodI,axiom,
% 16.19/9.00 ! [Z: nat,C: num > num > set_nat,A: num,B: num] :
% 16.19/9.00 ( ( member_nat @ Z @ ( C @ A @ B ) )
% 16.19/9.00 => ( member_nat @ Z @ ( produc1361121860356118632et_nat @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6308_mem__case__prodI,axiom,
% 16.19/9.00 ! [Z: real,C: num > num > set_real,A: num,B: num] :
% 16.19/9.00 ( ( member_real @ Z @ ( C @ A @ B ) )
% 16.19/9.00 => ( member_real @ Z @ ( produc8296048397933160132t_real @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6309_mem__case__prodI,axiom,
% 16.19/9.00 ! [Z: int,C: num > num > set_int,A: num,B: num] :
% 16.19/9.00 ( ( member_int @ Z @ ( C @ A @ B ) )
% 16.19/9.00 => ( member_int @ Z @ ( produc6406642877701697732et_int @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6310_mem__case__prodI,axiom,
% 16.19/9.00 ! [Z: complex,C: num > num > set_complex,A: num,B: num] :
% 16.19/9.00 ( ( member_complex @ Z @ ( C @ A @ B ) )
% 16.19/9.00 => ( member_complex @ Z @ ( produc2866383454006189126omplex @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6311_mem__case__prodI,axiom,
% 16.19/9.00 ! [Z: nat,C: nat > num > set_nat,A: nat,B: num] :
% 16.19/9.00 ( ( member_nat @ Z @ ( C @ A @ B ) )
% 16.19/9.00 => ( member_nat @ Z @ ( produc4130284055270567454et_nat @ C @ ( product_Pair_nat_num @ A @ B ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6312_mem__case__prodI,axiom,
% 16.19/9.00 ! [Z: real,C: nat > num > set_real,A: nat,B: num] :
% 16.19/9.00 ( ( member_real @ Z @ ( C @ A @ B ) )
% 16.19/9.00 => ( member_real @ Z @ ( produc1435849484188172666t_real @ C @ ( product_Pair_nat_num @ A @ B ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6313_mem__case__prodI2,axiom,
% 16.19/9.00 ! [P4: produc6271795597528267376eger_o,Z: nat,C: code_integer > $o > set_nat] :
% 16.19/9.00 ( ! [A3: code_integer,B2: $o] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( produc6677183202524767010eger_o @ A3 @ B2 ) )
% 16.19/9.00 => ( member_nat @ Z @ ( C @ A3 @ B2 ) ) )
% 16.19/9.00 => ( member_nat @ Z @ ( produc5431169771168744661et_nat @ C @ P4 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6314_mem__case__prodI2,axiom,
% 16.19/9.00 ! [P4: produc6271795597528267376eger_o,Z: real,C: code_integer > $o > set_real] :
% 16.19/9.00 ( ! [A3: code_integer,B2: $o] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( produc6677183202524767010eger_o @ A3 @ B2 ) )
% 16.19/9.00 => ( member_real @ Z @ ( C @ A3 @ B2 ) ) )
% 16.19/9.00 => ( member_real @ Z @ ( produc242741666403216561t_real @ C @ P4 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6315_mem__case__prodI2,axiom,
% 16.19/9.00 ! [P4: produc6271795597528267376eger_o,Z: int,C: code_integer > $o > set_int] :
% 16.19/9.00 ( ! [A3: code_integer,B2: $o] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( produc6677183202524767010eger_o @ A3 @ B2 ) )
% 16.19/9.00 => ( member_int @ Z @ ( C @ A3 @ B2 ) ) )
% 16.19/9.00 => ( member_int @ Z @ ( produc1253318751659547953et_int @ C @ P4 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6316_mem__case__prodI2,axiom,
% 16.19/9.00 ! [P4: produc6271795597528267376eger_o,Z: complex,C: code_integer > $o > set_complex] :
% 16.19/9.00 ( ! [A3: code_integer,B2: $o] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( produc6677183202524767010eger_o @ A3 @ B2 ) )
% 16.19/9.00 => ( member_complex @ Z @ ( C @ A3 @ B2 ) ) )
% 16.19/9.00 => ( member_complex @ Z @ ( produc1043322548047392435omplex @ C @ P4 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6317_mem__case__prodI2,axiom,
% 16.19/9.00 ! [P4: product_prod_num_num,Z: nat,C: num > num > set_nat] :
% 16.19/9.00 ( ! [A3: num,B2: num] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( product_Pair_num_num @ A3 @ B2 ) )
% 16.19/9.00 => ( member_nat @ Z @ ( C @ A3 @ B2 ) ) )
% 16.19/9.00 => ( member_nat @ Z @ ( produc1361121860356118632et_nat @ C @ P4 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6318_mem__case__prodI2,axiom,
% 16.19/9.00 ! [P4: product_prod_num_num,Z: real,C: num > num > set_real] :
% 16.19/9.00 ( ! [A3: num,B2: num] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( product_Pair_num_num @ A3 @ B2 ) )
% 16.19/9.00 => ( member_real @ Z @ ( C @ A3 @ B2 ) ) )
% 16.19/9.00 => ( member_real @ Z @ ( produc8296048397933160132t_real @ C @ P4 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6319_mem__case__prodI2,axiom,
% 16.19/9.00 ! [P4: product_prod_num_num,Z: int,C: num > num > set_int] :
% 16.19/9.00 ( ! [A3: num,B2: num] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( product_Pair_num_num @ A3 @ B2 ) )
% 16.19/9.00 => ( member_int @ Z @ ( C @ A3 @ B2 ) ) )
% 16.19/9.00 => ( member_int @ Z @ ( produc6406642877701697732et_int @ C @ P4 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6320_mem__case__prodI2,axiom,
% 16.19/9.00 ! [P4: product_prod_num_num,Z: complex,C: num > num > set_complex] :
% 16.19/9.00 ( ! [A3: num,B2: num] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( product_Pair_num_num @ A3 @ B2 ) )
% 16.19/9.00 => ( member_complex @ Z @ ( C @ A3 @ B2 ) ) )
% 16.19/9.00 => ( member_complex @ Z @ ( produc2866383454006189126omplex @ C @ P4 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6321_mem__case__prodI2,axiom,
% 16.19/9.00 ! [P4: product_prod_nat_num,Z: nat,C: nat > num > set_nat] :
% 16.19/9.00 ( ! [A3: nat,B2: num] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( product_Pair_nat_num @ A3 @ B2 ) )
% 16.19/9.00 => ( member_nat @ Z @ ( C @ A3 @ B2 ) ) )
% 16.19/9.00 => ( member_nat @ Z @ ( produc4130284055270567454et_nat @ C @ P4 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6322_mem__case__prodI2,axiom,
% 16.19/9.00 ! [P4: product_prod_nat_num,Z: real,C: nat > num > set_real] :
% 16.19/9.00 ( ! [A3: nat,B2: num] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( product_Pair_nat_num @ A3 @ B2 ) )
% 16.19/9.00 => ( member_real @ Z @ ( C @ A3 @ B2 ) ) )
% 16.19/9.00 => ( member_real @ Z @ ( produc1435849484188172666t_real @ C @ P4 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6323_case__prodI2_H,axiom,
% 16.19/9.00 ! [P4: product_prod_nat_nat,C: nat > nat > product_prod_nat_nat > $o,X: product_prod_nat_nat] :
% 16.19/9.00 ( ! [A3: nat,B2: nat] :
% 16.19/9.00 ( ( ( product_Pair_nat_nat @ A3 @ B2 )
% 16.19/9.00 = P4 )
% 16.19/9.00 => ( C @ A3 @ B2 @ X ) )
% 16.19/9.00 => ( produc8739625826339149834_nat_o @ C @ P4 @ X ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6324_semiring__norm_I89_J,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( bit1 @ M )
% 16.19/9.00 != ( bit0 @ N2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6325_semiring__norm_I88_J,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( bit0 @ M )
% 16.19/9.00 != ( bit1 @ N2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6326_semiring__norm_I86_J,axiom,
% 16.19/9.00 ! [M: num] :
% 16.19/9.00 ( ( bit1 @ M )
% 16.19/9.00 != one ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6327_semiring__norm_I84_J,axiom,
% 16.19/9.00 ! [N2: num] :
% 16.19/9.00 ( one
% 16.19/9.00 != ( bit1 @ N2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6328_abs__numeral,axiom,
% 16.19/9.00 ! [N2: num] :
% 16.19/9.00 ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N2 ) )
% 16.19/9.00 = ( numera6620942414471956472nteger @ N2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6329_abs__numeral,axiom,
% 16.19/9.00 ! [N2: num] :
% 16.19/9.00 ( ( abs_abs_real @ ( numeral_numeral_real @ N2 ) )
% 16.19/9.00 = ( numeral_numeral_real @ N2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6330_abs__numeral,axiom,
% 16.19/9.00 ! [N2: num] :
% 16.19/9.00 ( ( abs_abs_rat @ ( numeral_numeral_rat @ N2 ) )
% 16.19/9.00 = ( numeral_numeral_rat @ N2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6331_abs__numeral,axiom,
% 16.19/9.00 ! [N2: num] :
% 16.19/9.00 ( ( abs_abs_int @ ( numeral_numeral_int @ N2 ) )
% 16.19/9.00 = ( numeral_numeral_int @ N2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6332_abs__mult__self__eq,axiom,
% 16.19/9.00 ! [A: code_integer] :
% 16.19/9.00 ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
% 16.19/9.00 = ( times_3573771949741848930nteger @ A @ A ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6333_abs__mult__self__eq,axiom,
% 16.19/9.00 ! [A: real] :
% 16.19/9.00 ( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
% 16.19/9.00 = ( times_times_real @ A @ A ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6334_abs__mult__self__eq,axiom,
% 16.19/9.00 ! [A: rat] :
% 16.19/9.00 ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
% 16.19/9.00 = ( times_times_rat @ A @ A ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6335_abs__mult__self__eq,axiom,
% 16.19/9.00 ! [A: int] :
% 16.19/9.00 ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
% 16.19/9.00 = ( times_times_int @ A @ A ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6336_abs__add__abs,axiom,
% 16.19/9.00 ! [A: code_integer,B: code_integer] :
% 16.19/9.00 ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
% 16.19/9.00 = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6337_abs__add__abs,axiom,
% 16.19/9.00 ! [A: real,B: real] :
% 16.19/9.00 ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
% 16.19/9.00 = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6338_abs__add__abs,axiom,
% 16.19/9.00 ! [A: rat,B: rat] :
% 16.19/9.00 ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
% 16.19/9.00 = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6339_abs__add__abs,axiom,
% 16.19/9.00 ! [A: int,B: int] :
% 16.19/9.00 ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
% 16.19/9.00 = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6340_abs__1,axiom,
% 16.19/9.00 ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 16.19/9.00 = one_one_Code_integer ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6341_abs__1,axiom,
% 16.19/9.00 ( ( abs_abs_complex @ one_one_complex )
% 16.19/9.00 = one_one_complex ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6342_abs__1,axiom,
% 16.19/9.00 ( ( abs_abs_real @ one_one_real )
% 16.19/9.00 = one_one_real ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6343_abs__1,axiom,
% 16.19/9.00 ( ( abs_abs_rat @ one_one_rat )
% 16.19/9.00 = one_one_rat ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6344_abs__1,axiom,
% 16.19/9.00 ( ( abs_abs_int @ one_one_int )
% 16.19/9.00 = one_one_int ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6345_abs__divide,axiom,
% 16.19/9.00 ! [A: complex,B: complex] :
% 16.19/9.00 ( ( abs_abs_complex @ ( divide1717551699836669952omplex @ A @ B ) )
% 16.19/9.00 = ( divide1717551699836669952omplex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6346_abs__divide,axiom,
% 16.19/9.00 ! [A: real,B: real] :
% 16.19/9.00 ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 16.19/9.00 = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6347_abs__divide,axiom,
% 16.19/9.00 ! [A: rat,B: rat] :
% 16.19/9.00 ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 16.19/9.00 = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6348_tanh__real__less__iff,axiom,
% 16.19/9.00 ! [X: real,Y: real] :
% 16.19/9.00 ( ( ord_less_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) )
% 16.19/9.00 = ( ord_less_real @ X @ Y ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6349_tanh__real__le__iff,axiom,
% 16.19/9.00 ! [X: real,Y: real] :
% 16.19/9.00 ( ( ord_less_eq_real @ ( tanh_real @ X ) @ ( tanh_real @ Y ) )
% 16.19/9.00 = ( ord_less_eq_real @ X @ Y ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6350_semiring__norm_I80_J,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( ord_less_num @ M @ N2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6351_semiring__norm_I73_J,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( ord_less_eq_num @ M @ N2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6352_abs__sum__abs,axiom,
% 16.19/9.00 ! [F: int > int,A2: set_int] :
% 16.19/9.00 ( ( abs_abs_int
% 16.19/9.00 @ ( groups4538972089207619220nt_int
% 16.19/9.00 @ ^ [A5: int] : ( abs_abs_int @ ( F @ A5 ) )
% 16.19/9.00 @ A2 ) )
% 16.19/9.00 = ( groups4538972089207619220nt_int
% 16.19/9.00 @ ^ [A5: int] : ( abs_abs_int @ ( F @ A5 ) )
% 16.19/9.00 @ A2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6353_abs__sum__abs,axiom,
% 16.19/9.00 ! [F: nat > real,A2: set_nat] :
% 16.19/9.00 ( ( abs_abs_real
% 16.19/9.00 @ ( groups6591440286371151544t_real
% 16.19/9.00 @ ^ [A5: nat] : ( abs_abs_real @ ( F @ A5 ) )
% 16.19/9.00 @ A2 ) )
% 16.19/9.00 = ( groups6591440286371151544t_real
% 16.19/9.00 @ ^ [A5: nat] : ( abs_abs_real @ ( F @ A5 ) )
% 16.19/9.00 @ A2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6354_abs__le__zero__iff,axiom,
% 16.19/9.00 ! [A: code_integer] :
% 16.19/9.00 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 16.19/9.00 = ( A = zero_z3403309356797280102nteger ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6355_abs__le__zero__iff,axiom,
% 16.19/9.00 ! [A: real] :
% 16.19/9.00 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
% 16.19/9.00 = ( A = zero_zero_real ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6356_abs__le__zero__iff,axiom,
% 16.19/9.00 ! [A: rat] :
% 16.19/9.00 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
% 16.19/9.00 = ( A = zero_zero_rat ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6357_abs__le__zero__iff,axiom,
% 16.19/9.00 ! [A: int] :
% 16.19/9.00 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
% 16.19/9.00 = ( A = zero_zero_int ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6358_abs__le__self__iff,axiom,
% 16.19/9.00 ! [A: code_integer] :
% 16.19/9.00 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
% 16.19/9.00 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6359_abs__le__self__iff,axiom,
% 16.19/9.00 ! [A: real] :
% 16.19/9.00 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
% 16.19/9.00 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6360_abs__le__self__iff,axiom,
% 16.19/9.00 ! [A: rat] :
% 16.19/9.00 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
% 16.19/9.00 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6361_abs__le__self__iff,axiom,
% 16.19/9.00 ! [A: int] :
% 16.19/9.00 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
% 16.19/9.00 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6362_abs__of__nonneg,axiom,
% 16.19/9.00 ! [A: code_integer] :
% 16.19/9.00 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 16.19/9.00 => ( ( abs_abs_Code_integer @ A )
% 16.19/9.00 = A ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6363_abs__of__nonneg,axiom,
% 16.19/9.00 ! [A: real] :
% 16.19/9.00 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/9.00 => ( ( abs_abs_real @ A )
% 16.19/9.00 = A ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6364_abs__of__nonneg,axiom,
% 16.19/9.00 ! [A: rat] :
% 16.19/9.00 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/9.00 => ( ( abs_abs_rat @ A )
% 16.19/9.00 = A ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6365_abs__of__nonneg,axiom,
% 16.19/9.00 ! [A: int] :
% 16.19/9.00 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/9.00 => ( ( abs_abs_int @ A )
% 16.19/9.00 = A ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6366_zero__less__abs__iff,axiom,
% 16.19/9.00 ! [A: code_integer] :
% 16.19/9.00 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
% 16.19/9.00 = ( A != zero_z3403309356797280102nteger ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6367_zero__less__abs__iff,axiom,
% 16.19/9.00 ! [A: real] :
% 16.19/9.00 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
% 16.19/9.00 = ( A != zero_zero_real ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6368_zero__less__abs__iff,axiom,
% 16.19/9.00 ! [A: rat] :
% 16.19/9.00 ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
% 16.19/9.00 = ( A != zero_zero_rat ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6369_zero__less__abs__iff,axiom,
% 16.19/9.00 ! [A: int] :
% 16.19/9.00 ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
% 16.19/9.00 = ( A != zero_zero_int ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6370_abs__neg__numeral,axiom,
% 16.19/9.00 ! [N2: num] :
% 16.19/9.00 ( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 16.19/9.00 = ( numeral_numeral_real @ N2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6371_abs__neg__numeral,axiom,
% 16.19/9.00 ! [N2: num] :
% 16.19/9.00 ( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/9.00 = ( numeral_numeral_int @ N2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6372_abs__neg__numeral,axiom,
% 16.19/9.00 ! [N2: num] :
% 16.19/9.00 ( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 16.19/9.00 = ( numeral_numeral_rat @ N2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6373_abs__neg__numeral,axiom,
% 16.19/9.00 ! [N2: num] :
% 16.19/9.00 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
% 16.19/9.00 = ( numera6620942414471956472nteger @ N2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6374_abs__neg__one,axiom,
% 16.19/9.00 ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
% 16.19/9.00 = one_one_real ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6375_abs__neg__one,axiom,
% 16.19/9.00 ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
% 16.19/9.00 = one_one_int ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6376_abs__neg__one,axiom,
% 16.19/9.00 ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 16.19/9.00 = one_one_rat ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6377_abs__neg__one,axiom,
% 16.19/9.00 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 16.19/9.00 = one_one_Code_integer ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6378_abs__power__minus,axiom,
% 16.19/9.00 ! [A: real,N2: nat] :
% 16.19/9.00 ( ( abs_abs_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 ) )
% 16.19/9.00 = ( abs_abs_real @ ( power_power_real @ A @ N2 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6379_abs__power__minus,axiom,
% 16.19/9.00 ! [A: int,N2: nat] :
% 16.19/9.00 ( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 ) )
% 16.19/9.00 = ( abs_abs_int @ ( power_power_int @ A @ N2 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6380_abs__power__minus,axiom,
% 16.19/9.00 ! [A: rat,N2: nat] :
% 16.19/9.00 ( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 ) )
% 16.19/9.00 = ( abs_abs_rat @ ( power_power_rat @ A @ N2 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6381_abs__power__minus,axiom,
% 16.19/9.00 ! [A: code_integer,N2: nat] :
% 16.19/9.00 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 ) )
% 16.19/9.00 = ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6382_semiring__norm_I7_J,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( bit1 @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6383_semiring__norm_I9_J,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 16.19/9.00 = ( bit1 @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6384_semiring__norm_I14_J,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( bit0 @ ( times_times_num @ M @ ( bit1 @ N2 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6385_semiring__norm_I15_J,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 16.19/9.00 = ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N2 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6386_semiring__norm_I81_J,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 16.19/9.00 = ( ord_less_num @ M @ N2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6387_semiring__norm_I72_J,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( ord_less_eq_num @ M @ N2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6388_semiring__norm_I77_J,axiom,
% 16.19/9.00 ! [N2: num] : ( ord_less_num @ one @ ( bit1 @ N2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6389_semiring__norm_I70_J,axiom,
% 16.19/9.00 ! [M: num] :
% 16.19/9.00 ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6390_tanh__real__pos__iff,axiom,
% 16.19/9.00 ! [X: real] :
% 16.19/9.00 ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X ) )
% 16.19/9.00 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6391_tanh__real__neg__iff,axiom,
% 16.19/9.00 ! [X: real] :
% 16.19/9.00 ( ( ord_less_real @ ( tanh_real @ X ) @ zero_zero_real )
% 16.19/9.00 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6392_tanh__real__nonpos__iff,axiom,
% 16.19/9.00 ! [X: real] :
% 16.19/9.00 ( ( ord_less_eq_real @ ( tanh_real @ X ) @ zero_zero_real )
% 16.19/9.00 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6393_tanh__real__nonneg__iff,axiom,
% 16.19/9.00 ! [X: real] :
% 16.19/9.00 ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X ) )
% 16.19/9.00 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6394_sum__abs,axiom,
% 16.19/9.00 ! [F: int > int,A2: set_int] :
% 16.19/9.00 ( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F @ A2 ) )
% 16.19/9.00 @ ( groups4538972089207619220nt_int
% 16.19/9.00 @ ^ [I3: int] : ( abs_abs_int @ ( F @ I3 ) )
% 16.19/9.00 @ A2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6395_sum__abs,axiom,
% 16.19/9.00 ! [F: nat > real,A2: set_nat] :
% 16.19/9.00 ( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
% 16.19/9.00 @ ( groups6591440286371151544t_real
% 16.19/9.00 @ ^ [I3: nat] : ( abs_abs_real @ ( F @ I3 ) )
% 16.19/9.00 @ A2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6396_divide__le__0__abs__iff,axiom,
% 16.19/9.00 ! [A: real,B: real] :
% 16.19/9.00 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
% 16.19/9.00 = ( ( ord_less_eq_real @ A @ zero_zero_real )
% 16.19/9.00 | ( B = zero_zero_real ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6397_divide__le__0__abs__iff,axiom,
% 16.19/9.00 ! [A: rat,B: rat] :
% 16.19/9.00 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) @ zero_zero_rat )
% 16.19/9.00 = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 16.19/9.00 | ( B = zero_zero_rat ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6398_zero__le__divide__abs__iff,axiom,
% 16.19/9.00 ! [A: real,B: real] :
% 16.19/9.00 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
% 16.19/9.00 = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/9.00 | ( B = zero_zero_real ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6399_zero__le__divide__abs__iff,axiom,
% 16.19/9.00 ! [A: rat,B: rat] :
% 16.19/9.00 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) )
% 16.19/9.00 = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/9.00 | ( B = zero_zero_rat ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6400_abs__of__nonpos,axiom,
% 16.19/9.00 ! [A: real] :
% 16.19/9.00 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 16.19/9.00 => ( ( abs_abs_real @ A )
% 16.19/9.00 = ( uminus_uminus_real @ A ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6401_abs__of__nonpos,axiom,
% 16.19/9.00 ! [A: code_integer] :
% 16.19/9.00 ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
% 16.19/9.00 => ( ( abs_abs_Code_integer @ A )
% 16.19/9.00 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6402_abs__of__nonpos,axiom,
% 16.19/9.00 ! [A: rat] :
% 16.19/9.00 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 16.19/9.00 => ( ( abs_abs_rat @ A )
% 16.19/9.00 = ( uminus_uminus_rat @ A ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6403_abs__of__nonpos,axiom,
% 16.19/9.00 ! [A: int] :
% 16.19/9.00 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 16.19/9.00 => ( ( abs_abs_int @ A )
% 16.19/9.00 = ( uminus_uminus_int @ A ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6404_zdiv__numeral__Bit1,axiom,
% 16.19/9.00 ! [V: num,W: num] :
% 16.19/9.00 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 16.19/9.00 = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6405_semiring__norm_I10_J,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N2 ) @ one ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6406_semiring__norm_I8_J,axiom,
% 16.19/9.00 ! [M: num] :
% 16.19/9.00 ( ( plus_plus_num @ ( bit1 @ M ) @ one )
% 16.19/9.00 = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6407_semiring__norm_I5_J,axiom,
% 16.19/9.00 ! [M: num] :
% 16.19/9.00 ( ( plus_plus_num @ ( bit0 @ M ) @ one )
% 16.19/9.00 = ( bit1 @ M ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6408_semiring__norm_I4_J,axiom,
% 16.19/9.00 ! [N2: num] :
% 16.19/9.00 ( ( plus_plus_num @ one @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( bit0 @ ( plus_plus_num @ N2 @ one ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6409_semiring__norm_I3_J,axiom,
% 16.19/9.00 ! [N2: num] :
% 16.19/9.00 ( ( plus_plus_num @ one @ ( bit0 @ N2 ) )
% 16.19/9.00 = ( bit1 @ N2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6410_artanh__minus__real,axiom,
% 16.19/9.00 ! [X: real] :
% 16.19/9.00 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 16.19/9.00 => ( ( artanh_real @ ( uminus_uminus_real @ X ) )
% 16.19/9.00 = ( uminus_uminus_real @ ( artanh_real @ X ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6411_sum__abs__ge__zero,axiom,
% 16.19/9.00 ! [F: int > int,A2: set_int] :
% 16.19/9.00 ( ord_less_eq_int @ zero_zero_int
% 16.19/9.00 @ ( groups4538972089207619220nt_int
% 16.19/9.00 @ ^ [I3: int] : ( abs_abs_int @ ( F @ I3 ) )
% 16.19/9.00 @ A2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6412_sum__abs__ge__zero,axiom,
% 16.19/9.00 ! [F: nat > real,A2: set_nat] :
% 16.19/9.00 ( ord_less_eq_real @ zero_zero_real
% 16.19/9.00 @ ( groups6591440286371151544t_real
% 16.19/9.00 @ ^ [I3: nat] : ( abs_abs_real @ ( F @ I3 ) )
% 16.19/9.00 @ A2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6413_semiring__norm_I16_J,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N2 ) @ ( bit0 @ ( times_times_num @ M @ N2 ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6414_semiring__norm_I79_J,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( ord_less_eq_num @ M @ N2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6415_semiring__norm_I74_J,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 16.19/9.00 = ( ord_less_num @ M @ N2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6416_zero__less__power__abs__iff,axiom,
% 16.19/9.00 ! [A: code_integer,N2: nat] :
% 16.19/9.00 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N2 ) )
% 16.19/9.00 = ( ( A != zero_z3403309356797280102nteger )
% 16.19/9.00 | ( N2 = zero_zero_nat ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6417_zero__less__power__abs__iff,axiom,
% 16.19/9.00 ! [A: real,N2: nat] :
% 16.19/9.00 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) )
% 16.19/9.00 = ( ( A != zero_zero_real )
% 16.19/9.00 | ( N2 = zero_zero_nat ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6418_zero__less__power__abs__iff,axiom,
% 16.19/9.00 ! [A: rat,N2: nat] :
% 16.19/9.00 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N2 ) )
% 16.19/9.00 = ( ( A != zero_zero_rat )
% 16.19/9.00 | ( N2 = zero_zero_nat ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6419_zero__less__power__abs__iff,axiom,
% 16.19/9.00 ! [A: int,N2: nat] :
% 16.19/9.00 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N2 ) )
% 16.19/9.00 = ( ( A != zero_zero_int )
% 16.19/9.00 | ( N2 = zero_zero_nat ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6420_abs__power2,axiom,
% 16.19/9.00 ! [A: rat] :
% 16.19/9.00 ( ( abs_abs_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.00 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6421_abs__power2,axiom,
% 16.19/9.00 ! [A: code_integer] :
% 16.19/9.00 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.00 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6422_abs__power2,axiom,
% 16.19/9.00 ! [A: real] :
% 16.19/9.00 ( ( abs_abs_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.00 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6423_abs__power2,axiom,
% 16.19/9.00 ! [A: int] :
% 16.19/9.00 ( ( abs_abs_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.00 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6424_power2__abs,axiom,
% 16.19/9.00 ! [A: rat] :
% 16.19/9.00 ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.00 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6425_power2__abs,axiom,
% 16.19/9.00 ! [A: code_integer] :
% 16.19/9.00 ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.00 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6426_power2__abs,axiom,
% 16.19/9.00 ! [A: real] :
% 16.19/9.00 ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.00 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6427_power2__abs,axiom,
% 16.19/9.00 ! [A: int] :
% 16.19/9.00 ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.00 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6428_sum_Ocl__ivl__Suc,axiom,
% 16.19/9.00 ! [N2: nat,M: nat,G: nat > complex] :
% 16.19/9.00 ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 16.19/9.00 => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 16.19/9.00 = zero_zero_complex ) )
% 16.19/9.00 & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 16.19/9.00 => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 16.19/9.00 = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6429_sum_Ocl__ivl__Suc,axiom,
% 16.19/9.00 ! [N2: nat,M: nat,G: nat > rat] :
% 16.19/9.00 ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 16.19/9.00 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 16.19/9.00 = zero_zero_rat ) )
% 16.19/9.00 & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 16.19/9.00 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 16.19/9.00 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6430_sum_Ocl__ivl__Suc,axiom,
% 16.19/9.00 ! [N2: nat,M: nat,G: nat > int] :
% 16.19/9.00 ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 16.19/9.00 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 16.19/9.00 = zero_zero_int ) )
% 16.19/9.00 & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 16.19/9.00 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 16.19/9.00 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6431_sum_Ocl__ivl__Suc,axiom,
% 16.19/9.00 ! [N2: nat,M: nat,G: nat > nat] :
% 16.19/9.00 ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 16.19/9.00 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 16.19/9.00 = zero_zero_nat ) )
% 16.19/9.00 & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 16.19/9.00 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 16.19/9.00 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6432_sum_Ocl__ivl__Suc,axiom,
% 16.19/9.00 ! [N2: nat,M: nat,G: nat > real] :
% 16.19/9.00 ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 16.19/9.00 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 16.19/9.00 = zero_zero_real ) )
% 16.19/9.00 & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 16.19/9.00 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 16.19/9.00 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6433_dvd__numeral__simp,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( dvd_dvd_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 16.19/9.00 = ( unique6319869463603278526ux_int @ ( unique5052692396658037445od_int @ N2 @ M ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6434_dvd__numeral__simp,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
% 16.19/9.00 = ( unique6322359934112328802ux_nat @ ( unique5055182867167087721od_nat @ N2 @ M ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6435_dvd__numeral__simp,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) )
% 16.19/9.00 = ( unique5706413561485394159nteger @ ( unique3479559517661332726nteger @ N2 @ M ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6436_divmod__algorithm__code_I2_J,axiom,
% 16.19/9.00 ! [M: num] :
% 16.19/9.00 ( ( unique5052692396658037445od_int @ M @ one )
% 16.19/9.00 = ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6437_divmod__algorithm__code_I2_J,axiom,
% 16.19/9.00 ! [M: num] :
% 16.19/9.00 ( ( unique5055182867167087721od_nat @ M @ one )
% 16.19/9.00 = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6438_divmod__algorithm__code_I2_J,axiom,
% 16.19/9.00 ! [M: num] :
% 16.19/9.00 ( ( unique3479559517661332726nteger @ M @ one )
% 16.19/9.00 = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6439_sum__zero__power,axiom,
% 16.19/9.00 ! [A2: set_nat,C: nat > complex] :
% 16.19/9.00 ( ( ( ( finite_finite_nat @ A2 )
% 16.19/9.00 & ( member_nat @ zero_zero_nat @ A2 ) )
% 16.19/9.00 => ( ( groups2073611262835488442omplex
% 16.19/9.00 @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) )
% 16.19/9.00 @ A2 )
% 16.19/9.00 = ( C @ zero_zero_nat ) ) )
% 16.19/9.00 & ( ~ ( ( finite_finite_nat @ A2 )
% 16.19/9.00 & ( member_nat @ zero_zero_nat @ A2 ) )
% 16.19/9.00 => ( ( groups2073611262835488442omplex
% 16.19/9.00 @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) )
% 16.19/9.00 @ A2 )
% 16.19/9.00 = zero_zero_complex ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6440_sum__zero__power,axiom,
% 16.19/9.00 ! [A2: set_nat,C: nat > rat] :
% 16.19/9.00 ( ( ( ( finite_finite_nat @ A2 )
% 16.19/9.00 & ( member_nat @ zero_zero_nat @ A2 ) )
% 16.19/9.00 => ( ( groups2906978787729119204at_rat
% 16.19/9.00 @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) )
% 16.19/9.00 @ A2 )
% 16.19/9.00 = ( C @ zero_zero_nat ) ) )
% 16.19/9.00 & ( ~ ( ( finite_finite_nat @ A2 )
% 16.19/9.00 & ( member_nat @ zero_zero_nat @ A2 ) )
% 16.19/9.00 => ( ( groups2906978787729119204at_rat
% 16.19/9.00 @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) )
% 16.19/9.00 @ A2 )
% 16.19/9.00 = zero_zero_rat ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6441_sum__zero__power,axiom,
% 16.19/9.00 ! [A2: set_nat,C: nat > real] :
% 16.19/9.00 ( ( ( ( finite_finite_nat @ A2 )
% 16.19/9.00 & ( member_nat @ zero_zero_nat @ A2 ) )
% 16.19/9.00 => ( ( groups6591440286371151544t_real
% 16.19/9.00 @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) )
% 16.19/9.00 @ A2 )
% 16.19/9.00 = ( C @ zero_zero_nat ) ) )
% 16.19/9.00 & ( ~ ( ( finite_finite_nat @ A2 )
% 16.19/9.00 & ( member_nat @ zero_zero_nat @ A2 ) )
% 16.19/9.00 => ( ( groups6591440286371151544t_real
% 16.19/9.00 @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) )
% 16.19/9.00 @ A2 )
% 16.19/9.00 = zero_zero_real ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6442_power__even__abs__numeral,axiom,
% 16.19/9.00 ! [W: num,A: rat] :
% 16.19/9.00 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 16.19/9.00 => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ W ) )
% 16.19/9.00 = ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6443_power__even__abs__numeral,axiom,
% 16.19/9.00 ! [W: num,A: code_integer] :
% 16.19/9.00 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 16.19/9.00 => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ W ) )
% 16.19/9.00 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6444_power__even__abs__numeral,axiom,
% 16.19/9.00 ! [W: num,A: real] :
% 16.19/9.00 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 16.19/9.00 => ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ W ) )
% 16.19/9.00 = ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6445_power__even__abs__numeral,axiom,
% 16.19/9.00 ! [W: num,A: int] :
% 16.19/9.00 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 16.19/9.00 => ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ W ) )
% 16.19/9.00 = ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6446_div__Suc__eq__div__add3,axiom,
% 16.19/9.00 ! [M: nat,N2: nat] :
% 16.19/9.00 ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
% 16.19/9.00 = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6447_Suc__div__eq__add3__div__numeral,axiom,
% 16.19/9.00 ! [M: nat,V: num] :
% 16.19/9.00 ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 16.19/9.00 = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6448_divmod__algorithm__code_I3_J,axiom,
% 16.19/9.00 ! [N2: num] :
% 16.19/9.00 ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N2 ) )
% 16.19/9.00 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6449_divmod__algorithm__code_I3_J,axiom,
% 16.19/9.00 ! [N2: num] :
% 16.19/9.00 ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N2 ) )
% 16.19/9.00 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6450_divmod__algorithm__code_I3_J,axiom,
% 16.19/9.00 ! [N2: num] :
% 16.19/9.00 ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N2 ) )
% 16.19/9.00 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6451_mod__Suc__eq__mod__add3,axiom,
% 16.19/9.00 ! [M: nat,N2: nat] :
% 16.19/9.00 ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
% 16.19/9.00 = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6452_Suc__mod__eq__add3__mod__numeral,axiom,
% 16.19/9.00 ! [M: nat,V: num] :
% 16.19/9.00 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 16.19/9.00 = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6453_divmod__algorithm__code_I4_J,axiom,
% 16.19/9.00 ! [N2: num] :
% 16.19/9.00 ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6454_divmod__algorithm__code_I4_J,axiom,
% 16.19/9.00 ! [N2: num] :
% 16.19/9.00 ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6455_divmod__algorithm__code_I4_J,axiom,
% 16.19/9.00 ! [N2: num] :
% 16.19/9.00 ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6456_sum__zero__power_H,axiom,
% 16.19/9.00 ! [A2: set_nat,C: nat > complex,D: nat > complex] :
% 16.19/9.00 ( ( ( ( finite_finite_nat @ A2 )
% 16.19/9.00 & ( member_nat @ zero_zero_nat @ A2 ) )
% 16.19/9.00 => ( ( groups2073611262835488442omplex
% 16.19/9.00 @ ^ [I3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) ) @ ( D @ I3 ) )
% 16.19/9.00 @ A2 )
% 16.19/9.00 = ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 16.19/9.00 & ( ~ ( ( finite_finite_nat @ A2 )
% 16.19/9.00 & ( member_nat @ zero_zero_nat @ A2 ) )
% 16.19/9.00 => ( ( groups2073611262835488442omplex
% 16.19/9.00 @ ^ [I3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ zero_zero_complex @ I3 ) ) @ ( D @ I3 ) )
% 16.19/9.00 @ A2 )
% 16.19/9.00 = zero_zero_complex ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6457_sum__zero__power_H,axiom,
% 16.19/9.00 ! [A2: set_nat,C: nat > rat,D: nat > rat] :
% 16.19/9.00 ( ( ( ( finite_finite_nat @ A2 )
% 16.19/9.00 & ( member_nat @ zero_zero_nat @ A2 ) )
% 16.19/9.00 => ( ( groups2906978787729119204at_rat
% 16.19/9.00 @ ^ [I3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) ) @ ( D @ I3 ) )
% 16.19/9.00 @ A2 )
% 16.19/9.00 = ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 16.19/9.00 & ( ~ ( ( finite_finite_nat @ A2 )
% 16.19/9.00 & ( member_nat @ zero_zero_nat @ A2 ) )
% 16.19/9.00 => ( ( groups2906978787729119204at_rat
% 16.19/9.00 @ ^ [I3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ zero_zero_rat @ I3 ) ) @ ( D @ I3 ) )
% 16.19/9.00 @ A2 )
% 16.19/9.00 = zero_zero_rat ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6458_sum__zero__power_H,axiom,
% 16.19/9.00 ! [A2: set_nat,C: nat > real,D: nat > real] :
% 16.19/9.00 ( ( ( ( finite_finite_nat @ A2 )
% 16.19/9.00 & ( member_nat @ zero_zero_nat @ A2 ) )
% 16.19/9.00 => ( ( groups6591440286371151544t_real
% 16.19/9.00 @ ^ [I3: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) ) @ ( D @ I3 ) )
% 16.19/9.00 @ A2 )
% 16.19/9.00 = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 16.19/9.00 & ( ~ ( ( finite_finite_nat @ A2 )
% 16.19/9.00 & ( member_nat @ zero_zero_nat @ A2 ) )
% 16.19/9.00 => ( ( groups6591440286371151544t_real
% 16.19/9.00 @ ^ [I3: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ zero_zero_real @ I3 ) ) @ ( D @ I3 ) )
% 16.19/9.00 @ A2 )
% 16.19/9.00 = zero_zero_real ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6459_zmod__numeral__Bit1,axiom,
% 16.19/9.00 ! [V: num,W: num] :
% 16.19/9.00 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 16.19/9.00 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) @ one_one_int ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6460_divmod__algorithm__code_I8_J,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( ( ord_less_num @ M @ N2 )
% 16.19/9.00 => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
% 16.19/9.00 & ( ~ ( ord_less_num @ M @ N2 )
% 16.19/9.00 => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( unique5026877609467782581ep_nat @ ( bit1 @ N2 ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6461_divmod__algorithm__code_I8_J,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( ( ord_less_num @ M @ N2 )
% 16.19/9.00 => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
% 16.19/9.00 & ( ~ ( ord_less_num @ M @ N2 )
% 16.19/9.00 => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( unique5024387138958732305ep_int @ ( bit1 @ N2 ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6462_divmod__algorithm__code_I8_J,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( ( ord_less_num @ M @ N2 )
% 16.19/9.00 => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
% 16.19/9.00 & ( ~ ( ord_less_num @ M @ N2 )
% 16.19/9.00 => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( unique4921790084139445826nteger @ ( bit1 @ N2 ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6463_divmod__algorithm__code_I7_J,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( ( ord_less_eq_num @ M @ N2 )
% 16.19/9.00 => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
% 16.19/9.00 & ( ~ ( ord_less_eq_num @ M @ N2 )
% 16.19/9.00 => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( unique5026877609467782581ep_nat @ ( bit1 @ N2 ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6464_divmod__algorithm__code_I7_J,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( ( ord_less_eq_num @ M @ N2 )
% 16.19/9.00 => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
% 16.19/9.00 & ( ~ ( ord_less_eq_num @ M @ N2 )
% 16.19/9.00 => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( unique5024387138958732305ep_int @ ( bit1 @ N2 ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6465_divmod__algorithm__code_I7_J,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( ( ord_less_eq_num @ M @ N2 )
% 16.19/9.00 => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
% 16.19/9.00 & ( ~ ( ord_less_eq_num @ M @ N2 )
% 16.19/9.00 => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( unique4921790084139445826nteger @ ( bit1 @ N2 ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6466_signed__take__bit__Suc__bit1,axiom,
% 16.19/9.00 ! [N2: nat,K: num] :
% 16.19/9.00 ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 16.19/9.00 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6467_divmod__algorithm__code_I6_J,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 16.19/9.00 = ( produc4245557441103728435nt_int
% 16.19/9.00 @ ^ [Q4: int,R5: int] : ( product_Pair_int_int @ Q4 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) @ one_one_int ) )
% 16.19/9.00 @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6468_divmod__algorithm__code_I6_J,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 16.19/9.00 = ( produc2626176000494625587at_nat
% 16.19/9.00 @ ^ [Q4: nat,R5: nat] : ( product_Pair_nat_nat @ Q4 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) @ one_one_nat ) )
% 16.19/9.00 @ ( unique5055182867167087721od_nat @ M @ N2 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6469_divmod__algorithm__code_I6_J,axiom,
% 16.19/9.00 ! [M: num,N2: num] :
% 16.19/9.00 ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 16.19/9.00 = ( produc6916734918728496179nteger
% 16.19/9.00 @ ^ [Q4: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q4 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) @ one_one_Code_integer ) )
% 16.19/9.00 @ ( unique3479559517661332726nteger @ M @ N2 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6470_Compl__eq,axiom,
% 16.19/9.00 ( uminus612125837232591019t_real
% 16.19/9.00 = ( ^ [A4: set_real] :
% 16.19/9.00 ( collect_real
% 16.19/9.00 @ ^ [X2: real] :
% 16.19/9.00 ~ ( member_real @ X2 @ A4 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6471_Compl__eq,axiom,
% 16.19/9.00 ( uminus6221592323253981072nt_int
% 16.19/9.00 = ( ^ [A4: set_Pr958786334691620121nt_int] :
% 16.19/9.00 ( collec213857154873943460nt_int
% 16.19/9.00 @ ^ [X2: product_prod_int_int] :
% 16.19/9.00 ~ ( member5262025264175285858nt_int @ X2 @ A4 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6472_Compl__eq,axiom,
% 16.19/9.00 ( uminus8566677241136511917omplex
% 16.19/9.00 = ( ^ [A4: set_complex] :
% 16.19/9.00 ( collect_complex
% 16.19/9.00 @ ^ [X2: complex] :
% 16.19/9.00 ~ ( member_complex @ X2 @ A4 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6473_Compl__eq,axiom,
% 16.19/9.00 ( uminus613421341184616069et_nat
% 16.19/9.00 = ( ^ [A4: set_set_nat] :
% 16.19/9.00 ( collect_set_nat
% 16.19/9.00 @ ^ [X2: set_nat] :
% 16.19/9.00 ~ ( member_set_nat @ X2 @ A4 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6474_Compl__eq,axiom,
% 16.19/9.00 ( uminus5710092332889474511et_nat
% 16.19/9.00 = ( ^ [A4: set_nat] :
% 16.19/9.00 ( collect_nat
% 16.19/9.00 @ ^ [X2: nat] :
% 16.19/9.00 ~ ( member_nat @ X2 @ A4 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6475_Compl__eq,axiom,
% 16.19/9.00 ( uminus1532241313380277803et_int
% 16.19/9.00 = ( ^ [A4: set_int] :
% 16.19/9.00 ( collect_int
% 16.19/9.00 @ ^ [X2: int] :
% 16.19/9.00 ~ ( member_int @ X2 @ A4 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6476_Collect__neg__eq,axiom,
% 16.19/9.00 ! [P: product_prod_int_int > $o] :
% 16.19/9.00 ( ( collec213857154873943460nt_int
% 16.19/9.00 @ ^ [X2: product_prod_int_int] :
% 16.19/9.00 ~ ( P @ X2 ) )
% 16.19/9.00 = ( uminus6221592323253981072nt_int @ ( collec213857154873943460nt_int @ P ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6477_Collect__neg__eq,axiom,
% 16.19/9.00 ! [P: complex > $o] :
% 16.19/9.00 ( ( collect_complex
% 16.19/9.00 @ ^ [X2: complex] :
% 16.19/9.00 ~ ( P @ X2 ) )
% 16.19/9.00 = ( uminus8566677241136511917omplex @ ( collect_complex @ P ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6478_Collect__neg__eq,axiom,
% 16.19/9.00 ! [P: set_nat > $o] :
% 16.19/9.00 ( ( collect_set_nat
% 16.19/9.00 @ ^ [X2: set_nat] :
% 16.19/9.00 ~ ( P @ X2 ) )
% 16.19/9.00 = ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6479_Collect__neg__eq,axiom,
% 16.19/9.00 ! [P: nat > $o] :
% 16.19/9.00 ( ( collect_nat
% 16.19/9.00 @ ^ [X2: nat] :
% 16.19/9.00 ~ ( P @ X2 ) )
% 16.19/9.00 = ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6480_Collect__neg__eq,axiom,
% 16.19/9.00 ! [P: int > $o] :
% 16.19/9.00 ( ( collect_int
% 16.19/9.00 @ ^ [X2: int] :
% 16.19/9.00 ~ ( P @ X2 ) )
% 16.19/9.00 = ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6481_uminus__set__def,axiom,
% 16.19/9.00 ( uminus612125837232591019t_real
% 16.19/9.00 = ( ^ [A4: set_real] :
% 16.19/9.00 ( collect_real
% 16.19/9.00 @ ( uminus_uminus_real_o
% 16.19/9.00 @ ^ [X2: real] : ( member_real @ X2 @ A4 ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6482_uminus__set__def,axiom,
% 16.19/9.00 ( uminus6221592323253981072nt_int
% 16.19/9.00 = ( ^ [A4: set_Pr958786334691620121nt_int] :
% 16.19/9.00 ( collec213857154873943460nt_int
% 16.19/9.00 @ ( uminus7117520113953359693_int_o
% 16.19/9.00 @ ^ [X2: product_prod_int_int] : ( member5262025264175285858nt_int @ X2 @ A4 ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6483_uminus__set__def,axiom,
% 16.19/9.00 ( uminus8566677241136511917omplex
% 16.19/9.00 = ( ^ [A4: set_complex] :
% 16.19/9.00 ( collect_complex
% 16.19/9.00 @ ( uminus1680532995456772888plex_o
% 16.19/9.00 @ ^ [X2: complex] : ( member_complex @ X2 @ A4 ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6484_uminus__set__def,axiom,
% 16.19/9.00 ( uminus613421341184616069et_nat
% 16.19/9.00 = ( ^ [A4: set_set_nat] :
% 16.19/9.00 ( collect_set_nat
% 16.19/9.00 @ ( uminus6401447641752708672_nat_o
% 16.19/9.00 @ ^ [X2: set_nat] : ( member_set_nat @ X2 @ A4 ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6485_uminus__set__def,axiom,
% 16.19/9.00 ( uminus5710092332889474511et_nat
% 16.19/9.00 = ( ^ [A4: set_nat] :
% 16.19/9.00 ( collect_nat
% 16.19/9.00 @ ( uminus_uminus_nat_o
% 16.19/9.00 @ ^ [X2: nat] : ( member_nat @ X2 @ A4 ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6486_uminus__set__def,axiom,
% 16.19/9.00 ( uminus1532241313380277803et_int
% 16.19/9.00 = ( ^ [A4: set_int] :
% 16.19/9.00 ( collect_int
% 16.19/9.00 @ ( uminus_uminus_int_o
% 16.19/9.00 @ ^ [X2: int] : ( member_int @ X2 @ A4 ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6487_abs__ge__self,axiom,
% 16.19/9.00 ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6488_abs__ge__self,axiom,
% 16.19/9.00 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6489_abs__ge__self,axiom,
% 16.19/9.00 ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6490_abs__ge__self,axiom,
% 16.19/9.00 ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6491_abs__le__D1,axiom,
% 16.19/9.00 ! [A: real,B: real] :
% 16.19/9.00 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 16.19/9.00 => ( ord_less_eq_real @ A @ B ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6492_abs__le__D1,axiom,
% 16.19/9.00 ! [A: code_integer,B: code_integer] :
% 16.19/9.00 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 16.19/9.00 => ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6493_abs__le__D1,axiom,
% 16.19/9.00 ! [A: rat,B: rat] :
% 16.19/9.00 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 16.19/9.00 => ( ord_less_eq_rat @ A @ B ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6494_abs__le__D1,axiom,
% 16.19/9.00 ! [A: int,B: int] :
% 16.19/9.00 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 16.19/9.00 => ( ord_less_eq_int @ A @ B ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6495_abs__mult,axiom,
% 16.19/9.00 ! [A: code_integer,B: code_integer] :
% 16.19/9.00 ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 16.19/9.00 = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6496_abs__mult,axiom,
% 16.19/9.00 ! [A: real,B: real] :
% 16.19/9.00 ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 16.19/9.00 = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6497_abs__mult,axiom,
% 16.19/9.00 ! [A: rat,B: rat] :
% 16.19/9.00 ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
% 16.19/9.00 = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6498_abs__mult,axiom,
% 16.19/9.00 ! [A: int,B: int] :
% 16.19/9.00 ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
% 16.19/9.00 = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6499_abs__one,axiom,
% 16.19/9.00 ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 16.19/9.00 = one_one_Code_integer ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6500_abs__one,axiom,
% 16.19/9.00 ( ( abs_abs_real @ one_one_real )
% 16.19/9.00 = one_one_real ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6501_abs__one,axiom,
% 16.19/9.00 ( ( abs_abs_rat @ one_one_rat )
% 16.19/9.00 = one_one_rat ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6502_abs__one,axiom,
% 16.19/9.00 ( ( abs_abs_int @ one_one_int )
% 16.19/9.00 = one_one_int ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6503_abs__minus__commute,axiom,
% 16.19/9.00 ! [A: code_integer,B: code_integer] :
% 16.19/9.00 ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
% 16.19/9.00 = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6504_abs__minus__commute,axiom,
% 16.19/9.00 ! [A: real,B: real] :
% 16.19/9.00 ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
% 16.19/9.00 = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6505_abs__minus__commute,axiom,
% 16.19/9.00 ! [A: rat,B: rat] :
% 16.19/9.00 ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
% 16.19/9.00 = ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6506_abs__minus__commute,axiom,
% 16.19/9.00 ! [A: int,B: int] :
% 16.19/9.00 ( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
% 16.19/9.00 = ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6507_power__abs,axiom,
% 16.19/9.00 ! [A: rat,N2: nat] :
% 16.19/9.00 ( ( abs_abs_rat @ ( power_power_rat @ A @ N2 ) )
% 16.19/9.00 = ( power_power_rat @ ( abs_abs_rat @ A ) @ N2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6508_power__abs,axiom,
% 16.19/9.00 ! [A: code_integer,N2: nat] :
% 16.19/9.00 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) )
% 16.19/9.00 = ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6509_power__abs,axiom,
% 16.19/9.00 ! [A: real,N2: nat] :
% 16.19/9.00 ( ( abs_abs_real @ ( power_power_real @ A @ N2 ) )
% 16.19/9.00 = ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6510_power__abs,axiom,
% 16.19/9.00 ! [A: int,N2: nat] :
% 16.19/9.00 ( ( abs_abs_int @ ( power_power_int @ A @ N2 ) )
% 16.19/9.00 = ( power_power_int @ ( abs_abs_int @ A ) @ N2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6511_mem__case__prodE,axiom,
% 16.19/9.00 ! [Z: nat,C: code_integer > $o > set_nat,P4: produc6271795597528267376eger_o] :
% 16.19/9.00 ( ( member_nat @ Z @ ( produc5431169771168744661et_nat @ C @ P4 ) )
% 16.19/9.00 => ~ ! [X4: code_integer,Y4: $o] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( produc6677183202524767010eger_o @ X4 @ Y4 ) )
% 16.19/9.00 => ~ ( member_nat @ Z @ ( C @ X4 @ Y4 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6512_mem__case__prodE,axiom,
% 16.19/9.00 ! [Z: real,C: code_integer > $o > set_real,P4: produc6271795597528267376eger_o] :
% 16.19/9.00 ( ( member_real @ Z @ ( produc242741666403216561t_real @ C @ P4 ) )
% 16.19/9.00 => ~ ! [X4: code_integer,Y4: $o] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( produc6677183202524767010eger_o @ X4 @ Y4 ) )
% 16.19/9.00 => ~ ( member_real @ Z @ ( C @ X4 @ Y4 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6513_mem__case__prodE,axiom,
% 16.19/9.00 ! [Z: int,C: code_integer > $o > set_int,P4: produc6271795597528267376eger_o] :
% 16.19/9.00 ( ( member_int @ Z @ ( produc1253318751659547953et_int @ C @ P4 ) )
% 16.19/9.00 => ~ ! [X4: code_integer,Y4: $o] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( produc6677183202524767010eger_o @ X4 @ Y4 ) )
% 16.19/9.00 => ~ ( member_int @ Z @ ( C @ X4 @ Y4 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6514_mem__case__prodE,axiom,
% 16.19/9.00 ! [Z: complex,C: code_integer > $o > set_complex,P4: produc6271795597528267376eger_o] :
% 16.19/9.00 ( ( member_complex @ Z @ ( produc1043322548047392435omplex @ C @ P4 ) )
% 16.19/9.00 => ~ ! [X4: code_integer,Y4: $o] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( produc6677183202524767010eger_o @ X4 @ Y4 ) )
% 16.19/9.00 => ~ ( member_complex @ Z @ ( C @ X4 @ Y4 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6515_mem__case__prodE,axiom,
% 16.19/9.00 ! [Z: nat,C: num > num > set_nat,P4: product_prod_num_num] :
% 16.19/9.00 ( ( member_nat @ Z @ ( produc1361121860356118632et_nat @ C @ P4 ) )
% 16.19/9.00 => ~ ! [X4: num,Y4: num] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( product_Pair_num_num @ X4 @ Y4 ) )
% 16.19/9.00 => ~ ( member_nat @ Z @ ( C @ X4 @ Y4 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6516_mem__case__prodE,axiom,
% 16.19/9.00 ! [Z: real,C: num > num > set_real,P4: product_prod_num_num] :
% 16.19/9.00 ( ( member_real @ Z @ ( produc8296048397933160132t_real @ C @ P4 ) )
% 16.19/9.00 => ~ ! [X4: num,Y4: num] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( product_Pair_num_num @ X4 @ Y4 ) )
% 16.19/9.00 => ~ ( member_real @ Z @ ( C @ X4 @ Y4 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6517_mem__case__prodE,axiom,
% 16.19/9.00 ! [Z: int,C: num > num > set_int,P4: product_prod_num_num] :
% 16.19/9.00 ( ( member_int @ Z @ ( produc6406642877701697732et_int @ C @ P4 ) )
% 16.19/9.00 => ~ ! [X4: num,Y4: num] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( product_Pair_num_num @ X4 @ Y4 ) )
% 16.19/9.00 => ~ ( member_int @ Z @ ( C @ X4 @ Y4 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6518_mem__case__prodE,axiom,
% 16.19/9.00 ! [Z: complex,C: num > num > set_complex,P4: product_prod_num_num] :
% 16.19/9.00 ( ( member_complex @ Z @ ( produc2866383454006189126omplex @ C @ P4 ) )
% 16.19/9.00 => ~ ! [X4: num,Y4: num] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( product_Pair_num_num @ X4 @ Y4 ) )
% 16.19/9.00 => ~ ( member_complex @ Z @ ( C @ X4 @ Y4 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6519_mem__case__prodE,axiom,
% 16.19/9.00 ! [Z: nat,C: nat > num > set_nat,P4: product_prod_nat_num] :
% 16.19/9.00 ( ( member_nat @ Z @ ( produc4130284055270567454et_nat @ C @ P4 ) )
% 16.19/9.00 => ~ ! [X4: nat,Y4: num] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( product_Pair_nat_num @ X4 @ Y4 ) )
% 16.19/9.00 => ~ ( member_nat @ Z @ ( C @ X4 @ Y4 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6520_mem__case__prodE,axiom,
% 16.19/9.00 ! [Z: real,C: nat > num > set_real,P4: product_prod_nat_num] :
% 16.19/9.00 ( ( member_real @ Z @ ( produc1435849484188172666t_real @ C @ P4 ) )
% 16.19/9.00 => ~ ! [X4: nat,Y4: num] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( product_Pair_nat_num @ X4 @ Y4 ) )
% 16.19/9.00 => ~ ( member_real @ Z @ ( C @ X4 @ Y4 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6521_verit__eq__simplify_I14_J,axiom,
% 16.19/9.00 ! [X22: num,X32: num] :
% 16.19/9.00 ( ( bit0 @ X22 )
% 16.19/9.00 != ( bit1 @ X32 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6522_verit__eq__simplify_I12_J,axiom,
% 16.19/9.00 ! [X32: num] :
% 16.19/9.00 ( one
% 16.19/9.00 != ( bit1 @ X32 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6523_case__prodD,axiom,
% 16.19/9.00 ! [F: code_integer > $o > $o,A: code_integer,B: $o] :
% 16.19/9.00 ( ( produc7828578312038201481er_o_o @ F @ ( produc6677183202524767010eger_o @ A @ B ) )
% 16.19/9.00 => ( F @ A @ B ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6524_case__prodD,axiom,
% 16.19/9.00 ! [F: num > num > $o,A: num,B: num] :
% 16.19/9.00 ( ( produc5703948589228662326_num_o @ F @ ( product_Pair_num_num @ A @ B ) )
% 16.19/9.00 => ( F @ A @ B ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6525_case__prodD,axiom,
% 16.19/9.00 ! [F: nat > num > $o,A: nat,B: num] :
% 16.19/9.00 ( ( produc4927758841916487424_num_o @ F @ ( product_Pair_nat_num @ A @ B ) )
% 16.19/9.00 => ( F @ A @ B ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6526_case__prodD,axiom,
% 16.19/9.00 ! [F: nat > nat > $o,A: nat,B: nat] :
% 16.19/9.00 ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
% 16.19/9.00 => ( F @ A @ B ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6527_case__prodD,axiom,
% 16.19/9.00 ! [F: int > int > $o,A: int,B: int] :
% 16.19/9.00 ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
% 16.19/9.00 => ( F @ A @ B ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6528_case__prodE,axiom,
% 16.19/9.00 ! [C: code_integer > $o > $o,P4: produc6271795597528267376eger_o] :
% 16.19/9.00 ( ( produc7828578312038201481er_o_o @ C @ P4 )
% 16.19/9.00 => ~ ! [X4: code_integer,Y4: $o] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( produc6677183202524767010eger_o @ X4 @ Y4 ) )
% 16.19/9.00 => ~ ( C @ X4 @ Y4 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6529_case__prodE,axiom,
% 16.19/9.00 ! [C: num > num > $o,P4: product_prod_num_num] :
% 16.19/9.00 ( ( produc5703948589228662326_num_o @ C @ P4 )
% 16.19/9.00 => ~ ! [X4: num,Y4: num] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( product_Pair_num_num @ X4 @ Y4 ) )
% 16.19/9.00 => ~ ( C @ X4 @ Y4 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6530_case__prodE,axiom,
% 16.19/9.00 ! [C: nat > num > $o,P4: product_prod_nat_num] :
% 16.19/9.00 ( ( produc4927758841916487424_num_o @ C @ P4 )
% 16.19/9.00 => ~ ! [X4: nat,Y4: num] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( product_Pair_nat_num @ X4 @ Y4 ) )
% 16.19/9.00 => ~ ( C @ X4 @ Y4 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6531_case__prodE,axiom,
% 16.19/9.00 ! [C: nat > nat > $o,P4: product_prod_nat_nat] :
% 16.19/9.00 ( ( produc6081775807080527818_nat_o @ C @ P4 )
% 16.19/9.00 => ~ ! [X4: nat,Y4: nat] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( product_Pair_nat_nat @ X4 @ Y4 ) )
% 16.19/9.00 => ~ ( C @ X4 @ Y4 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6532_case__prodE,axiom,
% 16.19/9.00 ! [C: int > int > $o,P4: product_prod_int_int] :
% 16.19/9.00 ( ( produc4947309494688390418_int_o @ C @ P4 )
% 16.19/9.00 => ~ ! [X4: int,Y4: int] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( product_Pair_int_int @ X4 @ Y4 ) )
% 16.19/9.00 => ~ ( C @ X4 @ Y4 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6533_case__prodD_H,axiom,
% 16.19/9.00 ! [R2: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat,C: product_prod_nat_nat] :
% 16.19/9.00 ( ( produc8739625826339149834_nat_o @ R2 @ ( product_Pair_nat_nat @ A @ B ) @ C )
% 16.19/9.00 => ( R2 @ A @ B @ C ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6534_case__prodE_H,axiom,
% 16.19/9.00 ! [C: nat > nat > product_prod_nat_nat > $o,P4: product_prod_nat_nat,Z: product_prod_nat_nat] :
% 16.19/9.00 ( ( produc8739625826339149834_nat_o @ C @ P4 @ Z )
% 16.19/9.00 => ~ ! [X4: nat,Y4: nat] :
% 16.19/9.00 ( ( P4
% 16.19/9.00 = ( product_Pair_nat_nat @ X4 @ Y4 ) )
% 16.19/9.00 => ~ ( C @ X4 @ Y4 @ Z ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6535_abs__ge__zero,axiom,
% 16.19/9.00 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6536_abs__ge__zero,axiom,
% 16.19/9.00 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6537_abs__ge__zero,axiom,
% 16.19/9.00 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6538_abs__ge__zero,axiom,
% 16.19/9.00 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6539_abs__of__pos,axiom,
% 16.19/9.00 ! [A: code_integer] :
% 16.19/9.00 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 16.19/9.00 => ( ( abs_abs_Code_integer @ A )
% 16.19/9.00 = A ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6540_abs__of__pos,axiom,
% 16.19/9.00 ! [A: real] :
% 16.19/9.00 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/9.00 => ( ( abs_abs_real @ A )
% 16.19/9.00 = A ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6541_abs__of__pos,axiom,
% 16.19/9.00 ! [A: rat] :
% 16.19/9.00 ( ( ord_less_rat @ zero_zero_rat @ A )
% 16.19/9.00 => ( ( abs_abs_rat @ A )
% 16.19/9.00 = A ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6542_abs__of__pos,axiom,
% 16.19/9.00 ! [A: int] :
% 16.19/9.00 ( ( ord_less_int @ zero_zero_int @ A )
% 16.19/9.00 => ( ( abs_abs_int @ A )
% 16.19/9.00 = A ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6543_abs__not__less__zero,axiom,
% 16.19/9.00 ! [A: code_integer] :
% 16.19/9.00 ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6544_abs__not__less__zero,axiom,
% 16.19/9.00 ! [A: real] :
% 16.19/9.00 ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6545_abs__not__less__zero,axiom,
% 16.19/9.00 ! [A: rat] :
% 16.19/9.00 ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6546_abs__not__less__zero,axiom,
% 16.19/9.00 ! [A: int] :
% 16.19/9.00 ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6547_abs__triangle__ineq,axiom,
% 16.19/9.00 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6548_abs__triangle__ineq,axiom,
% 16.19/9.00 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6549_abs__triangle__ineq,axiom,
% 16.19/9.00 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6550_abs__triangle__ineq,axiom,
% 16.19/9.00 ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6551_abs__mult__less,axiom,
% 16.19/9.00 ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 16.19/9.00 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
% 16.19/9.00 => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D )
% 16.19/9.00 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6552_abs__mult__less,axiom,
% 16.19/9.00 ! [A: real,C: real,B: real,D: real] :
% 16.19/9.00 ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
% 16.19/9.00 => ( ( ord_less_real @ ( abs_abs_real @ B ) @ D )
% 16.19/9.00 => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C @ D ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6553_abs__mult__less,axiom,
% 16.19/9.00 ! [A: rat,C: rat,B: rat,D: rat] :
% 16.19/9.00 ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
% 16.19/9.00 => ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D )
% 16.19/9.00 => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6554_abs__mult__less,axiom,
% 16.19/9.00 ! [A: int,C: int,B: int,D: int] :
% 16.19/9.00 ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
% 16.19/9.00 => ( ( ord_less_int @ ( abs_abs_int @ B ) @ D )
% 16.19/9.00 => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6555_abs__triangle__ineq2,axiom,
% 16.19/9.00 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6556_abs__triangle__ineq2,axiom,
% 16.19/9.00 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6557_abs__triangle__ineq2,axiom,
% 16.19/9.00 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6558_abs__triangle__ineq2,axiom,
% 16.19/9.00 ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6559_abs__triangle__ineq3,axiom,
% 16.19/9.00 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6560_abs__triangle__ineq3,axiom,
% 16.19/9.00 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6561_abs__triangle__ineq3,axiom,
% 16.19/9.00 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6562_abs__triangle__ineq3,axiom,
% 16.19/9.00 ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6563_abs__triangle__ineq2__sym,axiom,
% 16.19/9.00 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6564_abs__triangle__ineq2__sym,axiom,
% 16.19/9.00 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6565_abs__triangle__ineq2__sym,axiom,
% 16.19/9.00 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6566_abs__triangle__ineq2__sym,axiom,
% 16.19/9.00 ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6567_nonzero__abs__divide,axiom,
% 16.19/9.00 ! [B: real,A: real] :
% 16.19/9.00 ( ( B != zero_zero_real )
% 16.19/9.00 => ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 16.19/9.00 = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6568_nonzero__abs__divide,axiom,
% 16.19/9.00 ! [B: rat,A: rat] :
% 16.19/9.00 ( ( B != zero_zero_rat )
% 16.19/9.00 => ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 16.19/9.00 = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6569_abs__ge__minus__self,axiom,
% 16.19/9.00 ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6570_abs__ge__minus__self,axiom,
% 16.19/9.00 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6571_abs__ge__minus__self,axiom,
% 16.19/9.00 ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6572_abs__ge__minus__self,axiom,
% 16.19/9.00 ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6573_abs__le__iff,axiom,
% 16.19/9.00 ! [A: real,B: real] :
% 16.19/9.00 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 16.19/9.00 = ( ( ord_less_eq_real @ A @ B )
% 16.19/9.00 & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6574_abs__le__iff,axiom,
% 16.19/9.00 ! [A: code_integer,B: code_integer] :
% 16.19/9.00 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 16.19/9.00 = ( ( ord_le3102999989581377725nteger @ A @ B )
% 16.19/9.00 & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6575_abs__le__iff,axiom,
% 16.19/9.00 ! [A: rat,B: rat] :
% 16.19/9.00 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 16.19/9.00 = ( ( ord_less_eq_rat @ A @ B )
% 16.19/9.00 & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6576_abs__le__iff,axiom,
% 16.19/9.00 ! [A: int,B: int] :
% 16.19/9.00 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 16.19/9.00 = ( ( ord_less_eq_int @ A @ B )
% 16.19/9.00 & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6577_abs__le__D2,axiom,
% 16.19/9.00 ! [A: real,B: real] :
% 16.19/9.00 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 16.19/9.00 => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6578_abs__le__D2,axiom,
% 16.19/9.00 ! [A: code_integer,B: code_integer] :
% 16.19/9.00 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 16.19/9.00 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6579_abs__le__D2,axiom,
% 16.19/9.00 ! [A: rat,B: rat] :
% 16.19/9.00 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 16.19/9.00 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6580_abs__le__D2,axiom,
% 16.19/9.00 ! [A: int,B: int] :
% 16.19/9.00 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 16.19/9.00 => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6581_abs__leI,axiom,
% 16.19/9.00 ! [A: real,B: real] :
% 16.19/9.00 ( ( ord_less_eq_real @ A @ B )
% 16.19/9.00 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 16.19/9.00 => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6582_abs__leI,axiom,
% 16.19/9.00 ! [A: code_integer,B: code_integer] :
% 16.19/9.00 ( ( ord_le3102999989581377725nteger @ A @ B )
% 16.19/9.00 => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 16.19/9.00 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6583_abs__leI,axiom,
% 16.19/9.00 ! [A: rat,B: rat] :
% 16.19/9.00 ( ( ord_less_eq_rat @ A @ B )
% 16.19/9.00 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 16.19/9.00 => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6584_abs__leI,axiom,
% 16.19/9.00 ! [A: int,B: int] :
% 16.19/9.00 ( ( ord_less_eq_int @ A @ B )
% 16.19/9.00 => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 16.19/9.00 => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6585_abs__less__iff,axiom,
% 16.19/9.00 ! [A: real,B: real] :
% 16.19/9.00 ( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
% 16.19/9.00 = ( ( ord_less_real @ A @ B )
% 16.19/9.00 & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6586_abs__less__iff,axiom,
% 16.19/9.00 ! [A: int,B: int] :
% 16.19/9.00 ( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
% 16.19/9.00 = ( ( ord_less_int @ A @ B )
% 16.19/9.00 & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6587_abs__less__iff,axiom,
% 16.19/9.00 ! [A: rat,B: rat] :
% 16.19/9.00 ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B )
% 16.19/9.00 = ( ( ord_less_rat @ A @ B )
% 16.19/9.00 & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6588_abs__less__iff,axiom,
% 16.19/9.00 ! [A: code_integer,B: code_integer] :
% 16.19/9.00 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 16.19/9.00 = ( ( ord_le6747313008572928689nteger @ A @ B )
% 16.19/9.00 & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6589_sum__cong__Suc,axiom,
% 16.19/9.00 ! [A2: set_nat,F: nat > nat,G: nat > nat] :
% 16.19/9.00 ( ~ ( member_nat @ zero_zero_nat @ A2 )
% 16.19/9.00 => ( ! [X4: nat] :
% 16.19/9.00 ( ( member_nat @ ( suc @ X4 ) @ A2 )
% 16.19/9.00 => ( ( F @ ( suc @ X4 ) )
% 16.19/9.00 = ( G @ ( suc @ X4 ) ) ) )
% 16.19/9.00 => ( ( groups3542108847815614940at_nat @ F @ A2 )
% 16.19/9.00 = ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6590_sum__cong__Suc,axiom,
% 16.19/9.00 ! [A2: set_nat,F: nat > real,G: nat > real] :
% 16.19/9.00 ( ~ ( member_nat @ zero_zero_nat @ A2 )
% 16.19/9.00 => ( ! [X4: nat] :
% 16.19/9.00 ( ( member_nat @ ( suc @ X4 ) @ A2 )
% 16.19/9.00 => ( ( F @ ( suc @ X4 ) )
% 16.19/9.00 = ( G @ ( suc @ X4 ) ) ) )
% 16.19/9.00 => ( ( groups6591440286371151544t_real @ F @ A2 )
% 16.19/9.00 = ( groups6591440286371151544t_real @ G @ A2 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6591_abs__real__def,axiom,
% 16.19/9.00 ( abs_abs_real
% 16.19/9.00 = ( ^ [A5: real] : ( if_real @ ( ord_less_real @ A5 @ zero_zero_real ) @ ( uminus_uminus_real @ A5 ) @ A5 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6592_num_Oexhaust,axiom,
% 16.19/9.00 ! [Y: num] :
% 16.19/9.00 ( ( Y != one )
% 16.19/9.00 => ( ! [X23: num] :
% 16.19/9.00 ( Y
% 16.19/9.00 != ( bit0 @ X23 ) )
% 16.19/9.00 => ~ ! [X33: num] :
% 16.19/9.00 ( Y
% 16.19/9.00 != ( bit1 @ X33 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6593_xor__num_Ocases,axiom,
% 16.19/9.00 ! [X: product_prod_num_num] :
% 16.19/9.00 ( ( X
% 16.19/9.00 != ( product_Pair_num_num @ one @ one ) )
% 16.19/9.00 => ( ! [N4: num] :
% 16.19/9.00 ( X
% 16.19/9.00 != ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) )
% 16.19/9.00 => ( ! [N4: num] :
% 16.19/9.00 ( X
% 16.19/9.00 != ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) )
% 16.19/9.00 => ( ! [M2: num] :
% 16.19/9.00 ( X
% 16.19/9.00 != ( product_Pair_num_num @ ( bit0 @ M2 ) @ one ) )
% 16.19/9.00 => ( ! [M2: num,N4: num] :
% 16.19/9.00 ( X
% 16.19/9.00 != ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit0 @ N4 ) ) )
% 16.19/9.00 => ( ! [M2: num,N4: num] :
% 16.19/9.00 ( X
% 16.19/9.00 != ( product_Pair_num_num @ ( bit0 @ M2 ) @ ( bit1 @ N4 ) ) )
% 16.19/9.00 => ( ! [M2: num] :
% 16.19/9.00 ( X
% 16.19/9.00 != ( product_Pair_num_num @ ( bit1 @ M2 ) @ one ) )
% 16.19/9.00 => ( ! [M2: num,N4: num] :
% 16.19/9.00 ( X
% 16.19/9.00 != ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit0 @ N4 ) ) )
% 16.19/9.00 => ~ ! [M2: num,N4: num] :
% 16.19/9.00 ( X
% 16.19/9.00 != ( product_Pair_num_num @ ( bit1 @ M2 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6594_sin__bound__lemma,axiom,
% 16.19/9.00 ! [X: real,Y: real,U: real,V: real] :
% 16.19/9.00 ( ( X = Y )
% 16.19/9.00 => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
% 16.19/9.00 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U ) @ Y ) ) @ V ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6595_sum__subtractf__nat,axiom,
% 16.19/9.00 ! [A2: set_real,G: real > nat,F: real > nat] :
% 16.19/9.00 ( ! [X4: real] :
% 16.19/9.00 ( ( member_real @ X4 @ A2 )
% 16.19/9.00 => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 16.19/9.00 => ( ( groups1935376822645274424al_nat
% 16.19/9.00 @ ^ [X2: real] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
% 16.19/9.00 @ A2 )
% 16.19/9.00 = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6596_sum__subtractf__nat,axiom,
% 16.19/9.00 ! [A2: set_int,G: int > nat,F: int > nat] :
% 16.19/9.00 ( ! [X4: int] :
% 16.19/9.00 ( ( member_int @ X4 @ A2 )
% 16.19/9.00 => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 16.19/9.00 => ( ( groups4541462559716669496nt_nat
% 16.19/9.00 @ ^ [X2: int] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
% 16.19/9.00 @ A2 )
% 16.19/9.00 = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6597_sum__subtractf__nat,axiom,
% 16.19/9.00 ! [A2: set_complex,G: complex > nat,F: complex > nat] :
% 16.19/9.00 ( ! [X4: complex] :
% 16.19/9.00 ( ( member_complex @ X4 @ A2 )
% 16.19/9.00 => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 16.19/9.00 => ( ( groups5693394587270226106ex_nat
% 16.19/9.00 @ ^ [X2: complex] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
% 16.19/9.00 @ A2 )
% 16.19/9.00 = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6598_sum__subtractf__nat,axiom,
% 16.19/9.00 ! [A2: set_set_nat,G: set_nat > nat,F: set_nat > nat] :
% 16.19/9.00 ( ! [X4: set_nat] :
% 16.19/9.00 ( ( member_set_nat @ X4 @ A2 )
% 16.19/9.00 => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 16.19/9.00 => ( ( groups8294997508430121362at_nat
% 16.19/9.00 @ ^ [X2: set_nat] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
% 16.19/9.00 @ A2 )
% 16.19/9.00 = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A2 ) @ ( groups8294997508430121362at_nat @ G @ A2 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6599_sum__subtractf__nat,axiom,
% 16.19/9.00 ! [A2: set_nat,G: nat > nat,F: nat > nat] :
% 16.19/9.00 ( ! [X4: nat] :
% 16.19/9.00 ( ( member_nat @ X4 @ A2 )
% 16.19/9.00 => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 16.19/9.00 => ( ( groups3542108847815614940at_nat
% 16.19/9.00 @ ^ [X2: nat] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
% 16.19/9.00 @ A2 )
% 16.19/9.00 = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6600_sum_Oshift__bounds__cl__Suc__ivl,axiom,
% 16.19/9.00 ! [G: nat > nat,M: nat,N2: nat] :
% 16.19/9.00 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
% 16.19/9.00 = ( groups3542108847815614940at_nat
% 16.19/9.00 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 16.19/9.00 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6601_sum_Oshift__bounds__cl__Suc__ivl,axiom,
% 16.19/9.00 ! [G: nat > real,M: nat,N2: nat] :
% 16.19/9.00 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
% 16.19/9.00 = ( groups6591440286371151544t_real
% 16.19/9.00 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 16.19/9.00 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6602_sum_Oshift__bounds__cl__nat__ivl,axiom,
% 16.19/9.00 ! [G: nat > nat,M: nat,K: nat,N2: nat] :
% 16.19/9.00 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
% 16.19/9.00 = ( groups3542108847815614940at_nat
% 16.19/9.00 @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
% 16.19/9.00 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6603_sum_Oshift__bounds__cl__nat__ivl,axiom,
% 16.19/9.00 ! [G: nat > real,M: nat,K: nat,N2: nat] :
% 16.19/9.00 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
% 16.19/9.00 = ( groups6591440286371151544t_real
% 16.19/9.00 @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
% 16.19/9.00 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6604_tanh__real__lt__1,axiom,
% 16.19/9.00 ! [X: real] : ( ord_less_real @ ( tanh_real @ X ) @ one_one_real ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6605_tanh__real__gt__neg1,axiom,
% 16.19/9.00 ! [X: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6606_dense__eq0__I,axiom,
% 16.19/9.00 ! [X: real] :
% 16.19/9.00 ( ! [E2: real] :
% 16.19/9.00 ( ( ord_less_real @ zero_zero_real @ E2 )
% 16.19/9.00 => ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E2 ) )
% 16.19/9.00 => ( X = zero_zero_real ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6607_dense__eq0__I,axiom,
% 16.19/9.00 ! [X: rat] :
% 16.19/9.00 ( ! [E2: rat] :
% 16.19/9.00 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 16.19/9.00 => ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E2 ) )
% 16.19/9.00 => ( X = zero_zero_rat ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6608_abs__mult__pos,axiom,
% 16.19/9.00 ! [X: code_integer,Y: code_integer] :
% 16.19/9.00 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 16.19/9.00 => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y ) @ X )
% 16.19/9.00 = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y @ X ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6609_abs__mult__pos,axiom,
% 16.19/9.00 ! [X: real,Y: real] :
% 16.19/9.00 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.00 => ( ( times_times_real @ ( abs_abs_real @ Y ) @ X )
% 16.19/9.00 = ( abs_abs_real @ ( times_times_real @ Y @ X ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6610_abs__mult__pos,axiom,
% 16.19/9.00 ! [X: rat,Y: rat] :
% 16.19/9.00 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 16.19/9.00 => ( ( times_times_rat @ ( abs_abs_rat @ Y ) @ X )
% 16.19/9.00 = ( abs_abs_rat @ ( times_times_rat @ Y @ X ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6611_abs__mult__pos,axiom,
% 16.19/9.00 ! [X: int,Y: int] :
% 16.19/9.00 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 16.19/9.00 => ( ( times_times_int @ ( abs_abs_int @ Y ) @ X )
% 16.19/9.00 = ( abs_abs_int @ ( times_times_int @ Y @ X ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6612_abs__eq__mult,axiom,
% 16.19/9.00 ! [A: code_integer,B: code_integer] :
% 16.19/9.00 ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 16.19/9.00 | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
% 16.19/9.00 & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 16.19/9.00 | ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
% 16.19/9.00 => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 16.19/9.00 = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6613_abs__eq__mult,axiom,
% 16.19/9.00 ! [A: real,B: real] :
% 16.19/9.00 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/9.00 | ( ord_less_eq_real @ A @ zero_zero_real ) )
% 16.19/9.00 & ( ( ord_less_eq_real @ zero_zero_real @ B )
% 16.19/9.00 | ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 16.19/9.00 => ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 16.19/9.00 = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6614_abs__eq__mult,axiom,
% 16.19/9.00 ! [A: rat,B: rat] :
% 16.19/9.00 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/9.00 | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 16.19/9.00 & ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 16.19/9.00 | ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 16.19/9.00 => ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
% 16.19/9.00 = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6615_abs__eq__mult,axiom,
% 16.19/9.00 ! [A: int,B: int] :
% 16.19/9.00 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/9.00 | ( ord_less_eq_int @ A @ zero_zero_int ) )
% 16.19/9.00 & ( ( ord_less_eq_int @ zero_zero_int @ B )
% 16.19/9.00 | ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 16.19/9.00 => ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
% 16.19/9.00 = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6616_abs__eq__iff_H,axiom,
% 16.19/9.00 ! [A: real,B: real] :
% 16.19/9.00 ( ( ( abs_abs_real @ A )
% 16.19/9.00 = B )
% 16.19/9.00 = ( ( ord_less_eq_real @ zero_zero_real @ B )
% 16.19/9.00 & ( ( A = B )
% 16.19/9.00 | ( A
% 16.19/9.00 = ( uminus_uminus_real @ B ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6617_abs__eq__iff_H,axiom,
% 16.19/9.00 ! [A: code_integer,B: code_integer] :
% 16.19/9.00 ( ( ( abs_abs_Code_integer @ A )
% 16.19/9.00 = B )
% 16.19/9.00 = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 16.19/9.00 & ( ( A = B )
% 16.19/9.00 | ( A
% 16.19/9.00 = ( uminus1351360451143612070nteger @ B ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6618_abs__eq__iff_H,axiom,
% 16.19/9.00 ! [A: rat,B: rat] :
% 16.19/9.00 ( ( ( abs_abs_rat @ A )
% 16.19/9.00 = B )
% 16.19/9.00 = ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 16.19/9.00 & ( ( A = B )
% 16.19/9.00 | ( A
% 16.19/9.00 = ( uminus_uminus_rat @ B ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6619_abs__eq__iff_H,axiom,
% 16.19/9.00 ! [A: int,B: int] :
% 16.19/9.00 ( ( ( abs_abs_int @ A )
% 16.19/9.00 = B )
% 16.19/9.00 = ( ( ord_less_eq_int @ zero_zero_int @ B )
% 16.19/9.00 & ( ( A = B )
% 16.19/9.00 | ( A
% 16.19/9.00 = ( uminus_uminus_int @ B ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6620_eq__abs__iff_H,axiom,
% 16.19/9.00 ! [A: real,B: real] :
% 16.19/9.00 ( ( A
% 16.19/9.00 = ( abs_abs_real @ B ) )
% 16.19/9.00 = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/9.00 & ( ( B = A )
% 16.19/9.00 | ( B
% 16.19/9.00 = ( uminus_uminus_real @ A ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6621_eq__abs__iff_H,axiom,
% 16.19/9.00 ! [A: code_integer,B: code_integer] :
% 16.19/9.00 ( ( A
% 16.19/9.00 = ( abs_abs_Code_integer @ B ) )
% 16.19/9.00 = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 16.19/9.00 & ( ( B = A )
% 16.19/9.00 | ( B
% 16.19/9.00 = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6622_eq__abs__iff_H,axiom,
% 16.19/9.00 ! [A: rat,B: rat] :
% 16.19/9.00 ( ( A
% 16.19/9.00 = ( abs_abs_rat @ B ) )
% 16.19/9.00 = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 16.19/9.00 & ( ( B = A )
% 16.19/9.00 | ( B
% 16.19/9.00 = ( uminus_uminus_rat @ A ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6623_eq__abs__iff_H,axiom,
% 16.19/9.00 ! [A: int,B: int] :
% 16.19/9.00 ( ( A
% 16.19/9.00 = ( abs_abs_int @ B ) )
% 16.19/9.00 = ( ( ord_less_eq_int @ zero_zero_int @ A )
% 16.19/9.00 & ( ( B = A )
% 16.19/9.00 | ( B
% 16.19/9.00 = ( uminus_uminus_int @ A ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6624_abs__minus__le__zero,axiom,
% 16.19/9.00 ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6625_abs__minus__le__zero,axiom,
% 16.19/9.00 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6626_abs__minus__le__zero,axiom,
% 16.19/9.00 ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6627_abs__minus__le__zero,axiom,
% 16.19/9.00 ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6628_zero__le__power__abs,axiom,
% 16.19/9.00 ! [A: code_integer,N2: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6629_zero__le__power__abs,axiom,
% 16.19/9.00 ! [A: real,N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6630_zero__le__power__abs,axiom,
% 16.19/9.00 ! [A: rat,N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6631_zero__le__power__abs,axiom,
% 16.19/9.00 ! [A: int,N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N2 ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6632_abs__div__pos,axiom,
% 16.19/9.00 ! [Y: real,X: real] :
% 16.19/9.00 ( ( ord_less_real @ zero_zero_real @ Y )
% 16.19/9.00 => ( ( divide_divide_real @ ( abs_abs_real @ X ) @ Y )
% 16.19/9.00 = ( abs_abs_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6633_abs__div__pos,axiom,
% 16.19/9.00 ! [Y: rat,X: rat] :
% 16.19/9.00 ( ( ord_less_rat @ zero_zero_rat @ Y )
% 16.19/9.00 => ( ( divide_divide_rat @ ( abs_abs_rat @ X ) @ Y )
% 16.19/9.00 = ( abs_abs_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6634_abs__if__raw,axiom,
% 16.19/9.00 ( abs_abs_real
% 16.19/9.00 = ( ^ [A5: real] : ( if_real @ ( ord_less_real @ A5 @ zero_zero_real ) @ ( uminus_uminus_real @ A5 ) @ A5 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6635_abs__if__raw,axiom,
% 16.19/9.00 ( abs_abs_int
% 16.19/9.00 = ( ^ [A5: int] : ( if_int @ ( ord_less_int @ A5 @ zero_zero_int ) @ ( uminus_uminus_int @ A5 ) @ A5 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6636_abs__if__raw,axiom,
% 16.19/9.00 ( abs_abs_rat
% 16.19/9.00 = ( ^ [A5: rat] : ( if_rat @ ( ord_less_rat @ A5 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A5 ) @ A5 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6637_abs__if__raw,axiom,
% 16.19/9.00 ( abs_abs_Code_integer
% 16.19/9.00 = ( ^ [A5: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A5 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A5 ) @ A5 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6638_abs__of__neg,axiom,
% 16.19/9.00 ! [A: real] :
% 16.19/9.00 ( ( ord_less_real @ A @ zero_zero_real )
% 16.19/9.00 => ( ( abs_abs_real @ A )
% 16.19/9.00 = ( uminus_uminus_real @ A ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6639_abs__of__neg,axiom,
% 16.19/9.00 ! [A: int] :
% 16.19/9.00 ( ( ord_less_int @ A @ zero_zero_int )
% 16.19/9.00 => ( ( abs_abs_int @ A )
% 16.19/9.00 = ( uminus_uminus_int @ A ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6640_abs__of__neg,axiom,
% 16.19/9.00 ! [A: rat] :
% 16.19/9.00 ( ( ord_less_rat @ A @ zero_zero_rat )
% 16.19/9.00 => ( ( abs_abs_rat @ A )
% 16.19/9.00 = ( uminus_uminus_rat @ A ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6641_abs__of__neg,axiom,
% 16.19/9.00 ! [A: code_integer] :
% 16.19/9.00 ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 16.19/9.00 => ( ( abs_abs_Code_integer @ A )
% 16.19/9.00 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6642_abs__if,axiom,
% 16.19/9.00 ( abs_abs_real
% 16.19/9.00 = ( ^ [A5: real] : ( if_real @ ( ord_less_real @ A5 @ zero_zero_real ) @ ( uminus_uminus_real @ A5 ) @ A5 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6643_abs__if,axiom,
% 16.19/9.00 ( abs_abs_int
% 16.19/9.00 = ( ^ [A5: int] : ( if_int @ ( ord_less_int @ A5 @ zero_zero_int ) @ ( uminus_uminus_int @ A5 ) @ A5 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6644_abs__if,axiom,
% 16.19/9.00 ( abs_abs_rat
% 16.19/9.00 = ( ^ [A5: rat] : ( if_rat @ ( ord_less_rat @ A5 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A5 ) @ A5 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6645_abs__if,axiom,
% 16.19/9.00 ( abs_abs_Code_integer
% 16.19/9.00 = ( ^ [A5: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A5 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A5 ) @ A5 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6646_abs__triangle__ineq4,axiom,
% 16.19/9.00 ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6647_abs__triangle__ineq4,axiom,
% 16.19/9.00 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6648_abs__triangle__ineq4,axiom,
% 16.19/9.00 ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6649_abs__triangle__ineq4,axiom,
% 16.19/9.00 ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6650_abs__diff__triangle__ineq,axiom,
% 16.19/9.00 ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ D ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6651_abs__diff__triangle__ineq,axiom,
% 16.19/9.00 ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6652_abs__diff__triangle__ineq,axiom,
% 16.19/9.00 ! [A: rat,B: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6653_abs__diff__triangle__ineq,axiom,
% 16.19/9.00 ! [A: int,B: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6654_abs__diff__le__iff,axiom,
% 16.19/9.00 ! [X: code_integer,A: code_integer,R: code_integer] :
% 16.19/9.00 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R )
% 16.19/9.00 = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R ) @ X )
% 16.19/9.00 & ( ord_le3102999989581377725nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6655_abs__diff__le__iff,axiom,
% 16.19/9.00 ! [X: real,A: real,R: real] :
% 16.19/9.00 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R )
% 16.19/9.00 = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R ) @ X )
% 16.19/9.00 & ( ord_less_eq_real @ X @ ( plus_plus_real @ A @ R ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6656_abs__diff__le__iff,axiom,
% 16.19/9.00 ! [X: rat,A: rat,R: rat] :
% 16.19/9.00 ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R )
% 16.19/9.00 = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R ) @ X )
% 16.19/9.00 & ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A @ R ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6657_abs__diff__le__iff,axiom,
% 16.19/9.00 ! [X: int,A: int,R: int] :
% 16.19/9.00 ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R )
% 16.19/9.00 = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R ) @ X )
% 16.19/9.00 & ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6658_abs__diff__less__iff,axiom,
% 16.19/9.00 ! [X: code_integer,A: code_integer,R: code_integer] :
% 16.19/9.00 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R )
% 16.19/9.00 = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R ) @ X )
% 16.19/9.00 & ( ord_le6747313008572928689nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6659_abs__diff__less__iff,axiom,
% 16.19/9.00 ! [X: real,A: real,R: real] :
% 16.19/9.00 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R )
% 16.19/9.00 = ( ( ord_less_real @ ( minus_minus_real @ A @ R ) @ X )
% 16.19/9.00 & ( ord_less_real @ X @ ( plus_plus_real @ A @ R ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6660_abs__diff__less__iff,axiom,
% 16.19/9.00 ! [X: rat,A: rat,R: rat] :
% 16.19/9.00 ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R )
% 16.19/9.00 = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R ) @ X )
% 16.19/9.00 & ( ord_less_rat @ X @ ( plus_plus_rat @ A @ R ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6661_abs__diff__less__iff,axiom,
% 16.19/9.00 ! [X: int,A: int,R: int] :
% 16.19/9.00 ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R )
% 16.19/9.00 = ( ( ord_less_int @ ( minus_minus_int @ A @ R ) @ X )
% 16.19/9.00 & ( ord_less_int @ X @ ( plus_plus_int @ A @ R ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6662_sum__eq__Suc0__iff,axiom,
% 16.19/9.00 ! [A2: set_int,F: int > nat] :
% 16.19/9.00 ( ( finite_finite_int @ A2 )
% 16.19/9.00 => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
% 16.19/9.00 = ( suc @ zero_zero_nat ) )
% 16.19/9.00 = ( ? [X2: int] :
% 16.19/9.00 ( ( member_int @ X2 @ A2 )
% 16.19/9.00 & ( ( F @ X2 )
% 16.19/9.00 = ( suc @ zero_zero_nat ) )
% 16.19/9.00 & ! [Y2: int] :
% 16.19/9.00 ( ( member_int @ Y2 @ A2 )
% 16.19/9.00 => ( ( X2 != Y2 )
% 16.19/9.00 => ( ( F @ Y2 )
% 16.19/9.00 = zero_zero_nat ) ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6663_sum__eq__Suc0__iff,axiom,
% 16.19/9.00 ! [A2: set_complex,F: complex > nat] :
% 16.19/9.00 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/9.00 => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
% 16.19/9.00 = ( suc @ zero_zero_nat ) )
% 16.19/9.00 = ( ? [X2: complex] :
% 16.19/9.00 ( ( member_complex @ X2 @ A2 )
% 16.19/9.00 & ( ( F @ X2 )
% 16.19/9.00 = ( suc @ zero_zero_nat ) )
% 16.19/9.00 & ! [Y2: complex] :
% 16.19/9.00 ( ( member_complex @ Y2 @ A2 )
% 16.19/9.00 => ( ( X2 != Y2 )
% 16.19/9.00 => ( ( F @ Y2 )
% 16.19/9.00 = zero_zero_nat ) ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6664_sum__eq__Suc0__iff,axiom,
% 16.19/9.00 ! [A2: set_nat,F: nat > nat] :
% 16.19/9.00 ( ( finite_finite_nat @ A2 )
% 16.19/9.00 => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
% 16.19/9.00 = ( suc @ zero_zero_nat ) )
% 16.19/9.00 = ( ? [X2: nat] :
% 16.19/9.00 ( ( member_nat @ X2 @ A2 )
% 16.19/9.00 & ( ( F @ X2 )
% 16.19/9.00 = ( suc @ zero_zero_nat ) )
% 16.19/9.00 & ! [Y2: nat] :
% 16.19/9.00 ( ( member_nat @ Y2 @ A2 )
% 16.19/9.00 => ( ( X2 != Y2 )
% 16.19/9.00 => ( ( F @ Y2 )
% 16.19/9.00 = zero_zero_nat ) ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6665_sum__SucD,axiom,
% 16.19/9.00 ! [F: nat > nat,A2: set_nat,N2: nat] :
% 16.19/9.00 ( ( ( groups3542108847815614940at_nat @ F @ A2 )
% 16.19/9.00 = ( suc @ N2 ) )
% 16.19/9.00 => ? [X4: nat] :
% 16.19/9.00 ( ( member_nat @ X4 @ A2 )
% 16.19/9.00 & ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6666_sum__eq__1__iff,axiom,
% 16.19/9.00 ! [A2: set_int,F: int > nat] :
% 16.19/9.00 ( ( finite_finite_int @ A2 )
% 16.19/9.00 => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
% 16.19/9.00 = one_one_nat )
% 16.19/9.00 = ( ? [X2: int] :
% 16.19/9.00 ( ( member_int @ X2 @ A2 )
% 16.19/9.00 & ( ( F @ X2 )
% 16.19/9.00 = one_one_nat )
% 16.19/9.00 & ! [Y2: int] :
% 16.19/9.00 ( ( member_int @ Y2 @ A2 )
% 16.19/9.00 => ( ( X2 != Y2 )
% 16.19/9.00 => ( ( F @ Y2 )
% 16.19/9.00 = zero_zero_nat ) ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6667_sum__eq__1__iff,axiom,
% 16.19/9.00 ! [A2: set_complex,F: complex > nat] :
% 16.19/9.00 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/9.00 => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
% 16.19/9.00 = one_one_nat )
% 16.19/9.00 = ( ? [X2: complex] :
% 16.19/9.00 ( ( member_complex @ X2 @ A2 )
% 16.19/9.00 & ( ( F @ X2 )
% 16.19/9.00 = one_one_nat )
% 16.19/9.00 & ! [Y2: complex] :
% 16.19/9.00 ( ( member_complex @ Y2 @ A2 )
% 16.19/9.00 => ( ( X2 != Y2 )
% 16.19/9.00 => ( ( F @ Y2 )
% 16.19/9.00 = zero_zero_nat ) ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6668_sum__eq__1__iff,axiom,
% 16.19/9.00 ! [A2: set_nat,F: nat > nat] :
% 16.19/9.00 ( ( finite_finite_nat @ A2 )
% 16.19/9.00 => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
% 16.19/9.00 = one_one_nat )
% 16.19/9.00 = ( ? [X2: nat] :
% 16.19/9.00 ( ( member_nat @ X2 @ A2 )
% 16.19/9.00 & ( ( F @ X2 )
% 16.19/9.00 = one_one_nat )
% 16.19/9.00 & ! [Y2: nat] :
% 16.19/9.00 ( ( member_nat @ Y2 @ A2 )
% 16.19/9.00 => ( ( X2 != Y2 )
% 16.19/9.00 => ( ( F @ Y2 )
% 16.19/9.00 = zero_zero_nat ) ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6669_numeral__Bit1,axiom,
% 16.19/9.00 ! [N2: num] :
% 16.19/9.00 ( ( numera6690914467698888265omplex @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) @ one_one_complex ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6670_numeral__Bit1,axiom,
% 16.19/9.00 ! [N2: num] :
% 16.19/9.00 ( ( numeral_numeral_real @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) @ one_one_real ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6671_numeral__Bit1,axiom,
% 16.19/9.00 ! [N2: num] :
% 16.19/9.00 ( ( numeral_numeral_rat @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) @ one_one_rat ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6672_numeral__Bit1,axiom,
% 16.19/9.00 ! [N2: num] :
% 16.19/9.00 ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) @ one_one_nat ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6673_numeral__Bit1,axiom,
% 16.19/9.00 ! [N2: num] :
% 16.19/9.00 ( ( numeral_numeral_int @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) @ one_one_int ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6674_eval__nat__numeral_I3_J,axiom,
% 16.19/9.00 ! [N2: num] :
% 16.19/9.00 ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6675_cong__exp__iff__simps_I13_J,axiom,
% 16.19/9.00 ! [M: num,Q2: num,N2: num] :
% 16.19/9.00 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 16.19/9.00 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 16.19/9.00 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 16.19/9.00 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6676_cong__exp__iff__simps_I13_J,axiom,
% 16.19/9.00 ! [M: num,Q2: num,N2: num] :
% 16.19/9.00 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 16.19/9.00 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 16.19/9.00 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 16.19/9.00 = ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6677_cong__exp__iff__simps_I13_J,axiom,
% 16.19/9.00 ! [M: num,Q2: num,N2: num] :
% 16.19/9.00 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 16.19/9.00 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 16.19/9.00 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 16.19/9.00 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6678_cong__exp__iff__simps_I12_J,axiom,
% 16.19/9.00 ! [M: num,Q2: num,N2: num] :
% 16.19/9.00 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 16.19/9.00 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6679_cong__exp__iff__simps_I12_J,axiom,
% 16.19/9.00 ! [M: num,Q2: num,N2: num] :
% 16.19/9.00 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 16.19/9.00 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6680_cong__exp__iff__simps_I12_J,axiom,
% 16.19/9.00 ! [M: num,Q2: num,N2: num] :
% 16.19/9.00 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 16.19/9.00 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6681_cong__exp__iff__simps_I10_J,axiom,
% 16.19/9.00 ! [M: num,Q2: num,N2: num] :
% 16.19/9.00 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 16.19/9.00 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6682_cong__exp__iff__simps_I10_J,axiom,
% 16.19/9.00 ! [M: num,Q2: num,N2: num] :
% 16.19/9.00 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 16.19/9.00 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6683_cong__exp__iff__simps_I10_J,axiom,
% 16.19/9.00 ! [M: num,Q2: num,N2: num] :
% 16.19/9.00 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 16.19/9.00 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6684_power__minus__Bit1,axiom,
% 16.19/9.00 ! [X: real,K: num] :
% 16.19/9.00 ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 16.19/9.00 = ( uminus_uminus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6685_power__minus__Bit1,axiom,
% 16.19/9.00 ! [X: int,K: num] :
% 16.19/9.00 ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 16.19/9.00 = ( uminus_uminus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6686_power__minus__Bit1,axiom,
% 16.19/9.00 ! [X: complex,K: num] :
% 16.19/9.00 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 16.19/9.00 = ( uminus1482373934393186551omplex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6687_power__minus__Bit1,axiom,
% 16.19/9.00 ! [X: rat,K: num] :
% 16.19/9.00 ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 16.19/9.00 = ( uminus_uminus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6688_power__minus__Bit1,axiom,
% 16.19/9.00 ! [X: code_integer,K: num] :
% 16.19/9.00 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 16.19/9.00 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6689_lemma__interval__lt,axiom,
% 16.19/9.00 ! [A: real,X: real,B: real] :
% 16.19/9.00 ( ( ord_less_real @ A @ X )
% 16.19/9.00 => ( ( ord_less_real @ X @ B )
% 16.19/9.00 => ? [D4: real] :
% 16.19/9.00 ( ( ord_less_real @ zero_zero_real @ D4 )
% 16.19/9.00 & ! [Y5: real] :
% 16.19/9.00 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D4 )
% 16.19/9.00 => ( ( ord_less_real @ A @ Y5 )
% 16.19/9.00 & ( ord_less_real @ Y5 @ B ) ) ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6690_sum__power__add,axiom,
% 16.19/9.00 ! [X: complex,M: nat,I6: set_nat] :
% 16.19/9.00 ( ( groups2073611262835488442omplex
% 16.19/9.00 @ ^ [I3: nat] : ( power_power_complex @ X @ ( plus_plus_nat @ M @ I3 ) )
% 16.19/9.00 @ I6 )
% 16.19/9.00 = ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ I6 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6691_sum__power__add,axiom,
% 16.19/9.00 ! [X: rat,M: nat,I6: set_nat] :
% 16.19/9.00 ( ( groups2906978787729119204at_rat
% 16.19/9.00 @ ^ [I3: nat] : ( power_power_rat @ X @ ( plus_plus_nat @ M @ I3 ) )
% 16.19/9.00 @ I6 )
% 16.19/9.00 = ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ I6 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6692_sum__power__add,axiom,
% 16.19/9.00 ! [X: int,M: nat,I6: set_nat] :
% 16.19/9.00 ( ( groups3539618377306564664at_int
% 16.19/9.00 @ ^ [I3: nat] : ( power_power_int @ X @ ( plus_plus_nat @ M @ I3 ) )
% 16.19/9.00 @ I6 )
% 16.19/9.00 = ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ I6 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6693_sum__power__add,axiom,
% 16.19/9.00 ! [X: real,M: nat,I6: set_nat] :
% 16.19/9.00 ( ( groups6591440286371151544t_real
% 16.19/9.00 @ ^ [I3: nat] : ( power_power_real @ X @ ( plus_plus_nat @ M @ I3 ) )
% 16.19/9.00 @ I6 )
% 16.19/9.00 = ( times_times_real @ ( power_power_real @ X @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ I6 ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6694_sum_OatLeastAtMost__rev,axiom,
% 16.19/9.00 ! [G: nat > nat,N2: nat,M: nat] :
% 16.19/9.00 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
% 16.19/9.00 = ( groups3542108847815614940at_nat
% 16.19/9.00 @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I3 ) )
% 16.19/9.00 @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6695_sum_OatLeastAtMost__rev,axiom,
% 16.19/9.00 ! [G: nat > real,N2: nat,M: nat] :
% 16.19/9.00 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
% 16.19/9.00 = ( groups6591440286371151544t_real
% 16.19/9.00 @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I3 ) )
% 16.19/9.00 @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6696_numeral__code_I3_J,axiom,
% 16.19/9.00 ! [N2: num] :
% 16.19/9.00 ( ( numera6690914467698888265omplex @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) @ one_one_complex ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6697_numeral__code_I3_J,axiom,
% 16.19/9.00 ! [N2: num] :
% 16.19/9.00 ( ( numeral_numeral_real @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) @ one_one_real ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6698_numeral__code_I3_J,axiom,
% 16.19/9.00 ! [N2: num] :
% 16.19/9.00 ( ( numeral_numeral_rat @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) @ one_one_rat ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6699_numeral__code_I3_J,axiom,
% 16.19/9.00 ! [N2: num] :
% 16.19/9.00 ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) @ one_one_nat ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6700_numeral__code_I3_J,axiom,
% 16.19/9.00 ! [N2: num] :
% 16.19/9.00 ( ( numeral_numeral_int @ ( bit1 @ N2 ) )
% 16.19/9.00 = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) @ one_one_int ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6701_power__numeral__odd,axiom,
% 16.19/9.00 ! [Z: complex,W: num] :
% 16.19/9.00 ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 16.19/9.00 = ( times_times_complex @ ( times_times_complex @ Z @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 16.19/9.00
% 16.19/9.00 thf(fact_6702_power__numeral__odd,axiom,
% 16.19/9.00 ! [Z: real,W: num] :
% 16.19/9.00 ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 16.19/9.01 = ( times_times_real @ ( times_times_real @ Z @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6703_power__numeral__odd,axiom,
% 16.19/9.01 ! [Z: rat,W: num] :
% 16.19/9.01 ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 16.19/9.01 = ( times_times_rat @ ( times_times_rat @ Z @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6704_power__numeral__odd,axiom,
% 16.19/9.01 ! [Z: nat,W: num] :
% 16.19/9.01 ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 16.19/9.01 = ( times_times_nat @ ( times_times_nat @ Z @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6705_power__numeral__odd,axiom,
% 16.19/9.01 ! [Z: int,W: num] :
% 16.19/9.01 ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 16.19/9.01 = ( times_times_int @ ( times_times_int @ Z @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6706_sum__roots__unity,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( ord_less_nat @ one_one_nat @ N2 )
% 16.19/9.01 => ( ( groups7754918857620584856omplex
% 16.19/9.01 @ ^ [X2: complex] : X2
% 16.19/9.01 @ ( collect_complex
% 16.19/9.01 @ ^ [Z3: complex] :
% 16.19/9.01 ( ( power_power_complex @ Z3 @ N2 )
% 16.19/9.01 = one_one_complex ) ) )
% 16.19/9.01 = zero_zero_complex ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6707_sum__nth__roots,axiom,
% 16.19/9.01 ! [N2: nat,C: complex] :
% 16.19/9.01 ( ( ord_less_nat @ one_one_nat @ N2 )
% 16.19/9.01 => ( ( groups7754918857620584856omplex
% 16.19/9.01 @ ^ [X2: complex] : X2
% 16.19/9.01 @ ( collect_complex
% 16.19/9.01 @ ^ [Z3: complex] :
% 16.19/9.01 ( ( power_power_complex @ Z3 @ N2 )
% 16.19/9.01 = C ) ) )
% 16.19/9.01 = zero_zero_complex ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6708_abs__add__one__gt__zero,axiom,
% 16.19/9.01 ! [X: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6709_abs__add__one__gt__zero,axiom,
% 16.19/9.01 ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6710_abs__add__one__gt__zero,axiom,
% 16.19/9.01 ! [X: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6711_abs__add__one__gt__zero,axiom,
% 16.19/9.01 ! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6712_sum__diff__nat,axiom,
% 16.19/9.01 ! [B3: set_int,A2: set_int,F: int > nat] :
% 16.19/9.01 ( ( finite_finite_int @ B3 )
% 16.19/9.01 => ( ( ord_less_eq_set_int @ B3 @ A2 )
% 16.19/9.01 => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ B3 ) )
% 16.19/9.01 = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B3 ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6713_sum__diff__nat,axiom,
% 16.19/9.01 ! [B3: set_complex,A2: set_complex,F: complex > nat] :
% 16.19/9.01 ( ( finite3207457112153483333omplex @ B3 )
% 16.19/9.01 => ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 16.19/9.01 => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ B3 ) )
% 16.19/9.01 = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B3 ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6714_sum__diff__nat,axiom,
% 16.19/9.01 ! [B3: set_nat,A2: set_nat,F: nat > nat] :
% 16.19/9.01 ( ( finite_finite_nat @ B3 )
% 16.19/9.01 => ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 16.19/9.01 => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ B3 ) )
% 16.19/9.01 = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ F @ B3 ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6715_sum__shift__lb__Suc0__0,axiom,
% 16.19/9.01 ! [F: nat > complex,K: nat] :
% 16.19/9.01 ( ( ( F @ zero_zero_nat )
% 16.19/9.01 = zero_zero_complex )
% 16.19/9.01 => ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 16.19/9.01 = ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6716_sum__shift__lb__Suc0__0,axiom,
% 16.19/9.01 ! [F: nat > rat,K: nat] :
% 16.19/9.01 ( ( ( F @ zero_zero_nat )
% 16.19/9.01 = zero_zero_rat )
% 16.19/9.01 => ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 16.19/9.01 = ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6717_sum__shift__lb__Suc0__0,axiom,
% 16.19/9.01 ! [F: nat > int,K: nat] :
% 16.19/9.01 ( ( ( F @ zero_zero_nat )
% 16.19/9.01 = zero_zero_int )
% 16.19/9.01 => ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 16.19/9.01 = ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6718_sum__shift__lb__Suc0__0,axiom,
% 16.19/9.01 ! [F: nat > nat,K: nat] :
% 16.19/9.01 ( ( ( F @ zero_zero_nat )
% 16.19/9.01 = zero_zero_nat )
% 16.19/9.01 => ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 16.19/9.01 = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6719_sum__shift__lb__Suc0__0,axiom,
% 16.19/9.01 ! [F: nat > real,K: nat] :
% 16.19/9.01 ( ( ( F @ zero_zero_nat )
% 16.19/9.01 = zero_zero_real )
% 16.19/9.01 => ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 16.19/9.01 = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6720_sum_OatLeast0__atMost__Suc,axiom,
% 16.19/9.01 ! [G: nat > rat,N2: nat] :
% 16.19/9.01 ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
% 16.19/9.01 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6721_sum_OatLeast0__atMost__Suc,axiom,
% 16.19/9.01 ! [G: nat > int,N2: nat] :
% 16.19/9.01 ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
% 16.19/9.01 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6722_sum_OatLeast0__atMost__Suc,axiom,
% 16.19/9.01 ! [G: nat > nat,N2: nat] :
% 16.19/9.01 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
% 16.19/9.01 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6723_sum_OatLeast0__atMost__Suc,axiom,
% 16.19/9.01 ! [G: nat > real,N2: nat] :
% 16.19/9.01 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
% 16.19/9.01 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6724_sum_OatLeast__Suc__atMost,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,G: nat > rat] :
% 16.19/9.01 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.01 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.01 = ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6725_sum_OatLeast__Suc__atMost,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,G: nat > int] :
% 16.19/9.01 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.01 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.01 = ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6726_sum_OatLeast__Suc__atMost,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,G: nat > nat] :
% 16.19/9.01 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.01 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.01 = ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6727_sum_OatLeast__Suc__atMost,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,G: nat > real] :
% 16.19/9.01 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.01 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.01 = ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6728_sum_Onat__ivl__Suc_H,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,G: nat > rat] :
% 16.19/9.01 ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 16.19/9.01 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 16.19/9.01 = ( plus_plus_rat @ ( G @ ( suc @ N2 ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6729_sum_Onat__ivl__Suc_H,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,G: nat > int] :
% 16.19/9.01 ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 16.19/9.01 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 16.19/9.01 = ( plus_plus_int @ ( G @ ( suc @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6730_sum_Onat__ivl__Suc_H,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,G: nat > nat] :
% 16.19/9.01 ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 16.19/9.01 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 16.19/9.01 = ( plus_plus_nat @ ( G @ ( suc @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6731_sum_Onat__ivl__Suc_H,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,G: nat > real] :
% 16.19/9.01 ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 16.19/9.01 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 16.19/9.01 = ( plus_plus_real @ ( G @ ( suc @ N2 ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6732_numeral__Bit1__div__2,axiom,
% 16.19/9.01 ! [N2: num] :
% 16.19/9.01 ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.01 = ( numeral_numeral_nat @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6733_numeral__Bit1__div__2,axiom,
% 16.19/9.01 ! [N2: num] :
% 16.19/9.01 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/9.01 = ( numeral_numeral_int @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6734_odd__numeral,axiom,
% 16.19/9.01 ! [N2: num] :
% 16.19/9.01 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6735_odd__numeral,axiom,
% 16.19/9.01 ! [N2: num] :
% 16.19/9.01 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6736_odd__numeral,axiom,
% 16.19/9.01 ! [N2: num] :
% 16.19/9.01 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6737_cong__exp__iff__simps_I3_J,axiom,
% 16.19/9.01 ! [N2: num,Q2: num] :
% 16.19/9.01 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 16.19/9.01 != zero_zero_nat ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6738_cong__exp__iff__simps_I3_J,axiom,
% 16.19/9.01 ! [N2: num,Q2: num] :
% 16.19/9.01 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 16.19/9.01 != zero_zero_int ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6739_cong__exp__iff__simps_I3_J,axiom,
% 16.19/9.01 ! [N2: num,Q2: num] :
% 16.19/9.01 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 16.19/9.01 != zero_z3403309356797280102nteger ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6740_power3__eq__cube,axiom,
% 16.19/9.01 ! [A: complex] :
% 16.19/9.01 ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 16.19/9.01 = ( times_times_complex @ ( times_times_complex @ A @ A ) @ A ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6741_power3__eq__cube,axiom,
% 16.19/9.01 ! [A: real] :
% 16.19/9.01 ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 16.19/9.01 = ( times_times_real @ ( times_times_real @ A @ A ) @ A ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6742_power3__eq__cube,axiom,
% 16.19/9.01 ! [A: rat] :
% 16.19/9.01 ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 16.19/9.01 = ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6743_power3__eq__cube,axiom,
% 16.19/9.01 ! [A: nat] :
% 16.19/9.01 ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 16.19/9.01 = ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6744_power3__eq__cube,axiom,
% 16.19/9.01 ! [A: int] :
% 16.19/9.01 ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 16.19/9.01 = ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6745_numeral__3__eq__3,axiom,
% 16.19/9.01 ( ( numeral_numeral_nat @ ( bit1 @ one ) )
% 16.19/9.01 = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6746_Suc3__eq__add__3,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( suc @ ( suc @ ( suc @ N2 ) ) )
% 16.19/9.01 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6747_lemma__interval,axiom,
% 16.19/9.01 ! [A: real,X: real,B: real] :
% 16.19/9.01 ( ( ord_less_real @ A @ X )
% 16.19/9.01 => ( ( ord_less_real @ X @ B )
% 16.19/9.01 => ? [D4: real] :
% 16.19/9.01 ( ( ord_less_real @ zero_zero_real @ D4 )
% 16.19/9.01 & ! [Y5: real] :
% 16.19/9.01 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D4 )
% 16.19/9.01 => ( ( ord_less_eq_real @ A @ Y5 )
% 16.19/9.01 & ( ord_less_eq_real @ Y5 @ B ) ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6748_sum_OSuc__reindex__ivl,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,G: nat > rat] :
% 16.19/9.01 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.01 => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
% 16.19/9.01 = ( plus_plus_rat @ ( G @ M )
% 16.19/9.01 @ ( groups2906978787729119204at_rat
% 16.19/9.01 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 16.19/9.01 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6749_sum_OSuc__reindex__ivl,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,G: nat > int] :
% 16.19/9.01 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.01 => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
% 16.19/9.01 = ( plus_plus_int @ ( G @ M )
% 16.19/9.01 @ ( groups3539618377306564664at_int
% 16.19/9.01 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 16.19/9.01 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6750_sum_OSuc__reindex__ivl,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,G: nat > nat] :
% 16.19/9.01 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.01 => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
% 16.19/9.01 = ( plus_plus_nat @ ( G @ M )
% 16.19/9.01 @ ( groups3542108847815614940at_nat
% 16.19/9.01 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 16.19/9.01 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6751_sum_OSuc__reindex__ivl,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,G: nat > real] :
% 16.19/9.01 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.01 => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
% 16.19/9.01 = ( plus_plus_real @ ( G @ M )
% 16.19/9.01 @ ( groups6591440286371151544t_real
% 16.19/9.01 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 16.19/9.01 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6752_sum__Suc__diff,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,F: nat > rat] :
% 16.19/9.01 ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 16.19/9.01 => ( ( groups2906978787729119204at_rat
% 16.19/9.01 @ ^ [I3: nat] : ( minus_minus_rat @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
% 16.19/9.01 @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.01 = ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6753_sum__Suc__diff,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,F: nat > int] :
% 16.19/9.01 ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 16.19/9.01 => ( ( groups3539618377306564664at_int
% 16.19/9.01 @ ^ [I3: nat] : ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
% 16.19/9.01 @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.01 = ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6754_sum__Suc__diff,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,F: nat > real] :
% 16.19/9.01 ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 16.19/9.01 => ( ( groups6591440286371151544t_real
% 16.19/9.01 @ ^ [I3: nat] : ( minus_minus_real @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
% 16.19/9.01 @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.01 = ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6755_mod__exhaust__less__4,axiom,
% 16.19/9.01 ! [M: nat] :
% 16.19/9.01 ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 16.19/9.01 = zero_zero_nat )
% 16.19/9.01 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 16.19/9.01 = one_one_nat )
% 16.19/9.01 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 16.19/9.01 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.01 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 16.19/9.01 = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6756_abs__le__square__iff,axiom,
% 16.19/9.01 ! [X: code_integer,Y: code_integer] :
% 16.19/9.01 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ ( abs_abs_Code_integer @ Y ) )
% 16.19/9.01 = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6757_abs__le__square__iff,axiom,
% 16.19/9.01 ! [X: real,Y: real] :
% 16.19/9.01 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) )
% 16.19/9.01 = ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6758_abs__le__square__iff,axiom,
% 16.19/9.01 ! [X: rat,Y: rat] :
% 16.19/9.01 ( ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ ( abs_abs_rat @ Y ) )
% 16.19/9.01 = ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6759_abs__le__square__iff,axiom,
% 16.19/9.01 ! [X: int,Y: int] :
% 16.19/9.01 ( ( ord_less_eq_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) )
% 16.19/9.01 = ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6760_abs__square__eq__1,axiom,
% 16.19/9.01 ! [X: code_integer] :
% 16.19/9.01 ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.01 = one_one_Code_integer )
% 16.19/9.01 = ( ( abs_abs_Code_integer @ X )
% 16.19/9.01 = one_one_Code_integer ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6761_abs__square__eq__1,axiom,
% 16.19/9.01 ! [X: rat] :
% 16.19/9.01 ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.01 = one_one_rat )
% 16.19/9.01 = ( ( abs_abs_rat @ X )
% 16.19/9.01 = one_one_rat ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6762_abs__square__eq__1,axiom,
% 16.19/9.01 ! [X: real] :
% 16.19/9.01 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.01 = one_one_real )
% 16.19/9.01 = ( ( abs_abs_real @ X )
% 16.19/9.01 = one_one_real ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6763_abs__square__eq__1,axiom,
% 16.19/9.01 ! [X: int] :
% 16.19/9.01 ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.01 = one_one_int )
% 16.19/9.01 = ( ( abs_abs_int @ X )
% 16.19/9.01 = one_one_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6764_num_Osize_I6_J,axiom,
% 16.19/9.01 ! [X32: num] :
% 16.19/9.01 ( ( size_size_num @ ( bit1 @ X32 ) )
% 16.19/9.01 = ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6765_power__even__abs,axiom,
% 16.19/9.01 ! [N2: nat,A: rat] :
% 16.19/9.01 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.01 => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ N2 )
% 16.19/9.01 = ( power_power_rat @ A @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6766_power__even__abs,axiom,
% 16.19/9.01 ! [N2: nat,A: code_integer] :
% 16.19/9.01 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.01 => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N2 )
% 16.19/9.01 = ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6767_power__even__abs,axiom,
% 16.19/9.01 ! [N2: nat,A: real] :
% 16.19/9.01 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.01 => ( ( power_power_real @ ( abs_abs_real @ A ) @ N2 )
% 16.19/9.01 = ( power_power_real @ A @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6768_power__even__abs,axiom,
% 16.19/9.01 ! [N2: nat,A: int] :
% 16.19/9.01 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.01 => ( ( power_power_int @ ( abs_abs_int @ A ) @ N2 )
% 16.19/9.01 = ( power_power_int @ A @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6769_sum_Oub__add__nat,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,G: nat > rat,P4: nat] :
% 16.19/9.01 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
% 16.19/9.01 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
% 16.19/9.01 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6770_sum_Oub__add__nat,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,G: nat > int,P4: nat] :
% 16.19/9.01 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
% 16.19/9.01 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
% 16.19/9.01 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6771_sum_Oub__add__nat,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,G: nat > nat,P4: nat] :
% 16.19/9.01 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
% 16.19/9.01 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
% 16.19/9.01 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6772_sum_Oub__add__nat,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,G: nat > real,P4: nat] :
% 16.19/9.01 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
% 16.19/9.01 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
% 16.19/9.01 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6773_cong__exp__iff__simps_I11_J,axiom,
% 16.19/9.01 ! [M: num,Q2: num] :
% 16.19/9.01 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 16.19/9.01 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 16.19/9.01 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 16.19/9.01 = zero_zero_nat ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6774_cong__exp__iff__simps_I11_J,axiom,
% 16.19/9.01 ! [M: num,Q2: num] :
% 16.19/9.01 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 16.19/9.01 = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 16.19/9.01 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 16.19/9.01 = zero_zero_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6775_cong__exp__iff__simps_I11_J,axiom,
% 16.19/9.01 ! [M: num,Q2: num] :
% 16.19/9.01 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 16.19/9.01 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 16.19/9.01 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 16.19/9.01 = zero_z3403309356797280102nteger ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6776_cong__exp__iff__simps_I7_J,axiom,
% 16.19/9.01 ! [Q2: num,N2: num] :
% 16.19/9.01 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 16.19/9.01 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 16.19/9.01 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q2 ) )
% 16.19/9.01 = zero_zero_nat ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6777_cong__exp__iff__simps_I7_J,axiom,
% 16.19/9.01 ! [Q2: num,N2: num] :
% 16.19/9.01 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 16.19/9.01 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 16.19/9.01 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q2 ) )
% 16.19/9.01 = zero_zero_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6778_cong__exp__iff__simps_I7_J,axiom,
% 16.19/9.01 ! [Q2: num,N2: num] :
% 16.19/9.01 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 16.19/9.01 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 16.19/9.01 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 16.19/9.01 = zero_z3403309356797280102nteger ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6779_Suc__div__eq__add3__div,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 )
% 16.19/9.01 = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6780_Suc__mod__eq__add3__mod,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 )
% 16.19/9.01 = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6781_set__encode__def,axiom,
% 16.19/9.01 ( nat_set_encode
% 16.19/9.01 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6782_divmod__int__def,axiom,
% 16.19/9.01 ( unique5052692396658037445od_int
% 16.19/9.01 = ( ^ [M6: num,N: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6783_abs__sqrt__wlog,axiom,
% 16.19/9.01 ! [P: code_integer > code_integer > $o,X: code_integer] :
% 16.19/9.01 ( ! [X4: code_integer] :
% 16.19/9.01 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X4 )
% 16.19/9.01 => ( P @ X4 @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/9.01 => ( P @ ( abs_abs_Code_integer @ X ) @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6784_abs__sqrt__wlog,axiom,
% 16.19/9.01 ! [P: real > real > $o,X: real] :
% 16.19/9.01 ( ! [X4: real] :
% 16.19/9.01 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 16.19/9.01 => ( P @ X4 @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/9.01 => ( P @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6785_abs__sqrt__wlog,axiom,
% 16.19/9.01 ! [P: rat > rat > $o,X: rat] :
% 16.19/9.01 ( ! [X4: rat] :
% 16.19/9.01 ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 16.19/9.01 => ( P @ X4 @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/9.01 => ( P @ ( abs_abs_rat @ X ) @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6786_abs__sqrt__wlog,axiom,
% 16.19/9.01 ! [P: int > int > $o,X: int] :
% 16.19/9.01 ( ! [X4: int] :
% 16.19/9.01 ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 16.19/9.01 => ( P @ X4 @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/9.01 => ( P @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6787_power2__le__iff__abs__le,axiom,
% 16.19/9.01 ! [Y: code_integer,X: code_integer] :
% 16.19/9.01 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
% 16.19/9.01 => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.01 = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ Y ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6788_power2__le__iff__abs__le,axiom,
% 16.19/9.01 ! [Y: real,X: real] :
% 16.19/9.01 ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/9.01 => ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.01 = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ Y ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6789_power2__le__iff__abs__le,axiom,
% 16.19/9.01 ! [Y: rat,X: rat] :
% 16.19/9.01 ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
% 16.19/9.01 => ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.01 = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ Y ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6790_power2__le__iff__abs__le,axiom,
% 16.19/9.01 ! [Y: int,X: int] :
% 16.19/9.01 ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 16.19/9.01 => ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.01 = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ Y ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6791_abs__square__le__1,axiom,
% 16.19/9.01 ! [X: code_integer] :
% 16.19/9.01 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 16.19/9.01 = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6792_abs__square__le__1,axiom,
% 16.19/9.01 ! [X: real] :
% 16.19/9.01 ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 16.19/9.01 = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6793_abs__square__le__1,axiom,
% 16.19/9.01 ! [X: rat] :
% 16.19/9.01 ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 16.19/9.01 = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6794_abs__square__le__1,axiom,
% 16.19/9.01 ! [X: int] :
% 16.19/9.01 ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 16.19/9.01 = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6795_abs__square__less__1,axiom,
% 16.19/9.01 ! [X: code_integer] :
% 16.19/9.01 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 16.19/9.01 = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6796_abs__square__less__1,axiom,
% 16.19/9.01 ! [X: real] :
% 16.19/9.01 ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 16.19/9.01 = ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6797_abs__square__less__1,axiom,
% 16.19/9.01 ! [X: rat] :
% 16.19/9.01 ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 16.19/9.01 = ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6798_abs__square__less__1,axiom,
% 16.19/9.01 ! [X: int] :
% 16.19/9.01 ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 16.19/9.01 = ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6799_divmod__def,axiom,
% 16.19/9.01 ( unique5052692396658037445od_int
% 16.19/9.01 = ( ^ [M6: num,N: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6800_divmod__def,axiom,
% 16.19/9.01 ( unique5055182867167087721od_nat
% 16.19/9.01 = ( ^ [M6: num,N: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6801_divmod__def,axiom,
% 16.19/9.01 ( unique3479559517661332726nteger
% 16.19/9.01 = ( ^ [M6: num,N: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6802_divmod_H__nat__def,axiom,
% 16.19/9.01 ( unique5055182867167087721od_nat
% 16.19/9.01 = ( ^ [M6: num,N: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6803_power__mono__even,axiom,
% 16.19/9.01 ! [N2: nat,A: code_integer,B: code_integer] :
% 16.19/9.01 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.01 => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
% 16.19/9.01 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( power_8256067586552552935nteger @ B @ N2 ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6804_power__mono__even,axiom,
% 16.19/9.01 ! [N2: nat,A: real,B: real] :
% 16.19/9.01 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.01 => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) )
% 16.19/9.01 => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6805_power__mono__even,axiom,
% 16.19/9.01 ! [N2: nat,A: rat,B: rat] :
% 16.19/9.01 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.01 => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
% 16.19/9.01 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6806_power__mono__even,axiom,
% 16.19/9.01 ! [N2: nat,A: int,B: int] :
% 16.19/9.01 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.01 => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
% 16.19/9.01 => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6807_convex__sum__bound__le,axiom,
% 16.19/9.01 ! [I6: set_nat,X: nat > code_integer,A: nat > code_integer,B: code_integer,Delta: code_integer] :
% 16.19/9.01 ( ! [I4: nat] :
% 16.19/9.01 ( ( member_nat @ I4 @ I6 )
% 16.19/9.01 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I4 ) ) )
% 16.19/9.01 => ( ( ( groups7501900531339628137nteger @ X @ I6 )
% 16.19/9.01 = one_one_Code_integer )
% 16.19/9.01 => ( ! [I4: nat] :
% 16.19/9.01 ( ( member_nat @ I4 @ I6 )
% 16.19/9.01 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I4 ) @ B ) ) @ Delta ) )
% 16.19/9.01 => ( ord_le3102999989581377725nteger
% 16.19/9.01 @ ( abs_abs_Code_integer
% 16.19/9.01 @ ( minus_8373710615458151222nteger
% 16.19/9.01 @ ( groups7501900531339628137nteger
% 16.19/9.01 @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
% 16.19/9.01 @ I6 )
% 16.19/9.01 @ B ) )
% 16.19/9.01 @ Delta ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6808_convex__sum__bound__le,axiom,
% 16.19/9.01 ! [I6: set_real,X: real > code_integer,A: real > code_integer,B: code_integer,Delta: code_integer] :
% 16.19/9.01 ( ! [I4: real] :
% 16.19/9.01 ( ( member_real @ I4 @ I6 )
% 16.19/9.01 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I4 ) ) )
% 16.19/9.01 => ( ( ( groups7713935264441627589nteger @ X @ I6 )
% 16.19/9.01 = one_one_Code_integer )
% 16.19/9.01 => ( ! [I4: real] :
% 16.19/9.01 ( ( member_real @ I4 @ I6 )
% 16.19/9.01 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I4 ) @ B ) ) @ Delta ) )
% 16.19/9.01 => ( ord_le3102999989581377725nteger
% 16.19/9.01 @ ( abs_abs_Code_integer
% 16.19/9.01 @ ( minus_8373710615458151222nteger
% 16.19/9.01 @ ( groups7713935264441627589nteger
% 16.19/9.01 @ ^ [I3: real] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
% 16.19/9.01 @ I6 )
% 16.19/9.01 @ B ) )
% 16.19/9.01 @ Delta ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6809_convex__sum__bound__le,axiom,
% 16.19/9.01 ! [I6: set_int,X: int > code_integer,A: int > code_integer,B: code_integer,Delta: code_integer] :
% 16.19/9.01 ( ! [I4: int] :
% 16.19/9.01 ( ( member_int @ I4 @ I6 )
% 16.19/9.01 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I4 ) ) )
% 16.19/9.01 => ( ( ( groups7873554091576472773nteger @ X @ I6 )
% 16.19/9.01 = one_one_Code_integer )
% 16.19/9.01 => ( ! [I4: int] :
% 16.19/9.01 ( ( member_int @ I4 @ I6 )
% 16.19/9.01 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I4 ) @ B ) ) @ Delta ) )
% 16.19/9.01 => ( ord_le3102999989581377725nteger
% 16.19/9.01 @ ( abs_abs_Code_integer
% 16.19/9.01 @ ( minus_8373710615458151222nteger
% 16.19/9.01 @ ( groups7873554091576472773nteger
% 16.19/9.01 @ ^ [I3: int] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
% 16.19/9.01 @ I6 )
% 16.19/9.01 @ B ) )
% 16.19/9.01 @ Delta ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6810_convex__sum__bound__le,axiom,
% 16.19/9.01 ! [I6: set_complex,X: complex > code_integer,A: complex > code_integer,B: code_integer,Delta: code_integer] :
% 16.19/9.01 ( ! [I4: complex] :
% 16.19/9.01 ( ( member_complex @ I4 @ I6 )
% 16.19/9.01 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I4 ) ) )
% 16.19/9.01 => ( ( ( groups6621422865394947399nteger @ X @ I6 )
% 16.19/9.01 = one_one_Code_integer )
% 16.19/9.01 => ( ! [I4: complex] :
% 16.19/9.01 ( ( member_complex @ I4 @ I6 )
% 16.19/9.01 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I4 ) @ B ) ) @ Delta ) )
% 16.19/9.01 => ( ord_le3102999989581377725nteger
% 16.19/9.01 @ ( abs_abs_Code_integer
% 16.19/9.01 @ ( minus_8373710615458151222nteger
% 16.19/9.01 @ ( groups6621422865394947399nteger
% 16.19/9.01 @ ^ [I3: complex] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X @ I3 ) )
% 16.19/9.01 @ I6 )
% 16.19/9.01 @ B ) )
% 16.19/9.01 @ Delta ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6811_convex__sum__bound__le,axiom,
% 16.19/9.01 ! [I6: set_real,X: real > real,A: real > real,B: real,Delta: real] :
% 16.19/9.01 ( ! [I4: real] :
% 16.19/9.01 ( ( member_real @ I4 @ I6 )
% 16.19/9.01 => ( ord_less_eq_real @ zero_zero_real @ ( X @ I4 ) ) )
% 16.19/9.01 => ( ( ( groups8097168146408367636l_real @ X @ I6 )
% 16.19/9.01 = one_one_real )
% 16.19/9.01 => ( ! [I4: real] :
% 16.19/9.01 ( ( member_real @ I4 @ I6 )
% 16.19/9.01 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I4 ) @ B ) ) @ Delta ) )
% 16.19/9.01 => ( ord_less_eq_real
% 16.19/9.01 @ ( abs_abs_real
% 16.19/9.01 @ ( minus_minus_real
% 16.19/9.01 @ ( groups8097168146408367636l_real
% 16.19/9.01 @ ^ [I3: real] : ( times_times_real @ ( A @ I3 ) @ ( X @ I3 ) )
% 16.19/9.01 @ I6 )
% 16.19/9.01 @ B ) )
% 16.19/9.01 @ Delta ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6812_convex__sum__bound__le,axiom,
% 16.19/9.01 ! [I6: set_int,X: int > real,A: int > real,B: real,Delta: real] :
% 16.19/9.01 ( ! [I4: int] :
% 16.19/9.01 ( ( member_int @ I4 @ I6 )
% 16.19/9.01 => ( ord_less_eq_real @ zero_zero_real @ ( X @ I4 ) ) )
% 16.19/9.01 => ( ( ( groups8778361861064173332t_real @ X @ I6 )
% 16.19/9.01 = one_one_real )
% 16.19/9.01 => ( ! [I4: int] :
% 16.19/9.01 ( ( member_int @ I4 @ I6 )
% 16.19/9.01 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I4 ) @ B ) ) @ Delta ) )
% 16.19/9.01 => ( ord_less_eq_real
% 16.19/9.01 @ ( abs_abs_real
% 16.19/9.01 @ ( minus_minus_real
% 16.19/9.01 @ ( groups8778361861064173332t_real
% 16.19/9.01 @ ^ [I3: int] : ( times_times_real @ ( A @ I3 ) @ ( X @ I3 ) )
% 16.19/9.01 @ I6 )
% 16.19/9.01 @ B ) )
% 16.19/9.01 @ Delta ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6813_convex__sum__bound__le,axiom,
% 16.19/9.01 ! [I6: set_complex,X: complex > real,A: complex > real,B: real,Delta: real] :
% 16.19/9.01 ( ! [I4: complex] :
% 16.19/9.01 ( ( member_complex @ I4 @ I6 )
% 16.19/9.01 => ( ord_less_eq_real @ zero_zero_real @ ( X @ I4 ) ) )
% 16.19/9.01 => ( ( ( groups5808333547571424918x_real @ X @ I6 )
% 16.19/9.01 = one_one_real )
% 16.19/9.01 => ( ! [I4: complex] :
% 16.19/9.01 ( ( member_complex @ I4 @ I6 )
% 16.19/9.01 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I4 ) @ B ) ) @ Delta ) )
% 16.19/9.01 => ( ord_less_eq_real
% 16.19/9.01 @ ( abs_abs_real
% 16.19/9.01 @ ( minus_minus_real
% 16.19/9.01 @ ( groups5808333547571424918x_real
% 16.19/9.01 @ ^ [I3: complex] : ( times_times_real @ ( A @ I3 ) @ ( X @ I3 ) )
% 16.19/9.01 @ I6 )
% 16.19/9.01 @ B ) )
% 16.19/9.01 @ Delta ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6814_convex__sum__bound__le,axiom,
% 16.19/9.01 ! [I6: set_nat,X: nat > rat,A: nat > rat,B: rat,Delta: rat] :
% 16.19/9.01 ( ! [I4: nat] :
% 16.19/9.01 ( ( member_nat @ I4 @ I6 )
% 16.19/9.01 => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I4 ) ) )
% 16.19/9.01 => ( ( ( groups2906978787729119204at_rat @ X @ I6 )
% 16.19/9.01 = one_one_rat )
% 16.19/9.01 => ( ! [I4: nat] :
% 16.19/9.01 ( ( member_nat @ I4 @ I6 )
% 16.19/9.01 => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I4 ) @ B ) ) @ Delta ) )
% 16.19/9.01 => ( ord_less_eq_rat
% 16.19/9.01 @ ( abs_abs_rat
% 16.19/9.01 @ ( minus_minus_rat
% 16.19/9.01 @ ( groups2906978787729119204at_rat
% 16.19/9.01 @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( X @ I3 ) )
% 16.19/9.01 @ I6 )
% 16.19/9.01 @ B ) )
% 16.19/9.01 @ Delta ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6815_convex__sum__bound__le,axiom,
% 16.19/9.01 ! [I6: set_real,X: real > rat,A: real > rat,B: rat,Delta: rat] :
% 16.19/9.01 ( ! [I4: real] :
% 16.19/9.01 ( ( member_real @ I4 @ I6 )
% 16.19/9.01 => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I4 ) ) )
% 16.19/9.01 => ( ( ( groups1300246762558778688al_rat @ X @ I6 )
% 16.19/9.01 = one_one_rat )
% 16.19/9.01 => ( ! [I4: real] :
% 16.19/9.01 ( ( member_real @ I4 @ I6 )
% 16.19/9.01 => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I4 ) @ B ) ) @ Delta ) )
% 16.19/9.01 => ( ord_less_eq_rat
% 16.19/9.01 @ ( abs_abs_rat
% 16.19/9.01 @ ( minus_minus_rat
% 16.19/9.01 @ ( groups1300246762558778688al_rat
% 16.19/9.01 @ ^ [I3: real] : ( times_times_rat @ ( A @ I3 ) @ ( X @ I3 ) )
% 16.19/9.01 @ I6 )
% 16.19/9.01 @ B ) )
% 16.19/9.01 @ Delta ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6816_convex__sum__bound__le,axiom,
% 16.19/9.01 ! [I6: set_int,X: int > rat,A: int > rat,B: rat,Delta: rat] :
% 16.19/9.01 ( ! [I4: int] :
% 16.19/9.01 ( ( member_int @ I4 @ I6 )
% 16.19/9.01 => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I4 ) ) )
% 16.19/9.01 => ( ( ( groups3906332499630173760nt_rat @ X @ I6 )
% 16.19/9.01 = one_one_rat )
% 16.19/9.01 => ( ! [I4: int] :
% 16.19/9.01 ( ( member_int @ I4 @ I6 )
% 16.19/9.01 => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I4 ) @ B ) ) @ Delta ) )
% 16.19/9.01 => ( ord_less_eq_rat
% 16.19/9.01 @ ( abs_abs_rat
% 16.19/9.01 @ ( minus_minus_rat
% 16.19/9.01 @ ( groups3906332499630173760nt_rat
% 16.19/9.01 @ ^ [I3: int] : ( times_times_rat @ ( A @ I3 ) @ ( X @ I3 ) )
% 16.19/9.01 @ I6 )
% 16.19/9.01 @ B ) )
% 16.19/9.01 @ Delta ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6817_sum__natinterval__diff,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,F: nat > complex] :
% 16.19/9.01 ( ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.01 => ( ( groups2073611262835488442omplex
% 16.19/9.01 @ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 16.19/9.01 @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.01 = ( minus_minus_complex @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
% 16.19/9.01 & ( ~ ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.01 => ( ( groups2073611262835488442omplex
% 16.19/9.01 @ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 16.19/9.01 @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.01 = zero_zero_complex ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6818_sum__natinterval__diff,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,F: nat > rat] :
% 16.19/9.01 ( ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.01 => ( ( groups2906978787729119204at_rat
% 16.19/9.01 @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 16.19/9.01 @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.01 = ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
% 16.19/9.01 & ( ~ ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.01 => ( ( groups2906978787729119204at_rat
% 16.19/9.01 @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 16.19/9.01 @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.01 = zero_zero_rat ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6819_sum__natinterval__diff,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,F: nat > int] :
% 16.19/9.01 ( ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.01 => ( ( groups3539618377306564664at_int
% 16.19/9.01 @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 16.19/9.01 @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.01 = ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
% 16.19/9.01 & ( ~ ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.01 => ( ( groups3539618377306564664at_int
% 16.19/9.01 @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 16.19/9.01 @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.01 = zero_zero_int ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6820_sum__natinterval__diff,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,F: nat > real] :
% 16.19/9.01 ( ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.01 => ( ( groups6591440286371151544t_real
% 16.19/9.01 @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 16.19/9.01 @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.01 = ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
% 16.19/9.01 & ( ~ ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.01 => ( ( groups6591440286371151544t_real
% 16.19/9.01 @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 16.19/9.01 @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.01 = zero_zero_real ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6821_sum__telescope_H_H,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,F: nat > rat] :
% 16.19/9.01 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.01 => ( ( groups2906978787729119204at_rat
% 16.19/9.01 @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
% 16.19/9.01 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
% 16.19/9.01 = ( minus_minus_rat @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6822_sum__telescope_H_H,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,F: nat > int] :
% 16.19/9.01 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.01 => ( ( groups3539618377306564664at_int
% 16.19/9.01 @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
% 16.19/9.01 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
% 16.19/9.01 = ( minus_minus_int @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6823_sum__telescope_H_H,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,F: nat > real] :
% 16.19/9.01 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.01 => ( ( groups6591440286371151544t_real
% 16.19/9.01 @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
% 16.19/9.01 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
% 16.19/9.01 = ( minus_minus_real @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6824_divmod__nat__def,axiom,
% 16.19/9.01 ( divmod_nat
% 16.19/9.01 = ( ^ [M6: nat,N: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M6 @ N ) @ ( modulo_modulo_nat @ M6 @ N ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6825_mask__eq__sum__exp,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int )
% 16.19/9.01 = ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/9.01 @ ( collect_nat
% 16.19/9.01 @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N2 ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6826_mask__eq__sum__exp,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat )
% 16.19/9.01 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.01 @ ( collect_nat
% 16.19/9.01 @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N2 ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6827_sum__gp__multiplied,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,X: complex] :
% 16.19/9.01 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.01 => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
% 16.19/9.01 = ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6828_sum__gp__multiplied,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,X: rat] :
% 16.19/9.01 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.01 => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
% 16.19/9.01 = ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6829_sum__gp__multiplied,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,X: int] :
% 16.19/9.01 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.01 => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
% 16.19/9.01 = ( minus_minus_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ ( suc @ N2 ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6830_sum__gp__multiplied,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,X: real] :
% 16.19/9.01 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.01 => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
% 16.19/9.01 = ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N2 ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6831_sum_Oin__pairs,axiom,
% 16.19/9.01 ! [G: nat > rat,M: nat,N2: nat] :
% 16.19/9.01 ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 16.19/9.01 = ( groups2906978787729119204at_rat
% 16.19/9.01 @ ^ [I3: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
% 16.19/9.01 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6832_sum_Oin__pairs,axiom,
% 16.19/9.01 ! [G: nat > int,M: nat,N2: nat] :
% 16.19/9.01 ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 16.19/9.01 = ( groups3539618377306564664at_int
% 16.19/9.01 @ ^ [I3: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
% 16.19/9.01 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6833_sum_Oin__pairs,axiom,
% 16.19/9.01 ! [G: nat > nat,M: nat,N2: nat] :
% 16.19/9.01 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 16.19/9.01 = ( groups3542108847815614940at_nat
% 16.19/9.01 @ ^ [I3: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
% 16.19/9.01 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6834_sum_Oin__pairs,axiom,
% 16.19/9.01 ! [G: nat > real,M: nat,N2: nat] :
% 16.19/9.01 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 16.19/9.01 = ( groups6591440286371151544t_real
% 16.19/9.01 @ ^ [I3: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
% 16.19/9.01 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6835_eq__diff__eq_H,axiom,
% 16.19/9.01 ! [X: real,Y: real,Z: real] :
% 16.19/9.01 ( ( X
% 16.19/9.01 = ( minus_minus_real @ Y @ Z ) )
% 16.19/9.01 = ( Y
% 16.19/9.01 = ( plus_plus_real @ X @ Z ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6836_mask__eq__sum__exp__nat,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( suc @ zero_zero_nat ) )
% 16.19/9.01 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.01 @ ( collect_nat
% 16.19/9.01 @ ^ [Q4: nat] : ( ord_less_nat @ Q4 @ N2 ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6837_gauss__sum__nat,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( groups3542108847815614940at_nat
% 16.19/9.01 @ ^ [X2: nat] : X2
% 16.19/9.01 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 16.19/9.01 = ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6838_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
% 16.19/9.01 ! [X: real] :
% 16.19/9.01 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.01 => ( ( ord_less_eq_real @ X @ one_one_real )
% 16.19/9.01 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6839_arith__series__nat,axiom,
% 16.19/9.01 ! [A: nat,D: nat,N2: nat] :
% 16.19/9.01 ( ( groups3542108847815614940at_nat
% 16.19/9.01 @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I3 @ D ) )
% 16.19/9.01 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 16.19/9.01 = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N2 @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6840_Sum__Icc__nat,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( groups3542108847815614940at_nat
% 16.19/9.01 @ ^ [X2: nat] : X2
% 16.19/9.01 @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.01 = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( plus_plus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6841_odd__mod__4__div__2,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 16.19/9.01 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 16.19/9.01 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6842_divmod__divmod__step,axiom,
% 16.19/9.01 ( unique5055182867167087721od_nat
% 16.19/9.01 = ( ^ [M6: num,N: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M6 @ N ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M6 ) ) @ ( unique5026877609467782581ep_nat @ N @ ( unique5055182867167087721od_nat @ M6 @ ( bit0 @ N ) ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6843_divmod__divmod__step,axiom,
% 16.19/9.01 ( unique5052692396658037445od_int
% 16.19/9.01 = ( ^ [M6: num,N: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M6 @ N ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M6 ) ) @ ( unique5024387138958732305ep_int @ N @ ( unique5052692396658037445od_int @ M6 @ ( bit0 @ N ) ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6844_divmod__divmod__step,axiom,
% 16.19/9.01 ( unique3479559517661332726nteger
% 16.19/9.01 = ( ^ [M6: num,N: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M6 @ N ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M6 ) ) @ ( unique4921790084139445826nteger @ N @ ( unique3479559517661332726nteger @ M6 @ ( bit0 @ N ) ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6845_one__div__minus__numeral,axiom,
% 16.19/9.01 ! [N2: num] :
% 16.19/9.01 ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/9.01 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6846_minus__one__div__numeral,axiom,
% 16.19/9.01 ! [N2: num] :
% 16.19/9.01 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N2 ) )
% 16.19/9.01 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6847_signed__take__bit__numeral__minus__bit1,axiom,
% 16.19/9.01 ! [L2: num,K: num] :
% 16.19/9.01 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 16.19/9.01 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6848_dbl__dec__simps_I4_J,axiom,
% 16.19/9.01 ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ one_one_real ) )
% 16.19/9.01 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6849_dbl__dec__simps_I4_J,axiom,
% 16.19/9.01 ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
% 16.19/9.01 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6850_dbl__dec__simps_I4_J,axiom,
% 16.19/9.01 ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 16.19/9.01 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6851_dbl__dec__simps_I4_J,axiom,
% 16.19/9.01 ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 16.19/9.01 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6852_dbl__dec__simps_I4_J,axiom,
% 16.19/9.01 ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 16.19/9.01 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6853_signed__take__bit__numeral__bit1,axiom,
% 16.19/9.01 ! [L2: num,K: num] :
% 16.19/9.01 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 16.19/9.01 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6854_arctan__double,axiom,
% 16.19/9.01 ! [X: real] :
% 16.19/9.01 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 16.19/9.01 => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X ) )
% 16.19/9.01 = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6855_split__part,axiom,
% 16.19/9.01 ! [P: $o,Q: int > int > $o] :
% 16.19/9.01 ( ( produc4947309494688390418_int_o
% 16.19/9.01 @ ^ [A5: int,B5: int] :
% 16.19/9.01 ( P
% 16.19/9.01 & ( Q @ A5 @ B5 ) ) )
% 16.19/9.01 = ( ^ [Ab: product_prod_int_int] :
% 16.19/9.01 ( P
% 16.19/9.01 & ( produc4947309494688390418_int_o @ Q @ Ab ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6856_zdvd1__eq,axiom,
% 16.19/9.01 ! [X: int] :
% 16.19/9.01 ( ( dvd_dvd_int @ X @ one_one_int )
% 16.19/9.01 = ( ( abs_abs_int @ X )
% 16.19/9.01 = one_one_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6857_dbl__dec__simps_I3_J,axiom,
% 16.19/9.01 ( ( neg_nu6511756317524482435omplex @ one_one_complex )
% 16.19/9.01 = one_one_complex ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6858_dbl__dec__simps_I3_J,axiom,
% 16.19/9.01 ( ( neg_nu6075765906172075777c_real @ one_one_real )
% 16.19/9.01 = one_one_real ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6859_dbl__dec__simps_I3_J,axiom,
% 16.19/9.01 ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
% 16.19/9.01 = one_one_rat ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6860_dbl__dec__simps_I3_J,axiom,
% 16.19/9.01 ( ( neg_nu3811975205180677377ec_int @ one_one_int )
% 16.19/9.01 = one_one_int ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6861_zabs__less__one__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
% 16.19/9.01 = ( Z = zero_zero_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6862_pred__numeral__simps_I1_J,axiom,
% 16.19/9.01 ( ( pred_numeral @ one )
% 16.19/9.01 = zero_zero_nat ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6863_Suc__eq__numeral,axiom,
% 16.19/9.01 ! [N2: nat,K: num] :
% 16.19/9.01 ( ( ( suc @ N2 )
% 16.19/9.01 = ( numeral_numeral_nat @ K ) )
% 16.19/9.01 = ( N2
% 16.19/9.01 = ( pred_numeral @ K ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6864_eq__numeral__Suc,axiom,
% 16.19/9.01 ! [K: num,N2: nat] :
% 16.19/9.01 ( ( ( numeral_numeral_nat @ K )
% 16.19/9.01 = ( suc @ N2 ) )
% 16.19/9.01 = ( ( pred_numeral @ K )
% 16.19/9.01 = N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6865_arctan__less__zero__iff,axiom,
% 16.19/9.01 ! [X: real] :
% 16.19/9.01 ( ( ord_less_real @ ( arctan @ X ) @ zero_zero_real )
% 16.19/9.01 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6866_zero__less__arctan__iff,axiom,
% 16.19/9.01 ! [X: real] :
% 16.19/9.01 ( ( ord_less_real @ zero_zero_real @ ( arctan @ X ) )
% 16.19/9.01 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6867_arctan__le__zero__iff,axiom,
% 16.19/9.01 ! [X: real] :
% 16.19/9.01 ( ( ord_less_eq_real @ ( arctan @ X ) @ zero_zero_real )
% 16.19/9.01 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6868_zero__le__arctan__iff,axiom,
% 16.19/9.01 ! [X: real] :
% 16.19/9.01 ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X ) )
% 16.19/9.01 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6869_less__Suc__numeral,axiom,
% 16.19/9.01 ! [N2: nat,K: num] :
% 16.19/9.01 ( ( ord_less_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
% 16.19/9.01 = ( ord_less_nat @ N2 @ ( pred_numeral @ K ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6870_less__numeral__Suc,axiom,
% 16.19/9.01 ! [K: num,N2: nat] :
% 16.19/9.01 ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
% 16.19/9.01 = ( ord_less_nat @ ( pred_numeral @ K ) @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6871_pred__numeral__simps_I3_J,axiom,
% 16.19/9.01 ! [K: num] :
% 16.19/9.01 ( ( pred_numeral @ ( bit1 @ K ) )
% 16.19/9.01 = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6872_le__numeral__Suc,axiom,
% 16.19/9.01 ! [K: num,N2: nat] :
% 16.19/9.01 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
% 16.19/9.01 = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6873_le__Suc__numeral,axiom,
% 16.19/9.01 ! [N2: nat,K: num] :
% 16.19/9.01 ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
% 16.19/9.01 = ( ord_less_eq_nat @ N2 @ ( pred_numeral @ K ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6874_diff__Suc__numeral,axiom,
% 16.19/9.01 ! [N2: nat,K: num] :
% 16.19/9.01 ( ( minus_minus_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
% 16.19/9.01 = ( minus_minus_nat @ N2 @ ( pred_numeral @ K ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6875_diff__numeral__Suc,axiom,
% 16.19/9.01 ! [K: num,N2: nat] :
% 16.19/9.01 ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
% 16.19/9.01 = ( minus_minus_nat @ ( pred_numeral @ K ) @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6876_max__Suc__numeral,axiom,
% 16.19/9.01 ! [N2: nat,K: num] :
% 16.19/9.01 ( ( ord_max_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
% 16.19/9.01 = ( suc @ ( ord_max_nat @ N2 @ ( pred_numeral @ K ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6877_max__numeral__Suc,axiom,
% 16.19/9.01 ! [K: num,N2: nat] :
% 16.19/9.01 ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
% 16.19/9.01 = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6878_dbl__dec__simps_I2_J,axiom,
% 16.19/9.01 ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
% 16.19/9.01 = ( uminus_uminus_real @ one_one_real ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6879_dbl__dec__simps_I2_J,axiom,
% 16.19/9.01 ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
% 16.19/9.01 = ( uminus_uminus_int @ one_one_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6880_dbl__dec__simps_I2_J,axiom,
% 16.19/9.01 ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
% 16.19/9.01 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6881_dbl__dec__simps_I2_J,axiom,
% 16.19/9.01 ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
% 16.19/9.01 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6882_dbl__dec__simps_I2_J,axiom,
% 16.19/9.01 ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
% 16.19/9.01 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6883_numeral__div__minus__numeral,axiom,
% 16.19/9.01 ! [M: num,N2: num] :
% 16.19/9.01 ( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/9.01 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6884_minus__numeral__div__numeral,axiom,
% 16.19/9.01 ! [M: num,N2: num] :
% 16.19/9.01 ( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
% 16.19/9.01 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6885_signed__take__bit__numeral__bit0,axiom,
% 16.19/9.01 ! [L2: num,K: num] :
% 16.19/9.01 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 16.19/9.01 = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6886_signed__take__bit__numeral__minus__bit0,axiom,
% 16.19/9.01 ! [L2: num,K: num] :
% 16.19/9.01 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 16.19/9.01 = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6887_prod_Odisc__eq__case,axiom,
% 16.19/9.01 ! [Prod: product_prod_int_int] :
% 16.19/9.01 ( produc4947309494688390418_int_o
% 16.19/9.01 @ ^ [Uu3: int,Uv3: int] : $true
% 16.19/9.01 @ Prod ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6888_arctan__less__iff,axiom,
% 16.19/9.01 ! [X: real,Y: real] :
% 16.19/9.01 ( ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) )
% 16.19/9.01 = ( ord_less_real @ X @ Y ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6889_arctan__monotone,axiom,
% 16.19/9.01 ! [X: real,Y: real] :
% 16.19/9.01 ( ( ord_less_real @ X @ Y )
% 16.19/9.01 => ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6890_arctan__le__iff,axiom,
% 16.19/9.01 ! [X: real,Y: real] :
% 16.19/9.01 ( ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y ) )
% 16.19/9.01 = ( ord_less_eq_real @ X @ Y ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6891_arctan__monotone_H,axiom,
% 16.19/9.01 ! [X: real,Y: real] :
% 16.19/9.01 ( ( ord_less_eq_real @ X @ Y )
% 16.19/9.01 => ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6892_abs__zmult__eq__1,axiom,
% 16.19/9.01 ! [M: int,N2: int] :
% 16.19/9.01 ( ( ( abs_abs_int @ ( times_times_int @ M @ N2 ) )
% 16.19/9.01 = one_one_int )
% 16.19/9.01 => ( ( abs_abs_int @ M )
% 16.19/9.01 = one_one_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6893_abs__div,axiom,
% 16.19/9.01 ! [Y: int,X: int] :
% 16.19/9.01 ( ( dvd_dvd_int @ Y @ X )
% 16.19/9.01 => ( ( abs_abs_int @ ( divide_divide_int @ X @ Y ) )
% 16.19/9.01 = ( divide_divide_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6894_numeral__eq__Suc,axiom,
% 16.19/9.01 ( numeral_numeral_nat
% 16.19/9.01 = ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6895_zabs__def,axiom,
% 16.19/9.01 ( abs_abs_int
% 16.19/9.01 = ( ^ [I3: int] : ( if_int @ ( ord_less_int @ I3 @ zero_zero_int ) @ ( uminus_uminus_int @ I3 ) @ I3 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6896_abs__mod__less,axiom,
% 16.19/9.01 ! [L2: int,K: int] :
% 16.19/9.01 ( ( L2 != zero_zero_int )
% 16.19/9.01 => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L2 ) ) @ ( abs_abs_int @ L2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6897_pred__numeral__def,axiom,
% 16.19/9.01 ( pred_numeral
% 16.19/9.01 = ( ^ [K3: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K3 ) @ one_one_nat ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6898_zdvd__mult__cancel1,axiom,
% 16.19/9.01 ! [M: int,N2: int] :
% 16.19/9.01 ( ( M != zero_zero_int )
% 16.19/9.01 => ( ( dvd_dvd_int @ ( times_times_int @ M @ N2 ) @ M )
% 16.19/9.01 = ( ( abs_abs_int @ N2 )
% 16.19/9.01 = one_one_int ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6899_even__abs__add__iff,axiom,
% 16.19/9.01 ! [K: int,L2: int] :
% 16.19/9.01 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L2 ) )
% 16.19/9.01 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6900_even__add__abs__iff,axiom,
% 16.19/9.01 ! [K: int,L2: int] :
% 16.19/9.01 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L2 ) ) )
% 16.19/9.01 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6901_Divides_Oadjust__div__def,axiom,
% 16.19/9.01 ( adjust_div
% 16.19/9.01 = ( produc8211389475949308722nt_int
% 16.19/9.01 @ ^ [Q4: int,R5: int] : ( plus_plus_int @ Q4 @ ( zero_n2684676970156552555ol_int @ ( R5 != zero_zero_int ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6902_nat__intermed__int__val,axiom,
% 16.19/9.01 ! [M: nat,N2: nat,F: nat > int,K: int] :
% 16.19/9.01 ( ! [I4: nat] :
% 16.19/9.01 ( ( ( ord_less_eq_nat @ M @ I4 )
% 16.19/9.01 & ( ord_less_nat @ I4 @ N2 ) )
% 16.19/9.01 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) ) ) @ one_one_int ) )
% 16.19/9.01 => ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.01 => ( ( ord_less_eq_int @ ( F @ M ) @ K )
% 16.19/9.01 => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
% 16.19/9.01 => ? [I4: nat] :
% 16.19/9.01 ( ( ord_less_eq_nat @ M @ I4 )
% 16.19/9.01 & ( ord_less_eq_nat @ I4 @ N2 )
% 16.19/9.01 & ( ( F @ I4 )
% 16.19/9.01 = K ) ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6903_dbl__dec__def,axiom,
% 16.19/9.01 ( neg_nu6511756317524482435omplex
% 16.19/9.01 = ( ^ [X2: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X2 @ X2 ) @ one_one_complex ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6904_dbl__dec__def,axiom,
% 16.19/9.01 ( neg_nu6075765906172075777c_real
% 16.19/9.01 = ( ^ [X2: real] : ( minus_minus_real @ ( plus_plus_real @ X2 @ X2 ) @ one_one_real ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6905_dbl__dec__def,axiom,
% 16.19/9.01 ( neg_nu3179335615603231917ec_rat
% 16.19/9.01 = ( ^ [X2: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6906_dbl__dec__def,axiom,
% 16.19/9.01 ( neg_nu3811975205180677377ec_int
% 16.19/9.01 = ( ^ [X2: int] : ( minus_minus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6907_decr__lemma,axiom,
% 16.19/9.01 ! [D: int,X: int,Z: int] :
% 16.19/9.01 ( ( ord_less_int @ zero_zero_int @ D )
% 16.19/9.01 => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6908_incr__lemma,axiom,
% 16.19/9.01 ! [D: int,Z: int,X: int] :
% 16.19/9.01 ( ( ord_less_int @ zero_zero_int @ D )
% 16.19/9.01 => ( ord_less_int @ Z @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6909_nat__ivt__aux,axiom,
% 16.19/9.01 ! [N2: nat,F: nat > int,K: int] :
% 16.19/9.01 ( ! [I4: nat] :
% 16.19/9.01 ( ( ord_less_nat @ I4 @ N2 )
% 16.19/9.01 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) ) ) @ one_one_int ) )
% 16.19/9.01 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 16.19/9.01 => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
% 16.19/9.01 => ? [I4: nat] :
% 16.19/9.01 ( ( ord_less_eq_nat @ I4 @ N2 )
% 16.19/9.01 & ( ( F @ I4 )
% 16.19/9.01 = K ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6910_nat0__intermed__int__val,axiom,
% 16.19/9.01 ! [N2: nat,F: nat > int,K: int] :
% 16.19/9.01 ( ! [I4: nat] :
% 16.19/9.01 ( ( ord_less_nat @ I4 @ N2 )
% 16.19/9.01 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I4 @ one_one_nat ) ) @ ( F @ I4 ) ) ) @ one_one_int ) )
% 16.19/9.01 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 16.19/9.01 => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
% 16.19/9.01 => ? [I4: nat] :
% 16.19/9.01 ( ( ord_less_eq_nat @ I4 @ N2 )
% 16.19/9.01 & ( ( F @ I4 )
% 16.19/9.01 = K ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6911_arctan__add,axiom,
% 16.19/9.01 ! [X: real,Y: real] :
% 16.19/9.01 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 16.19/9.01 => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 16.19/9.01 => ( ( plus_plus_real @ ( arctan @ X ) @ ( arctan @ Y ) )
% 16.19/9.01 = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X @ Y ) ) ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6912_sum__gp,axiom,
% 16.19/9.01 ! [N2: nat,M: nat,X: rat] :
% 16.19/9.01 ( ( ( ord_less_nat @ N2 @ M )
% 16.19/9.01 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.01 = zero_zero_rat ) )
% 16.19/9.01 & ( ~ ( ord_less_nat @ N2 @ M )
% 16.19/9.01 => ( ( ( X = one_one_rat )
% 16.19/9.01 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.01 = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
% 16.19/9.01 & ( ( X != one_one_rat )
% 16.19/9.01 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.01 = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6913_sum__gp,axiom,
% 16.19/9.01 ! [N2: nat,M: nat,X: complex] :
% 16.19/9.01 ( ( ( ord_less_nat @ N2 @ M )
% 16.19/9.01 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.01 = zero_zero_complex ) )
% 16.19/9.01 & ( ~ ( ord_less_nat @ N2 @ M )
% 16.19/9.01 => ( ( ( X = one_one_complex )
% 16.19/9.01 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.01 = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
% 16.19/9.01 & ( ( X != one_one_complex )
% 16.19/9.01 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.01 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6914_sum__gp,axiom,
% 16.19/9.01 ! [N2: nat,M: nat,X: real] :
% 16.19/9.01 ( ( ( ord_less_nat @ N2 @ M )
% 16.19/9.01 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.01 = zero_zero_real ) )
% 16.19/9.01 & ( ~ ( ord_less_nat @ N2 @ M )
% 16.19/9.01 => ( ( ( X = one_one_real )
% 16.19/9.01 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.01 = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
% 16.19/9.01 & ( ( X != one_one_real )
% 16.19/9.01 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.01 = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N2 ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6915_dbl__inc__simps_I3_J,axiom,
% 16.19/9.01 ( ( neg_nu8557863876264182079omplex @ one_one_complex )
% 16.19/9.01 = ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6916_dbl__inc__simps_I3_J,axiom,
% 16.19/9.01 ( ( neg_nu8295874005876285629c_real @ one_one_real )
% 16.19/9.01 = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6917_dbl__inc__simps_I3_J,axiom,
% 16.19/9.01 ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
% 16.19/9.01 = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6918_dbl__inc__simps_I3_J,axiom,
% 16.19/9.01 ( ( neg_nu5851722552734809277nc_int @ one_one_int )
% 16.19/9.01 = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6919_divmod__BitM__2__eq,axiom,
% 16.19/9.01 ! [M: num] :
% 16.19/9.01 ( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
% 16.19/9.01 = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6920_gauss__sum__from__Suc__0,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
% 16.19/9.01 = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6921_gauss__sum__from__Suc__0,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
% 16.19/9.01 = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6922_of__int__code__if,axiom,
% 16.19/9.01 ( ring_1_of_int_real
% 16.19/9.01 = ( ^ [K3: int] :
% 16.19/9.01 ( if_real @ ( K3 = zero_zero_int ) @ zero_zero_real
% 16.19/9.01 @ ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K3 ) ) )
% 16.19/9.01 @ ( if_real
% 16.19/9.01 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/9.01 = zero_zero_int )
% 16.19/9.01 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 16.19/9.01 @ ( 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 ) ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6923_of__int__code__if,axiom,
% 16.19/9.01 ( ring_1_of_int_int
% 16.19/9.01 = ( ^ [K3: int] :
% 16.19/9.01 ( if_int @ ( K3 = zero_zero_int ) @ zero_zero_int
% 16.19/9.01 @ ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K3 ) ) )
% 16.19/9.01 @ ( if_int
% 16.19/9.01 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/9.01 = zero_zero_int )
% 16.19/9.01 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 16.19/9.01 @ ( 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 ) ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6924_of__int__code__if,axiom,
% 16.19/9.01 ( ring_17405671764205052669omplex
% 16.19/9.01 = ( ^ [K3: int] :
% 16.19/9.01 ( if_complex @ ( K3 = zero_zero_int ) @ zero_zero_complex
% 16.19/9.01 @ ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K3 ) ) )
% 16.19/9.01 @ ( if_complex
% 16.19/9.01 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/9.01 = zero_zero_int )
% 16.19/9.01 @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 16.19/9.01 @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6925_of__int__code__if,axiom,
% 16.19/9.01 ( ring_1_of_int_rat
% 16.19/9.01 = ( ^ [K3: int] :
% 16.19/9.01 ( if_rat @ ( K3 = zero_zero_int ) @ zero_zero_rat
% 16.19/9.01 @ ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K3 ) ) )
% 16.19/9.01 @ ( if_rat
% 16.19/9.01 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/9.01 = zero_zero_int )
% 16.19/9.01 @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 16.19/9.01 @ ( 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 ) ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6926_of__int__code__if,axiom,
% 16.19/9.01 ( ring_18347121197199848620nteger
% 16.19/9.01 = ( ^ [K3: int] :
% 16.19/9.01 ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 16.19/9.01 @ ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K3 ) ) )
% 16.19/9.01 @ ( if_Code_integer
% 16.19/9.01 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/9.01 = zero_zero_int )
% 16.19/9.01 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 16.19/9.01 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6927_of__nat__eq__iff,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( ( semiri1314217659103216013at_int @ M )
% 16.19/9.01 = ( semiri1314217659103216013at_int @ N2 ) )
% 16.19/9.01 = ( M = N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6928_of__nat__eq__iff,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( ( semiri5074537144036343181t_real @ M )
% 16.19/9.01 = ( semiri5074537144036343181t_real @ N2 ) )
% 16.19/9.01 = ( M = N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6929_of__nat__eq__iff,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( ( semiri1316708129612266289at_nat @ M )
% 16.19/9.01 = ( semiri1316708129612266289at_nat @ N2 ) )
% 16.19/9.01 = ( M = N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6930_of__nat__eq__iff,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( ( semiri8010041392384452111omplex @ M )
% 16.19/9.01 = ( semiri8010041392384452111omplex @ N2 ) )
% 16.19/9.01 = ( M = N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6931_int__eq__iff__numeral,axiom,
% 16.19/9.01 ! [M: nat,V: num] :
% 16.19/9.01 ( ( ( semiri1314217659103216013at_int @ M )
% 16.19/9.01 = ( numeral_numeral_int @ V ) )
% 16.19/9.01 = ( M
% 16.19/9.01 = ( numeral_numeral_nat @ V ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6932_abs__of__nat,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N2 ) )
% 16.19/9.01 = ( semiri681578069525770553at_rat @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6933_abs__of__nat,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N2 ) )
% 16.19/9.01 = ( semiri4939895301339042750nteger @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6934_abs__of__nat,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N2 ) )
% 16.19/9.01 = ( semiri1314217659103216013at_int @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6935_abs__of__nat,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N2 ) )
% 16.19/9.01 = ( semiri5074537144036343181t_real @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6936_of__nat__0,axiom,
% 16.19/9.01 ( ( semiri681578069525770553at_rat @ zero_zero_nat )
% 16.19/9.01 = zero_zero_rat ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6937_of__nat__0,axiom,
% 16.19/9.01 ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 16.19/9.01 = zero_zero_int ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6938_of__nat__0,axiom,
% 16.19/9.01 ( ( semiri5074537144036343181t_real @ zero_zero_nat )
% 16.19/9.01 = zero_zero_real ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6939_of__nat__0,axiom,
% 16.19/9.01 ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
% 16.19/9.01 = zero_zero_nat ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6940_of__nat__0,axiom,
% 16.19/9.01 ( ( semiri8010041392384452111omplex @ zero_zero_nat )
% 16.19/9.01 = zero_zero_complex ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6941_of__nat__0__eq__iff,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( zero_zero_rat
% 16.19/9.01 = ( semiri681578069525770553at_rat @ N2 ) )
% 16.19/9.01 = ( zero_zero_nat = N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6942_of__nat__0__eq__iff,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( zero_zero_int
% 16.19/9.01 = ( semiri1314217659103216013at_int @ N2 ) )
% 16.19/9.01 = ( zero_zero_nat = N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6943_of__nat__0__eq__iff,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( zero_zero_real
% 16.19/9.01 = ( semiri5074537144036343181t_real @ N2 ) )
% 16.19/9.01 = ( zero_zero_nat = N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6944_of__nat__0__eq__iff,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( zero_zero_nat
% 16.19/9.01 = ( semiri1316708129612266289at_nat @ N2 ) )
% 16.19/9.01 = ( zero_zero_nat = N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6945_of__nat__0__eq__iff,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( zero_zero_complex
% 16.19/9.01 = ( semiri8010041392384452111omplex @ N2 ) )
% 16.19/9.01 = ( zero_zero_nat = N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6946_of__nat__eq__0__iff,axiom,
% 16.19/9.01 ! [M: nat] :
% 16.19/9.01 ( ( ( semiri681578069525770553at_rat @ M )
% 16.19/9.01 = zero_zero_rat )
% 16.19/9.01 = ( M = zero_zero_nat ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6947_of__nat__eq__0__iff,axiom,
% 16.19/9.01 ! [M: nat] :
% 16.19/9.01 ( ( ( semiri1314217659103216013at_int @ M )
% 16.19/9.01 = zero_zero_int )
% 16.19/9.01 = ( M = zero_zero_nat ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6948_of__nat__eq__0__iff,axiom,
% 16.19/9.01 ! [M: nat] :
% 16.19/9.01 ( ( ( semiri5074537144036343181t_real @ M )
% 16.19/9.01 = zero_zero_real )
% 16.19/9.01 = ( M = zero_zero_nat ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6949_of__nat__eq__0__iff,axiom,
% 16.19/9.01 ! [M: nat] :
% 16.19/9.01 ( ( ( semiri1316708129612266289at_nat @ M )
% 16.19/9.01 = zero_zero_nat )
% 16.19/9.01 = ( M = zero_zero_nat ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6950_of__nat__eq__0__iff,axiom,
% 16.19/9.01 ! [M: nat] :
% 16.19/9.01 ( ( ( semiri8010041392384452111omplex @ M )
% 16.19/9.01 = zero_zero_complex )
% 16.19/9.01 = ( M = zero_zero_nat ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6951_of__nat__numeral,axiom,
% 16.19/9.01 ! [N2: num] :
% 16.19/9.01 ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N2 ) )
% 16.19/9.01 = ( numeral_numeral_rat @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6952_of__nat__numeral,axiom,
% 16.19/9.01 ! [N2: num] :
% 16.19/9.01 ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N2 ) )
% 16.19/9.01 = ( numeral_numeral_int @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6953_of__nat__numeral,axiom,
% 16.19/9.01 ! [N2: num] :
% 16.19/9.01 ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N2 ) )
% 16.19/9.01 = ( numeral_numeral_real @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6954_of__nat__numeral,axiom,
% 16.19/9.01 ! [N2: num] :
% 16.19/9.01 ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N2 ) )
% 16.19/9.01 = ( numeral_numeral_nat @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6955_of__nat__numeral,axiom,
% 16.19/9.01 ! [N2: num] :
% 16.19/9.01 ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N2 ) )
% 16.19/9.01 = ( numera6690914467698888265omplex @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6956_of__nat__less__iff,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
% 16.19/9.01 = ( ord_less_nat @ M @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6957_of__nat__less__iff,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
% 16.19/9.01 = ( ord_less_nat @ M @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6958_of__nat__less__iff,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
% 16.19/9.01 = ( ord_less_nat @ M @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6959_of__nat__less__iff,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
% 16.19/9.01 = ( ord_less_nat @ M @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6960_of__nat__le__iff,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
% 16.19/9.01 = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6961_of__nat__le__iff,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
% 16.19/9.01 = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6962_of__nat__le__iff,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
% 16.19/9.01 = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6963_of__nat__le__iff,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
% 16.19/9.01 = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6964_of__nat__add,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N2 ) )
% 16.19/9.01 = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6965_of__nat__add,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) )
% 16.19/9.01 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6966_of__nat__add,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N2 ) )
% 16.19/9.01 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6967_of__nat__add,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N2 ) )
% 16.19/9.01 = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6968_of__nat__add,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ N2 ) )
% 16.19/9.01 = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6969_of__nat__mult,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N2 ) )
% 16.19/9.01 = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6970_of__nat__mult,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N2 ) )
% 16.19/9.01 = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6971_of__nat__mult,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N2 ) )
% 16.19/9.01 = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6972_of__nat__mult,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N2 ) )
% 16.19/9.01 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6973_of__nat__mult,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N2 ) )
% 16.19/9.01 = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6974_of__nat__eq__1__iff,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( ( semiri681578069525770553at_rat @ N2 )
% 16.19/9.01 = one_one_rat )
% 16.19/9.01 = ( N2 = one_one_nat ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6975_of__nat__eq__1__iff,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( ( semiri1314217659103216013at_int @ N2 )
% 16.19/9.01 = one_one_int )
% 16.19/9.01 = ( N2 = one_one_nat ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6976_of__nat__eq__1__iff,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( ( semiri5074537144036343181t_real @ N2 )
% 16.19/9.01 = one_one_real )
% 16.19/9.01 = ( N2 = one_one_nat ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6977_of__nat__eq__1__iff,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( ( semiri1316708129612266289at_nat @ N2 )
% 16.19/9.01 = one_one_nat )
% 16.19/9.01 = ( N2 = one_one_nat ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6978_of__nat__eq__1__iff,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( ( semiri8010041392384452111omplex @ N2 )
% 16.19/9.01 = one_one_complex )
% 16.19/9.01 = ( N2 = one_one_nat ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6979_of__nat__1__eq__iff,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( one_one_rat
% 16.19/9.01 = ( semiri681578069525770553at_rat @ N2 ) )
% 16.19/9.01 = ( N2 = one_one_nat ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6980_of__nat__1__eq__iff,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( one_one_int
% 16.19/9.01 = ( semiri1314217659103216013at_int @ N2 ) )
% 16.19/9.01 = ( N2 = one_one_nat ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6981_of__nat__1__eq__iff,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( one_one_real
% 16.19/9.01 = ( semiri5074537144036343181t_real @ N2 ) )
% 16.19/9.01 = ( N2 = one_one_nat ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6982_of__nat__1__eq__iff,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( one_one_nat
% 16.19/9.01 = ( semiri1316708129612266289at_nat @ N2 ) )
% 16.19/9.01 = ( N2 = one_one_nat ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6983_of__nat__1__eq__iff,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( one_one_complex
% 16.19/9.01 = ( semiri8010041392384452111omplex @ N2 ) )
% 16.19/9.01 = ( N2 = one_one_nat ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6984_of__nat__1,axiom,
% 16.19/9.01 ( ( semiri681578069525770553at_rat @ one_one_nat )
% 16.19/9.01 = one_one_rat ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6985_of__nat__1,axiom,
% 16.19/9.01 ( ( semiri1314217659103216013at_int @ one_one_nat )
% 16.19/9.01 = one_one_int ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6986_of__nat__1,axiom,
% 16.19/9.01 ( ( semiri5074537144036343181t_real @ one_one_nat )
% 16.19/9.01 = one_one_real ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6987_of__nat__1,axiom,
% 16.19/9.01 ( ( semiri1316708129612266289at_nat @ one_one_nat )
% 16.19/9.01 = one_one_nat ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6988_of__nat__1,axiom,
% 16.19/9.01 ( ( semiri8010041392384452111omplex @ one_one_nat )
% 16.19/9.01 = one_one_complex ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6989_of__nat__power,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N2 ) )
% 16.19/9.01 = ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6990_of__nat__power,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N2 ) )
% 16.19/9.01 = ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6991_of__nat__power,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N2 ) )
% 16.19/9.01 = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6992_of__nat__power,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N2 ) )
% 16.19/9.01 = ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6993_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 16.19/9.01 ! [B: nat,W: nat,X: nat] :
% 16.19/9.01 ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
% 16.19/9.01 = ( semiri1314217659103216013at_int @ X ) )
% 16.19/9.01 = ( ( power_power_nat @ B @ W )
% 16.19/9.01 = X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6994_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 16.19/9.01 ! [B: nat,W: nat,X: nat] :
% 16.19/9.01 ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
% 16.19/9.01 = ( semiri5074537144036343181t_real @ X ) )
% 16.19/9.01 = ( ( power_power_nat @ B @ W )
% 16.19/9.01 = X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6995_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 16.19/9.01 ! [B: nat,W: nat,X: nat] :
% 16.19/9.01 ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
% 16.19/9.01 = ( semiri1316708129612266289at_nat @ X ) )
% 16.19/9.01 = ( ( power_power_nat @ B @ W )
% 16.19/9.01 = X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6996_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 16.19/9.01 ! [B: nat,W: nat,X: nat] :
% 16.19/9.01 ( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W )
% 16.19/9.01 = ( semiri8010041392384452111omplex @ X ) )
% 16.19/9.01 = ( ( power_power_nat @ B @ W )
% 16.19/9.01 = X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6997_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 16.19/9.01 ! [X: nat,B: nat,W: nat] :
% 16.19/9.01 ( ( ( semiri1314217659103216013at_int @ X )
% 16.19/9.01 = ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 16.19/9.01 = ( X
% 16.19/9.01 = ( power_power_nat @ B @ W ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6998_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 16.19/9.01 ! [X: nat,B: nat,W: nat] :
% 16.19/9.01 ( ( ( semiri5074537144036343181t_real @ X )
% 16.19/9.01 = ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 16.19/9.01 = ( X
% 16.19/9.01 = ( power_power_nat @ B @ W ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_6999_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 16.19/9.01 ! [X: nat,B: nat,W: nat] :
% 16.19/9.01 ( ( ( semiri1316708129612266289at_nat @ X )
% 16.19/9.01 = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 16.19/9.01 = ( X
% 16.19/9.01 = ( power_power_nat @ B @ W ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7000_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 16.19/9.01 ! [X: nat,B: nat,W: nat] :
% 16.19/9.01 ( ( ( semiri8010041392384452111omplex @ X )
% 16.19/9.01 = ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W ) )
% 16.19/9.01 = ( X
% 16.19/9.01 = ( power_power_nat @ B @ W ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7001_of__int__numeral,axiom,
% 16.19/9.01 ! [K: num] :
% 16.19/9.01 ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
% 16.19/9.01 = ( numera6690914467698888265omplex @ K ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7002_of__int__numeral,axiom,
% 16.19/9.01 ! [K: num] :
% 16.19/9.01 ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
% 16.19/9.01 = ( numeral_numeral_real @ K ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7003_of__int__numeral,axiom,
% 16.19/9.01 ! [K: num] :
% 16.19/9.01 ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
% 16.19/9.01 = ( numeral_numeral_rat @ K ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7004_of__int__numeral,axiom,
% 16.19/9.01 ! [K: num] :
% 16.19/9.01 ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
% 16.19/9.01 = ( numeral_numeral_int @ K ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7005_of__int__eq__numeral__iff,axiom,
% 16.19/9.01 ! [Z: int,N2: num] :
% 16.19/9.01 ( ( ( ring_17405671764205052669omplex @ Z )
% 16.19/9.01 = ( numera6690914467698888265omplex @ N2 ) )
% 16.19/9.01 = ( Z
% 16.19/9.01 = ( numeral_numeral_int @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7006_of__int__eq__numeral__iff,axiom,
% 16.19/9.01 ! [Z: int,N2: num] :
% 16.19/9.01 ( ( ( ring_1_of_int_real @ Z )
% 16.19/9.01 = ( numeral_numeral_real @ N2 ) )
% 16.19/9.01 = ( Z
% 16.19/9.01 = ( numeral_numeral_int @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7007_of__int__eq__numeral__iff,axiom,
% 16.19/9.01 ! [Z: int,N2: num] :
% 16.19/9.01 ( ( ( ring_1_of_int_rat @ Z )
% 16.19/9.01 = ( numeral_numeral_rat @ N2 ) )
% 16.19/9.01 = ( Z
% 16.19/9.01 = ( numeral_numeral_int @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7008_of__int__eq__numeral__iff,axiom,
% 16.19/9.01 ! [Z: int,N2: num] :
% 16.19/9.01 ( ( ( ring_1_of_int_int @ Z )
% 16.19/9.01 = ( numeral_numeral_int @ N2 ) )
% 16.19/9.01 = ( Z
% 16.19/9.01 = ( numeral_numeral_int @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7009_of__int__le__iff,axiom,
% 16.19/9.01 ! [W: int,Z: int] :
% 16.19/9.01 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 16.19/9.01 = ( ord_less_eq_int @ W @ Z ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7010_of__int__le__iff,axiom,
% 16.19/9.01 ! [W: int,Z: int] :
% 16.19/9.01 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
% 16.19/9.01 = ( ord_less_eq_int @ W @ Z ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7011_of__int__le__iff,axiom,
% 16.19/9.01 ! [W: int,Z: int] :
% 16.19/9.01 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 16.19/9.01 = ( ord_less_eq_int @ W @ Z ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7012_of__int__less__iff,axiom,
% 16.19/9.01 ! [W: int,Z: int] :
% 16.19/9.01 ( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 16.19/9.01 = ( ord_less_int @ W @ Z ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7013_of__int__less__iff,axiom,
% 16.19/9.01 ! [W: int,Z: int] :
% 16.19/9.01 ( ( ord_less_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
% 16.19/9.01 = ( ord_less_int @ W @ Z ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7014_of__int__less__iff,axiom,
% 16.19/9.01 ! [W: int,Z: int] :
% 16.19/9.01 ( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 16.19/9.01 = ( ord_less_int @ W @ Z ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7015_of__int__eq__1__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ( ring_17405671764205052669omplex @ Z )
% 16.19/9.01 = one_one_complex )
% 16.19/9.01 = ( Z = one_one_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7016_of__int__eq__1__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ( ring_1_of_int_int @ Z )
% 16.19/9.01 = one_one_int )
% 16.19/9.01 = ( Z = one_one_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7017_of__int__eq__1__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ( ring_1_of_int_real @ Z )
% 16.19/9.01 = one_one_real )
% 16.19/9.01 = ( Z = one_one_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7018_of__int__eq__1__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ( ring_1_of_int_rat @ Z )
% 16.19/9.01 = one_one_rat )
% 16.19/9.01 = ( Z = one_one_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7019_of__int__1,axiom,
% 16.19/9.01 ( ( ring_17405671764205052669omplex @ one_one_int )
% 16.19/9.01 = one_one_complex ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7020_of__int__1,axiom,
% 16.19/9.01 ( ( ring_1_of_int_int @ one_one_int )
% 16.19/9.01 = one_one_int ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7021_of__int__1,axiom,
% 16.19/9.01 ( ( ring_1_of_int_real @ one_one_int )
% 16.19/9.01 = one_one_real ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7022_of__int__1,axiom,
% 16.19/9.01 ( ( ring_1_of_int_rat @ one_one_int )
% 16.19/9.01 = one_one_rat ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7023_of__int__mult,axiom,
% 16.19/9.01 ! [W: int,Z: int] :
% 16.19/9.01 ( ( ring_1_of_int_real @ ( times_times_int @ W @ Z ) )
% 16.19/9.01 = ( times_times_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7024_of__int__mult,axiom,
% 16.19/9.01 ! [W: int,Z: int] :
% 16.19/9.01 ( ( ring_1_of_int_rat @ ( times_times_int @ W @ Z ) )
% 16.19/9.01 = ( times_times_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7025_of__int__mult,axiom,
% 16.19/9.01 ! [W: int,Z: int] :
% 16.19/9.01 ( ( ring_1_of_int_int @ ( times_times_int @ W @ Z ) )
% 16.19/9.01 = ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7026_of__int__add,axiom,
% 16.19/9.01 ! [W: int,Z: int] :
% 16.19/9.01 ( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z ) )
% 16.19/9.01 = ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7027_of__int__add,axiom,
% 16.19/9.01 ! [W: int,Z: int] :
% 16.19/9.01 ( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z ) )
% 16.19/9.01 = ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7028_of__int__add,axiom,
% 16.19/9.01 ! [W: int,Z: int] :
% 16.19/9.01 ( ( ring_1_of_int_rat @ ( plus_plus_int @ W @ Z ) )
% 16.19/9.01 = ( plus_plus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7029_negative__zless,axiom,
% 16.19/9.01 ! [N2: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7030_of__int__diff,axiom,
% 16.19/9.01 ! [W: int,Z: int] :
% 16.19/9.01 ( ( ring_1_of_int_real @ ( minus_minus_int @ W @ Z ) )
% 16.19/9.01 = ( minus_minus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7031_of__int__diff,axiom,
% 16.19/9.01 ! [W: int,Z: int] :
% 16.19/9.01 ( ( ring_1_of_int_rat @ ( minus_minus_int @ W @ Z ) )
% 16.19/9.01 = ( minus_minus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7032_of__int__diff,axiom,
% 16.19/9.01 ! [W: int,Z: int] :
% 16.19/9.01 ( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z ) )
% 16.19/9.01 = ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7033_of__int__power,axiom,
% 16.19/9.01 ! [Z: int,N2: nat] :
% 16.19/9.01 ( ( ring_1_of_int_rat @ ( power_power_int @ Z @ N2 ) )
% 16.19/9.01 = ( power_power_rat @ ( ring_1_of_int_rat @ Z ) @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7034_of__int__power,axiom,
% 16.19/9.01 ! [Z: int,N2: nat] :
% 16.19/9.01 ( ( ring_1_of_int_real @ ( power_power_int @ Z @ N2 ) )
% 16.19/9.01 = ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7035_of__int__power,axiom,
% 16.19/9.01 ! [Z: int,N2: nat] :
% 16.19/9.01 ( ( ring_1_of_int_int @ ( power_power_int @ Z @ N2 ) )
% 16.19/9.01 = ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7036_of__int__power,axiom,
% 16.19/9.01 ! [Z: int,N2: nat] :
% 16.19/9.01 ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z @ N2 ) )
% 16.19/9.01 = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z ) @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7037_of__int__eq__of__int__power__cancel__iff,axiom,
% 16.19/9.01 ! [B: int,W: nat,X: int] :
% 16.19/9.01 ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W )
% 16.19/9.01 = ( ring_1_of_int_rat @ X ) )
% 16.19/9.01 = ( ( power_power_int @ B @ W )
% 16.19/9.01 = X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7038_of__int__eq__of__int__power__cancel__iff,axiom,
% 16.19/9.01 ! [B: int,W: nat,X: int] :
% 16.19/9.01 ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
% 16.19/9.01 = ( ring_1_of_int_real @ X ) )
% 16.19/9.01 = ( ( power_power_int @ B @ W )
% 16.19/9.01 = X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7039_of__int__eq__of__int__power__cancel__iff,axiom,
% 16.19/9.01 ! [B: int,W: nat,X: int] :
% 16.19/9.01 ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
% 16.19/9.01 = ( ring_1_of_int_int @ X ) )
% 16.19/9.01 = ( ( power_power_int @ B @ W )
% 16.19/9.01 = X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7040_of__int__eq__of__int__power__cancel__iff,axiom,
% 16.19/9.01 ! [B: int,W: nat,X: int] :
% 16.19/9.01 ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W )
% 16.19/9.01 = ( ring_17405671764205052669omplex @ X ) )
% 16.19/9.01 = ( ( power_power_int @ B @ W )
% 16.19/9.01 = X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7041_of__int__power__eq__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: int,B: int,W: nat] :
% 16.19/9.01 ( ( ( ring_1_of_int_rat @ X )
% 16.19/9.01 = ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 16.19/9.01 = ( X
% 16.19/9.01 = ( power_power_int @ B @ W ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7042_of__int__power__eq__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: int,B: int,W: nat] :
% 16.19/9.01 ( ( ( ring_1_of_int_real @ X )
% 16.19/9.01 = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 16.19/9.01 = ( X
% 16.19/9.01 = ( power_power_int @ B @ W ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7043_of__int__power__eq__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: int,B: int,W: nat] :
% 16.19/9.01 ( ( ( ring_1_of_int_int @ X )
% 16.19/9.01 = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 16.19/9.01 = ( X
% 16.19/9.01 = ( power_power_int @ B @ W ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7044_of__int__power__eq__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: int,B: int,W: nat] :
% 16.19/9.01 ( ( ( ring_17405671764205052669omplex @ X )
% 16.19/9.01 = ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W ) )
% 16.19/9.01 = ( X
% 16.19/9.01 = ( power_power_int @ B @ W ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7045_dbl__inc__simps_I2_J,axiom,
% 16.19/9.01 ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
% 16.19/9.01 = one_one_complex ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7046_dbl__inc__simps_I2_J,axiom,
% 16.19/9.01 ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
% 16.19/9.01 = one_one_real ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7047_dbl__inc__simps_I2_J,axiom,
% 16.19/9.01 ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
% 16.19/9.01 = one_one_rat ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7048_dbl__inc__simps_I2_J,axiom,
% 16.19/9.01 ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
% 16.19/9.01 = one_one_int ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7049_of__nat__of__bool,axiom,
% 16.19/9.01 ! [P: $o] :
% 16.19/9.01 ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
% 16.19/9.01 = ( zero_n3304061248610475627l_real @ P ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7050_of__nat__of__bool,axiom,
% 16.19/9.01 ! [P: $o] :
% 16.19/9.01 ( ( semiri8010041392384452111omplex @ ( zero_n2687167440665602831ol_nat @ P ) )
% 16.19/9.01 = ( zero_n1201886186963655149omplex @ P ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7051_of__nat__of__bool,axiom,
% 16.19/9.01 ! [P: $o] :
% 16.19/9.01 ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 16.19/9.01 = ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7052_of__nat__of__bool,axiom,
% 16.19/9.01 ! [P: $o] :
% 16.19/9.01 ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
% 16.19/9.01 = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7053_of__nat__of__bool,axiom,
% 16.19/9.01 ! [P: $o] :
% 16.19/9.01 ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
% 16.19/9.01 = ( zero_n356916108424825756nteger @ P ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7054_dbl__inc__simps_I4_J,axiom,
% 16.19/9.01 ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
% 16.19/9.01 = ( uminus_uminus_real @ one_one_real ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7055_dbl__inc__simps_I4_J,axiom,
% 16.19/9.01 ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
% 16.19/9.01 = ( uminus_uminus_int @ one_one_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7056_dbl__inc__simps_I4_J,axiom,
% 16.19/9.01 ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 16.19/9.01 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7057_dbl__inc__simps_I4_J,axiom,
% 16.19/9.01 ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 16.19/9.01 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7058_dbl__inc__simps_I4_J,axiom,
% 16.19/9.01 ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 16.19/9.01 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7059_dbl__inc__simps_I5_J,axiom,
% 16.19/9.01 ! [K: num] :
% 16.19/9.01 ( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) )
% 16.19/9.01 = ( numera6690914467698888265omplex @ ( bit1 @ K ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7060_dbl__inc__simps_I5_J,axiom,
% 16.19/9.01 ! [K: num] :
% 16.19/9.01 ( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) )
% 16.19/9.01 = ( numeral_numeral_real @ ( bit1 @ K ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7061_dbl__inc__simps_I5_J,axiom,
% 16.19/9.01 ! [K: num] :
% 16.19/9.01 ( ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) )
% 16.19/9.01 = ( numeral_numeral_rat @ ( bit1 @ K ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7062_dbl__inc__simps_I5_J,axiom,
% 16.19/9.01 ! [K: num] :
% 16.19/9.01 ( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
% 16.19/9.01 = ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7063_of__nat__sum,axiom,
% 16.19/9.01 ! [F: int > nat,A2: set_int] :
% 16.19/9.01 ( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F @ A2 ) )
% 16.19/9.01 = ( groups4538972089207619220nt_int
% 16.19/9.01 @ ^ [X2: int] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
% 16.19/9.01 @ A2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7064_of__nat__sum,axiom,
% 16.19/9.01 ! [F: complex > nat,A2: set_complex] :
% 16.19/9.01 ( ( semiri8010041392384452111omplex @ ( groups5693394587270226106ex_nat @ F @ A2 ) )
% 16.19/9.01 = ( groups7754918857620584856omplex
% 16.19/9.01 @ ^ [X2: complex] : ( semiri8010041392384452111omplex @ ( F @ X2 ) )
% 16.19/9.01 @ A2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7065_of__nat__sum,axiom,
% 16.19/9.01 ! [F: nat > nat,A2: set_nat] :
% 16.19/9.01 ( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 16.19/9.01 = ( groups3539618377306564664at_int
% 16.19/9.01 @ ^ [X2: nat] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
% 16.19/9.01 @ A2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7066_of__nat__sum,axiom,
% 16.19/9.01 ! [F: nat > nat,A2: set_nat] :
% 16.19/9.01 ( ( semiri8010041392384452111omplex @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 16.19/9.01 = ( groups2073611262835488442omplex
% 16.19/9.01 @ ^ [X2: nat] : ( semiri8010041392384452111omplex @ ( F @ X2 ) )
% 16.19/9.01 @ A2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7067_of__nat__sum,axiom,
% 16.19/9.01 ! [F: nat > nat,A2: set_nat] :
% 16.19/9.01 ( ( semiri1316708129612266289at_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 16.19/9.01 = ( groups3542108847815614940at_nat
% 16.19/9.01 @ ^ [X2: nat] : ( semiri1316708129612266289at_nat @ ( F @ X2 ) )
% 16.19/9.01 @ A2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7068_of__nat__sum,axiom,
% 16.19/9.01 ! [F: nat > nat,A2: set_nat] :
% 16.19/9.01 ( ( semiri5074537144036343181t_real @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 16.19/9.01 = ( groups6591440286371151544t_real
% 16.19/9.01 @ ^ [X2: nat] : ( semiri5074537144036343181t_real @ ( F @ X2 ) )
% 16.19/9.01 @ A2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7069_dbl__dec__simps_I5_J,axiom,
% 16.19/9.01 ! [K: num] :
% 16.19/9.01 ( ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) )
% 16.19/9.01 = ( numera6690914467698888265omplex @ ( bitM @ K ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7070_dbl__dec__simps_I5_J,axiom,
% 16.19/9.01 ! [K: num] :
% 16.19/9.01 ( ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) )
% 16.19/9.01 = ( numeral_numeral_real @ ( bitM @ K ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7071_dbl__dec__simps_I5_J,axiom,
% 16.19/9.01 ! [K: num] :
% 16.19/9.01 ( ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) )
% 16.19/9.01 = ( numeral_numeral_rat @ ( bitM @ K ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7072_dbl__dec__simps_I5_J,axiom,
% 16.19/9.01 ! [K: num] :
% 16.19/9.01 ( ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) )
% 16.19/9.01 = ( numeral_numeral_int @ ( bitM @ K ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7073_of__nat__le__0__iff,axiom,
% 16.19/9.01 ! [M: nat] :
% 16.19/9.01 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
% 16.19/9.01 = ( M = zero_zero_nat ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7074_of__nat__le__0__iff,axiom,
% 16.19/9.01 ! [M: nat] :
% 16.19/9.01 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
% 16.19/9.01 = ( M = zero_zero_nat ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7075_of__nat__le__0__iff,axiom,
% 16.19/9.01 ! [M: nat] :
% 16.19/9.01 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
% 16.19/9.01 = ( M = zero_zero_nat ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7076_of__nat__le__0__iff,axiom,
% 16.19/9.01 ! [M: nat] :
% 16.19/9.01 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
% 16.19/9.01 = ( M = zero_zero_nat ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7077_of__nat__Suc,axiom,
% 16.19/9.01 ! [M: nat] :
% 16.19/9.01 ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
% 16.19/9.01 = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7078_of__nat__Suc,axiom,
% 16.19/9.01 ! [M: nat] :
% 16.19/9.01 ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
% 16.19/9.01 = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7079_of__nat__Suc,axiom,
% 16.19/9.01 ! [M: nat] :
% 16.19/9.01 ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
% 16.19/9.01 = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7080_of__nat__Suc,axiom,
% 16.19/9.01 ! [M: nat] :
% 16.19/9.01 ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
% 16.19/9.01 = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7081_of__nat__Suc,axiom,
% 16.19/9.01 ! [M: nat] :
% 16.19/9.01 ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
% 16.19/9.01 = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7082_real__of__nat__less__numeral__iff,axiom,
% 16.19/9.01 ! [N2: nat,W: num] :
% 16.19/9.01 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( numeral_numeral_real @ W ) )
% 16.19/9.01 = ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ W ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7083_numeral__less__real__of__nat__iff,axiom,
% 16.19/9.01 ! [W: num,N2: nat] :
% 16.19/9.01 ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N2 ) )
% 16.19/9.01 = ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7084_numeral__le__real__of__nat__iff,axiom,
% 16.19/9.01 ! [N2: num,M: nat] :
% 16.19/9.01 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ ( semiri5074537144036343181t_real @ M ) )
% 16.19/9.01 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ M ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7085_of__int__sum,axiom,
% 16.19/9.01 ! [F: complex > int,A2: set_complex] :
% 16.19/9.01 ( ( ring_17405671764205052669omplex @ ( groups5690904116761175830ex_int @ F @ A2 ) )
% 16.19/9.01 = ( groups7754918857620584856omplex
% 16.19/9.01 @ ^ [X2: complex] : ( ring_17405671764205052669omplex @ ( F @ X2 ) )
% 16.19/9.01 @ A2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7086_of__int__sum,axiom,
% 16.19/9.01 ! [F: nat > int,A2: set_nat] :
% 16.19/9.01 ( ( ring_1_of_int_real @ ( groups3539618377306564664at_int @ F @ A2 ) )
% 16.19/9.01 = ( groups6591440286371151544t_real
% 16.19/9.01 @ ^ [X2: nat] : ( ring_1_of_int_real @ ( F @ X2 ) )
% 16.19/9.01 @ A2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7087_of__int__sum,axiom,
% 16.19/9.01 ! [F: int > int,A2: set_int] :
% 16.19/9.01 ( ( ring_1_of_int_real @ ( groups4538972089207619220nt_int @ F @ A2 ) )
% 16.19/9.01 = ( groups8778361861064173332t_real
% 16.19/9.01 @ ^ [X2: int] : ( ring_1_of_int_real @ ( F @ X2 ) )
% 16.19/9.01 @ A2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7088_of__int__sum,axiom,
% 16.19/9.01 ! [F: int > int,A2: set_int] :
% 16.19/9.01 ( ( ring_1_of_int_rat @ ( groups4538972089207619220nt_int @ F @ A2 ) )
% 16.19/9.01 = ( groups3906332499630173760nt_rat
% 16.19/9.01 @ ^ [X2: int] : ( ring_1_of_int_rat @ ( F @ X2 ) )
% 16.19/9.01 @ A2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7089_of__int__sum,axiom,
% 16.19/9.01 ! [F: int > int,A2: set_int] :
% 16.19/9.01 ( ( ring_1_of_int_int @ ( groups4538972089207619220nt_int @ F @ A2 ) )
% 16.19/9.01 = ( groups4538972089207619220nt_int
% 16.19/9.01 @ ^ [X2: int] : ( ring_1_of_int_int @ ( F @ X2 ) )
% 16.19/9.01 @ A2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7090_pred__numeral__simps_I2_J,axiom,
% 16.19/9.01 ! [K: num] :
% 16.19/9.01 ( ( pred_numeral @ ( bit0 @ K ) )
% 16.19/9.01 = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7091_of__nat__0__less__iff,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N2 ) )
% 16.19/9.01 = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7092_of__nat__0__less__iff,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) )
% 16.19/9.01 = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7093_of__nat__0__less__iff,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) )
% 16.19/9.01 = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7094_of__nat__0__less__iff,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) )
% 16.19/9.01 = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7095_dbl__inc__simps_I1_J,axiom,
% 16.19/9.01 ! [K: num] :
% 16.19/9.01 ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 16.19/9.01 = ( uminus_uminus_real @ ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7096_dbl__inc__simps_I1_J,axiom,
% 16.19/9.01 ! [K: num] :
% 16.19/9.01 ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 16.19/9.01 = ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7097_dbl__inc__simps_I1_J,axiom,
% 16.19/9.01 ! [K: num] :
% 16.19/9.01 ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 16.19/9.01 = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7098_dbl__inc__simps_I1_J,axiom,
% 16.19/9.01 ! [K: num] :
% 16.19/9.01 ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 16.19/9.01 = ( uminus_uminus_rat @ ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7099_dbl__inc__simps_I1_J,axiom,
% 16.19/9.01 ! [K: num] :
% 16.19/9.01 ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 16.19/9.01 = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7100_dbl__dec__simps_I1_J,axiom,
% 16.19/9.01 ! [K: num] :
% 16.19/9.01 ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 16.19/9.01 = ( uminus_uminus_real @ ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7101_dbl__dec__simps_I1_J,axiom,
% 16.19/9.01 ! [K: num] :
% 16.19/9.01 ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 16.19/9.01 = ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7102_dbl__dec__simps_I1_J,axiom,
% 16.19/9.01 ! [K: num] :
% 16.19/9.01 ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 16.19/9.01 = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7103_dbl__dec__simps_I1_J,axiom,
% 16.19/9.01 ! [K: num] :
% 16.19/9.01 ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 16.19/9.01 = ( uminus_uminus_rat @ ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7104_dbl__dec__simps_I1_J,axiom,
% 16.19/9.01 ! [K: num] :
% 16.19/9.01 ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 16.19/9.01 = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7105_numeral__power__eq__of__nat__cancel__iff,axiom,
% 16.19/9.01 ! [X: num,N2: nat,Y: nat] :
% 16.19/9.01 ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 )
% 16.19/9.01 = ( semiri681578069525770553at_rat @ Y ) )
% 16.19/9.01 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
% 16.19/9.01 = Y ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7106_numeral__power__eq__of__nat__cancel__iff,axiom,
% 16.19/9.01 ! [X: num,N2: nat,Y: nat] :
% 16.19/9.01 ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
% 16.19/9.01 = ( semiri1314217659103216013at_int @ Y ) )
% 16.19/9.01 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
% 16.19/9.01 = Y ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7107_numeral__power__eq__of__nat__cancel__iff,axiom,
% 16.19/9.01 ! [X: num,N2: nat,Y: nat] :
% 16.19/9.01 ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 )
% 16.19/9.01 = ( semiri5074537144036343181t_real @ Y ) )
% 16.19/9.01 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
% 16.19/9.01 = Y ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7108_numeral__power__eq__of__nat__cancel__iff,axiom,
% 16.19/9.01 ! [X: num,N2: nat,Y: nat] :
% 16.19/9.01 ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
% 16.19/9.01 = ( semiri1316708129612266289at_nat @ Y ) )
% 16.19/9.01 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
% 16.19/9.01 = Y ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7109_numeral__power__eq__of__nat__cancel__iff,axiom,
% 16.19/9.01 ! [X: num,N2: nat,Y: nat] :
% 16.19/9.01 ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N2 )
% 16.19/9.01 = ( semiri8010041392384452111omplex @ Y ) )
% 16.19/9.01 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
% 16.19/9.01 = Y ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7110_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [Y: nat,X: num,N2: nat] :
% 16.19/9.01 ( ( ( semiri681578069525770553at_rat @ Y )
% 16.19/9.01 = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) )
% 16.19/9.01 = ( Y
% 16.19/9.01 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7111_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [Y: nat,X: num,N2: nat] :
% 16.19/9.01 ( ( ( semiri1314217659103216013at_int @ Y )
% 16.19/9.01 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
% 16.19/9.01 = ( Y
% 16.19/9.01 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7112_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [Y: nat,X: num,N2: nat] :
% 16.19/9.01 ( ( ( semiri5074537144036343181t_real @ Y )
% 16.19/9.01 = ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
% 16.19/9.01 = ( Y
% 16.19/9.01 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7113_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [Y: nat,X: num,N2: nat] :
% 16.19/9.01 ( ( ( semiri1316708129612266289at_nat @ Y )
% 16.19/9.01 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
% 16.19/9.01 = ( Y
% 16.19/9.01 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7114_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [Y: nat,X: num,N2: nat] :
% 16.19/9.01 ( ( ( semiri8010041392384452111omplex @ Y )
% 16.19/9.01 = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N2 ) )
% 16.19/9.01 = ( Y
% 16.19/9.01 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7115_of__nat__less__of__nat__power__cancel__iff,axiom,
% 16.19/9.01 ! [B: nat,W: nat,X: nat] :
% 16.19/9.01 ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
% 16.19/9.01 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7116_of__nat__less__of__nat__power__cancel__iff,axiom,
% 16.19/9.01 ! [B: nat,W: nat,X: nat] :
% 16.19/9.01 ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
% 16.19/9.01 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7117_of__nat__less__of__nat__power__cancel__iff,axiom,
% 16.19/9.01 ! [B: nat,W: nat,X: nat] :
% 16.19/9.01 ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
% 16.19/9.01 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7118_of__nat__less__of__nat__power__cancel__iff,axiom,
% 16.19/9.01 ! [B: nat,W: nat,X: nat] :
% 16.19/9.01 ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
% 16.19/9.01 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7119_of__nat__power__less__of__nat__cancel__iff,axiom,
% 16.19/9.01 ! [X: nat,B: nat,W: nat] :
% 16.19/9.01 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 16.19/9.01 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7120_of__nat__power__less__of__nat__cancel__iff,axiom,
% 16.19/9.01 ! [X: nat,B: nat,W: nat] :
% 16.19/9.01 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 16.19/9.01 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7121_of__nat__power__less__of__nat__cancel__iff,axiom,
% 16.19/9.01 ! [X: nat,B: nat,W: nat] :
% 16.19/9.01 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 16.19/9.01 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7122_of__nat__power__less__of__nat__cancel__iff,axiom,
% 16.19/9.01 ! [X: nat,B: nat,W: nat] :
% 16.19/9.01 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 16.19/9.01 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7123_of__nat__le__of__nat__power__cancel__iff,axiom,
% 16.19/9.01 ! [B: nat,W: nat,X: nat] :
% 16.19/9.01 ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
% 16.19/9.01 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7124_of__nat__le__of__nat__power__cancel__iff,axiom,
% 16.19/9.01 ! [B: nat,W: nat,X: nat] :
% 16.19/9.01 ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
% 16.19/9.01 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7125_of__nat__le__of__nat__power__cancel__iff,axiom,
% 16.19/9.01 ! [B: nat,W: nat,X: nat] :
% 16.19/9.01 ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
% 16.19/9.01 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7126_of__nat__le__of__nat__power__cancel__iff,axiom,
% 16.19/9.01 ! [B: nat,W: nat,X: nat] :
% 16.19/9.01 ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
% 16.19/9.01 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7127_of__nat__power__le__of__nat__cancel__iff,axiom,
% 16.19/9.01 ! [X: nat,B: nat,W: nat] :
% 16.19/9.01 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 16.19/9.01 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7128_of__nat__power__le__of__nat__cancel__iff,axiom,
% 16.19/9.01 ! [X: nat,B: nat,W: nat] :
% 16.19/9.01 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 16.19/9.01 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7129_of__nat__power__le__of__nat__cancel__iff,axiom,
% 16.19/9.01 ! [X: nat,B: nat,W: nat] :
% 16.19/9.01 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 16.19/9.01 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7130_of__nat__power__le__of__nat__cancel__iff,axiom,
% 16.19/9.01 ! [X: nat,B: nat,W: nat] :
% 16.19/9.01 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 16.19/9.01 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7131_of__int__0__le__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 16.19/9.01 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7132_of__int__0__le__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
% 16.19/9.01 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7133_of__int__0__le__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 16.19/9.01 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7134_of__int__le__0__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 16.19/9.01 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7135_of__int__le__0__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
% 16.19/9.01 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7136_of__int__le__0__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 16.19/9.01 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7137_of__int__0__less__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 16.19/9.01 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7138_of__int__0__less__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
% 16.19/9.01 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7139_of__int__0__less__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 16.19/9.01 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7140_of__int__less__0__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 16.19/9.01 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7141_of__int__less__0__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
% 16.19/9.01 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7142_of__int__less__0__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 16.19/9.01 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7143_of__int__le__numeral__iff,axiom,
% 16.19/9.01 ! [Z: int,N2: num] :
% 16.19/9.01 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N2 ) )
% 16.19/9.01 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7144_of__int__le__numeral__iff,axiom,
% 16.19/9.01 ! [Z: int,N2: num] :
% 16.19/9.01 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N2 ) )
% 16.19/9.01 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7145_of__int__le__numeral__iff,axiom,
% 16.19/9.01 ! [Z: int,N2: num] :
% 16.19/9.01 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N2 ) )
% 16.19/9.01 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7146_of__int__numeral__le__iff,axiom,
% 16.19/9.01 ! [N2: num,Z: int] :
% 16.19/9.01 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ ( ring_1_of_int_real @ Z ) )
% 16.19/9.01 = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7147_of__int__numeral__le__iff,axiom,
% 16.19/9.01 ! [N2: num,Z: int] :
% 16.19/9.01 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ ( ring_1_of_int_rat @ Z ) )
% 16.19/9.01 = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7148_of__int__numeral__le__iff,axiom,
% 16.19/9.01 ! [N2: num,Z: int] :
% 16.19/9.01 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ ( ring_1_of_int_int @ Z ) )
% 16.19/9.01 = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7149_of__int__numeral__less__iff,axiom,
% 16.19/9.01 ! [N2: num,Z: int] :
% 16.19/9.01 ( ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ ( ring_1_of_int_real @ Z ) )
% 16.19/9.01 = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7150_of__int__numeral__less__iff,axiom,
% 16.19/9.01 ! [N2: num,Z: int] :
% 16.19/9.01 ( ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ ( ring_1_of_int_rat @ Z ) )
% 16.19/9.01 = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7151_of__int__numeral__less__iff,axiom,
% 16.19/9.01 ! [N2: num,Z: int] :
% 16.19/9.01 ( ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ ( ring_1_of_int_int @ Z ) )
% 16.19/9.01 = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7152_of__int__less__numeral__iff,axiom,
% 16.19/9.01 ! [Z: int,N2: num] :
% 16.19/9.01 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N2 ) )
% 16.19/9.01 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7153_of__int__less__numeral__iff,axiom,
% 16.19/9.01 ! [Z: int,N2: num] :
% 16.19/9.01 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N2 ) )
% 16.19/9.01 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7154_of__int__less__numeral__iff,axiom,
% 16.19/9.01 ! [Z: int,N2: num] :
% 16.19/9.01 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N2 ) )
% 16.19/9.01 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7155_of__int__1__le__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 16.19/9.01 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7156_of__int__1__le__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
% 16.19/9.01 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7157_of__int__1__le__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 16.19/9.01 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7158_of__int__le__1__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 16.19/9.01 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7159_of__int__le__1__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
% 16.19/9.01 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7160_of__int__le__1__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 16.19/9.01 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7161_of__int__less__1__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 16.19/9.01 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7162_of__int__less__1__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
% 16.19/9.01 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7163_of__int__less__1__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 16.19/9.01 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7164_of__int__1__less__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 16.19/9.01 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7165_of__int__1__less__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
% 16.19/9.01 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7166_of__int__1__less__iff,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 16.19/9.01 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7167_numeral__power__eq__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: num,N2: nat,Y: int] :
% 16.19/9.01 ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N2 )
% 16.19/9.01 = ( ring_17405671764205052669omplex @ Y ) )
% 16.19/9.01 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
% 16.19/9.01 = Y ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7168_numeral__power__eq__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: num,N2: nat,Y: int] :
% 16.19/9.01 ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 )
% 16.19/9.01 = ( ring_1_of_int_real @ Y ) )
% 16.19/9.01 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
% 16.19/9.01 = Y ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7169_numeral__power__eq__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: num,N2: nat,Y: int] :
% 16.19/9.01 ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 )
% 16.19/9.01 = ( ring_1_of_int_rat @ Y ) )
% 16.19/9.01 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
% 16.19/9.01 = Y ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7170_numeral__power__eq__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: num,N2: nat,Y: int] :
% 16.19/9.01 ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
% 16.19/9.01 = ( ring_1_of_int_int @ Y ) )
% 16.19/9.01 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
% 16.19/9.01 = Y ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7171_of__int__eq__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [Y: int,X: num,N2: nat] :
% 16.19/9.01 ( ( ( ring_17405671764205052669omplex @ Y )
% 16.19/9.01 = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N2 ) )
% 16.19/9.01 = ( Y
% 16.19/9.01 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7172_of__int__eq__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [Y: int,X: num,N2: nat] :
% 16.19/9.01 ( ( ( ring_1_of_int_real @ Y )
% 16.19/9.01 = ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
% 16.19/9.01 = ( Y
% 16.19/9.01 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7173_of__int__eq__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [Y: int,X: num,N2: nat] :
% 16.19/9.01 ( ( ( ring_1_of_int_rat @ Y )
% 16.19/9.01 = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) )
% 16.19/9.01 = ( Y
% 16.19/9.01 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7174_of__int__eq__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [Y: int,X: num,N2: nat] :
% 16.19/9.01 ( ( ( ring_1_of_int_int @ Y )
% 16.19/9.01 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
% 16.19/9.01 = ( Y
% 16.19/9.01 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7175_of__int__le__of__int__power__cancel__iff,axiom,
% 16.19/9.01 ! [B: int,W: nat,X: int] :
% 16.19/9.01 ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 16.19/9.01 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7176_of__int__le__of__int__power__cancel__iff,axiom,
% 16.19/9.01 ! [B: int,W: nat,X: int] :
% 16.19/9.01 ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
% 16.19/9.01 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7177_of__int__le__of__int__power__cancel__iff,axiom,
% 16.19/9.01 ! [B: int,W: nat,X: int] :
% 16.19/9.01 ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 16.19/9.01 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7178_of__int__power__le__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: int,B: int,W: nat] :
% 16.19/9.01 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 16.19/9.01 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7179_of__int__power__le__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: int,B: int,W: nat] :
% 16.19/9.01 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 16.19/9.01 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7180_of__int__power__le__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: int,B: int,W: nat] :
% 16.19/9.01 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 16.19/9.01 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7181_of__int__power__less__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: int,B: int,W: nat] :
% 16.19/9.01 ( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 16.19/9.01 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7182_of__int__power__less__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: int,B: int,W: nat] :
% 16.19/9.01 ( ( ord_less_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 16.19/9.01 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7183_of__int__power__less__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: int,B: int,W: nat] :
% 16.19/9.01 ( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 16.19/9.01 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7184_of__int__less__of__int__power__cancel__iff,axiom,
% 16.19/9.01 ! [B: int,W: nat,X: int] :
% 16.19/9.01 ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 16.19/9.01 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7185_of__int__less__of__int__power__cancel__iff,axiom,
% 16.19/9.01 ! [B: int,W: nat,X: int] :
% 16.19/9.01 ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
% 16.19/9.01 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7186_of__int__less__of__int__power__cancel__iff,axiom,
% 16.19/9.01 ! [B: int,W: nat,X: int] :
% 16.19/9.01 ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 16.19/9.01 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7187_of__nat__zero__less__power__iff,axiom,
% 16.19/9.01 ! [X: nat,N2: nat] :
% 16.19/9.01 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N2 ) )
% 16.19/9.01 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 16.19/9.01 | ( N2 = zero_zero_nat ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7188_of__nat__zero__less__power__iff,axiom,
% 16.19/9.01 ! [X: nat,N2: nat] :
% 16.19/9.01 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N2 ) )
% 16.19/9.01 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 16.19/9.01 | ( N2 = zero_zero_nat ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7189_of__nat__zero__less__power__iff,axiom,
% 16.19/9.01 ! [X: nat,N2: nat] :
% 16.19/9.01 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N2 ) )
% 16.19/9.01 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 16.19/9.01 | ( N2 = zero_zero_nat ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7190_of__nat__zero__less__power__iff,axiom,
% 16.19/9.01 ! [X: nat,N2: nat] :
% 16.19/9.01 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N2 ) )
% 16.19/9.01 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 16.19/9.01 | ( N2 = zero_zero_nat ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7191_numeral__power__less__of__nat__cancel__iff,axiom,
% 16.19/9.01 ! [I2: num,N2: nat,X: nat] :
% 16.19/9.01 ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N2 ) @ ( semiri681578069525770553at_rat @ X ) )
% 16.19/9.01 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7192_numeral__power__less__of__nat__cancel__iff,axiom,
% 16.19/9.01 ! [I2: num,N2: nat,X: nat] :
% 16.19/9.01 ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N2 ) @ ( semiri1314217659103216013at_int @ X ) )
% 16.19/9.01 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7193_numeral__power__less__of__nat__cancel__iff,axiom,
% 16.19/9.01 ! [I2: num,N2: nat,X: nat] :
% 16.19/9.01 ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N2 ) @ ( semiri5074537144036343181t_real @ X ) )
% 16.19/9.01 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7194_numeral__power__less__of__nat__cancel__iff,axiom,
% 16.19/9.01 ! [I2: num,N2: nat,X: nat] :
% 16.19/9.01 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X ) )
% 16.19/9.01 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7195_of__nat__less__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [X: nat,I2: num,N2: nat] :
% 16.19/9.01 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N2 ) )
% 16.19/9.01 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7196_of__nat__less__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [X: nat,I2: num,N2: nat] :
% 16.19/9.01 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N2 ) )
% 16.19/9.01 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7197_of__nat__less__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [X: nat,I2: num,N2: nat] :
% 16.19/9.01 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N2 ) )
% 16.19/9.01 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7198_of__nat__less__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [X: nat,I2: num,N2: nat] :
% 16.19/9.01 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) )
% 16.19/9.01 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7199_numeral__power__le__of__nat__cancel__iff,axiom,
% 16.19/9.01 ! [I2: num,N2: nat,X: nat] :
% 16.19/9.01 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N2 ) @ ( semiri5074537144036343181t_real @ X ) )
% 16.19/9.01 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7200_numeral__power__le__of__nat__cancel__iff,axiom,
% 16.19/9.01 ! [I2: num,N2: nat,X: nat] :
% 16.19/9.01 ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N2 ) @ ( semiri681578069525770553at_rat @ X ) )
% 16.19/9.01 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7201_numeral__power__le__of__nat__cancel__iff,axiom,
% 16.19/9.01 ! [I2: num,N2: nat,X: nat] :
% 16.19/9.01 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X ) )
% 16.19/9.01 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7202_numeral__power__le__of__nat__cancel__iff,axiom,
% 16.19/9.01 ! [I2: num,N2: nat,X: nat] :
% 16.19/9.01 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N2 ) @ ( semiri1314217659103216013at_int @ X ) )
% 16.19/9.01 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7203_of__nat__le__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [X: nat,I2: num,N2: nat] :
% 16.19/9.01 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N2 ) )
% 16.19/9.01 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7204_of__nat__le__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [X: nat,I2: num,N2: nat] :
% 16.19/9.01 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N2 ) )
% 16.19/9.01 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7205_of__nat__le__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [X: nat,I2: num,N2: nat] :
% 16.19/9.01 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) )
% 16.19/9.01 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7206_of__nat__le__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [X: nat,I2: num,N2: nat] :
% 16.19/9.01 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N2 ) )
% 16.19/9.01 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7207_even__of__nat,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ N2 ) )
% 16.19/9.01 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7208_even__of__nat,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N2 ) )
% 16.19/9.01 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7209_even__of__nat,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N2 ) )
% 16.19/9.01 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7210_numeral__power__le__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: num,N2: nat,A: int] :
% 16.19/9.01 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
% 16.19/9.01 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7211_numeral__power__le__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: num,N2: nat,A: int] :
% 16.19/9.01 ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
% 16.19/9.01 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7212_numeral__power__le__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: num,N2: nat,A: int] :
% 16.19/9.01 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
% 16.19/9.01 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7213_of__int__le__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [A: int,X: num,N2: nat] :
% 16.19/9.01 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
% 16.19/9.01 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7214_of__int__le__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [A: int,X: num,N2: nat] :
% 16.19/9.01 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) )
% 16.19/9.01 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7215_of__int__le__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [A: int,X: num,N2: nat] :
% 16.19/9.01 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
% 16.19/9.01 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7216_numeral__power__less__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: num,N2: nat,A: int] :
% 16.19/9.01 ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
% 16.19/9.01 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7217_numeral__power__less__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: num,N2: nat,A: int] :
% 16.19/9.01 ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
% 16.19/9.01 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7218_numeral__power__less__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: num,N2: nat,A: int] :
% 16.19/9.01 ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
% 16.19/9.01 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7219_of__int__less__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [A: int,X: num,N2: nat] :
% 16.19/9.01 ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N2 ) )
% 16.19/9.01 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7220_of__int__less__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [A: int,X: num,N2: nat] :
% 16.19/9.01 ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N2 ) )
% 16.19/9.01 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7221_of__int__less__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [A: int,X: num,N2: nat] :
% 16.19/9.01 ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) )
% 16.19/9.01 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7222_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: num,N2: nat,Y: int] :
% 16.19/9.01 ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 )
% 16.19/9.01 = ( ring_1_of_int_real @ Y ) )
% 16.19/9.01 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
% 16.19/9.01 = Y ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7223_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: num,N2: nat,Y: int] :
% 16.19/9.01 ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
% 16.19/9.01 = ( ring_1_of_int_int @ Y ) )
% 16.19/9.01 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
% 16.19/9.01 = Y ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7224_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: num,N2: nat,Y: int] :
% 16.19/9.01 ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N2 )
% 16.19/9.01 = ( ring_17405671764205052669omplex @ Y ) )
% 16.19/9.01 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
% 16.19/9.01 = Y ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7225_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: num,N2: nat,Y: int] :
% 16.19/9.01 ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 )
% 16.19/9.01 = ( ring_1_of_int_rat @ Y ) )
% 16.19/9.01 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
% 16.19/9.01 = Y ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7226_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: num,N2: nat,Y: int] :
% 16.19/9.01 ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 )
% 16.19/9.01 = ( ring_18347121197199848620nteger @ Y ) )
% 16.19/9.01 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 )
% 16.19/9.01 = Y ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7227_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [Y: int,X: num,N2: nat] :
% 16.19/9.01 ( ( ( ring_1_of_int_real @ Y )
% 16.19/9.01 = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) )
% 16.19/9.01 = ( Y
% 16.19/9.01 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7228_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [Y: int,X: num,N2: nat] :
% 16.19/9.01 ( ( ( ring_1_of_int_int @ Y )
% 16.19/9.01 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) )
% 16.19/9.01 = ( Y
% 16.19/9.01 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7229_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [Y: int,X: num,N2: nat] :
% 16.19/9.01 ( ( ( ring_17405671764205052669omplex @ Y )
% 16.19/9.01 = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N2 ) )
% 16.19/9.01 = ( Y
% 16.19/9.01 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7230_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [Y: int,X: num,N2: nat] :
% 16.19/9.01 ( ( ( ring_1_of_int_rat @ Y )
% 16.19/9.01 = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) )
% 16.19/9.01 = ( Y
% 16.19/9.01 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7231_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [Y: int,X: num,N2: nat] :
% 16.19/9.01 ( ( ( ring_18347121197199848620nteger @ Y )
% 16.19/9.01 = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) )
% 16.19/9.01 = ( Y
% 16.19/9.01 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7232_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: num,N2: nat,A: int] :
% 16.19/9.01 ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
% 16.19/9.01 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7233_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: num,N2: nat,A: int] :
% 16.19/9.01 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) @ ( ring_18347121197199848620nteger @ A ) )
% 16.19/9.01 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7234_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: num,N2: nat,A: int] :
% 16.19/9.01 ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
% 16.19/9.01 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7235_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: num,N2: nat,A: int] :
% 16.19/9.01 ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
% 16.19/9.01 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7236_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [A: int,X: num,N2: nat] :
% 16.19/9.01 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) )
% 16.19/9.01 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7237_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [A: int,X: num,N2: nat] :
% 16.19/9.01 ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) )
% 16.19/9.01 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7238_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [A: int,X: num,N2: nat] :
% 16.19/9.01 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) )
% 16.19/9.01 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7239_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [A: int,X: num,N2: nat] :
% 16.19/9.01 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) )
% 16.19/9.01 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7240_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [A: int,X: num,N2: nat] :
% 16.19/9.01 ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) )
% 16.19/9.01 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7241_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [A: int,X: num,N2: nat] :
% 16.19/9.01 ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) )
% 16.19/9.01 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7242_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [A: int,X: num,N2: nat] :
% 16.19/9.01 ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) )
% 16.19/9.01 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7243_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 16.19/9.01 ! [A: int,X: num,N2: nat] :
% 16.19/9.01 ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) )
% 16.19/9.01 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7244_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: num,N2: nat,A: int] :
% 16.19/9.01 ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
% 16.19/9.01 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7245_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: num,N2: nat,A: int] :
% 16.19/9.01 ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
% 16.19/9.01 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7246_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: num,N2: nat,A: int] :
% 16.19/9.01 ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
% 16.19/9.01 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7247_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 16.19/9.01 ! [X: num,N2: nat,A: int] :
% 16.19/9.01 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N2 ) @ ( ring_18347121197199848620nteger @ A ) )
% 16.19/9.01 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N2 ) @ A ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7248_of__nat__less__of__int__iff,axiom,
% 16.19/9.01 ! [N2: nat,X: int] :
% 16.19/9.01 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( ring_1_of_int_rat @ X ) )
% 16.19/9.01 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7249_of__nat__less__of__int__iff,axiom,
% 16.19/9.01 ! [N2: nat,X: int] :
% 16.19/9.01 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( ring_1_of_int_int @ X ) )
% 16.19/9.01 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7250_of__nat__less__of__int__iff,axiom,
% 16.19/9.01 ! [N2: nat,X: int] :
% 16.19/9.01 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( ring_1_of_int_real @ X ) )
% 16.19/9.01 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7251_mult__of__int__commute,axiom,
% 16.19/9.01 ! [X: int,Y: real] :
% 16.19/9.01 ( ( times_times_real @ ( ring_1_of_int_real @ X ) @ Y )
% 16.19/9.01 = ( times_times_real @ Y @ ( ring_1_of_int_real @ X ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7252_mult__of__int__commute,axiom,
% 16.19/9.01 ! [X: int,Y: rat] :
% 16.19/9.01 ( ( times_times_rat @ ( ring_1_of_int_rat @ X ) @ Y )
% 16.19/9.01 = ( times_times_rat @ Y @ ( ring_1_of_int_rat @ X ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7253_mult__of__int__commute,axiom,
% 16.19/9.01 ! [X: int,Y: int] :
% 16.19/9.01 ( ( times_times_int @ ( ring_1_of_int_int @ X ) @ Y )
% 16.19/9.01 = ( times_times_int @ Y @ ( ring_1_of_int_int @ X ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7254_mult__of__nat__commute,axiom,
% 16.19/9.01 ! [X: nat,Y: rat] :
% 16.19/9.01 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y )
% 16.19/9.01 = ( times_times_rat @ Y @ ( semiri681578069525770553at_rat @ X ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7255_mult__of__nat__commute,axiom,
% 16.19/9.01 ! [X: nat,Y: int] :
% 16.19/9.01 ( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
% 16.19/9.01 = ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7256_mult__of__nat__commute,axiom,
% 16.19/9.01 ! [X: nat,Y: real] :
% 16.19/9.01 ( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
% 16.19/9.01 = ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7257_mult__of__nat__commute,axiom,
% 16.19/9.01 ! [X: nat,Y: nat] :
% 16.19/9.01 ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
% 16.19/9.01 = ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7258_mult__of__nat__commute,axiom,
% 16.19/9.01 ! [X: nat,Y: complex] :
% 16.19/9.01 ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X ) @ Y )
% 16.19/9.01 = ( times_times_complex @ Y @ ( semiri8010041392384452111omplex @ X ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7259_int__diff__cases,axiom,
% 16.19/9.01 ! [Z: int] :
% 16.19/9.01 ~ ! [M2: nat,N4: nat] :
% 16.19/9.01 ( Z
% 16.19/9.01 != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7260_semiring__norm_I26_J,axiom,
% 16.19/9.01 ( ( bitM @ one )
% 16.19/9.01 = one ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7261_of__nat__0__le__iff,axiom,
% 16.19/9.01 ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7262_of__nat__0__le__iff,axiom,
% 16.19/9.01 ! [N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7263_of__nat__0__le__iff,axiom,
% 16.19/9.01 ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7264_of__nat__0__le__iff,axiom,
% 16.19/9.01 ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7265_of__nat__less__0__iff,axiom,
% 16.19/9.01 ! [M: nat] :
% 16.19/9.01 ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7266_of__nat__less__0__iff,axiom,
% 16.19/9.01 ! [M: nat] :
% 16.19/9.01 ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7267_of__nat__less__0__iff,axiom,
% 16.19/9.01 ! [M: nat] :
% 16.19/9.01 ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7268_of__nat__less__0__iff,axiom,
% 16.19/9.01 ! [M: nat] :
% 16.19/9.01 ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7269_of__nat__neq__0,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( semiri681578069525770553at_rat @ ( suc @ N2 ) )
% 16.19/9.01 != zero_zero_rat ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7270_of__nat__neq__0,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
% 16.19/9.01 != zero_zero_int ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7271_of__nat__neq__0,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( semiri5074537144036343181t_real @ ( suc @ N2 ) )
% 16.19/9.01 != zero_zero_real ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7272_of__nat__neq__0,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( semiri1316708129612266289at_nat @ ( suc @ N2 ) )
% 16.19/9.01 != zero_zero_nat ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7273_of__nat__neq__0,axiom,
% 16.19/9.01 ! [N2: nat] :
% 16.19/9.01 ( ( semiri8010041392384452111omplex @ ( suc @ N2 ) )
% 16.19/9.01 != zero_zero_complex ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7274_div__mult2__eq_H,axiom,
% 16.19/9.01 ! [A: int,M: nat,N2: nat] :
% 16.19/9.01 ( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 16.19/9.01 = ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7275_div__mult2__eq_H,axiom,
% 16.19/9.01 ! [A: nat,M: nat,N2: nat] :
% 16.19/9.01 ( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) )
% 16.19/9.01 = ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7276_of__nat__less__imp__less,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
% 16.19/9.01 => ( ord_less_nat @ M @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7277_of__nat__less__imp__less,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
% 16.19/9.01 => ( ord_less_nat @ M @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7278_of__nat__less__imp__less,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
% 16.19/9.01 => ( ord_less_nat @ M @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7279_of__nat__less__imp__less,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
% 16.19/9.01 => ( ord_less_nat @ M @ N2 ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7280_less__imp__of__nat__less,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.01 ( ( ord_less_nat @ M @ N2 )
% 16.19/9.01 => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).
% 16.19/9.01
% 16.19/9.01 thf(fact_7281_less__imp__of__nat__less,axiom,
% 16.19/9.01 ! [M: nat,N2: nat] :
% 16.19/9.02 ( ( ord_less_nat @ M @ N2 )
% 16.19/9.02 => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7282_less__imp__of__nat__less,axiom,
% 16.19/9.02 ! [M: nat,N2: nat] :
% 16.19/9.02 ( ( ord_less_nat @ M @ N2 )
% 16.19/9.02 => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7283_less__imp__of__nat__less,axiom,
% 16.19/9.02 ! [M: nat,N2: nat] :
% 16.19/9.02 ( ( ord_less_nat @ M @ N2 )
% 16.19/9.02 => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7284_of__nat__mono,axiom,
% 16.19/9.02 ! [I2: nat,J: nat] :
% 16.19/9.02 ( ( ord_less_eq_nat @ I2 @ J )
% 16.19/9.02 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I2 ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7285_of__nat__mono,axiom,
% 16.19/9.02 ! [I2: nat,J: nat] :
% 16.19/9.02 ( ( ord_less_eq_nat @ I2 @ J )
% 16.19/9.02 => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I2 ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7286_of__nat__mono,axiom,
% 16.19/9.02 ! [I2: nat,J: nat] :
% 16.19/9.02 ( ( ord_less_eq_nat @ I2 @ J )
% 16.19/9.02 => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7287_of__nat__mono,axiom,
% 16.19/9.02 ! [I2: nat,J: nat] :
% 16.19/9.02 ( ( ord_less_eq_nat @ I2 @ J )
% 16.19/9.02 => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7288_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 16.19/9.02 ! [M: nat,N2: nat] :
% 16.19/9.02 ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) )
% 16.19/9.02 = ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7289_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 16.19/9.02 ! [M: nat,N2: nat] :
% 16.19/9.02 ( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N2 ) )
% 16.19/9.02 = ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7290_of__nat__dvd__iff,axiom,
% 16.19/9.02 ! [M: nat,N2: nat] :
% 16.19/9.02 ( ( dvd_dvd_Code_integer @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) )
% 16.19/9.02 = ( dvd_dvd_nat @ M @ N2 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7291_of__nat__dvd__iff,axiom,
% 16.19/9.02 ! [M: nat,N2: nat] :
% 16.19/9.02 ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
% 16.19/9.02 = ( dvd_dvd_nat @ M @ N2 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7292_of__nat__dvd__iff,axiom,
% 16.19/9.02 ! [M: nat,N2: nat] :
% 16.19/9.02 ( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
% 16.19/9.02 = ( dvd_dvd_nat @ M @ N2 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7293_int__ops_I3_J,axiom,
% 16.19/9.02 ! [N2: num] :
% 16.19/9.02 ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N2 ) )
% 16.19/9.02 = ( numeral_numeral_int @ N2 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7294_int__of__nat__induct,axiom,
% 16.19/9.02 ! [P: int > $o,Z: int] :
% 16.19/9.02 ( ! [N4: nat] : ( P @ ( semiri1314217659103216013at_int @ N4 ) )
% 16.19/9.02 => ( ! [N4: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) )
% 16.19/9.02 => ( P @ Z ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7295_int__cases,axiom,
% 16.19/9.02 ! [Z: int] :
% 16.19/9.02 ( ! [N4: nat] :
% 16.19/9.02 ( Z
% 16.19/9.02 != ( semiri1314217659103216013at_int @ N4 ) )
% 16.19/9.02 => ~ ! [N4: nat] :
% 16.19/9.02 ( Z
% 16.19/9.02 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7296_nat__int__comparison_I2_J,axiom,
% 16.19/9.02 ( ord_less_nat
% 16.19/9.02 = ( ^ [A5: nat,B5: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7297_nat__int__comparison_I3_J,axiom,
% 16.19/9.02 ( ord_less_eq_nat
% 16.19/9.02 = ( ^ [A5: nat,B5: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7298_zle__int,axiom,
% 16.19/9.02 ! [M: nat,N2: nat] :
% 16.19/9.02 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
% 16.19/9.02 = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7299_of__nat__mod,axiom,
% 16.19/9.02 ! [M: nat,N2: nat] :
% 16.19/9.02 ( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N2 ) )
% 16.19/9.02 = ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7300_of__nat__mod,axiom,
% 16.19/9.02 ! [M: nat,N2: nat] :
% 16.19/9.02 ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) )
% 16.19/9.02 = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7301_of__nat__mod,axiom,
% 16.19/9.02 ! [M: nat,N2: nat] :
% 16.19/9.02 ( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N2 ) )
% 16.19/9.02 = ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7302_zadd__int__left,axiom,
% 16.19/9.02 ! [M: nat,N2: nat,Z: int] :
% 16.19/9.02 ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ Z ) )
% 16.19/9.02 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) ) @ Z ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7303_int__plus,axiom,
% 16.19/9.02 ! [N2: nat,M: nat] :
% 16.19/9.02 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N2 @ M ) )
% 16.19/9.02 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7304_int__ops_I5_J,axiom,
% 16.19/9.02 ! [A: nat,B: nat] :
% 16.19/9.02 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
% 16.19/9.02 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7305_int__ops_I7_J,axiom,
% 16.19/9.02 ! [A: nat,B: nat] :
% 16.19/9.02 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
% 16.19/9.02 = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7306_int__ops_I2_J,axiom,
% 16.19/9.02 ( ( semiri1314217659103216013at_int @ one_one_nat )
% 16.19/9.02 = one_one_int ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7307_not__int__zless__negative,axiom,
% 16.19/9.02 ! [N2: nat,M: nat] :
% 16.19/9.02 ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7308_zdiv__int,axiom,
% 16.19/9.02 ! [A: nat,B: nat] :
% 16.19/9.02 ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
% 16.19/9.02 = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7309_of__nat__max,axiom,
% 16.19/9.02 ! [X: nat,Y: nat] :
% 16.19/9.02 ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X @ Y ) )
% 16.19/9.02 = ( ord_max_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7310_of__nat__max,axiom,
% 16.19/9.02 ! [X: nat,Y: nat] :
% 16.19/9.02 ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X @ Y ) )
% 16.19/9.02 = ( ord_max_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ Y ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7311_of__nat__max,axiom,
% 16.19/9.02 ! [X: nat,Y: nat] :
% 16.19/9.02 ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X @ Y ) )
% 16.19/9.02 = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( semiri1316708129612266289at_nat @ Y ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7312_semiring__norm_I28_J,axiom,
% 16.19/9.02 ! [N2: num] :
% 16.19/9.02 ( ( bitM @ ( bit1 @ N2 ) )
% 16.19/9.02 = ( bit1 @ ( bit0 @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7313_semiring__norm_I27_J,axiom,
% 16.19/9.02 ! [N2: num] :
% 16.19/9.02 ( ( bitM @ ( bit0 @ N2 ) )
% 16.19/9.02 = ( bit1 @ ( bitM @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7314_nat__less__as__int,axiom,
% 16.19/9.02 ( ord_less_nat
% 16.19/9.02 = ( ^ [A5: nat,B5: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7315_nat__leq__as__int,axiom,
% 16.19/9.02 ( ord_less_eq_nat
% 16.19/9.02 = ( ^ [A5: nat,B5: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7316_of__nat__diff,axiom,
% 16.19/9.02 ! [N2: nat,M: nat] :
% 16.19/9.02 ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/9.02 => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N2 ) )
% 16.19/9.02 = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7317_of__nat__diff,axiom,
% 16.19/9.02 ! [N2: nat,M: nat] :
% 16.19/9.02 ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/9.02 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N2 ) )
% 16.19/9.02 = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7318_of__nat__diff,axiom,
% 16.19/9.02 ! [N2: nat,M: nat] :
% 16.19/9.02 ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/9.02 => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N2 ) )
% 16.19/9.02 = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7319_of__nat__diff,axiom,
% 16.19/9.02 ! [N2: nat,M: nat] :
% 16.19/9.02 ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/9.02 => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N2 ) )
% 16.19/9.02 = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7320_of__nat__diff,axiom,
% 16.19/9.02 ! [N2: nat,M: nat] :
% 16.19/9.02 ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/9.02 => ( ( semiri8010041392384452111omplex @ ( minus_minus_nat @ M @ N2 ) )
% 16.19/9.02 = ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7321_reals__Archimedean3,axiom,
% 16.19/9.02 ! [X: real] :
% 16.19/9.02 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.02 => ! [Y5: real] :
% 16.19/9.02 ? [N4: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ X ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7322_real__of__int__div4,axiom,
% 16.19/9.02 ! [N2: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ X ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7323_int__cases4,axiom,
% 16.19/9.02 ! [M: int] :
% 16.19/9.02 ( ! [N4: nat] :
% 16.19/9.02 ( M
% 16.19/9.02 != ( semiri1314217659103216013at_int @ N4 ) )
% 16.19/9.02 => ~ ! [N4: nat] :
% 16.19/9.02 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 16.19/9.02 => ( M
% 16.19/9.02 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7324_real__of__nat__div4,axiom,
% 16.19/9.02 ! [N2: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7325_int__Suc,axiom,
% 16.19/9.02 ! [N2: nat] :
% 16.19/9.02 ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
% 16.19/9.02 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7326_int__ops_I4_J,axiom,
% 16.19/9.02 ! [A: nat] :
% 16.19/9.02 ( ( semiri1314217659103216013at_int @ ( suc @ A ) )
% 16.19/9.02 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7327_zless__iff__Suc__zadd,axiom,
% 16.19/9.02 ( ord_less_int
% 16.19/9.02 = ( ^ [W3: int,Z3: int] :
% 16.19/9.02 ? [N: nat] :
% 16.19/9.02 ( Z3
% 16.19/9.02 = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7328_real__of__nat__div,axiom,
% 16.19/9.02 ! [D: nat,N2: nat] :
% 16.19/9.02 ( ( dvd_dvd_nat @ D @ N2 )
% 16.19/9.02 => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ D ) )
% 16.19/9.02 = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7329_real__of__int__div,axiom,
% 16.19/9.02 ! [D: int,N2: int] :
% 16.19/9.02 ( ( dvd_dvd_int @ D @ N2 )
% 16.19/9.02 => ( ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ D ) )
% 16.19/9.02 = ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7330_int__sum,axiom,
% 16.19/9.02 ! [F: int > nat,A2: set_int] :
% 16.19/9.02 ( ( semiri1314217659103216013at_int @ ( groups4541462559716669496nt_nat @ F @ A2 ) )
% 16.19/9.02 = ( groups4538972089207619220nt_int
% 16.19/9.02 @ ^ [X2: int] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
% 16.19/9.02 @ A2 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7331_int__sum,axiom,
% 16.19/9.02 ! [F: nat > nat,A2: set_nat] :
% 16.19/9.02 ( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 16.19/9.02 = ( groups3539618377306564664at_int
% 16.19/9.02 @ ^ [X2: nat] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
% 16.19/9.02 @ A2 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7332_eval__nat__numeral_I2_J,axiom,
% 16.19/9.02 ! [N2: num] :
% 16.19/9.02 ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
% 16.19/9.02 = ( suc @ ( numeral_numeral_nat @ ( bitM @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7333_BitM__plus__one,axiom,
% 16.19/9.02 ! [N2: num] :
% 16.19/9.02 ( ( plus_plus_num @ ( bitM @ N2 ) @ one )
% 16.19/9.02 = ( bit0 @ N2 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7334_one__plus__BitM,axiom,
% 16.19/9.02 ! [N2: num] :
% 16.19/9.02 ( ( plus_plus_num @ one @ ( bitM @ N2 ) )
% 16.19/9.02 = ( bit0 @ N2 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7335_of__int__nonneg,axiom,
% 16.19/9.02 ! [Z: int] :
% 16.19/9.02 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 16.19/9.02 => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7336_of__int__nonneg,axiom,
% 16.19/9.02 ! [Z: int] :
% 16.19/9.02 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 16.19/9.02 => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7337_of__int__nonneg,axiom,
% 16.19/9.02 ! [Z: int] :
% 16.19/9.02 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 16.19/9.02 => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7338_of__int__leD,axiom,
% 16.19/9.02 ! [N2: int,X: code_integer] :
% 16.19/9.02 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N2 ) ) @ X )
% 16.19/9.02 => ( ( N2 = zero_zero_int )
% 16.19/9.02 | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7339_of__int__leD,axiom,
% 16.19/9.02 ! [N2: int,X: real] :
% 16.19/9.02 ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N2 ) ) @ X )
% 16.19/9.02 => ( ( N2 = zero_zero_int )
% 16.19/9.02 | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7340_of__int__leD,axiom,
% 16.19/9.02 ! [N2: int,X: rat] :
% 16.19/9.02 ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N2 ) ) @ X )
% 16.19/9.02 => ( ( N2 = zero_zero_int )
% 16.19/9.02 | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7341_of__int__leD,axiom,
% 16.19/9.02 ! [N2: int,X: int] :
% 16.19/9.02 ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X )
% 16.19/9.02 => ( ( N2 = zero_zero_int )
% 16.19/9.02 | ( ord_less_eq_int @ one_one_int @ X ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7342_of__int__pos,axiom,
% 16.19/9.02 ! [Z: int] :
% 16.19/9.02 ( ( ord_less_int @ zero_zero_int @ Z )
% 16.19/9.02 => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7343_of__int__pos,axiom,
% 16.19/9.02 ! [Z: int] :
% 16.19/9.02 ( ( ord_less_int @ zero_zero_int @ Z )
% 16.19/9.02 => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7344_of__int__pos,axiom,
% 16.19/9.02 ! [Z: int] :
% 16.19/9.02 ( ( ord_less_int @ zero_zero_int @ Z )
% 16.19/9.02 => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7345_of__int__lessD,axiom,
% 16.19/9.02 ! [N2: int,X: code_integer] :
% 16.19/9.02 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N2 ) ) @ X )
% 16.19/9.02 => ( ( N2 = zero_zero_int )
% 16.19/9.02 | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7346_of__int__lessD,axiom,
% 16.19/9.02 ! [N2: int,X: real] :
% 16.19/9.02 ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N2 ) ) @ X )
% 16.19/9.02 => ( ( N2 = zero_zero_int )
% 16.19/9.02 | ( ord_less_real @ one_one_real @ X ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7347_of__int__lessD,axiom,
% 16.19/9.02 ! [N2: int,X: rat] :
% 16.19/9.02 ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N2 ) ) @ X )
% 16.19/9.02 => ( ( N2 = zero_zero_int )
% 16.19/9.02 | ( ord_less_rat @ one_one_rat @ X ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7348_of__int__lessD,axiom,
% 16.19/9.02 ! [N2: int,X: int] :
% 16.19/9.02 ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X )
% 16.19/9.02 => ( ( N2 = zero_zero_int )
% 16.19/9.02 | ( ord_less_int @ one_one_int @ X ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7349_mod__mult2__eq_H,axiom,
% 16.19/9.02 ! [A: code_integer,M: nat,N2: nat] :
% 16.19/9.02 ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) )
% 16.19/9.02 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7350_mod__mult2__eq_H,axiom,
% 16.19/9.02 ! [A: int,M: nat,N2: nat] :
% 16.19/9.02 ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 16.19/9.02 = ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7351_mod__mult2__eq_H,axiom,
% 16.19/9.02 ! [A: nat,M: nat,N2: nat] :
% 16.19/9.02 ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) )
% 16.19/9.02 = ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7352_of__int__neg__numeral,axiom,
% 16.19/9.02 ! [K: num] :
% 16.19/9.02 ( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 16.19/9.02 = ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7353_of__int__neg__numeral,axiom,
% 16.19/9.02 ! [K: num] :
% 16.19/9.02 ( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 16.19/9.02 = ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7354_of__int__neg__numeral,axiom,
% 16.19/9.02 ! [K: num] :
% 16.19/9.02 ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 16.19/9.02 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7355_of__int__neg__numeral,axiom,
% 16.19/9.02 ! [K: num] :
% 16.19/9.02 ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 16.19/9.02 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7356_of__int__neg__numeral,axiom,
% 16.19/9.02 ! [K: num] :
% 16.19/9.02 ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 16.19/9.02 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7357_field__char__0__class_Oof__nat__div,axiom,
% 16.19/9.02 ! [M: nat,N2: nat] :
% 16.19/9.02 ( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N2 ) )
% 16.19/9.02 = ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N2 ) ) ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7358_field__char__0__class_Oof__nat__div,axiom,
% 16.19/9.02 ! [M: nat,N2: nat] :
% 16.19/9.02 ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N2 ) )
% 16.19/9.02 = ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N2 ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7359_field__char__0__class_Oof__nat__div,axiom,
% 16.19/9.02 ! [M: nat,N2: nat] :
% 16.19/9.02 ( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M @ N2 ) )
% 16.19/9.02 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M @ N2 ) ) ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7360_int__le__real__less,axiom,
% 16.19/9.02 ( ord_less_eq_int
% 16.19/9.02 = ( ^ [N: int,M6: int] : ( ord_less_real @ ( ring_1_of_int_real @ N ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M6 ) @ one_one_real ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7361_int__less__real__le,axiom,
% 16.19/9.02 ( ord_less_int
% 16.19/9.02 = ( ^ [N: int,M6: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) @ ( ring_1_of_int_real @ M6 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7362_pos__int__cases,axiom,
% 16.19/9.02 ! [K: int] :
% 16.19/9.02 ( ( ord_less_int @ zero_zero_int @ K )
% 16.19/9.02 => ~ ! [N4: nat] :
% 16.19/9.02 ( ( K
% 16.19/9.02 = ( semiri1314217659103216013at_int @ N4 ) )
% 16.19/9.02 => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7363_zero__less__imp__eq__int,axiom,
% 16.19/9.02 ! [K: int] :
% 16.19/9.02 ( ( ord_less_int @ zero_zero_int @ K )
% 16.19/9.02 => ? [N4: nat] :
% 16.19/9.02 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 16.19/9.02 & ( K
% 16.19/9.02 = ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7364_int__cases3,axiom,
% 16.19/9.02 ! [K: int] :
% 16.19/9.02 ( ( K != zero_zero_int )
% 16.19/9.02 => ( ! [N4: nat] :
% 16.19/9.02 ( ( K
% 16.19/9.02 = ( semiri1314217659103216013at_int @ N4 ) )
% 16.19/9.02 => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) )
% 16.19/9.02 => ~ ! [N4: nat] :
% 16.19/9.02 ( ( K
% 16.19/9.02 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
% 16.19/9.02 => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7365_nat__less__real__le,axiom,
% 16.19/9.02 ( ord_less_nat
% 16.19/9.02 = ( ^ [N: nat,M6: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M6 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7366_nat__le__real__less,axiom,
% 16.19/9.02 ( ord_less_eq_nat
% 16.19/9.02 = ( ^ [N: nat,M6: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M6 ) @ one_one_real ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7367_zmult__zless__mono2__lemma,axiom,
% 16.19/9.02 ! [I2: int,J: int,K: nat] :
% 16.19/9.02 ( ( ord_less_int @ I2 @ J )
% 16.19/9.02 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 16.19/9.02 => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7368_not__zle__0__negative,axiom,
% 16.19/9.02 ! [N2: nat] :
% 16.19/9.02 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7369_negD,axiom,
% 16.19/9.02 ! [X: int] :
% 16.19/9.02 ( ( ord_less_int @ X @ zero_zero_int )
% 16.19/9.02 => ? [N4: nat] :
% 16.19/9.02 ( X
% 16.19/9.02 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7370_negative__zless__0,axiom,
% 16.19/9.02 ! [N2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ zero_zero_int ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7371_dbl__inc__def,axiom,
% 16.19/9.02 ( neg_nu8557863876264182079omplex
% 16.19/9.02 = ( ^ [X2: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X2 @ X2 ) @ one_one_complex ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7372_dbl__inc__def,axiom,
% 16.19/9.02 ( neg_nu8295874005876285629c_real
% 16.19/9.02 = ( ^ [X2: real] : ( plus_plus_real @ ( plus_plus_real @ X2 @ X2 ) @ one_one_real ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7373_dbl__inc__def,axiom,
% 16.19/9.02 ( neg_nu5219082963157363817nc_rat
% 16.19/9.02 = ( ^ [X2: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7374_dbl__inc__def,axiom,
% 16.19/9.02 ( neg_nu5851722552734809277nc_int
% 16.19/9.02 = ( ^ [X2: int] : ( plus_plus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7375_int__ops_I6_J,axiom,
% 16.19/9.02 ! [A: nat,B: nat] :
% 16.19/9.02 ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 16.19/9.02 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 16.19/9.02 = zero_zero_int ) )
% 16.19/9.02 & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 16.19/9.02 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 16.19/9.02 = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7376_real__of__int__div__aux,axiom,
% 16.19/9.02 ! [X: int,D: int] :
% 16.19/9.02 ( ( divide_divide_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ D ) )
% 16.19/9.02 = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7377_real__of__nat__div__aux,axiom,
% 16.19/9.02 ! [X: nat,D: nat] :
% 16.19/9.02 ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ D ) )
% 16.19/9.02 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7378_numeral__BitM,axiom,
% 16.19/9.02 ! [N2: num] :
% 16.19/9.02 ( ( numera6690914467698888265omplex @ ( bitM @ N2 ) )
% 16.19/9.02 = ( minus_minus_complex @ ( numera6690914467698888265omplex @ ( bit0 @ N2 ) ) @ one_one_complex ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7379_numeral__BitM,axiom,
% 16.19/9.02 ! [N2: num] :
% 16.19/9.02 ( ( numeral_numeral_real @ ( bitM @ N2 ) )
% 16.19/9.02 = ( minus_minus_real @ ( numeral_numeral_real @ ( bit0 @ N2 ) ) @ one_one_real ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7380_numeral__BitM,axiom,
% 16.19/9.02 ! [N2: num] :
% 16.19/9.02 ( ( numeral_numeral_rat @ ( bitM @ N2 ) )
% 16.19/9.02 = ( minus_minus_rat @ ( numeral_numeral_rat @ ( bit0 @ N2 ) ) @ one_one_rat ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7381_numeral__BitM,axiom,
% 16.19/9.02 ! [N2: num] :
% 16.19/9.02 ( ( numeral_numeral_int @ ( bitM @ N2 ) )
% 16.19/9.02 = ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ one_one_int ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7382_odd__numeral__BitM,axiom,
% 16.19/9.02 ! [W: num] :
% 16.19/9.02 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bitM @ W ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7383_odd__numeral__BitM,axiom,
% 16.19/9.02 ! [W: num] :
% 16.19/9.02 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bitM @ W ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7384_odd__numeral__BitM,axiom,
% 16.19/9.02 ! [W: num] :
% 16.19/9.02 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bitM @ W ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7385_of__nat__less__two__power,axiom,
% 16.19/9.02 ! [N2: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7386_of__nat__less__two__power,axiom,
% 16.19/9.02 ! [N2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7387_of__nat__less__two__power,axiom,
% 16.19/9.02 ! [N2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7388_inverse__of__nat__le,axiom,
% 16.19/9.02 ! [N2: nat,M: nat] :
% 16.19/9.02 ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/9.02 => ( ( N2 != zero_zero_nat )
% 16.19/9.02 => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7389_inverse__of__nat__le,axiom,
% 16.19/9.02 ! [N2: nat,M: nat] :
% 16.19/9.02 ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/9.02 => ( ( N2 != zero_zero_nat )
% 16.19/9.02 => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7390_real__archimedian__rdiv__eq__0,axiom,
% 16.19/9.02 ! [X: real,C: real] :
% 16.19/9.02 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.02 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 16.19/9.02 => ( ! [M2: nat] :
% 16.19/9.02 ( ( ord_less_nat @ zero_zero_nat @ M2 )
% 16.19/9.02 => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ X ) @ C ) )
% 16.19/9.02 => ( X = zero_zero_real ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7391_real__of__int__div2,axiom,
% 16.19/9.02 ! [N2: int,X: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ X ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7392_real__of__int__div3,axiom,
% 16.19/9.02 ! [N2: int,X: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ X ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ X ) ) ) @ one_one_real ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7393_neg__int__cases,axiom,
% 16.19/9.02 ! [K: int] :
% 16.19/9.02 ( ( ord_less_int @ K @ zero_zero_int )
% 16.19/9.02 => ~ ! [N4: nat] :
% 16.19/9.02 ( ( K
% 16.19/9.02 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
% 16.19/9.02 => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7394_zdiff__int__split,axiom,
% 16.19/9.02 ! [P: int > $o,X: nat,Y: nat] :
% 16.19/9.02 ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
% 16.19/9.02 = ( ( ( ord_less_eq_nat @ Y @ X )
% 16.19/9.02 => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
% 16.19/9.02 & ( ( ord_less_nat @ X @ Y )
% 16.19/9.02 => ( P @ zero_zero_int ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7395_real__of__nat__div2,axiom,
% 16.19/9.02 ! [N2: nat,X: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7396_real__of__nat__div3,axiom,
% 16.19/9.02 ! [N2: nat,X: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X ) ) ) @ one_one_real ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7397_ln__realpow,axiom,
% 16.19/9.02 ! [X: real,N2: nat] :
% 16.19/9.02 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.02 => ( ( ln_ln_real @ ( power_power_real @ X @ N2 ) )
% 16.19/9.02 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( ln_ln_real @ X ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7398_even__of__int__iff,axiom,
% 16.19/9.02 ! [K: int] :
% 16.19/9.02 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ K ) )
% 16.19/9.02 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7399_even__of__int__iff,axiom,
% 16.19/9.02 ! [K: int] :
% 16.19/9.02 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ K ) )
% 16.19/9.02 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7400_linear__plus__1__le__power,axiom,
% 16.19/9.02 ! [X: real,N2: nat] :
% 16.19/9.02 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.02 => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X @ one_one_real ) @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7401_Bernoulli__inequality,axiom,
% 16.19/9.02 ! [X: real,N2: nat] :
% 16.19/9.02 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 16.19/9.02 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7402_Bernoulli__inequality__even,axiom,
% 16.19/9.02 ! [N2: nat,X: real] :
% 16.19/9.02 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.02 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7403_double__gauss__sum,axiom,
% 16.19/9.02 ! [N2: nat] :
% 16.19/9.02 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 16.19/9.02 = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7404_double__gauss__sum,axiom,
% 16.19/9.02 ! [N2: nat] :
% 16.19/9.02 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 16.19/9.02 = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7405_double__gauss__sum,axiom,
% 16.19/9.02 ! [N2: nat] :
% 16.19/9.02 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 16.19/9.02 = ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7406_double__gauss__sum,axiom,
% 16.19/9.02 ! [N2: nat] :
% 16.19/9.02 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 16.19/9.02 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7407_double__gauss__sum,axiom,
% 16.19/9.02 ! [N2: nat] :
% 16.19/9.02 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 16.19/9.02 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7408_double__arith__series,axiom,
% 16.19/9.02 ! [A: rat,D: rat,N2: nat] :
% 16.19/9.02 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
% 16.19/9.02 @ ( groups2906978787729119204at_rat
% 16.19/9.02 @ ^ [I3: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I3 ) @ D ) )
% 16.19/9.02 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 16.19/9.02 = ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ D ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7409_double__arith__series,axiom,
% 16.19/9.02 ! [A: int,D: int,N2: nat] :
% 16.19/9.02 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
% 16.19/9.02 @ ( groups3539618377306564664at_int
% 16.19/9.02 @ ^ [I3: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I3 ) @ D ) )
% 16.19/9.02 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 16.19/9.02 = ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ D ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7410_double__arith__series,axiom,
% 16.19/9.02 ! [A: complex,D: complex,N2: nat] :
% 16.19/9.02 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
% 16.19/9.02 @ ( groups2073611262835488442omplex
% 16.19/9.02 @ ^ [I3: nat] : ( plus_plus_complex @ A @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I3 ) @ D ) )
% 16.19/9.02 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 16.19/9.02 = ( times_times_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ D ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7411_double__arith__series,axiom,
% 16.19/9.02 ! [A: nat,D: nat,N2: nat] :
% 16.19/9.02 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
% 16.19/9.02 @ ( groups3542108847815614940at_nat
% 16.19/9.02 @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ D ) )
% 16.19/9.02 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 16.19/9.02 = ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ D ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7412_double__arith__series,axiom,
% 16.19/9.02 ! [A: real,D: real,N2: nat] :
% 16.19/9.02 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 16.19/9.02 @ ( groups6591440286371151544t_real
% 16.19/9.02 @ ^ [I3: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I3 ) @ D ) )
% 16.19/9.02 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 16.19/9.02 = ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ D ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7413_gauss__sum,axiom,
% 16.19/9.02 ! [N2: nat] :
% 16.19/9.02 ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 16.19/9.02 = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7414_gauss__sum,axiom,
% 16.19/9.02 ! [N2: nat] :
% 16.19/9.02 ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 16.19/9.02 = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7415_arith__series,axiom,
% 16.19/9.02 ! [A: int,D: int,N2: nat] :
% 16.19/9.02 ( ( groups3539618377306564664at_int
% 16.19/9.02 @ ^ [I3: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I3 ) @ D ) )
% 16.19/9.02 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 16.19/9.02 = ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7416_arith__series,axiom,
% 16.19/9.02 ! [A: nat,D: nat,N2: nat] :
% 16.19/9.02 ( ( groups3542108847815614940at_nat
% 16.19/9.02 @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ D ) )
% 16.19/9.02 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 16.19/9.02 = ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7417_double__gauss__sum__from__Suc__0,axiom,
% 16.19/9.02 ! [N2: nat] :
% 16.19/9.02 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
% 16.19/9.02 = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7418_double__gauss__sum__from__Suc__0,axiom,
% 16.19/9.02 ! [N2: nat] :
% 16.19/9.02 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
% 16.19/9.02 = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7419_double__gauss__sum__from__Suc__0,axiom,
% 16.19/9.02 ! [N2: nat] :
% 16.19/9.02 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
% 16.19/9.02 = ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7420_double__gauss__sum__from__Suc__0,axiom,
% 16.19/9.02 ! [N2: nat] :
% 16.19/9.02 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
% 16.19/9.02 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7421_double__gauss__sum__from__Suc__0,axiom,
% 16.19/9.02 ! [N2: nat] :
% 16.19/9.02 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
% 16.19/9.02 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7422_sum__gp__offset,axiom,
% 16.19/9.02 ! [X: rat,M: nat,N2: nat] :
% 16.19/9.02 ( ( ( X = one_one_rat )
% 16.19/9.02 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
% 16.19/9.02 = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) )
% 16.19/9.02 & ( ( X != one_one_rat )
% 16.19/9.02 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
% 16.19/9.02 = ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N2 ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7423_sum__gp__offset,axiom,
% 16.19/9.02 ! [X: complex,M: nat,N2: nat] :
% 16.19/9.02 ( ( ( X = one_one_complex )
% 16.19/9.02 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
% 16.19/9.02 = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) )
% 16.19/9.02 & ( ( X != one_one_complex )
% 16.19/9.02 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
% 16.19/9.02 = ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N2 ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7424_sum__gp__offset,axiom,
% 16.19/9.02 ! [X: real,M: nat,N2: nat] :
% 16.19/9.02 ( ( ( X = one_one_real )
% 16.19/9.02 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
% 16.19/9.02 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) )
% 16.19/9.02 & ( ( X != one_one_real )
% 16.19/9.02 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
% 16.19/9.02 = ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X @ M ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N2 ) ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7425_of__nat__code__if,axiom,
% 16.19/9.02 ( semiri681578069525770553at_rat
% 16.19/9.02 = ( ^ [N: nat] :
% 16.19/9.02 ( if_rat @ ( N = zero_zero_nat ) @ zero_zero_rat
% 16.19/9.02 @ ( produc6207742614233964070at_rat
% 16.19/9.02 @ ^ [M6: nat,Q4: nat] : ( if_rat @ ( Q4 = zero_zero_nat ) @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M6 ) ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M6 ) ) @ one_one_rat ) )
% 16.19/9.02 @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7426_of__nat__code__if,axiom,
% 16.19/9.02 ( semiri1314217659103216013at_int
% 16.19/9.02 = ( ^ [N: nat] :
% 16.19/9.02 ( if_int @ ( N = zero_zero_nat ) @ zero_zero_int
% 16.19/9.02 @ ( produc6840382203811409530at_int
% 16.19/9.02 @ ^ [M6: nat,Q4: nat] : ( if_int @ ( Q4 = zero_zero_nat ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M6 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M6 ) ) @ one_one_int ) )
% 16.19/9.02 @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7427_of__nat__code__if,axiom,
% 16.19/9.02 ( semiri5074537144036343181t_real
% 16.19/9.02 = ( ^ [N: nat] :
% 16.19/9.02 ( if_real @ ( N = zero_zero_nat ) @ zero_zero_real
% 16.19/9.02 @ ( produc1703576794950452218t_real
% 16.19/9.02 @ ^ [M6: nat,Q4: nat] : ( if_real @ ( Q4 = zero_zero_nat ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M6 ) ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M6 ) ) @ one_one_real ) )
% 16.19/9.02 @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7428_of__nat__code__if,axiom,
% 16.19/9.02 ( semiri1316708129612266289at_nat
% 16.19/9.02 = ( ^ [N: nat] :
% 16.19/9.02 ( if_nat @ ( N = zero_zero_nat ) @ zero_zero_nat
% 16.19/9.02 @ ( produc6842872674320459806at_nat
% 16.19/9.02 @ ^ [M6: nat,Q4: nat] : ( if_nat @ ( Q4 = zero_zero_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M6 ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M6 ) ) @ one_one_nat ) )
% 16.19/9.02 @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7429_of__nat__code__if,axiom,
% 16.19/9.02 ( semiri8010041392384452111omplex
% 16.19/9.02 = ( ^ [N: nat] :
% 16.19/9.02 ( if_complex @ ( N = zero_zero_nat ) @ zero_zero_complex
% 16.19/9.02 @ ( produc1917071388513777916omplex
% 16.19/9.02 @ ^ [M6: nat,Q4: nat] : ( if_complex @ ( Q4 = zero_zero_nat ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M6 ) ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M6 ) ) @ one_one_complex ) )
% 16.19/9.02 @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7430_nat__approx__posE,axiom,
% 16.19/9.02 ! [E: rat] :
% 16.19/9.02 ( ( ord_less_rat @ zero_zero_rat @ E )
% 16.19/9.02 => ~ ! [N4: nat] :
% 16.19/9.02 ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N4 ) ) ) @ E ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7431_nat__approx__posE,axiom,
% 16.19/9.02 ! [E: real] :
% 16.19/9.02 ( ( ord_less_real @ zero_zero_real @ E )
% 16.19/9.02 => ~ ! [N4: nat] :
% 16.19/9.02 ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ E ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7432_floor__exists,axiom,
% 16.19/9.02 ! [X: real] :
% 16.19/9.02 ? [Z5: int] :
% 16.19/9.02 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z5 ) @ X )
% 16.19/9.02 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7433_floor__exists,axiom,
% 16.19/9.02 ! [X: rat] :
% 16.19/9.02 ? [Z5: int] :
% 16.19/9.02 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z5 ) @ X )
% 16.19/9.02 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7434_floor__exists1,axiom,
% 16.19/9.02 ! [X: real] :
% 16.19/9.02 ? [X4: int] :
% 16.19/9.02 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X4 ) @ X )
% 16.19/9.02 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X4 @ one_one_int ) ) )
% 16.19/9.02 & ! [Y5: int] :
% 16.19/9.02 ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y5 ) @ X )
% 16.19/9.02 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
% 16.19/9.02 => ( Y5 = X4 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7435_floor__exists1,axiom,
% 16.19/9.02 ! [X: rat] :
% 16.19/9.02 ? [X4: int] :
% 16.19/9.02 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X4 ) @ X )
% 16.19/9.02 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X4 @ one_one_int ) ) )
% 16.19/9.02 & ! [Y5: int] :
% 16.19/9.02 ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y5 ) @ X )
% 16.19/9.02 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
% 16.19/9.02 => ( Y5 = X4 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7436_monoseq__arctan__series,axiom,
% 16.19/9.02 ! [X: real] :
% 16.19/9.02 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 16.19/9.02 => ( topolo6980174941875973593q_real
% 16.19/9.02 @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7437_lemma__termdiff3,axiom,
% 16.19/9.02 ! [H2: real,Z: real,K5: real,N2: nat] :
% 16.19/9.02 ( ( H2 != zero_zero_real )
% 16.19/9.02 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K5 )
% 16.19/9.02 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H2 ) ) @ K5 )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N2 ) @ ( power_power_real @ Z @ N2 ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7438_lemma__termdiff3,axiom,
% 16.19/9.02 ! [H2: complex,Z: complex,K5: real,N2: nat] :
% 16.19/9.02 ( ( H2 != zero_zero_complex )
% 16.19/9.02 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K5 )
% 16.19/9.02 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H2 ) ) @ K5 )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7439_complex__mod__minus__le__complex__mod,axiom,
% 16.19/9.02 ! [X: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7440_complex__mod__triangle__ineq2,axiom,
% 16.19/9.02 ! [B: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B @ A ) ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7441_monoseq__realpow,axiom,
% 16.19/9.02 ! [X: real] :
% 16.19/9.02 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.02 => ( ( ord_less_eq_real @ X @ one_one_real )
% 16.19/9.02 => ( topolo6980174941875973593q_real @ ( power_power_real @ X ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7442_real__arch__simple,axiom,
% 16.19/9.02 ! [X: real] :
% 16.19/9.02 ? [N4: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N4 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7443_real__arch__simple,axiom,
% 16.19/9.02 ! [X: rat] :
% 16.19/9.02 ? [N4: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N4 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7444_reals__Archimedean2,axiom,
% 16.19/9.02 ! [X: rat] :
% 16.19/9.02 ? [N4: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N4 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7445_reals__Archimedean2,axiom,
% 16.19/9.02 ! [X: real] :
% 16.19/9.02 ? [N4: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N4 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7446_exists__least__lemma,axiom,
% 16.19/9.02 ! [P: nat > $o] :
% 16.19/9.02 ( ~ ( P @ zero_zero_nat )
% 16.19/9.02 => ( ? [X_12: nat] : ( P @ X_12 )
% 16.19/9.02 => ? [N4: nat] :
% 16.19/9.02 ( ~ ( P @ N4 )
% 16.19/9.02 & ( P @ ( suc @ N4 ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7447_ex__le__of__int,axiom,
% 16.19/9.02 ! [X: real] :
% 16.19/9.02 ? [Z5: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z5 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7448_ex__le__of__int,axiom,
% 16.19/9.02 ! [X: rat] :
% 16.19/9.02 ? [Z5: int] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z5 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7449_ex__of__int__less,axiom,
% 16.19/9.02 ! [X: real] :
% 16.19/9.02 ? [Z5: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z5 ) @ X ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7450_ex__of__int__less,axiom,
% 16.19/9.02 ! [X: rat] :
% 16.19/9.02 ? [Z5: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z5 ) @ X ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7451_ex__less__of__int,axiom,
% 16.19/9.02 ! [X: real] :
% 16.19/9.02 ? [Z5: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z5 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7452_ex__less__of__int,axiom,
% 16.19/9.02 ! [X: rat] :
% 16.19/9.02 ? [Z5: int] : ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z5 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7453_ex__less__of__nat__mult,axiom,
% 16.19/9.02 ! [X: rat,Y: rat] :
% 16.19/9.02 ( ( ord_less_rat @ zero_zero_rat @ X )
% 16.19/9.02 => ? [N4: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N4 ) @ X ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7454_ex__less__of__nat__mult,axiom,
% 16.19/9.02 ! [X: real,Y: real] :
% 16.19/9.02 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.02 => ? [N4: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ X ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7455_norm__divide__numeral,axiom,
% 16.19/9.02 ! [A: real,W: num] :
% 16.19/9.02 ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ W ) ) )
% 16.19/9.02 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7456_norm__divide__numeral,axiom,
% 16.19/9.02 ! [A: complex,W: num] :
% 16.19/9.02 ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 16.19/9.02 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7457_norm__mult__numeral2,axiom,
% 16.19/9.02 ! [A: real,W: num] :
% 16.19/9.02 ( ( real_V7735802525324610683m_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) )
% 16.19/9.02 = ( times_times_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7458_norm__mult__numeral2,axiom,
% 16.19/9.02 ! [A: complex,W: num] :
% 16.19/9.02 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 16.19/9.02 = ( times_times_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7459_norm__mult__numeral1,axiom,
% 16.19/9.02 ! [W: num,A: real] :
% 16.19/9.02 ( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
% 16.19/9.02 = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V7735802525324610683m_real @ A ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7460_norm__mult__numeral1,axiom,
% 16.19/9.02 ! [W: num,A: complex] :
% 16.19/9.02 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
% 16.19/9.02 = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V1022390504157884413omplex @ A ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7461_norm__neg__numeral,axiom,
% 16.19/9.02 ! [W: num] :
% 16.19/9.02 ( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 16.19/9.02 = ( numeral_numeral_real @ W ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7462_norm__neg__numeral,axiom,
% 16.19/9.02 ! [W: num] :
% 16.19/9.02 ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 16.19/9.02 = ( numeral_numeral_real @ W ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7463_norm__le__zero__iff,axiom,
% 16.19/9.02 ! [X: real] :
% 16.19/9.02 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
% 16.19/9.02 = ( X = zero_zero_real ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7464_norm__le__zero__iff,axiom,
% 16.19/9.02 ! [X: complex] :
% 16.19/9.02 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real )
% 16.19/9.02 = ( X = zero_zero_complex ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7465_norm__one,axiom,
% 16.19/9.02 ( ( real_V7735802525324610683m_real @ one_one_real )
% 16.19/9.02 = one_one_real ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7466_norm__one,axiom,
% 16.19/9.02 ( ( real_V1022390504157884413omplex @ one_one_complex )
% 16.19/9.02 = one_one_real ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7467_norm__numeral,axiom,
% 16.19/9.02 ! [W: num] :
% 16.19/9.02 ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
% 16.19/9.02 = ( numeral_numeral_real @ W ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7468_norm__numeral,axiom,
% 16.19/9.02 ! [W: num] :
% 16.19/9.02 ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
% 16.19/9.02 = ( numeral_numeral_real @ W ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7469_zero__less__norm__iff,axiom,
% 16.19/9.02 ! [X: real] :
% 16.19/9.02 ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
% 16.19/9.02 = ( X != zero_zero_real ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7470_zero__less__norm__iff,axiom,
% 16.19/9.02 ! [X: complex] :
% 16.19/9.02 ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) )
% 16.19/9.02 = ( X != zero_zero_complex ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7471_norm__minus__commute,axiom,
% 16.19/9.02 ! [A: real,B: real] :
% 16.19/9.02 ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) )
% 16.19/9.02 = ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7472_norm__minus__commute,axiom,
% 16.19/9.02 ! [A: complex,B: complex] :
% 16.19/9.02 ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) )
% 16.19/9.02 = ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ A ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7473_norm__not__less__zero,axiom,
% 16.19/9.02 ! [X: complex] :
% 16.19/9.02 ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7474_norm__ge__zero,axiom,
% 16.19/9.02 ! [X: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7475_norm__mult,axiom,
% 16.19/9.02 ! [X: real,Y: real] :
% 16.19/9.02 ( ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) )
% 16.19/9.02 = ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7476_norm__mult,axiom,
% 16.19/9.02 ! [X: complex,Y: complex] :
% 16.19/9.02 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) )
% 16.19/9.02 = ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7477_norm__divide,axiom,
% 16.19/9.02 ! [A: real,B: real] :
% 16.19/9.02 ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
% 16.19/9.02 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7478_norm__divide,axiom,
% 16.19/9.02 ! [A: complex,B: complex] :
% 16.19/9.02 ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 16.19/9.02 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7479_sum__norm__le,axiom,
% 16.19/9.02 ! [S: set_real,F: real > complex,G: real > real] :
% 16.19/9.02 ( ! [X4: real] :
% 16.19/9.02 ( ( member_real @ X4 @ S )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X4 ) ) @ ( G @ X4 ) ) )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F @ S ) ) @ ( groups8097168146408367636l_real @ G @ S ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7480_sum__norm__le,axiom,
% 16.19/9.02 ! [S: set_int,F: int > complex,G: int > real] :
% 16.19/9.02 ( ! [X4: int] :
% 16.19/9.02 ( ( member_int @ X4 @ S )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X4 ) ) @ ( G @ X4 ) ) )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups3049146728041665814omplex @ F @ S ) ) @ ( groups8778361861064173332t_real @ G @ S ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7481_sum__norm__le,axiom,
% 16.19/9.02 ! [S: set_set_nat,F: set_nat > complex,G: set_nat > real] :
% 16.19/9.02 ( ! [X4: set_nat] :
% 16.19/9.02 ( ( member_set_nat @ X4 @ S )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X4 ) ) @ ( G @ X4 ) ) )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups8255218700646806128omplex @ F @ S ) ) @ ( groups5107569545109728110t_real @ G @ S ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7482_sum__norm__le,axiom,
% 16.19/9.02 ! [S: set_nat,F: nat > complex,G: nat > real] :
% 16.19/9.02 ( ! [X4: nat] :
% 16.19/9.02 ( ( member_nat @ X4 @ S )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X4 ) ) @ ( G @ X4 ) ) )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ S ) ) @ ( groups6591440286371151544t_real @ G @ S ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7483_sum__norm__le,axiom,
% 16.19/9.02 ! [S: set_complex,F: complex > complex,G: complex > real] :
% 16.19/9.02 ( ! [X4: complex] :
% 16.19/9.02 ( ( member_complex @ X4 @ S )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X4 ) ) @ ( G @ X4 ) ) )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ S ) ) @ ( groups5808333547571424918x_real @ G @ S ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7484_sum__norm__le,axiom,
% 16.19/9.02 ! [S: set_nat,F: nat > real,G: nat > real] :
% 16.19/9.02 ( ! [X4: nat] :
% 16.19/9.02 ( ( member_nat @ X4 @ S )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X4 ) ) @ ( G @ X4 ) ) )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ S ) ) @ ( groups6591440286371151544t_real @ G @ S ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7485_norm__power,axiom,
% 16.19/9.02 ! [X: real,N2: nat] :
% 16.19/9.02 ( ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N2 ) )
% 16.19/9.02 = ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N2 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7486_norm__power,axiom,
% 16.19/9.02 ! [X: complex,N2: nat] :
% 16.19/9.02 ( ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N2 ) )
% 16.19/9.02 = ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N2 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7487_norm__sum,axiom,
% 16.19/9.02 ! [F: nat > complex,A2: set_nat] :
% 16.19/9.02 ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ A2 ) )
% 16.19/9.02 @ ( groups6591440286371151544t_real
% 16.19/9.02 @ ^ [I3: nat] : ( real_V1022390504157884413omplex @ ( F @ I3 ) )
% 16.19/9.02 @ A2 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7488_norm__sum,axiom,
% 16.19/9.02 ! [F: complex > complex,A2: set_complex] :
% 16.19/9.02 ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ A2 ) )
% 16.19/9.02 @ ( groups5808333547571424918x_real
% 16.19/9.02 @ ^ [I3: complex] : ( real_V1022390504157884413omplex @ ( F @ I3 ) )
% 16.19/9.02 @ A2 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7489_norm__sum,axiom,
% 16.19/9.02 ! [F: nat > real,A2: set_nat] :
% 16.19/9.02 ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
% 16.19/9.02 @ ( groups6591440286371151544t_real
% 16.19/9.02 @ ^ [I3: nat] : ( real_V7735802525324610683m_real @ ( F @ I3 ) )
% 16.19/9.02 @ A2 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7490_norm__uminus__minus,axiom,
% 16.19/9.02 ! [X: real,Y: real] :
% 16.19/9.02 ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ Y ) )
% 16.19/9.02 = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7491_norm__uminus__minus,axiom,
% 16.19/9.02 ! [X: complex,Y: complex] :
% 16.19/9.02 ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ Y ) )
% 16.19/9.02 = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7492_nonzero__norm__divide,axiom,
% 16.19/9.02 ! [B: real,A: real] :
% 16.19/9.02 ( ( B != zero_zero_real )
% 16.19/9.02 => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
% 16.19/9.02 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7493_nonzero__norm__divide,axiom,
% 16.19/9.02 ! [B: complex,A: complex] :
% 16.19/9.02 ( ( B != zero_zero_complex )
% 16.19/9.02 => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 16.19/9.02 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7494_power__eq__imp__eq__norm,axiom,
% 16.19/9.02 ! [W: real,N2: nat,Z: real] :
% 16.19/9.02 ( ( ( power_power_real @ W @ N2 )
% 16.19/9.02 = ( power_power_real @ Z @ N2 ) )
% 16.19/9.02 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.02 => ( ( real_V7735802525324610683m_real @ W )
% 16.19/9.02 = ( real_V7735802525324610683m_real @ Z ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7495_power__eq__imp__eq__norm,axiom,
% 16.19/9.02 ! [W: complex,N2: nat,Z: complex] :
% 16.19/9.02 ( ( ( power_power_complex @ W @ N2 )
% 16.19/9.02 = ( power_power_complex @ Z @ N2 ) )
% 16.19/9.02 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.02 => ( ( real_V1022390504157884413omplex @ W )
% 16.19/9.02 = ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7496_norm__mult__less,axiom,
% 16.19/9.02 ! [X: real,R: real,Y: real,S3: real] :
% 16.19/9.02 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R )
% 16.19/9.02 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S3 )
% 16.19/9.02 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ R @ S3 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7497_norm__mult__less,axiom,
% 16.19/9.02 ! [X: complex,R: real,Y: complex,S3: real] :
% 16.19/9.02 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R )
% 16.19/9.02 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S3 )
% 16.19/9.02 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ R @ S3 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7498_norm__mult__ineq,axiom,
% 16.19/9.02 ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7499_norm__mult__ineq,axiom,
% 16.19/9.02 ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7500_norm__add__less,axiom,
% 16.19/9.02 ! [X: real,R: real,Y: real,S3: real] :
% 16.19/9.02 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R )
% 16.19/9.02 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S3 )
% 16.19/9.02 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ R @ S3 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7501_norm__add__less,axiom,
% 16.19/9.02 ! [X: complex,R: real,Y: complex,S3: real] :
% 16.19/9.02 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R )
% 16.19/9.02 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S3 )
% 16.19/9.02 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ R @ S3 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7502_norm__triangle__lt,axiom,
% 16.19/9.02 ! [X: real,Y: real,E: real] :
% 16.19/9.02 ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
% 16.19/9.02 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7503_norm__triangle__lt,axiom,
% 16.19/9.02 ! [X: complex,Y: complex,E: real] :
% 16.19/9.02 ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
% 16.19/9.02 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7504_norm__power__ineq,axiom,
% 16.19/9.02 ! [X: real,N2: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N2 ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N2 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7505_norm__power__ineq,axiom,
% 16.19/9.02 ! [X: complex,N2: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N2 ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N2 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7506_norm__triangle__mono,axiom,
% 16.19/9.02 ! [A: real,R: real,B: real,S3: real] :
% 16.19/9.02 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R )
% 16.19/9.02 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S3 )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R @ S3 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7507_norm__triangle__mono,axiom,
% 16.19/9.02 ! [A: complex,R: real,B: complex,S3: real] :
% 16.19/9.02 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R )
% 16.19/9.02 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S3 )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R @ S3 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7508_norm__triangle__ineq,axiom,
% 16.19/9.02 ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7509_norm__triangle__ineq,axiom,
% 16.19/9.02 ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7510_norm__triangle__le,axiom,
% 16.19/9.02 ! [X: real,Y: real,E: real] :
% 16.19/9.02 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y ) ) @ E ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7511_norm__triangle__le,axiom,
% 16.19/9.02 ! [X: complex,Y: complex,E: real] :
% 16.19/9.02 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y ) ) @ E ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7512_norm__add__leD,axiom,
% 16.19/9.02 ! [A: real,B: real,C: real] :
% 16.19/9.02 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7513_norm__add__leD,axiom,
% 16.19/9.02 ! [A: complex,B: complex,C: real] :
% 16.19/9.02 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7514_norm__diff__triangle__less,axiom,
% 16.19/9.02 ! [X: real,Y: real,E1: real,Z: real,E22: real] :
% 16.19/9.02 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
% 16.19/9.02 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z ) ) @ E22 )
% 16.19/9.02 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7515_norm__diff__triangle__less,axiom,
% 16.19/9.02 ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
% 16.19/9.02 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
% 16.19/9.02 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z ) ) @ E22 )
% 16.19/9.02 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7516_norm__triangle__sub,axiom,
% 16.19/9.02 ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7517_norm__triangle__sub,axiom,
% 16.19/9.02 ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7518_norm__triangle__ineq4,axiom,
% 16.19/9.02 ! [A: real,B: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7519_norm__triangle__ineq4,axiom,
% 16.19/9.02 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7520_norm__diff__triangle__le,axiom,
% 16.19/9.02 ! [X: real,Y: real,E1: real,Z: real,E22: real] :
% 16.19/9.02 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E1 )
% 16.19/9.02 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z ) ) @ E22 )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7521_norm__diff__triangle__le,axiom,
% 16.19/9.02 ! [X: complex,Y: complex,E1: real,Z: complex,E22: real] :
% 16.19/9.02 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E1 )
% 16.19/9.02 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z ) ) @ E22 )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7522_norm__triangle__le__diff,axiom,
% 16.19/9.02 ! [X: real,Y: real,E: real] :
% 16.19/9.02 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y ) ) @ E ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7523_norm__triangle__le__diff,axiom,
% 16.19/9.02 ! [X: complex,Y: complex,E: real] :
% 16.19/9.02 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y ) ) @ E ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7524_norm__diff__ineq,axiom,
% 16.19/9.02 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7525_norm__diff__ineq,axiom,
% 16.19/9.02 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7526_norm__triangle__ineq2,axiom,
% 16.19/9.02 ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7527_norm__triangle__ineq2,axiom,
% 16.19/9.02 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7528_power__eq__1__iff,axiom,
% 16.19/9.02 ! [W: real,N2: nat] :
% 16.19/9.02 ( ( ( power_power_real @ W @ N2 )
% 16.19/9.02 = one_one_real )
% 16.19/9.02 => ( ( ( real_V7735802525324610683m_real @ W )
% 16.19/9.02 = one_one_real )
% 16.19/9.02 | ( N2 = zero_zero_nat ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7529_power__eq__1__iff,axiom,
% 16.19/9.02 ! [W: complex,N2: nat] :
% 16.19/9.02 ( ( ( power_power_complex @ W @ N2 )
% 16.19/9.02 = one_one_complex )
% 16.19/9.02 => ( ( ( real_V1022390504157884413omplex @ W )
% 16.19/9.02 = one_one_real )
% 16.19/9.02 | ( N2 = zero_zero_nat ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7530_norm__diff__triangle__ineq,axiom,
% 16.19/9.02 ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7531_norm__diff__triangle__ineq,axiom,
% 16.19/9.02 ! [A: complex,B: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ D ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7532_norm__triangle__ineq3,axiom,
% 16.19/9.02 ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7533_norm__triangle__ineq3,axiom,
% 16.19/9.02 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7534_square__norm__one,axiom,
% 16.19/9.02 ! [X: real] :
% 16.19/9.02 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.02 = one_one_real )
% 16.19/9.02 => ( ( real_V7735802525324610683m_real @ X )
% 16.19/9.02 = one_one_real ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7535_square__norm__one,axiom,
% 16.19/9.02 ! [X: complex] :
% 16.19/9.02 ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.02 = one_one_complex )
% 16.19/9.02 => ( ( real_V1022390504157884413omplex @ X )
% 16.19/9.02 = one_one_real ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7536_norm__power__diff,axiom,
% 16.19/9.02 ! [Z: real,W: real,M: nat] :
% 16.19/9.02 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
% 16.19/9.02 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W ) @ one_one_real )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( power_power_real @ Z @ M ) @ ( power_power_real @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Z @ W ) ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7537_norm__power__diff,axiom,
% 16.19/9.02 ! [Z: complex,W: complex,M: nat] :
% 16.19/9.02 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
% 16.19/9.02 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W ) @ one_one_real )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( power_power_complex @ Z @ M ) @ ( power_power_complex @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Z @ W ) ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7538_ln__series,axiom,
% 16.19/9.02 ! [X: real] :
% 16.19/9.02 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.02 => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 16.19/9.02 => ( ( ln_ln_real @ X )
% 16.19/9.02 = ( suminf_real
% 16.19/9.02 @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N ) ) ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7539_arctan__series,axiom,
% 16.19/9.02 ! [X: real] :
% 16.19/9.02 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 16.19/9.02 => ( ( arctan @ X )
% 16.19/9.02 = ( suminf_real
% 16.19/9.02 @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7540_round__unique,axiom,
% 16.19/9.02 ! [X: real,Y: int] :
% 16.19/9.02 ( ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y ) )
% 16.19/9.02 => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 16.19/9.02 => ( ( archim8280529875227126926d_real @ X )
% 16.19/9.02 = Y ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7541_round__unique,axiom,
% 16.19/9.02 ! [X: rat,Y: int] :
% 16.19/9.02 ( ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y ) )
% 16.19/9.02 => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
% 16.19/9.02 => ( ( archim7778729529865785530nd_rat @ X )
% 16.19/9.02 = Y ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7542_lemma__termdiff2,axiom,
% 16.19/9.02 ! [H2: rat,Z: rat,N2: nat] :
% 16.19/9.02 ( ( H2 != zero_zero_rat )
% 16.19/9.02 => ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ N2 ) @ ( power_power_rat @ Z @ N2 ) ) @ H2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
% 16.19/9.02 = ( times_times_rat @ H2
% 16.19/9.02 @ ( groups2906978787729119204at_rat
% 16.19/9.02 @ ^ [P5: nat] :
% 16.19/9.02 ( groups2906978787729119204at_rat
% 16.19/9.02 @ ^ [Q4: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ Q4 ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7543_lemma__termdiff2,axiom,
% 16.19/9.02 ! [H2: complex,Z: complex,N2: nat] :
% 16.19/9.02 ( ( H2 != zero_zero_complex )
% 16.19/9.02 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
% 16.19/9.02 = ( times_times_complex @ H2
% 16.19/9.02 @ ( groups2073611262835488442omplex
% 16.19/9.02 @ ^ [P5: nat] :
% 16.19/9.02 ( groups2073611262835488442omplex
% 16.19/9.02 @ ^ [Q4: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ Q4 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7544_lemma__termdiff2,axiom,
% 16.19/9.02 ! [H2: real,Z: real,N2: nat] :
% 16.19/9.02 ( ( H2 != zero_zero_real )
% 16.19/9.02 => ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N2 ) @ ( power_power_real @ Z @ N2 ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
% 16.19/9.02 = ( times_times_real @ H2
% 16.19/9.02 @ ( groups6591440286371151544t_real
% 16.19/9.02 @ ^ [P5: nat] :
% 16.19/9.02 ( groups6591440286371151544t_real
% 16.19/9.02 @ ^ [Q4: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ Q4 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q4 ) ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7545_summable__arctan__series,axiom,
% 16.19/9.02 ! [X: real] :
% 16.19/9.02 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 16.19/9.02 => ( summable_real
% 16.19/9.02 @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7546_lessThan__iff,axiom,
% 16.19/9.02 ! [I2: set_nat,K: set_nat] :
% 16.19/9.02 ( ( member_set_nat @ I2 @ ( set_or890127255671739683et_nat @ K ) )
% 16.19/9.02 = ( ord_less_set_nat @ I2 @ K ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7547_lessThan__iff,axiom,
% 16.19/9.02 ! [I2: rat,K: rat] :
% 16.19/9.02 ( ( member_rat @ I2 @ ( set_ord_lessThan_rat @ K ) )
% 16.19/9.02 = ( ord_less_rat @ I2 @ K ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7548_lessThan__iff,axiom,
% 16.19/9.02 ! [I2: num,K: num] :
% 16.19/9.02 ( ( member_num @ I2 @ ( set_ord_lessThan_num @ K ) )
% 16.19/9.02 = ( ord_less_num @ I2 @ K ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7549_lessThan__iff,axiom,
% 16.19/9.02 ! [I2: int,K: int] :
% 16.19/9.02 ( ( member_int @ I2 @ ( set_ord_lessThan_int @ K ) )
% 16.19/9.02 = ( ord_less_int @ I2 @ K ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7550_lessThan__iff,axiom,
% 16.19/9.02 ! [I2: nat,K: nat] :
% 16.19/9.02 ( ( member_nat @ I2 @ ( set_ord_lessThan_nat @ K ) )
% 16.19/9.02 = ( ord_less_nat @ I2 @ K ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7551_lessThan__iff,axiom,
% 16.19/9.02 ! [I2: real,K: real] :
% 16.19/9.02 ( ( member_real @ I2 @ ( set_or5984915006950818249n_real @ K ) )
% 16.19/9.02 = ( ord_less_real @ I2 @ K ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7552_lessThan__subset__iff,axiom,
% 16.19/9.02 ! [X: rat,Y: rat] :
% 16.19/9.02 ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X ) @ ( set_ord_lessThan_rat @ Y ) )
% 16.19/9.02 = ( ord_less_eq_rat @ X @ Y ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7553_lessThan__subset__iff,axiom,
% 16.19/9.02 ! [X: num,Y: num] :
% 16.19/9.02 ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X ) @ ( set_ord_lessThan_num @ Y ) )
% 16.19/9.02 = ( ord_less_eq_num @ X @ Y ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7554_lessThan__subset__iff,axiom,
% 16.19/9.02 ! [X: int,Y: int] :
% 16.19/9.02 ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y ) )
% 16.19/9.02 = ( ord_less_eq_int @ X @ Y ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7555_lessThan__subset__iff,axiom,
% 16.19/9.02 ! [X: nat,Y: nat] :
% 16.19/9.02 ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
% 16.19/9.02 = ( ord_less_eq_nat @ X @ Y ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7556_lessThan__subset__iff,axiom,
% 16.19/9.02 ! [X: real,Y: real] :
% 16.19/9.02 ( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X ) @ ( set_or5984915006950818249n_real @ Y ) )
% 16.19/9.02 = ( ord_less_eq_real @ X @ Y ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7557_round__numeral,axiom,
% 16.19/9.02 ! [N2: num] :
% 16.19/9.02 ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N2 ) )
% 16.19/9.02 = ( numeral_numeral_int @ N2 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7558_round__numeral,axiom,
% 16.19/9.02 ! [N2: num] :
% 16.19/9.02 ( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N2 ) )
% 16.19/9.02 = ( numeral_numeral_int @ N2 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7559_round__1,axiom,
% 16.19/9.02 ( ( archim8280529875227126926d_real @ one_one_real )
% 16.19/9.02 = one_one_int ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7560_round__1,axiom,
% 16.19/9.02 ( ( archim7778729529865785530nd_rat @ one_one_rat )
% 16.19/9.02 = one_one_int ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7561_sum_OlessThan__Suc,axiom,
% 16.19/9.02 ! [G: nat > rat,N2: nat] :
% 16.19/9.02 ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 16.19/9.02 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7562_sum_OlessThan__Suc,axiom,
% 16.19/9.02 ! [G: nat > int,N2: nat] :
% 16.19/9.02 ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 16.19/9.02 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7563_sum_OlessThan__Suc,axiom,
% 16.19/9.02 ! [G: nat > nat,N2: nat] :
% 16.19/9.02 ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 16.19/9.02 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7564_sum_OlessThan__Suc,axiom,
% 16.19/9.02 ! [G: nat > real,N2: nat] :
% 16.19/9.02 ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 16.19/9.02 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7565_round__neg__numeral,axiom,
% 16.19/9.02 ! [N2: num] :
% 16.19/9.02 ( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 16.19/9.02 = ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7566_round__neg__numeral,axiom,
% 16.19/9.02 ! [N2: num] :
% 16.19/9.02 ( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 16.19/9.02 = ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7567_powser__zero,axiom,
% 16.19/9.02 ! [F: nat > complex] :
% 16.19/9.02 ( ( suminf_complex
% 16.19/9.02 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) ) )
% 16.19/9.02 = ( F @ zero_zero_nat ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7568_powser__zero,axiom,
% 16.19/9.02 ! [F: nat > real] :
% 16.19/9.02 ( ( suminf_real
% 16.19/9.02 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ zero_zero_real @ N ) ) )
% 16.19/9.02 = ( F @ zero_zero_nat ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7569_lessThan__def,axiom,
% 16.19/9.02 ( set_or890127255671739683et_nat
% 16.19/9.02 = ( ^ [U2: set_nat] :
% 16.19/9.02 ( collect_set_nat
% 16.19/9.02 @ ^ [X2: set_nat] : ( ord_less_set_nat @ X2 @ U2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7570_lessThan__def,axiom,
% 16.19/9.02 ( set_ord_lessThan_rat
% 16.19/9.02 = ( ^ [U2: rat] :
% 16.19/9.02 ( collect_rat
% 16.19/9.02 @ ^ [X2: rat] : ( ord_less_rat @ X2 @ U2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7571_lessThan__def,axiom,
% 16.19/9.02 ( set_ord_lessThan_num
% 16.19/9.02 = ( ^ [U2: num] :
% 16.19/9.02 ( collect_num
% 16.19/9.02 @ ^ [X2: num] : ( ord_less_num @ X2 @ U2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7572_lessThan__def,axiom,
% 16.19/9.02 ( set_ord_lessThan_int
% 16.19/9.02 = ( ^ [U2: int] :
% 16.19/9.02 ( collect_int
% 16.19/9.02 @ ^ [X2: int] : ( ord_less_int @ X2 @ U2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7573_lessThan__def,axiom,
% 16.19/9.02 ( set_ord_lessThan_nat
% 16.19/9.02 = ( ^ [U2: nat] :
% 16.19/9.02 ( collect_nat
% 16.19/9.02 @ ^ [X2: nat] : ( ord_less_nat @ X2 @ U2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7574_lessThan__def,axiom,
% 16.19/9.02 ( set_or5984915006950818249n_real
% 16.19/9.02 = ( ^ [U2: real] :
% 16.19/9.02 ( collect_real
% 16.19/9.02 @ ^ [X2: real] : ( ord_less_real @ X2 @ U2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7575_powser__insidea,axiom,
% 16.19/9.02 ! [F: nat > real,X: real,Z: real] :
% 16.19/9.02 ( ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ X @ N ) ) )
% 16.19/9.02 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X ) )
% 16.19/9.02 => ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7576_powser__insidea,axiom,
% 16.19/9.02 ! [F: nat > complex,X: complex,Z: complex] :
% 16.19/9.02 ( ( summable_complex
% 16.19/9.02 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ X @ N ) ) )
% 16.19/9.02 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X ) )
% 16.19/9.02 => ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7577_lessThan__strict__subset__iff,axiom,
% 16.19/9.02 ! [M: rat,N2: rat] :
% 16.19/9.02 ( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N2 ) )
% 16.19/9.02 = ( ord_less_rat @ M @ N2 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7578_lessThan__strict__subset__iff,axiom,
% 16.19/9.02 ! [M: num,N2: num] :
% 16.19/9.02 ( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N2 ) )
% 16.19/9.02 = ( ord_less_num @ M @ N2 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7579_lessThan__strict__subset__iff,axiom,
% 16.19/9.02 ! [M: int,N2: int] :
% 16.19/9.02 ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N2 ) )
% 16.19/9.02 = ( ord_less_int @ M @ N2 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7580_lessThan__strict__subset__iff,axiom,
% 16.19/9.02 ! [M: nat,N2: nat] :
% 16.19/9.02 ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.02 = ( ord_less_nat @ M @ N2 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7581_lessThan__strict__subset__iff,axiom,
% 16.19/9.02 ! [M: real,N2: real] :
% 16.19/9.02 ( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N2 ) )
% 16.19/9.02 = ( ord_less_real @ M @ N2 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7582_sum__pos__lt__pair,axiom,
% 16.19/9.02 ! [F: nat > real,K: nat] :
% 16.19/9.02 ( ( summable_real @ F )
% 16.19/9.02 => ( ! [D4: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D4 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D4 ) @ one_one_nat ) ) ) ) )
% 16.19/9.02 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7583_round__mono,axiom,
% 16.19/9.02 ! [X: rat,Y: rat] :
% 16.19/9.02 ( ( ord_less_eq_rat @ X @ Y )
% 16.19/9.02 => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X ) @ ( archim7778729529865785530nd_rat @ Y ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7584_sum_Onat__diff__reindex,axiom,
% 16.19/9.02 ! [G: nat > nat,N2: nat] :
% 16.19/9.02 ( ( groups3542108847815614940at_nat
% 16.19/9.02 @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.02 = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7585_sum_Onat__diff__reindex,axiom,
% 16.19/9.02 ! [G: nat > real,N2: nat] :
% 16.19/9.02 ( ( groups6591440286371151544t_real
% 16.19/9.02 @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.02 = ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7586_sum__diff__distrib,axiom,
% 16.19/9.02 ! [Q: real > nat,P: real > nat,N2: real] :
% 16.19/9.02 ( ! [X4: real] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P @ X4 ) )
% 16.19/9.02 => ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P @ ( set_or5984915006950818249n_real @ N2 ) ) @ ( groups1935376822645274424al_nat @ Q @ ( set_or5984915006950818249n_real @ N2 ) ) )
% 16.19/9.02 = ( groups1935376822645274424al_nat
% 16.19/9.02 @ ^ [X2: real] : ( minus_minus_nat @ ( P @ X2 ) @ ( Q @ X2 ) )
% 16.19/9.02 @ ( set_or5984915006950818249n_real @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7587_sum__diff__distrib,axiom,
% 16.19/9.02 ! [Q: nat > nat,P: nat > nat,N2: nat] :
% 16.19/9.02 ( ! [X4: nat] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P @ X4 ) )
% 16.19/9.02 => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N2 ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N2 ) ) )
% 16.19/9.02 = ( groups3542108847815614940at_nat
% 16.19/9.02 @ ^ [X2: nat] : ( minus_minus_nat @ ( P @ X2 ) @ ( Q @ X2 ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7588_powser__inside,axiom,
% 16.19/9.02 ! [F: nat > real,X: real,Z: real] :
% 16.19/9.02 ( ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ X @ N ) ) )
% 16.19/9.02 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X ) )
% 16.19/9.02 => ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7589_powser__inside,axiom,
% 16.19/9.02 ! [F: nat > complex,X: complex,Z: complex] :
% 16.19/9.02 ( ( summable_complex
% 16.19/9.02 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ X @ N ) ) )
% 16.19/9.02 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X ) )
% 16.19/9.02 => ( summable_complex
% 16.19/9.02 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7590_sum_OlessThan__Suc__shift,axiom,
% 16.19/9.02 ! [G: nat > rat,N2: nat] :
% 16.19/9.02 ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 16.19/9.02 = ( plus_plus_rat @ ( G @ zero_zero_nat )
% 16.19/9.02 @ ( groups2906978787729119204at_rat
% 16.19/9.02 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7591_sum_OlessThan__Suc__shift,axiom,
% 16.19/9.02 ! [G: nat > int,N2: nat] :
% 16.19/9.02 ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 16.19/9.02 = ( plus_plus_int @ ( G @ zero_zero_nat )
% 16.19/9.02 @ ( groups3539618377306564664at_int
% 16.19/9.02 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7592_sum_OlessThan__Suc__shift,axiom,
% 16.19/9.02 ! [G: nat > nat,N2: nat] :
% 16.19/9.02 ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 16.19/9.02 = ( plus_plus_nat @ ( G @ zero_zero_nat )
% 16.19/9.02 @ ( groups3542108847815614940at_nat
% 16.19/9.02 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7593_sum_OlessThan__Suc__shift,axiom,
% 16.19/9.02 ! [G: nat > real,N2: nat] :
% 16.19/9.02 ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 16.19/9.02 = ( plus_plus_real @ ( G @ zero_zero_nat )
% 16.19/9.02 @ ( groups6591440286371151544t_real
% 16.19/9.02 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7594_sum__lessThan__telescope,axiom,
% 16.19/9.02 ! [F: nat > rat,M: nat] :
% 16.19/9.02 ( ( groups2906978787729119204at_rat
% 16.19/9.02 @ ^ [N: nat] : ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ M ) )
% 16.19/9.02 = ( minus_minus_rat @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7595_sum__lessThan__telescope,axiom,
% 16.19/9.02 ! [F: nat > int,M: nat] :
% 16.19/9.02 ( ( groups3539618377306564664at_int
% 16.19/9.02 @ ^ [N: nat] : ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ M ) )
% 16.19/9.02 = ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7596_sum__lessThan__telescope,axiom,
% 16.19/9.02 ! [F: nat > real,M: nat] :
% 16.19/9.02 ( ( groups6591440286371151544t_real
% 16.19/9.02 @ ^ [N: nat] : ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ M ) )
% 16.19/9.02 = ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7597_sum__lessThan__telescope_H,axiom,
% 16.19/9.02 ! [F: nat > rat,M: nat] :
% 16.19/9.02 ( ( groups2906978787729119204at_rat
% 16.19/9.02 @ ^ [N: nat] : ( minus_minus_rat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ M ) )
% 16.19/9.02 = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7598_sum__lessThan__telescope_H,axiom,
% 16.19/9.02 ! [F: nat > int,M: nat] :
% 16.19/9.02 ( ( groups3539618377306564664at_int
% 16.19/9.02 @ ^ [N: nat] : ( minus_minus_int @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ M ) )
% 16.19/9.02 = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7599_sum__lessThan__telescope_H,axiom,
% 16.19/9.02 ! [F: nat > real,M: nat] :
% 16.19/9.02 ( ( groups6591440286371151544t_real
% 16.19/9.02 @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ M ) )
% 16.19/9.02 = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7600_sumr__diff__mult__const2,axiom,
% 16.19/9.02 ! [F: nat > rat,N2: nat,R: rat] :
% 16.19/9.02 ( ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ R ) )
% 16.19/9.02 = ( groups2906978787729119204at_rat
% 16.19/9.02 @ ^ [I3: nat] : ( minus_minus_rat @ ( F @ I3 ) @ R )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7601_sumr__diff__mult__const2,axiom,
% 16.19/9.02 ! [F: nat > int,N2: nat,R: int] :
% 16.19/9.02 ( ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ R ) )
% 16.19/9.02 = ( groups3539618377306564664at_int
% 16.19/9.02 @ ^ [I3: nat] : ( minus_minus_int @ ( F @ I3 ) @ R )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7602_sumr__diff__mult__const2,axiom,
% 16.19/9.02 ! [F: nat > complex,N2: nat,R: complex] :
% 16.19/9.02 ( ( minus_minus_complex @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ R ) )
% 16.19/9.02 = ( groups2073611262835488442omplex
% 16.19/9.02 @ ^ [I3: nat] : ( minus_minus_complex @ ( F @ I3 ) @ R )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7603_sumr__diff__mult__const2,axiom,
% 16.19/9.02 ! [F: nat > real,N2: nat,R: real] :
% 16.19/9.02 ( ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ R ) )
% 16.19/9.02 = ( groups6591440286371151544t_real
% 16.19/9.02 @ ^ [I3: nat] : ( minus_minus_real @ ( F @ I3 ) @ R )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7604_sum_OatLeast1__atMost__eq,axiom,
% 16.19/9.02 ! [G: nat > nat,N2: nat] :
% 16.19/9.02 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
% 16.19/9.02 = ( groups3542108847815614940at_nat
% 16.19/9.02 @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7605_sum_OatLeast1__atMost__eq,axiom,
% 16.19/9.02 ! [G: nat > real,N2: nat] :
% 16.19/9.02 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
% 16.19/9.02 = ( groups6591440286371151544t_real
% 16.19/9.02 @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7606_power__diff__1__eq,axiom,
% 16.19/9.02 ! [X: complex,N2: nat] :
% 16.19/9.02 ( ( minus_minus_complex @ ( power_power_complex @ X @ N2 ) @ one_one_complex )
% 16.19/9.02 = ( times_times_complex @ ( minus_minus_complex @ X @ one_one_complex ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7607_power__diff__1__eq,axiom,
% 16.19/9.02 ! [X: rat,N2: nat] :
% 16.19/9.02 ( ( minus_minus_rat @ ( power_power_rat @ X @ N2 ) @ one_one_rat )
% 16.19/9.02 = ( times_times_rat @ ( minus_minus_rat @ X @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7608_power__diff__1__eq,axiom,
% 16.19/9.02 ! [X: int,N2: nat] :
% 16.19/9.02 ( ( minus_minus_int @ ( power_power_int @ X @ N2 ) @ one_one_int )
% 16.19/9.02 = ( times_times_int @ ( minus_minus_int @ X @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7609_power__diff__1__eq,axiom,
% 16.19/9.02 ! [X: real,N2: nat] :
% 16.19/9.02 ( ( minus_minus_real @ ( power_power_real @ X @ N2 ) @ one_one_real )
% 16.19/9.02 = ( times_times_real @ ( minus_minus_real @ X @ one_one_real ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7610_one__diff__power__eq,axiom,
% 16.19/9.02 ! [X: complex,N2: nat] :
% 16.19/9.02 ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N2 ) )
% 16.19/9.02 = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7611_one__diff__power__eq,axiom,
% 16.19/9.02 ! [X: rat,N2: nat] :
% 16.19/9.02 ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N2 ) )
% 16.19/9.02 = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7612_one__diff__power__eq,axiom,
% 16.19/9.02 ! [X: int,N2: nat] :
% 16.19/9.02 ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N2 ) )
% 16.19/9.02 = ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7613_one__diff__power__eq,axiom,
% 16.19/9.02 ! [X: real,N2: nat] :
% 16.19/9.02 ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N2 ) )
% 16.19/9.02 = ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7614_geometric__sum,axiom,
% 16.19/9.02 ! [X: complex,N2: nat] :
% 16.19/9.02 ( ( X != one_one_complex )
% 16.19/9.02 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.02 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ N2 ) @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7615_geometric__sum,axiom,
% 16.19/9.02 ! [X: rat,N2: nat] :
% 16.19/9.02 ( ( X != one_one_rat )
% 16.19/9.02 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.02 = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ N2 ) @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7616_geometric__sum,axiom,
% 16.19/9.02 ! [X: real,N2: nat] :
% 16.19/9.02 ( ( X != one_one_real )
% 16.19/9.02 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.02 = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ N2 ) @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7617_round__diff__minimal,axiom,
% 16.19/9.02 ! [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 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7618_round__diff__minimal,axiom,
% 16.19/9.02 ! [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 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7619_sum__gp__strict,axiom,
% 16.19/9.02 ! [X: rat,N2: nat] :
% 16.19/9.02 ( ( ( X = one_one_rat )
% 16.19/9.02 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.02 = ( semiri681578069525770553at_rat @ N2 ) ) )
% 16.19/9.02 & ( ( X != one_one_rat )
% 16.19/9.02 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.02 = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N2 ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7620_sum__gp__strict,axiom,
% 16.19/9.02 ! [X: complex,N2: nat] :
% 16.19/9.02 ( ( ( X = one_one_complex )
% 16.19/9.02 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.02 = ( semiri8010041392384452111omplex @ N2 ) ) )
% 16.19/9.02 & ( ( X != one_one_complex )
% 16.19/9.02 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.02 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N2 ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7621_sum__gp__strict,axiom,
% 16.19/9.02 ! [X: real,N2: nat] :
% 16.19/9.02 ( ( ( X = one_one_real )
% 16.19/9.02 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.02 = ( semiri5074537144036343181t_real @ N2 ) ) )
% 16.19/9.02 & ( ( X != one_one_real )
% 16.19/9.02 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.02 = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N2 ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7622_lemma__termdiff1,axiom,
% 16.19/9.02 ! [Z: complex,H2: complex,M: nat] :
% 16.19/9.02 ( ( groups2073611262835488442omplex
% 16.19/9.02 @ ^ [P5: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_complex @ Z @ P5 ) ) @ ( power_power_complex @ Z @ M ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ M ) )
% 16.19/9.02 = ( groups2073611262835488442omplex
% 16.19/9.02 @ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ Z @ P5 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7623_lemma__termdiff1,axiom,
% 16.19/9.02 ! [Z: rat,H2: rat,M: nat] :
% 16.19/9.02 ( ( groups2906978787729119204at_rat
% 16.19/9.02 @ ^ [P5: nat] : ( minus_minus_rat @ ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_rat @ Z @ P5 ) ) @ ( power_power_rat @ Z @ M ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ M ) )
% 16.19/9.02 = ( groups2906978787729119204at_rat
% 16.19/9.02 @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ Z @ P5 ) @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7624_lemma__termdiff1,axiom,
% 16.19/9.02 ! [Z: int,H2: int,M: nat] :
% 16.19/9.02 ( ( groups3539618377306564664at_int
% 16.19/9.02 @ ^ [P5: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_int @ Z @ P5 ) ) @ ( power_power_int @ Z @ M ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ M ) )
% 16.19/9.02 = ( groups3539618377306564664at_int
% 16.19/9.02 @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ Z @ P5 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_int @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7625_lemma__termdiff1,axiom,
% 16.19/9.02 ! [Z: real,H2: real,M: nat] :
% 16.19/9.02 ( ( groups6591440286371151544t_real
% 16.19/9.02 @ ^ [P5: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_real @ Z @ P5 ) ) @ ( power_power_real @ Z @ M ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ M ) )
% 16.19/9.02 = ( groups6591440286371151544t_real
% 16.19/9.02 @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ Z @ P5 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_real @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7626_diff__power__eq__sum,axiom,
% 16.19/9.02 ! [X: complex,N2: nat,Y: complex] :
% 16.19/9.02 ( ( minus_minus_complex @ ( power_power_complex @ X @ ( suc @ N2 ) ) @ ( power_power_complex @ Y @ ( suc @ N2 ) ) )
% 16.19/9.02 = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
% 16.19/9.02 @ ( groups2073611262835488442omplex
% 16.19/9.02 @ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ X @ P5 ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ N2 @ P5 ) ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7627_diff__power__eq__sum,axiom,
% 16.19/9.02 ! [X: rat,N2: nat,Y: rat] :
% 16.19/9.02 ( ( minus_minus_rat @ ( power_power_rat @ X @ ( suc @ N2 ) ) @ ( power_power_rat @ Y @ ( suc @ N2 ) ) )
% 16.19/9.02 = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
% 16.19/9.02 @ ( groups2906978787729119204at_rat
% 16.19/9.02 @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ X @ P5 ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ N2 @ P5 ) ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7628_diff__power__eq__sum,axiom,
% 16.19/9.02 ! [X: int,N2: nat,Y: int] :
% 16.19/9.02 ( ( minus_minus_int @ ( power_power_int @ X @ ( suc @ N2 ) ) @ ( power_power_int @ Y @ ( suc @ N2 ) ) )
% 16.19/9.02 = ( times_times_int @ ( minus_minus_int @ X @ Y )
% 16.19/9.02 @ ( groups3539618377306564664at_int
% 16.19/9.02 @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ X @ P5 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ N2 @ P5 ) ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7629_diff__power__eq__sum,axiom,
% 16.19/9.02 ! [X: real,N2: nat,Y: real] :
% 16.19/9.02 ( ( minus_minus_real @ ( power_power_real @ X @ ( suc @ N2 ) ) @ ( power_power_real @ Y @ ( suc @ N2 ) ) )
% 16.19/9.02 = ( times_times_real @ ( minus_minus_real @ X @ Y )
% 16.19/9.02 @ ( groups6591440286371151544t_real
% 16.19/9.02 @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ X @ P5 ) @ ( power_power_real @ Y @ ( minus_minus_nat @ N2 @ P5 ) ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7630_power__diff__sumr2,axiom,
% 16.19/9.02 ! [X: complex,N2: nat,Y: complex] :
% 16.19/9.02 ( ( minus_minus_complex @ ( power_power_complex @ X @ N2 ) @ ( power_power_complex @ Y @ N2 ) )
% 16.19/9.02 = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
% 16.19/9.02 @ ( groups2073611262835488442omplex
% 16.19/9.02 @ ^ [I3: nat] : ( times_times_complex @ ( power_power_complex @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_complex @ X @ I3 ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7631_power__diff__sumr2,axiom,
% 16.19/9.02 ! [X: rat,N2: nat,Y: rat] :
% 16.19/9.02 ( ( minus_minus_rat @ ( power_power_rat @ X @ N2 ) @ ( power_power_rat @ Y @ N2 ) )
% 16.19/9.02 = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
% 16.19/9.02 @ ( groups2906978787729119204at_rat
% 16.19/9.02 @ ^ [I3: nat] : ( times_times_rat @ ( power_power_rat @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_rat @ X @ I3 ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7632_power__diff__sumr2,axiom,
% 16.19/9.02 ! [X: int,N2: nat,Y: int] :
% 16.19/9.02 ( ( minus_minus_int @ ( power_power_int @ X @ N2 ) @ ( power_power_int @ Y @ N2 ) )
% 16.19/9.02 = ( times_times_int @ ( minus_minus_int @ X @ Y )
% 16.19/9.02 @ ( groups3539618377306564664at_int
% 16.19/9.02 @ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_int @ X @ I3 ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7633_power__diff__sumr2,axiom,
% 16.19/9.02 ! [X: real,N2: nat,Y: real] :
% 16.19/9.02 ( ( minus_minus_real @ ( power_power_real @ X @ N2 ) @ ( power_power_real @ Y @ N2 ) )
% 16.19/9.02 = ( times_times_real @ ( minus_minus_real @ X @ Y )
% 16.19/9.02 @ ( groups6591440286371151544t_real
% 16.19/9.02 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_real @ X @ I3 ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7634_real__sum__nat__ivl__bounded2,axiom,
% 16.19/9.02 ! [N2: nat,F: nat > rat,K5: rat,K: nat] :
% 16.19/9.02 ( ! [P7: nat] :
% 16.19/9.02 ( ( ord_less_nat @ P7 @ N2 )
% 16.19/9.02 => ( ord_less_eq_rat @ ( F @ P7 ) @ K5 ) )
% 16.19/9.02 => ( ( ord_less_eq_rat @ zero_zero_rat @ K5 )
% 16.19/9.02 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ K5 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7635_real__sum__nat__ivl__bounded2,axiom,
% 16.19/9.02 ! [N2: nat,F: nat > int,K5: int,K: nat] :
% 16.19/9.02 ( ! [P7: nat] :
% 16.19/9.02 ( ( ord_less_nat @ P7 @ N2 )
% 16.19/9.02 => ( ord_less_eq_int @ ( F @ P7 ) @ K5 ) )
% 16.19/9.02 => ( ( ord_less_eq_int @ zero_zero_int @ K5 )
% 16.19/9.02 => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ K5 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7636_real__sum__nat__ivl__bounded2,axiom,
% 16.19/9.02 ! [N2: nat,F: nat > nat,K5: nat,K: nat] :
% 16.19/9.02 ( ! [P7: nat] :
% 16.19/9.02 ( ( ord_less_nat @ P7 @ N2 )
% 16.19/9.02 => ( ord_less_eq_nat @ ( F @ P7 ) @ K5 ) )
% 16.19/9.02 => ( ( ord_less_eq_nat @ zero_zero_nat @ K5 )
% 16.19/9.02 => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ K5 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7637_real__sum__nat__ivl__bounded2,axiom,
% 16.19/9.02 ! [N2: nat,F: nat > real,K5: real,K: nat] :
% 16.19/9.02 ( ! [P7: nat] :
% 16.19/9.02 ( ( ord_less_nat @ P7 @ N2 )
% 16.19/9.02 => ( ord_less_eq_real @ ( F @ P7 ) @ K5 ) )
% 16.19/9.02 => ( ( ord_less_eq_real @ zero_zero_real @ K5 )
% 16.19/9.02 => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ K5 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7638_one__diff__power__eq_H,axiom,
% 16.19/9.02 ! [X: complex,N2: nat] :
% 16.19/9.02 ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N2 ) )
% 16.19/9.02 = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X )
% 16.19/9.02 @ ( groups2073611262835488442omplex
% 16.19/9.02 @ ^ [I3: nat] : ( power_power_complex @ X @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7639_one__diff__power__eq_H,axiom,
% 16.19/9.02 ! [X: rat,N2: nat] :
% 16.19/9.02 ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N2 ) )
% 16.19/9.02 = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X )
% 16.19/9.02 @ ( groups2906978787729119204at_rat
% 16.19/9.02 @ ^ [I3: nat] : ( power_power_rat @ X @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7640_one__diff__power__eq_H,axiom,
% 16.19/9.02 ! [X: int,N2: nat] :
% 16.19/9.02 ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N2 ) )
% 16.19/9.02 = ( times_times_int @ ( minus_minus_int @ one_one_int @ X )
% 16.19/9.02 @ ( groups3539618377306564664at_int
% 16.19/9.02 @ ^ [I3: nat] : ( power_power_int @ X @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7641_one__diff__power__eq_H,axiom,
% 16.19/9.02 ! [X: real,N2: nat] :
% 16.19/9.02 ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N2 ) )
% 16.19/9.02 = ( times_times_real @ ( minus_minus_real @ one_one_real @ X )
% 16.19/9.02 @ ( groups6591440286371151544t_real
% 16.19/9.02 @ ^ [I3: nat] : ( power_power_real @ X @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7642_sum__split__even__odd,axiom,
% 16.19/9.02 ! [F: nat > real,G: nat > real,N2: nat] :
% 16.19/9.02 ( ( groups6591440286371151544t_real
% 16.19/9.02 @ ^ [I3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( F @ I3 ) @ ( G @ I3 ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 16.19/9.02 = ( plus_plus_real
% 16.19/9.02 @ ( groups6591440286371151544t_real
% 16.19/9.02 @ ^ [I3: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.02 @ ( groups6591440286371151544t_real
% 16.19/9.02 @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ one_one_nat ) )
% 16.19/9.02 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7643_of__int__round__le,axiom,
% 16.19/9.02 ! [X: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7644_of__int__round__le,axiom,
% 16.19/9.02 ! [X: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7645_of__int__round__ge,axiom,
% 16.19/9.02 ! [X: real] : ( ord_less_eq_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7646_of__int__round__ge,axiom,
% 16.19/9.02 ! [X: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7647_of__int__round__gt,axiom,
% 16.19/9.02 ! [X: real] : ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7648_of__int__round__gt,axiom,
% 16.19/9.02 ! [X: rat] : ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7649_of__int__round__abs__le,axiom,
% 16.19/9.02 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X ) ) @ X ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7650_of__int__round__abs__le,axiom,
% 16.19/9.02 ! [X: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X ) ) @ X ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7651_round__unique_H,axiom,
% 16.19/9.02 ! [X: real,N2: int] :
% 16.19/9.02 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ ( ring_1_of_int_real @ N2 ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.02 => ( ( archim8280529875227126926d_real @ X )
% 16.19/9.02 = N2 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7652_round__unique_H,axiom,
% 16.19/9.02 ! [X: rat,N2: int] :
% 16.19/9.02 ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ ( ring_1_of_int_rat @ N2 ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 16.19/9.02 => ( ( archim7778729529865785530nd_rat @ X )
% 16.19/9.02 = N2 ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7653_summable__geometric__iff,axiom,
% 16.19/9.02 ! [C: real] :
% 16.19/9.02 ( ( summable_real @ ( power_power_real @ C ) )
% 16.19/9.02 = ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7654_summable__geometric__iff,axiom,
% 16.19/9.02 ! [C: complex] :
% 16.19/9.02 ( ( summable_complex @ ( power_power_complex @ C ) )
% 16.19/9.02 = ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7655_summable__If__finite,axiom,
% 16.19/9.02 ! [P: nat > $o,F: nat > complex] :
% 16.19/9.02 ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 16.19/9.02 => ( summable_complex
% 16.19/9.02 @ ^ [R5: nat] : ( if_complex @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_complex ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7656_summable__If__finite,axiom,
% 16.19/9.02 ! [P: nat > $o,F: nat > real] :
% 16.19/9.02 ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 16.19/9.02 => ( summable_real
% 16.19/9.02 @ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7657_summable__If__finite,axiom,
% 16.19/9.02 ! [P: nat > $o,F: nat > nat] :
% 16.19/9.02 ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 16.19/9.02 => ( summable_nat
% 16.19/9.02 @ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7658_summable__If__finite,axiom,
% 16.19/9.02 ! [P: nat > $o,F: nat > int] :
% 16.19/9.02 ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 16.19/9.02 => ( summable_int
% 16.19/9.02 @ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7659_summable__If__finite__set,axiom,
% 16.19/9.02 ! [A2: set_nat,F: nat > complex] :
% 16.19/9.02 ( ( finite_finite_nat @ A2 )
% 16.19/9.02 => ( summable_complex
% 16.19/9.02 @ ^ [R5: nat] : ( if_complex @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_complex ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7660_summable__If__finite__set,axiom,
% 16.19/9.02 ! [A2: set_nat,F: nat > real] :
% 16.19/9.02 ( ( finite_finite_nat @ A2 )
% 16.19/9.02 => ( summable_real
% 16.19/9.02 @ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7661_summable__If__finite__set,axiom,
% 16.19/9.02 ! [A2: set_nat,F: nat > nat] :
% 16.19/9.02 ( ( finite_finite_nat @ A2 )
% 16.19/9.02 => ( summable_nat
% 16.19/9.02 @ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7662_summable__If__finite__set,axiom,
% 16.19/9.02 ! [A2: set_nat,F: nat > int] :
% 16.19/9.02 ( ( finite_finite_nat @ A2 )
% 16.19/9.02 => ( summable_int
% 16.19/9.02 @ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7663_summable__divide__iff,axiom,
% 16.19/9.02 ! [F: nat > complex,C: complex] :
% 16.19/9.02 ( ( summable_complex
% 16.19/9.02 @ ^ [N: nat] : ( divide1717551699836669952omplex @ ( F @ N ) @ C ) )
% 16.19/9.02 = ( ( C = zero_zero_complex )
% 16.19/9.02 | ( summable_complex @ F ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7664_summable__divide__iff,axiom,
% 16.19/9.02 ! [F: nat > real,C: real] :
% 16.19/9.02 ( ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ C ) )
% 16.19/9.02 = ( ( C = zero_zero_real )
% 16.19/9.02 | ( summable_real @ F ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7665_suminf__geometric,axiom,
% 16.19/9.02 ! [C: real] :
% 16.19/9.02 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 16.19/9.02 => ( ( suminf_real @ ( power_power_real @ C ) )
% 16.19/9.02 = ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7666_suminf__geometric,axiom,
% 16.19/9.02 ! [C: complex] :
% 16.19/9.02 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 16.19/9.02 => ( ( suminf_complex @ ( power_power_complex @ C ) )
% 16.19/9.02 = ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7667_summable__zero,axiom,
% 16.19/9.02 ( summable_complex
% 16.19/9.02 @ ^ [N: nat] : zero_zero_complex ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7668_summable__zero,axiom,
% 16.19/9.02 ( summable_real
% 16.19/9.02 @ ^ [N: nat] : zero_zero_real ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7669_summable__zero,axiom,
% 16.19/9.02 ( summable_nat
% 16.19/9.02 @ ^ [N: nat] : zero_zero_nat ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7670_summable__zero,axiom,
% 16.19/9.02 ( summable_int
% 16.19/9.02 @ ^ [N: nat] : zero_zero_int ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7671_summable__single,axiom,
% 16.19/9.02 ! [I2: nat,F: nat > complex] :
% 16.19/9.02 ( summable_complex
% 16.19/9.02 @ ^ [R5: nat] : ( if_complex @ ( R5 = I2 ) @ ( F @ R5 ) @ zero_zero_complex ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7672_summable__single,axiom,
% 16.19/9.02 ! [I2: nat,F: nat > real] :
% 16.19/9.02 ( summable_real
% 16.19/9.02 @ ^ [R5: nat] : ( if_real @ ( R5 = I2 ) @ ( F @ R5 ) @ zero_zero_real ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7673_summable__single,axiom,
% 16.19/9.02 ! [I2: nat,F: nat > nat] :
% 16.19/9.02 ( summable_nat
% 16.19/9.02 @ ^ [R5: nat] : ( if_nat @ ( R5 = I2 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7674_summable__single,axiom,
% 16.19/9.02 ! [I2: nat,F: nat > int] :
% 16.19/9.02 ( summable_int
% 16.19/9.02 @ ^ [R5: nat] : ( if_int @ ( R5 = I2 ) @ ( F @ R5 ) @ zero_zero_int ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7675_suminf__zero,axiom,
% 16.19/9.02 ( ( suminf_complex
% 16.19/9.02 @ ^ [N: nat] : zero_zero_complex )
% 16.19/9.02 = zero_zero_complex ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7676_suminf__zero,axiom,
% 16.19/9.02 ( ( suminf_real
% 16.19/9.02 @ ^ [N: nat] : zero_zero_real )
% 16.19/9.02 = zero_zero_real ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7677_suminf__zero,axiom,
% 16.19/9.02 ( ( suminf_nat
% 16.19/9.02 @ ^ [N: nat] : zero_zero_nat )
% 16.19/9.02 = zero_zero_nat ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7678_suminf__zero,axiom,
% 16.19/9.02 ( ( suminf_int
% 16.19/9.02 @ ^ [N: nat] : zero_zero_int )
% 16.19/9.02 = zero_zero_int ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7679_summable__iff__shift,axiom,
% 16.19/9.02 ! [F: nat > real,K: nat] :
% 16.19/9.02 ( ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
% 16.19/9.02 = ( summable_real @ F ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7680_summable__iff__shift,axiom,
% 16.19/9.02 ! [F: nat > complex,K: nat] :
% 16.19/9.02 ( ( summable_complex
% 16.19/9.02 @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
% 16.19/9.02 = ( summable_complex @ F ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7681_summable__cmult__iff,axiom,
% 16.19/9.02 ! [C: complex,F: nat > complex] :
% 16.19/9.02 ( ( summable_complex
% 16.19/9.02 @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) ) )
% 16.19/9.02 = ( ( C = zero_zero_complex )
% 16.19/9.02 | ( summable_complex @ F ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7682_summable__cmult__iff,axiom,
% 16.19/9.02 ! [C: real,F: nat > real] :
% 16.19/9.02 ( ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) ) )
% 16.19/9.02 = ( ( C = zero_zero_real )
% 16.19/9.02 | ( summable_real @ F ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7683_summable__const__iff,axiom,
% 16.19/9.02 ! [C: complex] :
% 16.19/9.02 ( ( summable_complex
% 16.19/9.02 @ ^ [Uu3: nat] : C )
% 16.19/9.02 = ( C = zero_zero_complex ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7684_summable__const__iff,axiom,
% 16.19/9.02 ! [C: real] :
% 16.19/9.02 ( ( summable_real
% 16.19/9.02 @ ^ [Uu3: nat] : C )
% 16.19/9.02 = ( C = zero_zero_real ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7685_summable__mult2,axiom,
% 16.19/9.02 ! [F: nat > complex,C: complex] :
% 16.19/9.02 ( ( summable_complex @ F )
% 16.19/9.02 => ( summable_complex
% 16.19/9.02 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ C ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7686_summable__mult2,axiom,
% 16.19/9.02 ! [F: nat > real,C: real] :
% 16.19/9.02 ( ( summable_real @ F )
% 16.19/9.02 => ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ C ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7687_summable__mult,axiom,
% 16.19/9.02 ! [F: nat > complex,C: complex] :
% 16.19/9.02 ( ( summable_complex @ F )
% 16.19/9.02 => ( summable_complex
% 16.19/9.02 @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7688_summable__mult,axiom,
% 16.19/9.02 ! [F: nat > real,C: real] :
% 16.19/9.02 ( ( summable_real @ F )
% 16.19/9.02 => ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7689_summable__add,axiom,
% 16.19/9.02 ! [F: nat > complex,G: nat > complex] :
% 16.19/9.02 ( ( summable_complex @ F )
% 16.19/9.02 => ( ( summable_complex @ G )
% 16.19/9.02 => ( summable_complex
% 16.19/9.02 @ ^ [N: nat] : ( plus_plus_complex @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7690_summable__add,axiom,
% 16.19/9.02 ! [F: nat > real,G: nat > real] :
% 16.19/9.02 ( ( summable_real @ F )
% 16.19/9.02 => ( ( summable_real @ G )
% 16.19/9.02 => ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( plus_plus_real @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7691_summable__add,axiom,
% 16.19/9.02 ! [F: nat > nat,G: nat > nat] :
% 16.19/9.02 ( ( summable_nat @ F )
% 16.19/9.02 => ( ( summable_nat @ G )
% 16.19/9.02 => ( summable_nat
% 16.19/9.02 @ ^ [N: nat] : ( plus_plus_nat @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7692_summable__add,axiom,
% 16.19/9.02 ! [F: nat > int,G: nat > int] :
% 16.19/9.02 ( ( summable_int @ F )
% 16.19/9.02 => ( ( summable_int @ G )
% 16.19/9.02 => ( summable_int
% 16.19/9.02 @ ^ [N: nat] : ( plus_plus_int @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7693_summable__diff,axiom,
% 16.19/9.02 ! [F: nat > complex,G: nat > complex] :
% 16.19/9.02 ( ( summable_complex @ F )
% 16.19/9.02 => ( ( summable_complex @ G )
% 16.19/9.02 => ( summable_complex
% 16.19/9.02 @ ^ [N: nat] : ( minus_minus_complex @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7694_summable__diff,axiom,
% 16.19/9.02 ! [F: nat > real,G: nat > real] :
% 16.19/9.02 ( ( summable_real @ F )
% 16.19/9.02 => ( ( summable_real @ G )
% 16.19/9.02 => ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7695_summable__divide,axiom,
% 16.19/9.02 ! [F: nat > complex,C: complex] :
% 16.19/9.02 ( ( summable_complex @ F )
% 16.19/9.02 => ( summable_complex
% 16.19/9.02 @ ^ [N: nat] : ( divide1717551699836669952omplex @ ( F @ N ) @ C ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7696_summable__divide,axiom,
% 16.19/9.02 ! [F: nat > real,C: real] :
% 16.19/9.02 ( ( summable_real @ F )
% 16.19/9.02 => ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ C ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7697_summable__Suc__iff,axiom,
% 16.19/9.02 ! [F: nat > real] :
% 16.19/9.02 ( ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
% 16.19/9.02 = ( summable_real @ F ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7698_summable__Suc__iff,axiom,
% 16.19/9.02 ! [F: nat > complex] :
% 16.19/9.02 ( ( summable_complex
% 16.19/9.02 @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
% 16.19/9.02 = ( summable_complex @ F ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7699_summable__minus__iff,axiom,
% 16.19/9.02 ! [F: nat > real] :
% 16.19/9.02 ( ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( uminus_uminus_real @ ( F @ N ) ) )
% 16.19/9.02 = ( summable_real @ F ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7700_summable__minus__iff,axiom,
% 16.19/9.02 ! [F: nat > complex] :
% 16.19/9.02 ( ( summable_complex
% 16.19/9.02 @ ^ [N: nat] : ( uminus1482373934393186551omplex @ ( F @ N ) ) )
% 16.19/9.02 = ( summable_complex @ F ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7701_summable__minus,axiom,
% 16.19/9.02 ! [F: nat > real] :
% 16.19/9.02 ( ( summable_real @ F )
% 16.19/9.02 => ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( uminus_uminus_real @ ( F @ N ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7702_summable__minus,axiom,
% 16.19/9.02 ! [F: nat > complex] :
% 16.19/9.02 ( ( summable_complex @ F )
% 16.19/9.02 => ( summable_complex
% 16.19/9.02 @ ^ [N: nat] : ( uminus1482373934393186551omplex @ ( F @ N ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7703_summable__ignore__initial__segment,axiom,
% 16.19/9.02 ! [F: nat > real,K: nat] :
% 16.19/9.02 ( ( summable_real @ F )
% 16.19/9.02 => ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7704_summable__ignore__initial__segment,axiom,
% 16.19/9.02 ! [F: nat > complex,K: nat] :
% 16.19/9.02 ( ( summable_complex @ F )
% 16.19/9.02 => ( summable_complex
% 16.19/9.02 @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7705_summable__sum,axiom,
% 16.19/9.02 ! [I6: set_real,F: real > nat > real] :
% 16.19/9.02 ( ! [I4: real] :
% 16.19/9.02 ( ( member_real @ I4 @ I6 )
% 16.19/9.02 => ( summable_real @ ( F @ I4 ) ) )
% 16.19/9.02 => ( summable_real
% 16.19/9.02 @ ^ [N: nat] :
% 16.19/9.02 ( groups8097168146408367636l_real
% 16.19/9.02 @ ^ [I3: real] : ( F @ I3 @ N )
% 16.19/9.02 @ I6 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7706_summable__sum,axiom,
% 16.19/9.02 ! [I6: set_int,F: int > nat > real] :
% 16.19/9.02 ( ! [I4: int] :
% 16.19/9.02 ( ( member_int @ I4 @ I6 )
% 16.19/9.02 => ( summable_real @ ( F @ I4 ) ) )
% 16.19/9.02 => ( summable_real
% 16.19/9.02 @ ^ [N: nat] :
% 16.19/9.02 ( groups8778361861064173332t_real
% 16.19/9.02 @ ^ [I3: int] : ( F @ I3 @ N )
% 16.19/9.02 @ I6 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7707_summable__sum,axiom,
% 16.19/9.02 ! [I6: set_complex,F: complex > nat > real] :
% 16.19/9.02 ( ! [I4: complex] :
% 16.19/9.02 ( ( member_complex @ I4 @ I6 )
% 16.19/9.02 => ( summable_real @ ( F @ I4 ) ) )
% 16.19/9.02 => ( summable_real
% 16.19/9.02 @ ^ [N: nat] :
% 16.19/9.02 ( groups5808333547571424918x_real
% 16.19/9.02 @ ^ [I3: complex] : ( F @ I3 @ N )
% 16.19/9.02 @ I6 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7708_summable__sum,axiom,
% 16.19/9.02 ! [I6: set_nat,F: nat > nat > complex] :
% 16.19/9.02 ( ! [I4: nat] :
% 16.19/9.02 ( ( member_nat @ I4 @ I6 )
% 16.19/9.02 => ( summable_complex @ ( F @ I4 ) ) )
% 16.19/9.02 => ( summable_complex
% 16.19/9.02 @ ^ [N: nat] :
% 16.19/9.02 ( groups2073611262835488442omplex
% 16.19/9.02 @ ^ [I3: nat] : ( F @ I3 @ N )
% 16.19/9.02 @ I6 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7709_summable__sum,axiom,
% 16.19/9.02 ! [I6: set_real,F: real > nat > complex] :
% 16.19/9.02 ( ! [I4: real] :
% 16.19/9.02 ( ( member_real @ I4 @ I6 )
% 16.19/9.02 => ( summable_complex @ ( F @ I4 ) ) )
% 16.19/9.02 => ( summable_complex
% 16.19/9.02 @ ^ [N: nat] :
% 16.19/9.02 ( groups5754745047067104278omplex
% 16.19/9.02 @ ^ [I3: real] : ( F @ I3 @ N )
% 16.19/9.02 @ I6 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7710_summable__sum,axiom,
% 16.19/9.02 ! [I6: set_int,F: int > nat > complex] :
% 16.19/9.02 ( ! [I4: int] :
% 16.19/9.02 ( ( member_int @ I4 @ I6 )
% 16.19/9.02 => ( summable_complex @ ( F @ I4 ) ) )
% 16.19/9.02 => ( summable_complex
% 16.19/9.02 @ ^ [N: nat] :
% 16.19/9.02 ( groups3049146728041665814omplex
% 16.19/9.02 @ ^ [I3: int] : ( F @ I3 @ N )
% 16.19/9.02 @ I6 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7711_summable__sum,axiom,
% 16.19/9.02 ! [I6: set_int,F: int > nat > int] :
% 16.19/9.02 ( ! [I4: int] :
% 16.19/9.02 ( ( member_int @ I4 @ I6 )
% 16.19/9.02 => ( summable_int @ ( F @ I4 ) ) )
% 16.19/9.02 => ( summable_int
% 16.19/9.02 @ ^ [N: nat] :
% 16.19/9.02 ( groups4538972089207619220nt_int
% 16.19/9.02 @ ^ [I3: int] : ( F @ I3 @ N )
% 16.19/9.02 @ I6 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7712_summable__sum,axiom,
% 16.19/9.02 ! [I6: set_complex,F: complex > nat > complex] :
% 16.19/9.02 ( ! [I4: complex] :
% 16.19/9.02 ( ( member_complex @ I4 @ I6 )
% 16.19/9.02 => ( summable_complex @ ( F @ I4 ) ) )
% 16.19/9.02 => ( summable_complex
% 16.19/9.02 @ ^ [N: nat] :
% 16.19/9.02 ( groups7754918857620584856omplex
% 16.19/9.02 @ ^ [I3: complex] : ( F @ I3 @ N )
% 16.19/9.02 @ I6 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7713_summable__sum,axiom,
% 16.19/9.02 ! [I6: set_nat,F: nat > nat > nat] :
% 16.19/9.02 ( ! [I4: nat] :
% 16.19/9.02 ( ( member_nat @ I4 @ I6 )
% 16.19/9.02 => ( summable_nat @ ( F @ I4 ) ) )
% 16.19/9.02 => ( summable_nat
% 16.19/9.02 @ ^ [N: nat] :
% 16.19/9.02 ( groups3542108847815614940at_nat
% 16.19/9.02 @ ^ [I3: nat] : ( F @ I3 @ N )
% 16.19/9.02 @ I6 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7714_summable__sum,axiom,
% 16.19/9.02 ! [I6: set_nat,F: nat > nat > real] :
% 16.19/9.02 ( ! [I4: nat] :
% 16.19/9.02 ( ( member_nat @ I4 @ I6 )
% 16.19/9.02 => ( summable_real @ ( F @ I4 ) ) )
% 16.19/9.02 => ( summable_real
% 16.19/9.02 @ ^ [N: nat] :
% 16.19/9.02 ( groups6591440286371151544t_real
% 16.19/9.02 @ ^ [I3: nat] : ( F @ I3 @ N )
% 16.19/9.02 @ I6 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7715_summable__rabs__cancel,axiom,
% 16.19/9.02 ! [F: nat > real] :
% 16.19/9.02 ( ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( abs_abs_real @ ( F @ N ) ) )
% 16.19/9.02 => ( summable_real @ F ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7716_suminf__le,axiom,
% 16.19/9.02 ! [F: nat > real,G: nat > real] :
% 16.19/9.02 ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( G @ N4 ) )
% 16.19/9.02 => ( ( summable_real @ F )
% 16.19/9.02 => ( ( summable_real @ G )
% 16.19/9.02 => ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7717_suminf__le,axiom,
% 16.19/9.02 ! [F: nat > nat,G: nat > nat] :
% 16.19/9.02 ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( G @ N4 ) )
% 16.19/9.02 => ( ( summable_nat @ F )
% 16.19/9.02 => ( ( summable_nat @ G )
% 16.19/9.02 => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7718_suminf__le,axiom,
% 16.19/9.02 ! [F: nat > int,G: nat > int] :
% 16.19/9.02 ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ N4 ) @ ( G @ N4 ) )
% 16.19/9.02 => ( ( summable_int @ F )
% 16.19/9.02 => ( ( summable_int @ G )
% 16.19/9.02 => ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7719_summable__mult__D,axiom,
% 16.19/9.02 ! [C: complex,F: nat > complex] :
% 16.19/9.02 ( ( summable_complex
% 16.19/9.02 @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) ) )
% 16.19/9.02 => ( ( C != zero_zero_complex )
% 16.19/9.02 => ( summable_complex @ F ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7720_summable__mult__D,axiom,
% 16.19/9.02 ! [C: real,F: nat > real] :
% 16.19/9.02 ( ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) ) )
% 16.19/9.02 => ( ( C != zero_zero_real )
% 16.19/9.02 => ( summable_real @ F ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7721_summable__zero__power,axiom,
% 16.19/9.02 summable_real @ ( power_power_real @ zero_zero_real ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7722_summable__zero__power,axiom,
% 16.19/9.02 summable_int @ ( power_power_int @ zero_zero_int ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7723_summable__zero__power,axiom,
% 16.19/9.02 summable_complex @ ( power_power_complex @ zero_zero_complex ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7724_suminf__mult,axiom,
% 16.19/9.02 ! [F: nat > complex,C: complex] :
% 16.19/9.02 ( ( summable_complex @ F )
% 16.19/9.02 => ( ( suminf_complex
% 16.19/9.02 @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) ) )
% 16.19/9.02 = ( times_times_complex @ C @ ( suminf_complex @ F ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7725_suminf__mult,axiom,
% 16.19/9.02 ! [F: nat > real,C: real] :
% 16.19/9.02 ( ( summable_real @ F )
% 16.19/9.02 => ( ( suminf_real
% 16.19/9.02 @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) ) )
% 16.19/9.02 = ( times_times_real @ C @ ( suminf_real @ F ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7726_suminf__mult2,axiom,
% 16.19/9.02 ! [F: nat > complex,C: complex] :
% 16.19/9.02 ( ( summable_complex @ F )
% 16.19/9.02 => ( ( times_times_complex @ ( suminf_complex @ F ) @ C )
% 16.19/9.02 = ( suminf_complex
% 16.19/9.02 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ C ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7727_suminf__mult2,axiom,
% 16.19/9.02 ! [F: nat > real,C: real] :
% 16.19/9.02 ( ( summable_real @ F )
% 16.19/9.02 => ( ( times_times_real @ ( suminf_real @ F ) @ C )
% 16.19/9.02 = ( suminf_real
% 16.19/9.02 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ C ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7728_suminf__add,axiom,
% 16.19/9.02 ! [F: nat > complex,G: nat > complex] :
% 16.19/9.02 ( ( summable_complex @ F )
% 16.19/9.02 => ( ( summable_complex @ G )
% 16.19/9.02 => ( ( plus_plus_complex @ ( suminf_complex @ F ) @ ( suminf_complex @ G ) )
% 16.19/9.02 = ( suminf_complex
% 16.19/9.02 @ ^ [N: nat] : ( plus_plus_complex @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7729_suminf__add,axiom,
% 16.19/9.02 ! [F: nat > real,G: nat > real] :
% 16.19/9.02 ( ( summable_real @ F )
% 16.19/9.02 => ( ( summable_real @ G )
% 16.19/9.02 => ( ( plus_plus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
% 16.19/9.02 = ( suminf_real
% 16.19/9.02 @ ^ [N: nat] : ( plus_plus_real @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7730_suminf__add,axiom,
% 16.19/9.02 ! [F: nat > nat,G: nat > nat] :
% 16.19/9.02 ( ( summable_nat @ F )
% 16.19/9.02 => ( ( summable_nat @ G )
% 16.19/9.02 => ( ( plus_plus_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) )
% 16.19/9.02 = ( suminf_nat
% 16.19/9.02 @ ^ [N: nat] : ( plus_plus_nat @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7731_suminf__add,axiom,
% 16.19/9.02 ! [F: nat > int,G: nat > int] :
% 16.19/9.02 ( ( summable_int @ F )
% 16.19/9.02 => ( ( summable_int @ G )
% 16.19/9.02 => ( ( plus_plus_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) )
% 16.19/9.02 = ( suminf_int
% 16.19/9.02 @ ^ [N: nat] : ( plus_plus_int @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7732_suminf__diff,axiom,
% 16.19/9.02 ! [F: nat > complex,G: nat > complex] :
% 16.19/9.02 ( ( summable_complex @ F )
% 16.19/9.02 => ( ( summable_complex @ G )
% 16.19/9.02 => ( ( minus_minus_complex @ ( suminf_complex @ F ) @ ( suminf_complex @ G ) )
% 16.19/9.02 = ( suminf_complex
% 16.19/9.02 @ ^ [N: nat] : ( minus_minus_complex @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7733_suminf__diff,axiom,
% 16.19/9.02 ! [F: nat > real,G: nat > real] :
% 16.19/9.02 ( ( summable_real @ F )
% 16.19/9.02 => ( ( summable_real @ G )
% 16.19/9.02 => ( ( minus_minus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
% 16.19/9.02 = ( suminf_real
% 16.19/9.02 @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7734_suminf__divide,axiom,
% 16.19/9.02 ! [F: nat > complex,C: complex] :
% 16.19/9.02 ( ( summable_complex @ F )
% 16.19/9.02 => ( ( suminf_complex
% 16.19/9.02 @ ^ [N: nat] : ( divide1717551699836669952omplex @ ( F @ N ) @ C ) )
% 16.19/9.02 = ( divide1717551699836669952omplex @ ( suminf_complex @ F ) @ C ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7735_suminf__divide,axiom,
% 16.19/9.02 ! [F: nat > real,C: real] :
% 16.19/9.02 ( ( summable_real @ F )
% 16.19/9.02 => ( ( suminf_real
% 16.19/9.02 @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ C ) )
% 16.19/9.02 = ( divide_divide_real @ ( suminf_real @ F ) @ C ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7736_suminf__minus,axiom,
% 16.19/9.02 ! [F: nat > real] :
% 16.19/9.02 ( ( summable_real @ F )
% 16.19/9.02 => ( ( suminf_real
% 16.19/9.02 @ ^ [N: nat] : ( uminus_uminus_real @ ( F @ N ) ) )
% 16.19/9.02 = ( uminus_uminus_real @ ( suminf_real @ F ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7737_suminf__minus,axiom,
% 16.19/9.02 ! [F: nat > complex] :
% 16.19/9.02 ( ( summable_complex @ F )
% 16.19/9.02 => ( ( suminf_complex
% 16.19/9.02 @ ^ [N: nat] : ( uminus1482373934393186551omplex @ ( F @ N ) ) )
% 16.19/9.02 = ( uminus1482373934393186551omplex @ ( suminf_complex @ F ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7738_suminf__sum,axiom,
% 16.19/9.02 ! [I6: set_real,F: real > nat > real] :
% 16.19/9.02 ( ! [I4: real] :
% 16.19/9.02 ( ( member_real @ I4 @ I6 )
% 16.19/9.02 => ( summable_real @ ( F @ I4 ) ) )
% 16.19/9.02 => ( ( suminf_real
% 16.19/9.02 @ ^ [N: nat] :
% 16.19/9.02 ( groups8097168146408367636l_real
% 16.19/9.02 @ ^ [I3: real] : ( F @ I3 @ N )
% 16.19/9.02 @ I6 ) )
% 16.19/9.02 = ( groups8097168146408367636l_real
% 16.19/9.02 @ ^ [I3: real] : ( suminf_real @ ( F @ I3 ) )
% 16.19/9.02 @ I6 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7739_suminf__sum,axiom,
% 16.19/9.02 ! [I6: set_int,F: int > nat > real] :
% 16.19/9.02 ( ! [I4: int] :
% 16.19/9.02 ( ( member_int @ I4 @ I6 )
% 16.19/9.02 => ( summable_real @ ( F @ I4 ) ) )
% 16.19/9.02 => ( ( suminf_real
% 16.19/9.02 @ ^ [N: nat] :
% 16.19/9.02 ( groups8778361861064173332t_real
% 16.19/9.02 @ ^ [I3: int] : ( F @ I3 @ N )
% 16.19/9.02 @ I6 ) )
% 16.19/9.02 = ( groups8778361861064173332t_real
% 16.19/9.02 @ ^ [I3: int] : ( suminf_real @ ( F @ I3 ) )
% 16.19/9.02 @ I6 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7740_suminf__sum,axiom,
% 16.19/9.02 ! [I6: set_complex,F: complex > nat > real] :
% 16.19/9.02 ( ! [I4: complex] :
% 16.19/9.02 ( ( member_complex @ I4 @ I6 )
% 16.19/9.02 => ( summable_real @ ( F @ I4 ) ) )
% 16.19/9.02 => ( ( suminf_real
% 16.19/9.02 @ ^ [N: nat] :
% 16.19/9.02 ( groups5808333547571424918x_real
% 16.19/9.02 @ ^ [I3: complex] : ( F @ I3 @ N )
% 16.19/9.02 @ I6 ) )
% 16.19/9.02 = ( groups5808333547571424918x_real
% 16.19/9.02 @ ^ [I3: complex] : ( suminf_real @ ( F @ I3 ) )
% 16.19/9.02 @ I6 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7741_suminf__sum,axiom,
% 16.19/9.02 ! [I6: set_nat,F: nat > nat > complex] :
% 16.19/9.02 ( ! [I4: nat] :
% 16.19/9.02 ( ( member_nat @ I4 @ I6 )
% 16.19/9.02 => ( summable_complex @ ( F @ I4 ) ) )
% 16.19/9.02 => ( ( suminf_complex
% 16.19/9.02 @ ^ [N: nat] :
% 16.19/9.02 ( groups2073611262835488442omplex
% 16.19/9.02 @ ^ [I3: nat] : ( F @ I3 @ N )
% 16.19/9.02 @ I6 ) )
% 16.19/9.02 = ( groups2073611262835488442omplex
% 16.19/9.02 @ ^ [I3: nat] : ( suminf_complex @ ( F @ I3 ) )
% 16.19/9.02 @ I6 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7742_suminf__sum,axiom,
% 16.19/9.02 ! [I6: set_real,F: real > nat > complex] :
% 16.19/9.02 ( ! [I4: real] :
% 16.19/9.02 ( ( member_real @ I4 @ I6 )
% 16.19/9.02 => ( summable_complex @ ( F @ I4 ) ) )
% 16.19/9.02 => ( ( suminf_complex
% 16.19/9.02 @ ^ [N: nat] :
% 16.19/9.02 ( groups5754745047067104278omplex
% 16.19/9.02 @ ^ [I3: real] : ( F @ I3 @ N )
% 16.19/9.02 @ I6 ) )
% 16.19/9.02 = ( groups5754745047067104278omplex
% 16.19/9.02 @ ^ [I3: real] : ( suminf_complex @ ( F @ I3 ) )
% 16.19/9.02 @ I6 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7743_suminf__sum,axiom,
% 16.19/9.02 ! [I6: set_int,F: int > nat > complex] :
% 16.19/9.02 ( ! [I4: int] :
% 16.19/9.02 ( ( member_int @ I4 @ I6 )
% 16.19/9.02 => ( summable_complex @ ( F @ I4 ) ) )
% 16.19/9.02 => ( ( suminf_complex
% 16.19/9.02 @ ^ [N: nat] :
% 16.19/9.02 ( groups3049146728041665814omplex
% 16.19/9.02 @ ^ [I3: int] : ( F @ I3 @ N )
% 16.19/9.02 @ I6 ) )
% 16.19/9.02 = ( groups3049146728041665814omplex
% 16.19/9.02 @ ^ [I3: int] : ( suminf_complex @ ( F @ I3 ) )
% 16.19/9.02 @ I6 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7744_suminf__sum,axiom,
% 16.19/9.02 ! [I6: set_int,F: int > nat > int] :
% 16.19/9.02 ( ! [I4: int] :
% 16.19/9.02 ( ( member_int @ I4 @ I6 )
% 16.19/9.02 => ( summable_int @ ( F @ I4 ) ) )
% 16.19/9.02 => ( ( suminf_int
% 16.19/9.02 @ ^ [N: nat] :
% 16.19/9.02 ( groups4538972089207619220nt_int
% 16.19/9.02 @ ^ [I3: int] : ( F @ I3 @ N )
% 16.19/9.02 @ I6 ) )
% 16.19/9.02 = ( groups4538972089207619220nt_int
% 16.19/9.02 @ ^ [I3: int] : ( suminf_int @ ( F @ I3 ) )
% 16.19/9.02 @ I6 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7745_suminf__sum,axiom,
% 16.19/9.02 ! [I6: set_complex,F: complex > nat > complex] :
% 16.19/9.02 ( ! [I4: complex] :
% 16.19/9.02 ( ( member_complex @ I4 @ I6 )
% 16.19/9.02 => ( summable_complex @ ( F @ I4 ) ) )
% 16.19/9.02 => ( ( suminf_complex
% 16.19/9.02 @ ^ [N: nat] :
% 16.19/9.02 ( groups7754918857620584856omplex
% 16.19/9.02 @ ^ [I3: complex] : ( F @ I3 @ N )
% 16.19/9.02 @ I6 ) )
% 16.19/9.02 = ( groups7754918857620584856omplex
% 16.19/9.02 @ ^ [I3: complex] : ( suminf_complex @ ( F @ I3 ) )
% 16.19/9.02 @ I6 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7746_suminf__sum,axiom,
% 16.19/9.02 ! [I6: set_nat,F: nat > nat > nat] :
% 16.19/9.02 ( ! [I4: nat] :
% 16.19/9.02 ( ( member_nat @ I4 @ I6 )
% 16.19/9.02 => ( summable_nat @ ( F @ I4 ) ) )
% 16.19/9.02 => ( ( suminf_nat
% 16.19/9.02 @ ^ [N: nat] :
% 16.19/9.02 ( groups3542108847815614940at_nat
% 16.19/9.02 @ ^ [I3: nat] : ( F @ I3 @ N )
% 16.19/9.02 @ I6 ) )
% 16.19/9.02 = ( groups3542108847815614940at_nat
% 16.19/9.02 @ ^ [I3: nat] : ( suminf_nat @ ( F @ I3 ) )
% 16.19/9.02 @ I6 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7747_suminf__sum,axiom,
% 16.19/9.02 ! [I6: set_nat,F: nat > nat > real] :
% 16.19/9.02 ( ! [I4: nat] :
% 16.19/9.02 ( ( member_nat @ I4 @ I6 )
% 16.19/9.02 => ( summable_real @ ( F @ I4 ) ) )
% 16.19/9.02 => ( ( suminf_real
% 16.19/9.02 @ ^ [N: nat] :
% 16.19/9.02 ( groups6591440286371151544t_real
% 16.19/9.02 @ ^ [I3: nat] : ( F @ I3 @ N )
% 16.19/9.02 @ I6 ) )
% 16.19/9.02 = ( groups6591440286371151544t_real
% 16.19/9.02 @ ^ [I3: nat] : ( suminf_real @ ( F @ I3 ) )
% 16.19/9.02 @ I6 ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7748_summable__norm__cancel,axiom,
% 16.19/9.02 ! [F: nat > real] :
% 16.19/9.02 ( ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( F @ N ) ) )
% 16.19/9.02 => ( summable_real @ F ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7749_summable__norm__cancel,axiom,
% 16.19/9.02 ! [F: nat > complex] :
% 16.19/9.02 ( ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( F @ N ) ) )
% 16.19/9.02 => ( summable_complex @ F ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7750_suminf__nonneg,axiom,
% 16.19/9.02 ! [F: nat > real] :
% 16.19/9.02 ( ( summable_real @ F )
% 16.19/9.02 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
% 16.19/9.02 => ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7751_suminf__nonneg,axiom,
% 16.19/9.02 ! [F: nat > nat] :
% 16.19/9.02 ( ( summable_nat @ F )
% 16.19/9.02 => ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
% 16.19/9.02 => ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7752_suminf__nonneg,axiom,
% 16.19/9.02 ! [F: nat > int] :
% 16.19/9.02 ( ( summable_int @ F )
% 16.19/9.02 => ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
% 16.19/9.02 => ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7753_suminf__eq__zero__iff,axiom,
% 16.19/9.02 ! [F: nat > real] :
% 16.19/9.02 ( ( summable_real @ F )
% 16.19/9.02 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
% 16.19/9.02 => ( ( ( suminf_real @ F )
% 16.19/9.02 = zero_zero_real )
% 16.19/9.02 = ( ! [N: nat] :
% 16.19/9.02 ( ( F @ N )
% 16.19/9.02 = zero_zero_real ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7754_suminf__eq__zero__iff,axiom,
% 16.19/9.02 ! [F: nat > nat] :
% 16.19/9.02 ( ( summable_nat @ F )
% 16.19/9.02 => ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
% 16.19/9.02 => ( ( ( suminf_nat @ F )
% 16.19/9.02 = zero_zero_nat )
% 16.19/9.02 = ( ! [N: nat] :
% 16.19/9.02 ( ( F @ N )
% 16.19/9.02 = zero_zero_nat ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7755_suminf__eq__zero__iff,axiom,
% 16.19/9.02 ! [F: nat > int] :
% 16.19/9.02 ( ( summable_int @ F )
% 16.19/9.02 => ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
% 16.19/9.02 => ( ( ( suminf_int @ F )
% 16.19/9.02 = zero_zero_int )
% 16.19/9.02 = ( ! [N: nat] :
% 16.19/9.02 ( ( F @ N )
% 16.19/9.02 = zero_zero_int ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7756_suminf__pos,axiom,
% 16.19/9.02 ! [F: nat > real] :
% 16.19/9.02 ( ( summable_real @ F )
% 16.19/9.02 => ( ! [N4: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N4 ) )
% 16.19/9.02 => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7757_suminf__pos,axiom,
% 16.19/9.02 ! [F: nat > nat] :
% 16.19/9.02 ( ( summable_nat @ F )
% 16.19/9.02 => ( ! [N4: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N4 ) )
% 16.19/9.02 => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7758_suminf__pos,axiom,
% 16.19/9.02 ! [F: nat > int] :
% 16.19/9.02 ( ( summable_int @ F )
% 16.19/9.02 => ( ! [N4: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N4 ) )
% 16.19/9.02 => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7759_summable__0__powser,axiom,
% 16.19/9.02 ! [F: nat > complex] :
% 16.19/9.02 ( summable_complex
% 16.19/9.02 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7760_summable__0__powser,axiom,
% 16.19/9.02 ! [F: nat > real] :
% 16.19/9.02 ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ zero_zero_real @ N ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7761_summable__zero__power_H,axiom,
% 16.19/9.02 ! [F: nat > complex] :
% 16.19/9.02 ( summable_complex
% 16.19/9.02 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7762_summable__zero__power_H,axiom,
% 16.19/9.02 ! [F: nat > real] :
% 16.19/9.02 ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ zero_zero_real @ N ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7763_summable__zero__power_H,axiom,
% 16.19/9.02 ! [F: nat > int] :
% 16.19/9.02 ( summable_int
% 16.19/9.02 @ ^ [N: nat] : ( times_times_int @ ( F @ N ) @ ( power_power_int @ zero_zero_int @ N ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7764_powser__split__head_I3_J,axiom,
% 16.19/9.02 ! [F: nat > complex,Z: complex] :
% 16.19/9.02 ( ( summable_complex
% 16.19/9.02 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
% 16.19/9.02 => ( summable_complex
% 16.19/9.02 @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7765_powser__split__head_I3_J,axiom,
% 16.19/9.02 ! [F: nat > real,Z: real] :
% 16.19/9.02 ( ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
% 16.19/9.02 => ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7766_summable__powser__split__head,axiom,
% 16.19/9.02 ! [F: nat > complex,Z: complex] :
% 16.19/9.02 ( ( summable_complex
% 16.19/9.02 @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) ) )
% 16.19/9.02 = ( summable_complex
% 16.19/9.02 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7767_summable__powser__split__head,axiom,
% 16.19/9.02 ! [F: nat > real,Z: real] :
% 16.19/9.02 ( ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) ) )
% 16.19/9.02 = ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7768_summable__powser__ignore__initial__segment,axiom,
% 16.19/9.02 ! [F: nat > complex,M: nat,Z: complex] :
% 16.19/9.02 ( ( summable_complex
% 16.19/9.02 @ ^ [N: nat] : ( times_times_complex @ ( F @ ( plus_plus_nat @ N @ M ) ) @ ( power_power_complex @ Z @ N ) ) )
% 16.19/9.02 = ( summable_complex
% 16.19/9.02 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7769_summable__powser__ignore__initial__segment,axiom,
% 16.19/9.02 ! [F: nat > real,M: nat,Z: real] :
% 16.19/9.02 ( ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( times_times_real @ ( F @ ( plus_plus_nat @ N @ M ) ) @ ( power_power_real @ Z @ N ) ) )
% 16.19/9.02 = ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7770_summable__norm__comparison__test,axiom,
% 16.19/9.02 ! [F: nat > complex,G: nat > real] :
% 16.19/9.02 ( ? [N7: nat] :
% 16.19/9.02 ! [N4: nat] :
% 16.19/9.02 ( ( ord_less_eq_nat @ N7 @ N4 )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
% 16.19/9.02 => ( ( summable_real @ G )
% 16.19/9.02 => ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( F @ N ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7771_suminf__finite,axiom,
% 16.19/9.02 ! [N3: set_nat,F: nat > complex] :
% 16.19/9.02 ( ( finite_finite_nat @ N3 )
% 16.19/9.02 => ( ! [N4: nat] :
% 16.19/9.02 ( ~ ( member_nat @ N4 @ N3 )
% 16.19/9.02 => ( ( F @ N4 )
% 16.19/9.02 = zero_zero_complex ) )
% 16.19/9.02 => ( ( suminf_complex @ F )
% 16.19/9.02 = ( groups2073611262835488442omplex @ F @ N3 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7772_suminf__finite,axiom,
% 16.19/9.02 ! [N3: set_nat,F: nat > int] :
% 16.19/9.02 ( ( finite_finite_nat @ N3 )
% 16.19/9.02 => ( ! [N4: nat] :
% 16.19/9.02 ( ~ ( member_nat @ N4 @ N3 )
% 16.19/9.02 => ( ( F @ N4 )
% 16.19/9.02 = zero_zero_int ) )
% 16.19/9.02 => ( ( suminf_int @ F )
% 16.19/9.02 = ( groups3539618377306564664at_int @ F @ N3 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7773_suminf__finite,axiom,
% 16.19/9.02 ! [N3: set_nat,F: nat > nat] :
% 16.19/9.02 ( ( finite_finite_nat @ N3 )
% 16.19/9.02 => ( ! [N4: nat] :
% 16.19/9.02 ( ~ ( member_nat @ N4 @ N3 )
% 16.19/9.02 => ( ( F @ N4 )
% 16.19/9.02 = zero_zero_nat ) )
% 16.19/9.02 => ( ( suminf_nat @ F )
% 16.19/9.02 = ( groups3542108847815614940at_nat @ F @ N3 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7774_suminf__finite,axiom,
% 16.19/9.02 ! [N3: set_nat,F: nat > real] :
% 16.19/9.02 ( ( finite_finite_nat @ N3 )
% 16.19/9.02 => ( ! [N4: nat] :
% 16.19/9.02 ( ~ ( member_nat @ N4 @ N3 )
% 16.19/9.02 => ( ( F @ N4 )
% 16.19/9.02 = zero_zero_real ) )
% 16.19/9.02 => ( ( suminf_real @ F )
% 16.19/9.02 = ( groups6591440286371151544t_real @ F @ N3 ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7775_summable__rabs__comparison__test,axiom,
% 16.19/9.02 ! [F: nat > real,G: nat > real] :
% 16.19/9.02 ( ? [N7: nat] :
% 16.19/9.02 ! [N4: nat] :
% 16.19/9.02 ( ( ord_less_eq_nat @ N7 @ N4 )
% 16.19/9.02 => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
% 16.19/9.02 => ( ( summable_real @ G )
% 16.19/9.02 => ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( abs_abs_real @ ( F @ N ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7776_summable__rabs,axiom,
% 16.19/9.02 ! [F: nat > real] :
% 16.19/9.02 ( ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( abs_abs_real @ ( F @ N ) ) )
% 16.19/9.02 => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
% 16.19/9.02 @ ( suminf_real
% 16.19/9.02 @ ^ [N: nat] : ( abs_abs_real @ ( F @ N ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7777_suminf__pos__iff,axiom,
% 16.19/9.02 ! [F: nat > real] :
% 16.19/9.02 ( ( summable_real @ F )
% 16.19/9.02 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
% 16.19/9.02 => ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
% 16.19/9.02 = ( ? [I3: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7778_suminf__pos__iff,axiom,
% 16.19/9.02 ! [F: nat > nat] :
% 16.19/9.02 ( ( summable_nat @ F )
% 16.19/9.02 => ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
% 16.19/9.02 => ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
% 16.19/9.02 = ( ? [I3: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7779_suminf__pos__iff,axiom,
% 16.19/9.02 ! [F: nat > int] :
% 16.19/9.02 ( ( summable_int @ F )
% 16.19/9.02 => ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
% 16.19/9.02 => ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
% 16.19/9.02 = ( ? [I3: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I3 ) ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7780_suminf__pos2,axiom,
% 16.19/9.02 ! [F: nat > real,I2: nat] :
% 16.19/9.02 ( ( summable_real @ F )
% 16.19/9.02 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
% 16.19/9.02 => ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
% 16.19/9.02 => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7781_suminf__pos2,axiom,
% 16.19/9.02 ! [F: nat > nat,I2: nat] :
% 16.19/9.02 ( ( summable_nat @ F )
% 16.19/9.02 => ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
% 16.19/9.02 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
% 16.19/9.02 => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7782_suminf__pos2,axiom,
% 16.19/9.02 ! [F: nat > int,I2: nat] :
% 16.19/9.02 ( ( summable_int @ F )
% 16.19/9.02 => ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
% 16.19/9.02 => ( ( ord_less_int @ zero_zero_int @ ( F @ I2 ) )
% 16.19/9.02 => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7783_suminf__le__const,axiom,
% 16.19/9.02 ! [F: nat > int,X: int] :
% 16.19/9.02 ( ( summable_int @ F )
% 16.19/9.02 => ( ! [N4: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X )
% 16.19/9.02 => ( ord_less_eq_int @ ( suminf_int @ F ) @ X ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7784_suminf__le__const,axiom,
% 16.19/9.02 ! [F: nat > nat,X: nat] :
% 16.19/9.02 ( ( summable_nat @ F )
% 16.19/9.02 => ( ! [N4: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X )
% 16.19/9.02 => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7785_suminf__le__const,axiom,
% 16.19/9.02 ! [F: nat > real,X: real] :
% 16.19/9.02 ( ( summable_real @ F )
% 16.19/9.02 => ( ! [N4: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X )
% 16.19/9.02 => ( ord_less_eq_real @ ( suminf_real @ F ) @ X ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7786_summable__comparison__test_H,axiom,
% 16.19/9.02 ! [G: nat > real,N3: nat,F: nat > real] :
% 16.19/9.02 ( ( summable_real @ G )
% 16.19/9.02 => ( ! [N4: nat] :
% 16.19/9.02 ( ( ord_less_eq_nat @ N3 @ N4 )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
% 16.19/9.02 => ( summable_real @ F ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7787_summable__comparison__test_H,axiom,
% 16.19/9.02 ! [G: nat > real,N3: nat,F: nat > complex] :
% 16.19/9.02 ( ( summable_real @ G )
% 16.19/9.02 => ( ! [N4: nat] :
% 16.19/9.02 ( ( ord_less_eq_nat @ N3 @ N4 )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
% 16.19/9.02 => ( summable_complex @ F ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7788_summable__comparison__test,axiom,
% 16.19/9.02 ! [F: nat > real,G: nat > real] :
% 16.19/9.02 ( ? [N7: nat] :
% 16.19/9.02 ! [N4: nat] :
% 16.19/9.02 ( ( ord_less_eq_nat @ N7 @ N4 )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
% 16.19/9.02 => ( ( summable_real @ G )
% 16.19/9.02 => ( summable_real @ F ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7789_summable__comparison__test,axiom,
% 16.19/9.02 ! [F: nat > complex,G: nat > real] :
% 16.19/9.02 ( ? [N7: nat] :
% 16.19/9.02 ! [N4: nat] :
% 16.19/9.02 ( ( ord_less_eq_nat @ N7 @ N4 )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
% 16.19/9.02 => ( ( summable_real @ G )
% 16.19/9.02 => ( summable_complex @ F ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7790_summableI__nonneg__bounded,axiom,
% 16.19/9.02 ! [F: nat > int,X: int] :
% 16.19/9.02 ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
% 16.19/9.02 => ( ! [N4: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X )
% 16.19/9.02 => ( summable_int @ F ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7791_summableI__nonneg__bounded,axiom,
% 16.19/9.02 ! [F: nat > nat,X: nat] :
% 16.19/9.02 ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
% 16.19/9.02 => ( ! [N4: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X )
% 16.19/9.02 => ( summable_nat @ F ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7792_summableI__nonneg__bounded,axiom,
% 16.19/9.02 ! [F: nat > real,X: real] :
% 16.19/9.02 ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
% 16.19/9.02 => ( ! [N4: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X )
% 16.19/9.02 => ( summable_real @ F ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7793_complete__algebra__summable__geometric,axiom,
% 16.19/9.02 ! [X: real] :
% 16.19/9.02 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ one_one_real )
% 16.19/9.02 => ( summable_real @ ( power_power_real @ X ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7794_complete__algebra__summable__geometric,axiom,
% 16.19/9.02 ! [X: complex] :
% 16.19/9.02 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ one_one_real )
% 16.19/9.02 => ( summable_complex @ ( power_power_complex @ X ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7795_summable__geometric,axiom,
% 16.19/9.02 ! [C: real] :
% 16.19/9.02 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 16.19/9.02 => ( summable_real @ ( power_power_real @ C ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7796_summable__geometric,axiom,
% 16.19/9.02 ! [C: complex] :
% 16.19/9.02 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 16.19/9.02 => ( summable_complex @ ( power_power_complex @ C ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7797_suminf__split__head,axiom,
% 16.19/9.02 ! [F: nat > complex] :
% 16.19/9.02 ( ( summable_complex @ F )
% 16.19/9.02 => ( ( suminf_complex
% 16.19/9.02 @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
% 16.19/9.02 = ( minus_minus_complex @ ( suminf_complex @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7798_suminf__split__head,axiom,
% 16.19/9.02 ! [F: nat > real] :
% 16.19/9.02 ( ( summable_real @ F )
% 16.19/9.02 => ( ( suminf_real
% 16.19/9.02 @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
% 16.19/9.02 = ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7799_summable__norm,axiom,
% 16.19/9.02 ! [F: nat > real] :
% 16.19/9.02 ( ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( F @ N ) ) )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( suminf_real @ F ) )
% 16.19/9.02 @ ( suminf_real
% 16.19/9.02 @ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( F @ N ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7800_summable__norm,axiom,
% 16.19/9.02 ! [F: nat > complex] :
% 16.19/9.02 ( ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( F @ N ) ) )
% 16.19/9.02 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( suminf_complex @ F ) )
% 16.19/9.02 @ ( suminf_real
% 16.19/9.02 @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( F @ N ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7801_sum__le__suminf,axiom,
% 16.19/9.02 ! [F: nat > int,I6: set_nat] :
% 16.19/9.02 ( ( summable_int @ F )
% 16.19/9.02 => ( ( finite_finite_nat @ I6 )
% 16.19/9.02 => ( ! [N4: nat] :
% 16.19/9.02 ( ( member_nat @ N4 @ ( uminus5710092332889474511et_nat @ I6 ) )
% 16.19/9.02 => ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) ) )
% 16.19/9.02 => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I6 ) @ ( suminf_int @ F ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7802_sum__le__suminf,axiom,
% 16.19/9.02 ! [F: nat > nat,I6: set_nat] :
% 16.19/9.02 ( ( summable_nat @ F )
% 16.19/9.02 => ( ( finite_finite_nat @ I6 )
% 16.19/9.02 => ( ! [N4: nat] :
% 16.19/9.02 ( ( member_nat @ N4 @ ( uminus5710092332889474511et_nat @ I6 ) )
% 16.19/9.02 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) ) )
% 16.19/9.02 => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I6 ) @ ( suminf_nat @ F ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7803_sum__le__suminf,axiom,
% 16.19/9.02 ! [F: nat > real,I6: set_nat] :
% 16.19/9.02 ( ( summable_real @ F )
% 16.19/9.02 => ( ( finite_finite_nat @ I6 )
% 16.19/9.02 => ( ! [N4: nat] :
% 16.19/9.02 ( ( member_nat @ N4 @ ( uminus5710092332889474511et_nat @ I6 ) )
% 16.19/9.02 => ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) ) )
% 16.19/9.02 => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I6 ) @ ( suminf_real @ F ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7804_suminf__split__initial__segment,axiom,
% 16.19/9.02 ! [F: nat > complex,K: nat] :
% 16.19/9.02 ( ( summable_complex @ F )
% 16.19/9.02 => ( ( suminf_complex @ F )
% 16.19/9.02 = ( plus_plus_complex
% 16.19/9.02 @ ( suminf_complex
% 16.19/9.02 @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
% 16.19/9.02 @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7805_suminf__split__initial__segment,axiom,
% 16.19/9.02 ! [F: nat > real,K: nat] :
% 16.19/9.02 ( ( summable_real @ F )
% 16.19/9.02 => ( ( suminf_real @ F )
% 16.19/9.02 = ( plus_plus_real
% 16.19/9.02 @ ( suminf_real
% 16.19/9.02 @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
% 16.19/9.02 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7806_suminf__minus__initial__segment,axiom,
% 16.19/9.02 ! [F: nat > complex,K: nat] :
% 16.19/9.02 ( ( summable_complex @ F )
% 16.19/9.02 => ( ( suminf_complex
% 16.19/9.02 @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
% 16.19/9.02 = ( minus_minus_complex @ ( suminf_complex @ F ) @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7807_suminf__minus__initial__segment,axiom,
% 16.19/9.02 ! [F: nat > real,K: nat] :
% 16.19/9.02 ( ( summable_real @ F )
% 16.19/9.02 => ( ( suminf_real
% 16.19/9.02 @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
% 16.19/9.02 = ( minus_minus_real @ ( suminf_real @ F ) @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7808_sum__less__suminf,axiom,
% 16.19/9.02 ! [F: nat > int,N2: nat] :
% 16.19/9.02 ( ( summable_int @ F )
% 16.19/9.02 => ( ! [M2: nat] :
% 16.19/9.02 ( ( ord_less_eq_nat @ N2 @ M2 )
% 16.19/9.02 => ( ord_less_int @ zero_zero_int @ ( F @ M2 ) ) )
% 16.19/9.02 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_int @ F ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7809_sum__less__suminf,axiom,
% 16.19/9.02 ! [F: nat > nat,N2: nat] :
% 16.19/9.02 ( ( summable_nat @ F )
% 16.19/9.02 => ( ! [M2: nat] :
% 16.19/9.02 ( ( ord_less_eq_nat @ N2 @ M2 )
% 16.19/9.02 => ( ord_less_nat @ zero_zero_nat @ ( F @ M2 ) ) )
% 16.19/9.02 => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_nat @ F ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7810_sum__less__suminf,axiom,
% 16.19/9.02 ! [F: nat > real,N2: nat] :
% 16.19/9.02 ( ( summable_real @ F )
% 16.19/9.02 => ( ! [M2: nat] :
% 16.19/9.02 ( ( ord_less_eq_nat @ N2 @ M2 )
% 16.19/9.02 => ( ord_less_real @ zero_zero_real @ ( F @ M2 ) ) )
% 16.19/9.02 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_real @ F ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7811_powser__split__head_I1_J,axiom,
% 16.19/9.02 ! [F: nat > complex,Z: complex] :
% 16.19/9.02 ( ( summable_complex
% 16.19/9.02 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
% 16.19/9.02 => ( ( suminf_complex
% 16.19/9.02 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
% 16.19/9.02 = ( plus_plus_complex @ ( F @ zero_zero_nat )
% 16.19/9.02 @ ( times_times_complex
% 16.19/9.02 @ ( suminf_complex
% 16.19/9.02 @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) ) )
% 16.19/9.02 @ Z ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7812_powser__split__head_I1_J,axiom,
% 16.19/9.02 ! [F: nat > real,Z: real] :
% 16.19/9.02 ( ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
% 16.19/9.02 => ( ( suminf_real
% 16.19/9.02 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
% 16.19/9.02 = ( plus_plus_real @ ( F @ zero_zero_nat )
% 16.19/9.02 @ ( times_times_real
% 16.19/9.02 @ ( suminf_real
% 16.19/9.02 @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) ) )
% 16.19/9.02 @ Z ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7813_powser__split__head_I2_J,axiom,
% 16.19/9.02 ! [F: nat > complex,Z: complex] :
% 16.19/9.02 ( ( summable_complex
% 16.19/9.02 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
% 16.19/9.02 => ( ( times_times_complex
% 16.19/9.02 @ ( suminf_complex
% 16.19/9.02 @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) ) )
% 16.19/9.02 @ Z )
% 16.19/9.02 = ( minus_minus_complex
% 16.19/9.02 @ ( suminf_complex
% 16.19/9.02 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
% 16.19/9.02 @ ( F @ zero_zero_nat ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7814_powser__split__head_I2_J,axiom,
% 16.19/9.02 ! [F: nat > real,Z: real] :
% 16.19/9.02 ( ( summable_real
% 16.19/9.02 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
% 16.19/9.02 => ( ( times_times_real
% 16.19/9.02 @ ( suminf_real
% 16.19/9.02 @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) ) )
% 16.19/9.02 @ Z )
% 16.19/9.02 = ( minus_minus_real
% 16.19/9.02 @ ( suminf_real
% 16.19/9.02 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
% 16.19/9.02 @ ( F @ zero_zero_nat ) ) ) ) ).
% 16.19/9.02
% 16.19/9.02 thf(fact_7815_suminf__exist__split,axiom,
% 16.19/9.02 ! [R: real,F: nat > real] :
% 16.19/9.02 ( ( ord_less_real @ zero_zero_real @ R )
% 16.19/9.02 => ( ( summable_real @ F )
% 16.19/9.02 => ? [N8: nat] :
% 16.19/9.02 ! [N9: nat] :
% 16.19/9.02 ( ( ord_less_eq_nat @ N8 @ N9 )
% 16.19/9.02 => ( ord_less_real
% 16.19/9.02 @ ( real_V7735802525324610683m_real
% 16.19/9.03 @ ( suminf_real
% 16.19/9.03 @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N9 ) ) ) )
% 16.19/9.03 @ R ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7816_suminf__exist__split,axiom,
% 16.19/9.03 ! [R: real,F: nat > complex] :
% 16.19/9.03 ( ( ord_less_real @ zero_zero_real @ R )
% 16.19/9.03 => ( ( summable_complex @ F )
% 16.19/9.03 => ? [N8: nat] :
% 16.19/9.03 ! [N9: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ N8 @ N9 )
% 16.19/9.03 => ( ord_less_real
% 16.19/9.03 @ ( real_V1022390504157884413omplex
% 16.19/9.03 @ ( suminf_complex
% 16.19/9.03 @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N9 ) ) ) )
% 16.19/9.03 @ R ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7817_summable__partial__sum__bound,axiom,
% 16.19/9.03 ! [F: nat > complex,E: real] :
% 16.19/9.03 ( ( summable_complex @ F )
% 16.19/9.03 => ( ( ord_less_real @ zero_zero_real @ E )
% 16.19/9.03 => ~ ! [N8: nat] :
% 16.19/9.03 ~ ! [M5: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ N8 @ M5 )
% 16.19/9.03 => ! [N9: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M5 @ N9 ) ) ) @ E ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7818_summable__partial__sum__bound,axiom,
% 16.19/9.03 ! [F: nat > real,E: real] :
% 16.19/9.03 ( ( summable_real @ F )
% 16.19/9.03 => ( ( ord_less_real @ zero_zero_real @ E )
% 16.19/9.03 => ~ ! [N8: nat] :
% 16.19/9.03 ~ ! [M5: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ N8 @ M5 )
% 16.19/9.03 => ! [N9: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M5 @ N9 ) ) ) @ E ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7819_summable__power__series,axiom,
% 16.19/9.03 ! [F: nat > real,Z: real] :
% 16.19/9.03 ( ! [I4: nat] : ( ord_less_eq_real @ ( F @ I4 ) @ one_one_real )
% 16.19/9.03 => ( ! [I4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
% 16.19/9.03 => ( ( ord_less_eq_real @ zero_zero_real @ Z )
% 16.19/9.03 => ( ( ord_less_real @ Z @ one_one_real )
% 16.19/9.03 => ( summable_real
% 16.19/9.03 @ ^ [I3: nat] : ( times_times_real @ ( F @ I3 ) @ ( power_power_real @ Z @ I3 ) ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7820_Abel__lemma,axiom,
% 16.19/9.03 ! [R: real,R0: real,A: nat > complex,M7: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ zero_zero_real @ R )
% 16.19/9.03 => ( ( ord_less_real @ R @ R0 )
% 16.19/9.03 => ( ! [N4: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N4 ) ) @ ( power_power_real @ R0 @ N4 ) ) @ M7 )
% 16.19/9.03 => ( summable_real
% 16.19/9.03 @ ^ [N: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N ) ) @ ( power_power_real @ R @ N ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7821_summable__ratio__test,axiom,
% 16.19/9.03 ! [C: real,N3: nat,F: nat > real] :
% 16.19/9.03 ( ( ord_less_real @ C @ one_one_real )
% 16.19/9.03 => ( ! [N4: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ N3 @ N4 )
% 16.19/9.03 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N4 ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N4 ) ) ) ) )
% 16.19/9.03 => ( summable_real @ F ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7822_summable__ratio__test,axiom,
% 16.19/9.03 ! [C: real,N3: nat,F: nat > complex] :
% 16.19/9.03 ( ( ord_less_real @ C @ one_one_real )
% 16.19/9.03 => ( ! [N4: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ N3 @ N4 )
% 16.19/9.03 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N4 ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N4 ) ) ) ) )
% 16.19/9.03 => ( summable_complex @ F ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7823_sum__less__suminf2,axiom,
% 16.19/9.03 ! [F: nat > int,N2: nat,I2: nat] :
% 16.19/9.03 ( ( summable_int @ F )
% 16.19/9.03 => ( ! [M2: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ N2 @ M2 )
% 16.19/9.03 => ( ord_less_eq_int @ zero_zero_int @ ( F @ M2 ) ) )
% 16.19/9.03 => ( ( ord_less_eq_nat @ N2 @ I2 )
% 16.19/9.03 => ( ( ord_less_int @ zero_zero_int @ ( F @ I2 ) )
% 16.19/9.03 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_int @ F ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7824_sum__less__suminf2,axiom,
% 16.19/9.03 ! [F: nat > nat,N2: nat,I2: nat] :
% 16.19/9.03 ( ( summable_nat @ F )
% 16.19/9.03 => ( ! [M2: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ N2 @ M2 )
% 16.19/9.03 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M2 ) ) )
% 16.19/9.03 => ( ( ord_less_eq_nat @ N2 @ I2 )
% 16.19/9.03 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
% 16.19/9.03 => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_nat @ F ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7825_sum__less__suminf2,axiom,
% 16.19/9.03 ! [F: nat > real,N2: nat,I2: nat] :
% 16.19/9.03 ( ( summable_real @ F )
% 16.19/9.03 => ( ! [M2: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ N2 @ M2 )
% 16.19/9.03 => ( ord_less_eq_real @ zero_zero_real @ ( F @ M2 ) ) )
% 16.19/9.03 => ( ( ord_less_eq_nat @ N2 @ I2 )
% 16.19/9.03 => ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
% 16.19/9.03 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_real @ F ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7826_sum__bounds__lt__plus1,axiom,
% 16.19/9.03 ! [F: nat > nat,Mm: nat] :
% 16.19/9.03 ( ( groups3542108847815614940at_nat
% 16.19/9.03 @ ^ [K3: nat] : ( F @ ( suc @ K3 ) )
% 16.19/9.03 @ ( set_ord_lessThan_nat @ Mm ) )
% 16.19/9.03 = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7827_sum__bounds__lt__plus1,axiom,
% 16.19/9.03 ! [F: nat > real,Mm: nat] :
% 16.19/9.03 ( ( groups6591440286371151544t_real
% 16.19/9.03 @ ^ [K3: nat] : ( F @ ( suc @ K3 ) )
% 16.19/9.03 @ ( set_ord_lessThan_nat @ Mm ) )
% 16.19/9.03 = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7828_sumr__cos__zero__one,axiom,
% 16.19/9.03 ! [N2: nat] :
% 16.19/9.03 ( ( groups6591440286371151544t_real
% 16.19/9.03 @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ zero_zero_real @ M6 ) )
% 16.19/9.03 @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 16.19/9.03 = one_one_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7829_pi__series,axiom,
% 16.19/9.03 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 16.19/9.03 = ( suminf_real
% 16.19/9.03 @ ^ [K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7830_infinite__int__iff__unbounded,axiom,
% 16.19/9.03 ! [S: set_int] :
% 16.19/9.03 ( ( ~ ( finite_finite_int @ S ) )
% 16.19/9.03 = ( ! [M6: int] :
% 16.19/9.03 ? [N: int] :
% 16.19/9.03 ( ( ord_less_int @ M6 @ ( abs_abs_int @ N ) )
% 16.19/9.03 & ( member_int @ N @ S ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7831_geometric__deriv__sums,axiom,
% 16.19/9.03 ! [Z: real] :
% 16.19/9.03 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
% 16.19/9.03 => ( sums_real
% 16.19/9.03 @ ^ [N: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) )
% 16.19/9.03 @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7832_geometric__deriv__sums,axiom,
% 16.19/9.03 ! [Z: complex] :
% 16.19/9.03 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
% 16.19/9.03 => ( sums_complex
% 16.19/9.03 @ ^ [N: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) )
% 16.19/9.03 @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7833_sums__zero,axiom,
% 16.19/9.03 ( sums_complex
% 16.19/9.03 @ ^ [N: nat] : zero_zero_complex
% 16.19/9.03 @ zero_zero_complex ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7834_sums__zero,axiom,
% 16.19/9.03 ( sums_real
% 16.19/9.03 @ ^ [N: nat] : zero_zero_real
% 16.19/9.03 @ zero_zero_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7835_sums__zero,axiom,
% 16.19/9.03 ( sums_nat
% 16.19/9.03 @ ^ [N: nat] : zero_zero_nat
% 16.19/9.03 @ zero_zero_nat ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7836_sums__zero,axiom,
% 16.19/9.03 ( sums_int
% 16.19/9.03 @ ^ [N: nat] : zero_zero_int
% 16.19/9.03 @ zero_zero_int ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7837_cos__coeff__0,axiom,
% 16.19/9.03 ( ( cos_coeff @ zero_zero_nat )
% 16.19/9.03 = one_one_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7838_powser__sums__zero__iff,axiom,
% 16.19/9.03 ! [A: nat > complex,X: complex] :
% 16.19/9.03 ( ( sums_complex
% 16.19/9.03 @ ^ [N: nat] : ( times_times_complex @ ( A @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) )
% 16.19/9.03 @ X )
% 16.19/9.03 = ( ( A @ zero_zero_nat )
% 16.19/9.03 = X ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7839_powser__sums__zero__iff,axiom,
% 16.19/9.03 ! [A: nat > real,X: real] :
% 16.19/9.03 ( ( sums_real
% 16.19/9.03 @ ^ [N: nat] : ( times_times_real @ ( A @ N ) @ ( power_power_real @ zero_zero_real @ N ) )
% 16.19/9.03 @ X )
% 16.19/9.03 = ( ( A @ zero_zero_nat )
% 16.19/9.03 = X ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7840_sums__le,axiom,
% 16.19/9.03 ! [F: nat > real,G: nat > real,S3: real,T: real] :
% 16.19/9.03 ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( G @ N4 ) )
% 16.19/9.03 => ( ( sums_real @ F @ S3 )
% 16.19/9.03 => ( ( sums_real @ G @ T )
% 16.19/9.03 => ( ord_less_eq_real @ S3 @ T ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7841_sums__le,axiom,
% 16.19/9.03 ! [F: nat > nat,G: nat > nat,S3: nat,T: nat] :
% 16.19/9.03 ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( G @ N4 ) )
% 16.19/9.03 => ( ( sums_nat @ F @ S3 )
% 16.19/9.03 => ( ( sums_nat @ G @ T )
% 16.19/9.03 => ( ord_less_eq_nat @ S3 @ T ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7842_sums__le,axiom,
% 16.19/9.03 ! [F: nat > int,G: nat > int,S3: int,T: int] :
% 16.19/9.03 ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ N4 ) @ ( G @ N4 ) )
% 16.19/9.03 => ( ( sums_int @ F @ S3 )
% 16.19/9.03 => ( ( sums_int @ G @ T )
% 16.19/9.03 => ( ord_less_eq_int @ S3 @ T ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7843_sums__single,axiom,
% 16.19/9.03 ! [I2: nat,F: nat > complex] :
% 16.19/9.03 ( sums_complex
% 16.19/9.03 @ ^ [R5: nat] : ( if_complex @ ( R5 = I2 ) @ ( F @ R5 ) @ zero_zero_complex )
% 16.19/9.03 @ ( F @ I2 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7844_sums__single,axiom,
% 16.19/9.03 ! [I2: nat,F: nat > real] :
% 16.19/9.03 ( sums_real
% 16.19/9.03 @ ^ [R5: nat] : ( if_real @ ( R5 = I2 ) @ ( F @ R5 ) @ zero_zero_real )
% 16.19/9.03 @ ( F @ I2 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7845_sums__single,axiom,
% 16.19/9.03 ! [I2: nat,F: nat > nat] :
% 16.19/9.03 ( sums_nat
% 16.19/9.03 @ ^ [R5: nat] : ( if_nat @ ( R5 = I2 ) @ ( F @ R5 ) @ zero_zero_nat )
% 16.19/9.03 @ ( F @ I2 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7846_sums__single,axiom,
% 16.19/9.03 ! [I2: nat,F: nat > int] :
% 16.19/9.03 ( sums_int
% 16.19/9.03 @ ^ [R5: nat] : ( if_int @ ( R5 = I2 ) @ ( F @ R5 ) @ zero_zero_int )
% 16.19/9.03 @ ( F @ I2 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7847_sums__mult,axiom,
% 16.19/9.03 ! [F: nat > complex,A: complex,C: complex] :
% 16.19/9.03 ( ( sums_complex @ F @ A )
% 16.19/9.03 => ( sums_complex
% 16.19/9.03 @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) )
% 16.19/9.03 @ ( times_times_complex @ C @ A ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7848_sums__mult,axiom,
% 16.19/9.03 ! [F: nat > real,A: real,C: real] :
% 16.19/9.03 ( ( sums_real @ F @ A )
% 16.19/9.03 => ( sums_real
% 16.19/9.03 @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) )
% 16.19/9.03 @ ( times_times_real @ C @ A ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7849_sums__mult2,axiom,
% 16.19/9.03 ! [F: nat > complex,A: complex,C: complex] :
% 16.19/9.03 ( ( sums_complex @ F @ A )
% 16.19/9.03 => ( sums_complex
% 16.19/9.03 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ C )
% 16.19/9.03 @ ( times_times_complex @ A @ C ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7850_sums__mult2,axiom,
% 16.19/9.03 ! [F: nat > real,A: real,C: real] :
% 16.19/9.03 ( ( sums_real @ F @ A )
% 16.19/9.03 => ( sums_real
% 16.19/9.03 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ C )
% 16.19/9.03 @ ( times_times_real @ A @ C ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7851_sums__add,axiom,
% 16.19/9.03 ! [F: nat > complex,A: complex,G: nat > complex,B: complex] :
% 16.19/9.03 ( ( sums_complex @ F @ A )
% 16.19/9.03 => ( ( sums_complex @ G @ B )
% 16.19/9.03 => ( sums_complex
% 16.19/9.03 @ ^ [N: nat] : ( plus_plus_complex @ ( F @ N ) @ ( G @ N ) )
% 16.19/9.03 @ ( plus_plus_complex @ A @ B ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7852_sums__add,axiom,
% 16.19/9.03 ! [F: nat > real,A: real,G: nat > real,B: real] :
% 16.19/9.03 ( ( sums_real @ F @ A )
% 16.19/9.03 => ( ( sums_real @ G @ B )
% 16.19/9.03 => ( sums_real
% 16.19/9.03 @ ^ [N: nat] : ( plus_plus_real @ ( F @ N ) @ ( G @ N ) )
% 16.19/9.03 @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7853_sums__add,axiom,
% 16.19/9.03 ! [F: nat > nat,A: nat,G: nat > nat,B: nat] :
% 16.19/9.03 ( ( sums_nat @ F @ A )
% 16.19/9.03 => ( ( sums_nat @ G @ B )
% 16.19/9.03 => ( sums_nat
% 16.19/9.03 @ ^ [N: nat] : ( plus_plus_nat @ ( F @ N ) @ ( G @ N ) )
% 16.19/9.03 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7854_sums__add,axiom,
% 16.19/9.03 ! [F: nat > int,A: int,G: nat > int,B: int] :
% 16.19/9.03 ( ( sums_int @ F @ A )
% 16.19/9.03 => ( ( sums_int @ G @ B )
% 16.19/9.03 => ( sums_int
% 16.19/9.03 @ ^ [N: nat] : ( plus_plus_int @ ( F @ N ) @ ( G @ N ) )
% 16.19/9.03 @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7855_sums__diff,axiom,
% 16.19/9.03 ! [F: nat > complex,A: complex,G: nat > complex,B: complex] :
% 16.19/9.03 ( ( sums_complex @ F @ A )
% 16.19/9.03 => ( ( sums_complex @ G @ B )
% 16.19/9.03 => ( sums_complex
% 16.19/9.03 @ ^ [N: nat] : ( minus_minus_complex @ ( F @ N ) @ ( G @ N ) )
% 16.19/9.03 @ ( minus_minus_complex @ A @ B ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7856_sums__diff,axiom,
% 16.19/9.03 ! [F: nat > real,A: real,G: nat > real,B: real] :
% 16.19/9.03 ( ( sums_real @ F @ A )
% 16.19/9.03 => ( ( sums_real @ G @ B )
% 16.19/9.03 => ( sums_real
% 16.19/9.03 @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( G @ N ) )
% 16.19/9.03 @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7857_sums__divide,axiom,
% 16.19/9.03 ! [F: nat > complex,A: complex,C: complex] :
% 16.19/9.03 ( ( sums_complex @ F @ A )
% 16.19/9.03 => ( sums_complex
% 16.19/9.03 @ ^ [N: nat] : ( divide1717551699836669952omplex @ ( F @ N ) @ C )
% 16.19/9.03 @ ( divide1717551699836669952omplex @ A @ C ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7858_sums__divide,axiom,
% 16.19/9.03 ! [F: nat > real,A: real,C: real] :
% 16.19/9.03 ( ( sums_real @ F @ A )
% 16.19/9.03 => ( sums_real
% 16.19/9.03 @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ C )
% 16.19/9.03 @ ( divide_divide_real @ A @ C ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7859_sums__minus,axiom,
% 16.19/9.03 ! [F: nat > real,A: real] :
% 16.19/9.03 ( ( sums_real @ F @ A )
% 16.19/9.03 => ( sums_real
% 16.19/9.03 @ ^ [N: nat] : ( uminus_uminus_real @ ( F @ N ) )
% 16.19/9.03 @ ( uminus_uminus_real @ A ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7860_sums__minus,axiom,
% 16.19/9.03 ! [F: nat > complex,A: complex] :
% 16.19/9.03 ( ( sums_complex @ F @ A )
% 16.19/9.03 => ( sums_complex
% 16.19/9.03 @ ^ [N: nat] : ( uminus1482373934393186551omplex @ ( F @ N ) )
% 16.19/9.03 @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7861_sums__sum,axiom,
% 16.19/9.03 ! [I6: set_real,F: real > nat > real,X: real > real] :
% 16.19/9.03 ( ! [I4: real] :
% 16.19/9.03 ( ( member_real @ I4 @ I6 )
% 16.19/9.03 => ( sums_real @ ( F @ I4 ) @ ( X @ I4 ) ) )
% 16.19/9.03 => ( sums_real
% 16.19/9.03 @ ^ [N: nat] :
% 16.19/9.03 ( groups8097168146408367636l_real
% 16.19/9.03 @ ^ [I3: real] : ( F @ I3 @ N )
% 16.19/9.03 @ I6 )
% 16.19/9.03 @ ( groups8097168146408367636l_real @ X @ I6 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7862_sums__sum,axiom,
% 16.19/9.03 ! [I6: set_int,F: int > nat > real,X: int > real] :
% 16.19/9.03 ( ! [I4: int] :
% 16.19/9.03 ( ( member_int @ I4 @ I6 )
% 16.19/9.03 => ( sums_real @ ( F @ I4 ) @ ( X @ I4 ) ) )
% 16.19/9.03 => ( sums_real
% 16.19/9.03 @ ^ [N: nat] :
% 16.19/9.03 ( groups8778361861064173332t_real
% 16.19/9.03 @ ^ [I3: int] : ( F @ I3 @ N )
% 16.19/9.03 @ I6 )
% 16.19/9.03 @ ( groups8778361861064173332t_real @ X @ I6 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7863_sums__sum,axiom,
% 16.19/9.03 ! [I6: set_complex,F: complex > nat > real,X: complex > real] :
% 16.19/9.03 ( ! [I4: complex] :
% 16.19/9.03 ( ( member_complex @ I4 @ I6 )
% 16.19/9.03 => ( sums_real @ ( F @ I4 ) @ ( X @ I4 ) ) )
% 16.19/9.03 => ( sums_real
% 16.19/9.03 @ ^ [N: nat] :
% 16.19/9.03 ( groups5808333547571424918x_real
% 16.19/9.03 @ ^ [I3: complex] : ( F @ I3 @ N )
% 16.19/9.03 @ I6 )
% 16.19/9.03 @ ( groups5808333547571424918x_real @ X @ I6 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7864_sums__sum,axiom,
% 16.19/9.03 ! [I6: set_nat,F: nat > nat > complex,X: nat > complex] :
% 16.19/9.03 ( ! [I4: nat] :
% 16.19/9.03 ( ( member_nat @ I4 @ I6 )
% 16.19/9.03 => ( sums_complex @ ( F @ I4 ) @ ( X @ I4 ) ) )
% 16.19/9.03 => ( sums_complex
% 16.19/9.03 @ ^ [N: nat] :
% 16.19/9.03 ( groups2073611262835488442omplex
% 16.19/9.03 @ ^ [I3: nat] : ( F @ I3 @ N )
% 16.19/9.03 @ I6 )
% 16.19/9.03 @ ( groups2073611262835488442omplex @ X @ I6 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7865_sums__sum,axiom,
% 16.19/9.03 ! [I6: set_real,F: real > nat > complex,X: real > complex] :
% 16.19/9.03 ( ! [I4: real] :
% 16.19/9.03 ( ( member_real @ I4 @ I6 )
% 16.19/9.03 => ( sums_complex @ ( F @ I4 ) @ ( X @ I4 ) ) )
% 16.19/9.03 => ( sums_complex
% 16.19/9.03 @ ^ [N: nat] :
% 16.19/9.03 ( groups5754745047067104278omplex
% 16.19/9.03 @ ^ [I3: real] : ( F @ I3 @ N )
% 16.19/9.03 @ I6 )
% 16.19/9.03 @ ( groups5754745047067104278omplex @ X @ I6 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7866_sums__sum,axiom,
% 16.19/9.03 ! [I6: set_int,F: int > nat > complex,X: int > complex] :
% 16.19/9.03 ( ! [I4: int] :
% 16.19/9.03 ( ( member_int @ I4 @ I6 )
% 16.19/9.03 => ( sums_complex @ ( F @ I4 ) @ ( X @ I4 ) ) )
% 16.19/9.03 => ( sums_complex
% 16.19/9.03 @ ^ [N: nat] :
% 16.19/9.03 ( groups3049146728041665814omplex
% 16.19/9.03 @ ^ [I3: int] : ( F @ I3 @ N )
% 16.19/9.03 @ I6 )
% 16.19/9.03 @ ( groups3049146728041665814omplex @ X @ I6 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7867_sums__sum,axiom,
% 16.19/9.03 ! [I6: set_int,F: int > nat > int,X: int > int] :
% 16.19/9.03 ( ! [I4: int] :
% 16.19/9.03 ( ( member_int @ I4 @ I6 )
% 16.19/9.03 => ( sums_int @ ( F @ I4 ) @ ( X @ I4 ) ) )
% 16.19/9.03 => ( sums_int
% 16.19/9.03 @ ^ [N: nat] :
% 16.19/9.03 ( groups4538972089207619220nt_int
% 16.19/9.03 @ ^ [I3: int] : ( F @ I3 @ N )
% 16.19/9.03 @ I6 )
% 16.19/9.03 @ ( groups4538972089207619220nt_int @ X @ I6 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7868_sums__sum,axiom,
% 16.19/9.03 ! [I6: set_complex,F: complex > nat > complex,X: complex > complex] :
% 16.19/9.03 ( ! [I4: complex] :
% 16.19/9.03 ( ( member_complex @ I4 @ I6 )
% 16.19/9.03 => ( sums_complex @ ( F @ I4 ) @ ( X @ I4 ) ) )
% 16.19/9.03 => ( sums_complex
% 16.19/9.03 @ ^ [N: nat] :
% 16.19/9.03 ( groups7754918857620584856omplex
% 16.19/9.03 @ ^ [I3: complex] : ( F @ I3 @ N )
% 16.19/9.03 @ I6 )
% 16.19/9.03 @ ( groups7754918857620584856omplex @ X @ I6 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7869_sums__sum,axiom,
% 16.19/9.03 ! [I6: set_nat,F: nat > nat > nat,X: nat > nat] :
% 16.19/9.03 ( ! [I4: nat] :
% 16.19/9.03 ( ( member_nat @ I4 @ I6 )
% 16.19/9.03 => ( sums_nat @ ( F @ I4 ) @ ( X @ I4 ) ) )
% 16.19/9.03 => ( sums_nat
% 16.19/9.03 @ ^ [N: nat] :
% 16.19/9.03 ( groups3542108847815614940at_nat
% 16.19/9.03 @ ^ [I3: nat] : ( F @ I3 @ N )
% 16.19/9.03 @ I6 )
% 16.19/9.03 @ ( groups3542108847815614940at_nat @ X @ I6 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7870_sums__sum,axiom,
% 16.19/9.03 ! [I6: set_nat,F: nat > nat > real,X: nat > real] :
% 16.19/9.03 ( ! [I4: nat] :
% 16.19/9.03 ( ( member_nat @ I4 @ I6 )
% 16.19/9.03 => ( sums_real @ ( F @ I4 ) @ ( X @ I4 ) ) )
% 16.19/9.03 => ( sums_real
% 16.19/9.03 @ ^ [N: nat] :
% 16.19/9.03 ( groups6591440286371151544t_real
% 16.19/9.03 @ ^ [I3: nat] : ( F @ I3 @ N )
% 16.19/9.03 @ I6 )
% 16.19/9.03 @ ( groups6591440286371151544t_real @ X @ I6 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7871_sums__mult__iff,axiom,
% 16.19/9.03 ! [C: complex,F: nat > complex,D: complex] :
% 16.19/9.03 ( ( C != zero_zero_complex )
% 16.19/9.03 => ( ( sums_complex
% 16.19/9.03 @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) )
% 16.19/9.03 @ ( times_times_complex @ C @ D ) )
% 16.19/9.03 = ( sums_complex @ F @ D ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7872_sums__mult__iff,axiom,
% 16.19/9.03 ! [C: real,F: nat > real,D: real] :
% 16.19/9.03 ( ( C != zero_zero_real )
% 16.19/9.03 => ( ( sums_real
% 16.19/9.03 @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) )
% 16.19/9.03 @ ( times_times_real @ C @ D ) )
% 16.19/9.03 = ( sums_real @ F @ D ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7873_sums__mult2__iff,axiom,
% 16.19/9.03 ! [C: complex,F: nat > complex,D: complex] :
% 16.19/9.03 ( ( C != zero_zero_complex )
% 16.19/9.03 => ( ( sums_complex
% 16.19/9.03 @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ C )
% 16.19/9.03 @ ( times_times_complex @ D @ C ) )
% 16.19/9.03 = ( sums_complex @ F @ D ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7874_sums__mult2__iff,axiom,
% 16.19/9.03 ! [C: real,F: nat > real,D: real] :
% 16.19/9.03 ( ( C != zero_zero_real )
% 16.19/9.03 => ( ( sums_real
% 16.19/9.03 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ C )
% 16.19/9.03 @ ( times_times_real @ D @ C ) )
% 16.19/9.03 = ( sums_real @ F @ D ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7875_pi__not__less__zero,axiom,
% 16.19/9.03 ~ ( ord_less_real @ pi @ zero_zero_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7876_pi__gt__zero,axiom,
% 16.19/9.03 ord_less_real @ zero_zero_real @ pi ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7877_pi__ge__zero,axiom,
% 16.19/9.03 ord_less_eq_real @ zero_zero_real @ pi ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7878_sums__mult__D,axiom,
% 16.19/9.03 ! [C: complex,F: nat > complex,A: complex] :
% 16.19/9.03 ( ( sums_complex
% 16.19/9.03 @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) )
% 16.19/9.03 @ A )
% 16.19/9.03 => ( ( C != zero_zero_complex )
% 16.19/9.03 => ( sums_complex @ F @ ( divide1717551699836669952omplex @ A @ C ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7879_sums__mult__D,axiom,
% 16.19/9.03 ! [C: real,F: nat > real,A: real] :
% 16.19/9.03 ( ( sums_real
% 16.19/9.03 @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) )
% 16.19/9.03 @ A )
% 16.19/9.03 => ( ( C != zero_zero_real )
% 16.19/9.03 => ( sums_real @ F @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7880_sums__Suc__imp,axiom,
% 16.19/9.03 ! [F: nat > complex,S3: complex] :
% 16.19/9.03 ( ( ( F @ zero_zero_nat )
% 16.19/9.03 = zero_zero_complex )
% 16.19/9.03 => ( ( sums_complex
% 16.19/9.03 @ ^ [N: nat] : ( F @ ( suc @ N ) )
% 16.19/9.03 @ S3 )
% 16.19/9.03 => ( sums_complex @ F @ S3 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7881_sums__Suc__imp,axiom,
% 16.19/9.03 ! [F: nat > real,S3: real] :
% 16.19/9.03 ( ( ( F @ zero_zero_nat )
% 16.19/9.03 = zero_zero_real )
% 16.19/9.03 => ( ( sums_real
% 16.19/9.03 @ ^ [N: nat] : ( F @ ( suc @ N ) )
% 16.19/9.03 @ S3 )
% 16.19/9.03 => ( sums_real @ F @ S3 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7882_sums__Suc__iff,axiom,
% 16.19/9.03 ! [F: nat > complex,S3: complex] :
% 16.19/9.03 ( ( sums_complex
% 16.19/9.03 @ ^ [N: nat] : ( F @ ( suc @ N ) )
% 16.19/9.03 @ S3 )
% 16.19/9.03 = ( sums_complex @ F @ ( plus_plus_complex @ S3 @ ( F @ zero_zero_nat ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7883_sums__Suc__iff,axiom,
% 16.19/9.03 ! [F: nat > real,S3: real] :
% 16.19/9.03 ( ( sums_real
% 16.19/9.03 @ ^ [N: nat] : ( F @ ( suc @ N ) )
% 16.19/9.03 @ S3 )
% 16.19/9.03 = ( sums_real @ F @ ( plus_plus_real @ S3 @ ( F @ zero_zero_nat ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7884_sums__Suc,axiom,
% 16.19/9.03 ! [F: nat > complex,L2: complex] :
% 16.19/9.03 ( ( sums_complex
% 16.19/9.03 @ ^ [N: nat] : ( F @ ( suc @ N ) )
% 16.19/9.03 @ L2 )
% 16.19/9.03 => ( sums_complex @ F @ ( plus_plus_complex @ L2 @ ( F @ zero_zero_nat ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7885_sums__Suc,axiom,
% 16.19/9.03 ! [F: nat > real,L2: real] :
% 16.19/9.03 ( ( sums_real
% 16.19/9.03 @ ^ [N: nat] : ( F @ ( suc @ N ) )
% 16.19/9.03 @ L2 )
% 16.19/9.03 => ( sums_real @ F @ ( plus_plus_real @ L2 @ ( F @ zero_zero_nat ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7886_sums__Suc,axiom,
% 16.19/9.03 ! [F: nat > nat,L2: nat] :
% 16.19/9.03 ( ( sums_nat
% 16.19/9.03 @ ^ [N: nat] : ( F @ ( suc @ N ) )
% 16.19/9.03 @ L2 )
% 16.19/9.03 => ( sums_nat @ F @ ( plus_plus_nat @ L2 @ ( F @ zero_zero_nat ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7887_sums__Suc,axiom,
% 16.19/9.03 ! [F: nat > int,L2: int] :
% 16.19/9.03 ( ( sums_int
% 16.19/9.03 @ ^ [N: nat] : ( F @ ( suc @ N ) )
% 16.19/9.03 @ L2 )
% 16.19/9.03 => ( sums_int @ F @ ( plus_plus_int @ L2 @ ( F @ zero_zero_nat ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7888_sums__zero__iff__shift,axiom,
% 16.19/9.03 ! [N2: nat,F: nat > complex,S3: complex] :
% 16.19/9.03 ( ! [I4: nat] :
% 16.19/9.03 ( ( ord_less_nat @ I4 @ N2 )
% 16.19/9.03 => ( ( F @ I4 )
% 16.19/9.03 = zero_zero_complex ) )
% 16.19/9.03 => ( ( sums_complex
% 16.19/9.03 @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
% 16.19/9.03 @ S3 )
% 16.19/9.03 = ( sums_complex @ F @ S3 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7889_sums__zero__iff__shift,axiom,
% 16.19/9.03 ! [N2: nat,F: nat > real,S3: real] :
% 16.19/9.03 ( ! [I4: nat] :
% 16.19/9.03 ( ( ord_less_nat @ I4 @ N2 )
% 16.19/9.03 => ( ( F @ I4 )
% 16.19/9.03 = zero_zero_real ) )
% 16.19/9.03 => ( ( sums_real
% 16.19/9.03 @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
% 16.19/9.03 @ S3 )
% 16.19/9.03 = ( sums_real @ F @ S3 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7890_sums__finite,axiom,
% 16.19/9.03 ! [N3: set_nat,F: nat > complex] :
% 16.19/9.03 ( ( finite_finite_nat @ N3 )
% 16.19/9.03 => ( ! [N4: nat] :
% 16.19/9.03 ( ~ ( member_nat @ N4 @ N3 )
% 16.19/9.03 => ( ( F @ N4 )
% 16.19/9.03 = zero_zero_complex ) )
% 16.19/9.03 => ( sums_complex @ F @ ( groups2073611262835488442omplex @ F @ N3 ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7891_sums__finite,axiom,
% 16.19/9.03 ! [N3: set_nat,F: nat > int] :
% 16.19/9.03 ( ( finite_finite_nat @ N3 )
% 16.19/9.03 => ( ! [N4: nat] :
% 16.19/9.03 ( ~ ( member_nat @ N4 @ N3 )
% 16.19/9.03 => ( ( F @ N4 )
% 16.19/9.03 = zero_zero_int ) )
% 16.19/9.03 => ( sums_int @ F @ ( groups3539618377306564664at_int @ F @ N3 ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7892_sums__finite,axiom,
% 16.19/9.03 ! [N3: set_nat,F: nat > nat] :
% 16.19/9.03 ( ( finite_finite_nat @ N3 )
% 16.19/9.03 => ( ! [N4: nat] :
% 16.19/9.03 ( ~ ( member_nat @ N4 @ N3 )
% 16.19/9.03 => ( ( F @ N4 )
% 16.19/9.03 = zero_zero_nat ) )
% 16.19/9.03 => ( sums_nat @ F @ ( groups3542108847815614940at_nat @ F @ N3 ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7893_sums__finite,axiom,
% 16.19/9.03 ! [N3: set_nat,F: nat > real] :
% 16.19/9.03 ( ( finite_finite_nat @ N3 )
% 16.19/9.03 => ( ! [N4: nat] :
% 16.19/9.03 ( ~ ( member_nat @ N4 @ N3 )
% 16.19/9.03 => ( ( F @ N4 )
% 16.19/9.03 = zero_zero_real ) )
% 16.19/9.03 => ( sums_real @ F @ ( groups6591440286371151544t_real @ F @ N3 ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7894_sums__If__finite,axiom,
% 16.19/9.03 ! [P: nat > $o,F: nat > complex] :
% 16.19/9.03 ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 16.19/9.03 => ( sums_complex
% 16.19/9.03 @ ^ [R5: nat] : ( if_complex @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_complex )
% 16.19/9.03 @ ( groups2073611262835488442omplex @ F @ ( collect_nat @ P ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7895_sums__If__finite,axiom,
% 16.19/9.03 ! [P: nat > $o,F: nat > int] :
% 16.19/9.03 ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 16.19/9.03 => ( sums_int
% 16.19/9.03 @ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_int )
% 16.19/9.03 @ ( groups3539618377306564664at_int @ F @ ( collect_nat @ P ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7896_sums__If__finite,axiom,
% 16.19/9.03 ! [P: nat > $o,F: nat > nat] :
% 16.19/9.03 ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 16.19/9.03 => ( sums_nat
% 16.19/9.03 @ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_nat )
% 16.19/9.03 @ ( groups3542108847815614940at_nat @ F @ ( collect_nat @ P ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7897_sums__If__finite,axiom,
% 16.19/9.03 ! [P: nat > $o,F: nat > real] :
% 16.19/9.03 ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 16.19/9.03 => ( sums_real
% 16.19/9.03 @ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_real )
% 16.19/9.03 @ ( groups6591440286371151544t_real @ F @ ( collect_nat @ P ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7898_sums__If__finite__set,axiom,
% 16.19/9.03 ! [A2: set_nat,F: nat > complex] :
% 16.19/9.03 ( ( finite_finite_nat @ A2 )
% 16.19/9.03 => ( sums_complex
% 16.19/9.03 @ ^ [R5: nat] : ( if_complex @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_complex )
% 16.19/9.03 @ ( groups2073611262835488442omplex @ F @ A2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7899_sums__If__finite__set,axiom,
% 16.19/9.03 ! [A2: set_nat,F: nat > int] :
% 16.19/9.03 ( ( finite_finite_nat @ A2 )
% 16.19/9.03 => ( sums_int
% 16.19/9.03 @ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_int )
% 16.19/9.03 @ ( groups3539618377306564664at_int @ F @ A2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7900_sums__If__finite__set,axiom,
% 16.19/9.03 ! [A2: set_nat,F: nat > nat] :
% 16.19/9.03 ( ( finite_finite_nat @ A2 )
% 16.19/9.03 => ( sums_nat
% 16.19/9.03 @ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_nat )
% 16.19/9.03 @ ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7901_sums__If__finite__set,axiom,
% 16.19/9.03 ! [A2: set_nat,F: nat > real] :
% 16.19/9.03 ( ( finite_finite_nat @ A2 )
% 16.19/9.03 => ( sums_real
% 16.19/9.03 @ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_real )
% 16.19/9.03 @ ( groups6591440286371151544t_real @ F @ A2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7902_powser__sums__if,axiom,
% 16.19/9.03 ! [M: nat,Z: complex] :
% 16.19/9.03 ( sums_complex
% 16.19/9.03 @ ^ [N: nat] : ( times_times_complex @ ( if_complex @ ( N = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z @ N ) )
% 16.19/9.03 @ ( power_power_complex @ Z @ M ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7903_powser__sums__if,axiom,
% 16.19/9.03 ! [M: nat,Z: real] :
% 16.19/9.03 ( sums_real
% 16.19/9.03 @ ^ [N: nat] : ( times_times_real @ ( if_real @ ( N = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z @ N ) )
% 16.19/9.03 @ ( power_power_real @ Z @ M ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7904_powser__sums__if,axiom,
% 16.19/9.03 ! [M: nat,Z: int] :
% 16.19/9.03 ( sums_int
% 16.19/9.03 @ ^ [N: nat] : ( times_times_int @ ( if_int @ ( N = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z @ N ) )
% 16.19/9.03 @ ( power_power_int @ Z @ M ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7905_powser__sums__zero,axiom,
% 16.19/9.03 ! [A: nat > complex] :
% 16.19/9.03 ( sums_complex
% 16.19/9.03 @ ^ [N: nat] : ( times_times_complex @ ( A @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) )
% 16.19/9.03 @ ( A @ zero_zero_nat ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7906_powser__sums__zero,axiom,
% 16.19/9.03 ! [A: nat > real] :
% 16.19/9.03 ( sums_real
% 16.19/9.03 @ ^ [N: nat] : ( times_times_real @ ( A @ N ) @ ( power_power_real @ zero_zero_real @ N ) )
% 16.19/9.03 @ ( A @ zero_zero_nat ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7907_sums__iff__shift,axiom,
% 16.19/9.03 ! [F: nat > complex,N2: nat,S3: complex] :
% 16.19/9.03 ( ( sums_complex
% 16.19/9.03 @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
% 16.19/9.03 @ S3 )
% 16.19/9.03 = ( sums_complex @ F @ ( plus_plus_complex @ S3 @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7908_sums__iff__shift,axiom,
% 16.19/9.03 ! [F: nat > real,N2: nat,S3: real] :
% 16.19/9.03 ( ( sums_real
% 16.19/9.03 @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
% 16.19/9.03 @ S3 )
% 16.19/9.03 = ( sums_real @ F @ ( plus_plus_real @ S3 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7909_sums__split__initial__segment,axiom,
% 16.19/9.03 ! [F: nat > complex,S3: complex,N2: nat] :
% 16.19/9.03 ( ( sums_complex @ F @ S3 )
% 16.19/9.03 => ( sums_complex
% 16.19/9.03 @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
% 16.19/9.03 @ ( minus_minus_complex @ S3 @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7910_sums__split__initial__segment,axiom,
% 16.19/9.03 ! [F: nat > real,S3: real,N2: nat] :
% 16.19/9.03 ( ( sums_real @ F @ S3 )
% 16.19/9.03 => ( sums_real
% 16.19/9.03 @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
% 16.19/9.03 @ ( minus_minus_real @ S3 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7911_sums__iff__shift_H,axiom,
% 16.19/9.03 ! [F: nat > complex,N2: nat,S3: complex] :
% 16.19/9.03 ( ( sums_complex
% 16.19/9.03 @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
% 16.19/9.03 @ ( minus_minus_complex @ S3 @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) )
% 16.19/9.03 = ( sums_complex @ F @ S3 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7912_sums__iff__shift_H,axiom,
% 16.19/9.03 ! [F: nat > real,N2: nat,S3: real] :
% 16.19/9.03 ( ( sums_real
% 16.19/9.03 @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
% 16.19/9.03 @ ( minus_minus_real @ S3 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) )
% 16.19/9.03 = ( sums_real @ F @ S3 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7913_sums__If__finite__set_H,axiom,
% 16.19/9.03 ! [G: nat > complex,S: complex,A2: set_nat,S4: complex,F: nat > complex] :
% 16.19/9.03 ( ( sums_complex @ G @ S )
% 16.19/9.03 => ( ( finite_finite_nat @ A2 )
% 16.19/9.03 => ( ( S4
% 16.19/9.03 = ( plus_plus_complex @ S
% 16.19/9.03 @ ( groups2073611262835488442omplex
% 16.19/9.03 @ ^ [N: nat] : ( minus_minus_complex @ ( F @ N ) @ ( G @ N ) )
% 16.19/9.03 @ A2 ) ) )
% 16.19/9.03 => ( sums_complex
% 16.19/9.03 @ ^ [N: nat] : ( if_complex @ ( member_nat @ N @ A2 ) @ ( F @ N ) @ ( G @ N ) )
% 16.19/9.03 @ S4 ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7914_sums__If__finite__set_H,axiom,
% 16.19/9.03 ! [G: nat > real,S: real,A2: set_nat,S4: real,F: nat > real] :
% 16.19/9.03 ( ( sums_real @ G @ S )
% 16.19/9.03 => ( ( finite_finite_nat @ A2 )
% 16.19/9.03 => ( ( S4
% 16.19/9.03 = ( plus_plus_real @ S
% 16.19/9.03 @ ( groups6591440286371151544t_real
% 16.19/9.03 @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( G @ N ) )
% 16.19/9.03 @ A2 ) ) )
% 16.19/9.03 => ( sums_real
% 16.19/9.03 @ ^ [N: nat] : ( if_real @ ( member_nat @ N @ A2 ) @ ( F @ N ) @ ( G @ N ) )
% 16.19/9.03 @ S4 ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7915_pi__less__4,axiom,
% 16.19/9.03 ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7916_pi__ge__two,axiom,
% 16.19/9.03 ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7917_pi__half__neq__two,axiom,
% 16.19/9.03 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 16.19/9.03 != ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7918_pi__half__neq__zero,axiom,
% 16.19/9.03 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 16.19/9.03 != zero_zero_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7919_pi__half__less__two,axiom,
% 16.19/9.03 ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7920_pi__half__le__two,axiom,
% 16.19/9.03 ord_less_eq_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7921_pi__half__gt__zero,axiom,
% 16.19/9.03 ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7922_pi__half__ge__zero,axiom,
% 16.19/9.03 ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7923_m2pi__less__pi,axiom,
% 16.19/9.03 ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7924_arctan__ubound,axiom,
% 16.19/9.03 ! [Y: real] : ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7925_arctan__one,axiom,
% 16.19/9.03 ( ( arctan @ one_one_real )
% 16.19/9.03 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7926_geometric__sums,axiom,
% 16.19/9.03 ! [C: real] :
% 16.19/9.03 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 16.19/9.03 => ( sums_real @ ( power_power_real @ C ) @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7927_geometric__sums,axiom,
% 16.19/9.03 ! [C: complex] :
% 16.19/9.03 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 16.19/9.03 => ( sums_complex @ ( power_power_complex @ C ) @ ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7928_power__half__series,axiom,
% 16.19/9.03 ( sums_real
% 16.19/9.03 @ ^ [N: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N ) )
% 16.19/9.03 @ one_one_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7929_minus__pi__half__less__zero,axiom,
% 16.19/9.03 ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7930_arctan__lbound,axiom,
% 16.19/9.03 ! [Y: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7931_arctan__bounded,axiom,
% 16.19/9.03 ! [Y: real] :
% 16.19/9.03 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
% 16.19/9.03 & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7932_sums__if_H,axiom,
% 16.19/9.03 ! [G: nat > real,X: real] :
% 16.19/9.03 ( ( sums_real @ G @ X )
% 16.19/9.03 => ( sums_real
% 16.19/9.03 @ ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/9.03 @ X ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7933_unbounded__k__infinite,axiom,
% 16.19/9.03 ! [K: nat,S: set_nat] :
% 16.19/9.03 ( ! [M2: nat] :
% 16.19/9.03 ( ( ord_less_nat @ K @ M2 )
% 16.19/9.03 => ? [N9: nat] :
% 16.19/9.03 ( ( ord_less_nat @ M2 @ N9 )
% 16.19/9.03 & ( member_nat @ N9 @ S ) ) )
% 16.19/9.03 => ~ ( finite_finite_nat @ S ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7934_infinite__nat__iff__unbounded,axiom,
% 16.19/9.03 ! [S: set_nat] :
% 16.19/9.03 ( ( ~ ( finite_finite_nat @ S ) )
% 16.19/9.03 = ( ! [M6: nat] :
% 16.19/9.03 ? [N: nat] :
% 16.19/9.03 ( ( ord_less_nat @ M6 @ N )
% 16.19/9.03 & ( member_nat @ N @ S ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7935_sums__if,axiom,
% 16.19/9.03 ! [G: nat > real,X: real,F: nat > real,Y: real] :
% 16.19/9.03 ( ( sums_real @ G @ X )
% 16.19/9.03 => ( ( sums_real @ F @ Y )
% 16.19/9.03 => ( sums_real
% 16.19/9.03 @ ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( F @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/9.03 @ ( plus_plus_real @ X @ Y ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7936_infinite__nat__iff__unbounded__le,axiom,
% 16.19/9.03 ! [S: set_nat] :
% 16.19/9.03 ( ( ~ ( finite_finite_nat @ S ) )
% 16.19/9.03 = ( ! [M6: nat] :
% 16.19/9.03 ? [N: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M6 @ N )
% 16.19/9.03 & ( member_nat @ N @ S ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7937_machin__Euler,axiom,
% 16.19/9.03 ( ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 16.19/9.03 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7938_machin,axiom,
% 16.19/9.03 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 16.19/9.03 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7939_sin__cos__npi,axiom,
% 16.19/9.03 ! [N2: nat] :
% 16.19/9.03 ( ( sin_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7940_diffs__equiv,axiom,
% 16.19/9.03 ! [C: nat > real,X: real] :
% 16.19/9.03 ( ( summable_real
% 16.19/9.03 @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X @ N ) ) )
% 16.19/9.03 => ( sums_real
% 16.19/9.03 @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( C @ N ) ) @ ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
% 16.19/9.03 @ ( suminf_real
% 16.19/9.03 @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7941_diffs__equiv,axiom,
% 16.19/9.03 ! [C: nat > complex,X: complex] :
% 16.19/9.03 ( ( summable_complex
% 16.19/9.03 @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X @ N ) ) )
% 16.19/9.03 => ( sums_complex
% 16.19/9.03 @ ^ [N: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( C @ N ) ) @ ( power_power_complex @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
% 16.19/9.03 @ ( suminf_complex
% 16.19/9.03 @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X @ N ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7942_cos__pi__eq__zero,axiom,
% 16.19/9.03 ! [M: nat] :
% 16.19/9.03 ( ( cos_real @ ( divide_divide_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 = zero_zero_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7943_monoI1,axiom,
% 16.19/9.03 ! [X8: nat > real] :
% 16.19/9.03 ( ! [M2: nat,N4: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M2 @ N4 )
% 16.19/9.03 => ( ord_less_eq_real @ ( X8 @ M2 ) @ ( X8 @ N4 ) ) )
% 16.19/9.03 => ( topolo6980174941875973593q_real @ X8 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7944_monoI1,axiom,
% 16.19/9.03 ! [X8: nat > set_nat] :
% 16.19/9.03 ( ! [M2: nat,N4: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M2 @ N4 )
% 16.19/9.03 => ( ord_less_eq_set_nat @ ( X8 @ M2 ) @ ( X8 @ N4 ) ) )
% 16.19/9.03 => ( topolo7278393974255667507et_nat @ X8 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7945_monoI1,axiom,
% 16.19/9.03 ! [X8: nat > rat] :
% 16.19/9.03 ( ! [M2: nat,N4: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M2 @ N4 )
% 16.19/9.03 => ( ord_less_eq_rat @ ( X8 @ M2 ) @ ( X8 @ N4 ) ) )
% 16.19/9.03 => ( topolo4267028734544971653eq_rat @ X8 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7946_monoI1,axiom,
% 16.19/9.03 ! [X8: nat > num] :
% 16.19/9.03 ( ! [M2: nat,N4: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M2 @ N4 )
% 16.19/9.03 => ( ord_less_eq_num @ ( X8 @ M2 ) @ ( X8 @ N4 ) ) )
% 16.19/9.03 => ( topolo1459490580787246023eq_num @ X8 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7947_monoI1,axiom,
% 16.19/9.03 ! [X8: nat > nat] :
% 16.19/9.03 ( ! [M2: nat,N4: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M2 @ N4 )
% 16.19/9.03 => ( ord_less_eq_nat @ ( X8 @ M2 ) @ ( X8 @ N4 ) ) )
% 16.19/9.03 => ( topolo4902158794631467389eq_nat @ X8 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7948_monoI1,axiom,
% 16.19/9.03 ! [X8: nat > int] :
% 16.19/9.03 ( ! [M2: nat,N4: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M2 @ N4 )
% 16.19/9.03 => ( ord_less_eq_int @ ( X8 @ M2 ) @ ( X8 @ N4 ) ) )
% 16.19/9.03 => ( topolo4899668324122417113eq_int @ X8 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7949_monoI2,axiom,
% 16.19/9.03 ! [X8: nat > real] :
% 16.19/9.03 ( ! [M2: nat,N4: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M2 @ N4 )
% 16.19/9.03 => ( ord_less_eq_real @ ( X8 @ N4 ) @ ( X8 @ M2 ) ) )
% 16.19/9.03 => ( topolo6980174941875973593q_real @ X8 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7950_monoI2,axiom,
% 16.19/9.03 ! [X8: nat > set_nat] :
% 16.19/9.03 ( ! [M2: nat,N4: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M2 @ N4 )
% 16.19/9.03 => ( ord_less_eq_set_nat @ ( X8 @ N4 ) @ ( X8 @ M2 ) ) )
% 16.19/9.03 => ( topolo7278393974255667507et_nat @ X8 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7951_monoI2,axiom,
% 16.19/9.03 ! [X8: nat > rat] :
% 16.19/9.03 ( ! [M2: nat,N4: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M2 @ N4 )
% 16.19/9.03 => ( ord_less_eq_rat @ ( X8 @ N4 ) @ ( X8 @ M2 ) ) )
% 16.19/9.03 => ( topolo4267028734544971653eq_rat @ X8 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7952_monoI2,axiom,
% 16.19/9.03 ! [X8: nat > num] :
% 16.19/9.03 ( ! [M2: nat,N4: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M2 @ N4 )
% 16.19/9.03 => ( ord_less_eq_num @ ( X8 @ N4 ) @ ( X8 @ M2 ) ) )
% 16.19/9.03 => ( topolo1459490580787246023eq_num @ X8 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7953_monoI2,axiom,
% 16.19/9.03 ! [X8: nat > nat] :
% 16.19/9.03 ( ! [M2: nat,N4: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M2 @ N4 )
% 16.19/9.03 => ( ord_less_eq_nat @ ( X8 @ N4 ) @ ( X8 @ M2 ) ) )
% 16.19/9.03 => ( topolo4902158794631467389eq_nat @ X8 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7954_monoI2,axiom,
% 16.19/9.03 ! [X8: nat > int] :
% 16.19/9.03 ( ! [M2: nat,N4: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M2 @ N4 )
% 16.19/9.03 => ( ord_less_eq_int @ ( X8 @ N4 ) @ ( X8 @ M2 ) ) )
% 16.19/9.03 => ( topolo4899668324122417113eq_int @ X8 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7955_cos__zero,axiom,
% 16.19/9.03 ( ( cos_complex @ zero_zero_complex )
% 16.19/9.03 = one_one_complex ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7956_cos__zero,axiom,
% 16.19/9.03 ( ( cos_real @ zero_zero_real )
% 16.19/9.03 = one_one_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7957_sin__pi__minus,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( sin_real @ ( minus_minus_real @ pi @ X ) )
% 16.19/9.03 = ( sin_real @ X ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7958_cos__pi,axiom,
% 16.19/9.03 ( ( cos_real @ pi )
% 16.19/9.03 = ( uminus_uminus_real @ one_one_real ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7959_cos__periodic__pi,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( cos_real @ ( plus_plus_real @ X @ pi ) )
% 16.19/9.03 = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7960_cos__periodic__pi2,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( cos_real @ ( plus_plus_real @ pi @ X ) )
% 16.19/9.03 = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7961_sin__periodic__pi,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( sin_real @ ( plus_plus_real @ X @ pi ) )
% 16.19/9.03 = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7962_sin__periodic__pi2,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( sin_real @ ( plus_plus_real @ pi @ X ) )
% 16.19/9.03 = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7963_cos__minus__pi,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( cos_real @ ( minus_minus_real @ X @ pi ) )
% 16.19/9.03 = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7964_cos__pi__minus,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( cos_real @ ( minus_minus_real @ pi @ X ) )
% 16.19/9.03 = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7965_sin__minus__pi,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( sin_real @ ( minus_minus_real @ X @ pi ) )
% 16.19/9.03 = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7966_sin__cos__squared__add3,axiom,
% 16.19/9.03 ! [X: complex] :
% 16.19/9.03 ( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ X ) ) @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ X ) ) )
% 16.19/9.03 = one_one_complex ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7967_sin__cos__squared__add3,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ X ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ X ) ) )
% 16.19/9.03 = one_one_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7968_sin__npi2,axiom,
% 16.19/9.03 ! [N2: nat] :
% 16.19/9.03 ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 16.19/9.03 = zero_zero_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7969_sin__npi,axiom,
% 16.19/9.03 ! [N2: nat] :
% 16.19/9.03 ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) )
% 16.19/9.03 = zero_zero_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7970_sin__npi__int,axiom,
% 16.19/9.03 ! [N2: int] :
% 16.19/9.03 ( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N2 ) ) )
% 16.19/9.03 = zero_zero_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7971_cos__pi__half,axiom,
% 16.19/9.03 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 = zero_zero_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7972_sin__two__pi,axiom,
% 16.19/9.03 ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 16.19/9.03 = zero_zero_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7973_sin__pi__half,axiom,
% 16.19/9.03 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 = one_one_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7974_cos__two__pi,axiom,
% 16.19/9.03 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 16.19/9.03 = one_one_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7975_cos__periodic,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( cos_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 16.19/9.03 = ( cos_real @ X ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7976_sin__periodic,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( sin_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 16.19/9.03 = ( sin_real @ X ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7977_cos__2pi__minus,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
% 16.19/9.03 = ( cos_real @ X ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7978_cos__npi,axiom,
% 16.19/9.03 ! [N2: nat] :
% 16.19/9.03 ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) )
% 16.19/9.03 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7979_cos__npi2,axiom,
% 16.19/9.03 ! [N2: nat] :
% 16.19/9.03 ( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 16.19/9.03 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7980_sin__cos__squared__add,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( plus_plus_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.03 = one_one_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7981_sin__cos__squared__add,axiom,
% 16.19/9.03 ! [X: complex] :
% 16.19/9.03 ( ( plus_plus_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.03 = one_one_complex ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7982_sin__cos__squared__add2,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( plus_plus_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.03 = one_one_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7983_sin__cos__squared__add2,axiom,
% 16.19/9.03 ! [X: complex] :
% 16.19/9.03 ( ( plus_plus_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.03 = one_one_complex ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7984_sin__2npi,axiom,
% 16.19/9.03 ! [N2: nat] :
% 16.19/9.03 ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) )
% 16.19/9.03 = zero_zero_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7985_cos__2npi,axiom,
% 16.19/9.03 ! [N2: nat] :
% 16.19/9.03 ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) )
% 16.19/9.03 = one_one_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7986_sin__2pi__minus,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
% 16.19/9.03 = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7987_sin__int__2pin,axiom,
% 16.19/9.03 ! [N2: int] :
% 16.19/9.03 ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N2 ) ) )
% 16.19/9.03 = zero_zero_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7988_cos__int__2pin,axiom,
% 16.19/9.03 ! [N2: int] :
% 16.19/9.03 ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N2 ) ) )
% 16.19/9.03 = one_one_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7989_cos__3over2__pi,axiom,
% 16.19/9.03 ( ( cos_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 16.19/9.03 = zero_zero_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7990_sin__3over2__pi,axiom,
% 16.19/9.03 ( ( sin_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 16.19/9.03 = ( uminus_uminus_real @ one_one_real ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7991_cos__npi__int,axiom,
% 16.19/9.03 ! [N2: int] :
% 16.19/9.03 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.03 => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N2 ) ) )
% 16.19/9.03 = one_one_real ) )
% 16.19/9.03 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.03 => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N2 ) ) )
% 16.19/9.03 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7992_cos__one__sin__zero,axiom,
% 16.19/9.03 ! [X: complex] :
% 16.19/9.03 ( ( ( cos_complex @ X )
% 16.19/9.03 = one_one_complex )
% 16.19/9.03 => ( ( sin_complex @ X )
% 16.19/9.03 = zero_zero_complex ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7993_cos__one__sin__zero,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ( cos_real @ X )
% 16.19/9.03 = one_one_real )
% 16.19/9.03 => ( ( sin_real @ X )
% 16.19/9.03 = zero_zero_real ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7994_polar__Ex,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ? [R3: real,A3: real] :
% 16.19/9.03 ( ( X
% 16.19/9.03 = ( times_times_real @ R3 @ ( cos_real @ A3 ) ) )
% 16.19/9.03 & ( Y
% 16.19/9.03 = ( times_times_real @ R3 @ ( sin_real @ A3 ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7995_sin__diff,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( sin_real @ ( minus_minus_real @ X @ Y ) )
% 16.19/9.03 = ( minus_minus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7996_sin__add,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( sin_real @ ( plus_plus_real @ X @ Y ) )
% 16.19/9.03 = ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7997_cos__add,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( cos_real @ ( plus_plus_real @ X @ Y ) )
% 16.19/9.03 = ( minus_minus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7998_cos__diff,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( cos_real @ ( minus_minus_real @ X @ Y ) )
% 16.19/9.03 = ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_7999_sin__zero__norm__cos__one,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ( sin_real @ X )
% 16.19/9.03 = zero_zero_real )
% 16.19/9.03 => ( ( real_V7735802525324610683m_real @ ( cos_real @ X ) )
% 16.19/9.03 = one_one_real ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8000_sin__zero__norm__cos__one,axiom,
% 16.19/9.03 ! [X: complex] :
% 16.19/9.03 ( ( ( sin_complex @ X )
% 16.19/9.03 = zero_zero_complex )
% 16.19/9.03 => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X ) )
% 16.19/9.03 = one_one_real ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8001_sin__zero__abs__cos__one,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ( sin_real @ X )
% 16.19/9.03 = zero_zero_real )
% 16.19/9.03 => ( ( abs_abs_real @ ( cos_real @ X ) )
% 16.19/9.03 = one_one_real ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8002_sin__double,axiom,
% 16.19/9.03 ! [X: complex] :
% 16.19/9.03 ( ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 16.19/9.03 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ X ) ) @ ( cos_complex @ X ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8003_sin__double,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 16.19/9.03 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ X ) ) @ ( cos_real @ X ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8004_sincos__principal__value,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ? [Y4: real] :
% 16.19/9.03 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y4 )
% 16.19/9.03 & ( ord_less_eq_real @ Y4 @ pi )
% 16.19/9.03 & ( ( sin_real @ Y4 )
% 16.19/9.03 = ( sin_real @ X ) )
% 16.19/9.03 & ( ( cos_real @ Y4 )
% 16.19/9.03 = ( cos_real @ X ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8005_sin__x__le__x,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.03 => ( ord_less_eq_real @ ( sin_real @ X ) @ X ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8006_sin__le__one,axiom,
% 16.19/9.03 ! [X: real] : ( ord_less_eq_real @ ( sin_real @ X ) @ one_one_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8007_cos__le__one,axiom,
% 16.19/9.03 ! [X: real] : ( ord_less_eq_real @ ( cos_real @ X ) @ one_one_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8008_abs__sin__x__le__abs__x,axiom,
% 16.19/9.03 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ ( abs_abs_real @ X ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8009_diffs__minus,axiom,
% 16.19/9.03 ! [C: nat > real] :
% 16.19/9.03 ( ( diffs_real
% 16.19/9.03 @ ^ [N: nat] : ( uminus_uminus_real @ ( C @ N ) ) )
% 16.19/9.03 = ( ^ [N: nat] : ( uminus_uminus_real @ ( diffs_real @ C @ N ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8010_diffs__minus,axiom,
% 16.19/9.03 ! [C: nat > int] :
% 16.19/9.03 ( ( diffs_int
% 16.19/9.03 @ ^ [N: nat] : ( uminus_uminus_int @ ( C @ N ) ) )
% 16.19/9.03 = ( ^ [N: nat] : ( uminus_uminus_int @ ( diffs_int @ C @ N ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8011_diffs__minus,axiom,
% 16.19/9.03 ! [C: nat > complex] :
% 16.19/9.03 ( ( diffs_complex
% 16.19/9.03 @ ^ [N: nat] : ( uminus1482373934393186551omplex @ ( C @ N ) ) )
% 16.19/9.03 = ( ^ [N: nat] : ( uminus1482373934393186551omplex @ ( diffs_complex @ C @ N ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8012_diffs__minus,axiom,
% 16.19/9.03 ! [C: nat > rat] :
% 16.19/9.03 ( ( diffs_rat
% 16.19/9.03 @ ^ [N: nat] : ( uminus_uminus_rat @ ( C @ N ) ) )
% 16.19/9.03 = ( ^ [N: nat] : ( uminus_uminus_rat @ ( diffs_rat @ C @ N ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8013_diffs__minus,axiom,
% 16.19/9.03 ! [C: nat > code_integer] :
% 16.19/9.03 ( ( diffs_Code_integer
% 16.19/9.03 @ ^ [N: nat] : ( uminus1351360451143612070nteger @ ( C @ N ) ) )
% 16.19/9.03 = ( ^ [N: nat] : ( uminus1351360451143612070nteger @ ( diffs_Code_integer @ C @ N ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8014_sin__cos__le1,axiom,
% 16.19/9.03 ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) @ one_one_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8015_cos__squared__eq,axiom,
% 16.19/9.03 ! [X: complex] :
% 16.19/9.03 ( ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.03 = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8016_cos__squared__eq,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.03 = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8017_sin__squared__eq,axiom,
% 16.19/9.03 ! [X: complex] :
% 16.19/9.03 ( ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.03 = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8018_sin__squared__eq,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.03 = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8019_sin__gt__zero,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.03 => ( ( ord_less_real @ X @ pi )
% 16.19/9.03 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8020_sin__x__ge__neg__x,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.03 => ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ ( sin_real @ X ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8021_sin__ge__zero,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.03 => ( ( ord_less_eq_real @ X @ pi )
% 16.19/9.03 => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8022_sin__ge__minus__one,axiom,
% 16.19/9.03 ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8023_cos__inj__pi,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.03 => ( ( ord_less_eq_real @ X @ pi )
% 16.19/9.03 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/9.03 => ( ( ord_less_eq_real @ Y @ pi )
% 16.19/9.03 => ( ( ( cos_real @ X )
% 16.19/9.03 = ( cos_real @ Y ) )
% 16.19/9.03 => ( X = Y ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8024_cos__mono__le__eq,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.03 => ( ( ord_less_eq_real @ X @ pi )
% 16.19/9.03 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/9.03 => ( ( ord_less_eq_real @ Y @ pi )
% 16.19/9.03 => ( ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
% 16.19/9.03 = ( ord_less_eq_real @ Y @ X ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8025_cos__monotone__0__pi__le,axiom,
% 16.19/9.03 ! [Y: real,X: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/9.03 => ( ( ord_less_eq_real @ Y @ X )
% 16.19/9.03 => ( ( ord_less_eq_real @ X @ pi )
% 16.19/9.03 => ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8026_cos__ge__minus__one,axiom,
% 16.19/9.03 ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8027_abs__sin__le__one,axiom,
% 16.19/9.03 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ one_one_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8028_abs__cos__le__one,axiom,
% 16.19/9.03 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X ) ) @ one_one_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8029_cos__diff__cos,axiom,
% 16.19/9.03 ! [W: complex,Z: complex] :
% 16.19/9.03 ( ( minus_minus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 16.19/9.03 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ Z @ W ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8030_cos__diff__cos,axiom,
% 16.19/9.03 ! [W: real,Z: real] :
% 16.19/9.03 ( ( minus_minus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 16.19/9.03 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ Z @ W ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8031_sin__diff__sin,axiom,
% 16.19/9.03 ! [W: complex,Z: complex] :
% 16.19/9.03 ( ( minus_minus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 16.19/9.03 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8032_sin__diff__sin,axiom,
% 16.19/9.03 ! [W: real,Z: real] :
% 16.19/9.03 ( ( minus_minus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 16.19/9.03 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8033_sin__plus__sin,axiom,
% 16.19/9.03 ! [W: complex,Z: complex] :
% 16.19/9.03 ( ( plus_plus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 16.19/9.03 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8034_sin__plus__sin,axiom,
% 16.19/9.03 ! [W: real,Z: real] :
% 16.19/9.03 ( ( plus_plus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 16.19/9.03 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8035_cos__times__sin,axiom,
% 16.19/9.03 ! [W: complex,Z: complex] :
% 16.19/9.03 ( ( times_times_complex @ ( cos_complex @ W ) @ ( sin_complex @ Z ) )
% 16.19/9.03 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8036_cos__times__sin,axiom,
% 16.19/9.03 ! [W: real,Z: real] :
% 16.19/9.03 ( ( times_times_real @ ( cos_real @ W ) @ ( sin_real @ Z ) )
% 16.19/9.03 = ( divide_divide_real @ ( minus_minus_real @ ( sin_real @ ( plus_plus_real @ W @ Z ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8037_sin__times__cos,axiom,
% 16.19/9.03 ! [W: complex,Z: complex] :
% 16.19/9.03 ( ( times_times_complex @ ( sin_complex @ W ) @ ( cos_complex @ Z ) )
% 16.19/9.03 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8038_sin__times__cos,axiom,
% 16.19/9.03 ! [W: real,Z: real] :
% 16.19/9.03 ( ( times_times_real @ ( sin_real @ W ) @ ( cos_real @ Z ) )
% 16.19/9.03 = ( divide_divide_real @ ( plus_plus_real @ ( sin_real @ ( plus_plus_real @ W @ Z ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8039_sin__times__sin,axiom,
% 16.19/9.03 ! [W: complex,Z: complex] :
% 16.19/9.03 ( ( times_times_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 16.19/9.03 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8040_sin__times__sin,axiom,
% 16.19/9.03 ! [W: real,Z: real] :
% 16.19/9.03 ( ( times_times_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 16.19/9.03 = ( divide_divide_real @ ( minus_minus_real @ ( cos_real @ ( minus_minus_real @ W @ Z ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8041_cos__double,axiom,
% 16.19/9.03 ! [X: complex] :
% 16.19/9.03 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 16.19/9.03 = ( minus_minus_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8042_cos__double,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 16.19/9.03 = ( minus_minus_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8043_cos__double__sin,axiom,
% 16.19/9.03 ! [W: complex] :
% 16.19/9.03 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 16.19/9.03 = ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( sin_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8044_cos__double__sin,axiom,
% 16.19/9.03 ! [W: real] :
% 16.19/9.03 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 16.19/9.03 = ( minus_minus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( sin_real @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8045_cos__two__neq__zero,axiom,
% 16.19/9.03 ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 16.19/9.03 != zero_zero_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8046_cos__monotone__0__pi,axiom,
% 16.19/9.03 ! [Y: real,X: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/9.03 => ( ( ord_less_real @ Y @ X )
% 16.19/9.03 => ( ( ord_less_eq_real @ X @ pi )
% 16.19/9.03 => ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8047_cos__mono__less__eq,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.03 => ( ( ord_less_eq_real @ X @ pi )
% 16.19/9.03 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/9.03 => ( ( ord_less_eq_real @ Y @ pi )
% 16.19/9.03 => ( ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y ) )
% 16.19/9.03 = ( ord_less_real @ Y @ X ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8048_sin__eq__0__pi,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
% 16.19/9.03 => ( ( ord_less_real @ X @ pi )
% 16.19/9.03 => ( ( ( sin_real @ X )
% 16.19/9.03 = zero_zero_real )
% 16.19/9.03 => ( X = zero_zero_real ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8049_sin__zero__pi__iff,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_real @ ( abs_abs_real @ X ) @ pi )
% 16.19/9.03 => ( ( ( sin_real @ X )
% 16.19/9.03 = zero_zero_real )
% 16.19/9.03 = ( X = zero_zero_real ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8050_cos__monotone__minus__pi__0_H,axiom,
% 16.19/9.03 ! [Y: real,X: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
% 16.19/9.03 => ( ( ord_less_eq_real @ Y @ X )
% 16.19/9.03 => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 16.19/9.03 => ( ord_less_eq_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8051_sin__zero__iff__int2,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ( sin_real @ X )
% 16.19/9.03 = zero_zero_real )
% 16.19/9.03 = ( ? [I3: int] :
% 16.19/9.03 ( X
% 16.19/9.03 = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ pi ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8052_diffs__def,axiom,
% 16.19/9.03 ( diffs_rat
% 16.19/9.03 = ( ^ [C4: nat > rat,N: nat] : ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( C4 @ ( suc @ N ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8053_diffs__def,axiom,
% 16.19/9.03 ( diffs_int
% 16.19/9.03 = ( ^ [C4: nat > int,N: nat] : ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( C4 @ ( suc @ N ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8054_diffs__def,axiom,
% 16.19/9.03 ( diffs_real
% 16.19/9.03 = ( ^ [C4: nat > real,N: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( C4 @ ( suc @ N ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8055_diffs__def,axiom,
% 16.19/9.03 ( diffs_complex
% 16.19/9.03 = ( ^ [C4: nat > complex,N: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( C4 @ ( suc @ N ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8056_sincos__total__pi,axiom,
% 16.19/9.03 ! [Y: real,X: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/9.03 => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.03 = one_one_real )
% 16.19/9.03 => ? [T5: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 16.19/9.03 & ( ord_less_eq_real @ T5 @ pi )
% 16.19/9.03 & ( X
% 16.19/9.03 = ( cos_real @ T5 ) )
% 16.19/9.03 & ( Y
% 16.19/9.03 = ( sin_real @ T5 ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8057_sin__expansion__lemma,axiom,
% 16.19/9.03 ! [X: real,M: nat] :
% 16.19/9.03 ( ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 16.19/9.03 = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8058_cos__expansion__lemma,axiom,
% 16.19/9.03 ! [X: real,M: nat] :
% 16.19/9.03 ( ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 16.19/9.03 = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8059_termdiff__converges__all,axiom,
% 16.19/9.03 ! [C: nat > complex,X: complex] :
% 16.19/9.03 ( ! [X4: complex] :
% 16.19/9.03 ( summable_complex
% 16.19/9.03 @ ^ [N: nat] : ( times_times_complex @ ( C @ N ) @ ( power_power_complex @ X4 @ N ) ) )
% 16.19/9.03 => ( summable_complex
% 16.19/9.03 @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X @ N ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8060_termdiff__converges__all,axiom,
% 16.19/9.03 ! [C: nat > real,X: real] :
% 16.19/9.03 ( ! [X4: real] :
% 16.19/9.03 ( summable_real
% 16.19/9.03 @ ^ [N: nat] : ( times_times_real @ ( C @ N ) @ ( power_power_real @ X4 @ N ) ) )
% 16.19/9.03 => ( summable_real
% 16.19/9.03 @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X @ N ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8061_sin__gt__zero__02,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.03 => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 16.19/9.03 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8062_cos__two__less__zero,axiom,
% 16.19/9.03 ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8063_cos__two__le__zero,axiom,
% 16.19/9.03 ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8064_cos__is__zero,axiom,
% 16.19/9.03 ? [X4: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 16.19/9.03 & ( ord_less_eq_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 16.19/9.03 & ( ( cos_real @ X4 )
% 16.19/9.03 = zero_zero_real )
% 16.19/9.03 & ! [Y5: real] :
% 16.19/9.03 ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 16.19/9.03 & ( ord_less_eq_real @ Y5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 16.19/9.03 & ( ( cos_real @ Y5 )
% 16.19/9.03 = zero_zero_real ) )
% 16.19/9.03 => ( Y5 = X4 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8065_cos__monotone__minus__pi__0,axiom,
% 16.19/9.03 ! [Y: real,X: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
% 16.19/9.03 => ( ( ord_less_real @ Y @ X )
% 16.19/9.03 => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 16.19/9.03 => ( ord_less_real @ ( cos_real @ Y ) @ ( cos_real @ X ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8066_cos__total,axiom,
% 16.19/9.03 ! [Y: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 16.19/9.03 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 16.19/9.03 => ? [X4: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 16.19/9.03 & ( ord_less_eq_real @ X4 @ pi )
% 16.19/9.03 & ( ( cos_real @ X4 )
% 16.19/9.03 = Y )
% 16.19/9.03 & ! [Y5: real] :
% 16.19/9.03 ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 16.19/9.03 & ( ord_less_eq_real @ Y5 @ pi )
% 16.19/9.03 & ( ( cos_real @ Y5 )
% 16.19/9.03 = Y ) )
% 16.19/9.03 => ( Y5 = X4 ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8067_sincos__total__pi__half,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.03 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/9.03 => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.03 = one_one_real )
% 16.19/9.03 => ? [T5: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 16.19/9.03 & ( ord_less_eq_real @ T5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 & ( X
% 16.19/9.03 = ( cos_real @ T5 ) )
% 16.19/9.03 & ( Y
% 16.19/9.03 = ( sin_real @ T5 ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8068_sincos__total__2pi__le,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.03 = one_one_real )
% 16.19/9.03 => ? [T5: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 16.19/9.03 & ( ord_less_eq_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 16.19/9.03 & ( X
% 16.19/9.03 = ( cos_real @ T5 ) )
% 16.19/9.03 & ( Y
% 16.19/9.03 = ( sin_real @ T5 ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8069_sincos__total__2pi,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.03 = one_one_real )
% 16.19/9.03 => ~ ! [T5: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 16.19/9.03 => ( ( ord_less_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 16.19/9.03 => ( ( X
% 16.19/9.03 = ( cos_real @ T5 ) )
% 16.19/9.03 => ( Y
% 16.19/9.03 != ( sin_real @ T5 ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8070_sin__pi__divide__n__ge__0,axiom,
% 16.19/9.03 ! [N2: nat] :
% 16.19/9.03 ( ( N2 != zero_zero_nat )
% 16.19/9.03 => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8071_cos__plus__cos,axiom,
% 16.19/9.03 ! [W: complex,Z: complex] :
% 16.19/9.03 ( ( plus_plus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 16.19/9.03 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8072_cos__plus__cos,axiom,
% 16.19/9.03 ! [W: real,Z: real] :
% 16.19/9.03 ( ( plus_plus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 16.19/9.03 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8073_cos__times__cos,axiom,
% 16.19/9.03 ! [W: complex,Z: complex] :
% 16.19/9.03 ( ( times_times_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 16.19/9.03 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8074_cos__times__cos,axiom,
% 16.19/9.03 ! [W: real,Z: real] :
% 16.19/9.03 ( ( times_times_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 16.19/9.03 = ( divide_divide_real @ ( plus_plus_real @ ( cos_real @ ( minus_minus_real @ W @ Z ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8075_sin__gt__zero2,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.03 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8076_sin__lt__zero,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_real @ pi @ X )
% 16.19/9.03 => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 16.19/9.03 => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8077_cos__double__less__one,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.03 => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 16.19/9.03 => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8078_sin__30,axiom,
% 16.19/9.03 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 16.19/9.03 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8079_cos__gt__zero,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.03 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8080_sin__inj__pi,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 16.19/9.03 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 16.19/9.03 => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 => ( ( ( sin_real @ X )
% 16.19/9.03 = ( sin_real @ Y ) )
% 16.19/9.03 => ( X = Y ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8081_sin__mono__le__eq,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 16.19/9.03 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 16.19/9.03 => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 => ( ( ord_less_eq_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
% 16.19/9.03 = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8082_sin__monotone__2pi__le,axiom,
% 16.19/9.03 ! [Y: real,X: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 16.19/9.03 => ( ( ord_less_eq_real @ Y @ X )
% 16.19/9.03 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 => ( ord_less_eq_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8083_cos__60,axiom,
% 16.19/9.03 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 16.19/9.03 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8084_cos__one__2pi__int,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ( cos_real @ X )
% 16.19/9.03 = one_one_real )
% 16.19/9.03 = ( ? [X2: int] :
% 16.19/9.03 ( X
% 16.19/9.03 = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8085_cos__double__cos,axiom,
% 16.19/9.03 ! [W: complex] :
% 16.19/9.03 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 16.19/9.03 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8086_cos__double__cos,axiom,
% 16.19/9.03 ! [W: real] :
% 16.19/9.03 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 16.19/9.03 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( cos_real @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_real ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8087_cos__treble__cos,axiom,
% 16.19/9.03 ! [X: complex] :
% 16.19/9.03 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ X ) )
% 16.19/9.03 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ ( cos_complex @ X ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8088_cos__treble__cos,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ X ) )
% 16.19/9.03 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( cos_real @ X ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8089_termdiff__converges,axiom,
% 16.19/9.03 ! [X: real,K5: real,C: nat > real] :
% 16.19/9.03 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ K5 )
% 16.19/9.03 => ( ! [X4: real] :
% 16.19/9.03 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X4 ) @ K5 )
% 16.19/9.03 => ( summable_real
% 16.19/9.03 @ ^ [N: nat] : ( times_times_real @ ( C @ N ) @ ( power_power_real @ X4 @ N ) ) ) )
% 16.19/9.03 => ( summable_real
% 16.19/9.03 @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8090_termdiff__converges,axiom,
% 16.19/9.03 ! [X: complex,K5: real,C: nat > complex] :
% 16.19/9.03 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ K5 )
% 16.19/9.03 => ( ! [X4: complex] :
% 16.19/9.03 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X4 ) @ K5 )
% 16.19/9.03 => ( summable_complex
% 16.19/9.03 @ ^ [N: nat] : ( times_times_complex @ ( C @ N ) @ ( power_power_complex @ X4 @ N ) ) ) )
% 16.19/9.03 => ( summable_complex
% 16.19/9.03 @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X @ N ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8091_sin__le__zero,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ pi @ X )
% 16.19/9.03 => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 16.19/9.03 => ( ord_less_eq_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8092_sin__less__zero,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 16.19/9.03 => ( ( ord_less_real @ X @ zero_zero_real )
% 16.19/9.03 => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8093_sin__mono__less__eq,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 16.19/9.03 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 16.19/9.03 => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 => ( ( ord_less_real @ ( sin_real @ X ) @ ( sin_real @ Y ) )
% 16.19/9.03 = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8094_sin__monotone__2pi,axiom,
% 16.19/9.03 ! [Y: real,X: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 16.19/9.03 => ( ( ord_less_real @ Y @ X )
% 16.19/9.03 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 => ( ord_less_real @ ( sin_real @ Y ) @ ( sin_real @ X ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8095_sin__total,axiom,
% 16.19/9.03 ! [Y: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 16.19/9.03 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 16.19/9.03 => ? [X4: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 16.19/9.03 & ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 & ( ( sin_real @ X4 )
% 16.19/9.03 = Y )
% 16.19/9.03 & ! [Y5: real] :
% 16.19/9.03 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 16.19/9.03 & ( ord_less_eq_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 & ( ( sin_real @ Y5 )
% 16.19/9.03 = Y ) )
% 16.19/9.03 => ( Y5 = X4 ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8096_cos__gt__zero__pi,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 16.19/9.03 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8097_cos__ge__zero,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 16.19/9.03 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8098_cos__one__2pi,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ( cos_real @ X )
% 16.19/9.03 = one_one_real )
% 16.19/9.03 = ( ? [X2: nat] :
% 16.19/9.03 ( X
% 16.19/9.03 = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 16.19/9.03 | ? [X2: nat] :
% 16.19/9.03 ( X
% 16.19/9.03 = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8099_sin__pi__divide__n__gt__0,axiom,
% 16.19/9.03 ! [N2: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.03 => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8100_monoseq__minus,axiom,
% 16.19/9.03 ! [A: nat > int] :
% 16.19/9.03 ( ( topolo4899668324122417113eq_int @ A )
% 16.19/9.03 => ( topolo4899668324122417113eq_int
% 16.19/9.03 @ ^ [N: nat] : ( uminus_uminus_int @ ( A @ N ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8101_monoseq__minus,axiom,
% 16.19/9.03 ! [A: nat > rat] :
% 16.19/9.03 ( ( topolo4267028734544971653eq_rat @ A )
% 16.19/9.03 => ( topolo4267028734544971653eq_rat
% 16.19/9.03 @ ^ [N: nat] : ( uminus_uminus_rat @ ( A @ N ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8102_monoseq__minus,axiom,
% 16.19/9.03 ! [A: nat > code_integer] :
% 16.19/9.03 ( ( topolo2919662092509805066nteger @ A )
% 16.19/9.03 => ( topolo2919662092509805066nteger
% 16.19/9.03 @ ^ [N: nat] : ( uminus1351360451143612070nteger @ ( A @ N ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8103_monoseq__minus,axiom,
% 16.19/9.03 ! [A: nat > real] :
% 16.19/9.03 ( ( topolo6980174941875973593q_real @ A )
% 16.19/9.03 => ( topolo6980174941875973593q_real
% 16.19/9.03 @ ^ [N: nat] : ( uminus_uminus_real @ ( A @ N ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8104_sin__zero__iff__int,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ( sin_real @ X )
% 16.19/9.03 = zero_zero_real )
% 16.19/9.03 = ( ? [I3: int] :
% 16.19/9.03 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I3 )
% 16.19/9.03 & ( X
% 16.19/9.03 = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8105_cos__zero__iff__int,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ( cos_real @ X )
% 16.19/9.03 = zero_zero_real )
% 16.19/9.03 = ( ? [I3: int] :
% 16.19/9.03 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I3 )
% 16.19/9.03 & ( X
% 16.19/9.03 = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8106_sin__zero__lemma,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.03 => ( ( ( sin_real @ X )
% 16.19/9.03 = zero_zero_real )
% 16.19/9.03 => ? [N4: nat] :
% 16.19/9.03 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
% 16.19/9.03 & ( X
% 16.19/9.03 = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8107_sin__zero__iff,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ( sin_real @ X )
% 16.19/9.03 = zero_zero_real )
% 16.19/9.03 = ( ? [N: nat] :
% 16.19/9.03 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 16.19/9.03 & ( X
% 16.19/9.03 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 16.19/9.03 | ? [N: nat] :
% 16.19/9.03 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 16.19/9.03 & ( X
% 16.19/9.03 = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8108_cos__zero__lemma,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.03 => ( ( ( cos_real @ X )
% 16.19/9.03 = zero_zero_real )
% 16.19/9.03 => ? [N4: nat] :
% 16.19/9.03 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
% 16.19/9.03 & ( X
% 16.19/9.03 = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8109_cos__zero__iff,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ( cos_real @ X )
% 16.19/9.03 = zero_zero_real )
% 16.19/9.03 = ( ? [N: nat] :
% 16.19/9.03 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 16.19/9.03 & ( X
% 16.19/9.03 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 16.19/9.03 | ? [N: nat] :
% 16.19/9.03 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 16.19/9.03 & ( X
% 16.19/9.03 = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8110_mono__SucI1,axiom,
% 16.19/9.03 ! [X8: nat > real] :
% 16.19/9.03 ( ! [N4: nat] : ( ord_less_eq_real @ ( X8 @ N4 ) @ ( X8 @ ( suc @ N4 ) ) )
% 16.19/9.03 => ( topolo6980174941875973593q_real @ X8 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8111_mono__SucI1,axiom,
% 16.19/9.03 ! [X8: nat > set_nat] :
% 16.19/9.03 ( ! [N4: nat] : ( ord_less_eq_set_nat @ ( X8 @ N4 ) @ ( X8 @ ( suc @ N4 ) ) )
% 16.19/9.03 => ( topolo7278393974255667507et_nat @ X8 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8112_mono__SucI1,axiom,
% 16.19/9.03 ! [X8: nat > rat] :
% 16.19/9.03 ( ! [N4: nat] : ( ord_less_eq_rat @ ( X8 @ N4 ) @ ( X8 @ ( suc @ N4 ) ) )
% 16.19/9.03 => ( topolo4267028734544971653eq_rat @ X8 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8113_mono__SucI1,axiom,
% 16.19/9.03 ! [X8: nat > num] :
% 16.19/9.03 ( ! [N4: nat] : ( ord_less_eq_num @ ( X8 @ N4 ) @ ( X8 @ ( suc @ N4 ) ) )
% 16.19/9.03 => ( topolo1459490580787246023eq_num @ X8 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8114_mono__SucI1,axiom,
% 16.19/9.03 ! [X8: nat > nat] :
% 16.19/9.03 ( ! [N4: nat] : ( ord_less_eq_nat @ ( X8 @ N4 ) @ ( X8 @ ( suc @ N4 ) ) )
% 16.19/9.03 => ( topolo4902158794631467389eq_nat @ X8 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8115_mono__SucI1,axiom,
% 16.19/9.03 ! [X8: nat > int] :
% 16.19/9.03 ( ! [N4: nat] : ( ord_less_eq_int @ ( X8 @ N4 ) @ ( X8 @ ( suc @ N4 ) ) )
% 16.19/9.03 => ( topolo4899668324122417113eq_int @ X8 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8116_mono__SucI2,axiom,
% 16.19/9.03 ! [X8: nat > real] :
% 16.19/9.03 ( ! [N4: nat] : ( ord_less_eq_real @ ( X8 @ ( suc @ N4 ) ) @ ( X8 @ N4 ) )
% 16.19/9.03 => ( topolo6980174941875973593q_real @ X8 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8117_mono__SucI2,axiom,
% 16.19/9.03 ! [X8: nat > set_nat] :
% 16.19/9.03 ( ! [N4: nat] : ( ord_less_eq_set_nat @ ( X8 @ ( suc @ N4 ) ) @ ( X8 @ N4 ) )
% 16.19/9.03 => ( topolo7278393974255667507et_nat @ X8 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8118_mono__SucI2,axiom,
% 16.19/9.03 ! [X8: nat > rat] :
% 16.19/9.03 ( ! [N4: nat] : ( ord_less_eq_rat @ ( X8 @ ( suc @ N4 ) ) @ ( X8 @ N4 ) )
% 16.19/9.03 => ( topolo4267028734544971653eq_rat @ X8 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8119_mono__SucI2,axiom,
% 16.19/9.03 ! [X8: nat > num] :
% 16.19/9.03 ( ! [N4: nat] : ( ord_less_eq_num @ ( X8 @ ( suc @ N4 ) ) @ ( X8 @ N4 ) )
% 16.19/9.03 => ( topolo1459490580787246023eq_num @ X8 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8120_mono__SucI2,axiom,
% 16.19/9.03 ! [X8: nat > nat] :
% 16.19/9.03 ( ! [N4: nat] : ( ord_less_eq_nat @ ( X8 @ ( suc @ N4 ) ) @ ( X8 @ N4 ) )
% 16.19/9.03 => ( topolo4902158794631467389eq_nat @ X8 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8121_mono__SucI2,axiom,
% 16.19/9.03 ! [X8: nat > int] :
% 16.19/9.03 ( ! [N4: nat] : ( ord_less_eq_int @ ( X8 @ ( suc @ N4 ) ) @ ( X8 @ N4 ) )
% 16.19/9.03 => ( topolo4899668324122417113eq_int @ X8 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8122_monoseq__Suc,axiom,
% 16.19/9.03 ( topolo6980174941875973593q_real
% 16.19/9.03 = ( ^ [X3: nat > real] :
% 16.19/9.03 ( ! [N: nat] : ( ord_less_eq_real @ ( X3 @ N ) @ ( X3 @ ( suc @ N ) ) )
% 16.19/9.03 | ! [N: nat] : ( ord_less_eq_real @ ( X3 @ ( suc @ N ) ) @ ( X3 @ N ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8123_monoseq__Suc,axiom,
% 16.19/9.03 ( topolo7278393974255667507et_nat
% 16.19/9.03 = ( ^ [X3: nat > set_nat] :
% 16.19/9.03 ( ! [N: nat] : ( ord_less_eq_set_nat @ ( X3 @ N ) @ ( X3 @ ( suc @ N ) ) )
% 16.19/9.03 | ! [N: nat] : ( ord_less_eq_set_nat @ ( X3 @ ( suc @ N ) ) @ ( X3 @ N ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8124_monoseq__Suc,axiom,
% 16.19/9.03 ( topolo4267028734544971653eq_rat
% 16.19/9.03 = ( ^ [X3: nat > rat] :
% 16.19/9.03 ( ! [N: nat] : ( ord_less_eq_rat @ ( X3 @ N ) @ ( X3 @ ( suc @ N ) ) )
% 16.19/9.03 | ! [N: nat] : ( ord_less_eq_rat @ ( X3 @ ( suc @ N ) ) @ ( X3 @ N ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8125_monoseq__Suc,axiom,
% 16.19/9.03 ( topolo1459490580787246023eq_num
% 16.19/9.03 = ( ^ [X3: nat > num] :
% 16.19/9.03 ( ! [N: nat] : ( ord_less_eq_num @ ( X3 @ N ) @ ( X3 @ ( suc @ N ) ) )
% 16.19/9.03 | ! [N: nat] : ( ord_less_eq_num @ ( X3 @ ( suc @ N ) ) @ ( X3 @ N ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8126_monoseq__Suc,axiom,
% 16.19/9.03 ( topolo4902158794631467389eq_nat
% 16.19/9.03 = ( ^ [X3: nat > nat] :
% 16.19/9.03 ( ! [N: nat] : ( ord_less_eq_nat @ ( X3 @ N ) @ ( X3 @ ( suc @ N ) ) )
% 16.19/9.03 | ! [N: nat] : ( ord_less_eq_nat @ ( X3 @ ( suc @ N ) ) @ ( X3 @ N ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8127_monoseq__Suc,axiom,
% 16.19/9.03 ( topolo4899668324122417113eq_int
% 16.19/9.03 = ( ^ [X3: nat > int] :
% 16.19/9.03 ( ! [N: nat] : ( ord_less_eq_int @ ( X3 @ N ) @ ( X3 @ ( suc @ N ) ) )
% 16.19/9.03 | ! [N: nat] : ( ord_less_eq_int @ ( X3 @ ( suc @ N ) ) @ ( X3 @ N ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8128_monoseq__def,axiom,
% 16.19/9.03 ( topolo6980174941875973593q_real
% 16.19/9.03 = ( ^ [X3: nat > real] :
% 16.19/9.03 ( ! [M6: nat,N: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M6 @ N )
% 16.19/9.03 => ( ord_less_eq_real @ ( X3 @ M6 ) @ ( X3 @ N ) ) )
% 16.19/9.03 | ! [M6: nat,N: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M6 @ N )
% 16.19/9.03 => ( ord_less_eq_real @ ( X3 @ N ) @ ( X3 @ M6 ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8129_monoseq__def,axiom,
% 16.19/9.03 ( topolo7278393974255667507et_nat
% 16.19/9.03 = ( ^ [X3: nat > set_nat] :
% 16.19/9.03 ( ! [M6: nat,N: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M6 @ N )
% 16.19/9.03 => ( ord_less_eq_set_nat @ ( X3 @ M6 ) @ ( X3 @ N ) ) )
% 16.19/9.03 | ! [M6: nat,N: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M6 @ N )
% 16.19/9.03 => ( ord_less_eq_set_nat @ ( X3 @ N ) @ ( X3 @ M6 ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8130_monoseq__def,axiom,
% 16.19/9.03 ( topolo4267028734544971653eq_rat
% 16.19/9.03 = ( ^ [X3: nat > rat] :
% 16.19/9.03 ( ! [M6: nat,N: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M6 @ N )
% 16.19/9.03 => ( ord_less_eq_rat @ ( X3 @ M6 ) @ ( X3 @ N ) ) )
% 16.19/9.03 | ! [M6: nat,N: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M6 @ N )
% 16.19/9.03 => ( ord_less_eq_rat @ ( X3 @ N ) @ ( X3 @ M6 ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8131_monoseq__def,axiom,
% 16.19/9.03 ( topolo1459490580787246023eq_num
% 16.19/9.03 = ( ^ [X3: nat > num] :
% 16.19/9.03 ( ! [M6: nat,N: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M6 @ N )
% 16.19/9.03 => ( ord_less_eq_num @ ( X3 @ M6 ) @ ( X3 @ N ) ) )
% 16.19/9.03 | ! [M6: nat,N: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M6 @ N )
% 16.19/9.03 => ( ord_less_eq_num @ ( X3 @ N ) @ ( X3 @ M6 ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8132_monoseq__def,axiom,
% 16.19/9.03 ( topolo4902158794631467389eq_nat
% 16.19/9.03 = ( ^ [X3: nat > nat] :
% 16.19/9.03 ( ! [M6: nat,N: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M6 @ N )
% 16.19/9.03 => ( ord_less_eq_nat @ ( X3 @ M6 ) @ ( X3 @ N ) ) )
% 16.19/9.03 | ! [M6: nat,N: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M6 @ N )
% 16.19/9.03 => ( ord_less_eq_nat @ ( X3 @ N ) @ ( X3 @ M6 ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8133_monoseq__def,axiom,
% 16.19/9.03 ( topolo4899668324122417113eq_int
% 16.19/9.03 = ( ^ [X3: nat > int] :
% 16.19/9.03 ( ! [M6: nat,N: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M6 @ N )
% 16.19/9.03 => ( ord_less_eq_int @ ( X3 @ M6 ) @ ( X3 @ N ) ) )
% 16.19/9.03 | ! [M6: nat,N: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M6 @ N )
% 16.19/9.03 => ( ord_less_eq_int @ ( X3 @ N ) @ ( X3 @ M6 ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8134_Maclaurin__cos__expansion2,axiom,
% 16.19/9.03 ! [X: real,N2: nat] :
% 16.19/9.03 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.03 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.03 => ? [T5: real] :
% 16.19/9.03 ( ( ord_less_real @ zero_zero_real @ T5 )
% 16.19/9.03 & ( ord_less_real @ T5 @ X )
% 16.19/9.03 & ( ( cos_real @ X )
% 16.19/9.03 = ( plus_plus_real
% 16.19/9.03 @ ( groups6591440286371151544t_real
% 16.19/9.03 @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 16.19/9.03 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.03 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8135_Maclaurin__minus__cos__expansion,axiom,
% 16.19/9.03 ! [N2: nat,X: real] :
% 16.19/9.03 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.03 => ( ( ord_less_real @ X @ zero_zero_real )
% 16.19/9.03 => ? [T5: real] :
% 16.19/9.03 ( ( ord_less_real @ X @ T5 )
% 16.19/9.03 & ( ord_less_real @ T5 @ zero_zero_real )
% 16.19/9.03 & ( ( cos_real @ X )
% 16.19/9.03 = ( plus_plus_real
% 16.19/9.03 @ ( groups6591440286371151544t_real
% 16.19/9.03 @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 16.19/9.03 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.03 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8136_Maclaurin__cos__expansion,axiom,
% 16.19/9.03 ! [X: real,N2: nat] :
% 16.19/9.03 ? [T5: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 16.19/9.03 & ( ( cos_real @ X )
% 16.19/9.03 = ( plus_plus_real
% 16.19/9.03 @ ( groups6591440286371151544t_real
% 16.19/9.03 @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 16.19/9.03 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.03 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8137_sin__paired,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( sums_real
% 16.19/9.03 @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
% 16.19/9.03 @ ( sin_real @ X ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8138_tan__double,axiom,
% 16.19/9.03 ! [X: complex] :
% 16.19/9.03 ( ( ( cos_complex @ X )
% 16.19/9.03 != zero_zero_complex )
% 16.19/9.03 => ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 16.19/9.03 != zero_zero_complex )
% 16.19/9.03 => ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 16.19/9.03 = ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8139_tan__double,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ( cos_real @ X )
% 16.19/9.03 != zero_zero_real )
% 16.19/9.03 => ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 16.19/9.03 != zero_zero_real )
% 16.19/9.03 => ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 16.19/9.03 = ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8140_tan__periodic__pi,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( tan_real @ ( plus_plus_real @ X @ pi ) )
% 16.19/9.03 = ( tan_real @ X ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8141_fact__0,axiom,
% 16.19/9.03 ( ( semiri5044797733671781792omplex @ zero_zero_nat )
% 16.19/9.03 = one_one_complex ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8142_fact__0,axiom,
% 16.19/9.03 ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
% 16.19/9.03 = one_one_rat ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8143_fact__0,axiom,
% 16.19/9.03 ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
% 16.19/9.03 = one_one_int ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8144_fact__0,axiom,
% 16.19/9.03 ( ( semiri2265585572941072030t_real @ zero_zero_nat )
% 16.19/9.03 = one_one_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8145_fact__0,axiom,
% 16.19/9.03 ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
% 16.19/9.03 = one_one_nat ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8146_fact__1,axiom,
% 16.19/9.03 ( ( semiri5044797733671781792omplex @ one_one_nat )
% 16.19/9.03 = one_one_complex ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8147_fact__1,axiom,
% 16.19/9.03 ( ( semiri773545260158071498ct_rat @ one_one_nat )
% 16.19/9.03 = one_one_rat ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8148_fact__1,axiom,
% 16.19/9.03 ( ( semiri1406184849735516958ct_int @ one_one_nat )
% 16.19/9.03 = one_one_int ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8149_fact__1,axiom,
% 16.19/9.03 ( ( semiri2265585572941072030t_real @ one_one_nat )
% 16.19/9.03 = one_one_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8150_fact__1,axiom,
% 16.19/9.03 ( ( semiri1408675320244567234ct_nat @ one_one_nat )
% 16.19/9.03 = one_one_nat ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8151_fact__Suc__0,axiom,
% 16.19/9.03 ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
% 16.19/9.03 = one_one_complex ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8152_fact__Suc__0,axiom,
% 16.19/9.03 ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
% 16.19/9.03 = one_one_rat ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8153_fact__Suc__0,axiom,
% 16.19/9.03 ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
% 16.19/9.03 = one_one_int ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8154_fact__Suc__0,axiom,
% 16.19/9.03 ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
% 16.19/9.03 = one_one_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8155_fact__Suc__0,axiom,
% 16.19/9.03 ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
% 16.19/9.03 = one_one_nat ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8156_fact__Suc,axiom,
% 16.19/9.03 ! [N2: nat] :
% 16.19/9.03 ( ( semiri773545260158071498ct_rat @ ( suc @ N2 ) )
% 16.19/9.03 = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8157_fact__Suc,axiom,
% 16.19/9.03 ! [N2: nat] :
% 16.19/9.03 ( ( semiri1406184849735516958ct_int @ ( suc @ N2 ) )
% 16.19/9.03 = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8158_fact__Suc,axiom,
% 16.19/9.03 ! [N2: nat] :
% 16.19/9.03 ( ( semiri5044797733671781792omplex @ ( suc @ N2 ) )
% 16.19/9.03 = ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( semiri5044797733671781792omplex @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8159_fact__Suc,axiom,
% 16.19/9.03 ! [N2: nat] :
% 16.19/9.03 ( ( semiri2265585572941072030t_real @ ( suc @ N2 ) )
% 16.19/9.03 = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8160_fact__Suc,axiom,
% 16.19/9.03 ! [N2: nat] :
% 16.19/9.03 ( ( semiri1408675320244567234ct_nat @ ( suc @ N2 ) )
% 16.19/9.03 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N2 ) ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8161_tan__npi,axiom,
% 16.19/9.03 ! [N2: nat] :
% 16.19/9.03 ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) )
% 16.19/9.03 = zero_zero_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8162_tan__periodic__n,axiom,
% 16.19/9.03 ! [X: real,N2: num] :
% 16.19/9.03 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ N2 ) @ pi ) ) )
% 16.19/9.03 = ( tan_real @ X ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8163_tan__periodic__nat,axiom,
% 16.19/9.03 ! [X: real,N2: nat] :
% 16.19/9.03 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) ) )
% 16.19/9.03 = ( tan_real @ X ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8164_tan__periodic__int,axiom,
% 16.19/9.03 ! [X: real,I2: int] :
% 16.19/9.03 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ pi ) ) )
% 16.19/9.03 = ( tan_real @ X ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8165_fact__2,axiom,
% 16.19/9.03 ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.03 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8166_fact__2,axiom,
% 16.19/9.03 ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.03 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8167_fact__2,axiom,
% 16.19/9.03 ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.03 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8168_fact__2,axiom,
% 16.19/9.03 ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.03 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8169_fact__2,axiom,
% 16.19/9.03 ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.03 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8170_tan__periodic,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 16.19/9.03 = ( tan_real @ X ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8171_fact__ge__zero,axiom,
% 16.19/9.03 ! [N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8172_fact__ge__zero,axiom,
% 16.19/9.03 ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8173_fact__ge__zero,axiom,
% 16.19/9.03 ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8174_fact__ge__zero,axiom,
% 16.19/9.03 ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8175_fact__not__neg,axiom,
% 16.19/9.03 ! [N2: nat] :
% 16.19/9.03 ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ zero_zero_rat ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8176_fact__not__neg,axiom,
% 16.19/9.03 ! [N2: nat] :
% 16.19/9.03 ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N2 ) @ zero_zero_int ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8177_fact__not__neg,axiom,
% 16.19/9.03 ! [N2: nat] :
% 16.19/9.03 ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N2 ) @ zero_zero_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8178_fact__not__neg,axiom,
% 16.19/9.03 ! [N2: nat] :
% 16.19/9.03 ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ zero_zero_nat ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8179_fact__gt__zero,axiom,
% 16.19/9.03 ! [N2: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8180_fact__gt__zero,axiom,
% 16.19/9.03 ! [N2: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8181_fact__gt__zero,axiom,
% 16.19/9.03 ! [N2: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8182_fact__gt__zero,axiom,
% 16.19/9.03 ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8183_fact__ge__1,axiom,
% 16.19/9.03 ! [N2: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8184_fact__ge__1,axiom,
% 16.19/9.03 ! [N2: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8185_fact__ge__1,axiom,
% 16.19/9.03 ! [N2: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8186_fact__ge__1,axiom,
% 16.19/9.03 ! [N2: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8187_fact__mono,axiom,
% 16.19/9.03 ! [M: nat,N2: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.03 => ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8188_fact__mono,axiom,
% 16.19/9.03 ! [M: nat,N2: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.03 => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8189_fact__mono,axiom,
% 16.19/9.03 ! [M: nat,N2: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.03 => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8190_fact__mono,axiom,
% 16.19/9.03 ! [M: nat,N2: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.03 => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8191_fact__dvd,axiom,
% 16.19/9.03 ! [N2: nat,M: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/9.03 => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8192_fact__dvd,axiom,
% 16.19/9.03 ! [N2: nat,M: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/9.03 => ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N2 ) @ ( semiri3624122377584611663nteger @ M ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8193_fact__dvd,axiom,
% 16.19/9.03 ! [N2: nat,M: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/9.03 => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8194_fact__dvd,axiom,
% 16.19/9.03 ! [N2: nat,M: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/9.03 => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8195_fact__less__mono,axiom,
% 16.19/9.03 ! [M: nat,N2: nat] :
% 16.19/9.03 ( ( ord_less_nat @ zero_zero_nat @ M )
% 16.19/9.03 => ( ( ord_less_nat @ M @ N2 )
% 16.19/9.03 => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8196_fact__less__mono,axiom,
% 16.19/9.03 ! [M: nat,N2: nat] :
% 16.19/9.03 ( ( ord_less_nat @ zero_zero_nat @ M )
% 16.19/9.03 => ( ( ord_less_nat @ M @ N2 )
% 16.19/9.03 => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8197_fact__less__mono,axiom,
% 16.19/9.03 ! [M: nat,N2: nat] :
% 16.19/9.03 ( ( ord_less_nat @ zero_zero_nat @ M )
% 16.19/9.03 => ( ( ord_less_nat @ M @ N2 )
% 16.19/9.03 => ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8198_fact__less__mono,axiom,
% 16.19/9.03 ! [M: nat,N2: nat] :
% 16.19/9.03 ( ( ord_less_nat @ zero_zero_nat @ M )
% 16.19/9.03 => ( ( ord_less_nat @ M @ N2 )
% 16.19/9.03 => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8199_fact__fact__dvd__fact,axiom,
% 16.19/9.03 ! [K: nat,N2: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N2 ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8200_fact__fact__dvd__fact,axiom,
% 16.19/9.03 ! [K: nat,N2: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N2 ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8201_fact__fact__dvd__fact,axiom,
% 16.19/9.03 ! [K: nat,N2: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N2 ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8202_fact__fact__dvd__fact,axiom,
% 16.19/9.03 ! [K: nat,N2: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8203_fact__fact__dvd__fact,axiom,
% 16.19/9.03 ! [K: nat,N2: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8204_fact__mod,axiom,
% 16.19/9.03 ! [M: nat,N2: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.03 => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1406184849735516958ct_int @ M ) )
% 16.19/9.03 = zero_zero_int ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8205_fact__mod,axiom,
% 16.19/9.03 ! [M: nat,N2: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.03 => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N2 ) @ ( semiri3624122377584611663nteger @ M ) )
% 16.19/9.03 = zero_z3403309356797280102nteger ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8206_fact__mod,axiom,
% 16.19/9.03 ! [M: nat,N2: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.03 => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ M ) )
% 16.19/9.03 = zero_zero_nat ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8207_fact__le__power,axiom,
% 16.19/9.03 ! [N2: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N2 @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8208_fact__le__power,axiom,
% 16.19/9.03 ! [N2: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N2 @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8209_fact__le__power,axiom,
% 16.19/9.03 ! [N2: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N2 @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8210_fact__le__power,axiom,
% 16.19/9.03 ! [N2: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N2 @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8211_tan__def,axiom,
% 16.19/9.03 ( tan_complex
% 16.19/9.03 = ( ^ [X2: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ X2 ) @ ( cos_complex @ X2 ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8212_tan__def,axiom,
% 16.19/9.03 ( tan_real
% 16.19/9.03 = ( ^ [X2: real] : ( divide_divide_real @ ( sin_real @ X2 ) @ ( cos_real @ X2 ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8213_choose__dvd,axiom,
% 16.19/9.03 ! [K: nat,N2: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ K @ N2 )
% 16.19/9.03 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri3624122377584611663nteger @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8214_choose__dvd,axiom,
% 16.19/9.03 ! [K: nat,N2: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ K @ N2 )
% 16.19/9.03 => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8215_choose__dvd,axiom,
% 16.19/9.03 ! [K: nat,N2: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ K @ N2 )
% 16.19/9.03 => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8216_choose__dvd,axiom,
% 16.19/9.03 ! [K: nat,N2: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ K @ N2 )
% 16.19/9.03 => ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8217_choose__dvd,axiom,
% 16.19/9.03 ! [K: nat,N2: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ K @ N2 )
% 16.19/9.03 => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8218_fact__numeral,axiom,
% 16.19/9.03 ! [K: num] :
% 16.19/9.03 ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ K ) )
% 16.19/9.03 = ( times_times_complex @ ( numera6690914467698888265omplex @ K ) @ ( semiri5044797733671781792omplex @ ( pred_numeral @ K ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8219_fact__numeral,axiom,
% 16.19/9.03 ! [K: num] :
% 16.19/9.03 ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ K ) )
% 16.19/9.03 = ( times_times_rat @ ( numeral_numeral_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( pred_numeral @ K ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8220_fact__numeral,axiom,
% 16.19/9.03 ! [K: num] :
% 16.19/9.03 ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ K ) )
% 16.19/9.03 = ( times_times_int @ ( numeral_numeral_int @ K ) @ ( semiri1406184849735516958ct_int @ ( pred_numeral @ K ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8221_fact__numeral,axiom,
% 16.19/9.03 ! [K: num] :
% 16.19/9.03 ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ K ) )
% 16.19/9.03 = ( times_times_real @ ( numeral_numeral_real @ K ) @ ( semiri2265585572941072030t_real @ ( pred_numeral @ K ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8222_fact__numeral,axiom,
% 16.19/9.03 ! [K: num] :
% 16.19/9.03 ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ K ) )
% 16.19/9.03 = ( times_times_nat @ ( numeral_numeral_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( pred_numeral @ K ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8223_square__fact__le__2__fact,axiom,
% 16.19/9.03 ! [N2: nat] : ( ord_less_eq_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8224_tan__45,axiom,
% 16.19/9.03 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 16.19/9.03 = one_one_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8225_fact__num__eq__if,axiom,
% 16.19/9.03 ( semiri773545260158071498ct_rat
% 16.19/9.03 = ( ^ [M6: nat] : ( if_rat @ ( M6 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M6 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8226_fact__num__eq__if,axiom,
% 16.19/9.03 ( semiri1406184849735516958ct_int
% 16.19/9.03 = ( ^ [M6: nat] : ( if_int @ ( M6 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8227_fact__num__eq__if,axiom,
% 16.19/9.03 ( semiri5044797733671781792omplex
% 16.19/9.03 = ( ^ [M6: nat] : ( if_complex @ ( M6 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M6 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8228_fact__num__eq__if,axiom,
% 16.19/9.03 ( semiri2265585572941072030t_real
% 16.19/9.03 = ( ^ [M6: nat] : ( if_real @ ( M6 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M6 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8229_fact__num__eq__if,axiom,
% 16.19/9.03 ( semiri1408675320244567234ct_nat
% 16.19/9.03 = ( ^ [M6: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M6 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8230_fact__reduce,axiom,
% 16.19/9.03 ! [N2: nat] :
% 16.19/9.03 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.03 => ( ( semiri773545260158071498ct_rat @ N2 )
% 16.19/9.03 = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8231_fact__reduce,axiom,
% 16.19/9.03 ! [N2: nat] :
% 16.19/9.03 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.03 => ( ( semiri1406184849735516958ct_int @ N2 )
% 16.19/9.03 = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8232_fact__reduce,axiom,
% 16.19/9.03 ! [N2: nat] :
% 16.19/9.03 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.03 => ( ( semiri5044797733671781792omplex @ N2 )
% 16.19/9.03 = ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8233_fact__reduce,axiom,
% 16.19/9.03 ! [N2: nat] :
% 16.19/9.03 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.03 => ( ( semiri2265585572941072030t_real @ N2 )
% 16.19/9.03 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8234_fact__reduce,axiom,
% 16.19/9.03 ! [N2: nat] :
% 16.19/9.03 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.03 => ( ( semiri1408675320244567234ct_nat @ N2 )
% 16.19/9.03 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8235_tan__gt__zero,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.03 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 => ( ord_less_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8236_lemma__tan__total,axiom,
% 16.19/9.03 ! [Y: real] :
% 16.19/9.03 ( ( ord_less_real @ zero_zero_real @ Y )
% 16.19/9.03 => ? [X4: real] :
% 16.19/9.03 ( ( ord_less_real @ zero_zero_real @ X4 )
% 16.19/9.03 & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 & ( ord_less_real @ Y @ ( tan_real @ X4 ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8237_tan__total,axiom,
% 16.19/9.03 ! [Y: real] :
% 16.19/9.03 ? [X4: real] :
% 16.19/9.03 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 16.19/9.03 & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 & ( ( tan_real @ X4 )
% 16.19/9.03 = Y )
% 16.19/9.03 & ! [Y5: real] :
% 16.19/9.03 ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 16.19/9.03 & ( ord_less_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 & ( ( tan_real @ Y5 )
% 16.19/9.03 = Y ) )
% 16.19/9.03 => ( Y5 = X4 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8238_tan__monotone,axiom,
% 16.19/9.03 ! [Y: real,X: real] :
% 16.19/9.03 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 16.19/9.03 => ( ( ord_less_real @ Y @ X )
% 16.19/9.03 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 => ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8239_tan__monotone_H,axiom,
% 16.19/9.03 ! [Y: real,X: real] :
% 16.19/9.03 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 16.19/9.03 => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 16.19/9.03 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 => ( ( ord_less_real @ Y @ X )
% 16.19/9.03 = ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X ) ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8240_tan__mono__lt__eq,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 16.19/9.03 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 16.19/9.03 => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 => ( ( ord_less_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
% 16.19/9.03 = ( ord_less_real @ X @ Y ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8241_lemma__tan__total1,axiom,
% 16.19/9.03 ! [Y: real] :
% 16.19/9.03 ? [X4: real] :
% 16.19/9.03 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 16.19/9.03 & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 & ( ( tan_real @ X4 )
% 16.19/9.03 = Y ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8242_tan__minus__45,axiom,
% 16.19/9.03 ( ( tan_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) )
% 16.19/9.03 = ( uminus_uminus_real @ one_one_real ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8243_tan__inverse,axiom,
% 16.19/9.03 ! [Y: real] :
% 16.19/9.03 ( ( divide_divide_real @ one_one_real @ ( tan_real @ Y ) )
% 16.19/9.03 = ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8244_add__tan__eq,axiom,
% 16.19/9.03 ! [X: complex,Y: complex] :
% 16.19/9.03 ( ( ( cos_complex @ X )
% 16.19/9.03 != zero_zero_complex )
% 16.19/9.03 => ( ( ( cos_complex @ Y )
% 16.19/9.03 != zero_zero_complex )
% 16.19/9.03 => ( ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) )
% 16.19/9.03 = ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8245_add__tan__eq,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( ( cos_real @ X )
% 16.19/9.03 != zero_zero_real )
% 16.19/9.03 => ( ( ( cos_real @ Y )
% 16.19/9.03 != zero_zero_real )
% 16.19/9.03 => ( ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
% 16.19/9.03 = ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8246_tan__pos__pi2__le,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.03 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8247_tan__total__pos,axiom,
% 16.19/9.03 ! [Y: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/9.03 => ? [X4: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 16.19/9.03 & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 & ( ( tan_real @ X4 )
% 16.19/9.03 = Y ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8248_tan__less__zero,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 16.19/9.03 => ( ( ord_less_real @ X @ zero_zero_real )
% 16.19/9.03 => ( ord_less_real @ ( tan_real @ X ) @ zero_zero_real ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8249_tan__mono__le__eq,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 16.19/9.03 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
% 16.19/9.03 => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 => ( ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) )
% 16.19/9.03 = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8250_tan__mono__le,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 16.19/9.03 => ( ( ord_less_eq_real @ X @ Y )
% 16.19/9.03 => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 => ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8251_tan__bound__pi2,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 16.19/9.03 => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X ) ) @ one_one_real ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8252_arctan,axiom,
% 16.19/9.03 ! [Y: real] :
% 16.19/9.03 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
% 16.19/9.03 & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 & ( ( tan_real @ ( arctan @ Y ) )
% 16.19/9.03 = Y ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8253_arctan__tan,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 16.19/9.03 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 => ( ( arctan @ ( tan_real @ X ) )
% 16.19/9.03 = X ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8254_arctan__unique,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 16.19/9.03 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 => ( ( ( tan_real @ X )
% 16.19/9.03 = Y )
% 16.19/9.03 => ( ( arctan @ Y )
% 16.19/9.03 = X ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8255_Maclaurin__zero,axiom,
% 16.19/9.03 ! [X: real,N2: nat,Diff: nat > complex > real] :
% 16.19/9.03 ( ( X = zero_zero_real )
% 16.19/9.03 => ( ( N2 != zero_zero_nat )
% 16.19/9.03 => ( ( groups6591440286371151544t_real
% 16.19/9.03 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_complex ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
% 16.19/9.03 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.03 = ( Diff @ zero_zero_nat @ zero_zero_complex ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8256_Maclaurin__zero,axiom,
% 16.19/9.03 ! [X: real,N2: nat,Diff: nat > real > real] :
% 16.19/9.03 ( ( X = zero_zero_real )
% 16.19/9.03 => ( ( N2 != zero_zero_nat )
% 16.19/9.03 => ( ( groups6591440286371151544t_real
% 16.19/9.03 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
% 16.19/9.03 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.03 = ( Diff @ zero_zero_nat @ zero_zero_real ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8257_Maclaurin__zero,axiom,
% 16.19/9.03 ! [X: real,N2: nat,Diff: nat > rat > real] :
% 16.19/9.03 ( ( X = zero_zero_real )
% 16.19/9.03 => ( ( N2 != zero_zero_nat )
% 16.19/9.03 => ( ( groups6591440286371151544t_real
% 16.19/9.03 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_rat ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
% 16.19/9.03 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.03 = ( Diff @ zero_zero_nat @ zero_zero_rat ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8258_Maclaurin__zero,axiom,
% 16.19/9.03 ! [X: real,N2: nat,Diff: nat > nat > real] :
% 16.19/9.03 ( ( X = zero_zero_real )
% 16.19/9.03 => ( ( N2 != zero_zero_nat )
% 16.19/9.03 => ( ( groups6591440286371151544t_real
% 16.19/9.03 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_nat ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
% 16.19/9.03 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.03 = ( Diff @ zero_zero_nat @ zero_zero_nat ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8259_Maclaurin__zero,axiom,
% 16.19/9.03 ! [X: real,N2: nat,Diff: nat > int > real] :
% 16.19/9.03 ( ( X = zero_zero_real )
% 16.19/9.03 => ( ( N2 != zero_zero_nat )
% 16.19/9.03 => ( ( groups6591440286371151544t_real
% 16.19/9.03 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_int ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
% 16.19/9.03 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.03 = ( Diff @ zero_zero_nat @ zero_zero_int ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8260_Maclaurin__lemma,axiom,
% 16.19/9.03 ! [H2: real,F: real > real,J: nat > real,N2: nat] :
% 16.19/9.03 ( ( ord_less_real @ zero_zero_real @ H2 )
% 16.19/9.03 => ? [B9: real] :
% 16.19/9.03 ( ( F @ H2 )
% 16.19/9.03 = ( plus_plus_real
% 16.19/9.03 @ ( groups6591440286371151544t_real
% 16.19/9.03 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
% 16.19/9.03 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.03 @ ( times_times_real @ B9 @ ( divide_divide_real @ ( power_power_real @ H2 @ N2 ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8261_tan__add,axiom,
% 16.19/9.03 ! [X: complex,Y: complex] :
% 16.19/9.03 ( ( ( cos_complex @ X )
% 16.19/9.03 != zero_zero_complex )
% 16.19/9.03 => ( ( ( cos_complex @ Y )
% 16.19/9.03 != zero_zero_complex )
% 16.19/9.03 => ( ( ( cos_complex @ ( plus_plus_complex @ X @ Y ) )
% 16.19/9.03 != zero_zero_complex )
% 16.19/9.03 => ( ( tan_complex @ ( plus_plus_complex @ X @ Y ) )
% 16.19/9.03 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8262_tan__add,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( ( cos_real @ X )
% 16.19/9.03 != zero_zero_real )
% 16.19/9.03 => ( ( ( cos_real @ Y )
% 16.19/9.03 != zero_zero_real )
% 16.19/9.03 => ( ( ( cos_real @ ( plus_plus_real @ X @ Y ) )
% 16.19/9.03 != zero_zero_real )
% 16.19/9.03 => ( ( tan_real @ ( plus_plus_real @ X @ Y ) )
% 16.19/9.03 = ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8263_tan__diff,axiom,
% 16.19/9.03 ! [X: complex,Y: complex] :
% 16.19/9.03 ( ( ( cos_complex @ X )
% 16.19/9.03 != zero_zero_complex )
% 16.19/9.03 => ( ( ( cos_complex @ Y )
% 16.19/9.03 != zero_zero_complex )
% 16.19/9.03 => ( ( ( cos_complex @ ( minus_minus_complex @ X @ Y ) )
% 16.19/9.03 != zero_zero_complex )
% 16.19/9.03 => ( ( tan_complex @ ( minus_minus_complex @ X @ Y ) )
% 16.19/9.03 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8264_tan__diff,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( ( cos_real @ X )
% 16.19/9.03 != zero_zero_real )
% 16.19/9.03 => ( ( ( cos_real @ Y )
% 16.19/9.03 != zero_zero_real )
% 16.19/9.03 => ( ( ( cos_real @ ( minus_minus_real @ X @ Y ) )
% 16.19/9.03 != zero_zero_real )
% 16.19/9.03 => ( ( tan_real @ ( minus_minus_real @ X @ Y ) )
% 16.19/9.03 = ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8265_lemma__tan__add1,axiom,
% 16.19/9.03 ! [X: complex,Y: complex] :
% 16.19/9.03 ( ( ( cos_complex @ X )
% 16.19/9.03 != zero_zero_complex )
% 16.19/9.03 => ( ( ( cos_complex @ Y )
% 16.19/9.03 != zero_zero_complex )
% 16.19/9.03 => ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X ) @ ( tan_complex @ Y ) ) )
% 16.19/9.03 = ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ Y ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8266_lemma__tan__add1,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( ( cos_real @ X )
% 16.19/9.03 != zero_zero_real )
% 16.19/9.03 => ( ( ( cos_real @ Y )
% 16.19/9.03 != zero_zero_real )
% 16.19/9.03 => ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X ) @ ( tan_real @ Y ) ) )
% 16.19/9.03 = ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8267_tan__total__pi4,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 16.19/9.03 => ? [Z5: real] :
% 16.19/9.03 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z5 )
% 16.19/9.03 & ( ord_less_real @ Z5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 16.19/9.03 & ( ( tan_real @ Z5 )
% 16.19/9.03 = X ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8268_tan__half,axiom,
% 16.19/9.03 ( tan_complex
% 16.19/9.03 = ( ^ [X2: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X2 ) ) @ one_one_complex ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8269_tan__half,axiom,
% 16.19/9.03 ( tan_real
% 16.19/9.03 = ( ^ [X2: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) @ one_one_real ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8270_cos__coeff__def,axiom,
% 16.19/9.03 ( cos_coeff
% 16.19/9.03 = ( ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ zero_zero_real ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8271_cos__paired,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( sums_real
% 16.19/9.03 @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( power_power_real @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 16.19/9.03 @ ( cos_real @ X ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8272_Maclaurin__sin__expansion3,axiom,
% 16.19/9.03 ! [N2: nat,X: real] :
% 16.19/9.03 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.03 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.03 => ? [T5: real] :
% 16.19/9.03 ( ( ord_less_real @ zero_zero_real @ T5 )
% 16.19/9.03 & ( ord_less_real @ T5 @ X )
% 16.19/9.03 & ( ( sin_real @ X )
% 16.19/9.03 = ( plus_plus_real
% 16.19/9.03 @ ( groups6591440286371151544t_real
% 16.19/9.03 @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 16.19/9.03 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.03 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8273_Maclaurin__sin__expansion4,axiom,
% 16.19/9.03 ! [X: real,N2: nat] :
% 16.19/9.03 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.03 => ? [T5: real] :
% 16.19/9.03 ( ( ord_less_real @ zero_zero_real @ T5 )
% 16.19/9.03 & ( ord_less_eq_real @ T5 @ X )
% 16.19/9.03 & ( ( sin_real @ X )
% 16.19/9.03 = ( plus_plus_real
% 16.19/9.03 @ ( groups6591440286371151544t_real
% 16.19/9.03 @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 16.19/9.03 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.03 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8274_Maclaurin__sin__expansion2,axiom,
% 16.19/9.03 ! [X: real,N2: nat] :
% 16.19/9.03 ? [T5: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 16.19/9.03 & ( ( sin_real @ X )
% 16.19/9.03 = ( plus_plus_real
% 16.19/9.03 @ ( groups6591440286371151544t_real
% 16.19/9.03 @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 16.19/9.03 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.03 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8275_Maclaurin__sin__expansion,axiom,
% 16.19/9.03 ! [X: real,N2: nat] :
% 16.19/9.03 ? [T5: real] :
% 16.19/9.03 ( ( sin_real @ X )
% 16.19/9.03 = ( plus_plus_real
% 16.19/9.03 @ ( groups6591440286371151544t_real
% 16.19/9.03 @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 16.19/9.03 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.03 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8276_sin__coeff__def,axiom,
% 16.19/9.03 ( sin_coeff
% 16.19/9.03 = ( ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8277_fact__mono__nat,axiom,
% 16.19/9.03 ! [M: nat,N2: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.03 => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8278_fact__ge__self,axiom,
% 16.19/9.03 ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8279_fact__less__mono__nat,axiom,
% 16.19/9.03 ! [M: nat,N2: nat] :
% 16.19/9.03 ( ( ord_less_nat @ zero_zero_nat @ M )
% 16.19/9.03 => ( ( ord_less_nat @ M @ N2 )
% 16.19/9.03 => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8280_fact__ge__Suc__0__nat,axiom,
% 16.19/9.03 ! [N2: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8281_dvd__fact,axiom,
% 16.19/9.03 ! [M: nat,N2: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ one_one_nat @ M )
% 16.19/9.03 => ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.03 => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8282_fact__diff__Suc,axiom,
% 16.19/9.03 ! [N2: nat,M: nat] :
% 16.19/9.03 ( ( ord_less_nat @ N2 @ ( suc @ M ) )
% 16.19/9.03 => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) )
% 16.19/9.03 = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N2 ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8283_fact__div__fact__le__pow,axiom,
% 16.19/9.03 ! [R: nat,N2: nat] :
% 16.19/9.03 ( ( ord_less_eq_nat @ R @ N2 )
% 16.19/9.03 => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ R ) ) ) @ ( power_power_nat @ N2 @ R ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8284_sin__coeff__Suc,axiom,
% 16.19/9.03 ! [N2: nat] :
% 16.19/9.03 ( ( sin_coeff @ ( suc @ N2 ) )
% 16.19/9.03 = ( divide_divide_real @ ( cos_coeff @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8285_cos__coeff__Suc,axiom,
% 16.19/9.03 ! [N2: nat] :
% 16.19/9.03 ( ( cos_coeff @ ( suc @ N2 ) )
% 16.19/9.03 = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N2 ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8286_sin__tan,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 => ( ( sin_real @ X )
% 16.19/9.03 = ( divide_divide_real @ ( tan_real @ X ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8287_cos__tan,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.03 => ( ( cos_real @ X )
% 16.19/9.03 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8288_complex__unimodular__polar,axiom,
% 16.19/9.03 ! [Z: complex] :
% 16.19/9.03 ( ( ( real_V1022390504157884413omplex @ Z )
% 16.19/9.03 = one_one_real )
% 16.19/9.03 => ~ ! [T5: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 16.19/9.03 => ( ( ord_less_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 16.19/9.03 => ( Z
% 16.19/9.03 != ( complex2 @ ( cos_real @ T5 ) @ ( sin_real @ T5 ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8289_Maclaurin__exp__lt,axiom,
% 16.19/9.03 ! [X: real,N2: nat] :
% 16.19/9.03 ( ( X != zero_zero_real )
% 16.19/9.03 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.03 => ? [T5: real] :
% 16.19/9.03 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T5 ) )
% 16.19/9.03 & ( ord_less_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 16.19/9.03 & ( ( exp_real @ X )
% 16.19/9.03 = ( plus_plus_real
% 16.19/9.03 @ ( groups6591440286371151544t_real
% 16.19/9.03 @ ^ [M6: nat] : ( divide_divide_real @ ( power_power_real @ X @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) )
% 16.19/9.03 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.03 @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T5 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8290_pochhammer__double,axiom,
% 16.19/9.03 ! [Z: rat,N2: nat] :
% 16.19/9.03 ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/9.03 = ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8291_pochhammer__double,axiom,
% 16.19/9.03 ! [Z: real,N2: nat] :
% 16.19/9.03 ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/9.03 = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s7457072308508201937r_real @ Z @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8292_pochhammer__double,axiom,
% 16.19/9.03 ! [Z: complex,N2: nat] :
% 16.19/9.03 ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/9.03 = ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s2602460028002588243omplex @ Z @ N2 ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8293_real__sqrt__eq__iff,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( ( sqrt @ X )
% 16.19/9.03 = ( sqrt @ Y ) )
% 16.19/9.03 = ( X = Y ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8294_real__sqrt__eq__zero__cancel__iff,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ( sqrt @ X )
% 16.19/9.03 = zero_zero_real )
% 16.19/9.03 = ( X = zero_zero_real ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8295_real__sqrt__zero,axiom,
% 16.19/9.03 ( ( sqrt @ zero_zero_real )
% 16.19/9.03 = zero_zero_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8296_real__sqrt__less__iff,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
% 16.19/9.03 = ( ord_less_real @ X @ Y ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8297_real__sqrt__le__iff,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
% 16.19/9.03 = ( ord_less_eq_real @ X @ Y ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8298_real__sqrt__one,axiom,
% 16.19/9.03 ( ( sqrt @ one_one_real )
% 16.19/9.03 = one_one_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8299_real__sqrt__eq__1__iff,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ( sqrt @ X )
% 16.19/9.03 = one_one_real )
% 16.19/9.03 = ( X = one_one_real ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8300_exp__less__mono,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( ord_less_real @ X @ Y )
% 16.19/9.03 => ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8301_exp__less__cancel__iff,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
% 16.19/9.03 = ( ord_less_real @ X @ Y ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8302_exp__le__cancel__iff,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
% 16.19/9.03 = ( ord_less_eq_real @ X @ Y ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8303_exp__zero,axiom,
% 16.19/9.03 ( ( exp_complex @ zero_zero_complex )
% 16.19/9.03 = one_one_complex ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8304_exp__zero,axiom,
% 16.19/9.03 ( ( exp_real @ zero_zero_real )
% 16.19/9.03 = one_one_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8305_real__sqrt__lt__0__iff,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_real @ ( sqrt @ X ) @ zero_zero_real )
% 16.19/9.03 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8306_real__sqrt__gt__0__iff,axiom,
% 16.19/9.03 ! [Y: real] :
% 16.19/9.03 ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y ) )
% 16.19/9.03 = ( ord_less_real @ zero_zero_real @ Y ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8307_real__sqrt__ge__0__iff,axiom,
% 16.19/9.03 ! [Y: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y ) )
% 16.19/9.03 = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8308_real__sqrt__le__0__iff,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ ( sqrt @ X ) @ zero_zero_real )
% 16.19/9.03 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8309_real__sqrt__gt__1__iff,axiom,
% 16.19/9.03 ! [Y: real] :
% 16.19/9.03 ( ( ord_less_real @ one_one_real @ ( sqrt @ Y ) )
% 16.19/9.03 = ( ord_less_real @ one_one_real @ Y ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8310_real__sqrt__lt__1__iff,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_real @ ( sqrt @ X ) @ one_one_real )
% 16.19/9.03 = ( ord_less_real @ X @ one_one_real ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8311_real__sqrt__ge__1__iff,axiom,
% 16.19/9.03 ! [Y: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y ) )
% 16.19/9.03 = ( ord_less_eq_real @ one_one_real @ Y ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8312_real__sqrt__le__1__iff,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ ( sqrt @ X ) @ one_one_real )
% 16.19/9.03 = ( ord_less_eq_real @ X @ one_one_real ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8313_pochhammer__0,axiom,
% 16.19/9.03 ! [A: complex] :
% 16.19/9.03 ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
% 16.19/9.03 = one_one_complex ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8314_pochhammer__0,axiom,
% 16.19/9.03 ! [A: real] :
% 16.19/9.03 ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
% 16.19/9.03 = one_one_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8315_pochhammer__0,axiom,
% 16.19/9.03 ! [A: rat] :
% 16.19/9.03 ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
% 16.19/9.03 = one_one_rat ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8316_pochhammer__0,axiom,
% 16.19/9.03 ! [A: nat] :
% 16.19/9.03 ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
% 16.19/9.03 = one_one_nat ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8317_pochhammer__0,axiom,
% 16.19/9.03 ! [A: int] :
% 16.19/9.03 ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
% 16.19/9.03 = one_one_int ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8318_exp__eq__one__iff,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ( exp_real @ X )
% 16.19/9.03 = one_one_real )
% 16.19/9.03 = ( X = zero_zero_real ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8319_real__sqrt__mult__self,axiom,
% 16.19/9.03 ! [A: real] :
% 16.19/9.03 ( ( times_times_real @ ( sqrt @ A ) @ ( sqrt @ A ) )
% 16.19/9.03 = ( abs_abs_real @ A ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8320_real__sqrt__abs2,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( sqrt @ ( times_times_real @ X @ X ) )
% 16.19/9.03 = ( abs_abs_real @ X ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8321_real__sqrt__four,axiom,
% 16.19/9.03 ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 16.19/9.03 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8322_exp__less__one__iff,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_real @ ( exp_real @ X ) @ one_one_real )
% 16.19/9.03 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8323_one__less__exp__iff,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_real @ one_one_real @ ( exp_real @ X ) )
% 16.19/9.03 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8324_one__le__exp__iff,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X ) )
% 16.19/9.03 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8325_exp__le__one__iff,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ ( exp_real @ X ) @ one_one_real )
% 16.19/9.03 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8326_exp__ln,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.03 => ( ( exp_real @ ( ln_ln_real @ X ) )
% 16.19/9.03 = X ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8327_exp__ln__iff,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ( exp_real @ ( ln_ln_real @ X ) )
% 16.19/9.03 = X )
% 16.19/9.03 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8328_norm__cos__sin,axiom,
% 16.19/9.03 ! [T: real] :
% 16.19/9.03 ( ( real_V1022390504157884413omplex @ ( complex2 @ ( cos_real @ T ) @ ( sin_real @ T ) ) )
% 16.19/9.03 = one_one_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8329_real__sqrt__abs,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( sqrt @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.03 = ( abs_abs_real @ X ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8330_real__sqrt__pow2__iff,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.03 = X )
% 16.19/9.03 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8331_real__sqrt__pow2,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.03 => ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.03 = X ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8332_real__sqrt__sum__squares__mult__squared__eq,axiom,
% 16.19/9.03 ! [X: real,Y: real,Xa2: real,Ya: real] :
% 16.19/9.03 ( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.03 = ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8333_exp__times__arg__commute,axiom,
% 16.19/9.03 ! [A2: complex] :
% 16.19/9.03 ( ( times_times_complex @ ( exp_complex @ A2 ) @ A2 )
% 16.19/9.03 = ( times_times_complex @ A2 @ ( exp_complex @ A2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8334_exp__times__arg__commute,axiom,
% 16.19/9.03 ! [A2: real] :
% 16.19/9.03 ( ( times_times_real @ ( exp_real @ A2 ) @ A2 )
% 16.19/9.03 = ( times_times_real @ A2 @ ( exp_real @ A2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8335_real__sqrt__divide,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( sqrt @ ( divide_divide_real @ X @ Y ) )
% 16.19/9.03 = ( divide_divide_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8336_real__sqrt__mult,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( sqrt @ ( times_times_real @ X @ Y ) )
% 16.19/9.03 = ( times_times_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8337_norm__exp,axiom,
% 16.19/9.03 ! [X: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ X ) ) @ ( exp_real @ ( real_V7735802525324610683m_real @ X ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8338_norm__exp,axiom,
% 16.19/9.03 ! [X: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ X ) ) @ ( exp_real @ ( real_V1022390504157884413omplex @ X ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8339_real__sqrt__le__mono,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ X @ Y )
% 16.19/9.03 => ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8340_exp__less__cancel,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
% 16.19/9.03 => ( ord_less_real @ X @ Y ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8341_real__sqrt__less__mono,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( ord_less_real @ X @ Y )
% 16.19/9.03 => ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8342_real__sqrt__minus,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( sqrt @ ( uminus_uminus_real @ X ) )
% 16.19/9.03 = ( uminus_uminus_real @ ( sqrt @ X ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8343_real__sqrt__power,axiom,
% 16.19/9.03 ! [X: real,K: nat] :
% 16.19/9.03 ( ( sqrt @ ( power_power_real @ X @ K ) )
% 16.19/9.03 = ( power_power_real @ ( sqrt @ X ) @ K ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8344_complex__diff,axiom,
% 16.19/9.03 ! [A: real,B: real,C: real,D: real] :
% 16.19/9.03 ( ( minus_minus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
% 16.19/9.03 = ( complex2 @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8345_real__sqrt__gt__zero,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.03 => ( ord_less_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8346_not__exp__less__zero,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ~ ( ord_less_real @ ( exp_real @ X ) @ zero_zero_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8347_exp__gt__zero,axiom,
% 16.19/9.03 ! [X: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8348_exp__total,axiom,
% 16.19/9.03 ! [Y: real] :
% 16.19/9.03 ( ( ord_less_real @ zero_zero_real @ Y )
% 16.19/9.03 => ? [X4: real] :
% 16.19/9.03 ( ( exp_real @ X4 )
% 16.19/9.03 = Y ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8349_real__sqrt__ge__zero,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.03 => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8350_real__sqrt__eq__zero__cancel,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.03 => ( ( ( sqrt @ X )
% 16.19/9.03 = zero_zero_real )
% 16.19/9.03 => ( X = zero_zero_real ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8351_not__exp__le__zero,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ~ ( ord_less_eq_real @ ( exp_real @ X ) @ zero_zero_real ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8352_exp__ge__zero,axiom,
% 16.19/9.03 ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8353_real__sqrt__ge__one,axiom,
% 16.19/9.03 ! [X: real] :
% 16.19/9.03 ( ( ord_less_eq_real @ one_one_real @ X )
% 16.19/9.03 => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8354_exp__add__commuting,axiom,
% 16.19/9.03 ! [X: complex,Y: complex] :
% 16.19/9.03 ( ( ( times_times_complex @ X @ Y )
% 16.19/9.03 = ( times_times_complex @ Y @ X ) )
% 16.19/9.03 => ( ( exp_complex @ ( plus_plus_complex @ X @ Y ) )
% 16.19/9.03 = ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8355_exp__add__commuting,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( ( times_times_real @ X @ Y )
% 16.19/9.03 = ( times_times_real @ Y @ X ) )
% 16.19/9.03 => ( ( exp_real @ ( plus_plus_real @ X @ Y ) )
% 16.19/9.03 = ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8356_mult__exp__exp,axiom,
% 16.19/9.03 ! [X: complex,Y: complex] :
% 16.19/9.03 ( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) )
% 16.19/9.03 = ( exp_complex @ ( plus_plus_complex @ X @ Y ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8357_mult__exp__exp,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ Y ) )
% 16.19/9.03 = ( exp_real @ ( plus_plus_real @ X @ Y ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8358_exp__diff,axiom,
% 16.19/9.03 ! [X: complex,Y: complex] :
% 16.19/9.03 ( ( exp_complex @ ( minus_minus_complex @ X @ Y ) )
% 16.19/9.03 = ( divide1717551699836669952omplex @ ( exp_complex @ X ) @ ( exp_complex @ Y ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8359_exp__diff,axiom,
% 16.19/9.03 ! [X: real,Y: real] :
% 16.19/9.03 ( ( exp_real @ ( minus_minus_real @ X @ Y ) )
% 16.19/9.03 = ( divide_divide_real @ ( exp_real @ X ) @ ( exp_real @ Y ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8360_Complex__eq__numeral,axiom,
% 16.19/9.03 ! [A: real,B: real,W: num] :
% 16.19/9.03 ( ( ( complex2 @ A @ B )
% 16.19/9.03 = ( numera6690914467698888265omplex @ W ) )
% 16.19/9.03 = ( ( A
% 16.19/9.03 = ( numeral_numeral_real @ W ) )
% 16.19/9.03 & ( B = zero_zero_real ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8361_pochhammer__pos,axiom,
% 16.19/9.03 ! [X: real,N2: nat] :
% 16.19/9.03 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.03 => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8362_pochhammer__pos,axiom,
% 16.19/9.03 ! [X: rat,N2: nat] :
% 16.19/9.03 ( ( ord_less_rat @ zero_zero_rat @ X )
% 16.19/9.03 => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8363_pochhammer__pos,axiom,
% 16.19/9.03 ! [X: nat,N2: nat] :
% 16.19/9.03 ( ( ord_less_nat @ zero_zero_nat @ X )
% 16.19/9.03 => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8364_pochhammer__pos,axiom,
% 16.19/9.03 ! [X: int,N2: nat] :
% 16.19/9.03 ( ( ord_less_int @ zero_zero_int @ X )
% 16.19/9.03 => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N2 ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8365_pochhammer__neq__0__mono,axiom,
% 16.19/9.03 ! [A: complex,M: nat,N2: nat] :
% 16.19/9.03 ( ( ( comm_s2602460028002588243omplex @ A @ M )
% 16.19/9.03 != zero_zero_complex )
% 16.19/9.03 => ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/9.03 => ( ( comm_s2602460028002588243omplex @ A @ N2 )
% 16.19/9.03 != zero_zero_complex ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8366_pochhammer__neq__0__mono,axiom,
% 16.19/9.03 ! [A: real,M: nat,N2: nat] :
% 16.19/9.03 ( ( ( comm_s7457072308508201937r_real @ A @ M )
% 16.19/9.03 != zero_zero_real )
% 16.19/9.03 => ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/9.03 => ( ( comm_s7457072308508201937r_real @ A @ N2 )
% 16.19/9.03 != zero_zero_real ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8367_pochhammer__neq__0__mono,axiom,
% 16.19/9.03 ! [A: rat,M: nat,N2: nat] :
% 16.19/9.03 ( ( ( comm_s4028243227959126397er_rat @ A @ M )
% 16.19/9.03 != zero_zero_rat )
% 16.19/9.03 => ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/9.03 => ( ( comm_s4028243227959126397er_rat @ A @ N2 )
% 16.19/9.03 != zero_zero_rat ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8368_pochhammer__eq__0__mono,axiom,
% 16.19/9.03 ! [A: complex,N2: nat,M: nat] :
% 16.19/9.03 ( ( ( comm_s2602460028002588243omplex @ A @ N2 )
% 16.19/9.03 = zero_zero_complex )
% 16.19/9.03 => ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/9.03 => ( ( comm_s2602460028002588243omplex @ A @ M )
% 16.19/9.03 = zero_zero_complex ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8369_pochhammer__eq__0__mono,axiom,
% 16.19/9.03 ! [A: real,N2: nat,M: nat] :
% 16.19/9.03 ( ( ( comm_s7457072308508201937r_real @ A @ N2 )
% 16.19/9.03 = zero_zero_real )
% 16.19/9.03 => ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/9.03 => ( ( comm_s7457072308508201937r_real @ A @ M )
% 16.19/9.03 = zero_zero_real ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8370_pochhammer__eq__0__mono,axiom,
% 16.19/9.03 ! [A: rat,N2: nat,M: nat] :
% 16.19/9.03 ( ( ( comm_s4028243227959126397er_rat @ A @ N2 )
% 16.19/9.03 = zero_zero_rat )
% 16.19/9.03 => ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/9.03 => ( ( comm_s4028243227959126397er_rat @ A @ M )
% 16.19/9.03 = zero_zero_rat ) ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8371_pochhammer__fact,axiom,
% 16.19/9.03 ( semiri5044797733671781792omplex
% 16.19/9.03 = ( comm_s2602460028002588243omplex @ one_one_complex ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8372_pochhammer__fact,axiom,
% 16.19/9.03 ( semiri773545260158071498ct_rat
% 16.19/9.03 = ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8373_pochhammer__fact,axiom,
% 16.19/9.03 ( semiri1406184849735516958ct_int
% 16.19/9.03 = ( comm_s4660882817536571857er_int @ one_one_int ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8374_pochhammer__fact,axiom,
% 16.19/9.03 ( semiri2265585572941072030t_real
% 16.19/9.03 = ( comm_s7457072308508201937r_real @ one_one_real ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8375_pochhammer__fact,axiom,
% 16.19/9.03 ( semiri1408675320244567234ct_nat
% 16.19/9.03 = ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).
% 16.19/9.03
% 16.19/9.03 thf(fact_8376_complex__add,axiom,
% 16.19/9.03 ! [A: real,B: real,C: real,D: real] :
% 16.19/9.04 ( ( plus_plus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
% 16.19/9.04 = ( complex2 @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8377_complex__norm,axiom,
% 16.19/9.04 ! [X: real,Y: real] :
% 16.19/9.04 ( ( real_V1022390504157884413omplex @ ( complex2 @ X @ Y ) )
% 16.19/9.04 = ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8378_exp__gt__one,axiom,
% 16.19/9.04 ! [X: real] :
% 16.19/9.04 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.04 => ( ord_less_real @ one_one_real @ ( exp_real @ X ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8379_real__div__sqrt,axiom,
% 16.19/9.04 ! [X: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.04 => ( ( divide_divide_real @ X @ ( sqrt @ X ) )
% 16.19/9.04 = ( sqrt @ X ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8380_sqrt__add__le__add__sqrt,axiom,
% 16.19/9.04 ! [X: real,Y: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.04 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/9.04 => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X @ Y ) ) @ ( plus_plus_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8381_exp__ge__add__one__self,axiom,
% 16.19/9.04 ! [X: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8382_le__real__sqrt__sumsq,axiom,
% 16.19/9.04 ! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8383_exp__minus__inverse,axiom,
% 16.19/9.04 ! [X: real] :
% 16.19/9.04 ( ( times_times_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) )
% 16.19/9.04 = one_one_real ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8384_exp__minus__inverse,axiom,
% 16.19/9.04 ! [X: complex] :
% 16.19/9.04 ( ( times_times_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) )
% 16.19/9.04 = one_one_complex ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8385_exp__of__nat__mult,axiom,
% 16.19/9.04 ! [N2: nat,X: real] :
% 16.19/9.04 ( ( exp_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X ) )
% 16.19/9.04 = ( power_power_real @ ( exp_real @ X ) @ N2 ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8386_exp__of__nat__mult,axiom,
% 16.19/9.04 ! [N2: nat,X: complex] :
% 16.19/9.04 ( ( exp_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ X ) )
% 16.19/9.04 = ( power_power_complex @ ( exp_complex @ X ) @ N2 ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8387_exp__of__nat2__mult,axiom,
% 16.19/9.04 ! [X: real,N2: nat] :
% 16.19/9.04 ( ( exp_real @ ( times_times_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 16.19/9.04 = ( power_power_real @ ( exp_real @ X ) @ N2 ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8388_exp__of__nat2__mult,axiom,
% 16.19/9.04 ! [X: complex,N2: nat] :
% 16.19/9.04 ( ( exp_complex @ ( times_times_complex @ X @ ( semiri8010041392384452111omplex @ N2 ) ) )
% 16.19/9.04 = ( power_power_complex @ ( exp_complex @ X ) @ N2 ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8389_Complex__eq__neg__numeral,axiom,
% 16.19/9.04 ! [A: real,B: real,W: num] :
% 16.19/9.04 ( ( ( complex2 @ A @ B )
% 16.19/9.04 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 16.19/9.04 = ( ( A
% 16.19/9.04 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 16.19/9.04 & ( B = zero_zero_real ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8390_pochhammer__nonneg,axiom,
% 16.19/9.04 ! [X: real,N2: nat] :
% 16.19/9.04 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.04 => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8391_pochhammer__nonneg,axiom,
% 16.19/9.04 ! [X: rat,N2: nat] :
% 16.19/9.04 ( ( ord_less_rat @ zero_zero_rat @ X )
% 16.19/9.04 => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8392_pochhammer__nonneg,axiom,
% 16.19/9.04 ! [X: nat,N2: nat] :
% 16.19/9.04 ( ( ord_less_nat @ zero_zero_nat @ X )
% 16.19/9.04 => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8393_pochhammer__nonneg,axiom,
% 16.19/9.04 ! [X: int,N2: nat] :
% 16.19/9.04 ( ( ord_less_int @ zero_zero_int @ X )
% 16.19/9.04 => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8394_pochhammer__0__left,axiom,
% 16.19/9.04 ! [N2: nat] :
% 16.19/9.04 ( ( ( N2 = zero_zero_nat )
% 16.19/9.04 => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N2 )
% 16.19/9.04 = one_one_complex ) )
% 16.19/9.04 & ( ( N2 != zero_zero_nat )
% 16.19/9.04 => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N2 )
% 16.19/9.04 = zero_zero_complex ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8395_pochhammer__0__left,axiom,
% 16.19/9.04 ! [N2: nat] :
% 16.19/9.04 ( ( ( N2 = zero_zero_nat )
% 16.19/9.04 => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N2 )
% 16.19/9.04 = one_one_real ) )
% 16.19/9.04 & ( ( N2 != zero_zero_nat )
% 16.19/9.04 => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N2 )
% 16.19/9.04 = zero_zero_real ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8396_pochhammer__0__left,axiom,
% 16.19/9.04 ! [N2: nat] :
% 16.19/9.04 ( ( ( N2 = zero_zero_nat )
% 16.19/9.04 => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N2 )
% 16.19/9.04 = one_one_rat ) )
% 16.19/9.04 & ( ( N2 != zero_zero_nat )
% 16.19/9.04 => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N2 )
% 16.19/9.04 = zero_zero_rat ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8397_pochhammer__0__left,axiom,
% 16.19/9.04 ! [N2: nat] :
% 16.19/9.04 ( ( ( N2 = zero_zero_nat )
% 16.19/9.04 => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N2 )
% 16.19/9.04 = one_one_nat ) )
% 16.19/9.04 & ( ( N2 != zero_zero_nat )
% 16.19/9.04 => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N2 )
% 16.19/9.04 = zero_zero_nat ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8398_pochhammer__0__left,axiom,
% 16.19/9.04 ! [N2: nat] :
% 16.19/9.04 ( ( ( N2 = zero_zero_nat )
% 16.19/9.04 => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N2 )
% 16.19/9.04 = one_one_int ) )
% 16.19/9.04 & ( ( N2 != zero_zero_nat )
% 16.19/9.04 => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N2 )
% 16.19/9.04 = zero_zero_int ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8399_complex__mult,axiom,
% 16.19/9.04 ! [A: real,B: real,C: real,D: real] :
% 16.19/9.04 ( ( times_times_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
% 16.19/9.04 = ( complex2 @ ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8400_Complex__eq__1,axiom,
% 16.19/9.04 ! [A: real,B: real] :
% 16.19/9.04 ( ( ( complex2 @ A @ B )
% 16.19/9.04 = one_one_complex )
% 16.19/9.04 = ( ( A = one_one_real )
% 16.19/9.04 & ( B = zero_zero_real ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8401_one__complex_Ocode,axiom,
% 16.19/9.04 ( one_one_complex
% 16.19/9.04 = ( complex2 @ one_one_real @ zero_zero_real ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8402_sqrt2__less__2,axiom,
% 16.19/9.04 ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8403_exp__ge__add__one__self__aux,axiom,
% 16.19/9.04 ! [X: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.04 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8404_lemma__exp__total,axiom,
% 16.19/9.04 ! [Y: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ one_one_real @ Y )
% 16.19/9.04 => ? [X4: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 16.19/9.04 & ( ord_less_eq_real @ X4 @ ( minus_minus_real @ Y @ one_one_real ) )
% 16.19/9.04 & ( ( exp_real @ X4 )
% 16.19/9.04 = Y ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8405_ln__ge__iff,axiom,
% 16.19/9.04 ! [X: real,Y: real] :
% 16.19/9.04 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.04 => ( ( ord_less_eq_real @ Y @ ( ln_ln_real @ X ) )
% 16.19/9.04 = ( ord_less_eq_real @ ( exp_real @ Y ) @ X ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8406_ln__x__over__x__mono,axiom,
% 16.19/9.04 ! [X: real,Y: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X )
% 16.19/9.04 => ( ( ord_less_eq_real @ X @ Y )
% 16.19/9.04 => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y ) @ Y ) @ ( divide_divide_real @ ( ln_ln_real @ X ) @ X ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8407_Complex__sum_H,axiom,
% 16.19/9.04 ! [F: nat > real,S3: set_nat] :
% 16.19/9.04 ( ( groups2073611262835488442omplex
% 16.19/9.04 @ ^ [X2: nat] : ( complex2 @ ( F @ X2 ) @ zero_zero_real )
% 16.19/9.04 @ S3 )
% 16.19/9.04 = ( complex2 @ ( groups6591440286371151544t_real @ F @ S3 ) @ zero_zero_real ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8408_Complex__sum_H,axiom,
% 16.19/9.04 ! [F: complex > real,S3: set_complex] :
% 16.19/9.04 ( ( groups7754918857620584856omplex
% 16.19/9.04 @ ^ [X2: complex] : ( complex2 @ ( F @ X2 ) @ zero_zero_real )
% 16.19/9.04 @ S3 )
% 16.19/9.04 = ( complex2 @ ( groups5808333547571424918x_real @ F @ S3 ) @ zero_zero_real ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8409_pochhammer__rec,axiom,
% 16.19/9.04 ! [A: complex,N2: nat] :
% 16.19/9.04 ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N2 ) )
% 16.19/9.04 = ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8410_pochhammer__rec,axiom,
% 16.19/9.04 ! [A: real,N2: nat] :
% 16.19/9.04 ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
% 16.19/9.04 = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8411_pochhammer__rec,axiom,
% 16.19/9.04 ! [A: rat,N2: nat] :
% 16.19/9.04 ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
% 16.19/9.04 = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8412_pochhammer__rec,axiom,
% 16.19/9.04 ! [A: nat,N2: nat] :
% 16.19/9.04 ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
% 16.19/9.04 = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8413_pochhammer__rec,axiom,
% 16.19/9.04 ! [A: int,N2: nat] :
% 16.19/9.04 ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
% 16.19/9.04 = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8414_pochhammer__rec_H,axiom,
% 16.19/9.04 ! [Z: rat,N2: nat] :
% 16.19/9.04 ( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N2 ) )
% 16.19/9.04 = ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8415_pochhammer__rec_H,axiom,
% 16.19/9.04 ! [Z: int,N2: nat] :
% 16.19/9.04 ( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N2 ) )
% 16.19/9.04 = ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( comm_s4660882817536571857er_int @ Z @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8416_pochhammer__rec_H,axiom,
% 16.19/9.04 ! [Z: real,N2: nat] :
% 16.19/9.04 ( ( comm_s7457072308508201937r_real @ Z @ ( suc @ N2 ) )
% 16.19/9.04 = ( times_times_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N2 ) ) @ ( comm_s7457072308508201937r_real @ Z @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8417_pochhammer__rec_H,axiom,
% 16.19/9.04 ! [Z: nat,N2: nat] :
% 16.19/9.04 ( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N2 ) )
% 16.19/9.04 = ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N2 ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8418_pochhammer__rec_H,axiom,
% 16.19/9.04 ! [Z: complex,N2: nat] :
% 16.19/9.04 ( ( comm_s2602460028002588243omplex @ Z @ ( suc @ N2 ) )
% 16.19/9.04 = ( times_times_complex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N2 ) ) @ ( comm_s2602460028002588243omplex @ Z @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8419_pochhammer__Suc,axiom,
% 16.19/9.04 ! [A: rat,N2: nat] :
% 16.19/9.04 ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
% 16.19/9.04 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N2 ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8420_pochhammer__Suc,axiom,
% 16.19/9.04 ! [A: int,N2: nat] :
% 16.19/9.04 ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
% 16.19/9.04 = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N2 ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8421_pochhammer__Suc,axiom,
% 16.19/9.04 ! [A: real,N2: nat] :
% 16.19/9.04 ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
% 16.19/9.04 = ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N2 ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8422_pochhammer__Suc,axiom,
% 16.19/9.04 ! [A: nat,N2: nat] :
% 16.19/9.04 ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
% 16.19/9.04 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N2 ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8423_pochhammer__Suc,axiom,
% 16.19/9.04 ! [A: complex,N2: nat] :
% 16.19/9.04 ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N2 ) )
% 16.19/9.04 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ A @ N2 ) @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N2 ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8424_pochhammer__eq__0__iff,axiom,
% 16.19/9.04 ! [A: rat,N2: nat] :
% 16.19/9.04 ( ( ( comm_s4028243227959126397er_rat @ A @ N2 )
% 16.19/9.04 = zero_zero_rat )
% 16.19/9.04 = ( ? [K3: nat] :
% 16.19/9.04 ( ( ord_less_nat @ K3 @ N2 )
% 16.19/9.04 & ( A
% 16.19/9.04 = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K3 ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8425_pochhammer__eq__0__iff,axiom,
% 16.19/9.04 ! [A: real,N2: nat] :
% 16.19/9.04 ( ( ( comm_s7457072308508201937r_real @ A @ N2 )
% 16.19/9.04 = zero_zero_real )
% 16.19/9.04 = ( ? [K3: nat] :
% 16.19/9.04 ( ( ord_less_nat @ K3 @ N2 )
% 16.19/9.04 & ( A
% 16.19/9.04 = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K3 ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8426_pochhammer__eq__0__iff,axiom,
% 16.19/9.04 ! [A: complex,N2: nat] :
% 16.19/9.04 ( ( ( comm_s2602460028002588243omplex @ A @ N2 )
% 16.19/9.04 = zero_zero_complex )
% 16.19/9.04 = ( ? [K3: nat] :
% 16.19/9.04 ( ( ord_less_nat @ K3 @ N2 )
% 16.19/9.04 & ( A
% 16.19/9.04 = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K3 ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8427_pochhammer__of__nat__eq__0__iff,axiom,
% 16.19/9.04 ! [N2: nat,K: nat] :
% 16.19/9.04 ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K )
% 16.19/9.04 = zero_zero_rat )
% 16.19/9.04 = ( ord_less_nat @ N2 @ K ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8428_pochhammer__of__nat__eq__0__iff,axiom,
% 16.19/9.04 ! [N2: nat,K: nat] :
% 16.19/9.04 ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K )
% 16.19/9.04 = zero_z3403309356797280102nteger )
% 16.19/9.04 = ( ord_less_nat @ N2 @ K ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8429_pochhammer__of__nat__eq__0__iff,axiom,
% 16.19/9.04 ! [N2: nat,K: nat] :
% 16.19/9.04 ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
% 16.19/9.04 = zero_zero_int )
% 16.19/9.04 = ( ord_less_nat @ N2 @ K ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8430_pochhammer__of__nat__eq__0__iff,axiom,
% 16.19/9.04 ! [N2: nat,K: nat] :
% 16.19/9.04 ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ K )
% 16.19/9.04 = zero_zero_real )
% 16.19/9.04 = ( ord_less_nat @ N2 @ K ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8431_pochhammer__of__nat__eq__0__iff,axiom,
% 16.19/9.04 ! [N2: nat,K: nat] :
% 16.19/9.04 ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ K )
% 16.19/9.04 = zero_zero_complex )
% 16.19/9.04 = ( ord_less_nat @ N2 @ K ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8432_pochhammer__of__nat__eq__0__lemma,axiom,
% 16.19/9.04 ! [N2: nat,K: nat] :
% 16.19/9.04 ( ( ord_less_nat @ N2 @ K )
% 16.19/9.04 => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K )
% 16.19/9.04 = zero_zero_rat ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8433_pochhammer__of__nat__eq__0__lemma,axiom,
% 16.19/9.04 ! [N2: nat,K: nat] :
% 16.19/9.04 ( ( ord_less_nat @ N2 @ K )
% 16.19/9.04 => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K )
% 16.19/9.04 = zero_z3403309356797280102nteger ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8434_pochhammer__of__nat__eq__0__lemma,axiom,
% 16.19/9.04 ! [N2: nat,K: nat] :
% 16.19/9.04 ( ( ord_less_nat @ N2 @ K )
% 16.19/9.04 => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
% 16.19/9.04 = zero_zero_int ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8435_pochhammer__of__nat__eq__0__lemma,axiom,
% 16.19/9.04 ! [N2: nat,K: nat] :
% 16.19/9.04 ( ( ord_less_nat @ N2 @ K )
% 16.19/9.04 => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ K )
% 16.19/9.04 = zero_zero_real ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8436_pochhammer__of__nat__eq__0__lemma,axiom,
% 16.19/9.04 ! [N2: nat,K: nat] :
% 16.19/9.04 ( ( ord_less_nat @ N2 @ K )
% 16.19/9.04 => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ K )
% 16.19/9.04 = zero_zero_complex ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8437_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 16.19/9.04 ! [K: nat,N2: nat] :
% 16.19/9.04 ( ( ord_less_eq_nat @ K @ N2 )
% 16.19/9.04 => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K )
% 16.19/9.04 != zero_zero_rat ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8438_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 16.19/9.04 ! [K: nat,N2: nat] :
% 16.19/9.04 ( ( ord_less_eq_nat @ K @ N2 )
% 16.19/9.04 => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K )
% 16.19/9.04 != zero_z3403309356797280102nteger ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8439_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 16.19/9.04 ! [K: nat,N2: nat] :
% 16.19/9.04 ( ( ord_less_eq_nat @ K @ N2 )
% 16.19/9.04 => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
% 16.19/9.04 != zero_zero_int ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8440_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 16.19/9.04 ! [K: nat,N2: nat] :
% 16.19/9.04 ( ( ord_less_eq_nat @ K @ N2 )
% 16.19/9.04 => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ K )
% 16.19/9.04 != zero_zero_real ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8441_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 16.19/9.04 ! [K: nat,N2: nat] :
% 16.19/9.04 ( ( ord_less_eq_nat @ K @ N2 )
% 16.19/9.04 => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ K )
% 16.19/9.04 != zero_zero_complex ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8442_pochhammer__product_H,axiom,
% 16.19/9.04 ! [Z: rat,N2: nat,M: nat] :
% 16.19/9.04 ( ( comm_s4028243227959126397er_rat @ Z @ ( plus_plus_nat @ N2 @ M ) )
% 16.19/9.04 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ N2 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N2 ) ) @ M ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8443_pochhammer__product_H,axiom,
% 16.19/9.04 ! [Z: int,N2: nat,M: nat] :
% 16.19/9.04 ( ( comm_s4660882817536571857er_int @ Z @ ( plus_plus_nat @ N2 @ M ) )
% 16.19/9.04 = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ N2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N2 ) ) @ M ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8444_pochhammer__product_H,axiom,
% 16.19/9.04 ! [Z: real,N2: nat,M: nat] :
% 16.19/9.04 ( ( comm_s7457072308508201937r_real @ Z @ ( plus_plus_nat @ N2 @ M ) )
% 16.19/9.04 = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ N2 ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N2 ) ) @ M ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8445_pochhammer__product_H,axiom,
% 16.19/9.04 ! [Z: nat,N2: nat,M: nat] :
% 16.19/9.04 ( ( comm_s4663373288045622133er_nat @ Z @ ( plus_plus_nat @ N2 @ M ) )
% 16.19/9.04 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ N2 ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N2 ) ) @ M ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8446_pochhammer__product_H,axiom,
% 16.19/9.04 ! [Z: complex,N2: nat,M: nat] :
% 16.19/9.04 ( ( comm_s2602460028002588243omplex @ Z @ ( plus_plus_nat @ N2 @ M ) )
% 16.19/9.04 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ N2 ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N2 ) ) @ M ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8447_Complex__eq__neg__1,axiom,
% 16.19/9.04 ! [A: real,B: real] :
% 16.19/9.04 ( ( ( complex2 @ A @ B )
% 16.19/9.04 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 16.19/9.04 = ( ( A
% 16.19/9.04 = ( uminus_uminus_real @ one_one_real ) )
% 16.19/9.04 & ( B = zero_zero_real ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8448_real__less__rsqrt,axiom,
% 16.19/9.04 ! [X: real,Y: real] :
% 16.19/9.04 ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
% 16.19/9.04 => ( ord_less_real @ X @ ( sqrt @ Y ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8449_real__le__rsqrt,axiom,
% 16.19/9.04 ! [X: real,Y: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
% 16.19/9.04 => ( ord_less_eq_real @ X @ ( sqrt @ Y ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8450_sqrt__le__D,axiom,
% 16.19/9.04 ! [X: real,Y: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ ( sqrt @ X ) @ Y )
% 16.19/9.04 => ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8451_exp__le,axiom,
% 16.19/9.04 ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8452_exp__divide__power__eq,axiom,
% 16.19/9.04 ! [N2: nat,X: real] :
% 16.19/9.04 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.04 => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 )
% 16.19/9.04 = ( exp_real @ X ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8453_exp__divide__power__eq,axiom,
% 16.19/9.04 ! [N2: nat,X: complex] :
% 16.19/9.04 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.04 => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X @ ( semiri8010041392384452111omplex @ N2 ) ) ) @ N2 )
% 16.19/9.04 = ( exp_complex @ X ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8454_tanh__altdef,axiom,
% 16.19/9.04 ( tanh_real
% 16.19/9.04 = ( ^ [X2: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X2 ) @ ( exp_real @ ( uminus_uminus_real @ X2 ) ) ) @ ( plus_plus_real @ ( exp_real @ X2 ) @ ( exp_real @ ( uminus_uminus_real @ X2 ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8455_tanh__altdef,axiom,
% 16.19/9.04 ( tanh_complex
% 16.19/9.04 = ( ^ [X2: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X2 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X2 ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X2 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X2 ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8456_pochhammer__product,axiom,
% 16.19/9.04 ! [M: nat,N2: nat,Z: rat] :
% 16.19/9.04 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.04 => ( ( comm_s4028243227959126397er_rat @ Z @ N2 )
% 16.19/9.04 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8457_pochhammer__product,axiom,
% 16.19/9.04 ! [M: nat,N2: nat,Z: int] :
% 16.19/9.04 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.04 => ( ( comm_s4660882817536571857er_int @ Z @ N2 )
% 16.19/9.04 = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8458_pochhammer__product,axiom,
% 16.19/9.04 ! [M: nat,N2: nat,Z: real] :
% 16.19/9.04 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.04 => ( ( comm_s7457072308508201937r_real @ Z @ N2 )
% 16.19/9.04 = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8459_pochhammer__product,axiom,
% 16.19/9.04 ! [M: nat,N2: nat,Z: nat] :
% 16.19/9.04 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.04 => ( ( comm_s4663373288045622133er_nat @ Z @ N2 )
% 16.19/9.04 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8460_pochhammer__product,axiom,
% 16.19/9.04 ! [M: nat,N2: nat,Z: complex] :
% 16.19/9.04 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.04 => ( ( comm_s2602460028002588243omplex @ Z @ N2 )
% 16.19/9.04 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ M ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8461_real__sqrt__unique,axiom,
% 16.19/9.04 ! [Y: real,X: real] :
% 16.19/9.04 ( ( ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.04 = X )
% 16.19/9.04 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/9.04 => ( ( sqrt @ X )
% 16.19/9.04 = Y ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8462_real__le__lsqrt,axiom,
% 16.19/9.04 ! [X: real,Y: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.04 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/9.04 => ( ( ord_less_eq_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.04 => ( ord_less_eq_real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8463_lemma__real__divide__sqrt__less,axiom,
% 16.19/9.04 ! [U: real] :
% 16.19/9.04 ( ( ord_less_real @ zero_zero_real @ U )
% 16.19/9.04 => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8464_real__sqrt__sum__squares__eq__cancel2,axiom,
% 16.19/9.04 ! [X: real,Y: real] :
% 16.19/9.04 ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/9.04 = Y )
% 16.19/9.04 => ( X = zero_zero_real ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8465_real__sqrt__sum__squares__eq__cancel,axiom,
% 16.19/9.04 ! [X: real,Y: real] :
% 16.19/9.04 ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/9.04 = X )
% 16.19/9.04 => ( Y = zero_zero_real ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8466_real__sqrt__sum__squares__triangle__ineq,axiom,
% 16.19/9.04 ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A @ C ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B @ D ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8467_real__sqrt__sum__squares__ge2,axiom,
% 16.19/9.04 ! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8468_real__sqrt__sum__squares__ge1,axiom,
% 16.19/9.04 ! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8469_exp__half__le2,axiom,
% 16.19/9.04 ord_less_eq_real @ ( exp_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8470_sqrt__ge__absD,axiom,
% 16.19/9.04 ! [X: real,Y: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ Y ) )
% 16.19/9.04 => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8471_cos__45,axiom,
% 16.19/9.04 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 16.19/9.04 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8472_sin__45,axiom,
% 16.19/9.04 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 16.19/9.04 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8473_tan__60,axiom,
% 16.19/9.04 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 16.19/9.04 = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8474_exp__double,axiom,
% 16.19/9.04 ! [Z: complex] :
% 16.19/9.04 ( ( exp_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) )
% 16.19/9.04 = ( power_power_complex @ ( exp_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8475_exp__double,axiom,
% 16.19/9.04 ! [Z: real] :
% 16.19/9.04 ( ( exp_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) )
% 16.19/9.04 = ( power_power_real @ ( exp_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8476_pochhammer__absorb__comp,axiom,
% 16.19/9.04 ! [R: rat,K: nat] :
% 16.19/9.04 ( ( times_times_rat @ ( minus_minus_rat @ R @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R ) @ K ) )
% 16.19/9.04 = ( times_times_rat @ R @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R ) @ one_one_rat ) @ K ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8477_pochhammer__absorb__comp,axiom,
% 16.19/9.04 ! [R: code_integer,K: nat] :
% 16.19/9.04 ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R ) @ K ) )
% 16.19/9.04 = ( times_3573771949741848930nteger @ R @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R ) @ one_one_Code_integer ) @ K ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8478_pochhammer__absorb__comp,axiom,
% 16.19/9.04 ! [R: int,K: nat] :
% 16.19/9.04 ( ( times_times_int @ ( minus_minus_int @ R @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R ) @ K ) )
% 16.19/9.04 = ( times_times_int @ R @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R ) @ one_one_int ) @ K ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8479_pochhammer__absorb__comp,axiom,
% 16.19/9.04 ! [R: real,K: nat] :
% 16.19/9.04 ( ( times_times_real @ ( minus_minus_real @ R @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R ) @ K ) )
% 16.19/9.04 = ( times_times_real @ R @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R ) @ one_one_real ) @ K ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8480_pochhammer__absorb__comp,axiom,
% 16.19/9.04 ! [R: complex,K: nat] :
% 16.19/9.04 ( ( times_times_complex @ ( minus_minus_complex @ R @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R ) @ K ) )
% 16.19/9.04 = ( times_times_complex @ R @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R ) @ one_one_complex ) @ K ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8481_pochhammer__same,axiom,
% 16.19/9.04 ! [N2: nat] :
% 16.19/9.04 ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ N2 )
% 16.19/9.04 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8482_pochhammer__same,axiom,
% 16.19/9.04 ! [N2: nat] :
% 16.19/9.04 ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ N2 )
% 16.19/9.04 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( semiri3624122377584611663nteger @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8483_pochhammer__same,axiom,
% 16.19/9.04 ! [N2: nat] :
% 16.19/9.04 ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ N2 )
% 16.19/9.04 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8484_pochhammer__same,axiom,
% 16.19/9.04 ! [N2: nat] :
% 16.19/9.04 ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ N2 )
% 16.19/9.04 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( semiri5044797733671781792omplex @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8485_pochhammer__same,axiom,
% 16.19/9.04 ! [N2: nat] :
% 16.19/9.04 ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ N2 )
% 16.19/9.04 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8486_real__less__lsqrt,axiom,
% 16.19/9.04 ! [X: real,Y: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.04 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/9.04 => ( ( ord_less_real @ X @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.04 => ( ord_less_real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8487_sqrt__sum__squares__le__sum,axiom,
% 16.19/9.04 ! [X: real,Y: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.04 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/9.04 => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X @ Y ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8488_sqrt__even__pow2,axiom,
% 16.19/9.04 ! [N2: nat] :
% 16.19/9.04 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.04 => ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/9.04 = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8489_real__sqrt__ge__abs1,axiom,
% 16.19/9.04 ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8490_real__sqrt__ge__abs2,axiom,
% 16.19/9.04 ! [Y: real,X: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8491_sqrt__sum__squares__le__sum__abs,axiom,
% 16.19/9.04 ! [X: real,Y: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8492_ln__sqrt,axiom,
% 16.19/9.04 ! [X: real] :
% 16.19/9.04 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.04 => ( ( ln_ln_real @ ( sqrt @ X ) )
% 16.19/9.04 = ( divide_divide_real @ ( ln_ln_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8493_cos__30,axiom,
% 16.19/9.04 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 16.19/9.04 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8494_sin__60,axiom,
% 16.19/9.04 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 16.19/9.04 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8495_exp__bound__half,axiom,
% 16.19/9.04 ! [Z: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.04 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8496_exp__bound__half,axiom,
% 16.19/9.04 ! [Z: complex] :
% 16.19/9.04 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.04 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8497_pochhammer__minus,axiom,
% 16.19/9.04 ! [B: rat,K: nat] :
% 16.19/9.04 ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K )
% 16.19/9.04 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8498_pochhammer__minus,axiom,
% 16.19/9.04 ! [B: code_integer,K: nat] :
% 16.19/9.04 ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K )
% 16.19/9.04 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8499_pochhammer__minus,axiom,
% 16.19/9.04 ! [B: int,K: nat] :
% 16.19/9.04 ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K )
% 16.19/9.04 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8500_pochhammer__minus,axiom,
% 16.19/9.04 ! [B: real,K: nat] :
% 16.19/9.04 ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K )
% 16.19/9.04 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8501_pochhammer__minus,axiom,
% 16.19/9.04 ! [B: complex,K: nat] :
% 16.19/9.04 ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K )
% 16.19/9.04 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8502_pochhammer__minus_H,axiom,
% 16.19/9.04 ! [B: rat,K: nat] :
% 16.19/9.04 ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K )
% 16.19/9.04 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8503_pochhammer__minus_H,axiom,
% 16.19/9.04 ! [B: code_integer,K: nat] :
% 16.19/9.04 ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
% 16.19/9.04 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8504_pochhammer__minus_H,axiom,
% 16.19/9.04 ! [B: int,K: nat] :
% 16.19/9.04 ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
% 16.19/9.04 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8505_pochhammer__minus_H,axiom,
% 16.19/9.04 ! [B: real,K: nat] :
% 16.19/9.04 ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K )
% 16.19/9.04 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8506_pochhammer__minus_H,axiom,
% 16.19/9.04 ! [B: complex,K: nat] :
% 16.19/9.04 ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
% 16.19/9.04 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8507_arsinh__real__aux,axiom,
% 16.19/9.04 ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8508_real__sqrt__power__even,axiom,
% 16.19/9.04 ! [N2: nat,X: real] :
% 16.19/9.04 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.04 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.04 => ( ( power_power_real @ ( sqrt @ X ) @ N2 )
% 16.19/9.04 = ( power_power_real @ X @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8509_exp__bound,axiom,
% 16.19/9.04 ! [X: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.04 => ( ( ord_less_eq_real @ X @ one_one_real )
% 16.19/9.04 => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8510_real__sqrt__sum__squares__mult__ge__zero,axiom,
% 16.19/9.04 ! [X: real,Y: real,Xa2: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8511_arith__geo__mean__sqrt,axiom,
% 16.19/9.04 ! [X: real,Y: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.04 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/9.04 => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X @ Y ) ) @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8512_tan__30,axiom,
% 16.19/9.04 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 16.19/9.04 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8513_real__exp__bound__lemma,axiom,
% 16.19/9.04 ! [X: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.04 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.04 => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8514_cos__x__y__le__one,axiom,
% 16.19/9.04 ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8515_real__sqrt__sum__squares__less,axiom,
% 16.19/9.04 ! [X: real,U: real,Y: real] :
% 16.19/9.04 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 16.19/9.04 => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 16.19/9.04 => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8516_arcosh__real__def,axiom,
% 16.19/9.04 ! [X: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ one_one_real @ X )
% 16.19/9.04 => ( ( arcosh_real @ X )
% 16.19/9.04 = ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8517_exp__ge__one__plus__x__over__n__power__n,axiom,
% 16.19/9.04 ! [N2: nat,X: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ X )
% 16.19/9.04 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.04 => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 ) @ ( exp_real @ X ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8518_exp__ge__one__minus__x__over__n__power__n,axiom,
% 16.19/9.04 ! [X: real,N2: nat] :
% 16.19/9.04 ( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N2 ) )
% 16.19/9.04 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.04 => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8519_cos__arctan,axiom,
% 16.19/9.04 ! [X: real] :
% 16.19/9.04 ( ( cos_real @ ( arctan @ X ) )
% 16.19/9.04 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8520_sin__arctan,axiom,
% 16.19/9.04 ! [X: real] :
% 16.19/9.04 ( ( sin_real @ ( arctan @ X ) )
% 16.19/9.04 = ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8521_exp__bound__lemma,axiom,
% 16.19/9.04 ! [Z: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.04 => ( 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 ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8522_exp__bound__lemma,axiom,
% 16.19/9.04 ! [Z: complex] :
% 16.19/9.04 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.04 => ( 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 ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8523_Maclaurin__exp__le,axiom,
% 16.19/9.04 ! [X: real,N2: nat] :
% 16.19/9.04 ? [T5: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 16.19/9.04 & ( ( exp_real @ X )
% 16.19/9.04 = ( plus_plus_real
% 16.19/9.04 @ ( groups6591440286371151544t_real
% 16.19/9.04 @ ^ [M6: nat] : ( divide_divide_real @ ( power_power_real @ X @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) )
% 16.19/9.04 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.04 @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T5 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8524_sqrt__sum__squares__half__less,axiom,
% 16.19/9.04 ! [X: real,U: real,Y: real] :
% 16.19/9.04 ( ( ord_less_real @ X @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.04 => ( ( ord_less_real @ Y @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.04 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.04 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/9.04 => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8525_exp__lower__Taylor__quadratic,axiom,
% 16.19/9.04 ! [X: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.04 => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X ) @ ( divide_divide_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8526_sin__cos__sqrt,axiom,
% 16.19/9.04 ! [X: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) )
% 16.19/9.04 => ( ( sin_real @ X )
% 16.19/9.04 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8527_arctan__half,axiom,
% 16.19/9.04 ( arctan
% 16.19/9.04 = ( ^ [X2: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X2 @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8528_tanh__real__altdef,axiom,
% 16.19/9.04 ( tanh_real
% 16.19/9.04 = ( ^ [X2: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8529_fact__double,axiom,
% 16.19/9.04 ! [N2: nat] :
% 16.19/9.04 ( ( semiri5044797733671781792omplex @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/9.04 = ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( comm_s2602460028002588243omplex @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ N2 ) ) @ ( semiri5044797733671781792omplex @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8530_fact__double,axiom,
% 16.19/9.04 ! [N2: nat] :
% 16.19/9.04 ( ( semiri773545260158071498ct_rat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/9.04 = ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ N2 ) ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8531_fact__double,axiom,
% 16.19/9.04 ! [N2: nat] :
% 16.19/9.04 ( ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/9.04 = ( times_times_real @ ( times_times_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ N2 ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8532_pochhammer__times__pochhammer__half,axiom,
% 16.19/9.04 ! [Z: rat,N2: nat] :
% 16.19/9.04 ( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
% 16.19/9.04 = ( groups73079841787564623at_rat
% 16.19/9.04 @ ^ [K3: nat] : ( plus_plus_rat @ Z @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 16.19/9.04 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8533_pochhammer__times__pochhammer__half,axiom,
% 16.19/9.04 ! [Z: real,N2: nat] :
% 16.19/9.04 ( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ ( suc @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
% 16.19/9.04 = ( groups129246275422532515t_real
% 16.19/9.04 @ ^ [K3: nat] : ( plus_plus_real @ Z @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.04 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8534_pochhammer__times__pochhammer__half,axiom,
% 16.19/9.04 ! [Z: complex,N2: nat] :
% 16.19/9.04 ( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ ( suc @ N2 ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
% 16.19/9.04 = ( groups6464643781859351333omplex
% 16.19/9.04 @ ^ [K3: nat] : ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 16.19/9.04 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8535_pochhammer__code,axiom,
% 16.19/9.04 ( comm_s4028243227959126397er_rat
% 16.19/9.04 = ( ^ [A5: rat,N: nat] :
% 16.19/9.04 ( if_rat @ ( N = zero_zero_nat ) @ one_one_rat
% 16.19/9.04 @ ( set_fo1949268297981939178at_rat
% 16.19/9.04 @ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A5 @ ( semiri681578069525770553at_rat @ O ) ) )
% 16.19/9.04 @ zero_zero_nat
% 16.19/9.04 @ ( minus_minus_nat @ N @ one_one_nat )
% 16.19/9.04 @ one_one_rat ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8536_pochhammer__code,axiom,
% 16.19/9.04 ( comm_s4660882817536571857er_int
% 16.19/9.04 = ( ^ [A5: int,N: nat] :
% 16.19/9.04 ( if_int @ ( N = zero_zero_nat ) @ one_one_int
% 16.19/9.04 @ ( set_fo2581907887559384638at_int
% 16.19/9.04 @ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A5 @ ( semiri1314217659103216013at_int @ O ) ) )
% 16.19/9.04 @ zero_zero_nat
% 16.19/9.04 @ ( minus_minus_nat @ N @ one_one_nat )
% 16.19/9.04 @ one_one_int ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8537_pochhammer__code,axiom,
% 16.19/9.04 ( comm_s7457072308508201937r_real
% 16.19/9.04 = ( ^ [A5: real,N: nat] :
% 16.19/9.04 ( if_real @ ( N = zero_zero_nat ) @ one_one_real
% 16.19/9.04 @ ( set_fo3111899725591712190t_real
% 16.19/9.04 @ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A5 @ ( semiri5074537144036343181t_real @ O ) ) )
% 16.19/9.04 @ zero_zero_nat
% 16.19/9.04 @ ( minus_minus_nat @ N @ one_one_nat )
% 16.19/9.04 @ one_one_real ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8538_pochhammer__code,axiom,
% 16.19/9.04 ( comm_s2602460028002588243omplex
% 16.19/9.04 = ( ^ [A5: complex,N: nat] :
% 16.19/9.04 ( if_complex @ ( N = zero_zero_nat ) @ one_one_complex
% 16.19/9.04 @ ( set_fo1517530859248394432omplex
% 16.19/9.04 @ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A5 @ ( semiri8010041392384452111omplex @ O ) ) )
% 16.19/9.04 @ zero_zero_nat
% 16.19/9.04 @ ( minus_minus_nat @ N @ one_one_nat )
% 16.19/9.04 @ one_one_complex ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8539_pochhammer__code,axiom,
% 16.19/9.04 ( comm_s4663373288045622133er_nat
% 16.19/9.04 = ( ^ [A5: nat,N: nat] :
% 16.19/9.04 ( if_nat @ ( N = zero_zero_nat ) @ one_one_nat
% 16.19/9.04 @ ( set_fo2584398358068434914at_nat
% 16.19/9.04 @ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A5 @ ( semiri1316708129612266289at_nat @ O ) ) )
% 16.19/9.04 @ zero_zero_nat
% 16.19/9.04 @ ( minus_minus_nat @ N @ one_one_nat )
% 16.19/9.04 @ one_one_nat ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8540_arsinh__real__def,axiom,
% 16.19/9.04 ( arsinh_real
% 16.19/9.04 = ( ^ [X2: real] : ( ln_ln_real @ ( plus_plus_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8541_cos__arcsin,axiom,
% 16.19/9.04 ! [X: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 16.19/9.04 => ( ( ord_less_eq_real @ X @ one_one_real )
% 16.19/9.04 => ( ( cos_real @ ( arcsin @ X ) )
% 16.19/9.04 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8542_sin__arccos__abs,axiom,
% 16.19/9.04 ! [Y: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 16.19/9.04 => ( ( sin_real @ ( arccos @ Y ) )
% 16.19/9.04 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8543_prod_Oneutral__const,axiom,
% 16.19/9.04 ! [A2: set_nat] :
% 16.19/9.04 ( ( groups708209901874060359at_nat
% 16.19/9.04 @ ^ [Uu3: nat] : one_one_nat
% 16.19/9.04 @ A2 )
% 16.19/9.04 = one_one_nat ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8544_prod_Oneutral__const,axiom,
% 16.19/9.04 ! [A2: set_nat] :
% 16.19/9.04 ( ( groups705719431365010083at_int
% 16.19/9.04 @ ^ [Uu3: nat] : one_one_int
% 16.19/9.04 @ A2 )
% 16.19/9.04 = one_one_int ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8545_prod_Oneutral__const,axiom,
% 16.19/9.04 ! [A2: set_int] :
% 16.19/9.04 ( ( groups1705073143266064639nt_int
% 16.19/9.04 @ ^ [Uu3: int] : one_one_int
% 16.19/9.04 @ A2 )
% 16.19/9.04 = one_one_int ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8546_of__nat__prod,axiom,
% 16.19/9.04 ! [F: int > nat,A2: set_int] :
% 16.19/9.04 ( ( semiri1314217659103216013at_int @ ( groups1707563613775114915nt_nat @ F @ A2 ) )
% 16.19/9.04 = ( groups1705073143266064639nt_int
% 16.19/9.04 @ ^ [X2: int] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
% 16.19/9.04 @ A2 ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8547_of__nat__prod,axiom,
% 16.19/9.04 ! [F: nat > nat,A2: set_nat] :
% 16.19/9.04 ( ( semiri5074537144036343181t_real @ ( groups708209901874060359at_nat @ F @ A2 ) )
% 16.19/9.04 = ( groups129246275422532515t_real
% 16.19/9.04 @ ^ [X2: nat] : ( semiri5074537144036343181t_real @ ( F @ X2 ) )
% 16.19/9.04 @ A2 ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8548_of__nat__prod,axiom,
% 16.19/9.04 ! [F: nat > nat,A2: set_nat] :
% 16.19/9.04 ( ( semiri8010041392384452111omplex @ ( groups708209901874060359at_nat @ F @ A2 ) )
% 16.19/9.04 = ( groups6464643781859351333omplex
% 16.19/9.04 @ ^ [X2: nat] : ( semiri8010041392384452111omplex @ ( F @ X2 ) )
% 16.19/9.04 @ A2 ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8549_of__nat__prod,axiom,
% 16.19/9.04 ! [F: nat > nat,A2: set_nat] :
% 16.19/9.04 ( ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ F @ A2 ) )
% 16.19/9.04 = ( groups708209901874060359at_nat
% 16.19/9.04 @ ^ [X2: nat] : ( semiri1316708129612266289at_nat @ ( F @ X2 ) )
% 16.19/9.04 @ A2 ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8550_of__nat__prod,axiom,
% 16.19/9.04 ! [F: nat > nat,A2: set_nat] :
% 16.19/9.04 ( ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ F @ A2 ) )
% 16.19/9.04 = ( groups705719431365010083at_int
% 16.19/9.04 @ ^ [X2: nat] : ( semiri1314217659103216013at_int @ ( F @ X2 ) )
% 16.19/9.04 @ A2 ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8551_of__int__prod,axiom,
% 16.19/9.04 ! [F: nat > int,A2: set_nat] :
% 16.19/9.04 ( ( ring_1_of_int_real @ ( groups705719431365010083at_int @ F @ A2 ) )
% 16.19/9.04 = ( groups129246275422532515t_real
% 16.19/9.04 @ ^ [X2: nat] : ( ring_1_of_int_real @ ( F @ X2 ) )
% 16.19/9.04 @ A2 ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8552_of__int__prod,axiom,
% 16.19/9.04 ! [F: nat > int,A2: set_nat] :
% 16.19/9.04 ( ( ring_1_of_int_rat @ ( groups705719431365010083at_int @ F @ A2 ) )
% 16.19/9.04 = ( groups73079841787564623at_rat
% 16.19/9.04 @ ^ [X2: nat] : ( ring_1_of_int_rat @ ( F @ X2 ) )
% 16.19/9.04 @ A2 ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8553_of__int__prod,axiom,
% 16.19/9.04 ! [F: nat > int,A2: set_nat] :
% 16.19/9.04 ( ( ring_1_of_int_int @ ( groups705719431365010083at_int @ F @ A2 ) )
% 16.19/9.04 = ( groups705719431365010083at_int
% 16.19/9.04 @ ^ [X2: nat] : ( ring_1_of_int_int @ ( F @ X2 ) )
% 16.19/9.04 @ A2 ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8554_of__int__prod,axiom,
% 16.19/9.04 ! [F: int > int,A2: set_int] :
% 16.19/9.04 ( ( ring_1_of_int_real @ ( groups1705073143266064639nt_int @ F @ A2 ) )
% 16.19/9.04 = ( groups2316167850115554303t_real
% 16.19/9.04 @ ^ [X2: int] : ( ring_1_of_int_real @ ( F @ X2 ) )
% 16.19/9.04 @ A2 ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8555_of__int__prod,axiom,
% 16.19/9.04 ! [F: int > int,A2: set_int] :
% 16.19/9.04 ( ( ring_1_of_int_rat @ ( groups1705073143266064639nt_int @ F @ A2 ) )
% 16.19/9.04 = ( groups1072433553688619179nt_rat
% 16.19/9.04 @ ^ [X2: int] : ( ring_1_of_int_rat @ ( F @ X2 ) )
% 16.19/9.04 @ A2 ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8556_of__int__prod,axiom,
% 16.19/9.04 ! [F: int > int,A2: set_int] :
% 16.19/9.04 ( ( ring_1_of_int_int @ ( groups1705073143266064639nt_int @ F @ A2 ) )
% 16.19/9.04 = ( groups1705073143266064639nt_int
% 16.19/9.04 @ ^ [X2: int] : ( ring_1_of_int_int @ ( F @ X2 ) )
% 16.19/9.04 @ A2 ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8557_prod_Oempty,axiom,
% 16.19/9.04 ! [G: nat > complex] :
% 16.19/9.04 ( ( groups6464643781859351333omplex @ G @ bot_bot_set_nat )
% 16.19/9.04 = one_one_complex ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8558_prod_Oempty,axiom,
% 16.19/9.04 ! [G: nat > real] :
% 16.19/9.04 ( ( groups129246275422532515t_real @ G @ bot_bot_set_nat )
% 16.19/9.04 = one_one_real ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8559_prod_Oempty,axiom,
% 16.19/9.04 ! [G: nat > rat] :
% 16.19/9.04 ( ( groups73079841787564623at_rat @ G @ bot_bot_set_nat )
% 16.19/9.04 = one_one_rat ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8560_prod_Oempty,axiom,
% 16.19/9.04 ! [G: int > complex] :
% 16.19/9.04 ( ( groups7440179247065528705omplex @ G @ bot_bot_set_int )
% 16.19/9.04 = one_one_complex ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8561_prod_Oempty,axiom,
% 16.19/9.04 ! [G: int > real] :
% 16.19/9.04 ( ( groups2316167850115554303t_real @ G @ bot_bot_set_int )
% 16.19/9.04 = one_one_real ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8562_prod_Oempty,axiom,
% 16.19/9.04 ! [G: int > rat] :
% 16.19/9.04 ( ( groups1072433553688619179nt_rat @ G @ bot_bot_set_int )
% 16.19/9.04 = one_one_rat ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8563_prod_Oempty,axiom,
% 16.19/9.04 ! [G: int > nat] :
% 16.19/9.04 ( ( groups1707563613775114915nt_nat @ G @ bot_bot_set_int )
% 16.19/9.04 = one_one_nat ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8564_prod_Oempty,axiom,
% 16.19/9.04 ! [G: real > complex] :
% 16.19/9.04 ( ( groups713298508707869441omplex @ G @ bot_bot_set_real )
% 16.19/9.04 = one_one_complex ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8565_prod_Oempty,axiom,
% 16.19/9.04 ! [G: real > real] :
% 16.19/9.04 ( ( groups1681761925125756287l_real @ G @ bot_bot_set_real )
% 16.19/9.04 = one_one_real ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8566_prod_Oempty,axiom,
% 16.19/9.04 ! [G: real > rat] :
% 16.19/9.04 ( ( groups4061424788464935467al_rat @ G @ bot_bot_set_real )
% 16.19/9.04 = one_one_rat ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8567_prod_Oinfinite,axiom,
% 16.19/9.04 ! [A2: set_nat,G: nat > complex] :
% 16.19/9.04 ( ~ ( finite_finite_nat @ A2 )
% 16.19/9.04 => ( ( groups6464643781859351333omplex @ G @ A2 )
% 16.19/9.04 = one_one_complex ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8568_prod_Oinfinite,axiom,
% 16.19/9.04 ! [A2: set_int,G: int > complex] :
% 16.19/9.04 ( ~ ( finite_finite_int @ A2 )
% 16.19/9.04 => ( ( groups7440179247065528705omplex @ G @ A2 )
% 16.19/9.04 = one_one_complex ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8569_prod_Oinfinite,axiom,
% 16.19/9.04 ! [A2: set_complex,G: complex > complex] :
% 16.19/9.04 ( ~ ( finite3207457112153483333omplex @ A2 )
% 16.19/9.04 => ( ( groups3708469109370488835omplex @ G @ A2 )
% 16.19/9.04 = one_one_complex ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8570_prod_Oinfinite,axiom,
% 16.19/9.04 ! [A2: set_nat,G: nat > real] :
% 16.19/9.04 ( ~ ( finite_finite_nat @ A2 )
% 16.19/9.04 => ( ( groups129246275422532515t_real @ G @ A2 )
% 16.19/9.04 = one_one_real ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8571_prod_Oinfinite,axiom,
% 16.19/9.04 ! [A2: set_int,G: int > real] :
% 16.19/9.04 ( ~ ( finite_finite_int @ A2 )
% 16.19/9.04 => ( ( groups2316167850115554303t_real @ G @ A2 )
% 16.19/9.04 = one_one_real ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8572_prod_Oinfinite,axiom,
% 16.19/9.04 ! [A2: set_complex,G: complex > real] :
% 16.19/9.04 ( ~ ( finite3207457112153483333omplex @ A2 )
% 16.19/9.04 => ( ( groups766887009212190081x_real @ G @ A2 )
% 16.19/9.04 = one_one_real ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8573_prod_Oinfinite,axiom,
% 16.19/9.04 ! [A2: set_nat,G: nat > rat] :
% 16.19/9.04 ( ~ ( finite_finite_nat @ A2 )
% 16.19/9.04 => ( ( groups73079841787564623at_rat @ G @ A2 )
% 16.19/9.04 = one_one_rat ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8574_prod_Oinfinite,axiom,
% 16.19/9.04 ! [A2: set_int,G: int > rat] :
% 16.19/9.04 ( ~ ( finite_finite_int @ A2 )
% 16.19/9.04 => ( ( groups1072433553688619179nt_rat @ G @ A2 )
% 16.19/9.04 = one_one_rat ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8575_prod_Oinfinite,axiom,
% 16.19/9.04 ! [A2: set_complex,G: complex > rat] :
% 16.19/9.04 ( ~ ( finite3207457112153483333omplex @ A2 )
% 16.19/9.04 => ( ( groups225925009352817453ex_rat @ G @ A2 )
% 16.19/9.04 = one_one_rat ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8576_prod_Oinfinite,axiom,
% 16.19/9.04 ! [A2: set_int,G: int > nat] :
% 16.19/9.04 ( ~ ( finite_finite_int @ A2 )
% 16.19/9.04 => ( ( groups1707563613775114915nt_nat @ G @ A2 )
% 16.19/9.04 = one_one_nat ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8577_arccos__1,axiom,
% 16.19/9.04 ( ( arccos @ one_one_real )
% 16.19/9.04 = zero_zero_real ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8578_prod_Odelta,axiom,
% 16.19/9.04 ! [S: set_real,A: real,B: real > complex] :
% 16.19/9.04 ( ( finite_finite_real @ S )
% 16.19/9.04 => ( ( ( member_real @ A @ S )
% 16.19/9.04 => ( ( groups713298508707869441omplex
% 16.19/9.04 @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 16.19/9.04 @ S )
% 16.19/9.04 = ( B @ A ) ) )
% 16.19/9.04 & ( ~ ( member_real @ A @ S )
% 16.19/9.04 => ( ( groups713298508707869441omplex
% 16.19/9.04 @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 16.19/9.04 @ S )
% 16.19/9.04 = one_one_complex ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8579_prod_Odelta,axiom,
% 16.19/9.04 ! [S: set_nat,A: nat,B: nat > complex] :
% 16.19/9.04 ( ( finite_finite_nat @ S )
% 16.19/9.04 => ( ( ( member_nat @ A @ S )
% 16.19/9.04 => ( ( groups6464643781859351333omplex
% 16.19/9.04 @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 16.19/9.04 @ S )
% 16.19/9.04 = ( B @ A ) ) )
% 16.19/9.04 & ( ~ ( member_nat @ A @ S )
% 16.19/9.04 => ( ( groups6464643781859351333omplex
% 16.19/9.04 @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 16.19/9.04 @ S )
% 16.19/9.04 = one_one_complex ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8580_prod_Odelta,axiom,
% 16.19/9.04 ! [S: set_int,A: int,B: int > complex] :
% 16.19/9.04 ( ( finite_finite_int @ S )
% 16.19/9.04 => ( ( ( member_int @ A @ S )
% 16.19/9.04 => ( ( groups7440179247065528705omplex
% 16.19/9.04 @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 16.19/9.04 @ S )
% 16.19/9.04 = ( B @ A ) ) )
% 16.19/9.04 & ( ~ ( member_int @ A @ S )
% 16.19/9.04 => ( ( groups7440179247065528705omplex
% 16.19/9.04 @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 16.19/9.04 @ S )
% 16.19/9.04 = one_one_complex ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8581_prod_Odelta,axiom,
% 16.19/9.04 ! [S: set_complex,A: complex,B: complex > complex] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ S )
% 16.19/9.04 => ( ( ( member_complex @ A @ S )
% 16.19/9.04 => ( ( groups3708469109370488835omplex
% 16.19/9.04 @ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 16.19/9.04 @ S )
% 16.19/9.04 = ( B @ A ) ) )
% 16.19/9.04 & ( ~ ( member_complex @ A @ S )
% 16.19/9.04 => ( ( groups3708469109370488835omplex
% 16.19/9.04 @ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 16.19/9.04 @ S )
% 16.19/9.04 = one_one_complex ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8582_prod_Odelta,axiom,
% 16.19/9.04 ! [S: set_real,A: real,B: real > real] :
% 16.19/9.04 ( ( finite_finite_real @ S )
% 16.19/9.04 => ( ( ( member_real @ A @ S )
% 16.19/9.04 => ( ( groups1681761925125756287l_real
% 16.19/9.04 @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 16.19/9.04 @ S )
% 16.19/9.04 = ( B @ A ) ) )
% 16.19/9.04 & ( ~ ( member_real @ A @ S )
% 16.19/9.04 => ( ( groups1681761925125756287l_real
% 16.19/9.04 @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 16.19/9.04 @ S )
% 16.19/9.04 = one_one_real ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8583_prod_Odelta,axiom,
% 16.19/9.04 ! [S: set_nat,A: nat,B: nat > real] :
% 16.19/9.04 ( ( finite_finite_nat @ S )
% 16.19/9.04 => ( ( ( member_nat @ A @ S )
% 16.19/9.04 => ( ( groups129246275422532515t_real
% 16.19/9.04 @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 16.19/9.04 @ S )
% 16.19/9.04 = ( B @ A ) ) )
% 16.19/9.04 & ( ~ ( member_nat @ A @ S )
% 16.19/9.04 => ( ( groups129246275422532515t_real
% 16.19/9.04 @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 16.19/9.04 @ S )
% 16.19/9.04 = one_one_real ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8584_prod_Odelta,axiom,
% 16.19/9.04 ! [S: set_int,A: int,B: int > real] :
% 16.19/9.04 ( ( finite_finite_int @ S )
% 16.19/9.04 => ( ( ( member_int @ A @ S )
% 16.19/9.04 => ( ( groups2316167850115554303t_real
% 16.19/9.04 @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 16.19/9.04 @ S )
% 16.19/9.04 = ( B @ A ) ) )
% 16.19/9.04 & ( ~ ( member_int @ A @ S )
% 16.19/9.04 => ( ( groups2316167850115554303t_real
% 16.19/9.04 @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 16.19/9.04 @ S )
% 16.19/9.04 = one_one_real ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8585_prod_Odelta,axiom,
% 16.19/9.04 ! [S: set_complex,A: complex,B: complex > real] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ S )
% 16.19/9.04 => ( ( ( member_complex @ A @ S )
% 16.19/9.04 => ( ( groups766887009212190081x_real
% 16.19/9.04 @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 16.19/9.04 @ S )
% 16.19/9.04 = ( B @ A ) ) )
% 16.19/9.04 & ( ~ ( member_complex @ A @ S )
% 16.19/9.04 => ( ( groups766887009212190081x_real
% 16.19/9.04 @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 16.19/9.04 @ S )
% 16.19/9.04 = one_one_real ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8586_prod_Odelta,axiom,
% 16.19/9.04 ! [S: set_real,A: real,B: real > rat] :
% 16.19/9.04 ( ( finite_finite_real @ S )
% 16.19/9.04 => ( ( ( member_real @ A @ S )
% 16.19/9.04 => ( ( groups4061424788464935467al_rat
% 16.19/9.04 @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
% 16.19/9.04 @ S )
% 16.19/9.04 = ( B @ A ) ) )
% 16.19/9.04 & ( ~ ( member_real @ A @ S )
% 16.19/9.04 => ( ( groups4061424788464935467al_rat
% 16.19/9.04 @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
% 16.19/9.04 @ S )
% 16.19/9.04 = one_one_rat ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8587_prod_Odelta,axiom,
% 16.19/9.04 ! [S: set_nat,A: nat,B: nat > rat] :
% 16.19/9.04 ( ( finite_finite_nat @ S )
% 16.19/9.04 => ( ( ( member_nat @ A @ S )
% 16.19/9.04 => ( ( groups73079841787564623at_rat
% 16.19/9.04 @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
% 16.19/9.04 @ S )
% 16.19/9.04 = ( B @ A ) ) )
% 16.19/9.04 & ( ~ ( member_nat @ A @ S )
% 16.19/9.04 => ( ( groups73079841787564623at_rat
% 16.19/9.04 @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
% 16.19/9.04 @ S )
% 16.19/9.04 = one_one_rat ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8588_prod_Odelta_H,axiom,
% 16.19/9.04 ! [S: set_real,A: real,B: real > complex] :
% 16.19/9.04 ( ( finite_finite_real @ S )
% 16.19/9.04 => ( ( ( member_real @ A @ S )
% 16.19/9.04 => ( ( groups713298508707869441omplex
% 16.19/9.04 @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 16.19/9.04 @ S )
% 16.19/9.04 = ( B @ A ) ) )
% 16.19/9.04 & ( ~ ( member_real @ A @ S )
% 16.19/9.04 => ( ( groups713298508707869441omplex
% 16.19/9.04 @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 16.19/9.04 @ S )
% 16.19/9.04 = one_one_complex ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8589_prod_Odelta_H,axiom,
% 16.19/9.04 ! [S: set_nat,A: nat,B: nat > complex] :
% 16.19/9.04 ( ( finite_finite_nat @ S )
% 16.19/9.04 => ( ( ( member_nat @ A @ S )
% 16.19/9.04 => ( ( groups6464643781859351333omplex
% 16.19/9.04 @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 16.19/9.04 @ S )
% 16.19/9.04 = ( B @ A ) ) )
% 16.19/9.04 & ( ~ ( member_nat @ A @ S )
% 16.19/9.04 => ( ( groups6464643781859351333omplex
% 16.19/9.04 @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 16.19/9.04 @ S )
% 16.19/9.04 = one_one_complex ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8590_prod_Odelta_H,axiom,
% 16.19/9.04 ! [S: set_int,A: int,B: int > complex] :
% 16.19/9.04 ( ( finite_finite_int @ S )
% 16.19/9.04 => ( ( ( member_int @ A @ S )
% 16.19/9.04 => ( ( groups7440179247065528705omplex
% 16.19/9.04 @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 16.19/9.04 @ S )
% 16.19/9.04 = ( B @ A ) ) )
% 16.19/9.04 & ( ~ ( member_int @ A @ S )
% 16.19/9.04 => ( ( groups7440179247065528705omplex
% 16.19/9.04 @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 16.19/9.04 @ S )
% 16.19/9.04 = one_one_complex ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8591_prod_Odelta_H,axiom,
% 16.19/9.04 ! [S: set_complex,A: complex,B: complex > complex] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ S )
% 16.19/9.04 => ( ( ( member_complex @ A @ S )
% 16.19/9.04 => ( ( groups3708469109370488835omplex
% 16.19/9.04 @ ^ [K3: complex] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 16.19/9.04 @ S )
% 16.19/9.04 = ( B @ A ) ) )
% 16.19/9.04 & ( ~ ( member_complex @ A @ S )
% 16.19/9.04 => ( ( groups3708469109370488835omplex
% 16.19/9.04 @ ^ [K3: complex] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 16.19/9.04 @ S )
% 16.19/9.04 = one_one_complex ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8592_prod_Odelta_H,axiom,
% 16.19/9.04 ! [S: set_real,A: real,B: real > real] :
% 16.19/9.04 ( ( finite_finite_real @ S )
% 16.19/9.04 => ( ( ( member_real @ A @ S )
% 16.19/9.04 => ( ( groups1681761925125756287l_real
% 16.19/9.04 @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 16.19/9.04 @ S )
% 16.19/9.04 = ( B @ A ) ) )
% 16.19/9.04 & ( ~ ( member_real @ A @ S )
% 16.19/9.04 => ( ( groups1681761925125756287l_real
% 16.19/9.04 @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 16.19/9.04 @ S )
% 16.19/9.04 = one_one_real ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8593_prod_Odelta_H,axiom,
% 16.19/9.04 ! [S: set_nat,A: nat,B: nat > real] :
% 16.19/9.04 ( ( finite_finite_nat @ S )
% 16.19/9.04 => ( ( ( member_nat @ A @ S )
% 16.19/9.04 => ( ( groups129246275422532515t_real
% 16.19/9.04 @ ^ [K3: nat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 16.19/9.04 @ S )
% 16.19/9.04 = ( B @ A ) ) )
% 16.19/9.04 & ( ~ ( member_nat @ A @ S )
% 16.19/9.04 => ( ( groups129246275422532515t_real
% 16.19/9.04 @ ^ [K3: nat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 16.19/9.04 @ S )
% 16.19/9.04 = one_one_real ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8594_prod_Odelta_H,axiom,
% 16.19/9.04 ! [S: set_int,A: int,B: int > real] :
% 16.19/9.04 ( ( finite_finite_int @ S )
% 16.19/9.04 => ( ( ( member_int @ A @ S )
% 16.19/9.04 => ( ( groups2316167850115554303t_real
% 16.19/9.04 @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 16.19/9.04 @ S )
% 16.19/9.04 = ( B @ A ) ) )
% 16.19/9.04 & ( ~ ( member_int @ A @ S )
% 16.19/9.04 => ( ( groups2316167850115554303t_real
% 16.19/9.04 @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 16.19/9.04 @ S )
% 16.19/9.04 = one_one_real ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8595_prod_Odelta_H,axiom,
% 16.19/9.04 ! [S: set_complex,A: complex,B: complex > real] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ S )
% 16.19/9.04 => ( ( ( member_complex @ A @ S )
% 16.19/9.04 => ( ( groups766887009212190081x_real
% 16.19/9.04 @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 16.19/9.04 @ S )
% 16.19/9.04 = ( B @ A ) ) )
% 16.19/9.04 & ( ~ ( member_complex @ A @ S )
% 16.19/9.04 => ( ( groups766887009212190081x_real
% 16.19/9.04 @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 16.19/9.04 @ S )
% 16.19/9.04 = one_one_real ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8596_prod_Odelta_H,axiom,
% 16.19/9.04 ! [S: set_real,A: real,B: real > rat] :
% 16.19/9.04 ( ( finite_finite_real @ S )
% 16.19/9.04 => ( ( ( member_real @ A @ S )
% 16.19/9.04 => ( ( groups4061424788464935467al_rat
% 16.19/9.04 @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
% 16.19/9.04 @ S )
% 16.19/9.04 = ( B @ A ) ) )
% 16.19/9.04 & ( ~ ( member_real @ A @ S )
% 16.19/9.04 => ( ( groups4061424788464935467al_rat
% 16.19/9.04 @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
% 16.19/9.04 @ S )
% 16.19/9.04 = one_one_rat ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8597_prod_Odelta_H,axiom,
% 16.19/9.04 ! [S: set_nat,A: nat,B: nat > rat] :
% 16.19/9.04 ( ( finite_finite_nat @ S )
% 16.19/9.04 => ( ( ( member_nat @ A @ S )
% 16.19/9.04 => ( ( groups73079841787564623at_rat
% 16.19/9.04 @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
% 16.19/9.04 @ S )
% 16.19/9.04 = ( B @ A ) ) )
% 16.19/9.04 & ( ~ ( member_nat @ A @ S )
% 16.19/9.04 => ( ( groups73079841787564623at_rat
% 16.19/9.04 @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
% 16.19/9.04 @ S )
% 16.19/9.04 = one_one_rat ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8598_prod_OlessThan__Suc,axiom,
% 16.19/9.04 ! [G: nat > real,N2: nat] :
% 16.19/9.04 ( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 16.19/9.04 = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8599_prod_OlessThan__Suc,axiom,
% 16.19/9.04 ! [G: nat > rat,N2: nat] :
% 16.19/9.04 ( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 16.19/9.04 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8600_prod_OlessThan__Suc,axiom,
% 16.19/9.04 ! [G: nat > nat,N2: nat] :
% 16.19/9.04 ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 16.19/9.04 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8601_prod_OlessThan__Suc,axiom,
% 16.19/9.04 ! [G: nat > int,N2: nat] :
% 16.19/9.04 ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 16.19/9.04 = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8602_arccos__minus__1,axiom,
% 16.19/9.04 ( ( arccos @ ( uminus_uminus_real @ one_one_real ) )
% 16.19/9.04 = pi ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8603_cos__arccos,axiom,
% 16.19/9.04 ! [Y: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 16.19/9.04 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 16.19/9.04 => ( ( cos_real @ ( arccos @ Y ) )
% 16.19/9.04 = Y ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8604_sin__arcsin,axiom,
% 16.19/9.04 ! [Y: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 16.19/9.04 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 16.19/9.04 => ( ( sin_real @ ( arcsin @ Y ) )
% 16.19/9.04 = Y ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8605_prod_Ocl__ivl__Suc,axiom,
% 16.19/9.04 ! [N2: nat,M: nat,G: nat > complex] :
% 16.19/9.04 ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 16.19/9.04 => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 16.19/9.04 = one_one_complex ) )
% 16.19/9.04 & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 16.19/9.04 => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 16.19/9.04 = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8606_prod_Ocl__ivl__Suc,axiom,
% 16.19/9.04 ! [N2: nat,M: nat,G: nat > real] :
% 16.19/9.04 ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 16.19/9.04 => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 16.19/9.04 = one_one_real ) )
% 16.19/9.04 & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 16.19/9.04 => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 16.19/9.04 = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8607_prod_Ocl__ivl__Suc,axiom,
% 16.19/9.04 ! [N2: nat,M: nat,G: nat > rat] :
% 16.19/9.04 ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 16.19/9.04 => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 16.19/9.04 = one_one_rat ) )
% 16.19/9.04 & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 16.19/9.04 => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 16.19/9.04 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8608_prod_Ocl__ivl__Suc,axiom,
% 16.19/9.04 ! [N2: nat,M: nat,G: nat > nat] :
% 16.19/9.04 ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 16.19/9.04 => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 16.19/9.04 = one_one_nat ) )
% 16.19/9.04 & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 16.19/9.04 => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 16.19/9.04 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8609_prod_Ocl__ivl__Suc,axiom,
% 16.19/9.04 ! [N2: nat,M: nat,G: nat > int] :
% 16.19/9.04 ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 16.19/9.04 => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 16.19/9.04 = one_one_int ) )
% 16.19/9.04 & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 16.19/9.04 => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 16.19/9.04 = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8610_arccos__0,axiom,
% 16.19/9.04 ( ( arccos @ zero_zero_real )
% 16.19/9.04 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8611_arcsin__1,axiom,
% 16.19/9.04 ( ( arcsin @ one_one_real )
% 16.19/9.04 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8612_arcsin__minus__1,axiom,
% 16.19/9.04 ( ( arcsin @ ( uminus_uminus_real @ one_one_real ) )
% 16.19/9.04 = ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8613_prod_Oswap,axiom,
% 16.19/9.04 ! [G: nat > nat > nat,B3: set_nat,A2: set_nat] :
% 16.19/9.04 ( ( groups708209901874060359at_nat
% 16.19/9.04 @ ^ [I3: nat] : ( groups708209901874060359at_nat @ ( G @ I3 ) @ B3 )
% 16.19/9.04 @ A2 )
% 16.19/9.04 = ( groups708209901874060359at_nat
% 16.19/9.04 @ ^ [J3: nat] :
% 16.19/9.04 ( groups708209901874060359at_nat
% 16.19/9.04 @ ^ [I3: nat] : ( G @ I3 @ J3 )
% 16.19/9.04 @ A2 )
% 16.19/9.04 @ B3 ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8614_prod_Oswap,axiom,
% 16.19/9.04 ! [G: nat > nat > int,B3: set_nat,A2: set_nat] :
% 16.19/9.04 ( ( groups705719431365010083at_int
% 16.19/9.04 @ ^ [I3: nat] : ( groups705719431365010083at_int @ ( G @ I3 ) @ B3 )
% 16.19/9.04 @ A2 )
% 16.19/9.04 = ( groups705719431365010083at_int
% 16.19/9.04 @ ^ [J3: nat] :
% 16.19/9.04 ( groups705719431365010083at_int
% 16.19/9.04 @ ^ [I3: nat] : ( G @ I3 @ J3 )
% 16.19/9.04 @ A2 )
% 16.19/9.04 @ B3 ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8615_prod_Oswap,axiom,
% 16.19/9.04 ! [G: nat > int > int,B3: set_int,A2: set_nat] :
% 16.19/9.04 ( ( groups705719431365010083at_int
% 16.19/9.04 @ ^ [I3: nat] : ( groups1705073143266064639nt_int @ ( G @ I3 ) @ B3 )
% 16.19/9.04 @ A2 )
% 16.19/9.04 = ( groups1705073143266064639nt_int
% 16.19/9.04 @ ^ [J3: int] :
% 16.19/9.04 ( groups705719431365010083at_int
% 16.19/9.04 @ ^ [I3: nat] : ( G @ I3 @ J3 )
% 16.19/9.04 @ A2 )
% 16.19/9.04 @ B3 ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8616_prod_Oswap,axiom,
% 16.19/9.04 ! [G: int > nat > int,B3: set_nat,A2: set_int] :
% 16.19/9.04 ( ( groups1705073143266064639nt_int
% 16.19/9.04 @ ^ [I3: int] : ( groups705719431365010083at_int @ ( G @ I3 ) @ B3 )
% 16.19/9.04 @ A2 )
% 16.19/9.04 = ( groups705719431365010083at_int
% 16.19/9.04 @ ^ [J3: nat] :
% 16.19/9.04 ( groups1705073143266064639nt_int
% 16.19/9.04 @ ^ [I3: int] : ( G @ I3 @ J3 )
% 16.19/9.04 @ A2 )
% 16.19/9.04 @ B3 ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8617_prod_Oswap,axiom,
% 16.19/9.04 ! [G: int > int > int,B3: set_int,A2: set_int] :
% 16.19/9.04 ( ( groups1705073143266064639nt_int
% 16.19/9.04 @ ^ [I3: int] : ( groups1705073143266064639nt_int @ ( G @ I3 ) @ B3 )
% 16.19/9.04 @ A2 )
% 16.19/9.04 = ( groups1705073143266064639nt_int
% 16.19/9.04 @ ^ [J3: int] :
% 16.19/9.04 ( groups1705073143266064639nt_int
% 16.19/9.04 @ ^ [I3: int] : ( G @ I3 @ J3 )
% 16.19/9.04 @ A2 )
% 16.19/9.04 @ B3 ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8618_prod_Onot__neutral__contains__not__neutral,axiom,
% 16.19/9.04 ! [G: nat > complex,A2: set_nat] :
% 16.19/9.04 ( ( ( groups6464643781859351333omplex @ G @ A2 )
% 16.19/9.04 != one_one_complex )
% 16.19/9.04 => ~ ! [A3: nat] :
% 16.19/9.04 ( ( member_nat @ A3 @ A2 )
% 16.19/9.04 => ( ( G @ A3 )
% 16.19/9.04 = one_one_complex ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8619_prod_Onot__neutral__contains__not__neutral,axiom,
% 16.19/9.04 ! [G: real > complex,A2: set_real] :
% 16.19/9.04 ( ( ( groups713298508707869441omplex @ G @ A2 )
% 16.19/9.04 != one_one_complex )
% 16.19/9.04 => ~ ! [A3: real] :
% 16.19/9.04 ( ( member_real @ A3 @ A2 )
% 16.19/9.04 => ( ( G @ A3 )
% 16.19/9.04 = one_one_complex ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8620_prod_Onot__neutral__contains__not__neutral,axiom,
% 16.19/9.04 ! [G: int > complex,A2: set_int] :
% 16.19/9.04 ( ( ( groups7440179247065528705omplex @ G @ A2 )
% 16.19/9.04 != one_one_complex )
% 16.19/9.04 => ~ ! [A3: int] :
% 16.19/9.04 ( ( member_int @ A3 @ A2 )
% 16.19/9.04 => ( ( G @ A3 )
% 16.19/9.04 = one_one_complex ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8621_prod_Onot__neutral__contains__not__neutral,axiom,
% 16.19/9.04 ! [G: complex > complex,A2: set_complex] :
% 16.19/9.04 ( ( ( groups3708469109370488835omplex @ G @ A2 )
% 16.19/9.04 != one_one_complex )
% 16.19/9.04 => ~ ! [A3: complex] :
% 16.19/9.04 ( ( member_complex @ A3 @ A2 )
% 16.19/9.04 => ( ( G @ A3 )
% 16.19/9.04 = one_one_complex ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8622_prod_Onot__neutral__contains__not__neutral,axiom,
% 16.19/9.04 ! [G: nat > real,A2: set_nat] :
% 16.19/9.04 ( ( ( groups129246275422532515t_real @ G @ A2 )
% 16.19/9.04 != one_one_real )
% 16.19/9.04 => ~ ! [A3: nat] :
% 16.19/9.04 ( ( member_nat @ A3 @ A2 )
% 16.19/9.04 => ( ( G @ A3 )
% 16.19/9.04 = one_one_real ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8623_prod_Onot__neutral__contains__not__neutral,axiom,
% 16.19/9.04 ! [G: real > real,A2: set_real] :
% 16.19/9.04 ( ( ( groups1681761925125756287l_real @ G @ A2 )
% 16.19/9.04 != one_one_real )
% 16.19/9.04 => ~ ! [A3: real] :
% 16.19/9.04 ( ( member_real @ A3 @ A2 )
% 16.19/9.04 => ( ( G @ A3 )
% 16.19/9.04 = one_one_real ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8624_prod_Onot__neutral__contains__not__neutral,axiom,
% 16.19/9.04 ! [G: int > real,A2: set_int] :
% 16.19/9.04 ( ( ( groups2316167850115554303t_real @ G @ A2 )
% 16.19/9.04 != one_one_real )
% 16.19/9.04 => ~ ! [A3: int] :
% 16.19/9.04 ( ( member_int @ A3 @ A2 )
% 16.19/9.04 => ( ( G @ A3 )
% 16.19/9.04 = one_one_real ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8625_prod_Onot__neutral__contains__not__neutral,axiom,
% 16.19/9.04 ! [G: complex > real,A2: set_complex] :
% 16.19/9.04 ( ( ( groups766887009212190081x_real @ G @ A2 )
% 16.19/9.04 != one_one_real )
% 16.19/9.04 => ~ ! [A3: complex] :
% 16.19/9.04 ( ( member_complex @ A3 @ A2 )
% 16.19/9.04 => ( ( G @ A3 )
% 16.19/9.04 = one_one_real ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8626_prod_Onot__neutral__contains__not__neutral,axiom,
% 16.19/9.04 ! [G: nat > rat,A2: set_nat] :
% 16.19/9.04 ( ( ( groups73079841787564623at_rat @ G @ A2 )
% 16.19/9.04 != one_one_rat )
% 16.19/9.04 => ~ ! [A3: nat] :
% 16.19/9.04 ( ( member_nat @ A3 @ A2 )
% 16.19/9.04 => ( ( G @ A3 )
% 16.19/9.04 = one_one_rat ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8627_prod_Onot__neutral__contains__not__neutral,axiom,
% 16.19/9.04 ! [G: real > rat,A2: set_real] :
% 16.19/9.04 ( ( ( groups4061424788464935467al_rat @ G @ A2 )
% 16.19/9.04 != one_one_rat )
% 16.19/9.04 => ~ ! [A3: real] :
% 16.19/9.04 ( ( member_real @ A3 @ A2 )
% 16.19/9.04 => ( ( G @ A3 )
% 16.19/9.04 = one_one_rat ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8628_prod_Oneutral,axiom,
% 16.19/9.04 ! [A2: set_nat,G: nat > nat] :
% 16.19/9.04 ( ! [X4: nat] :
% 16.19/9.04 ( ( member_nat @ X4 @ A2 )
% 16.19/9.04 => ( ( G @ X4 )
% 16.19/9.04 = one_one_nat ) )
% 16.19/9.04 => ( ( groups708209901874060359at_nat @ G @ A2 )
% 16.19/9.04 = one_one_nat ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8629_prod_Oneutral,axiom,
% 16.19/9.04 ! [A2: set_nat,G: nat > int] :
% 16.19/9.04 ( ! [X4: nat] :
% 16.19/9.04 ( ( member_nat @ X4 @ A2 )
% 16.19/9.04 => ( ( G @ X4 )
% 16.19/9.04 = one_one_int ) )
% 16.19/9.04 => ( ( groups705719431365010083at_int @ G @ A2 )
% 16.19/9.04 = one_one_int ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8630_prod_Oneutral,axiom,
% 16.19/9.04 ! [A2: set_int,G: int > int] :
% 16.19/9.04 ( ! [X4: int] :
% 16.19/9.04 ( ( member_int @ X4 @ A2 )
% 16.19/9.04 => ( ( G @ X4 )
% 16.19/9.04 = one_one_int ) )
% 16.19/9.04 => ( ( groups1705073143266064639nt_int @ G @ A2 )
% 16.19/9.04 = one_one_int ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8631_prod_Odistrib,axiom,
% 16.19/9.04 ! [G: nat > nat,H2: nat > nat,A2: set_nat] :
% 16.19/9.04 ( ( groups708209901874060359at_nat
% 16.19/9.04 @ ^ [X2: nat] : ( times_times_nat @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 16.19/9.04 @ A2 )
% 16.19/9.04 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ A2 ) @ ( groups708209901874060359at_nat @ H2 @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8632_prod_Odistrib,axiom,
% 16.19/9.04 ! [G: nat > int,H2: nat > int,A2: set_nat] :
% 16.19/9.04 ( ( groups705719431365010083at_int
% 16.19/9.04 @ ^ [X2: nat] : ( times_times_int @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 16.19/9.04 @ A2 )
% 16.19/9.04 = ( times_times_int @ ( groups705719431365010083at_int @ G @ A2 ) @ ( groups705719431365010083at_int @ H2 @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8633_prod_Odistrib,axiom,
% 16.19/9.04 ! [G: int > int,H2: int > int,A2: set_int] :
% 16.19/9.04 ( ( groups1705073143266064639nt_int
% 16.19/9.04 @ ^ [X2: int] : ( times_times_int @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 16.19/9.04 @ A2 )
% 16.19/9.04 = ( times_times_int @ ( groups1705073143266064639nt_int @ G @ A2 ) @ ( groups1705073143266064639nt_int @ H2 @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8634_prod__power__distrib,axiom,
% 16.19/9.04 ! [F: nat > nat,A2: set_nat,N2: nat] :
% 16.19/9.04 ( ( power_power_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ N2 )
% 16.19/9.04 = ( groups708209901874060359at_nat
% 16.19/9.04 @ ^ [X2: nat] : ( power_power_nat @ ( F @ X2 ) @ N2 )
% 16.19/9.04 @ A2 ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8635_prod__power__distrib,axiom,
% 16.19/9.04 ! [F: nat > int,A2: set_nat,N2: nat] :
% 16.19/9.04 ( ( power_power_int @ ( groups705719431365010083at_int @ F @ A2 ) @ N2 )
% 16.19/9.04 = ( groups705719431365010083at_int
% 16.19/9.04 @ ^ [X2: nat] : ( power_power_int @ ( F @ X2 ) @ N2 )
% 16.19/9.04 @ A2 ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8636_prod__power__distrib,axiom,
% 16.19/9.04 ! [F: int > int,A2: set_int,N2: nat] :
% 16.19/9.04 ( ( power_power_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ N2 )
% 16.19/9.04 = ( groups1705073143266064639nt_int
% 16.19/9.04 @ ^ [X2: int] : ( power_power_int @ ( F @ X2 ) @ N2 )
% 16.19/9.04 @ A2 ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8637_prod_Oswap__restrict,axiom,
% 16.19/9.04 ! [A2: set_real,B3: set_nat,G: real > nat > nat,R2: real > nat > $o] :
% 16.19/9.04 ( ( finite_finite_real @ A2 )
% 16.19/9.04 => ( ( finite_finite_nat @ B3 )
% 16.19/9.04 => ( ( groups4696554848551431203al_nat
% 16.19/9.04 @ ^ [X2: real] :
% 16.19/9.04 ( groups708209901874060359at_nat @ ( G @ X2 )
% 16.19/9.04 @ ( collect_nat
% 16.19/9.04 @ ^ [Y2: nat] :
% 16.19/9.04 ( ( member_nat @ Y2 @ B3 )
% 16.19/9.04 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/9.04 @ A2 )
% 16.19/9.04 = ( groups708209901874060359at_nat
% 16.19/9.04 @ ^ [Y2: nat] :
% 16.19/9.04 ( groups4696554848551431203al_nat
% 16.19/9.04 @ ^ [X2: real] : ( G @ X2 @ Y2 )
% 16.19/9.04 @ ( collect_real
% 16.19/9.04 @ ^ [X2: real] :
% 16.19/9.04 ( ( member_real @ X2 @ A2 )
% 16.19/9.04 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/9.04 @ B3 ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8638_prod_Oswap__restrict,axiom,
% 16.19/9.04 ! [A2: set_int,B3: set_nat,G: int > nat > nat,R2: int > nat > $o] :
% 16.19/9.04 ( ( finite_finite_int @ A2 )
% 16.19/9.04 => ( ( finite_finite_nat @ B3 )
% 16.19/9.04 => ( ( groups1707563613775114915nt_nat
% 16.19/9.04 @ ^ [X2: int] :
% 16.19/9.04 ( groups708209901874060359at_nat @ ( G @ X2 )
% 16.19/9.04 @ ( collect_nat
% 16.19/9.04 @ ^ [Y2: nat] :
% 16.19/9.04 ( ( member_nat @ Y2 @ B3 )
% 16.19/9.04 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/9.04 @ A2 )
% 16.19/9.04 = ( groups708209901874060359at_nat
% 16.19/9.04 @ ^ [Y2: nat] :
% 16.19/9.04 ( groups1707563613775114915nt_nat
% 16.19/9.04 @ ^ [X2: int] : ( G @ X2 @ Y2 )
% 16.19/9.04 @ ( collect_int
% 16.19/9.04 @ ^ [X2: int] :
% 16.19/9.04 ( ( member_int @ X2 @ A2 )
% 16.19/9.04 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/9.04 @ B3 ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8639_prod_Oswap__restrict,axiom,
% 16.19/9.04 ! [A2: set_complex,B3: set_nat,G: complex > nat > nat,R2: complex > nat > $o] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/9.04 => ( ( finite_finite_nat @ B3 )
% 16.19/9.04 => ( ( groups861055069439313189ex_nat
% 16.19/9.04 @ ^ [X2: complex] :
% 16.19/9.04 ( groups708209901874060359at_nat @ ( G @ X2 )
% 16.19/9.04 @ ( collect_nat
% 16.19/9.04 @ ^ [Y2: nat] :
% 16.19/9.04 ( ( member_nat @ Y2 @ B3 )
% 16.19/9.04 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/9.04 @ A2 )
% 16.19/9.04 = ( groups708209901874060359at_nat
% 16.19/9.04 @ ^ [Y2: nat] :
% 16.19/9.04 ( groups861055069439313189ex_nat
% 16.19/9.04 @ ^ [X2: complex] : ( G @ X2 @ Y2 )
% 16.19/9.04 @ ( collect_complex
% 16.19/9.04 @ ^ [X2: complex] :
% 16.19/9.04 ( ( member_complex @ X2 @ A2 )
% 16.19/9.04 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/9.04 @ B3 ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8640_prod_Oswap__restrict,axiom,
% 16.19/9.04 ! [A2: set_real,B3: set_nat,G: real > nat > int,R2: real > nat > $o] :
% 16.19/9.04 ( ( finite_finite_real @ A2 )
% 16.19/9.04 => ( ( finite_finite_nat @ B3 )
% 16.19/9.04 => ( ( groups4694064378042380927al_int
% 16.19/9.04 @ ^ [X2: real] :
% 16.19/9.04 ( groups705719431365010083at_int @ ( G @ X2 )
% 16.19/9.04 @ ( collect_nat
% 16.19/9.04 @ ^ [Y2: nat] :
% 16.19/9.04 ( ( member_nat @ Y2 @ B3 )
% 16.19/9.04 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/9.04 @ A2 )
% 16.19/9.04 = ( groups705719431365010083at_int
% 16.19/9.04 @ ^ [Y2: nat] :
% 16.19/9.04 ( groups4694064378042380927al_int
% 16.19/9.04 @ ^ [X2: real] : ( G @ X2 @ Y2 )
% 16.19/9.04 @ ( collect_real
% 16.19/9.04 @ ^ [X2: real] :
% 16.19/9.04 ( ( member_real @ X2 @ A2 )
% 16.19/9.04 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/9.04 @ B3 ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8641_prod_Oswap__restrict,axiom,
% 16.19/9.04 ! [A2: set_complex,B3: set_nat,G: complex > nat > int,R2: complex > nat > $o] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/9.04 => ( ( finite_finite_nat @ B3 )
% 16.19/9.04 => ( ( groups858564598930262913ex_int
% 16.19/9.04 @ ^ [X2: complex] :
% 16.19/9.04 ( groups705719431365010083at_int @ ( G @ X2 )
% 16.19/9.04 @ ( collect_nat
% 16.19/9.04 @ ^ [Y2: nat] :
% 16.19/9.04 ( ( member_nat @ Y2 @ B3 )
% 16.19/9.04 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/9.04 @ A2 )
% 16.19/9.04 = ( groups705719431365010083at_int
% 16.19/9.04 @ ^ [Y2: nat] :
% 16.19/9.04 ( groups858564598930262913ex_int
% 16.19/9.04 @ ^ [X2: complex] : ( G @ X2 @ Y2 )
% 16.19/9.04 @ ( collect_complex
% 16.19/9.04 @ ^ [X2: complex] :
% 16.19/9.04 ( ( member_complex @ X2 @ A2 )
% 16.19/9.04 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/9.04 @ B3 ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8642_prod_Oswap__restrict,axiom,
% 16.19/9.04 ! [A2: set_real,B3: set_int,G: real > int > int,R2: real > int > $o] :
% 16.19/9.04 ( ( finite_finite_real @ A2 )
% 16.19/9.04 => ( ( finite_finite_int @ B3 )
% 16.19/9.04 => ( ( groups4694064378042380927al_int
% 16.19/9.04 @ ^ [X2: real] :
% 16.19/9.04 ( groups1705073143266064639nt_int @ ( G @ X2 )
% 16.19/9.04 @ ( collect_int
% 16.19/9.04 @ ^ [Y2: int] :
% 16.19/9.04 ( ( member_int @ Y2 @ B3 )
% 16.19/9.04 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/9.04 @ A2 )
% 16.19/9.04 = ( groups1705073143266064639nt_int
% 16.19/9.04 @ ^ [Y2: int] :
% 16.19/9.04 ( groups4694064378042380927al_int
% 16.19/9.04 @ ^ [X2: real] : ( G @ X2 @ Y2 )
% 16.19/9.04 @ ( collect_real
% 16.19/9.04 @ ^ [X2: real] :
% 16.19/9.04 ( ( member_real @ X2 @ A2 )
% 16.19/9.04 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/9.04 @ B3 ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8643_prod_Oswap__restrict,axiom,
% 16.19/9.04 ! [A2: set_complex,B3: set_int,G: complex > int > int,R2: complex > int > $o] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/9.04 => ( ( finite_finite_int @ B3 )
% 16.19/9.04 => ( ( groups858564598930262913ex_int
% 16.19/9.04 @ ^ [X2: complex] :
% 16.19/9.04 ( groups1705073143266064639nt_int @ ( G @ X2 )
% 16.19/9.04 @ ( collect_int
% 16.19/9.04 @ ^ [Y2: int] :
% 16.19/9.04 ( ( member_int @ Y2 @ B3 )
% 16.19/9.04 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/9.04 @ A2 )
% 16.19/9.04 = ( groups1705073143266064639nt_int
% 16.19/9.04 @ ^ [Y2: int] :
% 16.19/9.04 ( groups858564598930262913ex_int
% 16.19/9.04 @ ^ [X2: complex] : ( G @ X2 @ Y2 )
% 16.19/9.04 @ ( collect_complex
% 16.19/9.04 @ ^ [X2: complex] :
% 16.19/9.04 ( ( member_complex @ X2 @ A2 )
% 16.19/9.04 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/9.04 @ B3 ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8644_prod_Oswap__restrict,axiom,
% 16.19/9.04 ! [A2: set_nat,B3: set_real,G: nat > real > nat,R2: nat > real > $o] :
% 16.19/9.04 ( ( finite_finite_nat @ A2 )
% 16.19/9.04 => ( ( finite_finite_real @ B3 )
% 16.19/9.04 => ( ( groups708209901874060359at_nat
% 16.19/9.04 @ ^ [X2: nat] :
% 16.19/9.04 ( groups4696554848551431203al_nat @ ( G @ X2 )
% 16.19/9.04 @ ( collect_real
% 16.19/9.04 @ ^ [Y2: real] :
% 16.19/9.04 ( ( member_real @ Y2 @ B3 )
% 16.19/9.04 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/9.04 @ A2 )
% 16.19/9.04 = ( groups4696554848551431203al_nat
% 16.19/9.04 @ ^ [Y2: real] :
% 16.19/9.04 ( groups708209901874060359at_nat
% 16.19/9.04 @ ^ [X2: nat] : ( G @ X2 @ Y2 )
% 16.19/9.04 @ ( collect_nat
% 16.19/9.04 @ ^ [X2: nat] :
% 16.19/9.04 ( ( member_nat @ X2 @ A2 )
% 16.19/9.04 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/9.04 @ B3 ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8645_prod_Oswap__restrict,axiom,
% 16.19/9.04 ! [A2: set_nat,B3: set_int,G: nat > int > nat,R2: nat > int > $o] :
% 16.19/9.04 ( ( finite_finite_nat @ A2 )
% 16.19/9.04 => ( ( finite_finite_int @ B3 )
% 16.19/9.04 => ( ( groups708209901874060359at_nat
% 16.19/9.04 @ ^ [X2: nat] :
% 16.19/9.04 ( groups1707563613775114915nt_nat @ ( G @ X2 )
% 16.19/9.04 @ ( collect_int
% 16.19/9.04 @ ^ [Y2: int] :
% 16.19/9.04 ( ( member_int @ Y2 @ B3 )
% 16.19/9.04 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/9.04 @ A2 )
% 16.19/9.04 = ( groups1707563613775114915nt_nat
% 16.19/9.04 @ ^ [Y2: int] :
% 16.19/9.04 ( groups708209901874060359at_nat
% 16.19/9.04 @ ^ [X2: nat] : ( G @ X2 @ Y2 )
% 16.19/9.04 @ ( collect_nat
% 16.19/9.04 @ ^ [X2: nat] :
% 16.19/9.04 ( ( member_nat @ X2 @ A2 )
% 16.19/9.04 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/9.04 @ B3 ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8646_prod_Oswap__restrict,axiom,
% 16.19/9.04 ! [A2: set_nat,B3: set_complex,G: nat > complex > nat,R2: nat > complex > $o] :
% 16.19/9.04 ( ( finite_finite_nat @ A2 )
% 16.19/9.04 => ( ( finite3207457112153483333omplex @ B3 )
% 16.19/9.04 => ( ( groups708209901874060359at_nat
% 16.19/9.04 @ ^ [X2: nat] :
% 16.19/9.04 ( groups861055069439313189ex_nat @ ( G @ X2 )
% 16.19/9.04 @ ( collect_complex
% 16.19/9.04 @ ^ [Y2: complex] :
% 16.19/9.04 ( ( member_complex @ Y2 @ B3 )
% 16.19/9.04 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/9.04 @ A2 )
% 16.19/9.04 = ( groups861055069439313189ex_nat
% 16.19/9.04 @ ^ [Y2: complex] :
% 16.19/9.04 ( groups708209901874060359at_nat
% 16.19/9.04 @ ^ [X2: nat] : ( G @ X2 @ Y2 )
% 16.19/9.04 @ ( collect_nat
% 16.19/9.04 @ ^ [X2: nat] :
% 16.19/9.04 ( ( member_nat @ X2 @ A2 )
% 16.19/9.04 & ( R2 @ X2 @ Y2 ) ) ) )
% 16.19/9.04 @ B3 ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8647_mod__prod__eq,axiom,
% 16.19/9.04 ! [F: nat > nat,A: nat,A2: set_nat] :
% 16.19/9.04 ( ( modulo_modulo_nat
% 16.19/9.04 @ ( groups708209901874060359at_nat
% 16.19/9.04 @ ^ [I3: nat] : ( modulo_modulo_nat @ ( F @ I3 ) @ A )
% 16.19/9.04 @ A2 )
% 16.19/9.04 @ A )
% 16.19/9.04 = ( modulo_modulo_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ A ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8648_mod__prod__eq,axiom,
% 16.19/9.04 ! [F: nat > int,A: int,A2: set_nat] :
% 16.19/9.04 ( ( modulo_modulo_int
% 16.19/9.04 @ ( groups705719431365010083at_int
% 16.19/9.04 @ ^ [I3: nat] : ( modulo_modulo_int @ ( F @ I3 ) @ A )
% 16.19/9.04 @ A2 )
% 16.19/9.04 @ A )
% 16.19/9.04 = ( modulo_modulo_int @ ( groups705719431365010083at_int @ F @ A2 ) @ A ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8649_mod__prod__eq,axiom,
% 16.19/9.04 ! [F: int > int,A: int,A2: set_int] :
% 16.19/9.04 ( ( modulo_modulo_int
% 16.19/9.04 @ ( groups1705073143266064639nt_int
% 16.19/9.04 @ ^ [I3: int] : ( modulo_modulo_int @ ( F @ I3 ) @ A )
% 16.19/9.04 @ A2 )
% 16.19/9.04 @ A )
% 16.19/9.04 = ( modulo_modulo_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ A ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8650_prod__mono,axiom,
% 16.19/9.04 ! [A2: set_nat,F: nat > real,G: nat > real] :
% 16.19/9.04 ( ! [I4: nat] :
% 16.19/9.04 ( ( member_nat @ I4 @ A2 )
% 16.19/9.04 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
% 16.19/9.04 & ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 16.19/9.04 => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8651_prod__mono,axiom,
% 16.19/9.04 ! [A2: set_real,F: real > real,G: real > real] :
% 16.19/9.04 ( ! [I4: real] :
% 16.19/9.04 ( ( member_real @ I4 @ A2 )
% 16.19/9.04 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
% 16.19/9.04 & ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 16.19/9.04 => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8652_prod__mono,axiom,
% 16.19/9.04 ! [A2: set_int,F: int > real,G: int > real] :
% 16.19/9.04 ( ! [I4: int] :
% 16.19/9.04 ( ( member_int @ I4 @ A2 )
% 16.19/9.04 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
% 16.19/9.04 & ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 16.19/9.04 => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8653_prod__mono,axiom,
% 16.19/9.04 ! [A2: set_complex,F: complex > real,G: complex > real] :
% 16.19/9.04 ( ! [I4: complex] :
% 16.19/9.04 ( ( member_complex @ I4 @ A2 )
% 16.19/9.04 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
% 16.19/9.04 & ( ord_less_eq_real @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 16.19/9.04 => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A2 ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8654_prod__mono,axiom,
% 16.19/9.04 ! [A2: set_nat,F: nat > rat,G: nat > rat] :
% 16.19/9.04 ( ! [I4: nat] :
% 16.19/9.04 ( ( member_nat @ I4 @ A2 )
% 16.19/9.04 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) )
% 16.19/9.04 & ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 16.19/9.04 => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8655_prod__mono,axiom,
% 16.19/9.04 ! [A2: set_real,F: real > rat,G: real > rat] :
% 16.19/9.04 ( ! [I4: real] :
% 16.19/9.04 ( ( member_real @ I4 @ A2 )
% 16.19/9.04 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) )
% 16.19/9.04 & ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 16.19/9.04 => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8656_prod__mono,axiom,
% 16.19/9.04 ! [A2: set_int,F: int > rat,G: int > rat] :
% 16.19/9.04 ( ! [I4: int] :
% 16.19/9.04 ( ( member_int @ I4 @ A2 )
% 16.19/9.04 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) )
% 16.19/9.04 & ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 16.19/9.04 => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ ( groups1072433553688619179nt_rat @ G @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8657_prod__mono,axiom,
% 16.19/9.04 ! [A2: set_complex,F: complex > rat,G: complex > rat] :
% 16.19/9.04 ( ! [I4: complex] :
% 16.19/9.04 ( ( member_complex @ I4 @ A2 )
% 16.19/9.04 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I4 ) )
% 16.19/9.04 & ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 16.19/9.04 => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ ( groups225925009352817453ex_rat @ G @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8658_prod__mono,axiom,
% 16.19/9.04 ! [A2: set_real,F: real > nat,G: real > nat] :
% 16.19/9.04 ( ! [I4: real] :
% 16.19/9.04 ( ( member_real @ I4 @ A2 )
% 16.19/9.04 => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) )
% 16.19/9.04 & ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 16.19/9.04 => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8659_prod__mono,axiom,
% 16.19/9.04 ! [A2: set_int,F: int > nat,G: int > nat] :
% 16.19/9.04 ( ! [I4: int] :
% 16.19/9.04 ( ( member_int @ I4 @ A2 )
% 16.19/9.04 => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) )
% 16.19/9.04 & ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 16.19/9.04 => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8660_prod__nonneg,axiom,
% 16.19/9.04 ! [A2: set_nat,F: nat > nat] :
% 16.19/9.04 ( ! [X4: nat] :
% 16.19/9.04 ( ( member_nat @ X4 @ A2 )
% 16.19/9.04 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 16.19/9.04 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8661_prod__nonneg,axiom,
% 16.19/9.04 ! [A2: set_nat,F: nat > int] :
% 16.19/9.04 ( ! [X4: nat] :
% 16.19/9.04 ( ( member_nat @ X4 @ A2 )
% 16.19/9.04 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 16.19/9.04 => ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8662_prod__nonneg,axiom,
% 16.19/9.04 ! [A2: set_int,F: int > int] :
% 16.19/9.04 ( ! [X4: int] :
% 16.19/9.04 ( ( member_int @ X4 @ A2 )
% 16.19/9.04 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 16.19/9.04 => ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8663_prod__pos,axiom,
% 16.19/9.04 ! [A2: set_nat,F: nat > nat] :
% 16.19/9.04 ( ! [X4: nat] :
% 16.19/9.04 ( ( member_nat @ X4 @ A2 )
% 16.19/9.04 => ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 16.19/9.04 => ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8664_prod__pos,axiom,
% 16.19/9.04 ! [A2: set_nat,F: nat > int] :
% 16.19/9.04 ( ! [X4: nat] :
% 16.19/9.04 ( ( member_nat @ X4 @ A2 )
% 16.19/9.04 => ( ord_less_int @ zero_zero_int @ ( F @ X4 ) ) )
% 16.19/9.04 => ( ord_less_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8665_prod__pos,axiom,
% 16.19/9.04 ! [A2: set_int,F: int > int] :
% 16.19/9.04 ( ! [X4: int] :
% 16.19/9.04 ( ( member_int @ X4 @ A2 )
% 16.19/9.04 => ( ord_less_int @ zero_zero_int @ ( F @ X4 ) ) )
% 16.19/9.04 => ( ord_less_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8666_prod__ge__1,axiom,
% 16.19/9.04 ! [A2: set_nat,F: nat > real] :
% 16.19/9.04 ( ! [X4: nat] :
% 16.19/9.04 ( ( member_nat @ X4 @ A2 )
% 16.19/9.04 => ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
% 16.19/9.04 => ( ord_less_eq_real @ one_one_real @ ( groups129246275422532515t_real @ F @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8667_prod__ge__1,axiom,
% 16.19/9.04 ! [A2: set_real,F: real > real] :
% 16.19/9.04 ( ! [X4: real] :
% 16.19/9.04 ( ( member_real @ X4 @ A2 )
% 16.19/9.04 => ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
% 16.19/9.04 => ( ord_less_eq_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8668_prod__ge__1,axiom,
% 16.19/9.04 ! [A2: set_int,F: int > real] :
% 16.19/9.04 ( ! [X4: int] :
% 16.19/9.04 ( ( member_int @ X4 @ A2 )
% 16.19/9.04 => ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
% 16.19/9.04 => ( ord_less_eq_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8669_prod__ge__1,axiom,
% 16.19/9.04 ! [A2: set_complex,F: complex > real] :
% 16.19/9.04 ( ! [X4: complex] :
% 16.19/9.04 ( ( member_complex @ X4 @ A2 )
% 16.19/9.04 => ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
% 16.19/9.04 => ( ord_less_eq_real @ one_one_real @ ( groups766887009212190081x_real @ F @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8670_prod__ge__1,axiom,
% 16.19/9.04 ! [A2: set_nat,F: nat > rat] :
% 16.19/9.04 ( ! [X4: nat] :
% 16.19/9.04 ( ( member_nat @ X4 @ A2 )
% 16.19/9.04 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
% 16.19/9.04 => ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8671_prod__ge__1,axiom,
% 16.19/9.04 ! [A2: set_real,F: real > rat] :
% 16.19/9.04 ( ! [X4: real] :
% 16.19/9.04 ( ( member_real @ X4 @ A2 )
% 16.19/9.04 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
% 16.19/9.04 => ( ord_less_eq_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8672_prod__ge__1,axiom,
% 16.19/9.04 ! [A2: set_int,F: int > rat] :
% 16.19/9.04 ( ! [X4: int] :
% 16.19/9.04 ( ( member_int @ X4 @ A2 )
% 16.19/9.04 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
% 16.19/9.04 => ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8673_prod__ge__1,axiom,
% 16.19/9.04 ! [A2: set_complex,F: complex > rat] :
% 16.19/9.04 ( ! [X4: complex] :
% 16.19/9.04 ( ( member_complex @ X4 @ A2 )
% 16.19/9.04 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X4 ) ) )
% 16.19/9.04 => ( ord_less_eq_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8674_prod__ge__1,axiom,
% 16.19/9.04 ! [A2: set_real,F: real > nat] :
% 16.19/9.04 ( ! [X4: real] :
% 16.19/9.04 ( ( member_real @ X4 @ A2 )
% 16.19/9.04 => ( ord_less_eq_nat @ one_one_nat @ ( F @ X4 ) ) )
% 16.19/9.04 => ( ord_less_eq_nat @ one_one_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8675_prod__ge__1,axiom,
% 16.19/9.04 ! [A2: set_int,F: int > nat] :
% 16.19/9.04 ( ! [X4: int] :
% 16.19/9.04 ( ( member_int @ X4 @ A2 )
% 16.19/9.04 => ( ord_less_eq_nat @ one_one_nat @ ( F @ X4 ) ) )
% 16.19/9.04 => ( ord_less_eq_nat @ one_one_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8676_prod__atLeastAtMost__code,axiom,
% 16.19/9.04 ! [F: nat > complex,A: nat,B: nat] :
% 16.19/9.04 ( ( groups6464643781859351333omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 16.19/9.04 = ( set_fo1517530859248394432omplex
% 16.19/9.04 @ ^ [A5: nat] : ( times_times_complex @ ( F @ A5 ) )
% 16.19/9.04 @ A
% 16.19/9.04 @ B
% 16.19/9.04 @ one_one_complex ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8677_prod__atLeastAtMost__code,axiom,
% 16.19/9.04 ! [F: nat > real,A: nat,B: nat] :
% 16.19/9.04 ( ( groups129246275422532515t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 16.19/9.04 = ( set_fo3111899725591712190t_real
% 16.19/9.04 @ ^ [A5: nat] : ( times_times_real @ ( F @ A5 ) )
% 16.19/9.04 @ A
% 16.19/9.04 @ B
% 16.19/9.04 @ one_one_real ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8678_prod__atLeastAtMost__code,axiom,
% 16.19/9.04 ! [F: nat > rat,A: nat,B: nat] :
% 16.19/9.04 ( ( groups73079841787564623at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 16.19/9.04 = ( set_fo1949268297981939178at_rat
% 16.19/9.04 @ ^ [A5: nat] : ( times_times_rat @ ( F @ A5 ) )
% 16.19/9.04 @ A
% 16.19/9.04 @ B
% 16.19/9.04 @ one_one_rat ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8679_prod__atLeastAtMost__code,axiom,
% 16.19/9.04 ! [F: nat > nat,A: nat,B: nat] :
% 16.19/9.04 ( ( groups708209901874060359at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 16.19/9.04 = ( set_fo2584398358068434914at_nat
% 16.19/9.04 @ ^ [A5: nat] : ( times_times_nat @ ( F @ A5 ) )
% 16.19/9.04 @ A
% 16.19/9.04 @ B
% 16.19/9.04 @ one_one_nat ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8680_prod__atLeastAtMost__code,axiom,
% 16.19/9.04 ! [F: nat > int,A: nat,B: nat] :
% 16.19/9.04 ( ( groups705719431365010083at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 16.19/9.04 = ( set_fo2581907887559384638at_int
% 16.19/9.04 @ ^ [A5: nat] : ( times_times_int @ ( F @ A5 ) )
% 16.19/9.04 @ A
% 16.19/9.04 @ B
% 16.19/9.04 @ one_one_int ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8681_prod_Ointer__filter,axiom,
% 16.19/9.04 ! [A2: set_real,G: real > complex,P: real > $o] :
% 16.19/9.04 ( ( finite_finite_real @ A2 )
% 16.19/9.04 => ( ( groups713298508707869441omplex @ G
% 16.19/9.04 @ ( collect_real
% 16.19/9.04 @ ^ [X2: real] :
% 16.19/9.04 ( ( member_real @ X2 @ A2 )
% 16.19/9.04 & ( P @ X2 ) ) ) )
% 16.19/9.04 = ( groups713298508707869441omplex
% 16.19/9.04 @ ^ [X2: real] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_complex )
% 16.19/9.04 @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8682_prod_Ointer__filter,axiom,
% 16.19/9.04 ! [A2: set_nat,G: nat > complex,P: nat > $o] :
% 16.19/9.04 ( ( finite_finite_nat @ A2 )
% 16.19/9.04 => ( ( groups6464643781859351333omplex @ G
% 16.19/9.04 @ ( collect_nat
% 16.19/9.04 @ ^ [X2: nat] :
% 16.19/9.04 ( ( member_nat @ X2 @ A2 )
% 16.19/9.04 & ( P @ X2 ) ) ) )
% 16.19/9.04 = ( groups6464643781859351333omplex
% 16.19/9.04 @ ^ [X2: nat] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_complex )
% 16.19/9.04 @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8683_prod_Ointer__filter,axiom,
% 16.19/9.04 ! [A2: set_int,G: int > complex,P: int > $o] :
% 16.19/9.04 ( ( finite_finite_int @ A2 )
% 16.19/9.04 => ( ( groups7440179247065528705omplex @ G
% 16.19/9.04 @ ( collect_int
% 16.19/9.04 @ ^ [X2: int] :
% 16.19/9.04 ( ( member_int @ X2 @ A2 )
% 16.19/9.04 & ( P @ X2 ) ) ) )
% 16.19/9.04 = ( groups7440179247065528705omplex
% 16.19/9.04 @ ^ [X2: int] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_complex )
% 16.19/9.04 @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8684_prod_Ointer__filter,axiom,
% 16.19/9.04 ! [A2: set_complex,G: complex > complex,P: complex > $o] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/9.04 => ( ( groups3708469109370488835omplex @ G
% 16.19/9.04 @ ( collect_complex
% 16.19/9.04 @ ^ [X2: complex] :
% 16.19/9.04 ( ( member_complex @ X2 @ A2 )
% 16.19/9.04 & ( P @ X2 ) ) ) )
% 16.19/9.04 = ( groups3708469109370488835omplex
% 16.19/9.04 @ ^ [X2: complex] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_complex )
% 16.19/9.04 @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8685_prod_Ointer__filter,axiom,
% 16.19/9.04 ! [A2: set_real,G: real > real,P: real > $o] :
% 16.19/9.04 ( ( finite_finite_real @ A2 )
% 16.19/9.04 => ( ( groups1681761925125756287l_real @ G
% 16.19/9.04 @ ( collect_real
% 16.19/9.04 @ ^ [X2: real] :
% 16.19/9.04 ( ( member_real @ X2 @ A2 )
% 16.19/9.04 & ( P @ X2 ) ) ) )
% 16.19/9.04 = ( groups1681761925125756287l_real
% 16.19/9.04 @ ^ [X2: real] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_real )
% 16.19/9.04 @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8686_prod_Ointer__filter,axiom,
% 16.19/9.04 ! [A2: set_nat,G: nat > real,P: nat > $o] :
% 16.19/9.04 ( ( finite_finite_nat @ A2 )
% 16.19/9.04 => ( ( groups129246275422532515t_real @ G
% 16.19/9.04 @ ( collect_nat
% 16.19/9.04 @ ^ [X2: nat] :
% 16.19/9.04 ( ( member_nat @ X2 @ A2 )
% 16.19/9.04 & ( P @ X2 ) ) ) )
% 16.19/9.04 = ( groups129246275422532515t_real
% 16.19/9.04 @ ^ [X2: nat] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_real )
% 16.19/9.04 @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8687_prod_Ointer__filter,axiom,
% 16.19/9.04 ! [A2: set_int,G: int > real,P: int > $o] :
% 16.19/9.04 ( ( finite_finite_int @ A2 )
% 16.19/9.04 => ( ( groups2316167850115554303t_real @ G
% 16.19/9.04 @ ( collect_int
% 16.19/9.04 @ ^ [X2: int] :
% 16.19/9.04 ( ( member_int @ X2 @ A2 )
% 16.19/9.04 & ( P @ X2 ) ) ) )
% 16.19/9.04 = ( groups2316167850115554303t_real
% 16.19/9.04 @ ^ [X2: int] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_real )
% 16.19/9.04 @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8688_prod_Ointer__filter,axiom,
% 16.19/9.04 ! [A2: set_complex,G: complex > real,P: complex > $o] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/9.04 => ( ( groups766887009212190081x_real @ G
% 16.19/9.04 @ ( collect_complex
% 16.19/9.04 @ ^ [X2: complex] :
% 16.19/9.04 ( ( member_complex @ X2 @ A2 )
% 16.19/9.04 & ( P @ X2 ) ) ) )
% 16.19/9.04 = ( groups766887009212190081x_real
% 16.19/9.04 @ ^ [X2: complex] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_real )
% 16.19/9.04 @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8689_prod_Ointer__filter,axiom,
% 16.19/9.04 ! [A2: set_real,G: real > rat,P: real > $o] :
% 16.19/9.04 ( ( finite_finite_real @ A2 )
% 16.19/9.04 => ( ( groups4061424788464935467al_rat @ G
% 16.19/9.04 @ ( collect_real
% 16.19/9.04 @ ^ [X2: real] :
% 16.19/9.04 ( ( member_real @ X2 @ A2 )
% 16.19/9.04 & ( P @ X2 ) ) ) )
% 16.19/9.04 = ( groups4061424788464935467al_rat
% 16.19/9.04 @ ^ [X2: real] : ( if_rat @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_rat )
% 16.19/9.04 @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8690_prod_Ointer__filter,axiom,
% 16.19/9.04 ! [A2: set_nat,G: nat > rat,P: nat > $o] :
% 16.19/9.04 ( ( finite_finite_nat @ A2 )
% 16.19/9.04 => ( ( groups73079841787564623at_rat @ G
% 16.19/9.04 @ ( collect_nat
% 16.19/9.04 @ ^ [X2: nat] :
% 16.19/9.04 ( ( member_nat @ X2 @ A2 )
% 16.19/9.04 & ( P @ X2 ) ) ) )
% 16.19/9.04 = ( groups73079841787564623at_rat
% 16.19/9.04 @ ^ [X2: nat] : ( if_rat @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_rat )
% 16.19/9.04 @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8691_prod_Oshift__bounds__cl__Suc__ivl,axiom,
% 16.19/9.04 ! [G: nat > nat,M: nat,N2: nat] :
% 16.19/9.04 ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
% 16.19/9.04 = ( groups708209901874060359at_nat
% 16.19/9.04 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 16.19/9.04 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8692_prod_Oshift__bounds__cl__Suc__ivl,axiom,
% 16.19/9.04 ! [G: nat > int,M: nat,N2: nat] :
% 16.19/9.04 ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
% 16.19/9.04 = ( groups705719431365010083at_int
% 16.19/9.04 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 16.19/9.04 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8693_power__sum,axiom,
% 16.19/9.04 ! [C: real,F: nat > nat,A2: set_nat] :
% 16.19/9.04 ( ( power_power_real @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 16.19/9.04 = ( groups129246275422532515t_real
% 16.19/9.04 @ ^ [A5: nat] : ( power_power_real @ C @ ( F @ A5 ) )
% 16.19/9.04 @ A2 ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8694_power__sum,axiom,
% 16.19/9.04 ! [C: complex,F: nat > nat,A2: set_nat] :
% 16.19/9.04 ( ( power_power_complex @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 16.19/9.04 = ( groups6464643781859351333omplex
% 16.19/9.04 @ ^ [A5: nat] : ( power_power_complex @ C @ ( F @ A5 ) )
% 16.19/9.04 @ A2 ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8695_power__sum,axiom,
% 16.19/9.04 ! [C: nat,F: nat > nat,A2: set_nat] :
% 16.19/9.04 ( ( power_power_nat @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 16.19/9.04 = ( groups708209901874060359at_nat
% 16.19/9.04 @ ^ [A5: nat] : ( power_power_nat @ C @ ( F @ A5 ) )
% 16.19/9.04 @ A2 ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8696_power__sum,axiom,
% 16.19/9.04 ! [C: int,F: nat > nat,A2: set_nat] :
% 16.19/9.04 ( ( power_power_int @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 16.19/9.04 = ( groups705719431365010083at_int
% 16.19/9.04 @ ^ [A5: nat] : ( power_power_int @ C @ ( F @ A5 ) )
% 16.19/9.04 @ A2 ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8697_power__sum,axiom,
% 16.19/9.04 ! [C: int,F: int > nat,A2: set_int] :
% 16.19/9.04 ( ( power_power_int @ C @ ( groups4541462559716669496nt_nat @ F @ A2 ) )
% 16.19/9.04 = ( groups1705073143266064639nt_int
% 16.19/9.04 @ ^ [A5: int] : ( power_power_int @ C @ ( F @ A5 ) )
% 16.19/9.04 @ A2 ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8698_prod_Oshift__bounds__cl__nat__ivl,axiom,
% 16.19/9.04 ! [G: nat > nat,M: nat,K: nat,N2: nat] :
% 16.19/9.04 ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
% 16.19/9.04 = ( groups708209901874060359at_nat
% 16.19/9.04 @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
% 16.19/9.04 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8699_prod_Oshift__bounds__cl__nat__ivl,axiom,
% 16.19/9.04 ! [G: nat > int,M: nat,K: nat,N2: nat] :
% 16.19/9.04 ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
% 16.19/9.04 = ( groups705719431365010083at_int
% 16.19/9.04 @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
% 16.19/9.04 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8700_prod__le__1,axiom,
% 16.19/9.04 ! [A2: set_nat,F: nat > real] :
% 16.19/9.04 ( ! [X4: nat] :
% 16.19/9.04 ( ( member_nat @ X4 @ A2 )
% 16.19/9.04 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 16.19/9.04 & ( ord_less_eq_real @ ( F @ X4 ) @ one_one_real ) ) )
% 16.19/9.04 => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ one_one_real ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8701_prod__le__1,axiom,
% 16.19/9.04 ! [A2: set_real,F: real > real] :
% 16.19/9.04 ( ! [X4: real] :
% 16.19/9.04 ( ( member_real @ X4 @ A2 )
% 16.19/9.04 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 16.19/9.04 & ( ord_less_eq_real @ ( F @ X4 ) @ one_one_real ) ) )
% 16.19/9.04 => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ one_one_real ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8702_prod__le__1,axiom,
% 16.19/9.04 ! [A2: set_int,F: int > real] :
% 16.19/9.04 ( ! [X4: int] :
% 16.19/9.04 ( ( member_int @ X4 @ A2 )
% 16.19/9.04 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 16.19/9.04 & ( ord_less_eq_real @ ( F @ X4 ) @ one_one_real ) ) )
% 16.19/9.04 => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ one_one_real ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8703_prod__le__1,axiom,
% 16.19/9.04 ! [A2: set_complex,F: complex > real] :
% 16.19/9.04 ( ! [X4: complex] :
% 16.19/9.04 ( ( member_complex @ X4 @ A2 )
% 16.19/9.04 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 16.19/9.04 & ( ord_less_eq_real @ ( F @ X4 ) @ one_one_real ) ) )
% 16.19/9.04 => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A2 ) @ one_one_real ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8704_prod__le__1,axiom,
% 16.19/9.04 ! [A2: set_nat,F: nat > rat] :
% 16.19/9.04 ( ! [X4: nat] :
% 16.19/9.04 ( ( member_nat @ X4 @ A2 )
% 16.19/9.04 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
% 16.19/9.04 & ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
% 16.19/9.04 => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ one_one_rat ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8705_prod__le__1,axiom,
% 16.19/9.04 ! [A2: set_real,F: real > rat] :
% 16.19/9.04 ( ! [X4: real] :
% 16.19/9.04 ( ( member_real @ X4 @ A2 )
% 16.19/9.04 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
% 16.19/9.04 & ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
% 16.19/9.04 => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ one_one_rat ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8706_prod__le__1,axiom,
% 16.19/9.04 ! [A2: set_int,F: int > rat] :
% 16.19/9.04 ( ! [X4: int] :
% 16.19/9.04 ( ( member_int @ X4 @ A2 )
% 16.19/9.04 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
% 16.19/9.04 & ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
% 16.19/9.04 => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ one_one_rat ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8707_prod__le__1,axiom,
% 16.19/9.04 ! [A2: set_complex,F: complex > rat] :
% 16.19/9.04 ( ! [X4: complex] :
% 16.19/9.04 ( ( member_complex @ X4 @ A2 )
% 16.19/9.04 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) )
% 16.19/9.04 & ( ord_less_eq_rat @ ( F @ X4 ) @ one_one_rat ) ) )
% 16.19/9.04 => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ one_one_rat ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8708_prod__le__1,axiom,
% 16.19/9.04 ! [A2: set_real,F: real > nat] :
% 16.19/9.04 ( ! [X4: real] :
% 16.19/9.04 ( ( member_real @ X4 @ A2 )
% 16.19/9.04 => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) )
% 16.19/9.04 & ( ord_less_eq_nat @ ( F @ X4 ) @ one_one_nat ) ) )
% 16.19/9.04 => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ one_one_nat ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8709_prod__le__1,axiom,
% 16.19/9.04 ! [A2: set_int,F: int > nat] :
% 16.19/9.04 ( ! [X4: int] :
% 16.19/9.04 ( ( member_int @ X4 @ A2 )
% 16.19/9.04 => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) )
% 16.19/9.04 & ( ord_less_eq_nat @ ( F @ X4 ) @ one_one_nat ) ) )
% 16.19/9.04 => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ one_one_nat ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8710_prod_Orelated,axiom,
% 16.19/9.04 ! [R2: complex > complex > $o,S: set_nat,H2: nat > complex,G: nat > complex] :
% 16.19/9.04 ( ( R2 @ one_one_complex @ one_one_complex )
% 16.19/9.04 => ( ! [X16: complex,Y15: complex,X23: complex,Y23: complex] :
% 16.19/9.04 ( ( ( R2 @ X16 @ X23 )
% 16.19/9.04 & ( R2 @ Y15 @ Y23 ) )
% 16.19/9.04 => ( R2 @ ( times_times_complex @ X16 @ Y15 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
% 16.19/9.04 => ( ( finite_finite_nat @ S )
% 16.19/9.04 => ( ! [X4: nat] :
% 16.19/9.04 ( ( member_nat @ X4 @ S )
% 16.19/9.04 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 16.19/9.04 => ( R2 @ ( groups6464643781859351333omplex @ H2 @ S ) @ ( groups6464643781859351333omplex @ G @ S ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8711_prod_Orelated,axiom,
% 16.19/9.04 ! [R2: complex > complex > $o,S: set_int,H2: int > complex,G: int > complex] :
% 16.19/9.04 ( ( R2 @ one_one_complex @ one_one_complex )
% 16.19/9.04 => ( ! [X16: complex,Y15: complex,X23: complex,Y23: complex] :
% 16.19/9.04 ( ( ( R2 @ X16 @ X23 )
% 16.19/9.04 & ( R2 @ Y15 @ Y23 ) )
% 16.19/9.04 => ( R2 @ ( times_times_complex @ X16 @ Y15 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
% 16.19/9.04 => ( ( finite_finite_int @ S )
% 16.19/9.04 => ( ! [X4: int] :
% 16.19/9.04 ( ( member_int @ X4 @ S )
% 16.19/9.04 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 16.19/9.04 => ( R2 @ ( groups7440179247065528705omplex @ H2 @ S ) @ ( groups7440179247065528705omplex @ G @ S ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8712_prod_Orelated,axiom,
% 16.19/9.04 ! [R2: complex > complex > $o,S: set_complex,H2: complex > complex,G: complex > complex] :
% 16.19/9.04 ( ( R2 @ one_one_complex @ one_one_complex )
% 16.19/9.04 => ( ! [X16: complex,Y15: complex,X23: complex,Y23: complex] :
% 16.19/9.04 ( ( ( R2 @ X16 @ X23 )
% 16.19/9.04 & ( R2 @ Y15 @ Y23 ) )
% 16.19/9.04 => ( R2 @ ( times_times_complex @ X16 @ Y15 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
% 16.19/9.04 => ( ( finite3207457112153483333omplex @ S )
% 16.19/9.04 => ( ! [X4: complex] :
% 16.19/9.04 ( ( member_complex @ X4 @ S )
% 16.19/9.04 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 16.19/9.04 => ( R2 @ ( groups3708469109370488835omplex @ H2 @ S ) @ ( groups3708469109370488835omplex @ G @ S ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8713_prod_Orelated,axiom,
% 16.19/9.04 ! [R2: real > real > $o,S: set_nat,H2: nat > real,G: nat > real] :
% 16.19/9.04 ( ( R2 @ one_one_real @ one_one_real )
% 16.19/9.04 => ( ! [X16: real,Y15: real,X23: real,Y23: real] :
% 16.19/9.04 ( ( ( R2 @ X16 @ X23 )
% 16.19/9.04 & ( R2 @ Y15 @ Y23 ) )
% 16.19/9.04 => ( R2 @ ( times_times_real @ X16 @ Y15 ) @ ( times_times_real @ X23 @ Y23 ) ) )
% 16.19/9.04 => ( ( finite_finite_nat @ S )
% 16.19/9.04 => ( ! [X4: nat] :
% 16.19/9.04 ( ( member_nat @ X4 @ S )
% 16.19/9.04 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 16.19/9.04 => ( R2 @ ( groups129246275422532515t_real @ H2 @ S ) @ ( groups129246275422532515t_real @ G @ S ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8714_prod_Orelated,axiom,
% 16.19/9.04 ! [R2: real > real > $o,S: set_int,H2: int > real,G: int > real] :
% 16.19/9.04 ( ( R2 @ one_one_real @ one_one_real )
% 16.19/9.04 => ( ! [X16: real,Y15: real,X23: real,Y23: real] :
% 16.19/9.04 ( ( ( R2 @ X16 @ X23 )
% 16.19/9.04 & ( R2 @ Y15 @ Y23 ) )
% 16.19/9.04 => ( R2 @ ( times_times_real @ X16 @ Y15 ) @ ( times_times_real @ X23 @ Y23 ) ) )
% 16.19/9.04 => ( ( finite_finite_int @ S )
% 16.19/9.04 => ( ! [X4: int] :
% 16.19/9.04 ( ( member_int @ X4 @ S )
% 16.19/9.04 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 16.19/9.04 => ( R2 @ ( groups2316167850115554303t_real @ H2 @ S ) @ ( groups2316167850115554303t_real @ G @ S ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8715_prod_Orelated,axiom,
% 16.19/9.04 ! [R2: real > real > $o,S: set_complex,H2: complex > real,G: complex > real] :
% 16.19/9.04 ( ( R2 @ one_one_real @ one_one_real )
% 16.19/9.04 => ( ! [X16: real,Y15: real,X23: real,Y23: real] :
% 16.19/9.04 ( ( ( R2 @ X16 @ X23 )
% 16.19/9.04 & ( R2 @ Y15 @ Y23 ) )
% 16.19/9.04 => ( R2 @ ( times_times_real @ X16 @ Y15 ) @ ( times_times_real @ X23 @ Y23 ) ) )
% 16.19/9.04 => ( ( finite3207457112153483333omplex @ S )
% 16.19/9.04 => ( ! [X4: complex] :
% 16.19/9.04 ( ( member_complex @ X4 @ S )
% 16.19/9.04 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 16.19/9.04 => ( R2 @ ( groups766887009212190081x_real @ H2 @ S ) @ ( groups766887009212190081x_real @ G @ S ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8716_prod_Orelated,axiom,
% 16.19/9.04 ! [R2: rat > rat > $o,S: set_nat,H2: nat > rat,G: nat > rat] :
% 16.19/9.04 ( ( R2 @ one_one_rat @ one_one_rat )
% 16.19/9.04 => ( ! [X16: rat,Y15: rat,X23: rat,Y23: rat] :
% 16.19/9.04 ( ( ( R2 @ X16 @ X23 )
% 16.19/9.04 & ( R2 @ Y15 @ Y23 ) )
% 16.19/9.04 => ( R2 @ ( times_times_rat @ X16 @ Y15 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
% 16.19/9.04 => ( ( finite_finite_nat @ S )
% 16.19/9.04 => ( ! [X4: nat] :
% 16.19/9.04 ( ( member_nat @ X4 @ S )
% 16.19/9.04 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 16.19/9.04 => ( R2 @ ( groups73079841787564623at_rat @ H2 @ S ) @ ( groups73079841787564623at_rat @ G @ S ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8717_prod_Orelated,axiom,
% 16.19/9.04 ! [R2: rat > rat > $o,S: set_int,H2: int > rat,G: int > rat] :
% 16.19/9.04 ( ( R2 @ one_one_rat @ one_one_rat )
% 16.19/9.04 => ( ! [X16: rat,Y15: rat,X23: rat,Y23: rat] :
% 16.19/9.04 ( ( ( R2 @ X16 @ X23 )
% 16.19/9.04 & ( R2 @ Y15 @ Y23 ) )
% 16.19/9.04 => ( R2 @ ( times_times_rat @ X16 @ Y15 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
% 16.19/9.04 => ( ( finite_finite_int @ S )
% 16.19/9.04 => ( ! [X4: int] :
% 16.19/9.04 ( ( member_int @ X4 @ S )
% 16.19/9.04 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 16.19/9.04 => ( R2 @ ( groups1072433553688619179nt_rat @ H2 @ S ) @ ( groups1072433553688619179nt_rat @ G @ S ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8718_prod_Orelated,axiom,
% 16.19/9.04 ! [R2: rat > rat > $o,S: set_complex,H2: complex > rat,G: complex > rat] :
% 16.19/9.04 ( ( R2 @ one_one_rat @ one_one_rat )
% 16.19/9.04 => ( ! [X16: rat,Y15: rat,X23: rat,Y23: rat] :
% 16.19/9.04 ( ( ( R2 @ X16 @ X23 )
% 16.19/9.04 & ( R2 @ Y15 @ Y23 ) )
% 16.19/9.04 => ( R2 @ ( times_times_rat @ X16 @ Y15 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
% 16.19/9.04 => ( ( finite3207457112153483333omplex @ S )
% 16.19/9.04 => ( ! [X4: complex] :
% 16.19/9.04 ( ( member_complex @ X4 @ S )
% 16.19/9.04 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 16.19/9.04 => ( R2 @ ( groups225925009352817453ex_rat @ H2 @ S ) @ ( groups225925009352817453ex_rat @ G @ S ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8719_prod_Orelated,axiom,
% 16.19/9.04 ! [R2: nat > nat > $o,S: set_int,H2: int > nat,G: int > nat] :
% 16.19/9.04 ( ( R2 @ one_one_nat @ one_one_nat )
% 16.19/9.04 => ( ! [X16: nat,Y15: nat,X23: nat,Y23: nat] :
% 16.19/9.04 ( ( ( R2 @ X16 @ X23 )
% 16.19/9.04 & ( R2 @ Y15 @ Y23 ) )
% 16.19/9.04 => ( R2 @ ( times_times_nat @ X16 @ Y15 ) @ ( times_times_nat @ X23 @ Y23 ) ) )
% 16.19/9.04 => ( ( finite_finite_int @ S )
% 16.19/9.04 => ( ! [X4: int] :
% 16.19/9.04 ( ( member_int @ X4 @ S )
% 16.19/9.04 => ( R2 @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 16.19/9.04 => ( R2 @ ( groups1707563613775114915nt_nat @ H2 @ S ) @ ( groups1707563613775114915nt_nat @ G @ S ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8720_prod__dvd__prod__subset,axiom,
% 16.19/9.04 ! [B3: set_int,A2: set_int,F: int > nat] :
% 16.19/9.04 ( ( finite_finite_int @ B3 )
% 16.19/9.04 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 16.19/9.04 => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ F @ B3 ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8721_prod__dvd__prod__subset,axiom,
% 16.19/9.04 ! [B3: set_complex,A2: set_complex,F: complex > nat] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ B3 )
% 16.19/9.04 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 16.19/9.04 => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ F @ B3 ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8722_prod__dvd__prod__subset,axiom,
% 16.19/9.04 ! [B3: set_complex,A2: set_complex,F: complex > int] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ B3 )
% 16.19/9.04 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 16.19/9.04 => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ F @ B3 ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8723_prod__dvd__prod__subset,axiom,
% 16.19/9.04 ! [B3: set_int,A2: set_int,F: int > code_integer] :
% 16.19/9.04 ( ( finite_finite_int @ B3 )
% 16.19/9.04 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 16.19/9.04 => ( dvd_dvd_Code_integer @ ( groups3827104343326376752nteger @ F @ A2 ) @ ( groups3827104343326376752nteger @ F @ B3 ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8724_prod__dvd__prod__subset,axiom,
% 16.19/9.04 ! [B3: set_complex,A2: set_complex,F: complex > code_integer] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ B3 )
% 16.19/9.04 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 16.19/9.04 => ( dvd_dvd_Code_integer @ ( groups8682486955453173170nteger @ F @ A2 ) @ ( groups8682486955453173170nteger @ F @ B3 ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8725_prod__dvd__prod__subset,axiom,
% 16.19/9.04 ! [B3: set_nat,A2: set_nat,F: nat > code_integer] :
% 16.19/9.04 ( ( finite_finite_nat @ B3 )
% 16.19/9.04 => ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 16.19/9.04 => ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A2 ) @ ( groups3455450783089532116nteger @ F @ B3 ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8726_prod__dvd__prod__subset,axiom,
% 16.19/9.04 ! [B3: set_nat,A2: set_nat,F: nat > nat] :
% 16.19/9.04 ( ( finite_finite_nat @ B3 )
% 16.19/9.04 => ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 16.19/9.04 => ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ ( groups708209901874060359at_nat @ F @ B3 ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8727_prod__dvd__prod__subset,axiom,
% 16.19/9.04 ! [B3: set_nat,A2: set_nat,F: nat > int] :
% 16.19/9.04 ( ( finite_finite_nat @ B3 )
% 16.19/9.04 => ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 16.19/9.04 => ( dvd_dvd_int @ ( groups705719431365010083at_int @ F @ A2 ) @ ( groups705719431365010083at_int @ F @ B3 ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8728_prod__dvd__prod__subset,axiom,
% 16.19/9.04 ! [B3: set_int,A2: set_int,F: int > int] :
% 16.19/9.04 ( ( finite_finite_int @ B3 )
% 16.19/9.04 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 16.19/9.04 => ( dvd_dvd_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ ( groups1705073143266064639nt_int @ F @ B3 ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8729_prod__dvd__prod__subset2,axiom,
% 16.19/9.04 ! [B3: set_real,A2: set_real,F: real > nat,G: real > nat] :
% 16.19/9.04 ( ( finite_finite_real @ B3 )
% 16.19/9.04 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 16.19/9.04 => ( ! [A3: real] :
% 16.19/9.04 ( ( member_real @ A3 @ A2 )
% 16.19/9.04 => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 16.19/9.04 => ( dvd_dvd_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ B3 ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8730_prod__dvd__prod__subset2,axiom,
% 16.19/9.04 ! [B3: set_int,A2: set_int,F: int > nat,G: int > nat] :
% 16.19/9.04 ( ( finite_finite_int @ B3 )
% 16.19/9.04 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 16.19/9.04 => ( ! [A3: int] :
% 16.19/9.04 ( ( member_int @ A3 @ A2 )
% 16.19/9.04 => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 16.19/9.04 => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ B3 ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8731_prod__dvd__prod__subset2,axiom,
% 16.19/9.04 ! [B3: set_complex,A2: set_complex,F: complex > nat,G: complex > nat] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ B3 )
% 16.19/9.04 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 16.19/9.04 => ( ! [A3: complex] :
% 16.19/9.04 ( ( member_complex @ A3 @ A2 )
% 16.19/9.04 => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 16.19/9.04 => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ G @ B3 ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8732_prod__dvd__prod__subset2,axiom,
% 16.19/9.04 ! [B3: set_real,A2: set_real,F: real > int,G: real > int] :
% 16.19/9.04 ( ( finite_finite_real @ B3 )
% 16.19/9.04 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 16.19/9.04 => ( ! [A3: real] :
% 16.19/9.04 ( ( member_real @ A3 @ A2 )
% 16.19/9.04 => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 16.19/9.04 => ( dvd_dvd_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ ( groups4694064378042380927al_int @ G @ B3 ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8733_prod__dvd__prod__subset2,axiom,
% 16.19/9.04 ! [B3: set_complex,A2: set_complex,F: complex > int,G: complex > int] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ B3 )
% 16.19/9.04 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 16.19/9.04 => ( ! [A3: complex] :
% 16.19/9.04 ( ( member_complex @ A3 @ A2 )
% 16.19/9.04 => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 16.19/9.04 => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ G @ B3 ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8734_prod__dvd__prod__subset2,axiom,
% 16.19/9.04 ! [B3: set_real,A2: set_real,F: real > code_integer,G: real > code_integer] :
% 16.19/9.04 ( ( finite_finite_real @ B3 )
% 16.19/9.04 => ( ( ord_less_eq_set_real @ A2 @ B3 )
% 16.19/9.04 => ( ! [A3: real] :
% 16.19/9.04 ( ( member_real @ A3 @ A2 )
% 16.19/9.04 => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 16.19/9.04 => ( dvd_dvd_Code_integer @ ( groups6225526099057966256nteger @ F @ A2 ) @ ( groups6225526099057966256nteger @ G @ B3 ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8735_prod__dvd__prod__subset2,axiom,
% 16.19/9.04 ! [B3: set_int,A2: set_int,F: int > code_integer,G: int > code_integer] :
% 16.19/9.04 ( ( finite_finite_int @ B3 )
% 16.19/9.04 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 16.19/9.04 => ( ! [A3: int] :
% 16.19/9.04 ( ( member_int @ A3 @ A2 )
% 16.19/9.04 => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 16.19/9.04 => ( dvd_dvd_Code_integer @ ( groups3827104343326376752nteger @ F @ A2 ) @ ( groups3827104343326376752nteger @ G @ B3 ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8736_prod__dvd__prod__subset2,axiom,
% 16.19/9.04 ! [B3: set_complex,A2: set_complex,F: complex > code_integer,G: complex > code_integer] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ B3 )
% 16.19/9.04 => ( ( ord_le211207098394363844omplex @ A2 @ B3 )
% 16.19/9.04 => ( ! [A3: complex] :
% 16.19/9.04 ( ( member_complex @ A3 @ A2 )
% 16.19/9.04 => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 16.19/9.04 => ( dvd_dvd_Code_integer @ ( groups8682486955453173170nteger @ F @ A2 ) @ ( groups8682486955453173170nteger @ G @ B3 ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8737_prod__dvd__prod__subset2,axiom,
% 16.19/9.04 ! [B3: set_nat,A2: set_nat,F: nat > code_integer,G: nat > code_integer] :
% 16.19/9.04 ( ( finite_finite_nat @ B3 )
% 16.19/9.04 => ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 16.19/9.04 => ( ! [A3: nat] :
% 16.19/9.04 ( ( member_nat @ A3 @ A2 )
% 16.19/9.04 => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 16.19/9.04 => ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A2 ) @ ( groups3455450783089532116nteger @ G @ B3 ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8738_prod__dvd__prod__subset2,axiom,
% 16.19/9.04 ! [B3: set_nat,A2: set_nat,F: nat > nat,G: nat > nat] :
% 16.19/9.04 ( ( finite_finite_nat @ B3 )
% 16.19/9.04 => ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 16.19/9.04 => ( ! [A3: nat] :
% 16.19/9.04 ( ( member_nat @ A3 @ A2 )
% 16.19/9.04 => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
% 16.19/9.04 => ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ ( groups708209901874060359at_nat @ G @ B3 ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8739_prod_Oreindex__bij__witness__not__neutral,axiom,
% 16.19/9.04 ! [S4: set_real,T4: set_real,S: set_real,I2: real > real,J: real > real,T3: set_real,G: real > complex,H2: real > complex] :
% 16.19/9.04 ( ( finite_finite_real @ S4 )
% 16.19/9.04 => ( ( finite_finite_real @ T4 )
% 16.19/9.04 => ( ! [A3: real] :
% 16.19/9.04 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 16.19/9.04 => ( ( I2 @ ( J @ A3 ) )
% 16.19/9.04 = A3 ) )
% 16.19/9.04 => ( ! [A3: real] :
% 16.19/9.04 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 16.19/9.04 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 16.19/9.04 => ( ! [B2: real] :
% 16.19/9.04 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 16.19/9.04 => ( ( J @ ( I2 @ B2 ) )
% 16.19/9.04 = B2 ) )
% 16.19/9.04 => ( ! [B2: real] :
% 16.19/9.04 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 16.19/9.04 => ( member_real @ ( I2 @ B2 ) @ ( minus_minus_set_real @ S @ S4 ) ) )
% 16.19/9.04 => ( ! [A3: real] :
% 16.19/9.04 ( ( member_real @ A3 @ S4 )
% 16.19/9.04 => ( ( G @ A3 )
% 16.19/9.04 = one_one_complex ) )
% 16.19/9.04 => ( ! [B2: real] :
% 16.19/9.04 ( ( member_real @ B2 @ T4 )
% 16.19/9.04 => ( ( H2 @ B2 )
% 16.19/9.04 = one_one_complex ) )
% 16.19/9.04 => ( ! [A3: real] :
% 16.19/9.04 ( ( member_real @ A3 @ S )
% 16.19/9.04 => ( ( H2 @ ( J @ A3 ) )
% 16.19/9.04 = ( G @ A3 ) ) )
% 16.19/9.04 => ( ( groups713298508707869441omplex @ G @ S )
% 16.19/9.04 = ( groups713298508707869441omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8740_prod_Oreindex__bij__witness__not__neutral,axiom,
% 16.19/9.04 ! [S4: set_real,T4: set_int,S: set_real,I2: int > real,J: real > int,T3: set_int,G: real > complex,H2: int > complex] :
% 16.19/9.04 ( ( finite_finite_real @ S4 )
% 16.19/9.04 => ( ( finite_finite_int @ T4 )
% 16.19/9.04 => ( ! [A3: real] :
% 16.19/9.04 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 16.19/9.04 => ( ( I2 @ ( J @ A3 ) )
% 16.19/9.04 = A3 ) )
% 16.19/9.04 => ( ! [A3: real] :
% 16.19/9.04 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 16.19/9.04 => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 16.19/9.04 => ( ! [B2: int] :
% 16.19/9.04 ( ( member_int @ B2 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 16.19/9.04 => ( ( J @ ( I2 @ B2 ) )
% 16.19/9.04 = B2 ) )
% 16.19/9.04 => ( ! [B2: int] :
% 16.19/9.04 ( ( member_int @ B2 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 16.19/9.04 => ( member_real @ ( I2 @ B2 ) @ ( minus_minus_set_real @ S @ S4 ) ) )
% 16.19/9.04 => ( ! [A3: real] :
% 16.19/9.04 ( ( member_real @ A3 @ S4 )
% 16.19/9.04 => ( ( G @ A3 )
% 16.19/9.04 = one_one_complex ) )
% 16.19/9.04 => ( ! [B2: int] :
% 16.19/9.04 ( ( member_int @ B2 @ T4 )
% 16.19/9.04 => ( ( H2 @ B2 )
% 16.19/9.04 = one_one_complex ) )
% 16.19/9.04 => ( ! [A3: real] :
% 16.19/9.04 ( ( member_real @ A3 @ S )
% 16.19/9.04 => ( ( H2 @ ( J @ A3 ) )
% 16.19/9.04 = ( G @ A3 ) ) )
% 16.19/9.04 => ( ( groups713298508707869441omplex @ G @ S )
% 16.19/9.04 = ( groups7440179247065528705omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8741_prod_Oreindex__bij__witness__not__neutral,axiom,
% 16.19/9.04 ! [S4: set_real,T4: set_complex,S: set_real,I2: complex > real,J: real > complex,T3: set_complex,G: real > complex,H2: complex > complex] :
% 16.19/9.04 ( ( finite_finite_real @ S4 )
% 16.19/9.04 => ( ( finite3207457112153483333omplex @ T4 )
% 16.19/9.04 => ( ! [A3: real] :
% 16.19/9.04 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 16.19/9.04 => ( ( I2 @ ( J @ A3 ) )
% 16.19/9.04 = A3 ) )
% 16.19/9.04 => ( ! [A3: real] :
% 16.19/9.04 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 16.19/9.04 => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 16.19/9.04 => ( ! [B2: complex] :
% 16.19/9.04 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 16.19/9.04 => ( ( J @ ( I2 @ B2 ) )
% 16.19/9.04 = B2 ) )
% 16.19/9.04 => ( ! [B2: complex] :
% 16.19/9.04 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 16.19/9.04 => ( member_real @ ( I2 @ B2 ) @ ( minus_minus_set_real @ S @ S4 ) ) )
% 16.19/9.04 => ( ! [A3: real] :
% 16.19/9.04 ( ( member_real @ A3 @ S4 )
% 16.19/9.04 => ( ( G @ A3 )
% 16.19/9.04 = one_one_complex ) )
% 16.19/9.04 => ( ! [B2: complex] :
% 16.19/9.04 ( ( member_complex @ B2 @ T4 )
% 16.19/9.04 => ( ( H2 @ B2 )
% 16.19/9.04 = one_one_complex ) )
% 16.19/9.04 => ( ! [A3: real] :
% 16.19/9.04 ( ( member_real @ A3 @ S )
% 16.19/9.04 => ( ( H2 @ ( J @ A3 ) )
% 16.19/9.04 = ( G @ A3 ) ) )
% 16.19/9.04 => ( ( groups713298508707869441omplex @ G @ S )
% 16.19/9.04 = ( groups3708469109370488835omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8742_prod_Oreindex__bij__witness__not__neutral,axiom,
% 16.19/9.04 ! [S4: set_int,T4: set_real,S: set_int,I2: real > int,J: int > real,T3: set_real,G: int > complex,H2: real > complex] :
% 16.19/9.04 ( ( finite_finite_int @ S4 )
% 16.19/9.04 => ( ( finite_finite_real @ T4 )
% 16.19/9.04 => ( ! [A3: int] :
% 16.19/9.04 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 16.19/9.04 => ( ( I2 @ ( J @ A3 ) )
% 16.19/9.04 = A3 ) )
% 16.19/9.04 => ( ! [A3: int] :
% 16.19/9.04 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 16.19/9.04 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 16.19/9.04 => ( ! [B2: real] :
% 16.19/9.04 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 16.19/9.04 => ( ( J @ ( I2 @ B2 ) )
% 16.19/9.04 = B2 ) )
% 16.19/9.04 => ( ! [B2: real] :
% 16.19/9.04 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 16.19/9.04 => ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
% 16.19/9.04 => ( ! [A3: int] :
% 16.19/9.04 ( ( member_int @ A3 @ S4 )
% 16.19/9.04 => ( ( G @ A3 )
% 16.19/9.04 = one_one_complex ) )
% 16.19/9.04 => ( ! [B2: real] :
% 16.19/9.04 ( ( member_real @ B2 @ T4 )
% 16.19/9.04 => ( ( H2 @ B2 )
% 16.19/9.04 = one_one_complex ) )
% 16.19/9.04 => ( ! [A3: int] :
% 16.19/9.04 ( ( member_int @ A3 @ S )
% 16.19/9.04 => ( ( H2 @ ( J @ A3 ) )
% 16.19/9.04 = ( G @ A3 ) ) )
% 16.19/9.04 => ( ( groups7440179247065528705omplex @ G @ S )
% 16.19/9.04 = ( groups713298508707869441omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8743_prod_Oreindex__bij__witness__not__neutral,axiom,
% 16.19/9.04 ! [S4: set_int,T4: set_int,S: set_int,I2: int > int,J: int > int,T3: set_int,G: int > complex,H2: int > complex] :
% 16.19/9.04 ( ( finite_finite_int @ S4 )
% 16.19/9.04 => ( ( finite_finite_int @ T4 )
% 16.19/9.04 => ( ! [A3: int] :
% 16.19/9.04 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 16.19/9.04 => ( ( I2 @ ( J @ A3 ) )
% 16.19/9.04 = A3 ) )
% 16.19/9.04 => ( ! [A3: int] :
% 16.19/9.04 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 16.19/9.04 => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 16.19/9.04 => ( ! [B2: int] :
% 16.19/9.04 ( ( member_int @ B2 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 16.19/9.04 => ( ( J @ ( I2 @ B2 ) )
% 16.19/9.04 = B2 ) )
% 16.19/9.04 => ( ! [B2: int] :
% 16.19/9.04 ( ( member_int @ B2 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 16.19/9.04 => ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
% 16.19/9.04 => ( ! [A3: int] :
% 16.19/9.04 ( ( member_int @ A3 @ S4 )
% 16.19/9.04 => ( ( G @ A3 )
% 16.19/9.04 = one_one_complex ) )
% 16.19/9.04 => ( ! [B2: int] :
% 16.19/9.04 ( ( member_int @ B2 @ T4 )
% 16.19/9.04 => ( ( H2 @ B2 )
% 16.19/9.04 = one_one_complex ) )
% 16.19/9.04 => ( ! [A3: int] :
% 16.19/9.04 ( ( member_int @ A3 @ S )
% 16.19/9.04 => ( ( H2 @ ( J @ A3 ) )
% 16.19/9.04 = ( G @ A3 ) ) )
% 16.19/9.04 => ( ( groups7440179247065528705omplex @ G @ S )
% 16.19/9.04 = ( groups7440179247065528705omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8744_prod_Oreindex__bij__witness__not__neutral,axiom,
% 16.19/9.04 ! [S4: set_int,T4: set_complex,S: set_int,I2: complex > int,J: int > complex,T3: set_complex,G: int > complex,H2: complex > complex] :
% 16.19/9.04 ( ( finite_finite_int @ S4 )
% 16.19/9.04 => ( ( finite3207457112153483333omplex @ T4 )
% 16.19/9.04 => ( ! [A3: int] :
% 16.19/9.04 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 16.19/9.04 => ( ( I2 @ ( J @ A3 ) )
% 16.19/9.04 = A3 ) )
% 16.19/9.04 => ( ! [A3: int] :
% 16.19/9.04 ( ( member_int @ A3 @ ( minus_minus_set_int @ S @ S4 ) )
% 16.19/9.04 => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 16.19/9.04 => ( ! [B2: complex] :
% 16.19/9.04 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 16.19/9.04 => ( ( J @ ( I2 @ B2 ) )
% 16.19/9.04 = B2 ) )
% 16.19/9.04 => ( ! [B2: complex] :
% 16.19/9.04 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 16.19/9.04 => ( member_int @ ( I2 @ B2 ) @ ( minus_minus_set_int @ S @ S4 ) ) )
% 16.19/9.04 => ( ! [A3: int] :
% 16.19/9.04 ( ( member_int @ A3 @ S4 )
% 16.19/9.04 => ( ( G @ A3 )
% 16.19/9.04 = one_one_complex ) )
% 16.19/9.04 => ( ! [B2: complex] :
% 16.19/9.04 ( ( member_complex @ B2 @ T4 )
% 16.19/9.04 => ( ( H2 @ B2 )
% 16.19/9.04 = one_one_complex ) )
% 16.19/9.04 => ( ! [A3: int] :
% 16.19/9.04 ( ( member_int @ A3 @ S )
% 16.19/9.04 => ( ( H2 @ ( J @ A3 ) )
% 16.19/9.04 = ( G @ A3 ) ) )
% 16.19/9.04 => ( ( groups7440179247065528705omplex @ G @ S )
% 16.19/9.04 = ( groups3708469109370488835omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8745_prod_Oreindex__bij__witness__not__neutral,axiom,
% 16.19/9.04 ! [S4: set_complex,T4: set_real,S: set_complex,I2: real > complex,J: complex > real,T3: set_real,G: complex > complex,H2: real > complex] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ S4 )
% 16.19/9.04 => ( ( finite_finite_real @ T4 )
% 16.19/9.04 => ( ! [A3: complex] :
% 16.19/9.04 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S @ S4 ) )
% 16.19/9.04 => ( ( I2 @ ( J @ A3 ) )
% 16.19/9.04 = A3 ) )
% 16.19/9.04 => ( ! [A3: complex] :
% 16.19/9.04 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S @ S4 ) )
% 16.19/9.04 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 16.19/9.04 => ( ! [B2: real] :
% 16.19/9.04 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 16.19/9.04 => ( ( J @ ( I2 @ B2 ) )
% 16.19/9.04 = B2 ) )
% 16.19/9.04 => ( ! [B2: real] :
% 16.19/9.04 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 16.19/9.04 => ( member_complex @ ( I2 @ B2 ) @ ( minus_811609699411566653omplex @ S @ S4 ) ) )
% 16.19/9.04 => ( ! [A3: complex] :
% 16.19/9.04 ( ( member_complex @ A3 @ S4 )
% 16.19/9.04 => ( ( G @ A3 )
% 16.19/9.04 = one_one_complex ) )
% 16.19/9.04 => ( ! [B2: real] :
% 16.19/9.04 ( ( member_real @ B2 @ T4 )
% 16.19/9.04 => ( ( H2 @ B2 )
% 16.19/9.04 = one_one_complex ) )
% 16.19/9.04 => ( ! [A3: complex] :
% 16.19/9.04 ( ( member_complex @ A3 @ S )
% 16.19/9.04 => ( ( H2 @ ( J @ A3 ) )
% 16.19/9.04 = ( G @ A3 ) ) )
% 16.19/9.04 => ( ( groups3708469109370488835omplex @ G @ S )
% 16.19/9.04 = ( groups713298508707869441omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8746_prod_Oreindex__bij__witness__not__neutral,axiom,
% 16.19/9.04 ! [S4: set_complex,T4: set_int,S: set_complex,I2: int > complex,J: complex > int,T3: set_int,G: complex > complex,H2: int > complex] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ S4 )
% 16.19/9.04 => ( ( finite_finite_int @ T4 )
% 16.19/9.04 => ( ! [A3: complex] :
% 16.19/9.04 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S @ S4 ) )
% 16.19/9.04 => ( ( I2 @ ( J @ A3 ) )
% 16.19/9.04 = A3 ) )
% 16.19/9.04 => ( ! [A3: complex] :
% 16.19/9.04 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S @ S4 ) )
% 16.19/9.04 => ( member_int @ ( J @ A3 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 16.19/9.04 => ( ! [B2: int] :
% 16.19/9.04 ( ( member_int @ B2 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 16.19/9.04 => ( ( J @ ( I2 @ B2 ) )
% 16.19/9.04 = B2 ) )
% 16.19/9.04 => ( ! [B2: int] :
% 16.19/9.04 ( ( member_int @ B2 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 16.19/9.04 => ( member_complex @ ( I2 @ B2 ) @ ( minus_811609699411566653omplex @ S @ S4 ) ) )
% 16.19/9.04 => ( ! [A3: complex] :
% 16.19/9.04 ( ( member_complex @ A3 @ S4 )
% 16.19/9.04 => ( ( G @ A3 )
% 16.19/9.04 = one_one_complex ) )
% 16.19/9.04 => ( ! [B2: int] :
% 16.19/9.04 ( ( member_int @ B2 @ T4 )
% 16.19/9.04 => ( ( H2 @ B2 )
% 16.19/9.04 = one_one_complex ) )
% 16.19/9.04 => ( ! [A3: complex] :
% 16.19/9.04 ( ( member_complex @ A3 @ S )
% 16.19/9.04 => ( ( H2 @ ( J @ A3 ) )
% 16.19/9.04 = ( G @ A3 ) ) )
% 16.19/9.04 => ( ( groups3708469109370488835omplex @ G @ S )
% 16.19/9.04 = ( groups7440179247065528705omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8747_prod_Oreindex__bij__witness__not__neutral,axiom,
% 16.19/9.04 ! [S4: set_complex,T4: set_complex,S: set_complex,I2: complex > complex,J: complex > complex,T3: set_complex,G: complex > complex,H2: complex > complex] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ S4 )
% 16.19/9.04 => ( ( finite3207457112153483333omplex @ T4 )
% 16.19/9.04 => ( ! [A3: complex] :
% 16.19/9.04 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S @ S4 ) )
% 16.19/9.04 => ( ( I2 @ ( J @ A3 ) )
% 16.19/9.04 = A3 ) )
% 16.19/9.04 => ( ! [A3: complex] :
% 16.19/9.04 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S @ S4 ) )
% 16.19/9.04 => ( member_complex @ ( J @ A3 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 16.19/9.04 => ( ! [B2: complex] :
% 16.19/9.04 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 16.19/9.04 => ( ( J @ ( I2 @ B2 ) )
% 16.19/9.04 = B2 ) )
% 16.19/9.04 => ( ! [B2: complex] :
% 16.19/9.04 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 16.19/9.04 => ( member_complex @ ( I2 @ B2 ) @ ( minus_811609699411566653omplex @ S @ S4 ) ) )
% 16.19/9.04 => ( ! [A3: complex] :
% 16.19/9.04 ( ( member_complex @ A3 @ S4 )
% 16.19/9.04 => ( ( G @ A3 )
% 16.19/9.04 = one_one_complex ) )
% 16.19/9.04 => ( ! [B2: complex] :
% 16.19/9.04 ( ( member_complex @ B2 @ T4 )
% 16.19/9.04 => ( ( H2 @ B2 )
% 16.19/9.04 = one_one_complex ) )
% 16.19/9.04 => ( ! [A3: complex] :
% 16.19/9.04 ( ( member_complex @ A3 @ S )
% 16.19/9.04 => ( ( H2 @ ( J @ A3 ) )
% 16.19/9.04 = ( G @ A3 ) ) )
% 16.19/9.04 => ( ( groups3708469109370488835omplex @ G @ S )
% 16.19/9.04 = ( groups3708469109370488835omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8748_prod_Oreindex__bij__witness__not__neutral,axiom,
% 16.19/9.04 ! [S4: set_real,T4: set_real,S: set_real,I2: real > real,J: real > real,T3: set_real,G: real > real,H2: real > real] :
% 16.19/9.04 ( ( finite_finite_real @ S4 )
% 16.19/9.04 => ( ( finite_finite_real @ T4 )
% 16.19/9.04 => ( ! [A3: real] :
% 16.19/9.04 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 16.19/9.04 => ( ( I2 @ ( J @ A3 ) )
% 16.19/9.04 = A3 ) )
% 16.19/9.04 => ( ! [A3: real] :
% 16.19/9.04 ( ( member_real @ A3 @ ( minus_minus_set_real @ S @ S4 ) )
% 16.19/9.04 => ( member_real @ ( J @ A3 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 16.19/9.04 => ( ! [B2: real] :
% 16.19/9.04 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 16.19/9.04 => ( ( J @ ( I2 @ B2 ) )
% 16.19/9.04 = B2 ) )
% 16.19/9.04 => ( ! [B2: real] :
% 16.19/9.04 ( ( member_real @ B2 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 16.19/9.04 => ( member_real @ ( I2 @ B2 ) @ ( minus_minus_set_real @ S @ S4 ) ) )
% 16.19/9.04 => ( ! [A3: real] :
% 16.19/9.04 ( ( member_real @ A3 @ S4 )
% 16.19/9.04 => ( ( G @ A3 )
% 16.19/9.04 = one_one_real ) )
% 16.19/9.04 => ( ! [B2: real] :
% 16.19/9.04 ( ( member_real @ B2 @ T4 )
% 16.19/9.04 => ( ( H2 @ B2 )
% 16.19/9.04 = one_one_real ) )
% 16.19/9.04 => ( ! [A3: real] :
% 16.19/9.04 ( ( member_real @ A3 @ S )
% 16.19/9.04 => ( ( H2 @ ( J @ A3 ) )
% 16.19/9.04 = ( G @ A3 ) ) )
% 16.19/9.04 => ( ( groups1681761925125756287l_real @ G @ S )
% 16.19/9.04 = ( groups1681761925125756287l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8749_prod_Osetdiff__irrelevant,axiom,
% 16.19/9.04 ! [A2: set_int,G: int > complex] :
% 16.19/9.04 ( ( finite_finite_int @ A2 )
% 16.19/9.04 => ( ( groups7440179247065528705omplex @ G
% 16.19/9.04 @ ( minus_minus_set_int @ A2
% 16.19/9.04 @ ( collect_int
% 16.19/9.04 @ ^ [X2: int] :
% 16.19/9.04 ( ( G @ X2 )
% 16.19/9.04 = one_one_complex ) ) ) )
% 16.19/9.04 = ( groups7440179247065528705omplex @ G @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8750_prod_Osetdiff__irrelevant,axiom,
% 16.19/9.04 ! [A2: set_complex,G: complex > complex] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/9.04 => ( ( groups3708469109370488835omplex @ G
% 16.19/9.04 @ ( minus_811609699411566653omplex @ A2
% 16.19/9.04 @ ( collect_complex
% 16.19/9.04 @ ^ [X2: complex] :
% 16.19/9.04 ( ( G @ X2 )
% 16.19/9.04 = one_one_complex ) ) ) )
% 16.19/9.04 = ( groups3708469109370488835omplex @ G @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8751_prod_Osetdiff__irrelevant,axiom,
% 16.19/9.04 ! [A2: set_int,G: int > real] :
% 16.19/9.04 ( ( finite_finite_int @ A2 )
% 16.19/9.04 => ( ( groups2316167850115554303t_real @ G
% 16.19/9.04 @ ( minus_minus_set_int @ A2
% 16.19/9.04 @ ( collect_int
% 16.19/9.04 @ ^ [X2: int] :
% 16.19/9.04 ( ( G @ X2 )
% 16.19/9.04 = one_one_real ) ) ) )
% 16.19/9.04 = ( groups2316167850115554303t_real @ G @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8752_prod_Osetdiff__irrelevant,axiom,
% 16.19/9.04 ! [A2: set_complex,G: complex > real] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/9.04 => ( ( groups766887009212190081x_real @ G
% 16.19/9.04 @ ( minus_811609699411566653omplex @ A2
% 16.19/9.04 @ ( collect_complex
% 16.19/9.04 @ ^ [X2: complex] :
% 16.19/9.04 ( ( G @ X2 )
% 16.19/9.04 = one_one_real ) ) ) )
% 16.19/9.04 = ( groups766887009212190081x_real @ G @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8753_prod_Osetdiff__irrelevant,axiom,
% 16.19/9.04 ! [A2: set_int,G: int > rat] :
% 16.19/9.04 ( ( finite_finite_int @ A2 )
% 16.19/9.04 => ( ( groups1072433553688619179nt_rat @ G
% 16.19/9.04 @ ( minus_minus_set_int @ A2
% 16.19/9.04 @ ( collect_int
% 16.19/9.04 @ ^ [X2: int] :
% 16.19/9.04 ( ( G @ X2 )
% 16.19/9.04 = one_one_rat ) ) ) )
% 16.19/9.04 = ( groups1072433553688619179nt_rat @ G @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8754_prod_Osetdiff__irrelevant,axiom,
% 16.19/9.04 ! [A2: set_complex,G: complex > rat] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/9.04 => ( ( groups225925009352817453ex_rat @ G
% 16.19/9.04 @ ( minus_811609699411566653omplex @ A2
% 16.19/9.04 @ ( collect_complex
% 16.19/9.04 @ ^ [X2: complex] :
% 16.19/9.04 ( ( G @ X2 )
% 16.19/9.04 = one_one_rat ) ) ) )
% 16.19/9.04 = ( groups225925009352817453ex_rat @ G @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8755_prod_Osetdiff__irrelevant,axiom,
% 16.19/9.04 ! [A2: set_int,G: int > nat] :
% 16.19/9.04 ( ( finite_finite_int @ A2 )
% 16.19/9.04 => ( ( groups1707563613775114915nt_nat @ G
% 16.19/9.04 @ ( minus_minus_set_int @ A2
% 16.19/9.04 @ ( collect_int
% 16.19/9.04 @ ^ [X2: int] :
% 16.19/9.04 ( ( G @ X2 )
% 16.19/9.04 = one_one_nat ) ) ) )
% 16.19/9.04 = ( groups1707563613775114915nt_nat @ G @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8756_prod_Osetdiff__irrelevant,axiom,
% 16.19/9.04 ! [A2: set_complex,G: complex > nat] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/9.04 => ( ( groups861055069439313189ex_nat @ G
% 16.19/9.04 @ ( minus_811609699411566653omplex @ A2
% 16.19/9.04 @ ( collect_complex
% 16.19/9.04 @ ^ [X2: complex] :
% 16.19/9.04 ( ( G @ X2 )
% 16.19/9.04 = one_one_nat ) ) ) )
% 16.19/9.04 = ( groups861055069439313189ex_nat @ G @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8757_prod_Osetdiff__irrelevant,axiom,
% 16.19/9.04 ! [A2: set_complex,G: complex > int] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/9.04 => ( ( groups858564598930262913ex_int @ G
% 16.19/9.04 @ ( minus_811609699411566653omplex @ A2
% 16.19/9.04 @ ( collect_complex
% 16.19/9.04 @ ^ [X2: complex] :
% 16.19/9.04 ( ( G @ X2 )
% 16.19/9.04 = one_one_int ) ) ) )
% 16.19/9.04 = ( groups858564598930262913ex_int @ G @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8758_prod_Osetdiff__irrelevant,axiom,
% 16.19/9.04 ! [A2: set_nat,G: nat > complex] :
% 16.19/9.04 ( ( finite_finite_nat @ A2 )
% 16.19/9.04 => ( ( groups6464643781859351333omplex @ G
% 16.19/9.04 @ ( minus_minus_set_nat @ A2
% 16.19/9.04 @ ( collect_nat
% 16.19/9.04 @ ^ [X2: nat] :
% 16.19/9.04 ( ( G @ X2 )
% 16.19/9.04 = one_one_complex ) ) ) )
% 16.19/9.04 = ( groups6464643781859351333omplex @ G @ A2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8759_exp__sum,axiom,
% 16.19/9.04 ! [I6: set_int,F: int > real] :
% 16.19/9.04 ( ( finite_finite_int @ I6 )
% 16.19/9.04 => ( ( exp_real @ ( groups8778361861064173332t_real @ F @ I6 ) )
% 16.19/9.04 = ( groups2316167850115554303t_real
% 16.19/9.04 @ ^ [X2: int] : ( exp_real @ ( F @ X2 ) )
% 16.19/9.04 @ I6 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8760_exp__sum,axiom,
% 16.19/9.04 ! [I6: set_complex,F: complex > real] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ I6 )
% 16.19/9.04 => ( ( exp_real @ ( groups5808333547571424918x_real @ F @ I6 ) )
% 16.19/9.04 = ( groups766887009212190081x_real
% 16.19/9.04 @ ^ [X2: complex] : ( exp_real @ ( F @ X2 ) )
% 16.19/9.04 @ I6 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8761_exp__sum,axiom,
% 16.19/9.04 ! [I6: set_nat,F: nat > complex] :
% 16.19/9.04 ( ( finite_finite_nat @ I6 )
% 16.19/9.04 => ( ( exp_complex @ ( groups2073611262835488442omplex @ F @ I6 ) )
% 16.19/9.04 = ( groups6464643781859351333omplex
% 16.19/9.04 @ ^ [X2: nat] : ( exp_complex @ ( F @ X2 ) )
% 16.19/9.04 @ I6 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8762_exp__sum,axiom,
% 16.19/9.04 ! [I6: set_int,F: int > complex] :
% 16.19/9.04 ( ( finite_finite_int @ I6 )
% 16.19/9.04 => ( ( exp_complex @ ( groups3049146728041665814omplex @ F @ I6 ) )
% 16.19/9.04 = ( groups7440179247065528705omplex
% 16.19/9.04 @ ^ [X2: int] : ( exp_complex @ ( F @ X2 ) )
% 16.19/9.04 @ I6 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8763_exp__sum,axiom,
% 16.19/9.04 ! [I6: set_complex,F: complex > complex] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ I6 )
% 16.19/9.04 => ( ( exp_complex @ ( groups7754918857620584856omplex @ F @ I6 ) )
% 16.19/9.04 = ( groups3708469109370488835omplex
% 16.19/9.04 @ ^ [X2: complex] : ( exp_complex @ ( F @ X2 ) )
% 16.19/9.04 @ I6 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8764_exp__sum,axiom,
% 16.19/9.04 ! [I6: set_nat,F: nat > real] :
% 16.19/9.04 ( ( finite_finite_nat @ I6 )
% 16.19/9.04 => ( ( exp_real @ ( groups6591440286371151544t_real @ F @ I6 ) )
% 16.19/9.04 = ( groups129246275422532515t_real
% 16.19/9.04 @ ^ [X2: nat] : ( exp_real @ ( F @ X2 ) )
% 16.19/9.04 @ I6 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8765_prod_Onat__diff__reindex,axiom,
% 16.19/9.04 ! [G: nat > nat,N2: nat] :
% 16.19/9.04 ( ( groups708209901874060359at_nat
% 16.19/9.04 @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
% 16.19/9.04 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.04 = ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8766_prod_Onat__diff__reindex,axiom,
% 16.19/9.04 ! [G: nat > int,N2: nat] :
% 16.19/9.04 ( ( groups705719431365010083at_int
% 16.19/9.04 @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
% 16.19/9.04 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.04 = ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8767_prod_OatLeastAtMost__rev,axiom,
% 16.19/9.04 ! [G: nat > nat,N2: nat,M: nat] :
% 16.19/9.04 ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
% 16.19/9.04 = ( groups708209901874060359at_nat
% 16.19/9.04 @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I3 ) )
% 16.19/9.04 @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8768_prod_OatLeastAtMost__rev,axiom,
% 16.19/9.04 ! [G: nat > int,N2: nat,M: nat] :
% 16.19/9.04 ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
% 16.19/9.04 = ( groups705719431365010083at_int
% 16.19/9.04 @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I3 ) )
% 16.19/9.04 @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8769_less__1__prod2,axiom,
% 16.19/9.04 ! [I6: set_real,I2: real,F: real > real] :
% 16.19/9.04 ( ( finite_finite_real @ I6 )
% 16.19/9.04 => ( ( member_real @ I2 @ I6 )
% 16.19/9.04 => ( ( ord_less_real @ one_one_real @ ( F @ I2 ) )
% 16.19/9.04 => ( ! [I4: real] :
% 16.19/9.04 ( ( member_real @ I4 @ I6 )
% 16.19/9.04 => ( ord_less_eq_real @ one_one_real @ ( F @ I4 ) ) )
% 16.19/9.04 => ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I6 ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8770_less__1__prod2,axiom,
% 16.19/9.04 ! [I6: set_nat,I2: nat,F: nat > real] :
% 16.19/9.04 ( ( finite_finite_nat @ I6 )
% 16.19/9.04 => ( ( member_nat @ I2 @ I6 )
% 16.19/9.04 => ( ( ord_less_real @ one_one_real @ ( F @ I2 ) )
% 16.19/9.04 => ( ! [I4: nat] :
% 16.19/9.04 ( ( member_nat @ I4 @ I6 )
% 16.19/9.04 => ( ord_less_eq_real @ one_one_real @ ( F @ I4 ) ) )
% 16.19/9.04 => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I6 ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8771_less__1__prod2,axiom,
% 16.19/9.04 ! [I6: set_int,I2: int,F: int > real] :
% 16.19/9.04 ( ( finite_finite_int @ I6 )
% 16.19/9.04 => ( ( member_int @ I2 @ I6 )
% 16.19/9.04 => ( ( ord_less_real @ one_one_real @ ( F @ I2 ) )
% 16.19/9.04 => ( ! [I4: int] :
% 16.19/9.04 ( ( member_int @ I4 @ I6 )
% 16.19/9.04 => ( ord_less_eq_real @ one_one_real @ ( F @ I4 ) ) )
% 16.19/9.04 => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I6 ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8772_less__1__prod2,axiom,
% 16.19/9.04 ! [I6: set_complex,I2: complex,F: complex > real] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ I6 )
% 16.19/9.04 => ( ( member_complex @ I2 @ I6 )
% 16.19/9.04 => ( ( ord_less_real @ one_one_real @ ( F @ I2 ) )
% 16.19/9.04 => ( ! [I4: complex] :
% 16.19/9.04 ( ( member_complex @ I4 @ I6 )
% 16.19/9.04 => ( ord_less_eq_real @ one_one_real @ ( F @ I4 ) ) )
% 16.19/9.04 => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I6 ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8773_less__1__prod2,axiom,
% 16.19/9.04 ! [I6: set_real,I2: real,F: real > rat] :
% 16.19/9.04 ( ( finite_finite_real @ I6 )
% 16.19/9.04 => ( ( member_real @ I2 @ I6 )
% 16.19/9.04 => ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
% 16.19/9.04 => ( ! [I4: real] :
% 16.19/9.04 ( ( member_real @ I4 @ I6 )
% 16.19/9.04 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I4 ) ) )
% 16.19/9.04 => ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I6 ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8774_less__1__prod2,axiom,
% 16.19/9.04 ! [I6: set_nat,I2: nat,F: nat > rat] :
% 16.19/9.04 ( ( finite_finite_nat @ I6 )
% 16.19/9.04 => ( ( member_nat @ I2 @ I6 )
% 16.19/9.04 => ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
% 16.19/9.04 => ( ! [I4: nat] :
% 16.19/9.04 ( ( member_nat @ I4 @ I6 )
% 16.19/9.04 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I4 ) ) )
% 16.19/9.04 => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I6 ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8775_less__1__prod2,axiom,
% 16.19/9.04 ! [I6: set_int,I2: int,F: int > rat] :
% 16.19/9.04 ( ( finite_finite_int @ I6 )
% 16.19/9.04 => ( ( member_int @ I2 @ I6 )
% 16.19/9.04 => ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
% 16.19/9.04 => ( ! [I4: int] :
% 16.19/9.04 ( ( member_int @ I4 @ I6 )
% 16.19/9.04 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I4 ) ) )
% 16.19/9.04 => ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I6 ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8776_less__1__prod2,axiom,
% 16.19/9.04 ! [I6: set_complex,I2: complex,F: complex > rat] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ I6 )
% 16.19/9.04 => ( ( member_complex @ I2 @ I6 )
% 16.19/9.04 => ( ( ord_less_rat @ one_one_rat @ ( F @ I2 ) )
% 16.19/9.04 => ( ! [I4: complex] :
% 16.19/9.04 ( ( member_complex @ I4 @ I6 )
% 16.19/9.04 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I4 ) ) )
% 16.19/9.04 => ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I6 ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8777_less__1__prod2,axiom,
% 16.19/9.04 ! [I6: set_real,I2: real,F: real > int] :
% 16.19/9.04 ( ( finite_finite_real @ I6 )
% 16.19/9.04 => ( ( member_real @ I2 @ I6 )
% 16.19/9.04 => ( ( ord_less_int @ one_one_int @ ( F @ I2 ) )
% 16.19/9.04 => ( ! [I4: real] :
% 16.19/9.04 ( ( member_real @ I4 @ I6 )
% 16.19/9.04 => ( ord_less_eq_int @ one_one_int @ ( F @ I4 ) ) )
% 16.19/9.04 => ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I6 ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8778_less__1__prod2,axiom,
% 16.19/9.04 ! [I6: set_complex,I2: complex,F: complex > int] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ I6 )
% 16.19/9.04 => ( ( member_complex @ I2 @ I6 )
% 16.19/9.04 => ( ( ord_less_int @ one_one_int @ ( F @ I2 ) )
% 16.19/9.04 => ( ! [I4: complex] :
% 16.19/9.04 ( ( member_complex @ I4 @ I6 )
% 16.19/9.04 => ( ord_less_eq_int @ one_one_int @ ( F @ I4 ) ) )
% 16.19/9.04 => ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I6 ) ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8779_arccos__le__arccos,axiom,
% 16.19/9.04 ! [X: real,Y: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 16.19/9.04 => ( ( ord_less_eq_real @ X @ Y )
% 16.19/9.04 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 16.19/9.04 => ( ord_less_eq_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8780_less__1__prod,axiom,
% 16.19/9.04 ! [I6: set_complex,F: complex > real] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ I6 )
% 16.19/9.04 => ( ( I6 != bot_bot_set_complex )
% 16.19/9.04 => ( ! [I4: complex] :
% 16.19/9.04 ( ( member_complex @ I4 @ I6 )
% 16.19/9.04 => ( ord_less_real @ one_one_real @ ( F @ I4 ) ) )
% 16.19/9.04 => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I6 ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8781_less__1__prod,axiom,
% 16.19/9.04 ! [I6: set_nat,F: nat > real] :
% 16.19/9.04 ( ( finite_finite_nat @ I6 )
% 16.19/9.04 => ( ( I6 != bot_bot_set_nat )
% 16.19/9.04 => ( ! [I4: nat] :
% 16.19/9.04 ( ( member_nat @ I4 @ I6 )
% 16.19/9.04 => ( ord_less_real @ one_one_real @ ( F @ I4 ) ) )
% 16.19/9.04 => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I6 ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8782_less__1__prod,axiom,
% 16.19/9.04 ! [I6: set_int,F: int > real] :
% 16.19/9.04 ( ( finite_finite_int @ I6 )
% 16.19/9.04 => ( ( I6 != bot_bot_set_int )
% 16.19/9.04 => ( ! [I4: int] :
% 16.19/9.04 ( ( member_int @ I4 @ I6 )
% 16.19/9.04 => ( ord_less_real @ one_one_real @ ( F @ I4 ) ) )
% 16.19/9.04 => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I6 ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8783_less__1__prod,axiom,
% 16.19/9.04 ! [I6: set_real,F: real > real] :
% 16.19/9.04 ( ( finite_finite_real @ I6 )
% 16.19/9.04 => ( ( I6 != bot_bot_set_real )
% 16.19/9.04 => ( ! [I4: real] :
% 16.19/9.04 ( ( member_real @ I4 @ I6 )
% 16.19/9.04 => ( ord_less_real @ one_one_real @ ( F @ I4 ) ) )
% 16.19/9.04 => ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I6 ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8784_less__1__prod,axiom,
% 16.19/9.04 ! [I6: set_complex,F: complex > rat] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ I6 )
% 16.19/9.04 => ( ( I6 != bot_bot_set_complex )
% 16.19/9.04 => ( ! [I4: complex] :
% 16.19/9.04 ( ( member_complex @ I4 @ I6 )
% 16.19/9.04 => ( ord_less_rat @ one_one_rat @ ( F @ I4 ) ) )
% 16.19/9.04 => ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I6 ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8785_less__1__prod,axiom,
% 16.19/9.04 ! [I6: set_nat,F: nat > rat] :
% 16.19/9.04 ( ( finite_finite_nat @ I6 )
% 16.19/9.04 => ( ( I6 != bot_bot_set_nat )
% 16.19/9.04 => ( ! [I4: nat] :
% 16.19/9.04 ( ( member_nat @ I4 @ I6 )
% 16.19/9.04 => ( ord_less_rat @ one_one_rat @ ( F @ I4 ) ) )
% 16.19/9.04 => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I6 ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8786_less__1__prod,axiom,
% 16.19/9.04 ! [I6: set_int,F: int > rat] :
% 16.19/9.04 ( ( finite_finite_int @ I6 )
% 16.19/9.04 => ( ( I6 != bot_bot_set_int )
% 16.19/9.04 => ( ! [I4: int] :
% 16.19/9.04 ( ( member_int @ I4 @ I6 )
% 16.19/9.04 => ( ord_less_rat @ one_one_rat @ ( F @ I4 ) ) )
% 16.19/9.04 => ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I6 ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8787_less__1__prod,axiom,
% 16.19/9.04 ! [I6: set_real,F: real > rat] :
% 16.19/9.04 ( ( finite_finite_real @ I6 )
% 16.19/9.04 => ( ( I6 != bot_bot_set_real )
% 16.19/9.04 => ( ! [I4: real] :
% 16.19/9.04 ( ( member_real @ I4 @ I6 )
% 16.19/9.04 => ( ord_less_rat @ one_one_rat @ ( F @ I4 ) ) )
% 16.19/9.04 => ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I6 ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8788_less__1__prod,axiom,
% 16.19/9.04 ! [I6: set_complex,F: complex > int] :
% 16.19/9.04 ( ( finite3207457112153483333omplex @ I6 )
% 16.19/9.04 => ( ( I6 != bot_bot_set_complex )
% 16.19/9.04 => ( ! [I4: complex] :
% 16.19/9.04 ( ( member_complex @ I4 @ I6 )
% 16.19/9.04 => ( ord_less_int @ one_one_int @ ( F @ I4 ) ) )
% 16.19/9.04 => ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I6 ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8789_less__1__prod,axiom,
% 16.19/9.04 ! [I6: set_real,F: real > int] :
% 16.19/9.04 ( ( finite_finite_real @ I6 )
% 16.19/9.04 => ( ( I6 != bot_bot_set_real )
% 16.19/9.04 => ( ! [I4: real] :
% 16.19/9.04 ( ( member_real @ I4 @ I6 )
% 16.19/9.04 => ( ord_less_int @ one_one_int @ ( F @ I4 ) ) )
% 16.19/9.04 => ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I6 ) ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8790_arccos__eq__iff,axiom,
% 16.19/9.04 ! [X: real,Y: real] :
% 16.19/9.04 ( ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 16.19/9.04 & ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real ) )
% 16.19/9.04 => ( ( ( arccos @ X )
% 16.19/9.04 = ( arccos @ Y ) )
% 16.19/9.04 = ( X = Y ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8791_arccos__le__mono,axiom,
% 16.19/9.04 ! [X: real,Y: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 16.19/9.04 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 16.19/9.04 => ( ( ord_less_eq_real @ ( arccos @ X ) @ ( arccos @ Y ) )
% 16.19/9.04 = ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% 16.19/9.04
% 16.19/9.04 thf(fact_8792_arcsin__le__arcsin,axiom,
% 16.19/9.04 ! [X: real,Y: real] :
% 16.19/9.04 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 16.19/9.04 => ( ( ord_less_eq_real @ X @ Y )
% 16.19/9.04 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 16.19/9.05 => ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8793_arcsin__minus,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 16.19/9.05 => ( ( ord_less_eq_real @ X @ one_one_real )
% 16.19/9.05 => ( ( arcsin @ ( uminus_uminus_real @ X ) )
% 16.19/9.05 = ( uminus_uminus_real @ ( arcsin @ X ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8794_prod_Osubset__diff,axiom,
% 16.19/9.05 ! [B3: set_int,A2: set_int,G: int > real] :
% 16.19/9.05 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 16.19/9.05 => ( ( finite_finite_int @ A2 )
% 16.19/9.05 => ( ( groups2316167850115554303t_real @ G @ A2 )
% 16.19/9.05 = ( times_times_real @ ( groups2316167850115554303t_real @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups2316167850115554303t_real @ G @ B3 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8795_prod_Osubset__diff,axiom,
% 16.19/9.05 ! [B3: set_complex,A2: set_complex,G: complex > real] :
% 16.19/9.05 ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 16.19/9.05 => ( ( finite3207457112153483333omplex @ A2 )
% 16.19/9.05 => ( ( groups766887009212190081x_real @ G @ A2 )
% 16.19/9.05 = ( times_times_real @ ( groups766887009212190081x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups766887009212190081x_real @ G @ B3 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8796_prod_Osubset__diff,axiom,
% 16.19/9.05 ! [B3: set_int,A2: set_int,G: int > rat] :
% 16.19/9.05 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 16.19/9.05 => ( ( finite_finite_int @ A2 )
% 16.19/9.05 => ( ( groups1072433553688619179nt_rat @ G @ A2 )
% 16.19/9.05 = ( times_times_rat @ ( groups1072433553688619179nt_rat @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups1072433553688619179nt_rat @ G @ B3 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8797_prod_Osubset__diff,axiom,
% 16.19/9.05 ! [B3: set_complex,A2: set_complex,G: complex > rat] :
% 16.19/9.05 ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 16.19/9.05 => ( ( finite3207457112153483333omplex @ A2 )
% 16.19/9.05 => ( ( groups225925009352817453ex_rat @ G @ A2 )
% 16.19/9.05 = ( times_times_rat @ ( groups225925009352817453ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups225925009352817453ex_rat @ G @ B3 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8798_prod_Osubset__diff,axiom,
% 16.19/9.05 ! [B3: set_int,A2: set_int,G: int > nat] :
% 16.19/9.05 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 16.19/9.05 => ( ( finite_finite_int @ A2 )
% 16.19/9.05 => ( ( groups1707563613775114915nt_nat @ G @ A2 )
% 16.19/9.05 = ( times_times_nat @ ( groups1707563613775114915nt_nat @ G @ ( minus_minus_set_int @ A2 @ B3 ) ) @ ( groups1707563613775114915nt_nat @ G @ B3 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8799_prod_Osubset__diff,axiom,
% 16.19/9.05 ! [B3: set_complex,A2: set_complex,G: complex > nat] :
% 16.19/9.05 ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 16.19/9.05 => ( ( finite3207457112153483333omplex @ A2 )
% 16.19/9.05 => ( ( groups861055069439313189ex_nat @ G @ A2 )
% 16.19/9.05 = ( times_times_nat @ ( groups861055069439313189ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups861055069439313189ex_nat @ G @ B3 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8800_prod_Osubset__diff,axiom,
% 16.19/9.05 ! [B3: set_complex,A2: set_complex,G: complex > int] :
% 16.19/9.05 ( ( ord_le211207098394363844omplex @ B3 @ A2 )
% 16.19/9.05 => ( ( finite3207457112153483333omplex @ A2 )
% 16.19/9.05 => ( ( groups858564598930262913ex_int @ G @ A2 )
% 16.19/9.05 = ( times_times_int @ ( groups858564598930262913ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B3 ) ) @ ( groups858564598930262913ex_int @ G @ B3 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8801_prod_Osubset__diff,axiom,
% 16.19/9.05 ! [B3: set_nat,A2: set_nat,G: nat > real] :
% 16.19/9.05 ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 16.19/9.05 => ( ( finite_finite_nat @ A2 )
% 16.19/9.05 => ( ( groups129246275422532515t_real @ G @ A2 )
% 16.19/9.05 = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( minus_minus_set_nat @ A2 @ B3 ) ) @ ( groups129246275422532515t_real @ G @ B3 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8802_prod_Osubset__diff,axiom,
% 16.19/9.05 ! [B3: set_nat,A2: set_nat,G: nat > rat] :
% 16.19/9.05 ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 16.19/9.05 => ( ( finite_finite_nat @ A2 )
% 16.19/9.05 => ( ( groups73079841787564623at_rat @ G @ A2 )
% 16.19/9.05 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A2 @ B3 ) ) @ ( groups73079841787564623at_rat @ G @ B3 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8803_prod_Osubset__diff,axiom,
% 16.19/9.05 ! [B3: set_nat,A2: set_nat,G: nat > nat] :
% 16.19/9.05 ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 16.19/9.05 => ( ( finite_finite_nat @ A2 )
% 16.19/9.05 => ( ( groups708209901874060359at_nat @ G @ A2 )
% 16.19/9.05 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ A2 @ B3 ) ) @ ( groups708209901874060359at_nat @ G @ B3 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8804_prod_Omono__neutral__cong__right,axiom,
% 16.19/9.05 ! [T3: set_real,S: set_real,G: real > complex,H2: real > complex] :
% 16.19/9.05 ( ( finite_finite_real @ T3 )
% 16.19/9.05 => ( ( ord_less_eq_set_real @ S @ T3 )
% 16.19/9.05 => ( ! [X4: real] :
% 16.19/9.05 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_complex ) )
% 16.19/9.05 => ( ! [X4: real] :
% 16.19/9.05 ( ( member_real @ X4 @ S )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = ( H2 @ X4 ) ) )
% 16.19/9.05 => ( ( groups713298508707869441omplex @ G @ T3 )
% 16.19/9.05 = ( groups713298508707869441omplex @ H2 @ S ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8805_prod_Omono__neutral__cong__right,axiom,
% 16.19/9.05 ! [T3: set_int,S: set_int,G: int > complex,H2: int > complex] :
% 16.19/9.05 ( ( finite_finite_int @ T3 )
% 16.19/9.05 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/9.05 => ( ! [X4: int] :
% 16.19/9.05 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_complex ) )
% 16.19/9.05 => ( ! [X4: int] :
% 16.19/9.05 ( ( member_int @ X4 @ S )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = ( H2 @ X4 ) ) )
% 16.19/9.05 => ( ( groups7440179247065528705omplex @ G @ T3 )
% 16.19/9.05 = ( groups7440179247065528705omplex @ H2 @ S ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8806_prod_Omono__neutral__cong__right,axiom,
% 16.19/9.05 ! [T3: set_complex,S: set_complex,G: complex > complex,H2: complex > complex] :
% 16.19/9.05 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/9.05 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/9.05 => ( ! [X4: complex] :
% 16.19/9.05 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_complex ) )
% 16.19/9.05 => ( ! [X4: complex] :
% 16.19/9.05 ( ( member_complex @ X4 @ S )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = ( H2 @ X4 ) ) )
% 16.19/9.05 => ( ( groups3708469109370488835omplex @ G @ T3 )
% 16.19/9.05 = ( groups3708469109370488835omplex @ H2 @ S ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8807_prod_Omono__neutral__cong__right,axiom,
% 16.19/9.05 ! [T3: set_real,S: set_real,G: real > real,H2: real > real] :
% 16.19/9.05 ( ( finite_finite_real @ T3 )
% 16.19/9.05 => ( ( ord_less_eq_set_real @ S @ T3 )
% 16.19/9.05 => ( ! [X4: real] :
% 16.19/9.05 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_real ) )
% 16.19/9.05 => ( ! [X4: real] :
% 16.19/9.05 ( ( member_real @ X4 @ S )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = ( H2 @ X4 ) ) )
% 16.19/9.05 => ( ( groups1681761925125756287l_real @ G @ T3 )
% 16.19/9.05 = ( groups1681761925125756287l_real @ H2 @ S ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8808_prod_Omono__neutral__cong__right,axiom,
% 16.19/9.05 ! [T3: set_int,S: set_int,G: int > real,H2: int > real] :
% 16.19/9.05 ( ( finite_finite_int @ T3 )
% 16.19/9.05 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/9.05 => ( ! [X4: int] :
% 16.19/9.05 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_real ) )
% 16.19/9.05 => ( ! [X4: int] :
% 16.19/9.05 ( ( member_int @ X4 @ S )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = ( H2 @ X4 ) ) )
% 16.19/9.05 => ( ( groups2316167850115554303t_real @ G @ T3 )
% 16.19/9.05 = ( groups2316167850115554303t_real @ H2 @ S ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8809_prod_Omono__neutral__cong__right,axiom,
% 16.19/9.05 ! [T3: set_complex,S: set_complex,G: complex > real,H2: complex > real] :
% 16.19/9.05 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/9.05 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/9.05 => ( ! [X4: complex] :
% 16.19/9.05 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_real ) )
% 16.19/9.05 => ( ! [X4: complex] :
% 16.19/9.05 ( ( member_complex @ X4 @ S )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = ( H2 @ X4 ) ) )
% 16.19/9.05 => ( ( groups766887009212190081x_real @ G @ T3 )
% 16.19/9.05 = ( groups766887009212190081x_real @ H2 @ S ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8810_prod_Omono__neutral__cong__right,axiom,
% 16.19/9.05 ! [T3: set_real,S: set_real,G: real > rat,H2: real > rat] :
% 16.19/9.05 ( ( finite_finite_real @ T3 )
% 16.19/9.05 => ( ( ord_less_eq_set_real @ S @ T3 )
% 16.19/9.05 => ( ! [X4: real] :
% 16.19/9.05 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_rat ) )
% 16.19/9.05 => ( ! [X4: real] :
% 16.19/9.05 ( ( member_real @ X4 @ S )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = ( H2 @ X4 ) ) )
% 16.19/9.05 => ( ( groups4061424788464935467al_rat @ G @ T3 )
% 16.19/9.05 = ( groups4061424788464935467al_rat @ H2 @ S ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8811_prod_Omono__neutral__cong__right,axiom,
% 16.19/9.05 ! [T3: set_int,S: set_int,G: int > rat,H2: int > rat] :
% 16.19/9.05 ( ( finite_finite_int @ T3 )
% 16.19/9.05 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/9.05 => ( ! [X4: int] :
% 16.19/9.05 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_rat ) )
% 16.19/9.05 => ( ! [X4: int] :
% 16.19/9.05 ( ( member_int @ X4 @ S )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = ( H2 @ X4 ) ) )
% 16.19/9.05 => ( ( groups1072433553688619179nt_rat @ G @ T3 )
% 16.19/9.05 = ( groups1072433553688619179nt_rat @ H2 @ S ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8812_prod_Omono__neutral__cong__right,axiom,
% 16.19/9.05 ! [T3: set_complex,S: set_complex,G: complex > rat,H2: complex > rat] :
% 16.19/9.05 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/9.05 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/9.05 => ( ! [X4: complex] :
% 16.19/9.05 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_rat ) )
% 16.19/9.05 => ( ! [X4: complex] :
% 16.19/9.05 ( ( member_complex @ X4 @ S )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = ( H2 @ X4 ) ) )
% 16.19/9.05 => ( ( groups225925009352817453ex_rat @ G @ T3 )
% 16.19/9.05 = ( groups225925009352817453ex_rat @ H2 @ S ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8813_prod_Omono__neutral__cong__right,axiom,
% 16.19/9.05 ! [T3: set_real,S: set_real,G: real > nat,H2: real > nat] :
% 16.19/9.05 ( ( finite_finite_real @ T3 )
% 16.19/9.05 => ( ( ord_less_eq_set_real @ S @ T3 )
% 16.19/9.05 => ( ! [X4: real] :
% 16.19/9.05 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_nat ) )
% 16.19/9.05 => ( ! [X4: real] :
% 16.19/9.05 ( ( member_real @ X4 @ S )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = ( H2 @ X4 ) ) )
% 16.19/9.05 => ( ( groups4696554848551431203al_nat @ G @ T3 )
% 16.19/9.05 = ( groups4696554848551431203al_nat @ H2 @ S ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8814_prod_Omono__neutral__cong__left,axiom,
% 16.19/9.05 ! [T3: set_real,S: set_real,H2: real > complex,G: real > complex] :
% 16.19/9.05 ( ( finite_finite_real @ T3 )
% 16.19/9.05 => ( ( ord_less_eq_set_real @ S @ T3 )
% 16.19/9.05 => ( ! [X4: real] :
% 16.19/9.05 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 16.19/9.05 => ( ( H2 @ X4 )
% 16.19/9.05 = one_one_complex ) )
% 16.19/9.05 => ( ! [X4: real] :
% 16.19/9.05 ( ( member_real @ X4 @ S )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = ( H2 @ X4 ) ) )
% 16.19/9.05 => ( ( groups713298508707869441omplex @ G @ S )
% 16.19/9.05 = ( groups713298508707869441omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8815_prod_Omono__neutral__cong__left,axiom,
% 16.19/9.05 ! [T3: set_int,S: set_int,H2: int > complex,G: int > complex] :
% 16.19/9.05 ( ( finite_finite_int @ T3 )
% 16.19/9.05 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/9.05 => ( ! [X4: int] :
% 16.19/9.05 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/9.05 => ( ( H2 @ X4 )
% 16.19/9.05 = one_one_complex ) )
% 16.19/9.05 => ( ! [X4: int] :
% 16.19/9.05 ( ( member_int @ X4 @ S )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = ( H2 @ X4 ) ) )
% 16.19/9.05 => ( ( groups7440179247065528705omplex @ G @ S )
% 16.19/9.05 = ( groups7440179247065528705omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8816_prod_Omono__neutral__cong__left,axiom,
% 16.19/9.05 ! [T3: set_complex,S: set_complex,H2: complex > complex,G: complex > complex] :
% 16.19/9.05 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/9.05 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/9.05 => ( ! [X4: complex] :
% 16.19/9.05 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 16.19/9.05 => ( ( H2 @ X4 )
% 16.19/9.05 = one_one_complex ) )
% 16.19/9.05 => ( ! [X4: complex] :
% 16.19/9.05 ( ( member_complex @ X4 @ S )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = ( H2 @ X4 ) ) )
% 16.19/9.05 => ( ( groups3708469109370488835omplex @ G @ S )
% 16.19/9.05 = ( groups3708469109370488835omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8817_prod_Omono__neutral__cong__left,axiom,
% 16.19/9.05 ! [T3: set_real,S: set_real,H2: real > real,G: real > real] :
% 16.19/9.05 ( ( finite_finite_real @ T3 )
% 16.19/9.05 => ( ( ord_less_eq_set_real @ S @ T3 )
% 16.19/9.05 => ( ! [X4: real] :
% 16.19/9.05 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 16.19/9.05 => ( ( H2 @ X4 )
% 16.19/9.05 = one_one_real ) )
% 16.19/9.05 => ( ! [X4: real] :
% 16.19/9.05 ( ( member_real @ X4 @ S )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = ( H2 @ X4 ) ) )
% 16.19/9.05 => ( ( groups1681761925125756287l_real @ G @ S )
% 16.19/9.05 = ( groups1681761925125756287l_real @ H2 @ T3 ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8818_prod_Omono__neutral__cong__left,axiom,
% 16.19/9.05 ! [T3: set_int,S: set_int,H2: int > real,G: int > real] :
% 16.19/9.05 ( ( finite_finite_int @ T3 )
% 16.19/9.05 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/9.05 => ( ! [X4: int] :
% 16.19/9.05 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/9.05 => ( ( H2 @ X4 )
% 16.19/9.05 = one_one_real ) )
% 16.19/9.05 => ( ! [X4: int] :
% 16.19/9.05 ( ( member_int @ X4 @ S )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = ( H2 @ X4 ) ) )
% 16.19/9.05 => ( ( groups2316167850115554303t_real @ G @ S )
% 16.19/9.05 = ( groups2316167850115554303t_real @ H2 @ T3 ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8819_prod_Omono__neutral__cong__left,axiom,
% 16.19/9.05 ! [T3: set_complex,S: set_complex,H2: complex > real,G: complex > real] :
% 16.19/9.05 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/9.05 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/9.05 => ( ! [X4: complex] :
% 16.19/9.05 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 16.19/9.05 => ( ( H2 @ X4 )
% 16.19/9.05 = one_one_real ) )
% 16.19/9.05 => ( ! [X4: complex] :
% 16.19/9.05 ( ( member_complex @ X4 @ S )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = ( H2 @ X4 ) ) )
% 16.19/9.05 => ( ( groups766887009212190081x_real @ G @ S )
% 16.19/9.05 = ( groups766887009212190081x_real @ H2 @ T3 ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8820_prod_Omono__neutral__cong__left,axiom,
% 16.19/9.05 ! [T3: set_real,S: set_real,H2: real > rat,G: real > rat] :
% 16.19/9.05 ( ( finite_finite_real @ T3 )
% 16.19/9.05 => ( ( ord_less_eq_set_real @ S @ T3 )
% 16.19/9.05 => ( ! [X4: real] :
% 16.19/9.05 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 16.19/9.05 => ( ( H2 @ X4 )
% 16.19/9.05 = one_one_rat ) )
% 16.19/9.05 => ( ! [X4: real] :
% 16.19/9.05 ( ( member_real @ X4 @ S )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = ( H2 @ X4 ) ) )
% 16.19/9.05 => ( ( groups4061424788464935467al_rat @ G @ S )
% 16.19/9.05 = ( groups4061424788464935467al_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8821_prod_Omono__neutral__cong__left,axiom,
% 16.19/9.05 ! [T3: set_int,S: set_int,H2: int > rat,G: int > rat] :
% 16.19/9.05 ( ( finite_finite_int @ T3 )
% 16.19/9.05 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/9.05 => ( ! [X4: int] :
% 16.19/9.05 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/9.05 => ( ( H2 @ X4 )
% 16.19/9.05 = one_one_rat ) )
% 16.19/9.05 => ( ! [X4: int] :
% 16.19/9.05 ( ( member_int @ X4 @ S )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = ( H2 @ X4 ) ) )
% 16.19/9.05 => ( ( groups1072433553688619179nt_rat @ G @ S )
% 16.19/9.05 = ( groups1072433553688619179nt_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8822_prod_Omono__neutral__cong__left,axiom,
% 16.19/9.05 ! [T3: set_complex,S: set_complex,H2: complex > rat,G: complex > rat] :
% 16.19/9.05 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/9.05 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/9.05 => ( ! [X4: complex] :
% 16.19/9.05 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 16.19/9.05 => ( ( H2 @ X4 )
% 16.19/9.05 = one_one_rat ) )
% 16.19/9.05 => ( ! [X4: complex] :
% 16.19/9.05 ( ( member_complex @ X4 @ S )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = ( H2 @ X4 ) ) )
% 16.19/9.05 => ( ( groups225925009352817453ex_rat @ G @ S )
% 16.19/9.05 = ( groups225925009352817453ex_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8823_prod_Omono__neutral__cong__left,axiom,
% 16.19/9.05 ! [T3: set_real,S: set_real,H2: real > nat,G: real > nat] :
% 16.19/9.05 ( ( finite_finite_real @ T3 )
% 16.19/9.05 => ( ( ord_less_eq_set_real @ S @ T3 )
% 16.19/9.05 => ( ! [X4: real] :
% 16.19/9.05 ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S ) )
% 16.19/9.05 => ( ( H2 @ X4 )
% 16.19/9.05 = one_one_nat ) )
% 16.19/9.05 => ( ! [X4: real] :
% 16.19/9.05 ( ( member_real @ X4 @ S )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = ( H2 @ X4 ) ) )
% 16.19/9.05 => ( ( groups4696554848551431203al_nat @ G @ S )
% 16.19/9.05 = ( groups4696554848551431203al_nat @ H2 @ T3 ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8824_prod_Omono__neutral__right,axiom,
% 16.19/9.05 ! [T3: set_int,S: set_int,G: int > complex] :
% 16.19/9.05 ( ( finite_finite_int @ T3 )
% 16.19/9.05 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/9.05 => ( ! [X4: int] :
% 16.19/9.05 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_complex ) )
% 16.19/9.05 => ( ( groups7440179247065528705omplex @ G @ T3 )
% 16.19/9.05 = ( groups7440179247065528705omplex @ G @ S ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8825_prod_Omono__neutral__right,axiom,
% 16.19/9.05 ! [T3: set_complex,S: set_complex,G: complex > complex] :
% 16.19/9.05 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/9.05 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/9.05 => ( ! [X4: complex] :
% 16.19/9.05 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_complex ) )
% 16.19/9.05 => ( ( groups3708469109370488835omplex @ G @ T3 )
% 16.19/9.05 = ( groups3708469109370488835omplex @ G @ S ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8826_prod_Omono__neutral__right,axiom,
% 16.19/9.05 ! [T3: set_int,S: set_int,G: int > real] :
% 16.19/9.05 ( ( finite_finite_int @ T3 )
% 16.19/9.05 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/9.05 => ( ! [X4: int] :
% 16.19/9.05 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_real ) )
% 16.19/9.05 => ( ( groups2316167850115554303t_real @ G @ T3 )
% 16.19/9.05 = ( groups2316167850115554303t_real @ G @ S ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8827_prod_Omono__neutral__right,axiom,
% 16.19/9.05 ! [T3: set_complex,S: set_complex,G: complex > real] :
% 16.19/9.05 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/9.05 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/9.05 => ( ! [X4: complex] :
% 16.19/9.05 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_real ) )
% 16.19/9.05 => ( ( groups766887009212190081x_real @ G @ T3 )
% 16.19/9.05 = ( groups766887009212190081x_real @ G @ S ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8828_prod_Omono__neutral__right,axiom,
% 16.19/9.05 ! [T3: set_int,S: set_int,G: int > rat] :
% 16.19/9.05 ( ( finite_finite_int @ T3 )
% 16.19/9.05 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/9.05 => ( ! [X4: int] :
% 16.19/9.05 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_rat ) )
% 16.19/9.05 => ( ( groups1072433553688619179nt_rat @ G @ T3 )
% 16.19/9.05 = ( groups1072433553688619179nt_rat @ G @ S ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8829_prod_Omono__neutral__right,axiom,
% 16.19/9.05 ! [T3: set_complex,S: set_complex,G: complex > rat] :
% 16.19/9.05 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/9.05 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/9.05 => ( ! [X4: complex] :
% 16.19/9.05 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_rat ) )
% 16.19/9.05 => ( ( groups225925009352817453ex_rat @ G @ T3 )
% 16.19/9.05 = ( groups225925009352817453ex_rat @ G @ S ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8830_prod_Omono__neutral__right,axiom,
% 16.19/9.05 ! [T3: set_int,S: set_int,G: int > nat] :
% 16.19/9.05 ( ( finite_finite_int @ T3 )
% 16.19/9.05 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/9.05 => ( ! [X4: int] :
% 16.19/9.05 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_nat ) )
% 16.19/9.05 => ( ( groups1707563613775114915nt_nat @ G @ T3 )
% 16.19/9.05 = ( groups1707563613775114915nt_nat @ G @ S ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8831_prod_Omono__neutral__right,axiom,
% 16.19/9.05 ! [T3: set_complex,S: set_complex,G: complex > nat] :
% 16.19/9.05 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/9.05 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/9.05 => ( ! [X4: complex] :
% 16.19/9.05 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_nat ) )
% 16.19/9.05 => ( ( groups861055069439313189ex_nat @ G @ T3 )
% 16.19/9.05 = ( groups861055069439313189ex_nat @ G @ S ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8832_prod_Omono__neutral__right,axiom,
% 16.19/9.05 ! [T3: set_complex,S: set_complex,G: complex > int] :
% 16.19/9.05 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/9.05 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/9.05 => ( ! [X4: complex] :
% 16.19/9.05 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_int ) )
% 16.19/9.05 => ( ( groups858564598930262913ex_int @ G @ T3 )
% 16.19/9.05 = ( groups858564598930262913ex_int @ G @ S ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8833_prod_Omono__neutral__right,axiom,
% 16.19/9.05 ! [T3: set_nat,S: set_nat,G: nat > complex] :
% 16.19/9.05 ( ( finite_finite_nat @ T3 )
% 16.19/9.05 => ( ( ord_less_eq_set_nat @ S @ T3 )
% 16.19/9.05 => ( ! [X4: nat] :
% 16.19/9.05 ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_complex ) )
% 16.19/9.05 => ( ( groups6464643781859351333omplex @ G @ T3 )
% 16.19/9.05 = ( groups6464643781859351333omplex @ G @ S ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8834_prod_Omono__neutral__left,axiom,
% 16.19/9.05 ! [T3: set_int,S: set_int,G: int > complex] :
% 16.19/9.05 ( ( finite_finite_int @ T3 )
% 16.19/9.05 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/9.05 => ( ! [X4: int] :
% 16.19/9.05 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_complex ) )
% 16.19/9.05 => ( ( groups7440179247065528705omplex @ G @ S )
% 16.19/9.05 = ( groups7440179247065528705omplex @ G @ T3 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8835_prod_Omono__neutral__left,axiom,
% 16.19/9.05 ! [T3: set_complex,S: set_complex,G: complex > complex] :
% 16.19/9.05 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/9.05 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/9.05 => ( ! [X4: complex] :
% 16.19/9.05 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_complex ) )
% 16.19/9.05 => ( ( groups3708469109370488835omplex @ G @ S )
% 16.19/9.05 = ( groups3708469109370488835omplex @ G @ T3 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8836_prod_Omono__neutral__left,axiom,
% 16.19/9.05 ! [T3: set_int,S: set_int,G: int > real] :
% 16.19/9.05 ( ( finite_finite_int @ T3 )
% 16.19/9.05 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/9.05 => ( ! [X4: int] :
% 16.19/9.05 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_real ) )
% 16.19/9.05 => ( ( groups2316167850115554303t_real @ G @ S )
% 16.19/9.05 = ( groups2316167850115554303t_real @ G @ T3 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8837_prod_Omono__neutral__left,axiom,
% 16.19/9.05 ! [T3: set_complex,S: set_complex,G: complex > real] :
% 16.19/9.05 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/9.05 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/9.05 => ( ! [X4: complex] :
% 16.19/9.05 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_real ) )
% 16.19/9.05 => ( ( groups766887009212190081x_real @ G @ S )
% 16.19/9.05 = ( groups766887009212190081x_real @ G @ T3 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8838_prod_Omono__neutral__left,axiom,
% 16.19/9.05 ! [T3: set_int,S: set_int,G: int > rat] :
% 16.19/9.05 ( ( finite_finite_int @ T3 )
% 16.19/9.05 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/9.05 => ( ! [X4: int] :
% 16.19/9.05 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_rat ) )
% 16.19/9.05 => ( ( groups1072433553688619179nt_rat @ G @ S )
% 16.19/9.05 = ( groups1072433553688619179nt_rat @ G @ T3 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8839_prod_Omono__neutral__left,axiom,
% 16.19/9.05 ! [T3: set_complex,S: set_complex,G: complex > rat] :
% 16.19/9.05 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/9.05 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/9.05 => ( ! [X4: complex] :
% 16.19/9.05 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_rat ) )
% 16.19/9.05 => ( ( groups225925009352817453ex_rat @ G @ S )
% 16.19/9.05 = ( groups225925009352817453ex_rat @ G @ T3 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8840_prod_Omono__neutral__left,axiom,
% 16.19/9.05 ! [T3: set_int,S: set_int,G: int > nat] :
% 16.19/9.05 ( ( finite_finite_int @ T3 )
% 16.19/9.05 => ( ( ord_less_eq_set_int @ S @ T3 )
% 16.19/9.05 => ( ! [X4: int] :
% 16.19/9.05 ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_nat ) )
% 16.19/9.05 => ( ( groups1707563613775114915nt_nat @ G @ S )
% 16.19/9.05 = ( groups1707563613775114915nt_nat @ G @ T3 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8841_prod_Omono__neutral__left,axiom,
% 16.19/9.05 ! [T3: set_complex,S: set_complex,G: complex > nat] :
% 16.19/9.05 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/9.05 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/9.05 => ( ! [X4: complex] :
% 16.19/9.05 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_nat ) )
% 16.19/9.05 => ( ( groups861055069439313189ex_nat @ G @ S )
% 16.19/9.05 = ( groups861055069439313189ex_nat @ G @ T3 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8842_prod_Omono__neutral__left,axiom,
% 16.19/9.05 ! [T3: set_complex,S: set_complex,G: complex > int] :
% 16.19/9.05 ( ( finite3207457112153483333omplex @ T3 )
% 16.19/9.05 => ( ( ord_le211207098394363844omplex @ S @ T3 )
% 16.19/9.05 => ( ! [X4: complex] :
% 16.19/9.05 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_int ) )
% 16.19/9.05 => ( ( groups858564598930262913ex_int @ G @ S )
% 16.19/9.05 = ( groups858564598930262913ex_int @ G @ T3 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8843_prod_Omono__neutral__left,axiom,
% 16.19/9.05 ! [T3: set_nat,S: set_nat,G: nat > complex] :
% 16.19/9.05 ( ( finite_finite_nat @ T3 )
% 16.19/9.05 => ( ( ord_less_eq_set_nat @ S @ T3 )
% 16.19/9.05 => ( ! [X4: nat] :
% 16.19/9.05 ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S ) )
% 16.19/9.05 => ( ( G @ X4 )
% 16.19/9.05 = one_one_complex ) )
% 16.19/9.05 => ( ( groups6464643781859351333omplex @ G @ S )
% 16.19/9.05 = ( groups6464643781859351333omplex @ G @ T3 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8844_prod_Osame__carrierI,axiom,
% 16.19/9.05 ! [C2: set_real,A2: set_real,B3: set_real,G: real > complex,H2: real > complex] :
% 16.19/9.05 ( ( finite_finite_real @ C2 )
% 16.19/9.05 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 16.19/9.05 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 16.19/9.05 => ( ! [A3: real] :
% 16.19/9.05 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 16.19/9.05 => ( ( G @ A3 )
% 16.19/9.05 = one_one_complex ) )
% 16.19/9.05 => ( ! [B2: real] :
% 16.19/9.05 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 16.19/9.05 => ( ( H2 @ B2 )
% 16.19/9.05 = one_one_complex ) )
% 16.19/9.05 => ( ( ( groups713298508707869441omplex @ G @ C2 )
% 16.19/9.05 = ( groups713298508707869441omplex @ H2 @ C2 ) )
% 16.19/9.05 => ( ( groups713298508707869441omplex @ G @ A2 )
% 16.19/9.05 = ( groups713298508707869441omplex @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8845_prod_Osame__carrierI,axiom,
% 16.19/9.05 ! [C2: set_int,A2: set_int,B3: set_int,G: int > complex,H2: int > complex] :
% 16.19/9.05 ( ( finite_finite_int @ C2 )
% 16.19/9.05 => ( ( ord_less_eq_set_int @ A2 @ C2 )
% 16.19/9.05 => ( ( ord_less_eq_set_int @ B3 @ C2 )
% 16.19/9.05 => ( ! [A3: int] :
% 16.19/9.05 ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 16.19/9.05 => ( ( G @ A3 )
% 16.19/9.05 = one_one_complex ) )
% 16.19/9.05 => ( ! [B2: int] :
% 16.19/9.05 ( ( member_int @ B2 @ ( minus_minus_set_int @ C2 @ B3 ) )
% 16.19/9.05 => ( ( H2 @ B2 )
% 16.19/9.05 = one_one_complex ) )
% 16.19/9.05 => ( ( ( groups7440179247065528705omplex @ G @ C2 )
% 16.19/9.05 = ( groups7440179247065528705omplex @ H2 @ C2 ) )
% 16.19/9.05 => ( ( groups7440179247065528705omplex @ G @ A2 )
% 16.19/9.05 = ( groups7440179247065528705omplex @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8846_prod_Osame__carrierI,axiom,
% 16.19/9.05 ! [C2: set_complex,A2: set_complex,B3: set_complex,G: complex > complex,H2: complex > complex] :
% 16.19/9.05 ( ( finite3207457112153483333omplex @ C2 )
% 16.19/9.05 => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 16.19/9.05 => ( ( ord_le211207098394363844omplex @ B3 @ C2 )
% 16.19/9.05 => ( ! [A3: complex] :
% 16.19/9.05 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 16.19/9.05 => ( ( G @ A3 )
% 16.19/9.05 = one_one_complex ) )
% 16.19/9.05 => ( ! [B2: complex] :
% 16.19/9.05 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B3 ) )
% 16.19/9.05 => ( ( H2 @ B2 )
% 16.19/9.05 = one_one_complex ) )
% 16.19/9.05 => ( ( ( groups3708469109370488835omplex @ G @ C2 )
% 16.19/9.05 = ( groups3708469109370488835omplex @ H2 @ C2 ) )
% 16.19/9.05 => ( ( groups3708469109370488835omplex @ G @ A2 )
% 16.19/9.05 = ( groups3708469109370488835omplex @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8847_prod_Osame__carrierI,axiom,
% 16.19/9.05 ! [C2: set_real,A2: set_real,B3: set_real,G: real > real,H2: real > real] :
% 16.19/9.05 ( ( finite_finite_real @ C2 )
% 16.19/9.05 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 16.19/9.05 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 16.19/9.05 => ( ! [A3: real] :
% 16.19/9.05 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 16.19/9.05 => ( ( G @ A3 )
% 16.19/9.05 = one_one_real ) )
% 16.19/9.05 => ( ! [B2: real] :
% 16.19/9.05 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 16.19/9.05 => ( ( H2 @ B2 )
% 16.19/9.05 = one_one_real ) )
% 16.19/9.05 => ( ( ( groups1681761925125756287l_real @ G @ C2 )
% 16.19/9.05 = ( groups1681761925125756287l_real @ H2 @ C2 ) )
% 16.19/9.05 => ( ( groups1681761925125756287l_real @ G @ A2 )
% 16.19/9.05 = ( groups1681761925125756287l_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8848_prod_Osame__carrierI,axiom,
% 16.19/9.05 ! [C2: set_int,A2: set_int,B3: set_int,G: int > real,H2: int > real] :
% 16.19/9.05 ( ( finite_finite_int @ C2 )
% 16.19/9.05 => ( ( ord_less_eq_set_int @ A2 @ C2 )
% 16.19/9.05 => ( ( ord_less_eq_set_int @ B3 @ C2 )
% 16.19/9.05 => ( ! [A3: int] :
% 16.19/9.05 ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 16.19/9.05 => ( ( G @ A3 )
% 16.19/9.05 = one_one_real ) )
% 16.19/9.05 => ( ! [B2: int] :
% 16.19/9.05 ( ( member_int @ B2 @ ( minus_minus_set_int @ C2 @ B3 ) )
% 16.19/9.05 => ( ( H2 @ B2 )
% 16.19/9.05 = one_one_real ) )
% 16.19/9.05 => ( ( ( groups2316167850115554303t_real @ G @ C2 )
% 16.19/9.05 = ( groups2316167850115554303t_real @ H2 @ C2 ) )
% 16.19/9.05 => ( ( groups2316167850115554303t_real @ G @ A2 )
% 16.19/9.05 = ( groups2316167850115554303t_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8849_prod_Osame__carrierI,axiom,
% 16.19/9.05 ! [C2: set_complex,A2: set_complex,B3: set_complex,G: complex > real,H2: complex > real] :
% 16.19/9.05 ( ( finite3207457112153483333omplex @ C2 )
% 16.19/9.05 => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 16.19/9.05 => ( ( ord_le211207098394363844omplex @ B3 @ C2 )
% 16.19/9.05 => ( ! [A3: complex] :
% 16.19/9.05 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 16.19/9.05 => ( ( G @ A3 )
% 16.19/9.05 = one_one_real ) )
% 16.19/9.05 => ( ! [B2: complex] :
% 16.19/9.05 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B3 ) )
% 16.19/9.05 => ( ( H2 @ B2 )
% 16.19/9.05 = one_one_real ) )
% 16.19/9.05 => ( ( ( groups766887009212190081x_real @ G @ C2 )
% 16.19/9.05 = ( groups766887009212190081x_real @ H2 @ C2 ) )
% 16.19/9.05 => ( ( groups766887009212190081x_real @ G @ A2 )
% 16.19/9.05 = ( groups766887009212190081x_real @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8850_prod_Osame__carrierI,axiom,
% 16.19/9.05 ! [C2: set_real,A2: set_real,B3: set_real,G: real > rat,H2: real > rat] :
% 16.19/9.05 ( ( finite_finite_real @ C2 )
% 16.19/9.05 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 16.19/9.05 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 16.19/9.05 => ( ! [A3: real] :
% 16.19/9.05 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 16.19/9.05 => ( ( G @ A3 )
% 16.19/9.05 = one_one_rat ) )
% 16.19/9.05 => ( ! [B2: real] :
% 16.19/9.05 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 16.19/9.05 => ( ( H2 @ B2 )
% 16.19/9.05 = one_one_rat ) )
% 16.19/9.05 => ( ( ( groups4061424788464935467al_rat @ G @ C2 )
% 16.19/9.05 = ( groups4061424788464935467al_rat @ H2 @ C2 ) )
% 16.19/9.05 => ( ( groups4061424788464935467al_rat @ G @ A2 )
% 16.19/9.05 = ( groups4061424788464935467al_rat @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8851_prod_Osame__carrierI,axiom,
% 16.19/9.05 ! [C2: set_int,A2: set_int,B3: set_int,G: int > rat,H2: int > rat] :
% 16.19/9.05 ( ( finite_finite_int @ C2 )
% 16.19/9.05 => ( ( ord_less_eq_set_int @ A2 @ C2 )
% 16.19/9.05 => ( ( ord_less_eq_set_int @ B3 @ C2 )
% 16.19/9.05 => ( ! [A3: int] :
% 16.19/9.05 ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 16.19/9.05 => ( ( G @ A3 )
% 16.19/9.05 = one_one_rat ) )
% 16.19/9.05 => ( ! [B2: int] :
% 16.19/9.05 ( ( member_int @ B2 @ ( minus_minus_set_int @ C2 @ B3 ) )
% 16.19/9.05 => ( ( H2 @ B2 )
% 16.19/9.05 = one_one_rat ) )
% 16.19/9.05 => ( ( ( groups1072433553688619179nt_rat @ G @ C2 )
% 16.19/9.05 = ( groups1072433553688619179nt_rat @ H2 @ C2 ) )
% 16.19/9.05 => ( ( groups1072433553688619179nt_rat @ G @ A2 )
% 16.19/9.05 = ( groups1072433553688619179nt_rat @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8852_prod_Osame__carrierI,axiom,
% 16.19/9.05 ! [C2: set_complex,A2: set_complex,B3: set_complex,G: complex > rat,H2: complex > rat] :
% 16.19/9.05 ( ( finite3207457112153483333omplex @ C2 )
% 16.19/9.05 => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 16.19/9.05 => ( ( ord_le211207098394363844omplex @ B3 @ C2 )
% 16.19/9.05 => ( ! [A3: complex] :
% 16.19/9.05 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 16.19/9.05 => ( ( G @ A3 )
% 16.19/9.05 = one_one_rat ) )
% 16.19/9.05 => ( ! [B2: complex] :
% 16.19/9.05 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B3 ) )
% 16.19/9.05 => ( ( H2 @ B2 )
% 16.19/9.05 = one_one_rat ) )
% 16.19/9.05 => ( ( ( groups225925009352817453ex_rat @ G @ C2 )
% 16.19/9.05 = ( groups225925009352817453ex_rat @ H2 @ C2 ) )
% 16.19/9.05 => ( ( groups225925009352817453ex_rat @ G @ A2 )
% 16.19/9.05 = ( groups225925009352817453ex_rat @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8853_prod_Osame__carrierI,axiom,
% 16.19/9.05 ! [C2: set_real,A2: set_real,B3: set_real,G: real > nat,H2: real > nat] :
% 16.19/9.05 ( ( finite_finite_real @ C2 )
% 16.19/9.05 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 16.19/9.05 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 16.19/9.05 => ( ! [A3: real] :
% 16.19/9.05 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 16.19/9.05 => ( ( G @ A3 )
% 16.19/9.05 = one_one_nat ) )
% 16.19/9.05 => ( ! [B2: real] :
% 16.19/9.05 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 16.19/9.05 => ( ( H2 @ B2 )
% 16.19/9.05 = one_one_nat ) )
% 16.19/9.05 => ( ( ( groups4696554848551431203al_nat @ G @ C2 )
% 16.19/9.05 = ( groups4696554848551431203al_nat @ H2 @ C2 ) )
% 16.19/9.05 => ( ( groups4696554848551431203al_nat @ G @ A2 )
% 16.19/9.05 = ( groups4696554848551431203al_nat @ H2 @ B3 ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8854_prod_Osame__carrier,axiom,
% 16.19/9.05 ! [C2: set_real,A2: set_real,B3: set_real,G: real > complex,H2: real > complex] :
% 16.19/9.05 ( ( finite_finite_real @ C2 )
% 16.19/9.05 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 16.19/9.05 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 16.19/9.05 => ( ! [A3: real] :
% 16.19/9.05 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 16.19/9.05 => ( ( G @ A3 )
% 16.19/9.05 = one_one_complex ) )
% 16.19/9.05 => ( ! [B2: real] :
% 16.19/9.05 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 16.19/9.05 => ( ( H2 @ B2 )
% 16.19/9.05 = one_one_complex ) )
% 16.19/9.05 => ( ( ( groups713298508707869441omplex @ G @ A2 )
% 16.19/9.05 = ( groups713298508707869441omplex @ H2 @ B3 ) )
% 16.19/9.05 = ( ( groups713298508707869441omplex @ G @ C2 )
% 16.19/9.05 = ( groups713298508707869441omplex @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8855_prod_Osame__carrier,axiom,
% 16.19/9.05 ! [C2: set_int,A2: set_int,B3: set_int,G: int > complex,H2: int > complex] :
% 16.19/9.05 ( ( finite_finite_int @ C2 )
% 16.19/9.05 => ( ( ord_less_eq_set_int @ A2 @ C2 )
% 16.19/9.05 => ( ( ord_less_eq_set_int @ B3 @ C2 )
% 16.19/9.05 => ( ! [A3: int] :
% 16.19/9.05 ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 16.19/9.05 => ( ( G @ A3 )
% 16.19/9.05 = one_one_complex ) )
% 16.19/9.05 => ( ! [B2: int] :
% 16.19/9.05 ( ( member_int @ B2 @ ( minus_minus_set_int @ C2 @ B3 ) )
% 16.19/9.05 => ( ( H2 @ B2 )
% 16.19/9.05 = one_one_complex ) )
% 16.19/9.05 => ( ( ( groups7440179247065528705omplex @ G @ A2 )
% 16.19/9.05 = ( groups7440179247065528705omplex @ H2 @ B3 ) )
% 16.19/9.05 = ( ( groups7440179247065528705omplex @ G @ C2 )
% 16.19/9.05 = ( groups7440179247065528705omplex @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8856_prod_Osame__carrier,axiom,
% 16.19/9.05 ! [C2: set_complex,A2: set_complex,B3: set_complex,G: complex > complex,H2: complex > complex] :
% 16.19/9.05 ( ( finite3207457112153483333omplex @ C2 )
% 16.19/9.05 => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 16.19/9.05 => ( ( ord_le211207098394363844omplex @ B3 @ C2 )
% 16.19/9.05 => ( ! [A3: complex] :
% 16.19/9.05 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 16.19/9.05 => ( ( G @ A3 )
% 16.19/9.05 = one_one_complex ) )
% 16.19/9.05 => ( ! [B2: complex] :
% 16.19/9.05 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B3 ) )
% 16.19/9.05 => ( ( H2 @ B2 )
% 16.19/9.05 = one_one_complex ) )
% 16.19/9.05 => ( ( ( groups3708469109370488835omplex @ G @ A2 )
% 16.19/9.05 = ( groups3708469109370488835omplex @ H2 @ B3 ) )
% 16.19/9.05 = ( ( groups3708469109370488835omplex @ G @ C2 )
% 16.19/9.05 = ( groups3708469109370488835omplex @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8857_prod_Osame__carrier,axiom,
% 16.19/9.05 ! [C2: set_real,A2: set_real,B3: set_real,G: real > real,H2: real > real] :
% 16.19/9.05 ( ( finite_finite_real @ C2 )
% 16.19/9.05 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 16.19/9.05 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 16.19/9.05 => ( ! [A3: real] :
% 16.19/9.05 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 16.19/9.05 => ( ( G @ A3 )
% 16.19/9.05 = one_one_real ) )
% 16.19/9.05 => ( ! [B2: real] :
% 16.19/9.05 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 16.19/9.05 => ( ( H2 @ B2 )
% 16.19/9.05 = one_one_real ) )
% 16.19/9.05 => ( ( ( groups1681761925125756287l_real @ G @ A2 )
% 16.19/9.05 = ( groups1681761925125756287l_real @ H2 @ B3 ) )
% 16.19/9.05 = ( ( groups1681761925125756287l_real @ G @ C2 )
% 16.19/9.05 = ( groups1681761925125756287l_real @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8858_prod_Osame__carrier,axiom,
% 16.19/9.05 ! [C2: set_int,A2: set_int,B3: set_int,G: int > real,H2: int > real] :
% 16.19/9.05 ( ( finite_finite_int @ C2 )
% 16.19/9.05 => ( ( ord_less_eq_set_int @ A2 @ C2 )
% 16.19/9.05 => ( ( ord_less_eq_set_int @ B3 @ C2 )
% 16.19/9.05 => ( ! [A3: int] :
% 16.19/9.05 ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 16.19/9.05 => ( ( G @ A3 )
% 16.19/9.05 = one_one_real ) )
% 16.19/9.05 => ( ! [B2: int] :
% 16.19/9.05 ( ( member_int @ B2 @ ( minus_minus_set_int @ C2 @ B3 ) )
% 16.19/9.05 => ( ( H2 @ B2 )
% 16.19/9.05 = one_one_real ) )
% 16.19/9.05 => ( ( ( groups2316167850115554303t_real @ G @ A2 )
% 16.19/9.05 = ( groups2316167850115554303t_real @ H2 @ B3 ) )
% 16.19/9.05 = ( ( groups2316167850115554303t_real @ G @ C2 )
% 16.19/9.05 = ( groups2316167850115554303t_real @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8859_prod_Osame__carrier,axiom,
% 16.19/9.05 ! [C2: set_complex,A2: set_complex,B3: set_complex,G: complex > real,H2: complex > real] :
% 16.19/9.05 ( ( finite3207457112153483333omplex @ C2 )
% 16.19/9.05 => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 16.19/9.05 => ( ( ord_le211207098394363844omplex @ B3 @ C2 )
% 16.19/9.05 => ( ! [A3: complex] :
% 16.19/9.05 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 16.19/9.05 => ( ( G @ A3 )
% 16.19/9.05 = one_one_real ) )
% 16.19/9.05 => ( ! [B2: complex] :
% 16.19/9.05 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B3 ) )
% 16.19/9.05 => ( ( H2 @ B2 )
% 16.19/9.05 = one_one_real ) )
% 16.19/9.05 => ( ( ( groups766887009212190081x_real @ G @ A2 )
% 16.19/9.05 = ( groups766887009212190081x_real @ H2 @ B3 ) )
% 16.19/9.05 = ( ( groups766887009212190081x_real @ G @ C2 )
% 16.19/9.05 = ( groups766887009212190081x_real @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8860_prod_Osame__carrier,axiom,
% 16.19/9.05 ! [C2: set_real,A2: set_real,B3: set_real,G: real > rat,H2: real > rat] :
% 16.19/9.05 ( ( finite_finite_real @ C2 )
% 16.19/9.05 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 16.19/9.05 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 16.19/9.05 => ( ! [A3: real] :
% 16.19/9.05 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 16.19/9.05 => ( ( G @ A3 )
% 16.19/9.05 = one_one_rat ) )
% 16.19/9.05 => ( ! [B2: real] :
% 16.19/9.05 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 16.19/9.05 => ( ( H2 @ B2 )
% 16.19/9.05 = one_one_rat ) )
% 16.19/9.05 => ( ( ( groups4061424788464935467al_rat @ G @ A2 )
% 16.19/9.05 = ( groups4061424788464935467al_rat @ H2 @ B3 ) )
% 16.19/9.05 = ( ( groups4061424788464935467al_rat @ G @ C2 )
% 16.19/9.05 = ( groups4061424788464935467al_rat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8861_prod_Osame__carrier,axiom,
% 16.19/9.05 ! [C2: set_int,A2: set_int,B3: set_int,G: int > rat,H2: int > rat] :
% 16.19/9.05 ( ( finite_finite_int @ C2 )
% 16.19/9.05 => ( ( ord_less_eq_set_int @ A2 @ C2 )
% 16.19/9.05 => ( ( ord_less_eq_set_int @ B3 @ C2 )
% 16.19/9.05 => ( ! [A3: int] :
% 16.19/9.05 ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A2 ) )
% 16.19/9.05 => ( ( G @ A3 )
% 16.19/9.05 = one_one_rat ) )
% 16.19/9.05 => ( ! [B2: int] :
% 16.19/9.05 ( ( member_int @ B2 @ ( minus_minus_set_int @ C2 @ B3 ) )
% 16.19/9.05 => ( ( H2 @ B2 )
% 16.19/9.05 = one_one_rat ) )
% 16.19/9.05 => ( ( ( groups1072433553688619179nt_rat @ G @ A2 )
% 16.19/9.05 = ( groups1072433553688619179nt_rat @ H2 @ B3 ) )
% 16.19/9.05 = ( ( groups1072433553688619179nt_rat @ G @ C2 )
% 16.19/9.05 = ( groups1072433553688619179nt_rat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8862_prod_Osame__carrier,axiom,
% 16.19/9.05 ! [C2: set_complex,A2: set_complex,B3: set_complex,G: complex > rat,H2: complex > rat] :
% 16.19/9.05 ( ( finite3207457112153483333omplex @ C2 )
% 16.19/9.05 => ( ( ord_le211207098394363844omplex @ A2 @ C2 )
% 16.19/9.05 => ( ( ord_le211207098394363844omplex @ B3 @ C2 )
% 16.19/9.05 => ( ! [A3: complex] :
% 16.19/9.05 ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A2 ) )
% 16.19/9.05 => ( ( G @ A3 )
% 16.19/9.05 = one_one_rat ) )
% 16.19/9.05 => ( ! [B2: complex] :
% 16.19/9.05 ( ( member_complex @ B2 @ ( minus_811609699411566653omplex @ C2 @ B3 ) )
% 16.19/9.05 => ( ( H2 @ B2 )
% 16.19/9.05 = one_one_rat ) )
% 16.19/9.05 => ( ( ( groups225925009352817453ex_rat @ G @ A2 )
% 16.19/9.05 = ( groups225925009352817453ex_rat @ H2 @ B3 ) )
% 16.19/9.05 = ( ( groups225925009352817453ex_rat @ G @ C2 )
% 16.19/9.05 = ( groups225925009352817453ex_rat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8863_prod_Osame__carrier,axiom,
% 16.19/9.05 ! [C2: set_real,A2: set_real,B3: set_real,G: real > nat,H2: real > nat] :
% 16.19/9.05 ( ( finite_finite_real @ C2 )
% 16.19/9.05 => ( ( ord_less_eq_set_real @ A2 @ C2 )
% 16.19/9.05 => ( ( ord_less_eq_set_real @ B3 @ C2 )
% 16.19/9.05 => ( ! [A3: real] :
% 16.19/9.05 ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A2 ) )
% 16.19/9.05 => ( ( G @ A3 )
% 16.19/9.05 = one_one_nat ) )
% 16.19/9.05 => ( ! [B2: real] :
% 16.19/9.05 ( ( member_real @ B2 @ ( minus_minus_set_real @ C2 @ B3 ) )
% 16.19/9.05 => ( ( H2 @ B2 )
% 16.19/9.05 = one_one_nat ) )
% 16.19/9.05 => ( ( ( groups4696554848551431203al_nat @ G @ A2 )
% 16.19/9.05 = ( groups4696554848551431203al_nat @ H2 @ B3 ) )
% 16.19/9.05 = ( ( groups4696554848551431203al_nat @ G @ C2 )
% 16.19/9.05 = ( groups4696554848551431203al_nat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8864_arcsin__le__mono,axiom,
% 16.19/9.05 ! [X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 16.19/9.05 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 16.19/9.05 => ( ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
% 16.19/9.05 = ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8865_arcsin__eq__iff,axiom,
% 16.19/9.05 ! [X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 16.19/9.05 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 16.19/9.05 => ( ( ( arcsin @ X )
% 16.19/9.05 = ( arcsin @ Y ) )
% 16.19/9.05 = ( X = Y ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8866_prod_OatLeast0__atMost__Suc,axiom,
% 16.19/9.05 ! [G: nat > real,N2: nat] :
% 16.19/9.05 ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
% 16.19/9.05 = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8867_prod_OatLeast0__atMost__Suc,axiom,
% 16.19/9.05 ! [G: nat > rat,N2: nat] :
% 16.19/9.05 ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
% 16.19/9.05 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8868_prod_OatLeast0__atMost__Suc,axiom,
% 16.19/9.05 ! [G: nat > nat,N2: nat] :
% 16.19/9.05 ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
% 16.19/9.05 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8869_prod_OatLeast0__atMost__Suc,axiom,
% 16.19/9.05 ! [G: nat > int,N2: nat] :
% 16.19/9.05 ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
% 16.19/9.05 = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8870_prod_OatLeast__Suc__atMost,axiom,
% 16.19/9.05 ! [M: nat,N2: nat,G: nat > real] :
% 16.19/9.05 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.05 => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.05 = ( times_times_real @ ( G @ M ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8871_prod_OatLeast__Suc__atMost,axiom,
% 16.19/9.05 ! [M: nat,N2: nat,G: nat > rat] :
% 16.19/9.05 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.05 => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.05 = ( times_times_rat @ ( G @ M ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8872_prod_OatLeast__Suc__atMost,axiom,
% 16.19/9.05 ! [M: nat,N2: nat,G: nat > nat] :
% 16.19/9.05 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.05 => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.05 = ( times_times_nat @ ( G @ M ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8873_prod_OatLeast__Suc__atMost,axiom,
% 16.19/9.05 ! [M: nat,N2: nat,G: nat > int] :
% 16.19/9.05 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.05 => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 16.19/9.05 = ( times_times_int @ ( G @ M ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8874_prod_Onat__ivl__Suc_H,axiom,
% 16.19/9.05 ! [M: nat,N2: nat,G: nat > real] :
% 16.19/9.05 ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 16.19/9.05 => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 16.19/9.05 = ( times_times_real @ ( G @ ( suc @ N2 ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8875_prod_Onat__ivl__Suc_H,axiom,
% 16.19/9.05 ! [M: nat,N2: nat,G: nat > rat] :
% 16.19/9.05 ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 16.19/9.05 => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 16.19/9.05 = ( times_times_rat @ ( G @ ( suc @ N2 ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8876_prod_Onat__ivl__Suc_H,axiom,
% 16.19/9.05 ! [M: nat,N2: nat,G: nat > nat] :
% 16.19/9.05 ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 16.19/9.05 => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 16.19/9.05 = ( times_times_nat @ ( G @ ( suc @ N2 ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8877_prod_Onat__ivl__Suc_H,axiom,
% 16.19/9.05 ! [M: nat,N2: nat,G: nat > int] :
% 16.19/9.05 ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 16.19/9.05 => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 16.19/9.05 = ( times_times_int @ ( G @ ( suc @ N2 ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8878_prod_OlessThan__Suc__shift,axiom,
% 16.19/9.05 ! [G: nat > real,N2: nat] :
% 16.19/9.05 ( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 16.19/9.05 = ( times_times_real @ ( G @ zero_zero_nat )
% 16.19/9.05 @ ( groups129246275422532515t_real
% 16.19/9.05 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 16.19/9.05 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8879_prod_OlessThan__Suc__shift,axiom,
% 16.19/9.05 ! [G: nat > rat,N2: nat] :
% 16.19/9.05 ( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 16.19/9.05 = ( times_times_rat @ ( G @ zero_zero_nat )
% 16.19/9.05 @ ( groups73079841787564623at_rat
% 16.19/9.05 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 16.19/9.05 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8880_prod_OlessThan__Suc__shift,axiom,
% 16.19/9.05 ! [G: nat > nat,N2: nat] :
% 16.19/9.05 ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 16.19/9.05 = ( times_times_nat @ ( G @ zero_zero_nat )
% 16.19/9.05 @ ( groups708209901874060359at_nat
% 16.19/9.05 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 16.19/9.05 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8881_prod_OlessThan__Suc__shift,axiom,
% 16.19/9.05 ! [G: nat > int,N2: nat] :
% 16.19/9.05 ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 16.19/9.05 = ( times_times_int @ ( G @ zero_zero_nat )
% 16.19/9.05 @ ( groups705719431365010083at_int
% 16.19/9.05 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 16.19/9.05 @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8882_prod_OSuc__reindex__ivl,axiom,
% 16.19/9.05 ! [M: nat,N2: nat,G: nat > real] :
% 16.19/9.05 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.05 => ( ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
% 16.19/9.05 = ( times_times_real @ ( G @ M )
% 16.19/9.05 @ ( groups129246275422532515t_real
% 16.19/9.05 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 16.19/9.05 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8883_prod_OSuc__reindex__ivl,axiom,
% 16.19/9.05 ! [M: nat,N2: nat,G: nat > rat] :
% 16.19/9.05 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.05 => ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
% 16.19/9.05 = ( times_times_rat @ ( G @ M )
% 16.19/9.05 @ ( groups73079841787564623at_rat
% 16.19/9.05 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 16.19/9.05 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8884_prod_OSuc__reindex__ivl,axiom,
% 16.19/9.05 ! [M: nat,N2: nat,G: nat > nat] :
% 16.19/9.05 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.05 => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
% 16.19/9.05 = ( times_times_nat @ ( G @ M )
% 16.19/9.05 @ ( groups708209901874060359at_nat
% 16.19/9.05 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 16.19/9.05 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8885_prod_OSuc__reindex__ivl,axiom,
% 16.19/9.05 ! [M: nat,N2: nat,G: nat > int] :
% 16.19/9.05 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.05 => ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
% 16.19/9.05 = ( times_times_int @ ( G @ M )
% 16.19/9.05 @ ( groups705719431365010083at_int
% 16.19/9.05 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 16.19/9.05 @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8886_prod_OatLeast1__atMost__eq,axiom,
% 16.19/9.05 ! [G: nat > nat,N2: nat] :
% 16.19/9.05 ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
% 16.19/9.05 = ( groups708209901874060359at_nat
% 16.19/9.05 @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
% 16.19/9.05 @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8887_prod_OatLeast1__atMost__eq,axiom,
% 16.19/9.05 ! [G: nat > int,N2: nat] :
% 16.19/9.05 ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
% 16.19/9.05 = ( groups705719431365010083at_int
% 16.19/9.05 @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
% 16.19/9.05 @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8888_fact__prod,axiom,
% 16.19/9.05 ( semiri1406184849735516958ct_int
% 16.19/9.05 = ( ^ [N: nat] :
% 16.19/9.05 ( semiri1314217659103216013at_int
% 16.19/9.05 @ ( groups708209901874060359at_nat
% 16.19/9.05 @ ^ [X2: nat] : X2
% 16.19/9.05 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8889_fact__prod,axiom,
% 16.19/9.05 ( semiri5044797733671781792omplex
% 16.19/9.05 = ( ^ [N: nat] :
% 16.19/9.05 ( semiri8010041392384452111omplex
% 16.19/9.05 @ ( groups708209901874060359at_nat
% 16.19/9.05 @ ^ [X2: nat] : X2
% 16.19/9.05 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8890_fact__prod,axiom,
% 16.19/9.05 ( semiri2265585572941072030t_real
% 16.19/9.05 = ( ^ [N: nat] :
% 16.19/9.05 ( semiri5074537144036343181t_real
% 16.19/9.05 @ ( groups708209901874060359at_nat
% 16.19/9.05 @ ^ [X2: nat] : X2
% 16.19/9.05 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8891_fact__prod,axiom,
% 16.19/9.05 ( semiri1408675320244567234ct_nat
% 16.19/9.05 = ( ^ [N: nat] :
% 16.19/9.05 ( semiri1316708129612266289at_nat
% 16.19/9.05 @ ( groups708209901874060359at_nat
% 16.19/9.05 @ ^ [X2: nat] : X2
% 16.19/9.05 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8892_prod__mono__strict,axiom,
% 16.19/9.05 ! [A2: set_int,F: int > nat,G: int > nat] :
% 16.19/9.05 ( ( finite_finite_int @ A2 )
% 16.19/9.05 => ( ! [I4: int] :
% 16.19/9.05 ( ( member_int @ I4 @ A2 )
% 16.19/9.05 => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) )
% 16.19/9.05 & ( ord_less_nat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 16.19/9.05 => ( ( A2 != bot_bot_set_int )
% 16.19/9.05 => ( ord_less_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ A2 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8893_prod__mono__strict,axiom,
% 16.19/9.05 ! [A2: set_real,F: real > nat,G: real > nat] :
% 16.19/9.05 ( ( finite_finite_real @ A2 )
% 16.19/9.05 => ( ! [I4: real] :
% 16.19/9.05 ( ( member_real @ I4 @ A2 )
% 16.19/9.05 => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) )
% 16.19/9.05 & ( ord_less_nat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 16.19/9.05 => ( ( A2 != bot_bot_set_real )
% 16.19/9.05 => ( ord_less_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8894_prod__mono__strict,axiom,
% 16.19/9.05 ! [A2: set_set_nat,F: set_nat > int,G: set_nat > int] :
% 16.19/9.05 ( ( finite1152437895449049373et_nat @ A2 )
% 16.19/9.05 => ( ! [I4: set_nat] :
% 16.19/9.05 ( ( member_set_nat @ I4 @ A2 )
% 16.19/9.05 => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) )
% 16.19/9.05 & ( ord_less_int @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 16.19/9.05 => ( ( A2 != bot_bot_set_set_nat )
% 16.19/9.05 => ( ord_less_int @ ( groups4246057289670975065at_int @ F @ A2 ) @ ( groups4246057289670975065at_int @ G @ A2 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8895_prod__mono__strict,axiom,
% 16.19/9.05 ! [A2: set_complex,F: complex > int,G: complex > int] :
% 16.19/9.05 ( ( finite3207457112153483333omplex @ A2 )
% 16.19/9.05 => ( ! [I4: complex] :
% 16.19/9.05 ( ( member_complex @ I4 @ A2 )
% 16.19/9.05 => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) )
% 16.19/9.05 & ( ord_less_int @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 16.19/9.05 => ( ( A2 != bot_bot_set_complex )
% 16.19/9.05 => ( ord_less_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ G @ A2 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8896_prod__mono__strict,axiom,
% 16.19/9.05 ! [A2: set_real,F: real > int,G: real > int] :
% 16.19/9.05 ( ( finite_finite_real @ A2 )
% 16.19/9.05 => ( ! [I4: real] :
% 16.19/9.05 ( ( member_real @ I4 @ A2 )
% 16.19/9.05 => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) )
% 16.19/9.05 & ( ord_less_int @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 16.19/9.05 => ( ( A2 != bot_bot_set_real )
% 16.19/9.05 => ( ord_less_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ ( groups4694064378042380927al_int @ G @ A2 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8897_prod__mono__strict,axiom,
% 16.19/9.05 ! [A2: set_nat,F: nat > nat,G: nat > nat] :
% 16.19/9.05 ( ( finite_finite_nat @ A2 )
% 16.19/9.05 => ( ! [I4: nat] :
% 16.19/9.05 ( ( member_nat @ I4 @ A2 )
% 16.19/9.05 => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I4 ) )
% 16.19/9.05 & ( ord_less_nat @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 16.19/9.05 => ( ( A2 != bot_bot_set_nat )
% 16.19/9.05 => ( ord_less_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ ( groups708209901874060359at_nat @ G @ A2 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8898_prod__mono__strict,axiom,
% 16.19/9.05 ! [A2: set_nat,F: nat > int,G: nat > int] :
% 16.19/9.05 ( ( finite_finite_nat @ A2 )
% 16.19/9.05 => ( ! [I4: nat] :
% 16.19/9.05 ( ( member_nat @ I4 @ A2 )
% 16.19/9.05 => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) )
% 16.19/9.05 & ( ord_less_int @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 16.19/9.05 => ( ( A2 != bot_bot_set_nat )
% 16.19/9.05 => ( ord_less_int @ ( groups705719431365010083at_int @ F @ A2 ) @ ( groups705719431365010083at_int @ G @ A2 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8899_prod__mono__strict,axiom,
% 16.19/9.05 ! [A2: set_int,F: int > int,G: int > int] :
% 16.19/9.05 ( ( finite_finite_int @ A2 )
% 16.19/9.05 => ( ! [I4: int] :
% 16.19/9.05 ( ( member_int @ I4 @ A2 )
% 16.19/9.05 => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) )
% 16.19/9.05 & ( ord_less_int @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
% 16.19/9.05 => ( ( A2 != bot_bot_set_int )
% 16.19/9.05 => ( ord_less_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ ( groups1705073143266064639nt_int @ G @ A2 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8900_arccos__lbound,axiom,
% 16.19/9.05 ! [Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 16.19/9.05 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 16.19/9.05 => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8901_arccos__less__arccos,axiom,
% 16.19/9.05 ! [X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 16.19/9.05 => ( ( ord_less_real @ X @ Y )
% 16.19/9.05 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 16.19/9.05 => ( ord_less_real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8902_arccos__less__mono,axiom,
% 16.19/9.05 ! [X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 16.19/9.05 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 16.19/9.05 => ( ( ord_less_real @ ( arccos @ X ) @ ( arccos @ Y ) )
% 16.19/9.05 = ( ord_less_real @ Y @ X ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8903_arccos__ubound,axiom,
% 16.19/9.05 ! [Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 16.19/9.05 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 16.19/9.05 => ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8904_arccos__cos,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_eq_real @ X @ pi )
% 16.19/9.05 => ( ( arccos @ ( cos_real @ X ) )
% 16.19/9.05 = X ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8905_arcsin__less__arcsin,axiom,
% 16.19/9.05 ! [X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 16.19/9.05 => ( ( ord_less_real @ X @ Y )
% 16.19/9.05 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 16.19/9.05 => ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8906_arcsin__less__mono,axiom,
% 16.19/9.05 ! [X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 16.19/9.05 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 16.19/9.05 => ( ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
% 16.19/9.05 = ( ord_less_real @ X @ Y ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8907_cos__arccos__abs,axiom,
% 16.19/9.05 ! [Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
% 16.19/9.05 => ( ( cos_real @ ( arccos @ Y ) )
% 16.19/9.05 = Y ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8908_arccos__cos__eq__abs,axiom,
% 16.19/9.05 ! [Theta: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( abs_abs_real @ Theta ) @ pi )
% 16.19/9.05 => ( ( arccos @ ( cos_real @ Theta ) )
% 16.19/9.05 = ( abs_abs_real @ Theta ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8909_fact__eq__fact__times,axiom,
% 16.19/9.05 ! [N2: nat,M: nat] :
% 16.19/9.05 ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/9.05 => ( ( semiri1408675320244567234ct_nat @ M )
% 16.19/9.05 = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N2 )
% 16.19/9.05 @ ( groups708209901874060359at_nat
% 16.19/9.05 @ ^ [X2: nat] : X2
% 16.19/9.05 @ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8910_arccos__lt__bounded,axiom,
% 16.19/9.05 ! [Y: real] :
% 16.19/9.05 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 16.19/9.05 => ( ( ord_less_real @ Y @ one_one_real )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y ) )
% 16.19/9.05 & ( ord_less_real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8911_arccos__bounded,axiom,
% 16.19/9.05 ! [Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 16.19/9.05 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 16.19/9.05 => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
% 16.19/9.05 & ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8912_sin__arccos__nonzero,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 16.19/9.05 => ( ( ord_less_real @ X @ one_one_real )
% 16.19/9.05 => ( ( sin_real @ ( arccos @ X ) )
% 16.19/9.05 != zero_zero_real ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8913_arccos__cos2,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 16.19/9.05 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X )
% 16.19/9.05 => ( ( arccos @ ( cos_real @ X ) )
% 16.19/9.05 = ( uminus_uminus_real @ X ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8914_arccos__minus,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 16.19/9.05 => ( ( ord_less_eq_real @ X @ one_one_real )
% 16.19/9.05 => ( ( arccos @ ( uminus_uminus_real @ X ) )
% 16.19/9.05 = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8915_cos__arcsin__nonzero,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 16.19/9.05 => ( ( ord_less_real @ X @ one_one_real )
% 16.19/9.05 => ( ( cos_real @ ( arcsin @ X ) )
% 16.19/9.05 != zero_zero_real ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8916_fact__div__fact,axiom,
% 16.19/9.05 ! [N2: nat,M: nat] :
% 16.19/9.05 ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/9.05 => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) )
% 16.19/9.05 = ( groups708209901874060359at_nat
% 16.19/9.05 @ ^ [X2: nat] : X2
% 16.19/9.05 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8917_arccos,axiom,
% 16.19/9.05 ! [Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 16.19/9.05 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 16.19/9.05 => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
% 16.19/9.05 & ( ord_less_eq_real @ ( arccos @ Y ) @ pi )
% 16.19/9.05 & ( ( cos_real @ ( arccos @ Y ) )
% 16.19/9.05 = Y ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8918_arccos__minus__abs,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 16.19/9.05 => ( ( arccos @ ( uminus_uminus_real @ X ) )
% 16.19/9.05 = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8919_arccos__le__pi2,axiom,
% 16.19/9.05 ! [Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/9.05 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 16.19/9.05 => ( ord_less_eq_real @ ( arccos @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8920_arcsin__lt__bounded,axiom,
% 16.19/9.05 ! [Y: real] :
% 16.19/9.05 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 16.19/9.05 => ( ( ord_less_real @ Y @ one_one_real )
% 16.19/9.05 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
% 16.19/9.05 & ( ord_less_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8921_arcsin__lbound,axiom,
% 16.19/9.05 ! [Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 16.19/9.05 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 16.19/9.05 => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8922_arcsin__ubound,axiom,
% 16.19/9.05 ! [Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 16.19/9.05 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 16.19/9.05 => ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8923_arcsin__bounded,axiom,
% 16.19/9.05 ! [Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 16.19/9.05 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 16.19/9.05 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
% 16.19/9.05 & ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8924_arcsin__sin,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 16.19/9.05 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.05 => ( ( arcsin @ ( sin_real @ X ) )
% 16.19/9.05 = X ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8925_arcsin,axiom,
% 16.19/9.05 ! [Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 16.19/9.05 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 16.19/9.05 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
% 16.19/9.05 & ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.05 & ( ( sin_real @ ( arcsin @ Y ) )
% 16.19/9.05 = Y ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8926_arcsin__pi,axiom,
% 16.19/9.05 ! [Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
% 16.19/9.05 => ( ( ord_less_eq_real @ Y @ one_one_real )
% 16.19/9.05 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
% 16.19/9.05 & ( ord_less_eq_real @ ( arcsin @ Y ) @ pi )
% 16.19/9.05 & ( ( sin_real @ ( arcsin @ Y ) )
% 16.19/9.05 = Y ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8927_arcsin__le__iff,axiom,
% 16.19/9.05 ! [X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 16.19/9.05 => ( ( ord_less_eq_real @ X @ one_one_real )
% 16.19/9.05 => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
% 16.19/9.05 => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.05 => ( ( ord_less_eq_real @ ( arcsin @ X ) @ Y )
% 16.19/9.05 = ( ord_less_eq_real @ X @ ( sin_real @ Y ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8928_le__arcsin__iff,axiom,
% 16.19/9.05 ! [X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 16.19/9.05 => ( ( ord_less_eq_real @ X @ one_one_real )
% 16.19/9.05 => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
% 16.19/9.05 => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.05 => ( ( ord_less_eq_real @ Y @ ( arcsin @ X ) )
% 16.19/9.05 = ( ord_less_eq_real @ ( sin_real @ Y ) @ X ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8929_arccos__cos__eq__abs__2pi,axiom,
% 16.19/9.05 ! [Theta: real] :
% 16.19/9.05 ~ ! [K2: int] :
% 16.19/9.05 ( ( arccos @ ( cos_real @ Theta ) )
% 16.19/9.05 != ( abs_abs_real @ ( minus_minus_real @ Theta @ ( times_times_real @ ( ring_1_of_int_real @ K2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8930_sin__arccos,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 16.19/9.05 => ( ( ord_less_eq_real @ X @ one_one_real )
% 16.19/9.05 => ( ( sin_real @ ( arccos @ X ) )
% 16.19/9.05 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8931_Maclaurin__sin__bound,axiom,
% 16.19/9.05 ! [X: real,N2: nat] :
% 16.19/9.05 ( ord_less_eq_real
% 16.19/9.05 @ ( abs_abs_real
% 16.19/9.05 @ ( minus_minus_real @ ( sin_real @ X )
% 16.19/9.05 @ ( groups6591440286371151544t_real
% 16.19/9.05 @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
% 16.19/9.05 @ ( set_ord_lessThan_nat @ N2 ) ) ) )
% 16.19/9.05 @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( abs_abs_real @ X ) @ N2 ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8932_central__binomial__lower__bound,axiom,
% 16.19/9.05 ! [N2: nat] :
% 16.19/9.05 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.05 => ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8933_cot__less__zero,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 16.19/9.05 => ( ( ord_less_real @ X @ zero_zero_real )
% 16.19/9.05 => ( ord_less_real @ ( cot_real @ X ) @ zero_zero_real ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8934_binomial__Suc__n,axiom,
% 16.19/9.05 ! [N2: nat] :
% 16.19/9.05 ( ( binomial @ ( suc @ N2 ) @ N2 )
% 16.19/9.05 = ( suc @ N2 ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8935_binomial__n__n,axiom,
% 16.19/9.05 ! [N2: nat] :
% 16.19/9.05 ( ( binomial @ N2 @ N2 )
% 16.19/9.05 = one_one_nat ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8936_binomial__0__Suc,axiom,
% 16.19/9.05 ! [K: nat] :
% 16.19/9.05 ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
% 16.19/9.05 = zero_zero_nat ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8937_binomial__1,axiom,
% 16.19/9.05 ! [N2: nat] :
% 16.19/9.05 ( ( binomial @ N2 @ ( suc @ zero_zero_nat ) )
% 16.19/9.05 = N2 ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8938_binomial__eq__0__iff,axiom,
% 16.19/9.05 ! [N2: nat,K: nat] :
% 16.19/9.05 ( ( ( binomial @ N2 @ K )
% 16.19/9.05 = zero_zero_nat )
% 16.19/9.05 = ( ord_less_nat @ N2 @ K ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8939_binomial__Suc__Suc,axiom,
% 16.19/9.05 ! [N2: nat,K: nat] :
% 16.19/9.05 ( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) )
% 16.19/9.05 = ( plus_plus_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8940_binomial__n__0,axiom,
% 16.19/9.05 ! [N2: nat] :
% 16.19/9.05 ( ( binomial @ N2 @ zero_zero_nat )
% 16.19/9.05 = one_one_nat ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8941_zero__less__binomial__iff,axiom,
% 16.19/9.05 ! [N2: nat,K: nat] :
% 16.19/9.05 ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N2 @ K ) )
% 16.19/9.05 = ( ord_less_eq_nat @ K @ N2 ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8942_cot__npi,axiom,
% 16.19/9.05 ! [N2: nat] :
% 16.19/9.05 ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) )
% 16.19/9.05 = zero_zero_real ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8943_cot__periodic,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( cot_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 16.19/9.05 = ( cot_real @ X ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8944_choose__one,axiom,
% 16.19/9.05 ! [N2: nat] :
% 16.19/9.05 ( ( binomial @ N2 @ one_one_nat )
% 16.19/9.05 = N2 ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8945_real__sqrt__inverse,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( sqrt @ ( inverse_inverse_real @ X ) )
% 16.19/9.05 = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8946_binomial__eq__0,axiom,
% 16.19/9.05 ! [N2: nat,K: nat] :
% 16.19/9.05 ( ( ord_less_nat @ N2 @ K )
% 16.19/9.05 => ( ( binomial @ N2 @ K )
% 16.19/9.05 = zero_zero_nat ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8947_Suc__times__binomial__eq,axiom,
% 16.19/9.05 ! [N2: nat,K: nat] :
% 16.19/9.05 ( ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) )
% 16.19/9.05 = ( times_times_nat @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8948_Suc__times__binomial,axiom,
% 16.19/9.05 ! [K: nat,N2: nat] :
% 16.19/9.05 ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) )
% 16.19/9.05 = ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8949_choose__mult__lemma,axiom,
% 16.19/9.05 ! [M: nat,R: nat,K: nat] :
% 16.19/9.05 ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
% 16.19/9.05 = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R ) @ M ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8950_binomial__symmetric,axiom,
% 16.19/9.05 ! [K: nat,N2: nat] :
% 16.19/9.05 ( ( ord_less_eq_nat @ K @ N2 )
% 16.19/9.05 => ( ( binomial @ N2 @ K )
% 16.19/9.05 = ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8951_binomial__le__pow,axiom,
% 16.19/9.05 ! [R: nat,N2: nat] :
% 16.19/9.05 ( ( ord_less_eq_nat @ R @ N2 )
% 16.19/9.05 => ( ord_less_eq_nat @ ( binomial @ N2 @ R ) @ ( power_power_nat @ N2 @ R ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8952_divide__real__def,axiom,
% 16.19/9.05 ( divide_divide_real
% 16.19/9.05 = ( ^ [X2: real,Y2: real] : ( times_times_real @ X2 @ ( inverse_inverse_real @ Y2 ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8953_zero__less__binomial,axiom,
% 16.19/9.05 ! [K: nat,N2: nat] :
% 16.19/9.05 ( ( ord_less_eq_nat @ K @ N2 )
% 16.19/9.05 => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N2 @ K ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8954_Suc__times__binomial__add,axiom,
% 16.19/9.05 ! [A: nat,B: nat] :
% 16.19/9.05 ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
% 16.19/9.05 = ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8955_binomial__Suc__Suc__eq__times,axiom,
% 16.19/9.05 ! [N2: nat,K: nat] :
% 16.19/9.05 ( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) )
% 16.19/9.05 = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) @ ( suc @ K ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8956_choose__mult,axiom,
% 16.19/9.05 ! [K: nat,M: nat,N2: nat] :
% 16.19/9.05 ( ( ord_less_eq_nat @ K @ M )
% 16.19/9.05 => ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.05 => ( ( times_times_nat @ ( binomial @ N2 @ M ) @ ( binomial @ M @ K ) )
% 16.19/9.05 = ( times_times_nat @ ( binomial @ N2 @ K ) @ ( binomial @ ( minus_minus_nat @ N2 @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8957_binomial__absorb__comp,axiom,
% 16.19/9.05 ! [N2: nat,K: nat] :
% 16.19/9.05 ( ( times_times_nat @ ( minus_minus_nat @ N2 @ K ) @ ( binomial @ N2 @ K ) )
% 16.19/9.05 = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8958_prod__int__eq,axiom,
% 16.19/9.05 ! [I2: nat,J: nat] :
% 16.19/9.05 ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I2 @ J ) )
% 16.19/9.05 = ( groups1705073143266064639nt_int
% 16.19/9.05 @ ^ [X2: int] : X2
% 16.19/9.05 @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8959_binomial__absorption,axiom,
% 16.19/9.05 ! [K: nat,N2: nat] :
% 16.19/9.05 ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) )
% 16.19/9.05 = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8960_binomial__fact__lemma,axiom,
% 16.19/9.05 ! [K: nat,N2: nat] :
% 16.19/9.05 ( ( ord_less_eq_nat @ K @ N2 )
% 16.19/9.05 => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( binomial @ N2 @ K ) )
% 16.19/9.05 = ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8961_forall__pos__mono__1,axiom,
% 16.19/9.05 ! [P: real > $o,E: real] :
% 16.19/9.05 ( ! [D4: real,E2: real] :
% 16.19/9.05 ( ( ord_less_real @ D4 @ E2 )
% 16.19/9.05 => ( ( P @ D4 )
% 16.19/9.05 => ( P @ E2 ) ) )
% 16.19/9.05 => ( ! [N4: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ E )
% 16.19/9.05 => ( P @ E ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8962_real__arch__inverse,axiom,
% 16.19/9.05 ! [E: real] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ E )
% 16.19/9.05 = ( ? [N: nat] :
% 16.19/9.05 ( ( N != zero_zero_nat )
% 16.19/9.05 & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) )
% 16.19/9.05 & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) @ E ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8963_forall__pos__mono,axiom,
% 16.19/9.05 ! [P: real > $o,E: real] :
% 16.19/9.05 ( ! [D4: real,E2: real] :
% 16.19/9.05 ( ( ord_less_real @ D4 @ E2 )
% 16.19/9.05 => ( ( P @ D4 )
% 16.19/9.05 => ( P @ E2 ) ) )
% 16.19/9.05 => ( ! [N4: nat] :
% 16.19/9.05 ( ( N4 != zero_zero_nat )
% 16.19/9.05 => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) ) )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ E )
% 16.19/9.05 => ( P @ E ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8964_sqrt__divide__self__eq,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( divide_divide_real @ ( sqrt @ X ) @ X )
% 16.19/9.05 = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8965_ln__inverse,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ln_ln_real @ ( inverse_inverse_real @ X ) )
% 16.19/9.05 = ( uminus_uminus_real @ ( ln_ln_real @ X ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8966_prod__int__plus__eq,axiom,
% 16.19/9.05 ! [I2: nat,J: nat] :
% 16.19/9.05 ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I2 @ ( plus_plus_nat @ I2 @ J ) ) )
% 16.19/9.05 = ( groups1705073143266064639nt_int
% 16.19/9.05 @ ^ [X2: int] : X2
% 16.19/9.05 @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I2 @ J ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8967_binomial__maximum_H,axiom,
% 16.19/9.05 ! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8968_binomial__mono,axiom,
% 16.19/9.05 ! [K: nat,K6: nat,N2: nat] :
% 16.19/9.05 ( ( ord_less_eq_nat @ K @ K6 )
% 16.19/9.05 => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N2 )
% 16.19/9.05 => ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K6 ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8969_binomial__antimono,axiom,
% 16.19/9.05 ! [K: nat,K6: nat,N2: nat] :
% 16.19/9.05 ( ( ord_less_eq_nat @ K @ K6 )
% 16.19/9.05 => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
% 16.19/9.05 => ( ( ord_less_eq_nat @ K6 @ N2 )
% 16.19/9.05 => ( ord_less_eq_nat @ ( binomial @ N2 @ K6 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8970_binomial__maximum,axiom,
% 16.19/9.05 ! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8971_binomial__le__pow2,axiom,
% 16.19/9.05 ! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8972_choose__reduce__nat,axiom,
% 16.19/9.05 ! [N2: nat,K: nat] :
% 16.19/9.05 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.05 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 16.19/9.05 => ( ( binomial @ N2 @ K )
% 16.19/9.05 = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8973_times__binomial__minus1__eq,axiom,
% 16.19/9.05 ! [K: nat,N2: nat] :
% 16.19/9.05 ( ( ord_less_nat @ zero_zero_nat @ K )
% 16.19/9.05 => ( ( times_times_nat @ K @ ( binomial @ N2 @ K ) )
% 16.19/9.05 = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8974_binomial__altdef__nat,axiom,
% 16.19/9.05 ! [K: nat,N2: nat] :
% 16.19/9.05 ( ( ord_less_eq_nat @ K @ N2 )
% 16.19/9.05 => ( ( binomial @ N2 @ K )
% 16.19/9.05 = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8975_binomial__less__binomial__Suc,axiom,
% 16.19/9.05 ! [K: nat,N2: nat] :
% 16.19/9.05 ( ( ord_less_nat @ K @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.05 => ( ord_less_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8976_binomial__strict__antimono,axiom,
% 16.19/9.05 ! [K: nat,K6: nat,N2: nat] :
% 16.19/9.05 ( ( ord_less_nat @ K @ K6 )
% 16.19/9.05 => ( ( ord_less_eq_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
% 16.19/9.05 => ( ( ord_less_eq_nat @ K6 @ N2 )
% 16.19/9.05 => ( ord_less_nat @ ( binomial @ N2 @ K6 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8977_binomial__strict__mono,axiom,
% 16.19/9.05 ! [K: nat,K6: nat,N2: nat] :
% 16.19/9.05 ( ( ord_less_nat @ K @ K6 )
% 16.19/9.05 => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N2 )
% 16.19/9.05 => ( ord_less_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K6 ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8978_central__binomial__odd,axiom,
% 16.19/9.05 ! [N2: nat] :
% 16.19/9.05 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.05 => ( ( binomial @ N2 @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/9.05 = ( binomial @ N2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8979_binomial__addition__formula,axiom,
% 16.19/9.05 ! [N2: nat,K: nat] :
% 16.19/9.05 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.05 => ( ( binomial @ N2 @ ( suc @ K ) )
% 16.19/9.05 = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8980_exp__plus__inverse__exp,axiom,
% 16.19/9.05 ! [X: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8981_choose__two,axiom,
% 16.19/9.05 ! [N2: nat] :
% 16.19/9.05 ( ( binomial @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.05 = ( divide_divide_nat @ ( times_times_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8982_plus__inverse__ge__2,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8983_real__inv__sqrt__pow2,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.05 = ( inverse_inverse_real @ X ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8984_tan__cot,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
% 16.19/9.05 = ( inverse_inverse_real @ ( tan_real @ X ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8985_real__le__x__sinh,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ord_less_eq_real @ X @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8986_real__le__abs__sinh,axiom,
% 16.19/9.05 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8987_cot__gt__zero,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.05 => ( ord_less_real @ zero_zero_real @ ( cot_real @ X ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8988_tan__cot_H,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
% 16.19/9.05 = ( cot_real @ X ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8989_binomial__code,axiom,
% 16.19/9.05 ( binomial
% 16.19/9.05 = ( ^ [N: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N @ ( minus_minus_nat @ N @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N @ K3 ) @ one_one_nat ) @ N @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8990_divide__complex__def,axiom,
% 16.19/9.05 ( divide1717551699836669952omplex
% 16.19/9.05 = ( ^ [X2: complex,Y2: complex] : ( times_times_complex @ X2 @ ( invers8013647133539491842omplex @ Y2 ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8991_real__scaleR__def,axiom,
% 16.19/9.05 real_V1485227260804924795R_real = times_times_real ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8992_lessThan__Suc__atMost,axiom,
% 16.19/9.05 ! [K: nat] :
% 16.19/9.05 ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 16.19/9.05 = ( set_ord_atMost_nat @ K ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8993_complex__scaleR,axiom,
% 16.19/9.05 ! [R: real,A: real,B: real] :
% 16.19/9.05 ( ( real_V2046097035970521341omplex @ R @ ( complex2 @ A @ B ) )
% 16.19/9.05 = ( complex2 @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ B ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8994_sum__choose__upper,axiom,
% 16.19/9.05 ! [M: nat,N2: nat] :
% 16.19/9.05 ( ( groups3542108847815614940at_nat
% 16.19/9.05 @ ^ [K3: nat] : ( binomial @ K3 @ M )
% 16.19/9.05 @ ( set_ord_atMost_nat @ N2 ) )
% 16.19/9.05 = ( binomial @ ( suc @ N2 ) @ ( suc @ M ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8995_sum__choose__lower,axiom,
% 16.19/9.05 ! [R: nat,N2: nat] :
% 16.19/9.05 ( ( groups3542108847815614940at_nat
% 16.19/9.05 @ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R @ K3 ) @ K3 )
% 16.19/9.05 @ ( set_ord_atMost_nat @ N2 ) )
% 16.19/9.05 = ( binomial @ ( suc @ ( plus_plus_nat @ R @ N2 ) ) @ N2 ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8996_choose__rising__sum_I2_J,axiom,
% 16.19/9.05 ! [N2: nat,M: nat] :
% 16.19/9.05 ( ( groups3542108847815614940at_nat
% 16.19/9.05 @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N2 @ J3 ) @ N2 )
% 16.19/9.05 @ ( set_ord_atMost_nat @ M ) )
% 16.19/9.05 = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N2 @ M ) @ one_one_nat ) @ M ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8997_choose__rising__sum_I1_J,axiom,
% 16.19/9.05 ! [N2: nat,M: nat] :
% 16.19/9.05 ( ( groups3542108847815614940at_nat
% 16.19/9.05 @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N2 @ J3 ) @ N2 )
% 16.19/9.05 @ ( set_ord_atMost_nat @ M ) )
% 16.19/9.05 = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N2 @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8998_sum__choose__diagonal,axiom,
% 16.19/9.05 ! [M: nat,N2: nat] :
% 16.19/9.05 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.05 => ( ( groups3542108847815614940at_nat
% 16.19/9.05 @ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N2 @ K3 ) @ ( minus_minus_nat @ M @ K3 ) )
% 16.19/9.05 @ ( set_ord_atMost_nat @ M ) )
% 16.19/9.05 = ( binomial @ ( suc @ N2 ) @ M ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_8999_vandermonde,axiom,
% 16.19/9.05 ! [M: nat,N2: nat,R: nat] :
% 16.19/9.05 ( ( groups3542108847815614940at_nat
% 16.19/9.05 @ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M @ K3 ) @ ( binomial @ N2 @ ( minus_minus_nat @ R @ K3 ) ) )
% 16.19/9.05 @ ( set_ord_atMost_nat @ R ) )
% 16.19/9.05 = ( binomial @ ( plus_plus_nat @ M @ N2 ) @ R ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9000_choose__row__sum,axiom,
% 16.19/9.05 ! [N2: nat] :
% 16.19/9.05 ( ( groups3542108847815614940at_nat @ ( binomial @ N2 ) @ ( set_ord_atMost_nat @ N2 ) )
% 16.19/9.05 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9001_binomial,axiom,
% 16.19/9.05 ! [A: nat,B: nat,N2: nat] :
% 16.19/9.05 ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N2 )
% 16.19/9.05 = ( groups3542108847815614940at_nat
% 16.19/9.05 @ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N2 @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
% 16.19/9.05 @ ( set_ord_atMost_nat @ N2 ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9002_polynomial__product__nat,axiom,
% 16.19/9.05 ! [M: nat,A: nat > nat,N2: nat,B: nat > nat,X: nat] :
% 16.19/9.05 ( ! [I4: nat] :
% 16.19/9.05 ( ( ord_less_nat @ M @ I4 )
% 16.19/9.05 => ( ( A @ I4 )
% 16.19/9.05 = zero_zero_nat ) )
% 16.19/9.05 => ( ! [J2: nat] :
% 16.19/9.05 ( ( ord_less_nat @ N2 @ J2 )
% 16.19/9.05 => ( ( B @ J2 )
% 16.19/9.05 = zero_zero_nat ) )
% 16.19/9.05 => ( ( times_times_nat
% 16.19/9.05 @ ( groups3542108847815614940at_nat
% 16.19/9.05 @ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( power_power_nat @ X @ I3 ) )
% 16.19/9.05 @ ( set_ord_atMost_nat @ M ) )
% 16.19/9.05 @ ( groups3542108847815614940at_nat
% 16.19/9.05 @ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X @ J3 ) )
% 16.19/9.05 @ ( set_ord_atMost_nat @ N2 ) ) )
% 16.19/9.05 = ( groups3542108847815614940at_nat
% 16.19/9.05 @ ^ [R5: nat] :
% 16.19/9.05 ( times_times_nat
% 16.19/9.05 @ ( groups3542108847815614940at_nat
% 16.19/9.05 @ ^ [K3: nat] : ( times_times_nat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
% 16.19/9.05 @ ( set_ord_atMost_nat @ R5 ) )
% 16.19/9.05 @ ( power_power_nat @ X @ R5 ) )
% 16.19/9.05 @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9003_choose__square__sum,axiom,
% 16.19/9.05 ! [N2: nat] :
% 16.19/9.05 ( ( groups3542108847815614940at_nat
% 16.19/9.05 @ ^ [K3: nat] : ( power_power_nat @ ( binomial @ N2 @ K3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.05 @ ( set_ord_atMost_nat @ N2 ) )
% 16.19/9.05 = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9004_complex__inverse,axiom,
% 16.19/9.05 ! [A: real,B: real] :
% 16.19/9.05 ( ( invers8013647133539491842omplex @ ( complex2 @ A @ B ) )
% 16.19/9.05 = ( complex2 @ ( divide_divide_real @ A @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ B ) @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9005_binomial__r__part__sum,axiom,
% 16.19/9.05 ! [M: nat] :
% 16.19/9.05 ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
% 16.19/9.05 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9006_choose__linear__sum,axiom,
% 16.19/9.05 ! [N2: nat] :
% 16.19/9.05 ( ( groups3542108847815614940at_nat
% 16.19/9.05 @ ^ [I3: nat] : ( times_times_nat @ I3 @ ( binomial @ N2 @ I3 ) )
% 16.19/9.05 @ ( set_ord_atMost_nat @ N2 ) )
% 16.19/9.05 = ( times_times_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9007_i__even__power,axiom,
% 16.19/9.05 ! [N2: nat] :
% 16.19/9.05 ( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.05 = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9008_log__base__10__eq1,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
% 16.19/9.05 = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9009_sinh__real__less__iff,axiom,
% 16.19/9.05 ! [X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) )
% 16.19/9.05 = ( ord_less_real @ X @ Y ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9010_sinh__real__le__iff,axiom,
% 16.19/9.05 ! [X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( sinh_real @ X ) @ ( sinh_real @ Y ) )
% 16.19/9.05 = ( ord_less_eq_real @ X @ Y ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9011_of__nat__id,axiom,
% 16.19/9.05 ( semiri1316708129612266289at_nat
% 16.19/9.05 = ( ^ [N: nat] : N ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9012_sinh__real__neg__iff,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_real @ ( sinh_real @ X ) @ zero_zero_real )
% 16.19/9.05 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9013_sinh__real__pos__iff,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X ) )
% 16.19/9.05 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9014_sinh__real__nonneg__iff,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X ) )
% 16.19/9.05 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9015_sinh__real__nonpos__iff,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( sinh_real @ X ) @ zero_zero_real )
% 16.19/9.05 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9016_log__one,axiom,
% 16.19/9.05 ! [A: real] :
% 16.19/9.05 ( ( log @ A @ one_one_real )
% 16.19/9.05 = zero_zero_real ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9017_norm__ii,axiom,
% 16.19/9.05 ( ( real_V1022390504157884413omplex @ imaginary_unit )
% 16.19/9.05 = one_one_real ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9018_log__eq__one,axiom,
% 16.19/9.05 ! [A: real] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/9.05 => ( ( A != one_one_real )
% 16.19/9.05 => ( ( log @ A @ A )
% 16.19/9.05 = one_one_real ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9019_log__less__cancel__iff,axiom,
% 16.19/9.05 ! [A: real,X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_real @ one_one_real @ A )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ Y )
% 16.19/9.05 => ( ( ord_less_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
% 16.19/9.05 = ( ord_less_real @ X @ Y ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9020_log__less__one__cancel__iff,axiom,
% 16.19/9.05 ! [A: real,X: real] :
% 16.19/9.05 ( ( ord_less_real @ one_one_real @ A )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_real @ ( log @ A @ X ) @ one_one_real )
% 16.19/9.05 = ( ord_less_real @ X @ A ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9021_one__less__log__cancel__iff,axiom,
% 16.19/9.05 ! [A: real,X: real] :
% 16.19/9.05 ( ( ord_less_real @ one_one_real @ A )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_real @ one_one_real @ ( log @ A @ X ) )
% 16.19/9.05 = ( ord_less_real @ A @ X ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9022_log__less__zero__cancel__iff,axiom,
% 16.19/9.05 ! [A: real,X: real] :
% 16.19/9.05 ( ( ord_less_real @ one_one_real @ A )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_real @ ( log @ A @ X ) @ zero_zero_real )
% 16.19/9.05 = ( ord_less_real @ X @ one_one_real ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9023_zero__less__log__cancel__iff,axiom,
% 16.19/9.05 ! [A: real,X: real] :
% 16.19/9.05 ( ( ord_less_real @ one_one_real @ A )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X ) )
% 16.19/9.05 = ( ord_less_real @ one_one_real @ X ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9024_divide__i,axiom,
% 16.19/9.05 ! [X: complex] :
% 16.19/9.05 ( ( divide1717551699836669952omplex @ X @ imaginary_unit )
% 16.19/9.05 = ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9025_i__squared,axiom,
% 16.19/9.05 ( ( times_times_complex @ imaginary_unit @ imaginary_unit )
% 16.19/9.05 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9026_log__le__cancel__iff,axiom,
% 16.19/9.05 ! [A: real,X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_real @ one_one_real @ A )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ Y )
% 16.19/9.05 => ( ( ord_less_eq_real @ ( log @ A @ X ) @ ( log @ A @ Y ) )
% 16.19/9.05 = ( ord_less_eq_real @ X @ Y ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9027_log__le__one__cancel__iff,axiom,
% 16.19/9.05 ! [A: real,X: real] :
% 16.19/9.05 ( ( ord_less_real @ one_one_real @ A )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_eq_real @ ( log @ A @ X ) @ one_one_real )
% 16.19/9.05 = ( ord_less_eq_real @ X @ A ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9028_one__le__log__cancel__iff,axiom,
% 16.19/9.05 ! [A: real,X: real] :
% 16.19/9.05 ( ( ord_less_real @ one_one_real @ A )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X ) )
% 16.19/9.05 = ( ord_less_eq_real @ A @ X ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9029_log__le__zero__cancel__iff,axiom,
% 16.19/9.05 ! [A: real,X: real] :
% 16.19/9.05 ( ( ord_less_real @ one_one_real @ A )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_eq_real @ ( log @ A @ X ) @ zero_zero_real )
% 16.19/9.05 = ( ord_less_eq_real @ X @ one_one_real ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9030_zero__le__log__cancel__iff,axiom,
% 16.19/9.05 ! [A: real,X: real] :
% 16.19/9.05 ( ( ord_less_real @ one_one_real @ A )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X ) )
% 16.19/9.05 = ( ord_less_eq_real @ one_one_real @ X ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9031_divide__numeral__i,axiom,
% 16.19/9.05 ! [Z: complex,N2: num] :
% 16.19/9.05 ( ( divide1717551699836669952omplex @ Z @ ( times_times_complex @ ( numera6690914467698888265omplex @ N2 ) @ imaginary_unit ) )
% 16.19/9.05 = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) @ ( numera6690914467698888265omplex @ N2 ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9032_log__pow__cancel,axiom,
% 16.19/9.05 ! [A: real,B: nat] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/9.05 => ( ( A != one_one_real )
% 16.19/9.05 => ( ( log @ A @ ( power_power_real @ A @ B ) )
% 16.19/9.05 = ( semiri5074537144036343181t_real @ B ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9033_power2__i,axiom,
% 16.19/9.05 ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.05 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9034_sinh__less__cosh__real,axiom,
% 16.19/9.05 ! [X: real] : ( ord_less_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9035_sinh__le__cosh__real,axiom,
% 16.19/9.05 ! [X: real] : ( ord_less_eq_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9036_complex__i__not__one,axiom,
% 16.19/9.05 imaginary_unit != one_one_complex ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9037_log__def,axiom,
% 16.19/9.05 ( log
% 16.19/9.05 = ( ^ [A5: real,X2: real] : ( divide_divide_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ A5 ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9038_cosh__real__pos,axiom,
% 16.19/9.05 ! [X: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9039_cosh__real__nonpos__le__iff,axiom,
% 16.19/9.05 ! [X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 16.19/9.05 => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 16.19/9.05 => ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
% 16.19/9.05 = ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9040_cosh__real__nonneg__le__iff,axiom,
% 16.19/9.05 ! [X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/9.05 => ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
% 16.19/9.05 = ( ord_less_eq_real @ X @ Y ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9041_cosh__real__nonneg,axiom,
% 16.19/9.05 ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9042_cosh__real__ge__1,axiom,
% 16.19/9.05 ! [X: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9043_cosh__real__strict__mono,axiom,
% 16.19/9.05 ! [X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_real @ X @ Y )
% 16.19/9.05 => ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9044_cosh__real__nonneg__less__iff,axiom,
% 16.19/9.05 ! [X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/9.05 => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
% 16.19/9.05 = ( ord_less_real @ X @ Y ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9045_cosh__real__nonpos__less__iff,axiom,
% 16.19/9.05 ! [X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 16.19/9.05 => ( ( ord_less_eq_real @ Y @ zero_zero_real )
% 16.19/9.05 => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y ) )
% 16.19/9.05 = ( ord_less_real @ Y @ X ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9046_log__ln,axiom,
% 16.19/9.05 ( ln_ln_real
% 16.19/9.05 = ( log @ ( exp_real @ one_one_real ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9047_arcosh__cosh__real,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( arcosh_real @ ( cosh_real @ X ) )
% 16.19/9.05 = X ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9048_imaginary__unit_Ocode,axiom,
% 16.19/9.05 ( imaginary_unit
% 16.19/9.05 = ( complex2 @ zero_zero_real @ one_one_real ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9049_Complex__eq__i,axiom,
% 16.19/9.05 ! [X: real,Y: real] :
% 16.19/9.05 ( ( ( complex2 @ X @ Y )
% 16.19/9.05 = imaginary_unit )
% 16.19/9.05 = ( ( X = zero_zero_real )
% 16.19/9.05 & ( Y = one_one_real ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9050_log__base__change,axiom,
% 16.19/9.05 ! [A: real,B: real,X: real] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/9.05 => ( ( A != one_one_real )
% 16.19/9.05 => ( ( log @ B @ X )
% 16.19/9.05 = ( divide_divide_real @ ( log @ A @ X ) @ ( log @ A @ B ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9051_log__of__power__eq,axiom,
% 16.19/9.05 ! [M: nat,B: real,N2: nat] :
% 16.19/9.05 ( ( ( semiri5074537144036343181t_real @ M )
% 16.19/9.05 = ( power_power_real @ B @ N2 ) )
% 16.19/9.05 => ( ( ord_less_real @ one_one_real @ B )
% 16.19/9.05 => ( ( semiri5074537144036343181t_real @ N2 )
% 16.19/9.05 = ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9052_less__log__of__power,axiom,
% 16.19/9.05 ! [B: real,N2: nat,M: real] :
% 16.19/9.05 ( ( ord_less_real @ ( power_power_real @ B @ N2 ) @ M )
% 16.19/9.05 => ( ( ord_less_real @ one_one_real @ B )
% 16.19/9.05 => ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ M ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9053_log__mult,axiom,
% 16.19/9.05 ! [A: real,X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/9.05 => ( ( A != one_one_real )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ Y )
% 16.19/9.05 => ( ( log @ A @ ( times_times_real @ X @ Y ) )
% 16.19/9.05 = ( plus_plus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9054_log__divide,axiom,
% 16.19/9.05 ! [A: real,X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/9.05 => ( ( A != one_one_real )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ Y )
% 16.19/9.05 => ( ( log @ A @ ( divide_divide_real @ X @ Y ) )
% 16.19/9.05 = ( minus_minus_real @ ( log @ A @ X ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9055_le__log__of__power,axiom,
% 16.19/9.05 ! [B: real,N2: nat,M: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( power_power_real @ B @ N2 ) @ M )
% 16.19/9.05 => ( ( ord_less_real @ one_one_real @ B )
% 16.19/9.05 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ M ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9056_log__base__pow,axiom,
% 16.19/9.05 ! [A: real,N2: nat,X: real] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/9.05 => ( ( log @ ( power_power_real @ A @ N2 ) @ X )
% 16.19/9.05 = ( divide_divide_real @ ( log @ A @ X ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9057_log__nat__power,axiom,
% 16.19/9.05 ! [X: real,B: real,N2: nat] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( log @ B @ ( power_power_real @ X @ N2 ) )
% 16.19/9.05 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ X ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9058_log__inverse,axiom,
% 16.19/9.05 ! [A: real,X: real] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/9.05 => ( ( A != one_one_real )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( log @ A @ ( inverse_inverse_real @ X ) )
% 16.19/9.05 = ( uminus_uminus_real @ ( log @ A @ X ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9059_log2__of__power__eq,axiom,
% 16.19/9.05 ! [M: nat,N2: nat] :
% 16.19/9.05 ( ( M
% 16.19/9.05 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/9.05 => ( ( semiri5074537144036343181t_real @ N2 )
% 16.19/9.05 = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9060_log__of__power__less,axiom,
% 16.19/9.05 ! [M: nat,B: real,N2: nat] :
% 16.19/9.05 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N2 ) )
% 16.19/9.05 => ( ( ord_less_real @ one_one_real @ B )
% 16.19/9.05 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 16.19/9.05 => ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9061_log__eq__div__ln__mult__log,axiom,
% 16.19/9.05 ! [A: real,B: real,X: real] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/9.05 => ( ( A != one_one_real )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ B )
% 16.19/9.05 => ( ( B != one_one_real )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( log @ A @ X )
% 16.19/9.05 = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9062_log__of__power__le,axiom,
% 16.19/9.05 ! [M: nat,B: real,N2: nat] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N2 ) )
% 16.19/9.05 => ( ( ord_less_real @ one_one_real @ B )
% 16.19/9.05 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 16.19/9.05 => ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9063_less__log2__of__power,axiom,
% 16.19/9.05 ! [N2: nat,M: nat] :
% 16.19/9.05 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M )
% 16.19/9.05 => ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9064_le__log2__of__power,axiom,
% 16.19/9.05 ! [N2: nat,M: nat] :
% 16.19/9.05 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M )
% 16.19/9.05 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9065_log2__of__power__less,axiom,
% 16.19/9.05 ! [M: nat,N2: nat] :
% 16.19/9.05 ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/9.05 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 16.19/9.05 => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9066_cosh__ln__real,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( cosh_real @ ( ln_ln_real @ X ) )
% 16.19/9.05 = ( divide_divide_real @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9067_log2__of__power__le,axiom,
% 16.19/9.05 ! [M: nat,N2: nat] :
% 16.19/9.05 ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/9.05 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 16.19/9.05 => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9068_log__base__10__eq2,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
% 16.19/9.05 = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9069_sinh__ln__real,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( sinh_real @ ( ln_ln_real @ X ) )
% 16.19/9.05 = ( divide_divide_real @ ( minus_minus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9070_Arg__minus__ii,axiom,
% 16.19/9.05 ( ( arg @ ( uminus1482373934393186551omplex @ imaginary_unit ) )
% 16.19/9.05 = ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9071_ceiling__log__nat__eq__powr__iff,axiom,
% 16.19/9.05 ! [B: nat,K: nat,N2: nat] :
% 16.19/9.05 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 16.19/9.05 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 16.19/9.05 => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 16.19/9.05 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) )
% 16.19/9.05 = ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 16.19/9.05 & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9072_Arg__ii,axiom,
% 16.19/9.05 ( ( arg @ imaginary_unit )
% 16.19/9.05 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9073_ceiling__log2__div2,axiom,
% 16.19/9.05 ! [N2: nat] :
% 16.19/9.05 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.05 => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 16.19/9.05 = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9074_ceiling__divide__eq__div__numeral,axiom,
% 16.19/9.05 ! [A: num,B: num] :
% 16.19/9.05 ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
% 16.19/9.05 = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9075_ceiling__minus__divide__eq__div__numeral,axiom,
% 16.19/9.05 ! [A: num,B: num] :
% 16.19/9.05 ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
% 16.19/9.05 = ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9076_Arg__bounded,axiom,
% 16.19/9.05 ! [Z: complex] :
% 16.19/9.05 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 16.19/9.05 & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9077_ceiling__log__nat__eq__if,axiom,
% 16.19/9.05 ! [B: nat,N2: nat,K: nat] :
% 16.19/9.05 ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 16.19/9.05 => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) )
% 16.19/9.05 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 16.19/9.05 => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 16.19/9.05 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9078_cis__minus__pi__half,axiom,
% 16.19/9.05 ( ( cis @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 16.19/9.05 = ( uminus1482373934393186551omplex @ imaginary_unit ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9079_ceiling__log__eq__powr__iff,axiom,
% 16.19/9.05 ! [X: real,B: real,K: nat] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_real @ one_one_real @ B )
% 16.19/9.05 => ( ( ( archim7802044766580827645g_real @ ( log @ B @ X ) )
% 16.19/9.05 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
% 16.19/9.05 = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X )
% 16.19/9.05 & ( ord_less_eq_real @ X @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9080_floor__log__nat__eq__powr__iff,axiom,
% 16.19/9.05 ! [B: nat,K: nat,N2: nat] :
% 16.19/9.05 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 16.19/9.05 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 16.19/9.05 => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 16.19/9.05 = ( semiri1314217659103216013at_int @ N2 ) )
% 16.19/9.05 = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 16.19/9.05 & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9081_int__ge__less__than2__def,axiom,
% 16.19/9.05 ( int_ge_less_than2
% 16.19/9.05 = ( ^ [D2: int] :
% 16.19/9.05 ( collec213857154873943460nt_int
% 16.19/9.05 @ ( produc4947309494688390418_int_o
% 16.19/9.05 @ ^ [Z6: int,Z3: int] :
% 16.19/9.05 ( ( ord_less_eq_int @ D2 @ Z3 )
% 16.19/9.05 & ( ord_less_int @ Z6 @ Z3 ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9082_powr__gt__zero,axiom,
% 16.19/9.05 ! [X: real,A: real] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X @ A ) )
% 16.19/9.05 = ( X != zero_zero_real ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9083_powr__nonneg__iff,axiom,
% 16.19/9.05 ! [A: real,X: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( powr_real @ A @ X ) @ zero_zero_real )
% 16.19/9.05 = ( A = zero_zero_real ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9084_powr__less__cancel__iff,axiom,
% 16.19/9.05 ! [X: real,A: real,B: real] :
% 16.19/9.05 ( ( ord_less_real @ one_one_real @ X )
% 16.19/9.05 => ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 16.19/9.05 = ( ord_less_real @ A @ B ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9085_norm__cis,axiom,
% 16.19/9.05 ! [A: real] :
% 16.19/9.05 ( ( real_V1022390504157884413omplex @ ( cis @ A ) )
% 16.19/9.05 = one_one_real ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9086_cis__zero,axiom,
% 16.19/9.05 ( ( cis @ zero_zero_real )
% 16.19/9.05 = one_one_complex ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9087_powr__eq__one__iff,axiom,
% 16.19/9.05 ! [A: real,X: real] :
% 16.19/9.05 ( ( ord_less_real @ one_one_real @ A )
% 16.19/9.05 => ( ( ( powr_real @ A @ X )
% 16.19/9.05 = one_one_real )
% 16.19/9.05 = ( X = zero_zero_real ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9088_powr__one__gt__zero__iff,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ( powr_real @ X @ one_one_real )
% 16.19/9.05 = X )
% 16.19/9.05 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9089_powr__one,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( powr_real @ X @ one_one_real )
% 16.19/9.05 = X ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9090_powr__le__cancel__iff,axiom,
% 16.19/9.05 ! [X: real,A: real,B: real] :
% 16.19/9.05 ( ( ord_less_real @ one_one_real @ X )
% 16.19/9.05 => ( ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 16.19/9.05 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9091_numeral__powr__numeral__real,axiom,
% 16.19/9.05 ! [M: num,N2: num] :
% 16.19/9.05 ( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
% 16.19/9.05 = ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9092_cis__pi,axiom,
% 16.19/9.05 ( ( cis @ pi )
% 16.19/9.05 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9093_powr__log__cancel,axiom,
% 16.19/9.05 ! [A: real,X: real] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/9.05 => ( ( A != one_one_real )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( powr_real @ A @ ( log @ A @ X ) )
% 16.19/9.05 = X ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9094_log__powr__cancel,axiom,
% 16.19/9.05 ! [A: real,Y: real] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/9.05 => ( ( A != one_one_real )
% 16.19/9.05 => ( ( log @ A @ ( powr_real @ A @ Y ) )
% 16.19/9.05 = Y ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9095_floor__divide__eq__div__numeral,axiom,
% 16.19/9.05 ! [A: num,B: num] :
% 16.19/9.05 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
% 16.19/9.05 = ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9096_powr__numeral,axiom,
% 16.19/9.05 ! [X: real,N2: num] :
% 16.19/9.05 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( powr_real @ X @ ( numeral_numeral_real @ N2 ) )
% 16.19/9.05 = ( power_power_real @ X @ ( numeral_numeral_nat @ N2 ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9097_cis__pi__half,axiom,
% 16.19/9.05 ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.05 = imaginary_unit ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9098_floor__one__divide__eq__div__numeral,axiom,
% 16.19/9.05 ! [B: num] :
% 16.19/9.05 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) )
% 16.19/9.05 = ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9099_cis__2pi,axiom,
% 16.19/9.05 ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 16.19/9.05 = one_one_complex ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9100_floor__minus__divide__eq__div__numeral,axiom,
% 16.19/9.05 ! [A: num,B: num] :
% 16.19/9.05 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
% 16.19/9.05 = ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9101_square__powr__half,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( powr_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.05 = ( abs_abs_real @ X ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9102_floor__minus__one__divide__eq__div__numeral,axiom,
% 16.19/9.05 ! [B: num] :
% 16.19/9.05 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) ) )
% 16.19/9.05 = ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9103_powr__powr,axiom,
% 16.19/9.05 ! [X: real,A: real,B: real] :
% 16.19/9.05 ( ( powr_real @ ( powr_real @ X @ A ) @ B )
% 16.19/9.05 = ( powr_real @ X @ ( times_times_real @ A @ B ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9104_powr__less__mono2__neg,axiom,
% 16.19/9.05 ! [A: real,X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_real @ A @ zero_zero_real )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_real @ X @ Y )
% 16.19/9.05 => ( ord_less_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9105_powr__non__neg,axiom,
% 16.19/9.05 ! [A: real,X: real] :
% 16.19/9.05 ~ ( ord_less_real @ ( powr_real @ A @ X ) @ zero_zero_real ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9106_powr__ge__pzero,axiom,
% 16.19/9.05 ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X @ Y ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9107_powr__mono2,axiom,
% 16.19/9.05 ! [A: real,X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/9.05 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_eq_real @ X @ Y )
% 16.19/9.05 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9108_powr__less__mono,axiom,
% 16.19/9.05 ! [A: real,B: real,X: real] :
% 16.19/9.05 ( ( ord_less_real @ A @ B )
% 16.19/9.05 => ( ( ord_less_real @ one_one_real @ X )
% 16.19/9.05 => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9109_powr__less__cancel,axiom,
% 16.19/9.05 ! [X: real,A: real,B: real] :
% 16.19/9.05 ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 16.19/9.05 => ( ( ord_less_real @ one_one_real @ X )
% 16.19/9.05 => ( ord_less_real @ A @ B ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9110_powr__mono,axiom,
% 16.19/9.05 ! [A: real,B: real,X: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ A @ B )
% 16.19/9.05 => ( ( ord_less_eq_real @ one_one_real @ X )
% 16.19/9.05 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9111_cis__mult,axiom,
% 16.19/9.05 ! [A: real,B: real] :
% 16.19/9.05 ( ( times_times_complex @ ( cis @ A ) @ ( cis @ B ) )
% 16.19/9.05 = ( cis @ ( plus_plus_real @ A @ B ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9112_cis__divide,axiom,
% 16.19/9.05 ! [A: real,B: real] :
% 16.19/9.05 ( ( divide1717551699836669952omplex @ ( cis @ A ) @ ( cis @ B ) )
% 16.19/9.05 = ( cis @ ( minus_minus_real @ A @ B ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9113_powr__mono2_H,axiom,
% 16.19/9.05 ! [A: real,X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_eq_real @ X @ Y )
% 16.19/9.05 => ( ord_less_eq_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9114_powr__less__mono2,axiom,
% 16.19/9.05 ! [A: real,X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/9.05 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_real @ X @ Y )
% 16.19/9.05 => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9115_powr__inj,axiom,
% 16.19/9.05 ! [A: real,X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/9.05 => ( ( A != one_one_real )
% 16.19/9.05 => ( ( ( powr_real @ A @ X )
% 16.19/9.05 = ( powr_real @ A @ Y ) )
% 16.19/9.05 = ( X = Y ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9116_gr__one__powr,axiom,
% 16.19/9.05 ! [X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_real @ one_one_real @ X )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ Y )
% 16.19/9.05 => ( ord_less_real @ one_one_real @ ( powr_real @ X @ Y ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9117_ge__one__powr__ge__zero,axiom,
% 16.19/9.05 ! [X: real,A: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ one_one_real @ X )
% 16.19/9.05 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/9.05 => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9118_powr__mono__both,axiom,
% 16.19/9.05 ! [A: real,B: real,X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/9.05 => ( ( ord_less_eq_real @ A @ B )
% 16.19/9.05 => ( ( ord_less_eq_real @ one_one_real @ X )
% 16.19/9.05 => ( ( ord_less_eq_real @ X @ Y )
% 16.19/9.05 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ B ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9119_powr__le1,axiom,
% 16.19/9.05 ! [A: real,X: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 16.19/9.05 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_eq_real @ X @ one_one_real )
% 16.19/9.05 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ one_one_real ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9120_powr__divide,axiom,
% 16.19/9.05 ! [X: real,Y: real,A: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/9.05 => ( ( powr_real @ ( divide_divide_real @ X @ Y ) @ A )
% 16.19/9.05 = ( divide_divide_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9121_powr__mult,axiom,
% 16.19/9.05 ! [X: real,Y: real,A: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/9.05 => ( ( powr_real @ ( times_times_real @ X @ Y ) @ A )
% 16.19/9.05 = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9122_inverse__powr,axiom,
% 16.19/9.05 ! [Y: real,A: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/9.05 => ( ( powr_real @ ( inverse_inverse_real @ Y ) @ A )
% 16.19/9.05 = ( inverse_inverse_real @ ( powr_real @ Y @ A ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9123_divide__powr__uminus,axiom,
% 16.19/9.05 ! [A: real,B: real,C: real] :
% 16.19/9.05 ( ( divide_divide_real @ A @ ( powr_real @ B @ C ) )
% 16.19/9.05 = ( times_times_real @ A @ ( powr_real @ B @ ( uminus_uminus_real @ C ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9124_log__base__powr,axiom,
% 16.19/9.05 ! [A: real,B: real,X: real] :
% 16.19/9.05 ( ( A != zero_zero_real )
% 16.19/9.05 => ( ( log @ ( powr_real @ A @ B ) @ X )
% 16.19/9.05 = ( divide_divide_real @ ( log @ A @ X ) @ B ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9125_ln__powr,axiom,
% 16.19/9.05 ! [X: real,Y: real] :
% 16.19/9.05 ( ( X != zero_zero_real )
% 16.19/9.05 => ( ( ln_ln_real @ ( powr_real @ X @ Y ) )
% 16.19/9.05 = ( times_times_real @ Y @ ( ln_ln_real @ X ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9126_log__powr,axiom,
% 16.19/9.05 ! [X: real,B: real,Y: real] :
% 16.19/9.05 ( ( X != zero_zero_real )
% 16.19/9.05 => ( ( log @ B @ ( powr_real @ X @ Y ) )
% 16.19/9.05 = ( times_times_real @ Y @ ( log @ B @ X ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9127_floor__log__eq__powr__iff,axiom,
% 16.19/9.05 ! [X: real,B: real,K: int] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_real @ one_one_real @ B )
% 16.19/9.05 => ( ( ( archim6058952711729229775r_real @ ( log @ B @ X ) )
% 16.19/9.05 = K )
% 16.19/9.05 = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X )
% 16.19/9.05 & ( ord_less_real @ X @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9128_powr__realpow,axiom,
% 16.19/9.05 ! [X: real,N2: nat] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N2 ) )
% 16.19/9.05 = ( power_power_real @ X @ N2 ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9129_less__log__iff,axiom,
% 16.19/9.05 ! [B: real,X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_real @ one_one_real @ B )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_real @ Y @ ( log @ B @ X ) )
% 16.19/9.05 = ( ord_less_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9130_log__less__iff,axiom,
% 16.19/9.05 ! [B: real,X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_real @ one_one_real @ B )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_real @ ( log @ B @ X ) @ Y )
% 16.19/9.05 = ( ord_less_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9131_less__powr__iff,axiom,
% 16.19/9.05 ! [B: real,X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_real @ one_one_real @ B )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_real @ X @ ( powr_real @ B @ Y ) )
% 16.19/9.05 = ( ord_less_real @ ( log @ B @ X ) @ Y ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9132_powr__less__iff,axiom,
% 16.19/9.05 ! [B: real,X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_real @ one_one_real @ B )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_real @ ( powr_real @ B @ Y ) @ X )
% 16.19/9.05 = ( ord_less_real @ Y @ ( log @ B @ X ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9133_real__of__int__floor__add__one__gt,axiom,
% 16.19/9.05 ! [R: real] : ( ord_less_real @ R @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) @ one_one_real ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9134_floor__eq,axiom,
% 16.19/9.05 ! [N2: int,X: real] :
% 16.19/9.05 ( ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ X )
% 16.19/9.05 => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
% 16.19/9.05 => ( ( archim6058952711729229775r_real @ X )
% 16.19/9.05 = N2 ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9135_real__of__int__floor__add__one__ge,axiom,
% 16.19/9.05 ! [R: real] : ( ord_less_eq_real @ R @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) @ one_one_real ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9136_real__of__int__floor__gt__diff__one,axiom,
% 16.19/9.05 ! [R: real] : ( ord_less_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9137_real__of__int__floor__ge__diff__one,axiom,
% 16.19/9.05 ! [R: real] : ( ord_less_eq_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9138_DeMoivre,axiom,
% 16.19/9.05 ! [A: real,N2: nat] :
% 16.19/9.05 ( ( power_power_complex @ ( cis @ A ) @ N2 )
% 16.19/9.05 = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ A ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9139_powr__neg__one,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( powr_real @ X @ ( uminus_uminus_real @ one_one_real ) )
% 16.19/9.05 = ( divide_divide_real @ one_one_real @ X ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9140_powr__mult__base,axiom,
% 16.19/9.05 ! [X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( times_times_real @ X @ ( powr_real @ X @ Y ) )
% 16.19/9.05 = ( powr_real @ X @ ( plus_plus_real @ one_one_real @ Y ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9141_le__log__iff,axiom,
% 16.19/9.05 ! [B: real,X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_real @ one_one_real @ B )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_eq_real @ Y @ ( log @ B @ X ) )
% 16.19/9.05 = ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9142_log__le__iff,axiom,
% 16.19/9.05 ! [B: real,X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_real @ one_one_real @ B )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_eq_real @ ( log @ B @ X ) @ Y )
% 16.19/9.05 = ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9143_le__powr__iff,axiom,
% 16.19/9.05 ! [B: real,X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_real @ one_one_real @ B )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_eq_real @ X @ ( powr_real @ B @ Y ) )
% 16.19/9.05 = ( ord_less_eq_real @ ( log @ B @ X ) @ Y ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9144_powr__le__iff,axiom,
% 16.19/9.05 ! [B: real,X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_real @ one_one_real @ B )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X )
% 16.19/9.05 = ( ord_less_eq_real @ Y @ ( log @ B @ X ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9145_floor__eq2,axiom,
% 16.19/9.05 ! [N2: int,X: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N2 ) @ X )
% 16.19/9.05 => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
% 16.19/9.05 => ( ( archim6058952711729229775r_real @ X )
% 16.19/9.05 = N2 ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9146_floor__divide__real__eq__div,axiom,
% 16.19/9.05 ! [B: int,A: real] :
% 16.19/9.05 ( ( ord_less_eq_int @ zero_zero_int @ B )
% 16.19/9.05 => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B ) ) )
% 16.19/9.05 = ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9147_ln__powr__bound,axiom,
% 16.19/9.05 ! [X: real,A: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ one_one_real @ X )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/9.05 => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( divide_divide_real @ ( powr_real @ X @ A ) @ A ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9148_ln__powr__bound2,axiom,
% 16.19/9.05 ! [X: real,A: real] :
% 16.19/9.05 ( ( ord_less_real @ one_one_real @ X )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/9.05 => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9149_add__log__eq__powr,axiom,
% 16.19/9.05 ! [B: real,X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ B )
% 16.19/9.05 => ( ( B != one_one_real )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( plus_plus_real @ Y @ ( log @ B @ X ) )
% 16.19/9.05 = ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9150_log__add__eq__powr,axiom,
% 16.19/9.05 ! [B: real,X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ B )
% 16.19/9.05 => ( ( B != one_one_real )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( plus_plus_real @ ( log @ B @ X ) @ Y )
% 16.19/9.05 = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9151_minus__log__eq__powr,axiom,
% 16.19/9.05 ! [B: real,X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ B )
% 16.19/9.05 => ( ( B != one_one_real )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( minus_minus_real @ Y @ ( log @ B @ X ) )
% 16.19/9.05 = ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9152_log__minus__eq__powr,axiom,
% 16.19/9.05 ! [B: real,X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ B )
% 16.19/9.05 => ( ( B != one_one_real )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( minus_minus_real @ ( log @ B @ X ) @ Y )
% 16.19/9.05 = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ ( uminus_uminus_real @ Y ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9153_powr__half__sqrt,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.05 = ( sqrt @ X ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9154_powr__neg__numeral,axiom,
% 16.19/9.05 ! [X: real,N2: num] :
% 16.19/9.05 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( powr_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 16.19/9.05 = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9155_floor__log2__div2,axiom,
% 16.19/9.05 ! [N2: nat] :
% 16.19/9.05 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.05 => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 16.19/9.05 = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9156_floor__log__nat__eq__if,axiom,
% 16.19/9.05 ! [B: nat,N2: nat,K: nat] :
% 16.19/9.05 ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 16.19/9.05 => ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) )
% 16.19/9.05 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 16.19/9.05 => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 16.19/9.05 = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9157_int__ge__less__than__def,axiom,
% 16.19/9.05 ( int_ge_less_than
% 16.19/9.05 = ( ^ [D2: int] :
% 16.19/9.05 ( collec213857154873943460nt_int
% 16.19/9.05 @ ( produc4947309494688390418_int_o
% 16.19/9.05 @ ^ [Z6: int,Z3: int] :
% 16.19/9.05 ( ( ord_less_eq_int @ D2 @ Z6 )
% 16.19/9.05 & ( ord_less_int @ Z6 @ Z3 ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9158_bij__betw__roots__unity,axiom,
% 16.19/9.05 ! [N2: nat] :
% 16.19/9.05 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.05 => ( bij_betw_nat_complex
% 16.19/9.05 @ ^ [K3: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K3 ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 16.19/9.05 @ ( set_ord_lessThan_nat @ N2 )
% 16.19/9.05 @ ( collect_complex
% 16.19/9.05 @ ^ [Z3: complex] :
% 16.19/9.05 ( ( power_power_complex @ Z3 @ N2 )
% 16.19/9.05 = one_one_complex ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9159_summable__complex__of__real,axiom,
% 16.19/9.05 ! [F: nat > real] :
% 16.19/9.05 ( ( summable_complex
% 16.19/9.05 @ ^ [N: nat] : ( real_V4546457046886955230omplex @ ( F @ N ) ) )
% 16.19/9.05 = ( summable_real @ F ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9160_exp__pi__i,axiom,
% 16.19/9.05 ( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
% 16.19/9.05 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9161_exp__pi__i_H,axiom,
% 16.19/9.05 ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
% 16.19/9.05 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9162_exp__two__pi__i,axiom,
% 16.19/9.05 ( ( exp_complex @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( real_V4546457046886955230omplex @ pi ) ) @ imaginary_unit ) )
% 16.19/9.05 = one_one_complex ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9163_exp__two__pi__i_H,axiom,
% 16.19/9.05 ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) )
% 16.19/9.05 = one_one_complex ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9164_Complex__mult__complex__of__real,axiom,
% 16.19/9.05 ! [X: real,Y: real,R: real] :
% 16.19/9.05 ( ( times_times_complex @ ( complex2 @ X @ Y ) @ ( real_V4546457046886955230omplex @ R ) )
% 16.19/9.05 = ( complex2 @ ( times_times_real @ X @ R ) @ ( times_times_real @ Y @ R ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9165_complex__of__real__mult__Complex,axiom,
% 16.19/9.05 ! [R: real,X: real,Y: real] :
% 16.19/9.05 ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ ( complex2 @ X @ Y ) )
% 16.19/9.05 = ( complex2 @ ( times_times_real @ R @ X ) @ ( times_times_real @ R @ Y ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9166_Complex__add__complex__of__real,axiom,
% 16.19/9.05 ! [X: real,Y: real,R: real] :
% 16.19/9.05 ( ( plus_plus_complex @ ( complex2 @ X @ Y ) @ ( real_V4546457046886955230omplex @ R ) )
% 16.19/9.05 = ( complex2 @ ( plus_plus_real @ X @ R ) @ Y ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9167_complex__of__real__add__Complex,axiom,
% 16.19/9.05 ! [R: real,X: real,Y: real] :
% 16.19/9.05 ( ( plus_plus_complex @ ( real_V4546457046886955230omplex @ R ) @ ( complex2 @ X @ Y ) )
% 16.19/9.05 = ( complex2 @ ( plus_plus_real @ R @ X ) @ Y ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9168_cmod__unit__one,axiom,
% 16.19/9.05 ! [A: real] :
% 16.19/9.05 ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) )
% 16.19/9.05 = one_one_real ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9169_csqrt__ii,axiom,
% 16.19/9.05 ( ( csqrt @ imaginary_unit )
% 16.19/9.05 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ one_one_complex @ imaginary_unit ) @ ( real_V4546457046886955230omplex @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9170_arctan__def,axiom,
% 16.19/9.05 ( arctan
% 16.19/9.05 = ( ^ [Y2: real] :
% 16.19/9.05 ( the_real
% 16.19/9.05 @ ^ [X2: real] :
% 16.19/9.05 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 16.19/9.05 & ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.05 & ( ( tan_real @ X2 )
% 16.19/9.05 = Y2 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9171_arcsin__def,axiom,
% 16.19/9.05 ( arcsin
% 16.19/9.05 = ( ^ [Y2: real] :
% 16.19/9.05 ( the_real
% 16.19/9.05 @ ^ [X2: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 16.19/9.05 & ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.05 & ( ( sin_real @ X2 )
% 16.19/9.05 = Y2 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9172_modulo__int__unfold,axiom,
% 16.19/9.05 ! [L2: int,K: int,N2: nat,M: nat] :
% 16.19/9.05 ( ( ( ( ( sgn_sgn_int @ L2 )
% 16.19/9.05 = zero_zero_int )
% 16.19/9.05 | ( ( sgn_sgn_int @ K )
% 16.19/9.05 = zero_zero_int )
% 16.19/9.05 | ( N2 = zero_zero_nat ) )
% 16.19/9.05 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 16.19/9.05 = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
% 16.19/9.05 & ( ~ ( ( ( sgn_sgn_int @ L2 )
% 16.19/9.05 = zero_zero_int )
% 16.19/9.05 | ( ( sgn_sgn_int @ K )
% 16.19/9.05 = zero_zero_int )
% 16.19/9.05 | ( N2 = zero_zero_nat ) )
% 16.19/9.05 => ( ( ( ( sgn_sgn_int @ K )
% 16.19/9.05 = ( sgn_sgn_int @ L2 ) )
% 16.19/9.05 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 16.19/9.05 = ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) )
% 16.19/9.05 & ( ( ( sgn_sgn_int @ K )
% 16.19/9.05 != ( sgn_sgn_int @ L2 ) )
% 16.19/9.05 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 16.19/9.05 = ( times_times_int @ ( sgn_sgn_int @ L2 )
% 16.19/9.05 @ ( minus_minus_int
% 16.19/9.05 @ ( semiri1314217659103216013at_int
% 16.19/9.05 @ ( times_times_nat @ N2
% 16.19/9.05 @ ( zero_n2687167440665602831ol_nat
% 16.19/9.05 @ ~ ( dvd_dvd_nat @ N2 @ M ) ) ) )
% 16.19/9.05 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9173_csqrt__1,axiom,
% 16.19/9.05 ( ( csqrt @ one_one_complex )
% 16.19/9.05 = one_one_complex ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9174_csqrt__eq__1,axiom,
% 16.19/9.05 ! [Z: complex] :
% 16.19/9.05 ( ( ( csqrt @ Z )
% 16.19/9.05 = one_one_complex )
% 16.19/9.05 = ( Z = one_one_complex ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9175_power2__csqrt,axiom,
% 16.19/9.05 ! [Z: complex] :
% 16.19/9.05 ( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.05 = Z ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9176_div__eq__sgn__abs,axiom,
% 16.19/9.05 ! [K: int,L2: int] :
% 16.19/9.05 ( ( ( sgn_sgn_int @ K )
% 16.19/9.05 = ( sgn_sgn_int @ L2 ) )
% 16.19/9.05 => ( ( divide_divide_int @ K @ L2 )
% 16.19/9.05 = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9177_ln__real__def,axiom,
% 16.19/9.05 ( ln_ln_real
% 16.19/9.05 = ( ^ [X2: real] :
% 16.19/9.05 ( the_real
% 16.19/9.05 @ ^ [U2: real] :
% 16.19/9.05 ( ( exp_real @ U2 )
% 16.19/9.05 = X2 ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9178_sgn__mod,axiom,
% 16.19/9.05 ! [L2: int,K: int] :
% 16.19/9.05 ( ( L2 != zero_zero_int )
% 16.19/9.05 => ( ~ ( dvd_dvd_int @ L2 @ K )
% 16.19/9.05 => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L2 ) )
% 16.19/9.05 = ( sgn_sgn_int @ L2 ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9179_ln__neg__is__const,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 16.19/9.05 => ( ( ln_ln_real @ X )
% 16.19/9.05 = ( the_real
% 16.19/9.05 @ ^ [X2: real] : $false ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9180_zsgn__def,axiom,
% 16.19/9.05 ( sgn_sgn_int
% 16.19/9.05 = ( ^ [I3: int] : ( if_int @ ( I3 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9181_div__sgn__abs__cancel,axiom,
% 16.19/9.05 ! [V: int,K: int,L2: int] :
% 16.19/9.05 ( ( V != zero_zero_int )
% 16.19/9.05 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L2 ) ) )
% 16.19/9.05 = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9182_div__dvd__sgn__abs,axiom,
% 16.19/9.05 ! [L2: int,K: int] :
% 16.19/9.05 ( ( dvd_dvd_int @ L2 @ K )
% 16.19/9.05 => ( ( divide_divide_int @ K @ L2 )
% 16.19/9.05 = ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( sgn_sgn_int @ L2 ) ) @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9183_of__real__sqrt,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( real_V4546457046886955230omplex @ ( sqrt @ X ) )
% 16.19/9.05 = ( csqrt @ ( real_V4546457046886955230omplex @ X ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9184_arccos__def,axiom,
% 16.19/9.05 ( arccos
% 16.19/9.05 = ( ^ [Y2: real] :
% 16.19/9.05 ( the_real
% 16.19/9.05 @ ^ [X2: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 16.19/9.05 & ( ord_less_eq_real @ X2 @ pi )
% 16.19/9.05 & ( ( cos_real @ X2 )
% 16.19/9.05 = Y2 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9185_eucl__rel__int__remainderI,axiom,
% 16.19/9.05 ! [R: int,L2: int,K: int,Q2: int] :
% 16.19/9.05 ( ( ( sgn_sgn_int @ R )
% 16.19/9.05 = ( sgn_sgn_int @ L2 ) )
% 16.19/9.05 => ( ( ord_less_int @ ( abs_abs_int @ R ) @ ( abs_abs_int @ L2 ) )
% 16.19/9.05 => ( ( K
% 16.19/9.05 = ( plus_plus_int @ ( times_times_int @ Q2 @ L2 ) @ R ) )
% 16.19/9.05 => ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9186_eucl__rel__int_Ocases,axiom,
% 16.19/9.05 ! [A12: int,A23: int,A32: product_prod_int_int] :
% 16.19/9.05 ( ( eucl_rel_int @ A12 @ A23 @ A32 )
% 16.19/9.05 => ( ( ( A23 = zero_zero_int )
% 16.19/9.05 => ( A32
% 16.19/9.05 != ( product_Pair_int_int @ zero_zero_int @ A12 ) ) )
% 16.19/9.05 => ( ! [Q3: int] :
% 16.19/9.05 ( ( A32
% 16.19/9.05 = ( product_Pair_int_int @ Q3 @ zero_zero_int ) )
% 16.19/9.05 => ( ( A23 != zero_zero_int )
% 16.19/9.05 => ( A12
% 16.19/9.05 != ( times_times_int @ Q3 @ A23 ) ) ) )
% 16.19/9.05 => ~ ! [R3: int,Q3: int] :
% 16.19/9.05 ( ( A32
% 16.19/9.05 = ( product_Pair_int_int @ Q3 @ R3 ) )
% 16.19/9.05 => ( ( ( sgn_sgn_int @ R3 )
% 16.19/9.05 = ( sgn_sgn_int @ A23 ) )
% 16.19/9.05 => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A23 ) )
% 16.19/9.05 => ( A12
% 16.19/9.05 != ( plus_plus_int @ ( times_times_int @ Q3 @ A23 ) @ R3 ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9187_eucl__rel__int_Osimps,axiom,
% 16.19/9.05 ( eucl_rel_int
% 16.19/9.05 = ( ^ [A1: int,A22: int,A33: product_prod_int_int] :
% 16.19/9.05 ( ? [K3: int] :
% 16.19/9.05 ( ( A1 = K3 )
% 16.19/9.05 & ( A22 = zero_zero_int )
% 16.19/9.05 & ( A33
% 16.19/9.05 = ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
% 16.19/9.05 | ? [L: int,K3: int,Q4: int] :
% 16.19/9.05 ( ( A1 = K3 )
% 16.19/9.05 & ( A22 = L )
% 16.19/9.05 & ( A33
% 16.19/9.05 = ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
% 16.19/9.05 & ( L != zero_zero_int )
% 16.19/9.05 & ( K3
% 16.19/9.05 = ( times_times_int @ Q4 @ L ) ) )
% 16.19/9.05 | ? [R5: int,L: int,K3: int,Q4: int] :
% 16.19/9.05 ( ( A1 = K3 )
% 16.19/9.05 & ( A22 = L )
% 16.19/9.05 & ( A33
% 16.19/9.05 = ( product_Pair_int_int @ Q4 @ R5 ) )
% 16.19/9.05 & ( ( sgn_sgn_int @ R5 )
% 16.19/9.05 = ( sgn_sgn_int @ L ) )
% 16.19/9.05 & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L ) )
% 16.19/9.05 & ( K3
% 16.19/9.05 = ( plus_plus_int @ ( times_times_int @ Q4 @ L ) @ R5 ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9188_div__noneq__sgn__abs,axiom,
% 16.19/9.05 ! [L2: int,K: int] :
% 16.19/9.05 ( ( L2 != zero_zero_int )
% 16.19/9.05 => ( ( ( sgn_sgn_int @ K )
% 16.19/9.05 != ( sgn_sgn_int @ L2 ) )
% 16.19/9.05 => ( ( divide_divide_int @ K @ L2 )
% 16.19/9.05 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) )
% 16.19/9.05 @ ( zero_n2684676970156552555ol_int
% 16.19/9.05 @ ~ ( dvd_dvd_int @ L2 @ K ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9189_pi__half,axiom,
% 16.19/9.05 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 16.19/9.05 = ( the_real
% 16.19/9.05 @ ^ [X2: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 16.19/9.05 & ( ord_less_eq_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 16.19/9.05 & ( ( cos_real @ X2 )
% 16.19/9.05 = zero_zero_real ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9190_pi__def,axiom,
% 16.19/9.05 ( pi
% 16.19/9.05 = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 16.19/9.05 @ ( the_real
% 16.19/9.05 @ ^ [X2: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 16.19/9.05 & ( ord_less_eq_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 16.19/9.05 & ( ( cos_real @ X2 )
% 16.19/9.05 = zero_zero_real ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9191_divide__int__unfold,axiom,
% 16.19/9.05 ! [L2: int,K: int,N2: nat,M: nat] :
% 16.19/9.05 ( ( ( ( ( sgn_sgn_int @ L2 )
% 16.19/9.05 = zero_zero_int )
% 16.19/9.05 | ( ( sgn_sgn_int @ K )
% 16.19/9.05 = zero_zero_int )
% 16.19/9.05 | ( N2 = zero_zero_nat ) )
% 16.19/9.05 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 16.19/9.05 = zero_zero_int ) )
% 16.19/9.05 & ( ~ ( ( ( sgn_sgn_int @ L2 )
% 16.19/9.05 = zero_zero_int )
% 16.19/9.05 | ( ( sgn_sgn_int @ K )
% 16.19/9.05 = zero_zero_int )
% 16.19/9.05 | ( N2 = zero_zero_nat ) )
% 16.19/9.05 => ( ( ( ( sgn_sgn_int @ K )
% 16.19/9.05 = ( sgn_sgn_int @ L2 ) )
% 16.19/9.05 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 16.19/9.05 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) ) ) )
% 16.19/9.05 & ( ( ( sgn_sgn_int @ K )
% 16.19/9.05 != ( sgn_sgn_int @ L2 ) )
% 16.19/9.05 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 16.19/9.05 = ( uminus_uminus_int
% 16.19/9.05 @ ( semiri1314217659103216013at_int
% 16.19/9.05 @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N2 )
% 16.19/9.05 @ ( zero_n2687167440665602831ol_nat
% 16.19/9.05 @ ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9192_sgn__le__0__iff,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( sgn_sgn_real @ X ) @ zero_zero_real )
% 16.19/9.05 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9193_zero__le__sgn__iff,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X ) )
% 16.19/9.05 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9194_real__sgn__eq,axiom,
% 16.19/9.05 ( sgn_sgn_real
% 16.19/9.05 = ( ^ [X2: real] : ( divide_divide_real @ X2 @ ( abs_abs_real @ X2 ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9195_sgn__eq,axiom,
% 16.19/9.05 ( sgn_sgn_complex
% 16.19/9.05 = ( ^ [Z3: complex] : ( divide1717551699836669952omplex @ Z3 @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ Z3 ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9196_sgn__real__def,axiom,
% 16.19/9.05 ( sgn_sgn_real
% 16.19/9.05 = ( ^ [A5: real] : ( if_real @ ( A5 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A5 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9197_sgn__power__injE,axiom,
% 16.19/9.05 ! [A: real,N2: nat,X: real,B: real] :
% 16.19/9.05 ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) )
% 16.19/9.05 = X )
% 16.19/9.05 => ( ( X
% 16.19/9.05 = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N2 ) ) )
% 16.19/9.05 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.05 => ( A = B ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9198_cis__Arg__unique,axiom,
% 16.19/9.05 ! [Z: complex,X: real] :
% 16.19/9.05 ( ( ( sgn_sgn_complex @ Z )
% 16.19/9.05 = ( cis @ X ) )
% 16.19/9.05 => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
% 16.19/9.05 => ( ( ord_less_eq_real @ X @ pi )
% 16.19/9.05 => ( ( arg @ Z )
% 16.19/9.05 = X ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9199_floor__real__def,axiom,
% 16.19/9.05 ( archim6058952711729229775r_real
% 16.19/9.05 = ( ^ [X2: real] :
% 16.19/9.05 ( the_int
% 16.19/9.05 @ ^ [Z3: int] :
% 16.19/9.05 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ X2 )
% 16.19/9.05 & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9200_Arg__correct,axiom,
% 16.19/9.05 ! [Z: complex] :
% 16.19/9.05 ( ( Z != zero_zero_complex )
% 16.19/9.05 => ( ( ( sgn_sgn_complex @ Z )
% 16.19/9.05 = ( cis @ ( arg @ Z ) ) )
% 16.19/9.05 & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 16.19/9.05 & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9201_arctan__inverse,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( X != zero_zero_real )
% 16.19/9.05 => ( ( arctan @ ( divide_divide_real @ one_one_real @ X ) )
% 16.19/9.05 = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9202_bij__betw__nth__root__unity,axiom,
% 16.19/9.05 ! [C: complex,N2: nat] :
% 16.19/9.05 ( ( C != zero_zero_complex )
% 16.19/9.05 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.05 => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N2 @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) )
% 16.19/9.05 @ ( collect_complex
% 16.19/9.05 @ ^ [Z3: complex] :
% 16.19/9.05 ( ( power_power_complex @ Z3 @ N2 )
% 16.19/9.05 = one_one_complex ) )
% 16.19/9.05 @ ( collect_complex
% 16.19/9.05 @ ^ [Z3: complex] :
% 16.19/9.05 ( ( power_power_complex @ Z3 @ N2 )
% 16.19/9.05 = C ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9203_modulo__int__def,axiom,
% 16.19/9.05 ( modulo_modulo_int
% 16.19/9.05 = ( ^ [K3: int,L: int] :
% 16.19/9.05 ( if_int @ ( L = zero_zero_int ) @ K3
% 16.19/9.05 @ ( if_int
% 16.19/9.05 @ ( ( sgn_sgn_int @ K3 )
% 16.19/9.05 = ( sgn_sgn_int @ L ) )
% 16.19/9.05 @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) )
% 16.19/9.05 @ ( times_times_int @ ( sgn_sgn_int @ L )
% 16.19/9.05 @ ( minus_minus_int
% 16.19/9.05 @ ( times_times_int @ ( abs_abs_int @ L )
% 16.19/9.05 @ ( zero_n2684676970156552555ol_int
% 16.19/9.05 @ ~ ( dvd_dvd_int @ L @ K3 ) ) )
% 16.19/9.05 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9204_divide__int__def,axiom,
% 16.19/9.05 ( divide_divide_int
% 16.19/9.05 = ( ^ [K3: int,L: int] :
% 16.19/9.05 ( if_int @ ( L = zero_zero_int ) @ zero_zero_int
% 16.19/9.05 @ ( if_int
% 16.19/9.05 @ ( ( sgn_sgn_int @ K3 )
% 16.19/9.05 = ( sgn_sgn_int @ L ) )
% 16.19/9.05 @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) )
% 16.19/9.05 @ ( uminus_uminus_int
% 16.19/9.05 @ ( semiri1314217659103216013at_int
% 16.19/9.05 @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) )
% 16.19/9.05 @ ( zero_n2687167440665602831ol_nat
% 16.19/9.05 @ ~ ( dvd_dvd_int @ L @ K3 ) ) ) ) ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9205_mask__nat__positive__iff,axiom,
% 16.19/9.05 ! [N2: nat] :
% 16.19/9.05 ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N2 ) )
% 16.19/9.05 = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9206_real__root__zero,axiom,
% 16.19/9.05 ! [N2: nat] :
% 16.19/9.05 ( ( root @ N2 @ zero_zero_real )
% 16.19/9.05 = zero_zero_real ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9207_nat__numeral,axiom,
% 16.19/9.05 ! [K: num] :
% 16.19/9.05 ( ( nat2 @ ( numeral_numeral_int @ K ) )
% 16.19/9.05 = ( numeral_numeral_nat @ K ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9208_real__root__Suc__0,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( root @ ( suc @ zero_zero_nat ) @ X )
% 16.19/9.05 = X ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9209_real__root__eq__iff,axiom,
% 16.19/9.05 ! [N2: nat,X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.05 => ( ( ( root @ N2 @ X )
% 16.19/9.05 = ( root @ N2 @ Y ) )
% 16.19/9.05 = ( X = Y ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9210_root__0,axiom,
% 16.19/9.05 ! [X: real] :
% 16.19/9.05 ( ( root @ zero_zero_nat @ X )
% 16.19/9.05 = zero_zero_real ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9211_nat__1,axiom,
% 16.19/9.05 ( ( nat2 @ one_one_int )
% 16.19/9.05 = ( suc @ zero_zero_nat ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9212_real__root__eq__0__iff,axiom,
% 16.19/9.05 ! [N2: nat,X: real] :
% 16.19/9.05 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.05 => ( ( ( root @ N2 @ X )
% 16.19/9.05 = zero_zero_real )
% 16.19/9.05 = ( X = zero_zero_real ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9213_real__root__less__iff,axiom,
% 16.19/9.05 ! [N2: nat,X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.05 => ( ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) )
% 16.19/9.05 = ( ord_less_real @ X @ Y ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9214_real__root__le__iff,axiom,
% 16.19/9.05 ! [N2: nat,X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.05 => ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) )
% 16.19/9.05 = ( ord_less_eq_real @ X @ Y ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9215_zless__nat__conj,axiom,
% 16.19/9.05 ! [W: int,Z: int] :
% 16.19/9.05 ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 16.19/9.05 = ( ( ord_less_int @ zero_zero_int @ Z )
% 16.19/9.05 & ( ord_less_int @ W @ Z ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9216_real__root__eq__1__iff,axiom,
% 16.19/9.05 ! [N2: nat,X: real] :
% 16.19/9.05 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.05 => ( ( ( root @ N2 @ X )
% 16.19/9.05 = one_one_real )
% 16.19/9.05 = ( X = one_one_real ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9217_real__root__one,axiom,
% 16.19/9.05 ! [N2: nat] :
% 16.19/9.05 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.05 => ( ( root @ N2 @ one_one_real )
% 16.19/9.05 = one_one_real ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9218_real__root__lt__0__iff,axiom,
% 16.19/9.05 ! [N2: nat,X: real] :
% 16.19/9.05 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.05 => ( ( ord_less_real @ ( root @ N2 @ X ) @ zero_zero_real )
% 16.19/9.05 = ( ord_less_real @ X @ zero_zero_real ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9219_real__root__gt__0__iff,axiom,
% 16.19/9.05 ! [N2: nat,Y: real] :
% 16.19/9.05 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.05 => ( ( ord_less_real @ zero_zero_real @ ( root @ N2 @ Y ) )
% 16.19/9.05 = ( ord_less_real @ zero_zero_real @ Y ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9220_real__root__le__0__iff,axiom,
% 16.19/9.05 ! [N2: nat,X: real] :
% 16.19/9.05 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.05 => ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ zero_zero_real )
% 16.19/9.05 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9221_real__root__ge__0__iff,axiom,
% 16.19/9.05 ! [N2: nat,Y: real] :
% 16.19/9.05 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.05 => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ Y ) )
% 16.19/9.05 = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9222_zero__less__nat__eq,axiom,
% 16.19/9.05 ! [Z: int] :
% 16.19/9.05 ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
% 16.19/9.05 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9223_real__root__lt__1__iff,axiom,
% 16.19/9.05 ! [N2: nat,X: real] :
% 16.19/9.05 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.05 => ( ( ord_less_real @ ( root @ N2 @ X ) @ one_one_real )
% 16.19/9.05 = ( ord_less_real @ X @ one_one_real ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9224_real__root__gt__1__iff,axiom,
% 16.19/9.05 ! [N2: nat,Y: real] :
% 16.19/9.05 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.05 => ( ( ord_less_real @ one_one_real @ ( root @ N2 @ Y ) )
% 16.19/9.05 = ( ord_less_real @ one_one_real @ Y ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9225_real__root__le__1__iff,axiom,
% 16.19/9.05 ! [N2: nat,X: real] :
% 16.19/9.05 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.05 => ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ one_one_real )
% 16.19/9.05 = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9226_real__root__ge__1__iff,axiom,
% 16.19/9.05 ! [N2: nat,Y: real] :
% 16.19/9.05 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.05 => ( ( ord_less_eq_real @ one_one_real @ ( root @ N2 @ Y ) )
% 16.19/9.05 = ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9227_diff__nat__numeral,axiom,
% 16.19/9.05 ! [V: num,V3: num] :
% 16.19/9.05 ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V3 ) )
% 16.19/9.05 = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V3 ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9228_numeral__power__eq__nat__cancel__iff,axiom,
% 16.19/9.05 ! [X: num,N2: nat,Y: int] :
% 16.19/9.05 ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 )
% 16.19/9.05 = ( nat2 @ Y ) )
% 16.19/9.05 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 )
% 16.19/9.05 = Y ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9229_nat__eq__numeral__power__cancel__iff,axiom,
% 16.19/9.05 ! [Y: int,X: num,N2: nat] :
% 16.19/9.05 ( ( ( nat2 @ Y )
% 16.19/9.05 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
% 16.19/9.05 = ( Y
% 16.19/9.05 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9230_nat__ceiling__le__eq,axiom,
% 16.19/9.05 ! [X: real,A: nat] :
% 16.19/9.05 ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
% 16.19/9.05 = ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9231_one__less__nat__eq,axiom,
% 16.19/9.05 ! [Z: int] :
% 16.19/9.05 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
% 16.19/9.05 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9232_real__root__pow__pos2,axiom,
% 16.19/9.05 ! [N2: nat,X: real] :
% 16.19/9.05 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.05 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ( power_power_real @ ( root @ N2 @ X ) @ N2 )
% 16.19/9.05 = X ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9233_nat__numeral__diff__1,axiom,
% 16.19/9.05 ! [V: num] :
% 16.19/9.05 ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
% 16.19/9.05 = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9234_nat__less__numeral__power__cancel__iff,axiom,
% 16.19/9.05 ! [A: int,X: num,N2: nat] :
% 16.19/9.05 ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
% 16.19/9.05 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9235_numeral__power__less__nat__cancel__iff,axiom,
% 16.19/9.05 ! [X: num,N2: nat,A: int] :
% 16.19/9.05 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) @ ( nat2 @ A ) )
% 16.19/9.05 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9236_nat__le__numeral__power__cancel__iff,axiom,
% 16.19/9.05 ! [A: int,X: num,N2: nat] :
% 16.19/9.05 ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) )
% 16.19/9.05 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9237_numeral__power__le__nat__cancel__iff,axiom,
% 16.19/9.05 ! [X: num,N2: nat,A: int] :
% 16.19/9.05 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N2 ) @ ( nat2 @ A ) )
% 16.19/9.05 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N2 ) @ A ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9238_less__eq__mask,axiom,
% 16.19/9.05 ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9239_real__root__minus,axiom,
% 16.19/9.05 ! [N2: nat,X: real] :
% 16.19/9.05 ( ( root @ N2 @ ( uminus_uminus_real @ X ) )
% 16.19/9.05 = ( uminus_uminus_real @ ( root @ N2 @ X ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9240_real__root__mult,axiom,
% 16.19/9.05 ! [N2: nat,X: real,Y: real] :
% 16.19/9.05 ( ( root @ N2 @ ( times_times_real @ X @ Y ) )
% 16.19/9.05 = ( times_times_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9241_real__root__mult__exp,axiom,
% 16.19/9.05 ! [M: nat,N2: nat,X: real] :
% 16.19/9.05 ( ( root @ ( times_times_nat @ M @ N2 ) @ X )
% 16.19/9.05 = ( root @ M @ ( root @ N2 @ X ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9242_nat__mask__eq,axiom,
% 16.19/9.05 ! [N2: nat] :
% 16.19/9.05 ( ( nat2 @ ( bit_se2000444600071755411sk_int @ N2 ) )
% 16.19/9.05 = ( bit_se2002935070580805687sk_nat @ N2 ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9243_real__root__commute,axiom,
% 16.19/9.05 ! [M: nat,N2: nat,X: real] :
% 16.19/9.05 ( ( root @ M @ ( root @ N2 @ X ) )
% 16.19/9.05 = ( root @ N2 @ ( root @ M @ X ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9244_real__root__divide,axiom,
% 16.19/9.05 ! [N2: nat,X: real,Y: real] :
% 16.19/9.05 ( ( root @ N2 @ ( divide_divide_real @ X @ Y ) )
% 16.19/9.05 = ( divide_divide_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9245_real__root__inverse,axiom,
% 16.19/9.05 ! [N2: nat,X: real] :
% 16.19/9.05 ( ( root @ N2 @ ( inverse_inverse_real @ X ) )
% 16.19/9.05 = ( inverse_inverse_real @ ( root @ N2 @ X ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9246_sgn__root,axiom,
% 16.19/9.05 ! [N2: nat,X: real] :
% 16.19/9.05 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.05 => ( ( sgn_sgn_real @ ( root @ N2 @ X ) )
% 16.19/9.05 = ( sgn_sgn_real @ X ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9247_real__root__pos__pos__le,axiom,
% 16.19/9.05 ! [X: real,N2: nat] :
% 16.19/9.05 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.05 => ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9248_nat__numeral__as__int,axiom,
% 16.19/9.05 ( numeral_numeral_nat
% 16.19/9.05 = ( ^ [I3: num] : ( nat2 @ ( numeral_numeral_int @ I3 ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9249_nat__mono,axiom,
% 16.19/9.05 ! [X: int,Y: int] :
% 16.19/9.05 ( ( ord_less_eq_int @ X @ Y )
% 16.19/9.05 => ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9250_nat__one__as__int,axiom,
% 16.19/9.05 ( one_one_nat
% 16.19/9.05 = ( nat2 @ one_one_int ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9251_unset__bit__nat__def,axiom,
% 16.19/9.05 ( bit_se4205575877204974255it_nat
% 16.19/9.05 = ( ^ [M6: nat,N: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M6 @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9252_mask__nonnegative__int,axiom,
% 16.19/9.05 ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N2 ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9253_not__mask__negative__int,axiom,
% 16.19/9.05 ! [N2: nat] :
% 16.19/9.05 ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N2 ) @ zero_zero_int ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9254_real__root__less__mono,axiom,
% 16.19/9.05 ! [N2: nat,X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.05 => ( ( ord_less_real @ X @ Y )
% 16.19/9.05 => ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9255_real__root__le__mono,axiom,
% 16.19/9.05 ! [N2: nat,X: real,Y: real] :
% 16.19/9.05 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.05 => ( ( ord_less_eq_real @ X @ Y )
% 16.19/9.05 => ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ) ).
% 16.19/9.05
% 16.19/9.05 thf(fact_9256_real__root__power,axiom,
% 16.19/9.05 ! [N2: nat,X: real,K: nat] :
% 16.19/9.05 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.05 => ( ( root @ N2 @ ( power_power_real @ X @ K ) )
% 16.19/9.05 = ( power_power_real @ ( root @ N2 @ X ) @ K ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9257_real__root__abs,axiom,
% 16.19/9.06 ! [N2: nat,X: real] :
% 16.19/9.06 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.06 => ( ( root @ N2 @ ( abs_abs_real @ X ) )
% 16.19/9.06 = ( abs_abs_real @ ( root @ N2 @ X ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9258_nat__mono__iff,axiom,
% 16.19/9.06 ! [Z: int,W: int] :
% 16.19/9.06 ( ( ord_less_int @ zero_zero_int @ Z )
% 16.19/9.06 => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 16.19/9.06 = ( ord_less_int @ W @ Z ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9259_zless__nat__eq__int__zless,axiom,
% 16.19/9.06 ! [M: nat,Z: int] :
% 16.19/9.06 ( ( ord_less_nat @ M @ ( nat2 @ Z ) )
% 16.19/9.06 = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9260_nat__le__iff,axiom,
% 16.19/9.06 ! [X: int,N2: nat] :
% 16.19/9.06 ( ( ord_less_eq_nat @ ( nat2 @ X ) @ N2 )
% 16.19/9.06 = ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9261_nat__int__add,axiom,
% 16.19/9.06 ! [A: nat,B: nat] :
% 16.19/9.06 ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
% 16.19/9.06 = ( plus_plus_nat @ A @ B ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9262_nat__abs__mult__distrib,axiom,
% 16.19/9.06 ! [W: int,Z: int] :
% 16.19/9.06 ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z ) ) )
% 16.19/9.06 = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9263_int__minus,axiom,
% 16.19/9.06 ! [N2: nat,M: nat] :
% 16.19/9.06 ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ M ) )
% 16.19/9.06 = ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9264_nat__plus__as__int,axiom,
% 16.19/9.06 ( plus_plus_nat
% 16.19/9.06 = ( ^ [A5: nat,B5: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9265_nat__times__as__int,axiom,
% 16.19/9.06 ( times_times_nat
% 16.19/9.06 = ( ^ [A5: nat,B5: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9266_real__nat__ceiling__ge,axiom,
% 16.19/9.06 ! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9267_less__mask,axiom,
% 16.19/9.06 ! [N2: nat] :
% 16.19/9.06 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 16.19/9.06 => ( ord_less_nat @ N2 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9268_nat__minus__as__int,axiom,
% 16.19/9.06 ( minus_minus_nat
% 16.19/9.06 = ( ^ [A5: nat,B5: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9269_nat__div__as__int,axiom,
% 16.19/9.06 ( divide_divide_nat
% 16.19/9.06 = ( ^ [A5: nat,B5: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9270_nat__mod__as__int,axiom,
% 16.19/9.06 ( modulo_modulo_nat
% 16.19/9.06 = ( ^ [A5: nat,B5: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9271_real__root__gt__zero,axiom,
% 16.19/9.06 ! [N2: nat,X: real] :
% 16.19/9.06 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.06 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.06 => ( ord_less_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9272_real__root__strict__decreasing,axiom,
% 16.19/9.06 ! [N2: nat,N3: nat,X: real] :
% 16.19/9.06 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.06 => ( ( ord_less_nat @ N2 @ N3 )
% 16.19/9.06 => ( ( ord_less_real @ one_one_real @ X )
% 16.19/9.06 => ( ord_less_real @ ( root @ N3 @ X ) @ ( root @ N2 @ X ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9273_sqrt__def,axiom,
% 16.19/9.06 ( sqrt
% 16.19/9.06 = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9274_root__abs__power,axiom,
% 16.19/9.06 ! [N2: nat,Y: real] :
% 16.19/9.06 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.06 => ( ( abs_abs_real @ ( root @ N2 @ ( power_power_real @ Y @ N2 ) ) )
% 16.19/9.06 = ( abs_abs_real @ Y ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9275_nat__less__eq__zless,axiom,
% 16.19/9.06 ! [W: int,Z: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ W )
% 16.19/9.06 => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 16.19/9.06 = ( ord_less_int @ W @ Z ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9276_nat__le__eq__zle,axiom,
% 16.19/9.06 ! [W: int,Z: int] :
% 16.19/9.06 ( ( ( ord_less_int @ zero_zero_int @ W )
% 16.19/9.06 | ( ord_less_eq_int @ zero_zero_int @ Z ) )
% 16.19/9.06 => ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 16.19/9.06 = ( ord_less_eq_int @ W @ Z ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9277_split__nat,axiom,
% 16.19/9.06 ! [P: nat > $o,I2: int] :
% 16.19/9.06 ( ( P @ ( nat2 @ I2 ) )
% 16.19/9.06 = ( ! [N: nat] :
% 16.19/9.06 ( ( I2
% 16.19/9.06 = ( semiri1314217659103216013at_int @ N ) )
% 16.19/9.06 => ( P @ N ) )
% 16.19/9.06 & ( ( ord_less_int @ I2 @ zero_zero_int )
% 16.19/9.06 => ( P @ zero_zero_nat ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9278_le__nat__iff,axiom,
% 16.19/9.06 ! [K: int,N2: nat] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 16.19/9.06 => ( ( ord_less_eq_nat @ N2 @ ( nat2 @ K ) )
% 16.19/9.06 = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ K ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9279_nat__add__distrib,axiom,
% 16.19/9.06 ! [Z: int,Z7: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 16.19/9.06 => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
% 16.19/9.06 => ( ( nat2 @ ( plus_plus_int @ Z @ Z7 ) )
% 16.19/9.06 = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9280_nat__mult__distrib,axiom,
% 16.19/9.06 ! [Z: int,Z7: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 16.19/9.06 => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
% 16.19/9.06 = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9281_Suc__as__int,axiom,
% 16.19/9.06 ( suc
% 16.19/9.06 = ( ^ [A5: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ one_one_int ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9282_nat__diff__distrib,axiom,
% 16.19/9.06 ! [Z7: int,Z: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
% 16.19/9.06 => ( ( ord_less_eq_int @ Z7 @ Z )
% 16.19/9.06 => ( ( nat2 @ ( minus_minus_int @ Z @ Z7 ) )
% 16.19/9.06 = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9283_nat__diff__distrib_H,axiom,
% 16.19/9.06 ! [X: int,Y: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 16.19/9.06 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 16.19/9.06 => ( ( nat2 @ ( minus_minus_int @ X @ Y ) )
% 16.19/9.06 = ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9284_nat__abs__triangle__ineq,axiom,
% 16.19/9.06 ! [K: int,L2: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L2 ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9285_nat__div__distrib_H,axiom,
% 16.19/9.06 ! [Y: int,X: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 16.19/9.06 => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
% 16.19/9.06 = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9286_nat__div__distrib,axiom,
% 16.19/9.06 ! [X: int,Y: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 16.19/9.06 => ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
% 16.19/9.06 = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9287_nat__power__eq,axiom,
% 16.19/9.06 ! [Z: int,N2: nat] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 16.19/9.06 => ( ( nat2 @ ( power_power_int @ Z @ N2 ) )
% 16.19/9.06 = ( power_power_nat @ ( nat2 @ Z ) @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9288_nat__floor__neg,axiom,
% 16.19/9.06 ! [X: real] :
% 16.19/9.06 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 16.19/9.06 => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 16.19/9.06 = zero_zero_nat ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9289_div__abs__eq__div__nat,axiom,
% 16.19/9.06 ! [K: int,L2: int] :
% 16.19/9.06 ( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) )
% 16.19/9.06 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9290_nat__mod__distrib,axiom,
% 16.19/9.06 ! [X: int,Y: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 16.19/9.06 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 16.19/9.06 => ( ( nat2 @ ( modulo_modulo_int @ X @ Y ) )
% 16.19/9.06 = ( modulo_modulo_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9291_floor__eq3,axiom,
% 16.19/9.06 ! [N2: nat,X: real] :
% 16.19/9.06 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ X )
% 16.19/9.06 => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
% 16.19/9.06 => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 16.19/9.06 = N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9292_le__nat__floor,axiom,
% 16.19/9.06 ! [X: nat,A: real] :
% 16.19/9.06 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A )
% 16.19/9.06 => ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9293_real__root__pos__pos,axiom,
% 16.19/9.06 ! [N2: nat,X: real] :
% 16.19/9.06 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.06 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.06 => ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9294_real__root__strict__increasing,axiom,
% 16.19/9.06 ! [N2: nat,N3: nat,X: real] :
% 16.19/9.06 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.06 => ( ( ord_less_nat @ N2 @ N3 )
% 16.19/9.06 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.06 => ( ( ord_less_real @ X @ one_one_real )
% 16.19/9.06 => ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N3 @ X ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9295_real__root__decreasing,axiom,
% 16.19/9.06 ! [N2: nat,N3: nat,X: real] :
% 16.19/9.06 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.06 => ( ( ord_less_eq_nat @ N2 @ N3 )
% 16.19/9.06 => ( ( ord_less_eq_real @ one_one_real @ X )
% 16.19/9.06 => ( ord_less_eq_real @ ( root @ N3 @ X ) @ ( root @ N2 @ X ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9296_real__root__pow__pos,axiom,
% 16.19/9.06 ! [N2: nat,X: real] :
% 16.19/9.06 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.06 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.06 => ( ( power_power_real @ ( root @ N2 @ X ) @ N2 )
% 16.19/9.06 = X ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9297_odd__real__root__power__cancel,axiom,
% 16.19/9.06 ! [N2: nat,X: real] :
% 16.19/9.06 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.06 => ( ( root @ N2 @ ( power_power_real @ X @ N2 ) )
% 16.19/9.06 = X ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9298_odd__real__root__unique,axiom,
% 16.19/9.06 ! [N2: nat,Y: real,X: real] :
% 16.19/9.06 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.06 => ( ( ( power_power_real @ Y @ N2 )
% 16.19/9.06 = X )
% 16.19/9.06 => ( ( root @ N2 @ X )
% 16.19/9.06 = Y ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9299_odd__real__root__pow,axiom,
% 16.19/9.06 ! [N2: nat,X: real] :
% 16.19/9.06 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.06 => ( ( power_power_real @ ( root @ N2 @ X ) @ N2 )
% 16.19/9.06 = X ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9300_real__root__power__cancel,axiom,
% 16.19/9.06 ! [N2: nat,X: real] :
% 16.19/9.06 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.06 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.06 => ( ( root @ N2 @ ( power_power_real @ X @ N2 ) )
% 16.19/9.06 = X ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9301_real__root__pos__unique,axiom,
% 16.19/9.06 ! [N2: nat,Y: real,X: real] :
% 16.19/9.06 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.06 => ( ( ord_less_eq_real @ zero_zero_real @ Y )
% 16.19/9.06 => ( ( ( power_power_real @ Y @ N2 )
% 16.19/9.06 = X )
% 16.19/9.06 => ( ( root @ N2 @ X )
% 16.19/9.06 = Y ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9302_nat__2,axiom,
% 16.19/9.06 ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/9.06 = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9303_sgn__power__root,axiom,
% 16.19/9.06 ! [N2: nat,X: real] :
% 16.19/9.06 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.06 => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N2 @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N2 @ X ) ) @ N2 ) )
% 16.19/9.06 = X ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9304_root__sgn__power,axiom,
% 16.19/9.06 ! [N2: nat,Y: real] :
% 16.19/9.06 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.06 => ( ( root @ N2 @ ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) ) )
% 16.19/9.06 = Y ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9305_Suc__nat__eq__nat__zadd1,axiom,
% 16.19/9.06 ! [Z: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 16.19/9.06 => ( ( suc @ ( nat2 @ Z ) )
% 16.19/9.06 = ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9306_nat__less__iff,axiom,
% 16.19/9.06 ! [W: int,M: nat] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ W )
% 16.19/9.06 => ( ( ord_less_nat @ ( nat2 @ W ) @ M )
% 16.19/9.06 = ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9307_nat__mult__distrib__neg,axiom,
% 16.19/9.06 ! [Z: int,Z7: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 16.19/9.06 => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
% 16.19/9.06 = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z7 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9308_nat__abs__int__diff,axiom,
% 16.19/9.06 ! [A: nat,B: nat] :
% 16.19/9.06 ( ( ( ord_less_eq_nat @ A @ B )
% 16.19/9.06 => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 16.19/9.06 = ( minus_minus_nat @ B @ A ) ) )
% 16.19/9.06 & ( ~ ( ord_less_eq_nat @ A @ B )
% 16.19/9.06 => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 16.19/9.06 = ( minus_minus_nat @ A @ B ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9309_floor__eq4,axiom,
% 16.19/9.06 ! [N2: nat,X: real] :
% 16.19/9.06 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ X )
% 16.19/9.06 => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
% 16.19/9.06 => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 16.19/9.06 = N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9310_diff__nat__eq__if,axiom,
% 16.19/9.06 ! [Z7: int,Z: int] :
% 16.19/9.06 ( ( ( ord_less_int @ Z7 @ zero_zero_int )
% 16.19/9.06 => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
% 16.19/9.06 = ( nat2 @ Z ) ) )
% 16.19/9.06 & ( ~ ( ord_less_int @ Z7 @ zero_zero_int )
% 16.19/9.06 => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
% 16.19/9.06 = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z7 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z7 ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9311_real__root__increasing,axiom,
% 16.19/9.06 ! [N2: nat,N3: nat,X: real] :
% 16.19/9.06 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.06 => ( ( ord_less_eq_nat @ N2 @ N3 )
% 16.19/9.06 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.06 => ( ( ord_less_eq_real @ X @ one_one_real )
% 16.19/9.06 => ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N3 @ X ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9312_Suc__mask__eq__exp,axiom,
% 16.19/9.06 ! [N2: nat] :
% 16.19/9.06 ( ( suc @ ( bit_se2002935070580805687sk_nat @ N2 ) )
% 16.19/9.06 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9313_mask__nat__less__exp,axiom,
% 16.19/9.06 ! [N2: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9314_split__root,axiom,
% 16.19/9.06 ! [P: real > $o,N2: nat,X: real] :
% 16.19/9.06 ( ( P @ ( root @ N2 @ X ) )
% 16.19/9.06 = ( ( ( N2 = zero_zero_nat )
% 16.19/9.06 => ( P @ zero_zero_real ) )
% 16.19/9.06 & ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.06 => ! [Y2: real] :
% 16.19/9.06 ( ( ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N2 ) )
% 16.19/9.06 = X )
% 16.19/9.06 => ( P @ Y2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9315_ln__root,axiom,
% 16.19/9.06 ! [N2: nat,B: real] :
% 16.19/9.06 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.06 => ( ( ord_less_real @ zero_zero_real @ B )
% 16.19/9.06 => ( ( ln_ln_real @ ( root @ N2 @ B ) )
% 16.19/9.06 = ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9316_log__root,axiom,
% 16.19/9.06 ! [N2: nat,A: real,B: real] :
% 16.19/9.06 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.06 => ( ( ord_less_real @ zero_zero_real @ A )
% 16.19/9.06 => ( ( log @ B @ ( root @ N2 @ A ) )
% 16.19/9.06 = ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9317_log__base__root,axiom,
% 16.19/9.06 ! [N2: nat,B: real,X: real] :
% 16.19/9.06 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.06 => ( ( ord_less_real @ zero_zero_real @ B )
% 16.19/9.06 => ( ( log @ ( root @ N2 @ B ) @ X )
% 16.19/9.06 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ X ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9318_mask__nat__def,axiom,
% 16.19/9.06 ( bit_se2002935070580805687sk_nat
% 16.19/9.06 = ( ^ [N: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9319_mask__half__int,axiom,
% 16.19/9.06 ! [N2: nat] :
% 16.19/9.06 ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/9.06 = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9320_mask__int__def,axiom,
% 16.19/9.06 ( bit_se2000444600071755411sk_int
% 16.19/9.06 = ( ^ [N: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9321_root__powr__inverse,axiom,
% 16.19/9.06 ! [N2: nat,X: real] :
% 16.19/9.06 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.06 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.06 => ( ( root @ N2 @ X )
% 16.19/9.06 = ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9322_even__nat__iff,axiom,
% 16.19/9.06 ! [K: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 16.19/9.06 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K ) )
% 16.19/9.06 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9323_powr__real__of__int,axiom,
% 16.19/9.06 ! [X: real,N2: int] :
% 16.19/9.06 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.06 => ( ( ( ord_less_eq_int @ zero_zero_int @ N2 )
% 16.19/9.06 => ( ( powr_real @ X @ ( ring_1_of_int_real @ N2 ) )
% 16.19/9.06 = ( power_power_real @ X @ ( nat2 @ N2 ) ) ) )
% 16.19/9.06 & ( ~ ( ord_less_eq_int @ zero_zero_int @ N2 )
% 16.19/9.06 => ( ( powr_real @ X @ ( ring_1_of_int_real @ N2 ) )
% 16.19/9.06 = ( inverse_inverse_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ N2 ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9324_powr__int,axiom,
% 16.19/9.06 ! [X: real,I2: int] :
% 16.19/9.06 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.06 => ( ( ( ord_less_eq_int @ zero_zero_int @ I2 )
% 16.19/9.06 => ( ( powr_real @ X @ ( ring_1_of_int_real @ I2 ) )
% 16.19/9.06 = ( power_power_real @ X @ ( nat2 @ I2 ) ) ) )
% 16.19/9.06 & ( ~ ( ord_less_eq_int @ zero_zero_int @ I2 )
% 16.19/9.06 => ( ( powr_real @ X @ ( ring_1_of_int_real @ I2 ) )
% 16.19/9.06 = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ I2 ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9325_floor__rat__def,axiom,
% 16.19/9.06 ( archim3151403230148437115or_rat
% 16.19/9.06 = ( ^ [X2: rat] :
% 16.19/9.06 ( the_int
% 16.19/9.06 @ ^ [Z3: int] :
% 16.19/9.06 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X2 )
% 16.19/9.06 & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9326_Arg__def,axiom,
% 16.19/9.06 ( arg
% 16.19/9.06 = ( ^ [Z3: complex] :
% 16.19/9.06 ( if_real @ ( Z3 = zero_zero_complex ) @ zero_zero_real
% 16.19/9.06 @ ( fChoice_real
% 16.19/9.06 @ ^ [A5: real] :
% 16.19/9.06 ( ( ( sgn_sgn_complex @ Z3 )
% 16.19/9.06 = ( cis @ A5 ) )
% 16.19/9.06 & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A5 )
% 16.19/9.06 & ( ord_less_eq_real @ A5 @ pi ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9327_num_Osize__gen_I3_J,axiom,
% 16.19/9.06 ! [X32: num] :
% 16.19/9.06 ( ( size_num @ ( bit1 @ X32 ) )
% 16.19/9.06 = ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9328_concat__bit__of__zero__2,axiom,
% 16.19/9.06 ! [N2: nat,K: int] :
% 16.19/9.06 ( ( bit_concat_bit @ N2 @ K @ zero_zero_int )
% 16.19/9.06 = ( bit_se2923211474154528505it_int @ N2 @ K ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9329_take__bit__of__Suc__0,axiom,
% 16.19/9.06 ! [N2: nat] :
% 16.19/9.06 ( ( bit_se2925701944663578781it_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 16.19/9.06 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9330_sgn__rat__def,axiom,
% 16.19/9.06 ( sgn_sgn_rat
% 16.19/9.06 = ( ^ [A5: rat] : ( if_rat @ ( A5 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A5 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9331_take__bit__nat__eq,axiom,
% 16.19/9.06 ! [K: int,N2: nat] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 16.19/9.06 => ( ( bit_se2925701944663578781it_nat @ N2 @ ( nat2 @ K ) )
% 16.19/9.06 = ( nat2 @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9332_nat__take__bit__eq,axiom,
% 16.19/9.06 ! [K: int,N2: nat] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 16.19/9.06 => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N2 @ K ) )
% 16.19/9.06 = ( bit_se2925701944663578781it_nat @ N2 @ ( nat2 @ K ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9333_take__bit__tightened__less__eq__nat,axiom,
% 16.19/9.06 ! [M: nat,N2: nat,Q2: nat] :
% 16.19/9.06 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.06 => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q2 ) @ ( bit_se2925701944663578781it_nat @ N2 @ Q2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9334_take__bit__nat__less__eq__self,axiom,
% 16.19/9.06 ! [N2: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ M ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9335_less__eq__rat__def,axiom,
% 16.19/9.06 ( ord_less_eq_rat
% 16.19/9.06 = ( ^ [X2: rat,Y2: rat] :
% 16.19/9.06 ( ( ord_less_rat @ X2 @ Y2 )
% 16.19/9.06 | ( X2 = Y2 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9336_obtain__pos__sum,axiom,
% 16.19/9.06 ! [R: rat] :
% 16.19/9.06 ( ( ord_less_rat @ zero_zero_rat @ R )
% 16.19/9.06 => ~ ! [S2: rat] :
% 16.19/9.06 ( ( ord_less_rat @ zero_zero_rat @ S2 )
% 16.19/9.06 => ! [T5: rat] :
% 16.19/9.06 ( ( ord_less_rat @ zero_zero_rat @ T5 )
% 16.19/9.06 => ( R
% 16.19/9.06 != ( plus_plus_rat @ S2 @ T5 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9337_take__bit__minus,axiom,
% 16.19/9.06 ! [N2: nat,K: int] :
% 16.19/9.06 ( ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) )
% 16.19/9.06 = ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ K ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9338_take__bit__mult,axiom,
% 16.19/9.06 ! [N2: nat,K: int,L2: int] :
% 16.19/9.06 ( ( bit_se2923211474154528505it_int @ N2 @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( bit_se2923211474154528505it_int @ N2 @ L2 ) ) )
% 16.19/9.06 = ( bit_se2923211474154528505it_int @ N2 @ ( times_times_int @ K @ L2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9339_abs__rat__def,axiom,
% 16.19/9.06 ( abs_abs_rat
% 16.19/9.06 = ( ^ [A5: rat] : ( if_rat @ ( ord_less_rat @ A5 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A5 ) @ A5 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9340_take__bit__diff,axiom,
% 16.19/9.06 ! [N2: nat,K: int,L2: int] :
% 16.19/9.06 ( ( bit_se2923211474154528505it_int @ N2 @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( bit_se2923211474154528505it_int @ N2 @ L2 ) ) )
% 16.19/9.06 = ( bit_se2923211474154528505it_int @ N2 @ ( minus_minus_int @ K @ L2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9341_concat__bit__take__bit__eq,axiom,
% 16.19/9.06 ! [N2: nat,B: int] :
% 16.19/9.06 ( ( bit_concat_bit @ N2 @ ( bit_se2923211474154528505it_int @ N2 @ B ) )
% 16.19/9.06 = ( bit_concat_bit @ N2 @ B ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9342_concat__bit__eq__iff,axiom,
% 16.19/9.06 ! [N2: nat,K: int,L2: int,R: int,S3: int] :
% 16.19/9.06 ( ( ( bit_concat_bit @ N2 @ K @ L2 )
% 16.19/9.06 = ( bit_concat_bit @ N2 @ R @ S3 ) )
% 16.19/9.06 = ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
% 16.19/9.06 = ( bit_se2923211474154528505it_int @ N2 @ R ) )
% 16.19/9.06 & ( L2 = S3 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9343_take__bit__tightened__less__eq__int,axiom,
% 16.19/9.06 ! [M: nat,N2: nat,K: int] :
% 16.19/9.06 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.06 => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9344_take__bit__int__less__eq__self__iff,axiom,
% 16.19/9.06 ! [N2: nat,K: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ K )
% 16.19/9.06 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9345_take__bit__nonnegative,axiom,
% 16.19/9.06 ! [N2: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9346_not__take__bit__negative,axiom,
% 16.19/9.06 ! [N2: nat,K: int] :
% 16.19/9.06 ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ zero_zero_int ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9347_take__bit__int__greater__self__iff,axiom,
% 16.19/9.06 ! [K: int,N2: nat] :
% 16.19/9.06 ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N2 @ K ) )
% 16.19/9.06 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9348_take__bit__eq__mask__iff,axiom,
% 16.19/9.06 ! [N2: nat,K: int] :
% 16.19/9.06 ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
% 16.19/9.06 = ( bit_se2000444600071755411sk_int @ N2 ) )
% 16.19/9.06 = ( ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ K @ one_one_int ) )
% 16.19/9.06 = zero_zero_int ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9349_take__bit__decr__eq,axiom,
% 16.19/9.06 ! [N2: nat,K: int] :
% 16.19/9.06 ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
% 16.19/9.06 != zero_zero_int )
% 16.19/9.06 => ( ( bit_se2923211474154528505it_int @ N2 @ ( minus_minus_int @ K @ one_one_int ) )
% 16.19/9.06 = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ one_one_int ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9350_take__bit__nat__eq__self,axiom,
% 16.19/9.06 ! [M: nat,N2: nat] :
% 16.19/9.06 ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/9.06 => ( ( bit_se2925701944663578781it_nat @ N2 @ M )
% 16.19/9.06 = M ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9351_take__bit__nat__less__exp,axiom,
% 16.19/9.06 ! [N2: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9352_take__bit__nat__eq__self__iff,axiom,
% 16.19/9.06 ! [N2: nat,M: nat] :
% 16.19/9.06 ( ( ( bit_se2925701944663578781it_nat @ N2 @ M )
% 16.19/9.06 = M )
% 16.19/9.06 = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9353_take__bit__nat__def,axiom,
% 16.19/9.06 ( bit_se2925701944663578781it_nat
% 16.19/9.06 = ( ^ [N: nat,M6: nat] : ( modulo_modulo_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9354_take__bit__int__less__exp,axiom,
% 16.19/9.06 ! [N2: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9355_take__bit__int__def,axiom,
% 16.19/9.06 ( bit_se2923211474154528505it_int
% 16.19/9.06 = ( ^ [N: nat,K3: int] : ( modulo_modulo_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9356_num_Osize__gen_I1_J,axiom,
% 16.19/9.06 ( ( size_num @ one )
% 16.19/9.06 = zero_zero_nat ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9357_take__bit__nat__less__self__iff,axiom,
% 16.19/9.06 ! [N2: nat,M: nat] :
% 16.19/9.06 ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ M )
% 16.19/9.06 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9358_take__bit__Suc__minus__bit0,axiom,
% 16.19/9.06 ! [N2: nat,K: num] :
% 16.19/9.06 ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 16.19/9.06 = ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9359_take__bit__int__greater__eq__self__iff,axiom,
% 16.19/9.06 ! [K: int,N2: nat] :
% 16.19/9.06 ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N2 @ K ) )
% 16.19/9.06 = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9360_take__bit__int__less__self__iff,axiom,
% 16.19/9.06 ! [N2: nat,K: int] :
% 16.19/9.06 ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ K )
% 16.19/9.06 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9361_take__bit__int__eq__self__iff,axiom,
% 16.19/9.06 ! [N2: nat,K: int] :
% 16.19/9.06 ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
% 16.19/9.06 = K )
% 16.19/9.06 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 16.19/9.06 & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9362_take__bit__int__eq__self,axiom,
% 16.19/9.06 ! [K: int,N2: nat] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 16.19/9.06 => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/9.06 => ( ( bit_se2923211474154528505it_int @ N2 @ K )
% 16.19/9.06 = K ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9363_take__bit__numeral__minus__bit0,axiom,
% 16.19/9.06 ! [L2: num,K: num] :
% 16.19/9.06 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 16.19/9.06 = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9364_take__bit__incr__eq,axiom,
% 16.19/9.06 ! [N2: nat,K: int] :
% 16.19/9.06 ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
% 16.19/9.06 != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) )
% 16.19/9.06 => ( ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ K @ one_one_int ) )
% 16.19/9.06 = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9365_take__bit__eq__mask__iff__exp__dvd,axiom,
% 16.19/9.06 ! [N2: nat,K: int] :
% 16.19/9.06 ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
% 16.19/9.06 = ( bit_se2000444600071755411sk_int @ N2 ) )
% 16.19/9.06 = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9366_take__bit__int__less__eq,axiom,
% 16.19/9.06 ! [N2: nat,K: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K )
% 16.19/9.06 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.06 => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9367_take__bit__int__greater__eq,axiom,
% 16.19/9.06 ! [K: int,N2: nat] :
% 16.19/9.06 ( ( ord_less_int @ K @ zero_zero_int )
% 16.19/9.06 => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9368_signed__take__bit__eq__take__bit__shift,axiom,
% 16.19/9.06 ( bit_ri631733984087533419it_int
% 16.19/9.06 = ( ^ [N: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( plus_plus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9369_take__bit__minus__small__eq,axiom,
% 16.19/9.06 ! [K: int,N2: nat] :
% 16.19/9.06 ( ( ord_less_int @ zero_zero_int @ K )
% 16.19/9.06 => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/9.06 => ( ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ K ) )
% 16.19/9.06 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9370_num_Osize__gen_I2_J,axiom,
% 16.19/9.06 ! [X22: num] :
% 16.19/9.06 ( ( size_num @ ( bit0 @ X22 ) )
% 16.19/9.06 = ( plus_plus_nat @ ( size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9371_take__bit__numeral__minus__bit1,axiom,
% 16.19/9.06 ! [L2: num,K: num] :
% 16.19/9.06 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 16.19/9.06 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9372_pred__numeral__inc,axiom,
% 16.19/9.06 ! [K: num] :
% 16.19/9.06 ( ( pred_numeral @ ( inc @ K ) )
% 16.19/9.06 = ( numeral_numeral_nat @ K ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9373_diff__rat__def,axiom,
% 16.19/9.06 ( minus_minus_rat
% 16.19/9.06 = ( ^ [Q4: rat,R5: rat] : ( plus_plus_rat @ Q4 @ ( uminus_uminus_rat @ R5 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9374_num__induct,axiom,
% 16.19/9.06 ! [P: num > $o,X: num] :
% 16.19/9.06 ( ( P @ one )
% 16.19/9.06 => ( ! [X4: num] :
% 16.19/9.06 ( ( P @ X4 )
% 16.19/9.06 => ( P @ ( inc @ X4 ) ) )
% 16.19/9.06 => ( P @ X ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9375_add__inc,axiom,
% 16.19/9.06 ! [X: num,Y: num] :
% 16.19/9.06 ( ( plus_plus_num @ X @ ( inc @ Y ) )
% 16.19/9.06 = ( inc @ ( plus_plus_num @ X @ Y ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9376_inc_Osimps_I1_J,axiom,
% 16.19/9.06 ( ( inc @ one )
% 16.19/9.06 = ( bit0 @ one ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9377_inc_Osimps_I2_J,axiom,
% 16.19/9.06 ! [X: num] :
% 16.19/9.06 ( ( inc @ ( bit0 @ X ) )
% 16.19/9.06 = ( bit1 @ X ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9378_inc_Osimps_I3_J,axiom,
% 16.19/9.06 ! [X: num] :
% 16.19/9.06 ( ( inc @ ( bit1 @ X ) )
% 16.19/9.06 = ( bit0 @ ( inc @ X ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9379_add__One,axiom,
% 16.19/9.06 ! [X: num] :
% 16.19/9.06 ( ( plus_plus_num @ X @ one )
% 16.19/9.06 = ( inc @ X ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9380_inc__BitM__eq,axiom,
% 16.19/9.06 ! [N2: num] :
% 16.19/9.06 ( ( inc @ ( bitM @ N2 ) )
% 16.19/9.06 = ( bit0 @ N2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9381_BitM__inc__eq,axiom,
% 16.19/9.06 ! [N2: num] :
% 16.19/9.06 ( ( bitM @ ( inc @ N2 ) )
% 16.19/9.06 = ( bit1 @ N2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9382_mult__inc,axiom,
% 16.19/9.06 ! [X: num,Y: num] :
% 16.19/9.06 ( ( times_times_num @ X @ ( inc @ Y ) )
% 16.19/9.06 = ( plus_plus_num @ ( times_times_num @ X @ Y ) @ X ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9383_take__bit__Suc__minus__bit1,axiom,
% 16.19/9.06 ! [N2: nat,K: num] :
% 16.19/9.06 ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 16.19/9.06 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9384_signed__take__bit__eq__take__bit__minus,axiom,
% 16.19/9.06 ( bit_ri631733984087533419it_int
% 16.19/9.06 = ( ^ [N: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K3 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9385_and__int__unfold,axiom,
% 16.19/9.06 ( bit_se725231765392027082nd_int
% 16.19/9.06 = ( ^ [K3: int,L: int] :
% 16.19/9.06 ( if_int
% 16.19/9.06 @ ( ( K3 = zero_zero_int )
% 16.19/9.06 | ( L = zero_zero_int ) )
% 16.19/9.06 @ zero_zero_int
% 16.19/9.06 @ ( if_int
% 16.19/9.06 @ ( K3
% 16.19/9.06 = ( uminus_uminus_int @ one_one_int ) )
% 16.19/9.06 @ L
% 16.19/9.06 @ ( if_int
% 16.19/9.06 @ ( L
% 16.19/9.06 = ( uminus_uminus_int @ one_one_int ) )
% 16.19/9.06 @ K3
% 16.19/9.06 @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9386_and__nonnegative__int__iff,axiom,
% 16.19/9.06 ! [K: int,L2: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) )
% 16.19/9.06 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 16.19/9.06 | ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9387_and__negative__int__iff,axiom,
% 16.19/9.06 ! [K: int,L2: int] :
% 16.19/9.06 ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ zero_zero_int )
% 16.19/9.06 = ( ( ord_less_int @ K @ zero_zero_int )
% 16.19/9.06 & ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9388_signed__take__bit__nonnegative__iff,axiom,
% 16.19/9.06 ! [N2: nat,K: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) )
% 16.19/9.06 = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9389_signed__take__bit__negative__iff,axiom,
% 16.19/9.06 ! [N2: nat,K: int] :
% 16.19/9.06 ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ zero_zero_int )
% 16.19/9.06 = ( bit_se1146084159140164899it_int @ K @ N2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9390_bit__minus__numeral__Bit0__Suc__iff,axiom,
% 16.19/9.06 ! [W: num,N2: nat] :
% 16.19/9.06 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N2 ) )
% 16.19/9.06 = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9391_bit__minus__numeral__Bit1__Suc__iff,axiom,
% 16.19/9.06 ! [W: num,N2: nat] :
% 16.19/9.06 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N2 ) )
% 16.19/9.06 = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9392_and__minus__numerals_I2_J,axiom,
% 16.19/9.06 ! [N2: num] :
% 16.19/9.06 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
% 16.19/9.06 = one_one_int ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9393_and__minus__numerals_I6_J,axiom,
% 16.19/9.06 ! [N2: num] :
% 16.19/9.06 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) @ one_one_int )
% 16.19/9.06 = one_one_int ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9394_and__minus__numerals_I1_J,axiom,
% 16.19/9.06 ! [N2: num] :
% 16.19/9.06 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
% 16.19/9.06 = zero_zero_int ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9395_and__minus__numerals_I5_J,axiom,
% 16.19/9.06 ! [N2: num] :
% 16.19/9.06 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) @ one_one_int )
% 16.19/9.06 = zero_zero_int ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9396_bit__minus__numeral__int_I1_J,axiom,
% 16.19/9.06 ! [W: num,N2: num] :
% 16.19/9.06 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N2 ) )
% 16.19/9.06 = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9397_bit__minus__numeral__int_I2_J,axiom,
% 16.19/9.06 ! [W: num,N2: num] :
% 16.19/9.06 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N2 ) )
% 16.19/9.06 = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N2 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9398_bit__and__int__iff,axiom,
% 16.19/9.06 ! [K: int,L2: int,N2: nat] :
% 16.19/9.06 ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ N2 )
% 16.19/9.06 = ( ( bit_se1146084159140164899it_int @ K @ N2 )
% 16.19/9.06 & ( bit_se1146084159140164899it_int @ L2 @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9399_AND__upper2_H,axiom,
% 16.19/9.06 ! [Y: int,Z: int,X: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 16.19/9.06 => ( ( ord_less_eq_int @ Y @ Z )
% 16.19/9.06 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9400_AND__upper1_H,axiom,
% 16.19/9.06 ! [Y: int,Z: int,Ya: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 16.19/9.06 => ( ( ord_less_eq_int @ Y @ Z )
% 16.19/9.06 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9401_AND__upper2,axiom,
% 16.19/9.06 ! [Y: int,X: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 16.19/9.06 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Y ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9402_AND__upper1,axiom,
% 16.19/9.06 ! [X: int,Y: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 16.19/9.06 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ X ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9403_AND__lower,axiom,
% 16.19/9.06 ! [X: int,Y: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 16.19/9.06 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X @ Y ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9404_pow_Osimps_I1_J,axiom,
% 16.19/9.06 ! [X: num] :
% 16.19/9.06 ( ( pow @ X @ one )
% 16.19/9.06 = X ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9405_AND__upper2_H_H,axiom,
% 16.19/9.06 ! [Y: int,Z: int,X: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 16.19/9.06 => ( ( ord_less_int @ Y @ Z )
% 16.19/9.06 => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ Z ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9406_AND__upper1_H_H,axiom,
% 16.19/9.06 ! [Y: int,Z: int,Ya: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 16.19/9.06 => ( ( ord_less_int @ Y @ Z )
% 16.19/9.06 => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9407_and__less__eq,axiom,
% 16.19/9.06 ! [L2: int,K: int] :
% 16.19/9.06 ( ( ord_less_int @ L2 @ zero_zero_int )
% 16.19/9.06 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ K ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9408_bit__not__int__iff_H,axiom,
% 16.19/9.06 ! [K: int,N2: nat] :
% 16.19/9.06 ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N2 )
% 16.19/9.06 = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9409_even__and__iff__int,axiom,
% 16.19/9.06 ! [K: int,L2: int] :
% 16.19/9.06 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L2 ) )
% 16.19/9.06 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 16.19/9.06 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9410_bit__imp__take__bit__positive,axiom,
% 16.19/9.06 ! [N2: nat,M: nat,K: int] :
% 16.19/9.06 ( ( ord_less_nat @ N2 @ M )
% 16.19/9.06 => ( ( bit_se1146084159140164899it_int @ K @ N2 )
% 16.19/9.06 => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9411_bit__concat__bit__iff,axiom,
% 16.19/9.06 ! [M: nat,K: int,L2: int,N2: nat] :
% 16.19/9.06 ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L2 ) @ N2 )
% 16.19/9.06 = ( ( ( ord_less_nat @ N2 @ M )
% 16.19/9.06 & ( bit_se1146084159140164899it_int @ K @ N2 ) )
% 16.19/9.06 | ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.06 & ( bit_se1146084159140164899it_int @ L2 @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9412_signed__take__bit__eq__concat__bit,axiom,
% 16.19/9.06 ( bit_ri631733984087533419it_int
% 16.19/9.06 = ( ^ [N: nat,K3: int] : ( bit_concat_bit @ N @ K3 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9413_int__bit__bound,axiom,
% 16.19/9.06 ! [K: int] :
% 16.19/9.06 ~ ! [N4: nat] :
% 16.19/9.06 ( ! [M5: nat] :
% 16.19/9.06 ( ( ord_less_eq_nat @ N4 @ M5 )
% 16.19/9.06 => ( ( bit_se1146084159140164899it_int @ K @ M5 )
% 16.19/9.06 = ( bit_se1146084159140164899it_int @ K @ N4 ) ) )
% 16.19/9.06 => ~ ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 16.19/9.06 => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N4 @ one_one_nat ) )
% 16.19/9.06 = ( ~ ( bit_se1146084159140164899it_int @ K @ N4 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9414_bit__int__def,axiom,
% 16.19/9.06 ( bit_se1146084159140164899it_int
% 16.19/9.06 = ( ^ [K3: int,N: nat] :
% 16.19/9.06 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9415_and__int__rec,axiom,
% 16.19/9.06 ( bit_se725231765392027082nd_int
% 16.19/9.06 = ( ^ [K3: int,L: int] :
% 16.19/9.06 ( plus_plus_int
% 16.19/9.06 @ ( zero_n2684676970156552555ol_int
% 16.19/9.06 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 16.19/9.06 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 16.19/9.06 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9416_set__bit__eq,axiom,
% 16.19/9.06 ( bit_se7879613467334960850it_int
% 16.19/9.06 = ( ^ [N: nat,K3: int] :
% 16.19/9.06 ( plus_plus_int @ K3
% 16.19/9.06 @ ( times_times_int
% 16.19/9.06 @ ( zero_n2684676970156552555ol_int
% 16.19/9.06 @ ~ ( bit_se1146084159140164899it_int @ K3 @ N ) )
% 16.19/9.06 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9417_unset__bit__eq,axiom,
% 16.19/9.06 ( bit_se4203085406695923979it_int
% 16.19/9.06 = ( ^ [N: nat,K3: int] : ( minus_minus_int @ K3 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9418_take__bit__Suc__from__most,axiom,
% 16.19/9.06 ! [N2: nat,K: int] :
% 16.19/9.06 ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ K )
% 16.19/9.06 = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9419_and__int_Oelims,axiom,
% 16.19/9.06 ! [X: int,Xa2: int,Y: int] :
% 16.19/9.06 ( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
% 16.19/9.06 = Y )
% 16.19/9.06 => ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 16.19/9.06 & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 16.19/9.06 => ( Y
% 16.19/9.06 = ( uminus_uminus_int
% 16.19/9.06 @ ( zero_n2684676970156552555ol_int
% 16.19/9.06 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 16.19/9.06 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
% 16.19/9.06 & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 16.19/9.06 & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 16.19/9.06 => ( Y
% 16.19/9.06 = ( plus_plus_int
% 16.19/9.06 @ ( zero_n2684676970156552555ol_int
% 16.19/9.06 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 16.19/9.06 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
% 16.19/9.06 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9420_and__int_Osimps,axiom,
% 16.19/9.06 ( bit_se725231765392027082nd_int
% 16.19/9.06 = ( ^ [K3: int,L: int] :
% 16.19/9.06 ( if_int
% 16.19/9.06 @ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 16.19/9.06 & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 16.19/9.06 @ ( uminus_uminus_int
% 16.19/9.06 @ ( zero_n2684676970156552555ol_int
% 16.19/9.06 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 16.19/9.06 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) )
% 16.19/9.06 @ ( plus_plus_int
% 16.19/9.06 @ ( zero_n2684676970156552555ol_int
% 16.19/9.06 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 16.19/9.06 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 16.19/9.06 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9421_or__int__unfold,axiom,
% 16.19/9.06 ( bit_se1409905431419307370or_int
% 16.19/9.06 = ( ^ [K3: int,L: int] :
% 16.19/9.06 ( if_int
% 16.19/9.06 @ ( ( K3
% 16.19/9.06 = ( uminus_uminus_int @ one_one_int ) )
% 16.19/9.06 | ( L
% 16.19/9.06 = ( uminus_uminus_int @ one_one_int ) ) )
% 16.19/9.06 @ ( uminus_uminus_int @ one_one_int )
% 16.19/9.06 @ ( if_int @ ( K3 = zero_zero_int ) @ L @ ( if_int @ ( L = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9422_cis__multiple__2pi,axiom,
% 16.19/9.06 ! [N2: real] :
% 16.19/9.06 ( ( member_real @ N2 @ ring_1_Ints_real )
% 16.19/9.06 => ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N2 ) )
% 16.19/9.06 = one_one_complex ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9423_or__nonnegative__int__iff,axiom,
% 16.19/9.06 ! [K: int,L2: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) )
% 16.19/9.06 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 16.19/9.06 & ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9424_or__negative__int__iff,axiom,
% 16.19/9.06 ! [K: int,L2: int] :
% 16.19/9.06 ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) @ zero_zero_int )
% 16.19/9.06 = ( ( ord_less_int @ K @ zero_zero_int )
% 16.19/9.06 | ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9425_and__nat__numerals_I1_J,axiom,
% 16.19/9.06 ! [Y: num] :
% 16.19/9.06 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 16.19/9.06 = zero_zero_nat ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9426_and__nat__numerals_I3_J,axiom,
% 16.19/9.06 ! [X: num] :
% 16.19/9.06 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 16.19/9.06 = zero_zero_nat ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9427_or__minus__numerals_I6_J,axiom,
% 16.19/9.06 ! [N2: num] :
% 16.19/9.06 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) @ one_one_int )
% 16.19/9.06 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9428_or__minus__numerals_I2_J,axiom,
% 16.19/9.06 ! [N2: num] :
% 16.19/9.06 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
% 16.19/9.06 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9429_and__nat__numerals_I2_J,axiom,
% 16.19/9.06 ! [Y: num] :
% 16.19/9.06 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 16.19/9.06 = one_one_nat ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9430_and__nat__numerals_I4_J,axiom,
% 16.19/9.06 ! [X: num] :
% 16.19/9.06 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 16.19/9.06 = one_one_nat ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9431_and__Suc__0__eq,axiom,
% 16.19/9.06 ! [N2: nat] :
% 16.19/9.06 ( ( bit_se727722235901077358nd_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 16.19/9.06 = ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9432_Suc__0__and__eq,axiom,
% 16.19/9.06 ! [N2: nat] :
% 16.19/9.06 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 16.19/9.06 = ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9433_bit__or__int__iff,axiom,
% 16.19/9.06 ! [K: int,L2: int,N2: nat] :
% 16.19/9.06 ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) @ N2 )
% 16.19/9.06 = ( ( bit_se1146084159140164899it_int @ K @ N2 )
% 16.19/9.06 | ( bit_se1146084159140164899it_int @ L2 @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9434_bit__Suc__0__iff,axiom,
% 16.19/9.06 ! [N2: nat] :
% 16.19/9.06 ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 16.19/9.06 = ( N2 = zero_zero_nat ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9435_not__bit__Suc__0__Suc,axiom,
% 16.19/9.06 ! [N2: nat] :
% 16.19/9.06 ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9436_OR__lower,axiom,
% 16.19/9.06 ! [X: int,Y: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 16.19/9.06 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 16.19/9.06 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X @ Y ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9437_or__greater__eq,axiom,
% 16.19/9.06 ! [L2: int,K: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ L2 )
% 16.19/9.06 => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9438_plus__and__or,axiom,
% 16.19/9.06 ! [X: int,Y: int] :
% 16.19/9.06 ( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X @ Y ) @ ( bit_se1409905431419307370or_int @ X @ Y ) )
% 16.19/9.06 = ( plus_plus_int @ X @ Y ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9439_not__bit__Suc__0__numeral,axiom,
% 16.19/9.06 ! [N2: num] :
% 16.19/9.06 ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9440_and__nat__def,axiom,
% 16.19/9.06 ( bit_se727722235901077358nd_nat
% 16.19/9.06 = ( ^ [M6: nat,N: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9441_bit__nat__iff,axiom,
% 16.19/9.06 ! [K: int,N2: nat] :
% 16.19/9.06 ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N2 )
% 16.19/9.06 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 16.19/9.06 & ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9442_sin__times__pi__eq__0,axiom,
% 16.19/9.06 ! [X: real] :
% 16.19/9.06 ( ( ( sin_real @ ( times_times_real @ X @ pi ) )
% 16.19/9.06 = zero_zero_real )
% 16.19/9.06 = ( member_real @ X @ ring_1_Ints_real ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9443_atLeastAtMostPlus1__int__conv,axiom,
% 16.19/9.06 ! [M: int,N2: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N2 ) )
% 16.19/9.06 => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N2 ) )
% 16.19/9.06 = ( insert_int @ ( plus_plus_int @ one_one_int @ N2 ) @ ( set_or1266510415728281911st_int @ M @ N2 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9444_simp__from__to,axiom,
% 16.19/9.06 ( set_or1266510415728281911st_int
% 16.19/9.06 = ( ^ [I3: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I3 ) @ bot_bot_set_int @ ( insert_int @ I3 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9445_bit__nat__def,axiom,
% 16.19/9.06 ( bit_se1148574629649215175it_nat
% 16.19/9.06 = ( ^ [M6: nat,N: nat] :
% 16.19/9.06 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9446_OR__upper,axiom,
% 16.19/9.06 ! [X: int,N2: nat,Y: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 16.19/9.06 => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/9.06 => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/9.06 => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9447_sin__integer__2pi,axiom,
% 16.19/9.06 ! [N2: real] :
% 16.19/9.06 ( ( member_real @ N2 @ ring_1_Ints_real )
% 16.19/9.06 => ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N2 ) )
% 16.19/9.06 = zero_zero_real ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9448_cos__integer__2pi,axiom,
% 16.19/9.06 ! [N2: real] :
% 16.19/9.06 ( ( member_real @ N2 @ ring_1_Ints_real )
% 16.19/9.06 => ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N2 ) )
% 16.19/9.06 = one_one_real ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9449_and__nat__unfold,axiom,
% 16.19/9.06 ( bit_se727722235901077358nd_nat
% 16.19/9.06 = ( ^ [M6: nat,N: nat] :
% 16.19/9.06 ( if_nat
% 16.19/9.06 @ ( ( M6 = zero_zero_nat )
% 16.19/9.06 | ( N = zero_zero_nat ) )
% 16.19/9.06 @ zero_zero_nat
% 16.19/9.06 @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9450_and__nat__rec,axiom,
% 16.19/9.06 ( bit_se727722235901077358nd_nat
% 16.19/9.06 = ( ^ [M6: nat,N: nat] :
% 16.19/9.06 ( plus_plus_nat
% 16.19/9.06 @ ( zero_n2687167440665602831ol_nat
% 16.19/9.06 @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 )
% 16.19/9.06 & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 16.19/9.06 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9451_or__int__rec,axiom,
% 16.19/9.06 ( bit_se1409905431419307370or_int
% 16.19/9.06 = ( ^ [K3: int,L: int] :
% 16.19/9.06 ( plus_plus_int
% 16.19/9.06 @ ( zero_n2684676970156552555ol_int
% 16.19/9.06 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 16.19/9.06 | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 16.19/9.06 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9452_and__int_Opsimps,axiom,
% 16.19/9.06 ! [K: int,L2: int] :
% 16.19/9.06 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L2 ) )
% 16.19/9.06 => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 16.19/9.06 & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 16.19/9.06 => ( ( bit_se725231765392027082nd_int @ K @ L2 )
% 16.19/9.06 = ( uminus_uminus_int
% 16.19/9.06 @ ( zero_n2684676970156552555ol_int
% 16.19/9.06 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 16.19/9.06 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) ) ) )
% 16.19/9.06 & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 16.19/9.06 & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 16.19/9.06 => ( ( bit_se725231765392027082nd_int @ K @ L2 )
% 16.19/9.06 = ( plus_plus_int
% 16.19/9.06 @ ( zero_n2684676970156552555ol_int
% 16.19/9.06 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 16.19/9.06 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 16.19/9.06 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9453_and__int_Opelims,axiom,
% 16.19/9.06 ! [X: int,Xa2: int,Y: int] :
% 16.19/9.06 ( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
% 16.19/9.06 = Y )
% 16.19/9.06 => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa2 ) )
% 16.19/9.06 => ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 16.19/9.06 & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 16.19/9.06 => ( Y
% 16.19/9.06 = ( uminus_uminus_int
% 16.19/9.06 @ ( zero_n2684676970156552555ol_int
% 16.19/9.06 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 16.19/9.06 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
% 16.19/9.06 & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 16.19/9.06 & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 16.19/9.06 => ( Y
% 16.19/9.06 = ( plus_plus_int
% 16.19/9.06 @ ( zero_n2684676970156552555ol_int
% 16.19/9.06 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 16.19/9.06 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
% 16.19/9.06 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 16.19/9.06 => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9454_or__minus__numerals_I1_J,axiom,
% 16.19/9.06 ! [N2: num] :
% 16.19/9.06 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
% 16.19/9.06 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9455_or__nat__numerals_I2_J,axiom,
% 16.19/9.06 ! [Y: num] :
% 16.19/9.06 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 16.19/9.06 = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9456_or__nat__numerals_I4_J,axiom,
% 16.19/9.06 ! [X: num] :
% 16.19/9.06 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 16.19/9.06 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9457_or__nat__numerals_I3_J,axiom,
% 16.19/9.06 ! [X: num] :
% 16.19/9.06 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 16.19/9.06 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9458_or__nat__numerals_I1_J,axiom,
% 16.19/9.06 ! [Y: num] :
% 16.19/9.06 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 16.19/9.06 = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9459_or__minus__numerals_I8_J,axiom,
% 16.19/9.06 ! [N2: num,M: num] :
% 16.19/9.06 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) @ ( numeral_numeral_int @ M ) )
% 16.19/9.06 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9460_or__minus__numerals_I4_J,axiom,
% 16.19/9.06 ! [M: num,N2: num] :
% 16.19/9.06 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
% 16.19/9.06 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9461_or__minus__numerals_I7_J,axiom,
% 16.19/9.06 ! [N2: num,M: num] :
% 16.19/9.06 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) @ ( numeral_numeral_int @ M ) )
% 16.19/9.06 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9462_or__minus__numerals_I3_J,axiom,
% 16.19/9.06 ! [M: num,N2: num] :
% 16.19/9.06 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
% 16.19/9.06 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9463_set__encode__insert,axiom,
% 16.19/9.06 ! [A2: set_nat,N2: nat] :
% 16.19/9.06 ( ( finite_finite_nat @ A2 )
% 16.19/9.06 => ( ~ ( member_nat @ N2 @ A2 )
% 16.19/9.06 => ( ( nat_set_encode @ ( insert_nat @ N2 @ A2 ) )
% 16.19/9.06 = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9464_or__minus__numerals_I5_J,axiom,
% 16.19/9.06 ! [N2: num] :
% 16.19/9.06 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) @ one_one_int )
% 16.19/9.06 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9465_or__not__num__neg_Osimps_I1_J,axiom,
% 16.19/9.06 ( ( bit_or_not_num_neg @ one @ one )
% 16.19/9.06 = one ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9466_lessThan__Suc,axiom,
% 16.19/9.06 ! [K: nat] :
% 16.19/9.06 ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 16.19/9.06 = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9467_atMost__Suc,axiom,
% 16.19/9.06 ! [K: nat] :
% 16.19/9.06 ( ( set_ord_atMost_nat @ ( suc @ K ) )
% 16.19/9.06 = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9468_or__not__num__neg_Osimps_I4_J,axiom,
% 16.19/9.06 ! [N2: num] :
% 16.19/9.06 ( ( bit_or_not_num_neg @ ( bit0 @ N2 ) @ one )
% 16.19/9.06 = ( bit0 @ one ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9469_or__not__num__neg_Osimps_I6_J,axiom,
% 16.19/9.06 ! [N2: num,M: num] :
% 16.19/9.06 ( ( bit_or_not_num_neg @ ( bit0 @ N2 ) @ ( bit1 @ M ) )
% 16.19/9.06 = ( bit0 @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9470_or__not__num__neg_Osimps_I7_J,axiom,
% 16.19/9.06 ! [N2: num] :
% 16.19/9.06 ( ( bit_or_not_num_neg @ ( bit1 @ N2 ) @ one )
% 16.19/9.06 = one ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9471_or__not__num__neg_Osimps_I3_J,axiom,
% 16.19/9.06 ! [M: num] :
% 16.19/9.06 ( ( bit_or_not_num_neg @ one @ ( bit1 @ M ) )
% 16.19/9.06 = ( bit1 @ M ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9472_or__not__num__neg_Osimps_I5_J,axiom,
% 16.19/9.06 ! [N2: num,M: num] :
% 16.19/9.06 ( ( bit_or_not_num_neg @ ( bit0 @ N2 ) @ ( bit0 @ M ) )
% 16.19/9.06 = ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9473_or__not__num__neg_Osimps_I9_J,axiom,
% 16.19/9.06 ! [N2: num,M: num] :
% 16.19/9.06 ( ( bit_or_not_num_neg @ ( bit1 @ N2 ) @ ( bit1 @ M ) )
% 16.19/9.06 = ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9474_or__nat__def,axiom,
% 16.19/9.06 ( bit_se1412395901928357646or_nat
% 16.19/9.06 = ( ^ [M6: nat,N: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9475_atLeast0__atMost__Suc,axiom,
% 16.19/9.06 ! [N2: nat] :
% 16.19/9.06 ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) )
% 16.19/9.06 = ( insert_nat @ ( suc @ N2 ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9476_Icc__eq__insert__lb__nat,axiom,
% 16.19/9.06 ! [M: nat,N2: nat] :
% 16.19/9.06 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.06 => ( ( set_or1269000886237332187st_nat @ M @ N2 )
% 16.19/9.06 = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9477_atLeastAtMostSuc__conv,axiom,
% 16.19/9.06 ! [M: nat,N2: nat] :
% 16.19/9.06 ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 16.19/9.06 => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) )
% 16.19/9.06 = ( insert_nat @ ( suc @ N2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9478_atLeastAtMost__insertL,axiom,
% 16.19/9.06 ! [M: nat,N2: nat] :
% 16.19/9.06 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.06 => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
% 16.19/9.06 = ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9479_lessThan__nat__numeral,axiom,
% 16.19/9.06 ! [K: num] :
% 16.19/9.06 ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
% 16.19/9.06 = ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9480_atMost__nat__numeral,axiom,
% 16.19/9.06 ! [K: num] :
% 16.19/9.06 ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
% 16.19/9.06 = ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9481_or__not__num__neg_Osimps_I2_J,axiom,
% 16.19/9.06 ! [M: num] :
% 16.19/9.06 ( ( bit_or_not_num_neg @ one @ ( bit0 @ M ) )
% 16.19/9.06 = ( bit1 @ M ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9482_or__not__num__neg_Osimps_I8_J,axiom,
% 16.19/9.06 ! [N2: num,M: num] :
% 16.19/9.06 ( ( bit_or_not_num_neg @ ( bit1 @ N2 ) @ ( bit0 @ M ) )
% 16.19/9.06 = ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9483_or__not__num__neg_Oelims,axiom,
% 16.19/9.06 ! [X: num,Xa2: num,Y: num] :
% 16.19/9.06 ( ( ( bit_or_not_num_neg @ X @ Xa2 )
% 16.19/9.06 = Y )
% 16.19/9.06 => ( ( ( X = one )
% 16.19/9.06 => ( ( Xa2 = one )
% 16.19/9.06 => ( Y != one ) ) )
% 16.19/9.06 => ( ( ( X = one )
% 16.19/9.06 => ! [M2: num] :
% 16.19/9.06 ( ( Xa2
% 16.19/9.06 = ( bit0 @ M2 ) )
% 16.19/9.06 => ( Y
% 16.19/9.06 != ( bit1 @ M2 ) ) ) )
% 16.19/9.06 => ( ( ( X = one )
% 16.19/9.06 => ! [M2: num] :
% 16.19/9.06 ( ( Xa2
% 16.19/9.06 = ( bit1 @ M2 ) )
% 16.19/9.06 => ( Y
% 16.19/9.06 != ( bit1 @ M2 ) ) ) )
% 16.19/9.06 => ( ( ? [N4: num] :
% 16.19/9.06 ( X
% 16.19/9.06 = ( bit0 @ N4 ) )
% 16.19/9.06 => ( ( Xa2 = one )
% 16.19/9.06 => ( Y
% 16.19/9.06 != ( bit0 @ one ) ) ) )
% 16.19/9.06 => ( ! [N4: num] :
% 16.19/9.06 ( ( X
% 16.19/9.06 = ( bit0 @ N4 ) )
% 16.19/9.06 => ! [M2: num] :
% 16.19/9.06 ( ( Xa2
% 16.19/9.06 = ( bit0 @ M2 ) )
% 16.19/9.06 => ( Y
% 16.19/9.06 != ( bitM @ ( bit_or_not_num_neg @ N4 @ M2 ) ) ) ) )
% 16.19/9.06 => ( ! [N4: num] :
% 16.19/9.06 ( ( X
% 16.19/9.06 = ( bit0 @ N4 ) )
% 16.19/9.06 => ! [M2: num] :
% 16.19/9.06 ( ( Xa2
% 16.19/9.06 = ( bit1 @ M2 ) )
% 16.19/9.06 => ( Y
% 16.19/9.06 != ( bit0 @ ( bit_or_not_num_neg @ N4 @ M2 ) ) ) ) )
% 16.19/9.06 => ( ( ? [N4: num] :
% 16.19/9.06 ( X
% 16.19/9.06 = ( bit1 @ N4 ) )
% 16.19/9.06 => ( ( Xa2 = one )
% 16.19/9.06 => ( Y != one ) ) )
% 16.19/9.06 => ( ! [N4: num] :
% 16.19/9.06 ( ( X
% 16.19/9.06 = ( bit1 @ N4 ) )
% 16.19/9.06 => ! [M2: num] :
% 16.19/9.06 ( ( Xa2
% 16.19/9.06 = ( bit0 @ M2 ) )
% 16.19/9.06 => ( Y
% 16.19/9.06 != ( bitM @ ( bit_or_not_num_neg @ N4 @ M2 ) ) ) ) )
% 16.19/9.06 => ~ ! [N4: num] :
% 16.19/9.06 ( ( X
% 16.19/9.06 = ( bit1 @ N4 ) )
% 16.19/9.06 => ! [M2: num] :
% 16.19/9.06 ( ( Xa2
% 16.19/9.06 = ( bit1 @ M2 ) )
% 16.19/9.06 => ( Y
% 16.19/9.06 != ( bitM @ ( bit_or_not_num_neg @ N4 @ M2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9484_atLeast1__atMost__eq__remove0,axiom,
% 16.19/9.06 ! [N2: nat] :
% 16.19/9.06 ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 16.19/9.06 = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9485_set__decode__plus__power__2,axiom,
% 16.19/9.06 ! [N2: nat,Z: nat] :
% 16.19/9.06 ( ~ ( member_nat @ N2 @ ( nat_set_decode @ Z ) )
% 16.19/9.06 => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ Z ) )
% 16.19/9.06 = ( insert_nat @ N2 @ ( nat_set_decode @ Z ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9486_Suc__0__or__eq,axiom,
% 16.19/9.06 ! [N2: nat] :
% 16.19/9.06 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 16.19/9.06 = ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9487_or__Suc__0__eq,axiom,
% 16.19/9.06 ! [N2: nat] :
% 16.19/9.06 ( ( bit_se1412395901928357646or_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 16.19/9.06 = ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9488_or__nat__rec,axiom,
% 16.19/9.06 ( bit_se1412395901928357646or_nat
% 16.19/9.06 = ( ^ [M6: nat,N: nat] :
% 16.19/9.06 ( plus_plus_nat
% 16.19/9.06 @ ( zero_n2687167440665602831ol_nat
% 16.19/9.06 @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 )
% 16.19/9.06 | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 16.19/9.06 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9489_and__int_Opinduct,axiom,
% 16.19/9.06 ! [A0: int,A12: int,P: int > int > $o] :
% 16.19/9.06 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
% 16.19/9.06 => ( ! [K2: int,L4: int] :
% 16.19/9.06 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L4 ) )
% 16.19/9.06 => ( ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 16.19/9.06 & ( member_int @ L4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 16.19/9.06 => ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 16.19/9.06 => ( P @ K2 @ L4 ) ) )
% 16.19/9.06 => ( P @ A0 @ A12 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9490_or__nat__unfold,axiom,
% 16.19/9.06 ( bit_se1412395901928357646or_nat
% 16.19/9.06 = ( ^ [M6: nat,N: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N @ ( if_nat @ ( N = zero_zero_nat ) @ M6 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9491_upto_Opinduct,axiom,
% 16.19/9.06 ! [A0: int,A12: int,P: int > int > $o] :
% 16.19/9.06 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
% 16.19/9.06 => ( ! [I4: int,J2: int] :
% 16.19/9.06 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I4 @ J2 ) )
% 16.19/9.06 => ( ( ( ord_less_eq_int @ I4 @ J2 )
% 16.19/9.06 => ( P @ ( plus_plus_int @ I4 @ one_one_int ) @ J2 ) )
% 16.19/9.06 => ( P @ I4 @ J2 ) ) )
% 16.19/9.06 => ( P @ A0 @ A12 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9492_rat__inverse__code,axiom,
% 16.19/9.06 ! [P4: rat] :
% 16.19/9.06 ( ( quotient_of @ ( inverse_inverse_rat @ P4 ) )
% 16.19/9.06 = ( produc4245557441103728435nt_int
% 16.19/9.06 @ ^ [A5: int,B5: int] : ( if_Pro3027730157355071871nt_int @ ( A5 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A5 ) @ B5 ) @ ( abs_abs_int @ A5 ) ) )
% 16.19/9.06 @ ( quotient_of @ P4 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9493_quotient__of__number_I3_J,axiom,
% 16.19/9.06 ! [K: num] :
% 16.19/9.06 ( ( quotient_of @ ( numeral_numeral_rat @ K ) )
% 16.19/9.06 = ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9494_rat__one__code,axiom,
% 16.19/9.06 ( ( quotient_of @ one_one_rat )
% 16.19/9.06 = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9495_rat__zero__code,axiom,
% 16.19/9.06 ( ( quotient_of @ zero_zero_rat )
% 16.19/9.06 = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9496_quotient__of__number_I5_J,axiom,
% 16.19/9.06 ! [K: num] :
% 16.19/9.06 ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 16.19/9.06 = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9497_quotient__of__number_I4_J,axiom,
% 16.19/9.06 ( ( quotient_of @ ( uminus_uminus_rat @ one_one_rat ) )
% 16.19/9.06 = ( product_Pair_int_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9498_divide__rat__def,axiom,
% 16.19/9.06 ( divide_divide_rat
% 16.19/9.06 = ( ^ [Q4: rat,R5: rat] : ( times_times_rat @ Q4 @ ( inverse_inverse_rat @ R5 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9499_quotient__of__div,axiom,
% 16.19/9.06 ! [R: rat,N2: int,D: int] :
% 16.19/9.06 ( ( ( quotient_of @ R )
% 16.19/9.06 = ( product_Pair_int_int @ N2 @ D ) )
% 16.19/9.06 => ( R
% 16.19/9.06 = ( divide_divide_rat @ ( ring_1_of_int_rat @ N2 ) @ ( ring_1_of_int_rat @ D ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9500_quotient__of__denom__pos,axiom,
% 16.19/9.06 ! [R: rat,P4: int,Q2: int] :
% 16.19/9.06 ( ( ( quotient_of @ R )
% 16.19/9.06 = ( product_Pair_int_int @ P4 @ Q2 ) )
% 16.19/9.06 => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9501_rat__uminus__code,axiom,
% 16.19/9.06 ! [P4: rat] :
% 16.19/9.06 ( ( quotient_of @ ( uminus_uminus_rat @ P4 ) )
% 16.19/9.06 = ( produc4245557441103728435nt_int
% 16.19/9.06 @ ^ [A5: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A5 ) )
% 16.19/9.06 @ ( quotient_of @ P4 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9502_rat__less__code,axiom,
% 16.19/9.06 ( ord_less_rat
% 16.19/9.06 = ( ^ [P5: rat,Q4: rat] :
% 16.19/9.06 ( produc4947309494688390418_int_o
% 16.19/9.06 @ ^ [A5: int,C4: int] :
% 16.19/9.06 ( produc4947309494688390418_int_o
% 16.19/9.06 @ ^ [B5: int,D2: int] : ( ord_less_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ C4 @ B5 ) )
% 16.19/9.06 @ ( quotient_of @ Q4 ) )
% 16.19/9.06 @ ( quotient_of @ P5 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9503_rat__floor__code,axiom,
% 16.19/9.06 ( archim3151403230148437115or_rat
% 16.19/9.06 = ( ^ [P5: rat] : ( produc8211389475949308722nt_int @ divide_divide_int @ ( quotient_of @ P5 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9504_rat__abs__code,axiom,
% 16.19/9.06 ! [P4: rat] :
% 16.19/9.06 ( ( quotient_of @ ( abs_abs_rat @ P4 ) )
% 16.19/9.06 = ( produc4245557441103728435nt_int
% 16.19/9.06 @ ^ [A5: int] : ( product_Pair_int_int @ ( abs_abs_int @ A5 ) )
% 16.19/9.06 @ ( quotient_of @ P4 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9505_rat__less__eq__code,axiom,
% 16.19/9.06 ( ord_less_eq_rat
% 16.19/9.06 = ( ^ [P5: rat,Q4: rat] :
% 16.19/9.06 ( produc4947309494688390418_int_o
% 16.19/9.06 @ ^ [A5: int,C4: int] :
% 16.19/9.06 ( produc4947309494688390418_int_o
% 16.19/9.06 @ ^ [B5: int,D2: int] : ( ord_less_eq_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ C4 @ B5 ) )
% 16.19/9.06 @ ( quotient_of @ Q4 ) )
% 16.19/9.06 @ ( quotient_of @ P5 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9506_quotient__of__int,axiom,
% 16.19/9.06 ! [A: int] :
% 16.19/9.06 ( ( quotient_of @ ( of_int @ A ) )
% 16.19/9.06 = ( product_Pair_int_int @ A @ one_one_int ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9507_rat__minus__code,axiom,
% 16.19/9.06 ! [P4: rat,Q2: rat] :
% 16.19/9.06 ( ( quotient_of @ ( minus_minus_rat @ P4 @ Q2 ) )
% 16.19/9.06 = ( produc4245557441103728435nt_int
% 16.19/9.06 @ ^ [A5: int,C4: int] :
% 16.19/9.06 ( produc4245557441103728435nt_int
% 16.19/9.06 @ ^ [B5: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ B5 @ C4 ) ) @ ( times_times_int @ C4 @ D2 ) ) )
% 16.19/9.06 @ ( quotient_of @ Q2 ) )
% 16.19/9.06 @ ( quotient_of @ P4 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9508_rat__plus__code,axiom,
% 16.19/9.06 ! [P4: rat,Q2: rat] :
% 16.19/9.06 ( ( quotient_of @ ( plus_plus_rat @ P4 @ Q2 ) )
% 16.19/9.06 = ( produc4245557441103728435nt_int
% 16.19/9.06 @ ^ [A5: int,C4: int] :
% 16.19/9.06 ( produc4245557441103728435nt_int
% 16.19/9.06 @ ^ [B5: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ B5 @ C4 ) ) @ ( times_times_int @ C4 @ D2 ) ) )
% 16.19/9.06 @ ( quotient_of @ Q2 ) )
% 16.19/9.06 @ ( quotient_of @ P4 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9509_normalize__denom__zero,axiom,
% 16.19/9.06 ! [P4: int] :
% 16.19/9.06 ( ( normalize @ ( product_Pair_int_int @ P4 @ zero_zero_int ) )
% 16.19/9.06 = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9510_normalize__negative,axiom,
% 16.19/9.06 ! [Q2: int,P4: int] :
% 16.19/9.06 ( ( ord_less_int @ Q2 @ zero_zero_int )
% 16.19/9.06 => ( ( normalize @ ( product_Pair_int_int @ P4 @ Q2 ) )
% 16.19/9.06 = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P4 ) @ ( uminus_uminus_int @ Q2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9511_normalize__denom__pos,axiom,
% 16.19/9.06 ! [R: product_prod_int_int,P4: int,Q2: int] :
% 16.19/9.06 ( ( ( normalize @ R )
% 16.19/9.06 = ( product_Pair_int_int @ P4 @ Q2 ) )
% 16.19/9.06 => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9512_rat__times__code,axiom,
% 16.19/9.06 ! [P4: rat,Q2: rat] :
% 16.19/9.06 ( ( quotient_of @ ( times_times_rat @ P4 @ Q2 ) )
% 16.19/9.06 = ( produc4245557441103728435nt_int
% 16.19/9.06 @ ^ [A5: int,C4: int] :
% 16.19/9.06 ( produc4245557441103728435nt_int
% 16.19/9.06 @ ^ [B5: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A5 @ B5 ) @ ( times_times_int @ C4 @ D2 ) ) )
% 16.19/9.06 @ ( quotient_of @ Q2 ) )
% 16.19/9.06 @ ( quotient_of @ P4 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9513_rat__divide__code,axiom,
% 16.19/9.06 ! [P4: rat,Q2: rat] :
% 16.19/9.06 ( ( quotient_of @ ( divide_divide_rat @ P4 @ Q2 ) )
% 16.19/9.06 = ( produc4245557441103728435nt_int
% 16.19/9.06 @ ^ [A5: int,C4: int] :
% 16.19/9.06 ( produc4245557441103728435nt_int
% 16.19/9.06 @ ^ [B5: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ C4 @ B5 ) ) )
% 16.19/9.06 @ ( quotient_of @ Q2 ) )
% 16.19/9.06 @ ( quotient_of @ P4 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9514_Frct__code__post_I5_J,axiom,
% 16.19/9.06 ! [K: num] :
% 16.19/9.06 ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
% 16.19/9.06 = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9515_Frct__code__post_I3_J,axiom,
% 16.19/9.06 ( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
% 16.19/9.06 = one_one_rat ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9516_Frct__code__post_I4_J,axiom,
% 16.19/9.06 ! [K: num] :
% 16.19/9.06 ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
% 16.19/9.06 = ( numeral_numeral_rat @ K ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9517_Frct__code__post_I6_J,axiom,
% 16.19/9.06 ! [K: num,L2: num] :
% 16.19/9.06 ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L2 ) ) )
% 16.19/9.06 = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9518_xor__Suc__0__eq,axiom,
% 16.19/9.06 ! [N2: nat] :
% 16.19/9.06 ( ( bit_se6528837805403552850or_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 16.19/9.06 = ( minus_minus_nat @ ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 16.19/9.06 @ ( zero_n2687167440665602831ol_nat
% 16.19/9.06 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9519_Suc__0__xor__eq,axiom,
% 16.19/9.06 ! [N2: nat] :
% 16.19/9.06 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 16.19/9.06 = ( minus_minus_nat @ ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 16.19/9.06 @ ( zero_n2687167440665602831ol_nat
% 16.19/9.06 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9520_horner__sum__of__bool__2__less,axiom,
% 16.19/9.06 ! [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 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9521_xor__nat__numerals_I4_J,axiom,
% 16.19/9.06 ! [X: num] :
% 16.19/9.06 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 16.19/9.06 = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9522_xor__nat__numerals_I3_J,axiom,
% 16.19/9.06 ! [X: num] :
% 16.19/9.06 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 16.19/9.06 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9523_xor__nat__numerals_I2_J,axiom,
% 16.19/9.06 ! [Y: num] :
% 16.19/9.06 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
% 16.19/9.06 = ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9524_xor__nat__numerals_I1_J,axiom,
% 16.19/9.06 ! [Y: num] :
% 16.19/9.06 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
% 16.19/9.06 = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9525_xor__nat__unfold,axiom,
% 16.19/9.06 ( bit_se6528837805403552850or_nat
% 16.19/9.06 = ( ^ [M6: nat,N: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N @ ( if_nat @ ( N = zero_zero_nat ) @ M6 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9526_xor__nat__rec,axiom,
% 16.19/9.06 ( bit_se6528837805403552850or_nat
% 16.19/9.06 = ( ^ [M6: nat,N: nat] :
% 16.19/9.06 ( plus_plus_nat
% 16.19/9.06 @ ( zero_n2687167440665602831ol_nat
% 16.19/9.06 @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 ) )
% 16.19/9.06 != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 16.19/9.06 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9527_Cauchy__iff2,axiom,
% 16.19/9.06 ( topolo4055970368930404560y_real
% 16.19/9.06 = ( ^ [X3: nat > real] :
% 16.19/9.06 ! [J3: nat] :
% 16.19/9.06 ? [M8: nat] :
% 16.19/9.06 ! [M6: nat] :
% 16.19/9.06 ( ( ord_less_eq_nat @ M8 @ M6 )
% 16.19/9.06 => ! [N: nat] :
% 16.19/9.06 ( ( ord_less_eq_nat @ M8 @ N )
% 16.19/9.06 => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X3 @ M6 ) @ ( X3 @ N ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9528_push__bit__nonnegative__int__iff,axiom,
% 16.19/9.06 ! [N2: nat,K: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N2 @ K ) )
% 16.19/9.06 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9529_push__bit__negative__int__iff,axiom,
% 16.19/9.06 ! [N2: nat,K: int] :
% 16.19/9.06 ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N2 @ K ) @ zero_zero_int )
% 16.19/9.06 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9530_concat__bit__of__zero__1,axiom,
% 16.19/9.06 ! [N2: nat,L2: int] :
% 16.19/9.06 ( ( bit_concat_bit @ N2 @ zero_zero_int @ L2 )
% 16.19/9.06 = ( bit_se545348938243370406it_int @ N2 @ L2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9531_xor__nonnegative__int__iff,axiom,
% 16.19/9.06 ! [K: int,L2: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) )
% 16.19/9.06 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 16.19/9.06 = ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9532_xor__negative__int__iff,axiom,
% 16.19/9.06 ! [K: int,L2: int] :
% 16.19/9.06 ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) @ zero_zero_int )
% 16.19/9.06 = ( ( ord_less_int @ K @ zero_zero_int )
% 16.19/9.06 != ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9533_push__bit__of__Suc__0,axiom,
% 16.19/9.06 ! [N2: nat] :
% 16.19/9.06 ( ( bit_se547839408752420682it_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 16.19/9.06 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9534_bit__xor__int__iff,axiom,
% 16.19/9.06 ! [K: int,L2: int,N2: nat] :
% 16.19/9.06 ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) @ N2 )
% 16.19/9.06 = ( ( bit_se1146084159140164899it_int @ K @ N2 )
% 16.19/9.06 != ( bit_se1146084159140164899it_int @ L2 @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9535_flip__bit__int__def,axiom,
% 16.19/9.06 ( bit_se2159334234014336723it_int
% 16.19/9.06 = ( ^ [N: nat,K3: int] : ( bit_se6526347334894502574or_int @ K3 @ ( bit_se545348938243370406it_int @ N @ one_one_int ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9536_push__bit__nat__eq,axiom,
% 16.19/9.06 ! [N2: nat,K: int] :
% 16.19/9.06 ( ( bit_se547839408752420682it_nat @ N2 @ ( nat2 @ K ) )
% 16.19/9.06 = ( nat2 @ ( bit_se545348938243370406it_int @ N2 @ K ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9537_XOR__lower,axiom,
% 16.19/9.06 ! [X: int,Y: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 16.19/9.06 => ( ( ord_less_eq_int @ zero_zero_int @ Y )
% 16.19/9.06 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X @ Y ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9538_set__bit__nat__def,axiom,
% 16.19/9.06 ( bit_se7882103937844011126it_nat
% 16.19/9.06 = ( ^ [M6: nat,N: nat] : ( bit_se1412395901928357646or_nat @ N @ ( bit_se547839408752420682it_nat @ M6 @ one_one_nat ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9539_flip__bit__nat__def,axiom,
% 16.19/9.06 ( bit_se2161824704523386999it_nat
% 16.19/9.06 = ( ^ [M6: nat,N: nat] : ( bit_se6528837805403552850or_nat @ N @ ( bit_se547839408752420682it_nat @ M6 @ one_one_nat ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9540_bit__push__bit__iff__int,axiom,
% 16.19/9.06 ! [M: nat,K: int,N2: nat] :
% 16.19/9.06 ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N2 )
% 16.19/9.06 = ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.06 & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9541_xor__nat__def,axiom,
% 16.19/9.06 ( bit_se6528837805403552850or_nat
% 16.19/9.06 = ( ^ [M6: nat,N: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9542_bit__push__bit__iff__nat,axiom,
% 16.19/9.06 ! [M: nat,Q2: nat,N2: nat] :
% 16.19/9.06 ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q2 ) @ N2 )
% 16.19/9.06 = ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.06 & ( bit_se1148574629649215175it_nat @ Q2 @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9543_concat__bit__eq,axiom,
% 16.19/9.06 ( bit_concat_bit
% 16.19/9.06 = ( ^ [N: nat,K3: int,L: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N @ K3 ) @ ( bit_se545348938243370406it_int @ N @ L ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9544_concat__bit__def,axiom,
% 16.19/9.06 ( bit_concat_bit
% 16.19/9.06 = ( ^ [N: nat,K3: int,L: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N @ K3 ) @ ( bit_se545348938243370406it_int @ N @ L ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9545_set__bit__int__def,axiom,
% 16.19/9.06 ( bit_se7879613467334960850it_int
% 16.19/9.06 = ( ^ [N: nat,K3: int] : ( bit_se1409905431419307370or_int @ K3 @ ( bit_se545348938243370406it_int @ N @ one_one_int ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9546_push__bit__nat__def,axiom,
% 16.19/9.06 ( bit_se547839408752420682it_nat
% 16.19/9.06 = ( ^ [N: nat,M6: nat] : ( times_times_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9547_push__bit__int__def,axiom,
% 16.19/9.06 ( bit_se545348938243370406it_int
% 16.19/9.06 = ( ^ [N: nat,K3: int] : ( times_times_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9548_push__bit__minus__one,axiom,
% 16.19/9.06 ! [N2: nat] :
% 16.19/9.06 ( ( bit_se545348938243370406it_int @ N2 @ ( uminus_uminus_int @ one_one_int ) )
% 16.19/9.06 = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9549_XOR__upper,axiom,
% 16.19/9.06 ! [X: int,N2: nat,Y: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 16.19/9.06 => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/9.06 => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 16.19/9.06 => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9550_xor__int__rec,axiom,
% 16.19/9.06 ( bit_se6526347334894502574or_int
% 16.19/9.06 = ( ^ [K3: int,L: int] :
% 16.19/9.06 ( plus_plus_int
% 16.19/9.06 @ ( zero_n2684676970156552555ol_int
% 16.19/9.06 @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) )
% 16.19/9.06 != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) )
% 16.19/9.06 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9551_xor__int__unfold,axiom,
% 16.19/9.06 ( bit_se6526347334894502574or_int
% 16.19/9.06 = ( ^ [K3: int,L: int] :
% 16.19/9.06 ( if_int
% 16.19/9.06 @ ( K3
% 16.19/9.06 = ( uminus_uminus_int @ one_one_int ) )
% 16.19/9.06 @ ( bit_ri7919022796975470100ot_int @ L )
% 16.19/9.06 @ ( if_int
% 16.19/9.06 @ ( L
% 16.19/9.06 = ( uminus_uminus_int @ one_one_int ) )
% 16.19/9.06 @ ( bit_ri7919022796975470100ot_int @ K3 )
% 16.19/9.06 @ ( if_int @ ( K3 = zero_zero_int ) @ L @ ( if_int @ ( L = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9552_Sum__Ico__nat,axiom,
% 16.19/9.06 ! [M: nat,N2: nat] :
% 16.19/9.06 ( ( groups3542108847815614940at_nat
% 16.19/9.06 @ ^ [X2: nat] : X2
% 16.19/9.06 @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
% 16.19/9.06 = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9553_VEBT_Osize_I3_J,axiom,
% 16.19/9.06 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 16.19/9.06 ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 16.19/9.06 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9554_not__nonnegative__int__iff,axiom,
% 16.19/9.06 ! [K: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 16.19/9.06 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9555_not__negative__int__iff,axiom,
% 16.19/9.06 ! [K: int] :
% 16.19/9.06 ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
% 16.19/9.06 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9556_atLeastLessThan__singleton,axiom,
% 16.19/9.06 ! [M: nat] :
% 16.19/9.06 ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
% 16.19/9.06 = ( insert_nat @ M @ bot_bot_set_nat ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9557_and__minus__minus__numerals,axiom,
% 16.19/9.06 ! [M: num,N2: num] :
% 16.19/9.06 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/9.06 = ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9558_or__minus__minus__numerals,axiom,
% 16.19/9.06 ! [M: num,N2: num] :
% 16.19/9.06 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/9.06 = ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9559_bit__not__int__iff,axiom,
% 16.19/9.06 ! [K: int,N2: nat] :
% 16.19/9.06 ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ K ) @ N2 )
% 16.19/9.06 = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9560_all__nat__less__eq,axiom,
% 16.19/9.06 ! [N2: nat,P: nat > $o] :
% 16.19/9.06 ( ( ! [M6: nat] :
% 16.19/9.06 ( ( ord_less_nat @ M6 @ N2 )
% 16.19/9.06 => ( P @ M6 ) ) )
% 16.19/9.06 = ( ! [X2: nat] :
% 16.19/9.06 ( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
% 16.19/9.06 => ( P @ X2 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9561_ex__nat__less__eq,axiom,
% 16.19/9.06 ! [N2: nat,P: nat > $o] :
% 16.19/9.06 ( ( ? [M6: nat] :
% 16.19/9.06 ( ( ord_less_nat @ M6 @ N2 )
% 16.19/9.06 & ( P @ M6 ) ) )
% 16.19/9.06 = ( ? [X2: nat] :
% 16.19/9.06 ( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
% 16.19/9.06 & ( P @ X2 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9562_atLeastLessThanSuc__atLeastAtMost,axiom,
% 16.19/9.06 ! [L2: nat,U: nat] :
% 16.19/9.06 ( ( set_or4665077453230672383an_nat @ L2 @ ( suc @ U ) )
% 16.19/9.06 = ( set_or1269000886237332187st_nat @ L2 @ U ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9563_or__int__def,axiom,
% 16.19/9.06 ( bit_se1409905431419307370or_int
% 16.19/9.06 = ( ^ [K3: int,L: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K3 ) @ ( bit_ri7919022796975470100ot_int @ L ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9564_not__int__def,axiom,
% 16.19/9.06 ( bit_ri7919022796975470100ot_int
% 16.19/9.06 = ( ^ [K3: int] : ( minus_minus_int @ ( uminus_uminus_int @ K3 ) @ one_one_int ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9565_and__not__numerals_I1_J,axiom,
% 16.19/9.06 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 16.19/9.06 = zero_zero_int ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9566_or__not__numerals_I1_J,axiom,
% 16.19/9.06 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 16.19/9.06 = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9567_atLeast0__lessThan__Suc,axiom,
% 16.19/9.06 ! [N2: nat] :
% 16.19/9.06 ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) )
% 16.19/9.06 = ( insert_nat @ N2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9568_unset__bit__int__def,axiom,
% 16.19/9.06 ( bit_se4203085406695923979it_int
% 16.19/9.06 = ( ^ [N: nat,K3: int] : ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N @ one_one_int ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9569_xor__int__def,axiom,
% 16.19/9.06 ( bit_se6526347334894502574or_int
% 16.19/9.06 = ( ^ [K3: int,L: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ L ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K3 ) @ L ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9570_not__int__div__2,axiom,
% 16.19/9.06 ! [K: int] :
% 16.19/9.06 ( ( divide_divide_int @ ( bit_ri7919022796975470100ot_int @ K ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/9.06 = ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9571_even__not__iff__int,axiom,
% 16.19/9.06 ! [K: int] :
% 16.19/9.06 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ K ) )
% 16.19/9.06 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9572_and__not__numerals_I2_J,axiom,
% 16.19/9.06 ! [N2: num] :
% 16.19/9.06 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
% 16.19/9.06 = one_one_int ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9573_and__not__numerals_I4_J,axiom,
% 16.19/9.06 ! [M: num] :
% 16.19/9.06 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 16.19/9.06 = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9574_or__not__numerals_I2_J,axiom,
% 16.19/9.06 ! [N2: num] :
% 16.19/9.06 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
% 16.19/9.06 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9575_or__not__numerals_I4_J,axiom,
% 16.19/9.06 ! [M: num] :
% 16.19/9.06 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 16.19/9.06 = ( bit_ri7919022796975470100ot_int @ one_one_int ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9576_bit__minus__int__iff,axiom,
% 16.19/9.06 ! [K: int,N2: nat] :
% 16.19/9.06 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N2 )
% 16.19/9.06 = ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9577_numeral__or__not__num__eq,axiom,
% 16.19/9.06 ! [M: num,N2: num] :
% 16.19/9.06 ( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N2 ) )
% 16.19/9.06 = ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9578_int__numeral__not__or__num__neg,axiom,
% 16.19/9.06 ! [M: num,N2: num] :
% 16.19/9.06 ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
% 16.19/9.06 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9579_int__numeral__or__not__num__neg,axiom,
% 16.19/9.06 ! [M: num,N2: num] :
% 16.19/9.06 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/9.06 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N2 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9580_atLeastLessThanSuc,axiom,
% 16.19/9.06 ! [M: nat,N2: nat] :
% 16.19/9.06 ( ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.06 => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) )
% 16.19/9.06 = ( insert_nat @ N2 @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) )
% 16.19/9.06 & ( ~ ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.06 => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) )
% 16.19/9.06 = bot_bot_set_nat ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9581_prod__Suc__Suc__fact,axiom,
% 16.19/9.06 ! [N2: nat] :
% 16.19/9.06 ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
% 16.19/9.06 = ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9582_prod__Suc__fact,axiom,
% 16.19/9.06 ! [N2: nat] :
% 16.19/9.06 ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
% 16.19/9.06 = ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9583_and__not__numerals_I5_J,axiom,
% 16.19/9.06 ! [M: num,N2: num] :
% 16.19/9.06 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
% 16.19/9.06 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9584_and__not__numerals_I7_J,axiom,
% 16.19/9.06 ! [M: num] :
% 16.19/9.06 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 16.19/9.06 = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9585_or__not__numerals_I3_J,axiom,
% 16.19/9.06 ! [N2: num] :
% 16.19/9.06 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
% 16.19/9.06 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9586_and__not__numerals_I3_J,axiom,
% 16.19/9.06 ! [N2: num] :
% 16.19/9.06 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
% 16.19/9.06 = zero_zero_int ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9587_or__not__numerals_I7_J,axiom,
% 16.19/9.06 ! [M: num] :
% 16.19/9.06 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 16.19/9.06 = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9588_atLeastLessThan__nat__numeral,axiom,
% 16.19/9.06 ! [M: nat,K: num] :
% 16.19/9.06 ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 16.19/9.06 => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 16.19/9.06 = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
% 16.19/9.06 & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 16.19/9.06 => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 16.19/9.06 = bot_bot_set_nat ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9589_and__not__numerals_I6_J,axiom,
% 16.19/9.06 ! [M: num,N2: num] :
% 16.19/9.06 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
% 16.19/9.06 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9590_and__not__numerals_I9_J,axiom,
% 16.19/9.06 ! [M: num,N2: num] :
% 16.19/9.06 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
% 16.19/9.06 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9591_or__not__numerals_I6_J,axiom,
% 16.19/9.06 ! [M: num,N2: num] :
% 16.19/9.06 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
% 16.19/9.06 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9592_or__not__numerals_I5_J,axiom,
% 16.19/9.06 ! [M: num,N2: num] :
% 16.19/9.06 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
% 16.19/9.06 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9593_atLeast1__lessThan__eq__remove0,axiom,
% 16.19/9.06 ! [N2: nat] :
% 16.19/9.06 ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 16.19/9.06 = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9594_and__not__numerals_I8_J,axiom,
% 16.19/9.06 ! [M: num,N2: num] :
% 16.19/9.06 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
% 16.19/9.06 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9595_or__not__numerals_I9_J,axiom,
% 16.19/9.06 ! [M: num,N2: num] :
% 16.19/9.06 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
% 16.19/9.06 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9596_or__not__numerals_I8_J,axiom,
% 16.19/9.06 ! [M: num,N2: num] :
% 16.19/9.06 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
% 16.19/9.06 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9597_not__int__rec,axiom,
% 16.19/9.06 ( bit_ri7919022796975470100ot_int
% 16.19/9.06 = ( ^ [K3: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9598_sum__power2,axiom,
% 16.19/9.06 ! [K: nat] :
% 16.19/9.06 ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
% 16.19/9.06 = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9599_Chebyshev__sum__upper__nat,axiom,
% 16.19/9.06 ! [N2: nat,A: nat > nat,B: nat > nat] :
% 16.19/9.06 ( ! [I4: nat,J2: nat] :
% 16.19/9.06 ( ( ord_less_eq_nat @ I4 @ J2 )
% 16.19/9.06 => ( ( ord_less_nat @ J2 @ N2 )
% 16.19/9.06 => ( ord_less_eq_nat @ ( A @ I4 ) @ ( A @ J2 ) ) ) )
% 16.19/9.06 => ( ! [I4: nat,J2: nat] :
% 16.19/9.06 ( ( ord_less_eq_nat @ I4 @ J2 )
% 16.19/9.06 => ( ( ord_less_nat @ J2 @ N2 )
% 16.19/9.06 => ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I4 ) ) ) )
% 16.19/9.06 => ( ord_less_eq_nat
% 16.19/9.06 @ ( times_times_nat @ N2
% 16.19/9.06 @ ( groups3542108847815614940at_nat
% 16.19/9.06 @ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( B @ I3 ) )
% 16.19/9.06 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) )
% 16.19/9.06 @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9600_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
% 16.19/9.06 ! [L2: int,U: int] :
% 16.19/9.06 ( ( set_or4662586982721622107an_int @ L2 @ ( plus_plus_int @ U @ one_one_int ) )
% 16.19/9.06 = ( set_or1266510415728281911st_int @ L2 @ U ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9601_VEBT_Osize__gen_I1_J,axiom,
% 16.19/9.06 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 16.19/9.06 ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 16.19/9.06 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9602_valid__eq,axiom,
% 16.19/9.06 vEBT_VEBT_valid = vEBT_invar_vebt ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9603_valid__eq2,axiom,
% 16.19/9.06 ! [T: vEBT_VEBT,D: nat] :
% 16.19/9.06 ( ( vEBT_VEBT_valid @ T @ D )
% 16.19/9.06 => ( vEBT_invar_vebt @ T @ D ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9604_valid__eq1,axiom,
% 16.19/9.06 ! [T: vEBT_VEBT,D: nat] :
% 16.19/9.06 ( ( vEBT_invar_vebt @ T @ D )
% 16.19/9.06 => ( vEBT_VEBT_valid @ T @ D ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9605_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
% 16.19/9.06 ! [Uu: $o,Uv: $o,D: nat] :
% 16.19/9.06 ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D )
% 16.19/9.06 = ( D = one_one_nat ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9606_VEBT_Osize__gen_I2_J,axiom,
% 16.19/9.06 ! [X21: $o,X222: $o] :
% 16.19/9.06 ( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 16.19/9.06 = zero_zero_nat ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9607_csqrt_Osimps_I1_J,axiom,
% 16.19/9.06 ! [Z: complex] :
% 16.19/9.06 ( ( re @ ( csqrt @ Z ) )
% 16.19/9.06 = ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9608_divmod__step__integer__def,axiom,
% 16.19/9.06 ( unique4921790084139445826nteger
% 16.19/9.06 = ( ^ [L: num] :
% 16.19/9.06 ( produc6916734918728496179nteger
% 16.19/9.06 @ ^ [Q4: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R5 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9609_complex__Re__numeral,axiom,
% 16.19/9.06 ! [V: num] :
% 16.19/9.06 ( ( re @ ( numera6690914467698888265omplex @ V ) )
% 16.19/9.06 = ( numeral_numeral_real @ V ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9610_Re__divide__of__nat,axiom,
% 16.19/9.06 ! [Z: complex,N2: nat] :
% 16.19/9.06 ( ( re @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N2 ) ) )
% 16.19/9.06 = ( divide_divide_real @ ( re @ Z ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9611_Re__divide__of__real,axiom,
% 16.19/9.06 ! [Z: complex,R: real] :
% 16.19/9.06 ( ( re @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R ) ) )
% 16.19/9.06 = ( divide_divide_real @ ( re @ Z ) @ R ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9612_Re__sgn,axiom,
% 16.19/9.06 ! [Z: complex] :
% 16.19/9.06 ( ( re @ ( sgn_sgn_complex @ Z ) )
% 16.19/9.06 = ( divide_divide_real @ ( re @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9613_Re__divide__numeral,axiom,
% 16.19/9.06 ! [Z: complex,W: num] :
% 16.19/9.06 ( ( re @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
% 16.19/9.06 = ( divide_divide_real @ ( re @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9614_divmod__integer_H__def,axiom,
% 16.19/9.06 ( unique3479559517661332726nteger
% 16.19/9.06 = ( ^ [M6: num,N: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9615_sgn__integer__code,axiom,
% 16.19/9.06 ( sgn_sgn_Code_integer
% 16.19/9.06 = ( ^ [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 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9616_sums__Re,axiom,
% 16.19/9.06 ! [X8: nat > complex,A: complex] :
% 16.19/9.06 ( ( sums_complex @ X8 @ A )
% 16.19/9.06 => ( sums_real
% 16.19/9.06 @ ^ [N: nat] : ( re @ ( X8 @ N ) )
% 16.19/9.06 @ ( re @ A ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9617_minus__integer__code_I1_J,axiom,
% 16.19/9.06 ! [K: code_integer] :
% 16.19/9.06 ( ( minus_8373710615458151222nteger @ K @ zero_z3403309356797280102nteger )
% 16.19/9.06 = K ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9618_minus__integer__code_I2_J,axiom,
% 16.19/9.06 ! [L2: code_integer] :
% 16.19/9.06 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L2 )
% 16.19/9.06 = ( uminus1351360451143612070nteger @ L2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9619_Cauchy__Re,axiom,
% 16.19/9.06 ! [X8: nat > complex] :
% 16.19/9.06 ( ( topolo6517432010174082258omplex @ X8 )
% 16.19/9.06 => ( topolo4055970368930404560y_real
% 16.19/9.06 @ ^ [N: nat] : ( re @ ( X8 @ N ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9620_complex__Re__le__cmod,axiom,
% 16.19/9.06 ! [X: complex] : ( ord_less_eq_real @ ( re @ X ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9621_one__complex_Osimps_I1_J,axiom,
% 16.19/9.06 ( ( re @ one_one_complex )
% 16.19/9.06 = one_one_real ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9622_plus__complex_Osimps_I1_J,axiom,
% 16.19/9.06 ! [X: complex,Y: complex] :
% 16.19/9.06 ( ( re @ ( plus_plus_complex @ X @ Y ) )
% 16.19/9.06 = ( plus_plus_real @ ( re @ X ) @ ( re @ Y ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9623_scaleR__complex_Osimps_I1_J,axiom,
% 16.19/9.06 ! [R: real,X: complex] :
% 16.19/9.06 ( ( re @ ( real_V2046097035970521341omplex @ R @ X ) )
% 16.19/9.06 = ( times_times_real @ R @ ( re @ X ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9624_minus__complex_Osimps_I1_J,axiom,
% 16.19/9.06 ! [X: complex,Y: complex] :
% 16.19/9.06 ( ( re @ ( minus_minus_complex @ X @ Y ) )
% 16.19/9.06 = ( minus_minus_real @ ( re @ X ) @ ( re @ Y ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9625_summable__Re,axiom,
% 16.19/9.06 ! [F: nat > complex] :
% 16.19/9.06 ( ( summable_complex @ F )
% 16.19/9.06 => ( summable_real
% 16.19/9.06 @ ^ [X2: nat] : ( re @ ( F @ X2 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9626_abs__Re__le__cmod,axiom,
% 16.19/9.06 ! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9627_Re__csqrt,axiom,
% 16.19/9.06 ! [Z: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9628_one__integer_Orsp,axiom,
% 16.19/9.06 one_one_int = one_one_int ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9629_one__natural_Orsp,axiom,
% 16.19/9.06 one_one_nat = one_one_nat ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9630_cmod__plus__Re__le__0__iff,axiom,
% 16.19/9.06 ! [Z: complex] :
% 16.19/9.06 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ zero_zero_real )
% 16.19/9.06 = ( ( re @ Z )
% 16.19/9.06 = ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9631_cos__n__Re__cis__pow__n,axiom,
% 16.19/9.06 ! [N2: nat,A: real] :
% 16.19/9.06 ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ A ) )
% 16.19/9.06 = ( re @ ( power_power_complex @ ( cis @ A ) @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9632_csqrt_Ocode,axiom,
% 16.19/9.06 ( csqrt
% 16.19/9.06 = ( ^ [Z3: complex] :
% 16.19/9.06 ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z3 ) @ ( re @ Z3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.06 @ ( times_times_real
% 16.19/9.06 @ ( if_real
% 16.19/9.06 @ ( ( im @ Z3 )
% 16.19/9.06 = zero_zero_real )
% 16.19/9.06 @ one_one_real
% 16.19/9.06 @ ( sgn_sgn_real @ ( im @ Z3 ) ) )
% 16.19/9.06 @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z3 ) @ ( re @ Z3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9633_csqrt_Osimps_I2_J,axiom,
% 16.19/9.06 ! [Z: complex] :
% 16.19/9.06 ( ( im @ ( csqrt @ Z ) )
% 16.19/9.06 = ( times_times_real
% 16.19/9.06 @ ( if_real
% 16.19/9.06 @ ( ( im @ Z )
% 16.19/9.06 = zero_zero_real )
% 16.19/9.06 @ one_one_real
% 16.19/9.06 @ ( sgn_sgn_real @ ( im @ Z ) ) )
% 16.19/9.06 @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9634_integer__of__int__code,axiom,
% 16.19/9.06 ( code_integer_of_int
% 16.19/9.06 = ( ^ [K3: int] :
% 16.19/9.06 ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K3 ) ) )
% 16.19/9.06 @ ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 16.19/9.06 @ ( if_Code_integer
% 16.19/9.06 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 16.19/9.06 = zero_zero_int )
% 16.19/9.06 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 16.19/9.06 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9635_Im__divide__of__real,axiom,
% 16.19/9.06 ! [Z: complex,R: real] :
% 16.19/9.06 ( ( im @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R ) ) )
% 16.19/9.06 = ( divide_divide_real @ ( im @ Z ) @ R ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9636_Im__sgn,axiom,
% 16.19/9.06 ! [Z: complex] :
% 16.19/9.06 ( ( im @ ( sgn_sgn_complex @ Z ) )
% 16.19/9.06 = ( divide_divide_real @ ( im @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9637_Re__power__real,axiom,
% 16.19/9.06 ! [X: complex,N2: nat] :
% 16.19/9.06 ( ( ( im @ X )
% 16.19/9.06 = zero_zero_real )
% 16.19/9.06 => ( ( re @ ( power_power_complex @ X @ N2 ) )
% 16.19/9.06 = ( power_power_real @ ( re @ X ) @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9638_Im__divide__numeral,axiom,
% 16.19/9.06 ! [Z: complex,W: num] :
% 16.19/9.06 ( ( im @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
% 16.19/9.06 = ( divide_divide_real @ ( im @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9639_Im__divide__of__nat,axiom,
% 16.19/9.06 ! [Z: complex,N2: nat] :
% 16.19/9.06 ( ( im @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N2 ) ) )
% 16.19/9.06 = ( divide_divide_real @ ( im @ Z ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9640_csqrt__of__real__nonneg,axiom,
% 16.19/9.06 ! [X: complex] :
% 16.19/9.06 ( ( ( im @ X )
% 16.19/9.06 = zero_zero_real )
% 16.19/9.06 => ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) )
% 16.19/9.06 => ( ( csqrt @ X )
% 16.19/9.06 = ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9641_csqrt__minus,axiom,
% 16.19/9.06 ! [X: complex] :
% 16.19/9.06 ( ( ( ord_less_real @ ( im @ X ) @ zero_zero_real )
% 16.19/9.06 | ( ( ( im @ X )
% 16.19/9.06 = zero_zero_real )
% 16.19/9.06 & ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) ) ) )
% 16.19/9.06 => ( ( csqrt @ ( uminus1482373934393186551omplex @ X ) )
% 16.19/9.06 = ( times_times_complex @ imaginary_unit @ ( csqrt @ X ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9642_csqrt__of__real__nonpos,axiom,
% 16.19/9.06 ! [X: complex] :
% 16.19/9.06 ( ( ( im @ X )
% 16.19/9.06 = zero_zero_real )
% 16.19/9.06 => ( ( ord_less_eq_real @ ( re @ X ) @ zero_zero_real )
% 16.19/9.06 => ( ( csqrt @ X )
% 16.19/9.06 = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9643_divide__integer_Oabs__eq,axiom,
% 16.19/9.06 ! [Xa2: int,X: int] :
% 16.19/9.06 ( ( divide6298287555418463151nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
% 16.19/9.06 = ( code_integer_of_int @ ( divide_divide_int @ Xa2 @ X ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9644_less__integer_Oabs__eq,axiom,
% 16.19/9.06 ! [Xa2: int,X: int] :
% 16.19/9.06 ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
% 16.19/9.06 = ( ord_less_int @ Xa2 @ X ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9645_abs__integer__code,axiom,
% 16.19/9.06 ( abs_abs_Code_integer
% 16.19/9.06 = ( ^ [K3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K3 ) @ K3 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9646_less__integer__code_I1_J,axiom,
% 16.19/9.06 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9647_sums__Im,axiom,
% 16.19/9.06 ! [X8: nat > complex,A: complex] :
% 16.19/9.06 ( ( sums_complex @ X8 @ A )
% 16.19/9.06 => ( sums_real
% 16.19/9.06 @ ^ [N: nat] : ( im @ ( X8 @ N ) )
% 16.19/9.06 @ ( im @ A ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9648_Cauchy__Im,axiom,
% 16.19/9.06 ! [X8: nat > complex] :
% 16.19/9.06 ( ( topolo6517432010174082258omplex @ X8 )
% 16.19/9.06 => ( topolo4055970368930404560y_real
% 16.19/9.06 @ ^ [N: nat] : ( im @ ( X8 @ N ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9649_imaginary__unit_Osimps_I2_J,axiom,
% 16.19/9.06 ( ( im @ imaginary_unit )
% 16.19/9.06 = one_one_real ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9650_one__complex_Osimps_I2_J,axiom,
% 16.19/9.06 ( ( im @ one_one_complex )
% 16.19/9.06 = zero_zero_real ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9651_one__integer__def,axiom,
% 16.19/9.06 ( one_one_Code_integer
% 16.19/9.06 = ( code_integer_of_int @ one_one_int ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9652_minus__integer_Oabs__eq,axiom,
% 16.19/9.06 ! [Xa2: int,X: int] :
% 16.19/9.06 ( ( minus_8373710615458151222nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
% 16.19/9.06 = ( code_integer_of_int @ ( minus_minus_int @ Xa2 @ X ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9653_plus__complex_Osimps_I2_J,axiom,
% 16.19/9.06 ! [X: complex,Y: complex] :
% 16.19/9.06 ( ( im @ ( plus_plus_complex @ X @ Y ) )
% 16.19/9.06 = ( plus_plus_real @ ( im @ X ) @ ( im @ Y ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9654_scaleR__complex_Osimps_I2_J,axiom,
% 16.19/9.06 ! [R: real,X: complex] :
% 16.19/9.06 ( ( im @ ( real_V2046097035970521341omplex @ R @ X ) )
% 16.19/9.06 = ( times_times_real @ R @ ( im @ X ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9655_minus__complex_Osimps_I2_J,axiom,
% 16.19/9.06 ! [X: complex,Y: complex] :
% 16.19/9.06 ( ( im @ ( minus_minus_complex @ X @ Y ) )
% 16.19/9.06 = ( minus_minus_real @ ( im @ X ) @ ( im @ Y ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9656_sums__complex__iff,axiom,
% 16.19/9.06 ( sums_complex
% 16.19/9.06 = ( ^ [F3: nat > complex,X2: complex] :
% 16.19/9.06 ( ( sums_real
% 16.19/9.06 @ ^ [Y2: nat] : ( re @ ( F3 @ Y2 ) )
% 16.19/9.06 @ ( re @ X2 ) )
% 16.19/9.06 & ( sums_real
% 16.19/9.06 @ ^ [Y2: nat] : ( im @ ( F3 @ Y2 ) )
% 16.19/9.06 @ ( im @ X2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9657_summable__Im,axiom,
% 16.19/9.06 ! [F: nat > complex] :
% 16.19/9.06 ( ( summable_complex @ F )
% 16.19/9.06 => ( summable_real
% 16.19/9.06 @ ^ [X2: nat] : ( im @ ( F @ X2 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9658_abs__Im__le__cmod,axiom,
% 16.19/9.06 ! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9659_summable__complex__iff,axiom,
% 16.19/9.06 ( summable_complex
% 16.19/9.06 = ( ^ [F3: nat > complex] :
% 16.19/9.06 ( ( summable_real
% 16.19/9.06 @ ^ [X2: nat] : ( re @ ( F3 @ X2 ) ) )
% 16.19/9.06 & ( summable_real
% 16.19/9.06 @ ^ [X2: nat] : ( im @ ( F3 @ X2 ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9660_times__complex_Osimps_I2_J,axiom,
% 16.19/9.06 ! [X: complex,Y: complex] :
% 16.19/9.06 ( ( im @ ( times_times_complex @ X @ Y ) )
% 16.19/9.06 = ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9661_cmod__Im__le__iff,axiom,
% 16.19/9.06 ! [X: complex,Y: complex] :
% 16.19/9.06 ( ( ( re @ X )
% 16.19/9.06 = ( re @ Y ) )
% 16.19/9.06 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) )
% 16.19/9.06 = ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( abs_abs_real @ ( im @ Y ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9662_cmod__Re__le__iff,axiom,
% 16.19/9.06 ! [X: complex,Y: complex] :
% 16.19/9.06 ( ( ( im @ X )
% 16.19/9.06 = ( im @ Y ) )
% 16.19/9.06 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) )
% 16.19/9.06 = ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( abs_abs_real @ ( re @ Y ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9663_times__complex_Osimps_I1_J,axiom,
% 16.19/9.06 ! [X: complex,Y: complex] :
% 16.19/9.06 ( ( re @ ( times_times_complex @ X @ Y ) )
% 16.19/9.06 = ( minus_minus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9664_plus__complex_Ocode,axiom,
% 16.19/9.06 ( plus_plus_complex
% 16.19/9.06 = ( ^ [X2: complex,Y2: complex] : ( complex2 @ ( plus_plus_real @ ( re @ X2 ) @ ( re @ Y2 ) ) @ ( plus_plus_real @ ( im @ X2 ) @ ( im @ Y2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9665_scaleR__complex_Ocode,axiom,
% 16.19/9.06 ( real_V2046097035970521341omplex
% 16.19/9.06 = ( ^ [R5: real,X2: complex] : ( complex2 @ ( times_times_real @ R5 @ ( re @ X2 ) ) @ ( times_times_real @ R5 @ ( im @ X2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9666_minus__complex_Ocode,axiom,
% 16.19/9.06 ( minus_minus_complex
% 16.19/9.06 = ( ^ [X2: complex,Y2: complex] : ( complex2 @ ( minus_minus_real @ ( re @ X2 ) @ ( re @ Y2 ) ) @ ( minus_minus_real @ ( im @ X2 ) @ ( im @ Y2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9667_csqrt__principal,axiom,
% 16.19/9.06 ! [Z: complex] :
% 16.19/9.06 ( ( ord_less_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) )
% 16.19/9.06 | ( ( ( re @ ( csqrt @ Z ) )
% 16.19/9.06 = zero_zero_real )
% 16.19/9.06 & ( ord_less_eq_real @ zero_zero_real @ ( im @ ( csqrt @ Z ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9668_cmod__le,axiom,
% 16.19/9.06 ! [Z: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9669_sin__n__Im__cis__pow__n,axiom,
% 16.19/9.06 ! [N2: nat,A: real] :
% 16.19/9.06 ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ A ) )
% 16.19/9.06 = ( im @ ( power_power_complex @ ( cis @ A ) @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9670_Re__exp,axiom,
% 16.19/9.06 ! [Z: complex] :
% 16.19/9.06 ( ( re @ ( exp_complex @ Z ) )
% 16.19/9.06 = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( cos_real @ ( im @ Z ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9671_Im__exp,axiom,
% 16.19/9.06 ! [Z: complex] :
% 16.19/9.06 ( ( im @ ( exp_complex @ Z ) )
% 16.19/9.06 = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( sin_real @ ( im @ Z ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9672_times__complex_Ocode,axiom,
% 16.19/9.06 ( times_times_complex
% 16.19/9.06 = ( ^ [X2: complex,Y2: complex] : ( complex2 @ ( minus_minus_real @ ( times_times_real @ ( re @ X2 ) @ ( re @ Y2 ) ) @ ( times_times_real @ ( im @ X2 ) @ ( im @ Y2 ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X2 ) @ ( im @ Y2 ) ) @ ( times_times_real @ ( im @ X2 ) @ ( re @ Y2 ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9673_cmod__power2,axiom,
% 16.19/9.06 ! [Z: complex] :
% 16.19/9.06 ( ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.06 = ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9674_Im__power2,axiom,
% 16.19/9.06 ! [X: complex] :
% 16.19/9.06 ( ( im @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.06 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ X ) ) @ ( im @ X ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9675_Re__power2,axiom,
% 16.19/9.06 ! [X: complex] :
% 16.19/9.06 ( ( re @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.06 = ( minus_minus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9676_complex__eq__0,axiom,
% 16.19/9.06 ! [Z: complex] :
% 16.19/9.06 ( ( Z = zero_zero_complex )
% 16.19/9.06 = ( ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.06 = zero_zero_real ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9677_norm__complex__def,axiom,
% 16.19/9.06 ( real_V1022390504157884413omplex
% 16.19/9.06 = ( ^ [Z3: complex] : ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9678_inverse__complex_Osimps_I1_J,axiom,
% 16.19/9.06 ! [X: complex] :
% 16.19/9.06 ( ( re @ ( invers8013647133539491842omplex @ X ) )
% 16.19/9.06 = ( divide_divide_real @ ( re @ X ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9679_complex__neq__0,axiom,
% 16.19/9.06 ! [Z: complex] :
% 16.19/9.06 ( ( Z != zero_zero_complex )
% 16.19/9.06 = ( 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 ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9680_Re__divide,axiom,
% 16.19/9.06 ! [X: complex,Y: complex] :
% 16.19/9.06 ( ( re @ ( divide1717551699836669952omplex @ X @ Y ) )
% 16.19/9.06 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9681_csqrt__unique,axiom,
% 16.19/9.06 ! [W: complex,Z: complex] :
% 16.19/9.06 ( ( ( power_power_complex @ W @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 16.19/9.06 = Z )
% 16.19/9.06 => ( ( ( ord_less_real @ zero_zero_real @ ( re @ W ) )
% 16.19/9.06 | ( ( ( re @ W )
% 16.19/9.06 = zero_zero_real )
% 16.19/9.06 & ( ord_less_eq_real @ zero_zero_real @ ( im @ W ) ) ) )
% 16.19/9.06 => ( ( csqrt @ Z )
% 16.19/9.06 = W ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9682_csqrt__square,axiom,
% 16.19/9.06 ! [B: complex] :
% 16.19/9.06 ( ( ( ord_less_real @ zero_zero_real @ ( re @ B ) )
% 16.19/9.06 | ( ( ( re @ B )
% 16.19/9.06 = zero_zero_real )
% 16.19/9.06 & ( ord_less_eq_real @ zero_zero_real @ ( im @ B ) ) ) )
% 16.19/9.06 => ( ( csqrt @ ( power_power_complex @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.06 = B ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9683_inverse__complex_Osimps_I2_J,axiom,
% 16.19/9.06 ! [X: complex] :
% 16.19/9.06 ( ( im @ ( invers8013647133539491842omplex @ X ) )
% 16.19/9.06 = ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9684_Im__divide,axiom,
% 16.19/9.06 ! [X: complex,Y: complex] :
% 16.19/9.06 ( ( im @ ( divide1717551699836669952omplex @ X @ Y ) )
% 16.19/9.06 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9685_complex__abs__le__norm,axiom,
% 16.19/9.06 ! [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 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9686_complex__unit__circle,axiom,
% 16.19/9.06 ! [Z: complex] :
% 16.19/9.06 ( ( Z != zero_zero_complex )
% 16.19/9.06 => ( ( plus_plus_real @ ( power_power_real @ ( divide_divide_real @ ( re @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( divide_divide_real @ ( im @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.06 = one_one_real ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9687_inverse__complex_Ocode,axiom,
% 16.19/9.06 ( invers8013647133539491842omplex
% 16.19/9.06 = ( ^ [X2: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X2 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X2 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9688_Complex__divide,axiom,
% 16.19/9.06 ( divide1717551699836669952omplex
% 16.19/9.06 = ( ^ [X2: complex,Y2: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X2 ) @ ( re @ Y2 ) ) @ ( times_times_real @ ( im @ X2 ) @ ( im @ Y2 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X2 ) @ ( re @ Y2 ) ) @ ( times_times_real @ ( re @ X2 ) @ ( im @ Y2 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9689_Im__Reals__divide,axiom,
% 16.19/9.06 ! [R: complex,Z: complex] :
% 16.19/9.06 ( ( member_complex @ R @ real_V2521375963428798218omplex )
% 16.19/9.06 => ( ( im @ ( divide1717551699836669952omplex @ R @ Z ) )
% 16.19/9.06 = ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R ) ) @ ( im @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9690_Re__Reals__divide,axiom,
% 16.19/9.06 ! [R: complex,Z: complex] :
% 16.19/9.06 ( ( member_complex @ R @ real_V2521375963428798218omplex )
% 16.19/9.06 => ( ( re @ ( divide1717551699836669952omplex @ R @ Z ) )
% 16.19/9.06 = ( divide_divide_real @ ( times_times_real @ ( re @ R ) @ ( re @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9691_Re__divide__Reals,axiom,
% 16.19/9.06 ! [R: complex,Z: complex] :
% 16.19/9.06 ( ( member_complex @ R @ real_V2521375963428798218omplex )
% 16.19/9.06 => ( ( re @ ( divide1717551699836669952omplex @ Z @ R ) )
% 16.19/9.06 = ( divide_divide_real @ ( re @ Z ) @ ( re @ R ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9692_Im__divide__Reals,axiom,
% 16.19/9.06 ! [R: complex,Z: complex] :
% 16.19/9.06 ( ( member_complex @ R @ real_V2521375963428798218omplex )
% 16.19/9.06 => ( ( im @ ( divide1717551699836669952omplex @ Z @ R ) )
% 16.19/9.06 = ( divide_divide_real @ ( im @ Z ) @ ( re @ R ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9693_complex__diff__cnj,axiom,
% 16.19/9.06 ! [Z: complex] :
% 16.19/9.06 ( ( minus_minus_complex @ Z @ ( cnj @ Z ) )
% 16.19/9.06 = ( times_times_complex @ ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( im @ Z ) ) ) @ imaginary_unit ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9694_complex__mult__cnj,axiom,
% 16.19/9.06 ! [Z: complex] :
% 16.19/9.06 ( ( times_times_complex @ Z @ ( cnj @ Z ) )
% 16.19/9.06 = ( real_V4546457046886955230omplex @ ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9695_complex__cnj__divide,axiom,
% 16.19/9.06 ! [X: complex,Y: complex] :
% 16.19/9.06 ( ( cnj @ ( divide1717551699836669952omplex @ X @ Y ) )
% 16.19/9.06 = ( divide1717551699836669952omplex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9696_complex__cnj__one,axiom,
% 16.19/9.06 ( ( cnj @ one_one_complex )
% 16.19/9.06 = one_one_complex ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9697_complex__cnj__one__iff,axiom,
% 16.19/9.06 ! [Z: complex] :
% 16.19/9.06 ( ( ( cnj @ Z )
% 16.19/9.06 = one_one_complex )
% 16.19/9.06 = ( Z = one_one_complex ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9698_complex__cnj__diff,axiom,
% 16.19/9.06 ! [X: complex,Y: complex] :
% 16.19/9.06 ( ( cnj @ ( minus_minus_complex @ X @ Y ) )
% 16.19/9.06 = ( minus_minus_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9699_sums__cnj,axiom,
% 16.19/9.06 ! [F: nat > complex,L2: complex] :
% 16.19/9.06 ( ( sums_complex
% 16.19/9.06 @ ^ [X2: nat] : ( cnj @ ( F @ X2 ) )
% 16.19/9.06 @ ( cnj @ L2 ) )
% 16.19/9.06 = ( sums_complex @ F @ L2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9700_Re__complex__div__eq__0,axiom,
% 16.19/9.06 ! [A: complex,B: complex] :
% 16.19/9.06 ( ( ( re @ ( divide1717551699836669952omplex @ A @ B ) )
% 16.19/9.06 = zero_zero_real )
% 16.19/9.06 = ( ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) )
% 16.19/9.06 = zero_zero_real ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9701_Im__complex__div__eq__0,axiom,
% 16.19/9.06 ! [A: complex,B: complex] :
% 16.19/9.06 ( ( ( im @ ( divide1717551699836669952omplex @ A @ B ) )
% 16.19/9.06 = zero_zero_real )
% 16.19/9.06 = ( ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) )
% 16.19/9.06 = zero_zero_real ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9702_Re__complex__div__gt__0,axiom,
% 16.19/9.06 ! [A: complex,B: complex] :
% 16.19/9.06 ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 16.19/9.06 = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9703_Re__complex__div__lt__0,axiom,
% 16.19/9.06 ! [A: complex,B: complex] :
% 16.19/9.06 ( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 16.19/9.06 = ( ord_less_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9704_Re__complex__div__le__0,axiom,
% 16.19/9.06 ! [A: complex,B: complex] :
% 16.19/9.06 ( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 16.19/9.06 = ( ord_less_eq_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9705_Re__complex__div__ge__0,axiom,
% 16.19/9.06 ! [A: complex,B: complex] :
% 16.19/9.06 ( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 16.19/9.06 = ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9706_Im__complex__div__gt__0,axiom,
% 16.19/9.06 ! [A: complex,B: complex] :
% 16.19/9.06 ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 16.19/9.06 = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9707_Im__complex__div__lt__0,axiom,
% 16.19/9.06 ! [A: complex,B: complex] :
% 16.19/9.06 ( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 16.19/9.06 = ( ord_less_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9708_Im__complex__div__le__0,axiom,
% 16.19/9.06 ! [A: complex,B: complex] :
% 16.19/9.06 ( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 16.19/9.06 = ( ord_less_eq_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9709_Im__complex__div__ge__0,axiom,
% 16.19/9.06 ! [A: complex,B: complex] :
% 16.19/9.06 ( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 16.19/9.06 = ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9710_complex__mod__mult__cnj,axiom,
% 16.19/9.06 ! [Z: complex] :
% 16.19/9.06 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
% 16.19/9.06 = ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9711_complex__div__gt__0,axiom,
% 16.19/9.06 ! [A: complex,B: complex] :
% 16.19/9.06 ( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 16.19/9.06 = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) )
% 16.19/9.06 & ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 16.19/9.06 = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9712_complex__norm__square,axiom,
% 16.19/9.06 ! [Z: complex] :
% 16.19/9.06 ( ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 16.19/9.06 = ( times_times_complex @ Z @ ( cnj @ Z ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9713_complex__add__cnj,axiom,
% 16.19/9.06 ! [Z: complex] :
% 16.19/9.06 ( ( plus_plus_complex @ Z @ ( cnj @ Z ) )
% 16.19/9.06 = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ Z ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9714_complex__div__cnj,axiom,
% 16.19/9.06 ( divide1717551699836669952omplex
% 16.19/9.06 = ( ^ [A5: complex,B5: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A5 @ ( cnj @ B5 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9715_cnj__add__mult__eq__Re,axiom,
% 16.19/9.06 ! [Z: complex,W: complex] :
% 16.19/9.06 ( ( plus_plus_complex @ ( times_times_complex @ Z @ ( cnj @ W ) ) @ ( times_times_complex @ ( cnj @ Z ) @ W ) )
% 16.19/9.06 = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ ( times_times_complex @ Z @ ( cnj @ W ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9716_integer__of__num_I3_J,axiom,
% 16.19/9.06 ! [N2: num] :
% 16.19/9.06 ( ( code_integer_of_num @ ( bit1 @ N2 ) )
% 16.19/9.06 = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N2 ) @ ( code_integer_of_num @ N2 ) ) @ one_one_Code_integer ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9717_int__of__integer__code,axiom,
% 16.19/9.06 ( code_int_of_integer
% 16.19/9.06 = ( ^ [K3: code_integer] :
% 16.19/9.06 ( if_int @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K3 ) ) )
% 16.19/9.06 @ ( if_int @ ( K3 = zero_z3403309356797280102nteger ) @ zero_zero_int
% 16.19/9.06 @ ( produc1553301316500091796er_int
% 16.19/9.06 @ ^ [L: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L ) ) @ one_one_int ) )
% 16.19/9.06 @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9718_bit__cut__integer__def,axiom,
% 16.19/9.06 ( code_bit_cut_integer
% 16.19/9.06 = ( ^ [K3: code_integer] :
% 16.19/9.06 ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 16.19/9.06 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K3 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9719_one__integer_Orep__eq,axiom,
% 16.19/9.06 ( ( code_int_of_integer @ one_one_Code_integer )
% 16.19/9.06 = one_one_int ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9720_minus__integer_Orep__eq,axiom,
% 16.19/9.06 ! [X: code_integer,Xa2: code_integer] :
% 16.19/9.06 ( ( code_int_of_integer @ ( minus_8373710615458151222nteger @ X @ Xa2 ) )
% 16.19/9.06 = ( minus_minus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9721_divide__integer_Orep__eq,axiom,
% 16.19/9.06 ! [X: code_integer,Xa2: code_integer] :
% 16.19/9.06 ( ( code_int_of_integer @ ( divide6298287555418463151nteger @ X @ Xa2 ) )
% 16.19/9.06 = ( divide_divide_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9722_less__integer_Orep__eq,axiom,
% 16.19/9.06 ( ord_le6747313008572928689nteger
% 16.19/9.06 = ( ^ [X2: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9723_integer__less__iff,axiom,
% 16.19/9.06 ( ord_le6747313008572928689nteger
% 16.19/9.06 = ( ^ [K3: code_integer,L: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9724_integer__of__num__triv_I1_J,axiom,
% 16.19/9.06 ( ( code_integer_of_num @ one )
% 16.19/9.06 = one_one_Code_integer ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9725_integer__of__num_I2_J,axiom,
% 16.19/9.06 ! [N2: num] :
% 16.19/9.06 ( ( code_integer_of_num @ ( bit0 @ N2 ) )
% 16.19/9.06 = ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N2 ) @ ( code_integer_of_num @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9726_integer__of__num__triv_I2_J,axiom,
% 16.19/9.06 ( ( code_integer_of_num @ ( bit0 @ one ) )
% 16.19/9.06 = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9727_divmod__integer__def,axiom,
% 16.19/9.06 ( code_divmod_integer
% 16.19/9.06 = ( ^ [K3: code_integer,L: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K3 @ L ) @ ( modulo364778990260209775nteger @ K3 @ L ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9728_num__of__integer__code,axiom,
% 16.19/9.06 ( code_num_of_integer
% 16.19/9.06 = ( ^ [K3: code_integer] :
% 16.19/9.06 ( if_num @ ( ord_le3102999989581377725nteger @ K3 @ one_one_Code_integer ) @ one
% 16.19/9.06 @ ( produc7336495610019696514er_num
% 16.19/9.06 @ ^ [L: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L ) @ ( code_num_of_integer @ L ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L ) @ ( code_num_of_integer @ L ) ) @ one ) )
% 16.19/9.06 @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9729_bit__cut__integer__code,axiom,
% 16.19/9.06 ( code_bit_cut_integer
% 16.19/9.06 = ( ^ [K3: code_integer] :
% 16.19/9.06 ( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
% 16.19/9.06 @ ( produc9125791028180074456eger_o
% 16.19/9.06 @ ^ [R5: code_integer,S5: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S5 ) ) @ ( S5 = one_one_Code_integer ) )
% 16.19/9.06 @ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9730_nat__of__integer__code,axiom,
% 16.19/9.06 ( code_nat_of_integer
% 16.19/9.06 = ( ^ [K3: code_integer] :
% 16.19/9.06 ( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
% 16.19/9.06 @ ( produc1555791787009142072er_nat
% 16.19/9.06 @ ^ [L: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L ) @ ( code_nat_of_integer @ L ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L ) @ ( code_nat_of_integer @ L ) ) @ one_one_nat ) )
% 16.19/9.06 @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9731_nat__of__integer__code__post_I3_J,axiom,
% 16.19/9.06 ! [K: num] :
% 16.19/9.06 ( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ K ) )
% 16.19/9.06 = ( numeral_numeral_nat @ K ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9732_nat__of__integer__code__post_I2_J,axiom,
% 16.19/9.06 ( ( code_nat_of_integer @ one_one_Code_integer )
% 16.19/9.06 = one_one_nat ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9733_divmod__abs__def,axiom,
% 16.19/9.06 ( code_divmod_abs
% 16.19/9.06 = ( ^ [K3: code_integer,L: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9734_divmod__integer__code,axiom,
% 16.19/9.06 ( code_divmod_integer
% 16.19/9.06 = ( ^ [K3: code_integer,L: code_integer] :
% 16.19/9.06 ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 16.19/9.06 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L )
% 16.19/9.06 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L )
% 16.19/9.06 @ ( produc6916734918728496179nteger
% 16.19/9.06 @ ^ [R5: code_integer,S5: code_integer] : ( if_Pro6119634080678213985nteger @ ( S5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L @ S5 ) ) )
% 16.19/9.06 @ ( code_divmod_abs @ K3 @ L ) ) )
% 16.19/9.06 @ ( if_Pro6119634080678213985nteger @ ( L = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
% 16.19/9.06 @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
% 16.19/9.06 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L )
% 16.19/9.06 @ ( produc6916734918728496179nteger
% 16.19/9.06 @ ^ [R5: code_integer,S5: code_integer] : ( if_Pro6119634080678213985nteger @ ( S5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L ) @ S5 ) ) )
% 16.19/9.06 @ ( code_divmod_abs @ K3 @ L ) ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9735_card__Collect__less__nat,axiom,
% 16.19/9.06 ! [N2: nat] :
% 16.19/9.06 ( ( finite_card_nat
% 16.19/9.06 @ ( collect_nat
% 16.19/9.06 @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N2 ) ) )
% 16.19/9.06 = N2 ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9736_card__atMost,axiom,
% 16.19/9.06 ! [U: nat] :
% 16.19/9.06 ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
% 16.19/9.06 = ( suc @ U ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9737_card__atLeastLessThan,axiom,
% 16.19/9.06 ! [L2: nat,U: nat] :
% 16.19/9.06 ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L2 @ U ) )
% 16.19/9.06 = ( minus_minus_nat @ U @ L2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9738_card__Collect__le__nat,axiom,
% 16.19/9.06 ! [N2: nat] :
% 16.19/9.06 ( ( finite_card_nat
% 16.19/9.06 @ ( collect_nat
% 16.19/9.06 @ ^ [I3: nat] : ( ord_less_eq_nat @ I3 @ N2 ) ) )
% 16.19/9.06 = ( suc @ N2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9739_card__atLeastAtMost,axiom,
% 16.19/9.06 ! [L2: nat,U: nat] :
% 16.19/9.06 ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L2 @ U ) )
% 16.19/9.06 = ( minus_minus_nat @ ( suc @ U ) @ L2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9740_card__atLeastLessThan__int,axiom,
% 16.19/9.06 ! [L2: int,U: int] :
% 16.19/9.06 ( ( finite_card_int @ ( set_or4662586982721622107an_int @ L2 @ U ) )
% 16.19/9.06 = ( nat2 @ ( minus_minus_int @ U @ L2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9741_card__atLeastAtMost__int,axiom,
% 16.19/9.06 ! [L2: int,U: int] :
% 16.19/9.06 ( ( finite_card_int @ ( set_or1266510415728281911st_int @ L2 @ U ) )
% 16.19/9.06 = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L2 ) @ one_one_int ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9742_nat_Odisc__eq__case_I1_J,axiom,
% 16.19/9.06 ! [Nat: nat] :
% 16.19/9.06 ( ( Nat = zero_zero_nat )
% 16.19/9.06 = ( case_nat_o @ $true
% 16.19/9.06 @ ^ [Uu3: nat] : $false
% 16.19/9.06 @ Nat ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9743_nat_Odisc__eq__case_I2_J,axiom,
% 16.19/9.06 ! [Nat: nat] :
% 16.19/9.06 ( ( Nat != zero_zero_nat )
% 16.19/9.06 = ( case_nat_o @ $false
% 16.19/9.06 @ ^ [Uu3: nat] : $true
% 16.19/9.06 @ Nat ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9744_card__less,axiom,
% 16.19/9.06 ! [M7: set_nat,I2: nat] :
% 16.19/9.06 ( ( member_nat @ zero_zero_nat @ M7 )
% 16.19/9.06 => ( ( finite_card_nat
% 16.19/9.06 @ ( collect_nat
% 16.19/9.06 @ ^ [K3: nat] :
% 16.19/9.06 ( ( member_nat @ K3 @ M7 )
% 16.19/9.06 & ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) )
% 16.19/9.06 != zero_zero_nat ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9745_card__less__Suc,axiom,
% 16.19/9.06 ! [M7: set_nat,I2: nat] :
% 16.19/9.06 ( ( member_nat @ zero_zero_nat @ M7 )
% 16.19/9.06 => ( ( suc
% 16.19/9.06 @ ( finite_card_nat
% 16.19/9.06 @ ( collect_nat
% 16.19/9.06 @ ^ [K3: nat] :
% 16.19/9.06 ( ( member_nat @ ( suc @ K3 ) @ M7 )
% 16.19/9.06 & ( ord_less_nat @ K3 @ I2 ) ) ) ) )
% 16.19/9.06 = ( finite_card_nat
% 16.19/9.06 @ ( collect_nat
% 16.19/9.06 @ ^ [K3: nat] :
% 16.19/9.06 ( ( member_nat @ K3 @ M7 )
% 16.19/9.06 & ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9746_card__less__Suc2,axiom,
% 16.19/9.06 ! [M7: set_nat,I2: nat] :
% 16.19/9.06 ( ~ ( member_nat @ zero_zero_nat @ M7 )
% 16.19/9.06 => ( ( finite_card_nat
% 16.19/9.06 @ ( collect_nat
% 16.19/9.06 @ ^ [K3: nat] :
% 16.19/9.06 ( ( member_nat @ ( suc @ K3 ) @ M7 )
% 16.19/9.06 & ( ord_less_nat @ K3 @ I2 ) ) ) )
% 16.19/9.06 = ( finite_card_nat
% 16.19/9.06 @ ( collect_nat
% 16.19/9.06 @ ^ [K3: nat] :
% 16.19/9.06 ( ( member_nat @ K3 @ M7 )
% 16.19/9.06 & ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9747_subset__card__intvl__is__intvl,axiom,
% 16.19/9.06 ! [A2: set_nat,K: nat] :
% 16.19/9.06 ( ( ord_less_eq_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) )
% 16.19/9.06 => ( A2
% 16.19/9.06 = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9748_less__eq__nat_Osimps_I2_J,axiom,
% 16.19/9.06 ! [M: nat,N2: nat] :
% 16.19/9.06 ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
% 16.19/9.06 = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9749_max__Suc1,axiom,
% 16.19/9.06 ! [N2: nat,M: nat] :
% 16.19/9.06 ( ( ord_max_nat @ ( suc @ N2 ) @ M )
% 16.19/9.06 = ( case_nat_nat @ ( suc @ N2 )
% 16.19/9.06 @ ^ [M3: nat] : ( suc @ ( ord_max_nat @ N2 @ M3 ) )
% 16.19/9.06 @ M ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9750_max__Suc2,axiom,
% 16.19/9.06 ! [M: nat,N2: nat] :
% 16.19/9.06 ( ( ord_max_nat @ M @ ( suc @ N2 ) )
% 16.19/9.06 = ( case_nat_nat @ ( suc @ N2 )
% 16.19/9.06 @ ^ [M3: nat] : ( suc @ ( ord_max_nat @ M3 @ N2 ) )
% 16.19/9.06 @ M ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9751_subset__eq__atLeast0__lessThan__card,axiom,
% 16.19/9.06 ! [N3: set_nat,N2: nat] :
% 16.19/9.06 ( ( ord_less_eq_set_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
% 16.19/9.06 => ( ord_less_eq_nat @ ( finite_card_nat @ N3 ) @ N2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9752_card__sum__le__nat__sum,axiom,
% 16.19/9.06 ! [S: set_nat] :
% 16.19/9.06 ( ord_less_eq_nat
% 16.19/9.06 @ ( groups3542108847815614940at_nat
% 16.19/9.06 @ ^ [X2: nat] : X2
% 16.19/9.06 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S ) ) )
% 16.19/9.06 @ ( groups3542108847815614940at_nat
% 16.19/9.06 @ ^ [X2: nat] : X2
% 16.19/9.06 @ S ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9753_card__nth__roots,axiom,
% 16.19/9.06 ! [C: complex,N2: nat] :
% 16.19/9.06 ( ( C != zero_zero_complex )
% 16.19/9.06 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.06 => ( ( finite_card_complex
% 16.19/9.06 @ ( collect_complex
% 16.19/9.06 @ ^ [Z3: complex] :
% 16.19/9.06 ( ( power_power_complex @ Z3 @ N2 )
% 16.19/9.06 = C ) ) )
% 16.19/9.06 = N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9754_card__roots__unity__eq,axiom,
% 16.19/9.06 ! [N2: nat] :
% 16.19/9.06 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.06 => ( ( finite_card_complex
% 16.19/9.06 @ ( collect_complex
% 16.19/9.06 @ ^ [Z3: complex] :
% 16.19/9.06 ( ( power_power_complex @ Z3 @ N2 )
% 16.19/9.06 = one_one_complex ) ) )
% 16.19/9.06 = N2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9755_diff__Suc,axiom,
% 16.19/9.06 ! [M: nat,N2: nat] :
% 16.19/9.06 ( ( minus_minus_nat @ M @ ( suc @ N2 ) )
% 16.19/9.06 = ( case_nat_nat @ zero_zero_nat
% 16.19/9.06 @ ^ [K3: nat] : K3
% 16.19/9.06 @ ( minus_minus_nat @ M @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9756_pred__def,axiom,
% 16.19/9.06 ( pred
% 16.19/9.06 = ( case_nat_nat @ zero_zero_nat
% 16.19/9.06 @ ^ [X24: nat] : X24 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9757_binomial__def,axiom,
% 16.19/9.06 ( binomial
% 16.19/9.06 = ( ^ [N: nat,K3: nat] :
% 16.19/9.06 ( finite_card_set_nat
% 16.19/9.06 @ ( collect_set_nat
% 16.19/9.06 @ ^ [K7: set_nat] :
% 16.19/9.06 ( ( member_set_nat @ K7 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
% 16.19/9.06 & ( ( finite_card_nat @ K7 )
% 16.19/9.06 = K3 ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9758_firstsimp,axiom,
% 16.19/9.06 ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 16.19/9.06 = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 16.19/9.06 @ ( vEBT_Node
% 16.19/9.06 @ ( some_P7363390416028606310at_nat
% 16.19/9.06 @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 16.19/9.06 @ ( if_nat
% 16.19/9.06 @ ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 16.19/9.06 = ma )
% 16.19/9.06 @ ( if_nat
% 16.19/9.06 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 16.19/9.06 = none_nat )
% 16.19/9.06 @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 16.19/9.06 @ ( 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 @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( append_VEBT_VEBT @ ( take_VEBT_VEBT @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ treeList ) @ ( append_VEBT_VEBT @ ( cons_VEBT_VEBT @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ nil_VEBT_VEBT ) @ ( drop_VEBT_VEBT @ ( plus_plus_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ one_one_nat ) @ treeList ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) ) ) )
% 16.19/9.06 @ ma ) ) )
% 16.19/9.06 @ deg
% 16.19/9.06 @ ( append_VEBT_VEBT @ ( take_VEBT_VEBT @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ treeList ) @ ( append_VEBT_VEBT @ ( cons_VEBT_VEBT @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ nil_VEBT_VEBT ) @ ( drop_VEBT_VEBT @ ( plus_plus_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ one_one_nat ) @ treeList ) ) )
% 16.19/9.06 @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) )
% 16.19/9.06 @ ( vEBT_Node
% 16.19/9.06 @ ( some_P7363390416028606310at_nat
% 16.19/9.06 @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 16.19/9.06 @ ( if_nat
% 16.19/9.06 @ ( ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx )
% 16.19/9.06 = ma )
% 16.19/9.06 @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( 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 @ ( append_VEBT_VEBT @ ( take_VEBT_VEBT @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ treeList ) @ ( append_VEBT_VEBT @ ( cons_VEBT_VEBT @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ nil_VEBT_VEBT ) @ ( drop_VEBT_VEBT @ ( plus_plus_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ one_one_nat ) @ treeList ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) )
% 16.19/9.06 @ ma ) ) )
% 16.19/9.06 @ deg
% 16.19/9.06 @ ( append_VEBT_VEBT @ ( take_VEBT_VEBT @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ treeList ) @ ( append_VEBT_VEBT @ ( cons_VEBT_VEBT @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ nil_VEBT_VEBT ) @ ( drop_VEBT_VEBT @ ( plus_plus_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ one_one_nat ) @ treeList ) ) )
% 16.19/9.06 @ summary ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9759_bezw__0,axiom,
% 16.19/9.06 ! [X: nat] :
% 16.19/9.06 ( ( bezw @ X @ zero_zero_nat )
% 16.19/9.06 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9760_drop__bit__numeral__minus__bit1,axiom,
% 16.19/9.06 ! [L2: num,K: num] :
% 16.19/9.06 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 16.19/9.06 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9761_drop__bit__nonnegative__int__iff,axiom,
% 16.19/9.06 ! [N2: nat,K: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N2 @ K ) )
% 16.19/9.06 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9762_drop__bit__negative__int__iff,axiom,
% 16.19/9.06 ! [N2: nat,K: int] :
% 16.19/9.06 ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N2 @ K ) @ zero_zero_int )
% 16.19/9.06 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9763_drop__bit__minus__one,axiom,
% 16.19/9.06 ! [N2: nat] :
% 16.19/9.06 ( ( bit_se8568078237143864401it_int @ N2 @ ( uminus_uminus_int @ one_one_int ) )
% 16.19/9.06 = ( uminus_uminus_int @ one_one_int ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9764_drop__bit__Suc__minus__bit0,axiom,
% 16.19/9.06 ! [N2: nat,K: num] :
% 16.19/9.06 ( ( bit_se8568078237143864401it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 16.19/9.06 = ( bit_se8568078237143864401it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9765_drop__bit__numeral__minus__bit0,axiom,
% 16.19/9.06 ! [L2: num,K: num] :
% 16.19/9.06 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 16.19/9.06 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9766_drop__bit__Suc__minus__bit1,axiom,
% 16.19/9.06 ! [N2: nat,K: num] :
% 16.19/9.06 ( ( bit_se8568078237143864401it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 16.19/9.06 = ( bit_se8568078237143864401it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9767_drop__bit__push__bit__int,axiom,
% 16.19/9.06 ! [M: nat,N2: nat,K: int] :
% 16.19/9.06 ( ( bit_se8568078237143864401it_int @ M @ ( bit_se545348938243370406it_int @ N2 @ K ) )
% 16.19/9.06 = ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ M @ N2 ) @ ( bit_se545348938243370406it_int @ ( minus_minus_nat @ N2 @ M ) @ K ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9768_drop__bit__int__def,axiom,
% 16.19/9.06 ( bit_se8568078237143864401it_int
% 16.19/9.06 = ( ^ [N: nat,K3: int] : ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9769_Suc__0__mod__numeral,axiom,
% 16.19/9.06 ! [K: num] :
% 16.19/9.06 ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 16.19/9.06 = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9770_drop__bit__of__Suc__0,axiom,
% 16.19/9.06 ! [N2: nat] :
% 16.19/9.06 ( ( bit_se8570568707652914677it_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 16.19/9.06 = ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9771_drop__bit__nat__eq,axiom,
% 16.19/9.06 ! [N2: nat,K: int] :
% 16.19/9.06 ( ( bit_se8570568707652914677it_nat @ N2 @ ( nat2 @ K ) )
% 16.19/9.06 = ( nat2 @ ( bit_se8568078237143864401it_int @ N2 @ K ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9772_drop__bit__nat__def,axiom,
% 16.19/9.06 ( bit_se8570568707652914677it_nat
% 16.19/9.06 = ( ^ [N: nat,M6: nat] : ( divide_divide_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9773_upto__aux__rec,axiom,
% 16.19/9.06 ( upto_aux
% 16.19/9.06 = ( ^ [I3: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9774_quotient__of__denom__pos_H,axiom,
% 16.19/9.06 ! [R: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9775_one__mod__minus__numeral,axiom,
% 16.19/9.06 ! [N2: num] :
% 16.19/9.06 ( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/9.06 = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N2 ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9776_minus__one__mod__numeral,axiom,
% 16.19/9.06 ! [N2: num] :
% 16.19/9.06 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N2 ) )
% 16.19/9.06 = ( adjust_mod @ ( numeral_numeral_int @ N2 ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9777_Divides_Oadjust__mod__def,axiom,
% 16.19/9.06 ( adjust_mod
% 16.19/9.06 = ( ^ [L: int,R5: int] : ( if_int @ ( R5 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ L @ R5 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9778_bezw_Osimps,axiom,
% 16.19/9.06 ( bezw
% 16.19/9.06 = ( ^ [X2: nat,Y2: nat] : ( if_Pro3027730157355071871nt_int @ ( Y2 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X2 @ Y2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X2 @ Y2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y2 @ ( modulo_modulo_nat @ X2 @ Y2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Y2 ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9779_bezw_Oelims,axiom,
% 16.19/9.06 ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
% 16.19/9.06 ( ( ( bezw @ X @ Xa2 )
% 16.19/9.06 = Y )
% 16.19/9.06 => ( ( ( Xa2 = zero_zero_nat )
% 16.19/9.06 => ( Y
% 16.19/9.06 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 16.19/9.06 & ( ( Xa2 != zero_zero_nat )
% 16.19/9.06 => ( Y
% 16.19/9.06 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa2 ) ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9780_bezw__non__0,axiom,
% 16.19/9.06 ! [Y: nat,X: nat] :
% 16.19/9.06 ( ( ord_less_nat @ zero_zero_nat @ Y )
% 16.19/9.06 => ( ( bezw @ X @ Y )
% 16.19/9.06 = ( 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 ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9781_rat__sgn__code,axiom,
% 16.19/9.06 ! [P4: rat] :
% 16.19/9.06 ( ( quotient_of @ ( sgn_sgn_rat @ P4 ) )
% 16.19/9.06 = ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P4 ) ) ) @ one_one_int ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9782_take__bit__numeral__minus__numeral__int,axiom,
% 16.19/9.06 ! [M: num,N2: num] :
% 16.19/9.06 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/9.06 = ( case_option_int_num @ zero_zero_int
% 16.19/9.06 @ ^ [Q4: num] : ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_int @ Q4 ) ) )
% 16.19/9.06 @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9783_bezw_Opelims,axiom,
% 16.19/9.06 ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
% 16.19/9.06 ( ( ( bezw @ X @ Xa2 )
% 16.19/9.06 = Y )
% 16.19/9.06 => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
% 16.19/9.06 => ~ ( ( ( ( Xa2 = zero_zero_nat )
% 16.19/9.06 => ( Y
% 16.19/9.06 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 16.19/9.06 & ( ( Xa2 != zero_zero_nat )
% 16.19/9.06 => ( Y
% 16.19/9.06 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa2 ) ) ) ) ) ) ) )
% 16.19/9.06 => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9784_take__bit__num__simps_I1_J,axiom,
% 16.19/9.06 ! [M: num] :
% 16.19/9.06 ( ( bit_take_bit_num @ zero_zero_nat @ M )
% 16.19/9.06 = none_num ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9785_fst__divmod__nat,axiom,
% 16.19/9.06 ! [M: nat,N2: nat] :
% 16.19/9.06 ( ( product_fst_nat_nat @ ( divmod_nat @ M @ N2 ) )
% 16.19/9.06 = ( divide_divide_nat @ M @ N2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9786_fst__divmod__integer,axiom,
% 16.19/9.06 ! [K: code_integer,L2: code_integer] :
% 16.19/9.06 ( ( produc8508995932063986495nteger @ ( code_divmod_integer @ K @ L2 ) )
% 16.19/9.06 = ( divide6298287555418463151nteger @ K @ L2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9787_take__bit__num__simps_I2_J,axiom,
% 16.19/9.06 ! [N2: nat] :
% 16.19/9.06 ( ( bit_take_bit_num @ ( suc @ N2 ) @ one )
% 16.19/9.06 = ( some_num @ one ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9788_take__bit__num__simps_I5_J,axiom,
% 16.19/9.06 ! [R: num] :
% 16.19/9.06 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ one )
% 16.19/9.06 = ( some_num @ one ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9789_take__bit__num__simps_I3_J,axiom,
% 16.19/9.06 ! [N2: nat,M: num] :
% 16.19/9.06 ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit0 @ M ) )
% 16.19/9.06 = ( case_o6005452278849405969um_num @ none_num
% 16.19/9.06 @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 16.19/9.06 @ ( bit_take_bit_num @ N2 @ M ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9790_fst__divmod__abs,axiom,
% 16.19/9.06 ! [K: code_integer,L2: code_integer] :
% 16.19/9.06 ( ( produc8508995932063986495nteger @ ( code_divmod_abs @ K @ L2 ) )
% 16.19/9.06 = ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K ) @ ( abs_abs_Code_integer @ L2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9791_take__bit__num__simps_I4_J,axiom,
% 16.19/9.06 ! [N2: nat,M: num] :
% 16.19/9.06 ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit1 @ M ) )
% 16.19/9.06 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9792_take__bit__num__simps_I6_J,axiom,
% 16.19/9.06 ! [R: num,M: num] :
% 16.19/9.06 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ ( bit0 @ M ) )
% 16.19/9.06 = ( case_o6005452278849405969um_num @ none_num
% 16.19/9.06 @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 16.19/9.06 @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9793_take__bit__num__simps_I7_J,axiom,
% 16.19/9.06 ! [R: num,M: num] :
% 16.19/9.06 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ ( bit1 @ M ) )
% 16.19/9.06 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9794_Suc__0__div__numeral,axiom,
% 16.19/9.06 ! [K: num] :
% 16.19/9.06 ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 16.19/9.06 = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9795_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
% 16.19/9.06 ! [N2: nat,M: num] :
% 16.19/9.06 ( ( bit_take_bit_num @ N2 @ ( bit0 @ M ) )
% 16.19/9.06 = ( case_nat_option_num @ none_num
% 16.19/9.06 @ ^ [N: nat] :
% 16.19/9.06 ( case_o6005452278849405969um_num @ none_num
% 16.19/9.06 @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 16.19/9.06 @ ( bit_take_bit_num @ N @ M ) )
% 16.19/9.06 @ N2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9796_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
% 16.19/9.06 ! [N2: nat] :
% 16.19/9.06 ( ( bit_take_bit_num @ N2 @ one )
% 16.19/9.06 = ( case_nat_option_num @ none_num
% 16.19/9.06 @ ^ [N: nat] : ( some_num @ one )
% 16.19/9.06 @ N2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9797_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
% 16.19/9.06 ! [N2: nat,M: num] :
% 16.19/9.06 ( ( bit_take_bit_num @ N2 @ ( bit1 @ M ) )
% 16.19/9.06 = ( case_nat_option_num @ none_num
% 16.19/9.06 @ ^ [N: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) )
% 16.19/9.06 @ N2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9798_and__minus__numerals_I7_J,axiom,
% 16.19/9.06 ! [N2: num,M: num] :
% 16.19/9.06 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) @ ( numeral_numeral_int @ M ) )
% 16.19/9.06 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N2 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9799_and__minus__numerals_I3_J,axiom,
% 16.19/9.06 ! [M: num,N2: num] :
% 16.19/9.06 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
% 16.19/9.06 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N2 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9800_and__minus__numerals_I4_J,axiom,
% 16.19/9.06 ! [M: num,N2: num] :
% 16.19/9.06 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
% 16.19/9.06 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N2 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9801_and__minus__numerals_I8_J,axiom,
% 16.19/9.06 ! [N2: num,M: num] :
% 16.19/9.06 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) @ ( numeral_numeral_int @ M ) )
% 16.19/9.06 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N2 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9802_and__not__num_Osimps_I1_J,axiom,
% 16.19/9.06 ( ( bit_and_not_num @ one @ one )
% 16.19/9.06 = none_num ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9803_and__not__num_Osimps_I2_J,axiom,
% 16.19/9.06 ! [N2: num] :
% 16.19/9.06 ( ( bit_and_not_num @ one @ ( bit0 @ N2 ) )
% 16.19/9.06 = ( some_num @ one ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9804_and__not__num_Osimps_I4_J,axiom,
% 16.19/9.06 ! [M: num] :
% 16.19/9.06 ( ( bit_and_not_num @ ( bit0 @ M ) @ one )
% 16.19/9.06 = ( some_num @ ( bit0 @ M ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9805_and__not__num_Osimps_I3_J,axiom,
% 16.19/9.06 ! [N2: num] :
% 16.19/9.06 ( ( bit_and_not_num @ one @ ( bit1 @ N2 ) )
% 16.19/9.06 = none_num ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9806_and__not__num_Osimps_I7_J,axiom,
% 16.19/9.06 ! [M: num] :
% 16.19/9.06 ( ( bit_and_not_num @ ( bit1 @ M ) @ one )
% 16.19/9.06 = ( some_num @ ( bit0 @ M ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9807_and__not__num__eq__Some__iff,axiom,
% 16.19/9.06 ! [M: num,N2: num,Q2: num] :
% 16.19/9.06 ( ( ( bit_and_not_num @ M @ N2 )
% 16.19/9.06 = ( some_num @ Q2 ) )
% 16.19/9.06 = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/9.06 = ( numeral_numeral_int @ Q2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9808_and__not__num_Osimps_I8_J,axiom,
% 16.19/9.06 ! [M: num,N2: num] :
% 16.19/9.06 ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 16.19/9.06 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 16.19/9.06 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 16.19/9.06 @ ( bit_and_not_num @ M @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9809_and__not__num__eq__None__iff,axiom,
% 16.19/9.06 ! [M: num,N2: num] :
% 16.19/9.06 ( ( ( bit_and_not_num @ M @ N2 )
% 16.19/9.06 = none_num )
% 16.19/9.06 = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/9.06 = zero_zero_int ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9810_int__numeral__not__and__num,axiom,
% 16.19/9.06 ! [M: num,N2: num] :
% 16.19/9.06 ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
% 16.19/9.06 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N2 @ M ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9811_int__numeral__and__not__num,axiom,
% 16.19/9.06 ! [M: num,N2: num] :
% 16.19/9.06 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/9.06 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N2 ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9812_Bit__Operations_Otake__bit__num__code,axiom,
% 16.19/9.06 ( bit_take_bit_num
% 16.19/9.06 = ( ^ [N: nat,M6: num] :
% 16.19/9.06 ( produc478579273971653890on_num
% 16.19/9.06 @ ^ [A5: nat,X2: num] :
% 16.19/9.06 ( case_nat_option_num @ none_num
% 16.19/9.06 @ ^ [O: nat] :
% 16.19/9.06 ( case_num_option_num @ ( some_num @ one )
% 16.19/9.06 @ ^ [P5: num] :
% 16.19/9.06 ( case_o6005452278849405969um_num @ none_num
% 16.19/9.06 @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
% 16.19/9.06 @ ( bit_take_bit_num @ O @ P5 ) )
% 16.19/9.06 @ ^ [P5: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P5 ) ) )
% 16.19/9.06 @ X2 )
% 16.19/9.06 @ A5 )
% 16.19/9.06 @ ( product_Pair_nat_num @ N @ M6 ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9813_normalize__def,axiom,
% 16.19/9.06 ( normalize
% 16.19/9.06 = ( ^ [P5: product_prod_int_int] :
% 16.19/9.06 ( 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 ) ) ) )
% 16.19/9.06 @ ( if_Pro3027730157355071871nt_int
% 16.19/9.06 @ ( ( product_snd_int_int @ P5 )
% 16.19/9.06 = zero_zero_int )
% 16.19/9.06 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 16.19/9.06 @ ( 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 ) ) ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9814_gcd__1__int,axiom,
% 16.19/9.06 ! [M: int] :
% 16.19/9.06 ( ( gcd_gcd_int @ M @ one_one_int )
% 16.19/9.06 = one_one_int ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9815_gcd__pos__int,axiom,
% 16.19/9.06 ! [M: int,N2: int] :
% 16.19/9.06 ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N2 ) )
% 16.19/9.06 = ( ( M != zero_zero_int )
% 16.19/9.06 | ( N2 != zero_zero_int ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9816_gcd__le2__int,axiom,
% 16.19/9.06 ! [B: int,A: int] :
% 16.19/9.06 ( ( ord_less_int @ zero_zero_int @ B )
% 16.19/9.06 => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ B ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9817_gcd__le1__int,axiom,
% 16.19/9.06 ! [A: int,B: int] :
% 16.19/9.06 ( ( ord_less_int @ zero_zero_int @ A )
% 16.19/9.06 => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ A ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9818_gcd__non__0__int,axiom,
% 16.19/9.06 ! [Y: int,X: int] :
% 16.19/9.06 ( ( ord_less_int @ zero_zero_int @ Y )
% 16.19/9.06 => ( ( gcd_gcd_int @ X @ Y )
% 16.19/9.06 = ( gcd_gcd_int @ Y @ ( modulo_modulo_int @ X @ Y ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9819_upto_Opsimps,axiom,
% 16.19/9.06 ! [I2: int,J: int] :
% 16.19/9.06 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J ) )
% 16.19/9.06 => ( ( ( ord_less_eq_int @ I2 @ J )
% 16.19/9.06 => ( ( upto @ I2 @ J )
% 16.19/9.06 = ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J ) ) ) )
% 16.19/9.06 & ( ~ ( ord_less_eq_int @ I2 @ J )
% 16.19/9.06 => ( ( upto @ I2 @ J )
% 16.19/9.06 = nil_int ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9820_upto_Opelims,axiom,
% 16.19/9.06 ! [X: int,Xa2: int,Y: list_int] :
% 16.19/9.06 ( ( ( upto @ X @ Xa2 )
% 16.19/9.06 = Y )
% 16.19/9.06 => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa2 ) )
% 16.19/9.06 => ~ ( ( ( ( ord_less_eq_int @ X @ Xa2 )
% 16.19/9.06 => ( Y
% 16.19/9.06 = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa2 ) ) ) )
% 16.19/9.06 & ( ~ ( ord_less_eq_int @ X @ Xa2 )
% 16.19/9.06 => ( Y = nil_int ) ) )
% 16.19/9.06 => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa2 ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9821_take__bit__num__def,axiom,
% 16.19/9.06 ( bit_take_bit_num
% 16.19/9.06 = ( ^ [N: nat,M6: num] :
% 16.19/9.06 ( if_option_num
% 16.19/9.06 @ ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ M6 ) )
% 16.19/9.06 = zero_zero_nat )
% 16.19/9.06 @ none_num
% 16.19/9.06 @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ M6 ) ) ) ) ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9822_gcd__1__nat,axiom,
% 16.19/9.06 ! [M: nat] :
% 16.19/9.06 ( ( gcd_gcd_nat @ M @ one_one_nat )
% 16.19/9.06 = one_one_nat ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9823_gcd__Suc__0,axiom,
% 16.19/9.06 ! [M: nat] :
% 16.19/9.06 ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
% 16.19/9.06 = ( suc @ zero_zero_nat ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9824_gcd__pos__nat,axiom,
% 16.19/9.06 ! [M: nat,N2: nat] :
% 16.19/9.06 ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N2 ) )
% 16.19/9.06 = ( ( M != zero_zero_nat )
% 16.19/9.06 | ( N2 != zero_zero_nat ) ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9825_num__of__nat__numeral__eq,axiom,
% 16.19/9.06 ! [Q2: num] :
% 16.19/9.06 ( ( num_of_nat @ ( numeral_numeral_nat @ Q2 ) )
% 16.19/9.06 = Q2 ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9826_upto__Nil,axiom,
% 16.19/9.06 ! [I2: int,J: int] :
% 16.19/9.06 ( ( ( upto @ I2 @ J )
% 16.19/9.06 = nil_int )
% 16.19/9.06 = ( ord_less_int @ J @ I2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9827_upto__Nil2,axiom,
% 16.19/9.06 ! [I2: int,J: int] :
% 16.19/9.06 ( ( nil_int
% 16.19/9.06 = ( upto @ I2 @ J ) )
% 16.19/9.06 = ( ord_less_int @ J @ I2 ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9828_upto__empty,axiom,
% 16.19/9.06 ! [J: int,I2: int] :
% 16.19/9.06 ( ( ord_less_int @ J @ I2 )
% 16.19/9.06 => ( ( upto @ I2 @ J )
% 16.19/9.06 = nil_int ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9829_upto__single,axiom,
% 16.19/9.06 ! [I2: int] :
% 16.19/9.06 ( ( upto @ I2 @ I2 )
% 16.19/9.06 = ( cons_int @ I2 @ nil_int ) ) ).
% 16.19/9.06
% 16.19/9.06 thf(fact_9830_nth__upto,axiom,
% 16.19/9.06 ! [I2: int,K: nat,J: int] :
% 16.19/9.06 ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ ( semiri1314217659103216013at_int @ K ) ) @ J )
% 16.19/9.06 => ( ( nth_int @ ( upto @ I2 @ J ) @ K )
% 16.19/9.06 = ( plus_plus_int @ I2 @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9831_length__upto,axiom,
% 16.19/9.07 ! [I2: int,J: int] :
% 16.19/9.07 ( ( size_size_list_int @ ( upto @ I2 @ J ) )
% 16.19/9.07 = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I2 ) @ one_one_int ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9832_upto__rec__numeral_I1_J,axiom,
% 16.19/9.07 ! [M: num,N2: num] :
% 16.19/9.07 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 16.19/9.07 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 16.19/9.07 = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N2 ) ) ) ) )
% 16.19/9.07 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 16.19/9.07 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 16.19/9.07 = nil_int ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9833_upto__rec__numeral_I4_J,axiom,
% 16.19/9.07 ! [M: num,N2: num] :
% 16.19/9.07 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/9.07 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/9.07 = ( 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 @ N2 ) ) ) ) ) )
% 16.19/9.07 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/9.07 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/9.07 = nil_int ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9834_upto__rec__numeral_I3_J,axiom,
% 16.19/9.07 ! [M: num,N2: num] :
% 16.19/9.07 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
% 16.19/9.07 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
% 16.19/9.07 = ( 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 @ N2 ) ) ) ) )
% 16.19/9.07 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
% 16.19/9.07 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
% 16.19/9.07 = nil_int ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9835_upto__rec__numeral_I2_J,axiom,
% 16.19/9.07 ! [M: num,N2: num] :
% 16.19/9.07 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/9.07 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/9.07 = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) )
% 16.19/9.07 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/9.07 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/9.07 = nil_int ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9836_gcd__mult__distrib__nat,axiom,
% 16.19/9.07 ! [K: nat,M: nat,N2: nat] :
% 16.19/9.07 ( ( times_times_nat @ K @ ( gcd_gcd_nat @ M @ N2 ) )
% 16.19/9.07 = ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9837_atLeastAtMost__upto,axiom,
% 16.19/9.07 ( set_or1266510415728281911st_int
% 16.19/9.07 = ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ I3 @ J3 ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9838_gcd__diff2__nat,axiom,
% 16.19/9.07 ! [M: nat,N2: nat] :
% 16.19/9.07 ( ( ord_less_eq_nat @ M @ N2 )
% 16.19/9.07 => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N2 @ M ) @ N2 )
% 16.19/9.07 = ( gcd_gcd_nat @ M @ N2 ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9839_gcd__diff1__nat,axiom,
% 16.19/9.07 ! [N2: nat,M: nat] :
% 16.19/9.07 ( ( ord_less_eq_nat @ N2 @ M )
% 16.19/9.07 => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 )
% 16.19/9.07 = ( gcd_gcd_nat @ M @ N2 ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9840_gcd__le2__nat,axiom,
% 16.19/9.07 ! [B: nat,A: nat] :
% 16.19/9.07 ( ( B != zero_zero_nat )
% 16.19/9.07 => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9841_gcd__le1__nat,axiom,
% 16.19/9.07 ! [A: nat,B: nat] :
% 16.19/9.07 ( ( A != zero_zero_nat )
% 16.19/9.07 => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9842_distinct__upto,axiom,
% 16.19/9.07 ! [I2: int,J: int] : ( distinct_int @ ( upto @ I2 @ J ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9843_upto__code,axiom,
% 16.19/9.07 ( upto
% 16.19/9.07 = ( ^ [I3: int,J3: int] : ( upto_aux @ I3 @ J3 @ nil_int ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9844_upto__aux__def,axiom,
% 16.19/9.07 ( upto_aux
% 16.19/9.07 = ( ^ [I3: int,J3: int] : ( append_int @ ( upto @ I3 @ J3 ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9845_num__of__nat_Osimps_I1_J,axiom,
% 16.19/9.07 ( ( num_of_nat @ zero_zero_nat )
% 16.19/9.07 = one ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9846_bezout__nat,axiom,
% 16.19/9.07 ! [A: nat,B: nat] :
% 16.19/9.07 ( ( A != zero_zero_nat )
% 16.19/9.07 => ? [X4: nat,Y4: nat] :
% 16.19/9.07 ( ( times_times_nat @ A @ X4 )
% 16.19/9.07 = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9847_bezout__gcd__nat_H,axiom,
% 16.19/9.07 ! [B: nat,A: nat] :
% 16.19/9.07 ? [X4: nat,Y4: nat] :
% 16.19/9.07 ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y4 ) @ ( times_times_nat @ A @ X4 ) )
% 16.19/9.07 & ( ( minus_minus_nat @ ( times_times_nat @ A @ X4 ) @ ( times_times_nat @ B @ Y4 ) )
% 16.19/9.07 = ( gcd_gcd_nat @ A @ B ) ) )
% 16.19/9.07 | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y4 ) @ ( times_times_nat @ B @ X4 ) )
% 16.19/9.07 & ( ( minus_minus_nat @ ( times_times_nat @ B @ X4 ) @ ( times_times_nat @ A @ Y4 ) )
% 16.19/9.07 = ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9848_upto__split2,axiom,
% 16.19/9.07 ! [I2: int,J: int,K: int] :
% 16.19/9.07 ( ( ord_less_eq_int @ I2 @ J )
% 16.19/9.07 => ( ( ord_less_eq_int @ J @ K )
% 16.19/9.07 => ( ( upto @ I2 @ K )
% 16.19/9.07 = ( append_int @ ( upto @ I2 @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9849_upto__split1,axiom,
% 16.19/9.07 ! [I2: int,J: int,K: int] :
% 16.19/9.07 ( ( ord_less_eq_int @ I2 @ J )
% 16.19/9.07 => ( ( ord_less_eq_int @ J @ K )
% 16.19/9.07 => ( ( upto @ I2 @ K )
% 16.19/9.07 = ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9850_atLeastLessThan__upto,axiom,
% 16.19/9.07 ( set_or4662586982721622107an_int
% 16.19/9.07 = ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9851_numeral__num__of__nat,axiom,
% 16.19/9.07 ! [N2: nat] :
% 16.19/9.07 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.07 => ( ( numeral_numeral_nat @ ( num_of_nat @ N2 ) )
% 16.19/9.07 = N2 ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9852_num__of__nat__One,axiom,
% 16.19/9.07 ! [N2: nat] :
% 16.19/9.07 ( ( ord_less_eq_nat @ N2 @ one_one_nat )
% 16.19/9.07 => ( ( num_of_nat @ N2 )
% 16.19/9.07 = one ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9853_upto__rec1,axiom,
% 16.19/9.07 ! [I2: int,J: int] :
% 16.19/9.07 ( ( ord_less_eq_int @ I2 @ J )
% 16.19/9.07 => ( ( upto @ I2 @ J )
% 16.19/9.07 = ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9854_upto_Osimps,axiom,
% 16.19/9.07 ( upto
% 16.19/9.07 = ( ^ [I3: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I3 @ J3 ) @ ( cons_int @ I3 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9855_upto_Oelims,axiom,
% 16.19/9.07 ! [X: int,Xa2: int,Y: list_int] :
% 16.19/9.07 ( ( ( upto @ X @ Xa2 )
% 16.19/9.07 = Y )
% 16.19/9.07 => ( ( ( ord_less_eq_int @ X @ Xa2 )
% 16.19/9.07 => ( Y
% 16.19/9.07 = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa2 ) ) ) )
% 16.19/9.07 & ( ~ ( ord_less_eq_int @ X @ Xa2 )
% 16.19/9.07 => ( Y = nil_int ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9856_upto__rec2,axiom,
% 16.19/9.07 ! [I2: int,J: int] :
% 16.19/9.07 ( ( ord_less_eq_int @ I2 @ J )
% 16.19/9.07 => ( ( upto @ I2 @ J )
% 16.19/9.07 = ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9857_num__of__nat__double,axiom,
% 16.19/9.07 ! [N2: nat] :
% 16.19/9.07 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.07 => ( ( num_of_nat @ ( plus_plus_nat @ N2 @ N2 ) )
% 16.19/9.07 = ( bit0 @ ( num_of_nat @ N2 ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9858_num__of__nat__plus__distrib,axiom,
% 16.19/9.07 ! [M: nat,N2: nat] :
% 16.19/9.07 ( ( ord_less_nat @ zero_zero_nat @ M )
% 16.19/9.07 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.07 => ( ( num_of_nat @ ( plus_plus_nat @ M @ N2 ) )
% 16.19/9.07 = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N2 ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9859_upto__split3,axiom,
% 16.19/9.07 ! [I2: int,J: int,K: int] :
% 16.19/9.07 ( ( ord_less_eq_int @ I2 @ J )
% 16.19/9.07 => ( ( ord_less_eq_int @ J @ K )
% 16.19/9.07 => ( ( upto @ I2 @ K )
% 16.19/9.07 = ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9860_num__of__nat_Osimps_I2_J,axiom,
% 16.19/9.07 ! [N2: nat] :
% 16.19/9.07 ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.07 => ( ( num_of_nat @ ( suc @ N2 ) )
% 16.19/9.07 = ( inc @ ( num_of_nat @ N2 ) ) ) )
% 16.19/9.07 & ( ~ ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.07 => ( ( num_of_nat @ ( suc @ N2 ) )
% 16.19/9.07 = one ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9861_gcd__nat_Opelims,axiom,
% 16.19/9.07 ! [X: nat,Xa2: nat,Y: nat] :
% 16.19/9.07 ( ( ( gcd_gcd_nat @ X @ Xa2 )
% 16.19/9.07 = Y )
% 16.19/9.07 => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
% 16.19/9.07 => ~ ( ( ( ( Xa2 = zero_zero_nat )
% 16.19/9.07 => ( Y = X ) )
% 16.19/9.07 & ( ( Xa2 != zero_zero_nat )
% 16.19/9.07 => ( Y
% 16.19/9.07 = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) ) )
% 16.19/9.07 => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9862_prod__decode__aux_Oelims,axiom,
% 16.19/9.07 ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
% 16.19/9.07 ( ( ( nat_prod_decode_aux @ X @ Xa2 )
% 16.19/9.07 = Y )
% 16.19/9.07 => ( ( ( ord_less_eq_nat @ Xa2 @ X )
% 16.19/9.07 => ( Y
% 16.19/9.07 = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
% 16.19/9.07 & ( ~ ( ord_less_eq_nat @ Xa2 @ X )
% 16.19/9.07 => ( Y
% 16.19/9.07 = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9863_nat__descend__induct,axiom,
% 16.19/9.07 ! [N2: nat,P: nat > $o,M: nat] :
% 16.19/9.07 ( ! [K2: nat] :
% 16.19/9.07 ( ( ord_less_nat @ N2 @ K2 )
% 16.19/9.07 => ( P @ K2 ) )
% 16.19/9.07 => ( ! [K2: nat] :
% 16.19/9.07 ( ( ord_less_eq_nat @ K2 @ N2 )
% 16.19/9.07 => ( ! [I: nat] :
% 16.19/9.07 ( ( ord_less_nat @ K2 @ I )
% 16.19/9.07 => ( P @ I ) )
% 16.19/9.07 => ( P @ K2 ) ) )
% 16.19/9.07 => ( P @ M ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9864_prod__decode__aux_Osimps,axiom,
% 16.19/9.07 ( nat_prod_decode_aux
% 16.19/9.07 = ( ^ [K3: nat,M6: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M6 @ K3 ) @ ( product_Pair_nat_nat @ M6 @ ( minus_minus_nat @ K3 @ M6 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M6 @ ( suc @ K3 ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9865_prod__decode__aux_Opelims,axiom,
% 16.19/9.07 ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
% 16.19/9.07 ( ( ( nat_prod_decode_aux @ X @ Xa2 )
% 16.19/9.07 = Y )
% 16.19/9.07 => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
% 16.19/9.07 => ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X )
% 16.19/9.07 => ( Y
% 16.19/9.07 = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
% 16.19/9.07 & ( ~ ( ord_less_eq_nat @ Xa2 @ X )
% 16.19/9.07 => ( Y
% 16.19/9.07 = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) )
% 16.19/9.07 => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9866_vebt__mint_Opelims,axiom,
% 16.19/9.07 ! [X: vEBT_VEBT,Y: option_nat] :
% 16.19/9.07 ( ( ( vEBT_vebt_mint @ X )
% 16.19/9.07 = Y )
% 16.19/9.07 => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X )
% 16.19/9.07 => ( ! [A3: $o,B2: $o] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/9.07 => ( ( ( A3
% 16.19/9.07 => ( Y
% 16.19/9.07 = ( some_nat @ zero_zero_nat ) ) )
% 16.19/9.07 & ( ~ A3
% 16.19/9.07 => ( ( B2
% 16.19/9.07 => ( Y
% 16.19/9.07 = ( some_nat @ one_one_nat ) ) )
% 16.19/9.07 & ( ~ B2
% 16.19/9.07 => ( Y = none_nat ) ) ) ) )
% 16.19/9.07 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A3 @ B2 ) ) ) )
% 16.19/9.07 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 16.19/9.07 => ( ( Y = none_nat )
% 16.19/9.07 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 16.19/9.07 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 16.19/9.07 => ( ( Y
% 16.19/9.07 = ( some_nat @ Mi2 ) )
% 16.19/9.07 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9867_vebt__maxt_Opelims,axiom,
% 16.19/9.07 ! [X: vEBT_VEBT,Y: option_nat] :
% 16.19/9.07 ( ( ( vEBT_vebt_maxt @ X )
% 16.19/9.07 = Y )
% 16.19/9.07 => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X )
% 16.19/9.07 => ( ! [A3: $o,B2: $o] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( vEBT_Leaf @ A3 @ B2 ) )
% 16.19/9.07 => ( ( ( B2
% 16.19/9.07 => ( Y
% 16.19/9.07 = ( some_nat @ one_one_nat ) ) )
% 16.19/9.07 & ( ~ B2
% 16.19/9.07 => ( ( A3
% 16.19/9.07 => ( Y
% 16.19/9.07 = ( some_nat @ zero_zero_nat ) ) )
% 16.19/9.07 & ( ~ A3
% 16.19/9.07 => ( Y = none_nat ) ) ) ) )
% 16.19/9.07 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A3 @ B2 ) ) ) )
% 16.19/9.07 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 16.19/9.07 => ( ( Y = none_nat )
% 16.19/9.07 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 16.19/9.07 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 16.19/9.07 => ( ( Y
% 16.19/9.07 = ( some_nat @ Ma2 ) )
% 16.19/9.07 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9868_VEBT__internal_OminNull_Opelims_I1_J,axiom,
% 16.19/9.07 ! [X: vEBT_VEBT,Y: $o] :
% 16.19/9.07 ( ( ( vEBT_VEBT_minNull @ X )
% 16.19/9.07 = Y )
% 16.19/9.07 => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
% 16.19/9.07 => ( ( ( X
% 16.19/9.07 = ( vEBT_Leaf @ $false @ $false ) )
% 16.19/9.07 => ( Y
% 16.19/9.07 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
% 16.19/9.07 => ( ! [Uv2: $o] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( vEBT_Leaf @ $true @ Uv2 ) )
% 16.19/9.07 => ( ~ Y
% 16.19/9.07 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
% 16.19/9.07 => ( ! [Uu2: $o] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( vEBT_Leaf @ Uu2 @ $true ) )
% 16.19/9.07 => ( ~ Y
% 16.19/9.07 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
% 16.19/9.07 => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 16.19/9.07 => ( Y
% 16.19/9.07 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
% 16.19/9.07 => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 16.19/9.07 => ( ~ Y
% 16.19/9.07 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9869_VEBT__internal_OminNull_Opelims_I2_J,axiom,
% 16.19/9.07 ! [X: vEBT_VEBT] :
% 16.19/9.07 ( ( vEBT_VEBT_minNull @ X )
% 16.19/9.07 => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
% 16.19/9.07 => ( ( ( X
% 16.19/9.07 = ( vEBT_Leaf @ $false @ $false ) )
% 16.19/9.07 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
% 16.19/9.07 => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 16.19/9.07 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9870_VEBT__internal_OminNull_Opelims_I3_J,axiom,
% 16.19/9.07 ! [X: vEBT_VEBT] :
% 16.19/9.07 ( ~ ( vEBT_VEBT_minNull @ X )
% 16.19/9.07 => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
% 16.19/9.07 => ( ! [Uv2: $o] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( vEBT_Leaf @ $true @ Uv2 ) )
% 16.19/9.07 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
% 16.19/9.07 => ( ! [Uu2: $o] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( vEBT_Leaf @ Uu2 @ $true ) )
% 16.19/9.07 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
% 16.19/9.07 => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 16.19/9.07 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9871_finite__enumerate,axiom,
% 16.19/9.07 ! [S: set_nat] :
% 16.19/9.07 ( ( finite_finite_nat @ S )
% 16.19/9.07 => ? [R3: nat > nat] :
% 16.19/9.07 ( ( strict1292158309912662752at_nat @ R3 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S ) ) )
% 16.19/9.07 & ! [N9: nat] :
% 16.19/9.07 ( ( ord_less_nat @ N9 @ ( finite_card_nat @ S ) )
% 16.19/9.07 => ( member_nat @ ( R3 @ N9 ) @ S ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9872_divmod__integer__eq__cases,axiom,
% 16.19/9.07 ( code_divmod_integer
% 16.19/9.07 = ( ^ [K3: code_integer,L: code_integer] :
% 16.19/9.07 ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 16.19/9.07 @ ( if_Pro6119634080678213985nteger @ ( L = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
% 16.19/9.07 @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L
% 16.19/9.07 @ ( if_Pro6119634080678213985nteger
% 16.19/9.07 @ ( ( sgn_sgn_Code_integer @ K3 )
% 16.19/9.07 = ( sgn_sgn_Code_integer @ L ) )
% 16.19/9.07 @ ( code_divmod_abs @ K3 @ L )
% 16.19/9.07 @ ( produc6916734918728496179nteger
% 16.19/9.07 @ ^ [R5: code_integer,S5: code_integer] : ( if_Pro6119634080678213985nteger @ ( S5 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L ) @ S5 ) ) )
% 16.19/9.07 @ ( code_divmod_abs @ K3 @ L ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9873_card_Ocomp__fun__commute__on,axiom,
% 16.19/9.07 ( ( comp_nat_nat_nat @ suc @ suc )
% 16.19/9.07 = ( comp_nat_nat_nat @ suc @ suc ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9874_card__greaterThanLessThan__int,axiom,
% 16.19/9.07 ! [L2: int,U: int] :
% 16.19/9.07 ( ( finite_card_int @ ( set_or5832277885323065728an_int @ L2 @ U ) )
% 16.19/9.07 = ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L2 @ one_one_int ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9875_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
% 16.19/9.07 ! [L2: int,U: int] :
% 16.19/9.07 ( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L2 @ one_one_int ) @ U )
% 16.19/9.07 = ( set_or5832277885323065728an_int @ L2 @ U ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9876_greaterThanLessThan__upto,axiom,
% 16.19/9.07 ( set_or5832277885323065728an_int
% 16.19/9.07 = ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9877_and__not__num_Oelims,axiom,
% 16.19/9.07 ! [X: num,Xa2: num,Y: option_num] :
% 16.19/9.07 ( ( ( bit_and_not_num @ X @ Xa2 )
% 16.19/9.07 = Y )
% 16.19/9.07 => ( ( ( X = one )
% 16.19/9.07 => ( ( Xa2 = one )
% 16.19/9.07 => ( Y != none_num ) ) )
% 16.19/9.07 => ( ( ( X = one )
% 16.19/9.07 => ( ? [N4: num] :
% 16.19/9.07 ( Xa2
% 16.19/9.07 = ( bit0 @ N4 ) )
% 16.19/9.07 => ( Y
% 16.19/9.07 != ( some_num @ one ) ) ) )
% 16.19/9.07 => ( ( ( X = one )
% 16.19/9.07 => ( ? [N4: num] :
% 16.19/9.07 ( Xa2
% 16.19/9.07 = ( bit1 @ N4 ) )
% 16.19/9.07 => ( Y != none_num ) ) )
% 16.19/9.07 => ( ! [M2: num] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( bit0 @ M2 ) )
% 16.19/9.07 => ( ( Xa2 = one )
% 16.19/9.07 => ( Y
% 16.19/9.07 != ( some_num @ ( bit0 @ M2 ) ) ) ) )
% 16.19/9.07 => ( ! [M2: num] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( bit0 @ M2 ) )
% 16.19/9.07 => ! [N4: num] :
% 16.19/9.07 ( ( Xa2
% 16.19/9.07 = ( bit0 @ N4 ) )
% 16.19/9.07 => ( Y
% 16.19/9.07 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M2 @ N4 ) ) ) ) )
% 16.19/9.07 => ( ! [M2: num] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( bit0 @ M2 ) )
% 16.19/9.07 => ! [N4: num] :
% 16.19/9.07 ( ( Xa2
% 16.19/9.07 = ( bit1 @ N4 ) )
% 16.19/9.07 => ( Y
% 16.19/9.07 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M2 @ N4 ) ) ) ) )
% 16.19/9.07 => ( ! [M2: num] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( bit1 @ M2 ) )
% 16.19/9.07 => ( ( Xa2 = one )
% 16.19/9.07 => ( Y
% 16.19/9.07 != ( some_num @ ( bit0 @ M2 ) ) ) ) )
% 16.19/9.07 => ( ! [M2: num] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( bit1 @ M2 ) )
% 16.19/9.07 => ! [N4: num] :
% 16.19/9.07 ( ( Xa2
% 16.19/9.07 = ( bit0 @ N4 ) )
% 16.19/9.07 => ( Y
% 16.19/9.07 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 16.19/9.07 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 16.19/9.07 @ ( bit_and_not_num @ M2 @ N4 ) ) ) ) )
% 16.19/9.07 => ~ ! [M2: num] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( bit1 @ M2 ) )
% 16.19/9.07 => ! [N4: num] :
% 16.19/9.07 ( ( Xa2
% 16.19/9.07 = ( bit1 @ N4 ) )
% 16.19/9.07 => ( Y
% 16.19/9.07 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M2 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9878_card__greaterThanLessThan,axiom,
% 16.19/9.07 ! [L2: nat,U: nat] :
% 16.19/9.07 ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L2 @ U ) )
% 16.19/9.07 = ( minus_minus_nat @ U @ ( suc @ L2 ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9879_atLeastSucLessThan__greaterThanLessThan,axiom,
% 16.19/9.07 ! [L2: nat,U: nat] :
% 16.19/9.07 ( ( set_or4665077453230672383an_nat @ ( suc @ L2 ) @ U )
% 16.19/9.07 = ( set_or5834768355832116004an_nat @ L2 @ U ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9880_and__not__num_Osimps_I5_J,axiom,
% 16.19/9.07 ! [M: num,N2: num] :
% 16.19/9.07 ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
% 16.19/9.07 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9881_and__not__num_Osimps_I9_J,axiom,
% 16.19/9.07 ! [M: num,N2: num] :
% 16.19/9.07 ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 16.19/9.07 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9882_and__not__num_Osimps_I6_J,axiom,
% 16.19/9.07 ! [M: num,N2: num] :
% 16.19/9.07 ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 16.19/9.07 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9883_tanh__real__bounds,axiom,
% 16.19/9.07 ! [X: real] : ( member_real @ ( tanh_real @ X ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9884_and__num_Oelims,axiom,
% 16.19/9.07 ! [X: num,Xa2: num,Y: option_num] :
% 16.19/9.07 ( ( ( bit_un7362597486090784418nd_num @ X @ Xa2 )
% 16.19/9.07 = Y )
% 16.19/9.07 => ( ( ( X = one )
% 16.19/9.07 => ( ( Xa2 = one )
% 16.19/9.07 => ( Y
% 16.19/9.07 != ( some_num @ one ) ) ) )
% 16.19/9.07 => ( ( ( X = one )
% 16.19/9.07 => ( ? [N4: num] :
% 16.19/9.07 ( Xa2
% 16.19/9.07 = ( bit0 @ N4 ) )
% 16.19/9.07 => ( Y != none_num ) ) )
% 16.19/9.07 => ( ( ( X = one )
% 16.19/9.07 => ( ? [N4: num] :
% 16.19/9.07 ( Xa2
% 16.19/9.07 = ( bit1 @ N4 ) )
% 16.19/9.07 => ( Y
% 16.19/9.07 != ( some_num @ one ) ) ) )
% 16.19/9.07 => ( ( ? [M2: num] :
% 16.19/9.07 ( X
% 16.19/9.07 = ( bit0 @ M2 ) )
% 16.19/9.07 => ( ( Xa2 = one )
% 16.19/9.07 => ( Y != none_num ) ) )
% 16.19/9.07 => ( ! [M2: num] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( bit0 @ M2 ) )
% 16.19/9.07 => ! [N4: num] :
% 16.19/9.07 ( ( Xa2
% 16.19/9.07 = ( bit0 @ N4 ) )
% 16.19/9.07 => ( Y
% 16.19/9.07 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M2 @ N4 ) ) ) ) )
% 16.19/9.07 => ( ! [M2: num] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( bit0 @ M2 ) )
% 16.19/9.07 => ! [N4: num] :
% 16.19/9.07 ( ( Xa2
% 16.19/9.07 = ( bit1 @ N4 ) )
% 16.19/9.07 => ( Y
% 16.19/9.07 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M2 @ N4 ) ) ) ) )
% 16.19/9.07 => ( ( ? [M2: num] :
% 16.19/9.07 ( X
% 16.19/9.07 = ( bit1 @ M2 ) )
% 16.19/9.07 => ( ( Xa2 = one )
% 16.19/9.07 => ( Y
% 16.19/9.07 != ( some_num @ one ) ) ) )
% 16.19/9.07 => ( ! [M2: num] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( bit1 @ M2 ) )
% 16.19/9.07 => ! [N4: num] :
% 16.19/9.07 ( ( Xa2
% 16.19/9.07 = ( bit0 @ N4 ) )
% 16.19/9.07 => ( Y
% 16.19/9.07 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M2 @ N4 ) ) ) ) )
% 16.19/9.07 => ~ ! [M2: num] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( bit1 @ M2 ) )
% 16.19/9.07 => ! [N4: num] :
% 16.19/9.07 ( ( Xa2
% 16.19/9.07 = ( bit1 @ N4 ) )
% 16.19/9.07 => ( Y
% 16.19/9.07 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 16.19/9.07 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 16.19/9.07 @ ( bit_un7362597486090784418nd_num @ M2 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9885_and__num_Osimps_I1_J,axiom,
% 16.19/9.07 ( ( bit_un7362597486090784418nd_num @ one @ one )
% 16.19/9.07 = ( some_num @ one ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9886_and__num_Osimps_I5_J,axiom,
% 16.19/9.07 ! [M: num,N2: num] :
% 16.19/9.07 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
% 16.19/9.07 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9887_and__num_Osimps_I3_J,axiom,
% 16.19/9.07 ! [N2: num] :
% 16.19/9.07 ( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N2 ) )
% 16.19/9.07 = ( some_num @ one ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9888_and__num_Osimps_I7_J,axiom,
% 16.19/9.07 ! [M: num] :
% 16.19/9.07 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one )
% 16.19/9.07 = ( some_num @ one ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9889_and__num_Osimps_I4_J,axiom,
% 16.19/9.07 ! [M: num] :
% 16.19/9.07 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
% 16.19/9.07 = none_num ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9890_and__num_Osimps_I2_J,axiom,
% 16.19/9.07 ! [N2: num] :
% 16.19/9.07 ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N2 ) )
% 16.19/9.07 = none_num ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9891_and__num_Osimps_I6_J,axiom,
% 16.19/9.07 ! [M: num,N2: num] :
% 16.19/9.07 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 16.19/9.07 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9892_and__num_Osimps_I8_J,axiom,
% 16.19/9.07 ! [M: num,N2: num] :
% 16.19/9.07 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 16.19/9.07 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9893_and__num_Osimps_I9_J,axiom,
% 16.19/9.07 ! [M: num,N2: num] :
% 16.19/9.07 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 16.19/9.07 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 16.19/9.07 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 16.19/9.07 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9894_xor__num_Oelims,axiom,
% 16.19/9.07 ! [X: num,Xa2: num,Y: option_num] :
% 16.19/9.07 ( ( ( bit_un2480387367778600638or_num @ X @ Xa2 )
% 16.19/9.07 = Y )
% 16.19/9.07 => ( ( ( X = one )
% 16.19/9.07 => ( ( Xa2 = one )
% 16.19/9.07 => ( Y != none_num ) ) )
% 16.19/9.07 => ( ( ( X = one )
% 16.19/9.07 => ! [N4: num] :
% 16.19/9.07 ( ( Xa2
% 16.19/9.07 = ( bit0 @ N4 ) )
% 16.19/9.07 => ( Y
% 16.19/9.07 != ( some_num @ ( bit1 @ N4 ) ) ) ) )
% 16.19/9.07 => ( ( ( X = one )
% 16.19/9.07 => ! [N4: num] :
% 16.19/9.07 ( ( Xa2
% 16.19/9.07 = ( bit1 @ N4 ) )
% 16.19/9.07 => ( Y
% 16.19/9.07 != ( some_num @ ( bit0 @ N4 ) ) ) ) )
% 16.19/9.07 => ( ! [M2: num] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( bit0 @ M2 ) )
% 16.19/9.07 => ( ( Xa2 = one )
% 16.19/9.07 => ( Y
% 16.19/9.07 != ( some_num @ ( bit1 @ M2 ) ) ) ) )
% 16.19/9.07 => ( ! [M2: num] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( bit0 @ M2 ) )
% 16.19/9.07 => ! [N4: num] :
% 16.19/9.07 ( ( Xa2
% 16.19/9.07 = ( bit0 @ N4 ) )
% 16.19/9.07 => ( Y
% 16.19/9.07 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M2 @ N4 ) ) ) ) )
% 16.19/9.07 => ( ! [M2: num] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( bit0 @ M2 ) )
% 16.19/9.07 => ! [N4: num] :
% 16.19/9.07 ( ( Xa2
% 16.19/9.07 = ( bit1 @ N4 ) )
% 16.19/9.07 => ( Y
% 16.19/9.07 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M2 @ N4 ) ) ) ) ) )
% 16.19/9.07 => ( ! [M2: num] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( bit1 @ M2 ) )
% 16.19/9.07 => ( ( Xa2 = one )
% 16.19/9.07 => ( Y
% 16.19/9.07 != ( some_num @ ( bit0 @ M2 ) ) ) ) )
% 16.19/9.07 => ( ! [M2: num] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( bit1 @ M2 ) )
% 16.19/9.07 => ! [N4: num] :
% 16.19/9.07 ( ( Xa2
% 16.19/9.07 = ( bit0 @ N4 ) )
% 16.19/9.07 => ( Y
% 16.19/9.07 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M2 @ N4 ) ) ) ) ) )
% 16.19/9.07 => ~ ! [M2: num] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( bit1 @ M2 ) )
% 16.19/9.07 => ! [N4: num] :
% 16.19/9.07 ( ( Xa2
% 16.19/9.07 = ( bit1 @ N4 ) )
% 16.19/9.07 => ( Y
% 16.19/9.07 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M2 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9895_and__num__dict,axiom,
% 16.19/9.07 bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9896_xor__num_Osimps_I1_J,axiom,
% 16.19/9.07 ( ( bit_un2480387367778600638or_num @ one @ one )
% 16.19/9.07 = none_num ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9897_xor__num_Osimps_I5_J,axiom,
% 16.19/9.07 ! [M: num,N2: num] :
% 16.19/9.07 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
% 16.19/9.07 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9898_xor__num_Osimps_I9_J,axiom,
% 16.19/9.07 ! [M: num,N2: num] :
% 16.19/9.07 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 16.19/9.07 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9899_xor__num_Osimps_I2_J,axiom,
% 16.19/9.07 ! [N2: num] :
% 16.19/9.07 ( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N2 ) )
% 16.19/9.07 = ( some_num @ ( bit1 @ N2 ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9900_xor__num_Osimps_I3_J,axiom,
% 16.19/9.07 ! [N2: num] :
% 16.19/9.07 ( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N2 ) )
% 16.19/9.07 = ( some_num @ ( bit0 @ N2 ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9901_xor__num_Osimps_I4_J,axiom,
% 16.19/9.07 ! [M: num] :
% 16.19/9.07 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one )
% 16.19/9.07 = ( some_num @ ( bit1 @ M ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9902_xor__num_Osimps_I7_J,axiom,
% 16.19/9.07 ! [M: num] :
% 16.19/9.07 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one )
% 16.19/9.07 = ( some_num @ ( bit0 @ M ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9903_xor__num_Osimps_I8_J,axiom,
% 16.19/9.07 ! [M: num,N2: num] :
% 16.19/9.07 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 16.19/9.07 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9904_xor__num_Osimps_I6_J,axiom,
% 16.19/9.07 ! [M: num,N2: num] :
% 16.19/9.07 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 16.19/9.07 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9905_xor__num__dict,axiom,
% 16.19/9.07 bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9906_xor__minus__numerals_I2_J,axiom,
% 16.19/9.07 ! [K: int,N2: num] :
% 16.19/9.07 ( ( bit_se6526347334894502574or_int @ K @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 16.19/9.07 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K @ ( neg_numeral_sub_int @ N2 @ one ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9907_xor__minus__numerals_I1_J,axiom,
% 16.19/9.07 ! [N2: num,K: int] :
% 16.19/9.07 ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ K )
% 16.19/9.07 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N2 @ one ) @ K ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9908_sub__BitM__One__eq,axiom,
% 16.19/9.07 ! [N2: num] :
% 16.19/9.07 ( ( neg_numeral_sub_int @ ( bitM @ N2 ) @ one )
% 16.19/9.07 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N2 @ one ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9909_Suc__funpow,axiom,
% 16.19/9.07 ! [N2: nat] :
% 16.19/9.07 ( ( compow_nat_nat @ N2 @ suc )
% 16.19/9.07 = ( plus_plus_nat @ N2 ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9910_max__nat_Osemilattice__neutr__order__axioms,axiom,
% 16.19/9.07 ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
% 16.19/9.07 @ ^ [X2: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ X2 )
% 16.19/9.07 @ ^ [X2: nat,Y2: nat] : ( ord_less_nat @ Y2 @ X2 ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9911_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
% 16.19/9.07 ( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
% 16.19/9.07 @ ^ [M6: nat,N: nat] :
% 16.19/9.07 ( ( dvd_dvd_nat @ M6 @ N )
% 16.19/9.07 & ( M6 != N ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9912_Gcd__remove0__nat,axiom,
% 16.19/9.07 ! [M7: set_nat] :
% 16.19/9.07 ( ( finite_finite_nat @ M7 )
% 16.19/9.07 => ( ( gcd_Gcd_nat @ M7 )
% 16.19/9.07 = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M7 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9913_times__int_Oabs__eq,axiom,
% 16.19/9.07 ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
% 16.19/9.07 ( ( times_times_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
% 16.19/9.07 = ( abs_Integ
% 16.19/9.07 @ ( produc27273713700761075at_nat
% 16.19/9.07 @ ^ [X2: nat,Y2: nat] :
% 16.19/9.07 ( produc2626176000494625587at_nat
% 16.19/9.07 @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U2 ) @ ( times_times_nat @ Y2 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y2 @ U2 ) ) ) )
% 16.19/9.07 @ Xa2
% 16.19/9.07 @ X ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9914_eq__Abs__Integ,axiom,
% 16.19/9.07 ! [Z: int] :
% 16.19/9.07 ~ ! [X4: nat,Y4: nat] :
% 16.19/9.07 ( Z
% 16.19/9.07 != ( abs_Integ @ ( product_Pair_nat_nat @ X4 @ Y4 ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9915_Gcd__nat__eq__one,axiom,
% 16.19/9.07 ! [N3: set_nat] :
% 16.19/9.07 ( ( member_nat @ one_one_nat @ N3 )
% 16.19/9.07 => ( ( gcd_Gcd_nat @ N3 )
% 16.19/9.07 = one_one_nat ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9916_nat_Oabs__eq,axiom,
% 16.19/9.07 ! [X: product_prod_nat_nat] :
% 16.19/9.07 ( ( nat2 @ ( abs_Integ @ X ) )
% 16.19/9.07 = ( produc6842872674320459806at_nat @ minus_minus_nat @ X ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9917_zero__int__def,axiom,
% 16.19/9.07 ( zero_zero_int
% 16.19/9.07 = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9918_int__def,axiom,
% 16.19/9.07 ( semiri1314217659103216013at_int
% 16.19/9.07 = ( ^ [N: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N @ zero_zero_nat ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9919_uminus__int_Oabs__eq,axiom,
% 16.19/9.07 ! [X: product_prod_nat_nat] :
% 16.19/9.07 ( ( uminus_uminus_int @ ( abs_Integ @ X ) )
% 16.19/9.07 = ( abs_Integ
% 16.19/9.07 @ ( produc2626176000494625587at_nat
% 16.19/9.07 @ ^ [X2: nat,Y2: nat] : ( product_Pair_nat_nat @ Y2 @ X2 )
% 16.19/9.07 @ X ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9920_one__int__def,axiom,
% 16.19/9.07 ( one_one_int
% 16.19/9.07 = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9921_less__int_Oabs__eq,axiom,
% 16.19/9.07 ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
% 16.19/9.07 ( ( ord_less_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
% 16.19/9.07 = ( produc8739625826339149834_nat_o
% 16.19/9.07 @ ^ [X2: nat,Y2: nat] :
% 16.19/9.07 ( produc6081775807080527818_nat_o
% 16.19/9.07 @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) )
% 16.19/9.07 @ Xa2
% 16.19/9.07 @ X ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9922_less__eq__int_Oabs__eq,axiom,
% 16.19/9.07 ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
% 16.19/9.07 ( ( ord_less_eq_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
% 16.19/9.07 = ( produc8739625826339149834_nat_o
% 16.19/9.07 @ ^ [X2: nat,Y2: nat] :
% 16.19/9.07 ( produc6081775807080527818_nat_o
% 16.19/9.07 @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U2 @ Y2 ) ) )
% 16.19/9.07 @ Xa2
% 16.19/9.07 @ X ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9923_plus__int_Oabs__eq,axiom,
% 16.19/9.07 ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
% 16.19/9.07 ( ( plus_plus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
% 16.19/9.07 = ( abs_Integ
% 16.19/9.07 @ ( produc27273713700761075at_nat
% 16.19/9.07 @ ^ [X2: nat,Y2: nat] :
% 16.19/9.07 ( produc2626176000494625587at_nat
% 16.19/9.07 @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U2 ) @ ( plus_plus_nat @ Y2 @ V4 ) ) )
% 16.19/9.07 @ Xa2
% 16.19/9.07 @ X ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9924_minus__int_Oabs__eq,axiom,
% 16.19/9.07 ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
% 16.19/9.07 ( ( minus_minus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
% 16.19/9.07 = ( abs_Integ
% 16.19/9.07 @ ( produc27273713700761075at_nat
% 16.19/9.07 @ ^ [X2: nat,Y2: nat] :
% 16.19/9.07 ( produc2626176000494625587at_nat
% 16.19/9.07 @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y2 @ U2 ) ) )
% 16.19/9.07 @ Xa2
% 16.19/9.07 @ X ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9925_bij__betw__Suc,axiom,
% 16.19/9.07 ! [M7: set_nat,N3: set_nat] :
% 16.19/9.07 ( ( bij_betw_nat_nat @ suc @ M7 @ N3 )
% 16.19/9.07 = ( ( image_nat_nat @ suc @ M7 )
% 16.19/9.07 = N3 ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9926_image__Suc__atLeastAtMost,axiom,
% 16.19/9.07 ! [I2: nat,J: nat] :
% 16.19/9.07 ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I2 @ J ) )
% 16.19/9.07 = ( set_or1269000886237332187st_nat @ ( suc @ I2 ) @ ( suc @ J ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9927_image__Suc__atLeastLessThan,axiom,
% 16.19/9.07 ! [I2: nat,J: nat] :
% 16.19/9.07 ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I2 @ J ) )
% 16.19/9.07 = ( set_or4665077453230672383an_nat @ ( suc @ I2 ) @ ( suc @ J ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9928_Gcd__int__eq,axiom,
% 16.19/9.07 ! [N3: set_nat] :
% 16.19/9.07 ( ( gcd_Gcd_int @ ( image_nat_int @ semiri1314217659103216013at_int @ N3 ) )
% 16.19/9.07 = ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ N3 ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9929_zero__notin__Suc__image,axiom,
% 16.19/9.07 ! [A2: set_nat] :
% 16.19/9.07 ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9930_image__Suc__lessThan,axiom,
% 16.19/9.07 ! [N2: nat] :
% 16.19/9.07 ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.07 = ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9931_image__Suc__atMost,axiom,
% 16.19/9.07 ! [N2: nat] :
% 16.19/9.07 ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N2 ) )
% 16.19/9.07 = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N2 ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9932_atLeast0__atMost__Suc__eq__insert__0,axiom,
% 16.19/9.07 ! [N2: nat] :
% 16.19/9.07 ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) )
% 16.19/9.07 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9933_atLeast0__lessThan__Suc__eq__insert__0,axiom,
% 16.19/9.07 ! [N2: nat] :
% 16.19/9.07 ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) )
% 16.19/9.07 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9934_lessThan__Suc__eq__insert__0,axiom,
% 16.19/9.07 ! [N2: nat] :
% 16.19/9.07 ( ( set_ord_lessThan_nat @ ( suc @ N2 ) )
% 16.19/9.07 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9935_atMost__Suc__eq__insert__0,axiom,
% 16.19/9.07 ! [N2: nat] :
% 16.19/9.07 ( ( set_ord_atMost_nat @ ( suc @ N2 ) )
% 16.19/9.07 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9936_image__minus__const__atLeastLessThan__nat,axiom,
% 16.19/9.07 ! [C: nat,Y: nat,X: nat] :
% 16.19/9.07 ( ( ( ord_less_nat @ C @ Y )
% 16.19/9.07 => ( ( image_nat_nat
% 16.19/9.07 @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
% 16.19/9.07 @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 16.19/9.07 = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y @ C ) ) ) )
% 16.19/9.07 & ( ~ ( ord_less_nat @ C @ Y )
% 16.19/9.07 => ( ( ( ord_less_nat @ X @ Y )
% 16.19/9.07 => ( ( image_nat_nat
% 16.19/9.07 @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
% 16.19/9.07 @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 16.19/9.07 = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
% 16.19/9.07 & ( ~ ( ord_less_nat @ X @ Y )
% 16.19/9.07 => ( ( image_nat_nat
% 16.19/9.07 @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
% 16.19/9.07 @ ( set_or4665077453230672383an_nat @ X @ Y ) )
% 16.19/9.07 = bot_bot_set_nat ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9937_Gcd__abs__eq,axiom,
% 16.19/9.07 ! [K5: set_int] :
% 16.19/9.07 ( ( gcd_Gcd_int @ ( image_int_int @ abs_abs_int @ K5 ) )
% 16.19/9.07 = ( gcd_Gcd_int @ K5 ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9938_Gcd__nat__abs__eq,axiom,
% 16.19/9.07 ! [K5: set_int] :
% 16.19/9.07 ( ( gcd_Gcd_nat
% 16.19/9.07 @ ( image_int_nat
% 16.19/9.07 @ ^ [K3: int] : ( nat2 @ ( abs_abs_int @ K3 ) )
% 16.19/9.07 @ K5 ) )
% 16.19/9.07 = ( nat2 @ ( gcd_Gcd_int @ K5 ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9939_Inf__real__def,axiom,
% 16.19/9.07 ( comple4887499456419720421f_real
% 16.19/9.07 = ( ^ [X3: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X3 ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9940_image__add__int__atLeastLessThan,axiom,
% 16.19/9.07 ! [L2: int,U: int] :
% 16.19/9.07 ( ( image_int_int
% 16.19/9.07 @ ^ [X2: int] : ( plus_plus_int @ X2 @ L2 )
% 16.19/9.07 @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L2 ) ) )
% 16.19/9.07 = ( set_or4662586982721622107an_int @ L2 @ U ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9941_Gcd__int__def,axiom,
% 16.19/9.07 ( gcd_Gcd_int
% 16.19/9.07 = ( ^ [K7: set_int] : ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ K7 ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9942_suminf__eq__SUP__real,axiom,
% 16.19/9.07 ! [X8: nat > real] :
% 16.19/9.07 ( ( summable_real @ X8 )
% 16.19/9.07 => ( ! [I4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X8 @ I4 ) )
% 16.19/9.07 => ( ( suminf_real @ X8 )
% 16.19/9.07 = ( comple1385675409528146559p_real
% 16.19/9.07 @ ( image_nat_real
% 16.19/9.07 @ ^ [I3: nat] : ( groups6591440286371151544t_real @ X8 @ ( set_ord_lessThan_nat @ I3 ) )
% 16.19/9.07 @ top_top_set_nat ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9943_UN__atMost__UNIV,axiom,
% 16.19/9.07 ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atMost_nat @ top_top_set_nat ) )
% 16.19/9.07 = top_top_set_nat ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9944_UN__lessThan__UNIV,axiom,
% 16.19/9.07 ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
% 16.19/9.07 = top_top_set_nat ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9945_UNIV__nat__eq,axiom,
% 16.19/9.07 ( top_top_set_nat
% 16.19/9.07 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9946_range__mod,axiom,
% 16.19/9.07 ! [N2: nat] :
% 16.19/9.07 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.07 => ( ( image_nat_nat
% 16.19/9.07 @ ^ [M6: nat] : ( modulo_modulo_nat @ M6 @ N2 )
% 16.19/9.07 @ top_top_set_nat )
% 16.19/9.07 = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9947_card__UNIV__unit,axiom,
% 16.19/9.07 ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
% 16.19/9.07 = one_one_nat ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9948_card__UNIV__bool,axiom,
% 16.19/9.07 ( ( finite_card_o @ top_top_set_o )
% 16.19/9.07 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9949_range__mult,axiom,
% 16.19/9.07 ! [A: real] :
% 16.19/9.07 ( ( ( A = zero_zero_real )
% 16.19/9.07 => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 16.19/9.07 = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
% 16.19/9.07 & ( ( A != zero_zero_real )
% 16.19/9.07 => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 16.19/9.07 = top_top_set_real ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9950_root__def,axiom,
% 16.19/9.07 ( root
% 16.19/9.07 = ( ^ [N: nat,X2: real] :
% 16.19/9.07 ( if_real @ ( N = zero_zero_nat ) @ zero_zero_real
% 16.19/9.07 @ ( the_in5290026491893676941l_real @ top_top_set_real
% 16.19/9.07 @ ^ [Y2: real] : ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N ) )
% 16.19/9.07 @ X2 ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9951_card__UNIV__char,axiom,
% 16.19/9.07 ( ( finite_card_char @ top_top_set_char )
% 16.19/9.07 = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9952_UNIV__char__of__nat,axiom,
% 16.19/9.07 ( top_top_set_char
% 16.19/9.07 = ( image_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9953_nat__of__char__less__256,axiom,
% 16.19/9.07 ! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9954_range__nat__of__char,axiom,
% 16.19/9.07 ( ( image_char_nat @ comm_s629917340098488124ar_nat @ top_top_set_char )
% 16.19/9.07 = ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9955_integer__of__char__code,axiom,
% 16.19/9.07 ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
% 16.19/9.07 ( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
% 16.19/9.07 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ B72 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B62 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B52 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B42 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B32 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B22 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B1 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B0 ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9956_char__of__integer__code,axiom,
% 16.19/9.07 ( char_of_integer
% 16.19/9.07 = ( ^ [K3: code_integer] :
% 16.19/9.07 ( produc4188289175737317920o_char
% 16.19/9.07 @ ^ [Q0: code_integer,B02: $o] :
% 16.19/9.07 ( produc4188289175737317920o_char
% 16.19/9.07 @ ^ [Q1: code_integer,B12: $o] :
% 16.19/9.07 ( produc4188289175737317920o_char
% 16.19/9.07 @ ^ [Q22: code_integer,B23: $o] :
% 16.19/9.07 ( produc4188289175737317920o_char
% 16.19/9.07 @ ^ [Q32: code_integer,B33: $o] :
% 16.19/9.07 ( produc4188289175737317920o_char
% 16.19/9.07 @ ^ [Q42: code_integer,B43: $o] :
% 16.19/9.07 ( produc4188289175737317920o_char
% 16.19/9.07 @ ^ [Q52: code_integer,B53: $o] :
% 16.19/9.07 ( produc4188289175737317920o_char
% 16.19/9.07 @ ^ [Q62: code_integer,B63: $o] :
% 16.19/9.07 ( produc4188289175737317920o_char
% 16.19/9.07 @ ^ [Uu3: code_integer] : ( char2 @ B02 @ B12 @ B23 @ B33 @ B43 @ B53 @ B63 )
% 16.19/9.07 @ ( code_bit_cut_integer @ Q62 ) )
% 16.19/9.07 @ ( code_bit_cut_integer @ Q52 ) )
% 16.19/9.07 @ ( code_bit_cut_integer @ Q42 ) )
% 16.19/9.07 @ ( code_bit_cut_integer @ Q32 ) )
% 16.19/9.07 @ ( code_bit_cut_integer @ Q22 ) )
% 16.19/9.07 @ ( code_bit_cut_integer @ Q1 ) )
% 16.19/9.07 @ ( code_bit_cut_integer @ Q0 ) )
% 16.19/9.07 @ ( code_bit_cut_integer @ K3 ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9957_String_Ochar__of__ascii__of,axiom,
% 16.19/9.07 ! [C: char] :
% 16.19/9.07 ( ( comm_s629917340098488124ar_nat @ ( ascii_of @ C ) )
% 16.19/9.07 = ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( comm_s629917340098488124ar_nat @ C ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9958_DERIV__even__real__root,axiom,
% 16.19/9.07 ! [N2: nat,X: real] :
% 16.19/9.07 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.07 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.07 => ( ( ord_less_real @ X @ zero_zero_real )
% 16.19/9.07 => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9959_deriv__nonneg__imp__mono,axiom,
% 16.19/9.07 ! [A: real,B: real,G: real > real,G2: real > real] :
% 16.19/9.07 ( ! [X4: real] :
% 16.19/9.07 ( ( member_real @ X4 @ ( set_or1222579329274155063t_real @ A @ B ) )
% 16.19/9.07 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 16.19/9.07 => ( ! [X4: real] :
% 16.19/9.07 ( ( member_real @ X4 @ ( set_or1222579329274155063t_real @ A @ B ) )
% 16.19/9.07 => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X4 ) ) )
% 16.19/9.07 => ( ( ord_less_eq_real @ A @ B )
% 16.19/9.07 => ( ord_less_eq_real @ ( G @ A ) @ ( G @ B ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9960_DERIV__nonneg__imp__nondecreasing,axiom,
% 16.19/9.07 ! [A: real,B: real,F: real > real] :
% 16.19/9.07 ( ( ord_less_eq_real @ A @ B )
% 16.19/9.07 => ( ! [X4: real] :
% 16.19/9.07 ( ( ord_less_eq_real @ A @ X4 )
% 16.19/9.07 => ( ( ord_less_eq_real @ X4 @ B )
% 16.19/9.07 => ? [Y5: real] :
% 16.19/9.07 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 16.19/9.07 & ( ord_less_eq_real @ zero_zero_real @ Y5 ) ) ) )
% 16.19/9.07 => ( ord_less_eq_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9961_DERIV__nonpos__imp__nonincreasing,axiom,
% 16.19/9.07 ! [A: real,B: real,F: real > real] :
% 16.19/9.07 ( ( ord_less_eq_real @ A @ B )
% 16.19/9.07 => ( ! [X4: real] :
% 16.19/9.07 ( ( ord_less_eq_real @ A @ X4 )
% 16.19/9.07 => ( ( ord_less_eq_real @ X4 @ B )
% 16.19/9.07 => ? [Y5: real] :
% 16.19/9.07 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 16.19/9.07 & ( ord_less_eq_real @ Y5 @ zero_zero_real ) ) ) )
% 16.19/9.07 => ( ord_less_eq_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9962_DERIV__mirror,axiom,
% 16.19/9.07 ! [F: real > real,Y: real,X: real] :
% 16.19/9.07 ( ( has_fi5821293074295781190e_real @ F @ Y @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ X ) @ top_top_set_real ) )
% 16.19/9.07 = ( has_fi5821293074295781190e_real
% 16.19/9.07 @ ^ [X2: real] : ( F @ ( uminus_uminus_real @ X2 ) )
% 16.19/9.07 @ ( uminus_uminus_real @ Y )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9963_DERIV__const__ratio__const2,axiom,
% 16.19/9.07 ! [A: real,B: real,F: real > real,K: real] :
% 16.19/9.07 ( ( A != B )
% 16.19/9.07 => ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 16.19/9.07 => ( ( divide_divide_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( minus_minus_real @ B @ A ) )
% 16.19/9.07 = K ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9964_DERIV__const__ratio__const,axiom,
% 16.19/9.07 ! [A: real,B: real,F: real > real,K: real] :
% 16.19/9.07 ( ( A != B )
% 16.19/9.07 => ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 16.19/9.07 => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 16.19/9.07 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ K ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9965_has__real__derivative__pos__inc__right,axiom,
% 16.19/9.07 ! [F: real > real,L2: real,X: real,S: set_real] :
% 16.19/9.07 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ S ) )
% 16.19/9.07 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 16.19/9.07 => ? [D4: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ D4 )
% 16.19/9.07 & ! [H4: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ H4 )
% 16.19/9.07 => ( ( member_real @ ( plus_plus_real @ X @ H4 ) @ S )
% 16.19/9.07 => ( ( ord_less_real @ H4 @ D4 )
% 16.19/9.07 => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H4 ) ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9966_has__real__derivative__neg__dec__right,axiom,
% 16.19/9.07 ! [F: real > real,L2: real,X: real,S: set_real] :
% 16.19/9.07 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ S ) )
% 16.19/9.07 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 16.19/9.07 => ? [D4: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ D4 )
% 16.19/9.07 & ! [H4: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ H4 )
% 16.19/9.07 => ( ( member_real @ ( plus_plus_real @ X @ H4 ) @ S )
% 16.19/9.07 => ( ( ord_less_real @ H4 @ D4 )
% 16.19/9.07 => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9967_has__real__derivative__pos__inc__left,axiom,
% 16.19/9.07 ! [F: real > real,L2: real,X: real,S: set_real] :
% 16.19/9.07 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ S ) )
% 16.19/9.07 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 16.19/9.07 => ? [D4: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ D4 )
% 16.19/9.07 & ! [H4: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ H4 )
% 16.19/9.07 => ( ( member_real @ ( minus_minus_real @ X @ H4 ) @ S )
% 16.19/9.07 => ( ( ord_less_real @ H4 @ D4 )
% 16.19/9.07 => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9968_has__real__derivative__neg__dec__left,axiom,
% 16.19/9.07 ! [F: real > real,L2: real,X: real,S: set_real] :
% 16.19/9.07 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ S ) )
% 16.19/9.07 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 16.19/9.07 => ? [D4: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ D4 )
% 16.19/9.07 & ! [H4: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ H4 )
% 16.19/9.07 => ( ( member_real @ ( minus_minus_real @ X @ H4 ) @ S )
% 16.19/9.07 => ( ( ord_less_real @ H4 @ D4 )
% 16.19/9.07 => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H4 ) ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9969_DERIV__local__const,axiom,
% 16.19/9.07 ! [F: real > real,L2: real,X: real,D: real] :
% 16.19/9.07 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 16.19/9.07 => ( ( ord_less_real @ zero_zero_real @ D )
% 16.19/9.07 => ( ! [Y4: real] :
% 16.19/9.07 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
% 16.19/9.07 => ( ( F @ X )
% 16.19/9.07 = ( F @ Y4 ) ) )
% 16.19/9.07 => ( L2 = zero_zero_real ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9970_DERIV__ln,axiom,
% 16.19/9.07 ! [X: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.07 => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9971_DERIV__neg__dec__left,axiom,
% 16.19/9.07 ! [F: real > real,L2: real,X: real] :
% 16.19/9.07 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 16.19/9.07 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 16.19/9.07 => ? [D4: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ D4 )
% 16.19/9.07 & ! [H4: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ H4 )
% 16.19/9.07 => ( ( ord_less_real @ H4 @ D4 )
% 16.19/9.07 => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H4 ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9972_DERIV__pos__inc__left,axiom,
% 16.19/9.07 ! [F: real > real,L2: real,X: real] :
% 16.19/9.07 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 16.19/9.07 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 16.19/9.07 => ? [D4: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ D4 )
% 16.19/9.07 & ! [H4: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ H4 )
% 16.19/9.07 => ( ( ord_less_real @ H4 @ D4 )
% 16.19/9.07 => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9973_DERIV__neg__dec__right,axiom,
% 16.19/9.07 ! [F: real > real,L2: real,X: real] :
% 16.19/9.07 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 16.19/9.07 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 16.19/9.07 => ? [D4: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ D4 )
% 16.19/9.07 & ! [H4: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ H4 )
% 16.19/9.07 => ( ( ord_less_real @ H4 @ D4 )
% 16.19/9.07 => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9974_DERIV__pos__inc__right,axiom,
% 16.19/9.07 ! [F: real > real,L2: real,X: real] :
% 16.19/9.07 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 16.19/9.07 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 16.19/9.07 => ? [D4: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ D4 )
% 16.19/9.07 & ! [H4: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ H4 )
% 16.19/9.07 => ( ( ord_less_real @ H4 @ D4 )
% 16.19/9.07 => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H4 ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9975_DERIV__isconst3,axiom,
% 16.19/9.07 ! [A: real,B: real,X: real,Y: real,F: real > real] :
% 16.19/9.07 ( ( ord_less_real @ A @ B )
% 16.19/9.07 => ( ( member_real @ X @ ( set_or1633881224788618240n_real @ A @ B ) )
% 16.19/9.07 => ( ( member_real @ Y @ ( set_or1633881224788618240n_real @ A @ B ) )
% 16.19/9.07 => ( ! [X4: real] :
% 16.19/9.07 ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 16.19/9.07 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 16.19/9.07 => ( ( F @ X )
% 16.19/9.07 = ( F @ Y ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9976_DERIV__neg__imp__decreasing,axiom,
% 16.19/9.07 ! [A: real,B: real,F: real > real] :
% 16.19/9.07 ( ( ord_less_real @ A @ B )
% 16.19/9.07 => ( ! [X4: real] :
% 16.19/9.07 ( ( ord_less_eq_real @ A @ X4 )
% 16.19/9.07 => ( ( ord_less_eq_real @ X4 @ B )
% 16.19/9.07 => ? [Y5: real] :
% 16.19/9.07 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 16.19/9.07 & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
% 16.19/9.07 => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9977_DERIV__pos__imp__increasing,axiom,
% 16.19/9.07 ! [A: real,B: real,F: real > real] :
% 16.19/9.07 ( ( ord_less_real @ A @ B )
% 16.19/9.07 => ( ! [X4: real] :
% 16.19/9.07 ( ( ord_less_eq_real @ A @ X4 )
% 16.19/9.07 => ( ( ord_less_eq_real @ X4 @ B )
% 16.19/9.07 => ? [Y5: real] :
% 16.19/9.07 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 16.19/9.07 & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
% 16.19/9.07 => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9978_MVT2,axiom,
% 16.19/9.07 ! [A: real,B: real,F: real > real,F4: real > real] :
% 16.19/9.07 ( ( ord_less_real @ A @ B )
% 16.19/9.07 => ( ! [X4: real] :
% 16.19/9.07 ( ( ord_less_eq_real @ A @ X4 )
% 16.19/9.07 => ( ( ord_less_eq_real @ X4 @ B )
% 16.19/9.07 => ( has_fi5821293074295781190e_real @ F @ ( F4 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 16.19/9.07 => ? [Z5: real] :
% 16.19/9.07 ( ( ord_less_real @ A @ Z5 )
% 16.19/9.07 & ( ord_less_real @ Z5 @ B )
% 16.19/9.07 & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 16.19/9.07 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F4 @ Z5 ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9979_DERIV__const__average,axiom,
% 16.19/9.07 ! [A: real,B: real,V: real > real,K: real] :
% 16.19/9.07 ( ( A != B )
% 16.19/9.07 => ( ! [X4: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 16.19/9.07 => ( ( V @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 16.19/9.07 = ( divide_divide_real @ ( plus_plus_real @ ( V @ A ) @ ( V @ B ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9980_DERIV__local__max,axiom,
% 16.19/9.07 ! [F: real > real,L2: real,X: real,D: real] :
% 16.19/9.07 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 16.19/9.07 => ( ( ord_less_real @ zero_zero_real @ D )
% 16.19/9.07 => ( ! [Y4: real] :
% 16.19/9.07 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
% 16.19/9.07 => ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X ) ) )
% 16.19/9.07 => ( L2 = zero_zero_real ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9981_DERIV__local__min,axiom,
% 16.19/9.07 ! [F: real > real,L2: real,X: real,D: real] :
% 16.19/9.07 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 16.19/9.07 => ( ( ord_less_real @ zero_zero_real @ D )
% 16.19/9.07 => ( ! [Y4: real] :
% 16.19/9.07 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
% 16.19/9.07 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
% 16.19/9.07 => ( L2 = zero_zero_real ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9982_DERIV__ln__divide,axiom,
% 16.19/9.07 ! [X: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.07 => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9983_DERIV__pow,axiom,
% 16.19/9.07 ! [N2: nat,X: real,S3: set_real] :
% 16.19/9.07 ( has_fi5821293074295781190e_real
% 16.19/9.07 @ ^ [X2: real] : ( power_power_real @ X2 @ N2 )
% 16.19/9.07 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ X @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ S3 ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9984_DERIV__fun__pow,axiom,
% 16.19/9.07 ! [G: real > real,M: real,X: real,N2: nat] :
% 16.19/9.07 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 16.19/9.07 => ( has_fi5821293074295781190e_real
% 16.19/9.07 @ ^ [X2: real] : ( power_power_real @ ( G @ X2 ) @ N2 )
% 16.19/9.07 @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( G @ X ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) @ M )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9985_has__real__derivative__powr,axiom,
% 16.19/9.07 ! [Z: real,R: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ Z )
% 16.19/9.07 => ( has_fi5821293074295781190e_real
% 16.19/9.07 @ ^ [Z3: real] : ( powr_real @ Z3 @ R )
% 16.19/9.07 @ ( times_times_real @ R @ ( powr_real @ Z @ ( minus_minus_real @ R @ one_one_real ) ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9986_DERIV__log,axiom,
% 16.19/9.07 ! [X: real,B: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.07 => ( has_fi5821293074295781190e_real @ ( log @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9987_DERIV__fun__powr,axiom,
% 16.19/9.07 ! [G: real > real,M: real,X: real,R: real] :
% 16.19/9.07 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 16.19/9.07 => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
% 16.19/9.07 => ( has_fi5821293074295781190e_real
% 16.19/9.07 @ ^ [X2: real] : ( powr_real @ ( G @ X2 ) @ R )
% 16.19/9.07 @ ( times_times_real @ ( times_times_real @ R @ ( powr_real @ ( G @ X ) @ ( minus_minus_real @ R @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9988_DERIV__powr,axiom,
% 16.19/9.07 ! [G: real > real,M: real,X: real,F: real > real,R: real] :
% 16.19/9.07 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 16.19/9.07 => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
% 16.19/9.07 => ( ( has_fi5821293074295781190e_real @ F @ R @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 16.19/9.07 => ( has_fi5821293074295781190e_real
% 16.19/9.07 @ ^ [X2: real] : ( powr_real @ ( G @ X2 ) @ ( F @ X2 ) )
% 16.19/9.07 @ ( times_times_real @ ( powr_real @ ( G @ X ) @ ( F @ X ) ) @ ( plus_plus_real @ ( times_times_real @ R @ ( ln_ln_real @ ( G @ X ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X ) ) @ ( G @ X ) ) ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9989_DERIV__real__sqrt,axiom,
% 16.19/9.07 ! [X: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.07 => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9990_DERIV__series_H,axiom,
% 16.19/9.07 ! [F: real > nat > real,F4: real > nat > real,X0: real,A: real,B: real,L5: nat > real] :
% 16.19/9.07 ( ! [N4: nat] :
% 16.19/9.07 ( has_fi5821293074295781190e_real
% 16.19/9.07 @ ^ [X2: real] : ( F @ X2 @ N4 )
% 16.19/9.07 @ ( F4 @ X0 @ N4 )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
% 16.19/9.07 => ( ! [X4: real] :
% 16.19/9.07 ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 16.19/9.07 => ( summable_real @ ( F @ X4 ) ) )
% 16.19/9.07 => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 16.19/9.07 => ( ( summable_real @ ( F4 @ X0 ) )
% 16.19/9.07 => ( ( summable_real @ L5 )
% 16.19/9.07 => ( ! [N4: nat,X4: real,Y4: real] :
% 16.19/9.07 ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 16.19/9.07 => ( ( member_real @ Y4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 16.19/9.07 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X4 @ N4 ) @ ( F @ Y4 @ N4 ) ) ) @ ( times_times_real @ ( L5 @ N4 ) @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y4 ) ) ) ) ) )
% 16.19/9.07 => ( has_fi5821293074295781190e_real
% 16.19/9.07 @ ^ [X2: real] : ( suminf_real @ ( F @ X2 ) )
% 16.19/9.07 @ ( suminf_real @ ( F4 @ X0 ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9991_DERIV__arctan,axiom,
% 16.19/9.07 ! [X: real] : ( has_fi5821293074295781190e_real @ arctan @ ( inverse_inverse_real @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9992_arsinh__real__has__field__derivative,axiom,
% 16.19/9.07 ! [X: real,A2: set_real] : ( has_fi5821293074295781190e_real @ arsinh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9993_DERIV__real__sqrt__generic,axiom,
% 16.19/9.07 ! [X: real,D3: real] :
% 16.19/9.07 ( ( X != zero_zero_real )
% 16.19/9.07 => ( ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.07 => ( D3
% 16.19/9.07 = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 16.19/9.07 => ( ( ( ord_less_real @ X @ zero_zero_real )
% 16.19/9.07 => ( D3
% 16.19/9.07 = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 16.19/9.07 => ( has_fi5821293074295781190e_real @ sqrt @ D3 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9994_arcosh__real__has__field__derivative,axiom,
% 16.19/9.07 ! [X: real,A2: set_real] :
% 16.19/9.07 ( ( ord_less_real @ one_one_real @ X )
% 16.19/9.07 => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9995_artanh__real__has__field__derivative,axiom,
% 16.19/9.07 ! [X: real,A2: set_real] :
% 16.19/9.07 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 16.19/9.07 => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ A2 ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9996_DERIV__power__series_H,axiom,
% 16.19/9.07 ! [R2: real,F: nat > real,X0: real] :
% 16.19/9.07 ( ! [X4: real] :
% 16.19/9.07 ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R2 ) @ R2 ) )
% 16.19/9.07 => ( summable_real
% 16.19/9.07 @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( F @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) @ ( power_power_real @ X4 @ N ) ) ) )
% 16.19/9.07 => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R2 ) @ R2 ) )
% 16.19/9.07 => ( ( ord_less_real @ zero_zero_real @ R2 )
% 16.19/9.07 => ( has_fi5821293074295781190e_real
% 16.19/9.07 @ ^ [X2: real] :
% 16.19/9.07 ( suminf_real
% 16.19/9.07 @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ X2 @ ( suc @ N ) ) ) )
% 16.19/9.07 @ ( suminf_real
% 16.19/9.07 @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( F @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) @ ( power_power_real @ X0 @ N ) ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9997_DERIV__real__root,axiom,
% 16.19/9.07 ! [N2: nat,X: real] :
% 16.19/9.07 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.07 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.07 => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9998_DERIV__arccos,axiom,
% 16.19/9.07 ! [X: real] :
% 16.19/9.07 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 16.19/9.07 => ( ( ord_less_real @ X @ one_one_real )
% 16.19/9.07 => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_9999_DERIV__arcsin,axiom,
% 16.19/9.07 ! [X: real] :
% 16.19/9.07 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 16.19/9.07 => ( ( ord_less_real @ X @ one_one_real )
% 16.19/9.07 => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10000_Maclaurin__all__le,axiom,
% 16.19/9.07 ! [Diff: nat > real > real,F: real > real,X: real,N2: nat] :
% 16.19/9.07 ( ( ( Diff @ zero_zero_nat )
% 16.19/9.07 = F )
% 16.19/9.07 => ( ! [M2: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 16.19/9.07 => ? [T5: real] :
% 16.19/9.07 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 16.19/9.07 & ( ( F @ X )
% 16.19/9.07 = ( plus_plus_real
% 16.19/9.07 @ ( groups6591440286371151544t_real
% 16.19/9.07 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
% 16.19/9.07 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.07 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T5 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10001_Maclaurin__all__le__objl,axiom,
% 16.19/9.07 ! [Diff: nat > real > real,F: real > real,X: real,N2: nat] :
% 16.19/9.07 ( ( ( ( Diff @ zero_zero_nat )
% 16.19/9.07 = F )
% 16.19/9.07 & ! [M2: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 16.19/9.07 => ? [T5: real] :
% 16.19/9.07 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 16.19/9.07 & ( ( F @ X )
% 16.19/9.07 = ( plus_plus_real
% 16.19/9.07 @ ( groups6591440286371151544t_real
% 16.19/9.07 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
% 16.19/9.07 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.07 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T5 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10002_DERIV__odd__real__root,axiom,
% 16.19/9.07 ! [N2: nat,X: real] :
% 16.19/9.07 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.07 => ( ( X != zero_zero_real )
% 16.19/9.07 => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10003_Maclaurin__minus,axiom,
% 16.19/9.07 ! [H2: real,N2: nat,Diff: nat > real > real,F: real > real] :
% 16.19/9.07 ( ( ord_less_real @ H2 @ zero_zero_real )
% 16.19/9.07 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.07 => ( ( ( Diff @ zero_zero_nat )
% 16.19/9.07 = F )
% 16.19/9.07 => ( ! [M2: nat,T5: real] :
% 16.19/9.07 ( ( ( ord_less_nat @ M2 @ N2 )
% 16.19/9.07 & ( ord_less_eq_real @ H2 @ T5 )
% 16.19/9.07 & ( ord_less_eq_real @ T5 @ zero_zero_real ) )
% 16.19/9.07 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 16.19/9.07 => ? [T5: real] :
% 16.19/9.07 ( ( ord_less_real @ H2 @ T5 )
% 16.19/9.07 & ( ord_less_real @ T5 @ zero_zero_real )
% 16.19/9.07 & ( ( F @ H2 )
% 16.19/9.07 = ( plus_plus_real
% 16.19/9.07 @ ( groups6591440286371151544t_real
% 16.19/9.07 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
% 16.19/9.07 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.07 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T5 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10004_Maclaurin2,axiom,
% 16.19/9.07 ! [H2: real,Diff: nat > real > real,F: real > real,N2: nat] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ H2 )
% 16.19/9.07 => ( ( ( Diff @ zero_zero_nat )
% 16.19/9.07 = F )
% 16.19/9.07 => ( ! [M2: nat,T5: real] :
% 16.19/9.07 ( ( ( ord_less_nat @ M2 @ N2 )
% 16.19/9.07 & ( ord_less_eq_real @ zero_zero_real @ T5 )
% 16.19/9.07 & ( ord_less_eq_real @ T5 @ H2 ) )
% 16.19/9.07 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 16.19/9.07 => ? [T5: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ T5 )
% 16.19/9.07 & ( ord_less_eq_real @ T5 @ H2 )
% 16.19/9.07 & ( ( F @ H2 )
% 16.19/9.07 = ( plus_plus_real
% 16.19/9.07 @ ( groups6591440286371151544t_real
% 16.19/9.07 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
% 16.19/9.07 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.07 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T5 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10005_Maclaurin,axiom,
% 16.19/9.07 ! [H2: real,N2: nat,Diff: nat > real > real,F: real > real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ H2 )
% 16.19/9.07 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.07 => ( ( ( Diff @ zero_zero_nat )
% 16.19/9.07 = F )
% 16.19/9.07 => ( ! [M2: nat,T5: real] :
% 16.19/9.07 ( ( ( ord_less_nat @ M2 @ N2 )
% 16.19/9.07 & ( ord_less_eq_real @ zero_zero_real @ T5 )
% 16.19/9.07 & ( ord_less_eq_real @ T5 @ H2 ) )
% 16.19/9.07 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 16.19/9.07 => ? [T5: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ T5 )
% 16.19/9.07 & ( ord_less_real @ T5 @ H2 )
% 16.19/9.07 & ( ( F @ H2 )
% 16.19/9.07 = ( plus_plus_real
% 16.19/9.07 @ ( groups6591440286371151544t_real
% 16.19/9.07 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H2 @ M6 ) )
% 16.19/9.07 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.07 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T5 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10006_Maclaurin__all__lt,axiom,
% 16.19/9.07 ! [Diff: nat > real > real,F: real > real,N2: nat,X: real] :
% 16.19/9.07 ( ( ( Diff @ zero_zero_nat )
% 16.19/9.07 = F )
% 16.19/9.07 => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.07 => ( ( X != zero_zero_real )
% 16.19/9.07 => ( ! [M2: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 16.19/9.07 => ? [T5: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T5 ) )
% 16.19/9.07 & ( ord_less_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 16.19/9.07 & ( ( F @ X )
% 16.19/9.07 = ( plus_plus_real
% 16.19/9.07 @ ( groups6591440286371151544t_real
% 16.19/9.07 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
% 16.19/9.07 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.07 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T5 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10007_Maclaurin__bi__le,axiom,
% 16.19/9.07 ! [Diff: nat > real > real,F: real > real,N2: nat,X: real] :
% 16.19/9.07 ( ( ( Diff @ zero_zero_nat )
% 16.19/9.07 = F )
% 16.19/9.07 => ( ! [M2: nat,T5: real] :
% 16.19/9.07 ( ( ( ord_less_nat @ M2 @ N2 )
% 16.19/9.07 & ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) ) )
% 16.19/9.07 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 16.19/9.07 => ? [T5: real] :
% 16.19/9.07 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 16.19/9.07 & ( ( F @ X )
% 16.19/9.07 = ( plus_plus_real
% 16.19/9.07 @ ( groups6591440286371151544t_real
% 16.19/9.07 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
% 16.19/9.07 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.07 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T5 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10008_Taylor__down,axiom,
% 16.19/9.07 ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 16.19/9.07 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.07 => ( ( ( Diff @ zero_zero_nat )
% 16.19/9.07 = F )
% 16.19/9.07 => ( ! [M2: nat,T5: real] :
% 16.19/9.07 ( ( ( ord_less_nat @ M2 @ N2 )
% 16.19/9.07 & ( ord_less_eq_real @ A @ T5 )
% 16.19/9.07 & ( ord_less_eq_real @ T5 @ B ) )
% 16.19/9.07 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 16.19/9.07 => ( ( ord_less_real @ A @ C )
% 16.19/9.07 => ( ( ord_less_eq_real @ C @ B )
% 16.19/9.07 => ? [T5: real] :
% 16.19/9.07 ( ( ord_less_real @ A @ T5 )
% 16.19/9.07 & ( ord_less_real @ T5 @ C )
% 16.19/9.07 & ( ( F @ A )
% 16.19/9.07 = ( plus_plus_real
% 16.19/9.07 @ ( groups6591440286371151544t_real
% 16.19/9.07 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M6 ) )
% 16.19/9.07 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.07 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T5 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10009_Taylor__up,axiom,
% 16.19/9.07 ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 16.19/9.07 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.07 => ( ( ( Diff @ zero_zero_nat )
% 16.19/9.07 = F )
% 16.19/9.07 => ( ! [M2: nat,T5: real] :
% 16.19/9.07 ( ( ( ord_less_nat @ M2 @ N2 )
% 16.19/9.07 & ( ord_less_eq_real @ A @ T5 )
% 16.19/9.07 & ( ord_less_eq_real @ T5 @ B ) )
% 16.19/9.07 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 16.19/9.07 => ( ( ord_less_eq_real @ A @ C )
% 16.19/9.07 => ( ( ord_less_real @ C @ B )
% 16.19/9.07 => ? [T5: real] :
% 16.19/9.07 ( ( ord_less_real @ C @ T5 )
% 16.19/9.07 & ( ord_less_real @ T5 @ B )
% 16.19/9.07 & ( ( F @ B )
% 16.19/9.07 = ( plus_plus_real
% 16.19/9.07 @ ( groups6591440286371151544t_real
% 16.19/9.07 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M6 ) )
% 16.19/9.07 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.07 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T5 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10010_Taylor,axiom,
% 16.19/9.07 ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X: real] :
% 16.19/9.07 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.07 => ( ( ( Diff @ zero_zero_nat )
% 16.19/9.07 = F )
% 16.19/9.07 => ( ! [M2: nat,T5: real] :
% 16.19/9.07 ( ( ( ord_less_nat @ M2 @ N2 )
% 16.19/9.07 & ( ord_less_eq_real @ A @ T5 )
% 16.19/9.07 & ( ord_less_eq_real @ T5 @ B ) )
% 16.19/9.07 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 16.19/9.07 => ( ( ord_less_eq_real @ A @ C )
% 16.19/9.07 => ( ( ord_less_eq_real @ C @ B )
% 16.19/9.07 => ( ( ord_less_eq_real @ A @ X )
% 16.19/9.07 => ( ( ord_less_eq_real @ X @ B )
% 16.19/9.07 => ( ( X != C )
% 16.19/9.07 => ? [T5: real] :
% 16.19/9.07 ( ( ( ord_less_real @ X @ C )
% 16.19/9.07 => ( ( ord_less_real @ X @ T5 )
% 16.19/9.07 & ( ord_less_real @ T5 @ C ) ) )
% 16.19/9.07 & ( ~ ( ord_less_real @ X @ C )
% 16.19/9.07 => ( ( ord_less_real @ C @ T5 )
% 16.19/9.07 & ( ord_less_real @ T5 @ X ) ) )
% 16.19/9.07 & ( ( F @ X )
% 16.19/9.07 = ( plus_plus_real
% 16.19/9.07 @ ( groups6591440286371151544t_real
% 16.19/9.07 @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ M6 ) )
% 16.19/9.07 @ ( set_ord_lessThan_nat @ N2 ) )
% 16.19/9.07 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T5 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10011_Maclaurin__lemma2,axiom,
% 16.19/9.07 ! [N2: nat,H2: real,Diff: nat > real > real,K: nat,B3: real] :
% 16.19/9.07 ( ! [M2: nat,T5: real] :
% 16.19/9.07 ( ( ( ord_less_nat @ M2 @ N2 )
% 16.19/9.07 & ( ord_less_eq_real @ zero_zero_real @ T5 )
% 16.19/9.07 & ( ord_less_eq_real @ T5 @ H2 ) )
% 16.19/9.07 => ( has_fi5821293074295781190e_real @ ( Diff @ M2 ) @ ( Diff @ ( suc @ M2 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 16.19/9.07 => ( ( N2
% 16.19/9.07 = ( suc @ K ) )
% 16.19/9.07 => ! [M5: nat,T6: real] :
% 16.19/9.07 ( ( ( ord_less_nat @ M5 @ N2 )
% 16.19/9.07 & ( ord_less_eq_real @ zero_zero_real @ T6 )
% 16.19/9.07 & ( ord_less_eq_real @ T6 @ H2 ) )
% 16.19/9.07 => ( has_fi5821293074295781190e_real
% 16.19/9.07 @ ^ [U2: real] :
% 16.19/9.07 ( minus_minus_real @ ( Diff @ M5 @ U2 )
% 16.19/9.07 @ ( plus_plus_real
% 16.19/9.07 @ ( groups6591440286371151544t_real
% 16.19/9.07 @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M5 @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ U2 @ P5 ) )
% 16.19/9.07 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ M5 ) ) )
% 16.19/9.07 @ ( times_times_real @ B3 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N2 @ M5 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ M5 ) ) ) ) ) )
% 16.19/9.07 @ ( minus_minus_real @ ( Diff @ ( suc @ M5 ) @ T6 )
% 16.19/9.07 @ ( plus_plus_real
% 16.19/9.07 @ ( groups6591440286371151544t_real
% 16.19/9.07 @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M5 ) @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ T6 @ P5 ) )
% 16.19/9.07 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ M5 ) ) ) )
% 16.19/9.07 @ ( times_times_real @ B3 @ ( divide_divide_real @ ( power_power_real @ T6 @ ( minus_minus_nat @ N2 @ ( suc @ M5 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ ( suc @ M5 ) ) ) ) ) ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10012_DERIV__arctan__series,axiom,
% 16.19/9.07 ! [X: real] :
% 16.19/9.07 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 16.19/9.07 => ( has_fi5821293074295781190e_real
% 16.19/9.07 @ ^ [X9: real] :
% 16.19/9.07 ( suminf_real
% 16.19/9.07 @ ^ [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 ) ) ) ) )
% 16.19/9.07 @ ( suminf_real
% 16.19/9.07 @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10013_DERIV__real__root__generic,axiom,
% 16.19/9.07 ! [N2: nat,X: real,D3: real] :
% 16.19/9.07 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.07 => ( ( X != zero_zero_real )
% 16.19/9.07 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.07 => ( ( ord_less_real @ zero_zero_real @ X )
% 16.19/9.07 => ( D3
% 16.19/9.07 = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
% 16.19/9.07 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.07 => ( ( ord_less_real @ X @ zero_zero_real )
% 16.19/9.07 => ( D3
% 16.19/9.07 = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
% 16.19/9.07 => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.07 => ( D3
% 16.19/9.07 = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
% 16.19/9.07 => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ D3 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10014_isCont__Lb__Ub,axiom,
% 16.19/9.07 ! [A: real,B: real,F: real > real] :
% 16.19/9.07 ( ( ord_less_eq_real @ A @ B )
% 16.19/9.07 => ( ! [X4: real] :
% 16.19/9.07 ( ( ( ord_less_eq_real @ A @ X4 )
% 16.19/9.07 & ( ord_less_eq_real @ X4 @ B ) )
% 16.19/9.07 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F ) )
% 16.19/9.07 => ? [L6: real,M9: real] :
% 16.19/9.07 ( ! [X5: real] :
% 16.19/9.07 ( ( ( ord_less_eq_real @ A @ X5 )
% 16.19/9.07 & ( ord_less_eq_real @ X5 @ B ) )
% 16.19/9.07 => ( ( ord_less_eq_real @ L6 @ ( F @ X5 ) )
% 16.19/9.07 & ( ord_less_eq_real @ ( F @ X5 ) @ M9 ) ) )
% 16.19/9.07 & ! [Y5: real] :
% 16.19/9.07 ( ( ( ord_less_eq_real @ L6 @ Y5 )
% 16.19/9.07 & ( ord_less_eq_real @ Y5 @ M9 ) )
% 16.19/9.07 => ? [X4: real] :
% 16.19/9.07 ( ( ord_less_eq_real @ A @ X4 )
% 16.19/9.07 & ( ord_less_eq_real @ X4 @ B )
% 16.19/9.07 & ( ( F @ X4 )
% 16.19/9.07 = Y5 ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10015_LIM__fun__less__zero,axiom,
% 16.19/9.07 ! [F: real > real,L2: real,C: real] :
% 16.19/9.07 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 16.19/9.07 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 16.19/9.07 => ? [R3: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ R3 )
% 16.19/9.07 & ! [X5: real] :
% 16.19/9.07 ( ( ( X5 != C )
% 16.19/9.07 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
% 16.19/9.07 => ( ord_less_real @ ( F @ X5 ) @ zero_zero_real ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10016_LIM__fun__not__zero,axiom,
% 16.19/9.07 ! [F: real > real,L2: real,C: real] :
% 16.19/9.07 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 16.19/9.07 => ( ( L2 != zero_zero_real )
% 16.19/9.07 => ? [R3: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ R3 )
% 16.19/9.07 & ! [X5: real] :
% 16.19/9.07 ( ( ( X5 != C )
% 16.19/9.07 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
% 16.19/9.07 => ( ( F @ X5 )
% 16.19/9.07 != zero_zero_real ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10017_LIM__fun__gt__zero,axiom,
% 16.19/9.07 ! [F: real > real,L2: real,C: real] :
% 16.19/9.07 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 16.19/9.07 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 16.19/9.07 => ? [R3: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ R3 )
% 16.19/9.07 & ! [X5: real] :
% 16.19/9.07 ( ( ( X5 != C )
% 16.19/9.07 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R3 ) )
% 16.19/9.07 => ( ord_less_real @ zero_zero_real @ ( F @ X5 ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10018_isCont__real__sqrt,axiom,
% 16.19/9.07 ! [X: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ sqrt ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10019_isCont__real__root,axiom,
% 16.19/9.07 ! [X: real,N2: nat] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ( root @ N2 ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10020_isCont__inverse__function2,axiom,
% 16.19/9.07 ! [A: real,X: real,B: real,G: real > real,F: real > real] :
% 16.19/9.07 ( ( ord_less_real @ A @ X )
% 16.19/9.07 => ( ( ord_less_real @ X @ B )
% 16.19/9.07 => ( ! [Z5: real] :
% 16.19/9.07 ( ( ord_less_eq_real @ A @ Z5 )
% 16.19/9.07 => ( ( ord_less_eq_real @ Z5 @ B )
% 16.19/9.07 => ( ( G @ ( F @ Z5 ) )
% 16.19/9.07 = Z5 ) ) )
% 16.19/9.07 => ( ! [Z5: real] :
% 16.19/9.07 ( ( ord_less_eq_real @ A @ Z5 )
% 16.19/9.07 => ( ( ord_less_eq_real @ Z5 @ B )
% 16.19/9.07 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) @ F ) ) )
% 16.19/9.07 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10021_isCont__arcosh,axiom,
% 16.19/9.07 ! [X: real] :
% 16.19/9.07 ( ( ord_less_real @ one_one_real @ X )
% 16.19/9.07 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcosh_real ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10022_LIM__cos__div__sin,axiom,
% 16.19/9.07 ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( cos_real @ X2 ) @ ( sin_real @ X2 ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ top_top_set_real ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10023_DERIV__inverse__function,axiom,
% 16.19/9.07 ! [F: real > real,D3: real,G: real > real,X: real,A: real,B: real] :
% 16.19/9.07 ( ( has_fi5821293074295781190e_real @ F @ D3 @ ( topolo2177554685111907308n_real @ ( G @ X ) @ top_top_set_real ) )
% 16.19/9.07 => ( ( D3 != zero_zero_real )
% 16.19/9.07 => ( ( ord_less_real @ A @ X )
% 16.19/9.07 => ( ( ord_less_real @ X @ B )
% 16.19/9.07 => ( ! [Y4: real] :
% 16.19/9.07 ( ( ord_less_real @ A @ Y4 )
% 16.19/9.07 => ( ( ord_less_real @ Y4 @ B )
% 16.19/9.07 => ( ( F @ ( G @ Y4 ) )
% 16.19/9.07 = Y4 ) ) )
% 16.19/9.07 => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ G )
% 16.19/9.07 => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D3 ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10024_isCont__arccos,axiom,
% 16.19/9.07 ! [X: real] :
% 16.19/9.07 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 16.19/9.07 => ( ( ord_less_real @ X @ one_one_real )
% 16.19/9.07 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arccos ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10025_isCont__arcsin,axiom,
% 16.19/9.07 ! [X: real] :
% 16.19/9.07 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 16.19/9.07 => ( ( ord_less_real @ X @ one_one_real )
% 16.19/9.07 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcsin ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10026_LIM__less__bound,axiom,
% 16.19/9.07 ! [B: real,X: real,F: real > real] :
% 16.19/9.07 ( ( ord_less_real @ B @ X )
% 16.19/9.07 => ( ! [X4: real] :
% 16.19/9.07 ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ B @ X ) )
% 16.19/9.07 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 16.19/9.07 => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F )
% 16.19/9.07 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10027_isCont__artanh,axiom,
% 16.19/9.07 ! [X: real] :
% 16.19/9.07 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 16.19/9.07 => ( ( ord_less_real @ X @ one_one_real )
% 16.19/9.07 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ artanh_real ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10028_isCont__inverse__function,axiom,
% 16.19/9.07 ! [D: real,X: real,G: real > real,F: real > real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ D )
% 16.19/9.07 => ( ! [Z5: real] :
% 16.19/9.07 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z5 @ X ) ) @ D )
% 16.19/9.07 => ( ( G @ ( F @ Z5 ) )
% 16.19/9.07 = Z5 ) )
% 16.19/9.07 => ( ! [Z5: real] :
% 16.19/9.07 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z5 @ X ) ) @ D )
% 16.19/9.07 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) @ F ) )
% 16.19/9.07 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10029_GMVT_H,axiom,
% 16.19/9.07 ! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F4: real > real] :
% 16.19/9.07 ( ( ord_less_real @ A @ B )
% 16.19/9.07 => ( ! [Z5: real] :
% 16.19/9.07 ( ( ord_less_eq_real @ A @ Z5 )
% 16.19/9.07 => ( ( ord_less_eq_real @ Z5 @ B )
% 16.19/9.07 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) @ F ) ) )
% 16.19/9.07 => ( ! [Z5: real] :
% 16.19/9.07 ( ( ord_less_eq_real @ A @ Z5 )
% 16.19/9.07 => ( ( ord_less_eq_real @ Z5 @ B )
% 16.19/9.07 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) @ G ) ) )
% 16.19/9.07 => ( ! [Z5: real] :
% 16.19/9.07 ( ( ord_less_real @ A @ Z5 )
% 16.19/9.07 => ( ( ord_less_real @ Z5 @ B )
% 16.19/9.07 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z5 ) @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) ) ) )
% 16.19/9.07 => ( ! [Z5: real] :
% 16.19/9.07 ( ( ord_less_real @ A @ Z5 )
% 16.19/9.07 => ( ( ord_less_real @ Z5 @ B )
% 16.19/9.07 => ( has_fi5821293074295781190e_real @ F @ ( F4 @ Z5 ) @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) ) ) )
% 16.19/9.07 => ? [C3: real] :
% 16.19/9.07 ( ( ord_less_real @ A @ C3 )
% 16.19/9.07 & ( ord_less_real @ C3 @ B )
% 16.19/9.07 & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C3 ) )
% 16.19/9.07 = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F4 @ C3 ) ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10030_summable__Leibniz_I2_J,axiom,
% 16.19/9.07 ! [A: nat > real] :
% 16.19/9.07 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 16.19/9.07 => ( ( topolo6980174941875973593q_real @ A )
% 16.19/9.07 => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
% 16.19/9.07 => ! [N9: nat] :
% 16.19/9.07 ( member_real
% 16.19/9.07 @ ( suminf_real
% 16.19/9.07 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
% 16.19/9.07 @ ( set_or1222579329274155063t_real
% 16.19/9.07 @ ( groups6591440286371151544t_real
% 16.19/9.07 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 16.19/9.07 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) )
% 16.19/9.07 @ ( groups6591440286371151544t_real
% 16.19/9.07 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 16.19/9.07 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10031_summable__Leibniz_I3_J,axiom,
% 16.19/9.07 ! [A: nat > real] :
% 16.19/9.07 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 16.19/9.07 => ( ( topolo6980174941875973593q_real @ A )
% 16.19/9.07 => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
% 16.19/9.07 => ! [N9: nat] :
% 16.19/9.07 ( member_real
% 16.19/9.07 @ ( suminf_real
% 16.19/9.07 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
% 16.19/9.07 @ ( set_or1222579329274155063t_real
% 16.19/9.07 @ ( groups6591440286371151544t_real
% 16.19/9.07 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 16.19/9.07 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) )
% 16.19/9.07 @ ( groups6591440286371151544t_real
% 16.19/9.07 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 16.19/9.07 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10032_filterlim__Suc,axiom,
% 16.19/9.07 filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10033_mult__nat__right__at__top,axiom,
% 16.19/9.07 ! [C: nat] :
% 16.19/9.07 ( ( ord_less_nat @ zero_zero_nat @ C )
% 16.19/9.07 => ( filterlim_nat_nat
% 16.19/9.07 @ ^ [X2: nat] : ( times_times_nat @ X2 @ C )
% 16.19/9.07 @ at_top_nat
% 16.19/9.07 @ at_top_nat ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10034_mult__nat__left__at__top,axiom,
% 16.19/9.07 ! [C: nat] :
% 16.19/9.07 ( ( ord_less_nat @ zero_zero_nat @ C )
% 16.19/9.07 => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10035_monoseq__convergent,axiom,
% 16.19/9.07 ! [X8: nat > real,B3: real] :
% 16.19/9.07 ( ( topolo6980174941875973593q_real @ X8 )
% 16.19/9.07 => ( ! [I4: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X8 @ I4 ) ) @ B3 )
% 16.19/9.07 => ~ ! [L6: real] :
% 16.19/9.07 ~ ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10036_LIMSEQ__root,axiom,
% 16.19/9.07 ( filterlim_nat_real
% 16.19/9.07 @ ^ [N: nat] : ( root @ N @ ( semiri5074537144036343181t_real @ N ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ one_one_real )
% 16.19/9.07 @ at_top_nat ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10037_nested__sequence__unique,axiom,
% 16.19/9.07 ! [F: nat > real,G: nat > real] :
% 16.19/9.07 ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 16.19/9.07 => ( ! [N4: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N4 ) ) @ ( G @ N4 ) )
% 16.19/9.07 => ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( G @ N4 ) )
% 16.19/9.07 => ( ( filterlim_nat_real
% 16.19/9.07 @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( G @ N ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 16.19/9.07 @ at_top_nat )
% 16.19/9.07 => ? [L4: real] :
% 16.19/9.07 ( ! [N9: nat] : ( ord_less_eq_real @ ( F @ N9 ) @ L4 )
% 16.19/9.07 & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat )
% 16.19/9.07 & ! [N9: nat] : ( ord_less_eq_real @ L4 @ ( G @ N9 ) )
% 16.19/9.07 & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10038_LIMSEQ__inverse__zero,axiom,
% 16.19/9.07 ! [X8: nat > real] :
% 16.19/9.07 ( ! [R3: real] :
% 16.19/9.07 ? [N7: nat] :
% 16.19/9.07 ! [N4: nat] :
% 16.19/9.07 ( ( ord_less_eq_nat @ N7 @ N4 )
% 16.19/9.07 => ( ord_less_real @ R3 @ ( X8 @ N4 ) ) )
% 16.19/9.07 => ( filterlim_nat_real
% 16.19/9.07 @ ^ [N: nat] : ( inverse_inverse_real @ ( X8 @ N ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 16.19/9.07 @ at_top_nat ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10039_lim__inverse__n_H,axiom,
% 16.19/9.07 ( filterlim_nat_real
% 16.19/9.07 @ ^ [N: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 16.19/9.07 @ at_top_nat ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10040_LIMSEQ__root__const,axiom,
% 16.19/9.07 ! [C: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ C )
% 16.19/9.07 => ( filterlim_nat_real
% 16.19/9.07 @ ^ [N: nat] : ( root @ N @ C )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ one_one_real )
% 16.19/9.07 @ at_top_nat ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10041_LIMSEQ__inverse__real__of__nat,axiom,
% 16.19/9.07 ( filterlim_nat_real
% 16.19/9.07 @ ^ [N: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 16.19/9.07 @ at_top_nat ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10042_LIMSEQ__inverse__real__of__nat__add,axiom,
% 16.19/9.07 ! [R: real] :
% 16.19/9.07 ( filterlim_nat_real
% 16.19/9.07 @ ^ [N: nat] : ( plus_plus_real @ R @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ R )
% 16.19/9.07 @ at_top_nat ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10043_increasing__LIMSEQ,axiom,
% 16.19/9.07 ! [F: nat > real,L2: real] :
% 16.19/9.07 ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 16.19/9.07 => ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ L2 )
% 16.19/9.07 => ( ! [E2: real] :
% 16.19/9.07 ( ( ord_less_real @ zero_zero_real @ E2 )
% 16.19/9.07 => ? [N9: nat] : ( ord_less_eq_real @ L2 @ ( plus_plus_real @ ( F @ N9 ) @ E2 ) ) )
% 16.19/9.07 => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10044_LIMSEQ__realpow__zero,axiom,
% 16.19/9.07 ! [X: real] :
% 16.19/9.07 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.07 => ( ( ord_less_real @ X @ one_one_real )
% 16.19/9.07 => ( filterlim_nat_real @ ( power_power_real @ X ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10045_LIMSEQ__divide__realpow__zero,axiom,
% 16.19/9.07 ! [X: real,A: real] :
% 16.19/9.07 ( ( ord_less_real @ one_one_real @ X )
% 16.19/9.07 => ( filterlim_nat_real
% 16.19/9.07 @ ^ [N: nat] : ( divide_divide_real @ A @ ( power_power_real @ X @ N ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 16.19/9.07 @ at_top_nat ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10046_LIMSEQ__abs__realpow__zero2,axiom,
% 16.19/9.07 ! [C: real] :
% 16.19/9.07 ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 16.19/9.07 => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10047_LIMSEQ__abs__realpow__zero,axiom,
% 16.19/9.07 ! [C: real] :
% 16.19/9.07 ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 16.19/9.07 => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10048_LIMSEQ__inverse__realpow__zero,axiom,
% 16.19/9.07 ! [X: real] :
% 16.19/9.07 ( ( ord_less_real @ one_one_real @ X )
% 16.19/9.07 => ( filterlim_nat_real
% 16.19/9.07 @ ^ [N: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 16.19/9.07 @ at_top_nat ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10049_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
% 16.19/9.07 ! [R: real] :
% 16.19/9.07 ( filterlim_nat_real
% 16.19/9.07 @ ^ [N: nat] : ( plus_plus_real @ R @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ R )
% 16.19/9.07 @ at_top_nat ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10050_tendsto__exp__limit__sequentially,axiom,
% 16.19/9.07 ! [X: real] :
% 16.19/9.07 ( filterlim_nat_real
% 16.19/9.07 @ ^ [N: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 16.19/9.07 @ at_top_nat ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10051_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
% 16.19/9.07 ! [R: real] :
% 16.19/9.07 ( filterlim_nat_real
% 16.19/9.07 @ ^ [N: nat] : ( times_times_real @ R @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ R )
% 16.19/9.07 @ at_top_nat ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10052_summable__Leibniz_I1_J,axiom,
% 16.19/9.07 ! [A: nat > real] :
% 16.19/9.07 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 16.19/9.07 => ( ( topolo6980174941875973593q_real @ A )
% 16.19/9.07 => ( summable_real
% 16.19/9.07 @ ^ [N: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( A @ N ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10053_summable,axiom,
% 16.19/9.07 ! [A: nat > real] :
% 16.19/9.07 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 16.19/9.07 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 16.19/9.07 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 16.19/9.07 => ( summable_real
% 16.19/9.07 @ ^ [N: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( A @ N ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10054_cos__diff__limit__1,axiom,
% 16.19/9.07 ! [Theta: nat > real,Theta2: real] :
% 16.19/9.07 ( ( filterlim_nat_real
% 16.19/9.07 @ ^ [J3: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J3 ) @ Theta2 ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ one_one_real )
% 16.19/9.07 @ at_top_nat )
% 16.19/9.07 => ~ ! [K2: nat > int] :
% 16.19/9.07 ~ ( filterlim_nat_real
% 16.19/9.07 @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ Theta2 )
% 16.19/9.07 @ at_top_nat ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10055_cos__limit__1,axiom,
% 16.19/9.07 ! [Theta: nat > real] :
% 16.19/9.07 ( ( filterlim_nat_real
% 16.19/9.07 @ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ one_one_real )
% 16.19/9.07 @ at_top_nat )
% 16.19/9.07 => ? [K2: nat > int] :
% 16.19/9.07 ( filterlim_nat_real
% 16.19/9.07 @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 16.19/9.07 @ at_top_nat ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10056_summable__Leibniz_I4_J,axiom,
% 16.19/9.07 ! [A: nat > real] :
% 16.19/9.07 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 16.19/9.07 => ( ( topolo6980174941875973593q_real @ A )
% 16.19/9.07 => ( filterlim_nat_real
% 16.19/9.07 @ ^ [N: nat] :
% 16.19/9.07 ( groups6591440286371151544t_real
% 16.19/9.07 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 16.19/9.07 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real
% 16.19/9.07 @ ( suminf_real
% 16.19/9.07 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
% 16.19/9.07 @ at_top_nat ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10057_zeroseq__arctan__series,axiom,
% 16.19/9.07 ! [X: real] :
% 16.19/9.07 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 16.19/9.07 => ( filterlim_nat_real
% 16.19/9.07 @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 16.19/9.07 @ at_top_nat ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10058_summable__Leibniz_H_I3_J,axiom,
% 16.19/9.07 ! [A: nat > real] :
% 16.19/9.07 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 16.19/9.07 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 16.19/9.07 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 16.19/9.07 => ( filterlim_nat_real
% 16.19/9.07 @ ^ [N: nat] :
% 16.19/9.07 ( groups6591440286371151544t_real
% 16.19/9.07 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 16.19/9.07 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real
% 16.19/9.07 @ ( suminf_real
% 16.19/9.07 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
% 16.19/9.07 @ at_top_nat ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10059_summable__Leibniz_H_I2_J,axiom,
% 16.19/9.07 ! [A: nat > real,N2: nat] :
% 16.19/9.07 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 16.19/9.07 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 16.19/9.07 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 16.19/9.07 => ( ord_less_eq_real
% 16.19/9.07 @ ( groups6591440286371151544t_real
% 16.19/9.07 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 16.19/9.07 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 16.19/9.07 @ ( suminf_real
% 16.19/9.07 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10060_sums__alternating__upper__lower,axiom,
% 16.19/9.07 ! [A: nat > real] :
% 16.19/9.07 ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 16.19/9.07 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 16.19/9.07 => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 16.19/9.07 => ? [L4: real] :
% 16.19/9.07 ( ! [N9: nat] :
% 16.19/9.07 ( ord_less_eq_real
% 16.19/9.07 @ ( groups6591440286371151544t_real
% 16.19/9.07 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 16.19/9.07 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) ) )
% 16.19/9.07 @ L4 )
% 16.19/9.07 & ( filterlim_nat_real
% 16.19/9.07 @ ^ [N: nat] :
% 16.19/9.07 ( groups6591440286371151544t_real
% 16.19/9.07 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 16.19/9.07 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ L4 )
% 16.19/9.07 @ at_top_nat )
% 16.19/9.07 & ! [N9: nat] :
% 16.19/9.07 ( ord_less_eq_real @ L4
% 16.19/9.07 @ ( groups6591440286371151544t_real
% 16.19/9.07 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 16.19/9.07 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N9 ) @ one_one_nat ) ) ) )
% 16.19/9.07 & ( filterlim_nat_real
% 16.19/9.07 @ ^ [N: nat] :
% 16.19/9.07 ( groups6591440286371151544t_real
% 16.19/9.07 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 16.19/9.07 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ L4 )
% 16.19/9.07 @ at_top_nat ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10061_summable__Leibniz_I5_J,axiom,
% 16.19/9.07 ! [A: nat > real] :
% 16.19/9.07 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 16.19/9.07 => ( ( topolo6980174941875973593q_real @ A )
% 16.19/9.07 => ( filterlim_nat_real
% 16.19/9.07 @ ^ [N: nat] :
% 16.19/9.07 ( groups6591440286371151544t_real
% 16.19/9.07 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 16.19/9.07 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real
% 16.19/9.07 @ ( suminf_real
% 16.19/9.07 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
% 16.19/9.07 @ at_top_nat ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10062_summable__Leibniz_H_I4_J,axiom,
% 16.19/9.07 ! [A: nat > real,N2: nat] :
% 16.19/9.07 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 16.19/9.07 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 16.19/9.07 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 16.19/9.07 => ( ord_less_eq_real
% 16.19/9.07 @ ( suminf_real
% 16.19/9.07 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
% 16.19/9.07 @ ( groups6591440286371151544t_real
% 16.19/9.07 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 16.19/9.07 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10063_summable__Leibniz_H_I5_J,axiom,
% 16.19/9.07 ! [A: nat > real] :
% 16.19/9.07 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 16.19/9.07 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 16.19/9.07 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 16.19/9.07 => ( filterlim_nat_real
% 16.19/9.07 @ ^ [N: nat] :
% 16.19/9.07 ( groups6591440286371151544t_real
% 16.19/9.07 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
% 16.19/9.07 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real
% 16.19/9.07 @ ( suminf_real
% 16.19/9.07 @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
% 16.19/9.07 @ at_top_nat ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10064_real__bounded__linear,axiom,
% 16.19/9.07 ( real_V5970128139526366754l_real
% 16.19/9.07 = ( ^ [F3: real > real] :
% 16.19/9.07 ? [C4: real] :
% 16.19/9.07 ( F3
% 16.19/9.07 = ( ^ [X2: real] : ( times_times_real @ X2 @ C4 ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10065_tendsto__exp__limit__at__right,axiom,
% 16.19/9.07 ! [X: real] :
% 16.19/9.07 ( filterlim_real_real
% 16.19/9.07 @ ^ [Y2: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X @ Y2 ) ) @ ( divide_divide_real @ one_one_real @ Y2 ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10066_dist__real__def,axiom,
% 16.19/9.07 ( real_V975177566351809787t_real
% 16.19/9.07 = ( ^ [X2: real,Y2: real] : ( abs_abs_real @ ( minus_minus_real @ X2 @ Y2 ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10067_dist__complex__def,axiom,
% 16.19/9.07 ( real_V3694042436643373181omplex
% 16.19/9.07 = ( ^ [X2: complex,Y2: complex] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y2 ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10068_tendsto__arcosh__at__left__1,axiom,
% 16.19/9.07 filterlim_real_real @ arcosh_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5849166863359141190n_real @ one_one_real ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10069_INT__greaterThan__UNIV,axiom,
% 16.19/9.07 ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ set_or1210151606488870762an_nat @ top_top_set_nat ) )
% 16.19/9.07 = bot_bot_set_nat ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10070_greaterThan__0,axiom,
% 16.19/9.07 ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
% 16.19/9.07 = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10071_greaterThan__Suc,axiom,
% 16.19/9.07 ! [K: nat] :
% 16.19/9.07 ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
% 16.19/9.07 = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10072_filterlim__tan__at__right,axiom,
% 16.19/9.07 filterlim_real_real @ tan_real @ at_bot_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10073_tendsto__arctan__at__bot,axiom,
% 16.19/9.07 filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ at_bot_real ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10074_tanh__real__at__bot,axiom,
% 16.19/9.07 filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ one_one_real ) ) @ at_bot_real ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10075_ln__at__0,axiom,
% 16.19/9.07 filterlim_real_real @ ln_ln_real @ at_bot_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10076_filterlim__inverse__at__bot__neg,axiom,
% 16.19/9.07 filterlim_real_real @ inverse_inverse_real @ at_bot_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5984915006950818249n_real @ zero_zero_real ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10077_artanh__real__at__right__1,axiom,
% 16.19/9.07 filterlim_real_real @ artanh_real @ at_bot_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ one_one_real ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10078_DERIV__pos__imp__increasing__at__bot,axiom,
% 16.19/9.07 ! [B: real,F: real > real,Flim: real] :
% 16.19/9.07 ( ! [X4: real] :
% 16.19/9.07 ( ( ord_less_eq_real @ X4 @ B )
% 16.19/9.07 => ? [Y5: real] :
% 16.19/9.07 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 16.19/9.07 & ( ord_less_real @ zero_zero_real @ Y5 ) ) )
% 16.19/9.07 => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
% 16.19/9.07 => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10079_filterlim__pow__at__bot__odd,axiom,
% 16.19/9.07 ! [N2: nat,F: real > real,F5: filter_real] :
% 16.19/9.07 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.07 => ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
% 16.19/9.07 => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.07 => ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( power_power_real @ ( F @ X2 ) @ N2 )
% 16.19/9.07 @ at_bot_real
% 16.19/9.07 @ F5 ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10080_Gcd__eq__Max,axiom,
% 16.19/9.07 ! [M7: set_nat] :
% 16.19/9.07 ( ( finite_finite_nat @ M7 )
% 16.19/9.07 => ( ( M7 != bot_bot_set_nat )
% 16.19/9.07 => ( ~ ( member_nat @ zero_zero_nat @ M7 )
% 16.19/9.07 => ( ( gcd_Gcd_nat @ M7 )
% 16.19/9.07 = ( lattic8265883725875713057ax_nat
% 16.19/9.07 @ ( comple7806235888213564991et_nat
% 16.19/9.07 @ ( image_nat_set_nat
% 16.19/9.07 @ ^ [M6: nat] :
% 16.19/9.07 ( collect_nat
% 16.19/9.07 @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M6 ) )
% 16.19/9.07 @ M7 ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10081_Max__divisors__self__nat,axiom,
% 16.19/9.07 ! [N2: nat] :
% 16.19/9.07 ( ( N2 != zero_zero_nat )
% 16.19/9.07 => ( ( lattic8265883725875713057ax_nat
% 16.19/9.07 @ ( collect_nat
% 16.19/9.07 @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ N2 ) ) )
% 16.19/9.07 = N2 ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10082_card__le__Suc__Max,axiom,
% 16.19/9.07 ! [S: set_nat] :
% 16.19/9.07 ( ( finite_finite_nat @ S )
% 16.19/9.07 => ( ord_less_eq_nat @ ( finite_card_nat @ S ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10083_divide__nat__def,axiom,
% 16.19/9.07 ( divide_divide_nat
% 16.19/9.07 = ( ^ [M6: nat,N: nat] :
% 16.19/9.07 ( if_nat @ ( N = zero_zero_nat ) @ zero_zero_nat
% 16.19/9.07 @ ( lattic8265883725875713057ax_nat
% 16.19/9.07 @ ( collect_nat
% 16.19/9.07 @ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N ) @ M6 ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10084_gcd__is__Max__divisors__nat,axiom,
% 16.19/9.07 ! [N2: nat,M: nat] :
% 16.19/9.07 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.07 => ( ( gcd_gcd_nat @ M @ N2 )
% 16.19/9.07 = ( lattic8265883725875713057ax_nat
% 16.19/9.07 @ ( collect_nat
% 16.19/9.07 @ ^ [D2: nat] :
% 16.19/9.07 ( ( dvd_dvd_nat @ D2 @ M )
% 16.19/9.07 & ( dvd_dvd_nat @ D2 @ N2 ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10085_filterlim__pow__at__bot__even,axiom,
% 16.19/9.07 ! [N2: nat,F: real > real,F5: filter_real] :
% 16.19/9.07 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.07 => ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
% 16.19/9.07 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 16.19/9.07 => ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( power_power_real @ ( F @ X2 ) @ N2 )
% 16.19/9.07 @ at_top_real
% 16.19/9.07 @ F5 ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10086_eventually__sequentially__Suc,axiom,
% 16.19/9.07 ! [P: nat > $o] :
% 16.19/9.07 ( ( eventually_nat
% 16.19/9.07 @ ^ [I3: nat] : ( P @ ( suc @ I3 ) )
% 16.19/9.07 @ at_top_nat )
% 16.19/9.07 = ( eventually_nat @ P @ at_top_nat ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10087_eventually__sequentially__seg,axiom,
% 16.19/9.07 ! [P: nat > $o,K: nat] :
% 16.19/9.07 ( ( eventually_nat
% 16.19/9.07 @ ^ [N: nat] : ( P @ ( plus_plus_nat @ N @ K ) )
% 16.19/9.07 @ at_top_nat )
% 16.19/9.07 = ( eventually_nat @ P @ at_top_nat ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10088_Max__divisors__self__int,axiom,
% 16.19/9.07 ! [N2: int] :
% 16.19/9.07 ( ( N2 != zero_zero_int )
% 16.19/9.07 => ( ( lattic8263393255366662781ax_int
% 16.19/9.07 @ ( collect_int
% 16.19/9.07 @ ^ [D2: int] : ( dvd_dvd_int @ D2 @ N2 ) ) )
% 16.19/9.07 = ( abs_abs_int @ N2 ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10089_lhopital__at__top__at__top,axiom,
% 16.19/9.07 ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
% 16.19/9.07 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 16.19/9.07 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 16.19/9.07 => ( ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 16.19/9.07 @ at_top_real
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 16.19/9.07 => ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 16.19/9.07 @ at_top_real
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10090_ln__at__top,axiom,
% 16.19/9.07 filterlim_real_real @ ln_ln_real @ at_top_real @ at_top_real ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10091_sqrt__at__top,axiom,
% 16.19/9.07 filterlim_real_real @ sqrt @ at_top_real @ at_top_real ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10092_exp__at__top,axiom,
% 16.19/9.07 filterlim_real_real @ exp_real @ at_top_real @ at_top_real ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10093_eventually__False__sequentially,axiom,
% 16.19/9.07 ~ ( eventually_nat
% 16.19/9.07 @ ^ [N: nat] : $false
% 16.19/9.07 @ at_top_nat ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10094_sequentially__offset,axiom,
% 16.19/9.07 ! [P: nat > $o,K: nat] :
% 16.19/9.07 ( ( eventually_nat @ P @ at_top_nat )
% 16.19/9.07 => ( eventually_nat
% 16.19/9.07 @ ^ [I3: nat] : ( P @ ( plus_plus_nat @ I3 @ K ) )
% 16.19/9.07 @ at_top_nat ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10095_eventually__sequentiallyI,axiom,
% 16.19/9.07 ! [C: nat,P: nat > $o] :
% 16.19/9.07 ( ! [X4: nat] :
% 16.19/9.07 ( ( ord_less_eq_nat @ C @ X4 )
% 16.19/9.07 => ( P @ X4 ) )
% 16.19/9.07 => ( eventually_nat @ P @ at_top_nat ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10096_eventually__sequentially,axiom,
% 16.19/9.07 ! [P: nat > $o] :
% 16.19/9.07 ( ( eventually_nat @ P @ at_top_nat )
% 16.19/9.07 = ( ? [N6: nat] :
% 16.19/9.07 ! [N: nat] :
% 16.19/9.07 ( ( ord_less_eq_nat @ N6 @ N )
% 16.19/9.07 => ( P @ N ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10097_le__sequentially,axiom,
% 16.19/9.07 ! [F5: filter_nat] :
% 16.19/9.07 ( ( ord_le2510731241096832064er_nat @ F5 @ at_top_nat )
% 16.19/9.07 = ( ! [N6: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N6 ) @ F5 ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10098_eventually__at__right__to__top,axiom,
% 16.19/9.07 ! [P: real > $o] :
% 16.19/9.07 ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 16.19/9.07 = ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( P @ ( inverse_inverse_real @ X2 ) )
% 16.19/9.07 @ at_top_real ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10099_eventually__at__top__to__right,axiom,
% 16.19/9.07 ! [P: real > $o] :
% 16.19/9.07 ( ( eventually_real @ P @ at_top_real )
% 16.19/9.07 = ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( P @ ( inverse_inverse_real @ X2 ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10100_lhopital__right__at__top__at__top,axiom,
% 16.19/9.07 ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
% 16.19/9.07 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 16.19/9.07 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 16.19/9.07 => ( ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 16.19/9.07 @ at_top_real
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 16.19/9.07 => ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 16.19/9.07 @ at_top_real
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10101_lhopital__at__top__at__bot,axiom,
% 16.19/9.07 ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
% 16.19/9.07 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 16.19/9.07 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 16.19/9.07 => ( ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 16.19/9.07 @ at_bot_real
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 16.19/9.07 => ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 16.19/9.07 @ at_bot_real
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10102_lhopital__left__at__top__at__top,axiom,
% 16.19/9.07 ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
% 16.19/9.07 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 16.19/9.07 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 16.19/9.07 => ( ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 16.19/9.07 @ at_top_real
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 16.19/9.07 => ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 16.19/9.07 @ at_top_real
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10103_lhospital__at__top__at__top,axiom,
% 16.19/9.07 ! [G: real > real,G2: real > real,F: real > real,F4: real > real,X: real] :
% 16.19/9.07 ( ( filterlim_real_real @ G @ at_top_real @ at_top_real )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] :
% 16.19/9.07 ( ( G2 @ X2 )
% 16.19/9.07 != zero_zero_real )
% 16.19/9.07 @ at_top_real )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ at_top_real )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ at_top_real )
% 16.19/9.07 => ( ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ X )
% 16.19/9.07 @ at_top_real )
% 16.19/9.07 => ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ X )
% 16.19/9.07 @ at_top_real ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10104_lhopital__at__top,axiom,
% 16.19/9.07 ! [G: real > real,X: real,G2: real > real,F: real > real,F4: real > real,Y: real] :
% 16.19/9.07 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] :
% 16.19/9.07 ( ( G2 @ X2 )
% 16.19/9.07 != zero_zero_real )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 16.19/9.07 => ( ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ Y )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 16.19/9.07 => ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ Y )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10105_filterlim__real__sequentially,axiom,
% 16.19/9.07 filterlim_nat_real @ semiri5074537144036343181t_real @ at_top_real @ at_top_nat ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10106_filterlim__uminus__at__top__at__bot,axiom,
% 16.19/9.07 filterlim_real_real @ uminus_uminus_real @ at_top_real @ at_bot_real ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10107_filterlim__uminus__at__bot__at__top,axiom,
% 16.19/9.07 filterlim_real_real @ uminus_uminus_real @ at_bot_real @ at_top_real ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10108_lhopital__right__at__top__at__bot,axiom,
% 16.19/9.07 ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
% 16.19/9.07 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 16.19/9.07 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 16.19/9.07 => ( ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 16.19/9.07 @ at_bot_real
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 16.19/9.07 => ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 16.19/9.07 @ at_bot_real
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10109_lhopital__left__at__top__at__bot,axiom,
% 16.19/9.07 ! [F: real > real,A: real,G: real > real,F4: real > real,G2: real > real] :
% 16.19/9.07 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 16.19/9.07 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 16.19/9.07 => ( ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 16.19/9.07 @ at_bot_real
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 16.19/9.07 => ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 16.19/9.07 @ at_bot_real
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10110_gcd__is__Max__divisors__int,axiom,
% 16.19/9.07 ! [N2: int,M: int] :
% 16.19/9.07 ( ( N2 != zero_zero_int )
% 16.19/9.07 => ( ( gcd_gcd_int @ M @ N2 )
% 16.19/9.07 = ( lattic8263393255366662781ax_int
% 16.19/9.07 @ ( collect_int
% 16.19/9.07 @ ^ [D2: int] :
% 16.19/9.07 ( ( dvd_dvd_int @ D2 @ M )
% 16.19/9.07 & ( dvd_dvd_int @ D2 @ N2 ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10111_lhopital__right__0__at__top,axiom,
% 16.19/9.07 ! [G: real > real,G2: real > real,F: real > real,F4: real > real,X: real] :
% 16.19/9.07 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] :
% 16.19/9.07 ( ( G2 @ X2 )
% 16.19/9.07 != zero_zero_real )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 16.19/9.07 => ( ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ X )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 16.19/9.07 => ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ X )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10112_lhopital__right__at__top,axiom,
% 16.19/9.07 ! [G: real > real,X: real,G2: real > real,F: real > real,F4: real > real,Y: real] :
% 16.19/9.07 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] :
% 16.19/9.07 ( ( G2 @ X2 )
% 16.19/9.07 != zero_zero_real )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 16.19/9.07 => ( ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ Y )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 16.19/9.07 => ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ Y )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10113_lhopital__left__at__top,axiom,
% 16.19/9.07 ! [G: real > real,X: real,G2: real > real,F: real > real,F4: real > real,Y: real] :
% 16.19/9.07 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] :
% 16.19/9.07 ( ( G2 @ X2 )
% 16.19/9.07 != zero_zero_real )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 16.19/9.07 => ( ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ Y )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 16.19/9.07 => ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ Y )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10114_eventually__at__right__to__0,axiom,
% 16.19/9.07 ! [P: real > $o,A: real] :
% 16.19/9.07 ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 16.19/9.07 = ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( P @ ( plus_plus_real @ X2 @ A ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10115_tanh__real__at__top,axiom,
% 16.19/9.07 filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ one_one_real ) @ at_top_real ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10116_eventually__at__left__to__right,axiom,
% 16.19/9.07 ! [P: real > $o,A: real] :
% 16.19/9.07 ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 16.19/9.07 = ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( P @ ( uminus_uminus_real @ X2 ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ A ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ A ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10117_eventually__at__right__real,axiom,
% 16.19/9.07 ! [A: real,B: real] :
% 16.19/9.07 ( ( ord_less_real @ A @ B )
% 16.19/9.07 => ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( member_real @ X2 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10118_eventually__at__left__real,axiom,
% 16.19/9.07 ! [B: real,A: real] :
% 16.19/9.07 ( ( ord_less_real @ B @ A )
% 16.19/9.07 => ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( member_real @ X2 @ ( set_or1633881224788618240n_real @ B @ A ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10119_ln__x__over__x__tendsto__0,axiom,
% 16.19/9.07 ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( ln_ln_real @ X2 ) @ X2 )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 16.19/9.07 @ at_top_real ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10120_artanh__real__at__left__1,axiom,
% 16.19/9.07 filterlim_real_real @ artanh_real @ at_top_real @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5984915006950818249n_real @ one_one_real ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10121_filterlim__inverse__at__top__right,axiom,
% 16.19/9.07 filterlim_real_real @ inverse_inverse_real @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10122_filterlim__inverse__at__right__top,axiom,
% 16.19/9.07 filterlim_real_real @ inverse_inverse_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) @ at_top_real ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10123_tendsto__power__div__exp__0,axiom,
% 16.19/9.07 ! [K: nat] :
% 16.19/9.07 ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( power_power_real @ X2 @ K ) @ ( exp_real @ X2 ) )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 16.19/9.07 @ at_top_real ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10124_lhopital,axiom,
% 16.19/9.07 ! [F: real > real,X: real,G: real > real,G2: real > real,F4: real > real,F5: filter_real] :
% 16.19/9.07 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 16.19/9.07 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] :
% 16.19/9.07 ( ( G @ X2 )
% 16.19/9.07 != zero_zero_real )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] :
% 16.19/9.07 ( ( G2 @ X2 )
% 16.19/9.07 != zero_zero_real )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 16.19/9.07 => ( ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 16.19/9.07 @ F5
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 16.19/9.07 => ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 16.19/9.07 @ F5
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10125_tendsto__exp__limit__at__top,axiom,
% 16.19/9.07 ! [X: real] :
% 16.19/9.07 ( filterlim_real_real
% 16.19/9.07 @ ^ [Y2: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ Y2 ) ) @ Y2 )
% 16.19/9.07 @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 16.19/9.07 @ at_top_real ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10126_DERIV__neg__imp__decreasing__at__top,axiom,
% 16.19/9.07 ! [B: real,F: real > real,Flim: real] :
% 16.19/9.07 ( ! [X4: real] :
% 16.19/9.07 ( ( ord_less_eq_real @ B @ X4 )
% 16.19/9.07 => ? [Y5: real] :
% 16.19/9.07 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 16.19/9.07 & ( ord_less_real @ Y5 @ zero_zero_real ) ) )
% 16.19/9.07 => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
% 16.19/9.07 => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10127_tendsto__arctan__at__top,axiom,
% 16.19/9.07 filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ at_top_real ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10128_lhopital__right__0,axiom,
% 16.19/9.07 ! [F0: real > real,G0: real > real,G2: real > real,F4: real > real,F5: filter_real] :
% 16.19/9.07 ( ( filterlim_real_real @ F0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 16.19/9.07 => ( ( filterlim_real_real @ G0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] :
% 16.19/9.07 ( ( G0 @ X2 )
% 16.19/9.07 != zero_zero_real )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] :
% 16.19/9.07 ( ( G2 @ X2 )
% 16.19/9.07 != zero_zero_real )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F0 @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G0 @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 16.19/9.07 => ( ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 16.19/9.07 @ F5
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 16.19/9.07 => ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F0 @ X2 ) @ ( G0 @ X2 ) )
% 16.19/9.07 @ F5
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10129_lhopital__right,axiom,
% 16.19/9.07 ! [F: real > real,X: real,G: real > real,G2: real > real,F4: real > real,F5: filter_real] :
% 16.19/9.07 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 16.19/9.07 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] :
% 16.19/9.07 ( ( G @ X2 )
% 16.19/9.07 != zero_zero_real )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] :
% 16.19/9.07 ( ( G2 @ X2 )
% 16.19/9.07 != zero_zero_real )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 16.19/9.07 => ( ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 16.19/9.07 @ F5
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 16.19/9.07 => ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 16.19/9.07 @ F5
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10130_lhopital__left,axiom,
% 16.19/9.07 ! [F: real > real,X: real,G: real > real,G2: real > real,F4: real > real,F5: filter_real] :
% 16.19/9.07 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 16.19/9.07 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] :
% 16.19/9.07 ( ( G @ X2 )
% 16.19/9.07 != zero_zero_real )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] :
% 16.19/9.07 ( ( G2 @ X2 )
% 16.19/9.07 != zero_zero_real )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 16.19/9.07 => ( ( eventually_real
% 16.19/9.07 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 16.19/9.07 => ( ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F4 @ X2 ) @ ( G2 @ X2 ) )
% 16.19/9.07 @ F5
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 16.19/9.07 => ( filterlim_real_real
% 16.19/9.07 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 16.19/9.07 @ F5
% 16.19/9.07 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10131_filterlim__tan__at__left,axiom,
% 16.19/9.07 filterlim_real_real @ tan_real @ at_top_real @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( set_or5984915006950818249n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10132_Bseq__realpow,axiom,
% 16.19/9.07 ! [X: real] :
% 16.19/9.07 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 16.19/9.07 => ( ( ord_less_eq_real @ X @ one_one_real )
% 16.19/9.07 => ( bfun_nat_real @ ( power_power_real @ X ) @ at_top_nat ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10133_card__greaterThanAtMost,axiom,
% 16.19/9.07 ! [L2: nat,U: nat] :
% 16.19/9.07 ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L2 @ U ) )
% 16.19/9.07 = ( minus_minus_nat @ U @ L2 ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10134_GreatestI__nat,axiom,
% 16.19/9.07 ! [P: nat > $o,K: nat,B: nat] :
% 16.19/9.07 ( ( P @ K )
% 16.19/9.07 => ( ! [Y4: nat] :
% 16.19/9.07 ( ( P @ Y4 )
% 16.19/9.07 => ( ord_less_eq_nat @ Y4 @ B ) )
% 16.19/9.07 => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10135_Greatest__le__nat,axiom,
% 16.19/9.07 ! [P: nat > $o,K: nat,B: nat] :
% 16.19/9.07 ( ( P @ K )
% 16.19/9.07 => ( ! [Y4: nat] :
% 16.19/9.07 ( ( P @ Y4 )
% 16.19/9.07 => ( ord_less_eq_nat @ Y4 @ B ) )
% 16.19/9.07 => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10136_GreatestI__ex__nat,axiom,
% 16.19/9.07 ! [P: nat > $o,B: nat] :
% 16.19/9.07 ( ? [X_12: nat] : ( P @ X_12 )
% 16.19/9.07 => ( ! [Y4: nat] :
% 16.19/9.07 ( ( P @ Y4 )
% 16.19/9.07 => ( ord_less_eq_nat @ Y4 @ B ) )
% 16.19/9.07 => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10137_atLeastSucAtMost__greaterThanAtMost,axiom,
% 16.19/9.07 ! [L2: nat,U: nat] :
% 16.19/9.07 ( ( set_or1269000886237332187st_nat @ ( suc @ L2 ) @ U )
% 16.19/9.07 = ( set_or6659071591806873216st_nat @ L2 @ U ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10138_card__greaterThanAtMost__int,axiom,
% 16.19/9.07 ! [L2: int,U: int] :
% 16.19/9.07 ( ( finite_card_int @ ( set_or6656581121297822940st_int @ L2 @ U ) )
% 16.19/9.07 = ( nat2 @ ( minus_minus_int @ U @ L2 ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10139_atLeast__Suc__greaterThan,axiom,
% 16.19/9.07 ! [K: nat] :
% 16.19/9.07 ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 16.19/9.07 = ( set_or1210151606488870762an_nat @ K ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10140_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
% 16.19/9.07 ! [L2: int,U: int] :
% 16.19/9.07 ( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L2 @ one_one_int ) @ U )
% 16.19/9.07 = ( set_or6656581121297822940st_int @ L2 @ U ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10141_decseq__bounded,axiom,
% 16.19/9.07 ! [X8: nat > real,B3: real] :
% 16.19/9.07 ( ( order_9091379641038594480t_real @ X8 )
% 16.19/9.07 => ( ! [I4: nat] : ( ord_less_eq_real @ B3 @ ( X8 @ I4 ) )
% 16.19/9.07 => ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10142_greaterThanAtMost__upto,axiom,
% 16.19/9.07 ( set_or6656581121297822940st_int
% 16.19/9.07 = ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10143_decseq__convergent,axiom,
% 16.19/9.07 ! [X8: nat > real,B3: real] :
% 16.19/9.07 ( ( order_9091379641038594480t_real @ X8 )
% 16.19/9.07 => ( ! [I4: nat] : ( ord_less_eq_real @ B3 @ ( X8 @ I4 ) )
% 16.19/9.07 => ~ ! [L6: real] :
% 16.19/9.07 ( ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
% 16.19/9.07 => ~ ! [I: nat] : ( ord_less_eq_real @ L6 @ ( X8 @ I ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10144_UN__atLeast__UNIV,axiom,
% 16.19/9.07 ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atLeast_nat @ top_top_set_nat ) )
% 16.19/9.07 = top_top_set_nat ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10145_atLeast__Suc,axiom,
% 16.19/9.07 ! [K: nat] :
% 16.19/9.07 ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 16.19/9.07 = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10146_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
% 16.19/9.07 ! [X: vEBT_VEBT,Xa2: nat] :
% 16.19/9.07 ( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
% 16.19/9.07 => ( ( ? [Uu2: $o,Uv2: $o] :
% 16.19/9.07 ( X
% 16.19/9.07 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 16.19/9.07 => ( Xa2 = one_one_nat ) )
% 16.19/9.07 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 16.19/9.07 => ( ( Deg2 = Xa2 )
% 16.19/9.07 & ! [X4: vEBT_VEBT] :
% 16.19/9.07 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 16.19/9.07 => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/9.07 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/9.07 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 16.19/9.07 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/9.07 & ( case_o184042715313410164at_nat
% 16.19/9.07 @ ( ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X3 )
% 16.19/9.07 & ! [X2: vEBT_VEBT] :
% 16.19/9.07 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 16.19/9.07 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 16.19/9.07 @ ( produc6081775807080527818_nat_o
% 16.19/9.07 @ ^ [Mi3: nat,Ma3: nat] :
% 16.19/9.07 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 16.19/9.07 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 16.19/9.07 & ! [I3: nat] :
% 16.19/9.07 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/9.07 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X3 ) )
% 16.19/9.07 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 16.19/9.07 & ( ( Mi3 = Ma3 )
% 16.19/9.07 => ! [X2: vEBT_VEBT] :
% 16.19/9.07 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 16.19/9.07 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 16.19/9.07 & ( ( Mi3 != Ma3 )
% 16.19/9.07 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 16.19/9.07 & ! [X2: nat] :
% 16.19/9.07 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 16.19/9.07 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
% 16.19/9.07 => ( ( ord_less_nat @ Mi3 @ X2 )
% 16.19/9.07 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
% 16.19/9.07 @ Mima ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10147_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
% 16.19/9.07 ! [X: vEBT_VEBT,Xa2: nat] :
% 16.19/9.07 ( ( vEBT_VEBT_valid @ X @ Xa2 )
% 16.19/9.07 => ( ( ? [Uu2: $o,Uv2: $o] :
% 16.19/9.07 ( X
% 16.19/9.07 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 16.19/9.07 => ( Xa2 != one_one_nat ) )
% 16.19/9.07 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 16.19/9.07 => ~ ( ( Deg2 = Xa2 )
% 16.19/9.07 & ! [X5: vEBT_VEBT] :
% 16.19/9.07 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 16.19/9.07 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/9.07 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/9.07 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 16.19/9.07 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/9.07 & ( case_o184042715313410164at_nat
% 16.19/9.07 @ ( ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X3 )
% 16.19/9.07 & ! [X2: vEBT_VEBT] :
% 16.19/9.07 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 16.19/9.07 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 16.19/9.07 @ ( produc6081775807080527818_nat_o
% 16.19/9.07 @ ^ [Mi3: nat,Ma3: nat] :
% 16.19/9.07 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 16.19/9.07 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 16.19/9.07 & ! [I3: nat] :
% 16.19/9.07 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/9.07 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X3 ) )
% 16.19/9.07 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 16.19/9.07 & ( ( Mi3 = Ma3 )
% 16.19/9.07 => ! [X2: vEBT_VEBT] :
% 16.19/9.07 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 16.19/9.07 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 16.19/9.07 & ( ( Mi3 != Ma3 )
% 16.19/9.07 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 16.19/9.07 & ! [X2: nat] :
% 16.19/9.07 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 16.19/9.07 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
% 16.19/9.07 => ( ( ord_less_nat @ Mi3 @ X2 )
% 16.19/9.07 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
% 16.19/9.07 @ Mima ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10148_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
% 16.19/9.07 ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
% 16.19/9.07 ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg4 )
% 16.19/9.07 = ( ( Deg = Deg4 )
% 16.19/9.07 & ! [X2: vEBT_VEBT] :
% 16.19/9.07 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 16.19/9.07 => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/9.07 & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/9.07 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 16.19/9.07 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/9.07 & ( case_o184042715313410164at_nat
% 16.19/9.07 @ ( ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X3 )
% 16.19/9.07 & ! [X2: vEBT_VEBT] :
% 16.19/9.07 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 16.19/9.07 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 16.19/9.07 @ ( produc6081775807080527818_nat_o
% 16.19/9.07 @ ^ [Mi3: nat,Ma3: nat] :
% 16.19/9.07 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 16.19/9.07 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 16.19/9.07 & ! [I3: nat] :
% 16.19/9.07 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/9.07 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X3 ) )
% 16.19/9.07 = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
% 16.19/9.07 & ( ( Mi3 = Ma3 )
% 16.19/9.07 => ! [X2: vEBT_VEBT] :
% 16.19/9.07 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 16.19/9.07 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 16.19/9.07 & ( ( Mi3 != Ma3 )
% 16.19/9.07 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 16.19/9.07 & ! [X2: nat] :
% 16.19/9.07 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 16.19/9.07 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X2 )
% 16.19/9.07 => ( ( ord_less_nat @ Mi3 @ X2 )
% 16.19/9.07 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
% 16.19/9.07 @ Mima2 ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10149_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
% 16.19/9.07 ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 16.19/9.07 ( ( ( vEBT_VEBT_valid @ X @ Xa2 )
% 16.19/9.07 = Y )
% 16.19/9.07 => ( ( ? [Uu2: $o,Uv2: $o] :
% 16.19/9.07 ( X
% 16.19/9.07 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 16.19/9.07 => ( Y
% 16.19/9.07 = ( Xa2 != one_one_nat ) ) )
% 16.19/9.07 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 16.19/9.07 => ( Y
% 16.19/9.07 = ( ~ ( ( Deg2 = Xa2 )
% 16.19/9.07 & ! [X2: vEBT_VEBT] :
% 16.19/9.07 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 16.19/9.07 => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/9.07 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/9.07 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 16.19/9.07 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/9.07 & ( case_o184042715313410164at_nat
% 16.19/9.07 @ ( ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X3 )
% 16.19/9.07 & ! [X2: vEBT_VEBT] :
% 16.19/9.07 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 16.19/9.07 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 16.19/9.07 @ ( produc6081775807080527818_nat_o
% 16.19/9.07 @ ^ [Mi3: nat,Ma3: nat] :
% 16.19/9.07 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 16.19/9.07 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 16.19/9.07 & ! [I3: nat] :
% 16.19/9.07 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/9.07 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X3 ) )
% 16.19/9.07 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 16.19/9.07 & ( ( Mi3 = Ma3 )
% 16.19/9.07 => ! [X2: vEBT_VEBT] :
% 16.19/9.07 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 16.19/9.07 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 16.19/9.07 & ( ( Mi3 != Ma3 )
% 16.19/9.07 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 16.19/9.07 & ! [X2: nat] :
% 16.19/9.07 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 16.19/9.07 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
% 16.19/9.07 => ( ( ord_less_nat @ Mi3 @ X2 )
% 16.19/9.07 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
% 16.19/9.07 @ Mima ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10150_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
% 16.19/9.07 ! [X: vEBT_VEBT,Xa2: nat] :
% 16.19/9.07 ( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
% 16.19/9.07 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 16.19/9.07 => ( ! [Uu2: $o,Uv2: $o] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 16.19/9.07 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
% 16.19/9.07 => ( Xa2 = one_one_nat ) ) )
% 16.19/9.07 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 16.19/9.07 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
% 16.19/9.07 => ( ( Deg2 = Xa2 )
% 16.19/9.07 & ! [X4: vEBT_VEBT] :
% 16.19/9.07 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 16.19/9.07 => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/9.07 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/9.07 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 16.19/9.07 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/9.07 & ( case_o184042715313410164at_nat
% 16.19/9.07 @ ( ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X3 )
% 16.19/9.07 & ! [X2: vEBT_VEBT] :
% 16.19/9.07 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 16.19/9.07 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 16.19/9.07 @ ( produc6081775807080527818_nat_o
% 16.19/9.07 @ ^ [Mi3: nat,Ma3: nat] :
% 16.19/9.07 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 16.19/9.07 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 16.19/9.07 & ! [I3: nat] :
% 16.19/9.07 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/9.07 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X3 ) )
% 16.19/9.07 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 16.19/9.07 & ( ( Mi3 = Ma3 )
% 16.19/9.07 => ! [X2: vEBT_VEBT] :
% 16.19/9.07 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 16.19/9.07 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 16.19/9.07 & ( ( Mi3 != Ma3 )
% 16.19/9.07 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 16.19/9.07 & ! [X2: nat] :
% 16.19/9.07 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 16.19/9.07 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
% 16.19/9.07 => ( ( ord_less_nat @ Mi3 @ X2 )
% 16.19/9.07 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
% 16.19/9.07 @ Mima ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10151_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
% 16.19/9.07 ! [X: vEBT_VEBT,Xa2: nat] :
% 16.19/9.07 ( ( vEBT_VEBT_valid @ X @ Xa2 )
% 16.19/9.07 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 16.19/9.07 => ( ! [Uu2: $o,Uv2: $o] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 16.19/9.07 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
% 16.19/9.07 => ( Xa2 != one_one_nat ) ) )
% 16.19/9.07 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 16.19/9.07 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
% 16.19/9.07 => ~ ( ( Deg2 = Xa2 )
% 16.19/9.07 & ! [X5: vEBT_VEBT] :
% 16.19/9.07 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 16.19/9.07 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/9.07 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/9.07 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 16.19/9.07 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/9.07 & ( case_o184042715313410164at_nat
% 16.19/9.07 @ ( ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X3 )
% 16.19/9.07 & ! [X2: vEBT_VEBT] :
% 16.19/9.07 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 16.19/9.07 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 16.19/9.07 @ ( produc6081775807080527818_nat_o
% 16.19/9.07 @ ^ [Mi3: nat,Ma3: nat] :
% 16.19/9.07 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 16.19/9.07 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 16.19/9.07 & ! [I3: nat] :
% 16.19/9.07 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/9.07 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X3 ) )
% 16.19/9.07 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 16.19/9.07 & ( ( Mi3 = Ma3 )
% 16.19/9.07 => ! [X2: vEBT_VEBT] :
% 16.19/9.07 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 16.19/9.07 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 16.19/9.07 & ( ( Mi3 != Ma3 )
% 16.19/9.07 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 16.19/9.07 & ! [X2: nat] :
% 16.19/9.07 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 16.19/9.07 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
% 16.19/9.07 => ( ( ord_less_nat @ Mi3 @ X2 )
% 16.19/9.07 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
% 16.19/9.07 @ Mima ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10152_Sup__int__def,axiom,
% 16.19/9.07 ( complete_Sup_Sup_int
% 16.19/9.07 = ( ^ [X3: set_int] :
% 16.19/9.07 ( the_int
% 16.19/9.07 @ ^ [X2: int] :
% 16.19/9.07 ( ( member_int @ X2 @ X3 )
% 16.19/9.07 & ! [Y2: int] :
% 16.19/9.07 ( ( member_int @ Y2 @ X3 )
% 16.19/9.07 => ( ord_less_eq_int @ Y2 @ X2 ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10153_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
% 16.19/9.07 ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
% 16.19/9.07 ( ( ( vEBT_VEBT_valid @ X @ Xa2 )
% 16.19/9.07 = Y )
% 16.19/9.07 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
% 16.19/9.07 => ( ! [Uu2: $o,Uv2: $o] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 16.19/9.07 => ( ( Y
% 16.19/9.07 = ( Xa2 = one_one_nat ) )
% 16.19/9.07 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
% 16.19/9.07 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 16.19/9.07 ( ( X
% 16.19/9.07 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 16.19/9.07 => ( ( Y
% 16.19/9.07 = ( ( Deg2 = Xa2 )
% 16.19/9.07 & ! [X2: vEBT_VEBT] :
% 16.19/9.07 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 16.19/9.07 => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/9.07 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 16.19/9.07 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 16.19/9.07 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/9.07 & ( case_o184042715313410164at_nat
% 16.19/9.07 @ ( ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X3 )
% 16.19/9.07 & ! [X2: vEBT_VEBT] :
% 16.19/9.07 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 16.19/9.07 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 16.19/9.07 @ ( produc6081775807080527818_nat_o
% 16.19/9.07 @ ^ [Mi3: nat,Ma3: nat] :
% 16.19/9.07 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 16.19/9.07 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 16.19/9.07 & ! [I3: nat] :
% 16.19/9.07 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 16.19/9.07 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X3 ) )
% 16.19/9.07 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 16.19/9.07 & ( ( Mi3 = Ma3 )
% 16.19/9.07 => ! [X2: vEBT_VEBT] :
% 16.19/9.07 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 16.19/9.07 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X3 ) ) )
% 16.19/9.07 & ( ( Mi3 != Ma3 )
% 16.19/9.07 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 16.19/9.07 & ! [X2: nat] :
% 16.19/9.07 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 16.19/9.07 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
% 16.19/9.07 => ( ( ord_less_nat @ Mi3 @ X2 )
% 16.19/9.07 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
% 16.19/9.07 @ Mima ) ) )
% 16.19/9.07 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10154_GMVT,axiom,
% 16.19/9.07 ! [A: real,B: real,F: real > real,G: real > real] :
% 16.19/9.07 ( ( ord_less_real @ A @ B )
% 16.19/9.07 => ( ! [X4: real] :
% 16.19/9.07 ( ( ( ord_less_eq_real @ A @ X4 )
% 16.19/9.07 & ( ord_less_eq_real @ X4 @ B ) )
% 16.19/9.07 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F ) )
% 16.19/9.07 => ( ! [X4: real] :
% 16.19/9.07 ( ( ( ord_less_real @ A @ X4 )
% 16.19/9.07 & ( ord_less_real @ X4 @ B ) )
% 16.19/9.07 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 16.19/9.07 => ( ! [X4: real] :
% 16.19/9.07 ( ( ( ord_less_eq_real @ A @ X4 )
% 16.19/9.07 & ( ord_less_eq_real @ X4 @ B ) )
% 16.19/9.07 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ G ) )
% 16.19/9.07 => ( ! [X4: real] :
% 16.19/9.07 ( ( ( ord_less_real @ A @ X4 )
% 16.19/9.07 & ( ord_less_real @ X4 @ B ) )
% 16.19/9.07 => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 16.19/9.07 => ? [G_c: real,F_c: real,C3: real] :
% 16.19/9.07 ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
% 16.19/9.07 & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
% 16.19/9.07 & ( ord_less_real @ A @ C3 )
% 16.19/9.07 & ( ord_less_real @ C3 @ B )
% 16.19/9.07 & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ G_c )
% 16.19/9.07 = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10155_MVT,axiom,
% 16.19/9.07 ! [A: real,B: real,F: real > real] :
% 16.19/9.07 ( ( ord_less_real @ A @ B )
% 16.19/9.07 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 16.19/9.07 => ( ! [X4: real] :
% 16.19/9.07 ( ( ord_less_real @ A @ X4 )
% 16.19/9.07 => ( ( ord_less_real @ X4 @ B )
% 16.19/9.07 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 16.19/9.07 => ? [L4: real,Z5: real] :
% 16.19/9.07 ( ( ord_less_real @ A @ Z5 )
% 16.19/9.07 & ( ord_less_real @ Z5 @ B )
% 16.19/9.07 & ( has_fi5821293074295781190e_real @ F @ L4 @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) )
% 16.19/9.07 & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 16.19/9.07 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ L4 ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10156_continuous__on__arsinh_H,axiom,
% 16.19/9.07 ! [A2: set_real,F: real > real] :
% 16.19/9.07 ( ( topolo5044208981011980120l_real @ A2 @ F )
% 16.19/9.07 => ( topolo5044208981011980120l_real @ A2
% 16.19/9.07 @ ^ [X2: real] : ( arsinh_real @ ( F @ X2 ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10157_continuous__on__arcosh,axiom,
% 16.19/9.07 ! [A2: set_real] :
% 16.19/9.07 ( ( ord_less_eq_set_real @ A2 @ ( set_ord_atLeast_real @ one_one_real ) )
% 16.19/9.07 => ( topolo5044208981011980120l_real @ A2 @ arcosh_real ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10158_continuous__on__arcosh_H,axiom,
% 16.19/9.07 ! [A2: set_real,F: real > real] :
% 16.19/9.07 ( ( topolo5044208981011980120l_real @ A2 @ F )
% 16.19/9.07 => ( ! [X4: real] :
% 16.19/9.07 ( ( member_real @ X4 @ A2 )
% 16.19/9.07 => ( ord_less_eq_real @ one_one_real @ ( F @ X4 ) ) )
% 16.19/9.07 => ( topolo5044208981011980120l_real @ A2
% 16.19/9.07 @ ^ [X2: real] : ( arcosh_real @ ( F @ X2 ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10159_continuous__image__closed__interval,axiom,
% 16.19/9.07 ! [A: real,B: real,F: real > real] :
% 16.19/9.07 ( ( ord_less_eq_real @ A @ B )
% 16.19/9.07 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 16.19/9.07 => ? [C3: real,D4: real] :
% 16.19/9.07 ( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
% 16.19/9.07 = ( set_or1222579329274155063t_real @ C3 @ D4 ) )
% 16.19/9.07 & ( ord_less_eq_real @ C3 @ D4 ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10160_continuous__on__arccos_H,axiom,
% 16.19/9.07 topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arccos ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10161_continuous__on__arcsin_H,axiom,
% 16.19/9.07 topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arcsin ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10162_continuous__on__artanh,axiom,
% 16.19/9.07 ! [A2: set_real] :
% 16.19/9.07 ( ( ord_less_eq_set_real @ A2 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
% 16.19/9.07 => ( topolo5044208981011980120l_real @ A2 @ artanh_real ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10163_continuous__on__artanh_H,axiom,
% 16.19/9.07 ! [A2: set_real,F: real > real] :
% 16.19/9.07 ( ( topolo5044208981011980120l_real @ A2 @ F )
% 16.19/9.07 => ( ! [X4: real] :
% 16.19/9.07 ( ( member_real @ X4 @ A2 )
% 16.19/9.07 => ( member_real @ ( F @ X4 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) )
% 16.19/9.07 => ( topolo5044208981011980120l_real @ A2
% 16.19/9.07 @ ^ [X2: real] : ( artanh_real @ ( F @ X2 ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10164_Rolle__deriv,axiom,
% 16.19/9.07 ! [A: real,B: real,F: real > real,F4: real > real > real] :
% 16.19/9.07 ( ( ord_less_real @ A @ B )
% 16.19/9.07 => ( ( ( F @ A )
% 16.19/9.07 = ( F @ B ) )
% 16.19/9.07 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 16.19/9.07 => ( ! [X4: real] :
% 16.19/9.07 ( ( ord_less_real @ A @ X4 )
% 16.19/9.07 => ( ( ord_less_real @ X4 @ B )
% 16.19/9.07 => ( has_de1759254742604945161l_real @ F @ ( F4 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 16.19/9.07 => ? [Z5: real] :
% 16.19/9.07 ( ( ord_less_real @ A @ Z5 )
% 16.19/9.07 & ( ord_less_real @ Z5 @ B )
% 16.19/9.07 & ( ( F4 @ Z5 )
% 16.19/9.07 = ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10165_mvt,axiom,
% 16.19/9.07 ! [A: real,B: real,F: real > real,F4: real > real > real] :
% 16.19/9.07 ( ( ord_less_real @ A @ B )
% 16.19/9.07 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 16.19/9.07 => ( ! [X4: real] :
% 16.19/9.07 ( ( ord_less_real @ A @ X4 )
% 16.19/9.07 => ( ( ord_less_real @ X4 @ B )
% 16.19/9.07 => ( has_de1759254742604945161l_real @ F @ ( F4 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 16.19/9.07 => ~ ! [Xi: real] :
% 16.19/9.07 ( ( ord_less_real @ A @ Xi )
% 16.19/9.07 => ( ( ord_less_real @ Xi @ B )
% 16.19/9.07 => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 16.19/9.07 != ( F4 @ Xi @ ( minus_minus_real @ B @ A ) ) ) ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10166_DERIV__isconst__end,axiom,
% 16.19/9.07 ! [A: real,B: real,F: real > real] :
% 16.19/9.07 ( ( ord_less_real @ A @ B )
% 16.19/9.07 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 16.19/9.07 => ( ! [X4: real] :
% 16.19/9.07 ( ( ord_less_real @ A @ X4 )
% 16.19/9.07 => ( ( ord_less_real @ X4 @ B )
% 16.19/9.07 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 16.19/9.07 => ( ( F @ B )
% 16.19/9.07 = ( F @ A ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10167_DERIV__neg__imp__decreasing__open,axiom,
% 16.19/9.07 ! [A: real,B: real,F: real > real] :
% 16.19/9.07 ( ( ord_less_real @ A @ B )
% 16.19/9.07 => ( ! [X4: real] :
% 16.19/9.07 ( ( ord_less_real @ A @ X4 )
% 16.19/9.07 => ( ( ord_less_real @ X4 @ B )
% 16.19/9.07 => ? [Y5: real] :
% 16.19/9.07 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 16.19/9.07 & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
% 16.19/9.07 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 16.19/9.07 => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ) ).
% 16.19/9.07
% 16.19/9.07 thf(fact_10168_DERIV__pos__imp__increasing__open,axiom,
% 16.19/9.07 ! [A: real,B: real,F: real > real] :
% 16.19/9.07 ( ( ord_less_real @ A @ B )
% 16.19/9.07 => ( ! [X4: real] :
% 16.19/9.07 ( ( ord_less_real @ A @ X4 )
% 16.19/9.07 => ( ( ord_less_real @ X4 @ B )
% 16.19/9.07 => ? [Y5: real] :
% 16.19/9.07 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 16.19/9.07 & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
% 16.19/9.08 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 16.19/9.08 => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10169_DERIV__isconst2,axiom,
% 16.19/9.08 ! [A: real,B: real,F: real > real,X: real] :
% 16.19/9.08 ( ( ord_less_real @ A @ B )
% 16.19/9.08 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 16.19/9.08 => ( ! [X4: real] :
% 16.19/9.08 ( ( ord_less_real @ A @ X4 )
% 16.19/9.08 => ( ( ord_less_real @ X4 @ B )
% 16.19/9.08 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 16.19/9.08 => ( ( ord_less_eq_real @ A @ X )
% 16.19/9.08 => ( ( ord_less_eq_real @ X @ B )
% 16.19/9.08 => ( ( F @ X )
% 16.19/9.08 = ( F @ A ) ) ) ) ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10170_Rolle,axiom,
% 16.19/9.08 ! [A: real,B: real,F: real > real] :
% 16.19/9.08 ( ( ord_less_real @ A @ B )
% 16.19/9.08 => ( ( ( F @ A )
% 16.19/9.08 = ( F @ B ) )
% 16.19/9.08 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 16.19/9.08 => ( ! [X4: real] :
% 16.19/9.08 ( ( ord_less_real @ A @ X4 )
% 16.19/9.08 => ( ( ord_less_real @ X4 @ B )
% 16.19/9.08 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 16.19/9.08 => ? [Z5: real] :
% 16.19/9.08 ( ( ord_less_real @ A @ Z5 )
% 16.19/9.08 & ( ord_less_real @ Z5 @ B )
% 16.19/9.08 & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) ) ) ) ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10171_less__eq__int_Orep__eq,axiom,
% 16.19/9.08 ( ord_less_eq_int
% 16.19/9.08 = ( ^ [X2: int,Xa4: int] :
% 16.19/9.08 ( produc8739625826339149834_nat_o
% 16.19/9.08 @ ^ [Y2: nat,Z3: nat] :
% 16.19/9.08 ( produc6081775807080527818_nat_o
% 16.19/9.08 @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y2 @ V4 ) @ ( plus_plus_nat @ U2 @ Z3 ) ) )
% 16.19/9.08 @ ( rep_Integ @ X2 )
% 16.19/9.08 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10172_less__int_Orep__eq,axiom,
% 16.19/9.08 ( ord_less_int
% 16.19/9.08 = ( ^ [X2: int,Xa4: int] :
% 16.19/9.08 ( produc8739625826339149834_nat_o
% 16.19/9.08 @ ^ [Y2: nat,Z3: nat] :
% 16.19/9.08 ( produc6081775807080527818_nat_o
% 16.19/9.08 @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y2 @ V4 ) @ ( plus_plus_nat @ U2 @ Z3 ) ) )
% 16.19/9.08 @ ( rep_Integ @ X2 )
% 16.19/9.08 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10173_nat_Orep__eq,axiom,
% 16.19/9.08 ( nat2
% 16.19/9.08 = ( ^ [X2: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X2 ) ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10174_uniformity__real__def,axiom,
% 16.19/9.08 ( topolo1511823702728130853y_real
% 16.19/9.08 = ( comple2936214249959783750l_real
% 16.19/9.08 @ ( image_2178119161166701260l_real
% 16.19/9.08 @ ^ [E3: real] :
% 16.19/9.08 ( princi6114159922880469582l_real
% 16.19/9.08 @ ( collec3799799289383736868l_real
% 16.19/9.08 @ ( produc5414030515140494994real_o
% 16.19/9.08 @ ^ [X2: real,Y2: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X2 @ Y2 ) @ E3 ) ) ) )
% 16.19/9.08 @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10175_uniformity__complex__def,axiom,
% 16.19/9.08 ( topolo896644834953643431omplex
% 16.19/9.08 = ( comple8358262395181532106omplex
% 16.19/9.08 @ ( image_5971271580939081552omplex
% 16.19/9.08 @ ^ [E3: real] :
% 16.19/9.08 ( princi3496590319149328850omplex
% 16.19/9.08 @ ( collec8663557070575231912omplex
% 16.19/9.08 @ ( produc6771430404735790350plex_o
% 16.19/9.08 @ ^ [X2: complex,Y2: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X2 @ Y2 ) @ E3 ) ) ) )
% 16.19/9.08 @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10176_open__complex__def,axiom,
% 16.19/9.08 ( topolo4110288021797289639omplex
% 16.19/9.08 = ( ^ [U3: set_complex] :
% 16.19/9.08 ! [X2: complex] :
% 16.19/9.08 ( ( member_complex @ X2 @ U3 )
% 16.19/9.08 => ( eventu5826381225784669381omplex
% 16.19/9.08 @ ( produc6771430404735790350plex_o
% 16.19/9.08 @ ^ [X9: complex,Y2: complex] :
% 16.19/9.08 ( ( X9 = X2 )
% 16.19/9.08 => ( member_complex @ Y2 @ U3 ) ) )
% 16.19/9.08 @ topolo896644834953643431omplex ) ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10177_open__real__def,axiom,
% 16.19/9.08 ( topolo4860482606490270245n_real
% 16.19/9.08 = ( ^ [U3: set_real] :
% 16.19/9.08 ! [X2: real] :
% 16.19/9.08 ( ( member_real @ X2 @ U3 )
% 16.19/9.08 => ( eventu3244425730907250241l_real
% 16.19/9.08 @ ( produc5414030515140494994real_o
% 16.19/9.08 @ ^ [X9: real,Y2: real] :
% 16.19/9.08 ( ( X9 = X2 )
% 16.19/9.08 => ( member_real @ Y2 @ U3 ) ) )
% 16.19/9.08 @ topolo1511823702728130853y_real ) ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10178_eventually__prod__sequentially,axiom,
% 16.19/9.08 ! [P: product_prod_nat_nat > $o] :
% 16.19/9.08 ( ( eventu1038000079068216329at_nat @ P @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
% 16.19/9.08 = ( ? [N6: nat] :
% 16.19/9.08 ! [M6: nat] :
% 16.19/9.08 ( ( ord_less_eq_nat @ N6 @ M6 )
% 16.19/9.08 => ! [N: nat] :
% 16.19/9.08 ( ( ord_less_eq_nat @ N6 @ N )
% 16.19/9.08 => ( P @ ( product_Pair_nat_nat @ N @ M6 ) ) ) ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10179_incseq__bounded,axiom,
% 16.19/9.08 ! [X8: nat > real,B3: real] :
% 16.19/9.08 ( ( order_mono_nat_real @ X8 )
% 16.19/9.08 => ( ! [I4: nat] : ( ord_less_eq_real @ ( X8 @ I4 ) @ B3 )
% 16.19/9.08 => ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10180_mono__times__nat,axiom,
% 16.19/9.08 ! [N2: nat] :
% 16.19/9.08 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.08 => ( order_mono_nat_nat @ ( times_times_nat @ N2 ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10181_mono__Suc,axiom,
% 16.19/9.08 order_mono_nat_nat @ suc ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10182_filtermap__at__right__shift,axiom,
% 16.19/9.08 ! [D: real,A: real] :
% 16.19/9.08 ( ( filtermap_real_real
% 16.19/9.08 @ ^ [X2: real] : ( minus_minus_real @ X2 @ D )
% 16.19/9.08 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 16.19/9.08 = ( topolo2177554685111907308n_real @ ( minus_minus_real @ A @ D ) @ ( set_or5849166863359141190n_real @ ( minus_minus_real @ A @ D ) ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10183_incseq__convergent,axiom,
% 16.19/9.08 ! [X8: nat > real,B3: real] :
% 16.19/9.08 ( ( order_mono_nat_real @ X8 )
% 16.19/9.08 => ( ! [I4: nat] : ( ord_less_eq_real @ ( X8 @ I4 ) @ B3 )
% 16.19/9.08 => ~ ! [L6: real] :
% 16.19/9.08 ( ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
% 16.19/9.08 => ~ ! [I: nat] : ( ord_less_eq_real @ ( X8 @ I ) @ L6 ) ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10184_at__right__to__0,axiom,
% 16.19/9.08 ! [A: real] :
% 16.19/9.08 ( ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) )
% 16.19/9.08 = ( filtermap_real_real
% 16.19/9.08 @ ^ [X2: real] : ( plus_plus_real @ X2 @ A )
% 16.19/9.08 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10185_at__right__minus,axiom,
% 16.19/9.08 ! [A: real] :
% 16.19/9.08 ( ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) )
% 16.19/9.08 = ( filtermap_real_real @ uminus_uminus_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ A ) @ ( set_or5984915006950818249n_real @ ( uminus_uminus_real @ A ) ) ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10186_at__left__minus,axiom,
% 16.19/9.08 ! [A: real] :
% 16.19/9.08 ( ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) )
% 16.19/9.08 = ( filtermap_real_real @ uminus_uminus_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ A ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ A ) ) ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10187_mono__ge2__power__minus__self,axiom,
% 16.19/9.08 ! [K: nat] :
% 16.19/9.08 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 16.19/9.08 => ( order_mono_nat_nat
% 16.19/9.08 @ ^ [M6: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M6 ) @ M6 ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10188_tendsto__at__topI__sequentially__real,axiom,
% 16.19/9.08 ! [F: real > real,Y: real] :
% 16.19/9.08 ( ( order_mono_real_real @ F )
% 16.19/9.08 => ( ( filterlim_nat_real
% 16.19/9.08 @ ^ [N: nat] : ( F @ ( semiri5074537144036343181t_real @ N ) )
% 16.19/9.08 @ ( topolo2815343760600316023s_real @ Y )
% 16.19/9.08 @ at_top_nat )
% 16.19/9.08 => ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Y ) @ at_top_real ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10189_nonneg__incseq__Bseq__subseq__iff,axiom,
% 16.19/9.08 ! [F: nat > real,G: nat > nat] :
% 16.19/9.08 ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 16.19/9.08 => ( ( order_mono_nat_real @ F )
% 16.19/9.08 => ( ( order_5726023648592871131at_nat @ G )
% 16.19/9.08 => ( ( bfun_nat_real
% 16.19/9.08 @ ^ [X2: nat] : ( F @ ( G @ X2 ) )
% 16.19/9.08 @ at_top_nat )
% 16.19/9.08 = ( bfun_nat_real @ F @ at_top_nat ) ) ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10190_eventually__subseq,axiom,
% 16.19/9.08 ! [R: nat > nat,P: nat > $o] :
% 16.19/9.08 ( ( order_5726023648592871131at_nat @ R )
% 16.19/9.08 => ( ( eventually_nat @ P @ at_top_nat )
% 16.19/9.08 => ( eventually_nat
% 16.19/9.08 @ ^ [N: nat] : ( P @ ( R @ N ) )
% 16.19/9.08 @ at_top_nat ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10191_strict__mono__imp__increasing,axiom,
% 16.19/9.08 ! [F: nat > nat,N2: nat] :
% 16.19/9.08 ( ( order_5726023648592871131at_nat @ F )
% 16.19/9.08 => ( ord_less_eq_nat @ N2 @ ( F @ N2 ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10192_inj__sgn__power,axiom,
% 16.19/9.08 ! [N2: nat] :
% 16.19/9.08 ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 16.19/9.08 => ( inj_on_real_real
% 16.19/9.08 @ ^ [Y2: real] : ( times_times_real @ ( sgn_sgn_real @ Y2 ) @ ( power_power_real @ ( abs_abs_real @ Y2 ) @ N2 ) )
% 16.19/9.08 @ top_top_set_real ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10193_log__inj,axiom,
% 16.19/9.08 ! [B: real] :
% 16.19/9.08 ( ( ord_less_real @ one_one_real @ B )
% 16.19/9.08 => ( inj_on_real_real @ ( log @ B ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10194_pos__deriv__imp__strict__mono,axiom,
% 16.19/9.08 ! [F: real > real,F4: real > real] :
% 16.19/9.08 ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 16.19/9.08 => ( ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( F4 @ X4 ) )
% 16.19/9.08 => ( order_7092887310737990675l_real @ F ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10195_inj__on__diff__nat,axiom,
% 16.19/9.08 ! [N3: set_nat,K: nat] :
% 16.19/9.08 ( ! [N4: nat] :
% 16.19/9.08 ( ( member_nat @ N4 @ N3 )
% 16.19/9.08 => ( ord_less_eq_nat @ K @ N4 ) )
% 16.19/9.08 => ( inj_on_nat_nat
% 16.19/9.08 @ ^ [N: nat] : ( minus_minus_nat @ N @ K )
% 16.19/9.08 @ N3 ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10196_inj__Suc,axiom,
% 16.19/9.08 ! [N3: set_nat] : ( inj_on_nat_nat @ suc @ N3 ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10197_summable__reindex,axiom,
% 16.19/9.08 ! [F: nat > real,G: nat > nat] :
% 16.19/9.08 ( ( summable_real @ F )
% 16.19/9.08 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 16.19/9.08 => ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 16.19/9.08 => ( summable_real @ ( comp_nat_real_nat @ F @ G ) ) ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10198_suminf__reindex__mono,axiom,
% 16.19/9.08 ! [F: nat > real,G: nat > nat] :
% 16.19/9.08 ( ( summable_real @ F )
% 16.19/9.08 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 16.19/9.08 => ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 16.19/9.08 => ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G ) ) @ ( suminf_real @ F ) ) ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10199_inj__on__char__of__nat,axiom,
% 16.19/9.08 inj_on_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10200_suminf__reindex,axiom,
% 16.19/9.08 ! [F: nat > real,G: nat > nat] :
% 16.19/9.08 ( ( summable_real @ F )
% 16.19/9.08 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 16.19/9.08 => ( ! [X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) )
% 16.19/9.08 => ( ! [X4: nat] :
% 16.19/9.08 ( ~ ( member_nat @ X4 @ ( image_nat_nat @ G @ top_top_set_nat ) )
% 16.19/9.08 => ( ( F @ X4 )
% 16.19/9.08 = zero_zero_real ) )
% 16.19/9.08 => ( ( suminf_real @ ( comp_nat_real_nat @ F @ G ) )
% 16.19/9.08 = ( suminf_real @ F ) ) ) ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(fact_10201_prod__encode__def,axiom,
% 16.19/9.08 ( nat_prod_encode
% 16.19/9.08 = ( produc6842872674320459806at_nat
% 16.19/9.08 @ ^ [M6: nat,N: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M6 @ N ) ) @ M6 ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
% 16.19/9.08 ! [X: int,Y: int] :
% 16.19/9.08 ( ( if_int @ $false @ X @ Y )
% 16.19/9.08 = Y ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
% 16.19/9.08 ! [X: int,Y: int] :
% 16.19/9.08 ( ( if_int @ $true @ X @ Y )
% 16.19/9.08 = X ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 16.19/9.08 ! [X: nat,Y: nat] :
% 16.19/9.08 ( ( if_nat @ $false @ X @ Y )
% 16.19/9.08 = Y ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 16.19/9.08 ! [X: nat,Y: nat] :
% 16.19/9.08 ( ( if_nat @ $true @ X @ Y )
% 16.19/9.08 = X ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
% 16.19/9.08 ! [X: num,Y: num] :
% 16.19/9.08 ( ( if_num @ $false @ X @ Y )
% 16.19/9.08 = Y ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
% 16.19/9.08 ! [X: num,Y: num] :
% 16.19/9.08 ( ( if_num @ $true @ X @ Y )
% 16.19/9.08 = X ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
% 16.19/9.08 ! [X: rat,Y: rat] :
% 16.19/9.08 ( ( if_rat @ $false @ X @ Y )
% 16.19/9.08 = Y ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
% 16.19/9.08 ! [X: rat,Y: rat] :
% 16.19/9.08 ( ( if_rat @ $true @ X @ Y )
% 16.19/9.08 = X ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 16.19/9.08 ! [X: real,Y: real] :
% 16.19/9.08 ( ( if_real @ $false @ X @ Y )
% 16.19/9.08 = Y ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 16.19/9.08 ! [X: real,Y: real] :
% 16.19/9.08 ( ( if_real @ $true @ X @ Y )
% 16.19/9.08 = X ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
% 16.19/9.08 ! [P: real > $o] :
% 16.19/9.08 ( ( P @ ( fChoice_real @ P ) )
% 16.19/9.08 = ( ? [X3: real] : ( P @ X3 ) ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
% 16.19/9.08 ! [X: complex,Y: complex] :
% 16.19/9.08 ( ( if_complex @ $false @ X @ Y )
% 16.19/9.08 = Y ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
% 16.19/9.08 ! [X: complex,Y: complex] :
% 16.19/9.08 ( ( if_complex @ $true @ X @ Y )
% 16.19/9.08 = X ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 16.19/9.08 ! [X: code_integer,Y: code_integer] :
% 16.19/9.08 ( ( if_Code_integer @ $false @ X @ Y )
% 16.19/9.08 = Y ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 16.19/9.08 ! [X: code_integer,Y: code_integer] :
% 16.19/9.08 ( ( if_Code_integer @ $true @ X @ Y )
% 16.19/9.08 = X ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 16.19/9.08 ! [X: set_int,Y: set_int] :
% 16.19/9.08 ( ( if_set_int @ $false @ X @ Y )
% 16.19/9.08 = Y ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 16.19/9.08 ! [X: set_int,Y: set_int] :
% 16.19/9.08 ( ( if_set_int @ $true @ X @ Y )
% 16.19/9.08 = X ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
% 16.19/9.08 ! [X: set_nat,Y: set_nat] :
% 16.19/9.08 ( ( if_set_nat @ $false @ X @ Y )
% 16.19/9.08 = Y ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
% 16.19/9.08 ! [X: set_nat,Y: set_nat] :
% 16.19/9.08 ( ( if_set_nat @ $true @ X @ Y )
% 16.19/9.08 = X ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
% 16.19/9.08 ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
% 16.19/9.08 ( ( if_VEBT_VEBT @ $false @ X @ Y )
% 16.19/9.08 = Y ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
% 16.19/9.08 ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
% 16.19/9.08 ( ( if_VEBT_VEBT @ $true @ X @ Y )
% 16.19/9.08 = X ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 16.19/9.08 ! [X: list_int,Y: list_int] :
% 16.19/9.08 ( ( if_list_int @ $false @ X @ Y )
% 16.19/9.08 = Y ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 16.19/9.08 ! [X: list_int,Y: list_int] :
% 16.19/9.08 ( ( if_list_int @ $true @ X @ Y )
% 16.19/9.08 = X ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
% 16.19/9.08 ! [X: option_nat,Y: option_nat] :
% 16.19/9.08 ( ( if_option_nat @ $false @ X @ Y )
% 16.19/9.08 = Y ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
% 16.19/9.08 ! [X: option_nat,Y: option_nat] :
% 16.19/9.08 ( ( if_option_nat @ $true @ X @ Y )
% 16.19/9.08 = X ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 16.19/9.08 ! [X: option_num,Y: option_num] :
% 16.19/9.08 ( ( if_option_num @ $false @ X @ Y )
% 16.19/9.08 = Y ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 16.19/9.08 ! [X: option_num,Y: option_num] :
% 16.19/9.08 ( ( if_option_num @ $true @ X @ Y )
% 16.19/9.08 = X ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 16.19/9.08 ! [X: product_prod_int_int,Y: product_prod_int_int] :
% 16.19/9.08 ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
% 16.19/9.08 = Y ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 16.19/9.08 ! [X: product_prod_int_int,Y: product_prod_int_int] :
% 16.19/9.08 ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
% 16.19/9.08 = X ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 16.19/9.08 ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 16.19/9.08 ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
% 16.19/9.08 = Y ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 16.19/9.08 ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 16.19/9.08 ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
% 16.19/9.08 = X ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 16.19/9.08 ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
% 16.19/9.08 ( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y )
% 16.19/9.08 = Y ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 16.19/9.08 ! [X: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
% 16.19/9.08 ( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y )
% 16.19/9.08 = X ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 16.19/9.08 ! [P: $o] :
% 16.19/9.08 ( ( P = $true )
% 16.19/9.08 | ( P = $false ) ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 16.19/9.08 ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
% 16.19/9.08 ( ( if_Pro6119634080678213985nteger @ $false @ X @ Y )
% 16.19/9.08 = Y ) ).
% 16.19/9.08
% 16.19/9.08 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 16.19/9.08 ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
% 16.19/9.08 ( ( if_Pro6119634080678213985nteger @ $true @ X @ Y )
% 16.19/9.08 = X ) ).
% 16.19/9.08
% 16.19/9.08 thf(conj_0,conjecture,
% 16.19/9.08 ( ( 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 @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ summin @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) @ lx ) @ na ) ) ) ) ) ) ) )
% 16.19/9.08 = ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) @ maxs ) @ maxi ) ) ).
% 16.19/9.08 ------------------------
% 16.21/10.01 % (23162)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_23017 for (2990ds/275Mi)
% 16.21/10.01 % (23157)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_23017 for (2990ds/183Mi)
% 16.21/10.01 % (23159)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_23017 for (2990ds/27Mi)
% 16.21/10.01 % (23161)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_23017 for (2990ds/2Mi)
% 16.21/10.01 % (23158)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_23017 for (2990ds/4Mi)
% 16.21/10.01 % (23160)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_23017 for (2990ds/2Mi)
% 16.21/10.01 % (23163)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_23017 for (2990ds/18Mi)
% 16.21/10.02 % (23161)Instruction limit reached!
% 16.21/10.02 % (23161)------------------------------
% 16.21/10.02 % (23161)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.21/10.02 % (23161)Termination reason: Unknown
% 16.21/10.02 % (23161)Termination phase: shuffling
% 16.21/10.02
% 16.21/10.02 % (23161)Memory used [KB]: 13304
% 16.21/10.02 % (23161)Time elapsed: 0.003 s
% 16.21/10.02 % (23161)Instructions burned: 2 (million)
% 16.21/10.02 % (23161)------------------------------
% 16.21/10.02 % (23161)------------------------------
% 16.21/10.02 % (23160)Instruction limit reached!
% 16.21/10.02 % (23160)------------------------------
% 16.21/10.02 % (23160)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.21/10.02 % (23160)Termination reason: Unknown
% 16.21/10.02 % (23160)Termination phase: shuffling
% 16.21/10.02
% 16.21/10.02 % (23160)Memory used [KB]: 13304
% 16.21/10.02 % (23160)Time elapsed: 0.004 s
% 16.21/10.02 % (23160)Instructions burned: 2 (million)
% 16.21/10.02 % (23160)------------------------------
% 16.21/10.02 % (23160)------------------------------
% 16.21/10.02 % (23158)Instruction limit reached!
% 16.21/10.02 % (23158)------------------------------
% 16.21/10.02 % (23158)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.21/10.02 % (23158)Termination reason: Unknown
% 16.21/10.02 % (23158)Termination phase: shuffling
% 16.21/10.02
% 16.21/10.02 % (23158)Memory used [KB]: 13432
% 16.21/10.02 % (23158)Time elapsed: 0.006 s
% 16.21/10.02 % (23158)Instructions burned: 4 (million)
% 16.21/10.02 % (23158)------------------------------
% 16.21/10.02 % (23158)------------------------------
% 17.27/10.03 % (23163)Instruction limit reached!
% 17.27/10.03 % (23163)------------------------------
% 17.27/10.03 % (23163)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.27/10.03 % (23163)Termination reason: Unknown
% 17.27/10.03 % (23163)Termination phase: shuffling
% 17.27/10.03
% 17.27/10.03 % (23163)Memory used [KB]: 14200
% 17.27/10.03 % (23163)Time elapsed: 0.021 s
% 17.27/10.03 % (23163)Instructions burned: 19 (million)
% 17.27/10.03 % (23163)------------------------------
% 17.27/10.03 % (23163)------------------------------
% 17.27/10.04 % (23164)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_23017 for (2990ds/3Mi)
% 17.27/10.04 % (23165)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_23017 for (2990ds/37Mi)
% 17.27/10.04 % (23166)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_23017 for (2990ds/15Mi)
% 17.27/10.04 % (23164)Instruction limit reached!
% 17.27/10.04 % (23164)------------------------------
% 17.27/10.04 % (23164)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.27/10.04 % (23164)Termination reason: Unknown
% 17.27/10.04 % (23164)Termination phase: shuffling
% 17.27/10.04
% 17.27/10.04 % (23164)Memory used [KB]: 13304
% 17.27/10.04 % (23164)Time elapsed: 0.004 s
% 17.27/10.04 % (23159)Instruction limit reached!
% 17.27/10.04 % (23159)------------------------------
% 17.27/10.04 % (23159)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.27/10.04 % (23159)Termination reason: Unknown
% 17.27/10.04 % (23159)Termination phase: shuffling
% 17.27/10.04
% 17.27/10.04 % (23159)Memory used [KB]: 14200
% 17.27/10.04 % (23159)Time elapsed: 0.027 s
% 17.27/10.04 % (23159)Instructions burned: 28 (million)
% 17.27/10.04 % (23159)------------------------------
% 17.27/10.04 % (23159)------------------------------
% 17.27/10.04 % (23164)Instructions burned: 3 (million)
% 17.27/10.04 % (23164)------------------------------
% 17.27/10.04 % (23164)------------------------------
% 17.27/10.05 % (23166)Instruction limit reached!
% 17.27/10.05 % (23166)------------------------------
% 17.27/10.05 % (23166)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.27/10.05 % (23166)Termination reason: Unknown
% 17.27/10.05 % (23167)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_23017 for (2990ds/3Mi)
% 17.27/10.05 % (23166)Termination phase: shuffling
% 17.27/10.05
% 17.27/10.05 % (23166)Memory used [KB]: 14200
% 17.27/10.05 % (23166)Time elapsed: 0.019 s
% 17.27/10.05 % (23166)Instructions burned: 15 (million)
% 17.27/10.05 % (23166)------------------------------
% 17.27/10.05 % (23166)------------------------------
% 17.27/10.06 % (23167)Instruction limit reached!
% 17.27/10.06 % (23167)------------------------------
% 17.27/10.06 % (23167)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.27/10.06 % (23167)Termination reason: Unknown
% 17.27/10.06 % (23167)Termination phase: shuffling
% 17.27/10.06
% 17.27/10.06 % (23167)Memory used [KB]: 13432
% 17.27/10.06 % (23167)Time elapsed: 0.005 s
% 17.27/10.06 % (23167)Instructions burned: 3 (million)
% 17.27/10.06 % (23167)------------------------------
% 17.27/10.06 % (23167)------------------------------
% 17.27/10.06 % (23168)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_23017 for (2990ds/1041Mi)
% 17.27/10.06 % (23169)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_23017 for (2990ds/7Mi)
% 17.27/10.07 % (23165)Instruction limit reached!
% 17.27/10.07 % (23165)------------------------------
% 17.27/10.07 % (23165)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.27/10.07 % (23165)Termination reason: Unknown
% 17.27/10.07 % (23165)Termination phase: shuffling
% 17.27/10.07
% 17.27/10.07 % (23165)Memory used [KB]: 14328
% 17.27/10.07 % (23165)Time elapsed: 0.032 s
% 17.27/10.07 % (23165)Instructions burned: 38 (million)
% 17.27/10.07 % (23165)------------------------------
% 17.27/10.07 % (23165)------------------------------
% 17.27/10.07 % (23169)Instruction limit reached!
% 17.27/10.07 % (23169)------------------------------
% 17.27/10.07 % (23169)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.27/10.07 % (23169)Termination reason: Unknown
% 17.27/10.07 % (23169)Termination phase: shuffling
% 17.27/10.07
% 17.27/10.07 % (23169)Memory used [KB]: 13688
% 17.27/10.07 % (23169)Time elapsed: 0.009 s
% 17.27/10.07 % (23169)Instructions burned: 7 (million)
% 17.27/10.07 % (23169)------------------------------
% 17.27/10.07 % (23169)------------------------------
% 17.27/10.08 % (23170)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_23017 for (2990ds/16Mi)
% 17.27/10.08 % (23171)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_23017 for (2990ds/3Mi)
% 17.27/10.08 % (23171)Instruction limit reached!
% 17.27/10.08 % (23171)------------------------------
% 17.27/10.08 % (23171)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.27/10.08 % (23171)Termination reason: Unknown
% 17.27/10.08 % (23171)Termination phase: shuffling
% 17.27/10.08
% 17.27/10.08 % (23171)Memory used [KB]: 13304
% 17.27/10.08 % (23171)Time elapsed: 0.004 s
% 17.27/10.08 % (23171)Instructions burned: 3 (million)
% 17.27/10.08 % (23171)------------------------------
% 17.27/10.08 % (23171)------------------------------
% 17.27/10.09 % (23173)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_23017 for (2990ds/7Mi)
% 17.27/10.09 % (23172)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_23017 for (2990ds/3Mi)
% 17.27/10.09 % (23172)Instruction limit reached!
% 17.27/10.09 % (23172)------------------------------
% 17.27/10.09 % (23172)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.27/10.09 % (23172)Termination reason: Unknown
% 17.27/10.09 % (23172)Termination phase: shuffling
% 17.27/10.09
% 17.27/10.09 % (23172)Memory used [KB]: 13304
% 17.27/10.09 % (23172)Time elapsed: 0.005 s
% 17.27/10.09 % (23172)Instructions burned: 3 (million)
% 17.27/10.09 % (23172)------------------------------
% 17.27/10.09 % (23172)------------------------------
% 17.27/10.10 % (23170)Instruction limit reached!
% 17.27/10.10 % (23170)------------------------------
% 17.27/10.10 % (23170)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.27/10.10 % (23170)Termination reason: Unknown
% 17.27/10.10 % (23170)Termination phase: shuffling
% 17.27/10.10
% 17.27/10.10 % (23170)Memory used [KB]: 14200
% 17.27/10.10 % (23170)Time elapsed: 0.021 s
% 17.27/10.10 % (23170)Instructions burned: 16 (million)
% 17.27/10.10 % (23170)------------------------------
% 17.27/10.10 % (23170)------------------------------
% 17.27/10.10 % (23173)Instruction limit reached!
% 17.27/10.10 % (23173)------------------------------
% 17.27/10.10 % (23173)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.27/10.10 % (23173)Termination reason: Unknown
% 17.27/10.10 % (23173)Termination phase: shuffling
% 17.27/10.10
% 17.27/10.10 % (23173)Memory used [KB]: 13688
% 17.27/10.10 % (23173)Time elapsed: 0.010 s
% 17.27/10.10 % (23173)Instructions burned: 7 (million)
% 17.27/10.10 % (23173)------------------------------
% 17.27/10.10 % (23173)------------------------------
% 17.27/10.10 % (23174)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_23017 for (2990ds/3Mi)
% 17.27/10.10 % (23174)Instruction limit reached!
% 17.27/10.10 % (23174)------------------------------
% 17.27/10.10 % (23174)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.27/10.10 % (23174)Termination reason: Unknown
% 17.27/10.10 % (23174)Termination phase: shuffling
% 17.27/10.10
% 17.27/10.10 % (23174)Memory used [KB]: 13432
% 17.27/10.10 % (23174)Time elapsed: 0.005 s
% 17.27/10.10 % (23174)Instructions burned: 3 (million)
% 17.27/10.10 % (23174)------------------------------
% 17.27/10.10 % (23174)------------------------------
% 17.27/10.11 % (23175)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_23017 for (2990ds/4Mi)
% 17.27/10.11 % (23175)Instruction limit reached!
% 17.27/10.11 % (23175)------------------------------
% 17.27/10.11 % (23175)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.27/10.12 % (23175)Termination reason: Unknown
% 17.27/10.12 % (23175)Termination phase: shuffling
% 17.27/10.12
% 17.27/10.12 % (23175)Memory used [KB]: 13432
% 17.27/10.12 % (23175)Time elapsed: 0.006 s
% 17.27/10.12 % (23175)Instructions burned: 4 (million)
% 17.27/10.12 % (23175)------------------------------
% 17.27/10.12 % (23175)------------------------------
% 17.27/10.12 % (23176)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_23017 for (2989ds/18Mi)
% 17.27/10.12 % (23177)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_23017 for (2989ds/710Mi)
% 17.27/10.13 % (23178)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_23017 for (2989ds/6Mi)
% 17.27/10.13 % (23157)Instruction limit reached!
% 17.27/10.13 % (23157)------------------------------
% 17.27/10.13 % (23157)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.27/10.13 % (23157)Termination reason: Unknown
% 17.27/10.13 % (23157)Termination phase: shuffling
% 17.27/10.13
% 17.27/10.13 % (23157)Memory used [KB]: 16502
% 17.27/10.13 % (23157)Time elapsed: 0.114 s
% 17.27/10.13 % (23157)Instructions burned: 184 (million)
% 17.27/10.13 % (23157)------------------------------
% 17.27/10.13 % (23157)------------------------------
% 17.83/10.13 % (23178)Instruction limit reached!
% 17.83/10.13 % (23178)------------------------------
% 17.83/10.13 % (23178)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.83/10.13 % (23178)Termination reason: Unknown
% 17.83/10.13 % (23178)Termination phase: shuffling
% 17.83/10.13
% 17.83/10.13 % (23178)Memory used [KB]: 13560
% 17.83/10.13 % (23178)Time elapsed: 0.009 s
% 17.83/10.13 % (23178)Instructions burned: 6 (million)
% 17.83/10.13 % (23178)------------------------------
% 17.83/10.13 % (23178)------------------------------
% 17.83/10.14 % (23179)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_23017 for (2989ds/902Mi)
% 17.83/10.14 % (23176)Instruction limit reached!
% 17.83/10.14 % (23176)------------------------------
% 17.83/10.14 % (23176)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.83/10.14 % (23176)Termination reason: Unknown
% 17.83/10.14 % (23176)Termination phase: shuffling
% 17.83/10.14
% 17.83/10.14 % (23176)Memory used [KB]: 14200
% 17.83/10.14 % (23176)Time elapsed: 0.022 s
% 17.83/10.14 % (23176)Instructions burned: 19 (million)
% 17.83/10.14 % (23176)------------------------------
% 17.83/10.14 % (23176)------------------------------
% 17.83/10.15 % (23180)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_23017 for (2989ds/21Mi)
% 17.83/10.15 % (23181)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_23017 for (2989ds/5Mi)
% 17.83/10.16 % (23182)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_23017 for (2989ds/6Mi)
% 17.83/10.16 % (23181)Instruction limit reached!
% 17.83/10.16 % (23181)------------------------------
% 17.83/10.16 % (23181)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.83/10.16 % (23181)Termination reason: Unknown
% 17.83/10.16 % (23181)Termination phase: shuffling
% 17.83/10.16
% 17.83/10.16 % (23181)Memory used [KB]: 13560
% 17.83/10.16 % (23181)Time elapsed: 0.008 s
% 17.83/10.16 % (23181)Instructions burned: 5 (million)
% 17.83/10.16 % (23181)------------------------------
% 17.83/10.16 % (23181)------------------------------
% 17.83/10.17 % (23182)Instruction limit reached!
% 17.83/10.17 % (23182)------------------------------
% 17.83/10.17 % (23182)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.83/10.17 % (23182)Termination reason: Unknown
% 17.83/10.17 % (23182)Termination phase: shuffling
% 17.83/10.17
% 17.83/10.17 % (23182)Memory used [KB]: 13560
% 17.83/10.17 % (23182)Time elapsed: 0.009 s
% 17.83/10.17 % (23182)Instructions burned: 6 (million)
% 17.83/10.17 % (23182)------------------------------
% 17.83/10.17 % (23182)------------------------------
% 17.83/10.17 % (23180)Instruction limit reached!
% 17.83/10.17 % (23180)------------------------------
% 17.83/10.17 % (23180)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.83/10.17 % (23180)Termination reason: Unknown
% 17.83/10.17 % (23180)Termination phase: shuffling
% 17.83/10.17
% 17.83/10.17 % (23180)Memory used [KB]: 14200
% 17.83/10.17 % (23180)Time elapsed: 0.024 s
% 17.83/10.17 % (23180)Instructions burned: 22 (million)
% 17.83/10.17 % (23180)------------------------------
% 17.83/10.17 % (23180)------------------------------
% 17.83/10.18 % (23162)Instruction limit reached!
% 17.83/10.18 % (23162)------------------------------
% 17.83/10.18 % (23162)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.83/10.18 % (23162)Termination reason: Unknown
% 17.83/10.18 % (23162)Termination phase: shuffling
% 17.83/10.18
% 17.83/10.18 % (23162)Memory used [KB]: 17526
% 17.83/10.18 % (23162)Time elapsed: 0.163 s
% 17.83/10.18 % (23162)Instructions burned: 275 (million)
% 17.83/10.18 % (23162)------------------------------
% 17.83/10.18 % (23162)------------------------------
% 17.83/10.18 % (23183)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_23017 for (2989ds/377Mi)
% 17.83/10.19 % (23184)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_23017 for (2989ds/779Mi)
% 17.83/10.19 % (23185)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_23017 for (2989ds/19Mi)
% 18.08/10.20 % (23186)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_23017 for (2989ds/879Mi)
% 18.08/10.21 % (23185)Instruction limit reached!
% 18.08/10.21 % (23185)------------------------------
% 18.08/10.21 % (23185)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.08/10.21 % (23185)Termination reason: Unknown
% 18.08/10.21 % (23185)Termination phase: shuffling
% 18.08/10.21
% 18.08/10.21 % (23185)Memory used [KB]: 14200
% 18.08/10.21 % (23185)Time elapsed: 0.022 s
% 18.08/10.21 % (23185)Instructions burned: 19 (million)
% 18.08/10.21 % (23185)------------------------------
% 18.08/10.21 % (23185)------------------------------
% 18.08/10.24 % (23187)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_23017 for (2988ds/17Mi)
% 18.08/10.25 % (23187)Instruction limit reached!
% 18.08/10.25 % (23187)------------------------------
% 18.08/10.25 % (23187)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.08/10.25 % (23187)Termination reason: Unknown
% 18.08/10.25 % (23187)Termination phase: shuffling
% 18.08/10.25
% 18.08/10.25 % (23187)Memory used [KB]: 14200
% 18.08/10.25 % (23187)Time elapsed: 0.021 s
% 18.08/10.25 % (23187)Instructions burned: 17 (million)
% 18.08/10.25 % (23187)------------------------------
% 18.08/10.25 % (23187)------------------------------
% 18.08/10.28 % (23188)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_23017 for (2988ds/3Mi)
% 18.08/10.28 % (23188)Instruction limit reached!
% 18.08/10.28 % (23188)------------------------------
% 18.08/10.28 % (23188)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.08/10.28 % (23188)Termination reason: Unknown
% 18.08/10.28 % (23188)Termination phase: shuffling
% 18.08/10.28
% 18.08/10.28 % (23188)Memory used [KB]: 13432
% 18.08/10.28 % (23188)Time elapsed: 0.005 s
% 18.08/10.28 % (23188)Instructions burned: 3 (million)
% 18.08/10.28 % (23188)------------------------------
% 18.08/10.28 % (23188)------------------------------
% 18.18/10.30 % (23189)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_23017 for (2988ds/30Mi)
% 18.18/10.33 % (23189)Instruction limit reached!
% 18.18/10.33 % (23189)------------------------------
% 18.18/10.33 % (23189)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.18/10.33 % (23189)Termination reason: Unknown
% 18.18/10.33 % (23189)Termination phase: shuffling
% 18.18/10.33
% 18.18/10.33 % (23189)Memory used [KB]: 14200
% 18.18/10.33 % (23189)Time elapsed: 0.030 s
% 18.18/10.33 % (23189)Instructions burned: 31 (million)
% 18.18/10.33 % (23189)------------------------------
% 18.18/10.33 % (23189)------------------------------
% 18.18/10.35 % (23190)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_23017 for (2987ds/127Mi)
% 18.23/10.40 % (23183)Instruction limit reached!
% 18.23/10.40 % (23183)------------------------------
% 18.23/10.40 % (23183)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.23/10.40 % (23183)Termination reason: Unknown
% 18.23/10.40 % (23183)Termination phase: shuffling
% 18.23/10.40
% 18.23/10.40 % (23183)Memory used [KB]: 18677
% 18.23/10.40 % (23183)Time elapsed: 0.220 s
% 18.23/10.40 % (23183)Instructions burned: 377 (million)
% 18.23/10.40 % (23183)------------------------------
% 18.23/10.40 % (23183)------------------------------
% 18.23/10.42 % (23191)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_23017 for (2986ds/100Mi)
% 18.23/10.43 % (23190)Instruction limit reached!
% 18.23/10.43 % (23190)------------------------------
% 18.23/10.43 % (23190)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.23/10.43 % (23190)Termination reason: Unknown
% 18.23/10.43 % (23190)Termination phase: shuffling
% 18.23/10.43
% 18.23/10.43 % (23190)Memory used [KB]: 15735
% 18.23/10.43 % (23190)Time elapsed: 0.084 s
% 18.23/10.43 % (23190)Instructions burned: 127 (million)
% 18.23/10.43 % (23190)------------------------------
% 18.23/10.43 % (23190)------------------------------
% 18.23/10.46 % (23192)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_23017 for (2986ds/3Mi)
% 18.23/10.46 % (23192)Instruction limit reached!
% 18.23/10.46 % (23192)------------------------------
% 18.23/10.46 % (23192)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.23/10.46 % (23192)Termination reason: Unknown
% 18.23/10.46 % (23192)Termination phase: shuffling
% 18.23/10.46
% 18.23/10.46 % (23192)Memory used [KB]: 13304
% 18.23/10.46 % (23192)Time elapsed: 0.005 s
% 18.23/10.46 % (23192)Instructions burned: 3 (million)
% 18.23/10.46 % (23192)------------------------------
% 18.23/10.46 % (23192)------------------------------
% 18.23/10.48 % (23193)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_23017 for (2986ds/20Mi)
% 18.23/10.49 % (23191)Instruction limit reached!
% 18.23/10.49 % (23191)------------------------------
% 18.23/10.49 % (23191)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.23/10.49 % (23191)Termination reason: Unknown
% 18.23/10.49 % (23191)Termination phase: shuffling
% 18.23/10.49
% 18.23/10.49 % (23191)Memory used [KB]: 15351
% 18.23/10.49 % (23191)Time elapsed: 0.065 s
% 18.23/10.49 % (23191)Instructions burned: 100 (million)
% 18.23/10.49 % (23191)------------------------------
% 18.23/10.49 % (23191)------------------------------
% 18.68/10.51 % (23193)Instruction limit reached!
% 18.68/10.51 % (23193)------------------------------
% 18.68/10.51 % (23193)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.68/10.51 % (23193)Termination reason: Unknown
% 18.68/10.51 % (23193)Termination phase: shuffling
% 18.68/10.51
% 18.68/10.51 % (23193)Memory used [KB]: 14200
% 18.68/10.51 % (23193)Time elapsed: 0.024 s
% 18.68/10.51 % (23193)Instructions burned: 21 (million)
% 18.68/10.51 % (23193)------------------------------
% 18.68/10.51 % (23193)------------------------------
% 18.68/10.51 % (23194)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_23017 for (2986ds/86Mi)
% 18.68/10.52 % (23177)Instruction limit reached!
% 18.68/10.52 % (23177)------------------------------
% 18.68/10.52 % (23177)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.68/10.52 % (23177)Termination reason: Unknown
% 18.68/10.52 % (23177)Termination phase: shuffling
% 18.68/10.52
% 18.68/10.52 % (23177)Memory used [KB]: 22899
% 18.68/10.52 % (23177)Time elapsed: 0.406 s
% 18.68/10.52 % (23177)Instructions burned: 710 (million)
% 18.68/10.52 % (23177)------------------------------
% 18.68/10.52 % (23177)------------------------------
% 18.68/10.53 % (23195)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_23017 for (2985ds/107Mi)
% 18.68/10.55 % (23196)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_23017 for (2985ds/39Mi)
% 18.68/10.56 % (23194)Instruction limit reached!
% 18.68/10.56 % (23194)------------------------------
% 18.68/10.56 % (23194)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.68/10.56 % (23194)Termination reason: Unknown
% 18.68/10.56 % (23194)Termination phase: shuffling
% 18.68/10.56
% 18.68/10.56 % (23194)Memory used [KB]: 14967
% 18.68/10.56 % (23194)Time elapsed: 0.058 s
% 18.68/10.56 % (23194)Instructions burned: 86 (million)
% 18.68/10.56 % (23194)------------------------------
% 18.68/10.56 % (23194)------------------------------
% 18.68/10.58 % (23196)Instruction limit reached!
% 18.68/10.58 % (23196)------------------------------
% 18.68/10.58 % (23196)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.68/10.58 % (23196)Termination reason: Unknown
% 18.68/10.58 % (23196)Termination phase: shuffling
% 18.68/10.58
% 18.68/10.58 % (23196)Memory used [KB]: 14456
% 18.68/10.58 % (23196)Time elapsed: 0.033 s
% 18.68/10.58 % (23196)Instructions burned: 39 (million)
% 18.68/10.58 % (23196)------------------------------
% 18.68/10.58 % (23196)------------------------------
% 18.68/10.59 % (23197)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_23017 for (2985ds/448Mi)
% 19.30/10.60 % (23195)Instruction limit reached!
% 19.30/10.60 % (23195)------------------------------
% 19.30/10.60 % (23195)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.30/10.60 % (23195)Termination reason: Unknown
% 19.30/10.60 % (23195)Termination phase: shuffling
% 19.30/10.60
% 19.30/10.60 % (23195)Memory used [KB]: 15351
% 19.30/10.60 % (23195)Time elapsed: 0.071 s
% 19.30/10.60 % (23195)Instructions burned: 107 (million)
% 19.30/10.60 % (23195)------------------------------
% 19.30/10.60 % (23195)------------------------------
% 19.30/10.60 % (23198)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_23017 for (2985ds/46Mi)
% 19.30/10.61 % (23184)Instruction limit reached!
% 19.30/10.61 % (23184)------------------------------
% 19.30/10.61 % (23184)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.30/10.61 % (23184)Termination reason: Unknown
% 19.30/10.61 % (23184)Termination phase: shuffling
% 19.30/10.61
% 19.30/10.61 % (23184)Memory used [KB]: 26609
% 19.30/10.61 % (23184)Time elapsed: 0.423 s
% 19.30/10.61 % (23184)Instructions burned: 780 (million)
% 19.30/10.61 % (23184)------------------------------
% 19.30/10.61 % (23184)------------------------------
% 19.30/10.62 % (23168)Instruction limit reached!
% 19.30/10.62 % (23168)------------------------------
% 19.30/10.62 % (23168)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.30/10.62 % (23168)Termination reason: Unknown
% 19.30/10.62 % (23168)Termination phase: shuffling
% 19.30/10.62
% 19.30/10.62 % (23168)Memory used [KB]: 34157
% 19.30/10.62 % (23168)Time elapsed: 0.562 s
% 19.30/10.62 % (23168)Instructions burned: 1042 (million)
% 19.30/10.62 % (23168)------------------------------
% 19.30/10.62 % (23168)------------------------------
% 19.30/10.62 % (23199)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_23017 for (2984ds/98Mi)
% 19.30/10.63 % (23200)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_23017 for (2984ds/507Mi)
% 19.30/10.64 % (23198)Instruction limit reached!
% 19.30/10.64 % (23198)------------------------------
% 19.30/10.64 % (23198)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.30/10.64 % (23198)Termination reason: Unknown
% 19.30/10.64 % (23198)Termination phase: shuffling
% 19.30/10.64
% 19.30/10.64 % (23198)Memory used [KB]: 14456
% 19.30/10.64 % (23198)Time elapsed: 0.038 s
% 19.30/10.64 % (23198)Instructions burned: 46 (million)
% 19.30/10.64 % (23198)------------------------------
% 19.30/10.64 % (23198)------------------------------
% 19.30/10.64 % (23201)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_23017 for (2984ds/149Mi)
% 19.30/10.65 % (23179)Instruction limit reached!
% 19.30/10.65 % (23179)------------------------------
% 19.30/10.65 % (23179)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.30/10.65 % (23179)Termination reason: Unknown
% 19.30/10.65 % (23179)Termination phase: shuffling
% 19.30/10.65
% 19.30/10.65 % (23179)Memory used [KB]: 27632
% 19.30/10.65 % (23179)Time elapsed: 0.519 s
% 19.30/10.65 % (23179)Instructions burned: 903 (million)
% 19.30/10.65 % (23179)------------------------------
% 19.30/10.65 % (23179)------------------------------
% 19.30/10.66 % (23202)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_23017 for (2984ds/39Mi)
% 19.30/10.68 % (23203)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_23017 for (2984ds/6Mi)
% 19.30/10.68 % (23199)Instruction limit reached!
% 19.30/10.68 % (23199)------------------------------
% 19.30/10.68 % (23199)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.30/10.68 % (23199)Termination reason: Unknown
% 19.30/10.68 % (23199)Termination phase: shuffling
% 19.30/10.68
% 19.30/10.68 % (23199)Memory used [KB]: 15223
% 19.30/10.68 % (23199)Time elapsed: 0.063 s
% 19.30/10.68 % (23199)Instructions burned: 98 (million)
% 19.30/10.68 % (23199)------------------------------
% 19.30/10.68 % (23199)------------------------------
% 19.30/10.68 % (23203)Instruction limit reached!
% 19.30/10.68 % (23203)------------------------------
% 19.30/10.68 % (23203)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.30/10.68 % (23203)Termination reason: Unknown
% 19.30/10.68 % (23203)Termination phase: shuffling
% 19.30/10.68
% 19.30/10.68 % (23203)Memory used [KB]: 13560
% 19.30/10.68 % (23203)Time elapsed: 0.009 s
% 19.30/10.68 % (23203)Instructions burned: 6 (million)
% 19.30/10.68 % (23203)------------------------------
% 19.30/10.68 % (23203)------------------------------
% 19.30/10.69 % (23202)Instruction limit reached!
% 19.30/10.69 % (23202)------------------------------
% 19.30/10.69 % (23202)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.30/10.69 % (23202)Termination reason: Unknown
% 19.30/10.69 % (23202)Termination phase: shuffling
% 19.30/10.69
% 19.30/10.69 % (23202)Memory used [KB]: 14456
% 19.30/10.69 % (23202)Time elapsed: 0.034 s
% 19.30/10.69 % (23202)Instructions burned: 39 (million)
% 19.30/10.69 % (23202)------------------------------
% 19.30/10.69 % (23202)------------------------------
% 21.19/10.70 % (23186)Instruction limit reached!
% 21.19/10.70 % (23186)------------------------------
% 21.19/10.70 % (23186)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.19/10.70 % (23186)Termination reason: Unknown
% 21.19/10.70 % (23186)Termination phase: shuffling
% 21.19/10.70
% 21.19/10.70 % (23186)Memory used [KB]: 26993
% 21.19/10.70 % (23186)Time elapsed: 0.505 s
% 21.19/10.70 % (23186)Instructions burned: 879 (million)
% 21.19/10.70 % (23186)------------------------------
% 21.19/10.70 % (23186)------------------------------
% 21.19/10.70 % (23204)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_23017 for (2984ds/1087Mi)
% 21.19/10.71 % (23205)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_23017 for (2983ds/47Mi)
% 21.19/10.71 % (23206)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_23017 for (2983ds/44Mi)
% 21.19/10.73 % (23207)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_23017 for (2983ds/432Mi)
% 21.19/10.74 % (23201)Instruction limit reached!
% 21.19/10.74 % (23201)------------------------------
% 21.19/10.74 % (23201)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.19/10.74 % (23201)Termination reason: Unknown
% 21.19/10.74 % (23201)Termination phase: shuffling
% 21.19/10.74
% 21.19/10.74 % (23201)Memory used [KB]: 15863
% 21.19/10.74 % (23201)Time elapsed: 0.094 s
% 21.19/10.74 % (23201)Instructions burned: 149 (million)
% 21.19/10.74 % (23201)------------------------------
% 21.19/10.74 % (23201)------------------------------
% 21.19/10.74 % (23205)Instruction limit reached!
% 21.19/10.74 % (23205)------------------------------
% 21.19/10.74 % (23205)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.19/10.74 % (23205)Termination reason: Unknown
% 21.19/10.74 % (23205)Termination phase: shuffling
% 21.19/10.74
% 21.19/10.74 % (23205)Memory used [KB]: 14456
% 21.19/10.74 % (23205)Time elapsed: 0.037 s
% 21.19/10.74 % (23205)Instructions burned: 47 (million)
% 21.19/10.74 % (23205)------------------------------
% 21.19/10.74 % (23205)------------------------------
% 21.19/10.75 % (23206)Instruction limit reached!
% 21.19/10.75 % (23206)------------------------------
% 21.19/10.75 % (23206)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.19/10.75 % (23206)Termination reason: Unknown
% 21.19/10.75 % (23206)Termination phase: shuffling
% 21.19/10.75
% 21.19/10.75 % (23206)Memory used [KB]: 14456
% 21.19/10.75 % (23206)Time elapsed: 0.036 s
% 21.19/10.75 % (23206)Instructions burned: 44 (million)
% 21.19/10.75 % (23206)------------------------------
% 21.19/10.75 % (23206)------------------------------
% 21.19/10.76 % (23208)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_23017 for (2983ds/62Mi)
% 21.19/10.76 % (23209)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_23017 for (2983ds/31Mi)
% 21.19/10.77 % (23210)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_23017 for (2983ds/96Mi)
% 21.19/10.79 % (23209)Instruction limit reached!
% 21.19/10.79 % (23209)------------------------------
% 21.19/10.79 % (23209)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.19/10.79 % (23209)Termination reason: Unknown
% 21.19/10.79 % (23209)Termination phase: shuffling
% 21.19/10.79
% 21.19/10.79 % (23209)Memory used [KB]: 14200
% 21.19/10.79 % (23209)Time elapsed: 0.028 s
% 21.19/10.79 % (23209)Instructions burned: 31 (million)
% 21.19/10.79 % (23209)------------------------------
% 21.19/10.79 % (23209)------------------------------
% 21.99/10.80 % (23208)Instruction limit reached!
% 21.99/10.80 % (23208)------------------------------
% 21.99/10.80 % (23208)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.99/10.80 % (23208)Termination reason: Unknown
% 21.99/10.80 % (23208)Termination phase: shuffling
% 21.99/10.80
% 21.99/10.80 % (23208)Memory used [KB]: 14711
% 21.99/10.80 % (23208)Time elapsed: 0.045 s
% 21.99/10.80 % (23208)Instructions burned: 62 (million)
% 21.99/10.80 % (23208)------------------------------
% 21.99/10.80 % (23208)------------------------------
% 21.99/10.81 % (23211)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_23017 for (2982ds/453Mi)
% 21.99/10.82 % (23212)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_23017 for (2982ds/71Mi)
% 21.99/10.83 % (23210)Instruction limit reached!
% 21.99/10.83 % (23210)------------------------------
% 21.99/10.83 % (23210)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.99/10.83 % (23210)Termination reason: Unknown
% 21.99/10.83 % (23210)Termination phase: shuffling
% 21.99/10.83
% 21.99/10.83 % (23210)Memory used [KB]: 15223
% 21.99/10.83 % (23210)Time elapsed: 0.065 s
% 21.99/10.83 % (23210)Instructions burned: 96 (million)
% 21.99/10.83 % (23210)------------------------------
% 21.99/10.83 % (23210)------------------------------
% 21.99/10.83 % (23197)Instruction limit reached!
% 21.99/10.83 % (23197)------------------------------
% 21.99/10.83 % (23197)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.99/10.83 % (23197)Termination reason: Unknown
% 21.99/10.83 % (23197)Termination phase: shuffling
% 21.99/10.83
% 21.99/10.83 % (23197)Memory used [KB]: 22259
% 21.99/10.83 % (23197)Time elapsed: 0.248 s
% 21.99/10.83 % (23197)Instructions burned: 448 (million)
% 21.99/10.83 % (23197)------------------------------
% 21.99/10.83 % (23197)------------------------------
% 21.99/10.86 % (23213)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_23017 for (2982ds/83Mi)
% 21.99/10.86 % (23214)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_23017 for (2982ds/11Mi)
% 21.99/10.87 % (23214)Instruction limit reached!
% 21.99/10.87 % (23214)------------------------------
% 21.99/10.87 % (23214)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.99/10.87 % (23214)Termination reason: Unknown
% 21.99/10.87 % (23214)Termination phase: shuffling
% 21.99/10.87
% 21.99/10.87 % (23214)Memory used [KB]: 13944
% 21.99/10.87 % (23214)Time elapsed: 0.015 s
% 21.99/10.87 % (23214)Instructions burned: 11 (million)
% 21.99/10.87 % (23214)------------------------------
% 21.99/10.87 % (23214)------------------------------
% 21.99/10.87 % (23212)Instruction limit reached!
% 21.99/10.87 % (23212)------------------------------
% 21.99/10.87 % (23212)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.99/10.87 % (23212)Termination reason: Unknown
% 21.99/10.87 % (23212)Termination phase: shuffling
% 21.99/10.87
% 21.99/10.87 % (23212)Memory used [KB]: 14711
% 21.99/10.87 % (23212)Time elapsed: 0.050 s
% 21.99/10.87 % (23212)Instructions burned: 71 (million)
% 21.99/10.87 % (23212)------------------------------
% 21.99/10.87 % (23212)------------------------------
% 21.99/10.89 % (23215)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_23017 for (2982ds/41Mi)
% 21.99/10.90 % (23216)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_23017 for (2982ds/6Mi)
% 22.51/10.90 % (23216)Instruction limit reached!
% 22.51/10.90 % (23216)------------------------------
% 22.51/10.90 % (23216)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.51/10.90 % (23216)Termination reason: Unknown
% 22.51/10.90 % (23216)Termination phase: shuffling
% 22.51/10.90
% 22.51/10.90 % (23216)Memory used [KB]: 13560
% 22.51/10.90 % (23216)Time elapsed: 0.009 s
% 22.51/10.90 % (23216)Instructions burned: 6 (million)
% 22.51/10.90 % (23216)------------------------------
% 22.51/10.90 % (23216)------------------------------
% 22.51/10.91 % (23213)Instruction limit reached!
% 22.51/10.91 % (23213)------------------------------
% 22.51/10.91 % (23213)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.51/10.91 % (23213)Termination reason: Unknown
% 22.51/10.91 % (23213)Termination phase: shuffling
% 22.51/10.91
% 22.51/10.91 % (23213)Memory used [KB]: 15095
% 22.51/10.91 % (23213)Time elapsed: 0.059 s
% 22.51/10.91 % (23213)Instructions burned: 83 (million)
% 22.51/10.91 % (23213)------------------------------
% 22.51/10.91 % (23213)------------------------------
% 22.51/10.92 % (23200)Instruction limit reached!
% 22.51/10.92 % (23200)------------------------------
% 22.51/10.92 % (23200)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.51/10.92 % (23200)Termination reason: Unknown
% 22.51/10.92 % (23200)Termination phase: shuffling
% 22.51/10.92
% 22.51/10.92 % (23200)Memory used [KB]: 20340
% 22.51/10.92 % (23200)Time elapsed: 0.292 s
% 22.51/10.92 % (23200)Instructions burned: 508 (million)
% 22.51/10.92 % (23200)------------------------------
% 22.51/10.92 % (23200)------------------------------
% 22.51/10.92 % (23217)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_23017 for (2981ds/150Mi)
% 22.51/10.92 % (23215)Instruction limit reached!
% 22.51/10.92 % (23215)------------------------------
% 22.51/10.92 % (23215)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.51/10.92 % (23215)Termination reason: Unknown
% 22.51/10.92 % (23215)Termination phase: shuffling
% 22.51/10.92
% 22.51/10.92 % (23215)Memory used [KB]: 14456
% 22.51/10.92 % (23215)Time elapsed: 0.034 s
% 22.51/10.92 % (23215)Instructions burned: 41 (million)
% 22.51/10.92 % (23215)------------------------------
% 22.51/10.92 % (23215)------------------------------
% 22.51/10.93 % (23218)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_23017 for (2981ds/343Mi)
% 22.51/10.95 % (23220)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_23017 for (2981ds/253Mi)
% 22.51/10.95 % (23219)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_23017 for (2981ds/109Mi)
% 22.51/10.96 % (23207)Instruction limit reached!
% 22.51/10.96 % (23207)------------------------------
% 22.51/10.96 % (23207)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.51/10.96 % (23207)Termination reason: Unknown
% 22.51/10.96 % (23207)Termination phase: shuffling
% 22.51/10.96
% 22.51/10.96 % (23207)Memory used [KB]: 22003
% 22.51/10.96 % (23207)Time elapsed: 0.239 s
% 22.51/10.96 % (23207)Instructions burned: 432 (million)
% 22.51/10.96 % (23207)------------------------------
% 22.51/10.96 % (23207)------------------------------
% 22.51/10.99 % (23221)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_23017 for (2981ds/140Mi)
% 22.51/11.02 % (23219)Instruction limit reached!
% 22.51/11.02 % (23219)------------------------------
% 22.51/11.02 % (23219)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.51/11.02 % (23219)Termination reason: Unknown
% 22.51/11.02 % (23219)Termination phase: shuffling
% 22.51/11.02
% 22.51/11.02 % (23219)Memory used [KB]: 15479
% 22.51/11.02 % (23219)Time elapsed: 0.073 s
% 22.51/11.02 % (23219)Instructions burned: 110 (million)
% 22.51/11.02 % (23219)------------------------------
% 22.51/11.02 % (23219)------------------------------
% 22.51/11.02 % (23217)Instruction limit reached!
% 22.51/11.02 % (23217)------------------------------
% 22.51/11.02 % (23217)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.51/11.02 % (23217)Termination reason: Unknown
% 22.51/11.02 % (23217)Termination phase: shuffling
% 22.51/11.02
% 22.51/11.02 % (23217)Memory used [KB]: 15863
% 22.51/11.02 % (23217)Time elapsed: 0.095 s
% 22.51/11.02 % (23217)Instructions burned: 151 (million)
% 22.51/11.02 % (23217)------------------------------
% 22.51/11.02 % (23217)------------------------------
% 24.26/11.04 % (23222)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_23017 for (2980ds/614Mi)
% 24.26/11.04 % (23223)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_23017 for (2980ds/202Mi)
% 24.26/11.06 % (23211)Instruction limit reached!
% 24.26/11.06 % (23211)------------------------------
% 24.26/11.06 % (23211)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.26/11.06 % (23211)Termination reason: Unknown
% 24.26/11.06 % (23211)Termination phase: shuffling
% 24.26/11.06
% 24.26/11.06 % (23211)Memory used [KB]: 22259
% 24.26/11.06 % (23211)Time elapsed: 0.251 s
% 24.26/11.06 % (23211)Instructions burned: 453 (million)
% 24.26/11.06 % (23211)------------------------------
% 24.26/11.06 % (23211)------------------------------
% 24.26/11.07 % (23221)Instruction limit reached!
% 24.26/11.07 % (23221)------------------------------
% 24.26/11.07 % (23221)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.26/11.07 % (23221)Termination reason: Unknown
% 24.26/11.07 % (23221)Termination phase: shuffling
% 24.26/11.07
% 24.26/11.07 % (23221)Memory used [KB]: 15735
% 24.26/11.07 % (23221)Time elapsed: 0.089 s
% 24.26/11.07 % (23221)Instructions burned: 141 (million)
% 24.26/11.07 % (23221)------------------------------
% 24.26/11.07 % (23221)------------------------------
% 24.26/11.08 % (23224)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_23017 for (2980ds/339Mi)
% 24.26/11.10 % (23220)Instruction limit reached!
% 24.26/11.10 % (23220)------------------------------
% 24.26/11.10 % (23220)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.26/11.10 % (23220)Termination reason: Unknown
% 24.26/11.10 % (23220)Termination phase: shuffling
% 24.26/11.10
% 24.26/11.10 % (23220)Memory used [KB]: 17270
% 24.26/11.10 % (23220)Time elapsed: 0.152 s
% 24.26/11.10 % (23220)Instructions burned: 253 (million)
% 24.26/11.10 % (23220)------------------------------
% 24.26/11.10 % (23220)------------------------------
% 24.26/11.10 % (23225)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_23017 for (2980ds/283Mi)
% 24.26/11.12 % (23226)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_23017 for (2979ds/134Mi)
% 24.65/11.13 % (23218)Instruction limit reached!
% 24.65/11.13 % (23218)------------------------------
% 24.65/11.13 % (23218)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.65/11.13 % (23218)Termination reason: Unknown
% 24.65/11.13 % (23218)Termination phase: shuffling
% 24.65/11.13
% 24.65/11.13 % (23218)Memory used [KB]: 18293
% 24.65/11.13 % (23218)Time elapsed: 0.203 s
% 24.65/11.13 % (23218)Instructions burned: 344 (million)
% 24.65/11.13 % (23218)------------------------------
% 24.65/11.13 % (23218)------------------------------
% 24.65/11.16 % (23227)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_23017 for (2979ds/298Mi)
% 24.65/11.16 % (23223)Instruction limit reached!
% 24.65/11.16 % (23223)------------------------------
% 24.65/11.16 % (23223)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.65/11.16 % (23223)Termination reason: Unknown
% 24.65/11.16 % (23223)Termination phase: shuffling
% 24.65/11.16
% 24.65/11.16 % (23223)Memory used [KB]: 16630
% 24.65/11.16 % (23223)Time elapsed: 0.125 s
% 24.65/11.16 % (23223)Instructions burned: 202 (million)
% 24.65/11.16 % (23223)------------------------------
% 24.65/11.16 % (23223)------------------------------
% 24.65/11.19 % (23228)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_23017 for (2979ds/192Mi)
% 24.89/11.20 % (23226)Instruction limit reached!
% 24.89/11.20 % (23226)------------------------------
% 24.89/11.20 % (23226)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.89/11.20 % (23226)Termination reason: Unknown
% 24.89/11.20 % (23226)Termination phase: shuffling
% 24.89/11.20
% 24.89/11.20 % (23226)Memory used [KB]: 15863
% 24.89/11.20 % (23226)Time elapsed: 0.087 s
% 24.89/11.20 % (23226)Instructions burned: 134 (million)
% 24.89/11.20 % (23226)------------------------------
% 24.89/11.20 % (23226)------------------------------
% 24.89/11.23 % (23229)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_23017 for (2978ds/258Mi)
% 24.89/11.26 % (23225)Instruction limit reached!
% 24.89/11.26 % (23225)------------------------------
% 24.89/11.26 % (23225)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.89/11.26 % (23225)Termination reason: Unknown
% 24.89/11.26 % (23225)Termination phase: shuffling
% 24.89/11.26
% 24.89/11.26 % (23225)Memory used [KB]: 19957
% 24.89/11.26 % (23225)Time elapsed: 0.162 s
% 24.89/11.26 % (23225)Instructions burned: 284 (million)
% 24.89/11.26 % (23225)------------------------------
% 24.89/11.26 % (23225)------------------------------
% 24.89/11.28 % (23230)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_23017 for (2978ds/90Mi)
% 24.89/11.28 % (23224)Instruction limit reached!
% 24.89/11.28 % (23224)------------------------------
% 24.89/11.28 % (23224)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.89/11.28 % (23224)Termination reason: Unknown
% 24.89/11.28 % (23224)Termination phase: shuffling
% 24.89/11.28
% 24.89/11.28 % (23224)Memory used [KB]: 18293
% 24.89/11.28 % (23224)Time elapsed: 0.202 s
% 24.89/11.28 % (23224)Instructions burned: 340 (million)
% 24.89/11.28 % (23224)------------------------------
% 24.89/11.28 % (23224)------------------------------
% 25.51/11.30 % (23228)Instruction limit reached!
% 25.51/11.30 % (23228)------------------------------
% 25.51/11.30 % (23228)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.51/11.30 % (23228)Termination reason: Unknown
% 25.51/11.30 % (23228)Termination phase: shuffling
% 25.51/11.30
% 25.51/11.30 % (23228)Memory used [KB]: 16502
% 25.51/11.30 % (23228)Time elapsed: 0.118 s
% 25.51/11.30 % (23228)Instructions burned: 192 (million)
% 25.51/11.30 % (23228)------------------------------
% 25.51/11.30 % (23228)------------------------------
% 25.51/11.31 % (23231)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_23017 for (2977ds/231Mi)
% 25.51/11.32 % (23204)Instruction limit reached!
% 25.51/11.32 % (23204)------------------------------
% 25.51/11.32 % (23204)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.51/11.32 % (23204)Termination reason: Unknown
% 25.51/11.32 % (23204)Termination phase: Property scanning
% 25.51/11.32
% 25.51/11.32 % (23204)Memory used [KB]: 27760
% 25.51/11.32 % (23204)Time elapsed: 0.615 s
% 25.51/11.32 % (23204)Instructions burned: 1088 (million)
% 25.51/11.32 % (23204)------------------------------
% 25.51/11.32 % (23204)------------------------------
% 25.51/11.33 % (23227)Instruction limit reached!
% 25.51/11.33 % (23227)------------------------------
% 25.51/11.33 % (23227)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.51/11.33 % (23227)Termination reason: Unknown
% 25.51/11.33 % (23227)Termination phase: shuffling
% 25.51/11.33
% 25.51/11.33 % (23227)Memory used [KB]: 20084
% 25.51/11.33 % (23227)Time elapsed: 0.169 s
% 25.51/11.33 % (23227)Instructions burned: 298 (million)
% 25.51/11.33 % (23227)------------------------------
% 25.51/11.33 % (23227)------------------------------
% 25.51/11.33 % (23232)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_23017 for (2977ds/458Mi)
% 25.51/11.34 % (23230)Instruction limit reached!
% 25.51/11.34 % (23230)------------------------------
% 25.51/11.34 % (23230)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.51/11.34 % (23230)Termination reason: Unknown
% 25.51/11.34 % (23230)Termination phase: shuffling
% 25.51/11.34
% 25.51/11.34 % (23230)Memory used [KB]: 15223
% 25.51/11.34 % (23230)Time elapsed: 0.063 s
% 25.51/11.34 % (23230)Instructions burned: 91 (million)
% 25.51/11.34 % (23230)------------------------------
% 25.51/11.34 % (23230)------------------------------
% 25.51/11.34 % (23229)Instruction limit reached!
% 25.51/11.34 % (23229)------------------------------
% 25.51/11.34 % (23229)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.51/11.34 % (23229)Termination reason: Unknown
% 25.51/11.34 % (23229)Termination phase: shuffling
% 25.51/11.34
% 25.51/11.34 % (23229)Memory used [KB]: 17398
% 25.51/11.34 % (23229)Time elapsed: 0.118 s
% 25.51/11.34 % (23229)Instructions burned: 260 (million)
% 25.51/11.34 % (23229)------------------------------
% 25.51/11.34 % (23229)------------------------------
% 25.51/11.35 % (23234)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_23017 for (2977ds/297Mi)
% 25.51/11.35 % (23233)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_23017 for (2977ds/31Mi)
% 25.51/11.36 % (23236)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_23017 for (2977ds/182Mi)
% 25.51/11.37 % (23235)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_23017 for (2977ds/987Mi)
% 25.51/11.38 % (23233)Instruction limit reached!
% 25.51/11.38 % (23233)------------------------------
% 25.51/11.38 % (23233)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.51/11.38 % (23233)Termination reason: Unknown
% 25.51/11.38 % (23233)Termination phase: shuffling
% 25.51/11.38
% 25.51/11.38 % (23233)Memory used [KB]: 14328
% 25.51/11.38 % (23233)Time elapsed: 0.029 s
% 25.51/11.38 % (23233)Instructions burned: 32 (million)
% 25.51/11.38 % (23233)------------------------------
% 25.51/11.38 % (23233)------------------------------
% 25.51/11.39 % (23222)Instruction limit reached!
% 25.51/11.39 % (23222)------------------------------
% 25.51/11.39 % (23222)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.51/11.39 % (23222)Termination reason: Unknown
% 25.51/11.39 % (23222)Termination phase: shuffling
% 25.51/11.39
% 25.51/11.39 % (23222)Memory used [KB]: 21875
% 25.51/11.39 % (23222)Time elapsed: 0.355 s
% 25.51/11.39 % (23222)Instructions burned: 615 (million)
% 25.51/11.39 % (23222)------------------------------
% 25.51/11.39 % (23222)------------------------------
% 26.52/11.40 % (23237)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_23017 for (2977ds/182Mi)
% 26.52/11.42 % (23238)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_23017 for (2976ds/154Mi)
% 26.52/11.44 % (23231)Instruction limit reached!
% 26.52/11.44 % (23231)------------------------------
% 26.52/11.44 % (23231)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.52/11.44 % (23231)Termination reason: Unknown
% 26.52/11.44 % (23231)Termination phase: shuffling
% 26.52/11.44
% 26.52/11.44 % (23231)Memory used [KB]: 17142
% 26.52/11.44 % (23231)Time elapsed: 0.133 s
% 26.52/11.44 % (23231)Instructions burned: 231 (million)
% 26.52/11.44 % (23231)------------------------------
% 26.52/11.44 % (23231)------------------------------
% 26.52/11.46 % (23239)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_23017 for (2976ds/79Mi)
% 26.52/11.46 % (23236)Instruction limit reached!
% 26.52/11.46 % (23236)------------------------------
% 26.52/11.46 % (23236)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.52/11.46 % (23236)Termination reason: Unknown
% 26.52/11.46 % (23236)Termination phase: shuffling
% 26.52/11.46
% 26.52/11.46 % (23236)Memory used [KB]: 16119
% 26.52/11.46 % (23236)Time elapsed: 0.111 s
% 26.52/11.46 % (23236)Instructions burned: 182 (million)
% 26.52/11.46 % (23236)------------------------------
% 26.52/11.46 % (23236)------------------------------
% 26.52/11.49 % (23240)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_23017 for (2976ds/216Mi)
% 26.79/11.51 % (23238)Instruction limit reached!
% 26.79/11.51 % (23238)------------------------------
% 26.79/11.51 % (23238)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.79/11.51 % (23238)Termination reason: Unknown
% 26.79/11.51 % (23238)Termination phase: shuffling
% 26.79/11.51
% 26.79/11.51 % (23238)Memory used [KB]: 16119
% 26.79/11.51 % (23238)Time elapsed: 0.093 s
% 26.79/11.51 % (23238)Instructions burned: 154 (million)
% 26.79/11.51 % (23238)------------------------------
% 26.79/11.51 % (23238)------------------------------
% 26.79/11.51 % (23237)Instruction limit reached!
% 26.79/11.51 % (23237)------------------------------
% 26.79/11.51 % (23237)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.79/11.51 % (23237)Termination reason: Unknown
% 26.79/11.51 % (23237)Termination phase: shuffling
% 26.79/11.51
% 26.79/11.51 % (23237)Memory used [KB]: 16375
% 26.79/11.51 % (23237)Time elapsed: 0.113 s
% 26.79/11.51 % (23237)Instructions burned: 182 (million)
% 26.79/11.51 % (23237)------------------------------
% 26.79/11.51 % (23237)------------------------------
% 26.79/11.52 % (23234)Instruction limit reached!
% 26.79/11.52 % (23234)------------------------------
% 26.79/11.52 % (23234)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.79/11.52 % (23234)Termination reason: Unknown
% 26.79/11.52 % (23234)Termination phase: shuffling
% 26.79/11.52
% 26.79/11.52 % (23234)Memory used [KB]: 20212
% 26.79/11.52 % (23234)Time elapsed: 0.169 s
% 26.79/11.52 % (23234)Instructions burned: 298 (million)
% 26.79/11.52 % (23234)------------------------------
% 26.79/11.52 % (23234)------------------------------
% 26.79/11.52 % (23239)Instruction limit reached!
% 26.79/11.52 % (23239)------------------------------
% 26.79/11.52 % (23239)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.79/11.52 % (23239)Termination reason: Unknown
% 26.79/11.52 % (23239)Termination phase: shuffling
% 26.79/11.52
% 26.79/11.52 % (23239)Memory used [KB]: 14967
% 26.79/11.52 % (23239)Time elapsed: 0.056 s
% 26.79/11.52 % (23239)Instructions burned: 79 (million)
% 26.79/11.52 % (23239)------------------------------
% 26.79/11.52 % (23239)------------------------------
% 26.79/11.53 % (23241)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_23017 for (2975ds/114Mi)
% 26.79/11.53 % (23242)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_23017 for (2975ds/922Mi)
% 26.79/11.54 % (23243)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_23017 for (2975ds/182Mi)
% 26.79/11.54 % (23244)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_23017 for (2975ds/164Mi)
% 26.79/11.59 % (23232)Instruction limit reached!
% 26.79/11.59 % (23232)------------------------------
% 26.79/11.59 % (23232)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.79/11.59 % (23232)Termination reason: Unknown
% 26.79/11.59 % (23232)Termination phase: shuffling
% 26.79/11.59
% 26.79/11.59 % (23232)Memory used [KB]: 19701
% 26.79/11.59 % (23232)Time elapsed: 0.265 s
% 26.79/11.59 % (23232)Instructions burned: 459 (million)
% 26.79/11.59 % (23232)------------------------------
% 26.79/11.59 % (23232)------------------------------
% 27.60/11.60 % (23241)Instruction limit reached!
% 27.60/11.60 % (23241)------------------------------
% 27.60/11.60 % (23241)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.60/11.60 % (23241)Termination reason: Unknown
% 27.60/11.60 % (23241)Termination phase: shuffling
% 27.60/11.60
% 27.60/11.60 % (23241)Memory used [KB]: 15479
% 27.60/11.60 % (23241)Time elapsed: 0.076 s
% 27.60/11.60 % (23241)Instructions burned: 114 (million)
% 27.60/11.60 % (23241)------------------------------
% 27.60/11.60 % (23241)------------------------------
% 27.60/11.61 % (23240)Instruction limit reached!
% 27.60/11.61 % (23240)------------------------------
% 27.60/11.61 % (23240)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.60/11.61 % (23240)Termination reason: Unknown
% 27.60/11.61 % (23240)Termination phase: shuffling
% 27.60/11.61
% 27.60/11.61 % (23240)Memory used [KB]: 17014
% 27.60/11.61 % (23240)Time elapsed: 0.126 s
% 27.60/11.61 % (23240)Instructions burned: 217 (million)
% 27.60/11.61 % (23240)------------------------------
% 27.60/11.61 % (23240)------------------------------
% 27.60/11.62 % (23245)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_23017 for (2974ds/50Mi)
% 27.60/11.63 % (23246)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_23017 for (2974ds/368Mi)
% 27.60/11.63 % (23244)Instruction limit reached!
% 27.60/11.63 % (23244)------------------------------
% 27.60/11.63 % (23244)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.60/11.63 % (23244)Termination reason: Unknown
% 27.60/11.63 % (23244)Termination phase: shuffling
% 27.60/11.63
% 27.60/11.63 % (23244)Memory used [KB]: 16119
% 27.60/11.63 % (23244)Time elapsed: 0.098 s
% 27.60/11.63 % (23244)Instructions burned: 165 (million)
% 27.60/11.63 % (23244)------------------------------
% 27.60/11.63 % (23244)------------------------------
% 27.60/11.64 % (23247)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_23017 for (2974ds/1021Mi)
% 27.60/11.64 % (23243)Instruction limit reached!
% 27.60/11.64 % (23243)------------------------------
% 27.60/11.64 % (23243)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.60/11.64 % (23243)Termination reason: Unknown
% 27.60/11.64 % (23243)Termination phase: shuffling
% 27.60/11.64
% 27.60/11.64 % (23243)Memory used [KB]: 16375
% 27.60/11.64 % (23243)Time elapsed: 0.106 s
% 27.60/11.64 % (23243)Instructions burned: 183 (million)
% 27.60/11.64 % (23243)------------------------------
% 27.60/11.64 % (23243)------------------------------
% 27.60/11.65 % (23245)Instruction limit reached!
% 27.60/11.65 % (23245)------------------------------
% 27.60/11.65 % (23245)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.60/11.65 % (23245)Termination reason: Unknown
% 27.60/11.65 % (23245)Termination phase: shuffling
% 27.60/11.65
% 27.60/11.65 % (23245)Memory used [KB]: 14456
% 27.60/11.65 % (23245)Time elapsed: 0.039 s
% 27.60/11.65 % (23245)Instructions burned: 51 (million)
% 27.60/11.65 % (23245)------------------------------
% 27.60/11.65 % (23245)------------------------------
% 27.60/11.66 % (23248)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_23017 for (2974ds/267Mi)
% 27.60/11.67 % (23249)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_23017 for (2974ds/204Mi)
% 27.60/11.68 % (23250)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_23017 for (2974ds/26Mi)
% 28.14/11.70 % (23250)Instruction limit reached!
% 28.14/11.70 % (23250)------------------------------
% 28.14/11.70 % (23250)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.14/11.70 % (23250)Termination reason: Unknown
% 28.14/11.70 % (23250)Termination phase: shuffling
% 28.14/11.70
% 28.14/11.70 % (23250)Memory used [KB]: 14328
% 28.14/11.70 % (23250)Time elapsed: 0.027 s
% 28.14/11.70 % (23250)Instructions burned: 27 (million)
% 28.14/11.70 % (23250)------------------------------
% 28.14/11.70 % (23250)------------------------------
% 28.14/11.73 % (23251)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_23017 for (2973ds/53Mi)
% 28.14/11.76 % (23251)Instruction limit reached!
% 28.14/11.76 % (23251)------------------------------
% 28.14/11.76 % (23251)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.14/11.76 % (23251)Termination reason: Unknown
% 28.14/11.76 % (23251)Termination phase: shuffling
% 28.14/11.76
% 28.14/11.76 % (23251)Memory used [KB]: 14583
% 28.14/11.76 % (23251)Time elapsed: 0.041 s
% 28.14/11.76 % (23251)Instructions burned: 53 (million)
% 28.14/11.76 % (23251)------------------------------
% 28.14/11.76 % (23251)------------------------------
% 28.14/11.79 % (23252)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_23017 for (2973ds/4436Mi)
% 28.14/11.79 % (23249)Instruction limit reached!
% 28.14/11.79 % (23249)------------------------------
% 28.14/11.79 % (23249)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.14/11.79 % (23249)Termination reason: Unknown
% 28.14/11.79 % (23249)Termination phase: shuffling
% 28.14/11.79
% 28.14/11.79 % (23249)Memory used [KB]: 16630
% 28.14/11.79 % (23249)Time elapsed: 0.125 s
% 28.14/11.79 % (23249)Instructions burned: 204 (million)
% 28.14/11.79 % (23249)------------------------------
% 28.14/11.79 % (23249)------------------------------
% 28.39/11.81 % (23248)Instruction limit reached!
% 28.39/11.81 % (23248)------------------------------
% 28.39/11.81 % (23248)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.39/11.81 % (23248)Termination reason: Unknown
% 28.39/11.81 % (23248)Termination phase: shuffling
% 28.39/11.81
% 28.39/11.81 % (23248)Memory used [KB]: 19701
% 28.39/11.81 % (23248)Time elapsed: 0.153 s
% 28.39/11.81 % (23248)Instructions burned: 268 (million)
% 28.39/11.81 % (23248)------------------------------
% 28.39/11.81 % (23248)------------------------------
% 28.39/11.81 % (23253)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_23017 for (2972ds/1539Mi)
% 28.39/11.83 % (23254)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_23017 for (2972ds/408Mi)
% 28.39/11.84 % (23246)Instruction limit reached!
% 28.39/11.84 % (23246)------------------------------
% 28.39/11.84 % (23246)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.39/11.84 % (23246)Termination reason: Unknown
% 28.39/11.84 % (23246)Termination phase: shuffling
% 28.39/11.84
% 28.39/11.84 % (23246)Memory used [KB]: 18549
% 28.39/11.84 % (23246)Time elapsed: 0.215 s
% 28.39/11.84 % (23246)Instructions burned: 368 (million)
% 28.39/11.84 % (23246)------------------------------
% 28.39/11.84 % (23246)------------------------------
% 28.39/11.86 % (23255)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_23017 for (2972ds/5205Mi)
% 28.81/11.90 % (23235)Instruction limit reached!
% 28.81/11.90 % (23235)------------------------------
% 28.81/11.90 % (23235)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.81/11.90 % (23235)Termination reason: Unknown
% 28.81/11.90 % (23235)Termination phase: shuffling
% 28.81/11.90
% 28.81/11.90 % (23235)Memory used [KB]: 33389
% 28.81/11.90 % (23235)Time elapsed: 0.536 s
% 28.81/11.90 % (23235)Instructions burned: 987 (million)
% 28.81/11.90 % (23235)------------------------------
% 28.81/11.90 % (23235)------------------------------
% 28.81/11.93 % (23256)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_23017 for (2971ds/1913Mi)
% 31.33/12.07 % (23254)Instruction limit reached!
% 31.33/12.07 % (23254)------------------------------
% 31.33/12.07 % (23254)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 31.33/12.07 % (23254)Termination reason: Unknown
% 31.33/12.07 % (23254)Termination phase: shuffling
% 31.33/12.07
% 31.33/12.07 % (23254)Memory used [KB]: 19189
% 31.33/12.07 % (23254)Time elapsed: 0.237 s
% 31.33/12.07 % (23254)Instructions burned: 408 (million)
% 31.33/12.07 % (23254)------------------------------
% 31.33/12.07 % (23254)------------------------------
% 31.33/12.09 % (23242)Instruction limit reached!
% 31.33/12.09 % (23242)------------------------------
% 31.33/12.09 % (23242)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 31.33/12.09 % (23242)Termination reason: Unknown
% 31.33/12.09 % (23242)Termination phase: Property scanning
% 31.33/12.09
% 31.33/12.09 % (23242)Memory used [KB]: 27760
% 31.33/12.09 % (23242)Time elapsed: 0.548 s
% 31.33/12.09 % (23242)Instructions burned: 923 (million)
% 31.33/12.09 % (23242)------------------------------
% 31.33/12.09 % (23242)------------------------------
% 31.33/12.09 % (23257)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_23017 for (2970ds/4350Mi)
% 31.33/12.12 % (23258)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_23017 for (2969ds/68Mi)
% 31.95/12.16 % (23258)Instruction limit reached!
% 31.95/12.16 % (23258)------------------------------
% 31.95/12.16 % (23258)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 31.95/12.16 % (23258)Termination reason: Unknown
% 31.95/12.16 % (23258)Termination phase: shuffling
% 31.95/12.16
% 31.95/12.16 % (23258)Memory used [KB]: 14839
% 31.95/12.16 % (23258)Time elapsed: 0.050 s
% 31.95/12.16 % (23258)Instructions burned: 69 (million)
% 31.95/12.16 % (23258)------------------------------
% 31.95/12.16 % (23258)------------------------------
% 31.95/12.19 % (23259)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_23017 for (2969ds/358Mi)
% 32.28/12.22 % (23247)Instruction limit reached!
% 32.28/12.22 % (23247)------------------------------
% 32.28/12.22 % (23247)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.28/12.22 % (23247)Termination reason: Unknown
% 32.28/12.22 % (23247)Termination phase: Property scanning
% 32.28/12.22
% 32.28/12.22 % (23247)Memory used [KB]: 27760
% 32.28/12.22 % (23247)Time elapsed: 0.581 s
% 32.28/12.22 % (23247)Instructions burned: 1021 (million)
% 32.28/12.22 % (23247)------------------------------
% 32.28/12.22 % (23247)------------------------------
% 32.28/12.24 % (23260)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_23017 for (2968ds/31Mi)
% 32.28/12.27 % (23260)Instruction limit reached!
% 32.28/12.27 % (23260)------------------------------
% 32.28/12.27 % (23260)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.28/12.27 % (23260)Termination reason: Unknown
% 32.28/12.27 % (23260)Termination phase: shuffling
% 32.28/12.27
% 32.28/12.27 % (23260)Memory used [KB]: 14200
% 32.28/12.27 % (23260)Time elapsed: 0.030 s
% 32.28/12.27 % (23260)Instructions burned: 31 (million)
% 32.28/12.27 % (23260)------------------------------
% 32.28/12.27 % (23260)------------------------------
% 32.28/12.29 % (23261)lrs+10_1:1_cnfonf=lazy_simp:sgt=5:sos=on:ss=axioms:i=279:si=on:rtra=on_0 on DTF2THF_23017 for (2968ds/279Mi)
% 32.76/12.40 % (23259)Instruction limit reached!
% 32.76/12.40 % (23259)------------------------------
% 32.76/12.40 % (23259)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.76/12.40 % (23259)Termination reason: Unknown
% 32.76/12.40 % (23259)Termination phase: shuffling
% 32.76/12.40
% 32.76/12.40 % (23259)Memory used [KB]: 18549
% 32.76/12.40 % (23259)Time elapsed: 0.210 s
% 32.76/12.40 % (23259)Instructions burned: 359 (million)
% 32.76/12.40 % (23259)------------------------------
% 32.76/12.40 % (23259)------------------------------
% 33.17/12.42 % (23262)lrs+10_8:1_au=on:fde=none:slsq=on:i=258:si=on:rtra=on_0 on DTF2THF_23017 for (2966ds/258Mi)
% 33.17/12.45 % (23261)Instruction limit reached!
% 33.17/12.45 % (23261)------------------------------
% 33.17/12.45 % (23261)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.17/12.45 % (23261)Termination reason: Unknown
% 33.17/12.45 % (23261)Termination phase: shuffling
% 33.17/12.45
% 33.17/12.45 % (23261)Memory used [KB]: 19957
% 33.17/12.45 % (23261)Time elapsed: 0.161 s
% 33.17/12.45 % (23261)Instructions burned: 280 (million)
% 33.17/12.45 % (23261)------------------------------
% 33.17/12.45 % (23261)------------------------------
% 33.17/12.47 % (23263)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_23017 for (2966ds/26Mi)
% 33.98/12.50 % (23263)Instruction limit reached!
% 33.98/12.50 % (23263)------------------------------
% 33.98/12.50 % (23263)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.98/12.50 % (23263)Termination reason: Unknown
% 33.98/12.50 % (23263)Termination phase: shuffling
% 33.98/12.50
% 33.98/12.50 % (23263)Memory used [KB]: 14200
% 33.98/12.50 % (23263)Time elapsed: 0.026 s
% 33.98/12.50 % (23263)Instructions burned: 26 (million)
% 33.98/12.50 % (23263)------------------------------
% 33.98/12.50 % (23263)------------------------------
% 33.98/12.52 % (23264)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_23017 for (2965ds/995Mi)
% 33.98/12.57 % (23262)Instruction limit reached!
% 33.98/12.57 % (23262)------------------------------
% 33.98/12.57 % (23262)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.98/12.57 % (23262)Termination reason: Unknown
% 33.98/12.57 % (23262)Termination phase: shuffling
% 33.98/12.57
% 33.98/12.57 % (23262)Memory used [KB]: 17270
% 33.98/12.57 % (23262)Time elapsed: 0.156 s
% 33.98/12.57 % (23262)Instructions burned: 259 (million)
% 33.98/12.57 % (23262)------------------------------
% 33.98/12.57 % (23262)------------------------------
% 34.60/12.60 % (23265)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_23017 for (2965ds/373Mi)
% 34.60/12.66 % (23253)Instruction limit reached!
% 34.60/12.66 % (23253)------------------------------
% 34.60/12.66 % (23253)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.60/12.66 % (23253)Termination reason: Unknown
% 34.60/12.66 % (23253)Termination phase: Property scanning
% 34.60/12.66
% 34.60/12.66 % (23253)Memory used [KB]: 27632
% 34.60/12.66 % (23253)Time elapsed: 0.846 s
% 34.60/12.66 % (23253)Instructions burned: 1540 (million)
% 34.60/12.66 % (23253)------------------------------
% 34.60/12.66 % (23253)------------------------------
% 34.60/12.68 % (23266)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_23017 for (2964ds/253Mi)
% 35.26/12.82 % (23265)Instruction limit reached!
% 35.26/12.82 % (23265)------------------------------
% 35.26/12.82 % (23265)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.26/12.82 % (23265)Termination reason: Unknown
% 35.26/12.82 % (23265)Termination phase: shuffling
% 35.26/12.82
% 35.26/12.82 % (23265)Memory used [KB]: 18677
% 35.26/12.82 % (23265)Time elapsed: 0.221 s
% 35.26/12.82 % (23265)Instructions burned: 373 (million)
% 35.26/12.82 % (23265)------------------------------
% 35.26/12.82 % (23265)------------------------------
% 35.26/12.83 % (23266)Instruction limit reached!
% 35.26/12.83 % (23266)------------------------------
% 35.26/12.83 % (23266)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.26/12.83 % (23266)Termination reason: Unknown
% 35.26/12.83 % (23266)Termination phase: shuffling
% 35.26/12.83
% 35.26/12.83 % (23266)Memory used [KB]: 19573
% 35.26/12.83 % (23266)Time elapsed: 0.146 s
% 35.26/12.83 % (23266)Instructions burned: 253 (million)
% 35.26/12.83 % (23266)------------------------------
% 35.26/12.83 % (23266)------------------------------
% 35.26/12.84 % (23267)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_23017 for (2962ds/832Mi)
% 35.26/12.85 % (23268)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_23017 for (2962ds/668Mi)
% 35.98/13.00 % (23256)Instruction limit reached!
% 35.98/13.00 % (23256)------------------------------
% 35.98/13.00 % (23256)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.98/13.00 % (23256)Termination reason: Unknown
% 35.98/13.00 % (23256)Termination phase: SInE selection
% 35.98/13.00
% 35.98/13.00 % (23256)Memory used [KB]: 28528
% 35.98/13.00 % (23256)Time elapsed: 1.077 s
% 35.98/13.00 % (23256)Instructions burned: 1913 (million)
% 35.98/13.00 % (23256)------------------------------
% 35.98/13.00 % (23256)------------------------------
% 38.20/13.03 % (23269)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_23017 for (2960ds/40Mi)
% 38.20/13.06 % (23269)Instruction limit reached!
% 38.20/13.06 % (23269)------------------------------
% 38.20/13.06 % (23269)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 38.20/13.06 % (23269)Termination reason: Unknown
% 38.20/13.06 % (23269)Termination phase: shuffling
% 38.20/13.06
% 38.20/13.06 % (23269)Memory used [KB]: 14456
% 38.20/13.06 % (23269)Time elapsed: 0.034 s
% 38.20/13.06 % (23269)Instructions burned: 41 (million)
% 38.20/13.06 % (23264)Instruction limit reached!
% 38.20/13.06 % (23264)------------------------------
% 38.20/13.06 % (23264)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 38.20/13.06 % (23264)Termination reason: Unknown
% 38.20/13.06 % (23264)Termination phase: shuffling
% 38.20/13.06
% 38.20/13.06 % (23264)Memory used [KB]: 33517
% 38.20/13.06 % (23264)Time elapsed: 0.541 s
% 38.20/13.06 % (23264)Instructions burned: 995 (million)
% 38.20/13.06 % (23264)------------------------------
% 38.20/13.06 % (23264)------------------------------
% 38.20/13.06 % (23269)------------------------------
% 38.20/13.06 % (23269)------------------------------
% 38.20/13.08 % (23270)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_23017 for (2960ds/294Mi)
% 38.20/13.08 % (23271)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_23017 for (2960ds/187Mi)
% 38.82/13.19 % (23271)Instruction limit reached!
% 38.82/13.19 % (23271)------------------------------
% 38.82/13.19 % (23271)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 38.82/13.19 % (23271)Termination reason: Unknown
% 38.82/13.19 % (23271)Termination phase: shuffling
% 38.82/13.19
% 38.82/13.19 % (23271)Memory used [KB]: 16119
% 38.82/13.19 % (23271)Time elapsed: 0.112 s
% 38.82/13.19 % (23271)Instructions burned: 187 (million)
% 38.82/13.19 % (23271)------------------------------
% 38.82/13.19 % (23271)------------------------------
% 39.16/13.21 % (23268)Instruction limit reached!
% 39.16/13.21 % (23268)------------------------------
% 39.16/13.21 % (23268)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 39.16/13.21 % (23268)Termination reason: Unknown
% 39.16/13.21 % (23268)Termination phase: shuffling
% 39.16/13.21
% 39.16/13.21 % (23268)Memory used [KB]: 25202
% 39.16/13.21 % (23268)Time elapsed: 0.365 s
% 39.16/13.21 % (23268)Instructions burned: 668 (million)
% 39.16/13.21 % (23268)------------------------------
% 39.16/13.21 % (23268)------------------------------
% 39.16/13.22 % (23272)lrs+1010_2:3_au=on:fsr=off:sd=4:ss=axioms:st=1.5:i=371:si=on:rtra=on_0 on DTF2THF_23017 for (2958ds/371Mi)
% 39.16/13.24 % (23273)dis+1003_1:1_au=on:bd=preordered:fde=none:ixr=off:nwc=10.0:i=1827:si=on:rtra=on_0 on DTF2THF_23017 for (2958ds/1827Mi)
% 39.16/13.25 % (23270)Instruction limit reached!
% 39.16/13.25 % (23270)------------------------------
% 39.16/13.25 % (23270)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 39.16/13.25 % (23270)Termination reason: Unknown
% 39.16/13.25 % (23270)Termination phase: shuffling
% 39.16/13.25
% 39.16/13.25 % (23270)Memory used [KB]: 20084
% 39.16/13.25 % (23270)Time elapsed: 0.168 s
% 39.16/13.25 % (23270)Instructions burned: 294 (million)
% 39.16/13.25 % (23270)------------------------------
% 39.16/13.25 % (23270)------------------------------
% 39.16/13.27 % (23274)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_23017 for (2958ds/70Mi)
% 39.90/13.31 % (23274)Instruction limit reached!
% 39.90/13.31 % (23274)------------------------------
% 39.90/13.31 % (23274)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 39.90/13.31 % (23274)Termination reason: Unknown
% 39.90/13.31 % (23274)Termination phase: shuffling
% 39.90/13.31
% 39.90/13.31 % (23274)Memory used [KB]: 14839
% 39.90/13.31 % (23274)Time elapsed: 0.035 s
% 39.90/13.31 % (23274)Instructions burned: 70 (million)
% 39.90/13.31 % (23274)------------------------------
% 39.90/13.31 % (23274)------------------------------
% 39.90/13.32 % (23267)Instruction limit reached!
% 39.90/13.32 % (23267)------------------------------
% 39.90/13.32 % (23267)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 39.90/13.32 % (23267)Termination reason: Unknown
% 39.90/13.32 % (23267)Termination phase: shuffling
% 39.90/13.32
% 39.90/13.32 % (23267)Memory used [KB]: 26737
% 39.90/13.32 % (23267)Time elapsed: 0.480 s
% 39.90/13.32 % (23267)Instructions burned: 832 (million)
% 39.90/13.32 % (23267)------------------------------
% 39.90/13.32 % (23267)------------------------------
% 39.90/13.33 % (23275)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_23017 for (2957ds/316Mi)
% 39.90/13.34 % (23276)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_23017 for (2957ds/932Mi)
% 40.30/13.43 % (23272)Instruction limit reached!
% 40.30/13.43 % (23272)------------------------------
% 40.30/13.43 % (23272)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 40.30/13.43 % (23272)Termination reason: Unknown
% 40.30/13.43 % (23272)Termination phase: shuffling
% 40.30/13.43
% 40.30/13.43 % (23272)Memory used [KB]: 18677
% 40.30/13.43 % (23272)Time elapsed: 0.216 s
% 40.30/13.43 % (23272)Instructions burned: 371 (million)
% 40.30/13.43 % (23272)------------------------------
% 40.30/13.43 % (23272)------------------------------
% 40.30/13.46 % (23277)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_23017 for (2956ds/4714Mi)
% 40.83/13.51 % (23275)Instruction limit reached!
% 40.83/13.51 % (23275)------------------------------
% 40.83/13.51 % (23275)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 40.83/13.51 % (23275)Termination reason: Unknown
% 40.83/13.51 % (23275)Termination phase: shuffling
% 40.83/13.51
% 40.83/13.51 % (23275)Memory used [KB]: 20468
% 40.83/13.51 % (23275)Time elapsed: 0.180 s
% 40.83/13.51 % (23275)Instructions burned: 316 (million)
% 40.83/13.51 % (23275)------------------------------
% 40.83/13.51 % (23275)------------------------------
% 40.83/13.53 % (23278)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_23017 for (2955ds/8250Mi)
% 41.94/13.85 % (23276)Instruction limit reached!
% 41.94/13.85 % (23276)------------------------------
% 41.94/13.85 % (23276)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.94/13.85 % (23276)Termination reason: Unknown
% 41.94/13.85 % (23276)Termination phase: shuffling
% 41.94/13.85
% 41.94/13.85 % (23276)Memory used [KB]: 32750
% 41.94/13.85 % (23276)Time elapsed: 0.510 s
% 41.94/13.85 % (23276)Instructions burned: 933 (million)
% 41.94/13.85 % (23276)------------------------------
% 41.94/13.85 % (23276)------------------------------
% 41.94/13.88 % (23279)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_23017 for (2952ds/770Mi)
% 46.41/14.27 % (23273)Instruction limit reached!
% 46.41/14.27 % (23273)------------------------------
% 46.41/14.27 % (23273)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 46.41/14.27 % (23273)Termination reason: Unknown
% 46.41/14.27 % (23273)Termination phase: Preprocessing 3
% 46.41/14.27
% 46.41/14.27 % (23273)Memory used [KB]: 33517
% 46.41/14.27 % (23273)Time elapsed: 1.038 s
% 46.41/14.27 % (23273)Instructions burned: 1828 (million)
% 46.41/14.27 % (23273)------------------------------
% 46.41/14.27 % (23273)------------------------------
% 46.41/14.29 % (23280)ott+1010_5:1_av=off:fe=off:ins=1:kws=precedence:sp=reverse_arity:i=794:si=on:rtra=on_0 on DTF2THF_23017 for (2948ds/794Mi)
% 47.09/14.31 % (23279)Instruction limit reached!
% 47.09/14.31 % (23279)------------------------------
% 47.09/14.31 % (23279)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 47.09/14.31 % (23279)Termination reason: Unknown
% 47.09/14.31 % (23279)Termination phase: shuffling
% 47.09/14.31
% 47.09/14.31 % (23279)Memory used [KB]: 25202
% 47.09/14.31 % (23279)Time elapsed: 0.436 s
% 47.09/14.31 % (23279)Instructions burned: 770 (million)
% 47.09/14.31 % (23279)------------------------------
% 47.09/14.31 % (23279)------------------------------
% 47.09/14.32 % (23252)Instruction limit reached!
% 47.09/14.32 % (23252)------------------------------
% 47.09/14.32 % (23252)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 47.09/14.32 % (23252)Termination reason: Unknown
% 47.09/14.32 % (23252)Termination phase: Saturation
% 47.09/14.32
% 47.09/14.32 % (23252)Memory used [KB]: 36715
% 47.09/14.32 % (23252)Time elapsed: 2.538 s
% 47.09/14.32 % (23252)Instructions burned: 4437 (million)
% 47.09/14.32 % (23252)------------------------------
% 47.09/14.32 % (23252)------------------------------
% 47.09/14.33 % (23281)dis+1002_1:1_au=on:avsq=on:avsqr=1,16:cond=on:sp=frequency:i=464:si=on:rtra=on_0 on DTF2THF_23017 for (2947ds/464Mi)
% 47.09/14.35 % (23282)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_23017 for (2947ds/1376Mi)
% 47.96/14.52 % (23257)Instruction limit reached!
% 47.96/14.52 % (23257)------------------------------
% 47.96/14.52 % (23257)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 47.96/14.52 % (23257)Termination reason: Unknown
% 47.96/14.52 % (23257)Termination phase: Saturation
% 47.96/14.52
% 47.96/14.52 % (23257)Memory used [KB]: 35308
% 47.96/14.52 % (23257)Time elapsed: 2.429 s
% 47.96/14.52 % (23257)Instructions burned: 4350 (million)
% 47.96/14.52 % (23257)------------------------------
% 47.96/14.52 % (23257)------------------------------
% 47.96/14.55 % (23283)dis+10_1:1_etr=on:plsq=on:plsqc=1:plsqr=32,1:i=906:si=on:rtra=on_0 on DTF2THF_23017 for (2945ds/906Mi)
% 48.29/14.60 % (23281)Instruction limit reached!
% 48.29/14.60 % (23281)------------------------------
% 48.29/14.60 % (23281)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 48.29/14.60 % (23281)Termination reason: Unknown
% 48.29/14.60 % (23281)Termination phase: shuffling
% 48.29/14.60
% 48.29/14.60 % (23281)Memory used [KB]: 19701
% 48.29/14.60 % (23281)Time elapsed: 0.269 s
% 48.29/14.60 % (23281)Instructions burned: 465 (million)
% 48.29/14.60 % (23281)------------------------------
% 48.29/14.60 % (23281)------------------------------
% 48.29/14.60 % (23255)Instruction limit reached!
% 48.29/14.60 % (23255)------------------------------
% 48.29/14.60 % (23255)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 48.29/14.60 % (23255)Termination reason: Unknown
% 48.29/14.60 % (23255)Termination phase: Property scanning
% 48.29/14.60
% 48.29/14.60 % (23255)Memory used [KB]: 60766
% 48.29/14.60 % (23255)Time elapsed: 2.742 s
% 48.29/14.60 % (23255)Instructions burned: 5205 (million)
% 48.29/14.60 % (23255)------------------------------
% 48.29/14.60 % (23255)------------------------------
% 48.29/14.63 % (23284)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_23017 for (2944ds/434Mi)
% 48.29/14.63 % (23285)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_23017 for (2944ds/865Mi)
% 48.81/14.72 % (23280)Instruction limit reached!
% 48.81/14.72 % (23280)------------------------------
% 48.81/14.72 % (23280)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 48.81/14.72 % (23280)Termination reason: Unknown
% 48.81/14.72 % (23280)Termination phase: shuffling
% 48.81/14.72
% 48.81/14.72 % (23280)Memory used [KB]: 25969
% 48.81/14.72 % (23280)Time elapsed: 0.420 s
% 48.81/14.72 % (23280)Instructions burned: 795 (million)
% 48.81/14.72 % (23280)------------------------------
% 48.81/14.72 % (23280)------------------------------
% 48.81/14.75 % (23286)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_23017 for (2943ds/593Mi)
% 49.44/14.88 % (23284)Instruction limit reached!
% 49.44/14.88 % (23284)------------------------------
% 49.44/14.88 % (23284)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 49.44/14.88 % (23284)Termination reason: Unknown
% 49.44/14.88 % (23284)Termination phase: shuffling
% 49.44/14.88
% 49.44/14.88 % (23284)Memory used [KB]: 19701
% 49.44/14.88 % (23284)Time elapsed: 0.255 s
% 49.44/14.88 % (23284)Instructions burned: 434 (million)
% 49.44/14.88 % (23284)------------------------------
% 49.44/14.88 % (23284)------------------------------
% 49.91/14.90 % (23287)dis+10_1:1_sd=2:sgt=20:slsq=on:ss=axioms:st=6.0:i=443:si=on:rtra=on_0 on DTF2THF_23017 for (2942ds/443Mi)
% 52.16/15.06 % (23283)Instruction limit reached!
% 52.16/15.06 % (23283)------------------------------
% 52.16/15.06 % (23283)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 52.16/15.06 % (23283)Termination reason: Unknown
% 52.16/15.06 % (23283)Termination phase: shuffling
% 52.16/15.06
% 52.16/15.06 % (23283)Memory used [KB]: 27376
% 52.16/15.06 % (23283)Time elapsed: 0.518 s
% 52.16/15.06 % (23283)Instructions burned: 907 (million)
% 52.16/15.06 % (23283)------------------------------
% 52.16/15.06 % (23283)------------------------------
% 52.16/15.07 % (23286)Instruction limit reached!
% 52.16/15.07 % (23286)------------------------------
% 52.16/15.07 % (23286)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 52.16/15.07 % (23286)Termination reason: Unknown
% 52.16/15.07 % (23286)Termination phase: shuffling
% 52.16/15.07
% 52.16/15.07 % (23286)Memory used [KB]: 24178
% 52.16/15.07 % (23286)Time elapsed: 0.325 s
% 52.16/15.07 % (23286)Instructions burned: 593 (million)
% 52.16/15.07 % (23286)------------------------------
% 52.16/15.07 % (23286)------------------------------
% 52.16/15.09 % (23288)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_23017 for (2940ds/594Mi)
% 52.16/15.10 % (23282)Instruction limit reached!
% 52.16/15.10 % (23282)------------------------------
% 52.16/15.10 % (23282)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 52.16/15.10 % (23282)Termination reason: Unknown
% 52.16/15.10 % (23282)Termination phase: shuffling
% 52.16/15.10
% 52.16/15.10 % (23282)Memory used [KB]: 38506
% 52.16/15.10 % (23282)Time elapsed: 0.738 s
% 52.16/15.10 % (23282)Instructions burned: 1378 (million)
% 52.16/15.10 % (23282)------------------------------
% 52.16/15.10 % (23282)------------------------------
% 52.16/15.10 % (23289)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_23017 for (2940ds/327Mi)
% 52.16/15.10 % (23285)Instruction limit reached!
% 52.16/15.10 % (23285)------------------------------
% 52.16/15.10 % (23285)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 52.16/15.10 % (23285)Termination reason: Unknown
% 52.16/15.10 % (23285)Termination phase: shuffling
% 52.16/15.10
% 52.16/15.10 % (23285)Memory used [KB]: 31854
% 52.16/15.10 % (23285)Time elapsed: 0.468 s
% 52.16/15.10 % (23285)Instructions burned: 866 (million)
% 52.16/15.10 % (23285)------------------------------
% 52.16/15.10 % (23285)------------------------------
% 52.16/15.12 % (23291)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_23017 for (2939ds/1762Mi)
% 52.16/15.12 % (23290)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_23017 for (2939ds/5411Mi)
% 52.74/15.16 % (23287)Instruction limit reached!
% 52.74/15.16 % (23287)------------------------------
% 52.74/15.16 % (23287)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 52.74/15.16 % (23287)Termination reason: Unknown
% 52.74/15.16 % (23287)Termination phase: shuffling
% 52.74/15.16
% 52.74/15.16 % (23287)Memory used [KB]: 19573
% 52.74/15.16 % (23287)Time elapsed: 0.259 s
% 52.74/15.16 % (23287)Instructions burned: 443 (million)
% 52.74/15.16 % (23287)------------------------------
% 52.74/15.16 % (23287)------------------------------
% 52.74/15.18 % (23292)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_23017 for (2939ds/96Mi)
% 52.94/15.25 % (23292)Instruction limit reached!
% 52.94/15.25 % (23292)------------------------------
% 52.94/15.25 % (23292)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 52.94/15.25 % (23292)Termination reason: Unknown
% 52.94/15.25 % (23292)Termination phase: shuffling
% 52.94/15.25
% 52.94/15.25 % (23292)Memory used [KB]: 15351
% 52.94/15.25 % (23292)Time elapsed: 0.064 s
% 52.94/15.25 % (23292)Instructions burned: 97 (million)
% 52.94/15.25 % (23292)------------------------------
% 52.94/15.25 % (23292)------------------------------
% 52.94/15.27 % (23293)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_23017 for (2938ds/503Mi)
% 52.94/15.29 % (23289)Instruction limit reached!
% 52.94/15.29 % (23289)------------------------------
% 52.94/15.29 % (23289)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 52.94/15.29 % (23289)Termination reason: Unknown
% 52.94/15.29 % (23289)Termination phase: shuffling
% 52.94/15.29
% 52.94/15.29 % (23289)Memory used [KB]: 18421
% 52.94/15.29 % (23289)Time elapsed: 0.195 s
% 52.94/15.29 % (23289)Instructions burned: 328 (million)
% 52.94/15.29 % (23289)------------------------------
% 52.94/15.29 % (23289)------------------------------
% 53.44/15.32 % (23294)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_23017 for (2937ds/90Mi)
% 53.44/15.38 % (23294)Instruction limit reached!
% 53.44/15.38 % (23294)------------------------------
% 53.44/15.38 % (23294)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 53.44/15.38 % (23294)Termination reason: Unknown
% 53.44/15.38 % (23294)Termination phase: shuffling
% 53.44/15.38
% 53.44/15.38 % (23294)Memory used [KB]: 15095
% 53.44/15.38 % (23294)Time elapsed: 0.062 s
% 53.44/15.38 % (23294)Instructions burned: 90 (million)
% 53.44/15.38 % (23294)------------------------------
% 53.44/15.38 % (23294)------------------------------
% 53.72/15.40 % (23295)lrs+10_1:64_hud=5:plsq=on:plsqr=2011171,524288:i=593:si=on:rtra=on_0 on DTF2THF_23017 for (2937ds/593Mi)
% 53.72/15.43 % (23288)Instruction limit reached!
% 53.72/15.43 % (23288)------------------------------
% 53.72/15.43 % (23288)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 53.72/15.43 % (23288)Termination reason: Unknown
% 53.72/15.43 % (23288)Termination phase: shuffling
% 53.72/15.43
% 53.72/15.43 % (23288)Memory used [KB]: 21620
% 53.72/15.43 % (23288)Time elapsed: 0.341 s
% 53.72/15.43 % (23288)Instructions burned: 594 (million)
% 53.72/15.43 % (23288)------------------------------
% 53.72/15.43 % (23288)------------------------------
% 53.72/15.45 % (23296)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_23017 for (2936ds/1512Mi)
% 54.29/15.56 % (23293)Instruction limit reached!
% 54.29/15.56 % (23293)------------------------------
% 54.29/15.56 % (23293)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 54.29/15.56 % (23293)Termination reason: Unknown
% 54.29/15.56 % (23293)Termination phase: shuffling
% 54.29/15.56
% 54.29/15.56 % (23293)Memory used [KB]: 20724
% 54.29/15.56 % (23293)Time elapsed: 0.296 s
% 54.29/15.56 % (23293)Instructions burned: 503 (million)
% 54.29/15.56 % (23293)------------------------------
% 54.29/15.56 % (23293)------------------------------
% 54.29/15.59 % (23297)lrs+1002_1:128_cbe=off:cnfonf=lazy_not_gen_be_off:ins=1:i=569:si=on:rtra=on_0 on DTF2THF_23017 for (2935ds/569Mi)
% 55.01/15.74 % (23295)Instruction limit reached!
% 55.01/15.74 % (23295)------------------------------
% 55.01/15.74 % (23295)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 55.01/15.74 % (23295)Termination reason: Unknown
% 55.01/15.74 % (23295)Termination phase: shuffling
% 55.01/15.74
% 55.01/15.74 % (23295)Memory used [KB]: 21748
% 55.01/15.74 % (23295)Time elapsed: 0.346 s
% 55.01/15.74 % (23295)Instructions burned: 593 (million)
% 55.01/15.74 % (23295)------------------------------
% 55.01/15.74 % (23295)------------------------------
% 55.01/15.77 % (23298)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_23017 for (2933ds/501Mi)
% 55.78/15.90 % (23297)Instruction limit reached!
% 55.78/15.90 % (23297)------------------------------
% 55.78/15.90 % (23297)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 55.78/15.90 % (23297)Termination reason: Unknown
% 55.78/15.90 % (23297)Termination phase: shuffling
% 55.78/15.90
% 55.78/15.90 % (23297)Memory used [KB]: 23794
% 55.78/15.90 % (23297)Time elapsed: 0.313 s
% 55.78/15.90 % (23297)Instructions burned: 569 (million)
% 55.78/15.90 % (23297)------------------------------
% 55.78/15.90 % (23297)------------------------------
% 55.78/15.92 % (23299)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_23017 for (2931ds/562Mi)
% 55.78/15.94 % (23277)Instruction limit reached!
% 55.78/15.94 % (23277)------------------------------
% 55.78/15.94 % (23277)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 55.78/15.94 % (23277)Termination reason: Unknown
% 55.78/15.94 % (23277)Termination phase: Function definition elimination
% 55.78/15.94
% 55.78/15.94 % (23277)Memory used [KB]: 57184
% 55.78/15.94 % (23277)Time elapsed: 2.482 s
% 55.78/15.94 % (23277)Instructions burned: 4714 (million)
% 55.78/15.94 % (23277)------------------------------
% 55.78/15.94 % (23277)------------------------------
% 55.78/15.97 % (23300)lrs+10_1:1_au=on:cnfonf=off:fde=unused:kws=precedence:tgt=full:i=406:si=on:rtra=on_0 on DTF2THF_23017 for (2931ds/406Mi)
% 59.15/16.05 % (23291)Instruction limit reached!
% 59.15/16.05 % (23291)------------------------------
% 59.15/16.05 % (23291)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 59.15/16.05 % (23291)Termination reason: Unknown
% 59.15/16.05 % (23291)Termination phase: Property scanning
% 59.15/16.05
% 59.15/16.05 % (23291)Memory used [KB]: 42728
% 59.15/16.05 % (23291)Time elapsed: 0.931 s
% 59.15/16.05 % (23291)Instructions burned: 1763 (million)
% 59.15/16.05 % (23291)------------------------------
% 59.15/16.05 % (23291)------------------------------
% 59.15/16.07 % (23298)Instruction limit reached!
% 59.15/16.07 % (23298)------------------------------
% 59.15/16.07 % (23298)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 59.15/16.07 % (23298)Termination reason: Unknown
% 59.15/16.07 % (23298)Termination phase: shuffling
% 59.15/16.07
% 59.15/16.07 % (23298)Memory used [KB]: 20468
% 59.15/16.07 % (23298)Time elapsed: 0.291 s
% 59.15/16.07 % (23298)Instructions burned: 501 (million)
% 59.15/16.07 % (23298)------------------------------
% 59.15/16.07 % (23298)------------------------------
% 59.15/16.08 % (23301)dis+1004_1:128_au=on:cbe=off:e2e=on:sac=on:i=582:si=on:rtra=on_0 on DTF2THF_23017 for (2930ds/582Mi)
% 59.15/16.09 % (23302)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_23017 for (2930ds/456Mi)
% 60.10/16.20 % (23300)Instruction limit reached!
% 60.10/16.20 % (23300)------------------------------
% 60.10/16.20 % (23300)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 60.10/16.20 % (23300)Termination reason: Unknown
% 60.10/16.20 % (23300)Termination phase: shuffling
% 60.10/16.20
% 60.10/16.20 % (23300)Memory used [KB]: 19317
% 60.10/16.20 % (23300)Time elapsed: 0.236 s
% 60.10/16.20 % (23300)Instructions burned: 407 (million)
% 60.10/16.20 % (23300)------------------------------
% 60.10/16.20 % (23300)------------------------------
% 60.10/16.22 % (23303)lrs+1010_1:1_au=on:cbe=off:cnfonf=conj_eager:tgt=ground:i=393:si=on:rtra=on_0 on DTF2THF_23017 for (2928ds/393Mi)
% 60.10/16.23 % (23299)Instruction limit reached!
% 60.10/16.23 % (23299)------------------------------
% 60.10/16.23 % (23299)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 60.10/16.23 % (23299)Termination reason: Unknown
% 60.10/16.23 % (23299)Termination phase: shuffling
% 60.10/16.23
% 60.10/16.23 % (23299)Memory used [KB]: 23794
% 60.10/16.23 % (23299)Time elapsed: 0.310 s
% 60.10/16.23 % (23299)Instructions burned: 563 (million)
% 60.10/16.23 % (23299)------------------------------
% 60.10/16.23 % (23299)------------------------------
% 60.10/16.26 % (23304)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_23017 for (2928ds/1335Mi)
% 60.10/16.29 % (23296)Instruction limit reached!
% 60.10/16.29 % (23296)------------------------------
% 60.10/16.29 % (23296)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 60.10/16.29 % (23296)Termination reason: Unknown
% 60.10/16.29 % (23296)Termination phase: Property scanning
% 60.10/16.29
% 60.10/16.29 % (23296)Memory used [KB]: 30191
% 60.10/16.29 % (23296)Time elapsed: 0.845 s
% 60.10/16.29 % (23296)Instructions burned: 1513 (million)
% 60.10/16.29 % (23296)------------------------------
% 60.10/16.29 % (23296)------------------------------
% 59.60/16.32 % (23305)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_23017 for (2927ds/3373Mi)
% 59.60/16.36 % (23302)Instruction limit reached!
% 59.60/16.36 % (23302)------------------------------
% 59.60/16.36 % (23302)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 59.60/16.36 % (23302)Termination reason: Unknown
% 59.60/16.36 % (23302)Termination phase: shuffling
% 59.60/16.36
% 59.60/16.36 % (23302)Memory used [KB]: 19957
% 59.60/16.36 % (23302)Time elapsed: 0.266 s
% 59.60/16.36 % (23302)Instructions burned: 456 (million)
% 59.60/16.36 % (23302)------------------------------
% 59.60/16.36 % (23302)------------------------------
% 59.60/16.38 % (23306)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_23017 for (2927ds/474Mi)
% 60.95/16.41 % (23301)Instruction limit reached!
% 60.95/16.41 % (23301)------------------------------
% 60.95/16.41 % (23301)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 60.95/16.41 % (23301)Termination reason: Unknown
% 60.95/16.41 % (23301)Termination phase: shuffling
% 60.95/16.41
% 60.95/16.41 % (23301)Memory used [KB]: 21620
% 60.95/16.41 % (23301)Time elapsed: 0.338 s
% 60.95/16.41 % (23301)Instructions burned: 583 (million)
% 60.95/16.41 % (23301)------------------------------
% 60.95/16.41 % (23301)------------------------------
% 60.95/16.44 % (23303)Instruction limit reached!
% 60.95/16.44 % (23303)------------------------------
% 60.95/16.44 % (23303)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 60.95/16.44 % (23303)Termination reason: Unknown
% 60.95/16.44 % (23303)Termination phase: shuffling
% 60.95/16.44
% 60.95/16.44 % (23303)Memory used [KB]: 21364
% 60.95/16.44 % (23303)Time elapsed: 0.215 s
% 60.95/16.44 % (23303)Instructions burned: 393 (million)
% 60.95/16.44 % (23303)------------------------------
% 60.95/16.44 % (23303)------------------------------
% 60.95/16.44 % (23307)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_23017 for (2926ds/157Mi)
% 60.95/16.46 % (23308)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_23017 for (2926ds/590Mi)
% 61.76/16.53 % (23307)Instruction limit reached!
% 61.76/16.53 % (23307)------------------------------
% 61.76/16.53 % (23307)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 61.76/16.53 % (23307)Termination reason: Unknown
% 61.76/16.53 % (23307)Termination phase: shuffling
% 61.76/16.53
% 61.76/16.53 % (23307)Memory used [KB]: 16119
% 61.76/16.53 % (23307)Time elapsed: 0.097 s
% 61.76/16.53 % (23307)Instructions burned: 158 (million)
% 61.76/16.53 % (23307)------------------------------
% 61.76/16.53 % (23307)------------------------------
% 61.76/16.56 % (23309)dis+10_1:1024_kws=precedence:prag=on:rawr=on:sos=all:sp=unary_first:i=104:si=on:rtra=on_0 on DTF2THF_23017 for (2925ds/104Mi)
% 62.07/16.63 % (23309)Instruction limit reached!
% 62.07/16.63 % (23309)------------------------------
% 62.07/16.63 % (23309)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 62.07/16.63 % (23309)Termination reason: Unknown
% 62.07/16.63 % (23309)Termination phase: shuffling
% 62.07/16.63
% 62.07/16.63 % (23309)Memory used [KB]: 15351
% 62.07/16.63 % (23309)Time elapsed: 0.071 s
% 62.07/16.63 % (23309)Instructions burned: 104 (million)
% 62.07/16.63 % (23309)------------------------------
% 62.07/16.63 % (23309)------------------------------
% 62.07/16.64 % (23306)Instruction limit reached!
% 62.07/16.64 % (23306)------------------------------
% 62.07/16.64 % (23306)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 62.07/16.64 % (23306)Termination reason: Unknown
% 62.07/16.64 % (23306)Termination phase: shuffling
% 62.07/16.64
% 62.07/16.64 % (23306)Memory used [KB]: 22643
% 62.07/16.64 % (23306)Time elapsed: 0.260 s
% 62.07/16.64 % (23306)Instructions burned: 475 (million)
% 62.07/16.64 % (23306)------------------------------
% 62.07/16.64 % (23306)------------------------------
% 62.07/16.65 % (23310)lrs+1010_1:10_au=on:bd=off:cnfonf=lazy_pi_sigma_gen:sp=occurrence:i=414:si=on:rtra=on_0 on DTF2THF_23017 for (2924ds/414Mi)
% 62.07/16.66 % (23311)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_23017 for (2924ds/715Mi)
% 62.85/16.80 % (23308)Instruction limit reached!
% 62.85/16.80 % (23308)------------------------------
% 62.85/16.80 % (23308)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 62.85/16.80 % (23308)Termination reason: Unknown
% 62.85/16.80 % (23308)Termination phase: shuffling
% 62.85/16.80
% 62.85/16.80 % (23308)Memory used [KB]: 21492
% 62.85/16.80 % (23308)Time elapsed: 0.341 s
% 62.85/16.80 % (23308)Instructions burned: 590 (million)
% 62.85/16.80 % (23308)------------------------------
% 62.85/16.80 % (23308)------------------------------
% 62.85/16.82 % (23312)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_23017 for (2922ds/961Mi)
% 62.85/16.88 % (23310)Instruction limit reached!
% 62.85/16.88 % (23310)------------------------------
% 62.85/16.88 % (23310)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 62.85/16.88 % (23310)Termination reason: Unknown
% 62.85/16.88 % (23310)Termination phase: shuffling
% 62.85/16.88
% 62.85/16.88 % (23310)Memory used [KB]: 21875
% 62.85/16.88 % (23310)Time elapsed: 0.231 s
% 62.85/16.88 % (23310)Instructions burned: 415 (million)
% 62.85/16.88 % (23310)------------------------------
% 62.85/16.88 % (23310)------------------------------
% 63.67/16.90 % (23313)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_23017 for (2922ds/1213Mi)
% 63.67/16.97 % (23304)Instruction limit reached!
% 63.67/16.97 % (23304)------------------------------
% 63.67/16.97 % (23304)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 63.67/16.97 % (23304)Termination reason: Unknown
% 63.67/16.97 % (23304)Termination phase: shuffling
% 63.67/16.97
% 63.67/16.97 % (23304)Memory used [KB]: 37995
% 63.67/16.97 % (23304)Time elapsed: 0.716 s
% 63.67/16.97 % (23304)Instructions burned: 1335 (million)
% 63.67/16.97 % (23304)------------------------------
% 63.67/16.97 % (23304)------------------------------
% 63.67/17.00 % (23314)lrs+1002_1:12_bd=off:hud=1:ntd=on:piset=equals:prag=on:i=1979:si=on:rtra=on_0 on DTF2THF_23017 for (2921ds/1979Mi)
% 66.14/17.07 % (23311)Instruction limit reached!
% 66.14/17.07 % (23311)------------------------------
% 66.14/17.07 % (23311)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 66.14/17.07 % (23311)Termination reason: Unknown
% 66.14/17.07 % (23311)Termination phase: shuffling
% 66.14/17.07
% 66.14/17.07 % (23311)Memory used [KB]: 25202
% 66.14/17.07 % (23311)Time elapsed: 0.415 s
% 66.14/17.07 % (23311)Instructions burned: 715 (million)
% 66.14/17.07 % (23311)------------------------------
% 66.14/17.07 % (23311)------------------------------
% 66.14/17.10 % (23315)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_23017 for (2920ds/481Mi)
% 67.84/17.36 % (23315)Instruction limit reached!
% 67.84/17.36 % (23315)------------------------------
% 67.84/17.36 % (23315)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 67.84/17.36 % (23315)Termination reason: Unknown
% 67.84/17.36 % (23315)Termination phase: shuffling
% 67.84/17.36
% 67.84/17.36 % (23315)Memory used [KB]: 22771
% 67.84/17.36 % (23315)Time elapsed: 0.267 s
% 67.84/17.36 % (23315)Instructions burned: 481 (million)
% 67.84/17.36 % (23315)------------------------------
% 67.84/17.36 % (23315)------------------------------
% 67.84/17.37 % (23312)Instruction limit reached!
% 67.84/17.37 % (23312)------------------------------
% 67.84/17.37 % (23312)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 67.84/17.37 % (23312)Termination reason: Unknown
% 67.84/17.37 % (23312)Termination phase: Property scanning
% 67.84/17.37
% 67.84/17.37 % (23312)Memory used [KB]: 27632
% 67.84/17.37 % (23312)Time elapsed: 0.548 s
% 67.84/17.37 % (23312)Instructions burned: 961 (million)
% 67.84/17.37 % (23312)------------------------------
% 67.84/17.37 % (23312)------------------------------
% 67.84/17.39 % (23316)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_23017 for (2917ds/783Mi)
% 67.84/17.40 % (23317)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_23017 for (2917ds/432Mi)
% 69.21/17.58 % (23313)Instruction limit reached!
% 69.21/17.58 % (23313)------------------------------
% 69.21/17.58 % (23313)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 69.21/17.58 % (23313)Termination reason: Unknown
% 69.21/17.58 % (23313)Termination phase: Property scanning
% 69.21/17.58
% 69.21/17.58 % (23313)Memory used [KB]: 27760
% 69.21/17.58 % (23313)Time elapsed: 0.679 s
% 69.21/17.58 % (23313)Instructions burned: 1213 (million)
% 69.21/17.58 % (23313)------------------------------
% 69.21/17.58 % (23313)------------------------------
% 69.71/17.61 % (23318)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_23017 for (2914ds/1670Mi)
% 69.71/17.65 % (23317)Instruction limit reached!
% 69.71/17.65 % (23317)------------------------------
% 69.71/17.65 % (23317)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 69.71/17.65 % (23317)Termination reason: Unknown
% 69.71/17.65 % (23317)Termination phase: shuffling
% 69.71/17.65
% 69.71/17.65 % (23317)Memory used [KB]: 19701
% 69.71/17.65 % (23317)Time elapsed: 0.253 s
% 69.71/17.65 % (23317)Instructions burned: 432 (million)
% 69.71/17.65 % (23317)------------------------------
% 69.71/17.65 % (23317)------------------------------
% 69.71/17.67 % (23319)dis+2_1:1_au=on:hfaw=0:hfsq=on:sp=weighted_frequency:i=90:si=on:rtra=on_0 on DTF2THF_23017 for (2914ds/90Mi)
% 70.39/17.73 % (23319)Instruction limit reached!
% 70.39/17.73 % (23319)------------------------------
% 70.39/17.73 % (23319)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 70.39/17.73 % (23319)Termination reason: Unknown
% 70.39/17.73 % (23319)Termination phase: shuffling
% 70.39/17.73
% 70.39/17.73 % (23319)Memory used [KB]: 15095
% 70.39/17.73 % (23319)Time elapsed: 0.063 s
% 70.39/17.73 % (23319)Instructions burned: 91 (million)
% 70.39/17.73 % (23319)------------------------------
% 70.39/17.73 % (23319)------------------------------
% 70.39/17.76 % (23320)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_23017 for (2913ds/346Mi)
% 70.87/17.83 % (23316)Instruction limit reached!
% 70.87/17.83 % (23316)------------------------------
% 70.87/17.83 % (23316)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 70.87/17.83 % (23316)Termination reason: Unknown
% 70.87/17.83 % (23316)Termination phase: shuffling
% 70.87/17.83
% 70.87/17.83 % (23316)Memory used [KB]: 25713
% 70.87/17.83 % (23316)Time elapsed: 0.446 s
% 70.87/17.83 % (23316)Instructions burned: 783 (million)
% 70.87/17.83 % (23316)------------------------------
% 70.87/17.83 % (23316)------------------------------
% 70.87/17.86 % (23321)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_23017 for (2912ds/401Mi)
% 71.73/17.95 % (23320)Instruction limit reached!
% 71.73/17.95 % (23320)------------------------------
% 71.73/17.95 % (23320)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 71.73/17.95 % (23320)Termination reason: Unknown
% 71.73/17.95 % (23320)Termination phase: shuffling
% 71.73/17.95
% 71.73/17.95 % (23320)Memory used [KB]: 20852
% 71.73/17.95 % (23320)Time elapsed: 0.195 s
% 71.73/17.95 % (23320)Instructions burned: 347 (million)
% 71.73/17.95 % (23320)------------------------------
% 71.73/17.95 % (23320)------------------------------
% 71.73/17.97 % (23322)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=6454:si=on:rtra=on_0 on DTF2THF_23017 for (2911ds/6454Mi)
% 73.16/18.07 % (23321)Instruction limit reached!
% 73.16/18.07 % (23321)------------------------------
% 73.16/18.07 % (23321)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 73.16/18.07 % (23321)Termination reason: Unknown
% 73.16/18.07 % (23321)Termination phase: shuffling
% 73.16/18.07
% 73.16/18.07 % (23321)Memory used [KB]: 21748
% 73.16/18.07 % (23321)Time elapsed: 0.220 s
% 73.16/18.07 % (23321)Instructions burned: 402 (million)
% 73.16/18.07 % (23321)------------------------------
% 73.16/18.07 % (23321)------------------------------
% 73.16/18.10 % (23323)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_23017 for (2910ds/283Mi)
% 73.16/18.13 % (23314)Instruction limit reached!
% 73.16/18.13 % (23314)------------------------------
% 73.16/18.13 % (23314)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 73.16/18.13 % (23314)Termination reason: Unknown
% 73.16/18.13 % (23314)Termination phase: Preprocessing 3
% 73.16/18.13
% 73.16/18.13 % (23314)Memory used [KB]: 35436
% 73.16/18.13 % (23314)Time elapsed: 1.132 s
% 73.16/18.13 % (23314)Instructions burned: 1979 (million)
% 73.16/18.13 % (23314)------------------------------
% 73.16/18.13 % (23314)------------------------------
% 72.69/18.15 % (23324)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_23017 for (2909ds/865Mi)
% 72.69/18.17 % (23290)Instruction limit reached!
% 72.69/18.17 % (23290)------------------------------
% 72.69/18.17 % (23290)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 72.69/18.17 % (23290)Termination reason: Unknown
% 72.69/18.17 % (23290)Termination phase: Saturation
% 72.69/18.17
% 72.69/18.17 % (23290)Memory used [KB]: 38378
% 72.69/18.17 % (23290)Time elapsed: 3.048 s
% 72.69/18.17 % (23290)Instructions burned: 5412 (million)
% 72.69/18.17 % (23290)------------------------------
% 72.69/18.17 % (23290)------------------------------
% 72.69/18.18 % (23305)Instruction limit reached!
% 72.69/18.18 % (23305)------------------------------
% 72.69/18.18 % (23305)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 72.69/18.18 % (23305)Termination reason: Unknown
% 72.69/18.18 % (23305)Termination phase: Function definition elimination
% 72.69/18.18
% 72.69/18.18 % (23305)Memory used [KB]: 45670
% 72.69/18.18 % (23305)Time elapsed: 1.864 s
% 72.69/18.18 % (23305)Instructions burned: 3374 (million)
% 72.69/18.18 % (23305)------------------------------
% 72.69/18.18 % (23305)------------------------------
% 72.69/18.20 % (23325)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_23017 for (2909ds/4528Mi)
% 74.16/18.21 % (23326)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_23017 for (2909ds/1001Mi)
% 74.16/18.26 % (23323)Instruction limit reached!
% 74.16/18.26 % (23323)------------------------------
% 74.16/18.26 % (23323)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 74.16/18.26 % (23323)Termination reason: Unknown
% 74.16/18.26 % (23323)Termination phase: shuffling
% 74.16/18.26
% 74.16/18.26 % (23323)Memory used [KB]: 17398
% 74.16/18.26 % (23323)Time elapsed: 0.163 s
% 74.16/18.26 % (23323)Instructions burned: 283 (million)
% 74.16/18.26 % (23323)------------------------------
% 74.16/18.26 % (23323)------------------------------
% 74.16/18.28 % (23327)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=102:si=on:rtra=on_0 on DTF2THF_23017 for (2908ds/102Mi)
% 74.65/18.30 % (23278)Instruction limit reached!
% 74.65/18.30 % (23278)------------------------------
% 74.65/18.30 % (23278)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 74.65/18.31 % (23278)Termination reason: Unknown
% 74.65/18.31 % (23278)Termination phase: Saturation
% 74.65/18.31
% 74.65/18.31 % (23278)Memory used [KB]: 59103
% 74.65/18.31 % (23278)Time elapsed: 4.776 s
% 74.65/18.31 % (23278)Instructions burned: 8251 (million)
% 74.65/18.31 % (23278)------------------------------
% 74.65/18.31 % (23278)------------------------------
% 74.65/18.33 % (23328)ott+1010_3:1_nm=20:s2a=on:s2at=3.0:sp=const_frequency:i=952:si=on:rtra=on_0 on DTF2THF_23017 for (2907ds/952Mi)
% 74.65/18.35 % (23327)Instruction limit reached!
% 74.65/18.35 % (23327)------------------------------
% 74.65/18.35 % (23327)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 74.65/18.35 % (23327)Termination reason: Unknown
% 74.65/18.35 % (23327)Termination phase: shuffling
% 74.65/18.35
% 74.65/18.35 % (23327)Memory used [KB]: 15351
% 74.65/18.35 % (23327)Time elapsed: 0.066 s
% 74.65/18.35 % (23327)Instructions burned: 103 (million)
% 74.65/18.35 % (23327)------------------------------
% 74.65/18.35 % (23327)------------------------------
% 74.65/18.37 % (23329)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_23017 for (2907ds/578Mi)
% 75.17/18.50 % (23318)Instruction limit reached!
% 75.17/18.50 % (23318)------------------------------
% 75.17/18.50 % (23318)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 75.17/18.50 % (23318)Termination reason: Unknown
% 75.17/18.50 % (23318)Termination phase: shuffling
% 75.17/18.50
% 75.17/18.50 % (23318)Memory used [KB]: 42472
% 75.17/18.50 % (23318)Time elapsed: 0.896 s
% 75.17/18.50 % (23318)Instructions burned: 1671 (million)
% 75.17/18.50 % (23318)------------------------------
% 75.17/18.50 % (23318)------------------------------
% 75.17/18.53 % (23330)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_23017 for (2905ds/643Mi)
% 75.30/18.64 % (23324)Instruction limit reached!
% 75.30/18.64 % (23324)------------------------------
% 75.30/18.64 % (23324)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 75.30/18.64 % (23324)Termination reason: Unknown
% 75.30/18.64 % (23324)Termination phase: shuffling
% 75.30/18.64
% 75.30/18.64 % (23324)Memory used [KB]: 26609
% 75.30/18.64 % (23324)Time elapsed: 0.492 s
% 75.30/18.64 % (23324)Instructions burned: 865 (million)
% 75.30/18.64 % (23324)------------------------------
% 75.30/18.64 % (23324)------------------------------
% 75.30/18.67 % (23331)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_23017 for (2904ds/865Mi)
% 75.30/18.69 % (23329)Instruction limit reached!
% 75.30/18.69 % (23329)------------------------------
% 75.30/18.69 % (23329)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 75.30/18.69 % (23329)Termination reason: Unknown
% 75.30/18.69 % (23329)Termination phase: shuffling
% 75.30/18.69
% 75.30/18.69 % (23329)Memory used [KB]: 23922
% 75.30/18.69 % (23329)Time elapsed: 0.318 s
% 75.30/18.69 % (23329)Instructions burned: 578 (million)
% 75.30/18.69 % (23329)------------------------------
% 75.30/18.69 % (23329)------------------------------
% 76.25/18.71 % (23332)lrs+10_1:1_au=on:erd=off:s2a=on:s2at=3.0:sos=all:i=1174:si=on:rtra=on_0 on DTF2THF_23017 for (2903ds/1174Mi)
% 76.25/18.78 % (23326)Instruction limit reached!
% 76.25/18.78 % (23326)------------------------------
% 76.25/18.78 % (23326)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 76.25/18.78 % (23326)Termination reason: Unknown
% 76.25/18.78 % (23326)Termination phase: Property scanning
% 76.25/18.78
% 76.25/18.78 % (23326)Memory used [KB]: 27632
% 76.25/18.78 % (23326)Time elapsed: 0.571 s
% 76.25/18.78 % (23326)Instructions burned: 1001 (million)
% 76.25/18.78 % (23326)------------------------------
% 76.25/18.78 % (23326)------------------------------
% 76.94/18.81 % (23333)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_23017 for (2902ds/644Mi)
% 76.94/18.88 % (23328)Instruction limit reached!
% 76.94/18.88 % (23328)------------------------------
% 76.94/18.88 % (23328)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 76.94/18.88 % (23328)Termination reason: Unknown
% 76.94/18.88 % (23328)Termination phase: Property scanning
% 76.94/18.88
% 76.94/18.88 % (23328)Memory used [KB]: 27632
% 76.94/18.88 % (23328)Time elapsed: 0.545 s
% 76.94/18.88 % (23328)Instructions burned: 952 (million)
% 76.94/18.88 % (23328)------------------------------
% 76.94/18.88 % (23328)------------------------------
% 76.94/18.88 % (23330)Instruction limit reached!
% 76.94/18.88 % (23330)------------------------------
% 76.94/18.88 % (23330)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 76.94/18.88 % (23330)Termination reason: Unknown
% 76.94/18.88 % (23330)Termination phase: shuffling
% 76.94/18.88
% 76.94/18.88 % (23330)Memory used [KB]: 24818
% 76.94/18.88 % (23330)Time elapsed: 0.351 s
% 76.94/18.88 % (23330)Instructions burned: 644 (million)
% 76.94/18.88 % (23330)------------------------------
% 76.94/18.88 % (23330)------------------------------
% 77.99/18.90 % (23335)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_23017 for (2901ds/1046Mi)
% 77.99/18.90 % (23334)lrs+10_1:1_alpa=true:au=on:cs=on:tgt=ground:i=1201:si=on:rtra=on_0 on DTF2THF_23017 for (2901ds/1201Mi)
% 80.90/19.16 % (23331)Instruction limit reached!
% 80.90/19.16 % (23331)------------------------------
% 80.90/19.16 % (23331)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 80.90/19.16 % (23331)Termination reason: Unknown
% 80.90/19.16 % (23331)Termination phase: shuffling
% 80.90/19.16
% 80.90/19.16 % (23331)Memory used [KB]: 26865
% 80.90/19.16 % (23331)Time elapsed: 0.494 s
% 80.90/19.16 % (23331)Instructions burned: 866 (million)
% 80.90/19.16 % (23331)------------------------------
% 80.90/19.16 % (23331)------------------------------
% 80.90/19.17 % (23333)Instruction limit reached!
% 80.90/19.17 % (23333)------------------------------
% 80.90/19.17 % (23333)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 80.90/19.17 % (23333)Termination reason: Unknown
% 80.90/19.17 % (23333)Termination phase: shuffling
% 80.90/19.17
% 80.90/19.17 % (23333)Memory used [KB]: 24050
% 80.90/19.17 % (23333)Time elapsed: 0.370 s
% 80.90/19.17 % (23333)Instructions burned: 645 (million)
% 80.90/19.17 % (23333)------------------------------
% 80.90/19.17 % (23333)------------------------------
% 80.90/19.19 % (23336)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_23017 for (2899ds/222Mi)
% 80.90/19.20 % (23337)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_23017 for (2899ds/68Mi)
% 81.28/19.24 % (23337)Instruction limit reached!
% 81.28/19.24 % (23337)------------------------------
% 81.28/19.24 % (23337)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 81.28/19.24 % (23337)Termination reason: Unknown
% 81.28/19.24 % (23337)Termination phase: shuffling
% 81.28/19.24
% 81.28/19.24 % (23337)Memory used [KB]: 14711
% 81.28/19.24 % (23337)Time elapsed: 0.049 s
% 81.28/19.24 % (23337)Instructions burned: 68 (million)
% 81.28/19.24 % (23337)------------------------------
% 81.28/19.24 % (23337)------------------------------
% 81.28/19.27 % (23338)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_23017 for (2898ds/454Mi)
% 81.86/19.31 % (23336)Instruction limit reached!
% 81.86/19.31 % (23336)------------------------------
% 81.86/19.31 % (23336)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 81.86/19.31 % (23336)Termination reason: Unknown
% 81.86/19.31 % (23336)Termination phase: shuffling
% 81.86/19.31
% 81.86/19.31 % (23336)Memory used [KB]: 16886
% 81.86/19.31 % (23336)Time elapsed: 0.125 s
% 81.86/19.31 % (23336)Instructions burned: 222 (million)
% 81.86/19.31 % (23336)------------------------------
% 81.86/19.31 % (23336)------------------------------
% 81.86/19.33 % (23339)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_23017 for (2897ds/360Mi)
% 81.86/19.39 % (23332)Instruction limit reached!
% 81.86/19.39 % (23332)------------------------------
% 81.86/19.39 % (23332)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 81.86/19.39 % (23332)Termination reason: Unknown
% 81.86/19.39 % (23332)Termination phase: Property scanning
% 81.86/19.39
% 81.86/19.40 % (23332)Memory used [KB]: 27632
% 81.86/19.40 % (23332)Time elapsed: 0.665 s
% 81.86/19.40 % (23332)Instructions burned: 1175 (million)
% 81.86/19.40 % (23332)------------------------------
% 81.86/19.40 % (23332)------------------------------
% 81.83/19.42 % (23340)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_23017 for (2896ds/157Mi)
% 81.83/19.47 % (23335)Instruction limit reached!
% 81.83/19.47 % (23335)------------------------------
% 81.83/19.47 % (23335)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 81.83/19.47 % (23335)Termination reason: Unknown
% 81.83/19.47 % (23335)Termination phase: shuffling
% 81.83/19.47
% 81.83/19.47 % (23335)Memory used [KB]: 34285
% 81.83/19.47 % (23335)Time elapsed: 0.566 s
% 81.83/19.47 % (23335)Instructions burned: 1046 (million)
% 81.83/19.47 % (23335)------------------------------
% 81.83/19.47 % (23335)------------------------------
% 81.83/19.49 % (23341)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_23017 for (2896ds/5061Mi)
% 82.90/19.51 % (23340)Instruction limit reached!
% 82.90/19.51 % (23340)------------------------------
% 82.90/19.51 % (23340)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 82.90/19.51 % (23340)Termination reason: Unknown
% 82.90/19.51 % (23340)Termination phase: shuffling
% 82.90/19.51
% 82.90/19.51 % (23340)Memory used [KB]: 16119
% 82.90/19.51 % (23340)Time elapsed: 0.096 s
% 82.90/19.51 % (23340)Instructions burned: 157 (million)
% 82.90/19.51 % (23340)------------------------------
% 82.90/19.51 % (23340)------------------------------
% 82.90/19.52 % (23338)Instruction limit reached!
% 82.90/19.52 % (23338)------------------------------
% 82.90/19.52 % (23338)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 82.90/19.52 % (23338)Termination reason: Unknown
% 82.90/19.52 % (23338)Termination phase: shuffling
% 82.90/19.52
% 82.90/19.52 % (23338)Memory used [KB]: 22003
% 82.90/19.52 % (23338)Time elapsed: 0.251 s
% 82.90/19.52 % (23338)Instructions burned: 454 (million)
% 82.90/19.52 % (23338)------------------------------
% 82.90/19.52 % (23338)------------------------------
% 82.90/19.54 % (23339)Instruction limit reached!
% 82.90/19.54 % (23339)------------------------------
% 82.90/19.54 % (23339)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 82.90/19.54 % (23339)Termination reason: Unknown
% 82.90/19.54 % (23339)Termination phase: shuffling
% 82.90/19.54
% 82.90/19.54 % (23339)Memory used [KB]: 20980
% 82.90/19.54 % (23339)Time elapsed: 0.205 s
% 82.90/19.54 % (23339)Instructions burned: 361 (million)
% 82.90/19.54 % (23339)------------------------------
% 82.90/19.54 % (23339)------------------------------
% 82.90/19.54 % (23342)lrs+1002_1:8_av=off:fd=off:hud=5:tnu=1:i=2401:si=on:rtra=on_0 on DTF2THF_23017 for (2895ds/2401Mi)
% 82.90/19.54 % (23343)dis+10_1:1_av=off:e2e=on:nm=4:s2a=on:i=5798:si=on:rtra=on_0 on DTF2THF_23017 for (2895ds/5798Mi)
% 82.90/19.56 % (23344)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_23017 for (2895ds/346Mi)
% 82.90/19.57 % (23334)Instruction limit reached!
% 82.90/19.57 % (23334)------------------------------
% 82.90/19.57 % (23334)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 82.90/19.57 % (23334)Termination reason: Unknown
% 82.90/19.57 % (23334)Termination phase: Property scanning
% 82.90/19.57
% 82.90/19.57 % (23334)Memory used [KB]: 27632
% 82.90/19.57 % (23334)Time elapsed: 0.671 s
% 82.90/19.57 % (23334)Instructions burned: 1201 (million)
% 82.90/19.57 % (23334)------------------------------
% 82.90/19.57 % (23334)------------------------------
% 83.81/19.60 % (23345)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_23017 for (2895ds/1321Mi)
% 84.01/19.76 % (23344)Instruction limit reached!
% 84.01/19.76 % (23344)------------------------------
% 84.01/19.76 % (23344)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 84.01/19.76 % (23344)Termination reason: Unknown
% 84.01/19.76 % (23344)Termination phase: shuffling
% 84.01/19.76
% 84.01/19.76 % (23344)Memory used [KB]: 20724
% 84.01/19.76 % (23344)Time elapsed: 0.196 s
% 84.01/19.76 % (23344)Instructions burned: 347 (million)
% 84.01/19.76 % (23344)------------------------------
% 84.01/19.76 % (23344)------------------------------
% 84.01/19.77 % (23322)First to succeed.
% 84.01/19.78 % (23346)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:i=1312:si=on:rtra=on_0 on DTF2THF_23017 for (2893ds/1312Mi)
% 84.01/19.79 % (23322)Refutation found. Thanks to Tanya!
% 84.01/19.79 % SZS status Theorem for DTF2THF_23017
% 84.01/19.79 % SZS output start Proof for DTF2THF_23017
% 84.01/19.79 thf(type_def_5, type, real: $tType).
% 84.01/19.79 thf(type_def_6, type, int: $tType).
% 84.01/19.79 thf(type_def_8, type, rat: $tType).
% 84.01/19.79 thf(type_def_9, type, nat: $tType).
% 84.01/19.79 thf(type_def_10, type, product_prod_int_int: $tType).
% 84.01/19.79 thf(type_def_11, type, num: $tType).
% 84.01/19.79 thf(type_def_12, type, option_num: $tType).
% 84.01/19.79 thf(type_def_13, type, code_integer: $tType).
% 84.01/19.79 thf(type_def_14, type, produc6271795597528267376eger_o: $tType).
% 84.01/19.79 thf(type_def_15, type, produc8923325533196201883nteger: $tType).
% 84.01/19.79 thf(type_def_16, type, set_fi4554929511873752355omplex: $tType).
% 84.01/19.79 thf(type_def_17, type, filter6041513312241820739omplex: $tType).
% 84.01/19.79 thf(type_def_18, type, set_fi7789364187291644575l_real: $tType).
% 84.01/19.79 thf(type_def_19, type, filter2146258269922977983l_real: $tType).
% 84.01/19.79 thf(type_def_20, type, set_real: $tType).
% 84.01/19.79 thf(type_def_21, type, set_set_nat: $tType).
% 84.01/19.79 thf(type_def_22, type, set_nat: $tType).
% 84.01/19.79 thf(type_def_23, type, set_int: $tType).
% 84.01/19.79 thf(type_def_24, type, complex: $tType).
% 84.01/19.79 thf(type_def_25, type, filter_real: $tType).
% 84.01/19.79 thf(type_def_26, type, product_prod_nat_nat: $tType).
% 84.01/19.79 thf(type_def_27, type, filter_nat: $tType).
% 84.01/19.79 thf(type_def_28, type, produc4411394909380815293omplex: $tType).
% 84.01/19.79 thf(type_def_29, type, filter1242075044329608583at_nat: $tType).
% 84.01/19.79 thf(type_def_30, type, produc2422161461964618553l_real: $tType).
% 84.01/19.79 thf(type_def_31, type, set_Pr5085853215250843933omplex: $tType).
% 84.01/19.79 thf(type_def_32, type, set_Pr6218003697084177305l_real: $tType).
% 84.01/19.79 thf(type_def_33, type, set_o: $tType).
% 84.01/19.79 thf(type_def_34, type, set_complex: $tType).
% 84.01/19.79 thf(type_def_35, type, set_Product_unit: $tType).
% 84.01/19.79 thf(type_def_36, type, set_char: $tType).
% 84.01/19.79 thf(type_def_37, type, set_list_o: $tType).
% 84.01/19.79 thf(type_def_38, type, set_list_complex: $tType).
% 84.01/19.79 thf(type_def_39, type, set_list_int: $tType).
% 84.01/19.79 thf(type_def_40, type, set_list_nat: $tType).
% 84.01/19.79 thf(type_def_41, type, set_list_VEBT_VEBT: $tType).
% 84.01/19.79 thf(type_def_42, type, set_num: $tType).
% 84.01/19.79 thf(type_def_43, type, set_Pr958786334691620121nt_int: $tType).
% 84.01/19.79 thf(type_def_44, type, set_rat: $tType).
% 84.01/19.79 thf(type_def_45, type, set_set_complex: $tType).
% 84.01/19.79 thf(type_def_46, type, set_set_int: $tType).
% 84.01/19.79 thf(type_def_47, type, set_VEBT_VEBT: $tType).
% 84.01/19.79 thf(type_def_48, type, char: $tType).
% 84.01/19.79 thf(type_def_49, type, extended_enat: $tType).
% 84.01/19.79 thf(type_def_50, type, list_o: $tType).
% 84.01/19.79 thf(type_def_51, type, list_int: $tType).
% 84.01/19.79 thf(type_def_52, type, option_nat: $tType).
% 84.01/19.79 thf(type_def_53, type, vEBT_VEBT: $tType).
% 84.01/19.79 thf(type_def_54, type, list_VEBT_VEBT: $tType).
% 84.01/19.79 thf(type_def_55, type, list_complex: $tType).
% 84.01/19.79 thf(type_def_56, type, list_nat: $tType).
% 84.01/19.79 thf(type_def_57, type, list_real: $tType).
% 84.01/19.79 thf(type_def_58, type, list_set_nat: $tType).
% 84.01/19.79 thf(type_def_59, type, list_Code_integer: $tType).
% 84.01/19.79 thf(type_def_60, type, list_num: $tType).
% 84.01/19.79 thf(type_def_61, type, list_P4002435161011370285od_o_o: $tType).
% 84.01/19.79 thf(type_def_62, type, product_prod_o_o: $tType).
% 84.01/19.79 thf(type_def_63, type, list_P3795440434834930179_o_int: $tType).
% 84.01/19.79 thf(type_def_64, type, product_prod_o_int: $tType).
% 84.01/19.79 thf(type_def_65, type, list_P7495141550334521929T_VEBT: $tType).
% 84.01/19.79 thf(type_def_66, type, produc2504756804600209347T_VEBT: $tType).
% 84.01/19.79 thf(type_def_67, type, list_P8526636022914148096eger_o: $tType).
% 84.01/19.79 thf(type_def_68, type, list_P6011104703257516679at_nat: $tType).
% 84.01/19.79 thf(type_def_69, type, list_P1726324292696863441at_num: $tType).
% 84.01/19.79 thf(type_def_70, type, product_prod_nat_num: $tType).
% 84.01/19.79 thf(type_def_71, type, list_P3744719386663036955um_num: $tType).
% 84.01/19.79 thf(type_def_72, type, product_prod_num_num: $tType).
% 84.01/19.79 thf(type_def_73, type, list_P3126845725202233233VEBT_o: $tType).
% 84.01/19.79 thf(type_def_74, type, produc334124729049499915VEBT_o: $tType).
% 84.01/19.79 thf(type_def_75, type, list_P4547456442757143711BT_int: $tType).
% 84.01/19.79 thf(type_def_76, type, produc4894624898956917775BT_int: $tType).
% 84.01/19.79 thf(type_def_77, type, list_P7413028617227757229T_VEBT: $tType).
% 84.01/19.79 thf(type_def_78, type, produc8243902056947475879T_VEBT: $tType).
% 84.01/19.79 thf(type_def_79, type, list_P5087981734274514673_int_o: $tType).
% 84.01/19.79 thf(type_def_80, type, list_P5707943133018811711nt_int: $tType).
% 84.01/19.79 thf(type_def_81, type, list_P7524865323317820941T_VEBT: $tType).
% 84.01/19.79 thf(type_def_82, type, option4927543243414619207at_nat: $tType).
% 84.01/19.79 thf(type_def_83, type, set_Code_integer: $tType).
% 84.01/19.79 thf(type_def_84, type, produc4953844613479565601on_nat: $tType).
% 84.01/19.79 thf(type_def_85, type, produc2233624965454879586on_nat: $tType).
% 84.01/19.79 thf(type_def_86, type, produc7248412053542808358at_nat: $tType).
% 84.01/19.79 thf(type_def_87, type, produc4471711990508489141at_nat: $tType).
% 84.01/19.79 thf(type_def_88, type, produc8306885398267862888on_nat: $tType).
% 84.01/19.79 thf(type_def_89, type, produc2963631642982155120at_num: $tType).
% 84.01/19.79 thf(type_def_90, type, produc3368934014287244435at_num: $tType).
% 84.01/19.79 thf(type_def_91, type, produc3447558737645232053on_num: $tType).
% 84.01/19.79 thf(type_def_92, type, produc7036089656553540234on_num: $tType).
% 84.01/19.79 thf(type_def_93, type, produc1193250871479095198on_num: $tType).
% 84.01/19.79 thf(type_def_94, type, produc6121120109295599847at_nat: $tType).
% 84.01/19.79 thf(type_def_95, type, produc5491161045314408544at_nat: $tType).
% 84.01/19.79 thf(type_def_96, type, produc5542196010084753463at_nat: $tType).
% 84.01/19.79 thf(type_def_97, type, produc9072475918466114483BT_nat: $tType).
% 84.01/19.79 thf(func_def_0, type, produc5542196010084753463at_nat: $tType).
% 84.01/19.79 thf(func_def_1, type, produc5491161045314408544at_nat: $tType).
% 84.01/19.79 thf(func_def_2, type, produc1193250871479095198on_num: $tType).
% 84.01/19.79 thf(func_def_3, type, produc8306885398267862888on_nat: $tType).
% 84.01/19.79 thf(func_def_4, type, produc6121120109295599847at_nat: $tType).
% 84.01/19.79 thf(func_def_5, type, produc3368934014287244435at_num: $tType).
% 84.01/19.79 thf(func_def_6, type, produc4471711990508489141at_nat: $tType).
% 84.01/19.79 thf(func_def_7, type, produc7036089656553540234on_num: $tType).
% 84.01/19.79 thf(func_def_8, type, produc2233624965454879586on_nat: $tType).
% 84.01/19.79 thf(func_def_9, type, set_fi4554929511873752355omplex: $tType).
% 84.01/19.79 thf(func_def_10, type, list_P7413028617227757229T_VEBT: $tType).
% 84.01/19.79 thf(func_def_11, type, produc3447558737645232053on_num: $tType).
% 84.01/19.79 thf(func_def_12, type, produc4953844613479565601on_nat: $tType).
% 84.01/19.79 thf(func_def_13, type, produc2963631642982155120at_num: $tType).
% 84.01/19.79 thf(func_def_14, type, produc7248412053542808358at_nat: $tType).
% 84.01/19.79 thf(func_def_15, type, set_fi7789364187291644575l_real: $tType).
% 84.01/19.79 thf(func_def_16, type, filter6041513312241820739omplex: $tType).
% 84.01/19.79 thf(func_def_17, type, list_P4547456442757143711BT_int: $tType).
% 84.01/19.79 thf(func_def_18, type, list_P7524865323317820941T_VEBT: $tType).
% 84.01/19.79 thf(func_def_19, type, produc8243902056947475879T_VEBT: $tType).
% 84.01/19.79 thf(func_def_20, type, set_Pr5085853215250843933omplex: $tType).
% 84.01/19.79 thf(func_def_21, type, produc8923325533196201883nteger: $tType).
% 84.01/19.79 thf(func_def_22, type, list_P3126845725202233233VEBT_o: $tType).
% 84.01/19.79 thf(func_def_23, type, list_P7495141550334521929T_VEBT: $tType).
% 84.01/19.79 thf(func_def_24, type, filter2146258269922977983l_real: $tType).
% 84.01/19.79 thf(func_def_25, type, list_P8526636022914148096eger_o: $tType).
% 84.01/19.79 thf(func_def_26, type, option4927543243414619207at_nat: $tType).
% 84.01/19.79 thf(func_def_27, type, filter1242075044329608583at_nat: $tType).
% 84.01/19.79 thf(func_def_28, type, set_Pr6218003697084177305l_real: $tType).
% 84.01/19.79 thf(func_def_29, type, list_P3744719386663036955um_num: $tType).
% 84.01/19.79 thf(func_def_30, type, list_P1726324292696863441at_num: $tType).
% 84.01/19.79 thf(func_def_31, type, list_P6011104703257516679at_nat: $tType).
% 84.01/19.79 thf(func_def_32, type, list_P5707943133018811711nt_int: $tType).
% 84.01/19.79 thf(func_def_33, type, produc9072475918466114483BT_nat: $tType).
% 84.01/19.79 thf(func_def_34, type, produc4894624898956917775BT_int: $tType).
% 84.01/19.79 thf(func_def_35, type, set_Pr958786334691620121nt_int: $tType).
% 84.01/19.79 thf(func_def_36, type, produc4411394909380815293omplex: $tType).
% 84.01/19.79 thf(func_def_37, type, list_P5087981734274514673_int_o: $tType).
% 84.01/19.79 thf(func_def_38, type, list_P3795440434834930179_o_int: $tType).
% 84.01/19.79 thf(func_def_39, type, set_list_VEBT_VEBT: $tType).
% 84.01/19.79 thf(func_def_40, type, produc334124729049499915VEBT_o: $tType).
% 84.01/19.79 thf(func_def_41, type, produc2504756804600209347T_VEBT: $tType).
% 84.01/19.79 thf(func_def_42, type, produc6271795597528267376eger_o: $tType).
% 84.01/19.79 thf(func_def_43, type, produc2422161461964618553l_real: $tType).
% 84.01/19.79 thf(func_def_44, type, product_prod_num_num: $tType).
% 84.01/19.79 thf(func_def_45, type, product_prod_nat_num: $tType).
% 84.01/19.79 thf(func_def_46, type, product_prod_nat_nat: $tType).
% 84.01/19.79 thf(func_def_47, type, product_prod_int_int: $tType).
% 84.01/19.79 thf(func_def_48, type, list_P4002435161011370285od_o_o: $tType).
% 84.01/19.79 thf(func_def_49, type, set_list_complex: $tType).
% 84.01/19.79 thf(func_def_50, type, set_set_complex: $tType).
% 84.01/19.79 thf(func_def_51, type, list_VEBT_VEBT: $tType).
% 84.01/19.79 thf(func_def_52, type, set_list_nat: $tType).
% 84.01/19.79 thf(func_def_53, type, set_list_int: $tType).
% 84.01/19.79 thf(func_def_54, type, product_prod_o_int: $tType).
% 84.01/19.79 thf(func_def_55, type, list_set_nat: $tType).
% 84.01/19.79 thf(func_def_56, type, list_Code_integer: $tType).
% 84.01/19.79 thf(func_def_57, type, set_VEBT_VEBT: $tType).
% 84.01/19.79 thf(func_def_58, type, set_set_nat: $tType).
% 84.01/19.79 thf(func_def_59, type, set_set_int: $tType).
% 84.01/19.79 thf(func_def_60, type, set_Code_integer: $tType).
% 84.01/19.79 thf(func_def_61, type, set_Product_unit: $tType).
% 84.01/19.79 thf(func_def_62, type, list_complex: $tType).
% 84.01/19.79 thf(func_def_63, type, set_list_o: $tType).
% 84.01/19.79 thf(func_def_64, type, product_prod_o_o: $tType).
% 84.01/19.79 thf(func_def_65, type, set_complex: $tType).
% 84.01/19.79 thf(func_def_66, type, filter_real: $tType).
% 84.01/19.79 thf(func_def_67, type, option_num: $tType).
% 84.01/19.79 thf(func_def_68, type, option_nat: $tType).
% 84.01/19.79 thf(func_def_69, type, filter_nat: $tType).
% 84.01/19.79 thf(func_def_70, type, set_char: $tType).
% 84.01/19.79 thf(func_def_71, type, list_real: $tType).
% 84.01/19.79 thf(func_def_72, type, set_real: $tType).
% 84.01/19.79 thf(func_def_73, type, list_num: $tType).
% 84.01/19.79 thf(func_def_74, type, list_nat: $tType).
% 84.01/19.79 thf(func_def_75, type, list_int: $tType).
% 84.01/19.79 thf(func_def_76, type, vEBT_VEBT: $tType).
% 84.01/19.79 thf(func_def_77, type, set_rat: $tType).
% 84.01/19.79 thf(func_def_78, type, set_num: $tType).
% 84.01/19.79 thf(func_def_79, type, set_nat: $tType).
% 84.01/19.79 thf(func_def_80, type, set_int: $tType).
% 84.01/19.79 thf(func_def_81, type, code_integer: $tType).
% 84.01/19.79 thf(func_def_82, type, extended_enat: $tType).
% 84.01/19.79 thf(func_def_83, type, list_o: $tType).
% 84.01/19.79 thf(func_def_84, type, complex: $tType).
% 84.01/19.79 thf(func_def_85, type, set_o: $tType).
% 84.01/19.79 thf(func_def_86, type, char: $tType).
% 84.01/19.79 thf(func_def_87, type, real: $tType).
% 84.01/19.79 thf(func_def_88, type, rat: $tType).
% 84.01/19.79 thf(func_def_89, type, num: $tType).
% 84.01/19.79 thf(func_def_90, type, nat: $tType).
% 84.01/19.79 thf(func_def_91, type, int: $tType).
% 84.01/19.79 thf(func_def_92, type, archim7802044766580827645g_real: real > int).
% 84.01/19.79 thf(func_def_93, type, archim3151403230148437115or_rat: rat > int).
% 84.01/19.79 thf(func_def_94, type, archim6058952711729229775r_real: real > int).
% 84.01/19.79 thf(func_def_95, type, archim7778729529865785530nd_rat: rat > int).
% 84.01/19.79 thf(func_def_96, type, archim8280529875227126926d_real: real > int).
% 84.01/19.79 thf(func_def_97, type, binomial: nat > nat > nat).
% 84.01/19.79 thf(func_def_98, type, bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o).
% 84.01/19.79 thf(func_def_99, type, bit_and_not_num: num > num > option_num).
% 84.01/19.79 thf(func_def_100, type, bit_concat_bit: nat > int > int > int).
% 84.01/19.79 thf(func_def_101, type, bit_or_not_num_neg: num > num > num).
% 84.01/19.79 thf(func_def_102, type, bit_ri7919022796975470100ot_int: int > int).
% 84.01/19.79 thf(func_def_103, type, bit_ri6519982836138164636nteger: nat > code_integer > code_integer).
% 84.01/19.79 thf(func_def_104, type, bit_ri631733984087533419it_int: nat > int > int).
% 84.01/19.79 thf(func_def_105, type, bit_se725231765392027082nd_int: int > int > int).
% 84.01/19.79 thf(func_def_106, type, bit_se727722235901077358nd_nat: nat > nat > nat).
% 84.01/19.79 thf(func_def_107, type, bit_se8568078237143864401it_int: nat > int > int).
% 84.01/19.79 thf(func_def_108, type, bit_se8570568707652914677it_nat: nat > nat > nat).
% 84.01/19.79 thf(func_def_109, type, bit_se1345352211410354436nteger: nat > code_integer > code_integer).
% 84.01/19.79 thf(func_def_110, type, bit_se2159334234014336723it_int: nat > int > int).
% 84.01/19.79 thf(func_def_111, type, bit_se2161824704523386999it_nat: nat > nat > nat).
% 84.01/19.79 thf(func_def_112, type, bit_se2000444600071755411sk_int: nat > int).
% 84.01/19.79 thf(func_def_113, type, bit_se2002935070580805687sk_nat: nat > nat).
% 84.01/19.79 thf(func_def_114, type, bit_se1409905431419307370or_int: int > int > int).
% 84.01/19.79 thf(func_def_115, type, bit_se1412395901928357646or_nat: nat > nat > nat).
% 84.01/19.79 thf(func_def_116, type, bit_se545348938243370406it_int: nat > int > int).
% 84.01/19.79 thf(func_def_117, type, bit_se547839408752420682it_nat: nat > nat > nat).
% 84.01/19.79 thf(func_def_118, type, bit_se2793503036327961859nteger: nat > code_integer > code_integer).
% 84.01/19.79 thf(func_def_119, type, bit_se7879613467334960850it_int: nat > int > int).
% 84.01/19.79 thf(func_def_120, type, bit_se7882103937844011126it_nat: nat > nat > nat).
% 84.01/19.79 thf(func_def_121, type, bit_se2923211474154528505it_int: nat > int > int).
% 84.01/19.79 thf(func_def_122, type, bit_se2925701944663578781it_nat: nat > nat > nat).
% 84.01/19.79 thf(func_def_123, type, bit_se8260200283734997820nteger: nat > code_integer > code_integer).
% 84.01/19.79 thf(func_def_124, type, bit_se4203085406695923979it_int: nat > int > int).
% 84.01/19.79 thf(func_def_125, type, bit_se4205575877204974255it_nat: nat > nat > nat).
% 84.01/19.79 thf(func_def_126, type, bit_se6526347334894502574or_int: int > int > int).
% 84.01/19.79 thf(func_def_127, type, bit_se6528837805403552850or_nat: nat > nat > nat).
% 84.01/19.79 thf(func_def_128, type, bit_se1146084159140164899it_int: int > nat > $o).
% 84.01/19.79 thf(func_def_129, type, bit_se1148574629649215175it_nat: nat > nat > $o).
% 84.01/19.79 thf(func_def_130, type, bit_take_bit_num: nat > num > option_num).
% 84.01/19.79 thf(func_def_131, type, bit_un1837492267222099188nd_num: num > num > option_num).
% 84.01/19.79 thf(func_def_132, type, bit_un6178654185764691216or_num: num > num > option_num).
% 84.01/19.79 thf(func_def_133, type, bit_un7362597486090784418nd_num: num > num > option_num).
% 84.01/19.79 thf(func_def_134, type, bit_un2480387367778600638or_num: num > num > option_num).
% 84.01/19.79 thf(func_def_135, type, code_bit_cut_integer: code_integer > produc6271795597528267376eger_o).
% 84.01/19.79 thf(func_def_136, type, code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger).
% 84.01/19.79 thf(func_def_137, type, code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger).
% 84.01/19.79 thf(func_def_138, type, code_int_of_integer: code_integer > int).
% 84.01/19.79 thf(func_def_139, type, code_integer_of_int: int > code_integer).
% 84.01/19.79 thf(func_def_140, type, code_integer_of_num: num > code_integer).
% 84.01/19.79 thf(func_def_141, type, code_nat_of_integer: code_integer > nat).
% 84.01/19.79 thf(func_def_142, type, code_num_of_integer: code_integer > num).
% 84.01/19.79 thf(func_def_143, type, comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex).
% 84.01/19.79 thf(func_def_144, type, comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real).
% 84.01/19.79 thf(func_def_145, type, comple4887499456419720421f_real: set_real > real).
% 84.01/19.79 thf(func_def_146, type, comple7806235888213564991et_nat: set_set_nat > set_nat).
% 84.01/19.79 thf(func_def_147, type, complete_Sup_Sup_int: set_int > int).
% 84.01/19.79 thf(func_def_148, type, comple1385675409528146559p_real: set_real > real).
% 84.01/19.79 thf(func_def_149, type, comple7399068483239264473et_nat: set_set_nat > set_nat).
% 84.01/19.79 thf(func_def_150, type, arg: complex > real).
% 84.01/19.79 thf(func_def_151, type, cis: real > complex).
% 84.01/19.79 thf(func_def_152, type, cnj: complex > complex).
% 84.01/19.79 thf(func_def_153, type, complex2: real > real > complex).
% 84.01/19.79 thf(func_def_154, type, im: complex > real).
% 84.01/19.79 thf(func_def_155, type, re: complex > real).
% 84.01/19.79 thf(func_def_156, type, csqrt: complex > complex).
% 84.01/19.79 thf(func_def_157, type, imaginary_unit: complex).
% 84.01/19.79 thf(func_def_158, type, differ6690327859849518006l_real: (real > real) > filter_real > $o).
% 84.01/19.79 thf(func_def_159, type, has_de1759254742604945161l_real: (real > real) > (real > real) > filter_real > $o).
% 84.01/19.79 thf(func_def_160, type, has_fi5821293074295781190e_real: (real > real) > real > filter_real > $o).
% 84.01/19.79 thf(func_def_161, type, adjust_div: product_prod_int_int > int).
% 84.01/19.79 thf(func_def_162, type, adjust_mod: int > int > int).
% 84.01/19.79 thf(func_def_163, type, divmod_nat: nat > nat > product_prod_nat_nat).
% 84.01/19.79 thf(func_def_164, type, eucl_rel_int: int > int > product_prod_int_int > $o).
% 84.01/19.79 thf(func_def_165, type, unique5706413561485394159nteger: produc8923325533196201883nteger > $o).
% 84.01/19.79 thf(func_def_166, type, unique6319869463603278526ux_int: product_prod_int_int > $o).
% 84.01/19.79 thf(func_def_167, type, unique6322359934112328802ux_nat: product_prod_nat_nat > $o).
% 84.01/19.79 thf(func_def_168, type, unique3479559517661332726nteger: num > num > produc8923325533196201883nteger).
% 84.01/19.79 thf(func_def_169, type, unique5052692396658037445od_int: num > num > product_prod_int_int).
% 84.01/19.79 thf(func_def_170, type, unique5055182867167087721od_nat: num > num > product_prod_nat_nat).
% 84.01/19.79 thf(func_def_171, type, unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 84.01/19.79 thf(func_def_172, type, unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int).
% 84.01/19.79 thf(func_def_173, type, unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat).
% 84.01/19.79 thf(func_def_174, type, comm_s8582702949713902594nteger: code_integer > nat > code_integer).
% 84.01/19.79 thf(func_def_175, type, comm_s2602460028002588243omplex: complex > nat > complex).
% 84.01/19.79 thf(func_def_176, type, comm_s4660882817536571857er_int: int > nat > int).
% 84.01/19.79 thf(func_def_177, type, comm_s4663373288045622133er_nat: nat > nat > nat).
% 84.01/19.79 thf(func_def_178, type, comm_s4028243227959126397er_rat: rat > nat > rat).
% 84.01/19.79 thf(func_def_179, type, comm_s7457072308508201937r_real: real > nat > real).
% 84.01/19.79 thf(func_def_180, type, semiri3624122377584611663nteger: nat > code_integer).
% 84.01/19.79 thf(func_def_181, type, semiri5044797733671781792omplex: nat > complex).
% 84.01/19.79 thf(func_def_182, type, semiri1406184849735516958ct_int: nat > int).
% 84.01/19.79 thf(func_def_183, type, semiri1408675320244567234ct_nat: nat > nat).
% 84.01/19.79 thf(func_def_184, type, semiri773545260158071498ct_rat: nat > rat).
% 84.01/19.79 thf(func_def_185, type, semiri2265585572941072030t_real: nat > real).
% 84.01/19.79 thf(func_def_186, type, invers8013647133539491842omplex: complex > complex).
% 84.01/19.79 thf(func_def_187, type, inverse_inverse_rat: rat > rat).
% 84.01/19.79 thf(func_def_188, type, inverse_inverse_real: real > real).
% 84.01/19.79 thf(func_def_189, type, at_bot_real: filter_real).
% 84.01/19.79 thf(func_def_190, type, at_top_nat: filter_nat).
% 84.01/19.79 thf(func_def_191, type, at_top_real: filter_real).
% 84.01/19.79 thf(func_def_192, type, eventually_nat: (nat > $o) > filter_nat > $o).
% 84.01/19.79 thf(func_def_193, type, eventu5826381225784669381omplex: (produc4411394909380815293omplex > $o) > filter6041513312241820739omplex > $o).
% 84.01/19.79 thf(func_def_194, type, eventu1038000079068216329at_nat: (product_prod_nat_nat > $o) > filter1242075044329608583at_nat > $o).
% 84.01/19.79 thf(func_def_195, type, eventu3244425730907250241l_real: (produc2422161461964618553l_real > $o) > filter2146258269922977983l_real > $o).
% 84.01/19.79 thf(func_def_196, type, eventually_real: (real > $o) > filter_real > $o).
% 84.01/19.79 thf(func_def_197, type, filterlim_nat_nat: (nat > nat) > filter_nat > filter_nat > $o).
% 84.01/19.79 thf(func_def_198, type, filterlim_nat_real: (nat > real) > filter_real > filter_nat > $o).
% 84.01/19.79 thf(func_def_199, type, filterlim_real_real: (real > real) > filter_real > filter_real > $o).
% 84.01/19.79 thf(func_def_200, type, filtermap_real_real: (real > real) > filter_real > filter_real).
% 84.01/19.79 thf(func_def_201, type, princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex).
% 84.01/19.79 thf(func_def_202, type, princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real).
% 84.01/19.79 thf(func_def_203, type, prod_filter_nat_nat: filter_nat > filter_nat > filter1242075044329608583at_nat).
% 84.01/19.79 thf(func_def_204, type, finite_card_o: set_o > nat).
% 84.01/19.79 thf(func_def_205, type, finite_card_complex: set_complex > nat).
% 84.01/19.79 thf(func_def_206, type, finite_card_int: set_int > nat).
% 84.01/19.79 thf(func_def_207, type, finite_card_nat: set_nat > nat).
% 84.01/19.79 thf(func_def_208, type, finite410649719033368117t_unit: set_Product_unit > nat).
% 84.01/19.79 thf(func_def_209, type, finite_card_set_nat: set_set_nat > nat).
% 84.01/19.79 thf(func_def_210, type, finite_card_char: set_char > nat).
% 84.01/19.79 thf(func_def_211, type, finite_finite_o: set_o > $o).
% 84.01/19.79 thf(func_def_212, type, finite3207457112153483333omplex: set_complex > $o).
% 84.01/19.79 thf(func_def_213, type, finite_finite_int: set_int > $o).
% 84.01/19.79 thf(func_def_214, type, finite_finite_list_o: set_list_o > $o).
% 84.01/19.79 thf(func_def_215, type, finite8712137658972009173omplex: set_list_complex > $o).
% 84.01/19.79 thf(func_def_216, type, finite3922522038869484883st_int: set_list_int > $o).
% 84.01/19.79 thf(func_def_217, type, finite8100373058378681591st_nat: set_list_nat > $o).
% 84.01/19.79 thf(func_def_218, type, finite3004134309566078307T_VEBT: set_list_VEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_219, type, finite_finite_nat: set_nat > $o).
% 84.01/19.79 thf(func_def_220, type, finite_finite_num: set_num > $o).
% 84.01/19.79 thf(func_def_221, type, finite2998713641127702882nt_int: set_Pr958786334691620121nt_int > $o).
% 84.01/19.79 thf(func_def_222, type, finite_finite_rat: set_rat > $o).
% 84.01/19.79 thf(func_def_223, type, finite_finite_real: set_real > $o).
% 84.01/19.79 thf(func_def_224, type, finite6551019134538273531omplex: set_set_complex > $o).
% 84.01/19.79 thf(func_def_225, type, finite6197958912794628473et_int: set_set_int > $o).
% 84.01/19.79 thf(func_def_226, type, finite1152437895449049373et_nat: set_set_nat > $o).
% 84.01/19.79 thf(func_def_227, type, finite5795047828879050333T_VEBT: set_VEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_228, type, bij_be1856998921033663316omplex: (complex > complex) > set_complex > set_complex > $o).
% 84.01/19.79 thf(func_def_229, type, bij_betw_nat_complex: (nat > complex) > set_nat > set_complex > $o).
% 84.01/19.79 thf(func_def_230, type, bij_betw_nat_nat: (nat > nat) > set_nat > set_nat > $o).
% 84.01/19.79 thf(func_def_231, type, comp_C8797469213163452608nteger: ((code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > code_integer > code_integer) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 84.01/19.79 thf(func_def_232, type, comp_C1593894019821074884nteger: (code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > code_integer) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 84.01/19.79 thf(func_def_233, type, comp_int_nat_int: (int > nat) > (int > int) > int > nat).
% 84.01/19.79 thf(func_def_234, type, comp_nat_nat_nat: (nat > nat) > (nat > nat) > nat > nat).
% 84.01/19.79 thf(func_def_235, type, comp_nat_real_nat: (nat > real) > (nat > nat) > nat > real).
% 84.01/19.79 thf(func_def_236, type, inj_on_nat_nat: (nat > nat) > set_nat > $o).
% 84.01/19.79 thf(func_def_237, type, inj_on_nat_char: (nat > char) > set_nat > $o).
% 84.01/19.79 thf(func_def_238, type, inj_on_real_real: (real > real) > set_real > $o).
% 84.01/19.79 thf(func_def_239, type, strict1292158309912662752at_nat: (nat > nat) > set_nat > $o).
% 84.01/19.79 thf(func_def_240, type, the_in5290026491893676941l_real: set_real > (real > real) > real > real).
% 84.01/19.79 thf(func_def_241, type, gcd_Gcd_int: set_int > int).
% 84.01/19.79 thf(func_def_242, type, gcd_Gcd_nat: set_nat > nat).
% 84.01/19.79 thf(func_def_243, type, bezw: nat > nat > product_prod_int_int).
% 84.01/19.79 thf(func_def_244, type, bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 84.01/19.79 thf(func_def_245, type, gcd_gcd_int: int > int > int).
% 84.01/19.79 thf(func_def_246, type, gcd_gcd_nat: nat > nat > nat).
% 84.01/19.79 thf(func_def_247, type, gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 84.01/19.79 thf(func_def_248, type, abs_abs_Code_integer: code_integer > code_integer).
% 84.01/19.79 thf(func_def_249, type, abs_abs_complex: complex > complex).
% 84.01/19.79 thf(func_def_250, type, abs_abs_int: int > int).
% 84.01/19.79 thf(func_def_251, type, abs_abs_rat: rat > rat).
% 84.01/19.79 thf(func_def_252, type, abs_abs_real: real > real).
% 84.01/19.79 thf(func_def_253, type, minus_8727706125548526216plex_o: (complex > $o) > (complex > $o) > complex > $o).
% 84.01/19.79 thf(func_def_254, type, minus_minus_int_o: (int > $o) > (int > $o) > int > $o).
% 84.01/19.79 thf(func_def_255, type, minus_minus_nat_o: (nat > $o) > (nat > $o) > nat > $o).
% 84.01/19.79 thf(func_def_256, type, minus_711738161318947805_int_o: (product_prod_int_int > $o) > (product_prod_int_int > $o) > product_prod_int_int > $o).
% 84.01/19.79 thf(func_def_257, type, minus_minus_real_o: (real > $o) > (real > $o) > real > $o).
% 84.01/19.79 thf(func_def_258, type, minus_6910147592129066416_nat_o: (set_nat > $o) > (set_nat > $o) > set_nat > $o).
% 84.01/19.79 thf(func_def_259, type, minus_8373710615458151222nteger: code_integer > code_integer > code_integer).
% 84.01/19.79 thf(func_def_260, type, minus_minus_complex: complex > complex > complex).
% 84.01/19.79 thf(func_def_261, type, minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat).
% 84.01/19.79 thf(func_def_262, type, minus_minus_int: int > int > int).
% 84.01/19.79 thf(func_def_263, type, minus_minus_nat: nat > nat > nat).
% 84.01/19.79 thf(func_def_264, type, minus_minus_rat: rat > rat > rat).
% 84.01/19.79 thf(func_def_265, type, minus_minus_real: real > real > real).
% 84.01/19.79 thf(func_def_266, type, minus_811609699411566653omplex: set_complex > set_complex > set_complex).
% 84.01/19.79 thf(func_def_267, type, minus_minus_set_int: set_int > set_int > set_int).
% 84.01/19.79 thf(func_def_268, type, minus_minus_set_nat: set_nat > set_nat > set_nat).
% 84.01/19.79 thf(func_def_269, type, minus_1052850069191792384nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int).
% 84.01/19.79 thf(func_def_270, type, minus_minus_set_real: set_real > set_real > set_real).
% 84.01/19.79 thf(func_def_271, type, minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat).
% 84.01/19.79 thf(func_def_272, type, one_one_Code_integer: code_integer).
% 84.01/19.79 thf(func_def_273, type, one_one_complex: complex).
% 84.01/19.79 thf(func_def_274, type, one_on7984719198319812577d_enat: extended_enat).
% 84.01/19.79 thf(func_def_275, type, one_one_int: int).
% 84.01/19.79 thf(func_def_276, type, one_one_nat: nat).
% 84.01/19.79 thf(func_def_277, type, one_one_rat: rat).
% 84.01/19.79 thf(func_def_278, type, one_one_real: real).
% 84.01/19.79 thf(func_def_279, type, plus_p5714425477246183910nteger: code_integer > code_integer > code_integer).
% 84.01/19.79 thf(func_def_280, type, plus_plus_complex: complex > complex > complex).
% 84.01/19.79 thf(func_def_281, type, plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat).
% 84.01/19.79 thf(func_def_282, type, plus_plus_int: int > int > int).
% 84.01/19.79 thf(func_def_283, type, plus_plus_nat: nat > nat > nat).
% 84.01/19.79 thf(func_def_284, type, plus_plus_num: num > num > num).
% 84.01/19.79 thf(func_def_285, type, plus_plus_rat: rat > rat > rat).
% 84.01/19.79 thf(func_def_286, type, plus_plus_real: real > real > real).
% 84.01/19.79 thf(func_def_287, type, sgn_sgn_Code_integer: code_integer > code_integer).
% 84.01/19.79 thf(func_def_288, type, sgn_sgn_complex: complex > complex).
% 84.01/19.79 thf(func_def_289, type, sgn_sgn_int: int > int).
% 84.01/19.79 thf(func_def_290, type, sgn_sgn_rat: rat > rat).
% 84.01/19.79 thf(func_def_291, type, sgn_sgn_real: real > real).
% 84.01/19.79 thf(func_def_292, type, times_3573771949741848930nteger: code_integer > code_integer > code_integer).
% 84.01/19.79 thf(func_def_293, type, times_times_complex: complex > complex > complex).
% 84.01/19.79 thf(func_def_294, type, times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat).
% 84.01/19.79 thf(func_def_295, type, times_times_int: int > int > int).
% 84.01/19.79 thf(func_def_296, type, times_times_nat: nat > nat > nat).
% 84.01/19.79 thf(func_def_297, type, times_times_num: num > num > num).
% 84.01/19.79 thf(func_def_298, type, times_times_rat: rat > rat > rat).
% 84.01/19.79 thf(func_def_299, type, times_times_real: real > real > real).
% 84.01/19.79 thf(func_def_300, type, uminus1680532995456772888plex_o: (complex > $o) > complex > $o).
% 84.01/19.79 thf(func_def_301, type, uminus_uminus_int_o: (int > $o) > int > $o).
% 84.01/19.79 thf(func_def_302, type, uminus_uminus_nat_o: (nat > $o) > nat > $o).
% 84.01/19.79 thf(func_def_303, type, uminus7117520113953359693_int_o: (product_prod_int_int > $o) > product_prod_int_int > $o).
% 84.01/19.79 thf(func_def_304, type, uminus_uminus_real_o: (real > $o) > real > $o).
% 84.01/19.79 thf(func_def_305, type, uminus6401447641752708672_nat_o: (set_nat > $o) > set_nat > $o).
% 84.01/19.79 thf(func_def_306, type, uminus1351360451143612070nteger: code_integer > code_integer).
% 84.01/19.79 thf(func_def_307, type, uminus1482373934393186551omplex: complex > complex).
% 84.01/19.79 thf(func_def_308, type, uminus_uminus_int: int > int).
% 84.01/19.79 thf(func_def_309, type, uminus_uminus_rat: rat > rat).
% 84.01/19.79 thf(func_def_310, type, uminus_uminus_real: real > real).
% 84.01/19.79 thf(func_def_311, type, uminus8566677241136511917omplex: set_complex > set_complex).
% 84.01/19.79 thf(func_def_312, type, uminus1532241313380277803et_int: set_int > set_int).
% 84.01/19.79 thf(func_def_313, type, uminus5710092332889474511et_nat: set_nat > set_nat).
% 84.01/19.79 thf(func_def_314, type, uminus6221592323253981072nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int).
% 84.01/19.79 thf(func_def_315, type, uminus612125837232591019t_real: set_real > set_real).
% 84.01/19.79 thf(func_def_316, type, uminus613421341184616069et_nat: set_set_nat > set_set_nat).
% 84.01/19.79 thf(func_def_317, type, zero_z3403309356797280102nteger: code_integer).
% 84.01/19.79 thf(func_def_318, type, zero_zero_complex: complex).
% 84.01/19.79 thf(func_def_319, type, zero_z5237406670263579293d_enat: extended_enat).
% 84.01/19.79 thf(func_def_320, type, zero_zero_int: int).
% 84.01/19.79 thf(func_def_321, type, zero_zero_nat: nat).
% 84.01/19.79 thf(func_def_322, type, zero_zero_rat: rat).
% 84.01/19.79 thf(func_def_323, type, zero_zero_real: real).
% 84.01/19.79 thf(func_def_324, type, groups6621422865394947399nteger: (complex > code_integer) > set_complex > code_integer).
% 84.01/19.79 thf(func_def_325, type, groups7754918857620584856omplex: (complex > complex) > set_complex > complex).
% 84.01/19.79 thf(func_def_326, type, groups5690904116761175830ex_int: (complex > int) > set_complex > int).
% 84.01/19.79 thf(func_def_327, type, groups5693394587270226106ex_nat: (complex > nat) > set_complex > nat).
% 84.01/19.79 thf(func_def_328, type, groups5058264527183730370ex_rat: (complex > rat) > set_complex > rat).
% 84.01/19.79 thf(func_def_329, type, groups5808333547571424918x_real: (complex > real) > set_complex > real).
% 84.01/19.79 thf(func_def_330, type, groups7873554091576472773nteger: (int > code_integer) > set_int > code_integer).
% 84.01/19.79 thf(func_def_331, type, groups3049146728041665814omplex: (int > complex) > set_int > complex).
% 84.01/19.79 thf(func_def_332, type, groups4538972089207619220nt_int: (int > int) > set_int > int).
% 84.01/19.79 thf(func_def_333, type, groups4541462559716669496nt_nat: (int > nat) > set_int > nat).
% 84.01/19.79 thf(func_def_334, type, groups3906332499630173760nt_rat: (int > rat) > set_int > rat).
% 84.01/19.79 thf(func_def_335, type, groups8778361861064173332t_real: (int > real) > set_int > real).
% 84.01/19.79 thf(func_def_336, type, groups7501900531339628137nteger: (nat > code_integer) > set_nat > code_integer).
% 84.01/19.79 thf(func_def_337, type, groups2073611262835488442omplex: (nat > complex) > set_nat > complex).
% 84.01/19.79 thf(func_def_338, type, groups3539618377306564664at_int: (nat > int) > set_nat > int).
% 84.01/19.79 thf(func_def_339, type, groups3542108847815614940at_nat: (nat > nat) > set_nat > nat).
% 84.01/19.79 thf(func_def_340, type, groups2906978787729119204at_rat: (nat > rat) > set_nat > rat).
% 84.01/19.79 thf(func_def_341, type, groups6591440286371151544t_real: (nat > real) > set_nat > real).
% 84.01/19.79 thf(func_def_342, type, groups7713935264441627589nteger: (real > code_integer) > set_real > code_integer).
% 84.01/19.79 thf(func_def_343, type, groups5754745047067104278omplex: (real > complex) > set_real > complex).
% 84.01/19.79 thf(func_def_344, type, groups1932886352136224148al_int: (real > int) > set_real > int).
% 84.01/19.79 thf(func_def_345, type, groups1935376822645274424al_nat: (real > nat) > set_real > nat).
% 84.01/19.79 thf(func_def_346, type, groups1300246762558778688al_rat: (real > rat) > set_real > rat).
% 84.01/19.79 thf(func_def_347, type, groups8097168146408367636l_real: (real > real) > set_real > real).
% 84.01/19.79 thf(func_def_348, type, groups8255218700646806128omplex: (set_nat > complex) > set_set_nat > complex).
% 84.01/19.79 thf(func_def_349, type, groups8294997508430121362at_nat: (set_nat > nat) > set_set_nat > nat).
% 84.01/19.79 thf(func_def_350, type, groups5107569545109728110t_real: (set_nat > real) > set_set_nat > real).
% 84.01/19.79 thf(func_def_351, type, groups8682486955453173170nteger: (complex > code_integer) > set_complex > code_integer).
% 84.01/19.79 thf(func_def_352, type, groups3708469109370488835omplex: (complex > complex) > set_complex > complex).
% 84.01/19.79 thf(func_def_353, type, groups858564598930262913ex_int: (complex > int) > set_complex > int).
% 84.01/19.79 thf(func_def_354, type, groups861055069439313189ex_nat: (complex > nat) > set_complex > nat).
% 84.01/19.79 thf(func_def_355, type, groups225925009352817453ex_rat: (complex > rat) > set_complex > rat).
% 84.01/19.79 thf(func_def_356, type, groups766887009212190081x_real: (complex > real) > set_complex > real).
% 84.01/19.79 thf(func_def_357, type, groups3827104343326376752nteger: (int > code_integer) > set_int > code_integer).
% 84.01/19.79 thf(func_def_358, type, groups7440179247065528705omplex: (int > complex) > set_int > complex).
% 84.01/19.79 thf(func_def_359, type, groups1705073143266064639nt_int: (int > int) > set_int > int).
% 84.01/19.79 thf(func_def_360, type, groups1707563613775114915nt_nat: (int > nat) > set_int > nat).
% 84.01/19.79 thf(func_def_361, type, groups1072433553688619179nt_rat: (int > rat) > set_int > rat).
% 84.01/19.79 thf(func_def_362, type, groups2316167850115554303t_real: (int > real) > set_int > real).
% 84.01/19.79 thf(func_def_363, type, groups3455450783089532116nteger: (nat > code_integer) > set_nat > code_integer).
% 84.01/19.79 thf(func_def_364, type, groups6464643781859351333omplex: (nat > complex) > set_nat > complex).
% 84.01/19.79 thf(func_def_365, type, groups705719431365010083at_int: (nat > int) > set_nat > int).
% 84.01/19.79 thf(func_def_366, type, groups708209901874060359at_nat: (nat > nat) > set_nat > nat).
% 84.01/19.79 thf(func_def_367, type, groups73079841787564623at_rat: (nat > rat) > set_nat > rat).
% 84.01/19.79 thf(func_def_368, type, groups129246275422532515t_real: (nat > real) > set_nat > real).
% 84.01/19.79 thf(func_def_369, type, groups6225526099057966256nteger: (real > code_integer) > set_real > code_integer).
% 84.01/19.79 thf(func_def_370, type, groups713298508707869441omplex: (real > complex) > set_real > complex).
% 84.01/19.79 thf(func_def_371, type, groups4694064378042380927al_int: (real > int) > set_real > int).
% 84.01/19.79 thf(func_def_372, type, groups4696554848551431203al_nat: (real > nat) > set_real > nat).
% 84.01/19.79 thf(func_def_373, type, groups4061424788464935467al_rat: (real > rat) > set_real > rat).
% 84.01/19.79 thf(func_def_374, type, groups1681761925125756287l_real: (real > real) > set_real > real).
% 84.01/19.79 thf(func_def_375, type, groups4246057289670975065at_int: (set_nat > int) > set_set_nat > int).
% 84.01/19.79 thf(func_def_376, type, groups9116527308978886569_o_int: ($o > int) > int > list_o > int).
% 84.01/19.79 thf(func_def_377, type, the_int: (int > $o) > int).
% 84.01/19.79 thf(func_def_378, type, the_real: (real > $o) > real).
% 84.01/19.79 thf(func_def_379, type, if_Code_integer: $o > code_integer > code_integer > code_integer).
% 84.01/19.79 thf(func_def_380, type, if_complex: $o > complex > complex > complex).
% 84.01/19.79 thf(func_def_381, type, if_int: $o > int > int > int).
% 84.01/19.79 thf(func_def_382, type, if_list_int: $o > list_int > list_int > list_int).
% 84.01/19.79 thf(func_def_383, type, if_nat: $o > nat > nat > nat).
% 84.01/19.79 thf(func_def_384, type, if_num: $o > num > num > num).
% 84.01/19.79 thf(func_def_385, type, if_option_nat: $o > option_nat > option_nat > option_nat).
% 84.01/19.79 thf(func_def_386, type, if_option_num: $o > option_num > option_num > option_num).
% 84.01/19.79 thf(func_def_387, type, if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o).
% 84.01/19.79 thf(func_def_388, type, if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 84.01/19.79 thf(func_def_389, type, if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int).
% 84.01/19.79 thf(func_def_390, type, if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat).
% 84.01/19.79 thf(func_def_391, type, if_rat: $o > rat > rat > rat).
% 84.01/19.79 thf(func_def_392, type, if_real: $o > real > real > real).
% 84.01/19.79 thf(func_def_393, type, if_set_int: $o > set_int > set_int > set_int).
% 84.01/19.79 thf(func_def_394, type, if_set_nat: $o > set_nat > set_nat > set_nat).
% 84.01/19.79 thf(func_def_395, type, if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 84.01/19.79 thf(func_def_396, type, abs_Integ: product_prod_nat_nat > int).
% 84.01/19.79 thf(func_def_397, type, rep_Integ: int > product_prod_nat_nat).
% 84.01/19.79 thf(func_def_398, type, int_ge_less_than: int > set_Pr958786334691620121nt_int).
% 84.01/19.79 thf(func_def_399, type, int_ge_less_than2: int > set_Pr958786334691620121nt_int).
% 84.01/19.79 thf(func_def_400, type, nat2: int > nat).
% 84.01/19.79 thf(func_def_401, type, ring_1_Ints_real: set_real).
% 84.01/19.79 thf(func_def_402, type, ring_18347121197199848620nteger: int > code_integer).
% 84.01/19.79 thf(func_def_403, type, ring_17405671764205052669omplex: int > complex).
% 84.01/19.79 thf(func_def_404, type, ring_1_of_int_int: int > int).
% 84.01/19.79 thf(func_def_405, type, ring_1_of_int_rat: int > rat).
% 84.01/19.79 thf(func_def_406, type, ring_1_of_int_real: int > real).
% 84.01/19.79 thf(func_def_407, type, semila1623282765462674594er_nat: (nat > nat > nat) > nat > (nat > nat > $o) > (nat > nat > $o) > $o).
% 84.01/19.79 thf(func_def_408, type, lattic8263393255366662781ax_int: set_int > int).
% 84.01/19.79 thf(func_def_409, type, lattic8265883725875713057ax_nat: set_nat > nat).
% 84.01/19.79 thf(func_def_410, type, bfun_nat_real: (nat > real) > filter_nat > $o).
% 84.01/19.79 thf(func_def_411, type, append_int: list_int > list_int > list_int).
% 84.01/19.79 thf(func_def_412, type, append_VEBT_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_413, type, distinct_int: list_int > $o).
% 84.01/19.79 thf(func_def_414, type, drop_VEBT_VEBT: nat > list_VEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_415, type, cons_int: int > list_int > list_int).
% 84.01/19.79 thf(func_def_416, type, cons_VEBT_VEBT: vEBT_VEBT > list_VEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_417, type, nil_int: list_int).
% 84.01/19.79 thf(func_def_418, type, nil_VEBT_VEBT: list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_419, type, set_o2: list_o > set_o).
% 84.01/19.79 thf(func_def_420, type, set_complex2: list_complex > set_complex).
% 84.01/19.79 thf(func_def_421, type, set_int2: list_int > set_int).
% 84.01/19.79 thf(func_def_422, type, set_nat2: list_nat > set_nat).
% 84.01/19.79 thf(func_def_423, type, set_real2: list_real > set_real).
% 84.01/19.79 thf(func_def_424, type, set_set_nat2: list_set_nat > set_set_nat).
% 84.01/19.79 thf(func_def_425, type, set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT).
% 84.01/19.79 thf(func_def_426, type, size_list_VEBT_VEBT: (vEBT_VEBT > nat) > list_VEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_427, type, list_update_o: list_o > nat > $o > list_o).
% 84.01/19.79 thf(func_def_428, type, list_update_complex: list_complex > nat > complex > list_complex).
% 84.01/19.79 thf(func_def_429, type, list_update_int: list_int > nat > int > list_int).
% 84.01/19.79 thf(func_def_430, type, list_update_nat: list_nat > nat > nat > list_nat).
% 84.01/19.79 thf(func_def_431, type, list_update_real: list_real > nat > real > list_real).
% 84.01/19.79 thf(func_def_432, type, list_update_set_nat: list_set_nat > nat > set_nat > list_set_nat).
% 84.01/19.79 thf(func_def_433, type, list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_434, type, nth_o: list_o > nat > $o).
% 84.01/19.79 thf(func_def_435, type, nth_Code_integer: list_Code_integer > nat > code_integer).
% 84.01/19.79 thf(func_def_436, type, nth_complex: list_complex > nat > complex).
% 84.01/19.79 thf(func_def_437, type, nth_int: list_int > nat > int).
% 84.01/19.79 thf(func_def_438, type, nth_nat: list_nat > nat > nat).
% 84.01/19.79 thf(func_def_439, type, nth_num: list_num > nat > num).
% 84.01/19.79 thf(func_def_440, type, nth_Product_prod_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o).
% 84.01/19.79 thf(func_def_441, type, nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int).
% 84.01/19.79 thf(func_def_442, type, nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT).
% 84.01/19.79 thf(func_def_443, type, nth_Pr8522763379788166057eger_o: list_P8526636022914148096eger_o > nat > produc6271795597528267376eger_o).
% 84.01/19.79 thf(func_def_444, type, nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat).
% 84.01/19.79 thf(func_def_445, type, nth_Pr8326237132889035090at_num: list_P1726324292696863441at_num > nat > product_prod_nat_num).
% 84.01/19.79 thf(func_def_446, type, nth_Pr6456567536196504476um_num: list_P3744719386663036955um_num > nat > product_prod_num_num).
% 84.01/19.79 thf(func_def_447, type, nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o).
% 84.01/19.79 thf(func_def_448, type, nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int).
% 84.01/19.79 thf(func_def_449, type, nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT).
% 84.01/19.79 thf(func_def_450, type, nth_real: list_real > nat > real).
% 84.01/19.79 thf(func_def_451, type, nth_set_nat: list_set_nat > nat > set_nat).
% 84.01/19.79 thf(func_def_452, type, nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT).
% 84.01/19.79 thf(func_def_453, type, product_o_o: list_o > list_o > list_P4002435161011370285od_o_o).
% 84.01/19.79 thf(func_def_454, type, product_o_int: list_o > list_int > list_P3795440434834930179_o_int).
% 84.01/19.79 thf(func_def_455, type, product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT).
% 84.01/19.79 thf(func_def_456, type, produc3607205314601156340eger_o: list_Code_integer > list_o > list_P8526636022914148096eger_o).
% 84.01/19.79 thf(func_def_457, type, product_int_o: list_int > list_o > list_P5087981734274514673_int_o).
% 84.01/19.79 thf(func_def_458, type, product_int_int: list_int > list_int > list_P5707943133018811711nt_int).
% 84.01/19.79 thf(func_def_459, type, produc662631939642741121T_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT).
% 84.01/19.79 thf(func_def_460, type, product_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat).
% 84.01/19.79 thf(func_def_461, type, product_nat_num: list_nat > list_num > list_P1726324292696863441at_num).
% 84.01/19.79 thf(func_def_462, type, product_num_num: list_num > list_num > list_P3744719386663036955um_num).
% 84.01/19.79 thf(func_def_463, type, product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o).
% 84.01/19.79 thf(func_def_464, type, produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int).
% 84.01/19.79 thf(func_def_465, type, produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT).
% 84.01/19.79 thf(func_def_466, type, replicate_o: nat > $o > list_o).
% 84.01/19.79 thf(func_def_467, type, replicate_complex: nat > complex > list_complex).
% 84.01/19.79 thf(func_def_468, type, replicate_int: nat > int > list_int).
% 84.01/19.79 thf(func_def_469, type, replicate_nat: nat > nat > list_nat).
% 84.01/19.79 thf(func_def_470, type, replicate_real: nat > real > list_real).
% 84.01/19.79 thf(func_def_471, type, replicate_set_nat: nat > set_nat > list_set_nat).
% 84.01/19.79 thf(func_def_472, type, replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_473, type, take_VEBT_VEBT: nat > list_VEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_474, type, upto: int > int > list_int).
% 84.01/19.79 thf(func_def_475, type, upto_aux: int > int > list_int > list_int).
% 84.01/19.79 thf(func_def_476, type, upto_rel: product_prod_int_int > product_prod_int_int > $o).
% 84.01/19.79 thf(func_def_477, type, suc: nat > nat).
% 84.01/19.79 thf(func_def_478, type, compow_nat_nat: nat > (nat > nat) > nat > nat).
% 84.01/19.79 thf(func_def_479, type, case_nat_o: $o > (nat > $o) > nat > $o).
% 84.01/19.79 thf(func_def_480, type, case_nat_nat: nat > (nat > nat) > nat > nat).
% 84.01/19.79 thf(func_def_481, type, case_nat_option_num: option_num > (nat > option_num) > nat > option_num).
% 84.01/19.79 thf(func_def_482, type, pred: nat > nat).
% 84.01/19.79 thf(func_def_483, type, semiri4939895301339042750nteger: nat > code_integer).
% 84.01/19.79 thf(func_def_484, type, semiri8010041392384452111omplex: nat > complex).
% 84.01/19.79 thf(func_def_485, type, semiri1314217659103216013at_int: nat > int).
% 84.01/19.79 thf(func_def_486, type, semiri1316708129612266289at_nat: nat > nat).
% 84.01/19.79 thf(func_def_487, type, semiri681578069525770553at_rat: nat > rat).
% 84.01/19.79 thf(func_def_488, type, semiri5074537144036343181t_real: nat > real).
% 84.01/19.79 thf(func_def_489, type, size_size_list_o: list_o > nat).
% 84.01/19.79 thf(func_def_490, type, size_s3445333598471063425nteger: list_Code_integer > nat).
% 84.01/19.79 thf(func_def_491, type, size_s3451745648224563538omplex: list_complex > nat).
% 84.01/19.79 thf(func_def_492, type, size_size_list_int: list_int > nat).
% 84.01/19.79 thf(func_def_493, type, size_size_list_nat: list_nat > nat).
% 84.01/19.79 thf(func_def_494, type, size_size_list_num: list_num > nat).
% 84.01/19.79 thf(func_def_495, type, size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat).
% 84.01/19.79 thf(func_def_496, type, size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat).
% 84.01/19.79 thf(func_def_497, type, size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat).
% 84.01/19.79 thf(func_def_498, type, size_s4246224855604898693_int_o: list_P5087981734274514673_int_o > nat).
% 84.01/19.79 thf(func_def_499, type, size_s5157815400016825771nt_int: list_P5707943133018811711nt_int > nat).
% 84.01/19.79 thf(func_def_500, type, size_s6639371672096860321T_VEBT: list_P7524865323317820941T_VEBT > nat).
% 84.01/19.79 thf(func_def_501, type, size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat).
% 84.01/19.79 thf(func_def_502, type, size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat).
% 84.01/19.79 thf(func_def_503, type, size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat).
% 84.01/19.79 thf(func_def_504, type, size_size_list_real: list_real > nat).
% 84.01/19.79 thf(func_def_505, type, size_s3254054031482475050et_nat: list_set_nat > nat).
% 84.01/19.79 thf(func_def_506, type, size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_507, type, size_size_num: num > nat).
% 84.01/19.79 thf(func_def_508, type, size_size_option_nat: option_nat > nat).
% 84.01/19.79 thf(func_def_509, type, size_size_option_num: option_num > nat).
% 84.01/19.79 thf(func_def_510, type, size_s170228958280169651at_nat: option4927543243414619207at_nat > nat).
% 84.01/19.79 thf(func_def_511, type, size_size_VEBT_VEBT: vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_512, type, nat_prod_decode_aux: nat > nat > product_prod_nat_nat).
% 84.01/19.79 thf(func_def_513, type, nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 84.01/19.79 thf(func_def_514, type, nat_prod_encode: product_prod_nat_nat > nat).
% 84.01/19.79 thf(func_def_515, type, nat_set_decode: nat > set_nat).
% 84.01/19.79 thf(func_def_516, type, nat_set_encode: set_nat > nat).
% 84.01/19.79 thf(func_def_517, type, nat_triangle: nat > nat).
% 84.01/19.79 thf(func_def_518, type, root: nat > real > real).
% 84.01/19.79 thf(func_def_519, type, sqrt: real > real).
% 84.01/19.79 thf(func_def_520, type, bitM: num > num).
% 84.01/19.79 thf(func_def_521, type, inc: num > num).
% 84.01/19.79 thf(func_def_522, type, neg_nu8804712462038260780nteger: code_integer > code_integer).
% 84.01/19.79 thf(func_def_523, type, neg_nu7009210354673126013omplex: complex > complex).
% 84.01/19.79 thf(func_def_524, type, neg_numeral_dbl_int: int > int).
% 84.01/19.79 thf(func_def_525, type, neg_numeral_dbl_rat: rat > rat).
% 84.01/19.79 thf(func_def_526, type, neg_numeral_dbl_real: real > real).
% 84.01/19.79 thf(func_def_527, type, neg_nu7757733837767384882nteger: code_integer > code_integer).
% 84.01/19.79 thf(func_def_528, type, neg_nu6511756317524482435omplex: complex > complex).
% 84.01/19.79 thf(func_def_529, type, neg_nu3811975205180677377ec_int: int > int).
% 84.01/19.79 thf(func_def_530, type, neg_nu3179335615603231917ec_rat: rat > rat).
% 84.01/19.79 thf(func_def_531, type, neg_nu6075765906172075777c_real: real > real).
% 84.01/19.79 thf(func_def_532, type, neg_nu5831290666863070958nteger: code_integer > code_integer).
% 84.01/19.79 thf(func_def_533, type, neg_nu8557863876264182079omplex: complex > complex).
% 84.01/19.79 thf(func_def_534, type, neg_nu5851722552734809277nc_int: int > int).
% 84.01/19.79 thf(func_def_535, type, neg_nu5219082963157363817nc_rat: rat > rat).
% 84.01/19.79 thf(func_def_536, type, neg_nu8295874005876285629c_real: real > real).
% 84.01/19.79 thf(func_def_537, type, neg_numeral_sub_int: num > num > int).
% 84.01/19.79 thf(func_def_538, type, bit0: num > num).
% 84.01/19.79 thf(func_def_539, type, bit1: num > num).
% 84.01/19.79 thf(func_def_540, type, one: num).
% 84.01/19.79 thf(func_def_541, type, case_num_option_num: option_num > (num > option_num) > (num > option_num) > num > option_num).
% 84.01/19.79 thf(func_def_542, type, size_num: num > nat).
% 84.01/19.79 thf(func_def_543, type, num_of_nat: nat > num).
% 84.01/19.79 thf(func_def_544, type, numera6620942414471956472nteger: num > code_integer).
% 84.01/19.79 thf(func_def_545, type, numera6690914467698888265omplex: num > complex).
% 84.01/19.79 thf(func_def_546, type, numera1916890842035813515d_enat: num > extended_enat).
% 84.01/19.79 thf(func_def_547, type, numeral_numeral_int: num > int).
% 84.01/19.79 thf(func_def_548, type, numeral_numeral_nat: num > nat).
% 84.01/19.79 thf(func_def_549, type, numeral_numeral_rat: num > rat).
% 84.01/19.79 thf(func_def_550, type, numeral_numeral_real: num > real).
% 84.01/19.79 thf(func_def_551, type, pow: num > num > num).
% 84.01/19.79 thf(func_def_552, type, pred_numeral: num > nat).
% 84.01/19.79 thf(func_def_553, type, none_nat: option_nat).
% 84.01/19.79 thf(func_def_554, type, none_num: option_num).
% 84.01/19.79 thf(func_def_555, type, none_P5556105721700978146at_nat: option4927543243414619207at_nat).
% 84.01/19.79 thf(func_def_556, type, some_nat: nat > option_nat).
% 84.01/19.79 thf(func_def_557, type, some_num: num > option_num).
% 84.01/19.79 thf(func_def_558, type, some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat).
% 84.01/19.79 thf(func_def_559, type, case_o184042715313410164at_nat: $o > (product_prod_nat_nat > $o) > option4927543243414619207at_nat > $o).
% 84.01/19.79 thf(func_def_560, type, case_option_int_num: int > (num > int) > option_num > int).
% 84.01/19.79 thf(func_def_561, type, case_option_num_num: num > (num > num) > option_num > num).
% 84.01/19.79 thf(func_def_562, type, case_o6005452278849405969um_num: option_num > (num > option_num) > option_num > option_num).
% 84.01/19.79 thf(func_def_563, type, map_option_num_num: (num > num) > option_num > option_num).
% 84.01/19.79 thf(func_def_564, type, size_option_nat: (nat > nat) > option_nat > nat).
% 84.01/19.79 thf(func_def_565, type, size_option_num: (num > nat) > option_num > nat).
% 84.01/19.79 thf(func_def_566, type, size_o8335143837870341156at_nat: (product_prod_nat_nat > nat) > option4927543243414619207at_nat > nat).
% 84.01/19.79 thf(func_def_567, type, the_nat: option_nat > nat).
% 84.01/19.79 thf(func_def_568, type, the_num: option_num > num).
% 84.01/19.79 thf(func_def_569, type, the_Pr8591224930841456533at_nat: option4927543243414619207at_nat > product_prod_nat_nat).
% 84.01/19.79 thf(func_def_570, type, bot_bot_nat: nat).
% 84.01/19.79 thf(func_def_571, type, bot_bot_set_complex: set_complex).
% 84.01/19.79 thf(func_def_572, type, bot_bot_set_int: set_int).
% 84.01/19.79 thf(func_def_573, type, bot_bot_set_nat: set_nat).
% 84.01/19.79 thf(func_def_574, type, bot_bot_set_num: set_num).
% 84.01/19.79 thf(func_def_575, type, bot_bo1796632182523588997nt_int: set_Pr958786334691620121nt_int).
% 84.01/19.79 thf(func_def_576, type, bot_bot_set_rat: set_rat).
% 84.01/19.79 thf(func_def_577, type, bot_bot_set_real: set_real).
% 84.01/19.79 thf(func_def_578, type, bot_bot_set_set_nat: set_set_nat).
% 84.01/19.79 thf(func_def_579, type, ord_less_complex_o: (complex > $o) > (complex > $o) > $o).
% 84.01/19.79 thf(func_def_580, type, ord_less_int_o: (int > $o) > (int > $o) > $o).
% 84.01/19.79 thf(func_def_581, type, ord_less_nat_o: (nat > $o) > (nat > $o) > $o).
% 84.01/19.79 thf(func_def_582, type, ord_less_real_o: (real > $o) > (real > $o) > $o).
% 84.01/19.79 thf(func_def_583, type, ord_less_set_nat_o: (set_nat > $o) > (set_nat > $o) > $o).
% 84.01/19.79 thf(func_def_584, type, ord_le6747313008572928689nteger: code_integer > code_integer > $o).
% 84.01/19.79 thf(func_def_585, type, ord_le72135733267957522d_enat: extended_enat > extended_enat > $o).
% 84.01/19.79 thf(func_def_586, type, ord_less_int: int > int > $o).
% 84.01/19.79 thf(func_def_587, type, ord_less_nat: nat > nat > $o).
% 84.01/19.79 thf(func_def_588, type, ord_less_num: num > num > $o).
% 84.01/19.79 thf(func_def_589, type, ord_less_rat: rat > rat > $o).
% 84.01/19.79 thf(func_def_590, type, ord_less_real: real > real > $o).
% 84.01/19.79 thf(func_def_591, type, ord_le1307284697595431911nteger: set_Code_integer > set_Code_integer > $o).
% 84.01/19.79 thf(func_def_592, type, ord_less_set_complex: set_complex > set_complex > $o).
% 84.01/19.79 thf(func_def_593, type, ord_less_set_int: set_int > set_int > $o).
% 84.01/19.79 thf(func_def_594, type, ord_less_set_nat: set_nat > set_nat > $o).
% 84.01/19.79 thf(func_def_595, type, ord_less_set_num: set_num > set_num > $o).
% 84.01/19.79 thf(func_def_596, type, ord_less_set_rat: set_rat > set_rat > $o).
% 84.01/19.79 thf(func_def_597, type, ord_less_set_real: set_real > set_real > $o).
% 84.01/19.79 thf(func_def_598, type, ord_less_set_set_nat: set_set_nat > set_set_nat > $o).
% 84.01/19.79 thf(func_def_599, type, ord_le4573692005234683329plex_o: (complex > $o) > (complex > $o) > $o).
% 84.01/19.79 thf(func_def_600, type, ord_less_eq_int_o: (int > $o) > (int > $o) > $o).
% 84.01/19.79 thf(func_def_601, type, ord_less_eq_nat_o: (nat > $o) > (nat > $o) > $o).
% 84.01/19.79 thf(func_def_602, type, ord_less_eq_real_o: (real > $o) > (real > $o) > $o).
% 84.01/19.79 thf(func_def_603, type, ord_le3964352015994296041_nat_o: (set_nat > $o) > (set_nat > $o) > $o).
% 84.01/19.79 thf(func_def_604, type, ord_le3102999989581377725nteger: code_integer > code_integer > $o).
% 84.01/19.79 thf(func_def_605, type, ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o).
% 84.01/19.79 thf(func_def_606, type, ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o).
% 84.01/19.79 thf(func_def_607, type, ord_less_eq_int: int > int > $o).
% 84.01/19.79 thf(func_def_608, type, ord_less_eq_nat: nat > nat > $o).
% 84.01/19.79 thf(func_def_609, type, ord_less_eq_num: num > num > $o).
% 84.01/19.79 thf(func_def_610, type, ord_less_eq_rat: rat > rat > $o).
% 84.01/19.79 thf(func_def_611, type, ord_less_eq_real: real > real > $o).
% 84.01/19.79 thf(func_def_612, type, ord_less_eq_set_o: set_o > set_o > $o).
% 84.01/19.79 thf(func_def_613, type, ord_le7084787975880047091nteger: set_Code_integer > set_Code_integer > $o).
% 84.01/19.79 thf(func_def_614, type, ord_le211207098394363844omplex: set_complex > set_complex > $o).
% 84.01/19.79 thf(func_def_615, type, ord_less_eq_set_int: set_int > set_int > $o).
% 84.01/19.79 thf(func_def_616, type, ord_less_eq_set_nat: set_nat > set_nat > $o).
% 84.01/19.79 thf(func_def_617, type, ord_less_eq_set_num: set_num > set_num > $o).
% 84.01/19.79 thf(func_def_618, type, ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o).
% 84.01/19.79 thf(func_def_619, type, ord_less_eq_set_rat: set_rat > set_rat > $o).
% 84.01/19.79 thf(func_def_620, type, ord_less_eq_set_real: set_real > set_real > $o).
% 84.01/19.79 thf(func_def_621, type, ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o).
% 84.01/19.79 thf(func_def_622, type, ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_623, type, ord_max_Code_integer: code_integer > code_integer > code_integer).
% 84.01/19.79 thf(func_def_624, type, ord_max_int: int > int > int).
% 84.01/19.79 thf(func_def_625, type, ord_max_nat: nat > nat > nat).
% 84.01/19.79 thf(func_def_626, type, ord_max_num: num > num > num).
% 84.01/19.79 thf(func_def_627, type, ord_max_rat: rat > rat > rat).
% 84.01/19.79 thf(func_def_628, type, ord_max_real: real > real > real).
% 84.01/19.79 thf(func_def_629, type, ord_max_set_nat: set_nat > set_nat > set_nat).
% 84.01/19.79 thf(func_def_630, type, order_Greatest_nat: (nat > $o) > nat).
% 84.01/19.79 thf(func_def_631, type, order_9091379641038594480t_real: (nat > real) > $o).
% 84.01/19.79 thf(func_def_632, type, order_mono_nat_nat: (nat > nat) > $o).
% 84.01/19.79 thf(func_def_633, type, order_mono_nat_real: (nat > real) > $o).
% 84.01/19.79 thf(func_def_634, type, order_mono_real_real: (real > real) > $o).
% 84.01/19.79 thf(func_def_635, type, order_5726023648592871131at_nat: (nat > nat) > $o).
% 84.01/19.79 thf(func_def_636, type, order_7092887310737990675l_real: (real > real) > $o).
% 84.01/19.79 thf(func_def_637, type, top_top_set_o: set_o).
% 84.01/19.79 thf(func_def_638, type, top_top_set_nat: set_nat).
% 84.01/19.79 thf(func_def_639, type, top_to1996260823553986621t_unit: set_Product_unit).
% 84.01/19.79 thf(func_def_640, type, top_top_set_real: set_real).
% 84.01/19.79 thf(func_def_641, type, top_top_set_char: set_char).
% 84.01/19.79 thf(func_def_642, type, power_8256067586552552935nteger: code_integer > nat > code_integer).
% 84.01/19.79 thf(func_def_643, type, power_power_complex: complex > nat > complex).
% 84.01/19.79 thf(func_def_644, type, power_power_int: int > nat > int).
% 84.01/19.79 thf(func_def_645, type, power_power_nat: nat > nat > nat).
% 84.01/19.79 thf(func_def_646, type, power_power_rat: rat > nat > rat).
% 84.01/19.79 thf(func_def_647, type, power_power_real: real > nat > real).
% 84.01/19.79 thf(func_def_648, type, produc4035269172776083154on_nat: (nat > nat > $o) > produc4953844613479565601on_nat > produc2233624965454879586on_nat).
% 84.01/19.79 thf(func_def_649, type, produc3209952032786966637at_nat: (nat > nat > nat) > produc7248412053542808358at_nat > produc4471711990508489141at_nat).
% 84.01/19.79 thf(func_def_650, type, produc8929957630744042906on_nat: (nat > nat > nat) > produc4953844613479565601on_nat > produc8306885398267862888on_nat).
% 84.01/19.79 thf(func_def_651, type, produc851828971589881931at_num: (nat > num > num) > produc2963631642982155120at_num > produc3368934014287244435at_num).
% 84.01/19.79 thf(func_def_652, type, produc3576312749637752826on_num: (num > num > $o) > produc3447558737645232053on_num > produc7036089656553540234on_num).
% 84.01/19.79 thf(func_def_653, type, produc5778274026573060048on_num: (num > num > num) > produc3447558737645232053on_num > produc1193250871479095198on_num).
% 84.01/19.79 thf(func_def_654, type, produc3994169339658061776at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > produc6121120109295599847at_nat > produc5491161045314408544at_nat).
% 84.01/19.79 thf(func_def_655, type, produc2899441246263362727at_nat: (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > produc6121120109295599847at_nat > produc5542196010084753463at_nat).
% 84.01/19.79 thf(func_def_656, type, product_Pair_o_o: $o > $o > product_prod_o_o).
% 84.01/19.79 thf(func_def_657, type, product_Pair_o_int: $o > int > product_prod_o_int).
% 84.01/19.79 thf(func_def_658, type, produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT).
% 84.01/19.79 thf(func_def_659, type, produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o).
% 84.01/19.79 thf(func_def_660, type, produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger).
% 84.01/19.79 thf(func_def_661, type, product_Pair_int_int: int > int > product_prod_int_int).
% 84.01/19.79 thf(func_def_662, type, product_Pair_nat_nat: nat > nat > product_prod_nat_nat).
% 84.01/19.79 thf(func_def_663, type, product_Pair_nat_num: nat > num > product_prod_nat_num).
% 84.01/19.79 thf(func_def_664, type, produc487386426758144856at_nat: nat > product_prod_nat_nat > produc7248412053542808358at_nat).
% 84.01/19.79 thf(func_def_665, type, produc1195630363706982562at_num: nat > product_prod_nat_num > produc2963631642982155120at_num).
% 84.01/19.79 thf(func_def_666, type, product_Pair_num_num: num > num > product_prod_num_num).
% 84.01/19.79 thf(func_def_667, type, produc5098337634421038937on_nat: option_nat > option_nat > produc4953844613479565601on_nat).
% 84.01/19.79 thf(func_def_668, type, produc8585076106096196333on_num: option_num > option_num > produc3447558737645232053on_num).
% 84.01/19.79 thf(func_def_669, type, produc488173922507101015at_nat: option4927543243414619207at_nat > option4927543243414619207at_nat > produc6121120109295599847at_nat).
% 84.01/19.79 thf(func_def_670, type, produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o).
% 84.01/19.79 thf(func_def_671, type, produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int).
% 84.01/19.79 thf(func_def_672, type, produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat).
% 84.01/19.79 thf(func_def_673, type, produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT).
% 84.01/19.79 thf(func_def_674, type, produc6499014454317279255nteger: (code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 84.01/19.79 thf(func_def_675, type, produc7828578312038201481er_o_o: (code_integer > $o > $o) > produc6271795597528267376eger_o > $o).
% 84.01/19.79 thf(func_def_676, type, produc1043322548047392435omplex: (code_integer > $o > set_complex) > produc6271795597528267376eger_o > set_complex).
% 84.01/19.79 thf(func_def_677, type, produc1253318751659547953et_int: (code_integer > $o > set_int) > produc6271795597528267376eger_o > set_int).
% 84.01/19.79 thf(func_def_678, type, produc5431169771168744661et_nat: (code_integer > $o > set_nat) > produc6271795597528267376eger_o > set_nat).
% 84.01/19.79 thf(func_def_679, type, produc242741666403216561t_real: (code_integer > $o > set_real) > produc6271795597528267376eger_o > set_real).
% 84.01/19.79 thf(func_def_680, type, produc4188289175737317920o_char: (code_integer > $o > char) > produc6271795597528267376eger_o > char).
% 84.01/19.79 thf(func_def_681, type, produc1553301316500091796er_int: (code_integer > code_integer > int) > produc8923325533196201883nteger > int).
% 84.01/19.79 thf(func_def_682, type, produc1555791787009142072er_nat: (code_integer > code_integer > nat) > produc8923325533196201883nteger > nat).
% 84.01/19.79 thf(func_def_683, type, produc7336495610019696514er_num: (code_integer > code_integer > num) > produc8923325533196201883nteger > num).
% 84.01/19.79 thf(func_def_684, type, produc9125791028180074456eger_o: (code_integer > code_integer > produc6271795597528267376eger_o) > produc8923325533196201883nteger > produc6271795597528267376eger_o).
% 84.01/19.79 thf(func_def_685, type, produc6916734918728496179nteger: (code_integer > code_integer > produc8923325533196201883nteger) > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 84.01/19.79 thf(func_def_686, type, produc6771430404735790350plex_o: (complex > complex > $o) > produc4411394909380815293omplex > $o).
% 84.01/19.79 thf(func_def_687, type, produc4947309494688390418_int_o: (int > int > $o) > product_prod_int_int > $o).
% 84.01/19.79 thf(func_def_688, type, produc8211389475949308722nt_int: (int > int > int) > product_prod_int_int > int).
% 84.01/19.79 thf(func_def_689, type, produc4245557441103728435nt_int: (int > int > product_prod_int_int) > product_prod_int_int > product_prod_int_int).
% 84.01/19.79 thf(func_def_690, type, produc8739625826339149834_nat_o: (nat > nat > product_prod_nat_nat > $o) > product_prod_nat_nat > product_prod_nat_nat > $o).
% 84.01/19.79 thf(func_def_691, type, produc27273713700761075at_nat: (nat > nat > product_prod_nat_nat > product_prod_nat_nat) > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat).
% 84.01/19.79 thf(func_def_692, type, produc6081775807080527818_nat_o: (nat > nat > $o) > product_prod_nat_nat > $o).
% 84.01/19.79 thf(func_def_693, type, produc1917071388513777916omplex: (nat > nat > complex) > product_prod_nat_nat > complex).
% 84.01/19.79 thf(func_def_694, type, produc6840382203811409530at_int: (nat > nat > int) > product_prod_nat_nat > int).
% 84.01/19.79 thf(func_def_695, type, produc6842872674320459806at_nat: (nat > nat > nat) > product_prod_nat_nat > nat).
% 84.01/19.79 thf(func_def_696, type, produc2626176000494625587at_nat: (nat > nat > product_prod_nat_nat) > product_prod_nat_nat > product_prod_nat_nat).
% 84.01/19.79 thf(func_def_697, type, produc6207742614233964070at_rat: (nat > nat > rat) > product_prod_nat_nat > rat).
% 84.01/19.79 thf(func_def_698, type, produc1703576794950452218t_real: (nat > nat > real) > product_prod_nat_nat > real).
% 84.01/19.79 thf(func_def_699, type, produc4927758841916487424_num_o: (nat > num > $o) > product_prod_nat_num > $o).
% 84.01/19.79 thf(func_def_700, type, produc478579273971653890on_num: (nat > num > option_num) > product_prod_nat_num > option_num).
% 84.01/19.79 thf(func_def_701, type, produc4130284055270567454et_nat: (nat > num > set_nat) > product_prod_nat_num > set_nat).
% 84.01/19.79 thf(func_def_702, type, produc1435849484188172666t_real: (nat > num > set_real) > product_prod_nat_num > set_real).
% 84.01/19.79 thf(func_def_703, type, produc5703948589228662326_num_o: (num > num > $o) > product_prod_num_num > $o).
% 84.01/19.79 thf(func_def_704, type, produc2866383454006189126omplex: (num > num > set_complex) > product_prod_num_num > set_complex).
% 84.01/19.79 thf(func_def_705, type, produc6406642877701697732et_int: (num > num > set_int) > product_prod_num_num > set_int).
% 84.01/19.79 thf(func_def_706, type, produc1361121860356118632et_nat: (num > num > set_nat) > product_prod_num_num > set_nat).
% 84.01/19.79 thf(func_def_707, type, produc8296048397933160132t_real: (num > num > set_real) > product_prod_num_num > set_real).
% 84.01/19.79 thf(func_def_708, type, produc5414030515140494994real_o: (real > real > $o) > produc2422161461964618553l_real > $o).
% 84.01/19.79 thf(func_def_709, type, produc8508995932063986495nteger: produc8923325533196201883nteger > code_integer).
% 84.01/19.79 thf(func_def_710, type, product_fst_int_int: product_prod_int_int > int).
% 84.01/19.79 thf(func_def_711, type, product_fst_nat_nat: product_prod_nat_nat > nat).
% 84.01/19.79 thf(func_def_712, type, product_snd_int_int: product_prod_int_int > int).
% 84.01/19.79 thf(func_def_713, type, product_snd_nat_nat: product_prod_nat_nat > nat).
% 84.01/19.79 thf(func_def_714, type, frct: product_prod_int_int > rat).
% 84.01/19.79 thf(func_def_715, type, normalize: product_prod_int_int > product_prod_int_int).
% 84.01/19.79 thf(func_def_716, type, of_int: int > rat).
% 84.01/19.79 thf(func_def_717, type, quotient_of: rat > product_prod_int_int).
% 84.01/19.79 thf(func_def_718, type, real_V2521375963428798218omplex: set_complex).
% 84.01/19.79 thf(func_def_719, type, real_V5970128139526366754l_real: (real > real) > $o).
% 84.01/19.79 thf(func_def_720, type, real_V3694042436643373181omplex: complex > complex > real).
% 84.01/19.79 thf(func_def_721, type, real_V975177566351809787t_real: real > real > real).
% 84.01/19.79 thf(func_def_722, type, real_V1022390504157884413omplex: complex > real).
% 84.01/19.79 thf(func_def_723, type, real_V7735802525324610683m_real: real > real).
% 84.01/19.79 thf(func_def_724, type, real_V4546457046886955230omplex: real > complex).
% 84.01/19.79 thf(func_def_725, type, real_V2046097035970521341omplex: real > complex > complex).
% 84.01/19.79 thf(func_def_726, type, real_V1485227260804924795R_real: real > real > real).
% 84.01/19.79 thf(func_def_727, type, divide6298287555418463151nteger: code_integer > code_integer > code_integer).
% 84.01/19.79 thf(func_def_728, type, divide1717551699836669952omplex: complex > complex > complex).
% 84.01/19.79 thf(func_def_729, type, divide_divide_int: int > int > int).
% 84.01/19.79 thf(func_def_730, type, divide_divide_nat: nat > nat > nat).
% 84.01/19.79 thf(func_def_731, type, divide_divide_rat: rat > rat > rat).
% 84.01/19.79 thf(func_def_732, type, divide_divide_real: real > real > real).
% 84.01/19.79 thf(func_def_733, type, dvd_dvd_Code_integer: code_integer > code_integer > $o).
% 84.01/19.79 thf(func_def_734, type, dvd_dvd_complex: complex > complex > $o).
% 84.01/19.79 thf(func_def_735, type, dvd_dvd_int: int > int > $o).
% 84.01/19.79 thf(func_def_736, type, dvd_dvd_nat: nat > nat > $o).
% 84.01/19.79 thf(func_def_737, type, dvd_dvd_rat: rat > rat > $o).
% 84.01/19.79 thf(func_def_738, type, dvd_dvd_real: real > real > $o).
% 84.01/19.79 thf(func_def_739, type, modulo364778990260209775nteger: code_integer > code_integer > code_integer).
% 84.01/19.79 thf(func_def_740, type, modulo_modulo_int: int > int > int).
% 84.01/19.79 thf(func_def_741, type, modulo_modulo_nat: nat > nat > nat).
% 84.01/19.79 thf(func_def_742, type, zero_n356916108424825756nteger: $o > code_integer).
% 84.01/19.79 thf(func_def_743, type, zero_n1201886186963655149omplex: $o > complex).
% 84.01/19.79 thf(func_def_744, type, zero_n2684676970156552555ol_int: $o > int).
% 84.01/19.79 thf(func_def_745, type, zero_n2687167440665602831ol_nat: $o > nat).
% 84.01/19.79 thf(func_def_746, type, zero_n2052037380579107095ol_rat: $o > rat).
% 84.01/19.79 thf(func_def_747, type, zero_n3304061248610475627l_real: $o > real).
% 84.01/19.79 thf(func_def_748, type, suminf_complex: (nat > complex) > complex).
% 84.01/19.79 thf(func_def_749, type, suminf_int: (nat > int) > int).
% 84.01/19.79 thf(func_def_750, type, suminf_nat: (nat > nat) > nat).
% 84.01/19.79 thf(func_def_751, type, suminf_real: (nat > real) > real).
% 84.01/19.79 thf(func_def_752, type, summable_complex: (nat > complex) > $o).
% 84.01/19.79 thf(func_def_753, type, summable_int: (nat > int) > $o).
% 84.01/19.79 thf(func_def_754, type, summable_nat: (nat > nat) > $o).
% 84.01/19.79 thf(func_def_755, type, summable_real: (nat > real) > $o).
% 84.01/19.79 thf(func_def_756, type, sums_complex: (nat > complex) > complex > $o).
% 84.01/19.79 thf(func_def_757, type, sums_int: (nat > int) > int > $o).
% 84.01/19.79 thf(func_def_758, type, sums_nat: (nat > nat) > nat > $o).
% 84.01/19.79 thf(func_def_759, type, sums_real: (nat > real) > real > $o).
% 84.01/19.79 thf(func_def_760, type, collect_Code_integer: (code_integer > $o) > set_Code_integer).
% 84.01/19.79 thf(func_def_761, type, collect_complex: (complex > $o) > set_complex).
% 84.01/19.79 thf(func_def_762, type, collect_int: (int > $o) > set_int).
% 84.01/19.79 thf(func_def_763, type, collect_list_o: (list_o > $o) > set_list_o).
% 84.01/19.79 thf(func_def_764, type, collect_list_complex: (list_complex > $o) > set_list_complex).
% 84.01/19.79 thf(func_def_765, type, collect_list_int: (list_int > $o) > set_list_int).
% 84.01/19.79 thf(func_def_766, type, collect_list_nat: (list_nat > $o) > set_list_nat).
% 84.01/19.79 thf(func_def_767, type, collec5608196760682091941T_VEBT: (list_VEBT_VEBT > $o) > set_list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_768, type, collect_nat: (nat > $o) > set_nat).
% 84.01/19.79 thf(func_def_769, type, collect_num: (num > $o) > set_num).
% 84.01/19.79 thf(func_def_770, type, collec8663557070575231912omplex: (produc4411394909380815293omplex > $o) > set_Pr5085853215250843933omplex).
% 84.01/19.79 thf(func_def_771, type, collec213857154873943460nt_int: (product_prod_int_int > $o) > set_Pr958786334691620121nt_int).
% 84.01/19.79 thf(func_def_772, type, collec3799799289383736868l_real: (produc2422161461964618553l_real > $o) > set_Pr6218003697084177305l_real).
% 84.01/19.79 thf(func_def_773, type, collect_rat: (rat > $o) > set_rat).
% 84.01/19.79 thf(func_def_774, type, collect_real: (real > $o) > set_real).
% 84.01/19.79 thf(func_def_775, type, collect_set_complex: (set_complex > $o) > set_set_complex).
% 84.01/19.79 thf(func_def_776, type, collect_set_int: (set_int > $o) > set_set_int).
% 84.01/19.79 thf(func_def_777, type, collect_set_nat: (set_nat > $o) > set_set_nat).
% 84.01/19.79 thf(func_def_778, type, pow_nat: set_nat > set_set_nat).
% 84.01/19.79 thf(func_def_779, type, image_int_int: (int > int) > set_int > set_int).
% 84.01/19.79 thf(func_def_780, type, image_int_nat: (int > nat) > set_int > set_nat).
% 84.01/19.79 thf(func_def_781, type, image_nat_int: (nat > int) > set_nat > set_int).
% 84.01/19.79 thf(func_def_782, type, image_nat_nat: (nat > nat) > set_nat > set_nat).
% 84.01/19.79 thf(func_def_783, type, image_nat_real: (nat > real) > set_nat > set_real).
% 84.01/19.79 thf(func_def_784, type, image_nat_set_nat: (nat > set_nat) > set_nat > set_set_nat).
% 84.01/19.79 thf(func_def_785, type, image_nat_char: (nat > char) > set_nat > set_char).
% 84.01/19.79 thf(func_def_786, type, image_5971271580939081552omplex: (real > filter6041513312241820739omplex) > set_real > set_fi4554929511873752355omplex).
% 84.01/19.79 thf(func_def_787, type, image_2178119161166701260l_real: (real > filter2146258269922977983l_real) > set_real > set_fi7789364187291644575l_real).
% 84.01/19.79 thf(func_def_788, type, image_real_real: (real > real) > set_real > set_real).
% 84.01/19.79 thf(func_def_789, type, image_char_nat: (char > nat) > set_char > set_nat).
% 84.01/19.79 thf(func_def_790, type, insert_int: int > set_int > set_int).
% 84.01/19.79 thf(func_def_791, type, insert_nat: nat > set_nat > set_nat).
% 84.01/19.79 thf(func_def_792, type, insert_real: real > set_real > set_real).
% 84.01/19.79 thf(func_def_793, type, set_fo1517530859248394432omplex: (nat > complex > complex) > nat > nat > complex > complex).
% 84.01/19.79 thf(func_def_794, type, set_fo2581907887559384638at_int: (nat > int > int) > nat > nat > int > int).
% 84.01/19.79 thf(func_def_795, type, set_fo2584398358068434914at_nat: (nat > nat > nat) > nat > nat > nat > nat).
% 84.01/19.79 thf(func_def_796, type, set_fo1949268297981939178at_rat: (nat > rat > rat) > nat > nat > rat > rat).
% 84.01/19.79 thf(func_def_797, type, set_fo3111899725591712190t_real: (nat > real > real) > nat > nat > real > real).
% 84.01/19.79 thf(func_def_798, type, set_or1266510415728281911st_int: int > int > set_int).
% 84.01/19.79 thf(func_def_799, type, set_or1269000886237332187st_nat: nat > nat > set_nat).
% 84.01/19.79 thf(func_def_800, type, set_or7049704709247886629st_num: num > num > set_num).
% 84.01/19.79 thf(func_def_801, type, set_or633870826150836451st_rat: rat > rat > set_rat).
% 84.01/19.79 thf(func_def_802, type, set_or1222579329274155063t_real: real > real > set_real).
% 84.01/19.79 thf(func_def_803, type, set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat).
% 84.01/19.79 thf(func_def_804, type, set_or4662586982721622107an_int: int > int > set_int).
% 84.01/19.79 thf(func_def_805, type, set_or4665077453230672383an_nat: nat > nat > set_nat).
% 84.01/19.79 thf(func_def_806, type, set_ord_atLeast_nat: nat > set_nat).
% 84.01/19.79 thf(func_def_807, type, set_ord_atLeast_real: real > set_real).
% 84.01/19.79 thf(func_def_808, type, set_ord_atMost_nat: nat > set_nat).
% 84.01/19.79 thf(func_def_809, type, set_or6656581121297822940st_int: int > int > set_int).
% 84.01/19.79 thf(func_def_810, type, set_or6659071591806873216st_nat: nat > nat > set_nat).
% 84.01/19.79 thf(func_def_811, type, set_or5832277885323065728an_int: int > int > set_int).
% 84.01/19.79 thf(func_def_812, type, set_or5834768355832116004an_nat: nat > nat > set_nat).
% 84.01/19.79 thf(func_def_813, type, set_or1633881224788618240n_real: real > real > set_real).
% 84.01/19.79 thf(func_def_814, type, set_or1210151606488870762an_nat: nat > set_nat).
% 84.01/19.79 thf(func_def_815, type, set_or5849166863359141190n_real: real > set_real).
% 84.01/19.79 thf(func_def_816, type, set_ord_lessThan_int: int > set_int).
% 84.01/19.79 thf(func_def_817, type, set_ord_lessThan_nat: nat > set_nat).
% 84.01/19.79 thf(func_def_818, type, set_ord_lessThan_num: num > set_num).
% 84.01/19.79 thf(func_def_819, type, set_ord_lessThan_rat: rat > set_rat).
% 84.01/19.79 thf(func_def_820, type, set_or5984915006950818249n_real: real > set_real).
% 84.01/19.79 thf(func_def_821, type, set_or890127255671739683et_nat: set_nat > set_set_nat).
% 84.01/19.79 thf(func_def_822, type, ascii_of: char > char).
% 84.01/19.79 thf(func_def_823, type, char2: $o > $o > $o > $o > $o > $o > $o > $o > char).
% 84.01/19.79 thf(func_def_824, type, char_of_integer: code_integer > char).
% 84.01/19.79 thf(func_def_825, type, comm_s629917340098488124ar_nat: char > nat).
% 84.01/19.79 thf(func_def_826, type, integer_of_char: char > code_integer).
% 84.01/19.79 thf(func_def_827, type, unique3096191561947761185of_nat: nat > char).
% 84.01/19.79 thf(func_def_828, type, topolo4422821103128117721l_real: filter_real > (real > real) > $o).
% 84.01/19.79 thf(func_def_829, type, topolo5044208981011980120l_real: set_real > (real > real) > $o).
% 84.01/19.79 thf(func_def_830, type, topolo2919662092509805066nteger: (nat > code_integer) > $o).
% 84.01/19.79 thf(func_def_831, type, topolo4899668324122417113eq_int: (nat > int) > $o).
% 84.01/19.79 thf(func_def_832, type, topolo4902158794631467389eq_nat: (nat > nat) > $o).
% 84.01/19.79 thf(func_def_833, type, topolo1459490580787246023eq_num: (nat > num) > $o).
% 84.01/19.79 thf(func_def_834, type, topolo4267028734544971653eq_rat: (nat > rat) > $o).
% 84.01/19.79 thf(func_def_835, type, topolo6980174941875973593q_real: (nat > real) > $o).
% 84.01/19.79 thf(func_def_836, type, topolo7278393974255667507et_nat: (nat > set_nat) > $o).
% 84.01/19.79 thf(func_def_837, type, topolo4110288021797289639omplex: set_complex > $o).
% 84.01/19.79 thf(func_def_838, type, topolo4860482606490270245n_real: set_real > $o).
% 84.01/19.79 thf(func_def_839, type, topolo2177554685111907308n_real: real > set_real > filter_real).
% 84.01/19.79 thf(func_def_840, type, topolo2815343760600316023s_real: real > filter_real).
% 84.01/19.79 thf(func_def_841, type, topolo6517432010174082258omplex: (nat > complex) > $o).
% 84.01/19.79 thf(func_def_842, type, topolo4055970368930404560y_real: (nat > real) > $o).
% 84.01/19.79 thf(func_def_843, type, topolo896644834953643431omplex: filter6041513312241820739omplex).
% 84.01/19.79 thf(func_def_844, type, topolo1511823702728130853y_real: filter2146258269922977983l_real).
% 84.01/19.79 thf(func_def_845, type, arccos: real > real).
% 84.01/19.79 thf(func_def_846, type, arcosh_real: real > real).
% 84.01/19.79 thf(func_def_847, type, arcsin: real > real).
% 84.01/19.79 thf(func_def_848, type, arctan: real > real).
% 84.01/19.79 thf(func_def_849, type, arsinh_real: real > real).
% 84.01/19.79 thf(func_def_850, type, artanh_real: real > real).
% 84.01/19.79 thf(func_def_851, type, cos_complex: complex > complex).
% 84.01/19.79 thf(func_def_852, type, cos_real: real > real).
% 84.01/19.79 thf(func_def_853, type, cos_coeff: nat > real).
% 84.01/19.79 thf(func_def_854, type, cosh_real: real > real).
% 84.01/19.79 thf(func_def_855, type, cot_real: real > real).
% 84.01/19.79 thf(func_def_856, type, diffs_Code_integer: (nat > code_integer) > nat > code_integer).
% 84.01/19.79 thf(func_def_857, type, diffs_complex: (nat > complex) > nat > complex).
% 84.01/19.79 thf(func_def_858, type, diffs_int: (nat > int) > nat > int).
% 84.01/19.79 thf(func_def_859, type, diffs_rat: (nat > rat) > nat > rat).
% 84.01/19.79 thf(func_def_860, type, diffs_real: (nat > real) > nat > real).
% 84.01/19.79 thf(func_def_861, type, exp_complex: complex > complex).
% 84.01/19.79 thf(func_def_862, type, exp_real: real > real).
% 84.01/19.79 thf(func_def_863, type, ln_ln_real: real > real).
% 84.01/19.79 thf(func_def_864, type, log: real > real > real).
% 84.01/19.79 thf(func_def_865, type, pi: real).
% 84.01/19.79 thf(func_def_866, type, powr_real: real > real > real).
% 84.01/19.79 thf(func_def_867, type, sin_complex: complex > complex).
% 84.01/19.79 thf(func_def_868, type, sin_real: real > real).
% 84.01/19.79 thf(func_def_869, type, sin_coeff: nat > real).
% 84.01/19.79 thf(func_def_870, type, sinh_real: real > real).
% 84.01/19.79 thf(func_def_871, type, tan_complex: complex > complex).
% 84.01/19.79 thf(func_def_872, type, tan_real: real > real).
% 84.01/19.79 thf(func_def_873, type, tanh_complex: complex > complex).
% 84.01/19.79 thf(func_def_874, type, tanh_real: real > real).
% 84.01/19.79 thf(func_def_875, type, vEBT_Leaf: $o > $o > vEBT_VEBT).
% 84.01/19.79 thf(func_def_876, type, vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 84.01/19.79 thf(func_def_877, type, vEBT_size_VEBT: vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_878, type, vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o).
% 84.01/19.79 thf(func_def_879, type, vEBT_VEBT_high: nat > nat > nat).
% 84.01/19.79 thf(func_def_880, type, vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o).
% 84.01/19.79 thf(func_def_881, type, vEBT_VEBT_low: nat > nat > nat).
% 84.01/19.79 thf(func_def_882, type, vEBT_VEBT_membermima: vEBT_VEBT > nat > $o).
% 84.01/19.79 thf(func_def_883, type, vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 84.01/19.79 thf(func_def_884, type, vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o).
% 84.01/19.79 thf(func_def_885, type, vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 84.01/19.79 thf(func_def_886, type, vEBT_VEBT_valid: vEBT_VEBT > nat > $o).
% 84.01/19.79 thf(func_def_887, type, vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 84.01/19.79 thf(func_def_888, type, vEBT_invar_vebt: vEBT_VEBT > nat > $o).
% 84.01/19.79 thf(func_def_889, type, vEBT_set_vebt: vEBT_VEBT > set_nat).
% 84.01/19.79 thf(func_def_890, type, vEBT_vebt_buildup: nat > vEBT_VEBT).
% 84.01/19.79 thf(func_def_891, type, vEBT_v4011308405150292612up_rel: nat > nat > $o).
% 84.01/19.79 thf(func_def_892, type, vEBT_vebt_delete: vEBT_VEBT > nat > vEBT_VEBT).
% 84.01/19.79 thf(func_def_893, type, vEBT_vebt_delete_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 84.01/19.79 thf(func_def_894, type, vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT).
% 84.01/19.79 thf(func_def_895, type, vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 84.01/19.79 thf(func_def_896, type, vEBT_VEBT_bit_concat: nat > nat > nat > nat).
% 84.01/19.79 thf(func_def_897, type, vEBT_VEBT_minNull: vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_898, type, vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_899, type, vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat).
% 84.01/19.79 thf(func_def_900, type, vEBT_vebt_member: vEBT_VEBT > nat > $o).
% 84.01/19.79 thf(func_def_901, type, vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 84.01/19.79 thf(func_def_902, type, vEBT_VEBT_add: option_nat > option_nat > option_nat).
% 84.01/19.79 thf(func_def_903, type, vEBT_VEBT_greater: option_nat > option_nat > $o).
% 84.01/19.79 thf(func_def_904, type, vEBT_VEBT_less: option_nat > option_nat > $o).
% 84.01/19.79 thf(func_def_905, type, vEBT_VEBT_lesseq: option_nat > option_nat > $o).
% 84.01/19.79 thf(func_def_906, type, vEBT_VEBT_max_in_set: set_nat > nat > $o).
% 84.01/19.79 thf(func_def_907, type, vEBT_VEBT_min_in_set: set_nat > nat > $o).
% 84.01/19.79 thf(func_def_908, type, vEBT_VEBT_mul: option_nat > option_nat > option_nat).
% 84.01/19.79 thf(func_def_909, type, vEBT_V4262088993061758097ft_nat: (nat > nat > nat) > option_nat > option_nat > option_nat).
% 84.01/19.79 thf(func_def_910, type, vEBT_V819420779217536731ft_num: (num > num > num) > option_num > option_num > option_num).
% 84.01/19.79 thf(func_def_911, type, vEBT_V1502963449132264192at_nat: (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > option4927543243414619207at_nat > option4927543243414619207at_nat > option4927543243414619207at_nat).
% 84.01/19.79 thf(func_def_912, type, vEBT_VEBT_power: option_nat > option_nat > option_nat).
% 84.01/19.79 thf(func_def_913, type, vEBT_vebt_maxt: vEBT_VEBT > option_nat).
% 84.01/19.79 thf(func_def_914, type, vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_915, type, vEBT_vebt_mint: vEBT_VEBT > option_nat).
% 84.01/19.79 thf(func_def_916, type, vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_917, type, vEBT_is_pred_in_set: set_nat > nat > nat > $o).
% 84.01/19.79 thf(func_def_918, type, vEBT_vebt_pred: vEBT_VEBT > nat > option_nat).
% 84.01/19.79 thf(func_def_919, type, vEBT_vebt_pred_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 84.01/19.79 thf(func_def_920, type, vEBT_is_succ_in_set: set_nat > nat > nat > $o).
% 84.01/19.79 thf(func_def_921, type, vEBT_vebt_succ: vEBT_VEBT > nat > option_nat).
% 84.01/19.79 thf(func_def_922, type, vEBT_vebt_succ_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 84.01/19.79 thf(func_def_923, type, accp_nat: (nat > nat > $o) > nat > $o).
% 84.01/19.79 thf(func_def_924, type, accp_P1096762738010456898nt_int: (product_prod_int_int > product_prod_int_int > $o) > product_prod_int_int > $o).
% 84.01/19.79 thf(func_def_925, type, accp_P4275260045618599050at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > product_prod_nat_nat > $o).
% 84.01/19.79 thf(func_def_926, type, accp_P2887432264394892906BT_nat: (produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o) > produc9072475918466114483BT_nat > $o).
% 84.01/19.79 thf(func_def_927, type, accp_VEBT_VEBT: (vEBT_VEBT > vEBT_VEBT > $o) > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_928, type, fChoice_real: (real > $o) > real).
% 84.01/19.79 thf(func_def_929, type, member_o: $o > set_o > $o).
% 84.01/19.79 thf(func_def_930, type, member_complex: complex > set_complex > $o).
% 84.01/19.79 thf(func_def_931, type, member_int: int > set_int > $o).
% 84.01/19.79 thf(func_def_932, type, member_list_o: list_o > set_list_o > $o).
% 84.01/19.79 thf(func_def_933, type, member_list_int: list_int > set_list_int > $o).
% 84.01/19.79 thf(func_def_934, type, member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_935, type, member_nat: nat > set_nat > $o).
% 84.01/19.79 thf(func_def_936, type, member_num: num > set_num > $o).
% 84.01/19.79 thf(func_def_937, type, member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o).
% 84.01/19.79 thf(func_def_938, type, member_rat: rat > set_rat > $o).
% 84.01/19.79 thf(func_def_939, type, member_real: real > set_real > $o).
% 84.01/19.79 thf(func_def_940, type, member_set_nat: set_nat > set_set_nat > $o).
% 84.01/19.79 thf(func_def_941, type, member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_942, type, deg: nat).
% 84.01/19.79 thf(func_def_943, type, lx: nat).
% 84.01/19.79 thf(func_def_944, type, m: nat).
% 84.01/19.79 thf(func_def_945, type, ma: nat).
% 84.01/19.79 thf(func_def_946, type, maxi: nat).
% 84.01/19.79 thf(func_def_947, type, maxs: nat).
% 84.01/19.79 thf(func_def_948, type, mi: nat).
% 84.01/19.79 thf(func_def_949, type, na: nat).
% 84.01/19.79 thf(func_def_950, type, summary: vEBT_VEBT).
% 84.01/19.79 thf(func_def_951, type, summin: nat).
% 84.01/19.79 thf(func_def_952, type, treeList: list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_953, type, xa: nat).
% 84.01/19.79 thf(func_def_954, type, ya: nat).
% 84.01/19.79 thf(func_def_977, type, sP0: vEBT_VEBT > vEBT_VEBT > nat > $o).
% 84.01/19.79 thf(func_def_978, type, sP1: vEBT_VEBT > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_979, type, sP2: vEBT_VEBT > vEBT_VEBT > nat > $o).
% 84.01/19.79 thf(func_def_980, type, sP3: nat > vEBT_VEBT > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_981, type, sP4: vEBT_VEBT > nat > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_982, type, sP5: nat > vEBT_VEBT > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_983, type, sP6: vEBT_VEBT > option_nat > $o).
% 84.01/19.79 thf(func_def_984, type, sP7: option_nat > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_985, type, sP8: option_nat > vEBT_VEBT > nat > $o).
% 84.01/19.79 thf(func_def_986, type, sP9: vEBT_VEBT > option_nat > nat > $o).
% 84.01/19.79 thf(func_def_987, type, sP10: vEBT_VEBT > option_nat > nat > $o).
% 84.01/19.79 thf(func_def_988, type, sP11: nat > vEBT_VEBT > option_nat > $o).
% 84.01/19.79 thf(func_def_989, type, sP12: option_nat > nat > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_990, type, sP13: vEBT_VEBT > nat > option_nat > $o).
% 84.01/19.79 thf(func_def_991, type, sP14: nat > option_nat > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_992, type, sP15: nat > vEBT_VEBT > option_nat > $o).
% 84.01/19.79 thf(func_def_993, type, sP16: vEBT_VEBT > vEBT_VEBT > nat > $o).
% 84.01/19.79 thf(func_def_994, type, sP17: nat > vEBT_VEBT > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_995, type, sP18: nat > vEBT_VEBT > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_996, type, sP19: vEBT_VEBT > nat > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_997, type, sP20: nat > vEBT_VEBT > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_998, type, sP21: vEBT_VEBT > nat > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_999, type, sP22: nat > option_nat > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_1000, type, sP23: nat > vEBT_VEBT > option_nat > $o).
% 84.01/19.79 thf(func_def_1001, type, sP24: vEBT_VEBT > option_nat > nat > $o).
% 84.01/19.79 thf(func_def_1002, type, sP25: vEBT_VEBT > nat > option_nat > $o).
% 84.01/19.79 thf(func_def_1003, type, sP26: vEBT_VEBT > nat > option_nat > $o).
% 84.01/19.79 thf(func_def_1004, type, sP27: vEBT_VEBT > option_nat > nat > $o).
% 84.01/19.79 thf(func_def_1005, type, sP28: option_nat > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_1006, type, sP29: option_nat > vEBT_VEBT > nat > $o).
% 84.01/19.79 thf(func_def_1007, type, sP30: nat > vEBT_VEBT > option_nat > $o).
% 84.01/19.79 thf(func_def_1008, type, sP31: vEBT_VEBT > option_nat > nat > $o).
% 84.01/19.79 thf(func_def_1009, type, sP32: option_nat > nat > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_1010, type, sP33: vEBT_VEBT > nat > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_1011, type, sP34: nat > vEBT_VEBT > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_1012, type, sP35: vEBT_VEBT > nat > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_1013, type, sP36: vEBT_VEBT > vEBT_VEBT > nat > $o).
% 84.01/19.79 thf(func_def_1014, type, sK37: vEBT_VEBT > vEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1015, type, sK38: vEBT_VEBT > vEBT_VEBT > option4927543243414619207at_nat).
% 84.01/19.79 thf(func_def_1016, type, sK39: vEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1017, type, sK40: vEBT_VEBT > vEBT_VEBT > nat > $o).
% 84.01/19.79 thf(func_def_1018, type, sK41: vEBT_VEBT > vEBT_VEBT > nat > $o).
% 84.01/19.79 thf(func_def_1019, type, sK42: vEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1020, type, sK43: vEBT_VEBT > vEBT_VEBT > option4927543243414619207at_nat).
% 84.01/19.79 thf(func_def_1021, type, sK44: vEBT_VEBT > vEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1022, type, sK45: vEBT_VEBT > nat > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1023, type, sK46: vEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1024, type, sK47: vEBT_VEBT > nat > vEBT_VEBT > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1025, type, sK48: vEBT_VEBT > nat > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1026, type, sK49: vEBT_VEBT > nat > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1027, type, sK50: vEBT_VEBT > nat > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1028, type, sK51: vEBT_VEBT > nat > vEBT_VEBT > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1029, type, sK52: vEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1030, type, sK53: nat).
% 84.01/19.79 thf(func_def_1031, type, sK54: nat > nat > nat).
% 84.01/19.79 thf(func_def_1032, type, sK55: nat).
% 84.01/19.79 thf(func_def_1033, type, sK56: nat > vEBT_VEBT > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_1034, type, sK57: nat > vEBT_VEBT > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_1035, type, sK58: nat > nat).
% 84.01/19.79 thf(func_def_1036, type, sK59: vEBT_VEBT > nat > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_1037, type, sK60: vEBT_VEBT > nat > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_1038, type, sK61: nat > vEBT_VEBT > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_1039, type, sK62: nat > vEBT_VEBT > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_1040, type, sK63: vEBT_VEBT > vEBT_VEBT > nat > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1041, type, sK64: vEBT_VEBT > vEBT_VEBT > nat > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1042, type, sK65: vEBT_VEBT > vEBT_VEBT > nat > nat).
% 84.01/19.79 thf(func_def_1043, type, sK66: vEBT_VEBT > vEBT_VEBT > nat > nat).
% 84.01/19.79 thf(func_def_1044, type, sK67: vEBT_VEBT > vEBT_VEBT > nat > nat).
% 84.01/19.79 thf(func_def_1045, type, sK68: vEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1046, type, sK69: vEBT_VEBT > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1047, type, sK70: vEBT_VEBT > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1048, type, sK71: vEBT_VEBT > vEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1049, type, sK72: vEBT_VEBT > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1050, type, sK73: vEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1051, type, sK74: vEBT_VEBT > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1052, type, sK75: vEBT_VEBT > vEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1053, type, sK76: vEBT_VEBT > vEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1054, type, sK77: vEBT_VEBT > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1055, type, sK78: vEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1056, type, sK79: option_nat > vEBT_VEBT > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1057, type, sK80: option_nat > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1058, type, sK81: option_nat > vEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1059, type, sK82: option_nat > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1060, type, sK83: option_nat > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1061, type, sK84: vEBT_VEBT > option_nat > $o).
% 84.01/19.79 thf(func_def_1062, type, sK85: vEBT_VEBT > option_nat > $o).
% 84.01/19.79 thf(func_def_1063, type, sK86: vEBT_VEBT > option_nat > nat).
% 84.01/19.79 thf(func_def_1064, type, sK87: vEBT_VEBT > option_nat > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1065, type, sK88: vEBT_VEBT > option_nat > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1066, type, sK89: nat).
% 84.01/19.79 thf(func_def_1067, type, sK90: nat > nat > nat).
% 84.01/19.79 thf(func_def_1068, type, sK91: nat).
% 84.01/19.79 thf(func_def_1069, type, sK92: (nat > nat) > nat).
% 84.01/19.79 thf(func_def_1070, type, sK93: (nat > nat) > nat).
% 84.01/19.79 thf(func_def_1071, type, sK94: (nat > $o) > nat).
% 84.01/19.79 thf(func_def_1072, type, sK95: (nat > $o) > nat).
% 84.01/19.79 thf(func_def_1073, type, sK96: nat > nat).
% 84.01/19.79 thf(func_def_1074, type, sK97: vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_1075, type, sK98: vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_1076, type, sK99: vEBT_VEBT > option_nat > nat > nat).
% 84.01/19.79 thf(func_def_1077, type, sK100: vEBT_VEBT > option_nat > nat > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1078, type, sK101: vEBT_VEBT > option_nat > nat > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1079, type, sK102: vEBT_VEBT > option_nat > nat > nat).
% 84.01/19.79 thf(func_def_1080, type, sK103: vEBT_VEBT > option_nat > nat > nat).
% 84.01/19.79 thf(func_def_1081, type, sK104: option_nat > vEBT_VEBT > nat > $o).
% 84.01/19.79 thf(func_def_1082, type, sK105: option_nat > vEBT_VEBT > nat > $o).
% 84.01/19.79 thf(func_def_1083, type, sK106: vEBT_VEBT > product_prod_nat_nat).
% 84.01/19.79 thf(func_def_1084, type, sK107: vEBT_VEBT > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1085, type, sK108: vEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1086, type, sK109: vEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1087, type, sK110: vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1088, type, sK111: vEBT_VEBT > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1089, type, sK112: vEBT_VEBT > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1090, type, sK113: vEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1091, type, sK114: vEBT_VEBT > product_prod_nat_nat).
% 84.01/19.79 thf(func_def_1092, type, sK115: nat > vEBT_VEBT > option_nat > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1093, type, sK116: nat > vEBT_VEBT > option_nat > product_prod_nat_nat).
% 84.01/19.79 thf(func_def_1094, type, sK117: nat > vEBT_VEBT > option_nat > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1095, type, sK118: nat > option_nat > vEBT_VEBT > product_prod_nat_nat).
% 84.01/19.79 thf(func_def_1096, type, sK119: nat > option_nat > vEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1097, type, sK120: nat > option_nat > vEBT_VEBT > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1098, type, sK121: vEBT_VEBT > nat > option_nat > $o).
% 84.01/19.79 thf(func_def_1099, type, sK122: vEBT_VEBT > nat > option_nat > $o).
% 84.01/19.79 thf(func_def_1100, type, sK123: vEBT_VEBT > nat > option_nat > nat).
% 84.01/19.79 thf(func_def_1101, type, sK124: option_nat > nat > vEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1102, type, sK125: option_nat > nat > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1103, type, sK126: option_nat > nat > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1104, type, sK127: option_nat > nat > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1105, type, sK128: option_nat > nat > vEBT_VEBT > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1106, type, sK129: nat > vEBT_VEBT > option_nat > $o).
% 84.01/19.79 thf(func_def_1107, type, sK130: nat > vEBT_VEBT > option_nat > $o).
% 84.01/19.79 thf(func_def_1108, type, sK131: nat > option_nat > vEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1109, type, sK132: nat > option_nat > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1110, type, sK133: nat > option_nat > vEBT_VEBT > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1111, type, sK134: nat > nat).
% 84.01/19.79 thf(func_def_1112, type, sK135: nat > nat).
% 84.01/19.79 thf(func_def_1113, type, sK136: nat > nat > nat).
% 84.01/19.79 thf(func_def_1114, type, sK137: nat).
% 84.01/19.79 thf(func_def_1115, type, sK138: nat > nat > nat).
% 84.01/19.79 thf(func_def_1116, type, sK139: vEBT_VEBT > nat > vEBT_VEBT > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1117, type, sK140: vEBT_VEBT > nat > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1118, type, sK141: vEBT_VEBT > nat > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1119, type, sK142: vEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1120, type, sK143: nat > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1121, type, sK144: nat > vEBT_VEBT > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1122, type, sK145: nat > vEBT_VEBT > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1123, type, sK146: nat > vEBT_VEBT > vEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1124, type, sK147: vEBT_VEBT > nat > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_1125, type, sK148: vEBT_VEBT > nat > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_1126, type, sK149: nat > vEBT_VEBT > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_1127, type, sK150: nat > vEBT_VEBT > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_1128, type, sK151: nat > vEBT_VEBT > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_1129, type, sK152: nat > vEBT_VEBT > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_1130, type, sK153: nat > vEBT_VEBT > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1131, type, sK154: vEBT_VEBT > vEBT_VEBT > nat > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1132, type, sK155: vEBT_VEBT > vEBT_VEBT > nat > nat).
% 84.01/19.79 thf(func_def_1133, type, sK156: vEBT_VEBT > vEBT_VEBT > nat > nat).
% 84.01/19.79 thf(func_def_1134, type, sK157: vEBT_VEBT > vEBT_VEBT > nat > nat).
% 84.01/19.79 thf(func_def_1135, type, sK158: vEBT_VEBT > vEBT_VEBT > nat > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1136, type, sK159: nat > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1137, type, sK160: nat > vEBT_VEBT > vEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1138, type, sK161: nat > vEBT_VEBT > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1139, type, sK162: nat > nat > nat).
% 84.01/19.79 thf(func_def_1140, type, sK163: (nat > nat > $o) > nat).
% 84.01/19.79 thf(func_def_1141, type, sK164: (nat > nat > $o) > nat).
% 84.01/19.79 thf(func_def_1142, type, sK165: (nat > nat > $o) > nat).
% 84.01/19.79 thf(func_def_1143, type, sK166: (nat > nat > $o) > nat).
% 84.01/19.79 thf(func_def_1144, type, sK167: (nat > nat > $o) > nat).
% 84.01/19.79 thf(func_def_1145, type, sK168: nat > nat > nat).
% 84.01/19.79 thf(func_def_1146, type, sK169: vEBT_VEBT > option_nat > nat > product_prod_nat_nat).
% 84.01/19.79 thf(func_def_1147, type, sK170: vEBT_VEBT > option_nat > nat > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1148, type, sK171: vEBT_VEBT > option_nat > nat > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1149, type, sK172: vEBT_VEBT > nat > option_nat > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1150, type, sK173: vEBT_VEBT > nat > option_nat > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1151, type, sK174: vEBT_VEBT > nat > option_nat > nat).
% 84.01/19.79 thf(func_def_1152, type, sK175: vEBT_VEBT > nat > option_nat > $o).
% 84.01/19.79 thf(func_def_1153, type, sK176: vEBT_VEBT > nat > option_nat > $o).
% 84.01/19.79 thf(func_def_1154, type, sK177: vEBT_VEBT > option_nat > nat > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1155, type, sK178: vEBT_VEBT > option_nat > nat > nat).
% 84.01/19.79 thf(func_def_1156, type, sK179: vEBT_VEBT > option_nat > nat > nat).
% 84.01/19.79 thf(func_def_1157, type, sK180: vEBT_VEBT > option_nat > nat > nat).
% 84.01/19.79 thf(func_def_1158, type, sK181: vEBT_VEBT > option_nat > nat > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1159, type, sK182: nat > vEBT_VEBT > option_nat > $o).
% 84.01/19.79 thf(func_def_1160, type, sK183: nat > vEBT_VEBT > option_nat > $o).
% 84.01/19.79 thf(func_def_1161, type, sK184: nat > option_nat > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_1162, type, sK185: nat > option_nat > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_1163, type, sK186: nat > option_nat > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1164, type, sK187: option_nat > vEBT_VEBT > nat > product_prod_nat_nat).
% 84.01/19.79 thf(func_def_1165, type, sK188: option_nat > vEBT_VEBT > nat > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1166, type, sK189: option_nat > vEBT_VEBT > nat > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1167, type, sK190: nat).
% 84.01/19.79 thf(func_def_1168, type, sK191: option_nat > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_1169, type, sK192: option_nat > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_1170, type, sK193: option_nat > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1171, type, sK194: option_nat > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1172, type, sK195: option_nat > vEBT_VEBT > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1173, type, sK196: option_nat > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1174, type, sK197: option_nat > vEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1175, type, sK198: vEBT_VEBT > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1176, type, sK199: vEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1177, type, sK200: vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1178, type, sK201: option_nat > nat > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1179, type, sK202: option_nat > nat > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1180, type, sK203: option_nat > nat > vEBT_VEBT > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1181, type, sK204: option_nat > nat > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1182, type, sK205: option_nat > nat > vEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1183, type, sK206: vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_1184, type, sK207: vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_1185, type, sK208: nat > vEBT_VEBT > option_nat > $o).
% 84.01/19.79 thf(func_def_1186, type, sK209: nat > vEBT_VEBT > option_nat > $o).
% 84.01/19.79 thf(func_def_1187, type, sK210: option_nat > vEBT_VEBT > nat > $o).
% 84.01/19.79 thf(func_def_1188, type, sK211: option_nat > vEBT_VEBT > nat > $o).
% 84.01/19.79 thf(func_def_1189, type, sK212: nat > nat).
% 84.01/19.79 thf(func_def_1190, type, sK213: vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1191, type, sK214: vEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1192, type, sK215: vEBT_VEBT > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1193, type, sK216: vEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1194, type, sK217: vEBT_VEBT > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1195, type, sK218: vEBT_VEBT > product_prod_nat_nat).
% 84.01/19.79 thf(func_def_1196, type, sK219: vEBT_VEBT > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1197, type, sK220: vEBT_VEBT > product_prod_nat_nat).
% 84.01/19.79 thf(func_def_1198, type, sK221: vEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1199, type, sK222: vEBT_VEBT > vEBT_VEBT > nat > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1200, type, sK223: vEBT_VEBT > vEBT_VEBT > nat > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1201, type, sK224: vEBT_VEBT > vEBT_VEBT > nat > option4927543243414619207at_nat).
% 84.01/19.79 thf(func_def_1202, type, sK225: vEBT_VEBT > nat > vEBT_VEBT > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1203, type, sK226: vEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1204, type, sK227: vEBT_VEBT > nat > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1205, type, sK228: nat > vEBT_VEBT > vEBT_VEBT > option4927543243414619207at_nat).
% 84.01/19.79 thf(func_def_1206, type, sK229: nat > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1207, type, sK230: nat > vEBT_VEBT > vEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1208, type, sK231: vEBT_VEBT > nat > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_1209, type, sK232: vEBT_VEBT > nat > vEBT_VEBT > $o).
% 84.01/19.79 thf(func_def_1210, type, sK233: vEBT_VEBT > nat > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1211, type, sK234: vEBT_VEBT > nat > vEBT_VEBT > vEBT_VEBT).
% 84.01/19.79 thf(func_def_1212, type, sK235: vEBT_VEBT > nat > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1213, type, sK236: vEBT_VEBT > nat > vEBT_VEBT > nat).
% 84.01/19.79 thf(func_def_1214, type, sK237: vEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT).
% 84.01/19.79 thf(func_def_1215, type, sK238: nat > nat > nat).
% 84.01/19.79 thf(func_def_1216, type, sK239: nat > nat > nat).
% 84.01/19.79 thf(func_def_1218, type, sK241: nat > nat).
% 84.01/19.79 thf(f26869,plain,(
% 84.01/19.79 $false),
% 84.01/19.79 inference(trivial_inequality_removal,[status(thm)],[f26868])).
% 84.01/19.79 thf(f26868,plain,(
% 84.01/19.79 ((plus_plus_nat @ (times_times_nat @ (size_s6755466524823107622T_VEBT @ treeList) @ maxs) @ maxi) != (plus_plus_nat @ (times_times_nat @ (size_s6755466524823107622T_VEBT @ treeList) @ maxs) @ maxi))),
% 84.01/19.79 inference(forward_demodulation,[status(thm)],[f26867,f25368])).
% 84.01/19.79 thf(f25368,plain,(
% 84.01/19.79 ( ! [X0 : nat] : (((the_nat @ (some_nat @ X0)) = X0)) )),
% 84.01/19.79 inference(cnf_transformation,[status(thm)],[f24308])).
% 84.01/19.79 thf(f24308,plain,(
% 84.01/19.79 ! [X0 : nat] : ((the_nat @ (some_nat @ X0)) = X0)),
% 84.01/19.79 inference(rectify,[status(thm)],[f365])).
% 84.01/19.79 thf(f365,axiom,(
% 84.01/19.79 ! [X43 : nat] : ((the_nat @ (some_nat @ X43)) = X43)),
% 84.01/19.79 file('/export/starexec/sandbox/tmp/tmp.YjPWRkCLRl/DTF2THF_23017.p',fact_364_option_Osel)).
% 84.01/19.79 thf(f26867,plain,(
% 84.01/19.79 ((plus_plus_nat @ (times_times_nat @ (size_s6755466524823107622T_VEBT @ treeList) @ maxs) @ maxi) != (plus_plus_nat @ (times_times_nat @ (size_s6755466524823107622T_VEBT @ treeList) @ maxs) @ (the_nat @ (some_nat @ maxi))))),
% 84.01/19.79 inference(forward_demodulation,[status(thm)],[f26865,f25695])).
% 84.01/19.79 thf(f25695,plain,(
% 84.01/19.79 ((vEBT_vebt_maxt @ (nth_VEBT_VEBT @ treeList @ maxs)) = (some_nat @ maxi))),
% 84.01/19.79 inference(cnf_transformation,[status(thm)],[f17])).
% 84.01/19.79 thf(f17,axiom,(
% 84.01/19.79 ((vEBT_vebt_maxt @ (nth_VEBT_VEBT @ treeList @ maxs)) = (some_nat @ maxi))),
% 84.01/19.79 file('/export/starexec/sandbox/tmp/tmp.YjPWRkCLRl/DTF2THF_23017.p',fact_16__092_060open_062Some_Amaxi_A_061_Avebt__maxt_A_ItreeList_A_B_Amaxs_J_092_060close_062)).
% 84.01/19.79 thf(f26865,plain,(
% 84.01/19.79 ((plus_plus_nat @ (times_times_nat @ (size_s6755466524823107622T_VEBT @ treeList) @ maxs) @ maxi) != (plus_plus_nat @ (times_times_nat @ (size_s6755466524823107622T_VEBT @ treeList) @ maxs) @ (the_nat @ (vEBT_vebt_maxt @ (nth_VEBT_VEBT @ treeList @ maxs)))))),
% 84.01/19.79 inference(backward_demodulation,[status(thm)],[f26861,f26864])).
% 84.01/19.79 thf(f26864,plain,(
% 84.01/19.79 ((nth_VEBT_VEBT @ treeList @ maxs) = (nth_VEBT_VEBT @ (list_u1324408373059187874T_VEBT @ treeList @ (divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (size_s6755466524823107622T_VEBT @ treeList)) @ lx) @ (size_s6755466524823107622T_VEBT @ treeList)) @ (vEBT_vebt_delete @ (nth_VEBT_VEBT @ treeList @ (divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (size_s6755466524823107622T_VEBT @ treeList)) @ lx) @ (size_s6755466524823107622T_VEBT @ treeList))) @ (modulo_modulo_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (size_s6755466524823107622T_VEBT @ treeList)) @ lx) @ (size_s6755466524823107622T_VEBT @ treeList)))) @ maxs))),
% 84.01/19.79 inference(forward_demodulation,[status(thm)],[f26863,f25368])).
% 84.01/19.79 thf(f26863,plain,(
% 84.01/19.79 ((nth_VEBT_VEBT @ treeList @ maxs) = (nth_VEBT_VEBT @ (list_u1324408373059187874T_VEBT @ treeList @ (divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (size_s6755466524823107622T_VEBT @ treeList)) @ lx) @ (size_s6755466524823107622T_VEBT @ treeList)) @ (vEBT_vebt_delete @ (nth_VEBT_VEBT @ treeList @ (divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (size_s6755466524823107622T_VEBT @ treeList)) @ lx) @ (size_s6755466524823107622T_VEBT @ treeList))) @ (modulo_modulo_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (size_s6755466524823107622T_VEBT @ treeList)) @ lx) @ (size_s6755466524823107622T_VEBT @ treeList)))) @ (the_nat @ (some_nat @ maxs))))),
% 84.01/19.79 inference(forward_demodulation,[status(thm)],[f26862,f26782])).
% 84.01/19.79 thf(f26782,plain,(
% 84.01/19.79 ((some_nat @ maxs) = (vEBT_vebt_maxt @ (vEBT_vebt_delete @ summary @ (divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (size_s6755466524823107622T_VEBT @ treeList)) @ lx) @ (size_s6755466524823107622T_VEBT @ treeList)))))),
% 84.01/19.79 inference(forward_demodulation,[status(thm)],[f26558,f25940])).
% 84.01/19.79 thf(f25940,plain,(
% 84.01/19.79 ((size_s6755466524823107622T_VEBT @ treeList) = (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))),
% 84.01/19.79 inference(definition_unfolding,[status(thm)],[f25363,f25775])).
% 84.01/19.79 thf(f25775,plain,(
% 84.01/19.79 (m = (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))),
% 84.01/19.79 inference(definition_unfolding,[status(thm)],[f25198,f25523])).
% 84.01/19.79 thf(f25523,plain,(
% 84.01/19.79 (na = (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))),
% 84.01/19.79 inference(cnf_transformation,[status(thm)],[f7])).
% 84.01/19.79 thf(f7,axiom,(
% 84.01/19.79 (na = (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))),
% 84.01/19.79 file('/export/starexec/sandbox/tmp/tmp.YjPWRkCLRl/DTF2THF_23017.p',fact_6__092_060open_062deg_Adiv_A2_A_061_An_092_060close_062)).
% 84.01/19.79 thf(f25198,plain,(
% 84.01/19.79 (m = na)),
% 84.01/19.79 inference(cnf_transformation,[status(thm)],[f1])).
% 84.01/19.79 thf(f1,axiom,(
% 84.01/19.79 (m = na)),
% 84.01/19.79 file('/export/starexec/sandbox/tmp/tmp.YjPWRkCLRl/DTF2THF_23017.p',fact_0__C4_Ohyps_C_I3_J)).
% 84.01/19.79 thf(f25363,plain,(
% 84.01/19.79 ((size_s6755466524823107622T_VEBT @ treeList) = (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m))),
% 84.01/19.79 inference(cnf_transformation,[status(thm)],[f48])).
% 84.01/19.79 thf(f48,axiom,(
% 84.01/19.79 ((size_s6755466524823107622T_VEBT @ treeList) = (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m))),
% 84.01/19.79 file('/export/starexec/sandbox/tmp/tmp.YjPWRkCLRl/DTF2THF_23017.p',fact_47__C4_Ohyps_C_I2_J)).
% 84.01/19.79 thf(f26558,plain,(
% 84.01/19.79 ((some_nat @ maxs) = (vEBT_vebt_maxt @ (vEBT_vebt_delete @ summary @ (divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ lx) @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))))))),
% 84.01/19.79 inference(beta_eta_normalization,[status(thm)],[f26139])).
% 84.01/19.79 thf(f26139,plain,(
% 84.01/19.79 ((some_nat @ maxs) = (vEBT_vebt_maxt @ (vEBT_vebt_delete @ summary @ ((^[Y0 : nat]: ((^[Y1 : nat]: (divide_divide_nat @ Y0 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Y1))))) @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ lx) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))))),
% 84.01/19.79 inference(definition_unfolding,[status(thm)],[f25767,f25607,f25523,f25523])).
% 84.01/19.79 thf(f25607,plain,(
% 84.01/19.79 (vEBT_VEBT_high = (^[Y0 : nat]: ((^[Y1 : nat]: (divide_divide_nat @ Y0 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Y1))))))),
% 84.01/19.79 inference(cnf_transformation,[status(thm)],[f11172])).
% 84.01/19.79 thf(f11172,plain,(
% 84.01/19.79 (vEBT_VEBT_high = (^[Y0 : nat]: ((^[Y1 : nat]: (divide_divide_nat @ Y0 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Y1))))))),
% 84.01/19.79 inference(fool_elimination,[status(thm)],[f11171])).
% 84.01/19.79 thf(f11171,plain,(
% 84.01/19.79 (vEBT_VEBT_high = (^[X0 : nat, X1 : nat] : (divide_divide_nat @ X0 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ X1))))),
% 84.01/19.79 inference(rectify,[status(thm)],[f6])).
% 84.01/19.79 thf(f6,axiom,(
% 84.01/19.79 (vEBT_VEBT_high = (^[X5 : nat, X6 : nat] : (divide_divide_nat @ X5 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ X6))))),
% 84.01/19.79 file('/export/starexec/sandbox/tmp/tmp.YjPWRkCLRl/DTF2THF_23017.p',fact_5_high__def)).
% 84.01/19.79 thf(f25767,plain,(
% 84.01/19.79 ((vEBT_vebt_maxt @ (vEBT_vebt_delete @ summary @ (vEBT_VEBT_high @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na))) = (some_nat @ maxs))),
% 84.01/19.79 inference(cnf_transformation,[status(thm)],[f19])).
% 84.01/19.79 thf(f19,axiom,(
% 84.01/19.79 ((vEBT_vebt_maxt @ (vEBT_vebt_delete @ summary @ (vEBT_VEBT_high @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na))) = (some_nat @ maxs))),
% 84.01/19.79 file('/export/starexec/sandbox/tmp/tmp.YjPWRkCLRl/DTF2THF_23017.p',fact_18__092_060open_062Some_Amaxs_A_061_Avebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_092_060close_062)).
% 84.01/19.79 thf(f26862,plain,(
% 84.01/19.79 ((nth_VEBT_VEBT @ treeList @ maxs) = (nth_VEBT_VEBT @ (list_u1324408373059187874T_VEBT @ treeList @ (divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (size_s6755466524823107622T_VEBT @ treeList)) @ lx) @ (size_s6755466524823107622T_VEBT @ treeList)) @ (vEBT_vebt_delete @ (nth_VEBT_VEBT @ treeList @ (divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (size_s6755466524823107622T_VEBT @ treeList)) @ lx) @ (size_s6755466524823107622T_VEBT @ treeList))) @ (modulo_modulo_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (size_s6755466524823107622T_VEBT @ treeList)) @ lx) @ (size_s6755466524823107622T_VEBT @ treeList)))) @ (the_nat @ (vEBT_vebt_maxt @ (vEBT_vebt_delete @ summary @ (divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (size_s6755466524823107622T_VEBT @ treeList)) @ lx) @ (size_s6755466524823107622T_VEBT @ treeList)))))))),
% 84.01/19.79 inference(forward_demodulation,[status(thm)],[f26448,f25940])).
% 84.01/19.79 thf(f26448,plain,(
% 84.01/19.79 ((nth_VEBT_VEBT @ treeList @ maxs) = (nth_VEBT_VEBT @ (list_u1324408373059187874T_VEBT @ treeList @ (divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ lx) @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ (vEBT_vebt_delete @ (nth_VEBT_VEBT @ treeList @ (divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ lx) @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))) @ (modulo_modulo_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ lx) @ (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 @ (divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ lx) @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))))))))),
% 84.01/19.79 inference(beta_eta_normalization,[status(thm)],[f26017])).
% 84.01/19.79 thf(f26017,plain,(
% 84.01/19.79 ((nth_VEBT_VEBT @ treeList @ maxs) = (nth_VEBT_VEBT @ (list_u1324408373059187874T_VEBT @ treeList @ ((^[Y0 : nat]: ((^[Y1 : nat]: (divide_divide_nat @ Y0 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Y1))))) @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ lx) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))) @ (vEBT_vebt_delete @ (nth_VEBT_VEBT @ treeList @ ((^[Y0 : nat]: ((^[Y1 : nat]: (divide_divide_nat @ Y0 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Y1))))) @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ lx) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ ((^[Y0 : nat]: ((^[Y1 : nat]: (modulo_modulo_nat @ Y0 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Y1))))) @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ lx) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))) @ (the_nat @ (vEBT_vebt_maxt @ (vEBT_vebt_delete @ summary @ ((^[Y0 : nat]: ((^[Y1 : nat]: (divide_divide_nat @ Y0 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Y1))))) @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ lx) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))))))),
% 84.01/19.79 inference(definition_unfolding,[status(thm)],[f25546,f25607,f25523,f25523,f25607,f25523,f25523,f25153,f25523,f25523,f25607,f25523,f25523])).
% 84.01/19.79 thf(f25153,plain,(
% 84.01/19.79 (vEBT_VEBT_low = (^[Y0 : nat]: ((^[Y1 : nat]: (modulo_modulo_nat @ Y0 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Y1))))))),
% 84.01/19.79 inference(cnf_transformation,[status(thm)],[f17451])).
% 84.01/19.79 thf(f17451,plain,(
% 84.01/19.79 (vEBT_VEBT_low = (^[Y0 : nat]: ((^[Y1 : nat]: (modulo_modulo_nat @ Y0 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Y1))))))),
% 84.01/19.79 inference(fool_elimination,[status(thm)],[f17450])).
% 84.01/19.79 thf(f17450,plain,(
% 84.01/19.79 (vEBT_VEBT_low = (^[X0 : nat, X1 : nat] : (modulo_modulo_nat @ X0 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ X1))))),
% 84.01/19.79 inference(rectify,[status(thm)],[f1650])).
% 84.01/19.79 thf(f1650,axiom,(
% 84.01/19.79 (vEBT_VEBT_low = (^[X5 : nat, X6 : nat] : (modulo_modulo_nat @ X5 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ X6))))),
% 84.01/19.79 file('/export/starexec/sandbox/tmp/tmp.YjPWRkCLRl/DTF2THF_23017.p',fact_1649_low__def)).
% 84.01/19.79 thf(f25546,plain,(
% 84.01/19.79 ((nth_VEBT_VEBT @ treeList @ maxs) = (nth_VEBT_VEBT @ (list_u1324408373059187874T_VEBT @ treeList @ (vEBT_VEBT_high @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na) @ (vEBT_vebt_delete @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na)) @ (vEBT_VEBT_low @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na))) @ (the_nat @ (vEBT_vebt_maxt @ (vEBT_vebt_delete @ summary @ (vEBT_VEBT_high @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na))))))),
% 84.01/19.79 inference(cnf_transformation,[status(thm)],[f13])).
% 84.01/19.79 thf(f13,axiom,(
% 84.01/19.79 ((nth_VEBT_VEBT @ treeList @ maxs) = (nth_VEBT_VEBT @ (list_u1324408373059187874T_VEBT @ treeList @ (vEBT_VEBT_high @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na) @ (vEBT_vebt_delete @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na)) @ (vEBT_VEBT_low @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na))) @ (the_nat @ (vEBT_vebt_maxt @ (vEBT_vebt_delete @ summary @ (vEBT_VEBT_high @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na))))))),
% 84.01/19.79 file('/export/starexec/sandbox/tmp/tmp.YjPWRkCLRl/DTF2THF_23017.p',fact_12__092_060open_062treeList_A_091high_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_A_058_061_Avebt__delete_A_ItreeList_A_B_Ahigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_A_Ilow_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_093_A_B_Athe_A_Ivebt__maxt_A_Ivebt__delete_Asummary_A_Ihigh_A_Isummin_A_K_A2_A_094_An_A_L_Alx_J_An_J_J_J_A_061_AtreeList_A_B_Amaxs_092_060close_062)).
% 84.01/19.79 thf(f26861,plain,(
% 84.01/19.79 ((plus_plus_nat @ (times_times_nat @ (size_s6755466524823107622T_VEBT @ treeList) @ maxs) @ maxi) != (plus_plus_nat @ (times_times_nat @ (size_s6755466524823107622T_VEBT @ treeList) @ maxs) @ (the_nat @ (vEBT_vebt_maxt @ (nth_VEBT_VEBT @ (list_u1324408373059187874T_VEBT @ treeList @ (divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (size_s6755466524823107622T_VEBT @ treeList)) @ lx) @ (size_s6755466524823107622T_VEBT @ treeList)) @ (vEBT_vebt_delete @ (nth_VEBT_VEBT @ treeList @ (divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (size_s6755466524823107622T_VEBT @ treeList)) @ lx) @ (size_s6755466524823107622T_VEBT @ treeList))) @ (modulo_modulo_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (size_s6755466524823107622T_VEBT @ treeList)) @ lx) @ (size_s6755466524823107622T_VEBT @ treeList)))) @ maxs)))))),
% 84.01/19.79 inference(forward_demodulation,[status(thm)],[f26860,f25368])).
% 84.01/19.79 thf(f26860,plain,(
% 84.01/19.79 ((plus_plus_nat @ (times_times_nat @ (size_s6755466524823107622T_VEBT @ treeList) @ (the_nat @ (some_nat @ maxs))) @ (the_nat @ (vEBT_vebt_maxt @ (nth_VEBT_VEBT @ (list_u1324408373059187874T_VEBT @ treeList @ (divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (size_s6755466524823107622T_VEBT @ treeList)) @ lx) @ (size_s6755466524823107622T_VEBT @ treeList)) @ (vEBT_vebt_delete @ (nth_VEBT_VEBT @ treeList @ (divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (size_s6755466524823107622T_VEBT @ treeList)) @ lx) @ (size_s6755466524823107622T_VEBT @ treeList))) @ (modulo_modulo_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (size_s6755466524823107622T_VEBT @ treeList)) @ lx) @ (size_s6755466524823107622T_VEBT @ treeList)))) @ (the_nat @ (some_nat @ maxs)))))) != (plus_plus_nat @ (times_times_nat @ (size_s6755466524823107622T_VEBT @ treeList) @ maxs) @ maxi))),
% 84.01/19.79 inference(forward_demodulation,[status(thm)],[f26859,f26782])).
% 84.01/19.79 thf(f26859,plain,(
% 84.01/19.79 ((plus_plus_nat @ (times_times_nat @ (size_s6755466524823107622T_VEBT @ treeList) @ maxs) @ maxi) != (plus_plus_nat @ (times_times_nat @ (size_s6755466524823107622T_VEBT @ treeList) @ (the_nat @ (vEBT_vebt_maxt @ (vEBT_vebt_delete @ summary @ (divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (size_s6755466524823107622T_VEBT @ treeList)) @ lx) @ (size_s6755466524823107622T_VEBT @ treeList)))))) @ (the_nat @ (vEBT_vebt_maxt @ (nth_VEBT_VEBT @ (list_u1324408373059187874T_VEBT @ treeList @ (divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (size_s6755466524823107622T_VEBT @ treeList)) @ lx) @ (size_s6755466524823107622T_VEBT @ treeList)) @ (vEBT_vebt_delete @ (nth_VEBT_VEBT @ treeList @ (divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (size_s6755466524823107622T_VEBT @ treeList)) @ lx) @ (size_s6755466524823107622T_VEBT @ treeList))) @ (modulo_modulo_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (size_s6755466524823107622T_VEBT @ treeList)) @ lx) @ (size_s6755466524823107622T_VEBT @ treeList)))) @ (the_nat @ (vEBT_vebt_maxt @ (vEBT_vebt_delete @ summary @ (divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (size_s6755466524823107622T_VEBT @ treeList)) @ lx) @ (size_s6755466524823107622T_VEBT @ treeList))))))))))),
% 84.01/19.79 inference(forward_demodulation,[status(thm)],[f26449,f25940])).
% 84.01/19.79 thf(f26449,plain,(
% 84.01/19.79 ((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 @ (divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ lx) @ (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 @ treeList @ (divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ lx) @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ (vEBT_vebt_delete @ (nth_VEBT_VEBT @ treeList @ (divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ lx) @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))) @ (modulo_modulo_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ lx) @ (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 @ (divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ lx) @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))))))))) != (plus_plus_nat @ (times_times_nat @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))) @ maxs) @ maxi))),
% 84.01/19.79 inference(beta_eta_normalization,[status(thm)],[f25961])).
% 84.01/19.79 thf(f25961,plain,(
% 84.01/19.79 ((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 @ ((^[Y0 : nat]: ((^[Y1 : nat]: (divide_divide_nat @ Y0 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Y1))))) @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ lx) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))))) @ (the_nat @ (vEBT_vebt_maxt @ (nth_VEBT_VEBT @ (list_u1324408373059187874T_VEBT @ treeList @ ((^[Y0 : nat]: ((^[Y1 : nat]: (divide_divide_nat @ Y0 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Y1))))) @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ lx) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))) @ (vEBT_vebt_delete @ (nth_VEBT_VEBT @ treeList @ ((^[Y0 : nat]: ((^[Y1 : nat]: (divide_divide_nat @ Y0 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Y1))))) @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ lx) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ ((^[Y0 : nat]: ((^[Y1 : nat]: (modulo_modulo_nat @ Y0 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Y1))))) @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ lx) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))) @ (the_nat @ (vEBT_vebt_maxt @ (vEBT_vebt_delete @ summary @ ((^[Y0 : nat]: ((^[Y1 : nat]: (divide_divide_nat @ Y0 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Y1))))) @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ lx) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))))))))) != (plus_plus_nat @ (times_times_nat @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))) @ maxs) @ maxi))),
% 84.01/19.79 inference(definition_unfolding,[status(thm)],[f25411,f25607,f25523,f25523,f25607,f25523,f25523,f25607,f25523,f25523,f25153,f25523,f25523,f25607,f25523,f25523,f25523])).
% 84.01/19.79 thf(f25411,plain,(
% 84.01/19.79 ((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 @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na))))) @ (the_nat @ (vEBT_vebt_maxt @ (nth_VEBT_VEBT @ (list_u1324408373059187874T_VEBT @ treeList @ (vEBT_VEBT_high @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na) @ (vEBT_vebt_delete @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na)) @ (vEBT_VEBT_low @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na))) @ (the_nat @ (vEBT_vebt_maxt @ (vEBT_vebt_delete @ summary @ (vEBT_VEBT_high @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na)))))))) != (plus_plus_nat @ (times_times_nat @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na) @ maxs) @ maxi))),
% 84.01/19.79 inference(cnf_transformation,[status(thm)],[f24211])).
% 84.01/19.79 thf(f24211,plain,(
% 84.01/19.79 ((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 @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na))))) @ (the_nat @ (vEBT_vebt_maxt @ (nth_VEBT_VEBT @ (list_u1324408373059187874T_VEBT @ treeList @ (vEBT_VEBT_high @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na) @ (vEBT_vebt_delete @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na)) @ (vEBT_VEBT_low @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na))) @ (the_nat @ (vEBT_vebt_maxt @ (vEBT_vebt_delete @ summary @ (vEBT_VEBT_high @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na)))))))) != (plus_plus_nat @ (times_times_nat @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na) @ maxs) @ maxi))),
% 84.01/19.79 inference(flattening,[status(thm)],[f10240])).
% 84.01/19.79 thf(f10240,negated_conjecture,(
% 84.01/19.79 ~((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 @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na))))) @ (the_nat @ (vEBT_vebt_maxt @ (nth_VEBT_VEBT @ (list_u1324408373059187874T_VEBT @ treeList @ (vEBT_VEBT_high @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na) @ (vEBT_vebt_delete @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na)) @ (vEBT_VEBT_low @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na))) @ (the_nat @ (vEBT_vebt_maxt @ (vEBT_vebt_delete @ summary @ (vEBT_VEBT_high @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na)))))))) = (plus_plus_nat @ (times_times_nat @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na) @ maxs) @ maxi))),
% 84.01/19.79 inference(negated_conjecture,[status(cth)],[f10239])).
% 84.01/19.79 thf(f10239,conjecture,(
% 84.01/19.79 ((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 @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na))))) @ (the_nat @ (vEBT_vebt_maxt @ (nth_VEBT_VEBT @ (list_u1324408373059187874T_VEBT @ treeList @ (vEBT_VEBT_high @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na) @ (vEBT_vebt_delete @ (nth_VEBT_VEBT @ treeList @ (vEBT_VEBT_high @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na)) @ (vEBT_VEBT_low @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na))) @ (the_nat @ (vEBT_vebt_maxt @ (vEBT_vebt_delete @ summary @ (vEBT_VEBT_high @ (plus_plus_nat @ (times_times_nat @ summin @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na)) @ lx) @ na)))))))) = (plus_plus_nat @ (times_times_nat @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ na) @ maxs) @ maxi))),
% 84.01/19.79 file('/export/starexec/sandbox/tmp/tmp.YjPWRkCLRl/DTF2THF_23017.p',conj_0)).
% 84.01/19.79 % SZS output end Proof for DTF2THF_23017
% 84.01/19.79 % (23322)------------------------------
% 84.01/19.79 % (23322)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 84.01/19.79 % (23322)Termination reason: Refutation
% 84.01/19.79
% 84.01/19.79 % (23322)Memory used [KB]: 37483
% 84.01/19.79 % (23322)Time elapsed: 1.808 s
% 84.01/19.79 % (23322)Instructions burned: 3185 (million)
% 84.01/19.79 % (23322)------------------------------
% 84.01/19.79 % (23322)------------------------------
% 84.01/19.79 % (23156)Success in time 10.702 s
%------------------------------------------------------------------------------